# Sheaves of noncommutative smooth and holomorphic functions associated with
the non-Abelian two-dimensional Lie algebra

O. Yu. Aristov<EMAIL_ADDRESS>
###### Abstract.

Dosi and, quite recently, the author showed that, on the character space of a
nilpotent Lie algebra, there exists a sheaf of Fréchet–Arens–Michael algebras
(of noncommutative holomorphic functions in the complex case and of
noncommutative smooth functions in the real case). We construct similar
sheaves (both versions, holomorphic and smooth) on a special space of
representations for the Lie algebra of the group of affine transformations of
the real line (which is the simplest nonnilpotent solvable Lie algebra).

This work was supported by the Russian Foundation for Basic Research under
grant 19-01-00447.

With any triangular Lie algebra $\mathfrak{g}$ over $\mathbb{R}$ a
Fréchet–Arens–Michael algebra is associated, whose elements can be regarded as
noncommutative $C^{\infty}$ functions (see the author’s paper [1]). In[1], the
author considered a local version of this algebra in the case where
$\mathfrak{g}$ is nilpotent (by analogy with algebras of noncommutative
holomorphic functions defined by Dosi in [2] and [3]). Namely, for each open
subset $V$ of the character space of $\mathfrak{g}$, the multiplication in the
universal enveloping algebra $U(\mathfrak{g})$ can be extended to a
Fréche–Arens–Michael algebra of polynomial growth (it is denoted by
$C^{\infty}_{\mathfrak{g}}(V)$), and there arises a sheaf of algebras of this
type (which are noncommutative if $\mathfrak{g}$ is non-Abelian) [1, Theorems
5.1 and 5.5]. In a similar way, a sheaf of noncommutative holomorphic
functions for a nilpotent Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ is
constructed (see [3, Sec. 5.1, pp. 119–120], where certain constraints on
$\mathfrak{g}$ are imposed, and [1, Sec. 6], where the general case is
considered).

However, the definition in the form given in [1] cannot be applied to
nonnilpotent triangular Lie algebras. It is easy to check that, even for the
simplest nonnilpotent triangular Lie algebra $\mathfrak{a}\mathfrak{f}_{1}$
(of the group of affine transformations of the real line), the multiplication
in $U(\mathfrak{a}\mathfrak{f}_{1})$ does not generally extend to a continuous
operation on the required space [1, Sec. 5.2]. The same is true of
noncommutative holomorphic functions.

In this paper, we propose a method for overcoming this difficulty in the case
of $\mathfrak{a}\mathfrak{f}_{1}$. We consider a topological space consisting
of special indecomposable representations of $\mathfrak{a}\mathfrak{f}_{1}$
(the set of characters is its proper subset) and construct a
Fréchet–Arens–Michael algebra $C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$
of polynomial growth for each open set $V$ in this space. The proof that it
belongs to the required class uses curious algebras of triangular matrices;
see Sec. 1. As a result, we obtain a sheaf (Theorem 4), which can be called a
sheaf of noncommutative smooth functions by analogy with the Abelian case. In
a similar way, we construct a sheaf $\mathfrak{F}_{\mathfrak{g}}(-)$ of
noncommutative holomorphic functions for the complexification of the algebra
$\mathfrak{a}\mathfrak{f}_{1}$ (Theorem 7). The results obtained for this
simplest algebra give hope for the existence of sheaves for any triangular Lie
algebras.

### On the used classes of algebras

Polynomial growth is the most important property of both the globally defined
algebra $C^{\infty}_{\mathfrak{g}}$, where $\mathfrak{g}$ is a triangular Lie
$\mathbb{R}$-algebra, and the locally defined algebras
$C^{\infty}_{\mathfrak{g}}(V)$, where $\mathfrak{g}$ is nilpotent. In
particular, $C^{\infty}_{\mathfrak{g}}$ has a universal property in the class
of Banach algebras of polynomial growth [1, Theorem 4.4]. This is also true in
the case $\mathfrak{g}=\mathfrak{a}\mathfrak{f}_{1}$. Our first objective is
to specify locally defined $C^{\infty}$ algebras for this case. It is natural
to expect that they must be of polynomial growth, and this is indeed so
(Theorem 3).

Recall that a locally convex algebra (here and in what follows, all algebras
are assumed to be unital) over $\mathbb{R}$ or $\mathbb{C}$ is called an
_Arens–Michael algebra_ if its topology is determined by a family of
submultiplicative prenorms (i.e., prenorms for which
$\|ab\|\leqslant\|a\|\,\|b\|$). In other words, it is a projective limit of
Banach algebras (see [4]). Following [1, Definition 2.6], we say that an
Arens–Michael $\mathbb{R}$-algebra $B$ has _polynomial growth_ if, for any
$b\in B$ and any continuous submultiplicative prenorm $\|\cdot\|$ on $B$ there
exists a $K>0$ and an $\alpha\geqslant 0$ such that

$\|e^{isb}\|\leqslant K(1+|s|)^{\alpha}\qquad\text{for all $s\in\mathbb{R}$.}$

(here $\|\cdot\|$ is extended to the complexification). In particular, such
are the Banach algebras $C^{n}[a,b]$ of $n$-times continuously differentiable
functions for $n\in\mathbb{Z}_{+}$, the Arens–Michael algebras
$C^{\infty}[a,b]$ of infinitely differentiable functions, and their
generalizations. The basic properties of algebras of polynomial growth can be
found in [1] and [5].

The algebras of noncommutative $C^{\infty}$ functions defined in [1] were
introduced by analogy with the algebras of noncommutative holomorphic
functions considered previously by Dosi in [2] and [3] for nilpotent Lie
$\mathbb{C}$-algebras $\mathfrak{g}$, namely, with the globally defined
algebra $\mathfrak{F}_{\mathfrak{g}}$ and the locally defined algebras
$\mathfrak{F}_{\mathfrak{g}}(V)$. In [1], the definition of
$\mathfrak{F}_{\mathfrak{g}}$ was generalized to arbitrary finite-dimensional
solvable Lie $\mathbb{C}$-algebras (they are automatically triangular). Note
that, in the case $\mathfrak{g}=\mathfrak{a}\mathfrak{f}_{1}$, the
corresponding algebra $\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}$ has the
universal property in the class of all Banach algebras, i.e., is the
Arens–Michael envelope; however, this is a mere coincidence (this does not
hold for arbitrary $\mathfrak{g}$).

In [1, Remark 6.8], it was mentioned that the algebra
$\mathfrak{F}_{\mathfrak{g}}(V)$ for a nilpotent $\mathfrak{g}$ and the
algebra $\mathfrak{F}_{\mathfrak{g}}$ in the general case are projective
limits of Banach algebras satisfying a polynomial identity. Moreover, we can
restrict the set of polynomial identities, because, as essentially shown in
[1], these algebras are locally solvable Arens–Michael algebras. (We say that
an associative algebra is _solvable_ if it is solvable as a Lie algebra, and
we say that an Arens–Michael algebra is _locally solvable_ if it is a
projective limit of Banach solvable algebras.) Note also that any Banach
$\mathbb{R}$-algebra of polynomial growth is solvable.

As in the case of algebras of noncommutative smooth functions, it is natural
to expect that the properties of algebras
$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ defined in (10) are similar
to those in the nilpotent case. Indeed, the
$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ are locally solvable
Arens–Michael algebras (see Theorems 5 and 7).

##  1\. Algebras of triangular matrices

In this section, we define certain algebras of triangular matrices whose
entries belong to the spaces of infinitely smooth functions and of holomorphic
functions (with domains depending on the position in the matrix) and show that
they are Arens-Michael algebras of polynomial growth in the case of the field
$\mathbb{R}$ and locally solvable in the case of $\mathbb{C}$. These algebras
play an auxiliary role and are used in the proofs of the main theorems of Sec.
2, but they may also be of independent interest.

### The $C^{\infty}$-version: polynomial growth

Let $\mathrm{T}_{p}$ denote the algebra of upper triangular (including
diagonal) real matrices of order $p$, where $p\in\mathbb{N}$. In [1, Theorem
2.12], it was shown that, for any $p,m\in\mathbb{N}$ and any open set $W$ in
$\mathbb{R}^{m}$, the algebra $C^{\infty}(W,\mathrm{T}_{p})$ of matrix-valued
functions is of polynomial growth. This algebra can be identified with the
algebra $\mathrm{T}_{p}(C^{\infty}(W))$ of triangular matrices with entries in
$C^{\infty}(W)$. We need the following generalization.

We consider tuples

$\underline{K}=\\{K_{ij}\\!:\,1\leqslant i\leqslant j\leqslant p\\}$

of compact subsets of $\mathbb{R}^{m}$ such that the interior of each $K_{ij}$
is dense. We denote by $\mathcal{C}^{m}_{p}$ the family of all such sets
satisfying the condition $K_{ik}\subset K_{ij}\cap K_{jk}$ for all admissible
indices. Given $n\in\mathbb{Z}_{+}$, by $\mathrm{T}_{p}(C^{n}(\underline{K}))$
we denote the vector space of upper triangular matrices of the form

$\begin{pmatrix}f_{11}&f_{12}&\cdots&&f_{1p}\\\ &f_{22}&&&\\\
&&\ddots&&\vdots\\\ &&&f_{p-1,p-1}&f_{p-1,p}\\\ &&&&f_{pp}\end{pmatrix},$

in which the functions $f_{ij}$ belong to different spaces, namely, $f_{ij}\in
C^{n}(K_{ij})$. The set $K_{ij}$ is allowed to be empty; in this case, we set
$C^{n}(K_{ij})=0$. The space $\mathrm{T}_{p}(C^{n}(\underline{K}))$ treated as
a Cartesian product of function spaces is a Banach space.

Let $\underline{K}\in\mathcal{C}^{m}_{p}$. To simplify notation, for any $f\in
C^{n}(K_{ij})$ and $g\in C^{n}(K_{jk})$, we denote the product of the
restrictions of these functions to $K_{ik}$ by $fg$. Obviously, $fg\in
C^{n}(K_{ik})$. We define multiplication in
$\mathrm{T}_{p}(C^{n}(\underline{K}))$ by the standard formula: the product of
matrices $f=(f_{ij})$ and $g=(g_{jk})$ is the matrix with entries
$h_{ik}=\sum_{j}f_{ij}g_{jk}$. The associativity of multiplication is obvious.

######  Proposition 1.

If $n\in\mathbb{Z}_{+}$, $p,m\in\mathbb{N}$ and
$\underline{K}\in\mathcal{C}^{m}_{p}$, then
$\mathrm{T}_{p}(C^{n}(\underline{K}))$ is a Banach $\mathbb{R}$-algebra of
polynomial growth.

###### Proof.

It is easy to see that multiplication in
$\mathrm{T}_{p}(C^{n}(\underline{K}))$ is continuous; therefore, this is a
Banach algebra. Consider the following extension of Banach algebras:

$0\longleftarrow C^{n}(K_{11})\times\cdots\times
C^{n}(K_{pp})\longleftarrow\mathrm{T}_{p}(C^{n}(\underline{K}))\longleftarrow
I\longleftarrow 0,$

where $I$ is the ideal of matrices with zero diagonal. Obviously, it is split
and nilpotent. Since the class of Banach $\mathbb{R}$-algebras of polynomial
growth is stable with respect to the passage to closed subalgebras and finite
Cartesian products [1, Proposition 2.11] and all $C^{n}(K_{jj})$ are of
polynomial growth [1, Proposition 2.13], it follows that
$C^{n}(K_{11})\times\cdots\times C^{n}(K_{pp})$ is of polynomial growth as
well. Thus, $\mathrm{T}_{p}(C(\underline{K}))$ is a split nilpotent extension
of a Banach $\mathbb{R}$-algebra of polynomial growth; according to [1,
Theorem 2.14], it also has this property. ∎

Now let $\underline{W}=\\{W_{ij}\\!:\,1\leqslant i\leqslant j\leqslant p\\}$
be a tuple of open subsets $\mathbb{R}^{m}$ such that $W_{ik}\subset
W_{ij}\cap W_{jk}$ for all admissible indices. Consider the vector space
$\mathrm{T}_{p}(C^{\infty}(\underline{W}))$ of upper triangular matrices
$(h_{ij})$, where $h_{ij}\in C^{\infty}(W_{ij})$ for all $i\leqslant j$; we
endow it with the same multiplication as above and the product topology.

######  Theorem 1.

Let $p,m\in\mathbb{N}$, and let $\underline{W}=\\{W_{ij}\\!:\,1\leqslant
i\leqslant j\leqslant p\\}$ be a tuple of open subsets of $\mathbb{R}^{m}$
such that $W_{ik}\subset W_{ij}\cap W_{jk}$ for all admissible indices. Then
$\mathrm{T}_{p}(C^{\infty}(\underline{W}))$ is the projective limit of Banach
algebras of the form $\mathrm{T}_{p}(C^{n}(\underline{K}))$, where
$\underline{K}\in\mathcal{C}^{m}_{p}$ and $\underline{K}\subset\underline{W}$
(that is, $K_{ij}\subset W_{ij}$ for all admissible indices) and, therefore, a
Fréchet–Arens–Michael $\mathbb{R}$-algebra of polynomial growth.

###### Proof.

By Proposition 1, it suffices to prove the first assertion, i.e., that
$\mathrm{T}_{p}(C^{\infty}(\underline{W}))$ is the projective limit of Banach
algebras of the form $\mathrm{T}_{p}(C^{n}(\underline{K}))$, where
$\underline{K}\in\mathcal{C}^{m}_{p}$ and $\underline{K}\subset\underline{W}$.

Note that the sets $\underline{K}$ of compact subsets satisfying the
conditions $\underline{K}\in\mathcal{C}^{m}_{p}$ and
$\underline{K}\subset\underline{W}$ form a net with respect to inclusion.
Indeed, the first condition is preserved because

$K_{ik}\cup K^{\prime}_{ik}\subset(K_{ij}\cup K^{\prime}_{ij})\cap(K_{jk}\cup
K^{\prime}_{jk})$

if $K_{ik}\subset K_{ij}\cap K_{jk}$ and $K^{\prime}_{ik}\subset
K^{\prime}_{ij}\cap K^{\prime}_{jk}$, and the preservation of the second is
obvious.

If $\underline{K}\subset\underline{K}^{\prime}$ and $n^{\prime}\geqslant n$,
we have a continuous homomorphism

$\mathrm{T}_{p}(C^{n^{\prime}}(\underline{K}^{\prime}))\to\mathrm{T}_{p}(C^{n}(\underline{K})).$

There arises the projective system of Banach algebras

$(\mathrm{T}_{p}(C^{n}(\underline{K}))\\!:\underline{K}\in\mathcal{C}^{m}_{p},\,\underline{K}\subset\underline{W},\,n\in\mathbb{Z}_{+}).$

On the other hand, $\mathrm{T}_{p}(C^{\infty}(\underline{W}))$ is the
projective limit of spaces of the form $\mathrm{T}_{p}(C^{n}(\underline{K}))$,
where $\underline{K}\subset\underline{W}$, in the category of Banach spaces.
Thus, to complete the proof, it suffices, given any
$\underline{K}\subset\underline{W}$, to find a
$\underline{K}^{\prime}\in\mathcal{C}^{m}_{p}$ for which
$\underline{K}\subset\underline{K}^{\prime}\subset\underline{W}$.

First, note that, for fixed $i^{\prime}$ and $k^{\prime}$ satisfying the
inequality $i^{\prime}\leqslant k^{\prime}$ and any $x\in
W_{i^{\prime}k^{\prime}}$, there exists a $\underline{K}\subset\underline{W}$
such that $x\in K_{i^{\prime}k^{\prime}}$ and
$\underline{K}\in\mathcal{C}^{m}_{p}$. Indeed, we have $x\in
W_{i^{\prime}j}\cap W_{jk^{\prime}}$ for all $j$ satisfying the condition
$i^{\prime}\leqslant j\leqslant k^{\prime}$. Hence there exists a ball $S$ of
nonzero radius centered at $x$ which is contained in all $W_{i^{\prime}j}$ and
$W_{jk^{\prime}}$ for these $j$. We set $K_{ik}\\!:=S$ if $i^{\prime}\leqslant
i\leqslant k\leqslant k^{\prime}$ and $K_{ik}\\!:=\emptyset$ otherwise. It is
easy to check that the required condition $K_{ik}\subset K_{ij}\cap K_{jk}$
holds for all admissible indices, i.e., $\underline{K}\in\mathcal{C}^{m}_{p}$.

Consider the topological space $\coprod_{i\leqslant j}W_{ij}$ (disjoint union)
and its compact subset $\coprod_{i\leqslant j}K_{ij}$. According to what was
proved above, there exists a family $(\underline{K}_{\alpha})$ such that
$\bigcup_{\alpha}\underline{K}_{\alpha}=\underline{W}$ and
$\underline{K}_{\alpha}\in\mathcal{C}^{m}_{p}$ for each $\alpha$. Since
$\coprod_{i\leqslant j}K_{ij}$ is compact and each set in
$(\underline{K}_{\alpha})$ is a neighborhood, we can assume that this family
is finite. Since $\mathcal{C}^{m}_{p}$ is a net, it follows that, passing to
the union, we obtain the required $\underline{K}^{\prime}$. ∎

### The holomorphic version: locally solvable algebras

Now we turn to algebras of triangular matrices composed of holomorphic
functions.

We consider tuples $\underline{K}=\\{K_{ij}\\!:\,1\leqslant i\leqslant
j\leqslant p\\}$ of compact subsets of $\mathbb{C}^{m}$ and tuples

$\underline{W}=\\{W_{ij}\\!:\,1\leqslant i\leqslant j\leqslant p\\}$

of open subsets of $\mathbb{C}^{m}$ such that $W_{ik}\subset W_{ij}\cap
W_{jk}$ for all admissible indices. (The notation $\mathcal{C}^{m}_{p}$ is
used for the family of compact subsets satisfying the same conditions as in
the real case except the density of interior.)

Let $A(K_{ij})$ denote the Banach algebra of functions holomorphic inside
$K_{ij}$ and continuous on the closure of $K_{ij}$ (endowed with the max-
norm). Then $\mathrm{T}_{p}(A(\underline{K}))$ is a Banach space.

######  Proposition 2.

If $p,m\in\mathbb{N}$ and $\underline{K}\in\mathcal{C}^{m}_{p}$, then
$\mathrm{T}_{p}(A(\underline{K}))$ is a Banach solvable algebra.

###### Proof.

Since multiplication in $\mathrm{T}_{p}(A(\underline{K}))$ is continuous, it
follows that this is a Banach algebra. It is easy to see that the subspace
$[\mathrm{T}_{p}(A(\underline{K})),\mathrm{T}_{p}(A(\underline{K}))]$ is
nilpotent, and hence this algebra is solvable. ∎

Consider the vector space $\mathrm{T}_{p}(\mathcal{O}(\underline{W}))$ of
upper triangular matrices $(f_{ij})$ in which every $f_{ij}$ belongs to the
space $\mathcal{O}(W_{ij})$ of holomorphic functions; we endow it with the
same multiplication as above and with the product topology.

######  Theorem 2.

Let $p,m\in\mathbb{N}$, and let $\underline{W}=\\{W_{ij}\\!:\,1\leqslant
i\leqslant j\leqslant p\\}$ be a tuple of open subsets of $\mathbb{C}^{m}$
such that $W_{ik}\subset W_{ij}\cap W_{jk}$ for all admissible indices. Then
$\mathrm{T}_{p}(\mathcal{O}(\underline{W}))$ is the projective limit of Banach
algebras of the form $\mathrm{T}_{p}(A(\underline{K}))$, where
$\underline{K}\in\mathcal{C}^{m}_{p}$ and $\underline{K}\subset\underline{W}$;
therefore, this is a locally solvable Fréchet–Arens–Michael algebra.

The proof of this theorem is similar to that of Theorem 1 but uses Proposition
2 instead of Proposition 1.

##  2\. Locally defined algebras of noncommutative functions and sheaves

Let $\mathfrak{a}\mathfrak{f}_{1}$ denote the two-dimensional Lie algebra
(real or complex, depending on the context) with basis $e_{1},e_{2}$ and
multiplication defined by $[e_{1},e_{2}]=e_{2}$. In this section, we define a
topological space consisting of special indecomposable representations of
$\mathfrak{a}\mathfrak{f}_{1}$ and show that, for each open set $V$, we can
define a FréchetArens–Michael algebra
$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ of polynomial growth in the
real case and a locally solvable FréchetArens–Michael algebra
$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ in the complex case. We also
prove the main results of this paper, namely, that the corresponding functors
determine sheaves (Theorems 4 and 7).

### The sheaf of noncommutative $C^{\infty}$ functions

In the case where $\mathfrak{g}$ is a nilpotent Lie $\mathbb{C}$-algebra and
$V$ is an open subset of its character space, multiplication in the universal
enveloping algebra $U(\mathfrak{g})$ continuously extends to
$C^{\infty}_{\mathfrak{g}}(V)\\!:=C^{\infty}(V)^{\mathbb{Z}_{+}}$ (see [1,
Theorem 5.1]). For the algebra $\mathfrak{a}\mathfrak{f}_{1}$, which is not
nilpotent, the definition must be modified. The idea is to consider the
product of spaces of the form $C^{\infty}(V_{q})$, where the open sets $V_{q}$
are subject to certain relations, instead of the power of $C^{\infty}(V)$.

Consider the following family of indecomposable representations of the algebra
$\mathfrak{a}\mathfrak{f}_{1}$:

$\sigma_{r,q}(e_{1})\\!:=X_{q}+r\quad\text{and}\quad\sigma_{r,q}(e_{2})\\!:=Y_{q}\qquad(r\in\mathbb{R},\,q\in\mathbb{Z}_{+}),$

where

$X_{q}\\!:=\begin{pmatrix}q&&&&\\\ &q-1&&&\\\ &&\ddots&&\\\ &&&1&\\\
&&&&0\end{pmatrix}\quad\text{and}\quad Y_{q}\\!:=\begin{pmatrix}0&1&&&\\\
&0&1&&\\\ &&\ddots&\ddots&\\\ &&&0&1\\\ &&&&0\end{pmatrix}.$ (1)

(For brevity, we identify finite-dimensional operators with their matrices.)

Let us denote the set
$\\{\sigma_{r,q}\\!:\,r\in\mathbb{R},\,q\in\mathbb{Z}_{+}\\}$ by
$\Omega_{\mathbb{R}}$ and endow it with a topology as follows. Consider the
shift operation on the family of all subsets of $\mathbb{R}$:

$X+\mu\\!:=\\{x+\mu\\!:\,x\in
X\\}\qquad(X\subset\mathbb{R},\,\mu\in\mathbb{R}).$

We declare a subset of $\Omega_{\mathbb{R}}$ to be open if it is of the form

$\bigcup_{q=0}^{\infty}\\{\sigma_{r,q}\\!:\,r\in V_{q}\\},$ (2)

where $(V_{q})_{q\in\mathbb{Z}_{+}}$ is a sequence of open subsets of
$\mathbb{R}$ satisfying the condition

$V_{q}\cap(V_{q}+1)\supset V_{q+1}\qquad(q\in\mathbb{Z}_{+});$ (3)

in particular, the sequence $(V_{q})$ is nonincreasing. (The shift by $1$
appears because this number is an eigenvalue of the operator
$\mathop{\mathrm{ad}}\nolimits e_{1}$ in the adjoint representation.) What we
obtain is indeed a topology, because the shift commutes with the intersection
and union operations. (Note that this topology is not Hausdorff, because any
open set containing $\sigma_{r,q+1}$ contains also $\sigma_{r,q}$ and
$\sigma_{r-1,q}$.)

For an open subset $V$ of $\Omega_{\mathbb{R}}$ represented in the form (2),
we set

$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)\\!:=\prod_{q=0}^{\infty}C^{\infty}(V_{q}).$
(4)

(Note that if $V_{q}$ is empty for some $q$, then $C^{\infty}(V_{q})$ is the
zero algebra. Obviously, then the algebras for larger $q$ are zero as well; in
this case, the product can be regarded to be finite.)

In the case where $V=\Omega_{\mathbb{R}}$, we have $V_{q}=\mathbb{R}$ for all
$q$. The space
$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(\Omega_{\mathbb{R}})$ coincides
with the space $C^{\infty}_{\mathfrak{g}}$ defined in [1] (for
$\mathfrak{g}=\mathfrak{a}\mathfrak{f}_{1}$).

Writing the elements of the universal enveloping algebra
$U(\mathfrak{a}\mathfrak{f}_{1})$ in the form
$a=\sum_{j}f_{j}(e_{1})e_{2}^{j}$, where $f_{j}$ belongs to the polynomial
algebra $\mathbb{R}[\lambda]$, we obtain the linear map

$U(\mathfrak{a}\mathfrak{f}_{1})\to
C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)\\!:a\mapsto(f_{q}).$ (5)

######  Theorem 3.

Let $V$ be an open subset of $\Omega_{\mathbb{R}}$. Then multiplication in
$U(\mathfrak{a}\mathfrak{f}_{1})$ extends to a continuous multiplication in
$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$. Moreover,
$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ with this multiplication is a
projective limit of Banach $\mathbb{R}$-algebras of polynomial growth and,
therefore, a Fréchet–Arens–Michael algebra of polynomial growth.

###### Proof.

We argue as in [1, Example 4.7] with the only difference that, instead of
$\mathrm{T}_{q+1}(C^{\infty}(\mathbb{R}))$ we use an algebra of the form
$\mathrm{T}_{q+1}(C^{\infty}(\underline{W}))$ ) with a special
$\underline{W}$.

Let $\mathbb{R}[\lambda;\mathrm{T}_{q+1}]$ denote the algebra of matrix-valued
polynomials. For each $q\in\mathbb{Z}_{+}$, consider the homomorphism

$\widetilde{\pi}_{q}\\!:U(\mathfrak{a}\mathfrak{f}_{1})\to\mathbb{R}[\lambda;\mathrm{T}_{q+1}],\quad\widetilde{\pi}_{q}(a)\\!:=(\lambda\mapsto\sigma_{\lambda,q}(a)).$

According to the Poincaré-Birkhoff-Witt theorem, any element of
$U(\mathfrak{a}\mathfrak{f}_{1})$ can be represented in the form
$a=\sum_{j}f_{j}(e_{1})e_{2}^{j}$. It is easy to check that

$[\widetilde{\pi}_{q}(a)](\lambda)=\begin{pmatrix}f_{0}(\lambda+q)&f_{1}(\lambda+q)&\cdots&&f_{q}(\lambda+q)\\\
&f_{0}(\lambda+q-1)&&&\\\ &&\ddots&&\vdots\\\
&&&f_{0}(\lambda+1)&f_{1}(\lambda+1)\\\ &&&&f_{0}(\lambda)\end{pmatrix}.$ (6)

We set

$W^{q+1}_{ij}\\!:=V_{j-i}-q-1+i\qquad(1\leqslant i\leqslant j\leqslant q+1),$
(7)

and

$\underline{W}_{(q+1)}\\!:=\\{W^{q+1}_{ij}\\!:\,1\leqslant i\leqslant
j\leqslant q+1\\}.$

Obviously $\underline{W}_{(q+1)}$ is a tuple of open subsets of $\mathbb{R}$.

Let us check that $W^{q+1}_{ik}\subset W^{q+1}_{ij}\cap W^{q+1}_{jk}$ for
$1\leqslant i\leqslant j\leqslant k\leqslant q+1$. It follows from (3) that
$V_{k-i}\subset V_{j-i}$ and $V_{k-i}\subset V_{k-j}+j-i$ (because
$k-j=(k-i)-(j-i)$). Therefore,

$W^{q+1}_{ik}=V_{k-i}-q-1+i\subset(V_{j-i}-q-1+i)\cap(V_{k-j}-q-1+j)=W^{q+1}_{ij}\cap
W^{q+1}_{jk}.$

Thus, $\mathrm{T}_{q+1}(C^{\infty}(\underline{W}_{(q+1)}))$ satisfies the
assumptions of Theorem 1 and, therefore, is a projective limit of Banach
$\mathbb{R}$-algebras of polynomial growth and a Fréchet algebra.

Increasing the range of each $\widetilde{\pi}_{q}$ to
$\mathrm{T}_{q+1}(C^{\infty}(\underline{W}_{(q+1)}))$, we can consider the
homomorphism

$\rho_{V}\\!:U(\mathfrak{a}\mathfrak{f}_{1})\to\prod_{q=0}^{\infty}\mathrm{T}_{q+1}(C^{\infty}(\underline{W}_{(q+1)}))\\!:a\mapsto(\widetilde{\pi}_{q}(a)).$

Note that $U(\mathfrak{a}\mathfrak{f}_{1})$ is a dense subspace o
$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ (because, by Weierstrass’
theorem on approximation of continuously differentiable functions, the
polynomial algebra is dense in each $C^{\infty}(V_{q})$). Since the class of
Fréchet algebras is stable with respect to the passage to countable Cartesian
products and closed subalgebras and the same is true of the class of
projective limits of Banach algebras of polynomial growth [1, Proposition
2.11], it follows that, to complete the proof, it suffices to show that
$\rho_{V}$ is continuous and topologically injective (with respect to the
restriction of the topology of $C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$
to $U(\mathfrak{a}\mathfrak{f}_{1})$).

First, we prove the continuity. To this end, we need to show that the
homomorphism
$U(\mathfrak{a}\mathfrak{f}_{1})\to\mathrm{T}_{q+1}(C^{\infty}(\underline{W}_{(q+1)}))$
is continuous for each $q$. Recall that the topology of $C^{\infty}(U)$, where
$U$ is an open subset of $\mathbb{R}^{m}$, is determined by the prenorms

$|f|_{K,n}\\!:=\sum_{\beta\in\mathbb{Z}_{+}^{m},\,|\beta|=n}|f^{(\beta)}|_{K,0}\,,\qquad\text{where}\quad|f|_{K,0}\\!:=\max_{x\in
K}|f(x)|,$

$n\in\mathbb{Z}_{+}$ and $K$ is any compact subset of $U$. (Here
$|\beta|\\!:=\beta_{1}+\cdots\beta_{m}$ for
$\beta=(\beta_{1},\ldots,\beta_{m})\in\mathbb{Z}_{+}^{m}$ and $f^{(\beta)}$
denotes the partial derivative of the function $f$.)

On the one hand, the topology of
$\mathrm{T}_{q+1}(C^{\infty}(\underline{W}_{(q+1)}))$ is determined by the
prenorms

$h\mapsto|h_{ij}|_{K_{ij},n},\qquad
h=(h_{ij})\in\mathrm{T}_{q+1}(C^{\infty}(\underline{W}_{(q+1)})),$ (8)

where $1\leqslant i\leqslant j\leqslant q+1$, $K_{ij}$ is a compact subset of
$W_{ij}$, and $n\in\mathbb{Z}_{+}$. On the other hand, the topology of
$U(\mathfrak{a}\mathfrak{f}_{1})$, which is inherited from
$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$, is determined by the prenorms

$a\mapsto|f_{p}|_{M,l}\qquad\text{($p,l\in\mathbb{Z}_{+}$, $M\subset V_{p}$ is
compact)}.$ (9)

Estimating the matrix elements in (6), we obtain

$|\widetilde{\pi}_{q}(a)_{ij}|_{K_{ij},n}\leqslant|f_{j-i}|_{K_{ij}+q+1-i,\,n}\qquad(i\leqslant
j)$

for each $a\in U(\mathfrak{a}\mathfrak{f}_{1})$. Therefore, to prove the
continuity of $\rho_{V}$, it suffices, given any set $\underline{K}$ of
compact subsets such that $\underline{K}\subset\underline{W}_{(q+1)}$, to find
compact sets $M_{k}\subset V_{k}$, $k=0,\ldots,q$, such that
$K_{ij}+q+1-i\subset M_{j-i}$ for $i\leqslant j$. Since

$K_{ij}+q+1-i\subset W^{q+1}_{ik}+q+1-i=V_{j-i},$

it follows that the $M_{k}\\!:=\bigcup_{j-i=k}K_{ij}$ are as required. This
proves the continuity.

Now we show that $\rho_{V}$ is topologically injective. To this end, we must
estimate each prenorm in (9) (for given $p,l\in\mathbb{Z}_{+}$ and a compact
subset $M$ of $V_{p}$) in terms of prenorms of the form (8). According to (7),
we have $W^{p+1}_{1,p+1}\\!:=V_{p}-p$. Therefore, $M-p\subset
W^{p+1}_{1,p+1}$. Now consider the homomorphism $\widetilde{\pi}_{p}$. Since
$f_{p}$ corresponds to the upper right corner entry in (6) (with $q=p$), it
follows that

$|f_{p}|_{M,l}=|(\widetilde{\pi}_{p}(a))_{1,p+1}|_{M-p,l}.$

Hence $\rho$ is topologically injective, which completes the proof of the
theorem. ∎

Recall that a _sheaf of Fréchet spaces (Fréchet algebras)_ is defined as the
presheaf of Fréchet spaces (Fréchet algebras) which turns into a sheaf of sets
after the application of the forgetting functor (see the discussion in [1,
Remark 5.4]). In what follows, sheaves in subcategories of Fréchet algebras
are understood in the same sense.

Let $V$ and $W$ be open subsets of $\Omega_{\mathbb{R}}$, and let $W\subset
V$. Then the topology on $U(\mathfrak{a}\mathfrak{f}_{1})$, which is
determined by the embedding $U(\mathfrak{a}\mathfrak{f}_{1})\to
C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$, is finer than that determined
by the embedding $U(\mathfrak{a}\mathfrak{f}_{1})\to
C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(W)$. Since both algebras are
completions of $U(\mathfrak{a}\mathfrak{f}_{1})$, we have a continuous
homomorphism $\tau_{VW}\\!:C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)\to
C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(W)$.

######  Theorem 4.

_(cf.[1, Theorem 5.5])_ The correspondences

$V\mapsto
C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)\quad\text{and}\quad(W\subset
V)\mapsto\tau_{VW}$

determine a sheaf of Fréchet–Arens–Michael $\mathbb{R}$-algebras of polynomial
growth on $\Omega_{\mathbb{R}}$.

###### Proof.

The correspondence under consideration is a contravariant functor from the
category of open subsets of $\Omega_{\mathbb{R}}$ to the category of
Fréchet–Arens–Michael $\mathbb{R}$-algebras of polynomial growth, i.e., we
have a presheaf. By definition (see (4)) for each open set
$V\subset\Omega_{\mathbb{R}}$, the Fréchet space
$C^{\infty}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ is the countable product of the
spaces $C^{\infty}(V_{p})$. The gluing axiom for $C^{\infty}$ functions on
$\mathbb{R}$ implies the gluing axiom for the presheaf of sets on
$\Omega_{\mathbb{R}}$. ∎

### The sheaf of noncommutative holomorphic functions

In this section, we consider the complexification of the real Lie algebra
discussed above; to simplify notation, we denote it by the same symbol
$\mathfrak{a}\mathfrak{f}_{1}$. The basic constructions for it are the same as
in the real case. In particular, we need the family
$\\{\sigma_{\lambda,q}\\!:\,\lambda\in\mathbb{C},\,q\in\mathbb{Z}_{+}\\}$ of
representations of the form (1); we denote it by $\Omega_{\mathbb{C}}$. We
declare a subset of $\Omega_{\mathbb{C}}$ to be open if it has the form (2),
where $(V_{q})_{q\in\mathbb{Z}_{+}}$ is a sequence of open subsets of
$\mathbb{C}$ satisfying condition (3).

An important difference from the case of $C^{\infty}$ functions is that the
polynomial algebra is not necessarily dense in the algebra of holomorphic
functions on an open set. However, this constraint is of technical character
and can easily be overcome.

For an open set $V$ in $\Omega_{\mathbb{C}}$ of the form (2), we put

$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)\\!:=\prod_{q=0}^{\infty}\mathcal{O}(V_{q}),$
(10)

where $\mathcal{O}(V_{q})$ denotes the algebra of holomorphic functions. In
particular, the Fréchet space
$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(\Omega_{\mathbb{C}})$ coincides
with the space $\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}$ defined in [1,
Sec. 6].

Let $\mathcal{B}$ denote the family of all open subsets $V=(V_{q})$ of
$\Omega_{\mathbb{C}}$ satisfying the following conditions:

(1) there is a $p\in\mathbb{Z}_{+}$ such that the sets $V_{q}$ are empty if
$q>p$;

(2) for each $q=0,\ldots,p$, there exists an $\varepsilon\in(0,1)$ such that
$V_{q}$ is the union of open disks of radius $\varepsilon$ centered at
$\sigma_{\lambda,q},\ldots,\sigma_{\lambda-p+q,q}$ (in particular, $V_{p}$ is
a disk centered a $\sigma_{\lambda,p}$).

It is easy to see that $\mathcal{B}$ is a base for the topology of
$\Omega_{\mathbb{C}}$.

######  Lemma 1.

If $V$ belongs to $\mathcal{B}$, then the image of the linear map
$U(\mathfrak{a}\mathfrak{f}_{1})\to\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$
defined by (5) is dense.

###### Proof.

It suffices to show that the image of the map
$\mathbb{C}[\lambda]\to\mathcal{O}(V_{q})$ is dense for each $q$. Since $V$
belongs to $\mathcal{B}$, it follows that $V_{q}$ is either the empty set or a
union of disjoint open disks in $\mathbb{C}$. Thus, the required assertion
readily follows from Mergelyan’s theorem (see, e.g., [6, p. 386, Theorem
20.5]). ∎

The three theorems concluding the paper are similar to Theorems 6.2, 6.5, and
6.6 in [1].

######  Theorem 5.

Let $V=(V_{q})$ be an open subset of $\Omega_{\mathbb{C}}$ such that the
polynomial algebra is dense in $\mathcal{O}(V_{q})$ for each $q$ (e.g.,
$V\in\mathcal{B}$). Then the multiplication in
$U(\mathfrak{a}\mathfrak{f}_{1})$ can be extended to a continuous
multiplication in $\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$. With
respect to this multiplication,
$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$is a locally solvable
Fréchet–Arens–Michael $\mathbb{C}$-algebra.

We need the following proposition, which is verified directly.

######  Proposition 3.

The class of locally solvable Arens–Michael algebras is stable with respect to
the passage to Cartesian products and closed subalgebras.

###### Proof of Theorem 5.

This theorem is proved by the same argument as Theorem 3 with minor
modifications. We consider the homomorphism

$U(\mathfrak{a}\mathfrak{f}_{1})\to\prod_{q=0}^{\infty}\mathrm{T}_{q+1}(\mathcal{O}(\underline{W}_{(q+1)}))\\!:a\mapsto(\widetilde{\pi}_{q}(a)),$

where $\underline{W}_{(q+1)}$ is defined in the same way as previously. To
check that the algebra $\mathrm{T}_{q+1}(\mathcal{O}(\underline{W}_{(q+1)}))$
is a locally solvable Fréchet–Arens–Michael algebra, we use Theorem 2 instead
of Theorem 1. By assumption, the homomorphism has dense image, and according
to Proposition 3 it suffices to check the continuity and topological
injectivity. These properties can be proved in the same way as in Theorem 3,
because all estimates used in the proof involve only uniform norms (without
derivatives). ∎

If $V$ and $W$ belong to $\mathcal{B}$ and $W\subset V$, then we have the
continuous homomorphism
$\tau_{VW}\\!:\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)\to\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(W)$.

######  Theorem 6.

On $\Omega_{\mathbb{C}}$, the correspondences

$V\mapsto\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)\quad\text{and}\quad(W\subset
V)\mapsto\tau_{VW}$

determine a sheaf of locally solvable Fréchet–Arens–Michael
$\mathbb{C}$-algebras on the topology base $\mathcal{B}$.

The proof is similar to that of Theorem 4 with the obvious replacement of the
sheaf of smooth functions by the sheaf of holomorphic functions and of the
family of all open sets by the topology base.

######  Theorem 7.

For each open subset $V$ of $\Omega_{\mathbb{C}}$, there exists a
multiplication with respect to which
$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(V)$ is a locally solvable
Fréchet–Arens–Michael algebra. In the case where the polynomial algebra is
dense in $\mathcal{O}(V)$, this multiplication is a continuous extension of
that in $U(\mathfrak{a}\mathfrak{f}_{1})$. Moreover, the correspondences in
Theorem 6 determine the sheaf $\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(-)$
of locally solvable Fréchet–Arens–Michael $\mathbb{C}$-algebras on
$\Omega_{\mathbb{C}}$.

###### Proof.

Arguing as in the proof of Theorem 6.6 in [1], we show that
$\mathfrak{F}_{\mathfrak{a}\mathfrak{f}_{1}}(-)$ is an extension of the sheaf
on the base $\mathcal{B}$ constructed in Theorem 6. In particular, it turns
out to be a sheaf of Fréchet–Arens–Michael algebras. The local solvability
follows from Proposition 3. ∎

## References

  * [1] O. Yu. Aristov, _Functions of class $C^{\infty}$ in non-commuting variables in the context of triangular Lie algebras_, Izv. RAN. Ser. Mat. 86:6 (2022), 5–46 (in Russian), English trans. arXiv:2103.06143.
  * [2] A. A. Dosiev (Dosi), _Formally-radical functions in elements of a nilpotent Lie algebra and noncommutative localizations_ , Algebra Colloq., 17, Sp. Iss. 1 (2010), 749–788.
  * [3] A. A. Dosi, _Taylor functional calculus for supernilpotent Lie algebra of operators_ , Journal of Operator Theory, 63:1 (2010), 191–216.
  * [4] A. Ya. Khelemskii, _Banach and Polynormed Algebras: General Theory, Representations, Homology_ (Oxford University Press, Oxford, 1993).
  * [5] O. Yu. Aristov, _Envelopes in the class of Banach algebras of polynomial growth and $C^{\infty}$-functions of a finite number of free variables_, Preprint in Russian (2021).
  * [6] W. Rudin, _Real and Complex Analysis_ (McGraw-Hill Book, New York, 1987).


11institutetext: Center for Cosmology and Astrophysics, Alikhanian National
Laboratory and Yerevan State University, Yerevan, Armenia 22institutetext:
SIA, Sapienza Universita di Roma, Rome, Italy

# Cosmological constant, information and gedanken experiments with black hole
horizons

V.G. Gurzadyan 1122 A. Stepanian 11

(Received: date / Revised version: date)

###### Abstract

The cosmological constant if considered as a fundamental constant, provides an
information treatment for gravitation problems, both cosmological and of black
holes. The efficiency of that approach is shown via gedanken experiments for
the information behavior of the horizons for Schwarzschild-de Sitter and Kerr-
de Sitter metrics. A notion of entropy regarding any observer and in all
possible non-extreme black hole solutions is suggested, linked also to
Bekenstein bound. The suggested information approach forbids the existence of
naked singularities.

###### pacs:

98.80.-kCosmology

##  1 Introduction

Einstein denoted the cosmological constant $\Lambda$ as a ”universal constant”
in his original papers on the cosmological model E ; E1 . The recent
observational indications on the accelerated expansion of the Universe
essentially revived the multi-facet problem of the cosmological constant. Here
we will deal with the consequences of the consideration of cosmological
constant in the context of other physical constants Uz .

In GS1  it was shown that the cosmological constant can be considered as a
second gravitational constant, along with the Newtonian constant, as it is
dimension-independent and matter uncoupled. That description follows from the
weak-field General Relativity involving the cosmological constant GS1 ; G1 .
The latter is based on the general function satisfying the Newton’s theorem on
the equivalency of the gravity of sphere and point mass G . Then, the metric
involves both constants, $G$ and $\Lambda$, and is defined as follows

$g_{00}=1-\frac{2GM}{c^{2}r}-\frac{\Lambda r^{2}}{3};\quad
g_{rr}=(1-\frac{2GM}{c^{2}r}-\frac{\Lambda r^{2}}{3})^{-1},$ (1)

i.e. in the form of Schwarzschild-de Sitter spherically-symmetric metric. As
weak-field General Relativity it was applied to the dynamics of groups and
clusters of galaxies GS1 ; G1 ; GS2 , thus enabling one to describe the
observational data on the dark matter in those cosmic structures.

The consideration of the cosmological constant $\Lambda$ as the fourth
fundamental constant - along with $c$, $\hbar$ and $G$ \- opens new principal
possibilities to treat the cosmological evolution and the black hole event
horizons GS4 .

Namely, the combinations of four fundamental constants leads to the appearance
of dimensionless quantities GS4

$mI^{n}=m\frac{c^{3n}}{\hbar^{n}G^{n}\Lambda^{n}},\quad m,n\in\mathbb{R}.$ (2)

For $n=1$ and $m=3\pi$ this relation represents the notion of information - or
of entropy with Boltzmann constant $k_{B}$ \- of de Sitter (dS) universe HG ;
BekB .

According to Bekenstein Bek  the “simplest” particle in the Universe has an
area equal to $\frac{4G\hbar}{c^{3}}$ which contains exactly 1 unit of
information. That notion can be generalized to cosmological horizons too HG .
Thus, by considering the Hubble horizon $\frac{c}{H}$, where $H$ is the Hubble
constant obtained from Friedmann equations

$H(t)^{2}=\frac{8\pi G\rho}{3}+\frac{\Lambda
c^{2}}{3}-\frac{kc^{2}}{a(t)^{2}},$ (3)

one can conclude that once the total area of the Hubble horizon was
$4l_{p}^{2}$, its Information Content (IC) equals exactly 1. In addition, it
can be checked that inside the Hubble horizon the Bekenstein bound

$I_{BB}\leq\frac{2\pi RE}{c\hbar ln2}\quad bits,\quad S_{BB}\leq\frac{2\pi
REk_{B}}{c\hbar},$ (4)

where $R$ and $E$ represent the characteristic radius and energy of the
universe, respectively, is always saturated. Consequently, by considering
Eqs.(3, 4), one can conclude that once the total density of the Universe
inside the Hubble horizon becomes $\rho_{tot}=\frac{3c^{5}}{8i\hbar G^{2}}$,
the IC is increased by one unit i.e.

$I_{BB}[\frac{3c^{5}}{i8\hbar G^{2}}]=i,\quad i=1,...,3\pi I.$ (5)

In this sense, the evolution of Universe can be regarded as a discrete
process, when the first unit of information is created when the area of Hubble
horizon was $4l_{p}^{2}$ and its density was $\frac{3}{8}\frac{c^{5}}{\hbar
G^{2}}$, accordingly. Consequently, at the $i$-th step, where the area is
grown to $4il_{p}^{2}$ and the density becomes $\frac{3}{8i}\frac{c^{5}}{\hbar
G^{2}}$, the IC of universe becomes $i$. The process continues until the area
of Hubble horizon reaches $\frac{12\pi}{\Lambda}$ i.e. the dS cosmological
horizon and the density become equal to $\frac{\Lambda c^{2}}{8\pi G}$,
accordingly.

In this paper, considering $\Lambda$ as a fundamental constant, we investigate
the information aspects of the black hole event horizons. Within that
information treatment we use gedanken experiments to deal with the
Schwarzschild-de Sitter and Kerr-de Sitter black holes and Hawking effect. The
notion of entropy follows that approach and its link to Bekenstein bound
forbid the possibility of existence of naked singularities.

##  2 Horizons for Schwarzschild-de Sitter and Kerr-de Sitter black holes

Considering $\Lambda$ as one of the fundamental constants, it is clear that it
should enter in the BH solutions. Particularly, the two astrophysical
candidates of BHs i.e. Schwarzschild and Kerr solutions will be modified
accordingly. Namely, the metric of Schwarzschild BH changes to the
Schwarzschild-de Sitter (SdS)

$ds^{2}=(1-\frac{2GM}{c^{2}r}-\frac{\Lambda
r^{2}}{3})c^{2}dt^{2}-\frac{1}{(1-\frac{2GM}{c^{2}r}-\frac{\Lambda
r^{2}}{3})}dr^{2}-r^{2}d\Omega^{2},$ (6)

where instead of one, we will have two physical event horizons (EHs)

$r_{1}=\frac{2}{\sqrt{\Lambda}}cos(\frac{1}{3}cos^{-1}(\frac{3GM\sqrt{\Lambda}}{c^{2}})+\frac{\pi}{3}),\quad
r_{2}=\frac{2}{\sqrt{\Lambda}}cos(\frac{1}{3}cos^{-1}(\frac{3GM\sqrt{\Lambda}}{c^{2}})-\frac{\pi}{3}).$
(7)

There is also a third root defined as $r_{3}=-(r_{1}+r_{2})$ which is not
considered as a physical solution. Indeed, the negative solution is regarded
as the EH for the spacetime located on the other side of singularity, also
known as a “dual Universe”.

The Penrose diagram of this spacetime is shown in Fig.(1).

IIIIII$r_{2}$$r_{1}$$r_{1}$$r_{2}$$r_{1}$$r_{2}$$r_{2}$$r_{1}$$r_{1}$$r_{2}$$r_{2}$$r_{1}$$r_{1}$$r_{2}$$r_{2}$$r_{1}$$r=0$$r=0$$r=-\infty$$r=+\infty$$r=0$$r=0$$r=+\infty$$r=-\infty$$r=+\infty$$r=-\infty$
Figure 1: Penrose diagram for SdS BH.

Here the zigzag lines represent the spacelike singularity inside the BH. In
this sense a particle in region I, outside the BH, can either fall into BH
i.e. region II or stay outside. Meantime, the region III is the region beyond
the cosmological horizon.

The other possibility for BHs is the Kerr BH which might be formed during the
gravitational collapse of stars. In this sense, instead of Eq.(6), according
to “no-hair theorem” NH1 ; NH2 ; NH3 , one has the following metric

$ds^{2}=\frac{\Delta_{r}}{\rho^{2}L^{2}}(cdt-a\sin^{2}\theta
d\phi)^{2}-\frac{\rho^{2}}{\Delta_{r}}dr^{2}-\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}-\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}L^{2}}(acdt-(r^{2}+a^{2})d\phi)^{2},$
(8)

where

$\Delta_{r}=(1-\frac{\Lambda
r^{2}}{3})(r^{2}+a^{2})-\frac{2GMr}{c^{2}},\quad\Delta_{\theta}=(1+\frac{a^{2}\Lambda\cos^{2}\theta}{3}),\quad
L=(1+\frac{a^{2}\Lambda}{3}),\quad\rho^{2}=r^{2}+a^{2}\cos^{2}\theta.$ (9)

The above solution is known as Kerr-de Sitter (KdS) metric (see HG ; KS1 ; KS2
; KS3  for detailed analysis of mathematical and physical properties of this
metric and its further generalization to charged case). Comparing this metric
with Eq.(6) it turns out that there are two main differences:

-In contrast to SdS metric, the singularity in KdS is timelike and has a ring shape.

-Besides the cosmological horizon $r_{C}$, we have two other EHs i.e. $r^{\pm}_{H}$.

The Penrose diagram of this spacetime is shown in Fig.(2).

IIIIII$r^{\prime}_{C}$$r^{-}_{H}$$r^{-}_{H}$$r^{\prime}_{C}$$r^{-}_{H}$$r^{\prime}_{C}$$r^{\prime}_{C}$$r^{-}_{H}$$r^{-}_{H}$$r^{\prime}_{C}$$r^{\prime}_{C}$$r^{-}_{H}$$r^{\prime}_{C}$$r^{-}_{H}$$r^{-}_{H}$$r^{\prime}_{C}$$r_{C}$$r^{+}_{H}$$r^{+}_{H}$$r_{C}$$r^{+}_{H}$$r_{C}$$r_{C}$$r^{+}_{H}$$r^{+}_{H}$$r_{C}$$r_{C}$$r^{+}_{H}$$r_{C}$$r^{+}_{H}$$r^{+}_{H}$$r_{C}$
Figure 2: Penrose diagram for KdS BH.

Here the zigzag lines represent the timelike ring singularity and
$r^{\prime}_{C}$ denotes the cosmological horizon in $r<0$. Thus, a particle
outside the BH, in the region I, can enter into BH, by passing the
$r^{+}_{H}$, and appear in region II. Then, it will continue its path toward
the $r^{-}_{H}$. After passing $r^{-}_{H}$, the other fundamental difference
between SdS and KdS becomes evident which we will discuss in the next
subsection in more details.

##  3 Gedanken experiments with horizons

Concerning the nature of SdS and KdS solutions, it is possible to have
scenarios where several fundamental principles of physics are violated.
Namely,

$\bullet$ In case of SdS BHs, the two EHs will become identical if
$\frac{3GM\sqrt{\Lambda}}{c^{2}}=1$. Surely, considering the numerical values
of constants this critical mass is several orders of magnitude larger than
supermassive BHs. However, from theoretical point of view there is neither
fundamental restriction nor any other prevention to have such an extreme BH.
Indeed, according to “Area Theorem” BH , when a particle falls into SdS BH,
its information is destroyed in the central (spacelike) singularity. Thus, the
area of first horizon $A_{1}=4\pi r_{1}^{2}$ is increased while the area of
second one $A_{2}=4\pi r_{2}^{2}$ becomes smaller. Consequently, by
continuation of this process finally the two EHs will become identical i.e.

$r_{1}=r_{2}=\frac{1}{\sqrt{\Lambda}}.$ (10)

In addition, if the accretion to SdS BH continues, finally the extreme SdS BH
also will be changed into a full naked singularity.

$\bullet$ In another scenario, consider a BH with mass $M$ which accretes a
mass $m_{1}$. At the same time assume that according to Hawking effect Hw1 ;
Hw2  an amount of mass $m_{2}$ is evaporated. In this case, one of the
fundamental questions which arises is that

What is the output of this mixed process from the BH thermodynamics point of
view? Should one treat the process classical or quantum?.

However, for both options, we can propose a process which will be
contradictory. Namely, by considering the whole process as a classical effect,
we can assume a scenario where $m_{1}<m_{2}$. As a result of that we will have
a violation of second law of BH thermodynamics i.e.

$m_{1}<m_{2}:\quad
S_{initial}=\frac{A_{initial}k_{B}}{4l_{p}^{2}}>S_{final}=\frac{A_{final}k_{B}}{4l_{p}^{2}},$
(11)

where $A$ is the area of the first EH. On the other hand, if we consider the
whole scenario as a quantum effect, then some internal inconsistencies will
happen regarding the First law of BH thermodynamics. Indeed, it should be
recalled that the First Law is written as

$dE=TdS.$ (12)

Accordingly, in this case we will have

$m_{1}>m_{2}:\quad dE=c^{2}d(M+m_{1}-m_{2})\neq TdS.$ (13)

This is due to the fact that while the total amount of increase in both
energy/mass on the left hand side and the entropy on the right hand of side
Eq.(13) is related to the amount of $m_{1}-m_{2}$, the radiation is only due
to the mass $m_{2}$ evaporated due to Hawking effect as black body radiation
of temperature

$T=\frac{\hbar c^{3}}{8\pi Gm_{2}k_{B}}.$ (14)

$\bullet$ Considering the KdS BH, the situation can be more problematic as
there is the risk of violation of “determinism of GR” which is related to the
“strong cosmic censorship”. Namely, according to Fig.(2), since the
$r^{-}_{H}$ is the Cauchy horizon (CH), it is not possible to determine the
evolution of geodesics of infalling particles inside the CH. They can be
either destroyed in the singularity or continue their path toward outside
(either to region III or $r<0$). Furthermore, it can be easily checked that
due to special nature of ring singularity, it is possible that $r^{-}_{H}$
starts to increase more rapidly than $r^{+}_{H}$. As a result of such process,
we will have an extreme KdS with a timelike ring singularity equipped by one
EH which is a CH too. The perspective of this scenario will be such that the
acausal region (created during the gravitational collapse) i.e.
$0<r<r^{-}_{H}$ will grow and as a result of that eventually our universe will
become completely acausal. Here it should be noticed that, the problem of CH
and acausal regions is not restricted to KdS case. Indeed, it covers the whole
family of Kerr, Reissner-Nordström (RN) and Kerr-Newman (KN) BH solutions in
$\Lambda\geq 0$ asymptote.

##  4 Entropy vs horizons

The physics of BH with multiple horizons has been studied in several papers
MH1 ; MH2 ; MH3 . It has been shown in TSE , that considering $\Lambda$ in all
BH solutions i.e. SdS, RNdS, KdS and KNdS, the total sum of entropies (all
horizons, including the negative root) yields a fixed value

$\Sigma_{i}S_{i}=\frac{6\pi c^{3}k_{B}}{\hbar G\Lambda}.$ (15)

Meantime, it can be easily checked that by considering both positive and
negative cosmological horizons i.e. $r=\pm\sqrt{\frac{3}{\Lambda}}$, the above
relation will be valid for dS spacetime too. Thus, if we take the
$\Sigma_{i}S_{i}$ as the total entropy of the system (BH in dS asymptote),
then the Second law of thermodynamics will tell us that this sum must never be
decreased.

Such definition of entropy has the following advantages:

$\bullet$ First, for extreme and over-extreme cases (where two or even more
horizons coincide), the number of horizons are decreased. Thus for such cases
we will have

$\Sigma_{i}S_{i}<\frac{6\pi c^{3}k_{B}}{\hbar G\Lambda}.$ (16)

In this sense, the possible scenario for formation of all types of extreme BHs
including the full acausal universe and the naked singularities is dismissed.

$\bullet$ Second, the astrophysical BHs (originally formed as non-extreme
KdS), start to loose their rotational energy via Penrose process PenP  and
finally become SdS BHs. Since for both SdS and KdS cases, Eq.(15) is valid, no
violation is occurred during the conversion of KdS to SdS.

$\bullet$ Third, at the very late time of our Universe, when all KdS BHs
eventually become SdS, the Hawking evaporation starts to remove the spacelike
singularity of SdS which again does not violate the second law of
thermodynamics since as a final result we will obtain the final entropy of dS
universe. Namely, by considering Eq.(6), during the evaporation process we
will have

$M\to 0:\quad 4\pi r_{1}^{2}\to 0\quad 4\pi
r_{2}^{2}\to\frac{12\pi}{\Lambda}\quad 4\pi
r_{3}^{2}\to\frac{12\pi}{\Lambda},$ (17)

which means that

$M\to 0:\quad S_{1}\to 0\quad S_{2}\to\frac{3\pi c^{3}k_{B}}{\hbar
G\Lambda}\quad S_{3}\to\frac{3\pi c^{3}k_{B}}{\hbar G\Lambda}.$ (18)

Thus, when the BH singularity is fully removed and BH is evaporated, we will
have very low entropy photons (due to thermal radiation of BH evaporation)
while the second EH will become identical to horizon of dS universe. The above
results are in complete agreement with the Conformal Cyclic Cosmology (CCC),
where at the ending phase of an aeon, low entropy photons are produced which
can pass through the crossover and have their trace on the next aeon C1 ; C2 ;
C3 . Meantime, it should be noticed that in such cases, the negative root
$r_{3}$ of SdS solution will become identical to $-\sqrt{\frac{\Lambda}{3}}$.
Indeed, this negative root can be considered as the dual of our original dS
spacetime with the same amount of IC and entropy according to Eq.(4).

It should be noticed that, although in the previous section, we have
considered the BH solutions in dS asymptote, we can generalize them to FLRW
metric too. Indeed, considering the fact that the dS horizon is the special
case of Hubble horizon such that

$\rho\to 0:\quad\frac{c}{H}\to\sqrt{\frac{3}{\Lambda}},$ (19)

($\rho$ is the contribution of all forms of matter, radiation and spatial
curvature except the $\Lambda$ term) it is possible to look at the BH
solutions in different Cauchy hypersurfaces of FLRW metric. Namely, for each
step of IE, we can replace the notion of dS horizon
$\sqrt{\frac{3}{\Lambda}}$, by Hubble horizon. Consequently, for BH solutions,
once the $j$-th unit of information is created according to Eq.(5), we will
have

$\Sigma_{i}S_{i}=\frac{2\pi c^{5}k_{B}}{H_{j}^{2}\hbar G},\quad
H_{j}^{2}=\frac{8\pi G\rho_{j_{tot}}}{3}.$ (20)

Here the $\rho_{j_{tot}}$ is the total amount of density at the $j$-th step.
In this sense, at each step one should replace the notion of dS asymptote as
the background of BH solution by the background of FLRW which has a constant
density equal to $\rho_{j}=\frac{3c^{5}}{8j\hbar G^{2}}$; this treatment can
be linked to the Hubble diagram to describe the dark energy models KS .

The important consequence of Eq.(20) is that, it gives us a universal notion
of entropy, i.e. for all observers in all possible non-extreme BH solutions,
for all epochs of universe. Namely, if the observer is located outside the BH,
he/she will consider the total amount of entropy as the $S=\frac{2\pi
c^{5}k_{B}}{H_{j}^{2}\hbar G}$, which is in full agreement with Eq.(5)
(Additional factor 2 is appeared since we have also considered the negative
horizon i.e. $r_{j}=-\frac{c}{H_{j}}$.). On the other hand, if he/she is
inside the BH i.e. between EH and CH, in the acausal region or even in
$r^{\prime}_{C}<r<0$, then according to Eq.(20) again the same value will be
reported.

##  5 Bekenstein bound

Finally, it is worth to mention that originally the Bekenstein bound, Eq.(4),
was proposed for asymptotically flat spacetimes BekB2 . However, considering
the Hubble horizon and FLRW equations, we can generalize it for curved
spacetime too by revisiting the notion of mass and radius. Indeed, this point
is essential if we try to check the validity of Eq.(15) based on Eq.(4).
Namely for all BH solutions during the IE, if we consider the mass and the EHs
of BH i.e.

$\Sigma_{i}I_{i}=\frac{2\pi Mc}{\hbar ln2}(\Sigma_{i}r_{i}),$ (21)

since we have $\Sigma_{i}r_{i}=0$, the above relation will be equal to zero.
But, if we consider the contribution of the background, at the moment of
creation of $j$-th unit of information, then

$I=\Sigma_{i}I_{i}+\Sigma_{j}I_{j}=\frac{2\pi Mc}{\hbar
ln2}(\Sigma_{i}r_{i})+(\frac{2\pi}{\hbar ln2})2(\frac{\pi c^{5}}{2j\hbar
G^{2}})c(\frac{c}{H_{j}})^{4},$ (22)

which will be in agreement with Eq.(20). It should be noticed that for such
cases i.e. BHs in different eras of our Universe, we are dealing with two
different objects. First, there are BHs which are characterized by their mass
$M$ and their corresponding notions of horizons i.e. $r_{i}$. Second, there is
the notion of background with the mass $\pm\frac{\pi c^{8}}{2j\hbar
G^{2}H^{3}_{j}}$ and the characteristic radius i.e. $\pm\frac{c}{H_{j}}$. (For
both mass and radius cases the negative sign represents the corresponding
values of dual universe.) Considering both of them in Eq.(4), we will obtain
Eq.(22) where the first term is the contribution of BH in the total IC, while
the second term stands for IC of background. In this sense, the universality
of entropy for all observers at each epoch in Eq.(20) which is based on the
summation of areas of all horizons, can be obtained/verified according to
Eq.(21), based on characteristic mass and radii.

##  6 Conclusions

Considering the cosmological constant as a physical constant and the emergence
of the information scaling, we studied the entropy of the black holes with
multiple event horizons. Using gedanken experiments regarding the
Schwarzschild-de Sitter and Kerr-de Sitter black hole horizons, we showed that
the information treatment forbids the existence of naked singularities.

Then we showed that the suggested information approach leads to a universal
notion of entropy, i.e. for all observers and in all possible non-extreme BH
solutions and to its link to the Bekenstein bound. The cosmological evolution
of the Universe then is represented via a sequence of increasing information
defined by means of the physical constants, including the cosmological
constant.

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# Novel DNNs for Stiff ODEs with Applications to
Chemically Reacting Flows

Thomas S. Brown1,3, Harbir Antil3, Rainald Löhner1, Fumiya Togashi2,
Deepanshu Verma3 1Center for Computational Fluid Dynamics, College of Science,
George Mason University, Fairfax, VA 22030-4444, USA.
tbrown62<EMAIL_ADDRESS>2Applied Simulations, Inc., 1211 Pine Hill
Road, McLean, VA 22101, USA<EMAIL_ADDRESS>3Center for Mathematics
and Artificial Intelligence (CMAI), College of Science, George Mason
University, Fairfax, VA 22030-4444, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Chemically reacting flows are common in engineering, such as hypersonic flow,
combustion, explosions, manufacturing processes and environmental assessments.
For combustion, the number of reactions can be significant (over 100) and due
to the very large CPU requirements of chemical reactions (over 99%) a large
number of flow and combustion problems are presently beyond the capabilities
of even the largest supercomputers.
Motivated by this, novel Deep Neural Networks (DNNs) are introduced to
approximate stiff ODEs. Two approaches are compared, i.e., either learn the
solution or the derivative of the solution to these ODEs. These DNNs are
applied to multiple species and reactions common in chemically reacting flows.
Experimental results show that it is helpful to account for the physical
properties of species while designing DNNs. The proposed approach is shown to
generalize well.

This work was supported by the Defense Threat Reduction Agency (DTRA) under
contract HDTRA1-15-1-0068. Jacqueline Bell served as the technical monitor.

##  1\. Introduction

Chemically reacting flows are common in many fields of engineering, such as
hypersonic flow, combustion, explosions, manufacturing processes, and
environmental assessments [28, 26, 3, 16]. For hydrocarbon combustion and
explosions the numbers of species and reactions can reach into hundreds and
thousands respectively. Even with so-called reduced models [29, 11, 19, 14,
17, 27], which try to keep the main species and reactions while neglecting
those that are not important, typically over 100 reactions need to be updated.
An expedient (and widely used) way to compute flows with chemical reactions is
to separate the advection and diffusion of species from the actual reactions.
In this way, the vastly different timescales of the reactants can be treated
in a separate, stiff ODE solver. Such chemical reaction solvers take the given
species ${u}^{n-1}$ at the $n-1^{\rm th}$ time step and desired timestep
$\delta t$ and update the species to ${u}^{n}$. In terms of a ‘black box
solver’ this implies either:

${u}^{n}=Chem_{1}(\delta t,{u}^{n-1})\,,$ (1)

or

$\frac{u^{n}-u^{n-1}}{\delta t}=Chem_{2}(u^{n-1}),$ (2)

where $Chem_{1}$ stands for the ODE integrator of chemical reactions and
$Chem_{2}$ is the right-hand side of the system. This is the formulation to
solve the system of ODEs numerically using the standard Euler method. Compared
to a typical ‘cold’ flow case, the presence of these chemical reactions
implies an increase of computing requirements that can exceed factors of
1:100, i.e. 2 orders of magnitude. This makes many of these flow simulations
so expensive that entire classes of problems have been sidelined, waiting for
faster computers to be developed in years to come. The goal here is to replace
the ‘black box’ solvers (equivalently the functions $\emph{Chem}_{1}$ and
$\emph{Chem}_{2}$) given in (1) and (2) by novel, robust Deep Neural Networks
(DNNs) without sacrificing accuracy.

The list of references on using DNNs in computational fluid dynamics (CFD) is
growing fast, see for example, [18, 8]. However, the results on using DNNs in
chemical kinetics are scarce. A popular approach to solve PDEs and ODEs is
through the use of so-called Physics-Informed Neural Networks (PINNs) [23, 5].
The goal of PINNs is to minimize the PDE/ODE residual by using a neural
network as a PDE/ODE solution Ansatz. The inputs to the network are space-time
variables $(x,t)$ and all the derivatives are computed using automatic
differentiation. See [10] for an example of a PINN for stiff ODE systems where
the only input is time.

The approach presented here fundamentally differs from the aforementioned
approaches. Instead of _Physics-Informed-Neural-Networks_ , the goal is to
pursue _Learn-from-Physics/Chemistry_. For instance in (1) and (2), DNNs will
be used to learn $Chem_{1}$ and $Chem_{2}$ from a given dataset coming from
physics/chemistry simulations. Such an approach to learn $Chem_{1}$ has also
been considered recently in [21, 25, 20] where the authors employ an
architecture that is motivated by standard feed forward networks. The authors
of [30] consider a similar problem, but use an autoencoder, which is a type of
DNN used to reduce the dimension of the system. Notice that the proposed
approach will allow for the chemical reactions described by (1) and (2) to
start at any point in time without knowing a reference time. The latter is
crucial for the application under consideration.

The DNNs used in this paper have been motivated by the Residual Neural Network
(ResNet) architecture. ResNets have been introduced in [9, 24, 2] in the
context of data/image classification, see also [1] for parameterized PDEs and
[7] where the (related) so-called Neural ODE Nets [4] have been used to solve
stiff ODEs. The ResNet architecture is known to overcome the vanishing
gradient problem, which has been further analyzed using fractional order
derivatives in [2]. The key feature of a ResNet is that in the continuum
limit, it becomes an optimization problem constrained by an ODE. Such a
continuous representation further enables the analysis of the stability of the
network using nonlinear ODE theory. In addition, standard approaches from
optimization, such as the Lagrangian approach can be used to derive the
sensitivities with respect to the unknown weights and biases.

The main novelties in the DNNs presented in this work are the following:

  * •

These networks allow for learning both the solution (see (1)) and difference
quotients (see (2)). A similar approach to learn the solution in the context
of parameterized PDEs has been recently considered in [1].

  * •

Motivated by chemically reacting flows, the goal is to create networks that
can learn multiple reactions propagating multiple species. To accomplish this
task, parallel ResNets are constructed where the data corresponding to
multiple quantities is used as input for each network but the output is only a
single species. Similar approaches for chemical kinetics can be found in [21,
25], where the authors use standard feed forward networks.

  * •

The proposed DNNs are applied to non-trivial chemically reacting flows.
Experimental results show that it is helpful to know the underlying properties
of the species while designing the networks.

The remainder of the paper is organized as follows: In Section 2, the DNNs
used to approximate systems of type (1) and (2) are introduced, and training
strategies are discussed. Section 3 provides some information on the
implementation of the method. Several experimental results are discussed in
Section 4. Additional information on the data being used in the experiments is
provided in Appendix A.

##  2\. Deep Residual Neural Networks

###  2.1. Problem Formulation

Consider an input-to-output map

$u\mapsto S(u),$

where $S$ could be $Chem_{1}$ or $Chem_{2}$ in (1) and (2), respectively. The
goal is to learn an approximation $\widehat{S}$ of $S$ using Deep Neural
Networks (DNNs). In particular, the proposed DNNs are motivated by Deep
Residual Neural Networks (ResNets). See [24, 2] for examples of ResNets for
classification problems and [1] for an application to parameterized PDEs.

Given a training dataset $\\{(u^{n},S(u^{n}))\\}_{n=0}^{N-1}$, the DNN
approximation $\widehat{S}$ of $S$ is given by the output of the DNN (ResNet)

$\begin{cases}y_{1}=\tau\sigma(K_{0}y_{0}+b_{0}),\\\
y_{\ell}=y_{\ell-1}+\tau\sigma(K_{\ell-1}y_{\ell-1}+b_{\ell-1}),\quad&1<\ell\leq
L-1,\\\ y_{L}=K_{L-1}y_{L-1},\end{cases}$ (3)

i.e., $\widehat{S}(u)=y_{L}$. Here $\tau>0$ is a given parameter called the
skip connection parameter. The nonlinear function $\sigma$ denotes an
activation function. In this study, the activation function is taken to be a
smooth quadratic approximation of the ReLU function, i.e.,

$\sigma(x)=\begin{cases}\max\\{0,x\\}&|x|>\varepsilon,\\\
\frac{1}{4\varepsilon}x^{2}+\frac{1}{2}x+\frac{\varepsilon}{4}&|x|\leq\varepsilon.\end{cases}$

Figure 1 (left) shows that $\sigma(x)$ is a good approximation of ReLU. Here,
and in the experiments below, $\varepsilon$ is taken to be 0.1. The number of
layers (depth of the network) is denoted by $L$.

Figure 1. Left: Smooth ReLU and its derivative for $\varepsilon=0.1$. Right:
Example of a typical Deep ResNet used in the experiments.

Layer 0 is referred to as the input layer, layers 1 through $L-1$ as hidden
layers and layer $L$ as the output layer.

The quantities $\\{K_{\ell}\\}_{\ell=0}^{L-1},\\{b_{\ell}\\}_{\ell=0}^{L-2}$
denote the weights and biases, i.e. the parameters that need to be determined.
In this setting $K_{\ell}$ is a matrix and $b_{\ell}$ is a vector, together
they introduce an affine transformation. If $y_{\ell}\in\mathbb{R}^{n_{\ell}}$
for $\ell=0,\dots,L$, then $K_{\ell}\in\mathbb{R}^{n_{\ell+1}\times n_{\ell}}$
and $b_{\ell}\in\mathbb{R}^{n_{\ell+1}}$. The dimension $n_{\ell}$ is also
referred to as the width of the $\ell$-th layer. Notice that for $0<\ell<L$ in
(3), it is assumed that $n_{\ell}=n_{\ell+1}$, but this can be easily
generalized, see [24]. Nevertheless, in the current setup the dimension of the
input $y_{0}$ and output $y_{L}$ can be different.

An example of a typical DNN is shown in Figure 1 (right) with depth 3, width
8, input dimension 10, and output dimension 1. The values of the weights are
given by the color of the lines connecting the neurons, the values of the
biases are given by the color of the neurons, and the color of the input layer
has been set to zero.

The question remains of how to compute the weights
$\\{K_{\ell}\\}_{\ell=0}^{L-1}$ and biases $\\{b_{\ell}\\}_{\ell=0}^{L-2}$ in
order to obtain a good approximation $\widehat{S}$ of $S$. Following [24, 2],
these weights are computed by solving

$\min_{\\{K_{\ell}\\}_{\ell=1}^{L-1},\\{b_{\ell}\\}_{\ell=0}^{L-2}}\left\\{J(K_{\ell},b_{\ell}):=\frac{1}{2N}\sum_{n=0}^{N-1}\|y_{L}^{n}-S(u^{n})\|^{2}\right\\}\quad\mbox{subject
to constraints}\quad\eqref{eq:resnet},$ (4)

where $y_{L}^{n}=\widehat{S}(u^{n})$, that is, $y_{0}$ in (3) is taken to be
the training data $\\{u^{n}\\}_{n=0}^{N-1}$. The problem (4) is a constrained
optimization problem. To solve this problem, the gradient of $J$ with respect
to $K_{\ell}$ and $b_{\ell}$ needs to be evaluated. This requires introducing
the so-called adjoint equation (also known as back-propagation in machine
learning literature). See [24, 2, 1] for complete details. Having obtained the
gradient, an approximate Hessian is computed via the BFGS routine. Both the
gradient and the approximate Hessian are used to solve the minimization
problem.

###  2.2. Proposed Approach and Applications

As stated earlier, the goal is to replace $Chem_{1}$ and $Chem_{2}$ in (1) and
(2), respectively by DNNs. The continuous ODE is given by

$\frac{du}{dt}(t)=S(u(t)),\quad u(0)=u_{0}.$ (5)

The simplest time discretization of (5) is given by

$\frac{u^{n}-u^{n-1}}{\delta t}=S(u^{n-1}),\quad u^{0}=u_{0},$ (6)

for $n=1,\dots,N$. At first, the right-hand-side of (6) is approximated. This
is referred to as the _first approach_. In this case, the training data (input
and output) is $\left\\{\left(u^{n-1},\frac{u^{n}-u^{n-1}}{\delta
t}\right)\right\\}_{n=1}^{N}$. In the _second approach_ , the map to be
approximated is $u^{n+1}=S(u^{n})$, i.e., the setting of (1). Here $S$ marches
$u$ forward in time with time increments given by $\delta t$. The training
data for this approach is given by $\\{(u^{n-1},u^{n})\\}_{n=1}^{N}$.

####  Scaling.

The training data in the examples below has been computed using CHEMKIN [12].
Before training the networks, the data is scaled in the following manner: For
a data set $\\{x_{j}\\}_{j=0}^{N}$ corresponding to a single quantity (e.g.
temperature) the mean $\mu_{x}$ and standard deviation $\sigma_{x}$ of the set
are computed. Each entry $x_{i}$ is scaled as

$z_{i}:=\frac{x_{i}-\mu_{x}}{\sigma_{x}}~{}~{}.$ (7)

Using this new centered and normalized set, the training data is obtained by
defining

$\widehat{x}_{i}:=\frac{z_{i}-\min_{j}z_{j}}{\max_{j}z_{j}-\min_{j}z_{j}},$
(8)

so that the resulting data set $\\{\widehat{x}_{j}\\}_{j=1}^{N}$ lies in the
interval $[0,1]$. Given that chemical reactions follow an exponential
Arrhenius-type law, a logarithmic scaling was also tried. This implies
performing the above scaling on the dataset $\\{\log x_{j}\\}$ instead of
$\\{x_{j}\\}$.

####  Architecture.

For the DNNs implemented below, the input dimension will always be given by

$\displaystyle n_{0}=$ $\displaystyle 1$ $\displaystyle+$ $\displaystyle M$
$\displaystyle+$ $\displaystyle 1$ $\displaystyle=M+2.$
$\displaystyle(\text{temp}$ $\displaystyle\text{erature})\qquad$
$\displaystyle\qquad(\text{\\# of }$ $\displaystyle\text{species})\qquad$
$\displaystyle\qquad(\text{time inc}$ $\displaystyle\text{rement, $\delta
t$})$

The output dimension is $n_{L}=M+1$, which corresponds to temperature plus the
number of species.

Rather than using a single DNN to approximate the entire solution map $S$, in
many cases a parallel ResNet architecture is implemented in which a separate
ResNet is created to correspond to each desired output. With this structure
the output dimension for each ResNet is 1, but there are $M+1$ ResNets
implemented in parallel. The inputs to all of the parallel ResNets are the
same, and so the parallel architecture can also be thought of as a single
large ResNet (with $n_{L}=M+1$) that is not fully connected.

####  Loss Function and Training.

In the case of a parallel ResNet architecture, each parallel network is
associated to a separate loss function. The same form of the loss function is
used for each network. Letting $\theta^{(i)}$ represent the concatenation of
all weight and bias parameters associated to the $i$-th parallel network, the
loss function takes the form

$\frac{1}{2N}\sum_{n=0}^{N-1}\|y_{L}^{(i)}-S(u^{n})^{(i)}\|_{2}^{2}+\frac{\lambda}{2}\left(\|\theta^{(i)}\|_{1}+\|\theta^{(i)}\|_{2}^{2}\right),\qquad
i=1,\dots,M+1.$ (9)

In other words, the process of training each network is the process of finding
the parameters $\theta^{(i)}$ which minimize the mean squared error between
the network output and the training data, while also using both $\ell^{1}$ and
$\ell^{2}$ regularizations to penalize the size of the parameters. As
indicated in section 2, a gradient based method (BFGS in particular) is used
to solve the constrained optimization problem.

####  Validation.

DNNs are trained with validation data in order to overcome the overfitting
issue. The validation data is a subset of the training data that is not used
to compute the weights and biases. Instead, a separate loss function is formed
that computes the mean squared error between the ResNet output and the
validation data. This is a way to test how the ResNet performs on unseen data
during the training process itself. If the validation error increases, the
training continues for a certain number of additional iterations, called the
_patience_. During this time, if the validation error decreases to a new
minimum, the patience resets to zero and training continues. If the validation
error does not attain a new minimum and the full number of patience iterations
is reached, then the training process is terminated.

#### Testing.

After training and validation comes the testing phase in which the DNN
approximations are used to carry out time marching. Using the _first approach_
(see above) this involves implementing an Euler time-stepping commonly used to
numerically solve ODEs, with the usual right-hand-side of the ODE replaced by
the DNN output, that is,

$\widehat{u}^{n}=\widehat{u}^{n-1}+\delta
t\widehat{S}(\widehat{u}^{n-1})\qquad n=1,\dots,N,$ (10)

where $\widehat{u}^{0}=u^{0}$ is known data. For the _second approach_ ,
results are tested by using the following update step
$\widehat{u}^{n}=\widehat{S}(\widehat{u}^{n-1})$ for $n=1,\dots,N$, where
again $\widehat{u}^{0}=u^{0}$ is known.

#### Application.

The above methods are applied to a system of ODEs that model hydrogen-oxygen
reaction. In particular, the reduced hydrogen-air reaction model with 8
species and 18 reactions [22] is used. This model is simpler than the
hydrocarbon reaction model mentioned in section 1. However, it can still
capture several essential key features. More complicated DNNs which can handle
over 100 reactions will be part of future work.

##  3\. Implementation

To start, the _second approach_ was implemented in Keras [6] using stochastic
gradient descent. Libraries such as Keras have proven to be highly successful
for classification problems, but their success in physics based modeling is
still limited. Initially, it was observed that the results produced by using
Keras were unable to capture the ‘pulse’ in chemical reactions. After gaining
access to some of the Keras code, and seeing some improvement, the results
were still highly inconsistent. For example, when running the same experiment
multiple times, with the same hyperparameters, different solutions were
obtained. Such an inconsistency may not be critical for classification
problems (for example [13, 15]), but it is critical for ODE/PDE problems.
Implementation through Keras was abandoned as further customization of the
code proved to be a daunting task.

The DNN implementation used in this paper was carried out in MATLAB and a BFGS
routine with an Armijo line-search was used to solve the optimization problem.
Unlike many neural network software packages, this code uses the framework and
closely resembles code that is used to solve optimal control problems.

##  4\. Results

In this section a variety of results are presented that represent different
approaches and architectures. The results will be grouped together by the type
of data that was used to train the networks. Further details on the training
data are provided in Appendix A. The loss function is as given in (9) with
$\lambda=1e-7$, the skip connection parameter (in (3)) $\tau=2/(L-1)$, where
$L$ is the number of hidden layers, and a constant time increment of $\delta
t=1e-7$. Unless otherwise specified, all training data was scaled by the
method outlined in (7) and (8). All of the results presented below were
achieved by using the MATLAB implementation of the ResNets described above.
The blue curves in the plots below represent ‘true solution’ (data generated
using CHEMKIN), while the dashed red curves represent DNN output.

###  4.1. Training with a single data set.

Using the _first approach_ , a single ResNet was trained on data corresponding
to temperature, 8 species, and $\delta t$. In Figure 2, the results from a
ResNet with depth 9 and width 15 are shown. For this experiment only
$\ell^{2}$ regularization has been used in the loss function. After being
trained, the training data was used again to test the network, and these
results can be seen in the left plots of Figure 2. The initial condition from
the training data was marched forward in time as described in (10), and the
results are shown on the right of Figure 2 compared with the actual $u$.

Figure 2. A comparison between the actual $du/dt$ and the learned difference
quotient (left 9 plots) and the results of marching in time as in (10) (right
9 plots) using a Deep ResNet of depth 9 and width 15.

###  4.2. Training data sets with varying equivalence ratio and fixed initial
temperature.

Experimental results showed that while a single ResNet was able to handle data
corresponding to a single initial condition, it struggled with data containing
multiple initial conditions. To improve results, the parallel ResNet structure
described in Section 2.2 was created. The parallel DNNs were trained on a data
set containing multiple initial conditions with the same initial temperature
but varying equivalence ratio (see Appendix A). Figure 3 shows the results for
ResNets trained on this data without the use of validation data. Each ResNet
has depth 4 and width 10. For this result, the testing data was a subset of
the training data, and so these plots do not represent a generalization of the
networks to unseen data. Figure 4 shows the result of an experiment trained
with a configuration of data subsets from the same data set as Figure 3, (see
Appendix A for specific details). Additionally, validation data was
implemented during training with a patience of 250 iterations. The ResNets
used for this experiment have varying depths (3 to 5) but a fixed width of 10.
The plots in Figure 4 are the result of testing the DNNs on data that was not
used in training. For these results mass conservation is also implemented in
the sense that the sum of the densities of the 8 species is adjusted at the
end of each update step (10) to be equal to that of the previous step. This is
the only experiment presented in this work in which mass conservation is
enforced.

Figure 3. A comparison between the actual $du/dt$ and the learned difference
quotient (left 9 plots), and the results of marching in time as in (10) (right
9 plots) using parallel Deep ResNets of depth 4 and width 10.

Figure 4. A comparison between the actual $du/dt$ and the learned difference
quotient (left 9 plots), and the results of marching in time as in (10) (right
9 plots) using parallel Deep ResNets of different depths and width 10, trained
with validation data with patience of 250 iterations.

The next result uses the _second approach_ described in section 2.2, in which
the ResNets are trained to learn the next time step rather than the right-
hand-side of the ODE. For the results shown in Figure 5 parallel ResNets of
depth 7 and width 50 were used. These ResNets were trained using validation
data with a patience of 400. For the nine subplots on the left side of Figure
5, the ResNets are being used to predict only a single timestep. In other
words, the red dashed curves are produced by plotting
$\widehat{u}^{n+1}=\widehat{S}(u^{n})$ for all $n$ where $u^{n}$ comes from
known data. The right nine subplots show
$\widehat{u}^{n+1}=\widehat{S}(\widehat{u}^{n})$ where only $u^{0}$ is from
known data. The plots on the left are included as evidence that the networks
are close to learning the correct curves. The accumulation and propagation of
errors when the quantities are marched in time can significantly affect the
accuracy of the results as shown in the right plots. The results shown in
Figure 5 come from data on which the ResNets were not trained.

Figure 5. Results from using parallel ResNets to predict a single step (left 9
plots) and marching in time from an initial condition (9 right plots) using
DNNs with a depth of 7 and a width of 50.

In Figure 6, results are displayed in which the training data was log-scaled
(i.e. where $\log x_{i}$ is used rather than $x_{i}$ in (7)). The ResNets used
for these results all have depth 9 and width 20 and were trained using
validation data with a patience of 400 iterations. The plots in Figure 6 were
created using the same data set that was used to create the curves in Figure
5. From these results, and many others not shown here, it is unclear if log-
scaling provides any improvement in accuracy.

Figure 6. Predicting a single timestep (left 9 plots) and marching in time
(right 9 plots) from ResNets trained on log-scaled data. The ResNets used here
have a depth of 9 and a width of 20, and were trained with validation data
with patience of 400 iterations.

###  4.3. Grouping training data based on equivalence ratio.

For the next results the training data is split into three groups: fuel lean
(equivalence ratio $\leq 0.1$), fuel balanced (equivalence ratio between 0.1
and 2), and fuel rich (equivalence ratio $>2$). See Appendix A for more
details on these data sets. The results of an experiment using the second
approach, where a different group of parallel ResNets is trained for each of
the fuel-based data sets, are presented in Figure 7. In other words, there are
9 plots corresponding to results from ResNets trained only on fuel lean data
sets, 9 plots for ResNets trained only on fuel balanced data sets, and 9 plots
for ResNets trained only on fuel rich data sets. All ResNets have a depth of 6
and width of 30 and were trained using validation data with a patience of 400
iterations. Furthermore, these plots are presented with a logarithmically
scaled $x$-axis, as for some of the data the reactions happen very quickly.
All of the plots in Figure 7 show results of marching in time given only an
initial condition from known data that was not seen by the networks during
training. These results show that the ResNets trained on fuel-based grouping
is more successful than the previous attempts.

Figure 7. Results from marching initial conditions forward in time using
ResNets trained on fuel lean data (top left), fuel balanced (top right), and
fuel rich (bottom) data sets. The ResNets used have a depth of 6 and a width
of 30, and were trained with validation data with patience of 400 iterations.

###  4.4. Training data sets with varying initial temperatures, but fixed
equivalence ratio.

For all of the results shown in Figures 3 through 7 the ResNets were trained
on data sets where the equivalence ratio was varied and there were at most two
different initial temperatures (see Appendix A for more details). For this
final result, parallel ResNets were trained on data sets that had initial
conditions with the same equivalence ratio, but different initial
temperatures. The results of an experiment using DNNs trained on this data can
be seen in Figure 8. The 9 plots on the left show results from ResNets with a
depth of 6 and width of 30. On the right, the results came from ResNets with
different depths (3 to 8) all with width 30. All of the ResNets were trained
with validation data with patience of 400 iterations.

Figure 8. Results from ResNets trained on data that varies the initial
temperature. The plots on the left were produced from ResNets with a depth of
6 and a width of 30, and were trained with validation data with patience of
400 iterations. The plots on the right came from ResNets with varying depths
all with a width of 30, and were trained with validation data with patience of
400 iterations. All plots show the results of marching in time (known data in
blue, ResNet results in red).

##  5\. Conclusions and Future Directions

A number of ResNet based approaches to approximate the stiff ODEs arising in
chemical kinetics were developed. The experiments presented indicate that when
the equivalence ratio is fixed and initial temperature is varied (this case
has been considered in [25, 20]), the proposed DNNs can (almost) perfectly
generalize to unseen data. This capacity for generalization deteriorates,
however, when the initial temperature is kept fixed and the equivalence ratio
is varied. In order to overcome this issue, the training data was separated
into fuel lean, fuel balanced, and fuel rich based on the equivalence ratio.
This approach has led to encouraging results.

There are several questions that are currently being investigated:

  * •

All DNNs are dependent on the quality of training data. The data used for
experiments reported here were generated using CHEMKIN. In view of the high
dimensionality of the data, and the nature of chemical reactions (large
periods of inactivity followed by a sudden reaction followed by a convergence
to a steady state), quality criteria need to be developed to make sure
redundant data is avoided.

  * •

It will be interesting to see if further dividing the training data based on
equivalence ratio is helpful.

  * •

The usefulness of training with noisy data is also being explored. For some
classification problems, this approach is known to increase the robustness of
ResNets.

  * •

It is of interest to see how the proposed approach generalizes to more
reactions and species.

##  6\. Acknowledgements

This work was supported by the Defense Threat Reduction Agency (DTRA) under
contract HDTRA1-15-1-0068. Jacqueline Bell served as the technical monitor.
It is a pleasure to acknowledge a number of fruitful discussions with Drs.
Adam Moses, Alisha J. Sharma and Ryan F. Johnson from the Naval Research
Laboratory that lead to several improvements in the techniques discussed in
this paper.

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##  Appendix A Description of the Training Data Sets

In this appendix the data sets that were used to train the DNNs and create the
plots presented above are described in more detail. Specifically, information
about the number of points, the equivalence ratio, and initial temperature of
the individual data sets is provided. For the results in Figure 2, a single
ResNet was trained with a set containing data from a single initial condition.
This set has 4,999 points, and the initial condition has equivalence ratio 0.1
and initial temperature 1,200oK. Once the network was trained on this data, it
was tested with the same initial point from the set on which it was trained.

The parallel ResNets used to produce the plots in Figures 3 through 6 were
trained on a data set with subsets that correspond to 9 different initial
conditions, all with initial temperature 1,200oK. The data is described
further in Table 1, where the subsets are numbered for convenient reference.
This set also contains the data used to create Figure 2 (set 3). After being
trained on this entire data set, the ResNets that were used for the results in
Figure 3 were then tested on set 5, and therefore these results do not reflect
the ResNets generalizing to unseen data.

The subsets of data described in Table 1 are of different sizes. For the
results in Figure 4 the differences in sizes was compensated by training with
copies of the smaller sets. Specifically, the ResNets were trained with set 1,
two copies of sets 3 and 9, and eight copies of sets 5 and 7. The ResNets were
then tested by using the initial condition from set 6, and these are the
results that are shown in Figure 4. To train the ResNets that produced the
results in Figures 5 and 6 a single copy of sets 1, 3, 5, 7, and 9 from Table
1 was used. The trained networks were then tested on set 8. The difference
between these two experiments are the architecture of the ResNets and the way
that the training data was scaled prior to training. Also, as opposed to the
experiments represented by Figures 3 and 4, the experiments that resulted in
Figures 5 and 6 use the _second approach_ , where the ResNets are being
trained to learn the next timestep of the data. To create the training data in
this situation, the same data is used as both components (input and output).
This results in size of each subset being reduced by one, because there is no
next timestep to learn at the final time.

Set | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---
Number of points | 8,000 | 6,000 | 4,999 | 1,999 | 999 | 999 | 999 | 1,999 | 3,999
Equivalence eatio | 0.01 | 0.05 | 0.1 | 0.25 | 0.5 | 1 | 2 | 5 | 10
Initial temperature | 1,200 | 1,200 | 1,200 | 1,200 | 1,200 | 1,200 | 1,200 | 1,200 | 1,200
Table 1. Description of the training data used to produce Figures 3 through
Figure 6.

In Table 2 the data sets that have been separated into fuel lean, balanced,
and rich sets are described. The first thing to notice about this data is that
it contains the data sets from Table 1, where here the number of points is one
less for the same reason described above. This data was used to create the
results shown in Figure 7. The ResNets trained on fuel lean sets (top left set
of 9 plots) were trained with subsets 1, 3, 4, 6, and 9, and then tested with
the initial condition from set 8. The ResNets trained on the fuel balanced
sets were trained on subsets 1, 3, 5, 7, 9, and 12, and subsequently tested
with the initial condition from set 11. Finally the networks trained on fuel
rich sets were trained with subsets 1, 2, 3, 5, and 7, and then tested with
the initial condition from set 4.

Fuel Lean Sets
---
Set |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9
Number of points |  7,999 |  5,999 |  4,998 |  4,999 |  4,999 |  4,999 |  4,999 |  4,999 |  4,999
Equivalence ratio |  0.01 |  0.05 |  0.1 |  0.01 |  0.02 |  0.04 |  0.06 |  0.08 |  0.1
Initial temperature |  1,200 |  1,200 |  1,200 |  1,500 |  1,500 |  1,500 |  1,500 |  1,500 |  1,500

Fuel Balanced Sets
---
Set |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |  10 |  11 |  12
Number of points |  1,998 |  998 |  998 |  998 |  4,999 |  4,999 |  4,999 |  4,999 |  4,999 |  4,999 |  4,999 |  4,999
Equivalence ratio |  0.25 |  0.5 |  1 |  2 |  0.2 |  0.4 |  0.5 |  0.75 |  0.9 |  1 |  1.5 |  2
Initial temperature |  1,200 |  1,200 |  1,200 |  1,200 |  1,500 |  1,500 |  1,500 |  1,500 |  1,500 |  1,500 |  1,500 |  1,500

Fuel Lean Sets
---
Set |  1 |  2 |  3 |  4 |  5 |  6 |  7
Number of points |  1,998 |  3,998 |  4,999 |  4,999 |  4,999 |  4,999 |  4,999
Equivalence ratio |  5 |  10 |  2.5 |  3 |  3.5 |  4.5 |  5
Initial temperature |  1,200 |  1,200 |  1,500 |  1,500 |  1,500 |  1,500 |  1,500

Table 2. Description of the training data with fuel-based grouping that was
used for the results in Figure 7.

For the experiments that produced the plots in Figure 8, the training data
consists of 13 subsets. Each subset has 499 points and an equivalence ratio of
1. Furthermore, each subset has a different initial temperature beginning with
1,200oK and increasing by increments of 100oK to 2,400oK. For the experiment
shown in Figure 8, the ResNets were trained with the sets corresponding to
initial temperatures 1,200oK, 1,500oK, 1,800oK, 2,100oK, and 2,400oK. The
trained ResNets were then tested using the initial condition with temperature
2,200oK.