Pair Of Integers Research Articles

Characterization of hv-Convex Sequences

Reconstructing a discrete object by means of X-rays along a finite set U of (discrete) directions represents one of the main task in discrete tomography. Indeed, it is an ill-posed inverse problem, since different structures exist having the same projections along all lines whose directions range in U. Characteristic of ambiguous reconstructions are special configurations, called switching components, whose understanding represents a main issue in discrete tomography, and an independent interesting geometric problem as well. The investigation of switching component usually bases on some kind of prior knowledge that is incorporated in the tomographic problem. In this paper, we focus on switching components under the constraint of convexity along the horizontal and the vertical directions imposed to the unknown object. Moving from their geometric characterization in windows and curls, we provide a numerical description, by encoding them as special sequences of integers. A detailed study of these sequences leads to the complete understanding of their combinatorial structure, and to a polynomial-time algorithm that explicitly reconstructs any of them from a pair of integers arbitrarily given.

Boundary slopes for the Markov ordering on relatively prime pairs

Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs (q,p) with q≥p≥0. Our main theorem is a characterization of slopes along which the Markov ordering is monotone with respect to q, confirming conjectures of Lee-Li-Rabideau-Schiffler that refine conjectures of Aigner. The main tool is an explicit computation of the slopes at the corners of the stable norm ball for the modular torus.

A problem in comparative order theory

Write \(\mathrm {ord}_p(\cdot )\) for the multiplicative order in \({\mathbb {F}}_p^{\times }\). Recently, Matthew Just and the second author investigated the problem of classifying pairs \(\alpha , \beta \in {\mathbb {Q}}^{\times }\setminus \{\pm 1\}\) for which \(\mathrm {ord}_p(\alpha ) > \mathrm {ord}_p(\beta )\) holds for infinitely many primes p. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for \(\alpha ,\beta \) to be order-dominant. Via the large sieve, we show that almost all integer pairs \(\alpha ,\beta \) satisfy our condition, with a power savings on the size of the exceptional set.

Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c ≤ 12

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c ≥ 13 and d ≥ 1, we give first explicit constructions of c-crossing-critical graphs containing arbitrarily many vertices of degree greater than d. We also show that such unbounded degree constructions do not exist for c ≤ 12, precisely, that there exists a constant D such that every c-crossing-critical graph with c ≤ 12 has maximum degree at most D. Hence, the bounded maximum degree conjecture of c-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c ≤ 12.1

On the Divisibility a! + b! | (a + b)!

In this article, we investigate the pairs of positive integers for which sum of their factorials divides the factorial of their sum and establish a bound on their difference. We also solve the divisibility question over the set of Fibonacci numbers. We conclude by proving that there are infinitely many such pairs of positive integers with difference 2 and conjecture that for any positive integer k there are infinitely many such pairs (a, b) with

Construction of an Energy-Efficient Detour Non-Split Dominating Set in燱SN

Wireless sensor networks (WSNs) are one of the most important improvements due to their remarkable capacities and their continuous growth in various applications. However, the lifetime of WSNs is very confined because of the delimited energy limit of their sensor nodes. This is the reason why energy conservation is considered the main exploration worry for WSNs. For this energy-efficient routing is required to save energy and to subsequently drag out the lifetime of WSNs. In this report we use the Ant Colony Optimization (ACO) method and are evaluated using the Genetic Algorithm (GA), based on the Detour non-split dominant set (GA) In this research, we use the energy efficiency returnee non-split dominating set (DNSDS). A set S ⊆ V is supposed to be a DNSDS of G when the graph G = (V, E) is expressed as both detours as well as a non-split dominating set of G. Let the detour non-split domination number be addressed as γ_dns (G) and is the minimum order of its detour non-split dominating set. Any DNSDS of order is a -set of . Here, the γ_dns (G) of various standard graphs is resolved and some of its general properties are contemplated. A connected graph usually has an order n with detour non-split domination number as n or n – 1 are characterized. Also connected graphs of order and detour diameter with detour non-split dominating number or or are additionally portrayed. While considering any pair of positive integers to be specific and , there exists a connected graph which is normally indicated as , and () = a + b − 2, here indicates the detour domination number and indicates the detour number of a graph. The time is taken for the construction and the size of DNSDS are considered for examining the performance of the proposed method. The simulation result confirms that the DNSDS nodes are energy efficient.

Some new semiring structures

In this paper, we introduce some new semiring structures by considering the prime factors, sum of divisors and the number of relatively prime positive divisors of a pair of non-negative integers and solve a variety of system of linear equations over one of these semirings. We then discuss the properties of these structures and how they compare with classical algebra. The paper is concluded by providing a key-exchange scheme using one of the newly discovered semirings.

A fast vectorized sorting implementation based on the ARM scalable vector extension (SVE).

The way developers implement their algorithms and how these implementations behave on modern CPUs are governed by the design and organization of these. The vectorization units (SIMD) are among the few CPUs’ parts that can and must be explicitly controlled. In the HPC community, the x86 CPUs and their vectorization instruction sets were de-facto the standard for decades. Each new release of an instruction set was usually a doubling of the vector length coupled with new operations. Each generation was pushing for adapting and improving previous implementations. The release of the ARM scalable vector extension (SVE) changed things radically for several reasons. First, we expect ARM processors to equip many supercomputers in the next years. Second, SVE’s interface is different in several aspects from the x86 extensions as it provides different instructions, uses a predicate to control most operations, and has a vector size that is only known at execution time. Therefore, using SVE opens new challenges on how to adapt algorithms including the ones that are already well-optimized on x86. In this paper, we port a hybrid sort based on the well-known Quicksort and Bitonic-sort algorithms. We use a Bitonic sort to process small partitions/arrays and a vectorized partitioning implementation to divide the partitions. We explain how we use the predicates and how we manage the non-static vector size. We also explain how we efficiently implement the sorting kernels. Our approach only needs an array of O(log N) for the recursive calls in the partitioning phase, both in the sequential and in the parallel case. We test the performance of our approach on a modern ARMv8.2 (A64FX) CPU and assess the different layers of our implementation by sorting/partitioning integers, double floating-point numbers, and key/value pairs of integers. Our results show that our approach is faster than the GNU C++ sort algorithm by a speedup factor of 4 on average.

ON QUOTIENTS OF VALUES OF EULER’S FUNCTION ON FACTORIALS

AbstractWe investigate, for given positive integers a and b, the least positive integer $c=c(a,b)$ such that the quotient $\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$ is an integer. We derive results on the limit of $c(a,b)/(a+b)$ as a and b tend to infinity and show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ , with the exception of a set of density zero.

On optimal (Z6m×Z6n,4,1) and (Z2m×Z18n,4,1) difference packings and their related codes

AbstractWe give a direct construction of a relative difference family for and every prime with . These allow us to construct an optimal difference packing and an optimal difference packing for every pair of positive integers . The corresponding optimal optical orthogonal signature pattern codes are also obtained.

A numerical characterization of the extremal Betti numbers of t-spread strongly stable ideals

Let K be a field and let \(S=K[x_1,\ldots ,x_n]\) be a standard polynomial ring over a field K. We characterize the extremal Betti numbers, values as well as positions, of a t-spread strongly stable ideal of S. Our approach is constructive. Indeed, given some positive integers \(a_1,\dots ,a_r\) and some pairs of positive integers \((k_1,\ell _1),\ldots ,(k_r,\ell _r)\), we are able to determine under which conditions there exists a t-spread strongly stable ideal I of S with \(\beta _{k_i, k_i+\ell _i}(I)=a_i\), \(i=1, \ldots , r\), as extremal Betti numbers, and then to construct it.

Classifying three-character RCFTs with Wronskian index equalling 0 or 2

In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers [n,l]. n is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and l, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study [3,0] and [3,2] MLDEs in order to classify the corresponding CFTs. We reduce the problem to a “finite” problem: to classify CFTs with central charge 0 < c ≤ 96, we need to perform 6, 720 computations for the former and 20, 160 for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity.In the [3,0] case, for 0 < c ≤ 96, we obtain many character-like solutions: two infinite classes and a discrete set of 303. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify [3,2] CFTs for 0 < c ≤ 96: each CFT in this case is obtained by adjoining a constant character to a [2,0] CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.

Efficient Private Comparison Queries Over Encrypted Databases Using Fully Homomorphic Encryption With Finite Fields

To achieve security and privacy for data stored on the cloud, we need the ability to secure data in compute. Equality comparisons, “ <inline-formula><tex-math notation="LaTeX">$x=y, x\ne y$</tex-math></inline-formula> ”, have been widely studied with many proposals but there is much room for improvement for order comparisons, “ <inline-formula><tex-math notation="LaTeX">$x &lt; y,~x \leq y,~x &gt; y \text{ and } x \geq y$</tex-math></inline-formula> ”. Most protocols for order comparisons have some limitation, either leaking some information about the data or requiring several rounds of communication between client and server. In addition, little work has been done on retrieving with compound conditions, mixing several equality and order comparisons. Fully homomorphic encryption (FHE) promises the ability to compute arbitrary functions on encrypted data without sacrificing privacy and without communication, but its potential has yet to be fulfilled. Particularly, private comparisons for database queries using FHE are expensive to compute. In this article, we design an efficient private database query (PDQ) protocol which supports compound conditions with equality and order comparisons. To this end, we first present a private comparison algorithm on encrypted integers using FHE, which scales efficiently for the length of input integers, by applying techniques from finite field theory. Then, we consider a scenario for PDQ protocols, querying for values based on a conjunction of one order and four equality conditions on key columns. The proposed algorithm and protocol are implemented and tested to determine their performance in practice. The proposed comparison algorithm takes about <inline-formula><tex-math notation="LaTeX">$25.259$</tex-math></inline-formula> seconds to compare 697 pairs of 64-bit integers using Brakerski-Gentry-Vaikuntanathan's leveled FHE scheme with single instruction multiple data (SIMD) techniques at more than 138 bits of security. This yields an amortized rate of just 36 milliseconds per comparison. On top of that, we show that our techniques achieve an efficient PDQ protocol for one order and four equality comparisons, achieving an amortized time and communication cost of 57 milliseconds and 448 bytes per database element.

Increment of insecure RSA private exponent bound through perfect square RSA diophantine parameters cryptanalysis

The public parameters of the RSA cryptosystem are represented by the pair of integers N and e. In this work, first we show that if e satisfies the Diophantine equation of the form ex2−ϕ(N)y2=z for appropriate values of x,y and z under certain specified conditions, then one is able to factor N. That is, the unknown yx can be found amongst the convergents of eN via continued fractions algorithm. Consequently, Coppersmith’s theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given (Ni,ei) for i=1,2,⋯,k and a fixed x that fulfills the Diophantine equation eix2−yi2ϕ(Ni)=zi. This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.

On Spectra and Spectral Measures of Schreier and Cayley Graphs

Abstract We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the rooted $d$-regular tree, which provide examples of the following interesting phenomena. For $d=2$ and any $m\geq 2$, we get an uncountable family of non-quasi-isometric Cayley graphs with the same Laplacian spectrum, a union of two intervals, which we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another. For each $d\geq 3, m\geq 1$, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct, moreover, a complete system of eigenfunctions that are strongly localized.

A question of Joseph Ritt from the point of view of vertex algebras

Let k be a field of characteristic zero. This paper studies a problem proposed by Joseph F. Ritt in 1950. Precisely, we prove that(1)If p⩾2 is an integer, for every integer i∈N, the nilpotency index of the image of Ti in the ring k{T}/[Tp] equals (i+1)p−i.(2)For every pair of integers (i,j), the nilpotency index of the image of TiUj in the ring k{T}/[TU] equals i+j+1.

Pairing pythagorean pairs

A pair (a,b) of positive integers is a pythagorean pair if a2+b2=□ (i.e., a2+b2 is a square). A pythagorean pair (a,b) is called a double-pythapotent pair if there is another pythagorean pair (k,l) such that (ak,bl) is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair (k,l) which is not a multiple of (a,b), such that (a2k,b2l) is a pythagorean pair. To each pythagorean pair (a,b) we assign an elliptic curve Γa,b with torsion group Z/2Z×Z/4Z, such that Γa,b has positive rank over Q if and only if (a,b) is a double-pythapotent pair. Similarly, to each pythagorean pair (a,b) we assign an elliptic curve Γa2,b2 with torsion group Z/2Z×Z/8Z, such that Γa2,b2 has positive rank over Q if and only if (a,b) is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve Γ with torsion group Z/2Z×Z/8Z is isomorphic to a curve of the form Γa2,b2, where (a,b) is a pythagorean pair. As a side-result we get that if (a,b) is a double-pythapotent pair, then there are infinitely many pythagorean pairs (k,l), not multiples of each other, such that (ak,bl) is a pythagorean pair; the analogous result holds for quadratic pythapotent pairs.

A generalisation of uniform matroids

A matroid is uniform if and only if it has no minor isomorphic to U1,1⊕U0,1 and is paving if and only if it has no minor isomorphic to U2,2⊕U0,1. This paper considers, more generally, when a matroid M has no Uk,k⊕U0,ℓ-minor for a fixed pair of positive integers (k,ℓ). Calling such a matroid (k,ℓ)-uniform, it is shown that this is equivalent to the condition that every rank-(r(M)−k) flat of M has nullity less than ℓ. Generalising a result of Rajpal, we prove that for any pair (k,ℓ) of positive integers and prime power q, only finitely many simple cosimple GF(q)-representable matroids are (k,ℓ)-uniform. Consequently, if Rota's Conjecture holds, then for every prime power q, there exists a pair (kq,ℓq) of positive integers such that every excluded minor of GF(q)-representability is (kq,ℓq)-uniform. We also determine all binary (2,2)-uniform matroids and show the maximally 3-connected members to be Z5﹨t,AG(4,2),AG(4,2)⁎ and a particular self-dual matroid P10. Combined with results of Acketa and Rajpal, this completes the list of binary (k,ℓ)-uniform matroids for which k+ℓ≤4.

The onset of Rayleigh–Bénard convection and heat transfer under two‐frequency rotation modulation

AbstractThe impact of 16 combinations of sinusoidal (sine) and nonsinusoidal (square, triangular, and sawtooth) time‐periodic Coriolis force (rotation modulation) on Rayleigh–Bénard convection in a Newtonian liquid is studied in this paper. This consideration is made to capture the possible effects of two‐frequency rotation modulation on stability, that is, the onset of convection and the simultaneous amount of heat transfer in the system. The Venezian approach has been asserted on the linearized Lorenz model to derive the correction Rayleigh number as a function of the two frequencies. The Lorenz model with nonlinearities is evaluated numerically to assess the quantity of heat transfer in the system. The present study states that in comparison with existing studies of no‐modulation and single‐frequency rotation modulation, two‐frequency rotation modulation yields higher stability bounds and thus diminishes the heat transfer. Heat transfer is found to be enhanced by a pair of coprime integers associated with the harmonics in the system.

On epimorphisms and structurally regular semigroups

In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n,m)-regular semigroups for any pair (n,m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n,m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n,m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimorphisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1 ≤ k ≤ n.