Narrow Results

Valiant defined #P-completeness using Turing reductions. We propose a more restrictive definition, a variant of many-one reductions, and prove that the (0, 1)-permanent remains #P-complete under this definition.

The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class that is closed under log-lin reductions:

• All complete sets for the class under 2DFA reductions are also complete under one-one, length-increasing 2DFA reductions and are first-order isomorphic.

• The 2DFA-isomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions.

This paper will prove the uniqueness theorem for 3-layered complex-valued neural networks where the threshold parameters of the hidden neurons can take non-zeros. That is, if a 3-layered complex-valued neural network is irreducible, the 3-layered complex-valued neural network that approximates a given complex-valued function is uniquely determined up to a finite group on the transformations of the learnable parameters of the complex-valued neural network.

In this paper, we discuss a relationship between the surface symmetry and the spectral asymmetry. More precisely we show that an automorphism of the Macbeath surface of genus 7, or one of the three Hurwitz surfaces of genus 14 is reducible if and only if the η-invariant of the corresponding mapping torus vanishes.

The hypergeometric system $E(k,n)$ and the contiguity operators defined on the space of $k\times n$ matrices $Z_{k,n}$ induce the systems and the contiguity operators on the Grassmannian manifold $G_{k,n}:=G{L}_k\backslash Z_{k,n}$ and on the configuration space $X(k,n):=G{L}_k\backslash Z_{k,n}/H_n,$ where $H_n$ is the group consisting of diagonal matrices of size $n$. The first purpose of this paper is to give a rigorous treatment of these systems and contiguity operators, the second is to derive a condition that the system $E(k,n)$ is reducible, and the third is to derive the contiguity relations and the contiguous relations satisfied by the solutions of the system on $G_{k,n}$ or the solutions of the system on $X(k,n).$

In order to explore the properties depending only on structure such as structural controllability, the structured matrix (SM), and the rational function matrix (RFM) in multi-parameters (RFM) were used to describe the coefficient matrices of systems and networks. In this paper, Definitions 3.3 and 3.4 are given. On the basis of RLCM passive networks, some properties (separability, reducibility, and controllability) of a class of RLCM active networks over $F(z)$ are studied. Some criteria are obtained. Since these criteria depend only on network structures, their applications are straightforward. Finally, two illustrative examples are given.

Constructions that yield pseudorecursiveness in [I] (Int. J. Algebra Comput.6 (1996) 457–510) are extended in this article. Finitely based varieties of semigroups with increasingly strict expansions by additional unary operation symbols or individual constants are shown to have the pseudorecursive property: the equational theory is undecidable, but the subsets obtained by bounding the number of distinct variables are all recursive. The most stringent case considered here is the single unary operation or distinguished element. New techniques of stratified reducibility and interpretation via rewriting rules are employed to show the property inherits along a chain of theories. Pure semigroup varieties that are both finitely based and pseudorecursive will be discussed in a later paper.

In this paper, we investigate the reducibility property of semidirect products of the form $V\ast D$ relatively to (pointlike) systems of equations of the form $x_1=\cdots =x_n$, where $D$ d̃enotes the pseudovariety of definite semigroups. We establish a connection between pointlike reducibility of $V\ast D$ and the pointlike reducibility of the pseudovariety $V$. In particular, for the canonical signature $\kappa$ consisting of the multiplication and the $(\omega -1)$-power, we show that $V\ast D$ is pointlike $\kappa$-reducible when $V$ is pointlike $\kappa$-reducible.

This paper deals with the reducibility property of semidirect products of the form $V\ast D$ relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that if the pseudovariety $V$ is reducible with respect to the canonical signature $\kappa$ consisting of the multiplication and the $(\omega -1)$-power, then $V\ast D$ is also reducible with respect to $\kappa$.

We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\dots ,x)=x$ for all $x\in X$. We first provide a structure theorem for these symmetric $n$-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong $n$-ary semilattice of $n$-ary semigroups and we show that the symmetric $n$-ary bands are exactly the strong $n$-ary semilattices of $n$-ary extensions of Abelian groups whose exponents divide $n-1$. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric $n$-ary band to be reducible to a semigroup.

In this paper, we study the complexity of deciding code and monoid properties for regular sets specified by deterministic or nondeterministic finite automata. The results are as follows. The code problem for regular sets specified by deterministic or nondeterministic finite automata is NL-complete under NC⁽¹⁾ reducibilities. The problems of determining whether a regular set given by a deterministic finite automaton is a monoid or a free monoid or a finitely generated monoid are all NL-complete under NC⁽¹⁾ reducibilities. These monoid problems become PSPACE-complete if the regular sets are specified by nondeterministic finite automata instead. The maximal code problem for deterministic finite automata is shown to be in DET and NL-hard, while a PSPACE upper bound and NP-hardness lower bound hold for the case of nondeterministic finite automata.

Let K be a knot in S³ and suppose that K(r) is a reducible manifold. Howie has proved that the number of connected summands is at most 3 [10, Corollary 5.3]. Furthermore, he showed that if K(r) is the connected sum of exactly three irreducible 3-manifolds, then K(r) = L(p₁, q₁)♯L(p₂, q₂)♯M, where L(p₁, q₁) and L(p₂, q₂) are a lens spaces and M is a ℤ-homology sphere. In this paper we study this specific case and we show that |r| ≤ (b-2)(b-1), for any knot with bridge number b.

We introduce the $H^{\prime }$-splittings for 3-manifolds as follows. For a compact connected surface $F$ properly embedded in a compact connected orientable 3-manifold $M$, if $F$ decomposes $M$ into two handlebodies $H_1$ and $H_2$, then $H_1{\bigcup }_F{H}_2$ is called an $H^{\prime }$-splitting for $M$. Clearly, when $M$ is closed, this is just the Heegaard splitting for $M$; when $M$ is with boundary, the $H^{\prime }$-splitting for $M$ is different from the Heegaard splitting for $M$. In this paper, we first show that any compact connected orientable 3-manifold admits an $H^{\prime }$-splitting, then generalize Casson–Gordon theorem on weakly reducible Heegaard splitting to the $H^{\prime }$-splitting case in the following version: if $H_1{\bigcup }_F{H}_2$ is a weakly reducible $H^{\prime }$-splitting for a compact connected orientable 3-manifold $M$, then (1) $M$ contains an incompressible closed surface of positive genus or (2) the $H^{\prime }$-splitting $H_1{\bigcup }_F{H}_2$ is reducible or (3) there is an essential 2-sphere $Q$ in $M$ such that $Q\cap H_i$ is a collection of essential disks in $H_i$ and $Q\cap H_j$ is an incompressible and not boundary parallel planar surface in $H_j$ with at least two boundary components, where $\{i,j\}=\{1,2\}$ or (4) $H_1{\bigcup }_F{H}_2$ is stabilized.

We give a characterization for the reducibility of elements of any finite subgroup of the mapping class group of genus 2 surface in terms of the η-invariant of finite order mapping tori.

We study a family of quantum Markov semigroups with circulant structure. We obtain a complete description of the spectral representation for the Lindbladian and, putting it together with some purely probabilistic properties of a classical associated process, we can study asymptotic properties, invariant states, quantum-detailed balance conditions and reducibility. In particular, in the reducible case, we can construct a generator on a lower-dimensional space which can fully describe the original circulant semigroup.

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.

We show that there is an antichain basis for neither (1) the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor (2) the class of analytic graphs of uncountable Borel chromatic number under Borel reducibility.

The use of mathematics in physics is not restricted to non-formal use with emphasis on approximation and computability. There is also a domain of creative use of symbolic manipulation as well as of metaphoric imagination that makes applicability in the sciences so effective. The challenge to mathematical use in economics is to find ways to use it in these creative ways, including finding new discursive and writing strategies. For this to be possible, the nature of realism in economics has to be delineated since the way in which mathematics is used in economics seems to be in the "reductive" mode instead of the "applied" mode.

La${}_{1-x}$Sr_xMnO${}_{3-\partial }$ (LSMO) and LaMnO${}_{3+\partial }$ (LMO) nanoparticle catalysts have been synthesized via a one-step molten salt route. It was found that the partial substitution of lanthanum by strontium had a promoting effect on the catalytic performance for toluene oxidation. Under the condition of toluene $\mathrm{\text{concentration}}=1000$ppm, toluene/O₂$\mathrm{\text{molar ratio}}=1∕200$ and the space $\mathrm{\text{velocity}}=20000$mL/(g h), the temperature required for 50% and 90% toluene combustion conversion was 150${}^{\circ }$C and 205${}^{\circ }$C over LSMO catalyst, respectively. It is concluded that the oxygen vacancy, the molar ratio Mn${}^{4+}$/Mn${}^{3+}$ on the surface and the specific surface area contribute to the improved catalytic performance of the LSMO nanoparticle materials via a one-step molten salt method.

We present a simple method for computing factorizations for a large class of matrix equations (matrix pseudo-linear equations) that includes, in particular, systems of linear differential, difference and q–difference equations. The approach is based on the characterization of the properties (reducibility, decomposability, etc.) of matrix pseudo-linear equations in terms of their eigenrings.