Positive Integer Research Articles

New prime number theory

This paper introduces a novel approach to estimating the sum of prime numbers by leveraging insights from partition theory, prime number gaps, and the angles of triangles. The methodology is applied to infinite sums and the nth sum, and several ways of defining the nth sum of a prime number are proposed. By using the Ramanujan infinite series of natural numbers, it is possible to derive an infinite series of prime numbers.

Formalising the role of behaviour in neuroscience

AbstractWe develop a mathematical approach to formally proving that certain neural computations and representations exist based on patterns observed in an organism's behaviour. To illustrate, we provide a simple set of conditions under which an ant's ability to determine how far it is from its nest would logically imply neural structures isomorphic to the natural numbers . We generalise these results to arbitrary behaviours and representations and show what mathematical characterisation of neural computation and representation is simplest while being maximally predictive of behaviour. We develop this framework in detail using a path integration example, where an organism's ability to search for its nest in the correct location implies representational structures isomorphic to two‐dimensional coordinates under addition. We also study a system for processing strings common in comparative work. Our approach provides an objective way to determine what theory of a physical system is best, addressing a fundamental challenge in neuroscientific inference. These results motivate considering which neurobiological structures have the requisite formal structure and are otherwise physically plausible given relevant physical considerations such as generalisability, information density, thermodynamic stability and energetic cost.

Fourier coefficients of Jacobi forms of real weights

Let [Formula: see text] and [Formula: see text] be positive integers and [Formula: see text] be a negative real number. Let [Formula: see text] be a multiplier system of weight [Formula: see text] on [Formula: see text]. In this paper, we give an exact formula for the Fourier coefficients of the holomorphic part of a harmonic Maass–Jacobi form of weight [Formula: see text], index [Formula: see text], and multiplier system [Formula: see text] on [Formula: see text]. As an application, we investigate the number of ratios of the Fourier coefficients of the holomorphic parts of two harmonic Maass–Jacobi forms. Moreover, we obtain the equivalent conditions for the principal part of a harmonic Maass–Jacobi form to be the principal part of a weak Jacobi form with poles only at infinity.

A function related to the Mordell–Weil rank of elliptic curves

Let [Formula: see text] be a prime number. For each natural number [Formula: see text], we study the behavior of the function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square (mod [Formula: see text]). The study of this function is inspired by the cognate function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square. The descent theory of elliptic curves would show that if [Formula: see text] is unbounded for squarefree values of [Formula: see text], then there are elliptic curves over the rational number field with arbitrarily large rank. In this note, we show for every prime [Formula: see text], [Formula: see text] is unbounded as [Formula: see text] ranges over squarefree values, thus providing some evidence for the conjecture that [Formula: see text] is unbounded for squarefree [Formula: see text].

Note for the P versus NP Problem

P versus NP is considered as one of the most fundamental open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is NP-complete. It is well-known that P is equal to NP under the assumption of the existence of a polynomial time algorithm for some NP-complete. We show that the Monotone Weighted Xor 2-satisfiability problem (MWX2SAT) is NP-complete and P at the same time. Certainly, we make a polynomial time reduction from every directed graph and positive integer k in the K-CLOSURE problem to an instance of MWX2SAT. In this way, we show that MWX2SAT is also an NP-complete problem. Moreover, we create and implement a polynomial time algorithm which decides the instances of MWX2SAT. Consequently, we prove that P = NP.

Approximate Distance Sensitivity Oracles in Subquadratic Space

An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $\sigma \ge 1$ is a data structure that preprocesses a given undirected, unweighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $f$. When queried with a pair of vertices $s, t$ and a set $F$ of at most $f$ edges, it returns a $\sigma$-approximation of the $s$-$t$-distance in $G-F$. We study $f$-DSOs that take subquadratic space. Thorup and Zwick [JACM 2005] showed that this is only possible for $\sigma \ge 3$. We present, for any constant $f \ge 1$ and $\alpha \in (0, \frac{1}{2})$, and any $\varepsilon > 0$, a randomized $f$-DSO with stretch $ 3 + \varepsilon$ that w.h.p. takes $\widetilde{O}(n^{2-\frac{\alpha}{f+1}}) \cdot O(\log n/\varepsilon)^{f+2}$ space and has an $O(n^\alpha/\varepsilon^2)$ query time. The time to build the oracle is $\widetilde{O}(mn^{2-\frac{\alpha}{f+1}}) \cdot O(\log n/\varepsilon)^{f+1}$. We also give an improved construction for graphs with diameter at most $D$. For any positive integer $k$, we devise an $f$-DSO with stretch $2k-1$ that w.h.p. takes $O(D^{f+o(1)} n^{1+1/k})$ space and has $\widetilde{O}(D^{o(1)})$ query time, with a preprocessing time of $O(D^{f+o(1)} mn^{1/k})$. Chechik, Cohen, Fiat, and Kaplan [SODA 2017] devised an $f$-DSO with stretch $1{+}\varepsilon$ and preprocessing time $O(n^{5+o(1)}/\varepsilon^f)$, albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to $O(mn^{2+o(1)}/\varepsilon^f)$.

Simple 𝑝-adic Lie groups with abelian Lie algebras

Abstract For each prime 𝑝 and each positive integer 𝑑, we construct the first examples of second countable, topologically simple 𝑝-adic Lie groups of dimension 𝑑 whose Lie algebras are abelian. This answers several questions of Glöckner and Caprace–Monod. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.

On the computational complexity of the strong geodetic recognition problem

A vertex set $S$ of a graph $G$ is a strong geodetic set if it is possible to choose exactly one shortest path for each pair of vertices in $S$, and cover all the remaining vertices of the graph with these paths. In the \textsc{Strong Geodetic} problem (SG) a graph~$G$ and a positive integer~$k$ are given as input and one has to decide whether~$G$ has a strong geodetic set of cardinality at most~$k$. This problem is known to be \NP-complete for general graphs. In this work we introduce the \textsc{Strong Geodetic Recognition} problem (SGR), which consists in determining whether a given vertex set~$S \subseteq V(G)$ is strong geodetic. We demonstrate that this version is \NP-complete. We investigate and compare the computational complexity of both decision problems restricted to some graph classes, deriving polynomial-time algorithms, \NP-completeness proofs, and initial parameterized complexity results, including an answer to an open question in the literature for the complexity of SG for chordal graphs.

NOTES ON FERMAT-TYPE DIFFERENCE EQUATIONS

Abstract We consider the existence problem of meromorphic solutions of the Fermat-type difference equation $$ \begin{align*} f(z)^p+f(z+c)^q=h(z), \end{align*} $$ where $p,q$ are positive integers, and h has few zeros and poles in the sense that $N(r,h) + N(r,1/h) = S(r,h)$ . As a particular case, we consider $h=e^g$ , where g is an entire function. Additionally, we briefly discuss the case where h is small with respect to f in the standard sense $T(r,h)=S(r,f)$ .

The Covariety of Saturated Numerical Semigroups with Fixed Frobenius Number

In this work, we show that if F is a positive integer, then Sat(F)={S∣S is a saturated numerical semigroup with Frobenius number F} is a covariety. As a consequence, we present two algorithms: one that computes Sat(F), and another which computes all the elements of Sat(F) with a fixed genus. If X⊆S\Δ(F) for some S∈Sat(F), then we see that there exists the least element of Sat(F) containing X. This element is denoted by Sat(F)[X]. If S∈Sat(F), then we define the Sat(F)-rank of S as the minimum of {cardinality(X)∣S=Sat(F)[X]}. In this paper, we present an algorithm to compute all the elements of Sat(F) with a given Sat(F)-rank.

On the tails of [formula omitted]-modules

We study the end-behavior of integer-valued FI-modules. Our first result describes the high degrees of an FI-module in terms of newly defined tail invariants. Our main result provides an equivalence of categories between FI-tails and finitely supported modules for a new category that we call FJ. Objects of FJ are natural numbers, and morphisms are infinite series with summands drawn from certain modules of Lie brackets.

ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS

Abstract An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith.199 (2021), 33–53] introduced an analogue $\overline {a}_t(n)$ of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for $\overline {a}_t(n)$ where $t \in \{3,4,5,8\}$ and studied the arithmetic density of $\overline {a}_t(n)$ modulo $p_i^{\,j}$ where $t=p_1^{a_1}\cdots p_m^{a_m}$ and $p_i\geq 5$ are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq.27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by $\overline {a}_5(n)$ . We study the arithmetic densities of $\overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where $\gcd (m,6)$ =1. Also, employing a result of Ono and Taguchi [‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory1 (2005), 75–101] on the nilpotency of Hecke operators, we prove an infinite family of congruences for $\overline {a}_3(n)$ modulo arbitrary powers of 2.

Zarankiewicz numbers near the triple system threshold

AbstractFor positive integers and , the Zarankiewicz number can be defined as the maximum total degree of a linear hypergraph with vertices and edges. Guy determined for all . Here, we extend this by determining for all and, when is large, for all .

Counting connected partitions of graphs

AbstractMotivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph on vertices and edges determine the number of unordered solutions of positive integers such that every is realized by a connected subgraph of with edges. We also consider the vertex‐partition analogue. We prove various lower bounds on as a function of the number of vertices in , as a function of the average degree of , and also as the size of ‐partite connected maximum cuts of . Those three lower bounds are tight up to a multiplicative constant. We also prove that the number of unordered ‐tuples with , that are realizable by vertex partitions into connected parts, is at least .

Optimal L(3,2,1)-labeling of trees

Given a graph G, an L ( 3 , 2 , 1 ) -labeling of G is an assignment f of non-negative integers (labels) to the vertices of G such that | f ( u ) − f ( v ) | ≥ 4 − i if dist ( u , v ) = i (i = 1, 2, 3). For a non-negative integer k, a k- L ( 3 , 2 , 1 ) -labeling is an L ( 3 , 2 , 1 ) -labeling such that no label is greater than k. The L ( 3 , 2 , 1 ) -labeling number of G, denoted by λ 3 , 2 , 1 ( G ) , is the smallest number k such that G has a k- L ( 3 , 2 , 1 ) -labeling. Chia proved that the L ( 3 , 2 , 1 ) -labeling number of a tree T with maximum degree Δ can have one of three values: 2 Δ + 1 , 2 Δ + 2 and 2 Δ + 3 . This paper gives some sufficient conditions for λ 3 , 2 , 1 ( T ) ≥ 2 Δ + 2 and λ 3 , 2 , 1 ( T ) = 2 Δ + 3 , respectively. As a result, the L ( 3 , 2 , 1 ) -labeling numbers of complete m-ary trees, spiders and banana trees are completely determined.

Jet scheme and simple isolated hypersurface singularities

Let m≥2 be a positive integer. Let (H(f),o) be the analytic germ of hypersurface attached to a weighted-homogeneous polynomial f∈C[x1,…,xm]. In this article, we mainly characterize the simpleness of any isolated analytic quasi-homogeneous complex hypersurface singularities (H,o) in Cm in terms of the associated jet schemes. We also compute the dimension of the jet scheme Ln(H)o, formed by the n-jets centered at o, for every positive integer n>>0, which is the crucial technical part of our simpleness criterion.

On some applications of relative (p,q)-th order for rating the growths of composite entire functions

The main aim of this paper is to study some comparative growth properties of composite entire functions on the basis of relative (p,q)-th order and relative (p,q)-th lower order of entire function with respect to another entire function where p and q are any two positive integers.

On the Perron root and eigenvectors of a non-negative integer matrix

In this paper, we obtain a combinatorial expression for the Perron root and eigenvectors of a non-negative integer matrix using techniques from symbolic dynamics. We associate such a matrix with a multigraph and consider the edge shift corresponding to it. This gives rise to a collection of forbidden words F which correspond to the non-existence of an edge between two vertices, and a collection of repeated words R with multiplicities which correspond to multiple edges between two vertices. In general, for given collections F of forbidden words and R of repeated words with pre-assigned multiplicities, we construct a generalized language as a multiset. A combinatorial expression that enumerates the number of words of fixed length in this generalized language gives the Perron root and eigenvectors of the adjacency matrix. We also obtain conditions under which such a generalized language is a language of an edge shift.

Some results on nil-injective rings

Let R be a ring. A right R-module is called nil-injective if for any element w is belong to the set of nilpotent elements, and any right R-homomorphism can be extended to R to M. If RR is nil-injective, then R is called a right nil-injective ring. A right R-module is called Wnil-injective if for each non-zero nilpotent element w of R, there exists a positive integer n such that wn not zero that right R-homomorphism f:wnR to M can be extended to R to M. If RR is right Wnil-injective, then is called a right Wnil-injective ring. In the present work, we discuss some characterizations and properties of right nil-injective and Wnil-injective rings.

Hook length biases in ordinary and t-regular partitions

In this article, we study hook lengths of ordinary partitions and t-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in 2-regular partitions. For a positive integer k, let p(k)(n) denote the number of hooks of length k in all the partitions of n. We prove that p(k)(n)≥p(k+1)(n) for all n≥0 and n≠k+1; and p(k)(k+1)−p(k+1)(k+1)=−1 for k≥2. For integers t≥2 and k≥1, let bt,k(n) denote the number of hooks of length k in all the t-regular partitions of n. We find generating functions of bt,k(n) for certain values of t and k. Exploring hook length biases for bt,k(n), we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that b2,2(n)≥b2,1(n) for all n>4, whereas b2,2(n)≥b2,3(n) for all n≥0. We also propose some conjectures on biases among bt,k(n).