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In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation., Revised after referee comments

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces $$X\subset \mathbb {P}^{n+1}$$ . After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if $$X\subset \mathbb {P}^{n+1}$$ is a very general hypersurface of degree $$d\geqslant 2n+2$$ , the least gonality of a curve $$C\subset X$$ passing through a general point of X is $$\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor $$ , apart from some exceptions we list.

In this short paper, we construct a unipotent nearby cycle functor and show a p-adic analogue of Beilinson’s equivalence comparing two derived categories: the derived category of holonomic arithmetic $${\mathcal {D}}$$ -modules and the derived category of arithmetic $${\mathcal {D}}$$ -modules whose cohomologies are holonomic.

Understanding the structure of financial markets deals with suitably determining the functional relation between financial variables. In this respect, important variables are the trading activity, defined here as the number of trades N, the traded volume V, the asset price P, the squared volatility \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document}σ2, the bid-ask spread S and the cost of trading C. Different reasonings result in simple proportionality relations (“scaling laws”) between these variables. A basic proportionality is established between the trading activity and the squared volatility, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \sim \sigma ^2$$\end{document}N∼σ2. More sophisticated relations are the so called 3/2-law \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{3/2} \sim \sigma P V /C$$\end{document}N3/2∼σPV/C and the intriguing scaling \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \sim (\sigma P/S)^2$$\end{document}N∼(σP/S)2. We prove that these “scaling laws” are the only possible relations for considered sets of variables by means of a well-known argument from physics: dimensional analysis. Moreover, we provide empirical evidence based on data from the NASDAQ stock exchange showing that the sophisticated relations hold with a certain degree of universality. Finally, we discuss the time scaling of the volatility \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}σ, which turns out to be more subtle than one might naively expect.

In this paper, we apply the calculus of variations to solve the elastica problem. We examine a more general elastica problem in which the material under consideration need not be uniformly rigid. Using, the Euler–Lagrange equations, we derive a system of nonlinear differential equations whose solutions are given by these generalized elastica curves. We consider certain simplifying cases in which we can solve the system of differential equations. Finally, we use novel numerical techniques to approach solutions to the problem in full generality.

In this short paper, we extend the classical Hoffman-Meeks Halfspace Theorem [Hoffman and Meeks, 'The strong halfspace theorem for minimal surfaces', Invent. Math. 101 (1990) 373-377] to self-shrinkers, that is: Let $P$ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma $ contained in one of the closed half-space determined by $P$ is $\Sigma = P$. Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller [Kleene and Moller, 'Self-shrinkers with a rotational symmetry', Trans. Amer. Math. Soc. 366 (2014) 3943-3963]. Also, using a similar geometric idea, we obtain that the only self-shrinker properly immersed in an closed cylinder $ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1}$, for some $k\in \{1, \ldots, n\}$ and radius $R$, $R \leqslant \sqrt {2k}$, is the cylinder ${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$. We also extend the above results for $\lambda $-hypersurfaces.

Recently, Zou obtained the generalized results on the bounds for Tsallis relative operator entropy. In this short paper, we give precise bounds for Tsallis relative operator entropy. We also give precise bounds of relative operator entropy., To appear in Math. Ineq. Appl., 8 pages

In 1985, Odoni showed that in characteristic 0 0 the Galois group of the n n -th iterate of the generic polynomial with degree d d is as large as possible. That is, he showed that this Galois group is the n n -th wreath power of the symmetric group S d S_d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.

The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley's proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface., Comment: Final version to appear in Michigan Math. Journal. The last section of v1 has been removed and expanded in the paper "On singular Luroth quartics", arXiv:0911.2101v1

International audience; In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

Estimating power spectrum density (PSD) is essential in signal processing. This short paper gives a theorem to represent a smoothed PSD estimate with the Cauchy integral. It may be used for the approximation of the smoothed PSD estimate.

The objective of this research is to determinate the optimum conditions for the alcoholic fermentation process of aqueous jerivá pulp extract using the response surface methodology and simplex optimisation technique. The incomplete factorial design 3³ was applied with the yeast extract, NH4H2PO4 and yeast as the independent variables and the alcohol production yield as the response. The regression analysis indicated that the model is predictive, and the simplex optimisation generated a formulation containing 0.35 g L-1 yeast extract, 6.33 g L-1 yeast and 0.30 g L-1NH4H2PO4 for an optimum yield of 85.40% ethanol. To validate the predictive equation, the experiment was carried out in triplicate under optimum conditions, and an average yield of 87.15% was obtained. According to a t-test, no significant difference was observed (on the order of 5%) between the average value obtained and the value indicated by the simplex optimisation technique. The objective of this research is to determinate the optimum conditions for the alcoholic fermentation process of aqueous jerivá pulp extract using the response surface methodology and simplex optimisation technique. The incomplete factorial design 3³ was applied with the yeast extract, NH4H2PO4 and yeast as the independent variables and the alcohol production yield as the response. The regression analysis indicated that the model is predictive, and the simplex optimisation generated a formulation containing 0.35 g L-1 yeast extract, 6.33 g L-1 yeast and 0.30 g L-1 NH4H2PO4 for an optimum yield of 85.40% ethanol. To validate the predictive equation, the experiment was carried out in triplicate under optimum conditions, and an average yield of 87.15% was obtained. According to a t-test, no significant difference was observed (on the order of 5%) between the average value obtained and the value indicated by the simplex optimisation technique.

We prove optimal radially weighted L-2-norm inequalities for the Fourier extension operator associated to the unit sphere in R-n. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

Our experiments are intended to refocus attention on the study of geometry, in particular on ruled surfaces, their possible discrete forms and/or development in the plane. The study of geometry is oriented to the concept, formal and expressive control, and built form. We begin with the critical analysis of the geometric shape, go through the problems of its virtual representation – also in parametric form – and finally transform it into a physical object with simple prototyping equipment that, on a small scale, reproduces a process repeatable at the large scale of architectural building. This is therefore a digital approach with roots in descriptive geometry. During our experimentation several models were realized, both parametric and physical. Today models can assist in experimental testing and are useful instruments for showing how the research was carried out.

The point of departure of this paper is the idea that the development of mathematics is reflected in its publications. Hence, the existence of a nearly complete database renders possible general statistical accounts of the development of mathematical activities. To this end, the authors utilize the mathematical index of the Catalogue of Scientific Papers of the Royal Society of London dealing with the mathematical journal literature of the 19th century. The relation between the journal and book literature of that century is discussed, with the result that the size of the journal literature is presumably a valid indicator of the intensity of mathematical activities in particular areas. On the basis of this Catalogue, graphs of the publication activity of all of 19th-century mathematics and of 34 of its most important subareas are displayed; both the number of active contributors in each area and its share of 19th-century mathematics publications are exhibited. Furthermore, the share of mathematics of the total scientific journal literature of the 19th-century is estimated. Frequency distributions of publication activity and the specialization of 19th-century mathematicians conform to patterns well known in modern scientometrics. In dieser Arbeit wird davon ausgegangen, das sich die Entwicklung der Mathematik in ihren Publikationen widerspiegelt. Eine annahernd vollstandige bibliographische Datengrundlage gestattet daher globale statistische Beschreibungen der Entwicklung mathematischer Aktivitaten. Die Autoren werteten zu diesem Zweck den mathematischen Index des Catalogue of Scientific Papers der Royal Society of London aus, der die mathematische Zeitschriftenliteratur des 19. Jahrhunderts berucksichtigt. Sie diskutieren das Verhaltnis von Zeitschriften- zu Buchliteratur in diesem Jahrhundert mit dem Ergebnis, das der Umfang der Zeitschriftenliteratur vermutlich als Indikator der Intensitat mathematischer Aktivitaten auf einzelnen Gebieten gelten kann. Auf der Grundlage des Catalogue werden zur gesamten Mathematik sowie zu 34 der wichtigsten Teilgebiete Verlaufskurven der Publikationsaktivitaten gezeigt, zum einen als Publikationsanteile am Gesamtgebiet, zum anderen als absolute Zahl der auf einem Teilgebiet uberhaupt aktiven Mathematiker. Ferner wird der Anteil der Mathematik an der gesamten naturwissenschaftlichen Zeitschriftenliteratur des 19. Jahrhunderts geschatzt. Haufigkeitsverteilungen der Publikationsaktivitat und der Spezialisierung der mathematischen Autoren des 19. Jahrhunderts ergaben in der zeitgenossischen Szientometrie bekannte Verteilungsmuster. Utgangspunkten for denna artikel ar forestallningen, att matematikens utveckling aterspeglas i dess publikationer. Existensen av en sa gott som fullstandig databas mojliggor darfor allmanna statistiska beskrivningar av utvecklingen av matematiska aktiviteter. For detta andamal utnyttjade forfattarna det matematiska indexet till den Catalogue of Scientific Papers, som utgivits av Royal Society of London och som behandler 1800-talets matematiska tidskriftslitteratur. Forhallandet mellan det ahundradets tidskrifts- och boklitteratur diskuteras med resultatet, att tidskriftslitteraturens omfang formodligen gor, att den kan galla som indikator pa itensiteten hos matematiska aktiviteter pa ensklida omraden. Utgaende fran denna Catalogue visas kurvor pa 34 av dess viktigaste delomraden; i det senare fallet anges dels varje delomrades andel i publikationer av hela mathematiken, dels antalet aktiva matematiker pa omradet. Vidare uppskattas matematikens andel av hela den naturvetenskapliga tidskriftslitteraturen under 1800-talet. Frekvensfordelningar av publikationsaktiviteten och specialiseringen hos 1800-talets matematiker foljer monster, som ar bekanta i den moderna scientometrin.

This study provides a systematic literature review of research in the field of teaching and learning mathematics through Content and Language Integrated Learning (CLIL), Content-Based Instruction (CBI), and English Medium Instruction (EMI). The review aims to examine the most relevant literature with a focus on mathematics and CLIL, CBI, or EMI in Scopus and Web of Science per the Preferred Reporting Items for Systematic Reviews (PRISMA) guidelines. Based on 151 sources, 52 papers were selected according to predefined selection criteria. The papers were analysed and coded according to the following categories: (1) geographical productivity, (2) diachronic productivity growth, (3) main objectives, (4) methodology, and research design, (5) variables and measurement instruments, (6) context and sample details, and (7) main findings. The results reveal that most of the research analysed has been carried out in the Asian continent, followed by Europe. The analysis of diachronic productivity shows that the study of the application of CLIL, CBI, and EMI programs in the teaching-learning process of mathematics has notably increased in recent years, especially in the last triennium (2020–2022). Regarding the objectives of the selected corpus, the majority aimed at teaching practices and learning processes, and outcomes in mathematics and language proficiency in CLIL, CBI, and/or EMI classrooms. The corpus analysed fits within one or more of the following categories: empirical, qualitative, descriptive, correlational, and cross-sectional. The samples utilized in different studies differ significantly, both in terms of quantity –ranging from one person to 700– and in the educational level being studied, which would be primary, secondary, or tertiary education. The main variables studied in the corpus focus on mathematical competence, language proficiency, teaching practices, teacher training, science competence, and teachers’ perceptions. The most widely used instruments have been objective tests, such as questionnaires, together with standardized tests to measure some aspects related to mathematical competence and language proficiency. They are followed by an analysis of documents (academic records, teaching materials, official documents...), participant or non-participant observation, interviews, and video and audio recordings. In summary, in the scientific literature analysed, a positive or neutral view predominates on the effects of the CLIL, CBI, and EMI approaches on the learning of mathematics and the L2. This can be due to methodological issues fundamentally related to the methodology, research design, sample, and measurement instruments. Thus, we must highlight that some of the results from the selected papers must be interpreted with caution. Taking this factor into consideration, further comparative studies on a wider scale are required to examine thoroughly the effects of CLIL, CBI, and EMI on the teaching and learning of mathematics in an L2. Besides, it is important to study in greater depth the different levels of language acquisition since the research analysed shows that these have not been sufficiently addressed in the mathematical field of knowledge.

In this paper, we consider parallel-machine scheduling with release times and submodular penalties (P|rj,reject|Cmax+π(R)), in which each job can be accepted and processed on one of m identical parallel machines or rejected, but a penalty must paid if a job is rejected. Each job has a release time and a processing time, and the job can not be processed before its release time. The objective of P|rj,reject|Cmax+π(R) is to minimize the makespan of the accepted jobs plus the penalty of the rejected jobs, where the penalty is determined by a submodular function. This problem generalizes a multiprocessor scheduling problem with rejection, the parallel-machine scheduling with submodular penalties, and the single machine scheduling problem with release dates and submodular rejection penalties. In this paper, inspired by the primal-dual method, we present a combinatorial 2-approximation algorithm to P|rj,reject|Cmax+π(R). This ratio coincides with the best known ratio for the parallel-machine scheduling with submodular penalties and the single machine scheduling problem with release dates and submodular rejection penalties.

Swirling has a significant effect on the main characteristics of flow and can lead to its fundamental restructuring. On the flow axis, a stagnation point with zero velocity is possible, behind which a return flow zone is formed. The apparent instability leads to the formation of secondary vortex motions and can also be the cause of vortex breakdown. In the paper, a swirling flow with a velocity profile of the Batchelor vortex type has been studied on the basis of the linear hydrodynamic stability theory. An effective numerical method for solving the spectral problem has been developed. This method includes the asymptotic solutions at artificial and irregular singular points. The stability of flows was considered for the values of the Reynolds number in the range 10≤Re≤5×106. The calculations were carried out for the value of the azimuthal wavenumber parameter n=−1. As a result of the analysis of the solutions, the existence of up to eight simultaneously occurring unstable modes has been shown. The paper presents a classification of the detected modes. The critical parameters are calculated for each mode. For fixed values of the Reynolds numbers 60≤Re≤5000, the curves of neutral stability are plotted. Branching points of unstable modes are found. The maximum growth rates for each mode are determined. A new viscous instability mode is found. The performed calculations reveal the instability of the Batchelor vortex at large values of the swirl parameter for long-wave disturbances.

This paper investigates global dynamics in fractional-order dual inertial neural networks with time lags. Firstly, according to some crucial features of Mittag-Leffler functions and Banach contracting mapping principle, the existence and uniqueness of $ S $-asymptotically $ \omega $-periodic oscillation of the model are gained. Secondly, by using the comparison principle and the stability criteria of delayed Caputo fractional-order differential equations, global asymptotical stability of the model is studied. In the end, the feasibility and effectiveness of the obtained conclusions are supported by two numerical examples. There are few papers focus on $ S $-asymptotically $ \omega $-periodic dynamics in fractional-order dual inertial neural networks with time-varying lags, apparently, the works in this paper fill some of the gaps.

Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.

There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely $ \mathcal{I} $-$ {\delta_{\beta}}_{J} $-open sets and $ \mathcal{I} $-$ {\bigwedge_{\beta}}_{J} $-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.

In a recent paper [4], Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.

In this paper, we prove some new Simpson's type inequalities for partial $ (p, q) $-differentiable convex functions of two variables in the context of $ (p, q) $-calculus. We also show that the findings in this paper are generalizations of comparable findings in the literature.

In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

In this paper, we investigate the Hyers-Ulam stability of additive and Fréchet functional equations on restricted domains. We improve the bounds and thus the results obtained by S. M. Jung and J. M. Rassias. As a consequence, we obtain asymptotic behaviors of functional equations of different types. One of the objectives of this paper is to bring out the involvement of functional equations in various characterizations of inner product spaces.

The paper is devoted to the integration of the loaded modified Kortewegde Vries equation in the class of rapidly decreasing functions. It is well known that loaded differential equations in the literature are usually called equations containing in the coefficients or in the right-hand side any functionals of the solution, in particular, the values of the solution or its derivatives on manifolds of lower dimension. In this paper, we consider the Cauchy problem for the loaded modified Korteweg-de Vries equation. The problem is solved using the inverse scattering method, i.e. the evolution of the scattering data of a non-self-adjoint Dirac operator is derived, the potential of which is a solution to the loaded modified Korteweg-de Vries equation in the class of rapidly decreasing functions. A specific example is given to illustrate the application of the results obtained.

This paper has been designed to introduce the method for solving the Bi-level Multi-objective (BL-MO) Fully Quadratic Fractional Optimization Model through Fuzzy Goal Programming (FGP) approach by utilising non-linear programming. In Fully Quadratic Fractional Optimization Model, the objective functions are in fractional form, having quadratic functions in both numerator and denominator subject to quadratic constraints set. The motive behind this paper is to provide a solution to solve the BL-MO optimization model in which number of decision-makers (DM) exists at two levels in the hierarchy. First, the fractional functions with fuzzy demand, which are in the form of fuzzy numbers, are converted into crisp models by applying the concept of α-cuts. After that, membership functions are developed which are corresponding to each decision-maker’s objective and converted into simpler form to avoid complications due to calculations. Finally, the model is simplified by applying FGP approach, and a compromised solution to the initial model is obtained. An algorithm, flowchart and example are also given at the end to explain the study of the proposed model.

In this paper, it is explicitly determined the r-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph. Illustrative examples are shown for each one of the cases that are studied throughout the paper. Junta de Andalucía FQM-016

In this paper, it is derived a rich hierarchy for the Toda lattice with a selfconsistent source in the class of periodic functions. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated discrete Hill‘s equation with periodic coefficients. In particular, Dubrovintype equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy. At the end of the paper, we illustrate our theory on concrete example with analytical and numerical results.

The dynamic transitions of the Brusselator model has been recently analyzed in Y. Choi et’al (2021) and T. Ma, S. Wang (2011). Our aim in this paper is to address the relation between the pattern formation and dynamic transition results left open in those papers. We consider the problem in the setting of a 2D rectangular box where an instability of the homogeneous steady state occurs due to the perturbations in the direction of several modes becoming critical simultaneously. Our main results are two folds: (1) a rigorous characterization of the types and structure of the dynamic transitions of the model from basic homogeneous states and (2) the relation between the dynamic transitions and the pattern formations. We observe that the Brusselator model exhibits different transition types and patterns depending on the nonlinear interactions of the pattern of the critical modes.

When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages

This paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.

In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.

so-metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so-metrizable space if it has a σ \sigma -locally finite so-network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so-metrizable spaces by images of metric spaces. This paper gives such characterizations for so-metrizable spaces. More precisely, this paper introduces so-open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so-metrizable space if and only if it is an so-open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so-open, σ \sigma -image of a metric space. In addition, it is shown that so-open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so-metrizable spaces and establish some equivalent relations among so-open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.

This paper establishes a model of continuous-state branching processes with time inhomogeneous competition in Levy random environments. Some results on extinction are presented, including the distribution of the extinction time, the limiting distribution conditioned on large extinction times and the asymptotic behavior near extinction. This paper also provides a new time-space transformation which can be used for further exploration in similar models. The results are applied to an epidemic model to describe the dynamics of infectious population and a virus model to describe the dynamics of viral load.

Fractal is a common feature of many deterministic complex networks. The complex networks with fractal features have interesting structure and good performance. The network based on hypergraph is named hypernetwork. In this paper, we construct a hypernetwork model with fractal properties, and obtain its topological properties. Moreover, according to the exact controllability theory, we obtain the node controllability and the hyperedge controllability of the fractal hypernetwork. The simulation results show that the measure of hyperedge controllability is smaller than that of node in the fractal hypernetwork. In addition, We compare the controllability of three types of hypernetwork, which are easier to control by their hyperedges. It is shown the fractal hypernetwork constructed in this paper has the best controllability. Because of the good controllability of our fractal hypernetwork model, it is suitable for the topology structure of many real systems.

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.

The main objective of this work is to explore and characterize the idea of $ s $-type preinvex function and related inequalities. Some interesting algebraic properties and logical examples are given to support the newly introduced idea. In addition, we attain the novel version of Hermite-Hadamard type inequality utilizing the introduced preinvexity. Furthermore, we establish two new identities, and employing these, we present some refinements of Hermite-Hadamard-type inequality. Some special cases of the presented results for different preinvex functions are deduced as well. Finally, as applications, some new inequalities for the arithmetic, geometric and harmonic means are established. Results obtained in this paper can be viewed as a significant improvement of previously known results. The awe-inspiring concepts and formidable tools of this paper may invigorate and revitalize for additional research in this worthy and absorbing field.

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes., 12 pages

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction-diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper., Ministerio de Ciencia e Innovación and Regional Development European Funds, Grant/Award Number: PGC2018-101443-B-I00; Junta de Castilla y León and Feder, Grant/Award Number: VA169P20

In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.

The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.

The quantum calculus has emerged as a connection between mathematics and physics. It has wide applications, particularly in quantum mechanics, analytic number theory, combinatorial analysis, operation theory etc. The $ q $-calculus, which serves as a powerful tool to model quantum computing, has drawn attention of many researchers in the field of special functions and as a result the $ q $-analogues of certain special functions, especially hypergeometric function, 1-variable Hermite polynomials, Appell polynomials etc., are introduced and studied. In this paper, we introduce the 2-variable $ q $-Hermite polynomials by means of generating function. Also, its certain properties like series definition, recurrence relations, $ q $-differential equation and summation formulas are established. The operational definition and some integral representations of these polynomials are obtained. Some examples are also considerd to show the efficacy of the proposed method. Some concluding remarks are given. At the end of this paper, the graphical representations of these polynomials of certain degrees with specified values of $ q $ are given.

The main goal of this paper is to develop the significance of generalized fractional integral inequalities via convex functions. We obtain the new version of fractional integral inequalities with the extended Wright generalized Bessel function acting as a kernel for the convex function, which deals with the Hermite-Hadamard type and trapezoid type inequalities. Moreover, we establish new mid-point type and trapezoid type integral inequalities for $ (\eta_{1}, \eta_{2}) $-convex function related to Hermite-Hadamard type inequality. We establish new version of integral inequality for $ (\eta_{1}, \eta_{2}) $-convex function related to Fejer type. The results discussed in this paper are a generalized version of many inequalities in literature.

Lot sizing models involve operational and tactical decisions. These decisions may entail multi-level production processes such as assembly operations with multiple plants and limited capacities. Lot sizing problems are widely recognized as NP-hard problems therefore difficult to solve, especially the ones with multiple plants and capacity constraints. The level of complexity rises to an even higher level when there is an interplant transfer between the plants. This paper presents a Genetic Algorithm (GA) based solution methodology applied to large scale multi-plant capacitated lot sizing problem with interplant transfer (MPCLSP-IT). Although the GA has been a very effective and widely accepted meta-heuristic approach used to solve large scale complex problems, it has not been employed for MPCLSP-IT problem. This paper solves the MPCLSP-IT problem in large scale instances by using a genetic algorithm, and in doing so successfully obtains a better solution in terms of computation time when compared to the results obtained by the other methods such as Lagrangian relaxation, greedy randomized adaptive search procedure (GRASP) heuristics, and GRASP-path relinking techniques used in extant literature.