Positive Integer Research Articles

Explicit evaluation of triple convolution sums of the divisor functions

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums [Formula: see text] for odd integers [Formula: see text] and [Formula: see text], where [Formula: see text] is the sum of the [Formula: see text]th powers of the positive divisors of [Formula: see text]. We consider four cases, namely (i) [Formula: see text], (ii) [Formula: see text]; [Formula: see text] (iii) [Formula: see text]; [Formula: see text] and (iv) [Formula: see text], and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer [Formula: see text] by certain positive definite quadratic forms. The existing formulas for [Formula: see text] (in [20]), [Formula: see text] (in [7]), [Formula: see text] (in [35]), [Formula: see text], [Formula: see text] (in [30]) and [Formula: see text] (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

Cycle‐factors in oriented graphs

AbstractLet be a positive integer. A ‐cycle‐factor of an oriented graph is a set of disjoint cycles of length that covers all vertices of the graph. In this paper, we prove that there exists a positive constant such that for sufficiently large, any oriented graph on vertices with both minimum out‐degree and minimum in‐degree at least contains a ‐cycle‐factor for any . Additionally, under the same hypotheses, we also show that for any sequence with and the number of the equal to 3 is , where is any real number with , the oriented graph contains disjoint cycles of lengths . This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov.

Full Classification of Finite Singleton Local Rings

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants p,n,s, and t, where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of p6 and p7 that hold substantial importance in the field of coding theory.

On the gamma difference distribution

The gamma difference distribution is defined as the difference of two independent gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear differential equation specification of the probability density function is given. This is used to derive a Stein-type differential identity relating to the expectation with respect to the gamma difference distribution of a general twice differentiable function g(x). Choosing g(x)=xk gives a second order recurrence for the positive integer moments, which are also shown to permit evaluations in terms of 2F1 hypergeometric polynomials. A hypergeometric function evaluation is given for the absolute continuous moments. Specialising the gamma difference distribution gives the variance gamma distribution. Results of the type obtained herein have previously been obtained for this distribution, allowing for comparisons to be made.

Scaling spectrum of a class of self-similar measures with product form on ℝ

Abstract Let p, q, N ≥ 2 {N\geq 2} be three positive integers and let D = { 0 , 1 , … , N - 1 } ⊕ N p ⁢ { 0 , 1 , … , N - 1 } {D=\{0,1,\ldots,N-1\}\oplus N^{p}\{0,1,\ldots,N-1\}} be a product form digit set. It is well known that if q ∤ p {q\nmid p} , then the self-similar measure μ N q , D {\mu_{N^{q},D}} generated by the iterated function system { ( N q ) - 1 ⁢ ( x + d ) } d ∈ D , x ∈ ℝ {\{(N^{q})^{-1}(x+d)\}_{d\in D,x\in\mathbb{R}}} is a spectral measure with a spectrum Λ ⁢ ( N q , C ) = { ∑ i = 0 finite c i ⁢ N q ⁢ i : c i ∈ C } , \Lambda(N^{q},C)=\Bigg{\{}\sum_{i=0}^{\text{finite}}c_{i}N^{qi}:c_{i}\in C% \Bigg{\}}, where C = N q - p - 1 ⁢ D {C=N^{q-p-1}D} . In this paper, based on the properties of cyclic groups in number theory, we give some conditions on real number t under which the scaling set t ⁢ Λ ⁢ ( N q , C ) {t\Lambda(N^{q},C)} is also a spectrum of μ N q , D {\mu_{N^{q},D}} .

Adjacency preserving maps between tensor spaces

Let r and s be positive integers such that r⩾3. Let U1,…,Ur be vector spaces over a field F and V1,…,Vs be vector spaces over a field K such that dim⁡Ui,dim⁡Vj⩾2 for all i,j. In this paper, we characterize maps ψ:⨂i=1rUi→⨂i=1sVi that preserve adjacency in both directions, which extends Hua's fundamental theorem of geometry of rectangular matrices. We also characterize related results concerning locally full maps preserving adjacency in both directions between tensor spaces, maps preserving adjacency in both directions between tensor spaces over a field all whose nonzero endomorphisms are automorphisms, and injective continuous adjacency preserving maps on finite dimensional tensor spaces over the real field.

Deep learning with adaptive convolutions for classification of retinal diseases via optical coherence tomography

Optical coherence tomography (OCT) uses interferometry to capture high-resolution cross-sectional images of the retina to diagnose retinal diseases. Convolutional neural networks (CNNs) have become essential for developing efficient computer-aided diagnostic algorithms, but noisy images can hinder their performance. This study introduces an innovative image preprocessing strategy that involves a new method of representing images to reduce image noise and a new adaptive convolution layer. The adaptive convolution layer aims to replace traditional convolution layers for OCT image classification by relying on local’Feature Content’. The proposed image representation is based on Zeckendorf's theorem, which states that every positive integer may be split into a unique sum of distinct, non-adjacent Fibonacci numbers. The proposed approach enables the generation of two separate images, known as the ‘base’ and ‘fine,’ where the ‘base’ image is the denoised image. We assessed our methodology by evaluating against ten filters, comprising a Low-pass filter, Gaussian filter, Wiener filter, Wavelet filter, Guided filter, Lee filter, Frost filter, Kuan filter, Detail Preserving Anisotropic Diffusion (DPAD) filter, and Non-Local Means (NLM) filter. In our study, we found that the proposed filter produced the most favorable results in five of the six no-reference parameters (Blur Percent (BP), Blind/Referenceless Image Spatial Quality Evaluator (BRISQUE), Naturalness Image Quality Evaluator (NIQE), Wavelet Variance (WAVV), Wavelet Variance Radial (WAVR)) used to assess the effectiveness of the proposed image enhancement technique. The OCT dataset utilized in the study was compiled by the University of California San Diego. The proposed adaptive convolution layer and its accompanying activation function were tested using seven OCT image classification CNN architectures. The test architectures comprise OctNET, NT-CNN, AOCT-NET, M-CNN, LightOCT, RetiNet, and DeepOCT. Experiments were conducted to assess the impact of the new preprocessing algorithm and the placement of the adaptive convolution layer as a substitute for the conventional convolution layer. Implementing the proposed approaches resulted in accuracy improvements ranging from 0.44% to 2.44% across architectures. Our findings highlight the efficacy of the proposed indirect noise reduction technique and a texture-sensitive adaptive convolution layer.

Estimation of water inflows to reservoirs of irrigation dams in the Guinea Savannah ecological zone of Ghana

ABSTRACT The study was carried out in the northern Ghana to estimate water inflows to nine reservoirs of irrigation dams for a period of 20 years (1999–2018) using the USDA Natural Resources Conservation Service Curve Number model. The key input parameters of the model were hydrologic soil group, rainfall amount, landuse/landcover, weighted curve number, antecedent moisture condition, and potential maximum soil moisture retention. The annual rainfall within the reservoir catchments was found to vary between 617.20 and 1,382.30 mm. The estimated annual runoff depths ranged from 54.10 to 125.55 mm. A high degree of positive correlation was found between rainfall and runoff depths with a coefficient of determination (R2) ranging from 79.80 to 90.70%. Hydrologic soil groups accounted for about 67.2 and 62.5% of the variation in runoff depth and percentage runoff, respectively. The estimated annual water inflows to the reservoirs ranged from 145,206 ± 125,814 m3/y at Gambibgo to 47,074,634 ± 4,395,860 m3/y at Tono. The variation in the annual water inflows and annual storage capacity recharge among the various reservoirs was noted to be influenced by reservoir size, catchment size, rainfall, and soil characteristics in the reservoir catchments.

Free bicommutative superalgebras

We introduce the variety Bsup of bicommutative superalgebras over an arbitrary field of characteristic different from 2. The variety consists of all nonassociative Z2-graded algebras satisfying the polynomial super-identities of super- left- and right-commutativityx(yz)=(−1)x‾y‾y(xz) and (xy)z=(−1)y‾z‾(xz)y, where u‾∈{0,1} is the parity of the homogeneous element u.We present an explicit construction of the free bicommutative superalgebras, find their bases as vector spaces and show that they share many properties typical for ordinary bicommutative algebras and super-commutative associative superalgebras. In particular, in the case of free algebras of finite rank we compute the Hilbert series and find explicitly its coefficients. As a consequence we give a formula for the codimension sequence. We establish an analogue of the classical Hilbert Basissatz for two-sided ideals. We see that the Gröbner-Shirshov bases of these ideals are finite, the Gelfand-Kirillov dimensions of finitely generated bicommutative superalgebras are nonnegative integers and the Hilbert series of finitely generated graded bicommutative superalgebras are rational functions. Concerning problems studied in the theory of varieties of algebraic systems, we prove that the variety of bicommutative superalgebras satisfies the Specht property. In the case of characteristic 0 we compute the sequence of cocharacters.

Vertex degree sums for supereulerian bipartite digraphs

A digraph is considered supereulerian if it possesses a spanning closed ditrail. Let n be positive integer and Dn,n be a bipartite digraph, whose vertices are divided into two equal parts of size n. If Dn,n is strongly connected and δ−+δ+≥n+1, where δ− is its minimum in-degree and δ+ is its minimum out-degree, then Dn,n is supereulerian digraph.

On the Weihrauch degree of the additive Ramsey theorem

We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey-like theorems concerning colourings of the rationals and of the natural numbers. The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals respectively natural numbers satisfying properties determined by some additional algebraic structure on the set of colours. In the context of reverse mathematics, most of the principles we study are equivalent to Σ 2 0 -induction over RCA 0 . The associated problems in the Weihrauch lattice are related to TC N ∗ , ( LPO ′ ) ∗ or their product, depending on their precise formalizations.

A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space

The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in Rm spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects.

Surfaces of General Type with Maximal Picard Number Near the Noether Line

Abstract The first published non-trivial examples of algebraic surfaces of general type with maximal Picard number are due to Persson, who constructed surfaces with maximal Picard number on the Noether line $K^{2}=2\chi -6$ for every admissible pair $(K^{2},\chi )$ such that $\chi \not \equiv 0 \ \text {mod}\ 6$. In this note, given a non-negative integer $k$, algebraic surfaces of general type with maximal Picard number lying on the line $K^{2}=2\chi -6+k$ are constructed for every admissible pair $(K^{2},\chi )$ such that $\chi \geq 2k+10$. These constructions, obtained as bidouble covers of rational surfaces, not only allow to fill in Persson’s gap on the Noether line, but also provide infinitely many new examples of algebraic surfaces of general type with maximal Picard number above the Noether line.

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

The Turán and Laguerre inequalities for quasi-polynomial-like functions

This paper deals with both the higher order Turán inequalities and the Laguerre inequalities for quasi-polynomial-like functions that are expressions of the form f(n)=cl(n)nl+⋯+cd(n)nd+o(nd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(n)=c_l(n)n^l+\\cdots +c_d(n)n^d+o(n^d)$$\\end{document}, where d,l∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d,l\\in \\mathbb {N}$$\\end{document} and d⩽l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\leqslant l$$\\end{document}. A natural example of such a function is the A-partition function pA(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{A}(n)$$\\end{document}, which enumerates the number of partitions of n with parts in the fixed finite multiset A={a1,a2,…,ak}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A=\\{a_1,a_2,\\ldots ,a_k\\}$$\\end{document} of positive integers. For an arbitrary positive integer d, we present efficient criteria for both the order d Turán inequality and the dth Laguarre inequality for quasi-polynomial-like functions. In particular, we apply these results to deduce non-trivial analogues for pA(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_A(n)$$\\end{document}.

Graver bases of shifted numerical semigroups with 3 generators

A numerical semigroup [Formula: see text] is a subset of the non-negative integers that is closed under addition. A factorization of [Formula: see text] is an expression of [Formula: see text] as a sum of generators of [Formula: see text], and the Graver basis of [Formula: see text] is a collection [Formula: see text] of trades between the generators of [Formula: see text] that allows for efficient movement between factorizations. Given positive integers [Formula: see text], consider the family [Formula: see text] of “shifted” numerical semigroups whose generators are obtained by translating [Formula: see text] by an integer parameter [Formula: see text]. In this paper, we characterize the Graver basis [Formula: see text] of [Formula: see text] for sufficiently large [Formula: see text] in the case [Formula: see text], in the form of a recursive construction of [Formula: see text] from that of smaller values of [Formula: see text]. As a consequence of our result, the number of trades in [Formula: see text], when viewed as a function of [Formula: see text], is eventually quasilinear. We also obtain a sharp lower bound on the start of quasilinear behavior.

Restricted nominalism about number and its problems

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism.

L-functions and meromorphic functions satisfying the same Riemann-type functional equation and sharing sets

In 2023, Li–Du–Yi [Complex Var. Elliptic Equ., 68(10)2023, 1653–1677] proved that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and E L 1 ( S ) = E L 2 ( S ) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2 . We prove that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and share a set { α 1 , α 2 , … , α t } CM, where t ≥ 1 is a positive integer, and α 1 , α 2 , ··· , α t are t distinct finite complex numbers, then L 1 = L 2 . The result also improves the corresponding results from Khoai–An [The Ramanujan J., 58(1)(2022),253-267]. We also prove that if a non-constant meromorphic function f with finitely many poles in the complex plane shares a set { α 1 , α 2 , … , α t } CM and some finite value c IM with a non-constant L-function L having a positive degree, then f = L. Some examples are provided to show that the main results obtained in this paper, in a sense, are best possible.

On a conjecture of Graham on the p$p$‐divisibility of central binomial coefficients

AbstractLet be pairwise distinct primes. From a theorem of Kummer, each prime can divide at most times. We show that, for all , if are sufficiently large in terms of and , then there exist infinitely many positive integers such that each divides at most times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers such that is coprime to 105.

Some extensions of Collatz (periodic) conjecture

The Collatz conjecture is a famous conjecture aiming to estimate the behaviors of positive integers after a series of iterations of the Collatz function f(n), where f(n)={3n+12 ifnis odd,n2 ifnis even. Collatz conjecture has attracted a great deal of attention since it was put forward in 1930s due to the surprising nature. However, after strenuous efforts, many mathematicians think the current knowledge seems insufficient to solve Collatz conjecture, although it appears in a very simple form. As a replacement, a “periodic” (weaker) version of Collatz conjecture (called Collatz periodic conjecture) was proposed and discussed. In this paper, we would like to explore what properties Collatz (periodic) conjecture could imply. Maybe it is another possible direction to confirm or disprove Collatz (periodic) conjecture.