Some properties of the class of arithmetic functions
R. P. Pakshirajan (1963)
Annales Polonici Mathematici
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R. P. Pakshirajan (1963)
Annales Polonici Mathematici
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Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)
Czechoslovak Mathematical Journal
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Let and . Denote by the set of all integers whose canonical prime representation has all exponents being a multiple of or belonging to the arithmetic progression , . All integers in are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...
Antonio M. Oller-Marcén (2017)
Mathematica Bohemica
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A homothetic arithmetic function of ratio is a function such that for every . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of in terms of the period and the ratio of .
Melvyn B. Nathanson, Kevin O'Bryant (2015)
Acta Arithmetica
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A geometric progression of length k and integer ratio is a set of numbers of the form for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence of positive real numbers with a₁ = 1 such that the set contains no geometric progression of length k and integer ratio. Moreover, is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that...
Atsushi Moriwaki (2014)
Annales de la faculté des sciences de Toulouse Mathématiques
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In this paper, we give a numerical characterization of nef arithmetic -Cartier divisors of -type on an arithmetic surface. Namely an arithmetic -Cartier divisor of -type is nef if and only if is pseudo-effective and .
Itai Benjamini, Alain-Sol Sznitman (2008)
Journal of the European Mathematical Society
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We consider random walk on a discrete torus of side-length , in sufficiently high dimension . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time . We show that when is chosen small, as tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const . Moreover, this connected component occupies a...
Taras Banakh, Vesko Valov
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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...
Vladislav Vysotsky (2014)
Annales de l'I.H.P. Probabilités et statistiques
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Take a centered random walk and consider the sequence of its partial sums . Suppose is in the domain of normal attraction of an -stable law with . Assuming that is either right-exponential (i.e. for some and all ) or right-continuous (skip free), we prove that as , where depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.
Ghurumuruhan Ganesan (2013)
Annales de l'I.H.P. Probabilités et statistiques
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In this paper, we study the size of the giant component in the random geometric graph of nodes independently distributed each according to a certain density in satisfying . If for some positive constants , and as , we show that the giant component of contains at least nodes with probability at least for all and for some positive constant . We also obtain estimates on the diameter and number of the non-giant components of .
Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)
Studia Mathematica
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We show that, given an n-dimensional normed space X, a sequence of independent random vectors , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map defined by embeds X in with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into with asymptotically best possible relation between N, n, and ε.
Oleg Petrushov (2015)
Acta Arithmetica
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We consider the behavior of the power series as z tends to along a radius of the unit circle. If β is irrational with irrationality exponent 2 then . Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that .
Amir Dembo, Jian Ding, Fuchang Gao (2013)
Annales de l'I.H.P. Probabilités et statistiques
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Let denote the iterated partial sums. That is, , where . Assuming are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities with (and whenever is symmetric). The converse inequality holds whenever the non-zero is bounded or when it has only finite third moment and in addition is squared integrable. Furthermore, for any non-degenerate squared integrable, i.i.d., zero-mean . In contrast, we show that for any there exist integrable,...
Tomasz Łuczak (2011)
Banach Center Publications
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Let be a family of random independent k-element subsets of [n] = 1,2,...,n and let denote a family of ℓ-element subsets of [n] such that the event that S belongs to depends only on the edges of contained in S. Then, the edges of are ’weakly dependent’, say, the events that two given subsets S and T are in are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We...
Alejandro Ramírez, Vladas Sidoravicius (2004)
Journal of the European Mathematical Society
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We study a continuous time growth process on the -dimensional hypercubic lattice , which admits a phenomenological interpretation as the combustion reaction , where represents heat particles and inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...
Serguei Popov, Augusto Teixeira (2015)
Journal of the European Mathematical Society
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In this paper we establish a decoupling feature of the random interlacement process at level , . Roughly speaking, we show that observations of restricted to two disjoint subsets and of are approximately independent, once we add a sprinkling to the process by slightly increasing the parameter . Our results differ from previous ones in that we allow the mutual distance between the sets and to be much smaller than their diameters. We then provide an important application...
Júlia Komjáthy, Jason Miller, Yuval Peres (2014)
Annales de l'I.H.P. Probabilités et statistiques
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Suppose that is a finite, connected graph and is a lazy random walk on . The lamplighter chain associated with is the random walk on the wreath product , the graph whose vertices consist of pairs where is a labeling of the vertices of by elements of and is a vertex in . There is an edge between and in if and only if is adjacent to in and for all . In each step, moves from a configuration by updating to using the transition rule of and then...