Smooth torus quotients of Richardson varieties in the Grassmannian

Within the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of Sarymsakov matrices is the largest known subset that is closed under matrix multiplication, and more critically whose compact subsets are all consensus sets. This paper shows that a larger subset with these two properties can be obtained by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved by introducing the notion of the SIA index of a stochastic matrix, whose value is 1 for Sarymsakov matrices, and then exploring those stochastic matrices with larger SIA indices. In addition to constructing the larger set, this paper introduces another class of generalized Sarymsakov matrices, which contains matrices that are not SIA, and studies their products. Sufficient conditions are provided for an infinite product of matrices from this class, converging to a rank-one matrix. Finally, as an application of the results just described and to confirm their usefulness, a necessary and sufficient combinatorial condition, the “avoiding set condition,” for deciding whether or not a compact set of stochastic matrices is a consensus set is revisited. In addition, a necessary and sufficient combinatorial condition is established for deciding whether or not a compact set of doubly stochastic matrices is a consensus set.

Strong shift equivalence and shear adjacency of nonnegative square integer matrices

The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \to \mathbb{Z}_q$, where $q ≥q 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of $f$, which then allows us to derive sufficient conditions that $f$ is generalized bent for any even $q$. The case when $q$ is not a power of two, which has not been addressed previously, is treated separately and a suitable representation in terms of the component functions is employed. Consequently, the derived results lead to generic construction methods of this class of functions. The main remaining task, which is not answered in this article, is whether the sufficient conditions are also necessary. There are some indications that this might be true which is also formally confirmed for generalized bent functions that belong to the class of generalized Maiorana-McFarland functions (GMMF), but still we were unable to completely specify (in terms of necessity) gbent conditions.

The algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known. Based on this formulation, for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links, the output symbols at the sink nodes, at any given time instant, is a F(p)m-linear combination of the input symbols across different generations, where F(p)m denotes the field over which the network operates (p is prime and m is a positive integer). We use finite-field discrete Fourier transform to convert the output symbols at the sink nodes, at any given time instant, into a F(p)m-linear combination of the input symbols generated during the same generation without making use of memory at the intermediate nodes. We call this as transforming the acyclic network with delay into n-instantaneous networks (n is sufficiently large). We show that under certain conditions, there exists a network code satisfying sink demands in the usual (nontransform) approach if and only if there exists a network code satisfying sink demands in the transform approach. When the zero-interference conditions are not satisfied, we propose three precoding-based network alignment (PBNA) schemes for three-source three-destination multiple unicast network with delays (3-S 3-D MUN-D) termed as PBNA using transform approach and time-invariant local encoding coefficients (LECs), PBNA using time-varying LECs, and PBNA using transform approach and block time-varying LECs. We derive sets of necessary and sufficient conditions under which throughputs close to n' + 1/2n' + 1, n'/2n' + 1, and n'/2n' + 1 are achieved for the three source-destination pairs in a 3-S 3-D MUN-D employing PBNA using transform approach and time-invariant LECs, and PBNA using transform approach and block time-varying LECs, where n' is a positive integer. For PBNA using time-varying LECs, we obtain a sufficient condition under which a throughput demand of n(1)/n, n(2)/n, and n(3)/n can be met for the three source-destination pairs in a 3-S 3-D MUN-D, where n(1), n(2), and n(3) are positive integers less than or equal to the positive integer n. This condition is also necessary when n(1) + n(3) = n(1) + n(2) = n where n(1) >= n(2) >= n(3).

Balancing networks, originally introduced by Aspnes et al.(Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 348-358, May 1991), represent a new class of distributed, low-contention data structures suitable for solving many fundamental multi-processor coordination problems that can be expressed asbalancing problems. In this work, we present a mathematical study of the combinatorial structure of balancing networks, and a variety of its applications.Our study identifies important combinatorialtransfer parametersof balancing networks. In turn, necessary and sufficient combinatorial conditions are established, expressed in terms of transfer parameters, which precisely characterize many important and well studied classes of balancing networks such ascounting networksandsmoothing networks. We propose these combinatorial conditions to be “balancing analogs” of the well knownZero-One principleholding forsorting networksWithin the combinatorial framework we develop, our first application is in deriving combinatorial conditions, involving the transfer parameters, which precisely delimit the boundary between counting networks and sorting networks.

Balancing networks, originally introduced by Aspnes et al. (Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 348-358, May 1991), represent a new class of distributed, low-contention data structures suitable for solving many fundamental multi-processor coordination problems that can be expressed as balancing problems. In this work, we present a mathematical study of the combinatorial structure of balancing networks, and a variety of its applications.Our study identifies important combinatorial transfer parameters of balancing networks. In turn, necessary and sufficient combinatorial conditions are established, expressed in terms of transfer parameters, which precisely characterize many important and well studied classes of balancing networks such as counting networks and smoothing networks. We propose these combinatorial conditions to be “balancing analogs” of the well known Zero-One principle holding for sorting networksWithin the combinatorial framework we develop, our first application is in deriving combinatorial conditions, involving the transfer parameters, which precisely delimit the boundary between counting networks and sorting networks.

Multidimensional linear phase perfect reconstruction filter bank (MDLPPRFB) can be designed and implemented via lattice structure. The lattice structure for the MDLPPRFB with filter support N(MΞ) has been published by Muramatsu , where M is the decimation matrix, Ξ is a positive integer diagonal matrix, and N(N) denotes the set of integer vectors in the fundamental parallelepiped of the matrix N. Obviously, if Ξ is chosen to be other positive diagonal matrices instead of only positive integer ones, the corresponding lattice structure would provide more choices of filter banks, offering better trade-off between filter support and filter performance. We call such resulted filter bank as generalized-support MDLPPRFB (GSMDLPPRFB). The lattice structure for GSMDLPPRFB, however, cannot be designed by simply generalizing the process that Muramatsu employed. Furthermore, the related theories to assist the design also become different from those used by Muramatsu . Such issues will be addressed in this paper. To guide the design of GSMDLPPRFB, the necessary and sufficient conditions are established for a generalized-support multidimensional filter bank to be linear-phase. To determine the cases we can find a GSMDLPPRFB, the necessary conditions about the existence of it are proposed to be related with filter support and symmetry polarity (i.e., the number of symmetric filters ns and antisymmetric filters na). Based on a process (different from the one Muramatsu used) that combines several polyphase matrices to construct the starting block, one of the core building blocks of lattice structure, the lattice structure for GSMDLPPRFB is developed and shown to be minimal. Additionally, the result in this paper includes Muramatsu's as a special case.

In the author list, "Ferry Sansoto" should be Ferry

An algorithm for optimum common root functions of two digraphs

On constructive characterizations of ( k,l )-sparse graphs

On the computation of Seidel Laplacian eigenvalues for graph-based binary codes

On the representation of−1 as a sum of two squares in an algebraic number field

The oscillation of systems of difference equations

1. Main result. The purpose of this paper is to prove the theorem and corollary stated in this section. The corollary answers the question raised in Section 3 of [4] to which we also refer the reader for further motivation. THEOREM. Let X be a symmetric r.v., and Xi, i = 1, 2, , r.v.'s independently and identically distributed as X. The two conditions: (i) for any real number c, and positive integer n, there exist real numbers A = A (n, c) and B = B(n, c) for which En 1/(Xi + c) is distr-ibuted as A/(X + B), and (ii) for some c 0 0 the symmetric r.v. 1/(X1 + c) + 1/(X2 c) is distributed as A(c)/X, for some number A (c), are necessary and sufficient for X to have a Cauchy distribution. COROLLARY. Let Xi, i = 1, 2, * * be r.v.'s independently and identically distributed as a r.v. X. The necessary and sufficient condition that, for any real numbers ai 0, bi, i = 1, 2, * * and any positive integer n, there exist real numbers A and B for which En 1/(aiX, + bi) has the same distribution as A/(X + B), is that X have the Cauchy distribution.

<abstract><p>In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $ (a^2+2)x^2-y^2 = 2 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $.</p></abstract>