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1.
Let g>1 be an integer and sg(m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sg(n) and sg(kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sg(n)=sg(kn). In the second part of the paper, we show that for any K>0 the set of the integers n satisfying sg(n)?Ksg(kn) for all k∈N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt. 相似文献
2.
E. Fouvry 《Monatshefte für Mathematik》2006,58(1):117-135
Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that
?n £ x f(n)=o(x)\sum_{n\le x} f(n)=o(x)
. We propose sufficient conditions on the function f to deduce the equality
?n £ x f(s(n))=o(x)\sum_{n\le x} f(s(n))=o(x)
. Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))α, where α is a positive real number. 相似文献
3.
E. Fouvry 《Monatshefte für Mathematik》2006,147(2):117-135
Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that
. We propose sufficient conditions on the function f to deduce the equality
. Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))α, where α is a positive real number. 相似文献
4.
Janos Galambos 《Journal of Number Theory》1977,9(3):338-341
Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N?(N) → ∞ as N → + ∞. Let NνN(n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g(n; N, ?, z) be the number of those prime divisors p of n which satisfy NZ?(N) ? p ? N?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limvN(n : g(n; N, ?, z) ? g(n + 1; N, ?z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive. 相似文献
5.
Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L* consists of all permutations f∈L? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L? and L* coincide. 相似文献
6.
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
7.
Partial words, which are sequences that may have some undefined positions called holes, can be viewed as sequences over an extended alphabet A?=A∪{?}, where ? stands for a hole and matches (or is compatible with) every letter in A. The subword complexity of a partial word w, denoted by pw(n), is the number of distinct full words (those without holes) over the alphabet that are compatible with factors of length n of w. A function f:N→N is (k,h)-feasible if for each integer N≥1, there exists a k-ary partial word w with h holes such that pw(n)=f(n) for all n such that 1≤n≤N. We show that when dealing with feasibility in the context of finite binary partial words, the only affine functions that need investigation are f(n)=n+1 and f(n)=2n. It turns out that both are (2,h)-feasible for all non-negative integers h. We classify all minimal partial words with h holes of order N with respect to f(n)=n+1, called Sturmian, computing their lengths as well as their numbers, except when h=0 in which case we describe an algorithm that generates all minimal Sturmian full words. We show that up to reversal and complement, any minimal Sturmian partial word with one hole is of the form ai?ajbal, where i,j,l are integers satisfying some restrictions, that all minimal Sturmian partial words with two holes are one-periodic, and that up to complement, ?(aN−1?)h−1 is the only minimal Sturmian partial word with h≥3 holes. Finally, we give upper bounds on the lengths of minimal partial words with respect to f(n)=2n, showing them tight for h=0,1 or 2. 相似文献
8.
Richard Avery 《Journal of Mathematical Analysis and Applications》2003,277(2):395-404
We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem, (g(u′))′+a(t)f(u)=0, u(0)=0, and u(ν)=u(1), where g(v)=|v|p−2v, with p>1 and ν∈(0,1). 相似文献
9.
William D. Banks Kevin Ford Florian Luca Francesco Pappalardi Igor E. Shparlinski 《Monatshefte für Mathematik》2005,146(1):1-19
Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)r = λ(n)s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1. 相似文献
10.
Michel Mendes France 《Journal of Number Theory》1973,5(1):1-15
Suppose that the sequence (λn + nα) is equidistributed (mod 1) for every real α. It is shown that certain subsequences (λmn) are then equidistributed (mod 1). This result is applied to g-additive sequences and we show for example that if (mn) is the sequence of quadratfrei integers and if s(mn) represents the sum of the digits of mn in the decimal expansion, then the sequence (√2 s(mn)) is equidistributed (mod 1). 相似文献
11.
A total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set {1,…,k} in such a way that for any two different edges e and f their weights φ(f) and φ(e) are distinct. Here, the weight of an edge g=uv is φ(g)=ν(g)+ν(u)+ν(v), i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G.We have determined the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs. 相似文献
12.
Janne Kauhanen 《Journal of Mathematical Analysis and Applications》2006,315(2):656-665
Let f∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1)∈L1(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t)=∞, then there exists f such that K1/(n−1)/α(K)∈L1(Ω) and Jf vanishes in a set of positive measure. 相似文献
13.
Pak Tung Ho 《Mathematische Annalen》2010,348(2):319-332
Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N
n
, g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=Rm¥(g)=R(g)-2Dglog f-|?glog f|g2{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N
n
, g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane
\mathbbC{\mathbb{C}} or the cylinder
\mathbbR×\mathbbS1{\mathbb{R}\times\mathbb{S}^1}. 相似文献
14.
We give sufficient conditions for a positive-definite function to admit decomposition into a sum of positive-definite functions which are compactly supported within disks of increasing diameters Ln. More generally we consider positive-definite bilinear forms f→v(f,f) defined on . We say v has a finite range decomposition if v can be written as a sum v=∑Gn of positive-definite bilinear forms Gn such that Gn(f,g)=0 when the supports of the test functions f,g are separated by a distance greater or equal to Ln. We prove that such decompositions exist when v is dual to a bilinear form φ→∫2|Bφ| where B is a vector valued partial differential operator satisfying some regularity conditions. 相似文献
15.
James Demmel 《Journal of Pure and Applied Algebra》2007,209(1):189-200
This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S={x∈Rn:g1(x)≥0,…,gs(x)≥0} modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)>0 on S; furthermore, when the KKT ideal is radical, we argue that f(x) can be represented as a sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)≥0 on S. This is a generalization of results in [J. Nie, J. Demmel, B. Sturmfels, Minimizing polynomials via sum of squares over the gradient ideal, Mathematical Programming (in press)], which discusses the SOS representations of nonnegative polynomials over gradient ideals. 相似文献
16.
For a set A of nonnegative integers the representation functions R2(A,n), R3(A,n) are defined as the number of solutions of the equation n=a+a′,a,a′∈A with a<a′, a?a′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A0 be the set of all nonnegative integers a with even D(a) and A1 be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R2(A,n)=R2(N?A,n) for all n?2N−1, then R2(A,n)=R2(N?A,n)?1 for all n?12N2−10N−2 except for A=A0 or A=A1; (b) if R3(A,n)=R3(N?A,n) for all n?2N−1, then R3(A,n)=R3(N?A,n)?1 for all n?12N2+2N. Several problems are posed in this paper. 相似文献
17.
Florian Luca 《Monatshefte für Mathematik》2005,233(2):239-256
In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (an − 1,bn − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f1(x),g(x),g1(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality
gcd (f(n)an+g(n), f1(n)bn+g1(n)) < exp(ne){\rm gcd}\, (f(n)a^n+g(n), f_1(n)b^n+g_1(n)) < \exp(n\varepsilon)
holds for all but finitely many positive integers n. 相似文献
18.
In this paper we investigate the problem of the equiconvergence on T N = [-π, π) N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ L p (T N ) and g ∈ L p (? N ), where p > 1, N ≥ 3, g(x) = f(x) on T N , in the case when the “rectangular partial sums” of the indicated expansions, i.e.,– n (x; f) and J α(x; g), respectively, have indices n ∈ ? N and α ∈ ? N (n j = [α j ], j = 1,...,N, [t] is the integer part of t ∈ ?1), in those certain components are the elements of “lacunary sequences”. 相似文献
19.
Kamal Boussaf 《Bulletin des Sciences Mathématiques》2010,134(1):44
Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)n∈N in a disk d(0,R−) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions. 相似文献
20.
The functional equationf(x,y)+g(x)h(y)F(u/1?x,ν/1?y)=f(u,ν)+g(u)h(ν)F(x/1?u,y/1?ν) ... (1) forx, y, u, ν ∈ [0, 1) andx+u,y+ν ∈ [0,1) whereg andh satisfy the functional equationφ (x+y?xy)=φ(x)φ(y)... (2) has been solved for some non-constant solution of (2) in [0, 1] withφ (0)=1,φ(1)=0 and the solution is used in characterising some measures of information. 相似文献