Positive Integer Research Articles

HOW MANY TRIANGLES CAN A GRAPH HAVE?

Abstract We investigate the set $T_{n}$ of the possible number of triangles in a graph on n vertices. The first main result says that every natural number less than $\binom {n}{3} - (\sqrt {2}+o(1) )n^{3/2} $ belongs to $T_{n}$ . On the other hand, we show that there is a number $m = \binom {n}{3}-( \sqrt {2} +o(1))n^{3/2}$ that is not a member of $T_{n}$ . In addition, we prove that there are two interlacing sequences $\binom {n}{3} - ( \sqrt {2} + o(1) )n^{3/2} = c_{1} \le d_{1} \le c_{2} \le d_{2} \le \cdots \le c_{s} \le d_{s} = \binom {n}{3}$ with $| c_{t} - d_{t}| = n-2-\binom {s - t +1}{2}$ such that $( c_{t}, d_{t} ) \cap T_{n} = \emptyset $ for all t. Moreover, we obtain a generalisation of these results for the set of the possible number of copies of a connected graph H in a graph on n vertices.

FROBENIUS NUMBERS ASSOCIATED WITH DIOPHANTINE TRIPLES OF $x^2+y^2=z^3$

Abstract We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$ , that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$ . For the equation $x^2+y^2=z^2$ , the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$ .

Assessing climate change threats and urbanization impacts on surface runoff in Gdańsk (Poland): insights from remote sensing, machine learning and hydrological modeling

AbstractThis study investigates the impacts of Land Use/Land Cover (LULC) changes and climate change on surface runoff in Gdańsk, Poland, which is crucial for local LULC planning and urban flood risk management. The analysis employs two primary methodologies: remote sensing and hydrological modeling. Remote sensing was conducted using Google Earth Engine and Land Change Modeler in IDRISI Terrset software to analyze historical (1985–2022) and future (2050–2100) LULC. Hydrological modeling was performed using the Natural Resources Conservation Service curve number method to assess the overall impact of LULC changes on Gdańsk’s hydrology at the local scale. The Orunia basin, a critical area due to intensive LULC development, was selected for detailed hydrological analysis using the Hydrologic Modeling System (HEC-HMS). The analysis encompassed three scenarios: LULC changes, climate change, and combined LULC and climate change effects. The LULC analysis revealed a marked increase in urban area, a shift in forest and vegetation cover, and a reduction in agricultural land. HEC-HMS simulations showed an increase in the runoff coefficient across selected decades, which was attributed to the combined effect of LULC and climate change. The projected increases under the Representative Concentration Pathway (RCP) 4.5 and RCP 8.5 scenarios for 2050 and 2100 are projected to surpass those observed during the baseline period. The findings highlight that the synergistic effects of LULC and climate change have a more significant impact on Gdańsk’s hydrology at both local and basin scales than their separate effects. These insights into LULC shifts and urban hydrological responses hold implications for sustainable urban planning and effective flood risk management in Gdańsk and similar urban settings.

Completing an universal m-gonal form with a closing unary piece

AbstractIn this paper, we show that for any m-gonal form $$F_m({\textbf{x}})$$ F m ( x ) with $$m \ge 12$$ m ≥ 12 which represents every positive integer up to $$m-4$$ m - 4 , by putting together with only an unary m-gonal form, we may complete an universal form.

Degree-Constrained Steiner Problem in Graphs with Capacity Constraints

The degree-constrained Steiner problem in graphs is well known in the literature. In an undirected graph, positive integer degree bounds are associated with nodes and positive costs with the edges. The goal is to find the minimum cost tree spanning a given node set while respecting the degree bounds. As it is known, finding a tree satisfying the constraints is not always possible. The problem differs when the nodes can participate multiple times in the coverage and the constraints represent a limited degree (a capacity) for each occurrence of the nodes. The optimum corresponds to a graph-related structure, i.e., to a hierarchy. Finding the solution to this particular Steiner problem is NP-hard. We investigate the conditions of its existence and its exact computation. The gain of the hierarchies is demonstrated by solving ILPs to compute hierarchies and trees. The advantages of the spanning hierarchies are conclusive: (1) spanning hierarchies can be found in some cases where spanning trees matching the degree constraints do not exist; (2) the cost of the hierarchy can be lower even if the Steiner tree satisfying the constraints exists.

Some new congruences for simultaneously s-regular and t-distinct partition function

A partition of a positive integer n is said to be simultaneously s-regular and t-distinct partition if none of the parts is divisible by s and parts appear fewer than t times. In this paper, we present some new congruences for simultaneously s-regular and t-distinct partition function denoted by M d (n) with (s, t) (2, 5), (3, 4), (4, 9), (5\alpha, 5\beta ), (7\alpha, 7\beta ), (p, p), where\alpha and \beta are any positive integers and p is any prime.

Optimizing HX-Group Compositions Using C++: A Computational Approach to Dihedral Group Hyperstructures

The HX-groups represent a generalization of the group notion. The Chinese mathematicians Mi Honghai and Li Honxing analyzed this theory. Starting with a group (G,·), they constructed another group (G,∗)⊂P∗(G), where P∗(G) is the set of non-empty subsets of G. The hypercomposition “∗” is thus defined for any A, B from G, A∗B={a·b|a∈A,b∈B}. In this article, we consider a particular group, G, to be the dihedral group Dn,n is a natural number, greater than 3, and we analyze the HX-groups with the dihedral group Dn as a support. The HX-groups were studied algebraically, but the novelty of this article is that it is a computer analysis of the HX-groups by creating a program in C++. This code aims to improve the calculation time regarding the composition of the HX-groups. In the first part of the paper, we present some results from the hypergroup theory and HX-groups. We create another hyperstructure formed by reuniting all the HX-groups associated with a dihedral group Dn as a support for a natural fixed number n. In the second part, we present the C++ code created in the Microsoft Visual Studio program, and we provide concrete examples of the program’s application. We created this program because the code aims to improve the calculation time regarding the composition of HX-groups.

Application of the NRCS-curve number method in humid tropical basins of southeastern Nigeria: a statistical analysis

In the south-eastern region of Nigeria, the application of rainfall (P) and runoff (Q) data directly into the Natural Resources Conservation Service Curve Number (NRCS-CN) method hasn’t been thoroughly studied. This research aimed to determine representative values of the initial abstraction ratio (λ) and corresponding curve number (CN), fit the P and Q data using theoretical probability distributions, and establish confidence intervals for CN. The least squares minimization method and Kolmogorov-Smirnov test were employed on data from 129 sub-basins across 4 major basins. Findings revealed optimal initial abstraction ratio (λopt) = 0.24 and optimal CN (CNopt) = 80, with rainfall best fitted by Gamma, runoff by Weibull, and CN by Normal distributions. However, the study was limited to the available 8-year record period with 96 storm events. The 96 storm events over an 8-year period may seem limited for a humid tropical region, the available data were the most comprehensive and reliable dataset for this study area. Additional data collection over a longer time frame could enhance future studies. The localized CN value and associated confidence intervals can enhance runoff prediction accuracy for flood mitigation and water resources management in this humid tropical region, though further validation is recommended.

Infinite unrestricted sumsets of the form B+B$B+B$ in sets with large density

AbstractFor a set , we characterize the existence of an infinite set and such that , where , in terms of the density of the set . Specifically, when the lower density or the upper density , the existence of such a set and is assured. Furthermore, whenever or , we show that the shift is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three‐coloring of the natural numbers that does not contain a monochromatic for any infinite set and number .

A further generalization of sums of higher derivatives of the Riemann zeta function

We obtain a full asymptotic for the sum of [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th derivative of the Riemann zeta function, [Formula: see text] is a positive real number, and [Formula: see text] denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height [Formula: see text], as [Formula: see text]. We also specify what the asymptotic formula becomes when [Formula: see text] is a positive integer, highlighting the differences in the asymptotic expansions as [Formula: see text] changes its arithmetic nature.

Proof of a conjecture on congruences of the sum of minimal excludants

Very recently, Baruah et al. refined a result of Andrews and Newman on the sum of minimal excludants over all the partitions of a positive integer [Formula: see text] by establishing the generating functions of two functions [Formula: see text] and [Formula: see text] which denote the sum of odd minimal excludants and the sum of even minimal excludants, respectively. They proved some congruences on [Formula: see text] and [Formula: see text] modulo 4 and 8 and posed a conjecture on [Formula: see text] and [Formula: see text] at the end of their paper. In this paper, we confirm this conjecture by using theta function identities.

Log-Concavity with Respect to the Number of Orbits for Infinite Tuples of Commuting Permutations

AbstractLet A(p, n, k) be the number of p-tuples of commuting permutations of n elements whose permutation action results in exactly k orbits or connected components. We formulate the conjecture that, for every fixed p and n, the A(p, n, k) form a log-concave sequence with respect to k. For $$p=1$$ p = 1 this is a well-known property of unsigned Stirling numbers of the first kind. As the $$p=2$$ p = 2 case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov–Okounkov hook length polynomials. In this article, we prove the $$p=\infty $$ p = ∞ case of our conjecture. We start from an expression for the A(p, n, k), which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the $$p\rightarrow \infty $$ p → ∞ asymptotics. The last step essentially amounts to the log-concavity in k of a generalized Turán number, namely, the maximum product of k positive integers whose sum is n.

On a Problem of Douglass and Ono for the Plane Partition Function

AbstractIt is known that the plane partition function of n denoted $$\textrm{PL}(n)$$ PL ( n ) obeys Benford’s law in any integer base $$b\ge 2$$ b ≥ 2 . We give an upper bound for the smallest positive integer n such that $$\textrm{PL}(n)$$ PL ( n ) starts with a prescribed string f of digits in base b.

Answering a question on equally covered groups

Foguel, Moghaddamfar, Schmidt, and Velasquez-Berroteran asked in [Int. Electron. J. Algebra, 2024] whether there exists a positive integer $n$ with the property that, for every finite group $G$, the Cartesian power $G^n$ can be expressed as the union of a family of proper subgroups of the same order. We prove that the answer is negative.

The Bounds of Rigid Sphere Design

We give some necessary conditions for the existence of rigid sphere designs. In particular, we settle the conjecture proposed by Bannai (J Fac Sci Univ Tokyo Sect IA Math 34(3), 485-489, 1987). Given fixed positive integers t and d, we show that there exist only finitely many rigid t-designs on Sd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^d$$\\end{document}, up to orthogonal transformations.

Collatz Conjecture

This paper presents an analysis of the number of zeros in the binary representation of natural numbers. The primary method of analysis involves the use of the concept of the fractional part of a number, which naturally emerges in the determination of binary representation. This idea is grounded in the fundamental property of the Riemann zeta function, constructed using the fractional part of a number. Understanding that the ratio between the fractional and integer parts of a number, analogous to the Riemann zeta function, reflects the profound laws of numbers becomes the key insight of this work. The findings suggest a new perspective on the interrelation between elementary properties of numbers and more complex mathematical concepts, potentially opening new directions in number theory and analysis.This analysis has allowed to understand that the Collatz sequence initially tends towards a balanced symmetric arrangement of zeros and ones, and then it collapses, realizing the scenario of the Collatz conjecture.

Algorithms for representing positive odd integers as the sum of arithmetic progressions

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

On a union of homogeneous sequences

A sequence of positive integers is said to be quasi-primitive if there do not exist three distinct integers such that the greatest common divisor of two elements is not the third. It is clear that the sequence of prime numbers is quasi-primitive. The Erdős quasi-primitive conjecture states that the sum of the reciprocal of the product of an element by its logarithm over any quasi-primitive sequence is maximized by the sum over the sequence of all powers of prime numbers. Note that Erdős, before introduce his conjecture, showed that for any quasi-primitive sequence different from one, this sum is convergent and bounded by a real constant. In this article, by studying the analogous Erdős conjecture, we prove that when the real is sufficiently large, there exists a quasi-primitive sequence such that the sum of the reciprocal of the product of an element by the sum of its logarithm and , over this quasi-primitive sequence is very large than the sum over the sequence of all powers of prime numbers. Obviously, if the case where equals zero is also true, then the conjecture is false.

On singular foliations tangent to a given hypersurface

We consider a class of singular foliations in the sense of Androulidakis and Skandalis that we call transverse order k foliations. These have a finite number of leaves: one hypersurface (the singular leaf) together with the components of its complement (open leaves). The positive integer parameter k encodes the “order of tangency” of the leafwise vector fields to L . We show that a loop in the singular leaf induces a well-defined holonomy transformation at the level of (k-1) -jets. The resulting holonomy invariant can be used to give a complete classification of these foliations and obtain concrete descriptions of their associated groupoids and algebras.

Frobenius Local Rings of Order p4m

Suppose R is a finite commutative local ring, then it is known that R has four positive integers p,n,m,k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field Fpm and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p4m. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p,n,m,k. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.