Integer Research Articles

NATURAL LOG OF ZERO IS A COMPLEX NUMBER

We take the definition for n!, which takes on integer values for all n>=1, and, through a conjecture, generalize it to for all n> -infinity, from which we demonstrate that ln(0) is a complex number.

On the degeneracy of the energy levels of Schrödinger and Klein-Gordon equations on Riemannian coverings

Abstract We study the degeneracy of the energy levels of the Schrödinger equation with Kepler-Coulomb potential and of the Klein-Gordon equation on Riemannian coverings of the Euclidean space and of the Schwarzschild space-time respectively. Degeneracy of energy levels is a consequence of the superintegrability of the system. We see how the degree of degeneracy changes depending on the covering parameter k, the parameter that in space-times can be related with a cosmic string, and show examples of lower degeneracy in correspondence of non integer values of k.

Anderson localization of circularly polarized waves in a gyrotropic superlattice with correlated disorder

In this paper, we explore the transmission of circularly polarized electromagnetic waves through one-dimensional random periodic-on-average photonic crystals containing layers of magneto-optical material in Faraday geometry. Driven by evidence that long-range correlations crucially influence wave localization within certain spectral ranges, our study aims to harness these effects for the development of novel electromagnetic wave filters tunable via a dc magnetic field. We base our study on a model of light propagation through a finite array of alternating dielectric layers with random thickness variations and layers of gyrotropic material of equal thickness. Assuming weak positional disorder, we employ analytical and numerical methods to analyze the inverse localization length and assess filter performance. Our results demonstrate that specific correlated disorder introduced into periodic systems can enhance or suppress the transmissivity for a wave of a given frequency in any desired interval of the magneto-optical parameter q. Additionally, we show that the Anderson localization can be resonantly suppressed when the thickness of each gyrotropic layer accommodates an integer number of half-wavelengths.

On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.

Block occurrences in the binary expansion

The binary sum-of-digits function [Formula: see text] returns the number of ones in the binary expansion of a nonnegative integer. Cusick’s Hamming weight conjecture states that for all integers [Formula: see text] the set of integers [Formula: see text] such that [Formula: see text] has asymptotic density strictly larger than [Formula: see text]. So far, the strongest results are due to Spiegelhofer and Wallner, who showed that for [Formula: see text] having sufficiently many blocks of [Formula: see text]s in its binary expansion the conjecture holds and the difference [Formula: see text] is approximately normally distributed. In this paper, we are concerned with a related block-counting function [Formula: see text], returning the number of (overlapping) occurrences of the block [Formula: see text] in the binary expansion of [Formula: see text]. Our main result is a similar central limit-type theorem for the difference [Formula: see text], where we give a stronger estimate of the uniform error of Gaussian approximation. As a corollary, we obtain a partial result toward an analogue of Cusick’s conjecture for [Formula: see text]: the set of [Formula: see text] satisfying [Formula: see text] has density [Formula: see text] for [Formula: see text] belonging to a set of density [Formula: see text].

Sumset problem on dilated sets of integers

Let A be a non-empty finite set of integers. For integers m and k, let m⋅A+k⋅A={ma1+ka2:a1,a2∈A}. For m=2 and an odd prime k such that |A|>8kk, Hamidoune et al. [6] proved that |2⋅A+k⋅A|≥(k+2)|A|−k2−k+2. Ljujic [7] extended this result and obtained the same bound for k to be a power of an odd prime and product of two distinct odd primes. Balog et al. [1] proved that |p⋅A+q⋅A|≥(p+q)|A|−(pq)(p+q−3)(p+q)+1, where p<q are relatively primes. In this article, for any odd values of k and under some certain conditions on set A, we obtain that |2⋅A+k⋅A|≥(k+2)|A|−2k|Aˆ|, where Aˆ is the projection of A in Z/kZ. This obtained bound is better than the bound given by Balog et al. We also generalize this bound for |p⋅A+k⋅A|, where p is any odd prime and k be an odd positive integer with (p,k)=1.

First order Stickelberger modules over imaginary quadratic fields

Let K/k be a finite abelian extension of number fields of Galois group G with k imaginary quadratic. Let n≥2 be a rational integer, and for a certain finite set S of places of k, let OK,S be the ring of S-integers of K. We use generalized Stark elements to construct first order Stickelberger modules in odd higher algebraic K-groups of OK,S. We show that the Fitting ideal (resp. index) of these modules inside the corresponding odd K-groups is exactly the Fitting ideal (resp. cardinality) of the even higher algebraic K-group K2n−2(OK,S).

Brauer Configuration Algebras Induced by Integer Partitions and Their Applications in the Theory of Branched Coverings

Brauer configuration algebras are path algebras induced by appropriated multiset systems. Since their structures underlie combinatorial data, the general description of some of their algebraic invariants (e.g., their dimensions or the dimensions of their centers) is a hard problem. Integer partitions and compositions of a given integer number are examples of multiset systems which can be used to define Brauer configuration algebras. This paper gives formulas for the dimensions of Brauer configuration algebras (and their centers) induced by some integer partitions. As an application of these results, we give examples of Brauer configurations, which can be realized as branch data of suitable branched coverings over different surfaces.

Analytical approach in designing PID controller for complex fractional order transfer function

The study focuses on the fractional complex order plant model, which has gained popularity in applied mathematics, physics, and control systems. A significant contribution of this research lies in discussing the physical phenomena associated with complex plant models and their impact on system stability and robustness. The main purpose of the method presented in this paper is to tune the controller parameters to ensure the stability and robustness of the system. There are methods presented in the literature for this purpose. One of these methods is to keep the phase curve in the system frequency response flat within a certain range. However, this process is based on equating the derivative of the phase value to zero at a certain frequency and adds great mathematical complexity to the calculations. In this study, reliable analytical formulas are presented for the same purpose using a graphical approach. Since the fractional complex order plant model represents the most general mathematical form, it enables easy creation of other plant models, including integer order and fractional order plant. The reason why this plant is chosen is that this structure can be named as the universal plant, which all other structures can be built by making little variations. For instance, a transfer function having integer, real and/or complex number coefficients and/or orders can be obtained by proper determination of the parameters of the universal plant. A time delay can also be added towards researcher's desire. The main inspiration comes from studying on an inclusive plant. The method in this paper intends to tune the well‐known classical Proportional Integral Derivative controller. Thus, effectiveness of the integer order controller on various plants will be shown. This approach provides analytical calculation equations for the physical modifications of plants with integer, fractional, and/or complex coefficients and/or orders. The effectiveness of the method is demonstrated visually with different examples that include these different possible situations. The results observed in the changes of the parameters in the transfer functions were also examined. Thus, pros and cons of the variations of integer, fractional, and complex numbers on system parameters have been shown.

Signed integer-valued autoregressive model with time-varying coefficients

An integer time series is a sequence of discrete integers arranged chronologically to represent a specific event or phenomenon. It may include negative integers, and its behavior can be influenced by variations in covariates. To address these complexities, this paper introduces a p-order signed integer-valued autoregressive model with time-varying coefficients. Considering the presence of zero-inflated phenomena, we specifically employ the zero-inflated Skellam distribution. To estimate the unknown parameters and nonparametric functions in our proposed model, we adopt a two-step iterative process that combines local linear estimation with either conditional least squares or conditional maximum likelihood. Through a comprehensive simulation study, we evaluate the performance of our estimation method and demonstrate the usefulness of the proposed model using a real-life dataset.

Higher-order and fractional discrete time crystals in Floquet-driven Rydberg atoms

Higher-order and fractional discrete time crystals (DTCs) are exotic phases of matter where the discrete time translation symmetry is broken into higher-order and non-integer category. Generation of these unique DTCs has been widely studied theoretically in different systems. However, no current experimental methods can probe these higher-order and fractional DTCs in any quantum many-body systems. We demonstrate an experimental approach to observe higher-order and fractional DTCs in Floquet-driven Rydberg atomic gases. We have discovered multiple n-DTCs with integer values of n = 2, 3, and 4, and others ranging up to 14, along with fractional n-DTCs with n values beyond the integers. The system response can transition between adjacent integer DTCs, during which the fractional DTCs are investigated. Study of higher-order and fractional DTCs expands fundamental knowledge of non-equilibrium dynamics and is promising for discovery of more complex temporal symmetries beyond the single discrete time translation symmetry.

Sharp weak‐type estimate for maximal operators associated to Cartesian families under an arithmetic condition

AbstractGiven a set of integers , we consider the smallest family invariant by translations that contains the rectangles for any and where for integer. We prove that if contains arbitrary large arithmetic progression, then the maximal operator associated to the family satisfies a weak‐type estimate of the form but does not satisfy a weak‐type estimate of the form for any nonnegative convex increasing function such that as tends to infinity.

1/c deformations of AdS3 boundary conditions and the Dym hierarchy

This work introduces a novel family of boundary conditions for AdS3 General Relativity, constructed through a polynomial expansion in negative integer powers of the Brown-Henneaux central charge. The associated dynamics is governed by the Dym hierarchy of integrable equations. It is shown that the infinite set of Dym conserved charges generates an abelian asymptotic symmetry group. Additionally, these boundary conditions encompass black hole solutions, whose thermodynamic properties are examined.

Intersective sets for sparse sets of integers

Abstract For $E \subset \mathbb {N}$ , a subset $R \subset \mathbb {N}$ is E-intersective if for every $A \subset E$ having positive relative density, $R \cap (A - A) \neq \varnothing $ . We say that R is chromatically E-intersective if for every finite partition $E=\bigcup _{i=1}^k E_i$ , there exists i such that $R\cap (E_i-E_i)\neq \varnothing $ . When $E=\mathbb {N}$ , we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when $E = \mathbb {P}$ , the set of primes, or other sparse subsets of $\mathbb {N}$ . Among other things, we prove the following: (1) the set of shifted Chen primes $\mathbb {P}_{\mathrm {Chen}} + 1$ is both intersective and $\mathbb {P}$ -intersective; (2) there exists an intersective set that is not $\mathbb {P}$ -intersective; (3) every $\mathbb {P}$ -intersective set is intersective; (4) there exists a chromatically $\mathbb {P}$ -intersective set which is not intersective (and therefore not $\mathbb {P}$ -intersective).

Hilbert space of dyons

We revisit the construction of the Hilbert space of nonrelativistic particles moving in three spatial dimensions. This is given by the space of sections of a line bundle that can in general be topologically nontrivial. Such bundles are classified by a set of integers—one for each pair of particles—and arise physically when we describe the interactions of dyons, particles which carry both electric and magnetic charges. The choice of bundle fixes the representation of the Euclidean group carried by the Hilbert space. These representations are shown to recover the “pairwise helicity” formalism recently discussed in the literature. Published by the American Physical Society 2024

Investigating the Inharmonicity of Piano Strings

In classical physics, ideal string vibrations are modelled using the harmonic series, yet real mode frequencies deviate from the integer set of harmonics due to string stiffness, known as inharmonicity. This work investigates this phenomenon in five piano strings on a Yamaha GB1 Grand Piano. Using the audio software Audacity for spectral analysis, the inharmonicity coefficient is determined experimentally and theoretically based on string properties. These values are comparable with previous research, offering insights into tuning techniques to minimise dissonant inharmonicity. Possible innovations from inharmonicity research are explored, with suggestions such as the temperature-dependent self-tuning systems for acoustic pianos.

Most integers are not a sum of two palindromes

Abstract For $g \geqslant 2$ , we show that the number of positive integers at most X which can be written as sum of two base g palindromes is at most ${X}/{\log^c X}$ . This answers a question of Baxter, Cilleruelo and Luca.

On the size and structure of t-representable sumsets

Let A⊆Z≥0 be a finite set with minimum element 0, maximum element m, and ℓ elements strictly in between. Write (hA)(t) for the set of integers that can be written in at least t ways as a sum of h elements of A. We prove that (hA)(t) is “structured” forh≥(1+o(1))1emℓt1/ℓ (as ℓ→∞, t1/ℓ→∞), and prove a similar theorem on the size and structure of A⊆Zd for h sufficiently large. Moreover, we construct a family of sets A=A(m,ℓ,t)⊆Z≥0 for which (hA)(t) is not structured for h≪mℓt1/ℓ.

Linear waves in shallow water over an uneven bottom, slowing down near the shore

The exact solutions of the system of equations of the linear theory of shallow water are discussed, representing traveling waves with specific properties for the time propagation, which is infinite when approaching the shore and finite when leaving for deep water. These solutions are obtained by reducing one-dimensional shallow water equations to the Euler–Poisson–Darboux equation with a negative integer coefficient before the lower derivative. The analysis of the wave field dynamics is carried out. It is shown that the shape of a wave approaching the shore will be differentiated a certain number of times, which is illustrated by a number of examples. When a wave moves away from the shore, its profile is integrated. The solutions obtained in the framework of linear theory are valid only for a finite interval of depth variation.

A new procedure to find the optimum deconvolution kernel to deblur satellite images

ABSTRACT The digital number of a given pixel in a satellite image does not only measure the radiance originating from the area of the Earth’s surface represented by this pixel, but also measures radiance originating from surrounding areas represented by adjacent pixels. This adjacency effect, also known as image blurring, is a fundamental limitation of satellite images, and introduces errors in all image processing techniques that quantify the properties of the Earth’s surface on a per-pixel basis. Image deblurring via deconvolution, is a traditional technique to reduce the adjacency effect, which has proven effective in a variety of remote sensing applications. For example, it has provided an almost fourfold reduction of the adjacency error, when using support vector machine classifiers to estimate land cover proportion at a subpixel level. The remote sensing literature usually assumes that satellite images should be deconvolved with a kernel defined by the sensor’s Point Spread Function PSF(x, y), overlooking the results of three seminal empirical studies which systematically demonstrated that the optimum deconvolution kernel is a shrunk PSF of the form PSF(x/β, y/β), where β < 1 is the optimum shrink factor. The current empirical procedure to find the optimum shrink factor is laborious and restrictive since it requires suitable satellite images of much higher resolution, such that their Ground Sampling Distance (GSD) is equal to the GSD of the sensor of interest divided by an integer number much greater than one. A new theoretical procedure that uses synthetic edge images to find the optimum shrink factor is proposed and applied to the same cases considered by the current empirical procedure, showing that the same results are obtained. The new procedure is much simpler to apply and more accurate and versatile, opening a variety of research paths on satellite image deblurring.