Regular Matroids Have Polynomial Extension Complexity

We prove that the extension complexity of the independence polytope of every regular matroid on [Formula: see text] elements is [Formula: see text]. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a [Formula: see text] bound for the special case of (co)graphic matroids. However, the case of a general regular matroid was open, despite recent attempts. We also consider the extension complexity of circuit dominants of regular matroids, for which we give a [Formula: see text] bound.

A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. «Handshaking» Procedure

A new minimization method of logic functions of n variables in polynomial set-theoretical format has been considered. The method based on the so-called “handshaking” procedure. This procedure reflects the iterative polynomial extension of two conjuncterms of different ranks, the Hamming distance between which can be arbitrary. The advantages of the suggested method are illustrated by the examples.

Polynomial extension property in the classical Cartan domain $${\mathcal {R}_{II}}$$

In this work, it is shown that for the classical Cartan domain $$\mathcal {R}_{II}$$ R II consisting of symmetric $$2\times 2$$ 2 × 2 matrices, every algebraic subset of $$\mathcal {R}_{II}$$ R II , which admits the polynomial extension property, is a holomorphic retract.

Error-Correcting and Verifiable Parallel Inference in Graphical Models

We present a novel framework for parallel exact inference in graphical models. Our framework supports error-correction during inference and enables fast verification that the result of inference is correct, with probabilistic soundness. The computational complexity of inference essentially matches the cost of w-cutset conditioning, a known generalization of Pearl's classical loop-cutset conditioning for inference. Verifying the result for correctness can be done with as little as essentially the square root of the cost of inference. Our main technical contribution amounts to designing a low-degree polynomial extension of the cutset approach, and then reducing to a univariate polynomial employing techniques recently developed for noninteractive probabilistic proof systems.

Integral Domains in Which Every Nonzero w-Flat Ideal Is w-Invertible

Let D be an integral domain and w be the so-called w-operation on D. We define D to be a w-FF domain if every w-flat w-ideal of D is of w-finite type. This paper presents some properties of w-FF domains and related domains. Among other things, we study the w-FF property in the polynomial extension, the t-Nagata ring and the pullback construction.

Generalized Jacobsthal numbers and restricted k-ary words

We consider a generalization of the problem of counting ternary words of a given length which was recently investigated by Koshy and Grimaldi [10]. In particular, we use finite automata and ordinary generating functions in deriving a k-ary generalization. This approach allows us to obtain a general setting in which to study this problem over a k-ary language. The corresponding class of n-letter k-ary words is seen to be equinumerous with the closed walks of length n − 1 on the complete graph for k vertices as well as a restricted subset of colored square-and-domino tilings of the same length. A further polynomial extension of the k-ary case is introduced and its basic properties deduced. As a consequence, one obtains some apparently new binomial-type identities via a combinatorial argument.

The automorphism groups for a family of generalized Weyl algebras

In this paper, we study a family of generalized Weyl algebras [Formula: see text] and their polynomial extensions. We will show that the algebra [Formula: see text] has a simple localization [Formula: see text] when none of [Formula: see text] and [Formula: see text] is a root of unity. As an application, we determine all the height-one prime ideals and the center for [Formula: see text], and prove that [Formula: see text] is cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras [Formula: see text] and their polynomial extensions in the case where none of [Formula: see text] and [Formula: see text] is a root of unity. We will establish a quantum analogue of the Dixmier conjecture and compute the automorphism group for the simple localization [Formula: see text]. Moreover, we will completely determine the automorphism group for the algebra [Formula: see text] and its polynomial extension when [Formula: see text] and [Formula: see text].