M 0 Measures for the Walsh System

We show that the spaces M ₀(T) and M₀(D)M_{0}(\mathcal {D}) cannot be identified, D\mathcal {D} being the Cantor group ∏ _n=1^∞{±}. Our motivation is guided by the role M ₀ measures play in uniform distribution problems, via the Davenport, Erdòs, LeVeque theorem.     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

Subspaces where an immanant is convertible into its conjugate ∗

Let n≥5 and let be an irreducible nonlinear character of S_n such that whenever σ is a transposition or a cycle of length three; furthermore let T_n be the (0, 1)-matrix of order n that has ones exactly on and below the upper neighbours of the main diagonal and denote by E_ij the matrix of order n with 1 in position (i, j) and 0 elsewhere.

Given i,jε{1,…,n}, with i+1<j, we prove that if j?i≠3, then in the subspace M_n (T_n +E_ij there exist matrices for which the immanant is not convertible into the immanant by sign-affixing. Abusing language, we say that the space is -inconvertible, and show that spaces M_n (T_n +E₂₅ ) and M_n (T_n +E_n?3,n ). We also state some sufficient fonditions for the subspace M_n (T_n ) to be external convertible.

With some exceptions our theorems say that the coordinate subspaces found for the conversion of the permanent into the determinant by Gibson around 1970 are also best possible for other immanants.     

Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol

We perform a spectral analysis of the preconditioned Hermitian/skew‐Hermitian splitting (PHSS) method applied to multilevel block Toeplitz linear systems in which the coefficient matrix T_n(f) is associated with a Lebesgue integrable matrix‐valued function f. When the preconditioner is chosen as a Hermitian positive definite multilevel block Toeplitz matrix T_n(g), the resulting sequence of PHSS iteration matrices M_n belongs to the generalized locally Toeplitz class. In this case, we are able to compute the symbol ?(f,g) describing the asymptotic eigenvalue distribution of M_nwhen n→∞ and the matrix size diverges. By minimizing the infinity norm of the spectral radius of the symbol ?(f,g), we are also able to identify effective PHSS preconditioners T_n(g) for the matrix T_n(f). A number of numerical experiments are presented and commented, showing that the theoretical results are confirmed and that the spectral analysis leads to efficient PHSS methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

On linear operators strongly preserving invariants of Boolean matrices

Let U _k be the general Boolean algebra and T a linear operator on M _m,n (U _k ). If for any A in M _m,n (U _k ) (M _n (U _k ), respectively), A is regular (invertible, respectively) if and only if T(A) is regular (invertible, respectively), then T is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over U _k . Meanwhile, noting that a general Boolean algebra U _k is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over 169-7 _k from another point of view.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

For a general (real) parameter, let M _nbe the M-estimator and M _n ⁽¹⁾ be its one-step version (based on a suitable initial estimator M _n ⁽⁰⁾). It is known that, under certain regularity conditions, n(M _n ⁽¹⁾-M _n)=O _p(1). The asymptotic distribution of n(M _n ⁽¹⁾-M _n) is studied; it is typically non-normal and it reveals the role of the initial estimator M _n ⁽⁰⁾.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0387     

Finite Groups with S-permutable n-maximal Subgroups

Let G be a finite group and M _n (G) be the set of n-maximal subgroups of G, where n is an arbitrary given positive integer. Suppose that M _n (G) contains a nonidentity member and all members in M _n (G) are S-permutable in G. Then any of of the following conditions guarantees the supersolvability of G: (1) M _n (G) contains a nonidentity member whose order is not a prime; (2) all nonidentity members in M _n (G) are of prime order, and all cyclic members in M _n?1(G) of order 4 are S-permutable in G.     

Some Remarks on the Number of Solutions to the Equation f(X1) + … + f(Xn) = 0

Let S, T be finite sets, and let f be a function from S to T. Fix an element t in T, and let c_n denote the number of n-tuples (X₁,…,X_n) satisfying f(X₁) + … + f(X_n) = t here + denotes any binary operation on T. The sequence c₁, c₂,… satisfies a linear recurrence relation of degree at most |T|.     

Multiplicities ofp-finite modp Galois representations inJ 0 (Np)

LetM1 be an integer. LetJ ₀ (M) be the Jacobian Pic⁰(X ₀ (M)) of the modular curveX ₀ M _Q. Let T _M be the subring of End (J ₀ (M)) generated by the Hecke operatorsT _n withn1. Suppose thatp is a maximal ideal of T _M . The residue field T _M /p is a finite fieldk, whose characteristic will be denotedp. Attached top is a semisimple continuous degree-2 representation _p of Gal overk, such that _p () has characteristic polynomialX ²–T X+ (modp) for each prime numberr prime toM _p. We assume that _p is absolutely irreducible. By a result of Boston, Lenstra, and the author, the representation

Fundamental Groups of Closed Manifolds with Positive Curvature and Torus Actions

We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T^k (k large) acts effectively and isometrically. Our results are: (A) If k>(n − 3)/4 and n ≥ 17, then the fundamental group π₁(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k ≥ (n/6)+1 and n≠ 11, 15, 23, then any abelian subgroup of π₁(M) is cyclic. Moreover, if the T^k-fixed point set is empty, then π₁(M) is isomorphic to the fundamental group of a spherical 3-space form.Mathematics Subject Classification (2000). 53-XX*Supported partially by NSF Grant DMS 0203164 and by a reach found from Beijing normal university.**Supported partially by NSFC 10371008.     

Non-induced isomorphisms of matrix rings

We give examples of distinct integersi, j, and ringsT for which the matrix ringsM _i (T) andM _j (T) are isomorphic as rings, but for which the free modules _T T ⁽ⁱ⁾ and _T T ⁽ⁱ⁾ are non-isomorphic asT-modules.     

Matrix near-rings and 0-primitivity

In this article, we study the implication of the primitivity of a matrix near-ring ${\mathbb{M}_n(R) (n >1 )}${\mathbb{M}_n(R) (n >1 )} and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and \mathbbM_n(R){\mathbb{M}_n(R)} has the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)}-subgroups, then the 0-primitivity of \mathbbM_n(R){\mathbb{M}_n(R)} implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)} is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.     

On the stability of solutions to quadratic programming problems

We consider the parametric programming problem (Q _p ) of minimizing the quadratic function f(x,p):=x ^T Ax+b ^T x subject to the constraint Cx≤d, where x∈ℝ ⁿ , A∈ℝ ^n×n , b∈ℝ ⁿ , C∈ℝ ^m×n , d∈ℝ ^m , and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q _p ) are denoted by M(p) and M ^loc (p), respectively. It is proved that if the point-to-set mapping M ^loc (·) is lower semicontinuous at p then M ^loc (p) is a nonempty set which consists of at most ? _m,n points, where ? _m,n = is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones. Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000     

可换环上上三角矩阵李代数的局部自同构和局部导子

本文刻画了T_n(R)上的局部自同构和局部导子.利用关于T_n(R)的自同构和导子的主要结果和矩阵计算技巧,本文证明了T_n(R)上的每一个局部自同构是自同构,每一个局部导子是导子,这推广了文献关于T_n(R)的自同构和导子的主要结果.     

Pancyclic arcs and connectivity in tournaments

A tournament is an orientation of the edges of a complete graph. An arc is pancyclic in a tournament T if it is contained in a cycle of length l, for every 3 ≤ l ≤ |T|. Let p(T) denote the number of pancyclic arcs in a tournament T. In 4 , Moon showed that for every non‐trivial strong tournament T, p(T) ≥ 3. Actually, he proved a somewhat stronger result: for any non‐trivial strong tournament h(T) ≥ 3 where h(T) is the maximum number of pancyclic arcs contained in the same hamiltonian cycle of T. Moreover, Moon characterized the tournaments with h(T) = 3. All these tournaments are not 2‐strong. In this paper, we investigate relationship between the functions p(T) and h(T) and the connectivity of the tournament T. Let p_k(n) := min {p(T), T k‐strong tournament of order n} and h_k(n) := min{h(T), T k‐strong tournament of order n}. We conjecture that (for k ≥ 2) there exists a constant α_k> 0 such that p_k(n) ≥ α_kn and h_k(n) ≥ 2k+1. In this paper, we establish the later conjecture when k = 2. We then characterized the tournaments with h(T) = 4 and those with p(T) = 4. We also prove that for k ≥ 2, p_k(n) ≥ 2k+3. At last, we characterize the tournaments having exactly five pancyclic arcs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 87–110, 2004     

Sur l'approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifies

We study here a new kind of modified Bernstein polynomial operators on L¹(0, 1) introduced by J. L. Durrmeyer in [4]. We define for f integrable on [0, 1] the modified Bernstein polynomial M_n f: M_nf(x) = (n + 1) ∑ⁿ_{k = o}P_nk(x)∝¹₀ P_nk(t) f(t) dt. If the derivative d^r f/dx^r with r 0 is continuous on [0, 1], d^r/dx^rM_n f converge uniformly on [0,1] and sup_xε[0,1] ¦M_n f(x) − f(x)¦ 2ω_f(1/trn) if ω_f is the modulus of continuity of f. If f is in Sobolev space W_l,p(0, 1) with l 0, p 1, M_n f converge to f in w^l,p(0, 1).