---
author:
- 'Anton Izosimov[^1]'
bibliography:
- 'gasm.bib'
title: Generalized argument shift method and complete commutative subalgebras in polynomial Poisson algebras
---

Introduction
============

Let ${\mathfrak{g}}$ be a finite-dimensional Lie algebra. For the sake of simplicity we shall assume that the ground field is ${\mathbb{C}}$, however everything works over an arbitrary field of characteristic zero. The symmetric algebra $S({\mathfrak{g}})$ can be naturally identified with the algebra of polynomials on the dual space ${\mathfrak{g}}^*$ and carries a natural Poisson bracket (*Lie-Poisson bracket*) which is defined on linear functions by $\{\xi,\eta\} = [\xi,\eta]$ and is extended to all polynomials by the Leibnitz identity.

We will be interested in commutative subalgebras in $S({\mathfrak{g}})$. Let $C \subset S({\mathfrak{g}})$ be a commutative subalgebra. Then the maximal possible transcendence degree of $C$ is $$b({\mathfrak{g}}) =  \frac{1}{2}(\dim {\mathfrak{g}}+ {\mathrm{ind} \, {\mathfrak{g}}}).$$ If ${\mathrm{tr.deg.} \, C} = b({\mathfrak{g}})$, then $C$ is called a *complete commutative subalgebra*. Each complete commutative subalgebra in $S({\mathfrak{g}})$ can be interpreted as an integrable system on the Poisson manifold ${\mathfrak{g}}^*$.

One of the most universal methods for constructing “large” commutative subalgebras in $S({\mathfrak{g}})$ is the so-called *argument shift method*. This method was introduced by Mischenko and Fomenko [@MF] as a generalization of the Manakov construction [@Manakov] for the Lie algebra ${\mathfrak{so}}(n)$.

The argument shift method can be described as follows. Let $a \in {\mathfrak{g}}^*$ be an arbitrary regular element. Then there exist $m = {\mathrm{ind} \, {\mathfrak{g}}}$ analytic functionally-independent invariants of the coadjoint representation defined in a small neighborhood of $a$. Denote these invariants by $f_1, \dots, f_m$. For each $i = 1, \dots, m$ expand the function $f_i(a+\lambda x)$ in powers of $\lambda$: $$f_i(a+\lambda x) = \sum_{j=0}^\infty f_{ij}(x)\lambda^j$$ where all functions $f_{ij}(x)$ are polynomials. Denote the algebra generated by all these polynomials by ${\mathcal{F}}_a$ . Then, as was proved by Mischenko and Fomenko, ${\mathcal{F}}_a$ is a commutative subalgebra in $S({\mathfrak{g}})$. Moreover, if ${\mathfrak{g}}$ is semisimple, then ${\mathcal{F}}_a$ is complete.

Note that our description of the argument shift method is slightly different from the original description which required that the invariants $f_1, \dots, f_m$ are polynomial. The modification of the argument shift method presented here is due to Brailov (see Bolsinov [@BPC]).

The completeness criterion for sublagebras ${\mathcal{F}}_a$ was found by Bolsinov [@Bolsinov]. Namely, he proved that ${\mathcal{F}}_a$ is complete if and only if the set of singular elements ${\mathrm{Sing}}\subset {\mathfrak{g}}^*$ has codimension at least $2$ (if the ground field ${\mathbb{K}}$ is not algebraically closed, one should consider the singular set in ${\mathfrak{g}}^* \otimes \overline {\mathbb{K}}$ where $\overline {\mathbb{K}}$ is the algebraic closure of ${\mathbb{K}}$; see Bolsinov and Zhang [@BZ] for details).

Although the family ${\mathcal{F}}_a$ is in general not complete, Mischenko and Fomenko stated the following conjecture: for each finite-dimensional Lie algebra ${\mathfrak{g}}$, there exists a complete commutative subalgebra $C \subset S({\mathfrak{g}})$. This conjecture was proved by Sadetov [@Sadetov], see also Bolsinov [@bolsinov2012complete]. However, Sadetov’s construction is essentially different from the argument shift method. In particular, Sadetov’s subalgebras are commutative only with respect to the standard Poisson structure, while Mischenko-Fomenko subalgebras ${\mathcal{F}}_a$ have the following remarkable property: they are commutative with respect to two Poisson structures, one of which is standard, and the second one (the so-called *frozen argument bracket*) is defined as follows. It is given on linear functions by $\{\xi,\eta\}_a =  \langle a, [\xi,\eta] \rangle$, where $\langle \, ,\rangle$ denotes the pairing between ${\mathfrak{g}}$ and ${\mathfrak{g}}^*$, and is extended to all polynomials by the Leibnitz identity.

Our aim is to show that when the Mischenko-Fomenko subalgebra ${\mathcal{F}}_a$ is incomplete, i.e. when the singular set ${\mathrm{Sing}}$ has codimension one, then there is a natural extension $\widetilde {\mathcal{F}}_a \supset {\mathcal{F}}_a$ which is also commutative with respect to both brackets $\{f,g\}$ and $\{f,g\}_a$, and to give a completeness criterion for $\widetilde {\mathcal{F}}_a$.

In their paper [@BZh], Bolsinov and Zhang formulated the following “generalized argument shift” conjecture: [for each Lie algebra ${\mathfrak{g}}$ there exists a subalgebra in $S({\mathfrak{g}})$ which is commutative with respect to both brackets $\{f,g\}$ and $\{f,g\}_a$]{}. In this way, our note is a step towards the proof of this conjecture.

Generalized argument shift method
=================================

Let ${\mathfrak{g}}$ be a finite-dimensional Lie algebra over ${\mathbb{C}}$, and let $x \in {\mathfrak{g}}^*$. Let $${\mathfrak{g}}_x = \{  \xi \in {\mathfrak{g}}\mid {\mathrm{ad}}^*_\xi(x) = 0\}$$ be the stabilizer of $x$ w.r.t. the coadjoint representation. The set ${\mathrm{Sing}}$ of singular elements in ${\mathfrak{g}}^*$ is $${\mathrm{Sing}}= \{x \in {\mathfrak{g}}^* \mid \dim {\mathfrak{g}}_x > {\mathrm{ind} \, {\mathfrak{g}}}\}$$ where $${\mathrm{ind} \, {\mathfrak{g}}} = \min_{x \in {\mathfrak{g}}^*} \dim {\mathfrak{g}}_x$$ is the *index* of ${\mathfrak{g}}$.

We consider the case when ${\mathrm{Sing}}$ has codimension one. Let ${\mathrm{Sing}}_0$ be the union of all irreducible components of ${\mathrm{Sing}}$ which have maximal dimension. The set ${\mathrm{Sing}}_0$ is the zero set of a certain homogenous polynomial $ p_{\mathfrak{g}}(x)$. It is easy to see that $p_{\mathfrak{g}}$ is a semi-invariant of the coadjoint representation. Following Ooms and Van den Bergh [@Ooms], we call it the *fundamental semi-invariant* of ${\mathfrak{g}}$ (see also Joseph and Shafrir [@JS]).

More precisely, the fundamental semi-invariant is defined as follows. Let $t =  \dim {\mathfrak{g}}- {\mathrm{ind} \, {\mathfrak{g}}}$. Fix a basis in ${\mathfrak{g}}$, and let $c_{ij}^k$ be the structure constants in this basis. Then an element $x \in {\mathfrak{g}}^*$ is singular if and only if the rank of the matrix $\mathcal A_x = c_{ij}^kx_k$ is less than $t$, i.e. if Pfaffians of all principal $t \times t$ minors of $\mathcal A_x$ vanish. Define $p_{\mathfrak{g}}$ as the greatest common divisor of all these Pfaffians. Clealry, the zero set of $p_{\mathfrak{g}}$ coincides with ${\mathrm{Sing}}_0$. However, $p_{\mathfrak{g}}$ is not necessarily the minimal polynomial which define ${\mathrm{Sing}}_0$.

Now we use the fundamental semi-invariant $p_{\mathfrak{g}}$ to define the extended Mischenko-Fomenko subalgebra. Consider the polynomial $p_{\mathfrak{g}}(a+\lambda x)$ and expand it in powers of $\lambda$: $$p_{\mathfrak{g}}(a+\lambda x) =  \sum_{i=0}^{n} p_{i}(x)\lambda^i$$ where $n = \deg p_{\mathfrak{g}}$. Define the *extended Mischenko-Fomenko subalgebra* $\widetilde {\mathcal{F}}_a$ as a subalgebra in $S({\mathfrak{g}})$ generated by all elements of the classical Mischenko-Fomenko subalgebra $\mathcal F_a$ and the polynomials $p_1, \dots, p_{n}$.

\[thm1\] For each regular $a \in {\mathfrak{g}}^*$, the [extended Mischenko-Fomenko subalgebra]{} $\widetilde {\mathcal{F}}_a$ is commutative with respect to both brackets $\{f,g\}$ and $\{f,g\}_a$.

This statement is, in fact, not new. Firstly, it was proved by Arhangel’skiǐ[@Arh] that if two semi-invariants $f,g$ commute with respect to the Lie-Poisson bracket, then their shifts, i.e. functions of the form $f(x+\lambda a)$, $g(x + \mu a)$ where $a \in {\mathfrak{g}}^*$ is fixed, also commute. This statement easily implies that the extended Mischenko-Fomenko subalgebra $\widetilde {\mathcal{F}}_a$ is commutative with respect to the Lie-Poisson bracket. Moreover, the proof of Arhangel’skiǐ can be easily modified to show that $\widetilde {\mathcal{F}}_a$ is commutative with respect to the frozen argument bracket as well (see Section \[pt1\]). We also note that the assumption that $f$ and $g$ commute is in fact satisfied for arbitrary semi-invariants (see Ooms and Van den Bergh [@Ooms] and Section \[pt1\]).

Secondly, Theorem \[thm1\] can be viewed as a generalization of Proposition 7 of Bolsinov and Zhang [@BZh] which asserts that the functions $p_1, \dots, p_n$ are in involution with respect to both Lie-Poisson bracket and frozen argument bracket.

Thirdly, Theorem \[thm1\] follows from the following general construction from the theory of compatible Poisson brackets. Let $\mathcal A$ and $\mathcal B$ be compatible Poisson brackets. Consider the family $\mathcal F$ generated by Casimir functions of all generic linear combinations of $\mathcal A$ and $\mathcal B$. This family is commutative with respect to both $\mathcal A$ and $\mathcal B$ (see Reiman and Semenov-Tyan-Shanskii [@Reiman]), however it may be incomplete or even empty. In this case, ${\mathcal{F}}$ can be extended by adding eigenvalues of the operator $\mathcal A \mathcal B^{-1}$ (which is still well-defined on a certain quotient space even if $\mathcal B$ is degenerate, see e.g. [@sbs]). Denote the extended family by $\widetilde {\mathcal{F}}$.

\[prop1\] The family $\widetilde {\mathcal{F}}$ is commutative with respect to both brackets $\mathcal A$ and $\mathcal B$.

If we apply this construction to the Lie-Poisson bracket and the frozen argument bracket, we get Theorem \[thm1\].

Although we were not able to find the statement of Proposition \[prop1\] in the literature, we believe that it is well known to experts in the field. See [@Magri; @GD; @Reiman; @MM] where different constructions of integrable systems related to compatible Poisson brackets are discussed. Also note that the relation between the argument shift method and compatible Poisson brackets was probably first mentioned by Meshcheryakov [@Meshcheryakov].

Completeness criterion
======================

Let ${\mathrm{Sing}}_{sr} \subset {\mathrm{Sing}}_0$ be the subset of ${\mathrm{Sing}}_0$ which consists of subregular elements, i.e. $${\mathrm{Sing}}_{sr} = \{x \in {\mathrm{Sing}}_0 \mid \dim {\mathfrak{g}}_x = {\mathrm{ind} \, {\mathfrak{g}}} + 2\}.$$ It is clear that ${\mathrm{Sing}}_{sr}$ is open in ${\mathrm{Sing}}_0$, however it is not necessarily dense and may be empty. Let also ${\mathrm{Sing}}_{sm} \subset {\mathrm{Sing}}_0$ be the set of points where ${\mathrm{Sing}}_0$ is smooth. This set is open and dense in ${\mathrm{Sing}}_0$.

Denote the two-dimensional non-Abelian Lie algebra by $\mathfrak{b}_2$, and the $2n+1$-dimensional Heisenberg algebra by $\mathfrak{h}_{2n+1}$.

\[class\] Let $x \in {\mathrm{Sing}}_{sr} \cap {\mathrm{Sing}}_{sm}$. Then ${\mathfrak{g}}_x$ is isomorphic to one of the following Lie algebras:

1.  $\mathfrak{b}_2 \oplus \mbox{Abelian Lie algebra of dimension } {\mathrm{ind} \, {\mathfrak{g}}}$;

2.  $\mathfrak{h}_3  \oplus \mbox{Abelian Lie algebra of dimension } {\mathrm{ind} \, {\mathfrak{g}}} - 1$;

3.  Abelian Lie algebra of dimension ${\mathrm{ind} \, {\mathfrak{g}}} + 2$.

Corollary 2.1 of [@deral] implies that the derived algebra of ${\mathfrak{g}}_x$ is at most one-dimensional. It is easy to see that any Lie algebra with this property is either Abelian, or isomorphic to one of the following:

-   $\mathfrak{b}_2 \oplus \mbox{Abelian}$;

-   $\mathfrak{h}_{2n+1}  \oplus \mbox{Abelian}$;

Now, using that $\dim {\mathfrak{g}}_x = {\mathrm{ind} \, {\mathfrak{g}}}+2$ and the inequality ${\mathrm{ind} \, {\mathfrak{g}}}_x \geq {\mathrm{ind} \, {\mathfrak{g}}}$, we obtain the above list.

The inequality ${\mathrm{ind} \, {\mathfrak{g}}}_x \geq {\mathrm{ind} \, {\mathfrak{g}}}$ is true for any Lie algebra ${\mathfrak{g}}$ and any $x \in {\mathfrak{g}}^*$. It is known as the Vinberg inequality. See Panyushev [@Pan].

Consider the set $${\mathrm{Sing}}_{\mathfrak b} = \{ x \in {\mathrm{Sing}}_{sr} \cap {\mathrm{Sing}}_{sm} \mid {\mathfrak{g}}_x \simeq \mathfrak b_2 \oplus \mbox{Abelian}\} \subset {\mathrm{Sing}}_{sr} \cap {\mathrm{Sing}}_{sm} \subset {\mathrm{Sing}}_0.$$ It is easy to see that ${\mathrm{Sing}}_{\mathfrak b} $ is open in ${\mathrm{Sing}}_{sr} \cap {\mathrm{Sing}}_{sm}$ and thus in ${\mathrm{Sing}}_0$.

\[thm2\] The extended Mischenko-Fomenko subalgebra $\widetilde {\mathcal{F}}_a$ is complete if and only if the set $ {\mathrm{Sing}}_{\mathfrak b}$ is dense in ${\mathrm{Sing}}_0$.

Assume that for each irreducible component $S_i$ of the variety ${\mathrm{Sing}}_0$ there exists at least one $x \in S_i$ such that ${\mathfrak{g}}_x \simeq \mathfrak b_2 \oplus {\mathbb{C}}^{{\mathrm{ind} \, {\mathfrak{g}}}}$. Then the extended Mischenko-Fomenko subalgebra $\widetilde {\mathcal{F}}_a$ is complete. In particular, if ${\mathrm{Sing}}_0$ is irreducible, and there exists at least one $x \in {\mathrm{Sing}}_0$ such that ${\mathfrak{g}}_x \simeq \mathfrak b_2 \oplus {\mathbb{C}}^{{\mathrm{ind} \, {\mathfrak{g}}}}$, then $\widetilde {\mathcal{F}}_a$ is complete.

Theorem \[thm2\] can also be formulated as follows. For each irreducible component $S_i$ of the variety ${\mathrm{Sing}}_0$, there exists an open subset $\widetilde S_i$ such that all elements of this subset have isomorphic stabilizers. The extended Mischenko-Fomenko subalgebra $\widetilde {\mathcal{F}}_a$ is complete if and only if ${\mathfrak{g}}_x \simeq \mathfrak b_2 \oplus {\mathbb{C}}^{{\mathrm{ind} \, {\mathfrak{g}}}}$ for each $i$ and each $x \in \widetilde S_i$.

We also note that Theorem \[thm2\] can be generalized to the case of arbitrary compatible Poisson brackets.

Proof of Theorem 1 {#pt1}
==================

Though the statement of Theorem 1 follows from from considerations of Sections \[shiftsSec\] and \[LA\], we give an independent algebraic proof. As a matter of fact, we prove a stronger statement: if ${\mathfrak{g}}$ is a complex Lie algebra, and $a\in {\mathfrak{g}}^*$, then for any two semi-invariants $f,g \in S({\mathfrak{g}})$ and any $\lambda, \mu \in {\mathbb{C}}$, $$\begin{aligned}
\{f(a+\lambda x),g(a+\mu x) \}=0, \quad \{f(a+\lambda x),g(a+\mu x) \}_a = 0.\end{aligned}$$The proof given below follows the ideas of Arhangel’skiǐ[@Arh].

Recall that $f \in S({\mathfrak{g}})$ is called a semi-invariant of ${\mathfrak{g}}$ if there exists a character $\chi_f \colon {\mathfrak{g}}\to {\mathbb{C}}$ such that $$\{f,g\} = \chi_f({\mathrm{d}  g})f$$ for any $g \in S({\mathfrak{g}})$.

Assume that $\{f,g\}$ is divisible by $f$ for any $g \in S({\mathfrak{g}})$. Then $f$ is a semi-invariant.

Let $x_1, \dots, x_n$ be a basis in ${\mathfrak{g}}$. Then $$\begin{aligned}
\label{three}\{f,g\} = \sum_i\{f,x_i\}{ \dfrac{\partial g}{\partial x_i} }. \end{aligned}$$ Since $\{f,x_i\}$ is divisible by $f$ and has the same degree as $f$, there exists $c_i \in {\mathbb{C}}$ such that $ \{f,x_i\} = c_i f$. Define a linear function $\chi_f \colon {\mathfrak{g}}\to {\mathbb{C}}$ by setting $
  \chi_f(x_i) = c_i.
  $ Then can be rewritten as $$\{f,g\} = \chi_f({\mathrm{d}  g})f.$$ Further, $$\{f,\{x_i,x_j\}\} =   \{\{f,x_i\},x_j\} +   \{x_i,\{f,x_j\}\} = c_i\{f,x_j\} + c_j\{x_i,f\} = c^ic^jf-c^jc^if = 0,$$ so $$\chi_f(\{x_i,x_j\}) = \chi_f([x_i,x_j]) = 0,$$ i.e. $\chi_f$ is a character, q.e.d.

Any semi-invariant $f$ is a product of irreducible semi-invariants.

Let $f = f_1^{k_1}  \ldots  f_m^{k_m}$ where $f_1, \dots, f_m$ are irreducible and distinct. Then $$\begin{aligned}
\label{one}
 \{f,g\} = \{ f_1^{k_1}  \ldots  f_n^{k_m}, g\} = \sum_{i=1}^m k_i f_1^{k_1}  \ldots  f_i^{k_i - 1}\ldots  f_m^{k_m}\{f_i,g\}.\end{aligned}$$ On the other hand, $$\begin{aligned}
\label{two}
 \{f,g\} = \chi_f({\mathrm{d}  g})f =  \chi_f({\mathrm{d}  g}) f_1^{k_1}  \ldots  f_m^{k_m}.\end{aligned}$$ Comparing and , we conclude that $\{f_i,g\}$ is divisible by $f_i$, hence $f_i$ is a semi-invariant.

Let $f,g$ be two semi-invariants. Then $\{f,g\} = 0$.

Clearly, it suffices to prove the proposition for irreducible distinct $f$ and $g$. Let $h = \{f,g\}$. Then $h$ is divisible by both $f$ and $g$, and hence by $fg$. On the other hand, $$\deg h \leq \deg f + \deg  g- 1,$$ therefore $h=0$.

\[shift1\] Let $f,g$ be two semi-invariants. Then $\{f,g(a+\lambda x)\} = 0$.

Since $\{f,g\} = 0$, we have $\chi_f({\mathrm{d}  g}(x)) = 0$ for any $x$. Let $h(x) = g(a+\lambda x)$. Then $${\mathrm{d}  h}(x) = \lambda {\mathrm{d}  f}(a+\lambda x),$$ so $\chi_f({\mathrm{d}  h}(x)) = 0$, and $\{f,h\} = \chi_f({\mathrm{d}  h})f = 0$.

Let $f,g$ be two semi-invariants. Then $\{f,g(a+\lambda x)\}_a = 0$.

We shall use the following explicit formulas for the Lie-Poisson and frozen argument bracket: $$\{f,g\}(x) = \langle x,[{\mathrm{d}  f}(x), {\mathrm{d}  g}(x)]\rangle, \quad  \{f,g\}_a(x) = \langle a, [{\mathrm{d}  f}(x), {\mathrm{d}  g}(x)]\rangle.$$ Let $h(x) = g(a+\lambda x)$. Then $$\{f,h\}_a(x) = \langle a, [{\mathrm{d}  f}(x), {\mathrm{d}  h}(x)]\rangle = \lambda \langle a,[{\mathrm{d}  f}(x), {\mathrm{d}  g}(a+\lambda x)]\rangle.$$ On the other hand, $$\lambda \langle x, [{\mathrm{d}  f}(x), {\mathrm{d}  g}(a+\lambda x)] \rangle=   \{f,h\}(x) = 0,$$ so $$\{f,h\}_a(x) =  \lambda \langle a+\lambda x,[{\mathrm{d}  f}(x), {\mathrm{d}  g}(a+\lambda x)]\rangle,$$ and $$\{f,h\}_a\left(\frac{x-a}{\lambda} \right) =  \lambda \langle x ,[{\mathrm{d}  f}\left(\frac{x-a}{\lambda} \right), {\mathrm{d}  g}(x)] \rangle = \lambda^2\{f\left(\frac{x-a}{\lambda} \right),g\}(x).$$ The latter Poisson bracket vanishes by Proposition \[shift1\], so $$\{f,h\}_a\left(\frac{x-a}{\lambda} \right) = 0$$ for any $x$, and hence $\{f,h\}_a = 0$.

Let $f,g$ be two semi-invariants. We need to prove that $$\{f(a+\lambda x),g(a+\mu x)\} = \{f(a+\lambda x),g(a+\mu x)\}_a= 0.$$ Let $h(x) = f(a+\lambda x), k(x) = g(a+\mu x)$. Then $$\{h,k\}(x) = \langle x,[{\mathrm{d}  h}(x), {\mathrm{d}  k}(x)]\rangle= \lambda \mu \langle x, [{\mathrm{d}  f}(a+\lambda x), {\mathrm{d}  g}(a+\mu x)]\rangle,$$ so $$\begin{aligned}
  \{h,k\}\left(\frac{x-a}{\lambda} \right)  &= \lambda \mu \langle \frac{x-a}{\lambda}, [{\mathrm{d}  f}(x), {\mathrm{d}  g}\left(a + \frac{\mu}{\lambda} (x-a)\right)]\rangle = \\ &= \lambda\{f, g\left(a + \frac{\mu}{\lambda} (x-a)\right)\}(x) - \lambda \{f, g\left(a + \frac{\mu}{\lambda} (x-a)\right)\}_a(x) = 0,\end{aligned}$$ and hence $\{h,k\} = 0$. Analogously, $$\begin{aligned}
  \{h,k\}_a\left(\frac{x-a}{\lambda}\right)  = \lambda \mu \langle a ,[{\mathrm{d}  f}(x), {\mathrm{d}  g}\left(a + \frac{\mu}{\lambda} (x-a)\right)]\rangle = \lambda^2 \{f, g\left(a + \frac{\mu}{\lambda} (x-a)\right)\}_a(x) = 0,\end{aligned}$$ and hence $\{h,k\}_a = 0$.

Shifts of the fundamental semi-invariant {#shiftsSec}
========================================

Recall that the shifts $p_1, \dots, p_n$ of the fundamental semi-invariant are defined by the formula:

$$p_{\mathfrak{g}}(a+\lambda x) =  \sum_{i=0}^{n} p_{i}(x)\lambda^i$$ where $n = \deg p_{\mathfrak{g}}$. Consider the factorization of $p_{{\mathfrak{g}}}$ into irreducible factors: $$p_{{\mathfrak{g}}} = p_1^{k_1}\dots p_m^{k_m},$$ and let $d =\sum \deg p_i$. Let $x \in {\mathfrak{g}}^*$, and consider the equation $$p_{{\mathfrak{g}}}(x - \lambda a) = 0.$$ Obviously, this equation has at most $d$ distinct roots. Let us say that an element $x \in {\mathfrak{g}}^*$ is nice if it has the following properties:

1.  the equation $p_{{\mathfrak{g}}}(x - \lambda a) = 0$ has exactly $d$ distinct roots $\lambda_1, \dots, \lambda_d$;

2.  $\lambda_1, \dots, \lambda_d$ are locally analytic functions of $x$;

3.  for each $i$, the dimension of the stabilizer of ${x - \lambda_i(x)a}$ is locally constant;

4.  the line $x-\lambda a$ does not intersect the set ${\mathrm{Sing}}\setminus {\mathrm{Sing}}_0$.

It is clear that the set $\mathcal N$ of nice elements is Zariski dense in ${\mathfrak{g}}^*$.

\[spandiff\] Let $x \in \mathcal N$. Then the space spanned by the differentials of $\lambda_1, \dots, \lambda_d$ coincides with the space spanned by the differentials of $p_1, \dots p_n$.

Let $s_i$ be the multiplicity of $\lambda_i$. Then $$p_{{\mathfrak{g}}}(x - \lambda a) =p_{\mathfrak{g}}(a)\prod_{i=1}^d (\lambda_i(x) - \lambda)^{s_i},$$ so $$p_{{\mathfrak{g}}}(a+\lambda x) = \lambda^n p_{{\mathfrak{g}}}(x + \frac{1}{\lambda}a) = p_{\mathfrak{g}}(a)\prod_{i=1}^d (1 + \lambda_i(x)\lambda)^{s_i},$$ therefore the functions $p_1, \dots, p_n$ are, up to a constant factor, elementary symmetric polynomials of the functions $\lambda_1, \dots, \lambda_d$ taken with multiplicities, which easily implies the statement.

Let ${\mathfrak{g}}_i(x) = {\mathfrak{g}}_{x - \lambda_i(x)a}$ be the stabilizer of $x - \lambda_i(x)a$. The two following statements relate the differentials $\lambda_1, \dots, \lambda_d$ to the structure of stabilizers ${\mathfrak{g}}_1(x), \dots, {\mathfrak{g}}_d(x)$.

\[preFundId\] Let $x_0 \in \mathcal N$. Then ${\mathrm{d}  \lambda}_i(x_0) \in {\mathfrak{g}}_{i}(x_0)$.

Consider the coadjoint orbit $\mathcal O$ passing through $y_0 = x_0 - \lambda_i(x_0)a$, and let $\xi \in {\mathrm{T}}_{y_0}\mathcal O $ be a vector tangent to the orbit at $y_0$. Let also $y(t)$ be a curve such that $y(t) \in \mathcal O, y(0) = y_0,$ and $\dot y(0) = \xi$. Let also $x(t) =  y(t) + \lambda_i(x_0)a$. Obviously, $y(t) \in {\mathrm{Sing}}$, and $\lambda_i(x(t)) = \lambda_i(x_0)$. Differentiating this formula with respect to $t$ at $t = 0$, we obtain $$\langle \dot x(0), {\mathrm{d}  \lambda}_i(x_0) \rangle = 0,$$ and since $\dot x(0) = \dot y(0) = \xi$, we have $$\begin{aligned}
\label{ann}
 \langle \xi, {\mathrm{d}  \lambda}_i(x_0) \rangle = 0.\end{aligned}$$ Since is true for any $\xi \in {\mathrm{T}}_{y_0}\mathcal O $, we have $$\begin{aligned}
\langle  {\mathrm{T}}_{y_0}\mathcal O , {\mathrm{d}  \lambda}_i(x_0) \rangle = 0,\end{aligned}$$ which implies that ${\mathrm{d}  \lambda}_i(x_0) \in {\mathfrak{g}}_{y_0} = {\mathfrak{g}}_i(x_0)$.

The following simple formula is of fundamental significance for the present paper.

\[fundIdP\] Let $x_0 \in \mathcal N$, and let $\xi, \eta \in {\mathfrak{g}}_{i}(x_0)$. Then $$\begin{aligned}
\label{fundId}
  [\xi, \eta] = \langle a,  [\xi, \eta] \rangle \, {\mathrm{d}  \lambda}_i(x_0).\end{aligned}$$

Choose a neighborhood $U(x_0) \ni x_0$ such that the dimension of ${\mathfrak{g}}_{i}(x)$ is constant in $U(x_0)$. Then it is possible to define smooth mappings $\xi, \eta \colon U(x_0) \to {\mathfrak{g}}$ such that $\xi(x_0) = \xi, \eta(x_0) = \eta$, and $\xi(x), \eta(x) \in {\mathfrak{g}}_{i}(x)$ for each $x \in U(x_0)$. Differentiating the identity $$\langle x - \lambda_i(x)a, [\xi(x), \eta(x)]\rangle = 0.$$ at $x = x_0$, we obtain .

Linear algebra related to a pair of skew-symmetric forms {#LA}
========================================================

Let $P_0, P_\infty$ be two skew-symmetric forms on a vector space $V$, and let $$P_\lambda = P_0 - \lambda P_\infty.$$ Let also $$\begin{aligned}
 r =  \min_{\lambda \in \overline {\mathbb{C}}} {\mathrm{corank} \, P}_\lambda, \quad
  \Lambda = \{ \lambda \in \overline {\mathbb{C}}\mid {\mathrm{corank} \, P}_\lambda> r \}, \quad
  L = \sum_{\lambda \in \overline {\mathbb{C}}\setminus \Lambda} {\mathrm{Ker} \, P}_\lambda.\end{aligned}$$

\[LA2\] The space $L$ has the following properties:

1.  it is isotropic with respect to any form $P_\lambda$;

2.  the skew-orthogonal complement to $L$ given by $L^\bot = \{ \xi  \in V \mid P_\lambda(\xi,L) = 0\}$ does not depend on the choice of $\lambda \in \overline{\mathbb{C}}$;

3.  if $\lambda \notin \Lambda$, then $P_\lambda$ is non-degenerate on $L^\bot /L$;

4.  if $\lambda \in \Lambda$, then $\dim\, ({\mathrm{Ker} \, P}_\lambda \cap L) = r$;

5.  if $\lambda \in \Lambda$, and $\alpha \in \overline {\mathbb{C}}$, then $
         {\mathrm{Ker} \, \left}( P_\alpha \mid_{{\mathrm{Ker} \, P}_\lambda}  \right) \supset {\mathrm{Ker} \, P}_\lambda \cap L.
        $

Assume that $\infty \notin \Lambda$, and define the *recursion* operator $R \colon L^\bot / L \to L^\bot L$ by the formula $$R = P_\infty^{-1}P_0.$$

\[LA3\]The operator $R$ has the following properties:

1.  the spectrum of $R$ coincides with the set $\Lambda$; the multiplicity of each eigenvalue is at least two;

2.  the $\lambda$-eigenspace of $R$ coincides with the space $
            ({\mathrm{Ker} \, P}_{\lambda}) / ({\mathrm{Ker} \, P}_{\lambda} \cap L)
        $;

3.  the eigenspaces of $R$ are pairwise orthogonal with respect to $P_\lambda$ for each $\lambda$;

4.  the operator $R$ is diagonalizable if and only if for each $\lambda \in \Lambda$, the following identity holds $$\dim {\mathrm{Ker} \, \left}( P_\infty \mid_{{\mathrm{Ker} \, P}_\lambda}  \right) = r.$$

For the proof of Propositions \[LA2\], \[LA3\], see [@sbs]. They can also be easily deduced from the Jordan-Kronecker theorem [@Gurevich; @Thompson; @GZ2; @Kozlovj].

\[LA4\] Let $\Lambda = \{\lambda_1, \dots, \lambda_k\}$, and assume that $\xi_i \in {\mathrm{Ker} \, P}_{\lambda_i}$. Let $$U = L + \langle \xi_1, \dots, \xi_k \rangle.$$ Then

1.  $U$ is isotropic with respect to $P_\lambda$ for any $\lambda \in \overline {\mathbb{C}}$;

2.  if $\lambda \notin \Lambda$, then $U$ is maximal isotropic with respect to $P_\lambda$ if and only if all eigenvalues of $R$ have multiplicity two, and $\xi_i \notin L$ for each $i$.

The proof easily follows from Propositions \[LA2\] and \[LA3\].

Proof of Theorem 2
==================

Assume that $ {\mathrm{Sing}}_{\mathfrak b}$ is dense in ${\mathrm{Sing}}_0$, and prove that the extended Mischenko-Fomenko subalgebra $\widetilde{\mathcal F_a}$ is complete. Let us take $x \in \mathcal N$ such that $x - \lambda_i(x)a \in  {\mathrm{Sing}}_{\mathfrak b}$ for each $i$, and prove that the space $${\mathrm{d}  \widetilde}{\mathcal F}(x) = \{ {\mathrm{d}  f}(x) \mid f \in  \widetilde{\mathcal F}(x)\}$$ has dimension $$b({\mathfrak{g}}) =  \frac{1}{2}(\dim {\mathfrak{g}}+ {\mathrm{ind} \, {\mathfrak{g}}}),$$ which immediately implies the completeness of $\widetilde{\mathcal F_a}$.

Consider skew-symmetric forms $P_0 = \mathcal A_x$ and $P_\infty = \mathcal A_a$ on the cotangent space ${\mathrm{T}}^*_x {\mathfrak{g}}^* \simeq {\mathfrak{g}}$ which are given by $$A_x(\xi, \eta) = \langle x, [\xi,\eta] \rangle, \quad   A_a(\xi, \eta) = \langle a,[\xi,\eta]\rangle.$$ Let us apply the results of Section \[LA\] to these two forms. We note that $$\begin{aligned}
r = \mathrm{ind}\, {\mathfrak{g}}, \quad {\mathrm{Ker} \, P}_\lambda = {\mathfrak{g}}_{x - \lambda a}, \quad \Lambda = \{\lambda_1(x), \dots, \lambda_d(x) \}.\end{aligned}$$ The following lemma is due to Bolsinov [@Bolsinov].

Let $\mathcal F$ be the classical Mischenko-Fomenko subalgebra, and let $${\mathrm{d}  \mathcal F}(x) = \{ {\mathrm{d}  f}(x) \mid f \in  {\mathcal F}(x)\}.$$ Then $${\mathrm{d}  \mathcal F}(x) = L.$$

As follows from Proposition \[spandiff\], $${\mathrm{d}  \widetilde}{\mathcal F}(x) = {\mathrm{d}  \mathcal F}(x) + \langle {\mathrm{d}  \lambda}_1(x), \dots, {\mathrm{d}  \lambda}_d(x) \rangle.$$ By Proposition \[preFundId\], we have $${\mathrm{d}  \lambda}_i(x) \in {\mathfrak{g}}_{i}(x) = {\mathrm{Ker} \, P}_{\lambda_i},$$ so we can use Proposition \[LA4\] to show that ${\mathrm{d}  \widetilde}{\mathcal F}(x) $ is maximal isotropic with respect to $A_a$ and hence is of dimension $b({\mathfrak{g}})$. In order to do this, we need to show that the eigenvalues of the recursion operator $R$ have multiplicity two, and that ${\mathrm{d}  \lambda}_i(x) \notin L$.

Since $x - \lambda_i(x)a \in  {\mathrm{Sing}}_{\mathfrak b}$, we have $
\dim {\mathfrak{g}}_i =r + 2,
$ and by item 4 of Proposition \[LA2\], $\dim\left( {\mathfrak{g}}_i \cap L \right) = r$, so $$\dim \left( {\mathfrak{g}}_i / ( {\mathfrak{g}}_i \cap L) \right)= 2,$$ and all eigenspaces of $R$ are two-dimensional (see Proposition \[LA3\], item 2). Therefore, to prove that all eigenvalues of $R$ have multiplicity two, we need to show that $R$ has no Jordan blocks. By [item 4]{}\
of Proposition \[LA3\], it suffices to prove that $
\dim {\mathrm{Ker} \, (}A_a \mid {\mathfrak{g}}_i) = r.
$ We have $
\dim {\mathfrak{g}}_i =r + 2,
$ and $
\dim {\mathfrak{g}}_i \cap L = r.
$ By item 5 of Proposition \[LA2\], $ {\mathrm{Ker} \, (}A_a \mid {\mathfrak{g}}_i) \supset {\mathfrak{g}}_i \cap L$, so $\dim {\mathrm{Ker} \, (}A_a \mid {\mathfrak{g}}_i)$ can be either $r$ or $r+2$. Assume that it is $r+2$. Then $ {\mathrm{Ker} \, (}A_a \mid {\mathfrak{g}}_i) = {\mathfrak{g}}_i$, and $A_a \mid {\mathfrak{g}}_i = 0$. By Proposition \[fundIdP\], this implies that ${\mathfrak{g}}_i$ is Abelian, which is not the case.

Now, let us prove that ${\mathrm{d}  \lambda}_i(x) \notin L$. Since $ {\mathrm{Ker} \, (}A_a \mid {\mathfrak{g}}_i) \supset {\mathfrak{g}}_i \cap L$, Proposition \[fundIdP\] implies that $ {\mathfrak{g}}_i \cap L$ lies in the center $\mathcal Z({{\mathfrak{g}}_i})$. So, if ${\mathrm{d}  \lambda}_i(x) \in L$, then $${\mathrm{d}  \lambda}_i(x) \in \mathcal Z({{\mathfrak{g}}_i}).$$ On the other hand, since ${\mathfrak{g}}_i$ is not Abelian, Proposition \[fundIdP\] implies that $${\mathrm{d}  \lambda}_i(x) \in [{\mathfrak{g}}_i, {\mathfrak{g}}_i], \quad {\mathrm{d}  \lambda}_i(x) \neq 0,$$ where $ [{\mathfrak{g}}_i, {\mathfrak{g}}_i]$ is the derived subalgebra of ${\mathfrak{g}}_i$. But since ${\mathfrak{g}}_i \simeq \mathfrak b_2 \oplus \mbox{Abelian}$, we have $[{\mathfrak{g}}_i, {\mathfrak{g}}_i] \cap  \mathcal Z({{\mathfrak{g}}_i}) = 0$, so ${\mathrm{d}  \lambda}_i(x) \notin L$, which completes the proof of the if-part of the theorem. The proof of the only-if-part is analogous.

Acknowledgements
================

The author is grateful to Alexey Bolsinov for useful comments. This work was supported by the Dynasty Foundation Scholarship.

[^1]: Moscow State University and Higher School of Economics. E-mail: a.m.izosimov@gmail.com


Q:

Creating a form to send multiple emails to existing emails using Rails

I'm building an order management system for customers. I need to set it up so I can build a form that emails a brief of the order status to a customer. The trick is that a customer can have multiple emails i.e no limits on that and the form I need to set up would show the brief generated in a textbox(nothing hard there) as well as a list of checkboxes with the email addresses which to send to.
Clicking on/off the checkboxes determines which addresses would the brief be sent to. I have everything worked out except for one main thing i.e the form - I'm not sure how can I actually set this part up i.e I'm using simple_form here and I'm not sure which model would I be making the form against? Do I need to build another model here? I'm pretty stuck.

A:

May be you can create model like Email without database and use all ActiveRecord helper methods. These articles may be helpful.
http://yehudakatz.com/2010/01/10/activemodel-make-any-ruby-object-feel-like-activerecord/
http://railscasts.com/episodes/193-tableless-model?view=asciicast


# The following patch enables conversion webhook for CRD
# CRD conversion requires k8s 1.13 or later.
apiVersion: apiextensions.k8s.io/v1
kind: CustomResourceDefinition
metadata:
  name: azureclusters.infrastructure.cluster.x-k8s.io
spec:
  conversion:
    strategy: Webhook
    webhook:
      conversionReviewVersions: ["v1", "v1beta1"]
      clientConfig:
        # this is "\n" used as a placeholder, otherwise it will be rejected by the apiserver for being blank,
        # but we're going to set it later using the cert-manager (or potentially a patch if not using cert-manager)
        caBundle: Cg==
        service:
          namespace: system
          name: webhook-service
          path: /convert


Q:

Selling high, pay capital gains, re-purchase later

I'm trying to come up with a formula that tells me how far a stock must fall from a selling point before I re-purchase it with the assumption it will regain the selling price. In other words, if I sell stock and pay capital gains, at what point could I re-buy the stock to offset taxes paid with the following assumptions:
ASSUMPTIONS

Long term capital gain.
0.15 tax rate.
Cost basis/share < selling price/share
Assumes stock will go down.
Assumes stock will recover to selling price.

A:

Ignoring brokerage fees and the wash-sale rule (both of which are hazardous to your health), and since the 15% LTCG tax is only on the gain, the stock would have to drop 15% of the gain in price since you originally purchased it.
(SALEPRICE - PURCHPRICE) * .15 == Amount it would have to drop


A state bill to allow mid-rise, denser development near transit corridors was amended by its sponsors yesterday to address concerns raised by local planning agencies.

State Senator and former District 8 Supervisor Scott Weiner (D-San Francisco) issued his first round of amendments to SB 827 to clarify that local inclusionary housing requirements would still stand and that tenant protection provisions, such as the right to return, should be upheld.

Proponents say SB 827 would create incentives for residential developments near transit stops along “high-frequency corridors.” Along with Weiner, Senators Nancy Skinner (D-Berkeley), Ben Hueso (D-San Diego) and Assemblymember Phil Ting (D-San Francisco) are co-authors of the bill, which is sponsored by California YIMBY.

The bonus would allow developments to exceed local height and density limits, and remove requirements for a project to include a minimum number of parking spaces.

Under the bill, high frequency routes are defined as corridors that offer service more than once every 15 minutes during peak commute hours.

The proposed bill aims to build on a suite of affordable housing legislation adopted last year focused on streamlining and accelerating housing development statewide.

Yesterday's changes come after SF Planning issued an analysis raising concerns about its initial language. Prior to some of the clarifications proposed today, the bill seemed to invalidate height limitations on up to 96 percent of the city’s land parcels.

All of the beige areas are limited at 40 feet in height. | Map via SF Planning

When initially proposed, it seemed as if the bill would affect all land along the high-frequency routes themselves, rather than only the parcels within one-quarter of a mile from transit stops along those routes.

And while the bill's key goals—more housing near public transportation—align with San Francisco’s General Plan that calls for focusing growth near transit corridors, city planners were concerned that a blanket zoning standard across all transit corridors could degrade San Francisco’s open space, historic preservation, and other neighborhood-specific requirements.

The bill wouldn’t limit SF Planning’s ability to enforce zoning requirements aimed at protecting an area’s cultural, historic, or open space assets, Wiener spokesperson Jeff Cretan told us. SB 827 “only prevents them from denying a project based only on height, density, or lack of parking,” he added.

Under the proposed bill, any proposal for a mid-rise—55’-85’ tall— residential development in a transit corridor should be approved if it meets the city’s other requirements for that zoning area, even if its exceeds the density limitations and does not meet the minimum parking requirements.

Any development could still qualify for the state density bonus, which would increase both height and density allowances if on-site affordable housing is included.

SB 827 is unlikely to move forward exactly as proposed, or even as amended yesterday, as most state bills are amended throughout the legislative process.

Weiner said yesterday that he "will continue to work with a broad range of voices to make the bill as good as it can be.”

The SF Planning Commission is tentatively scheduled to review the most recent version of the legislation at its March 15th hearing.


Apartment Amenities

Apartment Amenities

Community Amenities

Community Amenities

Banks

Cleanliness

Community involvement

Gas Stations

Google Fiber Internet

Groceries

Off Street Parking

Restaurants and Shopping

Safety

Schools

Walkability

Overview

Santa Fe - BYU Men's Shrd Rms is located at 802 North 700 East #1 - #8 Provo, UT and is managed by Aspen Ridge Management, a reputable property management company with verified listings on http://www.rentcafe.com/. Santa Fe - BYU Men's Shrd Rms offers 2 bedroom apartments of size 700 sq.ft. Amenities include Air Conditioner, Business Center, Controlled Access/Gated, Covered Parking, Dishwasher and more. Property is located in the 84606 ZIP code. For more details, contact our office at (801) 607-1680 or use the online contact form and we will get back to you as soon as possible.

Pet Policy

Pets are not allowed

Office Hours

Why Choose Santa Fe - BYU Men's Shrd Rms

If you’re looking for a new home in the Joaquin Neighborhood in Provo, UT you’re in luck – the Santa Fe - BYU Men's Shrd Rms community has 1 apartment for rent available right now! This community offers amenities such as Banks, Cleanliness or Community involvement and is located on 802 North 700 East #1 - #8 in the 84606 zip code. The 700-square-foot 2 bedroom is equipped with amenities such as Central Air, Dishwasher and Disposal and has a rental rate of $300. Check out the 2 Bedroom floorplan and apply for a lease online! The Aspen Ridge Management property management company will verify your lease application and send you a timely response. Once you become a resident of Santa Fe - BYU Men's Shrd Rms, you will be able to pay your rent online, submit maintenance requests, and get community news by logging in securely to your Resident Portal.


Film Review: "The Wind Rises"

Fight or flight

Tools

“The Wind Rises”

(PG-13), directed by Hayao Miyazaki

Now playing

When legendary
Japanese director Hayao Miyazaki announced in
September of last year that "The Wind Rises" was to be his final film, it was
hard not to feel a sadness at what audiences would be
losing without any more of his imaginative, lushly animated films. True, the
director has promised retirement several times in the past, only to eventually
reconsider. Visually stunning and morally complex, his supposedly final film is
a fantastic achievement. If it does turn out that his retirement sticks this
time, "The Wind Rises" would make for a fitting end to a remarkable career.
Though the film contains none of the more fantastical elements that the
filmmaker behind "My Neighbor Totoro" and "Spirited
Away" is known for, it's clear that the story he's telling is a deeply personal
one. There's a wistful, occasionally mournful finality to the film that feels
very much like a farewell.

click to enlarge

PHOTO COURTESY TOUCHSTONE PICTURES

Mr. Caproni and Jiro Horikoshi in “The Wind Rises.”

"The Wind Rises" tells
a fictionalized version of the life story of Japanese aeronautics engineer JiroHorikoshi. As a child at the
dawn of the 20th century, he dreams of flying, but his poor eyesight prevents
him from becoming a pilot. He decides that if he can't fly planes, then he's
going to make a career out of designing them instead. He's a genius engineer,
always looking for ways to improve his work and create beautiful aircrafts,
like the ones he sees in his dreams. Jiro lands a job
working for the Mitsubishi corporation, and it's there
that his obsessive pursuit of perfection ultimately leads him to design the A6M
Zero bombers used by the Japanese army during World War II, most infamously in
their attack on Pearl Harbor.

Miyazaki's film has
not been without controversy, with a number of critics accusing the film of
whitewashing history by glossing over the deadly results of Jiro's
work. But to make that argument seems to miss the point of the film entirely.
The conflict at the heart of the film is whether or not realizing one's dreams
is worth it, if the results will cause harm to a great many people. Jiro is a genius who wants nothing more than to create
something beautiful. The rising tide of war creeps around the edges of his
life; he's always aware of it, and knows how his work is being put to use. But
he chooses to focus on the beauty and purity of his creations instead of the
destruction they'll bring.

Miyazaki's film most
clearly lays out these ideas during several dream sequences, as Jiro imagines himself in conversation with his boyhood
idol, Italian engineer Giovanni Caproni. It's Caproni who tells Jiro early on
that "the dream of aviation is cursed," with the wonder of flight constantly
being perverted and twisted into something that's so often used for evil. It's
hard not to flash back to those indelible images of 9/11 and realize how much
truth there is in that statement. It's in these dreams that we're allowed to
see the inner conflict Jiro keeps suppressed. "I choose
to live in a world with pyramids," Caproni says, with
the implication being that those wonders wouldn't have been made without the
manpower of an enslaved people.

As with the similar
criticisms raised against Martin Scorsese's "The Wolf of Wall Street" -- that
that film was glorifying the depraved behavior of its characters -- people seem
to have a problem when directors opt not to spell out for audiences exactly how
they should feel. The film depicting the ways in which Jiro
willingly puts blinders on when it comes to how his creations are being
utilized in the real world is very different from Miyazaki himself ignoring
those effects. Miyazaki is above all a humanist; he doesn't ask us to agree
with Jiro, but he does attempt to get us to
understand him. The director's work has always contained strong
environmentalist and anti-war messages, and "The Wind Rises" is no different.

The last third of
the film shifts gears slightly, allowing a love story that's introduced early in
the film to gradually take center stage. Jiro
develops a relationship with a young woman named Naoko. Jiro
learns that Naoko has tuberculosis, and though that doesn't stop the two from
falling in love, Jiro remains focused on his work
even as her sickness progresses. The love story feels underdeveloped, but it's
easy to see how Jiro's tunnel vision allowing his
career to trump his personal life may have personal resonance with the
Miyazaki.

The version of the
"The Wind Rises" playing in American theaters is dubbed in English and produced
by Disney. As with the American releases of many of Miyazaki's previous films,
the dubbing process was overseen by John Lasseter, Pixar's
chief creative officer, and it is impeccably done. The American vocal cast includes
Joseph Gordon-Levitt, John Krasinski, and Emily Blunt,
and while they all contribute fine work, they also have recognizable voices.
That occasionally distracts, as viewers try to mentally place whose voice they're
hearing.

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Abstract:

A piston rod and cylinder seal device includes a cylinder defining a
piston chamber extending between first and second cylinder heads. The
second cylinder head has a spud receiving bore, a pressure passage
communicating with the spud receiving bore, and a bore supply/vent
passage. A first piston is disposed in the piston chamber. A piston rod
is connected to the piston, the piston rod having a piston rod spud
extending beyond the first piston and including a blind shaft receiving
bore. A second piston slidably disposed in the blind shaft receiving bore
has a seal member connected thereto. A contact member connected to the
second cylinder head in the spud receiving bore has a central passage
extending therethrough in communication with the bore supply/vent
passage. The seal member when contacting the contact member acts to seal
the central passage.

Claims:

1. A piston rod and cylinder seal device, comprising: a cylinder creating
a piston chamber extending between opposed first and second cylinder
heads; a first piston slidably disposed in the piston chamber, the first
piston having a piston rod connected to the first piston; a piston rod
spud extending from the piston rod, the piston rod spud having a blind
bore and a second piston slidably received in the blind bore; and a
contact member connected to the second cylinder head, the contact member
having a central passage extending therethrough, a seal member disposed
in the second piston when contacting the contact member acting to seal
the central passage.

2. The piston rod and cylinder seal device of claim 1, wherein the second
cylinder head includes a spud receiving bore having the contact member
disposed therein, a pressure passage communicating with the spud
receiving bore, and a bore supply/vent passage communicating with the
central passage of the contact member.

3. The piston rod and cylinder seal device of claim 2, wherein when the
piston contacts the second cylinder head the piston rod spud is fully
received in the spud receiving bore and pressurized air in the bore
supply/vent passage communicating with the contact member is isolated by
the seal member from the pressure passage communicating with the spud
receiving bore.

4. The piston rod and cylinder seal device of claim 1, further including
a shaft bore having a threaded bore wall created in the second cylinder
head receiving a threaded portion of the contact member.

5. The piston rod and cylinder seal device of claim 1, further including
a biasing member positioned in the blind bore acting to continuously bias
the second piston toward the contact member.

6. The piston rod and cylinder seal device of claim 5, wherein the second
piston includes opposed first and second counterbores, the seal member
received in the first counterbore, and the biasing member partially
received in the second counterbore.

7. The piston rod and cylinder seal device of claim 5, wherein: the
second cylinder head includes a spud receiving bore having the contact
member disposed therein; and following contact between the contact member
and the seal member, continued displacement of the piston rod spud into
the spud receiving bore displaces the second piston into the blind bore
in a second piston contraction direction thereby compressing the biasing
member.

8. The piston rod and cylinder seal device of claim 1, wherein the
contact member includes a dome shaped curved contact end facing the seal
member.

9. The piston rod and cylinder seal device of claim 8, wherein the curved
contact end includes a circular apex acting to create a circular contact
seal when in direct contact with the seal member.

10. The piston rod and cylinder seal device of claim 1, further including
a retention ring assembly positioned in a ring slot created in the piston
rod spud, the second piston contacting the retention ring assembly in a
fully extended position of the second piston.

11. A piston rod and cylinder seal device, comprising: a cylinder
defining a piston chamber extending between opposed first and second
cylinder heads, the second cylinder head having a spud receiving bore, a
pressure passage communicating with the spud receiving bore, and a bore
supply/vent passage; a first piston slidably disposed in the piston
chamber; a piston rod connected to the piston, the piston rod having a
piston rod spud extending beyond the first piston and including a blind
shaft receiving bore; a second piston slidably disposed in the blind
shaft receiving bore, the second piston having a seal member connected
thereto; and a contact member connected to the second cylinder head in
the spud receiving bore, the contact member having a central passage
extending therethrough in communication with the bore supply/vent
passage, the seal member when contacting the contact member acting to
seal the central passage.

12. The piston rod and cylinder seal device of claim 11, further
including a biasing member positioned in the blind shaft receiving bore
acting to continuously bias the second piston in a second piston
extension direction toward the contact member.

13. The piston rod and cylinder seal device of claim 12, wherein
following contact between the contact member and the seal member
continued displacement of the first piston in a piston drive direction
causes the second piston to displace in an opposite second piston
contraction direction.

14. The piston rod and cylinder seal device of claim 11, wherein the
piston reciprocates in opposed piston return and drive directions in the
piston chamber, the piston rod spud is slidably received in the spud
receiving bore when the piston is displaced in the piston drive direction
to position the seal member in contact with the contact member thereby
preventing pressurized air in the bore supply/vent passage from entering
the spud receiving bore.

15. The piston rod and cylinder seal device of claim 11, further
including a cushion seal ring connected to the second cylinder head
creating an annular passage between the cushion seal ring and the piston
rod spud when the piston rod spud is received in the spud receiving bore.

16. The piston rod and cylinder seal device of claim 15, further
including a portion of the piston chamber defined between the piston and
the second cylinder wall, wherein the portion of the piston chamber is in
fluid communication with a flow path including the annular passage, the
spud receiving bore and the pressure passage prior to contact of the seal
member with the contact member.

17. The piston rod and cylinder seal device of claim 11, wherein the
second cylinder head further includes a first switch having a first
conductive biasing member extending into the piston chamber, the first
piston contacting the first conductive biasing member when the first
piston contacts the second cylinder head.

18. The piston rod and cylinder seal device of claim 17, wherein the
first cylinder head includes a second switch having a second conductive
biasing member extending into the piston chamber, the first piston
contacting the second conductive biasing member when the first piston
contacts the first cylinder head.

19. A crust breaker system, comprising: a piston rod and cylinder seal
device, including: a cylinder creating a piston chamber extending between
opposed first and second cylinder heads, the second cylinder head
including a pressure passage in communication with a spud receiving bore
and a bore supply/vent passage also in communication with the spud
receiving bore, a first piston slidably disposed in the piston chamber,
the first piston having a piston rod connected to the first piston; a
piston rod spud extending from the piston rod, the piston rod spud having
a blind bore and a second piston slidably received in the blind bore; and
a contact member connected to the second cylinder head in the spud
receiving bore, the contact member having a central passage extending
therethrough providing communication between the bore supply/vent passage
and the spud receiving bore, the contact member when contacted by a seal
member disposed in the second piston acting to seal the central passage;
and a pneumatic valve system having: a first control valve; and a valve
position control line connecting the first control valve to the pressure
passage.

20. The crust breaker system of claim 19, further including a crust
breaking rod connected to the piston rod opposite to the piston rod spud.

21. The crust breaker system of claim 20, wherein the pneumatic valve
system further includes a solenoid operated valve having a solenoid, the
solenoid energized or de-energized by a signal generated when the crust
breaking rod contacts a bath having a voltage completing a circuit
including a conductive seal of the piston and the cylinder.

22. The crust breaker system of claim 19, wherein the pressure passage
and the valve position control line are vented to atmosphere when the
piston rod spud is slidingly received in the spud receiving bore.

23. The crust breaker system of claim 19, wherein the pressure passage
and the valve position control line are both pressurized when the piston
rod spud is positioned outside of the spud receiving bore such that the
seal member is not in contact with the contact member.

24. The crust breaker system of claim 19, wherein the contact member
includes tool engagement flats.

25. The crust breaker system of claim 19, wherein the pneumatic valve
system further includes: a second control valve; a first pressure source
connected to a first portion of the piston chamber through the first
control valve; a second pressure source connected to a second portion of
the piston chamber through the second control valve; and a third pressure
source connected to the pressure transfer line.

Description:

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application is a continuation-in-part of U.S. patent
application Ser. No. 13/159,061 filed on Jun. 13, 2011. The entire
disclosure of the above application is incorporated herein by reference.

FIELD

[0002] The present disclosure relates to seal devices used in pneumatic
control systems for operating metal processing baths.

BACKGROUND

[0003] This section provides background information related to the present
disclosure which is not necessarily prior art.

[0004] Known systems used to control operations of metal processing baths
such as for aluminum processing can include pneumatic valves and piping
used to drive a crust breaking tool to create an aperture by breaking
through the hardened upper crust layer formed on the bath. The crust
breaking tool is intended to open the aperture to permit addition of
additional alumina material to the molten bath of aluminum. When creation
of the aperture has been confirmed, pressurized air directs the crust
breaking tool to retract from the crust layer. The drawbacks of such
systems include the large volumes of pressurized air which are used to
control a normal crust breaking operation, and particularly when crust
material forms on the crust breaking tool such that bath detection cannot
occur, and/or when the crust breaking tool cannot penetrate the crust
layer.

[0005] In these situations, the crust breaking tool can remain in the bath
for an undesirable length of time which can damage the crust breaking
tool, or render the detection system inoperative. Also in these
situations, the subsequent feeding of new alumina material into the bath
can be hindered, or the system may be unable to identify how many feed
events have occurred, thus leading to out-of-range conditions in the
bath. A further drawback of known control systems is the large volume of
high pressure air required significantly increases operating costs of the
system due to the size and volume of high pressure air system
requirements, power consumption and cost, the operating time of
pumps/compressors, and the number of air compressors and air dryers
required for operation.

SUMMARY

[0006] This section provides a general summary of the disclosure, and is
not a comprehensive disclosure of its full scope or all of its features.

[0007] According to several embodiments a piston rod and cylinder seal
device includes a cylinder creating a piston chamber extending between
opposed first and second cylinder heads. A first piston is slidably
disposed in the piston chamber. The first piston has a piston rod
connected to the first piston. A piston rod spud extends from the piston
rod, the piston rod spud having a blind bore and a second piston slidably
received in the blind bore. A contact member connected to the second
cylinder head has a central passage extending therethrough. The contact
member when contacted by a seal member disposed in the second piston acts
to seal the central passage.

[0008] According to other embodiments, a piston rod and cylinder seal
device includes a cylinder defining a piston chamber extending between
opposed first and second cylinder heads. The second cylinder head has a
spud receiving bore, a pressure passage communicating with the spud
receiving bore, and a bore supply/vent passage. A first piston is
slidably disposed in the piston chamber. A piston rod connected to the
piston has a piston rod spud extending beyond the first piston and having
a blind shaft receiving bore. A second piston slidably disposed in the
blind shaft has a seal member connected thereto. A contact member is
connected to the second cylinder head in the spud receiving bore. The
contact member has a central passage extending therethrough in
communication with the bore supply/vent passage. The seal member when
contacting the contact member acts to seal the central passage.

[0009] According to further embodiments, a crust breaker system includes a
piston rod and cylinder seal device, including a cylinder creating a
piston chamber extending between opposed first and second cylinder heads.
The second cylinder head includes a pressure passage in communication
with a spud receiving bore and a bore supply/vent passage also in
communication with the spud receiving bore. A first piston slidably
disposed in the piston chamber has a piston rod connected to the first
piston. A piston rod spud extending from the piston rod has a blind bore
and a second piston slidably received in the blind bore. A contact member
connected to the second cylinder head in the spud receiving bore has a
central passage extending therethrough providing communication between
the bore supply/vent passage and the spud receiving bore. The contact
member when contacted by a seal member disposed in the second piston acts
to seal the central passage. A pneumatic valve system includes a first
control valve and a valve position control line connecting the first
control valve to the pressure passage.

[0010] Further areas of applicability will become apparent from the
description provided herein. The description and specific examples in
this summary are intended for purposes of illustration only and are not
intended to limit the scope of the present disclosure.

DRAWINGS

[0011] The drawings described herein are for illustrative purposes only of
selected embodiments and not all possible implementations, and are not
intended to limit the scope of the present disclosure.

[0012]FIG. 1 is an end elevational view of an aluminum bath crust breaker
device having a piston rod and cylinder seal device of the present
disclosure;

[0013]FIG. 2 is a cross sectional front elevational view taken at section
2 of FIG. 1;

[0014]FIG. 3 is a cross sectional front elevational view taken at area 3
of FIG. 2;

[0015]FIG. 4 is a cross sectional front elevational view taken at area 4
of FIG. 2;

[0016]FIG. 5 is a cross sectional rear elevational view taken at section
5 of FIG. 1;

[0017]FIG. 6 is a cross sectional front elevational view taken at area 6
of FIG. 5;

[0018]FIG. 7 is a system diagram of a crust breaking system having the
piston rod and cylinder sealing device of FIG. 1;

[0019]FIG. 8 is a system diagram of the crust breaking system of FIG. 8
showing the crust breaker rod after breaking through the crust layer;

[0020]FIG. 9 is a cross sectional front elevational view modified from
the view taken at area 4 of FIG. 2 to include a further aspect of a
piston rod and cylinder seal device;

[0021]FIG. 10 is a cross sectional front elevational view modified from
FIG. 9 to show the piston in contact with the second cylinder wall;

[0022] FIG. 11 is a cross sectional front elevational view of area 11 of
FIG. 9;

[0023]FIG. 12 is a cross sectional front elevational view modified from
FIG. 9 to show the point of contact between the seal member and the
contact member;

[0024]FIG. 13 is a front perspective view of the contact member of FIG.
9;

[0025]FIG. 14 is a front elevational view of the contact member of FIG.
13;

[0026]FIG. 15 is a cross sectional side elevational view taken at section
15 of FIG. 14; and

[0027] FIG. 16 is a top front perspective view of the second piston of the
present disclosure.

[0028] Corresponding reference numerals indicate corresponding parts
throughout the several views of the drawings. For simplification, not all
parts are shown in all views of the drawings.

DETAILED DESCRIPTION

[0029] Example embodiments will now be described more fully with reference
to the accompanying drawings.

[0030] Referring to FIG. 1, a piston rod and cylinder sealing device 10
includes a cylinder 12 enclosed by a first cylinder head 14 defining a
first end of cylinder 12. A control portion 16 having one or more sensor
connectors 18 extending therefrom is also provided with piston rod and
cylinder sealing device 10.

[0031] Referring to FIG. 2, cylinder 12 defines a piston chamber 20 and
further includes a second cylinder head 22 creating a second end of
cylinder 12. Control portion 16 can be connected to second cylinder head
22. A piston 24 is slidably disposed within piston chamber 20 such that
piston chamber 20 is divided into a first portion 20a on a first side of
piston 24 and a second portion 20b on a second side of piston 24.

[0032] Piston 24 is connected to a piston rod 26 which can include a crust
breaker rod 28 connected to piston rod 26, or forming a free end of
piston rod 26. Piston rod 26 extends through first cylinder head 14 and
is slidably disposed using a rod bearing/seal 30 such that pressure
within piston chamber 20 is contained by rod bearing/seal 30. At an
opposite end of piston rod 26 is provided a piston rod spud 32 which is
slidingly disposed in a spud receiving bore 34 when the piston 24
contacts second cylinder head 22.

[0033] A hollow tubular shaft 36 is connected to second cylinder head 22
and is slidably received within piston rod spud 32 when piston rod spud
32 slidingly enters spud receiving bore 34. A fluid pressure such as
pressurized air can be introduced through hollow tubular shaft 36 from a
bore supply/vent passage 38 created in second cylinder head 22. A
pressure supply/vent port 40 is also provided with second cylinder head
22. Air pressure supplied at pressure supply/vent port 40 can be directed
into spud receiving bore 34.

[0034] Referring to FIG. 3, piston rod spud 32 includes a blind shaft
receiving bore 42 which is sized having a spud bore diameter "C" adapted
to slidingly receive a shaft diameter "D" of tubular shaft 36. When a
shaft free end 44 of tubular shaft 36 is initially received in shaft
receiving bore 42, the outer perimeter wall defining shaft diameter "D"
contacts a first seal member 46 which is positioned in a seal slot 48 of
piston rod spud 32. Continued displacement of piston rod spud 32 in the
piston return direction "A" provides continuous sealing contact between
tubular shaft 36 and first seal member 46 throughout the length of
tubular shaft 36. Tubular shaft 36 also provides a central passage 50
extending throughout a total length of tubular shaft 36. Central passage
50 therefore communicates with shaft receiving bore 42 of piston rod spud
32, therefore permitting fluid such as compressed air in shaft receiving
bore 42 to displace in the piston return direction "A" as piston rod spud
32 moves in the piston return direction "A". According to several
embodiments, a means for installing tubular shaft 36 is provided such as
the provision of a plurality of wrench engagement flats 52 which are
positioned proximate to shaft free end 44 and within central passage 50.
Wrench engagement flats 52 can be engaged by a tool (not shown) such as a
wrench used to rotate and therefore install tubular shaft 36.

[0035] It is further noted that an annular passage 53 is provided between
piston rod spud 32 and a cushion seal ring 54 which is connected to
second cylinder head 22. A sliding clearance is provided between piston
rod spud 32 and cushion seal ring 54. Cushion seal ring 54 as known in
the art allows pressurized fluid such as pressurized air in second
portion 20b of piston chamber 20 to pass from second portion 20b into
spud receiving bore 34 as the piston 24 and piston rod spud 32 both
travel in the piston return direction "A". During pressurized operation,
annular passage 53 also provides an opposite passageway for compressed or
pressurized air to pass between spud receiving bore 34 and into second
portion 20b.

[0036] Referring to FIG. 4, piston 24 is connected to piston rod 26 using
a piston retention fastener such as a nut 55 which is threadably engaged
with a threaded portion of piston rod 26. Piston retention nut 55 is
threadably engaged until piston retention nut 55 contacts an end face 56
of a nut receiving counter bore 58 created in piston 24. A width or
thickness of piston retention nut 55 is therefore substantially received
within nut receiving counter bore 58. Piston 24 further includes a
conductive seal 60 which is retained about a perimeter wall of piston 24
and slidingly contacts a cylinder inner wall 62 of cylinder 12 at any
sliding position of piston 24. As piston 24 moves in either of the piston
return direction "A" or piston drive direction "B", electrical contact is
therefore maintained between cylinder 12, conductive seal 60, piston 24
and piston rod 26. The use of conductive seal 60 therefore obviates the
need for a secondary connection between piston rod 26 and cylinder 12.

[0037] To displace piston 24 within piston chamber 20, a pressurized fluid
such as pressurized air is introduced for example into first portion 20a
which acts against a first piston face 64 displacing both piston 24 and
piston rod 26 in the piston return direction "A". This displacement of
piston 24 also co-displaces piston rod spud 32 into spud receiving bore
34. When piston rod spud 32 contacts and is sealingly engaged to tubular
shaft 36 using first seal member 46, any fluid in central passage 50 and
shaft receiving bore 42 is isolated from spud receiving bore 34.
Therefore, as piston 24 continues to move in the piston return direction
"A", fluid, such as pressurized air in second portion 20b of piston
chamber 20, is compressed between a second piston face 66 and a head face
68 of second cylinder head 22. Pressurized air in shaft receiving bore 42
is therefore displaced via a flow path including central passage 50 and
bore supply/vent passage 38. Pressurized air in spud receiving bore 34 is
outwardly displaced via a pressure passage 69 in communication with spud
receiving bore 34.

[0038] Tubular shaft 36 is connected to second cylinder head 22 using a
male threaded end 70 of tubular shaft 36 which is threadably engaged in
second cylinder head 22 in female threads created in a shaft receiving
bore 72. Bore supply/vent passage 38 is open to shaft receiving bore 72
via a connecting passage 74.

[0039] Referring to FIG. 5, piston 24 has been removed for clarity. When
piston rod 26 had been displaced in the piston return direction "A" to
the maximum extent, piston rod spud 32 is completely received within spud
receiving bore 34 and piston retention nut 55 is positioned proximate to
head face 68 of second cylinder head 22. To signal that the piston 24 is
at the returned or first piston contact position, a switch having a first
conductive biasing member 76 is contacted by second piston face 66 of
piston 24, thereby completing an electrical circuit indicating contact by
piston 24. A second switch having a second conductive biasing member 78
extends into piston chamber 20 from a head face 80 of first cylinder head
14. Contact between piston 24 and second conductive biasing member 78
would therefore create a second circuit signifying that piston 24 is at a
piston second contact position with first cylinder head 14.

[0040] Referring to FIG. 6, as previously noted, tubular shaft 36 includes
male threaded end 70 which is threadably engaged with a threaded bore
wall 82 of shaft receiving bore 72. To provide additional sealing
capability, tubular shaft 36 can further include a radially extending
flange 84 which contacts a flange contact face 86 created in second
cylinder head 22 proximate to threaded bore wall 82. A second seal member
88, such as an O-ring or D-ring, can be positioned between flange 84 and
flange contact face 86 to provide additional sealing capability. With
piston rod spud 32 completely extending into spud receiving bore 34, a
clearance can be maintained between a spud end face 90 of piston rod spud
32 and a bore end face 92 of spud receiving bore 34. This clearance
permits physical contact between piston 24 and head face 68 of second
cylinder head 22 as previously described in reference to FIG. 4.

[0041] Referring to FIG. 7 and again to FIGS. 1-6, piston rod and cylinder
sealing device 10 can be used in conjunction with a crust breaker system
94. Crust breaker system 94 can include a pneumatic valve system 96 which
is used to direct pressurized air into second portion 20b of piston
chamber 20 to direct piston 24 in the piston drive direction "B" such
that crust breaker rod 28 creates or maintains an aperture 98 through a
crust layer 100 of an aluminum melt bath 102. Aluminum melt bath 102 is
contained in a bath chamber 104. Aperture 98 is created through crust
layer 100 in order to add additional chemicals such as alumina material
to replenish aluminum melt bath 102.

[0042] Crust breaker system 94 can include a first pressure source 106
which can be aligned by control of a first control valve 108 and a second
control valve 110 to direct pressurized air from first pressure source
106 via a first air supply/vent line 112 into first portion 20a of piston
chamber 20 to hold piston 24 in the piston first contact position shown.
To displace piston 24 in the piston drive direction "B", first and second
control valves 108, 110 can be realigned such that pressurized air from a
second pressure source 114 can be directed through an air delivery/vent
line 116 and a second air supply/vent line 118 into spud receiving bore
34 to act on second piston face 66 while simultaneously first portion 20a
is vented to atmosphere via a path including first air supply/vent line
112 and second control valve 110.

[0043] When piston rod spud 32 is fully received within spud receiving
bore 34, air delivery/vent line 116 and second air supply/vent line 118
are both vented to atmosphere through second control valve 110. A valve
position control line 120 which connects air delivery/vent line 116 to a
first operating side of first control valve 108 is also vented to
atmosphere at this time. Piston chamber 20 is therefore not pressurized
to the full pressure range of first pressure source 106 because the
vented valve position control line 120 directs first control valve 108 to
isolate first pressure source 106 from piston chamber 20. Pressurized air
in a third pressure source 122 maintains this position of first control
valve 108 while maintaining a pressure in a pressure transfer line 124
which is connected to bore supply/vent passage 38 in second cylinder head
22. Pressure in pressure transfer line 124 also pressurizes shaft
receiving bore 42 but does not provide enough force to overcome the air
pressure in first portion 20a of piston chamber 20.

[0044] Pneumatic valve system 96 further includes a solenoid operated
valve 126 which directs pressure from a fourth pressure source 128 to
opposite ends of second control valve 110. By changing the orientation or
position of solenoid operated valve 126, second control valve 110 can be
positioned to pressurize either the first or second portion 20a, 20b of
piston chamber 20. Electronic signals used to coordinate the positioning
of solenoid operated valve 126 as well as feedback signals from contact
between crust breaker rod 28 and aluminum melt bath 102 are received
and/or generated using a control device 129.

[0045] Referring to FIG. 8 and again to FIG. 7, to displace piston 24 in
the piston drive direction "B" and away from the piston first contact
position shown in FIG. 7, second control valve 110 is repositioned using
pressurized air from fourth pressure source 128 after reorienting
solenoid operated valve 126 such that pressurized air from second
pressure source 114 is aligned with air delivery/vent line 116 and second
air supply/vent line 118 to pressurize second portion 20b of piston
chamber 20. Simultaneously, first portion 20a of piston chamber 20 is
vented to atmosphere by a path including first air supply/vent line 112
and second control valve 110. Pressurized air in second air supply/vent
line 118 enters spud receiving bore 34, pushing piston rod spud 32 out of
spud receiving bore 34 and further clearing a path for pressurized air in
pressure transfer line 124 to enter second portion 20b via tubular shaft
36. The combination of these two pressure sources acts on second piston
face 66 of piston 24 to displace piston 24 in the piston drive direction
"B". With pressurized air in second air supply/vent line 118, valve
position control line 120 is also pressurized, thereby repositioning
first control valve 108 to align first pressure source 106 to the supply
port of second control valve 110. The position of second control valve
110 temporarily prohibits pressurized air from first pressure source 106
from entering first portion 20a of piston chamber 20. It is noted that
the pressure in valve position control line 120 together with a biasing
member of first control valve 108 overcome the pressure from third
pressure source 122 acting on an opposite end of first control valve 108.
Therefore, even though pressurized air from third pressure source 122
flows through pressure transfer line 124, the biasing member of first
control valve 108 provides the additional force required to reposition
and hold first control valve 108 in the position shown.

[0046] As second piston face 66 of piston 24 displaces away from a contact
position with first conductive biasing member 76, a first switch 130
having first conductive biasing member 76 connected thereto, opens a
circuit signaling that piston 24 has left the piston first contact
position with head face 68. When first piston face 64 of piston 24 second
conductive biasing member 78, a second switch 132, having second
conductive biasing member 78 connected thereto closes a circuit signaling
that piston 24 is proximate to or has contacted first cylinder head 14,
defining a piston second contact position. These circuit signals are
received in control device 129.

[0047] When crust breaker rod 28 either creates or extends through
aperture 98 of crust layer 100 and enters aluminum melt bath 102, a
voltage V2 of the aluminum melt bath 102 is sensed and conducted via
an electrical path including crust breaker rod 28, piston rod 26, piston
24, conductive seal 60, cylinder 12 to control device 129. When the
voltage V2 of aluminum melt bath 102 is detected at control device
129, a signal is transmitted to reposition solenoid operated valve 126,
which subsequently repositions second control valve 110. This position
change of second control valve 110 isolates pressure from second pressure
source 114 and providing a flow path for pressure from first pressure
source 106 to re-enter first portion 20a of piston chamber 20. Piston 24
will thereafter return in the piston return direction "A" to the piston
first contact position shown in FIG. 7. As piston rod spud 32 engages and
seals against tubular shaft 36 pressurized air in pressure transfer line
124 is isolated from spud receiving bore 34, and second air supply/vent
line 118 is vented to atmosphere, thereby repositioning first control
valve 108. Piston rod spud 32, spud receiving bore 34, and tubular shaft
36 therefore provide the capability of redirecting pressurized air and/or
venting pressurized air such that the position of first control valve 108
can be pneumatically operated and repositioned, eliminating the need for
electronic control of either first or second control valves 108, 110.

[0048] Referring to FIG. 9 and again to FIGS. 1-4, according to additional
aspects a piston rod and cylinder sealing device 150 is modified from
piston rod and cylinder sealing device 10, therefore only the differences
will be further discussed herein. In piston rod and cylinder sealing
device 150 a first piston or piston 24 is connected to a piston rod 152
which is modified to include a piston rod spud 154 having a blind bore
156. Piston rod spud 154 is slidingly disposed in spud receiving bore 34
and is in sliding contact with cushion seal ring 54 as the piston 24
approaches and then contacts second cylinder head 22.

[0049] The hollow tubular shaft 36 of piston rod and cylinder sealing
device 10 is replaced in this embodiment with a hollow threaded contact
member 158 which is threadably connected to second cylinder head 22.
Similar to hollow tubular shaft 36, contact member 158 includes a central
passage 160 which in the piston rod sliding contact condition shown in
FIG. 9 opens into both spud receiving bore 34 and bore supply/vent
passage 38. Contact member 158 includes a curved or conical-shaped
contact end 162 which faces a resilient material seal member 164. Unlike
hollow tubular shaft 36 which slidingly enters spud receiving bore 34 and
creates a sliding external seal about hollow tubular shaft 36, the
contact end 162 directly contacts a planar face 166 of seal member 164 to
create a fluid tight seal, thereby preventing fluid/gas communication
between spud receiving bore 34 and bore supply/vent passage 38, and
further blocking pressurized air in bore supply/vent passage 38 from
exhausting through pressure passage 69. Similar to piston rod and
cylinder sealing device 10, a fluid pressure such as from pressurized air
can be introduced through the central passage 160 of contact member 158
from bore supply/vent passage 38 created in second cylinder head 22.

[0050] With continuing reference to FIGS. 9 and 1-4, seal member 164 is
retained in a first counterbore 168 created in a first end of a second
piston 170 which is slidably disposed in blind bore 156 of piston rod
spud 154. At an opposite, second end of second piston 170, a second
counterbore 172 receives a first end of a biasing member 174 which
continuously biases second piston 170 in a second piston extension
direction "E". According to several aspects, biasing member 174 is a
coiled compression spring. An air displacement passage 176 is created
between first and second counterbores 168, 172. During installation of
seal member 164 into first counterbore 168, air is displaced from behind
seal member 164 through air displacement passage 176 into second
counterbore 172, thereby allowing seal member 164 to fully seat within
first counterbore 168.

[0051] When contact end 162 of contact member 158 is spatially separated
from face 166 of seal member 164 as shown in FIG. 9, a biasing force of
biasing member 174 acts on second piston 170. The biasing force retains
second piston 170 at a fully extended position shown. The fully extended
position is reached when second piston 170 directly contacts a retention
ring assembly 178 which is seated in a ring slot 180 created in an inner
wall of piston rod spud 154. A second end of biasing member 174 directly
contacts a bore end wall 182 of blind bore 156. The second end of biasing
member 174 can also be slidably received in a biasing member bore 184
which is similar in diameter to the diameter of second counterbore 172.
Biasing member bore 184 slidably receives the second end of biasing
member 174, thereby helping to retain the alignment of biasing member 174
with second counterbore 172.

[0052] Referring to FIG. 10 and again to FIGS. 9 and 2-4, first piston 24
and piston rod 152 collectively move in the piston return direction "A",
and piston rod spud 154 displaces into spud receiving bore 34 until face
166 of seal member 164 contacts contact end 162 of contact member 158. A
fluid seal is thereby created between seal member 164 and contact member
158, which seals central passage 160 with respect to both blind bore 156
and portion 20b of piston chamber 20. At this time, second piston 170 is
located at its fully extended position in contact with retention ring
assembly 178, and is held at the fully extended position by the biasing
force of biasing member 174. As previously noted with respect to FIG. 9,
the fluid tight seal created when seal member 164 contacts contact end
162 of contact member 158 thereafter prevents fluid/gas communication
between spud receiving bore 34 and bore supply/vent passage 38, and
further blocks pressurized air in bore supply/vent passage 38 from
exhausting through pressure passage 69.

[0053] Referring to FIG. 11 and again to FIGS. 9-10, after contact between
contact end 162 of contact member 158 and seal member 164 occurs,
continued displacement of piston rod spud 154 and piston 24 in the piston
return direction "A" causes an opposite displacement of second piston 170
within blind bore 156 away from contact with the retention ring assembly
178, in second piston contraction direction "F". Sliding motion of second
piston 170 in the second piston contraction direction "F" compresses
biasing member 174. This increases the biasing force of biasing member
174, which thereafter acts to displace second piston 170 in the second
piston extension direction "E" when piston 24 is again displaced in the
piston drive direction "B". It is noted that second piston extension
direction "E" is parallel to piston return direction "A", and second
piston contraction direction "F" is parallel to piston drive direction
"B".

[0054] The fluid seal created when contact end 162 contacts seal member
164 is continuously and dynamically maintained during the sliding motion
of second piston 170. The curved or conical-shaped geometry of contact
end 162 provides the capability to maintain sealing contact between the
contact end 162 and seal member 164 while minimizing wear of seal member
164. Contact end 162 sealing contact with seal member 164 is maintained
at all sliding locations of second piston 170, thereby also accommodating
limited displacement of piston 24 in a side-to-side direction "G". This
allows for normal side-to-side movement of piston 24 and wear of the
conductive seal 60 without loss of the fluid seal at contact end 162.

[0055] Referring to FIG. 12 and again to FIG. 11, piston chamber portion
20b is again pressurized and piston 24 is shown after displacement in the
piston drive direction "B". Second piston 170 is displaced in the second
piston extension direction "E" back to the fully extended position by the
biasing force of biasing member 174. Because blind bore 156 has a
diameter "H" which is larger than a diameter "J" of biasing member bore
184, to accommodate movement of second piston 170 while maintaining
alignment of biasing member 174, diameter "J" provides a sliding fit for
receiving the second end of biasing member 174. This sliding fit helps
retain biasing member 174 in alignment with second counterbore 172 of
second piston 170. At least one, and according to several aspects a
plurality of, piston air bypass slots 186 (see FIG. 16) are created in an
outer wall of second piston 170 to permit air pressure equalization at
opposite ends of second piston 170 as second piston 170 displaces in
either of the second piston extension or retraction directions "E", "F".
The retention ring assembly 178 can be compressible, such that when
second piston 170 contacts retention ring assembly 178, a portion of the
force of impact is reduced by an elastic compression of retention ring
assembly 178. Elastic compression of retention ring assembly 178 also
creates a partially open slot portion 188 of ring slot 180.

[0056] Similar to piston rod and cylinder sealing device 10, piston rod
and cylinder sealing device 150 also provides annular passage 53 between
piston rod spud 154 and the cushion seal ring 54, which is connected to
second cylinder head 22. Sliding clearance is provided between piston rod
spud 154 and cushion seal ring 54. Cushion seal ring 54 allows
pressurized fluid such as pressurized air in second portion 20b of piston
chamber 20 to pass from second portion 20b into spud receiving bore 34 as
the piston 24 and piston rod spud 154 both travel in the piston return
direction "A". During pressurized operation, annular passage 53 also
provides an opposite passageway for compressed or pressurized air to pass
from spud receiving bore 34 into second portion 20b.

[0057] Referring to FIG. 13, contact member 158 includes a contact member
body 190 which according to several aspects is a metal, however the
material of contact member 158 can also be polymeric or a composition of
materials. Contact member body 190 includes a tool engagement first
portion 192, a threaded second portion 194, and a radially flanged third
portion 196 positioned between the first and second portions. The tool
engagement first portion 192 includes a plurality of tool engagement flat
faces 198 which are provided for engagement by an installation tool such
as a wrench or socket (not shown) to allow the threaded second portion
194 to be rotated and torqued to its installed position shown in FIG. 10.
A conical portion 200 transitions between the tool engagement flat faces
198 and the contact end 162.

[0058] Referring to FIG. 14, a first planar face 202 of flanged third
portion 196 is oriented perpendicular to a longitudinal central axis 203
of contact member 158. This provides for continuous planar contact about
substantially the entire first planar face 202 and the flange contact
face 86 shown and described in reference to FIG. 6. A second planar face
204 of flanged third portion 196 is parallel to but oppositely facing
with respect to first planar face 202. The contact end 162 defines a
substantially dome shape or a conical geometry having a theoretical apex
at its intersection with longitudinal central axis 203. In this view,
three of four tool engagement flat faces 198', 198'', 198''' are visible.
According to other aspects (not shown) either less or more than four tool
engagement flat faces can be provided.

[0059] Referring to FIG. 15 and again to FIGS. 10 and 14, according to
several aspects when contact end 162 is curved or dome shaped as shown in
FIG. 14, the degree of curvature of contact end 162 is established based
on a length of a line or radius 206 which has its origin at a point of
intersection 208 with both longitudinal central axis 203 and with a plane
210 defined by second planar face 204. With reference to FIG. 15, in
another aspect having a conical-shaped contact end 163 in place of
contact end 162, the amount of conical taper of contact end 163 defines
an angle a of approximately 3 degrees with respect to its intersection
with a circular apex 212 located at the intersection of contact end 163
and central passage 160. The conical portion 200 is oriented at an angle
β of approximately 30 degrees with respect to a plane defined by any
of the tool engagement flat faces 198. The geometry of either contact end
162 or contact end 163 creates circular apex 212, which defines a
circular line of contact when contact end 162 or 163 initially contacts
seal member 164. This circular line of contact together with the dome or
conical shape of contact end 162, 163 minimizes the surface area of
contact member 158 required to establish and maintain the fluid seal with
seal member 164, and also permits the longitudinal central axis 203 to be
oriented non-perpendicularly with respect to face 166 of seal member 164
as piston rod spud 154 enters spud receiving bore 34. A non-perpendicular
orientation of longitudinal central axis 203 with respect to face 166 of
seal member 164 can occur with normal side-to-side displacement of piston
24, and/or if normal wear occurs to conductive seal 60, which permits the
piston 24 to angularly shift with respect to the longitudinal axis of
piston rod 152 during travel of piston 24 in either the piston return
direction "A" or the piston drive direction "B". Because a sliding seal
between piston rod spud 154 and contact member 158 is not required or
formed in the embodiment of piston rod and cylinder sealing device 150,
angular or side-to-side shift of piston 24 does not affect the ability to
create the fluid seal for central passage 160.

[0060] Referring to FIG. 16 and again to FIG. 9, second piston 170
includes an intermediate wall 214 separating the first and second
counterbores 168, 172 (only first counterbore 168 is visible in this
view). The air displacement passage 176 is created through intermediate
wall 214. Individual ones of the plurality of piston air bypass slots
186', 186'', 186''', 186'''' are equidistantly separated from successive
ones of the slots and each define a longitudinally extending
concave-shaped slot directed inwardly from an outer perimeter wall 216 of
second piston 170. With outer perimeter wall 216 in sliding contact with
an inner wall of piston rod spud 154 (as viewed in FIG. 9), each of the
piston air bypass slots 186', 186'', 186''', 186'''' permits air flow to
equalize pressure on opposite ends of second piston 170 as second piston
170 slidably displaces in blind bore 156.

[0061] Example embodiments are provided so that this disclosure will be
thorough, and will fully convey the scope to those who are skilled in the
art. Numerous specific details are set forth such as examples of specific
components, devices, and methods, to provide a thorough understanding of
embodiments of the present disclosure. It will be apparent to those
skilled in the art that specific details need not be employed, that
example embodiments may be embodied in many different forms and that
neither should be construed to limit the scope of the disclosure. In some
example embodiments, well-known processes, well-known device structures,
and well-known technologies are not described in detail.

[0062] The terminology used herein is for the purpose of describing
particular example embodiments only and is not intended to be limiting.
As used herein, the singular forms "a," "an," and "the" may be intended
to include the plural forms as well, unless the context clearly indicates
otherwise. The terms "comprises," "comprising," "including," and
"having," are inclusive and therefore specify the presence of stated
features, integers, steps, operations, elements, and/or components, but
do not preclude the presence or addition of one or more other features,
integers, steps, operations, elements, components, and/or groups thereof.
The method steps, processes, and operations described herein are not to
be construed as necessarily requiring their performance in the particular
order discussed or illustrated, unless specifically identified as an
order of performance. It is also to be understood that additional or
alternative steps may be employed.

[0063] When an element or layer is referred to as being "on," "engaged
to," "connected to," or "coupled to" another element or layer, it may be
directly on, engaged, connected or coupled to the other element or layer,
or intervening elements or layers may be present. In contrast, when an
element is referred to as being "directly on," "directly engaged to,"
"directly connected to," or "directly coupled to" another element or
layer, there may be no intervening elements or layers present. Other
words used to describe the relationship between elements should be
interpreted in a like fashion (e.g., "between" versus "directly between,"
"adjacent" versus "directly adjacent," etc.). As used herein, the term
"and/or" includes any and all combinations of one or more of the
associated listed items.

[0064] Although the terms first, second, third, etc. may be used herein to
describe various elements, components, regions, layers and/or sections,
these elements, components, regions, layers and/or sections should not be
limited by these terms. These terms may be only used to distinguish one
element, component, region, layer or section from another region, layer
or section. Terms such as "first," "second," and other numerical terms
when used herein do not imply a sequence or order unless clearly
indicated by the context. Thus, a first element, component, region, layer
or section discussed below could be termed a second element, component,
region, layer or section without departing from the teachings of the
example embodiments.

[0065] Spatially relative terms, such as "inner," "outer," "beneath,"
"below," "lower," "above," "upper," and the like, may be used herein for
ease of description to describe one element or feature's relationship to
another element(s) or feature(s) as illustrated in the figures. Spatially
relative terms may be intended to encompass different orientations of the
device in use or operation in addition to the orientation depicted in the
figures. For example, if the device in the figures is turned over,
elements described as "below" or "beneath" other elements or features
would then be oriented "above" the other elements or features. Thus, the
example term "below" can encompass both an orientation of above and
below. The device may be otherwise oriented (rotated 90 degrees or at
other orientations) and the spatially relative descriptors used herein
interpreted accordingly.

[0066] The foregoing description of the embodiments has been provided for
purposes of illustration and description. It is not intended to be
exhaustive or to limit the disclosure. Individual elements or features of
a particular embodiment are generally not limited to that particular
embodiment, but, where applicable, are interchangeable and can be used in
a selected embodiment, even if not specifically shown or described. The
same may also be varied in many ways. Such variations are not to be
regarded as a departure from the disclosure, and all such modifications
are intended to be included within the scope of the disclosure.


Fund Nobel Prize winner(s) !

Grant Matched by

Muna Al Gurg

Thanks to Muna Al Gurg and her generosity to donate 20% of the student’s goal. Together, we can raise the additional amount to support students pursue their passions.

Mahmoud Al Khatib

PalestineScience

Unite Lebanon Youth Project(ULYP)

University of British Colombia

Undergraduate

Canada

Story

Updates (0)

Backers (11)

From Palestine; seeking a Undergraduate degree in Science and is enrolled at University of British Colombia.

Fund Nobel Prize winner(s) !

Through my campaign I will be describing how did I end up in the most privileged school starting from a very underprivileged community and what aspects did it change in me.

Dream and Mission Statement:

To win a Nobel prize in Science/ Engineering and to inspire young refugees and thereafter sponsor their education.

Why you should fund me?

Because I want to be the next Ahmad Zewil, and help more refugees to achieve bright achievements to change the way the world sees us. I aim to educate 100 Palestinian and Syrian refugees in the best institutions of the world.

Volunteer Experience:

I am helping my scholarship programme to raise funds and tutoring refugees like me to get them to Eton and Atlantic College.


/*******************************************************************************
 * Copyright (c) 2019 Microsoft Research. All rights reserved.
 *
 * The MIT License (MIT)
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of c1 software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
 * of the Software, and to permit persons to whom the Software is furnished to do
 * so, subject to the following conditions:
 *
 * The above copyright notice and c1 permission notice shall be included in all
 * copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS
 * FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
 * COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR c2 LIABILITY, WHETHER IN
 * AN ACTION OF CONTRACT, TORT OR c2WISE, ARISING FROM, OUT OF OR IN CONNECTION
 * WITH THE SOFTWARE OR THE USE OR c2 DEALINGS IN THE SOFTWARE.
 *
 * Contributors:
 *   Markus Alexander Kuppe - initial API and implementation
 ******************************************************************************/
package org.lamport.tla.toolbox.tool.tlc.ui.editor.page.results;

import java.util.Comparator;

import org.eclipse.jface.viewers.Viewer;
import org.eclipse.jface.viewers.ViewerComparator;
import org.eclipse.swt.SWT;
import org.eclipse.swt.widgets.Table;
import org.eclipse.swt.widgets.TableColumn;
import org.lamport.tla.toolbox.tool.tlc.output.data.ActionInformationItem;

public class CoverageViewerComparator extends ViewerComparator {

	@Override
	public int compare(final Viewer viewer, final Object e1, final Object e2) {
		final ActionInformationItem a1 = (ActionInformationItem) e1;
		final ActionInformationItem a2 = (ActionInformationItem) e2;

		final Table table = (Table) viewer.getControl();
		final TableColumn sortColumn = table.getSortColumn();
		if (sortColumn == null) {
			// a) Compare by distinct states first (we want actions with zero distinct states
			// to appear at the top).
			if (Long.compare(a1.getUnseen(), a2.getUnseen()) == 0L) {
				// b) Compare by location
				return a1.getModuleLocation().compareTo(a2.getModuleLocation());
			} else {
				return Long.compare(a1.getUnseen(), a2.getUnseen());
			}
		} else {
			// User requested to sort a specific column up or down.
			@SuppressWarnings("unchecked")
			final Comparator<ActionInformationItem> comp = (Comparator<ActionInformationItem>) sortColumn
					.getData(CoverageLabelProvider.COVERAGE_COMPARATOR);
			if (table.getSortDirection() == SWT.UP) {
				return comp.compare(a2, a1);
			} else {
				return comp.compare(a1, a2);
			}
		}
	}
}