Generalized matrix near-rings

Let R be a right near-ring with identity. The k×k matrix near-ring over R, Mat_k(R R), as defined by Meldrum and van der Walt, regards R as a left mod-ule over R. Let M be any faithful left R-module. Using the action of R on M, a generalized k×k matrix near-ring, Mat_k(R M), is defined. It is seen that Mat_k(R M) has many of the features of Mat_k(R R). Differences be-tween the two classes of near-rings are shown. In spe- cial cases there are relationships between Mat_k(R M) and Mat_k(R R). Generalized matrix near-rings Mat_k(R M) arise as the “right near-ring” of finite centraiizer near-rings of the form M _A{G)> where G is a finite group and A is a fixed point free automorphism group on G.     

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.     

Weighted inequalities for iterated maximal functions in Orlicz spaces

Let M be the classical Hardy‐Littlewood maximal operator. The object of our investigation in this paper is the iterated maximal function M^kf(x) = M(M^k?1f) (x) (k ≥ 2). Let Φ be a φ‐function which is not necessarily convex and Ψ be a Young function. Suppose that w is an A′_∞ weight and that k is a positive integer. If there exist positive constants C₁ and C₂ such that ((I)) then there exist positive constants C₃ and C₄ such that ((II)) where the functions a(t) and b(t) are the right derivatives of Φ(t) and Ψ(t), respectively. Conversely, if w is an A₁ weight, then (II) implies (I). Another necessary and sufficient condition will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Symplectic Action and Symplectomorphism Groups of Coadjoint Orbits

Let M be a quantizable symplectic manifold. If ψ_t is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κ_A(ψ)∊ U(1) around ψ_t(A), which is invariant under deformations of ψ, and such that κ_A(ψ) depends only on the homology class of A. Using properties of κ_A( ) we determine a lower bound for ♯π₁(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π₁(Ham(CPⁿ)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.     

Notes on flatness and the quot functor on rings

For flat modules M over a ring A we study the similarities between the three statements,dim _k (P) ( _k (P)? _A M =dfor all prime ideals P of A, the Ap-module M _p is free of rank d for all prime ideals P of A, and M is a locally free J4-module of rank d. We have particularly emphasized the case when there is an>l-algebra B, essentially of finite type, and M is a finitely generated B-module.     

The groups of points on abelian varieties over finite fields

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f _A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f _A (1 − t).     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ^ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring M_m(D(A ^ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c?Asuch that D(A[c^-1];ø)?M _m(D(A[c^-1]^ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.     

Wellposedness of abstract Cauchy problems for second order differential equations

This paper concerns the abstract Cauchy problem (ACP) for an evolution equation of second order in time. LetA be a closed linear operator with domainD(A) dense in a Banach spaceX. We first characterize the exponential wellposedness of ACP onD(A ^k+1),k teN. Next let {C(t);t teR} be a family of generalized solution operators, on [D(A ^k)] toX, associated with an exponentially wellposed ACP onD(A ^k+1). Then we define a new family {T(t); Ret>0} by the abstract Weierstrass formula. We show that {T(t)} forms a holomorphic semigroup of class (H _k) onX. Research of the second-named author was partially supported by Grant-in-Aid for Scientific Research (No. 63540139), Ministry of Education, Science and Culture.     

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by A_T the localised ring and let M be a primitive ideal of A_T Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QR_Q is generated by a regular g-centralising set of elements, then

(1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (R_Q 0 + ∣X_A (P)∣

(2)d gldim(A_T ) = Kdim(A_T ) = ht(M) = ht(P).     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

Blocks with nonabelian defect groups which have cyclic subgroups of index p

In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M _n+1(p) of order p ⁿ⁺¹ and exponent p ⁿ for n \geqslant 2{n \geqslant 2}, and where A is not necessarily a full defect p-block but its defect group P = M _n+1(p) is normal in G. The proof is independent of the classification of finite simple groups.     

On nonautonomous linear systems of differential and difference equations with -symmetric coefficient matrices

Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e^-2πi/k, d₀+d₁++d_k-1=n, and with . We say that is R-symmetric if R(t)A(t)R^-1(t)=A(t), , and we show that if A is R-symmetric then solving x^′=A(t)x or x^′=A(t)x+f(t) reduces to solving k independent d_ℓ×d_ℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations.     

Linear transformations which preserve fixed rank

Let M_{m, n}(F) denote the set of all m×n matrices over the algebraically closed field F. Let T denote a linear transformation, T:M_{m, n}(F)→M_{m, n}(F). Theorem: If max(m, n)?2k?2, k?1, and T preserves rank k matrices [i.e.?(A)=k implies ?(T(A))=k], then there exist nonsingular m×m and n×n matrices U and V respectively such that either (i) T:A→UAV for all A?M_{m, n}(F), or (ii) m=n and T:A→UA^tV for all A?M_n(F), where A^t denotes the transpose of A.     

On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow

Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.     

Matrix Near-rings over Centralizer Near-rings

For a finite group G and a subgroup A of Aut(G), let M_A(G) denote the centralizer near-ring determined by A and G. The group G is an M_A(G)-module. Using the action of M_A(G) on G, one has the n × n generalized matrix near-ring Mat_n(M_A(G);G). The correspondence between the ideals of M_A(G) and those of Mat_n(M_A(G);G) is investigated. It is shown that if every ideal of M_A(G) is an annihilator ideal, then there is a bijection between the ideals of M_A(G) and those of Mat_n(M_A(G);G).1991 Mathematics Subject Classification: 16Y30     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

Abelian Group Gradings on Full Matrix Rings

For positive integers ? and n, several authors studied ?_?-gradings of the full matrix ring M _n (k) over a field k. In this article, we show that every (G × H)-grading of M _n (k) can be constructed by a pair of compatible G-grading and H-grading of M _n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M _n (k).     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

On the diameter and radius of randon subgraphs of the cube

The n-dimensional cube Qⁿ is the graph whose vertices are the subsets of {1,…n}, with two vertices adjacent if and only if their symmetric difference is a singleton. Clearly Qⁿ has diameter and radious n. Write M = n2^n-1 = e(Qⁿ) for the size of Qⁿ. Let Q = (Q_t)_o^M be a random Qⁿ-process. Thus Q^t is a spanning subgraph of Qⁿ of size t, and Q_t is obtained from Q_t–1 by the random addition of an edge of Qⁿ not in Q_t–1, Let t^(k) = τ(Q;δ?k) be the hitting time of the property of having minimal degree at least k. We show that the diameter d_t = diam (Q_t) of Q_t in almost every Q? behaves as follows: d_t starts infinite and is first finite at time t⁽¹⁾, it equals n + 1 for t⁽¹⁾ ? t⁽²⁾ and d_t, = n for t ? t⁽²⁾. We also show that the radius of Q_t, is first finite for t = t⁽¹⁾, when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Q_t, for t = Ω(M). © 1994 John Wiley & Sons, Inc.