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First, we solve a crucial problem under which conditions increasing uniform K-monotonicity is equivalent to lower locally uniform K-monotonicity. Next, we investigate properties of substochastic operators on $L^1+L^\infty$ with applications. Namely, we show that a countable infinite combination of substochastic operators is also substochastic. Using K-monotonicity properties, we prove several theorems devoted to the convergence of the sequence of substochastic operators in the norm of a symmetric space E under addition assumption on E. In our final discussion we focus on compactness of admissible operators for arbitrary Banach couples., Comment: 18 pages

Let X be a real or complex Banach space and let F in X be a non-empty set. F is called an existence set of best coapproximation (existence set for brevity), if for any x in X, $R_F(x)$ is not the empty set, where $$ R_F (x) = \{ d \in F : \|d-c\| \leq \|x-c\| \hbox{ for any } c \in F \}.$$ It is clear that any existence set is a contractive subset of X. The aim of this paper is to present some conditions on F and X under which the notions of exsistence set and contractive set are equivalent.

The aim of this paper is to to show the admissibility of some class of Frechet spaces (see Definition 2.3). In particular, this generalizes the main results of [3]. As an application, we show the admissibility of a large class modular spaces equipped with F-norms determined in Theorem 4.1. It is worth noticing that F-norms introduced in Theorem 4.1 generalize the classical Luxemburg F-norm. Also a linear version of admissibility (so called metric approximation property) for order continuous symmetric spaces will be demonstrated (see Theorem 5.1).

This paper presents a fast and effective computer algebraic method for analyzing and verifying non-linear integer arithmetic circuits using a novel algebraic spectral model. It introduces a concept of algebraic spectrum, a numerical form of polynomial expression; it uses the distribution of coefficients of the monomials to determine the type of arithmetic function under verification. In contrast to previous works, the proof of functional correctness is achieved by computing an algebraic spectrum combined with a local rewriting of word-level polynomials. The speedup is achieved by propagating coefficients through the circuit using And-Inverter Graph (AIG) datastructure. The effectiveness of the method is demonstrated with experiments including standard and Booth multipliers, and other synthesized non-linear arithmetic circuits up to 1024 bits containing over 12 million gates., Comment: To appear at ASP-DAC'19

There are discussed some modifications of n-th von Neumann-Jordan constant considered by M. Kato, Y. Takahashi and K. Hashimoto. We study, among others, upper, lower, upper modified and lower modified n-th von Neumann-Jordan constant and relationships between them. There are characterized uniformly non-l_{n}^{1} Banach spaces in terms of the upper modified n-th von Neumann-Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces L^{p} and l^{p} (1\leq p\leq oo)., Comment: 18 pages

This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity and strict convexity in the sequence Lorentz spaces $\gamma_{p,w}$. Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space $\gamma_{1,w}$. We also establish a complete description with an isometry of the dual and predual spaces of the sequence Lorentz spaces $\gamma_{1,w}$ written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental application of geometric structure of $\gamma_{1,w}$ to one-complemented subspaces of $\gamma_{1,w}$., Comment: 21 pages

Let X be a linear space over K, K=R or K=C and let for n>1 \rho_i be s-convex semimodular defined on X for any i\in{1,...,n-1}. Put \rho=\max_{1\leq i \leq n-1}\{\rho_i\} and X_{\rho}= { x \in X: \rho(dx) < \infty for some d > 0 }. In this paper we define a new class of s-norms (norms if s=1) on X_{\rho}. In particular, our defintion generalizes in a natural way the Orlicz-Amemiya and Luxemburg norms defined for s-convex semimodulars. Then, we investigate order continuous, the Fatou Property and various monotonicity properties of semimodular spaces equipped with these s-norms., Comment: 28 pages

In this paper we investigate a relationship between fully k-rotundity properties, uniform K-monotonicity properties, reflexivity and K-order continuity in a symmetric spaces E. We also answer a crucial question whether fully k-rotundity properties might be restricted in definition to E^d the positive cone of all nonnegative and decreasing elements of E. We present a complete characterization of decreasing uniform K-monotonicity and K-order continuity in E. It is worth mentioning that we also establish several auxiliary results describing reflexivity in Lorentz spaces and K-order continuity in Orlicz spaces. Finally, we show an application of discussed geometric properties to the approximation theory., Comment: 24 pages

Galois field (GF) arithmetic circuits find numerous applications in communications, signal processing, and security engineering. Formal verification techniques of GF circuits are scarce and limited to circuits with known bit positions of the primary inputs and outputs. They also require knowledge of the irreducible polynomial $P(x)$, which affects final hardware implementation. This paper presents a computer algebra technique that performs verification and reverse engineering of GF($2^m$) multipliers directly from the gate-level implementation. The approach is based on extracting a unique irreducible polynomial in a parallel fashion and proceeds in three steps: 1) determine the bit position of the output bits; 2) determine the bit position of the input bits; and 3) extract the irreducible polynomial used in the design. We demonstrate that this method is able to reverse engineer GF($2^m$) multipliers in \textit{m} threads. Experiments performed on synthesized \textit{Mastrovito} and \textit{Montgomery} multipliers with different $P(x)$, including NIST-recommended polynomials, demonstrate high efficiency of the proposed method., Comment: To appear in IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. (TCAD'18); extended version. arXiv admin note: text overlap with arXiv:1611.05101

This paper presents an advanced DAG-based algorithm for datapath synthesis that targets area minimization using logic-level resource sharing. The problem of identifying common specification logic is formulated using unweighted graph isomorphism problem, in contrast to a weighted graph isomorphism using AIGs. In the context of gate-level datapath circuits, our algorithm solves the un- weighted graph isomorphism problem in linear time. The experiments are conducted within an industrial synthesis flow that includes the complete high-level synthesis, logic synthesis and placement and route procedures. Experimental results show a significant runtime improvements compared to the existing datapath synthesis algorithms., Comment: 6 pages, 8 figures. To appear in 2017 IEEE/ACM International Conference on Computer-Aided Design (ICCAD'17)

We research proximinality of $\mu$-sequentially compact sets and $\mu$-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and $\mu$-compactness of the sets in Banach function spaces. Also the property $S$ is investigated in Fr\'echet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fr\'echet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces $d(w,1)$., Comment: 26 pages

Using the local approach to the global structure of a symmetric space $E$ we establish a relationship between strict $K$- monotonicity, lower (resp. upper) local uniform $K$-monotonicity, order continuity and the Kadec-Klee property for global convergence in measure. We also answer the question under which condition upper local uniform $K$-monotonicity concludes upper local uniform monotonicity. Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$., Comment: 15 pages

This paper is devoted to strict $K$- monotonicity and $K$-order continuity in symmetric spaces. Using the local approach to the geometric structure in a symmetric space $E$ we investigate a connection between strict $K$-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of $K$-order continuity in a symmetric space $E$ on $[0,\infty)$ implies that the embedding $E\hookrightarrow{L^1}[0,\infty)$ does not hold. We finish this article with a complete characterization of $K$-order continuity in a symmetric space $E$ that is written using a notion of order continuity under some assumptions on the fundamental function of $E$.

Current techniques for formally verifying circuits implemented in Galois field (GF) arithmetic are limited to those with a known irreducible polynomial P(x). This paper presents a computer algebra based technique that extracts the irreducible polynomial P(x) used in the implementation of a multiplier in GF(2^m). The method is based on first extracting a unique polynomial in Galois field of each output bit independently. P(x) is then obtained by analyzing the algebraic expression in GF(2^m) of each output bit. We demonstrate that this method is able to reverse engineer the irreducible polynomial of an n-bit GF multiplier in n threads. Experiments were performed on Mastrovito and Montgomery multipliers with different P (x), including NIST-recommended polynomials and optimal polynomials for different microprocessor architectures., Comment: 6 pages, 4 figures, DATE 2017, Lausanne, Switzerland, March 27-31, 2017

Galois field (GF) arithmetic is used to implement critical arithmetic components in communication and security-related hardware, and verification of such components is of prime importance. Current techniques for formally verifying such components are based on computer algebra methods that proved successful in verification of integer arithmetic circuits. However, these methods are sequential in nature and do not offer any parallelism. This paper presents an algebraic functional verification technique of gate-level GF (2m ) multipliers, in which verification is performed in bit-parallel fashion. The method is based on extracting a unique polynomial in Galois field of each output bit independently. We demonstrate that this method is able to verify an n-bit GF multiplier in n threads. Experiments performed on pre- and post-synthesized Mastrovito and Montgomery multipliers show high efficiency up to 571 bits., Comment: 6 pages, 22nd Asia and South Pacific Design Automation Conference (ASP-DAC 2017), Japan

We show that the space of bounded and linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollob\'as property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an $L_1$-space., Comment: 12 pages