OD-Characterization of alternating and symmetric groups of degrees 16 and 22

Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p ₁, p ₂,...,p _k }, where p ₁ < p ₂ < ... < p _k , then the degree pattern of G is defined by D(G) = (deg(p ₁), deg(p ₂),...,deg(p _k )). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H| = |G| and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A ₂₂ is OD-characterizable. We also show that the automorphism groups of the alternating groups A ₁₆ and A ₂₂, i.e., the symmetric groups S ₁₆ and S ₂₂ are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.     

Regularity and Substructures of Hom

ABSTRACT

We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom _R (A,M) is regular if and only if Ker(f) ?^⊕ A and Im(f) ?^⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom _R (A,M). We study regularity in Hom _R (A ₁ ⊕ A ₂, M ₁ ⊕ M ₂). The existence of a regular function Hom _R (A,M) implies the existence of projective summands of Hom _R (A,M) _{End R (A)} and of _{End R ( M )} Hom _R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.     

Empirical wavelet analysis of tail and memory properties of LARCH and FIGARCH models

Using computationally efficient wavelet methods, we study two nonlinear models of financial returns {r _t }: linear ARCH (LARCH) and fractionally integrated GARCH (FIGARCH). We estimate the tail index α and the long memory parameter d of the squared returns X_t = r_t²{X_t= r_t^2} of LARCH, and of the powers X _t = |r _t | ^p of FIGARCH. We find that the X _t have infinite variance and long memory, and show how the estimates of α and d depend on the model parameters. These relationships are determined empirically, as the setting is quite complex, and no suitable theory has been developed so far. In particular, we provide empirical relationships between the estimates [^(d)]{\hat d} and the difference parameters in LARCH and FIGARCH. Our computational work uncovers tail and memory properties of LARCH and FIGARCH for practically relevant parameter ranges, and provides some guidance on modeling returns on speculative assets including FX rates, stocks and market indices.     

图和线图的谱性质

Let G be a simple connected graph with n vertices and m edges,Lo be the line graph of G and λ1(LG)≥λ2 (LG)≥...≥λm(LG) be the eigenvalues of the graph LG,.. In this paper, the range of eigenvalues of a line graph is considered. Some sharp upper bounds and sharp lower bounds of the eigenvalues of Lc. are obtained. In oarticular,it is oroved that-2cos(π/n)≤λn-1(LG)≤n-4 and λn(LG)=-2 if and only if G is bipartite.     

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B _H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ? ^d , like, for example, (B ₁(|C(t)|),…, B _d (|C(t)|)) and (C ₁(|C(t)|),…, C _d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B ₁(|B ₂(…|B _n+1(t)|…)|)|) are governed by fractional diffusion equations.     

Points of continuity and quasicontinuity

Let C(f), Q(f), E(f) and A(f) be the sets of all continuity, quasicontinuity, upper and lower quasicontinuity and cliquishness points of a real function f: X → ℝ, respectively. The triplets (C(f),Q(f),A(f)), (C(f),E(f),A(f) and (Q(f),E(f),A(f)are characterized for functions defined on Baire metric spaces without isolated points.     

Coordinates and automorphisms of polynomial and free associative algebras of rank three

We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K 〈x, y, z〉 over a field K, i.e., automorphisms that fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K 〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and algorithms that recognize z-tame automorphisms and z-tame coordinates.      

Finite-Dimensional Motives and the Conjectures of Beilinson and Murre

We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the conjectures of Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring CH*(X). We show (Theorem 3) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 4), Murre's conjecture for X ^m ×X ^m (for a suitable m) implies finite-dimensionality of M(X). We also show (Theorem 7) that, for a surface X with p _g = 0, the motive M(X) is finite-dimensional if and only if the Chow group of 0-cycles of X is finite-dimensional in the sense of Mumford, i.e. iff the Bloch conjecture holds for X.The second named author is a member of GNSAGA of CNR.     

Bicommutants and ranges of derivations

Let V be a vector space, V* its dual space and L(V) the algebra of all linear operators on V. For an operator a?∈?L(V) let a* be its adjoint acting on V*, and for a subset R of L(V) let R″ be its bicommutant. If R is a left noetherian subalgebra of L(V), then {a*: a?∈?R}″?=?{a*: a?∈?R″}. When R is singly generated R″ is described precisely. Further, for any two operators a, b?∈?L(V), b?∈?(a)″ if and only if the derivations d _a and d _b satisfy d _b (F(V))???d _a (F(V)), where F(V) is the set of all finite rank operators on V. In this case the inclusion d _b (L(V))???d _a (L(V)) also holds.     

Normal Families and Uniqueness of Entire Functions and Their Derivatives

Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f（z） = a→f′（z） = a, and f′（z） = a →f^（k）（z） = a, then either f = Ce^λz ＋ a or f = Ce^λz ＋ a（λ - 1）/）λ, where C and ）, are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.     

Laws of Iterated Logarithm and Related Asymptotics for Estimators of Conditional Density and Mode

Let (X _i , Y _i ) be a sequence of i.i.d. random vectors in R with an absolutely continuous distribution function H and let g _x (y), y R denote the conditional density of Y given X = x(F), the support of F, assuming that it exists. Also let M(x) be the (unique) conditional mode of Y given X = x defined by M(x) = arg max _y (y)). In this paper new classes of smoothed rank nearest neighbor (RNN) estimators of g _x (y), its derivatives and M(x) are proposed and the laws of iterated logarithm (pointwise), uniform a.s. convergence over – < y < and x a compact C(F) and the asymptotic normality for the proposed estimators are established. Our results and proofs also cover the Nadayara-Watson (NW) case. It is shown using the concept of the relative efficiency that the proposed RNN estimator is superior (asymtpotically) to the corresponding NW type estimator of M(x), considered earlier in literature.     

Bounded edge-connectivity and edge-persistence of Cartesian product of graphs

The bounded edge-connectivity λ_k(G) of a connected graph G with respect to is the minimum number of edges in G whose deletion from G results in a subgraph with diameter larger than k and the edge-persistence D⁺(G) is defined as λ_d(G)(G), where d(G) is the diameter of G. This paper considers the Cartesian product G₁×G₂, shows λ_k1+k2(G₁×G₂)≥λ_k1(G₁)+λ_k2(G₂) for k₁≥2 and k₂≥2, and determines the exact values of D⁺(G) for G=C_n×P_m, C_n×C_m, Q_n×P_m and Q_n×C_m.     

Polynomial interpolation and sums of powers of integers

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P_k(n) and Q_k(n), such that P_k(n) = Q_k(n) = f_k(n) for n = 1, 2,…?, k, where f_k(1), f_k(2),…?, f_k(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that f_k(n) is given by the sum of powers of the first n positive integers S_k(n) = 1^k + 2^k + ??? + n^k, and show that S_k(n) admits the polynomial representations S_k(n) = P_k(n) and S_k(n) = Q_k(n) for all n = 1, 2,…?, and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S_k(n) alternative to the well-known formula of Bernoulli.     

Null ideals and spanning ranks of matrices

Let Rbe a commutative ring and A?M _m×n . The spanning rank of Ais the smallest positive integer s for which A=PQ(m×s s×n) The spanning rank of the zero matrix is set equal to zero. If Ris a field, then the spanning rank of Ais just the classical rank of A. In the first section of this paper, various theorems and examples are given which indicate how much of the classical theory of rank is still valid for spanning rank over a commutative ring. If A= PQ(n×s s×n) is a spanning rank factorization of a square matrix and D= QP, then Dis called a spanning rank partner of A. In the second part of this paper, the null ideals N _Aand N _Dof Aand Drespectively are compared. For instance, we show N _A=N _Dif s= nand N _A= XN _Dif s<nwhenever Ris a PIDand A≠0. This result sometimes (e.g. s<<n) makes the computation of N _Aeasy.     

Herz spaces and summability of Fourier transforms

A general summability method is considered for functions from Herz spaces K^α_p,r (?^d ). The boundedness of the Hardy–Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ ‐means σ^θ _T f is also bounded on the corresponding Herz spaces and σ^θ _T f → f a.e. for all f ∈ K^{–d /p} _p,∞ (?^d ). Moreover, σ^θ _T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K^{–d /p} _p,∞ (?^d ) if and only if the Fourier transform of θ is in the Herz space K^{d /p} _{p ′,1} (?^d ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted L_p spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Normal families and uniqueness of entire functions and their derivatives

In this paper we use the theory of normal families to prove: Let a and b be two complex numbers such that b ≠ a, 0, and let f be a non-constant entire function. If f and f′ share the value a CM and if f = b whenever f′ = b, then f ≡ f′. This result improves a result due to Rubel and Yang. Received: 13 June 2005     

Central Factorial Numbers and Values of Bernoulli and Euler Polynomials at Rationals

The nth order derivatives of tan x and sec x may be represented by polynomials P _n (u) and Q _n (u) in u = tan x, which are known as the derivative polynomials for the tangent and secant and have occurred in diverse contexts. In this paper, explicit representations of P _n (u) and Q _n (u) are derived in terms of the central factorial numbers of the second kind, and the values of the Bernoulli and Euler polynomials at rationals are expressed by means of these polynomials.     

On orthogonal double covers of kn and a conjecture of chung and west

A collection ?? = {G₁, G₂,…,G_n} of spanning subgraphs of K_n is called an orthogonal double cover if (i) every edge of K_n belongs to exactly two of the G_i's and (ii) any two distinct G_i's intersect in exactly one edge. Chung and West conjectured that there exists an orthogonal double cover of K_n, for all n, in which each G_i has maximum degree 2, and proved this result for n in six of the residue classes modulo 12. In another context, Ganter and Gronau showed that for n ≡ 1 mod 3, n ≠ 10, there exists an orthogonal double cover of K_n in which each G_i consists of an isolated vertex and the vertex disjoint union of K₃'s (actually these orthogonal double covers result from the solution of the directed version of the problem, which reduces to the undirected case when the orientation of the arcs is ignored). Clearly the covers of Ganter and Gronau satisfy the Chung-West requirement. In this article, the configurations of Ganter and Gronau are generalized to include the cases n ≡ 0,2 mod 3, and the results are used to provide a unified solution of the Chung-West problem. For n ≠ 5 mod 6, all the spanning subgraphs in the collection ?? are isomorphic to each other; however, this is not the case when n ≡ 5 mod 6. In addition to solving the Chung-West problem, we also go on to show that for n ≡ 2 mod 3 and n > 287, there exists an orthogonal double cover of K_n in which each spanning subgraph G_i consists of the vertex-disjoint union of an isolated vertex, and quadrilateral, and (n?5)/3 triangles. Of the 96 cases with 2 ? n ? 287, 65 cases are resolved and 31 remain open. © 1995 John Wiley & Sons, Inc.     

Strong Compactness and a Global Version of a Theorem of Ben‐David and Magidor

Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ⁺ strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, □_γ+ω fails but □_{γ+ω, ω} holds. This generalizes a result of Ben‐David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness.     

The invertibility of the difference and the sum of commuting generalized and hypergeneralized projectors

In this article, we study necessary and sufficient conditions for the invertibility of a linear combination c ₁ A ^k ?+?c ₂ B ^l , in the case when A and B are both commuting generalized or hypergeneralized projectors. We present some results relating different matrix partial orderings and the invertibility of a linear combination c ₁ A ^k ?+?c ₂ B ^l when A and B are hypergeneralized projectors.