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1.
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. 相似文献
2.
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 ... 相似文献
3.
Juncheol Han 《代数通讯》2013,41(9):3551-3557
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). 相似文献
4.
Marilyn Breen 《Periodica Mathematica Hungarica》2012,64(1):29-37
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. 相似文献
5.
Let S={x1,…,xn} 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 GS(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(xi,xj)) denote the n×n matrix having f evaluated at the greatest common divisor (xi,xj) of xi and xj as its i,j-entry and let (f[xi,xj]) denote the n×n matrix having f evaluated at the least common multiple [xi,xj] of xi and xj 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(xi,xj)) is nonsingular, then the matrix (f(xi,xj)) divides the matrix (f[xi,xj]) in the ring Mn(Z) of n×n matrices over the integers. As a consequence, we show that (f(xi,xj)) divides (f[xi,xj]) in the ring Mn(Z) if (f∗μ)(d)∈Z whenever and f is a completely multiplicative function such that (f(xi,xj)) is nonsingular. This confirms a conjecture of Hong raised in 2004. 相似文献
6.
Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x1,…, xn) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r1,…, rn) ∈ Rn, then one of the following holds:
There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x1,…, xn)2 is central valued on R;
There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;
There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;
R satisfies s4 and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;
There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x1,…, xn)2 is central valued on R.
7.
Thomas Guédénon 《代数通讯》2013,41(7):3523-3533
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 AT the localised ring and let M be a primitive ideal of AT Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QRQ 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 (RQ 0 + ∣XA (P)∣ (2)d gldim(AT ) = Kdim(AT ) = ht(M) = ht(P). 相似文献
8.
Yasuo Kanai 《Archive for Mathematical Logic》2000,39(2):75-87
In this paper, the following are proved:
Theorem A.
The quotient algebra
${\cal P} (\kappa )/I$
is complete if and only if the only non-trivial
I
-closed ideals extending
I
are of the form
$I\lceil A$
for some
$A\in I^+$
.
Theorem B.
If
$\kappa$
is a stationary cardinal, then the quotient algebra
${\cal P} (\kappa )/ NS_\kappa$
is not complete.
Corollary. (1) If
$\kappa$
is a weak compact cardinal, then the quotient algebra
${\cal P} (\kappa )/NS_\kappa$
is not complete.
(2) If
$\kappa$
bears
$\kappa$
-saturated ideal, then the quotient algebra
${\cal P} (\kappa )/NS_\kappa$
is not complete.
Theorem C.
Assume that
$\kappa$
is a strongly compact cardinal,
I
is a non-trivial normal
$\kappa$
-complete ideal on
$\kappa$
and
B
is an
I
-regular complete Boolean algebra. Then if
${\cal P} (\kappa )/I$
is complete, it is
B
-valid that for some
$A\subseteq\check\kappa$
,
${\cal P} (\kappa )/({\bf J}\lceil A)$
is complete, where
J
is the ideal generated by
$\check I$
in
$V^B$
.
Corollary.
Let
M
be a transitive model of
ZFC
and in
M
, let
$\kappa$
be a strongly compact cardinal and
$\lambda$
a regular uncountable cardinal less than
$\kappa$
. Then there exists a generic extension
M
[
G
] in which
$\kappa =\lambda^+$
and
$\kappa$
carries a non-trivial
$\kappa$
-complete ideal
I
which is completive but not
$\kappa^+$
-saturated.
Received: 1 April 1997 / Revised version: 1 July 1998 相似文献
9.
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. 相似文献
10.
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 相似文献
11.
Cai Mao-Cheng 《Journal of Graph Theory》1991,15(3):283-301
Let a and b be integers with b ? a ? 0. A graph G is called an [a,b]-graph if a ? dG(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]. 相似文献
12.
Vincenzo De Filippis 《Proceedings Mathematical Sciences》2010,120(3):285-297
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. 相似文献
13.
Let 1 ≤ p ≤ ∞. A subset K of a Banach space X is said to be relatively p ‐compact if there is an 〈xn 〉 ∈ ls p (X) such that for every k ∈ K there is an 〈αn 〉 ∈ lp ′ such that k = σ∞n=1 αn xn . A linear operator T: X → Y is said to be p ‐compact if T (Ball (X)) is relatively p ‐compact in Y. The set of all p ‐compact operators Kp (X, Y) from X to Y is a Banach space with a suitable factorization norm κp and (Kp , κp ) is a Banach operator ideal. In this paper we investigate the dual operator ideal (Kd p , κd p ). It is shown that κd p (T) = πp (T) for all T ∈ B (X, Y) if either X or Y is finite‐dimensional. As a consequence it is proved that the adjoint ideal of Kd p is Ip ′, the ideal of p ′‐integral operators. Further, a composition/decomposition theorem Kd p = Πp K is proved which also yields that (Πmin p )inj = Kd p . Finally, we discuss the density of finite rank operators in Kd p and give some examples for different values of p in this respect. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
Kewen Zhao 《Monatshefte für Mathematik》2009,20(1):279-293
Let G be a simple graph with n vertices. For any v ? V(G){v \in V(G)} , let N(v)={u ? V(G): uv ? E(G)}{N(v)=\{u \in V(G): uv \in E(G)\}} , NC(G) = min{|N(u) èN(v)|: u, v ? V(G){NC(G)= \min \{|N(u) \cup N(v)|: u, v \in V(G)} and
uv \not ? E(G)}{uv \not \in E(G)\}} , and NC2(G) = min{|N(u) èN(v)|: u, v ? V(G){NC_2(G)= \min\{|N(u) \cup N(v)|: u, v \in V(G)} and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any u, v ? V(G){u, v \in V(G)} , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC
2(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC
2(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC
2(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected. 相似文献
15.
We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the discrete three point boundary value problem, ?(g(?u(t-1)))+a(t))f(u(t))=0, for t∈{a+1,…,b+1} and u(a)=0 with u(v)=u(b+2) where g(v)=|v| p-2 v, p>1, for some fixed v∈{a+1,…,b+1} and σ=(b+2+v)/2 is an integer. 相似文献
16.
Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A. 相似文献
17.
For any quasiordered set (`quoset') or topological space S, the set Sub
S of all nonempty subquosets or subspaces is quasiordered by embeddability. Given any cardinal number n, denote by p
n
and q
n
the smallest size of spaces S such that each poset, respectively, quoset with n points is embeddable in Sub
S. For finite n, we prove the inequalities n + 1 ≤p
n
≤q
n
≤p
n
+ l(n) + l(l(n)), where l(n) = min{k∈ℕ∣n≤2
k
}. For the smallest size b
n
of spaces S so that Sub
S contains a principal filter isomorphic to the power set ?(n), we show n + l(n) − 1 ≤b
n
≤n + l(n) + l(l(n))+2. Since p
n
≤b
n
, we thus improve recent results of McCluskey and McMaster who obtained p
n
≤n
2. For infinite n, we obtain the equation b
n
= p
n
= q
n
= n.
Received: April 19, 1999 Final version received: February 21, 2000 相似文献
18.
Grzegorz Lewicki 《Journal of Approximation Theory》1999,97(2):593
LetX,Ybe two separable Banach spaces and letVXandWYbe finite dimensional subspaces. Suppose thatVSX,WZYand letM
(S, V),N
(Z, W). We will prove that ifαis a reasonable, uniform crossnorm onXYthenλMN(VαW,XαY)=λM(V, X) λN(W, Y).Here for any Banach spaceX,VSXandM
(S, V)
Also some applications of the above mentioned result will be presented. 相似文献
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19.
Peter Gritzmann 《Israel Journal of Mathematics》1982,43(3):237-243
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. 相似文献
20.
Given a sequence of real or complex coefficients ci and a sequence of distinct nodes ti in a compact interval T, we prove the divergence and the unbounded divergence on superdense sets in the space C(T) of the simple quadrature formulas ∝Tx(t)du(t) = Qn(x) + Rn(x) and ∝Tw(t)x(t)dt = Qn(x) + Rn(x), where Qn(x)=∑i=1mn cix(ti), ε C(T).The divergence (not certainly unbounded) for at most one continuous function of the first simple quadrature formula, with mn = n and u(t) = t, was established by P. J. Davis in 1953. 相似文献