Axioms

In this paper, we consider certain quantities that arise in the images of the so-called graph-tree indexes of graph groupoids. In text, the graph groupoids are induced by connected finite-directed graphs with more than one vertex. If a graph groupoid $\mathbb{G}$G contains at least one loop-reduced finite path, then the order of $\mathbb{G}$ is infinity; hence, the canonical groupoid index $\left[\mathbb{G}:\mathbb{K}\right]$ of the inclusion $\mathbb{K}\subseteq \mathbb{G}$ is either ∞ or 1 (under the definition and a natural axiomatization) for the graph groupoids $\mathbb{K}$ of all “parts” K of G. A loop-reduced finite path generates a semicircular element in graph groupoid algebra. Thus, the existence of semicircular systems acting on the free-probabilistic structure of a given graph G is guaranteed by the existence of loop-reduced finite paths in $\mathbb{G}$. The non-semicircularity induced by graphs yields a new index-like notion called the graph-tree index $\Gamma$ of $\mathbb{G}$. We study the connections between our graph-tree index and non-semicircular cases. Hence, non-semicircularity also yields the classification of our graphs in terms of a certain type of trees. As an application, we construct towers of graph-groupoid-inclusions which preserve the graph-tree index. We further show that such classification applies to monoidal operads. Full article

In this study, first we establish a $\left(p,q\right)$-integral identity involving the second $\left(p,q\right)$-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-$\left(p,q\right)$-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature. Full article

The accurate modeling of curved graphene layers for time-domain electromagnetic simulations is discussed in the present work. Initially, the advanced properties of graphene are presented, focusing on the propagation of strongly confined surface plasmon polariton waves at the far-infrared regime. Then, the implementation of an unstructured triangular grid was examined, based on the Delaunay triangulation method. The electric-field components were placed at the edges of the triangles, while two different techniques were proposed for the sampling of the magnetic ones. Specifically, the first one suggests that the magnetic component is placed at the triangle’s circumcenter providing more accurate results, although instability may occur for nonacute triangles. On the other hand, the magnetic field was sampled at the triangle’s centroid, considering the second technique, ensuring the algorithm’s stability, but further approximations were required, leading to a slight accuracy reduction. Moreover, the updating equations in the time-domain were extracted via an appropriate approximation of Maxwell equations in their integral form. Finally, graphene was introduced in the computational domain as an equivalent surface current density, whose location matches the corresponding electric components. The validity of our methodology was successfully performed via the comparison of graphene surface wave propagation properties to their theoretical values, whereas the global error determination indicates the minimal triangle dimensions. Additionally, an instructive setup comprising a circular graphene scatterer was analyzed thoroughly, to reveal our technique’s advantages compared to the conventional staircase discretization. Full article

The classic model of Markowitz for designing investment portfolios is an optimization problem with two objectives: maximize returns and minimize risk. Various alternatives and improvements have been proposed by different authors, who have contributed to the theory of portfolio selection. One of the most important contributions is the Sharpe Ratio, which allows comparison of the expected return of portfolios. Another important concept for investors is diversification, measured through the average correlation. In this measure, a high correlation indicates a low level of diversification, while a low correlation represents a high degree of diversification. In this work, three algorithms developed to solve the portfolio problem are presented. These algorithms used the Sharpe Ratio as the main metric to solve the problem of the aforementioned two objectives into only one objective: maximization of the Sharpe Ratio. The first, GENPO, used a Genetic Algorithm (GA). In contrast, the second and third algorithms, SAIPO and TAIPO used Simulated Annealing and Threshold Accepting algorithms, respectively. We tested these algorithms using datasets taken from the Mexican Stock Exchange. The findings were compared with other mathematical models of related works, and obtained the best results with the proposed algorithms. Full article

We construct a countable algebraic basis of the algebra of all symmetric continuous polynomials on the Cartesian product ${\ell }_{p_1}\times \dots \times {\ell }_{p_n},$ where $p_1,\dots ,p_n\in [1,+\infty )$, and ${\ell }_p$ is the complex Banach space of all p-power summable sequences of complex numbers for $p\in [1,+\infty )$. Full article

This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based on the Robinson stability theorem is derived. The main novelty of this approach compared to the known results in this area is that only a local regularity with respect to the mixed constraints, that is, a regularity in an $\epsilon$-tube about the minimizer, is required instead of the conventional stronger global regularity hypothesis. This affects the maximum condition. Therefore, the normal set of Lagrange multipliers in question satisfies the maximum principle, albeit along with the modified maximum condition, in which the maximum is taken over a reduced feasible set. In the second part of this work, we address the case of abnormal minimizers, that is, when the full rank of controllability matrix condition is not valid. The same type of reduced maximum condition is obtained. Full article

We consider analytic functions in tubes ${\mathbb{R}}^n+iB\subset {\mathbb{C}}^n$ with values in Banach space or Hilbert space. The base of the tube B will be a proper open connected subset of ${\mathbb{R}}^n$, an open connected cone in ${\mathbb{R}}^n$, an open convex cone in ${\mathbb{R}}^n$, and a regular cone in ${\mathbb{R}}^n$, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in $L^p$ norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space $H^p$ in ${\mathbb{C}}^n$. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued $L^p$ functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions $H^p$. We obtain boundary value results on the distinguished boundary ${\mathbb{R}}^n+i\left\{\overline{0}\right\}$ and on the topological boundary ${\mathbb{R}}^n+i\partial B$ of the tube for the analytic functions in the $L^p$ and vector valued tempered distribution topologies. Suggestions for associated future research are given. Full article

In this paper, we establish the existence of a nontrivial weak solution to Schrödinger-kirchhoff type equations with the fractional magnetic field without Ambrosetti and Rabinowitz condition using mountain pass theorem under a suitable assumption of the external force. Furthermore, we prove the existence of infinitely many large- or small-energy solutions to this problem with Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem by applying the variational methods, that is, the fountain and the dual fountain theorem with Cerami condition. Full article

The goal of the present work is to develop and test in detail a numerical algorithm for solving the problem of complex heat transfer in hollow bricks. The finite-difference method is used to solve the governing equations. The article also provides a detailed description of the procedure for thickening the computational grid. The flow regime inside the hollow brick is turbulent, which is a distinctive feature of this work. As a rule, if the size of the cavities in the brick is greater than 20 cm and the temperature difference in the considered solution region is significant, then the numerical solution can be obtained in the turbulent approximation. The effect of surface emissivities of internal walls on the thermal transmission and air flow inside hollow brick is investigated. The distributions of isolines of the stream function and temperature are obtained. The results report that the emissivity of interior surfaces significantly affects the heat transfer through hollow bricks. Full article

The unknown friction factor from the implicit Colebrook equation cannot be expressed explicitly in an analytical way, and therefore to simplify the calculation, many explicit approximations can be used instead. The accuracy of such approximations should be evaluated only throughout the domain of interest in engineering practice where the number of test points can be chosen in many different ways, using uniform, quasi-uniform, random, and quasi-random patterns. To avoid picking points with undetected errors, a sufficient minimal number of such points should be chosen, and they should be distributed using proper patterns. A properly chosen pattern can minimize the required number of testing points that are sufficient to detect maximums of the error. The ability of the Sobol quasi-random vs. random distribution of testing points to capture the maximal relative error using a sufficiently small number of samples is evaluated. Sobol testing points that are quasi-randomly distributed can cover the domain of interest more evenly, avoiding large gaps. Sobol sequences are quasi-random and are always the same, which allows the exact repetition of scientific results. Full article

In this paper, we use the multi-stage network slacks-based measure model under uncertainty to evaluate the performance of Taiwanese financial holding companies (FHCs) from 2007 to 2012. We conceptualize the value creation process to be a two-stage framework, as the profitability and the marketability stages with a serial-linkage relationship, in order to solicit more information from the value creation process of FHCs. Under this framework, the profitability stage can be further divided into banking, insurance, and securities profitability sub-stages. In addition, we further extend the proposed model mentioned above to not only incorporate the NPLs as the undesirable output in the banking profitability stage but also use the fuzzy set approach to process its uncertainty. Results indicate that the discriminatory ability of our model is higher when we simplify the variables in the model specification. We also find that all Taiwanese FHCs in that particular period were “inefficient performers”, which is mainly attributed to the weak profitability performance, especially in banking service. We postulate that the global financial crisis, the US subprime crisis in particular, had a negative effect on the value creation performance of Taiwanese FHCs; however, they gradually recovered and improved their performance after the crisis. Full article

Brain tumors are most common in children and the elderly. It is a serious form of cancer caused by uncontrollable brain cell growth inside the skull. Tumor cells are notoriously difficult to classify due to their heterogeneity. Convolutional neural networks (CNNs) are the most widely used machine learning algorithm for visual learning and brain tumor recognition. This study proposed a CNN-based dense EfficientNet using min-max normalization to classify 3260 T1-weighted contrast-enhanced brain magnetic resonance images into four categories (glioma, meningioma, pituitary, and no tumor). The developed network is a variant of EfficientNet with dense and drop-out layers added. Similarly, the authors combined data augmentation with min-max normalization to increase the contrast of tumor cells. The benefit of the dense CNN model is that it can accurately categorize a limited database of pictures. As a result, the proposed approach provides exceptional overall performance. The experimental results indicate that the proposed model was 99.97% accurate during training and 98.78% accurate during testing. With high accuracy and a favorable F1 score, the newly designed EfficientNet CNN architecture can be a useful decision-making tool in the study of brain tumor diagnostic tests. Full article

Let G be a graph with a minimum degree $\delta$ of at least two. The inclusion chromatic index of G, denoted by ${\chi }_{\subset }^{\prime }\left(G\right)$, is the minimum number of colors needed to properly color the edges of G so that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. We prove that every connected subcubic graph G with $\delta \left(G\right)\ge 2$ either has an inclusion chromatic index of at most six, or G is isomorphic to ${\widehat{K}}_{2,3}$, where its inclusion chromatic index is seven. Full article

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective. Full article

The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where $\mathbf{L}:{\mathbb{C}}^n\to {\mathbb{R}}_{+}^n$ is a positive continuous function. For vector-valued functions from this class we prove some propositions describing their local properties. In particular, these functions possess the property that maximum of norm for some partial derivative at a skeleton of polydisc does not exceed norm of the derivative at the center of polydisc multiplied by some constant. The converse proposition is also true if the described inequality is satisfied for derivative in each variable. Full article

We establish some properties of the bilateral Riemann–Liouville fractional derivative $D^s$. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by $W^{s,1}(a,b)$, and the fractional bounded variation spaces of fractional order s, denoted by $B{V}^s(a,b)$. Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis. Full article

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials. Full article