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The aim of this paper is to exploit the structure of strongly continuous operator semigroups in order to formulate a categorical framework in which a fresh perspective can be applied to past operator theoretic results. In particular, we investigate the inverse-producing Arens extension for Banach algebras (Trans. Am. Math. Soc. 88:536–548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Oper. Theory 78(2):473–500, 2017) in this new framework, asking and answering questions using categorical language. We demonstrate that the Arens extension defines an extension functor in this setting and that it forms an adjunction with the suitably defined forgetful functor. As a by-product of this categorical framework, we also revisit the work on Banach direct sums by Lachowicz and Moszyński (Semigroup Forum 93(1):34–70, 2016). This paper can be considered as a brief exploration of the triple interface between operator semigroups, Banach algebras, and category theory. [ABSTRACT FROM AUTHOR]

There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of convergence for the parameters are found, while in three of them, optimal parameters are determined, and in one of the latter, many more cases, than in all the others, are distinguished, analyzed, and studied. It turns out that two of the optimal parameters coincide making the optimal three-parameter methods be equivalent to the optimal two-parameter known ones. Our aim in this work is manifold: (i) to show that the iterative methods we present are equivalent, (ii) to slightly change some statements in one of the main papers, (iii) to complete the analysis in another one, (iv) to explain how the transition from any of the methods to the others is made, (v) to extend the iterative method to cover the singular symmetric case, and (vi) to present a number of numerical examples in support of our theory. It would be an omission not to mention that the main material which all researchers in the area have inspired from and used is based on the one of the most cited papers by Bai et al. (Numer. Math. 102:1–38, 2005). [ABSTRACT FROM AUTHOR]

In this paper we study the large N limit of the O(N)-invariant linear sigma model, which is a vector-valued generalization of the Φ 4 quantum field theory, on the three dimensional torus. We study the problem via its stochastic quantization, which yields a coupled system of N interacting SPDEs. We prove tightness of the invariant measures in the large N limit. For large enough mass or small enough coupling constant, they converge to the (massive) Gaussian free field at a rate of order 1 / N with respect to the Wasserstein distance. We also obtain tightness results for certain O(N) invariant observables. These generalize some of the results in Shen et al. (Ann Probab 50(1):131–202, 2022) from two dimensions to three dimensions. The proof leverages the method recently developed by Gubinelli and Hofmanová (Commun Math Phys 384(1):1–75, 2021) and combines many new techniques such as uniform in N estimates on perturbative objects as well as the solutions. [ABSTRACT FROM AUTHOR]

Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers' knowledge of variance and invariance. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Specifically, we investigated if they even attend to invariant properties; what invariant properties they discern and discuss; and how DGE can support such discernment. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. We also saw that teachers' backgrounds and past experiences can play an important role in their descriptions of invariant properties. Possible future research directions and implications to teacher education are discussed. [ABSTRACT FROM AUTHOR]

This paper addresses an article by Felix Klein of 1886, in which he generalized his theory of polynomial equations of degree 5—comprehensively discussed in his Lectures on the Icosahedron two years earlier—to equations of degree 6 and 7. To do so, Klein used results previously established in line geometry. I review Klein's 1886 article, its diverse mathematical background, and its place within the broader history of mathematics. I argue that the program advanced by this article, although historically overlooked due to its eventual failure, offers a valuable insight into a time of crucial evolution of the subject. [ABSTRACT FROM AUTHOR]

Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro (Math Program 95(2):329–357, 2003) proposed to reduce the dimension of the problem by appealing to a low-rank factorization and solve the subsequent non-convex problem instead. In this paper, we present coordinate ascent based methods to solve this non-convex problem with provable convergence guarantees. More specifically, we prove that the block-coordinate maximization algorithm applied to the non-convex Burer–Monteiro method globally converges to a first-order stationary point with a sublinear rate without any assumptions on the problem. We further show that this algorithm converges linearly around a local maximum provided that the objective function exhibits quadratic decay. We establish that this condition generically holds when the rank of the factorization is sufficiently large. Furthermore, incorporating Lanczos method to the block-coordinate maximization, we propose an algorithm that is guaranteed to return a solution that provides 1 - O 1 / r approximation to the original SDP without any assumptions, where r is the rank of the factorization. This approximation ratio is known to be optimal (up to constants) under the unique games conjecture, and we can explicitly quantify the number of iterations to obtain such a solution. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which are not well-defined in the classical sense and are understood by using the paracontrolled distribution method introduced in (Gubinelli et al. in Forum Math Pi 3(6):1, 2015). By a new characterization of weighted Hölder spaces and Zvonkin's transformation we prove some new a priori estimates, and therefore establish the global well-posedness for singular HJB equations. As applications, we obtain global well-posedness in polynomial weighted Hölder spaces for KPZ type equations on the real line, as well as modified KPZ equations for which the Cole–Hopf transformation is not applicable. [ABSTRACT FROM AUTHOR]

This paper considers a coupled flow and mechanics problem in a fractured poro-elastic medium. The fracture geometry is explicitly treated as a possibly non-planar interface. The model equations are of mixed dimensional type where the flow equation on a d − 1 dimensional fracture surface is coupled to a d dimensional porous matrix. An extension of the widely used undrained split iterative coupling scheme to the fractured poro-elastic media is performed. The particularity is in the time-discretization where both the single rate scheme in which flow and mechanics share the exact same time step, and the multirate scheme in which flow takes multiple fine time steps within one coarse mechanics time step. In the coupled model considered here, the fracture flow model is a lubrication-type system (Girault et al. Math. Models Methods Appl. Sci. 25(4):587–645 2015) whereas in the porous matrix the linear quasi-static Biot equations are considered. Convergence analysis of the undrained split iterative scheme and Banach fixed-point contraction type results are obtained that establish the geometric convergence of the scheme and the uniqueness of the obtained solution. [ABSTRACT FROM AUTHOR]

In this paper, we show that Loewner spaces introduced by Heinonen and Koskela (Acta Math., 1998) are preserved under quasimöbius mappings between Ahlfors regular spaces. [ABSTRACT FROM AUTHOR]

This paper proves a theorem on convergence in mean of order p for double arrays of M-pairwise negatively dependent random variables, where 1 ≤ p < 2 . The main result extends some results in Thành (Int J Math Math Sci 27(8):1317–1320, 2005) and Hong and Hwang (Int J Math Math Sci 22(1):171–177, 1999). The proof is based on the von Bahr–Esseen inequality for M-pairwise negatively dependent random variables. The sharpness of the result is illustrated by two examples. [ABSTRACT FROM AUTHOR]

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in R n + 1 for all n ≥ 3 : we show that if a mean curvature flow { M t } in R n + 1 has an S n - 1 × R singularity at (x 0 , t 0) , then there exists an ε = ε (x 0 , t 0) > 0 such that M t ∩ B (x 0 , ε) is mean-convex for all t ∈ (t 0 - ε 2 , t 0 + ε 2) . As in the case n = 2 , which was resolved by the first three authors in Choi et al. (Acta Math, 2018), the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors (Hershkovits and White in Commun Pure Appl Math 73(3):558–580, 2020), we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when n ≥ 3 , smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call "moving plane method without assuming smoothness"—where smoothness and symmetry are established in tandem. [ABSTRACT FROM AUTHOR]

In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods. [ABSTRACT FROM AUTHOR]

In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and g ′ , where g ′ is a derived function of a 2π-periodic function g, in the generalized Zygmund class X z (η) , z ≥ 1 , using matrix-Cesàro (T C δ) means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1. [ABSTRACT FROM AUTHOR]

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers [Doc. Math., 21, 1607–1643 (2016)], [Taiwanese J. Math., 20(4), 723–741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number √q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author [Forum Math., 22(3), 565–582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations. [ABSTRACT FROM AUTHOR]

We consider the three dimensional free-boundary compressible elastodynamic system under the Rayleigh–Taylor sign condition. This describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin (J Differ Eq 264(3):1661–1715, 2018) by Nash–Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach. In the proof, we apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou–Lindblad (Commun Pure Appl Math 53(12):1536–1602, 2000) elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations despite a potential loss of derivative in the source term. To the best of our knowledge, we first establish the nonlinear energy estimate without loss of regularity for free-boundary compressible elastodynamics. The energy estimate is also uniform in sound speed which yields the incompressible limit, that is, the solutions of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in the boundary energy and analyze higher order wave equations as in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) can still be recovered for a slightly compressible elastic medium by further delicate analysis of the Alinhac good unknowns, which is completely different from Euler equations. [ABSTRACT FROM AUTHOR]

The higher regularity estimate has been a challenging question regarding the Boltzmann equation in bounded domains. Indeed, it is well-known to have "the non-existence of a second order derivative at the boundary" in Guo et al. (Invent Math 207:115–290, 2017) even for symmetric convex domains such as a disk or sphere. In this paper, we answer this question in the affirmative by constructing the C 1 , β solutions away from the grazing boundary, for any β < 1 , to the stationary Boltzmann equation with the non-isothermal diffuse boundary condition in strictly convex domains, as long as a smooth wall temperature has small fluctuation pointwisely. [ABSTRACT FROM AUTHOR]

We examine the family of all (at most) three-point symmetric quadratures on [ - 1 , 1 ] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020). [ABSTRACT FROM AUTHOR]

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the Lasserre moment-SOS hierarchy of relaxations for binary and box constrained POPs when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the DNN relaxation, and the design of efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Numerical results on various POPs are available in Ito et al. (ACM Trans Math Softw 45, Article 34, 2019). [ABSTRACT FROM AUTHOR]

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic (e.g. Giaquinto in J Philos 98(1):5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni (Talking about nothing: numbers, hallucinations and fictions, Oxford University Press, 2010), for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two systems that have so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic profiles, uniqueness and orbital stability of McCann's uniformly rotating binary stars (Houston J Math 32(2):603–631, 2006) governed by the Euler–Poisson system. A new uniqueness result will be importantly used in stability analysis. Moreover, we apply our framework to the study of uniformly rotating binary galaxies of the Vlasov–Poisson system through Rein's reduction (Handbook of differential equations: evolutionary equations, vol III, pp 383–476, 2007). [ABSTRACT FROM AUTHOR]

We extend the validity of Kiss's characterization of " [ α , β ] = 0 " from congruence modular varieties to varieties with a difference term. This fixes a recently discovered gap in our paper Kearnes et al. (Trans Am Math Soc 368:2115–2143, 2016). We also prove some related properties of Kiss terms in varieties with a difference term. [ABSTRACT FROM AUTHOR]

The concept of Tribonacci sequence spaces by the domain of a regular Tribonacci matrix was introduced by Yaying and Hazarika (Math. Slovaca 70(3):697–706, 2000). In this paper, by using the domain of regular Tribonacci matrix T = (t i k) and the concept of neutrosophic convergence, we introduce some neutrosophic normed space in Tribonacci convergent spaces and prove some topological and algebraic properties based results with respect to these spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove that Markov's theorem on variation of zeros of orthogonal polynomials on the real line (Markoff in Math Ann 27:177–182, 1886) remains essentially valid in the case of paraorthogonal polynomials on the unit circle. [ABSTRACT FROM AUTHOR]

Henning et al. (Discrete Appl Math 162:399–403, 2014) proved that if G is a bipartite, cubic graph of order n and of girth at least 6, then i (G) ≤ 4 11 n . In this paper, we improve the 4 11 -bound to a 5 14 -bound, and prove that if G is a bipartite, cubic graph of order n and of girth at least 6, then i (G) ≤ 5 14 n . [ABSTRACT FROM AUTHOR]

Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer t ≥ 1 , denote by MG t the class of graphs whose maximum multiplicity is at most t. A graph G is called strictly t-degenerate if every non-empty subgraph H of G contains a vertex v whose degree in H is at most t - 1 . The point partition number χ t (G) of G is the smallest number of colors needed to color the vertices of G so that each vertex receives a color and vertices with the same color induce a strictly t-degenerate subgraph of G. So χ 1 is the chromatic number, and χ 2 is known as the point aboricity. The point partition number χ t with t ≥ 1 was introduced by Lick and White (Can J Math 22:1082–1096, 1970). If H is a simple graph, then tH denotes the graph obtained from H by replacing each edge of H by t parallel edges. Then ω t (G) is the largest integer n such that G contains a t K n as a subgraph. Let G be a graph belonging to MG t . Then ω t (G) ≤ χ t (G) and we say that G is χ t -perfect if every induced subgraph H of G satisfies ω t (H) = χ t (H) . Based on the Strong Perfect Graph Theorem due to Chudnowsky, Robertson, Seymour and Thomas (Ann Math 164:51–229, 2006), we give a characterization of χ t -perfect graphs of MG t by a set of forbidden induced subgraphs (see Theorems 2 and 3). We also discuss some complexity problems for the class of χ t -critical graphs. [ABSTRACT FROM AUTHOR]

In this paper, we verify a relation between the symmetric continuity of lower semicontinuous extended seminorms on a locally convex cone with respect to the symmetric topology and barreledness of the locally convex cone. The assertion, whether a barreled cone is upper-barreled or not, was posed as a question in Ranjbari and Saiflu (J Math Anal Appl 332(2):1097–1108, 2007). We show that a barreled locally convex cone is not necessarily upper-barreled. [ABSTRACT FROM AUTHOR]

In this paper we show that for a generalized Berger metric g ^ on S 3 close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with (S 3 , [ g ^ ]) as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if g ^ is an SU (k + 1) -invariant metric on S 2 k + 1 for k ≥ 1 , the non-positively curved CCE metric on the (2 k + 1) -ball B 1 (0) with (S 2 k + 1 , [ g ^ ]) as its conformal infinity is unique up to isometries. In particular, since in Li (Trans Amer Math Soc 369(6): 4385–4413, 2017), we proved that if the Yamabe constant of the conformal infinity Y (S 2 k + 1 , [ g ^ ]) is close to that of the round sphere then any CCE manifold filled in must be negatively curved and simply connected, therefore if g ^ is an SU (k + 1) -invariant metric on S 2 k + 1 which is close to the round metric, the CCE metric filled in is unique up to isometries. Using the continuity method, we prove an existence result of the non-positively curved CCE metric with prescribed conformal infinity (S 2 k + 1 , [ g ^ ]) when the metric g ^ is SU (k + 1) -invariant. [ABSTRACT FROM AUTHOR]

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power and meta-theoretic properties when comparing first-order and second-order logic. [ABSTRACT FROM AUTHOR]

The paper is a continuation of Korolev and Laurinčikas (Aequ Math 93:859–873, 2019), where theorems on the approximation of analytic functions by shifts ζ (s + i h t k) , h > 0 , k ∈ N , and t k are the Gram points, were obtained. In this paper, it is proved, that the set of shifts ζ (s + i h t k) has a positive density in short intervals [ N , N + M ] with M = o (N) . [ABSTRACT FROM AUTHOR]

This paper provides results in the area of the analytical derivation of the efficient set of a mean-variance portfolio selection problem that has more than three criteria. By "analytical" we mean derived by formula as opposed to being computed by algorithm. By "more than three criteria", we mean that beyond the mean and variance of regular portfolio selection, the problems addressed have two or more additional linear objectives. The additional objectives might include sustainability, dividend yield, liquidity, and R&D as extra objectives like these are being seen with greater frequency. While not all multiple criteria portfolio selection problems lend themselves to an analytical derivation, a certain class does and the problems in this class are covered by the mathematics of this paper. [ABSTRACT FROM AUTHOR]

We continue the study of the rational Picard group of the moduli space of Hitchin spectral covers started in Korotkin and Zograf (J Math Phys 59(9):091412, 2018). In the first part of the paper we expand the "boundary", "Maxwell stratum" and "caustic" divisors introduced in Korotkin and Zograf (2018) via the set of standard generators of the rational Picard group. This generalizes the result of Korotkin and Zograf (2018), where the expansion of the full discriminant divisor (which is a linear combination of the classes mentioned above) was obtained. In the second part of the paper we derive a formula that relates two Hodge classes in the rational Picard group of the moduli space of Hitchin spectral covers. [ABSTRACT FROM AUTHOR]

A recursive algorithm for the determinant evaluation of general opposite-bordered tridiagonal matrices has been proposed by Jia et al. (J Comput Appl Math 290:423–432, 2015). Since the algorithm is a symbolic algorithm, it never suffers from breakdown. However, it may be time-consuming when many symbolic names emerge during the symbolic computation. In this paper, without using symbolic computation, first we present a novel breakdown-free numerical algorithm for computing the determinant of an n-by-n opposite-bordered tridiagonal matrix, which does not require any extra memory storage for the implementation. Then, we present a cost-efficient algorithm for the determinants of opposite-bordered tridiagonal matrices based on the use of the combination of an elementary column operation and Sylvester's determinant identity. Furthermore, we provide some numerical results with simulations in Matlab implementation in order to demonstrate the accuracy and efficiency of the proposed algorithms, and their competitiveness with other existing algorithms. The corresponding results in this paper can be readily obtained for computing the determinants of singly-bordered tridiagonal matrices. [ABSTRACT FROM AUTHOR]

The sixteen BIT papers that have been most frequently cited since 1981 are listed, along with citation counts collected from the ISI Web of Science citation database. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any pair of adjacent vertices have different sets consisting of colors assigned on their incident edges. The neighbor-distinguishing index of G , denoted by χ a ′ (G) , is the minimum number of colors in such an edge coloring of G . In this paper, we show that if G is a connected planar graph with maximum degree Δ ≥ 14 , then Δ ≤ χ a ′ (G) ≤ Δ + 1 , and χ a ′ (G) = Δ + 1 if and only if G contains a pair of adjacent vertices of maximum degree. This improves a result in [W. Wang, D. Huang, A characterization on the adjacent vertex distinguishing index of planar graphs with large maximum degree, SIAM J. Discrete Math. 29(2015), 2412–2431], which says that every connected planar graph G with Δ ≥ 16 has Δ ≤ χ a ′ (G) ≤ Δ + 1 , and χ a ′ (G) = Δ + 1 if and only if G contains a pair of adjacent vertices of maximum degree. [ABSTRACT FROM AUTHOR]

The SW26010 many-core processor is based on the Sunway architecture that is composed of management and computing processing elements (MPE and CPE, respectively), each of which is equipped with a stand-alone math library. The issue is that each Sunway Math Library (SML) version is written in assembly which is outside the power of compilers that take high-level languages as input; existing optimization approaches thus mainly rely on manual strategies, which are considered inefficient. In this paper, we leverage the concept of superblock scheduling, a well-known compilation technique, and present a tool named SMPOT to optimize the SML. SMPOT first builds a superblock using a novel tail duplication algorithm, and then uses code motion restrictions to avoid code compensation, followed by matching the machine model. Finally, it reorders instructions on the main path by an activation algorithm based on available computing resources. The experimental results show that SMPOT can effectively improve the performance of the SML. The main path performance of MPE functions is improved by 10.61% on average and overall performance by 5.40%. The main path performance of CPE functions is improved by 5.72% on average and overall performance by 2.98%. [ABSTRACT FROM AUTHOR]

The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space and time dependent logistic source, 0.1 ∂ t u = Δ u - χ ∇ · (u ∇ v) + u (a (x , t) - b (x , t) u) , x ∈ R N , 0 = Δ v - λ v + μ u , x ∈ R N , where N ≥ 1 is a positive integer, χ , λ and μ are positive constants, and the functions a(x, t) and b(x, t) are positive and bounded. In the first of the series (Salako and Shen in Math Models Methods Appl Sci 28(11):2237–2273, 2018), we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initial data, and the asymptotic spreading for solutions with compactly supported or front like initial data. In the second of the series (Salako and Shen in J Math Anal Appl 464(1):883–910, 2018), we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In particular, in the case of space homogeneous logistic source (i.e. a (x , t) ≡ a (t) and b (x , t) ≡ b (t) ), we proved in Salako and Shen (J Math Anal Appl 464(1):883–910, 2018) that the unique spatially homogeneous strictly positive entire solution (u ∗ (t) , v ∗ (t)) of (0.1) is uniformly and exponentially stable with respect to strictly positive perturbations when 0 < 2 χ μ < inf t ∈ R b (t) . In the current part of the series, we discuss the existence of transition front solutions of (0.1) connecting (0, 0) and (u ∗ (t) , v ∗ (t)) in the case of space homogeneous logistic source. We show that for every χ > 0 with χ μ (1 + sup t ∈ R a (t) inf t ∈ R a (t) ) < inf t ∈ R b (t) , there is a positive constant c χ ∗ such that for every c ̲ > c χ ∗ and every unit vector ξ , (0.1) has a transition front solution of the form (u (x , t) , v (x , t)) = (U (x · ξ - C (t) , t) , V (x · ξ - C (t) , t)) satisfying that C ′ (t) = a (t) + κ 2 κ for some positive number κ , lim inf t - s → ∞ C (t) - C (s) t - s = c ̲ , and lim x → - ∞ sup t ∈ R | U (x , t) - u ∗ (t) | = 0 and lim x → ∞ sup t ∈ R | U (x , t) e - κ x - 1 | = 0. Furthermore, we prove that there is no transition front solution (u (x , t) , v (x , t)) = (U (x · ξ - C (t) , t) , V (x · ξ - C (t) , t)) of (0.1) connecting (0, 0) and (u ∗ (t) , v ∗ (t)) with least mean speed less than 2 a ̲ , where a ̲ = lim inf t - s → ∞ 1 t - s ∫ s t a (τ) d τ . [ABSTRACT FROM AUTHOR]

In this paper, we consider the following one-dimensional, quasi-periodically forced, linear KdV equations u t + (1 + a 1 (ω t)) u xxx + a 2 (ω t , x) u xx + a 3 (ω t , x) u x + a 4 (ω t , x) u = 0 under the periodic boundary condition u (t , x + 2 π) = u (t , x) , where ω 's are frequency vectors lying in a bounded closed region Π ∗ ⊂ R b for some b > 1 , a 1 : T b → R , a i : T b × T → R , i = 2 , 3 , 4 , are real analytic, bounded from the above by a small parameter ϵ ∗ > 0 under a suitable norm, and a 1 , a 3 are even, a 2 , a 4 are odd. Under the real analyticity assumption of the coefficients, we re-visit a result of Baldi et al. (Math Ann 359(1–2):471–536, 2014) by showing that there exists a Cantor set Π ϵ ∗ ⊂ Π ∗ with | Π ∗ \ Π ϵ ∗ | = O (ϵ ∗ 1 100) such that for each ω ∈ Π ϵ ∗ , the corresponding equation is smoothly reducible to a constant-coefficient one. Our main result removes a condition originally assumed in Baldi et al. (2014) and thus can yield general existence and linear stability results for quasi-periodic solutions of a reversible, quasi-periodically forced, nonlinear KdV equation with much less restrictions on the nonlinearity. The proof of our reducibility result makes use of some special structures of the equations and is based on a refined Kuksin's estimate for solutions of homological equations with variable coefficients. [ABSTRACT FROM AUTHOR]

In a digraph with a source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some flow x that is not necessarily unsplittable but satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., y a ≈ x a for all arcs a. Twenty years ago, in a landmark paper, Dinitz et al. (Combinatorica 19:17–41, 1999) proved that there exists an unsplittable flow y such that y a ≤ x a + d max for all arcs a, where d max denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result, based upon an entirely new approach. Secondly, using a subtle variant of this approach, we obtain a new result: There is an unsplittable flow y such that y a ≥ x a - d max for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein (SIAM J Comput 31:919–946, 2002) and Skutella (Math Program 91:493–514, 2002), we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case when demands are integer multiples of each other. For arbitrary demand values, we prove the weaker simultaneous bounds x a / 2 - d max ≤ y a ≤ 2 x a + d max for all arcs a. [ABSTRACT FROM AUTHOR]

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the constraint matrix A ∈ Z r n × s + n t consists roughly of n repetitions of a matrix A ∈ Z r × s on the vertical line and n repetitions of a matrix B ∈ Z r × t on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and Δ , where Δ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time f (r , s , Δ) · poly (n , t) , where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps. [ABSTRACT FROM AUTHOR]

In this paper, we establish the full L p boundedness of noncommutative Bochner–Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in Chen et al. (Commun Math Phys 322(3):755–805, 2013) in the sense of the L p convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner–Riesz means. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if a R + b R = e R for some nonzero idempotent e in R. Let G ′ (R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G ′ (R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G ′ (R) is ≤ 2 . Finally, we show that the graph G ′ (R) is a line graph of some graph if and only if R is either a regular ring or a local ring. [ABSTRACT FROM AUTHOR]

In this paper, we establish the full L p boundedness of noncommutative Bochner–Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in Chen et al. (Commun Math Phys 322(3):755–805, 2013) in the sense of the L p convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner–Riesz means. [ABSTRACT FROM AUTHOR]

We consider the subclass of class B consisting of meromorphic functions f : C → C ^ for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class was introduced in Bergweiler and Kotus (Trans Am Math Soc 364(10):5369–5394, 2012) and the Hausdorff dimension HD (I (f)) of the set I (f) of all points escaping to infinity under forward iteration of f was estimated therein. In this paper we provide a closed formula for the exact value of HD (I (f)) identifying it with the critical exponent of the natural series introduced in Bergweiler and Kotus (2012). This exponent is very easy to calculate for many concrete functions. In particular, we construct a function from this class which is of infinite order and for which HD (I (f)) = 0 . [ABSTRACT FROM AUTHOR]

Teaching mathematical modeling is a demanding task. Thus, fostering teachers' competencies in this regard is an essential component of teacher education. Recent conceptualizations of teachers' competencies include situation-specific skills based on the concept of noticing, which is of particular interest for the spontaneous reactions needed when teaching mathematical modeling. The study described in this paper aims to analyze the development of a video-based instrument for measuring teachers' noticing competencies within a mathematical modeling context and obtain evidence for the validity of the instrument. Three kinds of validity are examined in three different studies: content validity, elemental validity and construct validity. Indicators for content validity could be found through different expert ratings and implementation with the target group, where participants were able to perceive all relevant aspects. The qualitative analysis of participants' reasoning, which is consistent with the coded level, indicates elemental validity. Moreover, the results of the confirmatory factor analysis suggest construct validity with one overall factor of noticing competence within a mathematical modeling context. Taken together, these studies imply a satisfactory validity of the video-based instrument. [ABSTRACT FROM AUTHOR]

This paper reports on empirical results about the influence of two different teaching designs on the development of tertiary students' modelling competency and attitudes towards modelling. A total of 144 first year engineering students were exposed to a diagnostic entrance test, a modelling unit consisting of five lessons with ten tasks, enframed by a pre- and a post-test, and at the end a questionnaire on attitudes towards mathematical modelling. Similar to the German DISUM study, in the modelling unit, one group of participants followed an independence-oriented teaching style, aiming at a balance between students' independent work and teacher's guidance, while two other groups were taught according to a more traditional teacher-guided style. Linear mixed regression models were used to compare pre- and post-test results. The results show that all groups had significant learning progress, although there is much room for further improvement, and that the group taught according to the independence-oriented design had the biggest competency growth. In addition, this group exhibited more positive attitudes than the other groups in five of six attitudinal aspects. [ABSTRACT FROM AUTHOR]

Although mathematical modeling plays an important role in many curricula worldwide, significant discrepancies persist in the importance of mathematical modeling in ordinary mathematics classrooms and teacher education. This paper compares pre-service mathematics teachers' professional mathematical modeling competencies in three different regions—Germany, Mainland China, and Hong Kong—where educational and cultural traditions differ, including the role of mathematical modeling. In total, 232 pre-service mathematics teachers from the three regions completed a modeling task covering mathematics content knowledge (MCK) of modeling and mathematical pedagogical content knowledge (MPCK) of modeling. The results show that pre-service teachers from Germany demonstrated the strongest MCK and MPCK of mathematical modeling; by contrast, pre-service mathematics teachers from Mainland China and Hong Kong demonstrated relatively weaker MCK and MPCK of mathematical modeling. MCK and MPCK of mathematical modeling were also found to be unevenly developed at different competence levels for the three regions. These differences may be attributed to the history of mathematical modeling in mathematics curricula, teacher education, and teaching culture in these three regions. [ABSTRACT FROM AUTHOR]

Modelling competencies are currently included in numerous curricula worldwide and are generally accepted as a complex, process-oriented construct. Therefore, effective measurement should include multiple dimensions, like the sub-competencies required throughout the modelling process. Departing from the characteristics of modelling problems as open and often underdetermined real-world problems, we propose to enrich the current conceptualisation of mathematical modelling competencies by including creativity, which plays an important role in numerous phases of the mathematical modelling process but has scarcely been considered in modelling discourse. In the study described in this paper, a new instrument for the evaluation of this enriched construct has been developed and implemented. The modelling competencies incorporating creativity of the students were evaluated based on the adequacy of the models and the modelling processes proposed, and the appropriateness and completeness of the approaches were evaluated in detail. Adapting measurement approaches for creativity that have been developed in the problem-solving discourse, certain criteria of creativity were selected to evaluate the creativity of the students' approaches in tackling modelling problems—namely, usefulness, fluency, and originality. The empirical study was conducted among 107 Chinese students at the upper secondary school level, who attended a modelling camp and independently solved three complex modelling problems. The results reveal significant correlations between fluency and originality in students' performances across all tasks; however, the relationships between usefulness and the other two creativity aspects were not consistent. Overall, the results of the study support the importance of the inclusion of creativity in the construct of modelling competencies. [ABSTRACT FROM AUTHOR]

It is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean N such that K is invariant with respect to the mean-type mapping M , N , which means that K ∘ M , N = K and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations. [ABSTRACT FROM AUTHOR]

Recently, Phuong and Thin (Ukrain Math J 67(7), 2015) proved some fundamental theorems for holomorphic curves on the annuli with target being fixed hyperplanes. In this paper, we consider the same problems in the case of target being fixed hypersurfaces. [ABSTRACT FROM AUTHOR]

Solovyov (Fuzzy Sets Syst 159:2567–2585, 2008) introduced the notion of stratified Q-topological spaces, which in turn gives rise to the category Str-Q-TOP of stratified Q-topological spaces (where Q is a fixed member of a fixed variety of Ω -algebras). Singh and Srivastava (Ann. Fuzzy Math. Inform. 12:539–546, 2016) introduced the Q-topological space (Q , ⟨ { i d Q } ∪ { q ̲ ∣ q ∈ Q } ⟩) which is a Sierpinski object in the category Str-Q-TOP. In this paper, motivated by Singh (Fuzzy Sets Syst 159:2611–2615, 2008), we determine the coreflective hull of (Q , ⟨ { i d Q } ∪ { q ̲ ∣ q ∈ Q } ⟩) in the category Str-Q-TOP. We also determine the coreflective hulls of the categories Str-Dis-Q-TOP of discrete Q-topological spaces and Str-Ind-Q-TOP of stratified indiscrete Q-topological spaces, in the category Str-Q-TOP, motivated by Hoffmann (Arch Math 33:258–262, 1979) and Singh and Srivastava (Quaest. Math. 36: 167–179, 2013) [ABSTRACT FROM AUTHOR]