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1.
Let e(x, y, ) be the spectral function and the unit spectral projection operator, with respect to the Laplace–Beltrami operator on a closed Riemannian manifold M. We generalize the one-term asymptotic expansion of e(x, x, ) by Hörmander (Acta Math.
88 (1968), 341–370) to that of
x
y
e(x,y,)|
x=y
for any multiindices , in a sufficiently small geodesic normal coordinate chart of M. Moreover, we extend the sharp (L
2,L
p) (2 p) estimates of by Sogge (J. Funct. Anal.
77 (1988), 123–134; London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, pp. 416–422) to the sharp (L
2, Sobolev L
p) estimates of . 相似文献
2.
M. Milman 《Analysis Mathematica》1978,4(3):215-223
X(Y) f -:X(Y)={fM(×): fX(Y)=f(x,.)YX< . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×). 相似文献
3.
V. N. Sudakov 《Mathematical Notes》1973,14(4):886-888
Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (x, y P (X)×P(Y), a probability measure P(F) such that
X
=
X
–1
,
Y
=
Y
–1
it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition
XA+
YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973. 相似文献
4.
Rangachari Venkataraman 《Monatshefte für Mathematik》1985,100(1):47-66
An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G
0 where 0 andG
0 is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L
1-algebra, Fourier transforms of abelian groups are given and their properties are studied. 相似文献
5.
M. B. Nathanson 《Acta Mathematica Hungarica》2000,87(3):179-195
Let A be a set of positive integers with gcd (A) = 1, and let p
A
(n) be the partition function of A. Let c
0 = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p
A
(n)/c
0 n and lim sup log p
A
(n)/c
0 n . In particular, if A has asymptotic density > 0, then log p
A
(n) c0n. Conversely, if > 0 and log p
A
(n) c
0 n, then the set A has asymptotic density . 相似文献
6.
Robert Blackburn 《Journal of Theoretical Probability》2000,13(3):825-842
For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e
iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L
x
t
denote the local time of X(t) and L*
t
=sup
xR
L
x
t
. Estimates are obtained for P(L
0
t
y) and P(L*
t
y) as y and t fixed. 相似文献
7.
In-Suk Wee 《Probability Theory and Related Fields》1990,85(4):469-488
Summary Let {X
t
} be a 1 process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x
–L
1
(x), v{yy<–x}=x
–L
2
(x) whereL
1,L
2 are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf
a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University 相似文献
8.
Helmut Karzel 《Aequationes Mathematicae》1978,17(1):83-90
A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and
–1
(x) =
–1
(x) implies
–1
=
–1
, andsharply 3-transitive, if for any two triples (x
1,x
2,x
3), (y
1,y
2,y
3) M
3 with|{x
1,x
2,x
3
}| = |{y
1,y
2,y
3
}| = 3 there is exactly one permutation with(x
1) =y
1,(x
2) =y
2,(x
3) =y
3. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic. 相似文献
9.
M. Sh. Shabozov 《Ukrainian Mathematical Journal》1995,47(11):1769-1778
For the classB
p
, 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L1. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995. 相似文献
10.
Xiang-Dong Hou 《Designs, Codes and Cryptography》1992,2(3):215-224
Let R(r, m) be the rth order Reed-Muller code of length 2
m
, and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University. 相似文献
11.
Joanna Janczewska 《Geometriae Dedicata》2002,91(1):7-21
The aim of this paper is to illustrate the use of topological degree for the study of bifurcation in von Kármán equations with two real positive parameters and for a thin elastic disk lying on the elastic base under the action of a compressing force, which may be written in the form of an operator equation F(x, , ) = 0 in some real Banach spaces X and Y. The bifurcation problem that we study is a mathematical model for a certain physical phenomenon and it is very important in the mechanics of elastic constructions. We reduce the bifurcation problem in the solution set of equation F(x, , ) = 0 at a point (0, 0, 0) X × IR
+
2 to the bifurcation problem in the solution set of a certain equation in IR
n
at a point (0, 0, 0) IR
n
× IR
+
2, where n = dim Ker F
x
(0, 0, 0) and F
x
(0, 0, 0): X Y is a Fréchet derivative of F with respect to x at (0, 0, 0). To solve the bifurcation problem obtained as a result of reduction, we apply homotopy and degree theory. 相似文献
12.
For each function f, f VMO, there exists a unique function f0, analytic in the circle
and such that f–f0=f{gVMOA}. We define the operator of best approximation (nonlinear) A, Af=f0, fVMO, In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes X, X VMO, AX X. One investigates the classes X containing unbounded functions. It is proved that if P_X is the space of the symbols of the Hankel operators from a Banach space E of functions into the Hardy space H2, then AX X. For E one can take almost any space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 5–17, 1985. 相似文献
13.
Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold
with the metricg and the second fundamental formB. The following theorems are proved. For
0 = +1 or –1,
1 = +1, –1 or 0 (2–2
0+
12n–2–2) and a positive constantk, every circlec inM withg(c, c) =
0 andg(
c
c,
c
c) =
1
k
2 is a circle in
iffM is an extrinsic sphere. For
0 = +1 or –1 (–0n–), every geodesicc inM withg(c, c) =
0 is a circle in
iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general. 相似文献
14.
J. Wu 《Monatshefte für Mathematik》1994,117(3-4):303-322
Let (n) be the number of all prime divisors ofn and (n) the number of distinct prime divisors ofn. We definev
q
(x)=|{nx(n)–(n)=q}|. In this paper, we give an asymptotic development ofv
q
(x); this improves on previous results. 相似文献
15.
Gikō Ikegami 《Inventiones Mathematicae》1989,95(2):215-246
Summary We define a constraint system
, [0,0), which is a kind of family of vector fields
on a manifold. This is a generalized version of the family of the equations
, [0,0),x
m
,y
n
. Finally, we prove a singular perturbation theorem for the system
, [0,0).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday 相似文献
16.
For a projective plane
n
of ordern, let(
n
) denote the minimum numberk, so that there is a coloring of the points of
n
ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 (
n
) 8 for every projective plane
n
of ordern.
Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation 相似文献
17.
Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.
Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - x , y normal stress in Cartesian coordinates - xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h 1, 1/h 2 stretch ratios - e x + y dilatation - 2 rotation - M bending moment 相似文献
Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - x , y normal stress in Cartesian coordinates - xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h 1, 1/h 2 stretch ratios - e x + y dilatation - 2 rotation - M bending moment 相似文献
18.
Dr. Krishna Balasundaram Athreya 《Probability Theory and Related Fields》1969,12(4):320-332
Summary Let X(t)=(X
1
(t), X
2
(t), , X
t
(t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m
ij
(t))) be the mean matrix where m
ij
(t)=E(X
j
(t)¦X
r
(0)=
ir
for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process
. It is shown that i) if 2 Re >1, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re 1 then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x)
–1
if 2 Re <1 and f(x)=(x log x)–1 if 2 Re =1, 1 the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University. 相似文献
19.
Let m= (1,..., m) denote an ordered field, where i+1>0 is infinitesimal relative to the elements of i, 0 < –i < m (by definition, 0= ). Given a system of inequalities f1 > 0, ..., fs > 0, fs+1 0, ..., fk 0, where fj m [X1,..., Xn] are polynomials such that, and the absolute value of any integer occurring in the coefficients of the fjs is at most 2M. An algorithm is constructed which tests the above system of inequalities for solvability over the real closure of m in polynomial time with respect to M, ((d)nd0)n+m. In the case m=, the algorithm explicitly constructs a family of real solutions of the system (provided the latter is consistent). Previously known algorithms for this problem had complexity of the order ofM(d d
0
m
2U(n)
.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 3–36, 1988. 相似文献
20.
For any subset A of the unit sphere of a Banach space X and for [0,2) the notion of -flatness is introduced as a measure of non-flatness of A. For any positive , construction of locally finite tilings of the unit sphere by -flat sets is carried out under suitable -renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed. 相似文献