This paper presents a new approach to the derivation of an existing numerical scheme for solving the one-dimensional heat equation. Two theorems are proposed and proven to establish the local truncation error and consistency of the scheme. The scheme’s accuracy is further investigated through step size refinement, and the stability of the scheme is rigorously analyzed using the matrix method. The scheme is implemented in a MATLAB environment, and its performance is evaluated by comparing the numerical solutions with the exact solution. Results demonstrate the superior accuracy of the proposed scheme. Furthermore, the \(L^2\) norm of the error is computed and visualized graphically, confirming the scheme’s effectiveness.
In this paper we construct indecomposable vector bundles associated to monads on multiprojective spaces. Specifically we establish the existence of monads on \(\mathbf P\)2n+1 × \(\mathbf P\)2n+1 × ⋯ × \(\mathbf P\)2n+1 and on \(\mathbf P\)a₁ × ⋯ × \(\mathbf P\)aₙ. We prove stability of the kernel bundle which is a dual of a generalized Schwarzenberger bundle associated to the monads on X = \(\mathbf P\)2n+1 × \(\mathbf P\)2n+1 × ⋯ × \(\mathbf P\)2n+1 and prove that the cohomology vector bundle which is simple, a generalization of special instanton bundles. We also prove stability of the kernel bundle and that the cohomology vector bundle associated to the monad on \(\mathbf P\)a₁ × ⋯ × \(\mathbf P\)aₙ is simple. Lastly, we construct explicitly the morphisms that establish the existence of monads on \(\mathbf P\)1 × ⋯ × \(\mathbf P\)1.
The Boolean logic of subsets, usually presented as `propositional logic,’ is considered as being “classical” while intuitionistic logic and the many sublogics and off-shoots are “non-classical.” But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of implication and negation in that other mathematical logic of partitions.
The Euler-Sombor index \(EU\) is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices \(u,v\) of the underlying graph, of the terms \(\sqrt{d_u^2+d_v^2+d_u\,d_v}\), where \(d_u\) and \(d_v\) are the degrees of the vertices \(u\) and \(v\), respectively. For a real number \(\lambda\), a variable version of \(EU\) is constructed, denoted by \(EU(\lambda)\), defined via \(\sqrt{d_u^2+d_v^2+\lambda\,d_u\,d_v}\). Its special cases for \(\lambda=2,\,-2,\,0\), and 1 are, respectively, the first Zagreb, Albertson, Sombor, and the ordinary Euler-Sombor indices. The basic properties of \(EU(\lambda)\) are determined, including a method for its approximate calculation and bounds in terms of minimum degree, maximum degree, order and size for several graph products. It is shown how to find values of \(\lambda\) for which \(EU(\lambda)\) is optimal with regard to predicting molecular properties.
Some new classes of nonconvex inverse variational inequalities are considered and studied. Using the projection technique, we establish the equivalence between the nonconvex inverse variational inequalities and the fixed point problems. This alternative equivalent formulation is used to study the existence of a solution of the nonconvex inverse variational inequalities. Several techniques including the projection, auxiliary principle, dynamical systems and nonexpansive mappings are explored for computing the approximate solution of nonconvex inverse variational inequalities. Convergence criteria of the proposed hybrid multi-step methods is investigated under suitable conditions. Our method of proofs is very simple as compared with other techniques. Some special cases are pointed are pointed as applications of the results. It is an open problem to explore the applications of the nonconvex inverse variational inequalities in various fields of mathematical and engineering sciences.
The purpose of this paper is to abstractly describe the notion of a generative mechanism that implements a code and to provide a number of examples including the DNA-RNA machinery that implements the genetic code, Chomsky’s Principles & Parameters model of a child acquiring a specific grammar given `chunks’ of linguistic experience (which play the role of the received code), and embryonic development where positional information in the developing embryo plays the role of the received code. A generative mechanism is distinguished from a selectionist mechanism that has heretofore played an important role in biological modeling (e.g., Darwinian evolution and the immune system).
Let \((X,d)\) be a metric space, \(D\subset (X,d)\) and \(f:D \longrightarrow \mathbb{R}\) a continuous function (with respect to the metric \(d\) and the Euclidean metric on \(\mathbb{R}\)) . Under suitable conditions on the function \(f\) and the set \(D\), we prove the existence of zeros of the function \(f\) on \(D\) by using the so-called \(\alpha\)-dense curves. To be more precise, we prove that if \(f(x_{1})f(x_{2})<0\) for some \(x_{1},x_{2}\in D\) and \(D\) is densifiable then \(f\) has some zero in \(D\). Moreover, from this result we derive a numerical method to approximate, with arbitrarily small error, at least one of these zeros. In particular, as a compact interval \([a,b]\) is densifiable, our method generalizes the well known bisection method. The feasibility and reliability of the proposed method is illustrated by several numerical examples.
The Hardy-Hilbert integral inequality is a classic result in mathematical analysis that has inspired numerous generalizations and modifications. In this article, we present two comprehensive frameworks that unify many of these developments. Our approach introduces kernel functions that take into account both the maximum and the product of the variables, controlled by three independent, adjustable parameters. Although the kernels are primarily inhomogeneous and somewhat complicated, they offer much greater generality. We provide detailed proofs, together with thorough discussions and context within the wider mathematical literature. In addition, several intermediate integral results emerge naturally from our framework and may serve as useful tools for further exploration.
We present an introduction to the mathematics of quantum physics and quantum computation which put emphasis on the basic mathematical aspects of definition and operations on qubits. We start by a comprehensive introduction of a qubit as a unit element of \( \mathbb{C}^2 \), and its representations on spheres in \( \mathbb{R}^3 \). This introduction leads to the interpretation of Pauli operators as basic rotations in \( \mathbb{R}^3 \). Then we study unitary operators. Their link to rotations in \( \mathbb{R}^3 \) is established using the density operator associated to a qubit. We complete this paper by some decomposition, or splitting, problems of unitary operators on \( \mathbb{C}^2 \) based on decomposition results of rotations in \( \mathbb{R}^3 \). These decomposition results are useful for the construction of quantum gates.
In this research article, the authors introduce the refinements of some special inequalities, like Lah-Ribarič type, Giaccardi, and Petrović’s inequalities. Also, the authors define Fejér, Giaccardi, and Petrović’s types of inequalities for different classes of convex functions.
