共查询到20条相似文献,搜索用时 15 毫秒
1.
Tauno Metsänkylä 《Journal of Number Theory》1978,10(4):510-522
Let f(x, χ) be the Iwasawa power series for the p-adic L-function Lp(s, χ), where χ is an even nonprincipal character with conductor not divisible by p2 (or by 8, when p = 2). The divisibility by p of the first p coefficients of f(x, χ) is characterized by Kummer type congruences of generalized Bernoulli numbers. Applications to Iwasawa invariants and class numbers of imaginary Abelian fields are discussed. 相似文献
2.
We address the analysis of the following problem: given a real Hölder potential f defined on the Bernoulli space and μ f its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by probabilities in periodic orbits associated to certain zeta functions. Given a Hölder function f > 0 and a value s such that 0 < s < 1, we can associate a shift-invariant probability ν s such that for each continuous function k we have $ \int {kd} v_s = \frac{{\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{x \in Fix_n } {e^{sf^n (x) - nP(f)\frac{{k^n (x)}} {n}} } } }} {{\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{x \in Fix_n } {e^{sf^n (x) - nP(f)} } } }}, $ , where P(f) is the pressure of f, Fix n is the set of solutions of σ n (x) = x, for any n ∈ ?, and f n (x) = f(x) + f(σ (x)) + … + f(σ n?1(x)). We call νs a zeta probability for f and s, because it can be obtained in a natural way from the dynamical zeta-functions. From the work of W. Parry and M. Pollicott it is known that ν s → µ f , when s → 1. We consider for each value c the potential c f and the corresponding equilibrium state μ cf . What happens with ν s when c goes to infinity and s goes to one? This question is related to the problem of how to approximate the maximizing probability for f by probabilities on periodic orbits. We study this question and also present here the deviation function I and Large Deviation Principle for this limit c → ∞, s → 1. We will make an assumption: for some fixed L we have lim c→∞, s→1 c(1 ? s) = L > 0. We do not assume here the maximizing probability for f is unique in order to get the L.D.P. 相似文献
3.
Jean Bourgain Moubariz Z. Garaev Sergei V. Konyagin Igor E. Shparlinski 《Journal d'Analyse Mathématique》2014,124(1):117-147
Recently, several bounds have been obtained on the number of solutions of congruences of the type $$({x_1} + s) \cdots ({x_v} + s) \equiv ({y_1} + s) \cdots ({y_v} + s)\not \equiv 0{\text{ (mod }}p{\text{),}}$$ where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval. 相似文献
4.
David Terman 《Journal of Differential Equations》1983,47(3):406-443
We consider the pure initial value problem for the system of equations , the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here , where H is the Heaviside step function and . This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if limt→∞ s(t) = ∞. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as . 相似文献
5.
V. G. Romanov 《Siberian Mathematical Journal》2006,47(1):135-151
In the space of variables (x, t) ∈ ? n+1, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? n+1 whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s 2(x, x 0) ? pt 2, where s(x, x 0) is the distance between points x and x 0 in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x 0 can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k 0 ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k 0 and sup x∈Ω s 2(x, x 0) satisfies some smallness condition. 相似文献
6.
《Comptes Rendus Mathematique》2003,336(12):991-996
We consider the equation ut=Δϕ(u), where ϕ∈C3(0,∞) is increasing. Under the condition ν·sϕ″(s)/ϕ′(s)⩾γ for some γ>0 and ν∈{−1;1}, we prove the estimate ν·du/dt⩾−u/γt. This result improves the estimates given by M.G. Crandall and M. Pierre (in J. Funct. Anal. 45 (1982) 194–212) for this equation. To cite this article: E. Chasseigne, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
7.
Letμ be any probability measure on ? with ∫|x|dμ(x)<∞ and letμ* denote the associated Hardy and Littlewood maximal p.m., the p.m. of the Hardy and Littlewood maximal function obtained fromμ. Dubins and Gilat [6] showed thatμ* is the least upper bound, in the usual stochastic order, of the collection of p.m.’sν on ? for which there is a martingale (X t )0≤t≤1 having distributions ofX 1 and sup0≤t≤1 X t given byμ andν respectively. In this paper, a type of ‘dual representation’ is given. Specifically, letν be any p.m. on ? with lim sup x →∞x ν[x,∞)=0xν[x, ∞)=0 and finitex 0=inf{z :ν(?∞,z]0}. Then there is a ‘minimal p.m.’ν Δ which is the greatest lower bound, in the usual convex order, of the collection of p.m.’sμ on ? for which there is a martingale (X t )0≤t≤1 having distributions ofX 1 and sup0≤t≤1 X t given byμ andν respectively. To demonstrate existence and to obtain identification of these minimal p.m.’s, we use, in particular, a lattice structure on the set of p.m.’s with the convex order, and an equivalence between a convex order of p.m.’s and the stochastic order of their maximal p.m.’s. Consequences of these order results include sharp expectation-based inequalities for martingales. These martingale inequalities form a new class of ‘prophet inequalities’ in the context of optimal stopping theory. 相似文献
8.
《Indagationes Mathematicae》2005,16(3-4):461-486
Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels.We consider the conformal group Conf(V) of a simple real Jordan algebra V. The maximal degenerate representations πs (s ε ℂ) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf(V). These representations πs can be realized on a space Is of smooth functions on V. There is an invariant bilinear form ℬs on the space Is. The problem we consider is to diagonalize this bilinear form ℬs, with respect to the action of a symmetric subgroup G of the conformal group Conf(V). This bilinear form can be written as an integral involving the Berezin kernel Bv an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of Bv. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: D(ν)Bν=b(ν)Bν+1, where D(ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I−s to Is. Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V). 相似文献
9.
Francisco Thaine 《Journal of Number Theory》1985,20(2):128-142
Let p be an odd prime and suppose that for some a, b, c ? \p we have that ap + bp + cp = 0. In Part I a simple new expression and a simple proof of the congruences of Mirimanoff which appeared in his papers of 1910 and 1911 are given. As is known, these congruences have Wieferich and Mirimanoff criteria (2p ? 1 ≡ 1 mod p2 and 3p ? 1 ≡ 1 mod p2) as immediate consequences. Mirimanoff's congruences are expressed in the form of polynomial congruences , 1 ≤ m ≤ p ? 1, and these polynomials Pm(X) are characterized by means of simple relations. In Part II a complement to Kummer-Mirimanoff congruences is given under the hypothesis that p does not divide the second factor of the class number of the p-cyclotomic field. 相似文献
10.
11.
Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Rédei zeta function of L is given by ?(s; L) = Σx∈Lμ(Ô, x) ν(Ô, x)?s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Hall's φ-function for a group and Rédei's zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Rédei zeta function. 相似文献
12.
We obtain asymptotic estimates for the quantity r = log P[Tf[rang]t] as t → ∞ where Tf = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d2(·)/dx2 and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫t0 λ1 (f(s) ds(1 + o(1)) where λ1(f) is the smallest eigenvalue for the process killed at ±f. 相似文献
13.
Our purpose in this paper is to delineate precisely the extent to which one can make explicit calculations involving the most basic linear feedback systems. Our results center around the Galois theory of the “root-locus” equation p(s) + kq(s) = 0 and the Lie symmetries associated with the related differential equation p(D)x + k(t)q(D)x = 0, D = d/dt. We show that the Galois theory leads to a more refined classification, but that these theories are related in a substantial way. Considerable insight into this is obtained through the study of the monodromy group associated with algebraic curve defined by p(s) + kq(s) = 0. 相似文献
14.
Let jνk, yνk and cνk denote the kth positive zeros of the Bessel functions Jν(x), Yν(x) and of the general cylinder function Cν(x), respectively. We show, among other things, that, for k = 2, 3,… and 0 < ν < ∞, cνk is a concave function of ν, cνk > ν + c0k and decreases as ν increases. In the cases of jνk and yνk, these results hold also for k = 1. 相似文献
15.
Christian Berg Helle Bjerg Petersen 《Journal of Fourier Analysis and Applications》2013,19(4):762-776
We show that the q-Digamma function ψ q for 0<q<1 appears in an iteration studied by Berg and Durán. This is connected with the determination of the probability measure ν q on the unit interval with moments $1/\sum_{k=1}^{n+1} (1-q)/(1-q^{k})$ , which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure ν q can be expressed in terms of the q-Digamma function. It is shown that ν q has a continuous density on ]0,1], which is piecewise C ∞ with kinks at the powers of q. Furthermore, (1?q)e ?x ν q (e ?x ) is a standard p-function from the theory of regenerative phenomena. 相似文献
16.
A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains
P.J. McKenna 《Journal of Functional Analysis》2007,244(1):220-246
A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems −Δu=f(x,u) on a bounded domain Ω⊂Rn with u=0 on ∂Ω are studied, where the nonlinearity 0?f(x,s) grows at most like sp. If Ω is a Lipschitz domain we exhibit two exponents p* and p*, which depend on the boundary behavior of the Green function and on the smallest interior opening angle of ∂Ω. We prove that for 1<p<p* all positive very weak solutions are a priori bounded in L∞. For p>p* we construct a nonlinearity f(x,s)=a(x)sp together with a positive very weak solution which does not belong to L∞. Finally we exhibit a class of domains for which p*=p*. For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49-81]. 相似文献
17.
Let be a family of elliptic differential operators with unbounded coefficients defined in RN+1. In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator G:=A−Ds generates a semigroup of positive contractions (Tp(t)) in Lp(RN+1,ν) for every 1?p<+∞, where ν is an infinitesimally invariant measure of (Tp(t)). Here, under some additional conditions on the growth of the coefficients of A, which cover also some growths with an exponential rate at ∞, we provide two different cores for the infinitesimal generator Gp of (Tp(t)) in Lp(RN+1,ν) for p∈[1,+∞), and we also give a partial characterization of D(Gp). Finally, we extend the results so far obtained to the case when the coefficients of the operator A are T-periodic with respect to the variable s for some T>0. 相似文献
18.
Zhong-hua Li 《Journal of Number Theory》2010,130(2):213-231
Let K be a field of characteristic 0, and R be a commutative K-algebra. Let Φ(x0,x1) be an element in R《x0,x1》 with regularized double shuffle relations. We define a gamma series ΓΦ(s)∈1+s2R?s? associated to Φ. We prove that the associated beta series is just the image of ΦY(x0,x1) in the commutative formal power series ring R?x0,x1?, where if Φ=1+Φ0x0+Φ1x1, then ΦY=1+Φ1x1. We also give some equivalent conditions for the reflection formula of the gamma series ΓΦ(s). 相似文献
19.
Let X be a convex subset of a finite-dimensional real vector space. A function M: X k → X is called a strict mean value, if M(x1,…, xk) lies in the convex hull of x1,…, xk), but does not coincide with one of its vertices. A sequence (xn)n∈ ? in X is called M-recursive if xn+k = M(xn, xn+1,…, xn+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05. 相似文献
20.
We obtain upper bounds for character sums modulo a composite number over sets of numbers with missing digits in a number system. We derive results on the solvability of congruences of the form x 1 ? x t ≡ λ (mod m) in the numbers with missing digits and also asymptotic formulas for the number of solutions. 相似文献