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1.
A comparative study of the functional equations f( x+ y) f( x– y)= f
2( x)– f
2( y), f( y){ f( x+ y)+ f( x– y)}= f( x) f(2 y) and f( x+ y)+ f( x– y)=2 f( x){1–2 f
2( y/2)} which characterise the sine function has been carried out. The zeros of the function f satisfying any one of the above equations play a vital role in the investigations. The relation of the equation f( x+ y)+ f( x– y)=2 f( x){1–2 f
2( y/2)} with D'Alembert's equation, f( x+ y)+ f( x– y)=2 f( x) f( y) and the sine-cosine equation g( x– y)= g( x) g( y) + f( x) f( y) has also been investigated. 相似文献
2.
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 = Π p K is proved which also yields that (Π min 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
3.
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
4.
We construct for all topological space X and all n∈ N a natural section e
n
X
: G
n
X→ G
n
G
n
X of the Ganea projection : G
n
G
n
X→ G
n
X and show that the triple ( G
n
, g
n
, e
n
) is a comonad on Top
*.
Received: 6 March 2000 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 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
7.
In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O( k2+ h2) for one, two and three space dimensional second-order hyperbolic equations utt= a( x, t) uxx+ α( x, t) ux-2 η2( x, t) u, utt= a( x, y, t) uxx+ b( x, y, t) uyy+ α( x, y, t) ux+ β( x, y, t) uy-2 η2( x, y, t) u, and utt= a( x, y, z, t) uxx+ b( x, y, z, t) uyy+ c( x, y, z, t) uzz+ α( x, y, z, t) ux+ β( x, y, z, t) uy+ γ( x, y, z, t) uz-2 η2( x, y, z, t) u,0< x, y, z<1, t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O( k2) in order to obtain numerical solution of u at first time step in a different manner. 相似文献
8.
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
9.
In this paper, we study orthogonal polynomials with respect to the bilinear form ( f, g)
S
= V( f) A
V( g)
T
+ < u, f
(N)
g
(N)V(f) =(f(c
0), f "(c
0), ..., f
(n – 1)
0(c
0), ..., f(c
p
), f "(c
p
), ..., f
(n – 1)
p(c
p
))
u is a regular linear functional on the linear space P of real polynomials, c
0, c
1, ..., c
p
are distinct real numbers, n
0, n
1, ..., n
p
are positive integer numbers, N=n
0+n
1+...+n
p
, and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation. 相似文献
10.
An SOLS (self-orthogonal latin square) of order v with ni missing sub-SOLS (holes) of order hi (1 ik), which are disjoint and spanning (i.e. ∑ i=1k nihi= v), is called a frame SOLS and denoted by FSOLS( h1n1h2n2 hknk). It has been proved that for b2 and n odd, an FSOLS( anb1) exists if and only if n4 and n1+2 b/ a. In this paper, we show the existence of FSOLS( anb1) for n even and FSOLS( an1 1) for n odd. 相似文献
11.
Abstract Let d 1 : k[ X] → k[ X] and d 2 : k[ Y] → k[ Y] be k-derivations, where k[ X] ? k[ x 1,…, x n ], k[ Y] ? k[ y 1,…, y m ] are polynomial algebras over a field k of characteristic zero. Denote by d 1 ⊕ d 2 the unique k-derivation of k[ X, Y] such that d| k[X] = d 1 and d| k[Y] = d 2. We prove that if d 1 and d 2 are positively homogeneous and if d 1 has no nontrivial Darboux polynomials, then every Darboux polynomial of d 1 ⊕ d 2 belongs to k[ Y] and is a Darboux polynomial of d 2. We prove a similar fact for the algebra of constants of d 1 ⊕ d 2 and present several applications of our results. 相似文献
12.
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. 相似文献
13.
For a graph G of order | V( G)| = n and a real-valued mapping
f: V( G)?\mathbb R{f:V(G)\rightarrow\mathbb{R}}, if S ì V( G){S\subset V(G)} then f( S)=? w ? S f( w){f(S)=\sum_{w\in S} f(w)} is called the weight of S under f. The closed ( respectively, open) neighborhood sum of f is the maximum weight of a closed (respectively, open) neighborhood under f, that is, NS[ f]=max{ f( N[ v])| v ? V( G)}{NS[f]={\rm max}\{f(N[v])|v \in V(G)\}} and NS( f)=max{ f( N( v))| v ? V( G)}{NS(f)={\rm max}\{f(N(v))|v \in V(G)\}}. The closed ( respectively, open) lower neighborhood sum of f is the minimum weight of a closed (respectively, open) neighborhood under f, that is, NS-[ f]=min{ f( N[ v])| v ? V( G)}{NS^{-}[f]={\rm min}\{f(N[v])|v\in V(G)\}} and NS-( f)=min{ f( N( v))| v ? V( G)}{NS^{-}(f)={\rm min}\{f(N(v))|v\in V(G)\}}. For
W ì \mathbb R{W\subset \mathbb{R}}, the closed and open neighborhood sum parameters are NSW[ G]=min{ NS[ f]| f: V( G)? W{NS_W[G]={\rm min}\{NS[f]|f:V(G)\rightarrow W} is a bijection} and NSW( G)=min{ NS( f)| f: V( G)? W{NS_W(G)={\rm min}\{NS(f)|f:V(G)\rightarrow W} is a bijection}. The lower neighbor sum parameters are NS-W[ G]=max NS-[ f]| f: V( G)? W{NS^{-}_W[G]={\rm max}NS^{-}[f]|f:V(G)\rightarrow W} is a bijection} and NS-W( G)=max NS-( f)| f: V( G)? W{NS^{-}_W(G)={\rm max}NS^{-}(f)|f:V(G)\rightarrow W} is a bijection}. For bijections f: V( G)? {1,2,?, n}{f:V(G)\rightarrow \{1,2,\ldots,n\}} we consider the parameters NS[ G], NS( G), NS
−[ G] and NS
−( G), as well as two parameters minimizing the maximum difference in neighborhood sums. 相似文献
14.
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
15.
Let k ? K be an extension of fields, and let A ? M n ( K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K n which are invariant under A. The space A ( m, n), whose R-rational points are A-invariant, free rank m summands of R n , is well known. We construct a distinct parameter space, A ( m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection A ( m, n) ∩ A ( m, n), which we denote by A ( m, n). Under suitable hypotheses on A, we construct affine open subschemes of A ( m, n) and A ( m, n) which cover their K-rational points. We conclude by using A ( m, n), A ( m, n), and A ( m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces. 相似文献
16.
Let Γ be a distance-regular graph of diameter d ≥ 3 with c
2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
|
(SC)
m
: For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them.
|
|
(BB)
m
: Let (x, y, z) be a triple of vertices with ∂
Γ
(x, y) = 1 and ∂
Γ
(x, z) = ∂
Γ
(y, z) = m. Then B(x, z) = B(y, z).
|
|
(CA)
m
: Let (x, y, z) be a triple of vertices with ∂
Γ
(x, y) = 2, ∂
Γ
(x, z) = ∂
Γ
(y, z) = m and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z).
|
Suppose that the condition ( SC)
m
holds. Then it has been known that the condition ( BB)
i
holds for all i with 1 ≤ i ≤ m. Similarly we can show that the condition ( CA)
i
holds for all i with 1 ≤ i ≤ m. In this paper we prove that if the conditions ( BB)
i
and ( CA)
i
hold for all i with 1 ≤ i ≤ m, then the condition ( SC)
m
holds. Applying this result we give a sufficient condition for the existence of a dual polar graph as a strongly closed subgraph
in Γ. 相似文献
17.
The Padé table of 2 F 1( a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z -, the denominator polynomial Q mn ( z) in the [ m/ n] Padé approximant P mn ( z)/ Q mn ( z) for 2 F 1( a, 1; c; z) and the remainder term Q mn ( z) 2 F 1( a, 1; c; z)- Pmn ( z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of Pmn ( z)/ Qmn ( z) lie on the cut (1,∞). We deduce that the sequence of approximants Pmn ( z)/ Qmn ( z) converges to 2 F 1( a, 1; c; z) as m → ∞, n/ m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc | z| < 1 for c > a > 0. 相似文献
18.
We study sums and products in a field. Let F be a field with ch(F) ≠ 2, where ch(F) is the characteristic of F. For any integer k ? 4, we show that any x ∈ F can be written as a1 + … + ak with a1, …, ak ∈ F and a1… ak = 1, and that for any α ∈ F {0} we can write every x ∈ F as a1 … ak with a1, …, ak ∈ F and a1 + … + ak = α. We also prove that for any x ∈ F and k ∈ {2, 3, …} there are a1, …, a2k ∈ F such that a1 + … + a2k = x = a1 … a2k. 相似文献
19.
Let F be a distribution and let f be a locally summable function. The distribution F( f) is defined as the neutrix limit of the sequence { F n ( f)}, where F n ( x) = F( x) * δ n ( x) and {δ n ( x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ( x). The composition of the distributions x ?s ln m | x| and x r is proved to exist and be equal to r m x ?rs ln m | x| for r, s, m = 2, 3…. 相似文献
20.
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
|