Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp (c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset $\{g_n : n\in\N\}$ that topologically generates the group and whose elements $g_n$ all have finite support. As a consequence, sufficient conditions are obtained that allow us to encode, by means of a topological group isomorphism, order controllable profinite abelian groups. Some applications of these results to group codes will appear subsequently \cite{FH:2021}.
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In this paper, some preliminaries about signal flow graph, linear time-invariant system on F(z) and computational complexity are first introduced in detail. In order to synthesize the necessary and sufficient condition on F(z) for a general 2-path problem, the sufficient condition on F(z) or R and necessary conditions on F(z) for a general 2-path problem are secondly analyzed respectively. Moreover, an equivalent sufficient and necessary condition on R whether there exists a general 2-path is deduced in detail. Finally, the computational complexity of the algorithm for this equivalent sufficient and necessary condition is introduced so that it means that the general 2-path problem is a P problem.
Determining the proper level of details to develop and solve physical models is usually difficult when one encounters new engineering problems. Such difficulty comes from how to balance the time (simulation cost) and accuracy for the physical model simulation afterwards. We propose a framework for automatic development of a family of surrogate models of physical systems that provide flexible cost-accuracy tradeoffs to assist making such determinations. We present both a model-based and a data-driven strategy to generate surrogate models. The former starts from a high-fidelity model generated from first principles and applies a bottom-up model order reduction (MOR) that preserves stability and convergence while providing a priori error bounds, although the resulting reduced-order model may lose its interpretability. The latter generates interpretable surrogate models by fitting artificial constitutive relations to a presupposed topological structure using experimental or simulation data. For the latter, we use Tonti diagrams to systematically produce differential equations from the assumed topological structure using algebraic topological semantics that are common to various lumped-parameter models (LPM). The parameter for the constitutive relations are estimated using standard system identification algorithms. Our framework is compatible with various spatial discretization schemes for distributed parameter models (DPM), and can supports solving engineering problems in different domains of physics.
The lazy algorithm for a real base $\beta$ is generalized to the setting of Cantor bases $\boldsymbol{\beta}=(\beta_n)_{n\in \mathbb{N}}$ introduced recently by Charlier and the author. To do so, let $x_{\boldsymbol{\beta}}$ be the greatest real number that has a $\boldsymbol{\beta}$-representation $a_0a_1a_2\cdots$ such that each letter $a_n$ belongs to $\{0,\ldots,\lceil \beta_n \rceil -1\}$. This paper is concerned with the combinatorial properties of the lazy $\boldsymbol{\beta}$-expansions, which are defined when $x_{\boldsymbol{\beta}}<+\infty$. As an illustration, Cantor bases following the Thue-Morse sequence are studied and a formula giving their corresponding value of $x_{\boldsymbol{\beta}}$ is proved. First, it is shown that the lazy $\boldsymbol{\beta}$-expansions are obtained by "flipping" the digits of the greedy $\boldsymbol{\beta}$-expansions. Next, a Parry-like criterion characterizing the sequences of non-negative integers that are the lazy $\boldsymbol{\beta}$-expansions of some real number in $(x_{\boldsymbol{\beta}}-1,x_{\boldsymbol{\beta}}]$ is proved. Moreover, the lazy $\boldsymbol{\beta}$-shift is studied and in the particular case of alternate bases, that is the periodic Cantor bases, an analogue of Bertrand-Mathis' theorem in the lazy framework is proved: the lazy $\boldsymbol{\beta}$-shift is sofic if and only if all quasi-lazy $\boldsymbol{\beta}^{(i)}$-expansions of $x_{\boldsymbol{\beta}^{(i)}}-1$ are ultimately periodic, where $\boldsymbol{\beta}^{(i)}$ is the $i$-th shift of the alternate base $\boldsymbol{\beta}$.
To overcome topological constraints and improve the expressiveness of normalizing flow architectures, Wu, K\"ohler and No\'e introduced stochastic normalizing flows which combine deterministic, learnable flow transformations with stochastic sampling methods. In this paper, we consider stochastic normalizing flows from a Markov chain point of view. In particular, we replace transition densities by general Markov kernels and establish proofs via Radon-Nikodym derivatives which allows to incorporate distributions without densities in a sound way. Further, we generalize the results for sampling from posterior distributions as required in inverse problems. The performance of the proposed conditional stochastic normalizing flow is demonstrated by numerical examples.
Using the nonholonomic exponential map, we generalize the well-known family of Newmark methods for nonholonomic systems. We give numerical examples including a test problem where the structure of reversible integrability responsible for good energy behaviour as described in [16] is lost. We observe that the composition of two Newmark methods is able to produce good energy behaviour on this test problem.
In this paper, we show several parameterized problems to be complete for the class XNLP: parameterized problems that can be solved with a non-deterministic algorithm that uses $f(k)\log n$ space and $f(k)n^c$ time, with $f$ a computable function, $n$ the input size, $k$ the parameter and $c$ a constant. The problems include Maximum Regular Induced Subgraph and Max Cut parameterized by linear clique-width, Capacitated (Red-Blue) Dominating Set parameterized by pathwidth, Odd Cycle Transversal parameterized by a new parameter we call logarithmic linear clique-width (defined as $k/\log n$ for an $n$-vertex graph of linear clique-width $k$), and Bipartite Bandwidth.
Past work shows that one can associate a notion of Shannon entropy to a Dirichlet polynomial, regarded as an empirical distribution. Indeed, entropy can be extracted from any $d\in\mathsf{Dir}$ by a two-step process, where the first step is a rig homomorphism out of $\mathsf{Dir}$, the \emph{set} of Dirichlet polynomials, with rig structure given by standard addition and multiplication. In this short note, we show that this rig homomorphism can be upgraded to a rig \emph{functor}, when we replace the set of Dirichlet polynomials by the \emph{category} of ordinary (Cartesian) polynomials. In the Cartesian case, the process has three steps. The first step is a rig functor $\mathbf{Poly}^{\mathbf{Cart}}\to\mathbf{Poly}$ sending a polynomial $p$ to $\dot{p}\mathcal{y}$, where $\dot{p}$ is the derivative of $p$. The second is a rig functor $\mathbf{Poly}\to\mathbf{Set}\times\mathbf{Set}^{\text{op}}$, sending a polynomial $q$ to the pair $(q(1),\Gamma(q))$, where $\Gamma(q)=\mathbf{Poly}(q,\mathcal{y})$ can be interpreted as the global sections of $q$ viewed as a bundle, and $q(1)$ as its base. To make this precise we define what appears to be a new distributive monoidal structure on $\mathbf{Set}\times\mathbf{Set}^{\text{op}}$, which can be understood geometrically in terms of rectangles. The last step, as for Dirichlet polynomials, is simply to extract the entropy as a real number from a pair of sets $(A,B)$; it is given by $\log A-\log \sqrt[A]{B}$ and can be thought of as the log aspect ratio of the rectangle.
We study Martin-L\"{o}f random (ML-random) points on computable probability measures on sample and parameter spaces (Bayes models). We consider four variants of conditional random sequences with respect to the conditional distributions: two of them are defined by ML-randomness on Bayes models and the others are defined by blind tests for conditional distributions. We consider a weak criterion for conditional ML-randomness and show that only variants of ML-randomness on Bayes models satisfy the criterion. We show that these four variants of conditional randomness are identical when the conditional probability measure is computable and the posterior distribution converges weakly to almost all parameters. We compare ML-randomness on Bayes models with randomness for uniformly computable parametric models. It is known that two computable probability measures are orthogonal if and only if their ML-random sets are disjoint. We extend these results for uniformly computable parametric models. Finally, we present an algorithmic solution to a classical problem in Bayes statistics, i.e.~the posterior distributions converge weakly to almost all parameters if and only if the posterior distributions converge weakly to all ML-random parameters.
In this paper, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned Runge-Kutta Munthe-Kaas (RKMK) methods on Lie groups. A homogeneous space $M$ is a manifold where a group $G$ acts transitively. Such a space can be understood as a quotient $M \cong G/H$, where $H$ a closed Lie subgroup, is the isotropy group of each point of $M$. The Lie algebra of $G$ may be decomposed into $\mathfrak{g} = \mathfrak{m} \oplus \mathfrak{h}$, where $\mathfrak{h}$ is the subalgebra that generates $H$ and $\mathfrak{m}$ is a subspace. Thus, variational problems on $M$ can be treated as nonholonomically constrained problems on $G$, by requiring variations to remain on $\mathfrak{m}$. Nonholonomic partitioned RKMK integrators are derived as a modification of those obtained by a discrete variational principle on Lie groups, and can be interpreted as obeying a discrete Chetaev principle. These integrators tend to preserve several properties of their purely variational counterparts.
A toric code, introduced by Hansen to extend the Reed-Solomon code as a $k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P \cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword; and the relative minimum distance, which measures how many errors can be corrected relative to how many bits each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.
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