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1.
In this paper, we consider equations of the form
, where
is a function with values in the Hilbert space
, the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in
. The linear operator
generating the C
0-semigroup in the energy space
is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms. 相似文献
2.
For a set of integers
, we define a q-ary
-cycle to be an assignment of the symbols
1 through q to the integers modulo
q
n
so that every word appears on some translate of
. This
definition generalizes that of de Bruijn cycles, and opens up a multitude of questions. We address
the existence of such cycles, discuss reduced cycles (ones in which the all-zeroes string need
not appear), and provide general bounds on the shortest sequence which contains all words on
some translate of
. We also prove a variant on recent results concerning decompositions of
complete graphs into cycles and employ it to resolve the case of
completely.AMS Subject Classification: 94A55, 05C70. 相似文献
3.
Andrei Moroianu 《Annals of Global Analysis and Geometry》1997,15(3):235-242
In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold
of even complex dimension satisfies the inequality
, where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T
2×N, where T
2 is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases. 相似文献
4.
Mervyn J. Silvapulle Pranab K. Sen 《Annals of the Institute of Statistical Mathematics》1993,45(1):159-171
Consider the linear modelY=X+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let
and
be anM- and the least squares estimator of respectively which are asymptotically normal about with covariance matrices 2(X
t
X)–1 and 2(X
t
X)–1 respectively. Let the Wald-type statistics based on
and
be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (2/2). Thus, the asymptotic efficiency-robustness properties of
relative to
translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that
is efficiency-robust compared to
. The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power. 相似文献
5.
6.
Given
, a compact abelian group G and a function
, we identify the maximal (i.e. optimal) domain of the convolution
operator
(as an operator from Lp(G) to itself). This is the
largest Banach function space (with order continuous norm) into which Lp(G)
is embedded and to which
has a continuous extension, still with values
in Lp(G). Of course, the optimal domain depends on p and g. Whereas
is compact, this is not always so for the extension of
to its optimal domain.
Several characterizations of precisely when this is the case are presented. 相似文献
7.
M. F. Gamal' 《Journal of Mathematical Sciences》2004,120(5):1672-1679
A contraction T acting on a Hilbert space H is called a weak contraction if the spectrum of T does not cover the unit disk
and the operator I-T
*
T is of trace class. Operators T1:H1
H1 and T2:H2
H2 are called quasisimilar if there exist operators >X:H1
H2 and Y:H2
H1 such that T2X=XT1, YT2=T1Y, and X and Y have zero kernels and dense ranges. It is proved that if two weak contractions T1 and T2 acting on separable spaces H1 and H2 are quasisimilar, then there exists an operator X:H1
H2 such that XT1=T2X and the mapping
, where
E=clos XE for E
Lat T1, is a lattice isomorphism. An example is given of two quasisimilar weak contractions such that for any isomorphism
, its inverse is not equal to
for a (bounded) operator Y. Bibliography: 4 titles. 相似文献
8.
We realize the lamplighter group
/2
as a group defined by a 2-state automaton. We study the corresponding action of this group on a binary tree and on its boundary. The final goal is the computation for a special system of generators of the spectrum of the Markov (or the random walk) operator which is [–1,1] in this case and of the spectral measure which is a discrete measure concentrated on a dense countable set of points in [–1,1] (a new effect unseen before for Markovian operators on groups which leads to a counterexample to the Strong Atiyah Conjecture). This is done by the computation of spectra of finite-dimensional approximations of the operator and uses an idea of fractalness in a similar way it was used by Bartholdi and Grigorchuk for the computation of the spectra of some branch groups. We also obtain the asymptotic of type e–1/1–x
of the spectral measure in the neighborhood of 1 and show that Følner sets grow exponentially. 相似文献
9.
We prove the following theorem. Let m and n be any positive integers with mn, and let
be a subset of the n-dimensional Euclidean space
n
. For each i=1, . . . , m, there is a class
of subsets M
i
j
of
Tn
. Assume that
for each i=1, . . . , m, that M
i
j
is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that
and its jth component
xjB(i, j)
imply
. Then, there exists a partition
of {1, . . . , n} such that
for all i and
We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem. 相似文献
10.
Zhen-Qing Chen 《Potential Analysis》1996,5(4):383-401
Let D be an open set in d and E be a relatively closed subset of D having zero Lebesgue measure. A necessary and sufficient integral condition is given for the Sobolev spaces W
1,2 (D) and W
1,2(D\E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on
being indistinguishable (in distribution) from RBM on
. This integral condition is satisfied, for example, when E has zero (d–1)-dimensional Hausdorff measure. Therefore it is possible to delete from D a relatively closed subset E having positive capacity but nevertheless the RBM on
is indistinguishable from the RBM on
, or equivalently, W
1,2(D\E)=W1,2(D). An example of such kind is: D=2 and E is the Cantor set. In the proof of above mentioned results, a detailed study of RBMs on general open sets is given. In particular, a semimartingale decomposition and approximation result previously proved in [3] for RBMs on bounded open sets is extended to the case of unbounded open sets.Research supported in part by NSF Grant DMS 86-57483. 相似文献
11.
S. Zhang 《Algebra Universalis》1996,35(4):485-505
Let
be a variety of completely regular semigroups. Define
C
* to be the class of all completely regular semigroupsS whose least full and self-conjugate subsemigroupC
*(S) belongs to
. ThenC
* is an operator on the lattice
of varieties of completely regular semigroups. In this note we show that the order ofC
* is infinite. This fact yields that the Mal'cev project is not associative on
. We describe
(C
*)1,
andi 0, in terms of -invariant normal subgroups of the free group over a countably infinite set. The lattice theoretic properties ofC
* are also studied.Presented by W. Taylor. 相似文献
12.
Ariyadasa Aluthge 《Integral Equations and Operator Theory》1996,24(4):497-501
A bounded linear operatorT is calledp-Hyponormal if (T
*T)p(TT
*)p, 0<p1. In Aluthge [1], we studied the properties of p-hyponormal operators using the operator
. In this work we consider a more general operator
, and generalize some properties of p-hyponormal operators obtained in [1]. 相似文献
13.
The problem (P) of optimizing a linear functiond
T
x over the efficient set for a multiple-objective linear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function directiond and the set of domination directionsD, ifd
T
0 for all nonzero D, then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point
, if there is no adjacent efficient edge yielding an increase ind
T
x, then a cutting plane
is used to obtain a multiple-objective linear program (
) with a reduced feasible set and an efficient set
. To find a better efficient point, we solve the problem (Ii) of maximizingc
i
T
x over the reduced feasible set in (
) sequentially fori. If there is a
that is an optimal solution of (Ii) for somei and
, then we can choosex
i
as a current efficient point. Pivoting on the reduced feasible set allows us to find a better efficient point or to show that the current efficient point
is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.The authors would like to thank an anonymous referee, H. P. Benson, and P. L. Yu for numerous helpful comments on this paper. 相似文献
14.
It is shown that an initial-boundary-value problem for Stokes' system, in which on the boundary one prescribes the vector field of velocities
, or the stress field, or the normal component of the velocity and the tangential stresses, reduces to an initial-boundary-value problem for a system of the form
, where the operator A contains a nonlocal term (the so-called singular Green operator). For the solutions of these problems, coercive estimates in the spaces W2
l, l/2 and also estimates of the norm of the resolving operator in W2
r are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 37–48, 1987. 相似文献
15.
In this paper we characterize the situation wherein a subspaceS of a separable Hilbert state space is holdable under the abstract linear autonomous control system
, whereA is the infinitesimal generator of aC
0-semigroup of operators and whereB is a bounded linear operator mapping a Hilbert space intoX. WhenS
D(A*) is dense inS
, it is shown that a necessary (but insufficient) condition for holdability is (1):
. A stronger condition than (1) is shown to be sufficient for a type of approximate holdability. In the finite dimensional setting, (1) reduces to (A, B)-invariance, which is known to be equivalent to the existence of a (bounded) linear feedback control law which achieves holdability inS. We prove that this equivalence holds in infinite dimensions as well, whenA is bounded and the linear spacesS, B andS+ B are closed.In the unbounded case, our results are illustrated by the shift semigroup and by the heat equation on an infinite rod with distributed controls. In the bounded case, our example is an integro-differential control system.Research sponsored by the National Research Council of Canada under Grant A7271.Research sponsored by the National Research Council of Canada under Grant A4641. 相似文献
16.
Let be a triangle in
and let
be the set of its three medians. We construct interpolants to smooth functions using transfinite (or blending) interpolation on
The interpolants are of type f(1)+g(2)+h(3), where (1,2,3) are the barycentric coordinates with respect to the vertices of . Based on an error representation formula, we prove that the interpolant is the unique best L1-approximant by functions of this type subject the function to be approximated is from a certain convexity cone in C3().Received: 17 December 2003 相似文献
17.
The necessary and sufficient conditions are found for the weight function v, which provide the boundedness and compactness of the Riemann–Liouville operator R
from L
p to
. The criteria are also established for the weight function w, which guarantee the boundedness and compactness of the Weyl operator W
from
to L
q. 相似文献
18.
For self-adjoint second-order elliptic differential operators that satisfy the non-trapping condition on the n-dimensional hyperbolic space
H
n
and coincide with the operator
in a neighborhood of infinity, where is the Laplace-Beltrami operator on
H
n
,we obtain the complete asymptotic expansion of the spectral function as +.For self-adjoint operators of the form (–)
+Q
m–r,where Q
m–r is a pseudodifferential operator of order m–r that is automorphic with respect to a discrete group of isometries of the spaceH
n
whose fundamental domain has finite volume, we introduce the spectral distribution function N(),which is the analog of the integrated state density, and we find its asymptotics up to order O((n–r)/m)as +.Bibliography: 49 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 4–32, 1991. 相似文献
19.
We study the spectrum of the perturbed shift operator
on the space
assuming that the set of values of the function a(n) is finite. It is shown that if the values of a(n) are distributed on the axis Z with a uniform frequency, then the essential part of the spectrum fills a generalized lemniscate. Bibliography: 9 titles. 相似文献
20.
For a Hopf algebra
, we define the structures of differential complexes on two dual exterior Hopf algebras: (1) an exterior extension of
and (2) an exterior extension of the dual algebra
*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on
. The first differential complex is an analogue of the de Rham complex. When
* is a universal enveloping algebra of a Lie (super)algebra, the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST operator Q. We give a recursive relation that uniquely defines the operator Q. We construct the BRST and anti-BRST operators explicitly and formulate the Hodge decomposition theorem for the case of the quantum Lie algebra U
q(gl(N)). 相似文献