Number Of Positive Integers Research Articles

On a system of difference equations of third order solved in closed form

In this work, we show that the system of difference equationsxn+1=(ayn-2xn-1yn+bxn-1yn-2+cyn-2+d)/(yn-2xn-1yn),yn+1=(axn-2yn-1xn+byn-1xn-2+cxn-2+d)/(xn-2yn-1xn),where n belongs to the set of positive integer numbers, x-2, x-1, x0, y-2, y-1 and y0 are arbitrary nonzero real numbers, and the parameters a, b, c and d are arbitrary real numbers with d nonzero can be solved in a closed form.We will see that when a = b = c = d = 1, the solutions are expressed using the famous Tetranacci numbers. In particular, the results obtained here extend those in our recent work.

On the number of semismooth integers

Let [Formula: see text] be a fixed positive number. Define [Formula: see text] as the number of positive integers [Formula: see text] having no prime factors [Formula: see text], and define [Formula: see text] as the number of positive integers [Formula: see text] having [Formula: see text] prime factors [Formula: see text], with all the other prime factors [Formula: see text]. In this paper, we give an asymptotic estimate for the ratio [Formula: see text], provided that [Formula: see text], [Formula: see text], and [Formula: see text] as [Formula: see text]. Also, combining this estimate with conventional ones for [Formula: see text], we provide sharp estimates for [Formula: see text].

The graph of a base power b, associated to a positive integer number

The graph of a base power b, associated to a positive integer number

An asymptotic version of the prime power conjecture for perfect difference sets

We show that the number of positive integers $$n\le N$$ such that $${\mathbb {Z}}/(n^2+n+1){\mathbb {Z}}$$ contains a perfect difference set is asymptotically $$\frac{N}{\log {N}}$$ .

On local metric dimensions of m-neighbourhood corona graphs

For a nontrivial connected graph G, let VG be a vertex set {v1, v2, …, vn }, J be an index set {1,2, …, n}, and H be any graph. Representation r(x|Z) of a vertex x in VG with respect to an ordered subset Z = {z 1, z 2, …, zk } of VG is k-tuple (dG (x, z 1), dG (x, z 2), …, dG (x, zk )), with dG (x, Zj ) is distance from x to zj for every j in J. If each of two adjacent vertices in VG has different representation, then Z is the local resolving set for G. Local base for G is resolving set Z with minimum number of vertices. Cardinality of base of G, is called the local metric dimension for G, dim l (G). For a positive integer m, an m-neighbourhood-corona of G and H, G * mH, is obtained by taking G and as many as | VG | graph mHj , where j ∈ {1, 2, …, |VG |} and Hj is the jth copy of the graph H, then making each vertex on the mHj graph adjacent to the neighbours of vertex vj in G. This article provides the exact values and characteristics of the local metric dimensions of the graphs generated from the m-neighbourhood-corona operations of G and H graphs, dim l (G*mH), with G ∈ { Pn, Cn, Kn, Ks,t ; with s + t = n ≥2, and s and t be a positive integer number, and Wn } and H=K 1 and their proofs. We gave dim l (Kn *mK 1) = n – 1, and dim l (Cn*mK 1) = 2; for odd n. Furthermore, dim l (G*mK 1) = 1 if only if G ∈ { Pn, Cn for even n, Ks,t; s + t = n ≥ 2}. For a graph operating result Wn , we gave dim l (Wn *mK 1) = n - 4, n ≥ 7.

On sum of prime factors of composite positive integers

Let $${\mathfrak {B}}(x)$$ be the number of composite positive integers up to x whose sum of distinct prime factors is a prime number. Luca and Moodley proved that there exist two positive constants $$a_1$$ and $$a_2$$ such that $$\begin{aligned} a_1x/\log ^3x\le {\mathfrak {B}}(x)\le a_2x/\log x. \end{aligned}$$ Assuming a uniform version of the Bateman–Horn conjecture, they gave a conditional proof of a lower bound of the same order of magnitude as the upper bound. In this paper, we offer an unconditional proof of the this result, i.e., $$\begin{aligned} {\mathfrak {B}}(x)\asymp \frac{x}{\log x}. \end{aligned}$$

IMPLEMENTASI ALGORITMA TANDA TANGAN DIGITAL BERBASIS KRIPTOGRAFI KURVA ELIPTIK DIFFIE-HELLMAN

In data communication systems, digital signatures are a form of electronic signature security services based on the Elliptic Curve Digital Signature Algorithm (ECDSA) which are considered resistant to certain types of attacks. Attacks on digital signature schemes aim to fake a signature or are called forgery which is said to be successful if the key pair and signature generated by the attacker are accepted by the verifier. Mathematical schemes used to prove the authenticity of messages or digital documents or guarantees that the data and information actually come from the correct source. ECDSA-based digital signatures rely on discrete logarithmic problems as the basis for mathematical calculations. Q = kP where Q and P are the points of the elliptic curve in the finite field or and k is a positive integer number. The hash function generated from the algorithm process is then encoded (encrypted) with an asymmetric key cryptographic algorithm. In this work use p = 149 to encrypt plain text by converting the original message using dots on a curve with the help of Python programs.

Yau type gradient estimates for Δu + au(log⁡u)p + bu = 0 on Riemannian manifolds

In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold (M,g)Δu+au(log⁡u)p+bu=0, where a,b∈R and p is a rational number with p=k12k2+1≥2 where k1 and k2 are positive integer numbers. We obtain the gradient bound of a positive solution to the equation which does not depend on the bounds of the solution and the Laplacian of the distance function on (M,g). Our results can be viewed as a natural extension of Yau's estimates on positive harmonic function.

Analytic continuation of harmonic sums near the integer values

We present a simple method for analytic continuation of harmonic sums near negative and positive integer numbers. We provide a precomputed database for the exact expansion of harmonic sums over a small parameter near these integer numbers, along with MATHEMATICA code, which shows the application of the database for actual problems. We also provide the FORM code that was used to obtain the database mentioned above. The applications of the obtained database for the study of evolution equations in the quantum field theory are discussed.

A branch-and-price algorithm for the Minimum Sum Coloring Problem

A proper coloring of a given graph is an assignment of a positive integer number (color) to each vertex such that two adjacent vertices receive different colors. This paper studies the Minimum Sum Coloring Problem (MSCP), which asks for finding a proper coloring while minimizing the sum of the colors assigned to the vertices. We propose the first branch-and-price algorithm to solve the MSCP to proven optimality. The newly developed exact approach is based on an Integer Programming (IP) formulation with an exponential number of variables which is tackled by column generation. We present extensive computational experiments, on synthetic and benchmark DIMACS graphs from the literature, to compare the performance of our newly developed branch-and-price algorithm against three compact IP formulations. On synthetic graphs, our algorithm outperforms the compact formulations in terms of: (i) number of solved instances, (ii) running times and (iii) exit gaps obtained when optimality is not achieved. For the DIMACS instances, our algorithm is competitive with the best compact formulation and provides very strong dual bounds.

A Numerical Study on the Regularity of d-Primes via Informational Entropy and Visibility Algorithms

Let a d -prime be a positive integer number with d divisors. From this definition, the usual prime numbers correspond to the particular case d = 2 . Here, the seemingly random sequence of gaps between consecutive d -primes is numerically investigated. First, the variability of the gap sequences for d ∈ 2,3 , … , 11 is evaluated by calculating the informational entropy. Then, these sequences are mapped into graphs by employing two visibility algorithms. Computer simulations reveal that the degree distribution of most of these graphs follows a power law. Conjectures on how some topological features of these graphs depend on d are proposed.

Tropical Geometry: new directions

The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q -numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems.

Odd harmonious labeling on squid graph and double squid graph

An injective function f from set of vertices in graph G to a set of {0,1,…,|E| − 1} is called an odd harmonious labeling if the function f induced the edge function f* from the set of edges of G to a set of odd positive integer number {1,3,5,…,2|E| − 1} with f*(xy) = f(x) + f(y) for every edge xy in E. Graph that has an odd harmonious labeling is called odd harmonious graph. The squid graph Tn,k is a graph which is obtained from a cycle Cn and we add k pendant to one vertex of the cycle. It is known that Cn is an odd harmonious graph if and only if n = 0 mod 4. However, by adding at least one pendant in the cycle graph, we can label the new graph odd harmoniously for all even number of vertices. In this paper, we showed that the graph Tn,k and T 2n,k are an odd harmonious graph, for n = 0 (mod 2), n ≥ 4 and k ≥ 1. The construction of the odd harmonious labeling of the graph Tn,k and T 2n,k are inspired by the odd harmonious labeling of Cn for n = 0(mod 4).

THE DISTRIBUTION OF DIVISORS OF POLYNOMIALS

Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for $y+y/\log^C y < z\le y^2$ and $y\le x^{1-\delta}$, showing that the order is the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$ with a divisor in $(y,z]$. Here $C$ is an arbitrarily large constant and $\delta>0$ is arbitrarily small.

Waring–Goldbach problem: two squares and three biquadrates

Let ℛ(n) denote the number of representations of the positive integer n as the sum of two squares and three biquadrates of primes and we write ℰ(N) for the number of positive integers n satisfying n≤N, n≡5,53,101(mod120) and | ℛ ( n ) − Γ 2 ( 1 2 ) Γ 3 ( 1 4 ) Γ ( 7 4 ) 𝔖 ( n ) n 3 4 log 5 n | ≥ n 3 4 log 1 1 2 n , where 0<𝔖(n)≪1 is the singular series. In this paper, we prove ℰ ( N ) ≪ N 1 5 3 2 + 𝜀 for any 𝜀>0. This result constitutes a refinement upon that of Friedlander and Wooley (2014).

The total H-irregularity strength of triangular ladder graphs

Let G be a graph which has V(G) as a vertex set and E(G) as an edge set. The G graph contain subgraph H that isomorphic with Hj, j = 1, 2, … s, if every e ∈ E(G) include only in one of the edge set of H subgraph (e ∈ E(H)). The total l-labeling is a graph labeling that give label positive integer number until l into vertices and edges. The total H-irregular labeling is a total l-labeling with condition that the sum of vertex labels and edge labels in two distinct subgraphs H 1 and H 2 isomorphic to H is different. We define H-weight as wtφ (H) = ∑ v∈V(H) φ(v) + ∑ e∈E(H) φ(e), for the subgraph H ⊆ G under the total l-labeling (φ). The total H-irregularity strength of G graph (ths(G, H)) is the smallest l value in label of G graph has total H-irregular labeling. We used plane graphs such as triangular ladder graph and windmill graph.

(A-n) – Potent operators on Hilbert space

In this paper, we introduce a new class of operators on a complex Hilbert space which is called (A-n)-potent operator. An operator is called (A-n)-potent operator if where is positive integer number greater than or equal 2. We investigate some basic properties of such operators and study the relation between (A-n)-potent operators and some kinds of operators.

Beurling numbers whose number of prime factors lies in a specified residue class

We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers.

Tuning band structures of photonic multilayers with positive and negative refractive index materials according to generalized Fibonacci and Thue–Morse sequences

We investigate the photonic band structure, using the transfer-matrix method, in one-dimensional structures composed of a dispersive metamaterial A juxtaposed with a non-dispersive dielectric B according to the generalized Fibonacci and Thue–Morse sequences, which are ruled and characterized by two positive integer numbers, p and q. We present the band structures of different generations of these sequences for both TE and TM modes and discuss how they are affected by the p and q parameters and the ratio between the thicknesses of the building blocks. We obtain a new and very important analytical expression for the frequency in which the average refraction index vanishes, i.e. when the gap condition is satisfied, and we investigate, in detail, the behavior of this emergent scale insensitive gap, as a function of the thicknesses ratio, for all structures considered. For the metamaterial considered, we show that the reduced frequency for converges faster or slower to 0.41 depending on the sequence. By adjusting the p and q parameters, we can obtain a higher concentration of the pass bands inside (outside) the nA < 0 region when there are more (less) blocks A than B in the unit cell. Our results also show the presence of omnidirectional and complete band gaps, which have various practical and technological applications.

Design of Planar Differential Microphone Arrays With Fractional Orders

Differential microphone arrays (DMAs) often encounter white noise amplification, especially at low frequencies. If the array geometry and the number of microphones are fixed, one can improve the white noise amplification problem by reducing the DMA order. With the existing differential beamforming methods, the DMA order can only be a positive integer number. Consequently, with a specified beampattern (or a kind of beampattern), reducing this order may easily lead to over compensation of the white noise gain (WNG) and too much reduction of the directivity factor (DF), which is not optimal. To deal with this problem, we present in this article a general approach to the design of DMAs with fractional orders. The major contributions of this article include but are not limited to: 1) we first define a directivity pattern that can achieve a continuous compromise between the pattern corresponding to the maximum DMA order and the omnidirectional pattern; 2) by approximating the beamformer's beampattern with the Jacobi-Anger expansion, we present a method to find the proper differential beamforming filter so that its beampattern matches closely the target directivity pattern of fractional orders; and 3) we show how to determine analytically the proper fractional order of the DMA with a given target beampattern when either the value of the DF or WNG is specified, which is useful in practice to achieve the desired beampattern and spatial gain while maintaining the robustness of the DMA system.