Search

The stabilization properties of dissipative Timoshenko systems have been attracted the attention and efforts of researchers over the years. In the past 20 years, the studies in this scenario distinguished primarily by the nature of the coupling and the type or strength of damping. Particularly, under the premise that the Timoshenko beam model is a two-by-two system of hyperbolic equations, a large number of papers have been devoted to the study of the so-called partially damped Timoshenko systems by assuming damping effects acting only on the angle rotation or vertical displacement (Almeida Júnior et al. in Math Methods Appl Sci 36:1965–1976, 2013; in Z Angew Math Phys 65:1233–1249, 2014; Alves et al. in SIAM J Math Anal 51(6):4520–4543, 2019; Ammar-Khodja et al. in J Differ Equ 194:82–115, 2003; Muñoz Rivera and Racke in Discrete Contin Dyn Syst Ser B 9:1625–1639, 2003; J Math Anal Appl 341:1068–1083, 2008; Santos et al. in J Differ Equ 253(9):2715–2733, 2012). In these cases, the desired exponential decay property of the energy solutions is achieved when the non-physical equal wave speed assumption plays the role to stabilization according since the pioneering Soufyane's paper (C R Acad Sci Paris 328(8):731–734, 1999). Recent results due to Almeida Júnior et al. (Z Angew Math Phys 68(145):1–31, 2017; Z Angew Math Mech 98(8):1320–1333, 2018; IMA J Appl Math 84(4):763–796, 2019; Acta Mech 231:3565–3581, 2020) show that the second vibration mode or simply second spectrum of frequency and it's damaging consequences appears as a lost element in analysis of stabilization and now it's more clear that the damping importance into stabilization scenario of Timoshenko type systems. This paper considers a one-dimensional viscoelastic Timoshenko type system in the light of the second spectrum of frequency where the equal wave speed assumption is not needed for getting the exponential decay property. Precisely, we consider the so-called truncated version for the Timoshenko system according studies due to Elishakoff (Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010; ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11, 2015; Int J Solids Struct 109:143–151, 2017; J Sound Vib 435:409–430, 2017; Int J Eng Sci 116:58–73, 2017; Acta Mech 229:1649–1686, 2018; Z Angew Math Mech 98(8):1334–1368, 2018; Math Mech Solids, 2019) and we added a viscoelastic damping acting on shear force. We firstly prove the global well-posedness of the system by Faedo–Galerkin approximation. By assuming minimal conditions on the relaxation function, we establish an optimal explicit and energy decay rate for which exponential and polynomial rates are special cases. This result is new and substantially improves earlier results in the literature where the equal wave speeds plays the role for getting the stability properties. It is likely to open more research areas to Timoshenko system and probably others. [ABSTRACT FROM AUTHOR]

In this paper, we characterize that the boundedness, compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space. This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng's paper [Trans. Amer. Math. Soc., 354, 2495–2520 (2002)]. [ABSTRACT FROM AUTHOR]

In the present paper, the full range Strichartz estimates for homogeneous Schrödinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained. [ABSTRACT FROM AUTHOR]

Taking as its point of departure, situations arising in classrooms exposed to a heavy use of CAS, exemplified by the Danish upper secondary school mathematics program, the paper discusses the use of "old" cognitive frameworks of mathematics education as an alternative to more recent theoretical developments in the field. The paper draws on three rich examples of CAS use in relation to the following: mathematical concept formation; mathematical proofs and proving; and classroom teaching. Each of these examples illustrates aspects of CAS use that appear better captured and addressed through "old" frameworks. This includes the mathematics specific learning difficulties that a heavy CAS use gives rise to; the inexpedient use of CAS in proofs in textbooks; and the tacit norms and contracts surrounding CAS use in the classroom. The paper thus serves as a call to continuously consulting our research back catalogue, also when addressing problems and challenges which at first sight appear "new", such as the constantly evolving digital technologies for learning and teaching mathematics. [ABSTRACT FROM AUTHOR]

Decomposition is an important tool that is used in many differential systems for solving real engineering problems and improving the stability of a system. It involves breaking down of high-order linear systems into lower-order commutative pairs. Commutativity plays an essential role in mathematics, and its applications are extended in physical science and engineering. This paper explicitly expresses all form of necessary and sufficient conditions for decomposition of any kind of fourth-order linear time-varying system as commutative pairs of two second-order systems. Regarding the nonzero initial conditions, additional requirements are derived in order to satisfy the decomposition process. In this paper, explicit method for reducing fourth-order linear time-varying systems (LTVS) into two second-order commutative pairs is derived and solved. The method points out the effect of disturbance and sensitivity on the systems and also highlights the necessary and sufficient conditions for commutativity of the decomposed systems. The results are illustrated by solving some examples. [ABSTRACT FROM AUTHOR]

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]

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]

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 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]

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]

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]

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 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]

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 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]

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]

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]

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]

As argumentation is an activity at the heart of mathematics, (not only) German school curricula request students to construct mathematical arguments, which get evaluated by teachers. However, it remains unclear which criteria teachers use to decide on a specific grade in a summative assessment setting. In this paper, we draw on two sources for these criteria: First, we present theoretically derived dimensions along which arguments can be assessed. Second, a qualitative interview study with 16 teachers from German secondary schools provides insights in their criteria developed in practice. Based on the detailed presentation of the case of one teacher, the paper then illustrates how criteria developed in practice take a variety of different aspects into account and also correspond with the theoretically identified dimensions. The findings are discussed in terms of implications for the teaching and learning about mathematical argumentation in school and university: An emphasis on more pedagogical criteria in high school offers one explanation to the perceived gap between school and university level mathematics. [ABSTRACT FROM AUTHOR]

The metaphors that students form and encounter have been shown to exert a powerful influence on how they think about mathematics. In this paper, we explore the linguistic metaphors about learning and doing mathematics that were prevalent in 11 advanced mathematics lectures. We present four metaphor clusters that were common in the corpus that we analyzed: Learning Mathematics is a Journey, Doing Mathematics is Work, Mathematics is Discovery, and Presenting Mathematics is a Story. We conclude the paper by discussing the implications of the metaphors that mathematicians use and students hear in their mathematics courses. [ABSTRACT FROM AUTHOR]

In the direction towards the question when mixed multiplicities are equal to the Hilbert–Samuel multiplicity of joint reductions, this paper not only generalizes Viet et al. (Proc Am Math Soc 142:1861–1873, 2014, Theorem 3.1) that covers the Rees's theorem Rees (J Lond Math Soc 29:397–414, 1984, Theorem 2.4), but also removes the hypothesis that joint reductions are systems of parameters in Viet et al. (2014, Theorem 3.1). The results of the paper seem to make the problem of expressing mixed multiplicities into the Hilbert–Samuel multiplicity of joint reductions become closer. [ABSTRACT FROM AUTHOR]

The aim of the paper is to argue for the cognitive unity of the mathematical results ascribed by ancient authors to Thales. These results are late ascriptions and so it is difficult to say anything certain about them on philological grounds. I will seek characteristic features of the cognitive unity of the mathematical results ascribed to Thales by comparing them with Galilean physics. This might seem at a first sight a rather unusual move. Nevertheless, I suggest viewing the process of turning geometry into an axiomatic-deductive science as a process of idealization in mathematics that is parallel to the process of idealization in physics. In Kvasz (Acta Phys Slovaca 62:519–614, 2012) I offered an epistemological reconstruction of the process of idealization in physics during the scientific revolution of the seventeenth century. In the present paper I try to employ these epistemological insights in the process of idealization in physics and propose a reconstruction of the cognitive unity of the mathematical results ascribed to Thales, who can, on the basis of these ascriptions, be seen as one of the initiators of idealization in mathematics. [ABSTRACT FROM AUTHOR]

An old problem since Leray (J Math Pure Appl (French) 9:1–82, 1933) asks whether homogeneous D-solutions of the 3 dimensional Navier–Stokes equation in R 3 or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two cases: (1) the full 3 dimensional slab case R 2 × 0 , 1 with Dirichlet boundary condition (Theorem 1.1); (2) when the solution is axially symmetric and periodic in the vertical variable (Theorem 1.3). Also, for the slab case, we prove that even if the Dirichlet integral has some growth, axially symmetric solutions with Dirichlet boundary condition must be swirl free, namely u θ = 0 , thus reducing the problem to essentially a "2 dimensional" problem. In addition, a general D-solution (without the axial symmetry assumption) vanishes in R 3 if, in spherical coordinates, the positive radial component of the velocity decays at order -1 of the distance. The paper is self contained comparing with (Carrillo et al. in Funct Anal, 2020. 10.1016/j.jfa.2020.108504) although the general idea is related. [ABSTRACT FROM AUTHOR]

In this paper, we will generalize some results in Manin's paper (Invent Math 104:223–243, 1991) to the supergeometric setting. More precisely, viewing C 1 | 1 as the boundary of the hyperbolic superspace H 3 | 2 , we reexpress the super-Green functions on the supersphere C ^ 1 | 1 and the supertorus T 1 | 1 by some data derived from the supergeodesics in H 3 | 2 . [ABSTRACT FROM AUTHOR]

Recent evidence demonstrates that mathematics knowledge in the early years is an important contributor to students' learning and later achievement in mathematics. As early mathematics education has assumed heightened importance, the quality of early mathematics teaching and learning experiences has attracted global attention. Several researchers have designed and used a number of instruments to measure and portray the quality of mathematics teaching in early childhood classrooms. As this body of work is growing fast, it is imperative to explore similarities and differences among the existing research tools that aim to measure the quality of early mathematics instruction. This paper takes a step towards exploring such similarities and differences by reviewing six observation instruments designed to measure early mathematics teaching quality. Therefore, the aim of this paper is twofold: (1) to provide an up-to-date overview by examining these measures' theoretical bases, foci, and psychometrics; and (2) to examine the similarities and differences between the reviewed tools. [ABSTRACT FROM AUTHOR]

This paper discusses the place of precision in mathematics education by exploring its role in curricular guidelines and in classroom life. By means of a joke on precision delivered by a school student in South Sweden, our study focuses on student participation in mathematical tasks that require precision in processes of measuring and reasoning. The paper uses theories on humour and inferentialism to revisit the normative place of "precision" in mathematics classroom discourse. [ABSTRACT FROM AUTHOR]

In the literature different concepts of compatibility between a projective structure P and a conformal structure C on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. In an often cited paper (Ehlers et al. in: O'Raifertaigh (ed) General Relativity, Papers in Honour of J.L. Synge, Clarendon Press, Oxford, 2012) Ehlers/Pirani/Schild introduce still another criterion which is natural from the physical point of view: every light like geodesics of C is a geodesics of P . Their claim that this type of compatibility is sufficient for introducing a Weylian metric has recently been questioned (Trautman in Gen Relativ Gravit 44:1581–1586, 2012); (Vladimir in Commun Math Phys 329:821–825, 2014); as reported by Scholz (in: A scalar field inducing a non-metrical contribution to gravitational acceleration and a compatible add-on to light deflection, 2019). Here it is proved that the conjecture of EPS is correct. [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]

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 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]