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We consider infinite sequences from a field and all matrices whose rows consist of distinct consecutive subsequences. We show that these matrices have bounded rank if and only if the sequence is a homogeneous linear recurrence; moreover, it is a -term linear recurrence if and only if the maximum rank is . This means, in particular, that the ranks of matrices from the sequence of primes are unbounded. Though not all matrices from the primes have full rank, because of the Green–Tao theorem, we conjecture that square matrices whose entries are a consecutive sequence of primes do have full rank.
We study a two-point boundary-value problem describing steady states of a population dynamics model with diffusion, logistic growth, and nonlinear density-dependent dispersal on the boundary. In particular, we focus on a model in which the population exhibits hump-shaped density-dependent dispersal on the boundary, and explore its effects on existence, uniqueness and multiplicity of steady states.
The -move for classical braids extends naturally to trivalent braids. We follow the -move approach to the Markov theorem to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this -move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we provide a proof of the Alexander theorem analogue for spatial trivalent graphs and trivalent braids.
We study the beginning of the Taylor tower, supplied by manifold calculus of functors, for the space of -immersions, which are immersions without -fold self-intersections. We describe the first layers of the tower and discuss the connectivities of the associated maps. We also prove several results about -immersions that are of independent interest.
This paper introduces an independent-proposal Metropolis–Hastings sampler for Bayesian probit regression. The Gibbs sampler of Albert and Chib has been the default sampler since its introduction in 1993. We conduct a simulation study comparing the two methods under various combinations of predictor variables and sample sizes. The proposed sampler is shown to outperform that of Albert and Chib in terms of efficiency measured through autocorrelation, effective sample size, and computation time. We then demonstrate performance of the samplers on real data applications with analogous results.
We determine the maximum number of points in which form exactly distinct triangles, where we restrict ourselves to the case of . We denote this quantity by . It is known from the work of Epstein et al. (Integers 18 (2018), art. id. A16) that . Here we show somewhat surprisingly that and , whenever , and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erdős and Fishburn (Discrete Math.160:1-3 (1996), 115–125).
We deal with a resonant boundary value problem involving a second-order differential equation with periodic boundary conditions. First, we modify the problem at resonance and consider an equivalent nonresonant boundary value problem. Next, we obtain sufficient conditions for the existence of solutions of the modified boundary value problem, using fixed-point theory. Consequently, these conditions suffice for the existence of solutions of the original boundary value problem. We demonstrate the applicability of established results through examples.
A well-known open problem in graph theory asks whether Stanley’s chromaticsymmetric function, a generalization of the chromatic polynomial of a graph, distinguishes between any two nonisomorphic trees. Previous work has proven the conjecture for a class of trees called spiders. This paper generalizes the class of spiders to -spiders, where normal spiders correspond to , and verifies the conjecture for .
We study basic properties of one-parametric families of the -metric, the scale-invariant Cassinian metric and the half-Apollonian metric on locally compact, noncomplete metric spaces. We first establish basic properties of these metrics on once-punctured general metric spaces and obtain sharp estimates between these metrics, and then we show that all these properties, except for -hyperbolicity, extend to the settings of locally compact noncomplete metric spaces. We also show that these metrics are -hyperbolic only if the underlying space is a once-punctured metric space.
One of the most important and challenging problems in coding theory is explicit construction of linear codes with the best possible parameters. It is well known that the class of quasitwisted (QT) codes is asymptotically good and contains many linear codes with best known parameters (BKLCs). A search algorithm (ASR) on QT codes has been particularly effective to construct such codes. Recently, the ASR algorithm was generalized based on the notion of code equivalence. In this work, we introduce a new generalization of the ASR algorithm to include a broader scope of QT codes. As a result of implementing this algorithm, we have found eight new linear codes over the field . Furthermore, we have found seven additional new codes from the standard constructions of puncturing, shortening or Construction X. We also introduce a new search algorithm that can be viewed as a further generalization of ASR into the class multitwisted (MT) codes. Using this method, we have found many codes with best known parameters with more direct and desirable constructions than what is currently available in the database of BKLCs.
We investigate conditions under which the identity matrix can be continuously factorized through a continuous matrix function with domain in . We study the relationship of the dimension , the diagonal entries of , and the norm of to the dimension and the norms of the matrices that witness the factorization of through .
Let be a complete local (Noetherian) domain such that . In addition, suppose contains the rationals, , and the set of all principal height-1 prime ideals of has the same cardinality as . We construct a universally catenary local unique factorization domain such that the completion of is and such that there exist uncountably many height-1 prime ideals of such that is a field. Furthermore, in the case where is a normal domain, we can make “close” to excellent in the following sense: the formal fiber at every prime ideal of of height not equal to 1 is geometrically regular, and uncountably many height-1 prime ideals of have geometrically regular formal fibers.
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