from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

"""BrainF**k interpreter.

Language info: https://en.wikipedia.org/wiki/Brainfuck

Based on public implementation:
https://github.com/pocmo/Python-Brainfuck/blob/master/brainfuck.py
"""

from collections import namedtuple
import time

EvalResult = namedtuple(
    'EvalResult', ['output', 'success', 'failure_reason', 'steps', 'time',
                   'memory', 'program_trace'])

ExecutionSnapshot = namedtuple(
    'ExecutionSnapshot',
    ['codeptr', 'codechar', 'memptr', 'memval', 'memory', 'next_input',
     'output_buffer'])

class Status(object):
  SUCCESS = 'success'
  TIMEOUT = 'timeout'
  STEP_LIMIT = 'step-limit'
  SYNTAX_ERROR = 'syntax-error'

CHARS = INT_TO_CHAR = ['>', '<', '+', '-', '[', ']', '.', ',']
CHAR_TO_INT = dict([(c, i) for i, c in enumerate(INT_TO_CHAR)])

class LookAheadIterator(object):
  """Same API as Python iterator, with additional peek method."""

  def __init__(self, iterable):
    self._it = iter(iterable)
    self._current_element = None
    self._done = False
    self._preload_next()

  def _preload_next(self):
    try:
      self._current_element = self._it.next()
    except StopIteration:
      self._done = True

  def next(self):
    if self._done:
      raise StopIteration
    element = self._current_element
    self._preload_next()
    return element

  def peek(self, default_value=None):
    if self._done:
      if default_value is None:
        raise StopIteration
      return default_value
    return self._current_element

def buildbracemap(code):
  """Build jump map.

  Args:
    code: List or string or BF chars.

  Returns:
    bracemap: dict mapping open and close brace positions in the code to their
        destination jumps. Specifically, positions of matching open/close braces
        if they exist.
    correct_syntax: True if all braces match. False if there are unmatched
        braces in the code. Even if there are unmatched braces, a bracemap will
        be built, and unmatched braces will map to themselves.
  """
  bracestack, bracemap = [], {}

  correct_syntax = True
  for position, command in enumerate(code):
    if command == '[':
      bracestack.append(position)
    if command == ']':
      if not bracestack:  # Unmatched closing brace.
        bracemap[position] = position  # Don't jump to any position.
        correct_syntax = False
        continue
      start = bracestack.pop()
      bracemap[start] = position
      bracemap[position] = start
  if bracestack:  # Unmatched opening braces.
    for pos in bracestack:
      bracemap[pos] = pos  # Don't jump to any position.
      correct_syntax = False
  return bracemap, correct_syntax

def evaluate(code, input_buffer=None, init_memory=None, base=256, timeout=1.0,
             max_steps=None, require_correct_syntax=True, output_memory=False,
             debug=False):
  """Execute BF code.

  Args:
    code: String or list of BF characters. Any character not in CHARS will be
        ignored.
    input_buffer: A list of ints which will be used as the program's input
        stream. Each read op "," will read an int from this list. 0's will be
        read once the end of the list is reached, or if no input buffer is
        given.
    init_memory: A list of ints. Memory for first k positions will be
        initialized to this list (where k = len(init_memory)). Memory positions
        are initialized to 0 by default.
    base: Integer base for the memory. When a memory value is incremented to
        `base` it will overflow to 0. When a memory value is decremented to -1
        it will underflow to `base` - 1.
    timeout: Time limit for program execution in seconds. Set to None to
        disable.
    max_steps: Execution step limit. An execution step is the execution of one
        operation (code character), even if that op has been executed before.
        Execution exits when this many steps are reached. Set to None to
        disable. Disabled by default.
    require_correct_syntax: If True, unmatched braces will cause `evaluate` to
        return without executing the code. The failure reason will be
        `Status.SYNTAX_ERROR`. If False, unmatched braces are ignored
        and execution will continue.
    output_memory: If True, the state of the memory at the end of execution is
        returned.
    debug: If True, then a full program trace will be returned.

  Returns:
    EvalResult namedtuple containing
      output: List of ints which were written out by the program with the "."
          operation.
      success: Boolean. Whether execution completed successfully.
      failure_reason: One of the attributes of `Status`. Gives extra info
          about why execution was not successful.
      steps: Number of execution steps the program ran for.
      time: Amount of time in seconds the program ran for.
      memory: If `output_memory` is True, a list of memory cells up to the last
          one written to. otherwise, None.
  """
  input_iter = (
      LookAheadIterator(input_buffer) if input_buffer is not None
      else LookAheadIterator([]))

  # Null memory value. This is the value of an empty memory. Also the value
  # returned by the read operation when the input buffer is empty, or the
  # end of the buffer is reached.
  null_value = 0

  code = list(code)
  bracemap, correct_syntax = buildbracemap(code)  # will modify code list
  if require_correct_syntax and not correct_syntax:
    return EvalResult([], False, Status.SYNTAX_ERROR, 0, 0.0,
                      [] if output_memory else None, [] if debug else None)

  output_buffer = []

  codeptr, cellptr = 0, 0

  cells = list(init_memory) if init_memory else [0]

  program_trace = [] if debug else None
  success = True
  reason = Status.SUCCESS
  start_time = time.time()
  steps = 0
  while codeptr < len(code):
    command = code[codeptr]

    if debug:
      # Add step to program trace.
      program_trace.append(ExecutionSnapshot(
          codeptr=codeptr, codechar=command, memptr=cellptr,
          memval=cells[cellptr], memory=list(cells),
          next_input=input_iter.peek(null_value),
          output_buffer=list(output_buffer)))

    if command == '>':
      cellptr += 1
      if cellptr == len(cells): cells.append(null_value)

    if command == '<':
      cellptr = 0 if cellptr <= 0 else cellptr - 1

    if command == '+':
      cells[cellptr] = cells[cellptr] + 1 if cells[cellptr] < (base - 1) else 0

    if command == '-':
      cells[cellptr] = cells[cellptr] - 1 if cells[cellptr] > 0 else (base - 1)

    if command == '[' and cells[cellptr] == 0: codeptr = bracemap[codeptr]
    if command == ']' and cells[cellptr] != 0: codeptr = bracemap[codeptr]

    if command == '.': output_buffer.append(cells[cellptr])
    if command == ',': cells[cellptr] = next(input_iter, null_value)

    codeptr += 1
    steps += 1

    if timeout is not None and time.time() - start_time > timeout:
      success = False
      reason = Status.TIMEOUT
      break
    if max_steps is not None and steps >= max_steps:
      success = False
      reason = Status.STEP_LIMIT
      break

  if debug:
    # Add step to program trace.
    command = code[codeptr] if codeptr < len(code) else ''
    program_trace.append(ExecutionSnapshot(
        codeptr=codeptr, codechar=command, memptr=cellptr,
        memval=cells[cellptr], memory=list(cells),
        next_input=input_iter.peek(null_value),
        output_buffer=list(output_buffer)))

  return EvalResult(
      output=output_buffer,
      success=success,
      failure_reason=reason,
      steps=steps,
      time=time.time() - start_time,
      memory=cells if output_memory else None,
      program_trace=program_trace)


BUCCI DONUT BOWL, FISH-BELLY GLAZE

SKU:

$2,400.00

$2,400.00

Unavailable

This definitive design was introduced by Franco Bucci in the ealry 1970s. The bowl is sensual to the touch and pleasing to the eye. The fish-belly glaze which added character and depth to the object was one of the three most intricate and beautiful glazes developed by Bucci early on in his Laboratorio Pesaro career. The other two were Celadon and Cobalt Blue. His black and white glazes were perfected early on with lustrous satin sheen that was delicate and subtle. This bowl has a tight crack on one side that can be mended invisibly from the underside. It was hand signed by Franco Bucci in the 1970s. 10"diam. x 3.5"hNet weight is about 3 lbs.


I was just looking around, but I did not find any section considering the most important bit of magic, the psychology and performance behind it. I am just wondering whether there can be established some kind of section considering how to practise, how to motivate yourself, how to present, and all that stuff that has to do with the mind:)

Hope something will be established/ does already exist:)

"A magician has the skill to conceal the skill"
- Paul Wilson
"Magic is driven by emotion and conflict"
- Tommy Wonder

Dave ScribnerAssistant Manager
Lake Hopatcong, NJ
4977 Posts

Posted: May 17, 2007 10:30 am

0

We do have a forum for discussion on psychology. It's called Food for thought and this is it's description.

Quote:

This is the area to discuss creativity,theory,and psychology in our magic


Chromolite

Chromogenex Chromolite IPL Pigmentation/Sun Damage removal

Treatment to remove pigment patches from most areas of the body. Brown marks can appear on the skin due to exposure to the sun, most common on the face and back of hands. These can be removed safely by intense pulsed light, 1-3 treatments , every two weeks for best results.


                id., 115 Nev. at 390, 990 P.2d at 1262, we conclude that this claim lacks
                merit.
                              Second, Gubbine contends that the State failed to give proper
                notice that a victim impact statement would reference a prior bad act and
                the district court abused its discretion by considering the prior bad act.
                The defense objected on hearsay grounds to the witness' mention of the act
                and the district court noted that, although not hearsay, the statement was
                inappropriate. The district court stated that it would not consider any
                uncharged crimes and specified that it based Gubbine's sentence upon the
                acts to which he pleaded no contest, the effect of the crimes on his children
                and parents, and his inability to conform his conduct to the requirements
                of the law. See Buschauer v. State, 106 Nev. 890, 894, 804 P.2d 1046, 1049
                (1990) (a district court can disclaim reliance on prior bad acts when
                imposing sentence). Gubbine's consecutive sentences of 48-120 months on
                count I, 12 months on count II, 24-60 months on count III, and 24-60
                months on count IV, are within the parameters provided by relevant
                statutes, NRS 193.140; NRS 200.508(1)(b)(1); NRS 200.5099(1)(a); NRS
                205.060(2), and we conclude that the district court did not abuse its
                discretion at sentencing, see NRS 176.035(1); Parrish v. State, 116 Nev.
                982, 989, 12 P.3d 953, 957 (2000). Accordingly, we
                              ORDER the judgment of conviction AFFIRMED.

                                             i   c--14cA4             ,J.
                                         Hardesty

                Parraguirre

SUPREME COURT
        OF
     NEVADA
                                                            2
(0) 1947A
                cc: Hon. Steve L. Dobrescu, District Judge
                     State Public Defender/Ely
                     State Public Defender/Carson City
                     Attorney General/Carson City
                     White Pine County District Attorney
                     White Pine County Clerk

SUPREME COURT
        OF
     NEVADA
                                                   3
(0) 1947A


Labels

Tuesday, 21 March 2017

Space Pug - Genghis and the Space Station

Some Pre-Viz work that focused on seeing how the lighting between the Genghis model and the Space Station would interact. I ended up having to change quite a few of Genghis' texture nodes in order to make his body in tune with the lighting of the Space Station. These modifications mostly focused on the ambient occlusion and rim lighting nodes. So far its come out nicely with a few exceptions; the eyes are too dark and the shadows beneath Genghis' body look unnatural.


New England TE heads south

Mike Farrell

Recruiting Analyst

Dartmouth, Massachusetts tight end Arthur Fontaine has made his college decision and this time he does not expect to change his mind. The 6-foot-5, 240-pound prospect is confident in his choice and ready to move forward.

Fontaine had previously been committed to Boston College before opening the recruiting process back up to check out more options. Where will he be spending his college years?

"I committed to Georgia," Fontaine said. "It was a hard decision. It came down to Georgia and BC, as well as Virginia and Maryland. In the end it became mostly about getting away to experience college away from home. It's about growing up and becoming an adult and experiencing new things."

Fontaine's mother and sister attended Boston College, but the four-star prospect said the opportunity to forge a new path for himself and experience life in a different part of the country was very attractive to him.

"I like Georgia a lot," he explained. "The coaches are great. Coach John Lilly was my recruiting coach and I really like Coach Richt. I took two visits down there. I felt very comfortable and the second visit was even better than the first visit. That's why I went a second time -- to see if I had that same feeling.

"I fit in well with what they want to do offensively, but this was more about the college experience than football."

Though he is moving in another direction, Fontaine had nothing but positive things to say about the other schools recruiting him.

"I love BC," he said. "I know I could be successful in that offense as well. I wish them well and I hope they continue their success. I like all the coaches that recruited me. I liked Maryland and Virginia as well. I could have been successful at all four. It doesn't have anything to do with tight end tradition or anything like that. This wasn't as much a football decision as it was a college decision."

Fontaine's mother, sister and grandfather are on board with his decision and Fontaine expects to stay solid this time. He had previously committed to Boston College but de-committed shortly after when he realized he rushed his decision. He is ranked as the No. 4 tight end in the nation and is a member of the Rivals250.

As a junior Fontaine caught 10 passes for 160 yards and two touchdowns while recording 63 tackles and four sacks on defense. With a different offensive approach this season, those numbers should increase.

"We're going to throw the ball around a lot more," he said. "So that's a good thing. I'm looking forward to my senior season now that this decision is behind me."

Fontaine becomes Georgia's 13th commitment but just the fourth from outside the Peach State.


Embassy Theatre (London)

The Embassy Theatre is a theatre at 64, Eton Avenue, Swiss Cottage, London.

Early years
The Embassy Theatre was opened as a repertory company in September 1928 on the initiative of Sybil Arundale and Herbert Jay., when the premises of Hampstead Conservatoire of Music were adapted by architect Andrew Mather. The following were some of its productions:
The Yellow Streak, September 1928. This was the opening production, featuring Jeanne de Casalis, Martita Hunt and Cecil Parker. The play was praised by the writer Dorothy Richardson.
The Seventh Guest, October 1928, a mystery melodrama with Cecil Parker and Margaret Rawlings

From September 1930 to March 1932 the theatre was directed by Alec L. Rea and A. R. Whatmore. Productions included:
Black Coffee (premiere), by Agatha Christie (her first play), December 1930, produced by André van Gyseghem, with Francis L. Sullivan as Poirot and also featuring Donald Wolfit.
Carpet Slippers, December 1930, with Griffith Jones (his debut) and Sebastian Shaw
Mary Broome, by Allan Monkhouse, December 1931. with Robert Donat and Herbert Lomas. This was a sudden and (still) unexplained substitution for the play originally announced, namely Chimneys, by Agatha Christie
Romeo and Juliet, February 1932, produced by A. R. Whatmore, with Sebastian Shaw as Romeo, Cecil Parker as Mercutio and George Coulouris as Tybalt

Ronald Adam years
Control than passed to Ronald Adam (also known as Ronald Adams), who remained at the helm until 1939. During that time he made over 150 new productions and revivals, of which over thirty were then transferred to various theatres in the West End. The Embassy school of acting was opened in the theatre in 1932. Some of the more notable productions at the theatre were:
Miracle at Verdun by Hans Chlumberg (translated by Edward Crankshaw), September 1932, produced by André van Gyseghem, with Derrick de Marney, George Howe, Alan Wheatley. The production was then transferred to the Comedy.
Ten Minute Alibi (premiere), by Anthony Armstrong, January 1933, with Robert Douglas and Celia Johnson. The production then transferred to the Haymarket.
The Glass Wall (premiere), by E. M. Delafield, February 1933, produced by André van Gyseghem, with Max Adrian.
All God's Chillun Got Wings, March 1933, produced by André van Gyseghem, with Paul Robeson and Flora Robson.
Sometimes Even Now (premiere), by Warren Chetham-Strode, May 1933, with Jack Hawkins, Celia Johnson, Marie Lohr
Napoleon, September 1934, produced by André van Gyseghem, with Edward Chapman, John Clements, Violet Farebrother, Eric Portman, Margaret Rawlings.
The Dominant Sex (premiere), by Michael Egan, December 1934, with Diana Churchill, Richard Bird, René Ray. The production then transferred to the Shaftesbury.
Stevedore, by Paul Peters and George Sklar, May 1935, produced by André van Gyseghem, with Paul Robeson, Robert Adams, Kathleen Davis.
This Desirable Residence, by A. R. Rawlinson, May 1935, with Coral Browne.
Close Quarters (premiere), by W. O. Somin and Gilbert Lennox, June 1935, with Oskar Homolka (London debut), Flora Robson. The production then transferred to the Haymarket.
Professor Bernhardi, June 1936, with Abraham Sofaer in the title role. and Max Adrian, Noel Howlett The production then transferred to the Phoenix.
Judgment Day (London premiere), by Elmer Rice, May 1937, with Glynis Johns, Catherine Lacey,<ref name=wwLacey>[https://archive.org/stream/dramaticlistwhos007920mbp#page/n895/mode/2up Who's Who in the Theatre: Lacey, Catherine]</ref> George Woodbridge The production then transferred to the Strand.Three Set Out, by Philip Leaver, June 1937, directed by Margaret Webster, with Constance Cummings and Michael RedgraveProfit and Loss, May 1938, produced by André van Gyseghem, with Mabel Love (final appearance)Julius Caesar, November 1939, in modern dress, with Joseph O'Conor (debut), Peter Copley, Hugh Griffith, Eric Portman

Ronald Adam's own list of significant transfers in that period was Ten Minute Alibi, Close Quarters, The Dominant Sex, Professor Bernhardi, Judgment Day.

 Post-war period
After war damage, the building was reopened in 1945, with a capacity of 678.
It was then run until 1954 by Anthony Hawtrey.Quality Street, February 1945, directed by Anthony Hawtrey, with Joyce Blair (debut, aged 13), Ursula Howells (London debut), Geoffrey Toone, Bryan Forbes, Gwendoline Watford, Linden TraversMyself a Stranger, August 1945, with Jack Allen, Hugh Burden, Cecil RamageFit for Heroes, September 1945, directed by Henry Kendall, with Irene Vanbrugh, Helen Cherry, Jack Allen, Raymond Lovell, Olaf Pooley.The Gambler, adapted by Norman Ginsbury from Dostoevsky, November 1945, directed by Sebastian Shaw, with Hugh Burden, Ferdy Mayne, Gwendoline Watford.Red Roses for Me, by Seán O'Casey, February 1946, with Kieron O'Hanrahan, Eddie ByrneNational Velvet, 1946.Sense and Sensibility, 1946.Mrs Dane's Defence, 1946, with Mary Ellis.Portrait of Hickory, April 1948, with Judy Campbell.Miranda, June 1947, directed by Richard Bird, with Nora Swinburne, Ronald Ward, Diane HartThe Father (August Strindberg), November 1948, with Michael RedgraveA Woman in Love ("Amoureuse"), April 1949, adapted and directed by Michael Redgrave, with Margaret RawlingsOn Monday Next (premiere), by Philip King, April 1949, with Henry Kendall, directed by him and Shaun Sutton, also with Leslie PhillipsOthello, July 1949, produced by André van Gyseghem, with Michael Aldridge in the title role, and Peter Wyngarde, Maxine Audley.Caro William, premiere 1952, with Robert Shaw (London debut), Rachel Gurney.The Merchant of Yonkers, 1952, directed by André van Gyseghem, with Robert Eddison, Raymond Lovell, Sophie Stewart, Alfie Bass, Esma Cannon, Peter Baylis, Nigel HawthorneUranium 235, by Ewan MacColl, May 1952, produced by Joan Littlewood with Harry H. Corbett, George A. Cooper, Avis Bunnage.Hamlet, March 1953 with Laurence Payne in the title role, George Coulouris, Christine FinnTwelfth Night 1953 with George Coulouris, Christine FinnThe Boy Friend (premiere for full version) 1953 with Hugh Paddick.

In 1953 it was sold to Sidney Bernstein with management by screenwriter and playwright Wolf Mankowitz.The Bespoke Overcoat, 1954The Lion in the Lighthouse, June 1955, with Henry Kendall (actor)The World of Sholem Aleichem, ca. 1955The Boychik, ca. 1956

 Central School
The theatre was sold to the Royal Central School of Speech and Drama, who continue as the resident company, in 1956.Mad Forest'' (premiere) 1990

References

General reference

Rob Wilton Theatricalia: Other Plays 1900-1939
Rob Wilton Theatricalia: Other Plays 1940-1949
Rob Wilton Theatricalia: Other Plays 1950-1959

External links
 Cullum and Nightingale, Architects' Journal, September 2003

Category:Theatres in the London Borough of Camden


/*-
 *  * Copyright 2016 Skymind, Inc.
 *  *
 *  *    Licensed under the Apache License, Version 2.0 (the "License");
 *  *    you may not use this file except in compliance with the License.
 *  *    You may obtain a copy of the License at
 *  *
 *  *        http://www.apache.org/licenses/LICENSE-2.0
 *  *
 *  *    Unless required by applicable law or agreed to in writing, software
 *  *    distributed under the License is distributed on an "AS IS" BASIS,
 *  *    WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  *    See the License for the specific language governing permissions and
 *  *    limitations under the License.
 */

package org.datavec.spark.transform.quality;

import lombok.AllArgsConstructor;
import org.apache.spark.api.java.function.Function2;
import org.datavec.api.transform.ColumnType;
import org.datavec.api.transform.metadata.*;
import org.datavec.api.transform.schema.Schema;
import org.datavec.api.writable.Writable;
import org.datavec.spark.transform.quality.bytes.BytesQualityAnalysisState;
import org.datavec.spark.transform.quality.categorical.CategoricalQualityAnalysisState;
import org.datavec.spark.transform.quality.integer.IntegerQualityAnalysisState;
import org.datavec.spark.transform.quality.longq.LongQualityAnalysisState;
import org.datavec.spark.transform.quality.real.RealQualityAnalysisState;
import org.datavec.spark.transform.quality.string.StringQualityAnalysisState;
import org.datavec.spark.transform.quality.time.TimeQualityAnalysisState;

import java.util.ArrayList;
import java.util.List;

/**
 * Add function used for undertaking quality analysis of a data set via Spark
 *
 * @author Alex Black
 */
@AllArgsConstructor
public class QualityAnalysisAddFunction
                implements Function2<List<QualityAnalysisState>, List<Writable>, List<QualityAnalysisState>> {

    private Schema schema;

    @Override
    public List<QualityAnalysisState> call(List<QualityAnalysisState> analysisStates, List<Writable> writables)
                    throws Exception {
        if (analysisStates == null) {
            analysisStates = new ArrayList<>();
            List<ColumnType> columnTypes = schema.getColumnTypes();
            List<ColumnMetaData> columnMetaDatas = schema.getColumnMetaData();
            for (int i = 0; i < columnTypes.size(); i++) {
                switch (columnTypes.get(i)) {
                    case String:
                        analysisStates.add(new StringQualityAnalysisState((StringMetaData) columnMetaDatas.get(i)));
                        break;
                    case Integer:
                        analysisStates.add(new IntegerQualityAnalysisState((IntegerMetaData) columnMetaDatas.get(i)));
                        break;
                    case Long:
                        analysisStates.add(new LongQualityAnalysisState((LongMetaData) columnMetaDatas.get(i)));
                        break;
                    case Double:
                        analysisStates.add(new RealQualityAnalysisState((DoubleMetaData) columnMetaDatas.get(i)));
                        break;
                    case Categorical:
                        analysisStates.add(new CategoricalQualityAnalysisState((CategoricalMetaData) columnMetaDatas.get(i)));
                        break;
                    case Time:
                        analysisStates.add(new TimeQualityAnalysisState((TimeMetaData)columnMetaDatas.get(i)));
                        break;
                    case Bytes:
                        analysisStates.add(new BytesQualityAnalysisState()); //TODO
                        break;
                    default:
                        throw new IllegalArgumentException("Unknown column type: " + columnTypes.get(i));
                }
            }
        }

        int size = analysisStates.size();
        if (size != writables.size())
            throw new IllegalStateException("Writables list and number of states does not match (" + writables.size()
                            + " vs " + size + ")");
        for (int i = 0; i < size; i++) {
            analysisStates.get(i).add(writables.get(i));
        }

        return analysisStates;
    }
}


---
abstract: 'We consider a viscous approximation for a nonlinear degenerate convection-diffusion equations in two space dimensions, and prove an $L^1$ error estimate. Precisely, we show that the $L^1_{\mathrm{loc}}$ difference between the approximate solution and the unique entropy solution converges at a rate $\mathcal{O}({\varepsilon}^{1/2})$, where ${\varepsilon}$ is the viscous parameter.'
title: 'An error estimate for viscous approximate solutions to degenerate anisotropic convection-diffusion equations'
---

Christian Klingenberg

Ujjwal Koley

(Communicated by the associate editor name)

Introduction {#sec:intro}
============

In this paper, we are interested in certain “viscous” approximations of entropy solutions of the following Cauchy problem $$\label{eq:main}
  \begin{cases}
    u_t + f(u)_y = u_{xx}, &(x,y,t)\in {\Pi_T},\\
    u(x,y,0)=u_0(x,y), & (x,y)\in {\mathbb{R}}^2,
  \end{cases}$$ where ${\Pi_T}={\mathbb{R}}\times {\mathbb{R}}\times (0,T)$ with $T>0$ fixed, $u:{\Pi_T}\to {\mathbb{R}}$ is the unknown function and $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ is the convective flux function. The main characteristics of this type of equations is that it has mixed parabolic-hyperbolic type, due to the directional separation of the diffusion and convection effects: while matter is convected along the $y$ axis, it is simultaneously diffused along orthogonal direction.

The existence of solutions of equation can be obtained by the classical method of adding a vanishing viscosity, in other terms a diffusion, in the missing direction (along the $y$ axis). Since the most characteristics of Equation is that it has mixed parabolic-hyperbolic type, or in other words, it has strong degeneracy due to the lack of diffusion in the $x$-direction, it is difficult to establish the uniqueness of solutions of . In [@ez1], authors have proved the existence as well as uniqueness of such problems.

In this paper we are interested in certain approximate solutions of coming from solving the uniformly parabolic problem $$\begin{aligned}
 u^{{\varepsilon}}_t + f(u^{{\varepsilon}})_y =  u^{{\varepsilon}}_{xx} + {\varepsilon}u^{{\varepsilon}}_{yy}.
\end{aligned}
\label{eq:regular}$$ We refer to $u^{{\varepsilon}}$ as a “viscous” approximate solution of . Since the convergence of $u^{{\varepsilon}}$ to the unique entropy solution $u$ of as ${\varepsilon}{\downarrow}0$ is well known, so our interest here is to give an explicit rate of convergence for $u^{{\varepsilon}}$ as ${\varepsilon}{\downarrow}0$, i.e., an $L^1$ error estimate for viscous approximate solutions. There are several ways to prove such an error estimate. One way is to view it as a consequence of a continuous dependence estimate. Combining the ideas of [@ez1] with a variant of Kruzkov’s “doubling of variables” device for , we prove that ${\left\|u^{{\varepsilon}} -u\right\|}_{L^1({\Pi_T})} = \mathcal{O}({\varepsilon}^{1/2})$. Although our proof is of independent interest, it may also shed some light on how to obtain an error estimates for numerical methods.

Preliminaries {#sec:prelim}
=============

Independently of the smoothness of the initial data, due to the lack of diffusion in the $x$-direction, jumps may form in the solution $u$. Therefore we consider solutions in the weak sense, i.e.,

Set ${\Pi_T}={\mathbb{R}}\times {\mathbb{R}}\times (0,T)$. A function $$u(t,x) \in C\left([0,T];L^1({\mathbb{R}}^2)\right) \cap L^\infty({\Pi_T})$$ is a weak solution of the initial value problem if it satisfies:

-   For all test functions ${\varphi}\in
        \mathcal{D}({\mathbb{R}}^2 \times [0,T))$ \[def:w2\] $$\label{eq:weaksol}
          \iiint_{{\Pi_T}} \left( u{\varphi}_t + f(u) {\varphi}_y + u {\varphi}_{xx} \right)\,dx \,dy \,dt
          + \iint_{{\mathbb{R}}^2} u_0(x,y) {\varphi}(x,y,0) \,dx \,dy= 0.$$

-   $u = u(x,y,t)$ is continuous at $t=0$ as a function: $[0,T) \rightarrow L^1({\mathbb{R}}^2)$.

Since weak solutions are not uniquely determined by their initial data, one must impose an additional entropy condition to single out the physically relevant solution.

A weak solution $u$ of the initial value problem is called an entropy solution, if the following entropy inequality holds for all test functions $0 \le {\varphi}\in \mathcal{D}({\mathbb{R}}^2 \times (0,T))$: $$\label{eq:entropysol}
 \begin{aligned}
 &\iiint_{{\Pi_T}} {\left|u -\psi(x)\right|}{\varphi}_t +
   {\mathrm{sign}}(u -\psi(x)) (f(u) -f(\psi(x))) {\varphi}_y \,dxdydt \\
  &\qquad \ge \iiint_{{\Pi_T}} - {\left|u- \psi(x)\right|} {\varphi}_{xx}  - {\mathrm{sign}}(u- \psi(x)) \psi_{xx} {\varphi}\,dxdydt.
 \end{aligned}$$

Note that the above entropy condition is inspired by Kružkov’s entropy condition for scalar conservation laws. However, due to the presence of diffusion term $u_{xx}$ in , Kružkov’s entropy condition has to be modified in this case. This is done by introducing as entropy test functions all functions of the form ${\left|u -\psi(x)\right|}$ and $\psi$ smooth, while Kružkov’s definition asks for $\psi$ to be a constant.

Regarding the regularized problem , it is well known [@ez1] that for all ${\varepsilon}> 0$ this problem has a unique solution $$\begin{aligned}
 u^{{\varepsilon}} \in C([0,T]; L^1({\mathbb{R}}^2)) \cap L^{\infty}({\Pi_T}).\end{aligned}$$ In fact, $$\begin{aligned}
 u^{{\varepsilon}} \in C([0,T]; W^{2,p}({\mathbb{R}}^2)) \cap C^{1}([0,T]; L^{p}({\mathbb{R}}^2)),\end{aligned}$$ for all $1 < p < \infty$. In addition, $u^{{\varepsilon}}$ satisfies the following properties

-   $\iint_{{\mathbb{R}}^2} u^{{\varepsilon}}(x,y,t) \,dxdy = \iint_{{\mathbb{R}}^2} u_0(x,y) \,dxdy$,

-   $ \text{TV}\, (u^{{\varepsilon}}(\cdot,\cdot, t)) \le \text{TV}\, (u_0) $,

-   $ {\left\|\partial_t u^{{\varepsilon}}(t)\right\|}_1 \le C [\text{TV}\, (u_0) + \text{TV}\, ((u_0)_x)]$.

Now we are in a position to state our main result, which is the following

Let $u$ be the unique entropy solution to and $u^{{\varepsilon}}$ be as defined by . Assume that $u_0 \in BV$, $f$ is Lipschitz continuous. Choose a constant $$M>\max_{{\left|u\right|}<{\left\|u_0\right\|}_{L^\infty({\mathbb{R}}^2)}} {\left|f'(u)\right|},$$ and another constant $L>MT$, where $T>0$. Then there exists a constant $C$, independent of ${\varepsilon}$, but depending on $f$, $L$, $T$ and $u_0$, such that $$\int_{{\mathbb{R}}}\int_{-L+Mt}^{L-Mt}{\left|u(t,x,y) - u^{{\varepsilon}}(t,x,y)\right|}\,dxdy  \le C {\varepsilon}^{1/2}
    \quad\text{for $t\le T$.}$$

Proof of the main theorem {#sec:proof}
=========================

The theorem will be proved by a “doubling of the variables” argument, which was introduced by Kruzhkov [@kr1; @kr2] in the context of hyperbolic conservation laws.

First, observe that from we have for any test function ${\varphi}$ with compact support in ${\mathbb{R}}\times {\mathbb{R}}\times (0,T)$ and $u = u(x,z,s)$, $$\begin{aligned}
 &\iiint_{{\Pi_T}} {\left|u -\psi(x)\right|}{\varphi}_s +
   {\mathrm{sign}}(u -\psi(x)) (f(u) -f(\psi(x))) {\varphi}_z \,dxdzds \\
  &\qquad \ge \iiint_{{\Pi_T}} - {\left|u- \psi(x)\right|} {\varphi}_{xx}  - {\mathrm{sign}}(u- \psi(x)) \psi_{xx} {\varphi}\,dxdzds.
 \end{aligned}
\label{eq:main_1}$$ For the regularized equation , we start not with the entropy condition, but in the argument leading up to this condition. To do that, first define the regularized counterpart of the signum function as $${\mathrm{sign}}_\eta(\sigma) =
\begin{cases}
  {\mathrm{sign}}(\sigma) & {\left|\sigma\right|}>\eta,\\
  \sin\left(\frac{\pi\sigma}{2\eta}\right) &\text{otherwise,}
\end{cases}$$ where $\eta>0$ and the signum function is defined as $${\mathrm{sign}}(\sigma) =
\begin{cases}
  -1 & \sigma<0,\\
  0 & \sigma = 0,\\
  1 & \sigma>0,
\end{cases}$$

Set $$\psi_\eta(u,\psi)=\int_{\psi}^{u} {\mathrm{sign}}_\eta (z-\psi)\,dz.$$ This is a convex entropy for all $\psi$. Set $u^{{\varepsilon}}=u^{{\varepsilon}}(x,y,t)$ and rewrite as $$u^{{\varepsilon}}_t + \left(f(u^{{\varepsilon}})-f(\psi)\right)_y = u^{{\varepsilon}}_{xx} + {\varepsilon}(u^{{\varepsilon}} -\psi(x))_{yy},$$ and multiply this with $(\psi_\eta)_u(u,\psi) \, {\varphi}$ where ${\varphi}$ is a test function with compact support in ${\mathbb{R}}\times {\mathbb{R}}\times (0,T)$. Observe that the solution $u^{{\varepsilon}}$ of is smooth. Hence after a partial integration, we arrive at $$\begin{aligned}
  &\iiint_{{\Pi_T}} \psi_\eta(u^{{\varepsilon}},\psi) {\varphi}_t +
   Q_{\eta} (u^{{\varepsilon}},\psi) {\varphi}_y \,dxdydt \\
  &\qquad = \iiint_{{\Pi_T}} - {\mathrm{sign}}_\eta (u^{{\varepsilon}}- \psi) u^{{\varepsilon}}_{xx}{\varphi}- {\varepsilon}{\mathrm{sign}}_\eta (u^{{\varepsilon}}- \psi) (u^{{\varepsilon}} -\psi(x))_{yy} {\varphi}\,dxdydt,\end{aligned}$$ where we have used $Q'_{\eta}(u,\psi) = \psi'_\eta(u,\psi) f'(u)$. Next, taking limit as $\eta \rightarrow 0$, we end up with the parabolic equality $$\begin{aligned}
  &\iiint_{{\Pi_T}} {\left|u^{{\varepsilon}} -\psi(x)\right|}{\varphi}_t +
   {\mathrm{sign}}(u^{{\varepsilon}} -\psi(x)) (f(u^{{\varepsilon}}) -f(\psi(x))) {\varphi}_y \,dxdydt \\
  &\qquad = \iiint_{{\Pi_T}} - {\mathrm{sign}}(u^{{\varepsilon}}- \psi(x)) u^{{\varepsilon}}_{xx} {\varphi}- {\varepsilon}\, {\mathrm{sign}}(u^{{\varepsilon}}- \psi(x)) (u^{{\varepsilon}} -\psi(x))_{yy} {\varphi}\,dxdydt, \\
& \qquad = \iiint_{{\Pi_T}} - {\mathrm{sign}}(u^{{\varepsilon}}- \psi(x)) u^{{\varepsilon}}_{xx} {\varphi}- {\varepsilon}\, {\left|u^{{\varepsilon}} -\psi(x)\right|}_{yy} {\varphi}\,dxdydt.
\end{aligned}
\label{eq:main_2}$$ At this point we are ready to use “doubling of the variables” device. First, using the entropy inequality for the solution $u=u(x,z,s)$ with $\psi(x)= u^{{\varepsilon}}(x,y,t)$, we get for $(y,t) \in {\mathbb{R}}\times (0,T)$ $$\begin{aligned}
 &\iiint_{{\Pi_T}} {\left|u -u^{{\varepsilon}}\right|}{\varphi}_s +
   {\mathrm{sign}}(u -u^{{\varepsilon}}) (f(u) -f(u^{{\varepsilon}})) {\varphi}_z \,dxdzds \\
  &\qquad \ge \iiint_{{\Pi_T}} - {\left|u- u^{{\varepsilon}}\right|} {\varphi}_{xx}  - {\mathrm{sign}}(u- u^{{\varepsilon}}) u^{{\varepsilon}}_{xx} {\varphi}\,dxdzds.
 \end{aligned}
\label{eq:main_3}$$ Similarly, from the parabolic equality for the solution $u^{{\varepsilon}} = u^{{\varepsilon}}(x,y,t)$ with $\psi(x)= u(x,z,s)$, we get for $(z,s) \in {\mathbb{R}}\times (0,T)$ $$\begin{aligned}
  &\iiint_{{\Pi_T}} {\left|u^{{\varepsilon}} -u\right|}{\varphi}_t +
   {\mathrm{sign}}(u^{{\varepsilon}} -u) (f(u^{{\varepsilon}}) -f(u)) {\varphi}_y \,dxdydt \\
  & \qquad = \iiint_{{\Pi_T}} - {\mathrm{sign}}(u^{{\varepsilon}}- u) u^{{\varepsilon}}_{xx} {\varphi}- {\varepsilon}\, {\left|u^{{\varepsilon}} -u\right|}_{yy} {\varphi}\,dxdydt.
\end{aligned}
\label{eq:main_4}$$ We now integrate over $(y,t) \in {\mathbb{R}}\times (0,T)$ and over $(z,s) \in {\mathbb{R}}\times (0,T)$. Addition of those two results yields $$\begin{aligned}
 & \int\iiiint_{Q_T} {\left| u^{{\varepsilon}} -u \right|} ({\varphi}_t +{\varphi}_s) +
   {\mathrm{sign}}( u^{{\varepsilon}} -u ) ( f(u^{{\varepsilon}}) -f(u)) ( {\varphi}_y + {\varphi}_z) \,dX \\
  &\qquad \ge \int\iiiint_{Q_T} - {\left|u^{{\varepsilon}} - u\right|} {\varphi}_{xx}  - {\varepsilon}\, {\left|u^{{\varepsilon}} -u\right|}_{yy} {\varphi}\,dX,\\
& \qquad := \int \iiiint_{Q_T} \mathcal{Q}_1 + \mathcal{Q}_2 \,dX,
 \end{aligned}
\label{eq:main_5}$$ where $dX=dx\,dy\,dz\,dt\,ds$ and $Q_T = {\mathbb{R}}\times {\mathbb{R}}\times {\mathbb{R}}\times (0,T) \times (0,T)$.

Following Kruzkov and Kuznetsov [@kr1; @kr2] we now specify a nonnegative test function $ {\varphi}=
{\varphi}(x,y,t,z,s)$ defined in $Q_T$. To this end, let $\omega \in
C_{0}^{\infty} ({\mathbb{R}})$ be a function satisfying $$\mathrm{supp}(\omega) \subset [-1,1], \qquad
  \omega(\sigma) \ge 0 , \qquad \int_{{\mathbb{R}}} \omega(\sigma)\, d\sigma = 1,$$ and define $\omega_r(x)=\omega(x/r)/r$. Next, let us choose $\phi \in C_c^{\infty} ({\mathbb{R}})$ such that $$\phi =
\begin{cases}
 1, & {\left|x\right|}<1,\\ 0 & {\left|x\right|}\ge 2,
\end{cases}$$ and $ 0  \le \phi \le 1$ when $1 \le {\left|x\right|} \le 2$. Then we define $K_{\beta}(x) = \phi(x/\beta)$. We will let $\beta \rightarrow \infty$ later. Furthermore, let $h(z)$ be defined as $$h(z)=
  \begin{cases}
    0, &z<-1,\\ z+1 & z\in [-1,0],\\ 1 &z>0.
  \end{cases}$$ and set $h_\alpha(z)=h(\alpha z)$. Let $\nu<\tau$ be two numbers in $(0,T)$, for any $\alpha> 0$ define $$\begin{gathered}
    H_{\alpha}(t)= \int_{-\infty}^{t}
    \omega_{\alpha}(\xi)\, d\xi, \\
    \begin{aligned}
      \Psi(y,t) &= \left(H_{\alpha_0} (t - \nu) - H_{\alpha_0} (t -
        \tau)\right) \left(h_\alpha(y-L_l(t)) -
        h_\alpha(y-L_r(t)-\frac{1}{\alpha})\right)\\
      &=:{\chi^{\alpha_0}_{(\nu,\tau)}(t)}\,{\chi^\alpha_{(L_l,L_r)}(y,t)}
    \end{aligned}
  \end{gathered}$$ where the lines $L_{l,r}$ are given by $$L_l(t)=-L+Mt,\ L_r(t)=L-Mt$$ where $M$ and $L$ are positive numbers, $M$ will be specified below. With $0<r< \min{\left\{\nu, T - \tau\right\}}$ and $\alpha_0\in(0,\min{\left\{\nu-r,T-\tau-r\right\}})$ we set $$\label{eq:testfn}
  {\varphi}(x,y,t,z,s)= K_{\beta}(x)\, \Psi(y,t)\,\omega_r(y-z)\,\omega_{r_0}(t-s).$$ We note that ${\varphi}$ has compact support and also that we have, $$\begin{aligned}
  {\varphi}_t + {\varphi}_s &= K_{\beta}(x) \, \Psi_t(y,t)\,
  \omega_r(y-z)\,\omega_{r_0}(t-s),\\
  {\varphi}_y+{\varphi}_z&= K_{\beta}(x)\, \Psi_y(y,t)\, \omega_r(y-z)\,\omega_{r_0}(t-s).\end{aligned}$$ For the record, we note that $$\label{eq:Psider}
  \begin{aligned}
    \Psi_t(y,t)&=-\chi^{\alpha_0}_{(\nu,\tau)}(t) M
    \left(h_\alpha'(y-L_l(t)) + h_\alpha'(y-L_r(t)-\frac{1}{\alpha})\right) \\
    & \qquad +
    \left(\omega_{\alpha_0}(t-\nu) - \omega_{\alpha_0}(t-\tau)\right)
    \chi^\alpha_{(L_l,L_r)}(y,t),\\
    \Psi_y(y,t)  &= \chi^{\alpha_0}_{(\nu,\tau)}(t)
    \left(h_\alpha'(y-L_l(t)) - h_\alpha'(y-L_r(t)-\frac{1}{\alpha})\right).
  \end{aligned}$$ We shall let all the “small parameters” $ \alpha$, $\alpha_0$, $r$, $r_0$, ${\varepsilon}$ and ${{\Delta x}}$ be sufficiently small and the “large parameter” $\beta$ be sufficiently big, but fixed.

Starting the first term on the left of , we write $$\begin{aligned}
  \int_{Q_T} & {\left|u^{{\varepsilon}}-u\right|}  \left({\varphi}_s  +{\varphi}_t\right)\,dX \le
  \underbrace{\int_{{\Pi_T}} {\left|u^{{\varepsilon}}(x,y,t)-u(x,y,t)\right|} K_{\beta}(x)\Psi_t \,dxdydt}_\delta \\
  &\quad + \underbrace{\int_{{\Pi_T}}\int_{{\mathbb{R}}} {\left|u(x,y,t)-u(x,y,s)\right|} K_{\beta}(x) {\left|\Psi_t(x,t)\right|}
  \omega_{r_0}(t-s) \,dxdydsdt}_{\beta} \\
  &\quad + \underbrace{\int_{Q_T}  {\left|u(x,y,s)-u(x,z,s)\right|} K_{\beta}(x) {\left|\Psi_t(x,t)\right|}
  \omega_{r_0}(t-s)\,\omega_r(x-y)\, dX}_\gamma . \end{aligned}$$ Following [@uk1], it is easy to find that $$\beta+\gamma \le C\left(r_0+r\right).\label{eq:betagamma}$$ To continue the estimate with the first term on the left of , we split $\delta$ as follows $$\begin{aligned}
  \delta &= -\iiint_{{\Pi_T}} \chi_{(\nu,\tau)}^{\alpha_0}(t) M
    \left(h'_\alpha(y-L_l(t)) + h'_\alpha(y-L_r(t)-\frac{1}{\alpha})\right) \\
& \qquad \qquad \qquad \qquad K_{\beta}(x) {\left|u^{{\varepsilon}}(x,y,t)-u(x,y,t)\right|} \, dxdydt \\
 &\quad \quad + \iiint_{{\Pi_T}} \chi^\alpha_{(L_l,L_r)}(y,t)
  {\left|u^{{\varepsilon}}(x,y,t)-u(x,y,t)\right|} \\
& \qquad \qquad \qquad \qquad K_{\beta}(x)
  \left(\omega_{\alpha_0}(t-\nu)-\omega_{\alpha_0}(t-\tau)\right) \,dxdydt, \\
& := \delta_1 + \delta_2.\end{aligned}$$ The term $\delta_1$ will be balanced against the first order derivative term on the left hand side of . To estimate $\delta_2$ we set $e(x,y,t)={\left|u^{{\varepsilon}}(x,y,t)-u(x,y,t)\right|}$ and following [@uk1], we find $$\label{eq:delta2est}
  \begin{aligned}
    \delta_2 &\le \iint \chi^\alpha_{(L_l,L_r)}(y,\nu) K_{\beta}(x) {\left|u^{{\varepsilon}}(x,y,\nu)-u(x,y,\nu)\right|}
    \,dxdy \\ & \qquad - \iint \chi^\alpha_{(L_l,L_r)}(y,\tau) K_{\beta}(x)
    {\left|u^{{\varepsilon}}(x,y,\tau)-u(x,y,\tau)\right|} \,dxdy +
    C\alpha_0.
  \end{aligned}$$ Now we rewrite the “first derivative term” on the left hand side of . Doing this, we get $$\begin{aligned}
  \int_{Q_T} & K_{\beta}(x) {\mathrm{sign}}(u^{{\varepsilon}} -u)
    \left(f(u^{{\varepsilon}})-f(u)\right)
    \left({\varphi}_y+{\varphi}_z\right) \,dX \\ &=
    \int_{Q_T} K_{\beta}(x) \operatorname*{sg}(x,y,z,t,s)\left(f(u^{{\varepsilon}}(x,y,t))-f(u(x,y,t))\right) \\
   & \qquad \qquad \qquad \qquad  \Psi_y(y,t)\,\omega_r(y-z)\, \omega_{r_0}(t-s)\,dX\\
    &\quad \quad +
    \int_{Q_T} K_{\beta}(x) \operatorname*{sg}(x,y,z,t,s)\left(f(u(x,y,t))-f(u(x,z,s))\right) \\
 & \qquad \qquad \qquad \qquad \Psi_y(y,t)\,
    \omega_r(y-z)\,\omega_{r_0}(t-s)\,dX \\
    &=:\delta_3 + \delta_4,\end{aligned}$$ where we have set $\operatorname*{sg}(x,y,z,t,s)={\mathrm{sign}}(u^{{\varepsilon}}(x,y,t)- u(x,z,s))$. We proceed as follows $$\begin{aligned}
  {\left|\delta_4\right|} \le \int_{Q_T} &{\left|f(u(x,y,t))-f(u(x,z,s))\right|}
  \chi^{\alpha_0}_{(\nu,\tau)}(t) K_{\beta}(x) \\
  &\qquad \omega_{r_0}(t-s)\,\omega_r(y-z)
  \left(
    h'_\alpha(y-L_l(t)) + h'_\alpha(y-L_r(t)-\frac{1}{\alpha})\right) \,dX.\end{aligned}$$ We follow [@uk1] to estimate each of these two terms to conclude $$\label{eq:delta4bnd}
  {\left|\delta_4\right|} \le C\left(r_0+r\right).$$ Again, choosing $M$ larger than the Lipschitz norm of $f$ implies that $$\label{eq:dl1dl2bnd}
  \delta_1+\delta_3\le 0.$$

Collecting all the terms we see that $$\label{eq:estimate-middle}
  \begin{aligned}
    \iint K_{\beta}(x) \chi^\alpha_{(L_l,L_r)}&(y,\tau) {\left|u^{{\varepsilon}}(x,y,\tau)-u(x,y,\tau)\right|}
    \,dxdy \\
    &\le \iint K_{\beta}(x) \chi^\alpha_{(L_l,L_r)}(y,\nu)
    {\left|u^{{\varepsilon}}(x,y,\nu)-u(x,y,\nu)\right|} \,dxdy\\
    &\qquad + C\left(r_0+ r +
      \alpha_0 + \alpha\right) + \biggl|{\int_{Q_T} \mathcal{Q}_1 + \mathcal{Q}_2 \, dX }\biggr|.
  \end{aligned}$$ In order to estimate $\mathcal{Q}_1$ and $\mathcal{Q}_2$, we proceed as follows $$\begin{aligned}
 {\left|\int_{Q_T}\mathcal{Q}_1\right|} \,dX & = \int_{Q_T} {\left|u^{{\varepsilon}} - u\right|}{\varphi}_{xx} \,dX \\
& = \int_{Q_T} {\left|u^{{\varepsilon}} - u\right|} K_{\beta}''(x) \Psi(y,t)\,\omega_r(y-z)\,\omega_{r_0}(t-s) \,dX \\
& \le {\left\|u^{{\varepsilon}} +u\right\|}_{L^{\infty}} \int_{Q_T} \frac{1}{\beta^2} \phi''(\frac{x}{\beta}) \Psi(y,t)\,\omega_r(y-z)\,\omega_{r_0}(t-s) \,dX \\
& \le \frac{C}{\beta^2} \iiint \phi''(\frac{x}{\beta}) {\chi^{\alpha_0}_{(\nu,\tau)}(t)}\,{\chi^\alpha_{(L_l,L_r)}(y,t)} \,dxdydt \\
& \le \frac{CK}{\beta \alpha},\end{aligned}$$ where $K := \int_{{\mathbb{R}}} {\left|\phi''(y)\right|} \,dy$.

Next, $$\begin{aligned}
& {\left|\int_{Q_T}\mathcal{Q}_2\right|} \,dX  = {\varepsilon}\int_{Q_T} {\left|u^{{\varepsilon}} - u\right|}_y {\varphi}_{y} \,dX \\
& = {\varepsilon}\int_{Q_T} {\mathrm{sign}}({u^{{\varepsilon}} - u}) (u^{{\varepsilon}})_y  K_{\beta}(x)\, \Psi_y(y,t)\,\omega_r(y-z)\,\omega_{r_0}(t-s) \\
& + {\varepsilon}\int_{Q_T} {\mathrm{sign}}({u^{{\varepsilon}} - u}) (u^{{\varepsilon}})_y K_{\beta}(x)\, \Psi(y,t)\,\omega_r'(y-z)\,\omega_{r_0}(t-s) \,dX \\
& := \mathcal{Q}_{2,1} + \mathcal{Q}_{2,2}.\end{aligned}$$ Each of the above terms can be approximated as follows $$\begin{aligned}
& {\left|\int_{Q_T}\mathcal{Q}_{2,1}\right|} \,dX   = {\varepsilon}\int_{Q_T} {\mathrm{sign}}({u^{{\varepsilon}} - u}) (u^{{\varepsilon}})_y K_{\beta}(x)\,\Psi_y(y,t)\,\omega_r(y-z)\,\omega_{r_0}(t-s) \,dX \\
& \le {\varepsilon}K \int_{{\Pi_T}}(u^{{\varepsilon}})_y (x,y,t) \chi^{\alpha_0}_{(\nu,\tau)}(t)
    \left(h_\alpha'(y-L_l(t)) - h_\alpha'(y-L_r(t)-\frac{1}{\alpha})\right) \,dxdydt \\
& \le K {\varepsilon}\alpha \iint {\left|u^{{\varepsilon}}_x\right|} \,dxdy \\
& \le C K \alpha {\varepsilon},\end{aligned}$$ where $K = {\left\|K_{\beta}\right\|}_{\infty}$ and $C = \text{TV}\, (u^{{\varepsilon}})$. Similarly for the other term $$\begin{aligned}
{\left|\int_{Q_T}\mathcal{Q}_{2,2}\right|} \,dX & = {\varepsilon}\int_{Q_T} {\mathrm{sign}}({u^{{\varepsilon}} - u}) (u^{{\varepsilon}})_y  K_{\beta}(x)\, \Psi(y,t)\,\omega_r'(y-z)\,\omega_{r_0}(t-s) \,dX \\
& \qquad \le \frac{K {\varepsilon}}{r} \int_{{\Pi_T}}(u^{{\varepsilon}})_y (x,y,t) {\chi^{\alpha_0}_{(\nu,\tau)}(t)}\,{\chi^\alpha_{(L_l,L_r)}(y,t)}\,dxdydt \\
& \qquad  \le C K \frac{{\varepsilon}}{r},\end{aligned}$$ where again $K = {\left\|K_{\beta}\right\|}_{\infty}$ and $C = \text{TV}\, (u^{{\varepsilon}})$.

Therefore $$\label{eq:qsum}
  {\left|\int_{{\Pi_T}^2} \mathcal{Q}_1 + \mathcal{Q}_2 \,dX \right|} \le
  C {\varepsilon}+ \frac{C}{\beta \alpha } + \frac{C {\varepsilon}}{r} .$$ where $C$ depends on (among other things) $L$ and $T$, but not on the parameters $\alpha_0$, $\alpha$, $r_0$, $r$, $\beta$ or ${\varepsilon}$.

Now we have proved the follwoing Lemma:

\[lem:estimates\] Assume that $u$ and $u^{{\varepsilon}}$ take values in the interval $[-K,K]$ for some positive $K$. Let $M>\max_{v\in
    [-K,K]} {\left|f'(v)\right|}$. Then if $T\ge\tau>\nu\ge 0$ and $L-M \tau >0$, we have $$\label{eq:estimates}
    \begin{aligned}
    \int_{{\mathbb{R}}}\int_{-L+M\tau}^{L-M\tau} &{\left|u^{{\varepsilon}}(x,y,\tau)-u(x,y,\tau)\right|}\,dxdy \\&\le
    \int_{{\mathbb{R}}^2} {\left|u^{{\varepsilon}}(x,y,\nu)-u(x,y,\nu)\right|} \,dxdy\\
    &\quad + C\biggl[
    r_0 + r  +\alpha + {\varepsilon}+ \frac{1}{\alpha \beta} + \frac{{\varepsilon}}{r}\biggr].
    \end{aligned}$$

This follows from and , observing that we can send $\alpha_0$ to zero. Now we let $u(x,y,t)$ be the unique entropy solution of . Also note that since we let $\beta$ tends to $\infty$ so $ \gamma = \frac{1}{\beta}$ is small. We set $\alpha=r=r_0$ and ${\varepsilon}= \gamma$, and assume that $\alpha$ is sufficiently small, then $$\label{eq:vudx-estimate}
  \int_{{\mathbb{R}}}\int_{-L+Mt}^{L-Mt} {\left|{u_{{{\Delta x}}}}(x,t)-v(x,t)\right|}\,dx \,dy\le
  C\left(\alpha+\frac{{\varepsilon}}{\alpha}\right),$$ for some constant $C$ which is independent of the small parameters $\alpha$, ${\varepsilon}$. This follows from . Then setting ${\varepsilon}=\alpha^2$ proves the main theorem.

Acknowledgments {#acknowledgments .unnumbered}
===============

The second author acknowledges support from the Alexander von Humboldt Foundation, through a Humboldt Research Fellowship for postdoctoral researchers.

[99]{}

M.  Escobedo, J.  L. Vazquez, and E.  Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity. , 343(2):829–842, 1994.

K.  H. Karlsen, U.  Koley, and N.  H. Risebro, An error estimate for the finite difference approximation to degenerate convection-diffusion equations.

S. N. Kru[ž]{}kov, First order quasilinear equations with several independent variables. , 81 (123):228–255, 1970.

N. N. Kuznetsov, The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation. , 16(6):1489–1502, 1627, 1976.

Received xxxx 20xx; revised xxxx 20xx.


###### Strengths and limitations of this study

-   Provides insight into the surgical capacity of Somalia.

-   Identifies significant gaps and areas for improvement towards evidence-based planning to improve surgical infrastructure, skills training and equipment and supplies.

-   Does not capture data from every first-referral health facility of the country.

-   In some cases, measures captured are estimates.

Introduction {#s1}
============

Conditions that can be treated with surgery account for 11% of the world's disability-adjusted life years.[@R1] Despite recent data estimating the global volume of surgery at 234 million surgical procedures annually and significant disparities between procedures performed in high-income and low-income countries, global public health initiatives have traditionally neglected the necessity for the provision of surgical services.[@R2] Poor access to surgical services, particularly at rural facilities, results in excess morbidity and mortality from a broad range of treatable surgical conditions including injuries, complications of pregnancy, sequelae of infectious diseases, acute abdominal conditions and congenital abnormalities. Improving access to surgical services in low-income countries requires addressing gaps in infrastructure, training and skills of personnel, appropriate equipment and medications.

Background {#s1a}
----------

Somalia's 21 years of war, armed conflict and insecurity have had a devastating effect on the health sector. The current situation is at its worst since the beginning of the civil war in 1991, with unprecedented levels of child malnutrition and internal displacement, as well as conflict and insecurity rife in parts of the country.[@R3] Continuing conflict has left 90% of health infrastructure looted, damaged or destroyed.[@R4] The low health workforce ratios before the war have depreciated through high levels of healthcare worker emigration, leaving an estimated 3 physicians per 100 000 population (total 253 physicians), 11 nurses per 100 000 population (861 nurses) and 2 midwives per 100 000 population (116 midwives) serving the whole country.[@R5]

Rates of death and disability from treatable surgical conditions continue to be unacceptably high. Somalia's maternal mortality rate is estimated to be 1044 per 100 000 live-births, one of the highest in the region.[@R6] This corresponds to a lifetime risk of one maternal death for every 10 women. The leading causes of maternal death (antepartum and postpartum haemorrhage, obstructed labour and eclampsia) can all be addressed with appropriate emergency obstetric care, which often require surgical and anaesthetic interventions. Trauma, as a consequence of road traffic accidents, armed fighting and landmine explosions, is also common. In 2010 and 2011, WHO reported more than 7500 war-wounded civilians who were admitted to the three main hospitals in Mogadishu, one in five being children and one in three women (WHO. Emergency and Essential Surgical Care. 2013. <http://www.who.int/surgery/en/>).

The health sector faces overwhelming challenges in a country where two decades of lawlessness have resulted in the collapse of central government and three almost autonomously ruled zones: northwest Somalia (NWS), known as Somaliland; northeast Somalia (NES), known as Puntland; and south/central Somalia (SCS). Somalia's public health system is tiered, comprising regional referral hospitals, district hospitals, maternal and child health centres and health posts. However, hospitals and health facilities are limited in number, inadequately distributed, operate using vastly different standards and often cannot provide a minimum package of primary healthcare (PHC) services. As funders and public health experts adopt the expansion of primary healthcare services, the inclusion of surgical services at the first-referral level is critical. The purpose of this survey was to collect knowledge gained from comprehensive quantitative assessments of surgical capacity in Somalia in order to assist in planning strategies for universal access to life-saving and disability-preventing surgical services (WHO. Emergency and Essential Surgical Care. 2013. <http://www.who.int/surgery/en/>).

Similar studies using the WHO Situational Analysis tool have been conducted in Tanzania,[@R7] Solomon Islands,[@R8] Gambia,[@R9] Liberia,[@R10] Ghana,[@R11] Afghanistan,[@R12] Sri Lanka,[@R13] Sierra Leone[@R14] and Mongolia.[@R15]

Objective {#s2}
=========

The objective of this study was to assess life-saving and disability-preventing surgical services (including emergency, trauma, obstetrics, anaesthesia) of health facilities in Somalia and to assist in the planning of strategies for strengthening surgical care systems.

The aim of the study was also to assess the current capacity for essential anaesthesia and surgical services in Somalia with the purpose of providing a benchmark for critical areas needing improvement.

Methods {#s3}
=======

The WHO Tool for Situational Analysis to Assess Emergency and Essential Surgical Care (EESC) was employed to capture a health facility\'s capacity to perform basic surgical (including obstetrics and trauma) and anaesthesia interventions by investigating four categories of data: infrastructure, human resources, interventions provided and equipment availability. The tool queries the availability of eight types of care providers, 35 surgical interventions and 67 items of equipment.[@R7] The answer 'available' was defined as 'fully available for all patients all of the time', while 'sometimes' was defined as 'available with frequent shortages or difficulties' (WHO EESC Situational Analysis Tool, 2012, <http://www.who.int/surgery/publications/WHO_EESC_SituationAnalysisTool.pdf>).

Facilities were also asked the size of the 'population served', intending to quantify the population living in the catchment area. This value represents the number of residents who might use the facility as their first-referral health facility, not the number of patients seen.

The WHO Tool for Situational Analysis to Assess EESC was distributed by the Ministry of Health and WHO Somalia to 20 first-referral level health facilities around the country and was completed at 14 health facilities representing 10 of the 18 regions of Somalia and all three administrative zones. The survey was extended to regional medical officers and facility administrators. Participation in the survey and reporting to the WHO was voluntary. The physical surveys were then obtained either by post directly to the WHO Somalia office (Nairobi, Kenya) or by site visits from the Somalia Ministry of Health (MoH), the WHO country office and members of the Global Initiative for EESC (GIEESC) between September 2011 and April 2012.

Despite efforts to standardise the reporting and collection of much of the data by site visits performed by agents affiliated to the authors, it must be noted that an inherent reporting bias may be present in the data, given that survey data were compiled by different agents at each site. Furthermore, in a limited number of cases, facilities did not respond to individual questions. Blank answers were considered as neither a positive or negative answer, but simply deemed as unreported.

The collected data were then entered into and analysed from WHO Global DataCol Database for EESC.

Results {#s4}
=======

Data were gathered from 14 health facilities in 10 regions (of 18) across Somalia, as featured in [figure 1](#BMJOPEN2013004360F1){ref-type="fig"} and [table 1](#BMJOPEN2013004360TB1){ref-type="table"}. The six facilities that received the survey but did not return information are also listed in [table 1](#BMJOPEN2013004360TB1){ref-type="table"}. Non-reporting can be attributed to lack of consistent operation, inability to gather data, and pervasive political instability and high-security risk throughout the country, making the transmission of this data difficult and costly.

######

Surveyed health facilities (N=14)

  Type of facility (self-identified)    Name of facility          Region            Number of beds   Operating theatres
  ------------------------------------- ------------------------- ----------------- ---------------- --------------------
  General hospital                      Garowe Hospital           Nugaal            51--80           11--20
  Gardo General Hospital                Bari                      21--50            1
  Hergeisa Group Hospital               Woqooyi Galbeed           301--400          5--10
  Provincial hospital                   Borama General Hospital   Awdal             201--300         1
  Burao Regional Hospital               Togdeer                   201--300          1
  Berbera Referral Hospital             Woqooyi Galbeed           101--200          2
  Erigavo Regional Hospital             Sanaag                    81--100           1
  Sool Regional Hospital                Sool                      81--100           1
  Health centre                         Doolow Health Centre      Gedo              5--10            1
  Private/NGO/mission hospital          Galkayo Medical Centre    Mudug             51--80           3--4
  Galkayo Specialist Hospital           Mudug                     3--4              1
  Medina Hospital                       Banadir                   201--300          5--10
  Maani Hospital                        Banadir                   21--50            1
  Manhal Specialty Hospital             Woqooyi Galbeed           51--80            5--10
  Facilities not responding to survey   Shikh Hospital            Woqooyi Galbeed   NA               NA
  Puntland Regional Hospital            Bossaso                   NA                NA
  General Medical Service Hospital      Mudug                     NA                NA
  Wajed Hospital                        Bakool                    NA                NA
  Dinsor Hospital                       Bakool                    NA                NA
  Sama Hospital                         Bay                       NA                NA

NA, not available.

![Participating regions of Somalia.](bmjopen2013004360f01){#BMJOPEN2013004360F1}

Health facility characteristics {#s4a}
-------------------------------

Of the 14 facilities that responded to the survey, 5 were provincial hospitals, 5 were private/mission/NGO hospitals, 3 were general hospitals and 1 was a health centre. Each facility reported having at least 1 operating theatre, with a maximum of 11--20 operating theatres. The populations served by each facility ranged from 100 000 to 1 000 000 people, with the average population of the reporting facilities at 331 250 people.

Infrastructure {#s4b}
--------------

The emergency and surgical infrastructure of the 14 reporting facilities were largely inadequate, with only 22% of the health facilities having oxygen cylinder supplies consistently available. 28% of the facilities never or only sometimes had access to running water and only 50% had a consistently available electricity source. Only 50% of the facilities had fully available access to running water and electricity. Access to a functional anaesthesia machine was only present in 2 (15%) facilities, one of those facilities being a local health centre. Only three facilities reported access to a functioning pulse-oximeter.

Fifty per cent of the facilities had permanent areas designated for emergency care and only 38% of the facilities had the same for postoperative care.

A minority of facilities had formal management guidelines for emergency care (3/14=21%), surgery (5/14=36%), anaesthesia (4/14=29%) or pain relief (1/14=7%).

[Table 2](#BMJOPEN2013004360TB2){ref-type="table"} depicts the key infrastructural elements available in the health facilities assessed.

######

Availability of infrastructure and health resources in Somalia, 2012 (N=14)

                                               Always   Sometimes   Not available
  -------------------------------------------- -------- ----------- ---------------
  Oxygen cylinder                              3        1           10
  Running water                                10       2           2
  Electricity source                           7        5           2
  Operational power generator                  2        2           9
  Blood bank                                   5        3           6
  Haemoglobin and urine testing                12       2           0
  X-ray machine                                8        4           2
  Medical records                              10       3           1
  *Pulse oximeter*                             3        0           8
  *Area designated for emergency care*         5        1           7
  *Area designated for postoperative care*     5        1           7
  *Management guidelines for emergency care*   3        1           10
  *Management guidelines for surgery*          5        0           9
  *Management guidelines for anaesthesia*      4        1           9
  *Management guidelines for pain relief*      1        3           10
  *Functioning anaesthesia machine*            7        5           2

Number of facilities where element is always, sometimes or never available. Italicised items designate those that are specific to quality surgical care delivery (N=14).

Human resources {#s4c}
---------------

More data are needed to comment on human resources within healthcare in Somalia. In this section of the survey, blank answers to staffing questions impeded our ability to comment on the state of the workforce completely. We did, however, summarise the results of the survey questions to which all or a majority of facilities supplied data.

The 14 facilities together reported a total of 137 healthcare providers. Only 15 (11%) individuals were identified as full-time trained surgical specialists, located only at 9 of 14 of the facilities. A total of 14 (11%) were identified as obstetricians, and only 4 (3%) individuals were identified as qualified anaesthesiologists and were located in only 3 of the 14 facilities, with 27 (20%) unlicensed medical staff (nurses, medical assistants, etc) providing anaesthetic services.

Interventions {#s4d}
-------------

Of the 35 interventions queried, only abscess incision and drainage were offered at all 14 facilities. Acute burn management, wound debridement, hydrocoele treatment and male circumcision were widely provided by the surveyed facilities, being offered at 85--86% of all facilities. Caesarean section was offered at 93% of all facilities.

Provision of anaesthesia services was severely lacking, however, with only 36% of facilities able to perform general anaesthesia inhalational procedures. Of the eight facilities that answered why they could not provide this service, one (Doolow Health Center) attributed it to lack of skill and four (Borama General Hospital, Doolow Health Center, Erigavo Regional Hospital, Burao Regional Hospital) to lack of supplies or drugs. One facility (Gardo General Hospital) attributed it to lack of skill and lack of equipment.

The ability of each health facility to provide basic surgical and anaesthesia procedures was assessed ([tables 3](#BMJOPEN2013004360TB3){ref-type="table"} and [4](#BMJOPEN2013004360TB4){ref-type="table"}).

######

Availability of surgical (including obstetrics and trauma) interventions

  General surgery and trauma                                                                                Available
  --------------------------------------------------------------------------------------------------------------- ----------- ------
  Resuscitation (airway, haemorrhage, peripheral percutaneous intravenous access, peripheral venous cut down)     11
  Cricothyroidotomy/tracheostomy                                                                                10
  Chest tube insertion                                                                                9
  Removal of foreign body (throat/ear/eye/nose)                                                                   11
  Acute burn management                                                                                12
  Incision and drainage of abscess                                                                                14
  Suturing (for wounds, episiotomy, cervical and vaginal lacerations)                                             13
  Wound debridement                                                                                11
  Caesarean section                                                                                13
  Dilation and curettage (gynaecology/obstetrics)                                                                 12
  Obstetric fistula repair                                                                                7
  Appendicectomy                                                                                11
  Hernia repair (strangulated, elective)                                                                          10
  Hydrocoele                                                                                12
  Cystostomy                                                                                9
  Urethral stricture dilation                                                                                11
  Laparotomy (uterine rupture, ectopic pregnancy, acute abdomen, intestinal obstruction, perforation, injuries)   12
  Male circumcision                                                                                12
  Congenital hernia repair                                                                                9
  Neonatal surgery (abdominal wall defect, colostomy imperforate anus, intussusceptions)                          5
  Cleft lip repair                                                                                10
  Clubfoot repair                                                                                9
  Contracture release/ skin grafting                                                                              10
  Fracture treatment                                                                                Closed      Open
  10                                                                                8
  Joint dislocation treatment                                                                                9
  Drainage of osteomyelitis/septic arthritis                                                                      9
  Amputation                                                                                11
  Biopsy (lymph node, mass, other)                                                                                8
  Tubal ligation/vasectomy                                                                                9
  Cataract surgery                                                                                4

Number of facilities at which the procedure is available (N=14).

######

Availability of anaesthesia interventions

  Anaesthesia
  ---------------------------------- ----
  Regional anaesthesia blocks        6
  Spinal anaesthesia                 13
  Ketamine intravenous anaesthesia   13
  General anaesthesia inhalational   5

Number of facilities at which the procedure is available (N=14).

Emergency equipment and supplies for resuscitation {#s4e}
--------------------------------------------------

Basic equipment access was also lacking. No facility had access to all basic and essential surgical equipment listed by the tool. Only one hospital always had adult resuscitator bag valves and masks available, and only 2 of 14 facilities reported the same for paediatric bag valves and masks. Sixty-four per cent of the facilities had access to oxygen only sometimes or never, and less than 30% had oropharyngeal airways always available for adults and children. Thirty-nine per cent of institutions were without an examination table.

Basic equipment such as suction catheters, intravenous cannulas and Foley catheters were consistently available in less than 30% of all facilities.

Fifty-seven per cent of the facilities did not have sterile gloves consistently available, and 64% reported the same for examination gloves. Only 64% of the reporting institutions had a steriliser available all the time. Seventy-one per cent did not have face masks available, and 86% were without consistent availability of eye protection. Availability of soap and wash basins was reported in less than two-thirds of the 14 facilities.

The availability of emergency equipment and supplies for resuscitation was assessed in each of the health facilities (depicted in [tables 5](#BMJOPEN2013004360TB5){ref-type="table"}[](#BMJOPEN2013004360TB6){ref-type="table"}--[7](#BMJOPEN2013004360TB7){ref-type="table"}).

######

Availability of capital outlays for resuscitation

  Capital outlays                                    Always   Sometimes   Never
  -------------------------------------------------- -------- ----------- -------
  Resuscitator bag valve and mask
   Adult                                             1        7           6
   Paediatric                                        2        5           7
  Oxygen                                             5        4           5
  Stethoscope                                        9        3           0
  Batteries for flash light                          8        2           4
  Suction pump                                       11       3           0
  Blood pressure monitoring equipment                10       3           1
  Thermometer                                        10       3           1
  Scalpel handle with blade                          9        4           1
  Retractor                                          7        6           0
  Scissors straight 12 cm                            8        5           1
  Scissors blunt 14 cm                               7        6           0
  Oropharyngeal airway
   Adult                                             4        6           4
   Paediatric                                        4        5           5
  Forceps Kocher no teeth                            5        7           1
  Forceps, artery                                    8        6           0
  Kidney dishes stainless steel                      6        6           1
  Capped bottle, alcohol-based solutions             2        5           3
  Gloves (sterile)                                   6        7           1
  Gloves (examination)                               5        8           1
  Needle holder                                      9        5           0
  Steriliser                                         9        5           0
  Nail brush, scrubbing surgeon\'s                   7        5           1
  Vaginal speculum                                   9        2           3
  Bucket, plastic                                    4        4           4
  Drum for sterile compresses, bandages, dressings   8        4           1
  Examination table                                  8        4           1

Number of facilities where element is always, sometimes or never available (N=14).

######

Availability of renewable supplies for resuscitation

  Renewable
  ------------------------------------------------------------- --- ---- ---
  Suction catheter sizes 16 FG                                  2   5    6
  Tongue depressor wooden disposable                            3   7    3
  Nasogastric tubes 10--16 FG                                   2   10   2
  Light source (lamp and flash light)                           4   9    1
  Intravenous fluid infusion set                                3   9    2
  Intravenous cannulas                                          4   8    2
  Scalp vein infusion set                                       3   7    3
  Syringes 2 mL                                                 4   9    1
  Syringes 10 mL                                                4   10   0
  Disposable needles                                            5   8    1
  Sharps disposal container                                     5   8    1
  Tourniquet                                                    4   8    1
  Sterile gauze dressing                                        3   8    2
  Bandages sterile                                              3   7    3
  Adhesive tape                                                 3   9    1
  Needles, cutting and round bodied (for suturing)              4   8    2
  Suture synthetic absorbable                                   4   6    3
  Splints for arm, leg                                          2   8    4
  Towel cloth                                                   2   8    3
  Absorbent cotton wool                                         2   10   1
  Urinary catheter Foleys disposable \#12, 14 and 18 with bag   3   7    3
  Sheeting, plastic for examination table                       4   7    3
  Waste disposal container                                      4   8    2
  Face masks                                                    4   8    2
  Eye protection                                                2   5    7
  Apron, plastic reusable                                       3   9    2
  Soap                                                          5   9    0
  Wash basin                                                    4   8    1

Number of facilities where element is always, sometimes or never available (N=14).

######

Availability of supplementary equipment for resuscitation

  Supplementary equipment
  --------------------------------- --- --- ----
  Magills forceps
   Adult                            3   2   9
   Paediatric                       1   2   10
  Endotracheal tubes                1   2   10
  Intravenous infusor bags          1   2   10
  Chest tubes insertion equipment   1   3   9
  Laryngoscope
   Adult                            3   3   7
   Paediatric                       1   3   9
  Cricothyroidotomy set             1   1   11

Number of facilities where element is always, sometimes or never available (N=14).

Limitations {#s5}
===========

There are several limitations to this survey. It provides only a brief overview of the capacity for surgical care and cannot be used for detailed programme planning. Although not every first-referral health facility in Somalia was surveyed, the majority identified by the MoH and WHO Somalia did reply and this allows a credible insight into the surgical capacity of the country.

Some of the data was taken after the WHO IMEESC toolkit was implemented in the country, and results should be qualified by the possibility of interpretation constraints of the survey.

Discussion {#s6}
==========

Surgically treatable diseases are among the top 15 causes of disability,[@R16] and conservative estimates show that 11% of the world\'s disability-adjusted life years (DALYs) stem from surgically treatable conditions.[@R1] In 2004, South East Asia and Africa accounted for 54% of the global burden,[@R17] and since then the burden of surgical disease is thought to be increasing.[@R16]

More than two billion people globally are without access to surgical care.[@R18] Of the 243 million surgical procedures performed globally each year, 34.8% of the world\'s population living in low-income countries only has access to 8.1 million (3.5%) of such procedures.[@R2] As a result, rates of maternal mortality are high, minor surgical pathologies become lethal and treatable trauma progresses to death.

Despite the great, unmet need and inclusion in policy-level dialogue on primary healthcare, the delivery of emergency surgical care has long been sidelined in global financing and policy debates, largely due to perceived complexity and cost. Human resources, metrics and science focused on global surgery, and sustained financing mechanisms for surgical infrastructure lag behind other public health priorities, despite the growing need for access to surgical care.[@R19]

Enormous shortfalls in infrastructure, supplies and procedures undertaken are common at district-level health facilities in LMICs.[@R20] Every \$1 spent strengthening local surgical capacity generates \$10 through improved health and increased productivity.[@R21] In 2008, the Copenhagen Consensus Center\'s research identified strengthening surgical capacity, particularly at the district hospital level, as a highly cost-efficient solution to global diseases. Notably, strengthening local surgical capacity is an approach that would provide a high degree of financial protection to populations and address the DALYs in a cost-effective manner.[@R1] In fact, the benefit--cost ratio for expansion of surgical capacity at district hospitals was found to be 10:1.[@R22] Furthermore, due to the implementation of 'vertical' programmes in public health (ie, child health, maternal health, cancer and trauma), in which most programmes contain some surgical component, strengthening surgical services may improve the health system overall.[@R23]

This study provides an overview of the capacity for surgical care representing health facilities in all three administrative zones of Somalia, including 10 of the 18 regions and demonstrates the significant gaps in infrastructure, life-saving and disability-preventing surgical interventions and essential equipment. Of the 14 reporting health facilities, 12 identified as full hospitals, serving an average patient population of over 300 000 people, further indicating the need for surgical care system strengthening.

Of the 35 basic surgical and anaesthesia interventions queried, most hospitals were unable to provide all of the basic surgical services. Major essential procedures such as caesarean section, appendicectomy, elective hernia and hydrocoele repair and laparotomy were offered at a majority of hospitals, but this conflicted directly with the continuous availability of basic and necessary supplies and infrastructures. Only three facilities had consistent access to oxygen cylinders, and even less reported access to tubing, valves and masks.

Necessary procedures including resuscitation and airway management were offered at a majority (79%) of the facilities. Facilities that weren\'t able to provide resuscitation, however, all reported provision of major surgical procedures. Such surgical procedures require provision and knowledge of resuscitation in order to be carried out safely. The facilities in question could not be contacted for confirmation.

Many facilities lacked access to key elements such as provisional oxygen equipment, oropharyngeal airways and intravenous cannulas. Only two facilities reported access to functional anaesthesia machines, and few facilities noted the ability to provide general anaesthesia inhalation. Oxygen and anaesthesia are essential medicines, (WHO Model List of Essential Medicines, WHO, 2013. <http://www.who.int/medicines/publications/essentialmedicines/18th_EML_Final_web_8Jul13.pdf>) and the existing deficit limits the safety and efficacy of life-saving surgeries in Somalia. It can also result in significant complications or patient suffering when these medicines are not available.[@R24]

A surprisingly low number of facilities reported having access to equipment necessary for prevention of HIV transmission and other infection control. Less than half of the facilities reported access to sterile gloves, waste disposal containers, face masks, eye masks, aprons and even soap, all key components in the reduction of HIV transmission (WHO Prevention of Transmission of HIV, 2007, <http://www.who.int/surgery/publications/HIVprevention.pdf>)

A majority of the facilities lacked any formal guidelines for the management of emergency care, surgery, pain relief or anaesthesia, providing a significant barrier to the strengthening of emergency and surgical care services and severely limiting their ability to respond to large-scale disaster and crisis situations.

The severe gaps in infrastructure and capital are also obstacles for the delivery of basic primary care in these facilities. Absence of key infrastructural elements such as water, electricity, and oxygen was common in the surveyed facilities. Lack of sterilisation tools and supplies for infection prevention was also a common feature. Establishing management guidelines for surgical infrastructure and targeting strategies to address the aforementioned lacks in infrastructure is a priority. This would not only increase the quality of surgical services, but also that of basic primary care services offered by these facilities.

Strengthening of surgical systems will reduce the disease burden of surgically related issues and increase progress towards achieving the 2015 Millennium Development Goals (MDGs). Strengthening surgical care delivery can help achieve MDGs 1 (eradication of poverty),[@R18] 4 (child health), 5 (maternal health) and 6 (HIV/AIDS prevention).[@R22] Investing in oxygen and basic infrastructure, such as electricity and running water, will also serve to benefit patients suffering from a range of conditions.

To meet the needs of LMICs, the WHO EESC programme, with its partners worldwide, developed the *Integrated Management for Emergency and Essential Surgical Care* (IMEESC) toolkit (WHO IMEESC Toolkit: <http://www.who.int/surgery/en>). This toolkit guides policies, research for evidence-based planning (WHO EESC Situation Analysis Tool to assess emergency, trauma, obstetrics, anaesthesia and surgical services, WHO EESC Global Database), training, best practice protocols monitoring and evaluation of the progress of surgical care systems. The IMEESC toolkit provides guidance for policy-makers, managers and providers at various levels of care. It also includes instructional videos, training modules on emergency and trauma for frontline health workers, disaster management resources and monitoring and evaluation.

This study provides insight into the surgical capacity of Somalia, using the WHO Tool for Situational Analysis to Assess EESC. Addressing the gaps identified in this study is needed not only to strengthen surgical care, but also holistic health systems in Somalia.[@R25] Further research is needed to determine the burden of surgical disease in Somalia and to determine the cost--benefit of specific interventions to improve surgical services.

Supplementary Material
======================

###### Author\'s manuscript

###### Reviewer comments

The following members of GIEESC in Somalia were instrumental in obtaining the reported data: Dr Yassin Arab Abdi, Dr Nima Abdi Hassan, Dr Mohamed Samy Salama, Dr Mahmoud Mostafa El-Antably, Dr Jamal Mohamed Jama, Dr Abdulcadir Mohamud Giama, Dr Nura Mohamed Ahmed, Dr Saeed Hussien Legeh, Dr Risaaq Mohamoud Yousuf, Dr Cabdi Qaadu Bikri, Dr Ahmed Jama, Dr Muse Geele and Dr Noor M Noor.

**Contributors:** NE and AS analysed and interpreted the data, drafted the article, revised it and finally approved the submitted version. MC contributed to the conception and design, acquisition and interpretation of the data, helped draft the article, made critical revisions and finally approved the submitted version. OS, ME, GP and AI contributed to acquisition of the data, made critical revisions and finally approved the submitted version.

**Funding:** This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

**Disclaimer:** The authors include staff members of WHO. They are responsible for the views expressed in this publication and do not necessarily represent the decisions or stated policy of WHO.

**Competing interests:** None.

**Provenance and peer review:** Not commissioned; externally peer reviewed.

**Data sharing statement:** No additional data are available.


Q:

Why is the Lagrangian for space-like geodesics equal to 1?

In Schwarzschild spacetime, the Lagrangian can be defined as
$$ L = -\left( 1 - \frac{2M}{r} \right) \dot{t}^2 + \left( 1- \frac{2M}{r} \right)^{-1} \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta \dot{\phi}^2 $$
where the dot denotes a derivative $\frac{\mathrm d}{\mathrm d\lambda}$ for an affine parameter $\lambda$.
From Noether's theorem, one can easily prove that the Lagrangian itself if an integral of motion. One can also prove that $\theta = \text{const} =  \pi/2$ without loss of generality. Hence, the quantity $$Q = -\left( 1 - \frac{2M}{r} \right) \dot{t}^2 + \left( 1- \frac{2M}{r} \right)^{-1} \dot{r}^2 + r^2 \dot{\phi}^2$$ is a constant of motion.
In my lecture notes, it is claimed without proof that $Q = 1$ for space-like geodesics. It seems like this is rather obvious from the way it is stated but I can't seem to find any proof of this anywhere in my lecture notes nor from Google (after a little bit of searching). Hence, my question is why is this the case and how can one prove this?

A:

The solutions to the Euler-Lagrange (EL) equations for OP's non-square root Lagrangian $L$ are affinely parametrized geodesics.
Noether's theorem yields that $L$ is a constant (of motion) along a geodesic, but it is not necessarily $1$. For more information, see e.g. my Phys.SE answer here.
The constant is $1$ if one picks the parameter $\lambda$ to be arclength $s$.


Back in February Environment Minister Rob Stokes announced funding for a new chicken pen and storage shed at John Therry Catholic High School to help students learn about sustainable living. So what are the benefits of having a chicken pen?

Backyard chickens are healthier Factory farmed chickens are kept in confined and unsuitable areas and are often fed an unnatural diet with hormones to increase growth on the cheap. Unhappy surroundings add to the stress of the chickens, this affects the taste and nutrients of the eggs and also limits the amount of eggs each hen can produce. If you're not convinced, check out our blog on chicken slaughterhouses. The eggs are more nutritious Eggs from the backyard have 25% more vitamin E, a third more vitamin A, and 75% more beta carotene. They also contain more omega-3 fatty acids than factory or farmed eggs.

“This new chicken pen will provide an authentic animal husbandry learning experience for students. It will also help the whole school community share social skills and build an understanding of sustainable living.” - Bryan Doyle, Member for Campbelltown

Chickens do great things for your compost Chicken poo is high in nitrogen which is really good for your compost bin. The best part? You can even compost their egg shells. The eggs taste better Eggs in the grocery store can sit there for weeks! Air can seep into the porous eggshell and it affects the nutrients, taste and consistency of the eggs. Fresh eggs should have firm whites and really bright orange yolks. Another great thing: no more bulky egg cartons in your fridge!

They are a great lesson for your kids I believe it's important for kids (and adults!) to have a connection to their food, and understand where it came from: the farm-to-plate chain. They will also love collecting the eggs, and learn how to care for animals in the process.

“By encouraging social responsibility and modelling sustainable practices to students, this project will have a positive impact on the wider community.” - Doyle on the Student Chicken Pen


Conference: Energy policy must take into account the global, European and national aspects

World Energy Council (WEC) Estonia held the annual energy policy conference in Tallinn; the conference was titled “Energy Trilemma – Estonia’s position in the World”. Conference was attended by representatives of Estonian parliament, ministries and many different stakeholders. In was concluded that energy policy must be synchronized to a greater degree (both regionally and in the EU) and a wide variety of different resources and technologies must be used.

Current Estonian Energy Policy

In this year’s WEC energy sustainability index Estonia placed 35 out of 94 WEC member states. Compared to last year Estonia climbed 3 places in the index. The figures are equally mediocre in all the categories: security of supply, environmental impact mitigation, and social equity.

There is ample room for improvement in all these categories. The index also illustrates the fact that Estonian energy policy has been very balanced throughout the years, none of the aspects is lagging behind too much. On a side note, it has to be mentioned that Estonia’s relatively good position was achieved thanks to the overall development of our society. Had it been only the energy related indicators, Estonia would have fared far worse – somewhere in the second half of the list.

Being a small country, Estonia has certain limitations which influence our performance in the index. The bigger the GDP of a country, the better the score in affordability of energy, and the lower the emissions per unit of GDP. Looking at Estonia’s neighbours it can be seen that almost all of them are ranked higher in the index: Latvia is 37th, Lithuania 31st, Finland 5th and Sweden 1st. Out of 5 Nordic-Baltic countries we are second to last.

Estonia must diversify its electricity production. Today most of electricity produced in Estonia comes from oil shale fired power plants. In addition to not being diverse and therefore posing a risk to security of supply, oil shale fired power plants also have quite a big environmental impact, which lowers our overall score in the sustainability index.

Our neighbours on the other hand use a wide range of different technologies and energy sources: coal, nuclear energy, natural gas, hydro and wind. Having this kind of variety is not only good for security of supply, but it also limits country’s vulnerability to outside supply and price risks.

Estonia as a small country faces the challenges how to diversify its energy portfolio, how to limit the environmental impacts, and how to keep costs to the final consumer down. “We have been focusing on one resource, oil shale, for a very long time. This must change, but we must also guarantee that the equilibrium wouldn’t shift totally to the other extreme. This wouldn’t be neither in the interest of the consumers nor the environment. If we speak about our energy portfolio, then diversification must be our goal,” said the Minister of the Environment Keit Pentus-Rosimannus.

Conflicting Energy PoliciesEinari Kisel, Senior Fellow of European Policies in the World Energy Council gave a keynote presentation at the conference. He said that conflicting policies are not unique to Estonian policy – this is a common trait for all European nations. “Unlike other regions of the World, Europe is very liberal. We have a large number of stakeholders trying to shift the balance in their favour. The final outcome is highly dependent on the stakeholders involved,” Kisel commented.

Einari Kisel described Estonia as a country with a very open economy, which in turn makes us very dependent of the global changes. “The global developments will influence us in the future even more than today. This is something we must remember and take into account. People might think that we are constantly taking U-turns with our policy, but in reality this is just the government reacting to global changes,” Kisel said.

Keit Pentus-Rosimannus was happy to see that Estonia’s position has improved during the years. “Already the fact that environmental impact is under discussion today, is a great step in the right direction. Not too long ago environment was a topic that was largely ignored. If there was a possibility to talk only of the positive, this was done,” Pentus-Rosimannus concluded.

Juhan Parts, the Minister of Economic Affairs and Communication focused on three aspects of Estonian energy policy: global context, national aspects, and balancing the energy trilemma.

“The global context is full of contradicting beliefs – some believe we are running out of oil, others say there has never been more; some say a climate catastrophe is coming, others say just the opposite,” said Parts. He added: “Estonia cannot copy its energy policy from other countries; we have our own national aspects that we must adhere to. We have oil shale, border with Russia and vast infrastructure already in place.” Third point Parts made was regarding the balance between having clean but also affordable energy.

Emissions Trading and Environmental Taxes

Einari Kisel finds that the principles of European climate policy are right, but how to implement this should be rethought so it would give right signals to the market participants and to the wider audience.

“The question is if emissions trading is built on the right foundations? Maybe we should have tax on CO2 or the minimum price pre-set, these are things discussed in Europe right now. For example Germany is having a lot of problems because of high energy prices and because of the uncertainties posed by CO2 price in the future. Companies are closing shop and moving elsewhere,” Kisel added.

Sandor Liive, CEO of Eesti Energia proposed having higher resource taxes for oil shale and setting up a fund for investing income from our national resources, as has been done in Norway and Denmark. Einari Kisel agreed with the proposal but also cautiously added that the government should also have a plan what to do if there is no more income from oil shale.

“For example, common electricity market with the Nordic countries means that power plants in Narva might not be competitive. This would mean lower environmental impact, but at the same time smaller income for the government. These possibilities must be taken into account,” said Kisel. Keit Pentus-Rosimannus mentioned that in Estonia all the environmental fees and taxes are invested to combat the impact to the environment.

Long-Term Plan

Einari Kisel said that Estonia has set itself a number of goals, all of which have been achieved and now we should think what should be the next goal.

“Sometimes national goals have undesired outcomes, companies don’t use the methods hoped by the government, instead companies look for the most profitable way to achieve these goals. One concrete example is Germany where high subsidies for PV were thought to boost local industry, the real outcome is that German PV manufacturers are bankrupt and the Chinese have taken over the market. So Estonia must be sure to find measures that actually boost our development,” Kisel elaborated.

Juhan Parts on the other hand found, that a country as small as Estonia must always follow the lead of others and must always remain flexible to react to the changes on the global market. “We are not rich enough to experiment with immature technologies,” Parts added.

Deputy Secretary General for Energy in the Ministry of Economic Affairs and Communication, Ando Leppiman, said that the ministry has plans that are dependent on the regional and European context. Estonia wants to be able to import its surplus of renewable energy via the mechanism of statistical transfers. Ando Leppiman continued that Estonia must always be flexible; this is one of the greatest strengths a small country has.

“We must have a long-term strategy, but this strategy must be such that we keep ourselves options to choose the best alternative available. We mustn’t limit ourselves to certain technologies and we mustn’t set ourselves unnecessary restrictions. We should be ready to support off-shore wind parks if this is the best option available or nuclear if this should happen to be the best technology on the market,” Leppiman said.

Martin Kruus, head of Estonian Wind Energy Association said that Estonia, and Europe as a whole, must become more integrated. “What’s the use of investing into cross-border transmission lines, if these are not used?” Kruus rhetorically asked. He added that the main point of having an integrated system is to use the strengths of each country so that in the end everybody would win.

Raul Kotov, Board Member in Eesti Gaas, said that having plans makes sense, but plans should be universal and not limit competition between energy sources and technologies. “It is important to give everybody a fair opportunity on the market. The government shouldn’t subsidize energy sources but instead should invest into R&D to find new and better ways to fuel our development,” found Kotov.

CEO of Tallinna Elektrijaam Andres Taukar thought it necessary to have long-term plans. “District heating infrastructure built today must remain in operation for the next 50 years. One must have long-term plans for such investments,” he firmly added.

Founded in 1923, the World Energy Council is the only truly global and inclusive forum for thought-leadership and tangible engagement committed to our sustainable energy future. Estonia joined WEC in 1937 in Paris, where Estonia was represented by the National Power Committee headed by Director J.Veerus and Professor P. Kogerman. In 1998, the 17th World Energy Congress in Houston formally reinstated Estonia’s membership of WEC. Today the Secretary General of WEC Estonia is Mihkel Härm.