Integer Research Articles

Bernstein–Jacobi-type operators preserving derivatives

A general frame for Bernstein-type operators that preserve derivatives is given. We introduce Bernstein-type operators based in the weighted classical Jacobi inner product on the interval [0, 1] that extend the well known Bernstein–Durrmeyer operator as well as some other types of Bernstein operators that appear in the literature. Apart from standard results, we deduce properties about the preservation of derivatives and prove that classical Jacobi orthogonal polynomials on [0, 1] are the eigenfunctions of these operators. We also study the limit cases when one of the parameters of the Jacobi polynomials is a negative integer. Finally, we study several numerical examples.

On a sumset problem for dilated integer sets

Let [Formula: see text] be a non-empty finite set of integers. For any integer [Formula: see text], let [Formula: see text]. One of the important problems in additive combinatorics is to find the optimal lower bound for the cardinality of the sumset [Formula: see text]. Cilleruelo, Silva, and Vinuesa conjectured that, if [Formula: see text] is a positive integer and [Formula: see text] is a finite set of integers with sufficiently large cardinality, then [Formula: see text]. For some values of [Formula: see text] this conjecture is already confirmed. In this article, under certain restrictions on set [Formula: see text], we confirm this conjecture for all positive integers [Formula: see text] except for those of the form [Formula: see text] or [Formula: see text] where [Formula: see text] is an odd prime number and [Formula: see text] for [Formula: see text].

Temperature and pressure dependencies of Sm valence in SmB[formula omitted] Kondo insulator probed by x-ray absorption spectroscopy

Pressure and temperature induced valence changes in the Kondo insulator SmB6 have been investigated using x-ray absorption spectroscopy (XAS) along several isobaric cooling runs and one isothermal compression. The average valence of Sm at various pressure and temperature conditions is extracted from high quality XAS data by peak fitting procedures. The temperature dependence of the Sm valence at ambient pressure shows changes of slope near the known characteristic temperatures of gap formation (120 K) and the in-gap state (50 K). At higher pressures the average valence also shows similar temperature dependent trends upon cooling but the slope changes are less pronounced. This behavior is discussed in relation to the possible existence of an underlying first order valence transition and its critical end point. At constant temperature (T=20 K), the Sm valence increases rapidly with pressure; however, it remains far below the expected integer value (3+) near the onset of the magnetic ordering and gap closure. In addition, the local structure of SmB6 at ambient conditions has been analyzed using extended x-ray absorption fine structure.

Higher-order beyond-band discrete solitons in binary waveguide arrays

We study higher-order beyond-band discrete solitons (HOBBDSs) and quasi-HOBBDSs, which can be constructed by multiplying the solutions of fundamental single-peaked beyond-band discrete solitons by a soliton order parameter larger than unity. In the quasi-continuous regime when the HOBBDS peak amplitude is low (thus its width is large) and the soliton order parameter is a small integer number, HOBBDSs periodically evolve during propagation and their dynamics are similar to those of higher-order solitons governed by the nonlinear Schrödinger equation in an optical fiber, including the periodicity, pattern evolution, and independence of the period length on the soliton order parameter. If the soliton order parameter is still small but not an integer, then one can obtain the quasi-HOBBDSs whose profiles almost periodically evolve during propagation. The breathing length of quasi-HOBBDSs decreases if the soliton order parameter increases. Moreover, the breathing length of quasi-HOBBDSs is approximately inversely proportional to the square values of the peak amplitude of the fundamental beyond-band discrete solitons, just like what happens with the period length of the higher-order solitons governed by the nonlinear Schrödinger equation. If the fundamental beyond-band discrete solitons are intense enough and/or the soliton order parameter is large enough, then most of the energy of the beams is eventually trapped in a single waveguide.

Oscillations of Fourier coefficients over the sparse set of integers

Oscillations of Fourier coefficients over the sparse set of integers

From cavitation to astrophysics: Explicit solution of the spherical collapse equation.

Differential equationsof the form R[over ̈]=-kR^{γ}, with a positive constant k and real parameter γ, are fundamental in describing phenomena such as the spherical gravitational collapse (γ=-2), the implosion of cavitation bubbles (γ=-4), and the orbital decay in binary black holes (γ=-7). While explicit elemental solutions exist for select integer values of γ, more comprehensive solutions encompassing larger subsets of γ have been independently developed in hydrostatics (see Lane-Emden equation) and hydrodynamics (see Rayleigh-Plesset equation). I here present a universal explicit solution for all real γ, invoking the beta distribution. Although standard numerical ordinary differential equationsolvers can readily evaluate more general second-order differential equations, this explicit solution reveals a hidden connection between collapse motions and probability theory that enables further analytical manipulations, it conceptually unifies distinct fields, and it offers insights into symmetry properties, thereby enhancing our understanding of these pervasive differential equations.

Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices.

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape-rectangle-on a square lattice and show that, for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well as for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This "continuum" limit with the lattice spacing tending to zero only along a subset of the possible directions in d dimensions results in a "semicontinuum" percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units is made to diverge.

HJ Semi Symmetric Cryptographic Algorithm

This paper presents an algorithm for performing encryption and decryption using modular exponentiation mainly based on Euler’s Theorem. The algorithm uses the properties of modular arithmetic to ensure secure communication. The gcd function is employed to find coprime numbers within a given range, which are then utilized as bases for encryption and decryption. The modular Exponentiation function efficiently computes the exponentiation of a base to a power modulo a given modulus. Through a step-by-step process, the algorithm encrypts a numerical message using the selected base and then decrypts it, ensuring the original message is recovered accurately. Experimental results demonstrate the effectiveness and reliability of the algorithm for secure data transmission. In the conclusion we discussed the effectiveness of the algorithm compared to the existing ones such as Elgamal algorithm and RSA Cryptography. The algorithm presented in this paper publishes only the modulus (integer) number and protocols of key exchanges kept secret between sender and receiver; therefore, the algorithm is not fully asymmetric but semi symmetric.

BVP with a Load in the Form of a Fractional Integral

A boundary value problem for a nonhomogeneous heat equation with a load in the form of a fractional Riemann–Liouville integral of an order β∈0,1 is considered. By inverting the differential part, the problem is reduced to an integral equation with a kernel with a special function. The special function is presented as a generalized hypergeometric function. The limiting cases of the order β of the fractional derivative are studied: it is shown that the interval for changing the order of the fractional derivative can be expanded to integer values β∈0,1. The results of the study remain unchanged. The kernel of the integral equation is estimated. Conditions for the solvability of the integral equation are obtained.

On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences

AbstractWe show the existence of a constant such that, for all positive integers , there exist integers such that there are at least distinct integers of the form with . This answers a question of Erdős and Graham. We also prove a non‐trivial upper bound on the maximum number of distinct integers of this form and address several open problems.

Rationality of four-valued families of Weil sums of binomials

We investigate the rationality of Weil sums of binomials of the form WuK,s=∑x∈Kψ(xs−ux), where K is a finite field whose canonical additive character is ψ, and where u is an element of K× and s is a positive integer relatively prime to |K×|, so that x↦xs is a permutation of K. The Weil spectrum for K and s, which is the family of values WuK,s as u runs through K×, is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if s is nondegenerate (i.e., if s is not a power of p modulo |K×|, where p is the characteristic of K). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where |K|=5 and s≡3(mod4).

Every Tree is an Integral Sum Graph

A finite simple graph G is called an integral sum graph (respectively, sum graph) if there is a bijection f from the vertices of G to a set of integers S (respectively, a set of positive integers S) such that uv is an edge of G if and only if f (u)+f (v) ∈ S. In 1999, Liaw et al (Ars Comb., Vol.54, 259-268) posed the conjecture that every tree is an integral sum graph. In this note, we prove that all trees are integral sum graphs. Further, we prove that every bipartite graph is an induced subgraph of a sum graph G with sum number σ(G) = 1.

Radical bound for Zaremba's conjecture

AbstractFamous Zaremba's conjecture (1971) states that for each positive integer , there exists a positive integer , coprime to , such that if you expand a fraction into a continued fraction , all of the coefficients ’s are bounded by some absolute constant , independent of . Zaremba conjectured that this should hold for . In 1986, Niederreiter proved Zaremba's conjecture for numbers of the form with and for with . In this paper, we prove that for each number , there exists , coprime to , such that all of the partial quotients in the continued fraction of are bounded by , where is the radical of an integer number, that is, the product of all distinct prime numbers dividing . In particular, this means that Zaremba's conjecture holds for numbers of the form with , generalizing Neiderreiter's result. Our result also improves upon the recent result by Moshchevitin, Murphy and Shkredov on numbers of the form , where is an arbitrary prime and is sufficiently large.

From discrete to continuous: Monochromatic 3-term arithmetic progressions

We prove a known 2-coloring of the integers [ N ] ≔ { 1 , 2 , 3 , … , N } [N] ≔\{1,2,3,\dots ,N\} minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large N N that involves 12 12 colored blocks. Here, we prove that the conjectured coloring minimizes monochromatic arithmetic 3 3 -progressions among anti-symmetric colorings with 12 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval [ 0 , 1 ] [0,1] used by Parrilo, Robertson, and Saracino as well as by Butler, Costello and Graham. Our proof identifies classes of colorings with permutations, then counts the permutations using mixed integer linear programming.

Nearness Cosets of the Nearness Groups

In 2019, Öztürk et al. [Gamma semigroups on weak nearness approximation spaces, Journal of the International Mathematical Virtual Institute 9 (2019) 53–72] defined the first algebraic structure on weak nearness approximation spaces which is the gamma semigroup. After this study, the view on the nearness of algebraic structures has completely changed. This view was first expressed in a previous study in 2021 [M. A. Öztürk, Nearness [Formula: see text]-algebras, Journal of Algebraic Hyperstructures and Logical Algebras 2 (2021) 73–84]. Our aim is to give the concept of modulo in the nearness group [Formula: see text] as in the nearness group [Formula: see text], where [Formula: see text] is the set of integers. Afterward, based on the concept of modulo in the nearness group [Formula: see text], the nearness cosets of the nearness group [Formula: see text] are constructed and its basic properties are examined. Lastly, it is seen that Lagrange theorem is not valid for the nearness subgroups as compared to the usual subgroups.

On the Validity of Granger Causality for Ecological Count Time Series

Knowledge of causal relationships is fundamental for understanding the dynamic mechanisms of ecological systems. To detect such relationships from multivariate time series, Granger causality, an idea first developed in econometrics, has been formulated in terms of vector autoregressive (VAR) models. Granger causality for count time series, often seen in ecology, has rarely been explored, and this may be due to the difficulty in estimating autoregressive models on multivariate count time series. The present research investigates the appropriateness of VAR-based Granger causality for ecological count time series by conducting a simulation study using several systems of different numbers of variables and time series lengths. VAR-based Granger causality for count time series (DVAR) seems to be estimated efficiently even for two counts in long time series. For all the studied time series lengths, DVAR for more than eight counts matches the Granger causality effects obtained by VAR on the continuous-valued time series well. The positive results, also in two ecological time series, suggest the use of VAR-based Granger causality for assessing causal relationships in real-world count time series even with few distinct integer values or many zeros.

Existance of solutions for some class of system quasilinear differentials with nonlocal boundary conditions and with nonlinearity depandant on the first derivate in time scales

This work is conserned with the construction of minimal and maximal solution for some equation quasilinear differentials with nonlocal boundary condition with nonlinearity is continuous function dependent on the first derivate in the time scale. In a previous study, see[11] it was found that there exists the extrimals solutions of a quasilinear elliptic system with intergral boundary conditions in the continuous case, and from here we asked the following question using the upper and lower solutions method coupled with monotone iterative technique and we asked the following question: do we have the existence of extrimals solutions for certain classes of systems of differential equations with nonlocal boundary conditions in time scales by same method. We can classify the problem following the monotonicity of the two functions and in the true types.Type 1: Increasing qasimonotonic systems.Type 2: Decreasing qasimonotonic systems. Type 3: Mexed qasimonotonic systems. In this paper we show the existence minimuls-maximuls solutions if the problem is of the type 1. If the problem is of the type 2 , we show the existence of maximuls-minimuls solutions. If the problem is of the type 3 , we show existence of least quasilear-solution. In the end, we gave examples of the results presented in this work to prove their validity through two cases: the continuous case, represented by the set of real numbers, and the discontinuous case, represented by the set of integers.

A distant descendant of the six-vertex model

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

Properties of arithmetics progressions in increasing sequence of T0-topologies on the set of positive integers

In this paper we continue researches deal with increasing sequence {Tm} of T0-topologies on the set of positive integers focusing on the properties of arithmetic progressions in topologies Tm. We characterize the closures of arithmetic progressions in all topologies Tm on N. Additionally, we present the characterization of the closures of arithmetic progressions in the common division topology T on N. Moreover, for each m∈N we characterize regular open arithmetic progressions in (N,Tm) and we examine which of these spaces are semiregular.

Investigation of structural, magnetic, mechanical, electronic and thermal properties of new quaternary Heusler: KMgNZ (Z=O or S)

The structural stability as well as the mechanical, magnetoelectronic and thermal properties of the quaternary Heusler KMgNZ (Z = O or S) were studied using first principles approach based on the DFT implemented in Wien2k code. The exchange-correlation potential is treated by PBE-GGA. The calculated negative formation energies for both materials corroborate their thermodynamic stability. Moreover, the derived elastic constants support the mechanical stabilities of the latter compounds. On the other hand, the density of states of the studied materials shows that both compounds are half-metallic ferromagnets. The total magnetic moments for both compounds are mainly originated from N atom, and its integer value followed the Slater-Pauling rule. The calculated half-metallic gaps are found for two systems. The three-dimensional plots of different mechanical moduli exhibit that the bulk modulus is isotropic, whereas Young's modulus, Poisson's ratio and shear modulus of both compounds are highly anisotropic. The effects of temperature and pressure on the heat capacity, thermal expansion coefficient and Debye temperature are also studied.