The purpose of the present paper is to find the hypersurface of a Finsler space with exponential change of (α, β) metric L= αe β/α + β given by b(x) = constant. We shall find the conditions under which the hypersurface be a hyperplane of the first or second kinds have been obtained. This hypersurface is not a hyperplane of third kind.
We construct explicit examples of bounded sequences \(\{a_n\}_{n=1}^\infty\) in \(\mathbb{R}\) with prescribed behaviors for their accumulation properties. Specifically, we present one sequence whose set of subsequential limits $S = \{\displaystyle\lim_{k\to\infty} a_{n_k} \mid \{a_{n_k}\}$ is a convergent subsequence of $\{a_n\} \}$ has cardinality \(\aleph_0\), the smallest infinite cardinality. We also construct a different example where the set of limit points $T = \{x \in \mathbb{R} \mid x$ is a limit point of $\{a_n\} \}$ has cardinality \(\aleph_0\) as well. These examples illustrate that not only can \(S\) and \(T\) differ in structure, but that both sets can be countably infinite---a possibility not often emphasized in introductory analysis. This work contributes to a deeper understanding of the diversity of limiting behavior in sequences and highlights the subtle distinctions between subsequential limits and limit points.
A decomposition of a graph $G$ is a set consisting of edge-disjoint subgraphs of $G$ whose union is $G$. A Hamiltonian cycle in $G$, is a cycle containing all of the vertices of $G$. A $m$-star is a star containing $m$ edges and denoted by $S_m$, which is isomorphic to the complete bipartite graph $K_{1,m}$. In this paper, the necessary and sufficient conditions for decomposing the balanced complete bipartite graph $K_{n,n}$ into $p$ copies of Hamiltonian cycles and $q$ copies of $3$-stars ($4$-stars) are given.
The theory of numerical semigroups is important for its applications in algebra. Research on symmetric numerical semigroups, in particular, has attracted attention in recent years with its applications in various fields. In this original work, one of these studies, we consider a class of symmetric numerical semigroups with no maximal embedding dimension and half of this class. Here, we will give some results on some non MED symmetric numerical semigroups. Also, we will examine the relationships between these semigroups.
University mathematics education in the new era is undergoing profound transformation, centered on breaking away from traditional knowledge transmission models and shifting toward an integrated, innovative approach grounded in competency-based learning and competency-oriented education. To this end, this paper articulates the author’s reflections and practices in response to the demands of university mathematics education in the new era.
We give another proof of our previous results stating the one-to-one correspondence between circulant Hadamard matrices and Hermitian circulant complex Hadamard matrices and the nonexistence of Hermitian circulant $q$-Butson Hadamard matrices of order $n>4$. In addition, we prove the nonexistence of skew-Hermitian circulant $q$-Butson Hadamard matrices of order $n>4$.
In this paper, we propose an efficient hybrid meta-heuristic algorithm combining the Nelder–Mead (NM) simplex search method with the Bat Algorithm (BA), termed as the Hybrid Simplex-Bat Algorithm (HSBA). The proposed method aims to leverage the local search strength of NM and the global exploration ability of BA to solve complex nonlinear global optimization problems. HSBA is tested on a suite of standard benchmark functions and compared with existing meta-heuristic and hybrid algorithms. Experimental results demonstrate that HSBA outperforms the standalone BA and many other existing hybrid approaches in terms of solution quality, convergence speed, and robustness.
In the present paper, we study the Pointwise Biprojectibility of Banach Algebras. We indicate that a Pointwise Biprojective Banach Algebra is a super-amenable if and only if it has an identity. In addition, we investigate other Pointwise Biprojective properties including, the relationship between Pointwise Biprojectibility and amenability for Banach Algebras.We also maintain what kind of relationship is between Pointwise Biprojectibility L1(G) and G. Finally,we define the concept of Pointwise projecttibility and investigate the relationship between Pointwise Projevtibility and Pointwise Biprojectibility.we consider any conditions for proof that biprojective and projective are two definition similar to pointwise projective and pointwise biprojective in extension of banach algebras.the srveral instructures, we proof that almost every where, banach algebras satisfyes another situations. In Future we will find that we can develop all theorems and lemmas of this paper for Pointwise amenability. We Recommend authors show that there is a Banach algebra that it dos not apply to the conditions mentioned in this article.
In this paper, we investigate the isomorphic decomposition problem of the elementary crown $C_{n,n-1}$ into trees of small size, and give the necessary and sufficient condition of the decomposition.
In present paper we have studied the Finslerian hypersurfaces and first approximate exponential change of Finsler metric. We have also proved that first approximate exponential change makes three type of hypersurfaces invariant under certain condition.
The purpose of the present paper is to find the necessary and sufficient conditions under which a Z-Shen square change of Finsler metric becomes a projective change. The condition under which a Z-Shen square change of Finsler metric of Douglas space becomes a Douglas space have been also found.
We have considered the Z-Shen Square metric L given by L^*=(L+β)^2/L . We have obtained three types of hypersurfaces and hyperplanes of first, second and third kind invariant under certain condition.
In this paper we study module and weak module amenability of the module extension Banach algebra $A\oplus X$ of a Banach algebra A by a Banach A-module X. As an example we show that for an inverse semigroup S with set of idempotents E, the module extension ${\ell ^{1}}(E)\oplus {\ell ^{1}}(S)$ is amenable as an ${\ell ^{1}}(E)$-module iff S is amenable. We also study module biflatness and module biprojectivity of module extensions.
Set E_(4(16k+11)) and E_(-4(16k+11)) as elliptic curves y^2=x^3+4px and y^2=x^3-4px with prime as p≡11(mod 16) and E_(-3p) and E_(-p) and E_(-2p) and E_pq as elliptic curves y^2=x^3-3px and y^2=x^3-px and y^2=x^3-2px and y^2=x^3+pqx then, we shall calculate the ranks of it and correlate the results to matrices and treat the basis and relate that results to determinant of partitioned matrix.
We determine completely the structure of finite groups all of whose nonnormal subgroups are of prime power order.
We have considered the conformal β – change of Finsler metric L given by L ̅ = e^σ f(L,β ), where f is any positively homogeneous function of degree one in L and 1-form β. We have obtained that due to this change of Finsler metric, the imbedding class of their tangent Riemannian space is increased at most by two.
Assign 𝐸 ∓𝑝 as elliptic curves 𝑦 2 = 𝑥 3 ∓ 𝑝𝑥 then, we will treat the ranks of these curves. Assume that 𝐸 −2𝑝 is an elliptic curve 𝑦 2 = 𝑥 3 − 2𝑝𝑥 then, we shall compute the ranks of it and compare the results with that of 𝑦 2 = 𝑥 3 ∓ 𝑝𝑥
Many statistical models, be it deterministic or stochastic, usually contain a number of parameters that make up the model(s). Ordinarily, the maximum likelihood estimation (MLE) and least squares estimation (LSE) methods are the most applied methods of estimation. However, in the two approaches, the main focus is on the estimation of the parameters since parameter estimation is a key step that cannot be avoided as far as modelling or model building is concerned. In this paper, parameter estimation in the updated vector autoregressive model is shown. We consider estimation of the parameters by use of the dual estimation approach, precisely using joint estimation which can estimate both the state and the parameters, applied to some VAR models in one dimen-sion and in two dimension. From the results, it is observed that there is convergence of the parameters to the true parameter values as time evolves.
