共查询到20条相似文献,搜索用时 31 毫秒
1.
G. V. Rozenblyum 《Mathematical Notes》1977,21(3):222-227
The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L2(Rm) for m?3. The asymptotic formula $$N(\lambda ,V) \sim \Upsilon _m \int {(\lambda - V(x))_ + ^{{m \mathord{\left/ {\vphantom {m {2_{dx} }}} \right. \kern-\nulldelimiterspace} {2_{dx} }}} ,} \lambda \to ---0,$$ is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = Vo + V1; Vo < 0, ¦Vo =o (¦Vo¦3/2) as ¦x¦ → ∞; σ (t/2, Vo) ?cσ (t. Vo) and V1∈Lm/2,loc, σ(t, V1) =o (σ (t, Vo)), where σ (t,f)= mes {x:¦f (x) ¦ > t). 相似文献
2.
D. Li 《Journal of Mathematical Sciences》2010,171(1):116-129
We consider a class of nonlinear recurrent systems of the form \( {\Lambda_p} = \frac{1}{p}\sum\limits_{{p_1} = 1}^{p - 1} {f\left( {\frac{{{p_1}}}{p}} \right){\Lambda_{{p_1}}}{\Lambda_{p - {p_1}}}} \), p > 1, where f is a given function on the interval [0, 1] and Λ1 = x is an adjustable real-valued parameter. Under some suitable assumptions on the function f, we show that there exists an initial value x * for which Λ p = Λ p (x * ) → const as p → ∞. More precise asymptotics of Λ p is also derived. 相似文献
3.
A. A. Illarionov 《Proceedings of the Steklov Institute of Mathematics》2017,299(1):96-108
Functional equations of the form f(x + y)g(x ? y) = Σ j=1 n α j (x)β j (y) as well as of the form f1(x + z)f2(y + z)f3(x + y ? z) = Σ j=1 m φ j (x, y)ψ j (z) are solved for unknown entire functions f, g,α j , β j : ? → ? and f1, f2, f3, ψ j : ? → ?, φ j : ?2 → ? in the cases of n = 3 and m = 4. 相似文献
4.
A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20. 相似文献
5.
Jeffrey D. Vaaler 《Journal of Number Theory》1977,9(1):71-78
Let {xn} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σn=1Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t0 for which A(N)?1 Σn=1Na(n)e?it0xn → 0 as N → ∞. In case f(s) = Σn=1∞a(n)e?sxn, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)?1f(σ + it0) → 0 as σ → 0 + if and only if A(N)?1 Σn=1Na(n)e?it0xn → 0 as N → ∞. 相似文献
6.
G. I. Gusev 《Mathematical Notes》1975,17(2):142-147
Let Nα denote the number of solutions to the congruence F(xi,..., xm) ≡ 0 (mod pα) for a polynomial F(xi,..., xm) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?n(x, y) + ?n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F3(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t. 相似文献
7.
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
8.
R.S. Singh 《Journal of multivariate analysis》1976,6(1):111-122
On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators fn(p) of f(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on Rm, p = (p1,…,pm), pj ≥ 0, fixed integers, and for x = (x1,…,xm) in Rm, f(p)(x) = ?p1+…+pm f(x)/(?p1x1 … ?pmxm). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of fn(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of fn(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out. 相似文献
9.
In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:
f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)