Integer Research Articles

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).

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.

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.

Local Orientation Transitions to a Lying Helix State in Negative Dielectric Anisotropy Cholesteric Liquid Crystal

Orientation transitions in a cholesteric liquid crystal (CLC) layer with negative dielectric anisotropy, under the influence of a non-uniform spatially periodic electric field created using a planar system of interdigitated electrodes, were studied experimentally and numerically. In the interelectrode space, transitions are observed from a planar Grandjean texture, with the helix axis perpendicular to the layer plane, to states with a lying helix, when the helix axis is parallel to the layer plane and perpendicular to the electrode stripes. It was found that the relaxation time of the induced state in the Grandjean zones, corresponding to two or more half-turns of the helix, significantly exceeded the relaxation time for the first Grandjean zone with one half-turn. An analysis of experimentally observed and numerically simulated textures shows that slow relaxation to the initial state in the second Grandjean zone, as well as in higher-order zones, is associated with the formation of local topologically equivalent states. In these states, the helix has a reduced integer number of helix half-turns throughout the layer thickness or unwound into the planar alignment state.

An Enhanced RSA Algorithm to Counter Repetitive Ciphertext Threats Empowering User-centric Security

The technology-driven modern age emphasizes the security and privacy of communication. Through this paper, we delve deep into the need for user-centric security within cloud-based environments. The need for enhancement in encryption arises due to the increasing cases of data breaches and insider threats in cloud-based environments recently. The focus is laid upon the use of RSA encryption, end-to-end encryption, and anonymous messaging to address security-based concerns. The primary focus of this research is to develop a comprehensive security system to ensure the confidentiality and authenticity of text-based messages shared in the cloud. The proposed improved RSA algorithm, as suggested, incorporates three prime numbers in the key generation process. To address repetitive ciphertext, the proposed algorithm involves adding the index of each character in the plaintext string to the character’s integer value before encryption. Conversely, during decryption, the same index is subtracted. This proposed algorithm has been utilized in a practical scenario, specifically in the implementation of a chat application. This paper presents a proof-of-concept for the proposed enhanced version of the RSA algorithm, accompanied by a thorough comparison and analysis of computational times across various bit lengths. Increase in data security at a cost of minor increase in computation time was observed through this research.

A predictive score for atrial fibrillation in poststroke patients.

Atrial fibrillation (AF) is a risk factor for cerebral ischemia. Identifying the presence of AF, especially in paroxysmal cases, may take time and lacks clear support in the literature regarding the optimal investigative approach; in resource-limited settings, identifying a higher-risk group for AF can assist in planning further investigation. To develop a scoring tool to predict the risk of incident AF in the poststroke follow-up. A retrospective longitudinal study with data collected from electronic medical records of patients hospitalized and followed up for cerebral ischemia from 2014 to 2021 at a tertiary stroke center. Demographic, clinical, laboratory, electrocardiogram, and echocardiogram data, as well as neuroimaging data, were collected. Stepwise logistic regression was employed to identify associated variables. A score with integer numbers was created based on beta coefficients. Calibration and validation were performed to evaluate accuracy. We included 872 patients in the final analysis. The score was created with left atrial diameter ≥ 42 mm (2 points), age ≥ 70 years (1 point), presence of septal aneurysm (2 points), and score ≥ 6 points at admission on the National Institutes of Health Stroke Scale (NIHSS; 1 point). The score ranges from 0 to 6. Patients with a score ≥ 2 points had a fivefold increased risk of having AF detected in the follow-up. The area under the curve (AUC) was of 0.77 (0.72-0.85). We were able structure an accurate risk score tool for incident AF, which could be validated in multicenter samples in future studies.

A Long Short Term Memory Model for Character-based Analysis of DNS Tunneling Detection

DNS tunneling is the attempt to create a hidden tunnel through a domain name service. Such a tunnel would jeopardize the targeted network and open the door for illegal access, control, and data exfiltration. The information security research community showed the variety of techniques that have been proposed to detect the tunnel. The majority of these efforts were relying on machine learning techniques where features of tunneling are considered such as length of DNS query, size, and entropy of the query. However, an additional analysis of the lexical information of the DNS query has been depicted recently and showed remarkable performance. This paper aims to examine the role of Long Short Term Memory (LSTM) model in terms of DNS lexical analysis. Two benchmark datasets related to DNS have been used. In addition, a character mapping mechanism has been used to replace every possible character with an integer number. Consequentially, the mapped representation has been fed into an LSTM model for DNS tunneling detection. Results showed that the proposed method was able to obtain a weighted average F1-score of 98% for both datasets respectively. Such results are competitive in the context of the state of the art and demonstrate the efficacy of the lexical analysis within the DNS tunneling detection task.

Realizing regular representations of finite groups

Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences E(X) and its group of homeomorphisms Aut(X) are isomorphic to G, and the action of G on the n-th homology group Hn(X) is the regular representation. We also discuss other representations.

Algebraic and Geometric Methods for Construction of Topological Quantum Codes from Lattices

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices Λ′/Λ of index m combined with geometric technique of the projections of vector basis Λ′ over vector basis Λ, we reproduce surface codes found in the literature with parameter [[2m2,2,|a|+|b|]] for the case Λ=Z2 and m=a2+b2, where a and b are integers that are not null, simultaneously. We also obtain a new class of surface code with parameters [[2m,2,|a|+|b|]] from the Λ=A2-lattice when m can be expressed as m=a2+ab+b2, where a and b are integer values. Finally, we will show how this technique can be extended to the construction of color codes with parameters [[18m,4,6(|a|+|b|)]] by considering honeycomb lattices partition A2/Λ′ of index m=9(a2+ab+b2) where a and b are not null integers.

Coupled torsion and inflation of functionally graded and residually stressed Mooney–Rivlin hollow cylinders

Functionally graded structures have material properties continuously varying in one or more directions. Examples include human teeth, sea shells, bamboo stems, and mammal organs in which the varying volume fraction and orientation of fibers optimize their functionalities. Here we analytically study large deformations due to combined torsion and inflation of residually stressed and radially graded Mooney–Rivlin hollow circular cylinders to provide insights into how grading their material moduli according to a power law function of the radius can be advantageously used. We simulate residual stresses in a thin wall hollow cylinder by inverting it inside out and in a thick wall cylinder by assuming that a longitudinal wedge opening parallel to the cylinder axis is closed by deforming it axisymmetrically. It is found that positive integer values of the power law exponent are generally advantageous but its negative values can have deleterious effects on stress distributions. Furthermore, a hollow cylinder with residual stresses generated by one of the foregoing methods in the reference configuration should be modeled as orthotropic rather than transversely isotropic for subsequent deformations. The analytical solutions provided here should help numerical analysts verify their algorithms for large deformations of rubber-like materials that are modeled by the Mooney–Rivlin relation, and design engineers for exploiting the radial gradation of the two material moduli to reduce structure’s mass without sacrificing its performance.

Mathematics MOVES Me—Digital Solutions for Co-ordinating Enactive and Symbolic Resources: The Case of Positive and Negative Integer Arithmetic

Mathematics MOVES Me—Digital Solutions for Co-ordinating Enactive and Symbolic Resources: The Case of Positive and Negative Integer Arithmetic

Ramanujan Sums of All Natural Numbers with Grandi Series

We are all know that the Ramanujan’s theory of calculating the sum of all the natural number. When we came to known that sum of all-natural number is ,that time everyone thinks about two things, one is how it possible that sum of positive number is negative &amp; other is how it is very close to zero. Another research paper gives value of all natural number is then I started the study of this sum, then I find it is as zero. Quit interestingly but this value has more important than Ramanujan’s sum because I did not consider the Grandi series for calculating this sum. And then it is very easy to calculate the sum of negative integer &amp; sum of all odd natural number &amp; sum of all even natural number.

Entanglement in Lifshitz fermion theories

We study the static entanglement structure in (1+1)-dimensional free Dirac-fermion theory with Lifshitz symmetry and arbitrary integer dynamical critical exponent. This model is different from the one introduced in [Hartmann et al., SciPost Phys.11 (2021) 031] due to a proper treatment of the square Laplace operator. Dirac fermion Lifshitz theory is local as opposed to its scalar counterpart which strongly affects its entanglement structure. We show that there is quantum entanglement across arbitrary subregions in various pure (including the vacuum) and mixed states of this theory for arbitrary integer values of the dynamical critical exponent. Our numerical investigations show that quantum entanglement in this theory is tightly bounded from above. Such a bound and other physical properties of quantum entanglement are carefully explained from the correlation structure in these theories. A generalization to (2+1)-dimensions where the entanglement structure is seriously different is addressed.

Review of Vedic Mathematics Yavadunam Tavadunikritya Varganicha Yojayet Sutra for Cubing a Number

This paper discloses secrets and logics of the short formula of ancient Vedic Mathematics Yavadunam Tavadunikritya Varganicha Yojayet Sutra in finding the cube of a positive number. Through the formalization process, we have aimed to bridge the gap between ancient mathematical wisdom and modern theoretical foundations, showcasing the timeless utility of Vedic techniques in mathematical calculations. The sutra has been reviewed using the algebraic explanation in terms of decimal number system of the multiplication process by applying Binomial theorem for the positive integral exponent 3. Some more short-cut techniques have been discussed to explain the sutra and it has been found that they are producing the same result. There are shortcomings that have been found in applying the sutra, which has been solved using some other techniques as a special rule. Later the sutra has been discussed for negative integers also with a note. Before coming to the conclusion, we have discussed some limitations of the sutra, although that has been solved in the paper but with a different process. We have discussed the implications of the sutra for modern computational methods and educational practices, suggesting potential areas for further research.

P-Numerical Semigroups of Triples from the Three-Term Recurrence Relations

Many people, including Horadam, have studied the numbers Wn, satisfying the recurrence relation Wn=uWn−1+vWn−2 (n≥2) with W0=0 and W1=1. In this paper, we study the p-numerical semigroups of the triple (Wi,Wi+2,Wi+k) for integers i,k(≥3). For a nonnegative integer p, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a1,a2,…,aκ with gcd(a1,a2,…,aκ)=1 are expressed in more than p ways. When p=0, S=S0 is the original numerical semigroup. The largest element and the cardinality of N0∖Sp are called the p-Frobenius number and the p-genus, respectively.

Detailed DFT based exploration on spin-gapless features of IrCoNbX (X=Al, Ga, In) alloys

An investigation has been conducted on the spin-polarized structural, electronic, charge transfer, −COHP and magnetic behaviour of IrCoNbX (X=Al, Ga, In) alloys. To corroborate the studied properties the generalized gradient approximation (GGA), modified Becke Johnson (mBJ) and DFT+U schemes within the framework of density functional theory. The structures found comparatively more stable in spin-polarized phase than non-spin polarized phase. Based on electronic band structure and electronic density of states’ analysis IrCoNbX alloys have demonstrated spin-gapless features in spin ↑ channel and semiconducting characteristics in spin ↓ channel. The −COHP examination revealed that Nb-Ir pair exhibited significantly stronger bonding interaction in comparison to XCo (XAl, Co, In, Nb) bonding pairs in IrCoNbX alloys. Each of the studied alloys manifested an integer value of the total magnetic moment of 2.0 μB/f.u under mBJ scheme. The findings of current study proposed the experimental investigation of these alloys for potential applications in spintronics field.

On the size of finite Sidon sets

UDC 519.1 A Sidon set (also called a Golomb ruler) is a B 2 sequence and a 1 -thin set is a set of integers containing no nontrivial solutions to the equation a + b = c + d . We improve the lower bound for the diameter of a Sidon set with k elements, namely, if k is sufficiently large and 𝒜 is a Sidon set with k elements, then ⅆ i a m ( 𝒜 ) ≥ k 2 - 1.99405 k 3 / 2 . Alternatively, if n is sufficiently large, then the cardinality of the largest subset of { 1,2 , … , n } , which is a Sidon set, does not exceed n 1 / 2 + 0.99703 n 1 / 4 .

A REFINED CLASSIFICATION OF THE SUBSET SUM PROBLEM IN POLYNOMIAL TIME COMPLEXITY

The problem considered is the selection of at least one subset from a set (array) of distinct positive integers, such that the sum of the subset's elements exactly matches a given target sum (target certificate). According to R. M. Karp, this problem belongs to the class of NP-complete problems. Diophantine equations and an auxiliary problem, which facilitates the solution of the original problem and has independent scientific interest, have been introduced. A novel method has been developed, which includes proven lemmas and theorems. These results enable the development of efficient and straightforward algorithms for solving the subset sum problem. The time and space complexity for selecting the required subsets do not exceed the square of the length of the original set. An analytical framework has been proposed for managing indices within the original set. These algorithms are applicable to solving problems related to the independent set of cardinality k and the k-vertex cover problem. Additionally, we present examples to confirm claimed results. It should be noted that the time complexity of sorting an array of integers is proportional to the square of the array's size, and this problem belongs to class P. Therefore, based on the newly developed method, it can be inferred that the subset sum problem, originally classified as NP-complete within the NP class, also belongs to P.