Integer Research Articles

On the \(D(4)\)-pairs \(\{a, ka\}\) with \(k\in \{2,3,6\}\)

Let \(a\) and \(b=ka\) be positive integers with \(k\in \{2, 3, 6\},\) such that \(ab+4\) is a perfect square. In this paper, we study the extensibility of the \(D(4)\)-pairs \(\{a, ka\}.\) More precisely, we prove that by considering families of positive integers \(c\) depending on \(a,\) if \(\{a, b, c, d\}\) is a set of positive integers which has the property that the product of any two of its elements increased by \(4\) is a perfect square, then \(d\) is given by d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))). As a corollary, we prove that any \(D(4)\)-quadruple tht contains the pair \(\{a, ka\}\) is regular.

On global extrema of power Takagi functions

By construction, power Takagi functions Sp are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions Sp(x) have one real parameter p>0 . They are defined on the real axis R by the series Sp(x)=∑∞n=0(S0(2nx)/2n)p , where S0(x) is the distance from real number x to the nearest integer number. We show that for every p>0 , the functions Sp are everywhere continuous, but nowhere differentiable on R . Next, we derive functional equations for Takagi power functions. With these, it is possible, in particular, to calculate the values Sp(x) at rational points x . In addition, for all values of the parameter p from the interval (0;1) , we find the global extrema of the functions Sp , as well as the points where they are reached. It turns out that the global maximum of Sp equals to 2p/(3p(2p−1)) and is reached only at points q+1/3 and q+2/3 , where q is an arbitrary integer. The global minimum of the functions Sp equals to 0 and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions Sp and find the points at which these estimates are reached.

A theory of dark energy that matches dark matter

In this paper, a theory of dark energy is proposed that matches dark matter. The relativistic quantum mechanics equations reveal that free particles can have negative energies. We think that the negative energy is the dark energy, which behaves as dark photons with negative energies. In this work, the photon number states are extended to the cases where the photon number can be negative integers, called negative integer photon states, the physical meaning of which are that the photons in such a state are of negative energy, i.e., dark photons. The dark photons constitute dark radiation, also called negative radiation. The formalism of the statistical mechanics and thermodynamics of the dark radiation is presented. This version of dark energy is of negative temperature and negative pressure, the latter regarded as responsible for the accelerate expansion of the universe. It is believed that there is a symmetry of energy-dark energy in the universe. In our previous work, the theory of the motion of the matters with negative kinetic energy was presented. In our opinion, the negative kinetic energy matter is dark matter. In the present work, we demonstrate that the dark substances absorb and release dark energy. In this view, the dark matter and dark energy match. Therefore, there is a symmetry of matter-energy match and dark matter-dark energy match in the universe. We present the reasons why the negative kinetic energy systems and negative radiation are dark to us.

The Order of Numbers and the Goldbach Conjecture

The Order of Numbers and the Goldbach Conjecture

The sequence of the repetitive motif influences the frequency of multistep mutations in Short Tandem Repeats

Microsatellites, or Short Tandem Repeats (STRs), are subject to frequent length mutations that involve the loss or gain of an integer number of repeats. This work aimed to investigate the correlation between STRs’ specific repetitive motif composition and mutational dynamics, specifically the occurrence of single- or multistep mutations. Allelic transmission data, comprising 323,818 allele transfers and 1,297 mutations, were gathered for 35 Y-chromosomal STRs with simple structure. Six structure groups were established: ATT, CTT, TCTA/GATA, GAAA/CTTT, CTTTT, and AGAGAT, according to the repetitive motif present in the DNA leading strand of the markers. Results show that the occurrence of multistep mutations varies significantly among groups of markers defined by the repetitive motif. The group of markers with the highest frequency of multistep mutations was the one with repetitive motif CTTTT (25% of the detected mutations) and the lowest frequency corresponding to the group with repetitive motifs TCTA/GATA (0.93%). Statistically significant differences (α = 0.05) were found between groups with repetitive motifs with different lengths, as is the case of TCTA/GATA and ATT (p = 0.0168), CTT (p < 0.0001) and CTTTT (p < 0.0001), as well as between GAAA/CTTT and CTTTT (p = 0.0102). The same occurred between the two tetrameric groups GAAA/CTTT and TCTA/GATA (p < 0.0001) – the first showing 5.7 times more multistep mutations than the second. When considering the number of repeats of the mutated paternal alleles, statistically significant differences were found for alleles with 10 or 12 repeats, between GATA and ATT structure groups. These results, which demonstrate the heterogeneity of mutational dynamics across repeat motifs, have implications in the fields of population genetics, epidemiology, or phylogeography, and whenever STR mutation models are used in evolutionary studies in general.

Stochastic Approach to the Quantum Noise of a Single-Emitter Nanolaser.

It is shown that the quantum intensity noise of a single-emitter nanolaser can be accurately computed by adopting a stochastic interpretation of the standard rate equation model. The only assumption made is that the emitter excitation and photon number are stochastic variables with integer values. This extends the validity of rate equations beyond the mean-field limit and avoids using the standard Langevin approach, which is shown to fail for few emitters. The model is validated by comparison to full quantum simulations of the relative intensity noise and second-order intensity correlation function, g^{(2)}(0). Surprisingly, even when the full quantum model displays vacuum Rabi oscillations, which are not accounted for by rate equations, the intensity quantum noise is correctly predicted by the stochastic approach. Adopting a simple discretization of the emitter and photon populations, thus, goes a long way in describing quantum noise in lasers. Besides providing a versatile and easy-to-use tool for modeling emerging nanolasers, these results provide insight into the fundamental nature of quantum noise in lasers.

(10,10)-THRESHOLD SECRET-SHARING SCHEME USING EQUIVALENCE CLASSES OF INTEGERS MODULO A PRIME

Introduction: The quest for increasing the security of data in secret-sharing schemes has attracted much attention in the world of cryptography. Several methods have been applied, and the application of the new technique, (n,n)-threshold secretsharing scheme based on equivalence classes, will be a perfect solution. Aims: The aim is to explore and construct strong knowledge in the theory and structure of the secret-sharing scheme on (n,n)- threshold secret-sharing scheme based on equivalence classes; (𝑋) = (𝑦 𝜖 𝑋: (𝑥, 𝑦)𝜖 ℝ) And achieve the following objectives: i. To investigate an (𝑛, 𝑛)-threshold secret-sharing scheme based on the equivalence classes of the prime over ℤ. ii. Investigation of the accuracy of such a scheme. Materials and Methods: The study uses the set of integers modulo a prime and modulo arithmetic on the set of integers. Results: The statistics on the coalition, security analysis, and information-theoretic efficiency are also discussed. Conclusion: The secret sharing scheme on (10,10)-threshold secret-sharing scheme based on equivalence classes of integers modulo a prime is perfect in terms of only qualified coalitions can obtain the secret and it is reliable by means of security. We have used the property of these classes to provide the reconstruction algorithms, and access structures and calculated the number of minimal coalitions of the scheme. This new system is ideal in that the size of the secret is equal to the size of the share.

Optimasi Pengurutan Data Bilangan dengan Menggabungkan Algoritma Selection Sort Hybrid dan Bucket Sort

Sorting algorithms are crucial in data processing, particularly for integer data. However, as the number of integers to be sorted increases, the sorting algorithm takes longer to complete, especially for algorithms with O(n2) complexity. This article discusses optimizing integer data sorting by combining the Selection Sort Hybrid and Bucket Sort algorithms. The study aims to test the performance of the Selection Sort Hybrid and Bucket Sort algorithms and compare them with other data sorting algorithms. The research method used is experimental quantitative research, using randomly generated data using Python. The data were tested using the Combined Selection Sort Hybrid with Bucket Sort algorithm, Selection Sort Hybrid, Quick Sort, and Merge Sort. Data analysis was done by calculating the execution time of each sorting algorithm. The results show that the Selection Sort Hybrid and Bucket Sort algorithms are faster than other sorting algorithms in testing with large and complex integer data. Therefore, combining Selection Sort Hybrid and Bucket Sort algorithms can improve the efficiency and speed of sorting complex integer data.

Exceptional Jacobi polynomials which are deformations of Jacobi polynomials

In [10] we constructed new examples of exceptional Jacobi polynomials. The difference with the so called standard families is that the examples constructed in [10] depend on an arbitrary number of continuous parameters. Besides these continuous parameters, denoted by M={M0,M1,…}, each one of the new families of exceptional Jacobi polynomials is associated to two negative integers a≤b≤−1 and a finite set of positive integers F satisfying that {−b,…,−a−b−1}⊂F. In this paper we prove that when F={u,u+1,…,−a−b−1}, with 1≤u≤−b, the exceptional Jacobi polynomials are deformations of Jacobi polynomials (when the continuous parameters Mi→1). Otherwise, the exceptional Jacobi polynomials are deformations of standard exceptional Jacobi polynomials (again when the continuous parameters Mi→1).

Some remarks on the Erdős Distinct subset sums problem

Let [Formula: see text] be a set of positive integers, [Formula: see text] denoting the largest element, so that for any two of the [Formula: see text] subsets the sum of all elements is distinct. Erdős asked whether this implies [Formula: see text] for some universal [Formula: see text]. We prove, slightly extending a result of Elkies, that for any [Formula: see text], [Formula: see text] with equality if and only if all subset sums are [Formula: see text]-separated. This leads to a new proof of the currently best lower bound [Formula: see text]. The main new insight is that having distinct subset sums and [Formula: see text] small requires the random variable [Formula: see text] to be close to Gaussian in a precise sense.

Almost all sets of nonnegative integers and their small perturbations are not sumsets

Fix α ∈ ( 0 , 1 / 3 ) \alpha \in (0,1/3) . We show that, from a topological point of view, almost all sets A ⊆ N A\subseteq \mathbb {N} have the property that, if A ′ = A A^\prime =A for all but o ( n α ) o(n^{\alpha }) elements, then A ′ A^\prime is not a nontrivial sumset B + C B+C . In particular, almost all A A are totally irreducible. In addition, we prove that the measure analogue holds with α = 1 \alpha =1 .

On integer values of sum and product of three positive rational numbers

On integer values of sum and product of three positive rational numbers

Notes on the Equation d(n) = d(φ(n)) and Related Inequalities

ABSTRACT Let d(n) denotes the number of positive integers dividing the positive integer n, and let φ(n) denotes Euler’s function representing the number of numbers less than and prime to n. In this paper, we present some notes on the equation d(n) = d(φ(n)). Several results on the related inequalities are also obtained.

Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}

ABSTRACT Let M be a set of positive integers. We study the maximal density μ(M) of the sets of nonnegative integers S whose elements do not differ by an element in M. In 1973, Cantor and Gordon established a formula for μ(M) for |M| ≤ 2. Since then, many researchers have worked upon the problem and found several partial results in the case |M| ≥ 3, including some results in the case when M is an infinite set. In this paper, we study the maximal density problem for the families M = {a, a+1, 2a+1, n} and M = {a, a+1, 2a+1, 3a+1, n}, where a and n are positive integers and n is sufficiently large. In most of the cases, we find bounds for the parameter kappa, denoted by κ(M), which actually serves as a lower bound for μ(M). The parameter κ(M) has already got its importance due to its rich connection with problems such as the “lonely runner conjecture” in Diophantine approximation and coloring parameters such as “circular coloring” and “fractional coloring” in graph theory. We also give some partial results for the general family M = {a, a +1, 2a + 1,…, (s – 2)a + 1, n}, where s ≥ 5 and mention related problems in the remaining cases for future work.

3-Vertex friendly index sets of operations of two graphs

The process of assigning integer values either to vertices or edges of a graph is known as graph labeling. A mapping f:VG→{0,1,2} is called ternary vertex labeling and fv represents the vertex label for v. The set FI3vG={ef∗i-ef∗j:f∗ runs over all 3-vertex friendly labeling f for all i,j∈{0,1,2}} is referred as 3-vertex friendly index set. In order to achieve FI3vG, number of vertices are partitioned into V0,V1,V2 such that Vi-Vj≤1for alli,j=0,1,2 with i≠j and label the edge uv by |f(u)-f(v)|wheref(u),f(v)∈{0,1,2}. A study of 3-vertex friendly index sets of operations of two graphs such as union, join and cartesian product are presented in this paper.

Coupling covariance matrix adaptation with continuum modeling for determination of kinetic parameters associated with electrochemical CO2 reduction

Coupling covariance matrix adaptation with continuum modeling for determination of kinetic parameters associated with electrochemical CO2 reduction

Destructive relativity.

The description of dynamics for high-energy particles requires an application of the special relativity theory framework, and analysis of properties of the corresponding equations of motion is very important. Here, we analyze Hamilton equations of motion in the limit of weak external field when potential satisfies the condition 2V(q)≪mc2. We formulate very strong necessary integrability conditions for the case when the potential is a homogeneous function of coordinates of integer non-zero degrees. If Hamilton equations are integrable in the Liouville sense, then eigenvalues of the scaled Hessian matrix γ-1V″(d) at any non-zero solution d of the algebraic system V'(d)=γd must be integer numbers of appropriate form depending on k. As it turns out, these conditions are much stronger than those for the corresponding non-relativistic Hamilton equations. According to our best knowledge, the obtained results are the first general integrability necessary conditions for relativistic systems. Moreover, a relation between the integrability of these systems and corresponding non-relativistic systems is discussed. The obtained integrability conditions are very easy to use because the calculations reduce to linear algebra. We show their strength in the example of Hamiltonian systems with two degrees of freedom with polynomial homogeneous potentials. It seems that the only integrable relativistic systems with such potentials are those depending only on one coordinate or having a radial form.

Spectrum-Efficient Analog Pulse Index Modulation for High-Volume Wireless Data Telemetry

Spectrum-efficient, high-density, and high-speed data transmission are essential to store and exchange sensitive information through Internet-of-Things (IoT) networks. However, the resource-constrained IoT devices and the limited existing frequency spectrum pose a significant challenge to employing the more efficient technique for the next-generation technology. This paper presents an innovative pulse index modulation (PIM) architecture with a randomization technique for analog pulse-based packet-less high-volume data transmission. The pulse-based compression simplifies the encoding and decoding schemes for short-range wireless telemetry. The pulse randomization technique creates the pulse blinding by reordering the pulse positions and adds an extra level of physical layer security. This paper describes the design, simulation, and prototype development of PIM to validate the feasibility and compatibility of pulse-based data telemetry using the standard architecture. Simulation results show that the proposed PIM increases the data compression ratio by 2.61 times and improves the processing time by 5.56 time compared to the Huffman coding for a 3-bit system. Test results show that the BER is 7.5×10-4 at 10.0 dB SNR, which satisfies the lower bound for data communication and increases the transmission speed k-times by supporting 2k*k bits data using 2k number of pulses, where k is a non-negative integer number. The proposed PIM has the potential to improve data rate with added security to support the next-generation IoT networks.

Monophonic pebbling number of some families of cycles

Let [Formula: see text] be a simple connected graph. A configuration of pebbles is a function from [Formula: see text] to a set of integers. Consider a configuration of pebbles on [Formula: see text]. A pebbling move consists of removing two pebbles off a vertex and placing one on an adjacent vertex. The monophonic pebbling number of [Formula: see text] is the smallest integer [Formula: see text] from which we can put one pebble to a target using monophonic path through pebbling moves. The monophonic [Formula: see text]-pebbling number of [Formula: see text] is the smallest positive integer [Formula: see text] such that from any configuration of [Formula: see text] pebbles we can put [Formula: see text] pebbles on a target using monophonic path. Here we discuss these concepts for families of cycles.

A Bound for a Sum of Products of Two Characters and Its Application

Using the exponent pair method, a bound is derived for the sum ∑manb≤xχ1a(m)χ2b(n), where a,b are fixed positive integers, χ1,χ2 are primitive Dirichlet characters modulo q1 and q2, respectively, and χ1a,χ2b are not principal characters. As an application, an estimate for the error term in an asymptotic formula for the number of square-full integers simultaneously belonging to two arithmetic progressions is obtained.