Academic literature on the topic 'K free integers'

Let E be an elliptic curve defined over the field of rational numbers ℚ. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed(ℚ) is zero. Let E(k) denote the elliptic curve y2 = x3 + k. Then the quadratic twist E(1)d of E(1) by d is the elliptic curve [Formula: see text]. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d ≡ r (mod 24) such that rank [Formula: see text], using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E(k3)(ℚ) has rank zero.

Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to arithmetical functions associated with k-free integers and k-th powers under regular convolution.

Fix integers k , b , q with k ≥ 2 , b ≥ 0 , q ≥ 2 . Define the function p to be: p ( x ) = k x + b . We call a set S of integers \emph{ ( k , b , q ) -linear-free} if x ∈ S implies p i ( x ) ∉ S for all i = 1 , 2 , … , q − 1 , where p i ( x ) = p ( p i − 1 ( x ) ) and p 0 ( x ) = x . Such a set S is maximal in [ n ] := { 1 , 2 , … , n } if S ∪ { t } , ∀ t ∈ [ n ] ∖ S is not ( k , b , q ) -linear-free. Let M k , b , q ( n ) be the set of all maximal ( k , b , q ) -linear-free subsets of [ n ] , and define g k , b , q ( n ) = min S ∈ M k , b , q ( n ) | S | and f k , b , q ( n ) = max S ∈ M k , b , q ( n ) | S | . In this paper, formulae for g k , b , q ( n ) and f k , b , q ( n ) are proposed. Also, it is proven that there is at least one maximal ( k , b , q ) -linear-free subset of [ n ] with exactly x elements for any integer x between g k , b , q ( n ) and f k , b , q ( n ) , inclusively.

Abstract Let k k be a nonnegative integer. Let K K be a number field and O K {{\mathcal{O}}}_{K} be the ring of integers of K K . Let ℓ ≥ 5 \ell \ge 5 be a prime and v v be a prime ideal of O K {{\mathcal{O}}}_{K} over ℓ \ell . Let f f be a modular form of weight k + 1 2 k+\frac{1}{2} on Γ 0 {\Gamma }_{0} (4) such that its Fourier coefficients are in O K {{\mathcal{O}}}_{K} . In this article, we study sufficient conditions that if f f has the form f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v ) f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers s i {s}_{i} , then f f is congruent to a linear combination of iterated derivatives of a single theta function modulo v v .

AbstractThe Chowla conjecture states that if t is any given positive integer, there are infinitely many prime positive integers N such that Per() = t, where Per() is the period length of the continued fraction expansion for . C. Friesen proved that, for any k ∈ ℕ, there are infinitely many square-free integers N, where the continued fraction expansion of has a fixed period. In this paper, we describe all polynomials for which the continued fraction expansion of has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials Q such that deg Q = 2d and Per =t.

The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).

Cette thèse aborde plusieurs questions liées à la fonction somme des chiffres et aux entiers friables. Le premier chapitre est consacré à une introduction qui rassemble les origines des thèmes principaux abordés dans cette thèse, ainsi que les rappels théoriques et les notations nécessaires pour la suite du travail. Les principaux résultats obtenus au cours de cette recherche y seront également présentés. Le deuxième chapitre est consacré à l'étude des propriétés de l'ensemble ({ n leq x : n ext{ est } k ext{-libre}, , s_q(Q(n)) equiv a pmod{m} }), où ( a in mathbb{Z} ), ( k ), et ( m ) désignent des entiers naturels supérieurs ou égaux à 2. La fonction ( s_q ) représente la somme des chiffres en base ( q ), les entiers ( k )-libres sont ceux qui ne sont pas divisibles par la ( k )-ième puissance d'un nombre premier, et ( Q ) est un polynôme de degré supérieur ou égal à 2. Afin de montrer notre résultat principal, nous évaluons des sommes exponentielles du type (sum_{n leq x atop{ n ext{ est } k ext{-libre}}} e(alpha s_q(Q(n)))), où ( alpha ) est tel que ((q - 1)alpha in mathbb{R} setminus mathbb{Z}). À la fin, nous montrons un résultat d'équirépartition modulo 1. Le troisième chapitre se concentre sur l'équirépartition de Zeckendorf et la somme des chiffres des entiers friables dans des classes de congruence. Un entier est dit ( y )-friable si tous ses facteurs premiers sont inférieurs ou égaux à ( y ). Nous utiliserons systématiquement la notation ( P(n) ) pour désigner le plus grand facteur premier de ( n ), et ( S(x, y) := { n leq x : P(n) leq y } ) pour désigner l'ensemble des entiers ( y )-friables inférieurs ou égaux à ( x ). L'objectif principal de ce chapitre est d'évaluer l'ensemble ( { n in S(x, y) : s_varphi(n) equiv a pmod{m} } ), où ( a in mathbb{Z} ) et ( m ) désigne un entier naturel supérieur ou égal à 2. Ici, ( s_varphi ) est la fonction de la somme des chiffres en base Fibonacci. Comme nous le faisons dans le deuxième chapitre, pour prouver le résultat principal, nous utilisons les sommes exponentielles, ainsi, nous profiterons de la propriété de décomposition des entiers friables dans des intervalles pour nos démonstrations afin d'évaluer la somme exponentielle(sum_{n in S(x, y)} e(vartheta s_varphi(n))), où ( vartheta in mathbb{R} setminus mathbb{Z} ). Le quatrième chapitre porte sur la moyenne des sommes de certaines fonctions multiplicatives sur les entiers friables. Dans ce chapitre notre objectif est de déterminer des estimations pour les expressions suivantes : sigma_s(n) = sum_{d mid n} d^s, varphi(n) = sum_{d mid n} mu(d) n/d, et psi(n) = sum_{d mid n} mu^2(n/d) d, où ( s ) est un nombre réel non nul, lorsque n parcourt l'ensemble S(x,y). Le dernier chapitre présente une application de l'inégalité de Turán-Kubilius. Il est bien connu que cette inégalité traite des fonctions additives et qu'elle a également permis de démontrer le théorème de Hardy-Ramanujan pour la fonction additive (omega(n)), qui compte les diviseurs premiers de l'entier (n). Dans ce chapitre, nous nous déplaçons dans l'espace des entiers friables et nous nous intéressons à la fonction additive ilde{omega}(n) = sum_{p mid n atop{s_q(p) equiv a pmod{b}}} 1,où ( a in mathbb{Z} ) et ( b geq 2 ) sont des entiers. Nous fournissons une estimation de (ilde{omega}(n)), lorsque (n) parcourt l'ensemble (S(x,y)), puis nous utilisons l'inégalité de Turán-Kubilius dans l'espace des entiers friables proposée par Tenenbaum et de la Bretèche, et présentons quelques applications
This thesis addresses some questions related to the sum of digits function and friable integers. The first chapter is dedicated to an introduction that gathers the origins of the main topics covered in this thesis, as well as a background and the necessary notations for the rest of the work. The main results obtained during this research will also be presented. The second chapter focuses on the behaviour of the set ({ n leq x : n ext{ is } k ext{-free}, , s_q(Q(n)) equiv a pmod{m} }), where ( a in mathbb{Z} ), ( k ), and ( m ) are natural numbers greater than or equal to 2. The function ( s_q ) represents the sum of digits in base ( q ), ( k )-free integers are those not divisible by the ( k )-th power of a prime number, and ( Q ) is a polynomial of degree greater than or equal to 2. To show our main result, we evaluate exponential sums of the type(sum_{n leq x atop{ n ext{ is } k ext{-free}}} e(alpha s_q(Q(n)))), where ( alpha ) is a real number such that ((q - 1)alpha in mathbb{R} setminus mathbb{Z}). In the end, we establish an equidistribution result modulo 1. The third chapter, we focus on the distribution of the Zeckendorf sum of digits over friable integers in congruence classes. An integer is called ( y )-friable if all its prime factors are less than or equal to ( y ). We use the notation ( P(n) ) to denote the largest prime factor of ( n ), and ( S(x, y) := { n leq x : P(n) leq y } ) to denote the set of ( y )-friable integers less than or equal to ( x ). The main objective of this chapter is to evaluate the set ( { n in S(x, y) : s_varphi(n) equiv a pmod{m} } ), where ( a in mathbb{Z} ) and ( m ) is a natural number greater than or equal to 2. Here, ( s_varphi ) is the sum of digits function in the Fibonacci base. As in the second chapter, to prove the main result, we use exponential sums, and we utilize the property of decomposition of friable integers into intervals for our demonstration to evaluate the exponential sum(sum_{n in S(x, y)} e(vartheta s_varphi(n))), where ( vartheta in mathbb{R} setminus mathbb{Z} ). The fourth chapter deals with the average of sums of certain multiplicative functions over friable integers. In this chapter, our goal is to determine estimates for the following expressions: sigma_s(n) = sum_{d mid n} d^s, varphi(n) = sum_{d mid n} mu(d) n/d, and psi(n) = sum_{d mid n} mu^2(n/d) d, where ( s ) is a non-zero real number, when (n) runs over the set (S(x,y)). The last chapter presents an application of the Turán-Kubilius inequality. It is well known that this inequality deals with additive functions and has also been used to prove the Hardy-Ramanujan theorem for the additive function (omega(n)), which counts the prime divisors of the integer (n). In this chapter, we move into the space of friable integers and focus on the additive function ilde{omega}(n) = sum_{p mid n atop{s_q(p) equiv a pmod{b}}} 1, where ( a in mathbb{Z} ) and ( b geq 2 ) are integers. Firstly, we provide an estimate of ( ilde{omega}(n)) when (n) runs through the set (S(x,y)), we then use the Turán-Kubilius inequality in the space of friable integers established by Tenenbaum and de la Bretèche to present few applications

For a nontrivial connected and simple graphs G= (V(G), E(G)), a set S E(G) is called edge geodetic set of G if every edge of G it’s in S or is contained in a geodesic joining some pair of edges in S. The edge geodetic number eds(G) of G is the minimum order of its edge geodetic sets. We prove that it is NP-complete to decide for a given bipartiti graphs G and a given integer k whether G has a edge geodetic set of cardinality at most k. A set M V(G) is called P3 set of G if all vertices of G have two neighbors in M. The P3 number of G is the minimum order of its P3 sets. We prove that it is NP-complete to decide for a given graphs G(diamond,odd-hole) free and a given integer k whether G has a P3 set of cardinality at most k.

Suppose we are given an integer k and n boxes, labeled 1,2,…,n by an adversary, each containing a single number chosen from an unknown distribution; the n distributions not necessarily identical. We have to choose an order to sequentially open the boxes, and each time we open the next box in this order, we learn the number inside. If we reject a number in a box, the box cannot be recalled. Our goal is to accept k of these numbers, without necessarily opening all boxes, such that the accepted numbers are the k largest numbers in the boxes (thus their sum is maximized). This problem, sometimes called a free order multiple-choice secretary problem, is one of the classic examples of online decision making problems. A natural approach to solve such problems is to sample elements in random order; however, as indicated in several sources, e.g., Turan et al. NIST 2015 [35], Bierhorst et al. Nature 2018 [10], pure randomness is hard to get in reality. Thus, pseudorandomness has to be used, with a small entropy. We show that with a very small O(log log n) entropy an almost-optimal approximation of the value of k largest numbers can be selected, with only a polynomially small additive error, for k < log n / log log n. Our solution works for exponentially larger range of parameter k compared to previously known algorithms (STOC 2015 [22]). We also prove a corresponding lower bound on the entropy of optimal (and even close to optimal, with respect to competitive ratio) solutions for this problem of choosing k largest numbers, matching the entropy of our algorithm. No previous lower bound on entropy was known for this problem if k > 1.

This paper is to study free response of a spinning, cyclic symmetric rotor assembled to a flexible housing via multiple bearings. In particular, the rotor spins at a constant speed ω3, and the housing is excited via a set of initial displacements. The focus is to study ground-based response of the rotor through theoretical and numerical analyses. The paper consists of three parts. The first part is to briefly summarize an equation of motion of the coupled rotor-bearing-housing systems for the subsequent analyses. The equation of motion, obtained from prior research [1], employs a ground-based and a rotor-based coordinate system to the housing and the rotor, respectively. As a result, the equation of motion takes the form of a set of ordinary differential equations with periodic coefficients of frequency ω3. To better understand its solutions, a numerical model is introduced as an example. In this example, the rotor is a disk with four radial slots and the housing is a square plate with a central shaft. The rotor and housing are connected via two ball bearings. The second part of the paper is to analyze the rotor’s response in the rotor-based coordinate system theoretically. When the rotor is at rest, let ωH be the natural frequency of a coupled rotor-bearing-housing mode whose response is dominated by the housing. The theoretical analysis then indicates that response of the spinning rotor will possess frequency components ωH ± ω3 demonstrating the interaction of the spinning rotor and the housing. The theoretical analysis further shows that this splitting phenomenon results from the periodic coefficients in the equation of motion. The numerical example also confirms this splitting phenomenon. The last part of the paper is to analyze the rotor’s response in the ground-based coordinate system. A coordinate transformation shows that the ground-based response of the spinning rotor consists of two major frequency branches ωH − (k + 1) ω3 and ωH − (k − 1) ω3, where k is an integer determined by the cyclic symmetry and vibration modes of interest. The numerical example also confirms this derivation.

Coarse typological studies on urban program and density defined by various urban energy conversion technologies in Singapore. Zhongming Shi1,2, Shanshan Hsieh1,2,3, Bhargava Krishna Sreepathi1,2, Jimeno A. Fonseca1,2, François Maréchal1,3, Arno Schlueter1,2 1 Future Cities Laboratory, Singapore-ETH Centre, 1 Create Way, CREATE Tower, 138602 Singapore 2 Architecture and Building Systems, Institute of Technology in Architecture, ETH Zurich, John-von-Neumann-Weg 9, CH-8093 Zurich, Switzerland 3 Industrial Process and Energy Systems Engineering Group, Ecole Polytechnique Federale de Lausanne, Lausanne 1015, Switzerland E-mail: shi@arch.ethz.ch, nils.schueler@epfl.ch, hsieh@arch.ethz.ch, sebastien.cajot@epfl.ch, fonseca@arch.ethz.ch, francois.marechal@epfl.ch, schlueter@arch.ethz.ch Keywords: Urban typology, urban form, energy technology, urban program, density Conference topics and scale: Efficient use of resources in sustainable cities Cities consume about three quarters of global primary energy. Compared to the beginning of the Twentieth Century, the urban area is expected to triple by 2030. The future urban energy performance is substantially influenced by how the urban area is planned, designed, and built. New energy technologies have enabled new possibilities of the urban form. For example, a district cooling system can free the building rooftops for more architectural design options, like an infinity pool or a sky garden. Vice versa, to maximize the energy performance, some new energy technologies enforce some specific requirements on the urban forms, like the urban form and density. We apply a Mixed Integer Linear Programming (MILP) formulation to identify the optimal allocation of energy demand density and energy systems (e.g. district cooling network) subject to resource availability and energy (or environmental) performance targets (e.g. renewable share). The optimized energy demand density can be translated into urban program combinations and density ranges and gradients. To build the model, we survey the prevailing energy conversion technologies and their costs. Based on the local standards of Singapore, we derive the energy profiles and demand densities of buildings with different programs. We adopt a real case study in Singapore to test the target energy technologies. Adjacent to the existing central business district, the site, currently a container terminal, has an area around 1,000 hectares. Upon the relocation of the terminal in 10 years, the energy technologies, the density, and the program of the site have a variety of possibilities. This paper builds a series of coarse urban typologies in terms of urban program and density when adopting different urban energy conversion technologies in Singapore. Furthermore, the general density and the density gradient may vary when the size of these energy infrastructures alters. In an integrated urban design process involving energy considerations, the urban designer can refer these urban typologies for rules on the general density, the density gradient, and the urban program combination based on the selected energy technologies. On the other way, these urban typologies can also help on the selection of energy technologies to accommodate the target urban density and program. References (100 words) Ratti, C., Baker, N., and Steemers, K. (2005). Energy consumption and urban texture. Energy Build. 37, 762–776. Salat, S. (2009). Energy loads, CO2 emissions and building stocks: morphologies, typologies, energy systems and behaviour. Build. Res. Inf. 37, 598–609. Seto, K.C., Güneralp, B., and Hutyra, L.R. (2012). Global forecasts of urban expansion to 2030 and direct impacts on biodiversity and carbon pools. Proc. Natl. Acad. Sci. U. S. A. 109, 16083–16088. UN-Habitat (2012). Energy. [Online]. Available: http://unhabitat.org/urban-themes/energy. [Accessed:08-Nov-2016].