Number Of Positive Integers Research Articles

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.

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.

Numerically explicit estimates for the distribution of rough numbers

For x≥y>1 and u:=log⁡x/log⁡y, let Φ(x,y) denote the number of positive integers up to x free of prime divisors less than or equal to y. In 1950 de Bruijn [4] studied the approximation of Φ(x,y) by the quantityμy(u)eγxlog⁡y∏p≤y(1−1p), where γ=0.5772156... is Euler's constant andμy(u):=∫1uyt−uω(t)dt. He showed that the asymptotic formulaΦ(x,y)=μy(u)eγxlog⁡y∏p≤y(1−1p)+O(xR(y)log⁡y) holds uniformly for all x≥y≥2, where R(y) is a positive decreasing function related to the error estimates in the Prime Number Theorem. In this paper we obtain numerically explicit versions of de Bruijn's result.

Optical properties of Eu3+ - doped Sr2TiO4 phosphor powder synthesised by solid-phase reaction method

Fluorescent powder Sr2TiO4 doped with Eu3+ ions was synthesised by the solid-phase reaction method with a sintering temperature of 1200oC in the air with a 1-6% doping concentration. The resulting material has a perovskite structure of I4/mmm space group, and the Eu3+ ion doping concentration does not change the material’s structure. This characteristic space group of the material was also investigated through the Raman scattering spectrum with characteristic peaks at positions 145, 240, 448, and 699 cm-1 corresponding to vibrational modes with 2A1g + 2Egcharacteristic states of Sr2TiO4 material. The results showed that the material absorbs strongly in the ultraviolet and blue light regions with characteristic peaks of the Sr2TiO4 matrix and Eu3+ ions to emit strongly in the red light region with characteristic emission peaks of Eu3+ ions in the background network when moving from the excited state 5D0 to state 7Fj (j is a positive integer number). Fluorescence quenching was also observed at a doping concentration of 4% Eu3+ ions. The fabricated material is suitable for coating applications on nUV-LED LED chips.

On the Hurwitz stability of noninteger Hadamard powers of stable polynomials

Consider a polynomial f(z)=anzn+...+a1z+a0 of positive coefficients that is stable (in the Hurwitz sense), i.e., every root of f lies in the open left half-plane of C. Due to Garloff and Wagner [J. Math. Anal. Appl. 202 (1996)], the pth Hadamard power of f: f[p](z):=anpzn+...+a1pz+a0p is stable if p is a positive integer number. However, it turns out that f[p] does not need to be stable for all real p>1. A counterexample is known for n=8 and p=1.139. On the other hand, f[p] is stable for n=1,2,3,4, and every p>1. In this paper we fill the gap by showing that f[p] is stable for n=5 and constructing counterexamples for n≥6. Moreover, by means of Rouché's Theorem, we give some stability conditions for polynomials and two examples that complete and illustrate the results.

Design of Low-Density Parity-Check Code Pair for Joint Source-Channel Coding Systems Based on Graph Theory.

In this article, a graph-theoretic method (taking advantage of constraints among sets associated with the corresponding parity-check matrices) is applied for the construction of a double low-density parity-check (D-LDPC) code (also known as LDPC code pair) in a joint source-channel coding (JSCC) system. Specifically, we pre-set the girth of the parity-check matrix for the LDPC code pair when jointly designing the two LDPC codes, which are constructed by following the set constraints. The constructed parity-check matrices for channel codes comprise an identity submatrix and an additional submatrix, whose column weights can be pre-set to be any positive integer numbers. Simulation results illustrate that the constructed D-LDPC codes exhibit significant performance improvement and enhanced flexible frame length (i.e., adaptability under various channel conditions) compared with the benchmark code pair.

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.

An Idempotent Cryptarithm

Summary Notice that the square of 9376 is 87909376 which has as its rightmost four digits 9376. To generalize this remarkable fact, we show that, for each integer , there exists at least one and at most two positive integers x with exactly n-digits in base 10 (meaning the leftmost or nth digit from the right is nonzero) such that squaring the integer results in an integer whose rightmost n digits form the integer x. We then generalize the argument to prove that, in an arbitrary number base with exactly m distinct prime factors, an upper bound is and a lower bound is for the number of such n-digit positive integers. For , there are exactly solutions, including 1 and excluding 0.

Quasi-normal Operator of order n

In this paper, we introduce a new class of operators acting on a complex Hilbert space H which is called quasi-normal operator of order n. An operator T∈B(H) is called quasi-normal operator of order n if T(T*n Tn)=(T*n Tn)T, where n is positive integer number greater than 1 and T* is the adjoint of the operator T, We investigate some basic properties of such operators and study relations among quasi-normal operator of order n and some other operators.

Integral points on cubic twists of Mordell curves

Fix a non-square integer $$k\ne 0$$ . We show that the number of curves $$E_B:y^2=x^3+kB^2$$ containing an integral point, where B ranges over positive integers less than N, is bounded by $$\ll _k N(\log N)^{-\frac{1}{2}+\epsilon }$$ . In particular, this implies that the number of positive integers $$B\le N$$ such that $$-3kB^2$$ is the discriminant of an elliptic curve over $$\mathbb {Q}$$ is o(N). The proof involves a discriminant-lowering procedure on integral binary cubic forms.

A FRAMEWORK FOR PROCESS-RESOURCE REQUIREMENTS PREDICTION IN A FACTORED OPERATING SYSTEMS USING MACHINE LEARNING

The evolution of multicore technology has come with new challenges to process scheduling. An intelligent process resource requirement prediction framework in a factored operating system (FOS) was developed using machine learning. To simulate this system, an array of positive integer numbers in the interval of 1 to 10, with a maximum length of 20, was used to represent process. The properties of this array (array length and sum of the array elements) were used as the parameters of the application processes. These served as input into the machine learning network to predict the process’ size and hence, the resource requirements for each process. The framework was simulated using Java 2EE. Experimental results of the framework showed that prediction of processes’ sizes with enhanced resource requirement allocation, and hence, required faster and efficient scheduling of processes to multiple resource cores, with an increased system throughout.

On the average value of the first $ n $ values of the sum-of-divisors function

We prove that the number of positive integers $ n \leq x $ dividing $ \sigma (1) + \cdots + \sigma (n)$ is less than $x/(\log x)^{0.15742} $ for all sufficiently large $ x $, where $ \sigma $ stands for the sum-of-divisors function.

Advances in additive number theory

We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of \pi(x) for enough large real x, being \pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to \pi(x).

A new theorem on quadratic residues modulo primes

where N p (a,b) is the number of positive integers x<p/2 with {x 2 +b} p >{ax 2 +b} p , and {m} p with m∈ℤ is the least nonnegative residue of m modulo p.

Extended bifocal depth imaging with modified generalized composite kinoform Fibonacci lenses

A modified generalized composite kinoform Fibonacci lens (MGCKFiL) is proposed to produce two foci with the same resolution and long focal depths. The MGCKFiL is based on a generalized composite kinoform Fibonacci lens (GCKFiL) and a phase generated by the weighted Gerchberg-Saxton algorithm. This phase expands the focal depths of two foci for the GCKFiL. Compared with the conventional kinoform Fibonacci lens (KFiL), the GCKFiL has the two equal-intensity foci with higher intensity. The GCKFiL with the positive integer number p = 4q + 2 has the two equal-intensity foci along the optic axis. When p is big enough, the bifocal positions of the GCKFiL with the bigger order S have the ratio rounding toward one. Furthermore, the MGCKFiL has the two images with the same size at the two main focal positions, and achieves the extended bifocal depth imaging. The proposed zone plates are applicable to generate the two same-size images, and achieve extended bifocal depth imaging and multi-planar optical trapping.

An investigation on the triple ideal convergent sequences in fuzzy metric spaces

The notion of ideal convergence is a process of generalizing of statistical convergence which is dependent on the idea of the ideal $I$ of subsets of the set positive integer numbers. In this study we also present the concept of ideal convergence for triple sequences in fuzzy metric spaces (FMS) in the manner of George and Veeramani and the terms of ideal Cauchy sequence and $I^{∗}$-Cauchy sequence in FMS and study their certain properties.

The balanced metrics and cscK metrics on certain holomorphic ball bundles

In this paper, we use the well-known Calabi ansatz, further generalized by Hwang-Singer, to study the existence of balanced metrics and constant scalar curvature Kähler (cscK for short) metrics on certain holomorphic ball bundles M which are locally expressed as M={(z,u,w)∈Ω×Cd1×Cd2:eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2<1}. Let g be the Kähler metric on M associated with the Kähler forms locally expressed as ω=−1∂∂‾(ϕ(z)+F(eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2)). Firstly, we obtain sufficient and necessary conditions for g to be cscK metrics. Secondly, using this result, we obtain necessary and sufficient conditions for mg to be balanced metrics for all sufficiently large positive integer numbers m. Finally, we obtain complete cscK metrics and balanced metrics on the ball bundles over simply connected Riemann surfaces. The main contribution of this paper is the explicit construction of complete, non-compact cscK metrics and balanced metrics.

Elements of prime order in Tate–Shafarevich groups of abelian varieties over

Abstract For each prime p, we show that there exist geometrically simple abelian varieties A over ${\mathbb Q}$ with . Specifically, for any prime $N\equiv 1 \ \pmod p$ , let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point P of order p, and let $B = A_f/\langle P \rangle $ . Then the number of positive integers $d \leq X$ with is $ \gg X/\log X$ , where $\widehat B_d$ is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of $\operatorname {\mathrm {GL}}_2$ -type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.

ENCRYPTION BASED ON CONFERENCE MATRIX

In this article, an encryption scheme based on conference matrix has been developed. An easier algorithm of formation of encryption/decryption keys have been discussed. The decryption key comprising of fixed number of positive integers with prime power yields a high level security of message. Some popular attacks have been discussed in the context of cryptoanalysis and observed that it is robust against the popular cipher attack and the security of the information does not compromise.

Finite groups with a nilpotency condition

Let m and n be positive integer numbers. In this paper, we study [Formula: see text], the class of all groups G that for all subsets m and n of G containing m and n elements, respectively, there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] is nilpotent, which introduced by Zarrin in 2012. We improve some results of Zarrin and find some sharp bounds for m and n such that [Formula: see text] implies that G is nilpotent. Also we will characterize all finite [Formula: see text]-groups in [Formula: see text], which [Formula: see text].