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Andrews introduced the singular overpartition function C ‾ k , i ( n ) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ± i ( mod k ) may be overlined. In this article, we study the divisibility properties of C ‾ 4 k , k ( n ) and C ‾ 6 k , k ( n ) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that C ‾ 4 k , k ( n ) is almost always divisible by arbitrary powers of 2. We also prove that C ‾ 6 k , k ( n ) is almost always divisible by arbitrary powers of 3 for an infinite family of k. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by C ‾ 4 ⋅ 2 α , 2 α ( n ) and C ‾ 4 ⋅ 3 ⋅ 2 α , 3 ⋅ 2 α ( n ) .

We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,\alpha,\epsilon)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}\alpha n^{k}$ we have $$e(A_1,\dots, A_k)=(1\pm\epsilon)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $\alpha=\alpha(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big(p,\alpha, o(1)\big)$-pseudo-random $n$-vertex hypergraph $H$ with a mild minimum vertex degree condition contains an $F$-factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudo-randomness such as jumbledness or large spectral gap. As a consequence, $\big(p,\alpha, o(1)\big)$-pseudo-random $k$-graphs as above contain: $(i)$ a perfect matching if $\alpha=o(p^{k})$ and $(ii)$ a loose Hamilton cycle if $\alpha=o(p^{k-1})$. This extends the works of Lenz--Mubayi, and Lenz--Mubayi--Mycroft who studied the analogous problems in the dense setting.

The bankruptcy problem is to divide a homogeneous divisible good (the “estate”) between claimants, when the sum of the claims exceeds the value of the estate. When the problem is looked at from an ex-ante point of view (i.e. before the size of the estate is revealed), it is possible to formulate a notion of Pareto efficiency that is stronger than when the more common ex-post perspective is taken. Under the assumption of common beliefs, the strong notion of efficiency leads, in combination with the requirement that all claims should be fulfilled when the value of the estate is equal to the sum of the claims, to a uniquely defined division rule when utility functions for all agents are given. The resulting rule can be represented in the form of a parametric function. For the case in which all agents are equipped with the same utility function, the class of parametric functions that can be obtained in this way is characterized. In particular, it is shown that two well-known division rules for the bankruptcy problem, namely Constrained Equal Losses and Proportional Division, can be rationalized under strong Pareto efficiency by constant absolute risk aversion and constant relative risk aversion respectively.

Linear time invariant (LTI) systems are often represented by rational forms or rational matrix functions. Such forms exhibit important properties of a system, but these properties are often thought to be due to time invariance. Are there such forms for linear time-variant (LTV) systems with comparable properties, generalizing the LTI case? One major obstacle to derive them has been the lack of a divisibility theory for general block lower-triangular matrices, in particular the lack of a Euclidean algorithm or, in its wake, Smith forms and Smith-McMillan forms. But do rational forms for matrices, generalizing the time-invariant properties, exist at all? It turns out they do. The theory presented here produces representations for block lower-triangular matrices as ratios of (block) staircase (or echelon) matrices, and shows how central properties such as co-prime factorization or Bezout identities hold. Instead of the Euclidean algorithm, the theory uses a technique derived from a paper of Paul Van Dooren, namely ‘dead-beat control’. The theory finds good applications in various domains (computational efficiency, determining pre-conditioners, control theory and model reduction via generalized forms of interpolation).

A finite set of prime numbers [Formula: see text] is called unavoidable with respect to [Formula: see text] if for each [Formula: see text] the sequence of integer parts [Formula: see text], [Formula: see text] contains infinitely many elements divisible by at least one prime number [Formula: see text] from the set [Formula: see text]. It is known that an unavoidable set exists with respect to [Formula: see text] and that it does not exist if [Formula: see text] is an integer such that [Formula: see text] is not square free. In this paper, we show that no finite unavoidable sets exist with respect to [Formula: see text] if [Formula: see text] is a prime number or [Formula: see text] belongs to some explicitly given arithmetic progressions, for instance, [Formula: see text] and [Formula: see text], [Formula: see text]

This paper is the first part of a new proof of decidability of the existential theory of the structure 〈 $$\mathbb{Z}$$ ; 0, 1, +, –, ≤, |〉, where | corresponds to the binary divisibility relation. The decidability was proved independently in 1976 by A.P. Bel’tyukov and L. Lipshitz. In 1977, V.I. Mart’yanov proved an equivalent result by considering the ternary GCD predicate instead of divisibility (the predicates are interchangeable with respect to existential definability). Generalizing in some sense the notion of quantifier elimination (QE) algorithm, we construct a quasi-QE algorithm to prove decidability of the positive existential theory of the structure 〈 $${{\mathbb{Z}}_{{ > 0}}}$$ ; 1, $${{\{ a{\kern 1pt} \cdot \} }_{{a \in {{\mathbb{Z}}_{{ > 0}}}}}}$$ , GCD〉. We reduce to the decision problem for this theory the decision problem for the existential theory of the structure 〈 $$\mathbb{Z}$$ ; 0, 1, +, –, ≤, GCD〉. A quasi-QE algorithm, which performs this reduction, will be constructed in the second part of the proof. Transformations of formulas are based on a generalization of the Chinese remainder theorem to systems of the form GCD(ai, bi + x) = di for every $$i \in [1..m]$$ , where ai, bi, di are some integers such that $${{a}_{i}} \ne 0$$ , $${{d}_{i}} > 0$$ .

Let f ∈ Z [ x ] and l ∈ N . Consider the set of all ( a 0 , a 1 , … , a l ) ∈ N l + 1 with a i a i + 1 and f ( a i ) | f ( a i + 1 ) for all 0 ≤ i ≤ l − 1 . We say that f satisfies the gap principle of order l if lim ⁡ a l / a 0 = ∞ as a 0 → ∞ . We also define the gap order of f ( x ) to be the smallest positive integer l such that f ( x ) satisfies the gap principle of order l. If such l does not exist, we say that f ( x ) does not satisfy the gap principle. In this article, we prove a conjecture by Chan, Choi and Lam that f ( x ) does not satisfy the gap principle if and only if f ( x ) is in the form of f ( x ) = A ( B x + C ) n for some A , B , C ∈ Z . Moreover, we completely determine the gap order of any polynomial. We will show that if f ( x ) is not in the form of A ( B x + C ) n , then f ( x ) has gap order 2 if f ( x ) is a quadratic polynomial or a power of a quadratic polynomial; and has gap order 1 otherwise.

Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p 2 + 1 = q 2 + r 2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22 n + k (n = 1, 2, . . . ) are composite.

The Fibonacci sequence, whose first terms are f0; 1; 1; 2; 3; 5; : : :g, is generated using the recursive formula Fn+2 = Fn+1 + Fn with F0 = 0 and F1 = 1. This sequence is one of the most famous integer sequences because of its fascinating mathematical properties and connections with other fields such as biology, art, and music. Closely related to the Fibonacci sequence is the Lucas sequence. The Lucas sequence, whose first terms are f2; 1; 3; 4; 7; 11; : : :g, is generated using the recursive formula Ln+2 = Ln+1 + Ln with L0 = 2 and L1 = 1. In this paper, patterns in the prime factors of sums of powers of Fibonacci and Lucas numbers are examined. For example, F2 3n+4 + F2 3n+2 is even for all n 2 N0. To prove these results, techniques from modular arithmetic and facts about the divisibility of Fibonacci and Lucas numbers are utilized. KEYWORDS: Fibonacci Sequence; Lucas Sequence; Modular Arithmetic; Divisibility Sequence

We continue the research of an extension (Formula presented.) of the divisibility relation to the Stone-Čech compactification (Formula presented.). First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in (Formula presented.) and nonstandard extensions of (Formula presented.) are answered, providing a few more equivalent conditions for divisibility in (Formula presented.). Results on uncountable chains in (Formula presented.) are proved and used in a construction of a well-ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type (Formula presented.) in (Formula presented.). Finally, we consider ultrafilters without divisors in (Formula presented.) and among them find the greatest class.

Given a vectorial function ${F}:\mathbb {F}_{2}^{{n}} \mapsto \mathbb {F}_{2}^{{m}}$ , the indicator $1_{{\mathcal {G}}_{{F}}}$ of its graph ${\mathcal{ G}}_{{F}}=\{({x},{F}({x})); {x}\in \mathbb {F}_{2}^{{n}}\}$ allows to express the algebraic degree of $F$ in a simple way. Exploiting the formula, obtained in a previous article, for the graph indicator of a composite function ${G}\circ {F}$ , that involves only a sum of products of $1_{{\mathcal {G}}_{{F}}}$ and $1_{{\mathcal {G}}_{{G}}}$ , we deduce the exact expression as well as bounds on the algebraic degree of ${G}\circ {F}$ , whose efficiency comes from the fact that the algebraic degree of the product of two Boolean functions is bounded above by the sum of their algebraic degrees, while for a composition, it is bounded above by their product. One of these bounds, that depends on the algebraic degrees of $G$ and $1_{{\mathcal {G}}_{{F}}}$ , is tight, general, simple, and most often efficient (for the case where it is not efficient, we give an improved bound, that is a little more complex). As far as we know, it is the first efficient upper bound ever found, that works without any condition on the vectorial functions. It provides a new criterion for the choice of S-boxes in block ciphers. It implies as a corollary a known bound assuming the divisibility of the Walsh transform values by a power of 2. It gives a better view why this latter bound works. All the bounds generalize to more than two functions and this represents also an improvement over the state of the art. When $F$ is a permutation, our expression of the algebraic degree of ${G}\circ {F}$ simplifies into a formula involving the algebraic degrees of the products of a coordinate function of $G$ and coordinate functions of ${F}^{-1}$ . This implies another known bound showing that the algebraic degree of ${F}^{-1}$ has more impact on that of ${G}\circ {F}$ than that of $F$ itself. Our approach by graph indicators gives a more complete explanation to this interesting fact. Our results include all the known efficient bounds as particular cases, and clarify the reasons why they work. We also deduce the exact expression of the algebraic degree of the composition of any number of functions, leading to a bound that is much more efficient than what we obtain by applying the known bound several times. We also obtain two bounds on the algebraic degree of ${G} \circ {F}$ , where $F$ is a permutation, given the divisibility by powers of 2 of some Walsh transform values of component functions of $F$ and their sums with a coordinate function of $G$ . We compare all the bounds of this kind obtained so far and show how they are complementary, and we study the generalizations of all (known and new) bounds of this kind to the composition of more than two functions.

In this paper, we examine properties of the tensor powers of quantum mappings Φ. In particular, we review positivity properties of unitary and nonunitary qubit mappings Φ⊗2. For arbitrary finite-dimensional systems, we present the relationship between the positive and completely positive divisibility of dynamical mappings $$ {\Phi}_t^{\otimes 2} $$ and Φt. A criterion of annihilation of entanglement by an arbitrary qubit mapping Φ⊗2 is found.

We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such a lattice can be converted into a lattice effect algebra and every lattice effct algebra can be reconstructed form its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced recently by Dvurecenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such a lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.

In this paper we relate t-designs to a forbidden configuration problem in extremal set theory. Let 1_t 0_l denote a column of t 1's on top of l 0's. We assume t>l. Let q. (1_t 0_l) denote the (t+l)xq matrix consisting of t rows of q 1's and l rows of q 0's. We consider extremal problems for matrices avoiding certain submatrices. Let A be a (0,1)-matrix forbidding any (t+l)x(\lambda+2) submatrix (\lambda+2). (1_t 0_l) . Assume A is m-rowed and only columns of sum t+1,t+2,... ,m-l are allowed to be repeated. Assume that A has the maximum number of columns subject to the given restrictions. Assume m is sufficiently large. Then A has each column of sum 0,1,... ,t and m-l+1,m-l+2,..., m exactly once and, given the appropriate divisibility condition, the columns of sum t+1 correspond to a t-design with block size t+1 and parameter \lambda and there are no other columns. The proof derives a basic upper bound on the number of columns of A by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design based constructions. We extend in a few directions., Comment: arXiv admin note: text overlap with arXiv:1909.07580

In this paper we introduce a class of sequences connected with the m–ary partition function and investigate their congruence properties. In particular, we get facts about the sequences of m–ary partitions ( b m ( n ) ) n ∈ N and m–ary partitions with no gaps ( c m ( n ) ) n ∈ N . We prove, for example, that for any natural number 2 h ≤ m each residue class modulo h appears infinitely many times in both sequences ( b m ( n ) ) n ∈ N and ( c m ( n ) ) n ∈ N . Moreover, we give new proofs of characterisations modulo m in terms of base–m representation of n for sequences ( b m ( n ) ) n ∈ N and ( c m ( n ) ) n ∈ N . We also present a general method of finding such characterisations modulo any power of m. Using our approach we get a description of ( b m ( n ) mod μ 2 ) n ∈ N , where μ 2 = m 2 if m is odd and μ 2 = m 2 / 2 if m is even.

We confirm three conjectures of Guo and Zeng on Franel numbers and binomial coefficients. Furthermore, we prove more refined divisibility results for a large class of polynomial sequences.

In 1974, P. Hartung proved the existence of infinitely many imaginary quadratic fields with class numbers not divisible by 3. We generalize this result to show the existence of infinitely many imag...

Kenelm Digby’s Two Treatises, of the Nature of Bodies and of the Nature of Mans Soule (1644) defends quite an idiosyncratic approach to mind-body dualism. In his use of the divisibility argument to prove that the human soul cannot be a material substance, Digby takes an uncompromising stand for merely potential material parts. In his Treatise of Bodies the present article focuses on the mode of construction of the definition of quantity as divisibility and on its links to two distinct fundamental arguments against the actual material parts doctrine. The first, positive, argument consists of a semantic reason drawn from Digby’s general doctrine of meaning, whereas the second, negative, argument, addresses the traditional question of the composition of the continuum. The latter, the author contends, does not build on the medieval controversy itself, but on Digby’s opposition to Galileo’s claim of indivisibilism in his Dialogues Concerning Two Sciences (1638).

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K × . We consider Kummer extensions of G of the form K ( ζ 2 m , G 2 n ) / K ( ζ 2 m ) , where n ⩽ m . In the paper by Debry and Perucca (2016) [1] , the degrees of those extensions have been evaluated in terms of divisibility parameters over K ( ζ 4 ) . We prove how properties of G over K explicitly determine the divisibility parameters over K ( ζ 4 ) . This result yields a clear computational advantage, since no field extension is required.

We derive the existence of p-adic Hurwitz–Lerch L-function by means of a method provided by Washington. This function is a generalization of the one-variable p-adic L-function of Kubota and Leopoldt, and two-variable p-adic L-function of Fox. We also deduce divisibility properties of generalized Apostol–Bernoulli polynomials, in particular establish Kummer-type congruences for them.

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n≥1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k≥1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

We consider five divisibility orders on the Stone-Cech compactification ?N. We find some possible lengths of chains and antichains and number of maximal and minimal elements, as well as some other ordering properties of these relations.

We study the structure algebra $\mathcal{Z}$ of the stable moment graph for the case of the affine root system $A_{1}$. The structure algebra $\mathcal{Z}$ is an algebra over a symmetric algebra and in particular, it is a module over a symmetric algebra. We study this module structure on $\mathcal{Z}$ and we construct a basis. By "setting $c$ equal to zero" in $\mathcal{Z}$, we obtain the module $\mathcal{Z}_{c=0}$. This module can be described in terms of the finite root system $A_{1}$ and we show that it is determined by a set of certain divisibility relations. These relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs, and because of this we call them higher-order congruence relations., 23 pages, 4 figures, final version

In this paper, using the rational generating for the second form of the q-analogue of r-Whitney numbers of the second kind, certain divisibility property for this form is established. Moreover, the Hankel transform for the second form of the q-analogue of r-Dowling numbers is derived.Â

We consider a generalized type of unique factorization of the positive integers with restrictions on the exponents and view them as a family of arithmetic convolutions and divisibility relations, similar to the convolutions defined by Narkewicz [On a class of arithmetical convolutions, Colloq. Math. 10 (1963) 81–94]. We introduce special types of multiplicativity corresponding to these convolutions, and discuss algebraic properties of the associated arithmetic convolutions and analogs of the Möbius functions. We also prove asymptotics for analogs of the totient function, totient summatory function, and divisor summatory function.

In this study, it was aimed to reveal the explanatory strategies that preservice teachers use in the process of explaining the concept of divisibility by zero. It was investigated how the concept of divisibility by zero, which can be used in expressing the case where the denominator is present in the definition of important concepts of the secondary school curriculum such as the fraction and rational number, is defined and explained. A scale consisting of three open-ended questions, in which it was questioned what the definition of the concept of divisibility by zero is and how this concept can be explained to the secondary school/high school student, was used as a data collection instrument. The data were collected through this scale and the content analysis method was adopted in the data analysis. As a result of the analyses made, it was determined that the preservice teachers use the rule strategy the most on the subject of divisibility by zero.

In this paper we consider the curves C k ( p , a ) : y p − y = x p k + 1 + a x defined over F p and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves C k ( p , a ) . Our proof involves finding an exact formula for the number of F p n -rational points on C k ( p , a ) for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.

We study quantum processes, as one parameter families of differentiable completely positive and trace preserving (CPTP) maps. Using different representations of the generator, and the Sylvester criterion for positive semi-definite matrices, we obtain conditions for the divisibility of the process into completely positive (CP-divisibility) and positive (P-divisibility) infinitesimal maps. Both concepts are directly related to the definition of quantum non-Markovianity. For the single qubit case we show that CP- and P-divisibility only depend on the dissipation matrix in the master equation form of the generator. We then discuss three classes of processes where the criteria for the different types of divisibility result in simple geometric inequalities, among these the class of non-unital anisotropic Pauli channels., Comment: 12 pages, 4 figures

We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base~$q$ and the $A_j$'s are positive integers relatively prime to $q$. We do this in fact within the framework of quasi-additive functions. As application, we show that most elements of "Catalan-like" sequences - by which we mean integer sequences defined by products/quotients of factorials - are divisible by any given positive integer., Comment: AmS-LaTeX, 11 pages; minor corrections and improvements

In this paper, we prove two conjectures of Z.-W. Sun: $$2n\binom{2n}n\big|\sum_{k=0}^{n-1}(3k+1)\binom{2k}k^3{16}^{n-1-k}\ \mbox{for}\ \mbox{all}\ n=2,3,\cdots,$$ and $$\sum_{k=0}^{(p-1)/2}\frac{3k+1}{16^k}\binom{2k}{k}^3\equiv p+2\left(\frac{-1}{p}\right)p^3E_{p-3}\pmod{p^4},$$ where $p>3$ is a prime and $E_0,E_1,E_2,\cdots$ are Euler numbers., 11 pages

We give explicit constructions for incomplete pairwise balanced designs IPBD((v; w), K), or, equivalently, edge-decompositions of a difference of two cliques $$K_v \setminus K_w$$ into cliques whose sizes belong to the set K. Our constructions produce such designs whenever v and w satisfy the usual divisibility conditions, have ratio v / w bounded away from the smallest value in K minus one, say $$v/w > k-1+\epsilon $$ , for $$k =\min K$$ and $$\epsilon >0$$ , and are sufficiently large (depending on K and $$\epsilon $$ ). As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as ‘templates’.

Group testing is the process of pooling arbitrary subsets from a set of $ {n}$ items so as to identify, with a minimal number of tests, a “small” subset of $ {d}$ defective items. In “classical” non-adaptive group testing, it is known that when $ {d}$ is substantially smaller than $ {n}$ , $\Theta ( {d}\log ( {n}))$ tests are both information-theoretically necessary and sufficient to guarantee recovery with high probability. Group testing schemes in the literature that meet this bound require most items to be tested $ {\Omega }(\log ( {n}))$ times, and most tests to incorporate $ {\Omega }({{n/d}})$ items. Motivated by physical considerations, we study group testing models in which the testing procedure is constrained to be “sparse.” Specifically, we consider (separately) scenarios in which 1) items are finitely divisible and hence may participate in at most $ {\gamma } \in {o}(\log ( {n}))$ tests; or 2) tests are size-constrained to pool no more than $\rho \in {o}({{n/d}})$ items per test. For both scenarios, we provide information-theoretic lower bounds on the number of tests required to guarantee high probability recovery. In particular, one of our main results shows that $ {\gamma }$ -finite divisibility of items forces any non-adaptive group testing algorithm with the probability of recovery error at most $ {\epsilon }$ to perform at least $ {\gamma } {d}({ {n/d}})^{({1}-{5} {\epsilon })/ {\gamma }}$ tests. Analogously, for $ {\rho }$ -sized constrained tests, we show an information-theoretic lower bound of $ {\Omega }( {n}/ {\rho })$ tests for high-probability recovery–hence in both settings the number of tests required grows dramatically (relative to the classical setting) as a function of $ {n}$ . In both scenarios, we provide both randomized constructions and explicit constructions of designs with computationally efficient reconstruction algorithms that require a number of tests that is optimal up to constant or small polynomial factors in some regimes of ${{n, d,}} {\gamma }$ , and $ {\rho }$ . The randomized design/reconstruction algorithm in the $ {\rho }$ -sized test scenario is universal –independent of the value of $ {d}$ , as long as $ {\rho } \in {o}({\textbf {n/d}})$ . We also investigate the effect of unreliability/noise in test outcomes, and show that whereas the impact of noise in test outcomes can be obviated with a small (constant factor) penalty in the number of tests in the $ {\rho }$ -sized tests scenario, there is no group-testing procedure, regardless of the number of tests, that can combat noise in the $ {\gamma }$ -divisible scenario.

We report that under some specific conditions a single qubit model weakly interacting with information environments can be referred to as a quantum classifier. We exploit the additivity and the divisibility properties of the completely positive (CP) quantum dynamical maps in order to obtain an open quantum classifier. The steady state response of the system with respect to the input parameters was numerically investigated and it's found that the response of the open quantum dynamics at steady state acts non-linearly with respect to the input data parameters. We also demonstrate the linear separation of the quantum data instances that reflects the success of the functionality of the proposed model both for ideal and experimental conditions. Superconducting circuits were pointed out as the physical model to implement the theoretical model with possible imperfections., Comment: 11 pages, 7 figures

In this paper, we establish some new divisibility results involving the Franel numbers [Formula: see text] [Formula: see text] and the polynomials [Formula: see text] [Formula: see text]. For example, we show that for any positive integer [Formula: see text] we have [Formula: see text] and [Formula: see text] and for any prime [Formula: see text] we have [Formula: see text]

Let S be a multiset of integers. We say S is a zero-sum sequence if the sum of its elements is 0. We study zero-sum sequences whose elements lie in the interval [ − k , k ] such that no subsequence of length t is also zero-sum. Given these restrictions, Augspurger, Minter, Shoukry, Sissokho, Voss show that there are arbitrarily long t -avoiding, k -bounded zero-sum sequences unless t is divisible by lcm ( 2 , 3 , 4 , … , max ( 2 , 2 k − 1 ) ) . We confirm a conjecture of these authors that for k and t such that this divisibility condition holds, every zero-sum sequence of length at least t + k 2 − k contains a zero-sum subsequence of length t , and that this is the minimal length for which this property holds.

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced.

[EN] Divisibility networks of natural numbers present a scale-free distribution as many other process in real life due to human interventions. This was quite unexpected since it is hard to find patterns concerning anything related with prime numbers. However, it is by now unclear if this behavior can also be found in other networks of mathematical nature. Even more, it was yet unknown if such patterns are present in other divisibility networks. We study networks of rational numbers in the unit interval where the edges are defined via the divisibility relation. Since we are dealing with infinite sets, we need to define an increasing covering of subnetworks. This requires an order of the numbers different from the canonical one. Therefore, we propose the construction of four different orders of the rational numbers in the unit interval inspired in Cantor's diagonal argument. We motivate why these orders are chosen and we compare the topologies of the corresponding divisibility networks showing that all of them have a free-scale distribution. We also discuss which of the four networks should be more suitable for these analyses, JAC was funded by MEC grant number MTM2016-75963-P. PASH acknowledges the support of MESCyT-RD, Casa Brugal, and Fundacion Proyecto Escuela Hoy Inc. for his PhD grants. MAGM acknowledges funding from the Spanish Ministry of Education and Vocational Training (MEFP) through the Beatriz Galindo program 2018 (BEAGAL18/00203) and Spanish Ministry MINECO FIDEUA PID2019-106901GB-I00/10.13039/501100011033.

This paper addresses project portfolio selection with divisibility, where cash flow and opportunity cost are simultaneously considered for the first time. If a project is selected, fixed assets req...

Understanding decoherence processes is crucial in the study of open quantum systems. In this paper, we discuss the mechanism of pure-dephasing process with a newly proposed boson-boson model, namely, a bosonic field coupled to another bosonic bath in thermal equilibrium. Our model is fully solvable and can reproduce the pure-dephasing process which is usually described by the well-known spin-boson model, therefore offering a new perspective to understanding decoherence processes in open quantum systems of high dimension. We also show that this model admits a generically non-Markovian dynamics with respect to various different non-Markovian measures., 8 pages, 8 figures

The paper first covers several properties of the extension of the divisibility relation to a set *N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone–Cech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible. Received 16 July 2018 AMS subject classification: Primary 54D80; Secondary 11U10, 03H15, 54D35 Keywords: divisibility, nonstandard integer, Stone-Cech compactification, ultrafilter

It is known that no additive division rules exist for bankruptcy problems. In this paper, we study a restricted additivity property which we call “feasible set additivity” (FSA). This property requires division rules to be additive when the set of feasible allocation vectors for a sum of problems does not include allocations that were unfeasible when considering each problem separately. In addition, we characterize the random arrival rule as the only division rule satisfying FSA and equal treatment of equals for two and three-agent cases. We also show that this characterization holds when the endowment is small enough in relation to the claims, while the question of whether it holds in general remains open.

This paper attempts to rethink difference and divisibility as conditions of (im)possibility for love and survival in the wake of Derrida's newly discovered—and just recently published—Geschlecht III. I argue that Derrida's deconstruction of what he calls ‘the grand logic of philosophy’ allows us to think love and survival without positing unicity as a sine qua non. This hypothesis is tested in and through a deconstructive reading of Heidegger's second essay on Trakl in On the Way to Language, where Heidegger's phonocentrism and surreptitious nationalism converge in an effort to ‘save the earth’ from a ‘degenerate’ Geschlecht that cannot survive the internal diremption between Geschlechter. I show that one way of problematizing Heidegger's claim is to point to the blank spaces in the ‘E i n’ of Trakl's ‘E i n Geschlecht’, an internal fissuring in the very word Heidegger mobilizes in order to secure the future of mankind.

It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

For nonnegative integers $j$ and $n$ let $\Theta(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are not divisible by $2^{j+1}$. In this paper we prove that the family $j\mapsto\Theta(j,n)$ usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of $\Theta(j,n)$, and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016)., Comment: 15 pages

The Department of Basic Education in South Africa has identified factorisation as a problem area in Mathematics teaching. Learners in earlier grades are exposed to mathematical tasks, involving con...

Let [Formula: see text] with [Formula: see text] and [Formula: see text] be an odd prime. We find a supercongruence for [Formula: see text] and related sums of powers of binomial coefficients. These results complement prior results for [Formula: see text] with [Formula: see text] obtained recently by the author.

Summary Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye [7]. Although the use of parity allowed to discover irrational numbers [6], there is a common opinion that this property is “not rich enough to become the main content focus of any particular research” [9]. On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes. The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in [5], [4], and [3] branches, formalized in Mizar [1], [2].