The group is a well-known algebraic object and one of the most fundamental structures within mathematics. For a given group, we may correspond the elements to those of a group of invertible matrices, forming a matrix representation of the group. Associated with these representations are characters, obtained by taking the traces of each matrix. Studying these characters, we find that they contain much information about their underlying group. One of the main goals in character theory is to determine what properties about the group can be determined by looking at its characters. We will discuss several recent conjectures on this topic that study properties of certain subgroups known as defect groups. In particular, we prove some such conjectures regarding generation properties of defect 2-groups for the symplectic group Sp4(q), which is a particular group of 4 & times; 4 matrices over any finite field Fq of odd characteristic.
Adaptive procedures using t and Wilcoxon-based tests for testing for treatment difference in mixed paired and two-sample designs are studied. Previous studies have found that t-based tests perform better for normal data, while Wilcoxon-based tests perform better for nonnormal data. The proposed adaptive procedures use tail index measures to distinguish between normal and nonnormal data and select the more powerful test. A simulation study is conducted to estimate the power and type I error rate of the proposed adaptive tests. The results suggest that the adaptive tests provide an objective approach to choose the more appropriate test.
We determine the number and the exact form of the involutions (a(2) = 1), idempotents (a(2) = a), and tridempotents (a(3) = a) in Z(n), in the ring of Gaussian integers modulo n, Z(n)[i], and in the ring of Eisenstein integers modulo n, Z(n)[omega], where omega is a primitive cube root of unity. We also provide two useful characterizations of tridempotents.
We study positive solutions to the ecological model of the form ( -u ''= lambda f (u), (0, 1 ), u ( 0 ) = u ( 1 ) = 0 , where lambda is a positive parameter proportional to the square of the habitat size, and f ( u ) = - au 3 + bu 2 + u , where a and b are positive parameters. This type of reaction-diffusion equation specifically models the population density of a species living in a habitat surrounded by a hostile exterior region. In it, our reaction term, f ( u ) = - au 3 + bu 2 + u , represents a weak Allee growth of the population. This means the per-capita growth rate, f ( u )/ u , is increasing at smaller population densities and decreasing at larger ones. The variable u represents the population density that is a function of the spatial variable x . In our model, the exterior region is extremely hostile, to the point that the population on the boundary has a density of zero. In this study, we discuss the existence of a weak Allee effect for any choices of a and b as well as the structure of positive steady states using a quadrature method. Specifically, we discuss a bifurcation curve modeling this scenario. Further, we present certain numerical results that we obtain using a quadrature method and Mathematica computations.
The Towse-Campbell construction of integer matrices with integer eigenvalues is generalized for matrices over dual numbers. Some properties of eigenvalues and eigenvectors are discussed as well.
We extend the degree/diameter problem to the degree/Steiner k-diameter problem. That is, given integers k >= 2, d >= k-1, and Delta >= 1, we ask for the largest possible order of a graph with maximum degree 1 and Steiner k-diameter d. We then show that, for any tree with order n, Steiner k-diameter d, and maximum degree Delta, the order is bounded by n <= {if k = 2 and either Delta = 1 or d = 1, d + 1 if Delta = 2 and d >= 2, Delta(Delta - 1)(l) -2/Delta - 2 +(d - k l)(Delta - 1)(l) if Delta >= 3 and d >= k - 1, where l = min{[d/k - k - Delta/k(Delta - 2], [d/k]}, and establish that this bound is tight.
We investigate the impossibility of certain (n(2 )+ n + k(n+1)) configurations. Firstly, for k = 2, the result of Gropp [9] that n(2)+n/2 is even and n + 1 is a perfect square or n(2)+n/2 is odd and n - 1 is a perfect square is reproved using the incidence matrix N and analyzing the form of N-T N . Then, for all k, configurations where parallelism is a transitive property are considered. It is then analogously established that if n equivalent to 0 or n equivalent to k - 1 mod k for k even, then n(2)+n/k is even and n + 1 is a perfect square or n(2)+n/k is odd and n- (k - 1) is a perfect square. Finally, the case k = 3 is investigated in full generality.
We study the horofunction compactification of the f1-product of proper geodesic metric spaces. We provide a complete characterisation of the horofunction compactification of the product space in terms of the horofunctions of the constituent spaces, and provide a complete characterisation of the Busemann points in terms of the Busemann points of the constituent spaces. We also identify the parts of the horofunction boundary and the detour distance. The results are applied to show that the horofunction compactification of the f1-product of finite-dimensional normed spaces with polyhedral or smooth unit balls is naturally homeomorphic to the closed dual unit ball.
Cyclotomic polynomials are a classical and fundamental topic in number theory, and still an active field of research. The aim of this work is providing a formula for the family of ternary cyclotomic polynomials Phi(3 pq), where p < q are prime numbers greater than 3 such that q equivalent to +/- 1, +/- 2 mod 3 p and q > 3 p. We can derive various properties from our formula. In particular, we prove a conjecture of Zhang on the number of maximum gaps for the coefficients.
We prove that, for all constants a is an element of N, b is an element of Z, c,d is an element of R, c not equal 0, the fractions phi(an+b)/(cn+d) lie dense in the interval ]0,D] (and in [D,0[ if c<0), where D=a phi(gcd(a,b))/(cgcd(a,b)). This interval is the largest possible, since it may happen that isolated fractions lie outside of the interval: we prove a complete determination of the case where this happens, which yields an algorithm that calculates the number of n such that rad(an+b)|g for coprime a,b and any g. Furthermore, this leads to an interesting open question which is a generalization of a famous problem raised by V. Arnold. We prove that the fractions phi(an+b)/phi(cn+d) with constants a,c is an element of N, b,d is an element of Z lie dense in ]0,infinity[ exactly when ad not equal bc.
The aim of this paper is twofold. Firstly, we give easy-to-handle criteria to determine whether a given family of subsets of a vector space is a neighbourhood basis of the origin for a complete vector topology. Then, we apply these criteria to construct quite general complete topological vector spaces of measurable functions.
We compute the truncated point schemes of subalgebras of Fomin-Kirillov algebras associated with certain graphs. While Fomin-Kirillov algebras do not admit any truncated point modules, we prove a tight bound on the degrees of truncated point modules over generalized Fomin-Kirillov algebras associated with trees.
We explore properties of generalized Paley graphs and we extend a result of Lim and Praeger by providing a more precise description of the connected components of disconnected generalized Paley graphs. This result leads to a new characterization of when generalized Paley graphs are disconnected. We also provide necessary and sufficient divisibility conditions for the multiplicative group of the prime subfield of certain finite fields to be contained in the multiplicative subgroup of nonzero k-th powers. This latter result plays a crucial role in our development of a sorting algorithm on generalized Paley graphs that exploits the vector space structure of finite fields to partition certain subsets of vertices in a manner that decomposes the induced bipartite subgraph between them into complete balanced bipartite subgraphs. As a consequence, we establish a matching condition between these subsets of vertices that results in an explicit formula for the condensed Ricci curvature on certain Paley graphs and their generalizations.
The special Euclidean group on the plane SE(2) has a left-invariant sub-Riemannian structure. Every sub-Riemannian manifold possesses a Hamiltonian function governing the sub-Riemannian geodesic flow. Two natural questions are: What are the necessary conditions for sub-Riemannian geodesics to be periodic? What type of geodesics are the metric lines in SE(2)? In this article, we answer both questions, and our method to answering the second is using the Hamilton-Jacobi theory.
The bad science matrix problem consists in finding, among all matrices A ∈ℝ^n × n with rows having unit ℓ^2 norm, one that maximizes β(A) = 1/2^n∑_x ∈{-1, 1}^nAx_∞. Our main contribution is an explicit construction of an n × n matrix A showing that β(A) ≥√(log_2(n+1)), which is only 18
We investigate the standard graded k-algebras over a field k of characteristic zero for which general linear forms are exact zero divisors. We formulate a conjecture regarding the Hilbert function of such rings. We prove our conjecture in the case when the ring is a quotient of a polynomial ring by a monomial ideal, and also in the case when the ideal is generated in degree 2 and all but one of the generators are monomials.
Working in the general context of "modules with an additive dimension," we complete the determination of the minimal dimension of a faithful Alt(n)-module and classify those modules in three of the exceptional cases: 2-dimensional Alt(5)-modules in characteristic 2, 3-dimensional Alt(5)-modules in characteristic 5, and 3-dimensional Alt(6)-modules in characteristic 3. We also highlight the remaining work needed to complete the classification of the faithful Alt(n)-modules of minimal dimension for all n; these open problems seem well suited as projects for advanced undergraduate or master's students.
We study a simplified version of a density-dependent first-order mean field game, in which the players face a penalization equal to the population density at their final position. We consider the problem of finding an equilibrium when the initial distribution is a discrete measure. We show that the problem becomes finite-dimensional: the final piecewise smooth density is completely determined by the weights and positions of the initial measure. We establish existence and uniqueness of a solution using classical fixed point theorems. Finally, we show that Newton's method provides an effective way to compute the solution. Our numerical simulations provide an illustration of how density penalization in a mean field game tends to the smoothen the initial distribution.
Zero forcing is a dynamic coloring process on graphs. Initially, each vertex of a graph is assigned a color of either blue or white, and then a process begins by which blue vertices force white vertices to become blue. The zero forcing number is the cardinality of the smallest set of initially blue vertices which can force the entire graph to become blue, and the propagation time is the minimum number of steps in such a zero forcing process. In this paper we will determine the zero forcing numbers and propagation times of two infinite classes of graphs called gear graphs and helm graphs.
