共查询到20条相似文献,搜索用时 15 毫秒
1.
William D. Banks Derrick N. Hart Pieter Moree C. Wesley Nevans 《Monatshefte für Mathematik》2009,1(1):303-322
In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e
γ
n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets S{\mathcal {S}} of the natural numbers such that the Robin inequality holds for all but finitely many n ? S{n \in \mathcal {S}} . As a special case, we determine the finitely many numbers of the form n = a
2 + b
2 that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e
γ
log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once. 相似文献
2.
Yu-Chung Chang 《Monatshefte für Mathematik》2009,158(1):1-22
In this paper we consider a compact oriented hypersurface M
n
with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S
n+1. Denote by the trace free part of the second fundamental form of M
n
, and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B
H,m
is the square of the positive root of P
H,m
(x) = 0. We show that if M
n
is a compact oriented hypersurface immersed in the sphere S
n+1 with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M
n
is the hypersurface .
相似文献
3.
Ali Ben Amor 《Mathematische Zeitschrift》2007,255(3):627-647
Let
be a regular Dirichlet form on L
2(X,m), μ a positive Radon measure charging no sets of zero capacity and Φ an N-function. We prove that the Sobolev-Orlicz inequality(SOI)
for every
is equivalent to a capacitary-type inequality. Further we show that if
is continuously embedded into L
2(X,μ), the latter one implies some integrability condition, which is nothing else but the classical uniform integrability condition
if μ is finite. We also prove that a SOI for
yields a Nash-type inequality and if further μ = m and Φ is admissible, it yields the ultracontractivity of the corresponding semigroup. After, in the spirit of SOIs, we derive
criteria for
to be compactly embedded into L
2(μ), provided μ is finite. As an illustration of the theory, we shall relate the compactness of the latter embedding to the
discreteness of the spectrum of the time changed Dirichlet form and shall derive lower bounds for its eigenvalues in term
of Φ.
This work has been supported by the Deutsche Forschungsgemeinschaft. 相似文献
4.
Yutaka Hiramine 《Designs, Codes and Cryptography》2009,50(3):285-290
Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set = H \ {1, ω}, where H = G/N and . Using D we define a two-to-one map g from to N. The map g satisfies g(σ
m
) = g(σ)
m
and g(σ) = g(σ
−1) for any multiplier m of D and any element σ ∈ . As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.
相似文献
5.
Kewen Zhao 《Monatshefte für Mathematik》2009,156(3):279-293
Let G be a simple graph with n vertices. For any , let , and , and 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 , 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
6.
We study the semiflow defined by a semilinear parabolic equation with a singular square potential . It is known that the Hardy-Poincaré inequality and its improved versions, have a prominent role on the definition of the
natural phase space. Our study concerns the case 0 < μ ≤ μ*, where μ* is the optimal constant for the Hardy-Poincaré inequality. On a bounded domain of , we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term f(s) = λs − |s|2γ
s, with λ as a bifurcation parameter. We remark some qualitative differences of the branches in the subcritical case μ < μ* and the critical case μ = μ*. The global bifurcation result is used to show that any solution , initiating form initial data tends to the unique nonnegative equilibrium. 相似文献
7.
Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that
This inequality is a spinorial analogue of Aubin’s inequality, an important inequality in the solution of the Yamabe problem.
The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D ≠ {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with , where denotes the first positive eigenvalue of the Laplace operator. 相似文献
8.
James Kennedy 《Journal of Evolution Equations》2008,8(3):557-582
We give an extension of the Faber-Krahn inequality to the Laplacian Δ on bounded Lipschitz domains , with generalised Wentzell boundary conditions on ∂Ω, where β, γ are nonzero real constants. We prove that when β, γ > 0, the ball B minimises the first eigenvalue with respect to all Lipschitz domains Ω of the same volume as B, and that B is the unique minimiser amongst C
2-domains. We also consider β, γ not both positive, and slightly extend what is known about the associated Wentzell operator and its resolvent in addition
to considering an analogue of the Faber-Krahn inequality. This is based on the recent extension of the Faber-Krahn inequality
to the Robin Laplacian. We also give a version of Cheeger’s inequality for the Wentzell Laplacian when β, γ > 0.
相似文献
9.
Georg Hein 《Monatshefte für Mathematik》2009,257(3):233-245
We construct vector bundles on a smooth projective curve X having the property that for all sheaves E of slope μ and rank rk on X we have an equivalence: E is a semistable vector bundle
. As a byproduct of our construction we obtain effective bounds on r such that the linear system |R·Θ| has base points on U
X
(r, r(g − 1)).
相似文献
10.
Abstract This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF
(the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here is a language with a distinguished linear order <, and REF
consists of formulas of the form
where φ is an -formula, φ
<x
is the -formula obtained by restricting all the quantifiers of φ to the initial segment determined by x, and x is a variable that does not appear in φ. Our results include:
Theorem
The following five conditions are equivalent for a complete first order theory T in a countable language
with a distinguished linear order:
Moreover, if κ is a regular cardinal satisfying κ = κ
<κ
, then each of the above conditions is equivalent to:
相似文献
(1) | Some model of T has an elementary end extension with a first new element. |
(2) | T ⊢ REF . |
(3) | T has an ω 1-like model that continuously embeds ω 1. |
(4) | For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ. |
(5) | For some regular uncountable cardinal κ, T has a κ-like model that has an elementary extension in which the supremum of M exists. |
(6) | T has a κ + -like model that continuously embeds a stationary subset of κ. |
11.
Feng-Yu Wang 《Potential Analysis》2008,28(4):321-334
For μ: = e
V(x)dx a probability measure on a complete connected Riemannian manifold, we establish a correspondence between the Entropy-Information
inequality and the transportation-cost inequality for μ(f
2) = 1, where Φ and Ψ are increasing functions. Moreover, under the curvature–dimension condition, a Sobolev type HWI (entropy-cost-information)
inequality is established. As applications, explicit estimates are obtained for the Sobolev constant and the diameter of a
compact manifold, which either extend or improve some corresponding known results.
Supported in part by NNSFC(10721091) and the 973-project in China. 相似文献
12.
Jun Wu 《Monatshefte für Mathematik》2008,19(1):83-87
For an irrational number x and n ≥ 1, we denote by k
n
(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x. G. Lochs proved that for almost all x, with respect to the Lebesgue measure
In this paper, we prove that an iterated logarithm law for {k
n
(x): n ≥ 1}, more precisely, for almost all x,
for some constant σ > 0.
Author’s address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China 相似文献
13.
Kate Okikiolu 《Geometric And Functional Analysis》2008,17(5):1655-1684
Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g
Γ, denote the area element by dV and the Laplace–Beltrami operator by Δ
g
. We define the Robin mass m(x) at the point x
M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of Δ
g
−1 is then defined by trace Δ−1 = ∫
M
m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let be the Laplace–Beltrami operator on the round sphere of volume V. We show that if there exists g
Γ with trace Δ
g
−1 < trace then the minimum of trace Δ−1 over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace Δ−1 over Γ is equal to trace . In fact we prove these results in the general setting where M is an n-dimensional closed, connected manifold and the Laplace–Beltrami operator is replaced by any non-negative elliptic operator
A of degree n which is conformally covariant in the sense that for the metric g we have . In this case the role of is assumed by the Paneitz or GJMS operator on the round n-sphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the Onofri–Beckner theorem.
Received: February 2006, Accepted: March 2006 相似文献
14.
Bassam Shayya 《Mathematische Zeitschrift》2009,262(1):41-55
Suppose dμ is affine surface measure on a convex radial surface Γ(x) = (x, γ(|x|)), a ≤ |x| < b, in . Under appropriate smoothness and growth conditions on γ, we prove and Fourier restriction estimates for Γ. 相似文献
15.
It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type and defined over the finite field all have the maximum number of -rational points allowed by the Weil “explicit formulas”, and that these curves are -maximal curves over infinitely many algebraic extensions of . Serre showed that an -rational curve which is -covered by an -maximal curve is also -maximal. This has posed the problem of the existence of -maximal curves other than the Deligne–Lusztig curves and their -subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications
to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal
curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n
3 with n = p
r
> 2, p ≥ 2 prime, we give a simple, explicit construction of an -maximal curve that is not -covered by any -maximal Deligne–Lusztig curve. Furthermore, the -automorphism group Aut has size n
3(n
3 + 1)(n
2 − 1)(n
2 − n + 1). Interestingly, has a very large -automorphism group with respect to its genus .
Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni, PRIN 2006–2007. 相似文献
16.
The intersection of two Steiner triple systems and is the set . The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection satisfies and . We show that for v ≡ 1 or 3 (mod 6), |I(v)| = Θ(v
3), where previous results only imply that |I(v)| = Ω(v
2).
Received: January 23, 2006. Final Version received: September 2, 2006 相似文献
17.
Brucker et al. (Math Methods Oper Res 56: 407–412, 2003) have given an O(n
2)-time algorithm for the problems
, outtree
and
, outtree
. In this note, we show that their algorithm admits an O(n log n)-time implementation. 相似文献
18.
We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C
n
depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore,
it is clarified that the L
∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality. 相似文献
19.
Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g
λ = g(· , λ) restricted to the sphere S
n
(r) is an immersion for every 0 < r < r (λ), so that the intersection number I(g
λ|S
n
(r)) is defined. Then is an algebraically constructible function.
I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7. 相似文献
20.
In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f
j
) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t
2log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.
相似文献