Positive Integer Research Articles

Complete Stability of Delayed Recurrent Neural Networks With New Wave-Type Activation Functions.

Activation functions have a significant effect on the dynamics of neural networks (NNs). This study proposes new nonmonotonic wave-type activation functions and examines the complete stability of delayed recurrent NNs (DRNNs) with these activation functions. Using the geometrical properties of the wave-type activation function and subsequent iteration scheme, sufficient conditions are provided to ensure that a DRNN with n neurons has exactly (2m + 3)n equilibria, where (m + 2)n equilibria are locally exponentially stable, the remainder (2m + 3)n - (m + 2)n equilibria are unstable, and a positive integer m is related to wave-type activation functions. Furthermore, the DRNN with the proposed activation function is completely stable. Compared with the previous literature, the total number of equilibria and the stable equilibria significantly increase, thereby enhancing the memory storage capacity of DRNN. Finally, several examples are presented to demonstrate our proposed results.

Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems

A new generalized definition of Mersenne numbers is proposed of the form an−a−1n, called global generalized Mersenne numbers and noted GMa,n with base a and exponent n positive integers. The properties are investigated for prime n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any a, GMa,n−1 is even and divisible by n, a and a−1 for any prime n>2, and by aa−1+1 for any prime n>5. The remaining factor is a function of triangular numbers of a−1, specific for each prime n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that GMa,n≡1or7mod12 depending on the congruence of amod4; second, that GMa,n−1 are divisible by 10 if n≡1mod4 and, if n≡3mod4, GMa,n≡1or7or9mod10, depending on the congruence of amod5; third, that all factors ci of GMa,n are of the form 2nfi+1 such that ci is either prime or the product of primes of the form 2nj+1, with fi,j natural integers; fourth, that for prime n>2, all GMa,n are periodically congruent to ±1or±3mod8 depending on the congruence of amod8; and fifth, that the factors of a composite GMa,n are of the form 2nfi+1 with fi≡umod4 with u=0, 1, 2 or 3 depending on the congruences of nmod4 and of amod8. The potential use of generalized Mersenne primes in cryptography is shortly addressed.

Further results on the hop domination number of a graph

A hop dominating set S in a connected graph G is called a minimal hop dominating set if no proper subset of S is a hop dominating set of G. The upper hop domination number γ+h (G) of G is the maximum cardinality of a minimal hop dominating set of G. Some general properties satisfied by this concept are studied. It is shown that for every two positive integers a and b where 2 ≤ a ≤ b, there exists a connected graph G such that γh(G) = a and γ+ h (G) = b. It is proved that minimal hop dominating set is NP-complete. It is proved that γh(G) and γ(G) are in general incomparable. It is shown that for every pair of positive integers a and b with a ≥ 2 and b ≥ 1, there exists a connected graph G such that γh(G) = a and γ(G) = b. We present an algorithm to compute minimal hop dominating set of G. Finally, we formulate an Integer linear programming problem to compute the hop domination number of G.

Katětov order between Hindman, Ramsey and summable ideals

AbstractA family $$\mathcal {I}$$ I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal $$\mathcal {I}$$ I on X is below an ideal $$\mathcal {J}$$ J on Y in the Katětov order if there is a function $$f{: }Y\rightarrow X$$ f : Y → X such that $$f^{-1}[A]\in \mathcal {J}$$ f - 1 [ A ] ∈ J for every $$A\in \mathcal {I}$$ A ∈ I . We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.

Extending Abelian Rings: A Generalized Approach

We introduce a novel framework for assessing the centrality of idempotents within a ring by presenting a general concept that assigns a degree of centrality. This approach aligns with the previously established notions of semicentral and q-central idempotents by Birkenmeier and Lam. Specifically, we define an idempotent $e$ in a ring $R$ to be $n$-central, where $n$ is a positive integer, if $[e, R]^ne=0$, where $[x,y]$ represents the additive commutator $xy-yx$. If every idempotent in a ring $R$ is $n$-central, we refer to $R$ as $n$-Abelian. Our study lays the groundwork by presenting foundational results that support this concept and examines key features of $n$-central idempotents essential for appropriately categorizing $n$-Abelian rings among various generalizations of Abelian rings introduced in prior literature. We provide examples of $n$-central idempotents that do not fall under the categories of semicentral or $q$-central. Furthermore, we demonstrate that the ring of upper matrices $\mathbb{T}_n(R)$, where $R$ is Abelian, is an $n$-abelian. We also prove that a ring where all of its idempotents are $n$-central is an exchange ring if and only if the ring is clean.

A Note on the Laplacian Energy of the Power Graph of a Finite Cyclic Group

In this study, the Laplacian matrix concept for the power graph of a finite cyclic group is redefined by considering the block matrix structure. Then, with the help of the eigenvalues of the Laplacian matrix in question, the concept of Laplacian energy for the power graphs of finite cyclic groups was defined and introduced into the literature. In addition, boundary studies were carried out for the Laplacian energy in question using the concepts the trace of a matrix, the Cauchy-Schwarz inequality, the relationship between the arithmetic mean and geometric mean, and determinant. Later, various results were obtained for the Laplacian energy in question for cases where the order of a cyclic group is the positive integer power of a prime.

Mathematical Concepts Exploration of Gamelan Musical Instruments and Dance Movements in Kuda Lumping Wanareja

This research explored mathematical concepts of Gamelan musical instruments and dance movements in Kuda Lumping Wanareja. This research used a qualitative approach with an ethnographic design. Located of research at Cihandiwung Lor, Adimulya, Wanareja, Cilacap, Central Java. Data was collected through observation, interviews, and documentation. Data were analyzed through reduction, presentation, and verification of conclusions. The research results showed that Wanareja's Kuda Lumping gamelan musical instruments consisted of kendang, demung, saron, calapita, kenong, bonang, kethuk, drum, gong, and kempul. Based on the shape of gamelan musical instruments, mathematical concepts include numbers, measurement, and geometry. The saron, demung, bonang, and kempul contain the concept of natural numbers, while kempul shows the concept of quantity. The kendang, demung, saron, calapita, kenong, bonang, kethuk, drum, gong, and kempul show the concept of a rectangle, circle, trapezoid, and tube. The layout of gamelan musical instruments (especially bonang) involves a matrix concept. The Kuda Lumping dance consists of ngatur sembah, pecak pedhang, losing to the enemy, fighting against the enemy from the right, fighting against the enemy from the left, ending the battle, mejang beksa, and janturan/mendem. The mathematical concept related to dance movement patterns is geometric transformation (reflection, translation, and rotation). Several dance movements for praying, breaking the sword, losing to the enemy, fighting against the enemy, and ending the battle show reflection and translation. The janturan and mejang beksa dance movements show rotation. Concerning Gamelan musical instruments, there are kethuk and saron which show two-dimensional geometric concepts such as circles and trapezoids. The results of this research are applied as context in mathematical problems.

The significance of basin slope for curve number estimation and the impact on flood prediction in arid basins

The Natural Resources Conservation Service (NRCS) method was developed in agricultural basins with slopes less than or equal to 5%. Extension of the method to arid regions and at higher slopes is needed. This research aims to investigate the relationship between the basin slope and runoff depth in arid basins by using the Natural Resources Conservation Service Curve Number (NRCS-CN). The results show that there is a clear effect of the proposed curve number equation with slope (the slope adjusted CN, CNIIα ) on the runoff depth in the studied basins, and this is consistent with the literature. However, large basin areas with a high percentage of alluvium area, vegetation, and high infiltration rates countereffects the slope of the basin, and consequently, the basin slope will not have a significant effect on runoff. The correlation coefficient of various developed equations varies between (0.89-0.93) and the root mean square error of runoff depth varies between 11 mm for λ=0.2 and 5.2 mm for λ=0.01.

N-Dual Square Free R-Module

In this Article we introduce and study a new module which is called n-dual-Square free modules (n- DSF for a short ) and is de_ned as: For a ring R and a positive integer n, an R-module M is called n

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.

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.

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.

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