共查询到20条相似文献,搜索用时 31 毫秒
1.
Qiaoling Wang Changyu Xia 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,69(1):171-175
In this paper, generalizing an earlier result by Payne–Rayner, we prove an isoperimetric lower bound for the first eigenvalue
of the Laplacian in the fixed membrane problem on a compact minimal surface in a Euclidean space R
n
with weakly connected boundary. We also prove an isoperimetric upper bound for the first eigenvalue of the Laplacian of an
embedded closed hypersurface in R
n
. 相似文献
2.
In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ
n
which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal
vector fields in ℍ
n
.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean
curvature (CMC) hypersurface. Our definition coincides with previous ones.
Our main result describes which are the CMC hypersurfaces of revolution in ℍ
n
.The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential
equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart
in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean
space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ
n
. 相似文献
3.
We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is
fully non linear and naturally arises in the study of real hypersurfaces in ℂn+1, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real
hypersurface ℂn+1 and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate
elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques
such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem
we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness
of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface. 相似文献
4.
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler’s constant γ and its alternating analog ln (4/π), and on the other hand the infinite products of the first author for e, of the second author for π, and of Ser for e
γ
. We obtain new double integral and infinite product representations of many classical constants, as well as a generalization
to Lerch’s transcendent of Hadjicostas’s double integral formula for the Riemann zeta function, and logarithmic series for
the digamma and Euler beta functions. The main tools are analytic continuations of Lerch’s function, including Hasse’s series.
We also use Ramanujan’s polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations
between certain dilogarithm values.
相似文献
5.
Some inequalities for Gaussian processes and applications 总被引:3,自引:0,他引:3
Yehoram Gordon 《Israel Journal of Mathematics》1985,50(4):265-289
We present a generalization of Slepian's lemma and Fernique's theorem. We show how these can be easily applied to give a new
proof, with improved estimates, of Dvoretzky’s theorem on the existence of “almost” spherical sections for arbitrary convex
bodies inR
N, while avoiding the isoperimetric inequality.
Supported by Technion V.P.R. grant #100–526, and fund for the promotion of research at the Technion #100–559. 相似文献
6.
Vivek V. Rane 《Proceedings Mathematical Sciences》2003,113(3):213-221
Euler-Maclaurin and Poisson analogues of the summations ε
a <n ≤b
χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula
in several variables. 相似文献
7.
Nestor Guillen 《Calculus of Variations and Partial Differential Equations》2009,36(4):533-546
We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve
the optimal regularity C
1,1/2. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary C
1,β
hypersurface, β > 1/2, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main
ingredients of the proof are a version of Almgren’s monotonicity formula and the optimal regularity of global solutions. 相似文献
8.
Stefano Nardulli 《Annals of Global Analysis and Geometry》2009,36(2):111-131
We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature.
We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles
which contains all the pseudo-bubbles C
2,α
-close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in
finite dimension.
Work supported by a grant from INDAM “Istituto Nazionale di Alta Matematica Francesco Severi” (Roma). 相似文献
9.
Nonholonomic systems are described by the Lagrange-D’Alembert’s principle. The presence of symmetry leads, upon the choice
of an arbitrary principal connection, to a reduced D’Alembert’s principle and to the Lagrange-D’Alembert-Poincaré reduced
equations. The case of rolling constraints has a long history and it has been the purpose of many works in recent times. In
this paper we find reduced equations for the case of a thick disk rolling on a rough surface, sometimes called Euler’s disk, using a 3-dimensional abelian group of symmetry. We also show how the reduced system can be transformed into a single second
order equation, which is an hypergeometric equation. 相似文献
10.
Axel Ferrari 《manuscripta mathematica》2007,124(3):363-390
Our purpose is to obtain a geometric formula as explicit as possible for the L
2 index of a Dirac operator over a locally symmetric space of finite volume, generalizing Arthur’s formula for the Euler–Poincaré
caracteristic (Arthur in Invent Math 97:257–290, 1989). 相似文献
11.
The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values
of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for
γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main
tools are Euler-type integrals and hypergeometric series.
2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15
The work of the second author is supported by an Alexander von Humboldt research fellowship
Dedication: To Leonhard Euler on his 300th birthday. 相似文献
12.
We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan
of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds
to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group
and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler
hydrodynamics (via Arnold’s approach) to shock formation in the multidimensional Burgers equation and the Kantorovich-Wasserstein
geometry of the space of densities. 相似文献
13.
We give an “elementary” proof of an inequality due to Maz’ya. As a prerequisite we prove an approximation property for the Hausdorff measure. We also comment on the relations between Maz’ya’s inequality, the isoperimetric inequality, and the Sobolev inequality. 相似文献
14.
We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan
of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds
to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group
and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler
hydrodynamics (via Arnold’s approach) to shock formation in the multidimensional Burgers equation and the Kantorovich-Wasserstein
geometry of the space of densities.
To Vladimir Igorevich Arnold on the occasion of his 70th birthday 相似文献
15.
Mathias Schulze 《manuscripta mathematica》2007,123(4):373-379
From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal
multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that
all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism
group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities
[Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4]. 相似文献
16.
We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function. 相似文献
17.
Various approaches are used to derive the Aronsson–Euler equations for L
∞ calculus of variations problems with constraints. The problems considered involve holonomic, nonholonomic, isoperimetric,
and isosupremic constraints on the minimizer. In addition, we derive the Aronsson–Euler equation for the basic L
∞ problem with a running cost and then consider properties of an absolute minimizer. Many open problems are introduced for
further study. 相似文献
18.
The phenomenon of “numerical extraneous roots” of Euler’s iteration has been found. By systematic searching, some polynomials
and the corresponding initial values are given, which make the fixed points of Euler’s iteration not the roots of the polynomials.
For those repelling extraneous fixed points, the adjoint dynamical types of Sullivan’s basins are also studied. Finally, the
fractal pictures are produced. 相似文献
19.
F. Carbonell K. J. Worsley L. Galan 《Annals of the Institute of Statistical Mathematics》2011,63(1):1-27
The statistical problem addressed in this paper is to approximate the P value of the maximum of a smooth random field of Wilks’s Λ statistics. So far results are only available for the usual univariate
statistics (Z, t, χ2, F) and a few multivariate statistics (Hotelling’s T
2, maximum canonical correlation, Roy’s maximum root). We derive results for any differentiable scalar function of two independent
Wishart random fields, such as Wilks’s Λ random field. We apply our results to a problem in brain shape analysis. 相似文献
20.
T. Androshchuk 《Journal of Mathematical Sciences》2007,147(4):6847-6863
We consider the Unit Root Bilinear model with a sequence of innovations given by a fractional Gaussian noise (increases of
a fractional Brownian motion). For such a model, we prove a variant of the Donsker-Prokhorov limit theorem and establish the
convergence of the model in probability to a solution of a proper stochastic differential equation with FBM. The proof is
based on a result on convergence of the Euler’s scheme with “small perturbations” for SDE with FBM, which is also proved.
Bibliography: 20 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 5–33. 相似文献