A comparative study of functional equations characterising sine and cosine

A comparative study of the functional equationsf(x+y)f(x–y)=f ²(x)–f ²(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

Additive Set of Idempotents in Rings

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f ∈ I(R) (e ≠ f), e + f ∈ I(R), and M(R) is additive in I(R) if for all e, f ∈ M(R)(e ≠ f), e + f ∈ I(R). In this article, the following points are shown: (1) I(R) is additive if and only if I(R) is multiplicative and the characteristic of R is 2; M(R) is additive in I(R) if and only if M(R) is orthogonal. If 0 ≠ ef ∈ I(R) for some e ∈ M(R) and f ∈ I(R), then ef ∈ M(R), (2) If R has a complete set of primitive idempotents, then R is a finite product of connected rings if and only if I(R) is multiplicative if and only if M(R) is additive in I(R).     

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).     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

Decomposition tree of a lexicographic product of binary structures

Given a set S and a positive integer k, a binary structure is a function . The set S is denoted by V(B) and the integer k is denoted by . With each subset X of V(B) associate the binary substructure B[X] of B induced by X defined by B[X](x,y)=B(x,y) for any x≠y∈X. A subset X of V(B) is a clan of B if for any x,y∈X and v∈V(B)?X, B(x,v)=B(y,v) and B(v,x)=B(v,y). A subset X of V(B) is a hyperclan of B if X is a clan of B satisfying: for every clan Y of B, if X∩Y≠0?, then X⊆Y or Y⊆X. With each binary structure B associate the family Π(B) of the maximal proper and nonempty hyperclans under inclusion of B. The decomposition tree of a binary structure B is constituted by the hyperclans X of B such that Π(B[X])≠0? and by the elements of Π(B[X]). Given binary structures B and C such that , the lexicographic product B⌊C⌋ of C by B is defined on V(B)×V(C) as follows. For any (x,y)≠(x^′,y^′)∈V(B)×V(C), B⌊C⌋((x,x^′),(y,y^′))=B(x,y) if x≠y and B⌊C⌋((x,x^′),(y,y^′))=C(x^′,y^′) if x=y. The decomposition tree of the lexicographic product B⌊C⌋ is described from the decomposition trees of B and C.     

Generating the kernel of a staircase starshaped set from certain staircase convex subsets

Let S be an orthogonal polygon in the plane. Assume that S is starshaped via staircase paths, and let K be any component of Ker S, the staircase kernel of S, where K ≠ S. For every x in S\K, define W _K (x) = {s: s lies on some staircase path in S from x to a point of K}. There is a minimal (finite) collection W(K) of W _K (x) sets whose union is S. Further, each set W _K (x) may be associated with a finite family U _K (x) of staircase convex subsets, each containing x and K, with ∪{U: U in U _K (x)} = W _K (x). If W(K) = {W _K (x ₁), ..., W _K (x _n )}, then K ⊆ V _K ≡ ∩{U: U in some family U _K (x _i ), 1 ≤ i ≤ n} ⊆ Ker S. It follows that each set V _K is staircase convex and ∪{V _k : K a component of Ker S} = Ker S.     

Upper and lower bounds of the valence-functional

For the non-negative integerg let (M, g) denote the closed orientable 2-dimensional manifold of genusg. K-realizationsP of (M, g) are geometric cell-complexes inP with convex facets such that set (P) is homeomorphic toM. ForK-realizationsP of (M, g) and verticesv ofP, val (v,P) denotes the number of edges ofP incident withv and the weighted vertex-number Σ(val(v, P)-3) taken over all vertices ofP is called valence-valuev (P) ofP. The valence-functionalV, which is important for the determination of all possiblef-vectors ofK-realisations of (M, g), in connection with Eberhard's problem etc., is defined byV(g):=min[v(P)|P is aK-realization of (M,g)]. The aim of the note is to prove the inequality 2g+1≦V(g)≦3g+3 for every positive integerg.     

Power cocentralizing generalized derivations on prime rings

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that (H(u)u − uG(u)) ⁿ = 0, for all u ∈ L, then one of the following holds: (1) there exists c ∈ U such that H(x) = xc, G(x) = cx; (2) R satisfies the standard identity s ₄ and char (R) = 2; (3) R satisfies s ₄ and there exist a, b, c ∈ U, such that H(x) = ax+xc, G(x) = cx+xb and (a − b) ⁿ = 0.     

On Isolated Submodules

Let R be a ring with identity and let M be a unital left R-module. A proper submodule L of M is radical if L is an intersection of prime submodules of M. Moreover, a submodule L of M is isolated if, for each proper submodule N of L, there exists a prime submodule K of M such that N ? K but L ? K. It is proved that every proper submodule of M is radical (and hence every submodule of M is isolated) if and only if N ∩ IM = IN for every submodule N of M and every (left primitive) ideal I of R. In case, R/P is an Artinian ring for every left primitive ideal P of R it is proved that a finitely generated submodule N of a nonzero left R-module M is isolated if and only if PN = N ∩ PM for every left primitive ideal P of R. If R is a commutative ring, then a finitely generated submodule N of a projective R-module M is isolated if and only if N is a direct summand of M.     

Vertex coloring complete multipartite graphs from random lists of size 2

Let K_s×m be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of K_s×m a list L(v) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size σ(m). In this paper we determine, for all fixed s and growing m, the asymptotic probability of the existence of a proper coloring φ, such that φ(v)∈L(v) for all v∈V(K_s×m). We show that this property exhibits a sharp threshold at σ(m)=2(s−1)m.     

Coinvariants for a coadjoint action of quantum matrices

Let K be a (algebraically closed ) field. A morphism A ⟼ g ⁻¹ Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational GL _q × GL _q -modules is a highest weight category.     

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))?Γ(M _m (E)), then n = m and |F|=|E|.     

Wave equations with time-dependent spatial operators of higher order

We study the initial-boundary value problem for ?_t²u(t,x)+A(t)u(t,x)+B(t)?_tu(t,x)=f(t,x) on [0,T]×Ω(Ω??ⁿ) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given.     

Divisibility of matrices associated with multiplicative functions

Let S={x₁,…,x_n} be a set of n distinct positive integers. For x,y∈S and y<x, we say the y is a greatest-type divisor of x in S if y∣x and it can be deduced that z=y from y∣z,z∣x,z<x and z∈S. For x∈S, let G_S(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(x_i,x_j)) denote the n×n matrix having f evaluated at the greatest common divisor (x_i,x_j) of x_i and x_j as its i,j-entry and let (f[x_i,x_j]) denote the n×n matrix having f evaluated at the least common multiple [x_i,x_j] of x_i and x_j as its i,j-entry. In this paper, we assume that S is a gcd-closed set and . We show that if f is a multiplicative function such that (f∗μ)(d)∈Z whenever and f(a)|f(b) whenever a|b and a,b∈S and (f(x_i,x_j)) is nonsingular, then the matrix (f(x_i,x_j)) divides the matrix (f[x_i,x_j]) in the ring M_n(Z) of n×n matrices over the integers. As a consequence, we show that (f(x_i,x_j)) divides (f[x_i,x_j]) in the ring M_n(Z) if (f∗μ)(d)∈Z whenever and f is a completely multiplicative function such that (f(x_i,x_j)) is nonsingular. This confirms a conjecture of Hong raised in 2004.     

On edge star sets in trees

Let A be a Hermitian matrix whose graph is G (i.e. there is an edge between the vertices i and j in G if and only if the (i,j) entry of A is non-zero). Let λ be an eigenvalue of A with multiplicity m_A(λ). An edge e=ij is said to be Parter (resp., neutral, downer) for λ,A if m_A(λ)−m_A−e(λ) is negative (resp., 0, positive ), where A−e is the matrix resulting from making the (i,j) and (j,i) entries of A zero. For a tree T with adjacency matrix A a subset S of the edge set of G is called an edge star set for an eigenvalue λ of A, if |S|=m_A(λ) and A−S has no eigenvalue λ. In this paper the existence of downer edges and edge star sets for non-zero eigenvalues of the adjacency matrix of a tree is proved. We prove that neutral edges always exist for eigenvalues of multiplicity more than 1. It is also proved that an edge e=uv is a downer edge for λ,A if and only if u and v are both downer vertices for λ,A; and e=uv is a neutral edge if u and v are neutral vertices. Among other results, it is shown that any edge star set for each eigenvalue of a tree is a matching.     

[a,b]-factorizations of graphs

Let a and b be integers with b ? a ? 0. A graph G is called an [a,b]-graph if a ? d_G(v) ? b for each vertex v ∈ V(G), and an [a,b]-factor of a graph G is a spanning [a,b]-subgraph of G. A graph is [a,b]-factorable if its edges can be decomposed into [a,b]-factors. The purpose of this paper is to prove the following three theorems: (i) if 1 ? b ? 2a, every [(12a + 2)m + 2an,(12b + 4)m + 2bn]-graph is [2a, 2b + 1]-factorable; (ii) if b ? 2a ?1, every [(12a ?4)m + 2an, (12b ?2)m + 2bn]-graph is [2a ?1,2b]-factorable; and (iii) if b ? 2a ?1, every [(6a ?2)m + 2an, (6b + 2)m + 2bn]-graph is [2a ?1,2b + 1]-factorable, where m and n are nonnegative integers. They generalize some [a,b]-factorization results of Akiyama and Kano [3], Kano [6], and Era [5].     

A-acceptability of derivatives of rational approximations to EXP(Z)

The question of A-acceptability in regard to derivatives of R_m/n, the [m/n] Padé approximation to the exponential, is examined for a range of values of m and n. It is proven that R′_{n − 1/n}, R′_n/n, R′_{n + 1/n}and R″_n/n are A-acceptable and that numerous other choices of m and n lead to non-A-acceptability. The results seem to indicate that the A-acceptability pattern of R_m/n^(k) displays an intriguing generalization of the Wanner-Hairer-Nørsett theorem on the A-acceptability of R_m/n.     

A constructive approach to minimal projections in Banach spaces

Let X be a Banach space and Y a finite-dimensional subspace of X. Let P be a minimal projection of X onto Y. It is shown (Theorem 1.1) that under certain conditions there exist sequences of finite-dimensional “approximating subspaces” X_m and Y_m of X with corresponding minimal projections P_m: X_m → Y_m, such that lim_m→∞ P_m = P. Moreover, a certain related sequence of projections i_m○P_m○π_m: X → Y has cluster points in the strong operator topology, each of which is a minimal projection of X onto Y. When X = C[a, b] the result reduces to a theorem of [7.]. It is shown (Corollary 1.11) that the hypothesis of Theorem 1.1 holds in many important Banach spaces, including C[a, b], L^P[a, b] and l^P for 1 p < ∞, and c₀, the space of sequences converging to zero in the sup norm.     

Nearly m-embedded subgroups and solvability of finite groups

A subgroup A of a finite group G is called {1≤G}-embedded in G if for each two subgroups K≤H of G, where K is a maximal subgroup of H, A either covers the pair (K,H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1≤G}-embedded subgroup C such that G?=?HT and H∩T≤C≤H. In this paper, we mainly prove that G is solvable if and only if its Sylow 3-subgroups, Sylow 5-subgroups and Sylow 7-subgroups are nearly m-embedded in G.