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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, I argue that there are cases of explanatory induction in mathematics. To do so, I first introduce the notion of explanatory definition in the context of mathematical explanation. A large part of the paper is dedicated to introducing and analyzing this notion of explanatory definition and the role it plays in mathematics. After doing so, I discuss a particular inductive definition in advanced mathematics— CW -complexes—and argue that it is explanatory. With this, we see that there are cases of explanatory induction. [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 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 introduces two central questions (1) what are available methodologies for making claims about mathematical practice and (2) when and how do claims about mathematical practice influence mathematics instruction. To motivate these questions, we critically analyze the relationship between research into mathematicians' practice and the design of mathematics instruction in three ways. First, we describe three influential research programs in mathematics education and illustrate how each research program uses claims about mathematical practice to inform their instructional goals. Second, by examining these important works, we highlight intrinsic difficulties in investigating mathematical practice. Our conclusion is that every research methodology for investigating mathematical practice is fundamentally limited and we require triangulation from multiple methods and theoretical lenses to fully understand mathematical practice. Third, we highlight reasons for why mathematical practice sometimes should not inform mathematics instruction. We conclude this paper by discussing how the articles in this special issue address our two central questions. [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]

Growing use of the internet in educational contexts has been prominent in recent years. In this survey paper, we describe how the internet is transforming the mathematics classroom and mathematics teacher education. We use as references several reviews of use of the internet in mathematics education settings made in recent years to determine how the field has evolved. We identify three domains in which new approaches are being generated by mathematic educators: principles of design of new settings; social interaction and construction knowledge; and tools and resources. The papers in this issue reflect different perspectives developed in the last decade in these three domains, providing evidence of the advances in theoretical frameworks and support in the generation of new meanings for old constructs such as 'tool', 'resources' or 'learning setting'. We firstly highlight the different ways in which the use of digital technologies generates new ways of thinking about mathematics and the settings in which it is learnt, and how mathematics teacher educators frame the new initiatives of initial training and professional development. In this survey paper, we identify trends for future research regarding theoretical and methodological aspects, and recognise new opportunities requiring further engagement. [ABSTRACT FROM AUTHOR]

This paper focuses on MOOCs (massive open online courses), a fairly recent paradigm in e-learning educational projects. Despite the high dropout rate, and the impossibility of benefiting from the opportunities that bring with it a face-to-face dialogue, several factors make MOOCs a good option for ongoing teacher professional learning. The MOOCs on which we focus address mathematics teacher education. In particular, we illustrate our experience based on four MOOCs that we organized in Italy in the last 4 years within the Math MOOC UniTo project, the aim of which is the professional development of mathematics in-service teachers. In the paper we articulate the conceptual framework that has guided the design, data collection, analysis and interpretation of findings for the project. It is mainly based on the hybridization of the meta-didactical transposition (MDT) model with the instrumentation/instrumentalization processes of the instrumental approach and the network of knowledge of Connectivism. We offer this new framework (called MOOC-MDT) to stimulate discussion with colleagues in the mathematics education research community about ways in which it might be refined and extended, towards building a shared understanding of the process of mathematics teacher education through MOOCs. [ABSTRACT FROM AUTHOR]

A correction to the article "Neuroscience Perspectives for Science and Mathematics Learning in Technology-Enhanced Learning Environments" that was published in the 2013 issue is presented.

In this paper, we investigate the 3D incompressible Hall-magnetohydrodynamics with partial dissipation. Based on the results in (Du in Bound. Value Probl. 2022:6, 2022; Du and Liu in Acta Math. Sci. 42A:5, 2022; Fei and Xiang in J. Math. Phys. 56:051504, 2015), we establish an improved blow-up criterion for classical solutions. Furthermore, using the blow-up criterion, we also obtain the existence of the classical solutions only under the condition that the initial data ∥ V 0 ∥ H 1 + ∥ B 0 ∥ H 2 are sufficiently small. [ABSTRACT FROM AUTHOR]

Decision-making under uncertainty involves three fundamental components: acts, states of nature, and consequences, as first introduced by Savage (The foundations of statistics, Wiley, New York, 1954). An act is uniquely determined by a number of random consequences that are associated with different states of nature. If the consequences are identical across all states of nature, then the act is state-independent. Prior research on distortion risk measures (DRMs) has primarily focused on state-independent acts. In this paper, we extend the research to state-dependent acts by introducing a state-dependent DRM (SDRM) under the Anscombe–Aumann's framework (Anscombe and Aumannin in Ann Math Stat 34(1):199–205, 1963). The proposed SDRM is the weighted average of DRMs at each state, where the weights are determined by the decision maker's (DM's) subjective probabilities of the states. In situations where there is incomplete information about the DM's true distortion function and/or the true subjective probabilities of the states, we introduce a preference robust SDRM (PRSDRM) for acts. The PRSDRM is based on the worst-case state-dependent distortion function and the worst-case subjective probabilities over a dependent joint ambiguity set constructed with partially available information. To compute the PRSDRM numerically, we show that when the distortion functions are concave, it can be formulated as a biconvex program and further as a convex program by changing some variables. As a motivation and application, we use the PRSDRM for decision-making problems and propose an alternating iterative algorithm for solving it. Finally, we conduct numerical experiments to assess the performance of our proposed model and computational scheme. [ABSTRACT FROM AUTHOR]

Background: Blended mathematical sensemaking in science ("Math-Sci sensemaking") involves deep conceptual understanding of quantitative relationships describing scientific phenomena and has been studied in various disciplines. However, no unified characterization of blended Math-Sci sensemaking exists. Results: We developed a theoretical cognitive model for blended Math-Sci sensemaking grounded in prior work. The model contains three broad levels representing increasingly sophisticated ways of engaging in blended Math-Sci sensemaking: (1) developing qualitative relationships among relevant variables in mathematical equations describing a phenomenon ("qualitative level"); (2) developing mathematical relationships among these variables ("quantitative level"); and (3) explaining how the mathematical operations used in the formula relate to the phenomenon ("conceptual level"). Each level contains three sublevels. We used PhET simulations to design dynamic assessment scenarios in various disciplines to test the model. We used these assessments to interview undergraduate students with a wide range of Math skills. Interview analysis provided validity evidence for the categories and preliminary evidence for the ordering of the categories comprising the cognitive model. It also revealed that students tend to perform at the same level across different disciplinary contexts, suggesting that blended Math-Sci sensemaking is a distinct cognitive construct, independent of specific disciplinary context. Conclusion: This paper presents a first-ever published validated cognitive model describing proficiency in blended Math-Sci sensemaking which can guide instruction, curriculum, and assessment development. [ABSTRACT FROM AUTHOR]

Poincaré's conventionalism has been interpreted in many writings as a philosophical position emerged by reflection on certain scientific problems, such as the applicability of geometry to physical space or the status of certain scientific principles. In this paper I would like to consider conventionalism as a philosophical position that emerged from Poincaré's scientific practice. But not so much from dealing with scientific problems, as from the use of two specific methodologies proper to modern mathematics and the modern natural sciences: methodological structuralism and the hypothetical-deductive method—thus, as a philosophical position which emerged from a way (or rather, two ways) of doing science. With this approach, I try to deepen the analysis of connections between Poincaré's scientific practice and his philosophy. [ABSTRACT FROM AUTHOR]

Philosophy of science in the twentieth century tends to emphasize either the logic of science (e.g., Popper and Hempel on explanation, confirmation, etc.) or its history/sociology (e.g., Kuhn on revolutions, holism, etc.). This dichotomy, however, is neither exhaustive nor exclusive. Questions regarding scientific understanding and mathematical explanation do not fit neatly inside either category, and addressing them has drawn from both logic and history. Additionally, interest in scientific and mathematical practice has led to work that falls between the two sides of the dichotomy. We might call these less narrow approaches 'holistic'. Prior to 1918, work in the philosophy of science was also somewhat more holistic and interdisciplinary in nature. An exemplar of this type of approach is found in the work of Henri Poincaré. In this paper I will consider one particularly flexible tool Poincaré uses throughout his philosophical writings about science and mathematics: the tool of the unifying concept. More specifically, I will focus on one unifying concept: that of structure. I aim to show first, that structure is a unifying scientific concept for Poincaré; second, a focus on structure helps explain his scientific and mathematical success; lastly, this case study may provide a model for how unifying concepts can facilitate progress in mathematics and science. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the range of a certain class of ASH algebras in [An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202], which is under the scheme of the Elliott program in the setting of real rank zero C*-algebras. As a reduction theorem, they prove that all these ASH algebras are still the AD algebras studied in [Dadarlat, M. and Loring, T. A., Classifying C*-algebras via ordered, mod-p K-theory, Math. Ann., 305, 1996, 601–616]. [ABSTRACT FROM AUTHOR]

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of unit-trace n × n PSD matrices, denoted by D, using at most N number of k × k PSD constraints?" In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace n × n symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in R n . We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any k = o (n) . Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial S + k -extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have S + k -extension complexity at least exp (C · min { n , n / k }) where C is some absolute constant. In addition, we show that any set S such that D ⊆ S and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have S + k -extension complexity at least exp (C · min { n 1 / 3 , n / k }) . These results imply that the cone of n × n PSD matrices cannot be approximated by a polynomial number of k × k PSD constraints for any k = o (n / log 2 n) . These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of S + n , which corresponds to the special case with k = 1 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate normalized solutions for the Chern–Simons–Schrödinger system with a trapping potential V (x) = ω | x | 2 and a exponential critical growth f(u). The solutions correspond to critical points of the underlying energy functional subject to the L 2 -norm constraint, namely, ∫ R 2 | u | 2 d x = c for c > 0 given. Under some suitable assumptions on f, we show that the system has at least two normalized solutions u c , u ^ c ∈ H 1 (R 2) , depending on the trapping frequency ω and the mass c, where u c is a ground state with positive energy and orbitally stable, while u ^ c is a high-energy solution with positive energy. In addition, the asymptotic behavior of the solution u c as c → 0 is described. [ABSTRACT FROM AUTHOR]

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in B ˙ 2 d , r - 3 2 × ( B ˙ 2 d , r - 1 2 ) d when 1 ≤ r < d due to the lack of continuity of the solution. [ABSTRACT FROM AUTHOR]

Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang et al. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge–Ampère type equation on the sphere. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the locally strongly convex affine hypersurfaces with semi-parallel cubic form relative to the Levi-Civita connection of affine metric. We obtain two results on such hypersurfaces which admit at most one affine principal curvature of multiplicity one: (1) classify these being not affine hyperspheres; (2) classify these affine hyperspheres with constant scalar curvature. For the latter, by proving the parallelism of their cubic forms we translate the classification into that of affine hypersurfaces with parallel cubic form, which has been completed by Hu-Li-Vrancken (J Differ Geom 87:239–307, 2011). [ABSTRACT FROM AUTHOR]

In this paper, we consider the multiplicity and concentration phenomenon of positive solutions to the following Choquard equation - ε 2 Δ u + V (x) u = ε - α Q (x) (I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ - 2 u + f (u) in R N , where N ≥ 3 , (N - 4) + < α < N , I α is the Riesz potential, ε is a small parameter, V (x) ∈ C (R N) ∩ L ∞ (R N) is a positive potential, f ∈ C 1 (R + , R) is a subcritical nonlinear term and 2 α ∗ = N + α N - 2 is the upper-critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. By means of variational methods and delicate energy estimates, we establish the relationship between the number of solutions and the profiles of potentials V and Q, and the concentration behavior of positive solutions is also obtained for ε > 0 small. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Morse index of the critical Möbius band in the 4-dimensional Euclidean ball B 4 equals 5. It is conjectured that this is the only embedded non-orientable free boundary minimal surface of index 5 in B 4 . One of the ingredients in the proof is a comparison theorem between the spectral index of the Steklov problem and the energy index. The latter also enables us to give another proof of the well-known result that the index of the critical catenoid in B 3 equals 4. [ABSTRACT FROM AUTHOR]

In this paper, we provide new examples of Levi-Civita Ricci-flat Hermitian metrics on certain compact non-Kähler Calabi-Yau manifolds, including compact Hermitian Weyl-Einstein manifolds, compact locally conformal hyperKähler manifolds, certain suspensions of Brieskorn manifolds, and every generalized Hopf manifold provided by suspensions of exotic spheres. These examples generalize previous constructions on Hopf manifolds. Additionally, we also construct new examples of compact Hermitian manifolds with nonnegative first Chern class that admit constant strictly negative Riemannian scalar curvature. Further, we remark some applications of our main results in the study of the Chern-Ricci flow on compact Hermitian Weyl-Einstein manifolds. In particular, we describe the Gromov-Hausdorff limit for certain explicit finite-time collapsing solutions which generalize previous constructions on Hopf manifolds. [ABSTRACT FROM AUTHOR]

On the recently suggested loophole view of causal closure, nothing in a universe has its cause coming from another universe. It is allowed, though, that something, especially a first thing, can have its cause situated in an interface between universes. However, the possibility of such an interface does not mean that there is any actual interface. In fact, there are several major obstacles to be managed before an interface should be hoped for. One such obstacle, the need for an account of how objects in one universe relate to objects in another universe, is addressed in this paper. The outcome is positive and suggests that objects of different ontological status relate as the kind 'object'. Some positive and negative properties of such higher-order objects are noted. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimal time decay rates for the higher order spatial derivatives of solutions to a diffusive Oldroyd-B model. As pointed out in the Sect. 1.2 of Huang et al. (J Differ Equ 306:456–491, 2022), how to establish the optimal decay estimate for the highest-order spatial derivatives of the solution to this model is still an open problem. Motivated by Wang and Wen (Sci China Math 65:1199–1228, 2022), we give a positive answer to this problem via some delicate analyses on the low- and high-frequency parts of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the idea of Song, Zhao and Huang [Czechoslov. Math. J., 70, 483–504 (2020)] and introduce the notion of right (left) Gorenstein subcategory r G (C , D) (l G (C , D)) , relative to two additive full subcategories C and D of an abelian category A . Under the assumption that C ⊆ D , we prove that the right Gorenstein subcategory r G (C , D) possesses many nice properties that it is closed under extensions, kernels of epimorphisms and direct summands. When C ⊆ D and C ⊥ D , we show that the right Gorenstein subcategory r G (C , D) admits some kind of stability. Then we discuss a resolution dimension for an object in A , called r G (C , D) -projective dimension. Finally, we prove that if (U , V) is a hereditary cotorsion pair with kernel C has enough injectives, such that U ⊆ D and U ⊥ D , then (r G (C , D) , r e s C ^ , C) is a weak Auslander—Buchweitz context, where res C ^ is the subcategory of A consisting of objects with finite C -projective dimension. [ABSTRACT FROM AUTHOR]

Wittgenstein harshly criticized what he considered "mythologies" infecting our understanding of how mathematics work. He was especially reluctant to the idea that they describe a realm of abstract entities influencing our cognitive capacities. Several Wittgenstein scholars have drawn the conclusion that his analysis results in a sociological interpretation of mathematics. The present paper aims at demonstrating that although mathematical practices deserve to be called "social" this doesn't necessarily leads us to the conclusion that they are offered to sociological explanation. [ABSTRACT FROM AUTHOR]

In this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all admissible interpretations of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman (Mathematics without numbers: towards a modal-structural interpretation. Oxford University Press: Oxford, 1989), arguing that it provides us with an easier epistemology of statements in arithmetic. [ABSTRACT FROM AUTHOR]

Daubechies first used localization operators as a mathematical tool to localized a signal in the time frequency plane. They have been a subject of research in many domains ever since. In this paper, we introduce the notion of Weinstein two-wavelet and we define the two-wavelet localization operators in the setting of the Weinstein theory. Then, we give a host of sufficient conditions for the boundedness and compactness of the two-wavelet localization operator on L α p (R + d + 1) for all 1 ≤ p ≤ ∞ , in terms of properties of the symbol σ and the functions φ and ψ . In the end, we study some typical examples of the Weinstein two-wavelet localization operators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of sets and a new class of functions called E-univex sets and E-univex functions, respectively. For an E-differentiable function, the concept of E-univexity is introduced by generalizing several concepts of generalized convexity earlier defined into optimization theory. In addition, some properties of E-differentiable E-univex functions are investigated. Further, also concepts of E-differentiable generalized E-univexity are introduced. Then, the sufficiency of the so-called E-Karush–Kuhn–Tucker necessary optimality conditions are proved for an E-differentiable nonlinear optimization problem in which the involved functions are E-univex and/or generalized E-univex. [ABSTRACT FROM AUTHOR]

In this paper, we study vacuum static spaces. We firstly derive a Bochner-type formula for the Weyl tensor to vacuum static space. Based on a global argument, under the condition of zero radial Weyl curvature, we then obtain a pointwise identity and use it to prove that each closed vacuum static space of dimension n ≥ 5 with scalar curvature n (n - 1) and zero radial Weyl curvature is Bach flat. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear Dirac equation - i ħ ∂ t ψ = i c ħ ∑ k = 1 3 α k ∂ k ψ - m c 2 β ψ + K (x) | ψ | p - 2 ψ , (N D E) where ψ : R × R 3 → C 4 , K ∈ L ∞ (R 3) , c denotes the speed of light, m > 0 , the mass of the electron, ħ is Planck's constant, ∂ k = ∂ ∂ x k , α 1 , α 2 , α 3 , β are 4 × 4 Pauli–Dirac matrices and p ∈ (2 , 8 3) . We present a new approach which is based on a perturbation argument and show the existence of L 2 -normalized solitary wave solutions of (NDE) using variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Schwarz lemma for the pentablock. The pentablock P is defined by P = { (a 21 , tr A , det A) : A = [ a ij ] i , j = 1 2 ∈ B 2 × 2 } where B 2 × 2 denotes the open unit ball in the space of 2 × 2 complex matrices. The pentablock is a bounded non-convex domain in C 3 which arises naturally in connection with a certain problem of μ -synthesis. We develop a concrete structure theory for the rational maps from the unit disc D to the closed pentablock P ¯ that map the unit circle T to the distinguished boundary b P ¯ of P ¯ . Such maps are called rational P ¯ -inner functions. We give relations between P ¯ -inner functions and inner functions from D to the symmetrized bidisc. We describe the construction of rational P ¯ -inner functions x = (a , s , p) : D → P ¯ of prescribed degree from the zeroes of a, s and s 2 - 4 p . The proof of this theorem is constructive: it gives an algorithm for the construction of a family of such functions x subject to the computation of Fejér–Riesz factorizations of certain non-negative trigonometric functions on the circle. We use properties and the construction of rational P ¯ -inner functions to prove a Schwarz lemma for the pentablock. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multidimensional wave equation with boundary fractional damping acting on part of the boundary on a Riemannian manifold. Firstly, using a general criterion of Arendt–Batty, we prove the strong stability of the system under feedback control of fractional order 1 < α < 2 . Next, using the frequency domain approach combined with the multiplier method, we obtain a polynomial decay rate for the above system under some geometric assumptions on the domain. In addition, we give an example to illustrate the effectiveness of the boundary fractional feedback. [ABSTRACT FROM AUTHOR]

Our main goal in the present paper is to expand the known class of noncompact manifolds over which the L 2 -spectrum of a general Dirac operator and its square is maximal. To achieve this, we first find sufficient conditions on the manifold so that the L p -spectrum of the Dirac operator and its square is independent of p for p ≥ 1 . Using the L 1 -spectrum, which is simpler to compute, we generalize the class of manifolds over which the L p -spectrum of the Dirac operator is the real line for all p. We also show that by applying the generalized Weyl criterion, we can find large classes of manifolds with asymptotically nonnegative Ricci curvature, or which are asymptotically flat, such that the L 2 -spectrum of a general Dirac operator and its square is maximal. [ABSTRACT FROM AUTHOR]

In this paper, we consider stabilization of the transmission wave equation with variable coefficients, which has an acceleration term and interior delay. By using Riemannian geometry method and Lyapunov functional, we obtain the exponential decay for the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of nontrivial solutions for the following Schrödinger–Poisson system - Δ u + u + λ ϕ u = | u | p - 2 u , - Δ ϕ = u 2 , lim | x | → + ∞ ϕ (x) = 0 , where λ > 0 and p ∈ (2 , 6) . In the case when 2 < p ≤ 3 , we obtain the existence of nontrivial radial solutions and the nonexistence of nontrivial solutions for the above system under weaker assumptions on the parameter λ than the ones in Ruiz (J Funct Anal 237:655–674, 2006). [ABSTRACT FROM AUTHOR]

Facilitators of professional development (PD) for mathematics teachers currently gain increasing attention, as their practices are crucial for the success of spreading mathematics educational ideas and innovations into schools and strengthening the professional expertise of teachers. So far, mainly two components of facilitation have been empirically researched: facilitators' knowledge resources and facilitation moves. In this paper, we propose to develop a more comprehensive conceptual framework in order to explain facilitators' practices and underlying decisions. For this purpose, we lift Schoenfeld's ROG framework for teachers' decision-making (comprising resources, orientations, and goals) to the facilitator level, and add identity as a fourth component to obtain what we call the ROGI framework. We draw on data collected simultaneously in two PD projects for mathematics teachers in Germany and in Israel. Through three cases of facilitation, we illustrate how the adapted ROGI framework helps to understand processes of decision-making made by facilitators during mathematics professional development sessions. [ABSTRACT FROM AUTHOR]

After about two years of emergency remote teaching during the pandemic, the teaching of mathematics is slowly returning to (what used to be called) normal. However, after the period of mostly teaching online, there is uncertainty about the extent to which we will return to the way we were teaching before. In this survey paper we attempt to give some background to the impact that emergency remote teaching may have had on teaching mathematics. We examine the possible social implications and then focus on the changing mathematics classroom, focusing on the actual mathematics curriculum, learning design and assessment, the role of collaborative activities and social media, educational videos, and the role of family and parents in future. There are indicators from the literature that educators may not return to the traditional way of teaching entirely, especially in secondary and higher education. We conclude with describing some possible new research areas that have developed through emergency remote teaching, including online education for younger learners, local learning ecosystems, the role of family and parents, instructional design, and the mathematics content of curricula. [ABSTRACT FROM AUTHOR]

Teacher noticing of student thinking represents a key aspect of teacher expertise as it informs teachers' actions 'on the fly' during a lesson. In science and mathematics education, teacher noticing has been conceptualised as being driven by attending to and making sense of student thinking. A range of models of noticing developed in professional fields such as aviation posit that perceptual interaction is also a key driver of noticing. This paper presents an alternate, ecological model of teacher noticing that begins with the premise that teachers have limited capacity to make sense of noteworthy events mid-lesson. Multimodal data collected from a primary science and mathematics lesson, taught by the same teacher, is analysed using the ecological lens. The analysis draws research attention to aspects of teacher/classroom interaction such as attention deployment and the role that classroom environments play in facilitating mid-lesson noticing of student thinking. Differences in teacher noticing between the science and mathematics lesson can be accounted for in terms of variance in environmental structure rather than solely in terms of teacher cognition. [ABSTRACT FROM AUTHOR]

Theory of quasi-orthogonal polynomials is significantly related to constructing vector Padé approximations. The present paper introduces an efficient procedure to compute adjacent families of quasi-orthogonal polynomials as defined in Sadaka (Appl. Numer. Math. 21:57–70, 1996) and Sadaka (Appl. Numer. Math. 24:483–499, 1997). The derived algorithm uses short recursive relations whose coefficients are written in terms of lower triangular block determinants. The strategy of computing such determinants is given and analyzed. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the behaviors of commutator μΩb generated by the Marcinkiewicz integral μΩ with b ∈ BMO(ℝn) on the weighted Hardy spaces. The authors show that there is a non-trivial subspace of BMO(ℝn), when b belongs to this subspaces, such that μΩb is bounded from the weighted Hardy spaces Hωp(ℝn) to the weighted Lebesgue spaces Lωp(ℝn) for some 0 < p ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the Dirichlet problem for a class of fully nonlinear elliptic partial differential equations. Under structural conditions, we establish boundary estimates for the lower bound of convex solutions to the problem. There are two major ingredients in our proof: one is the introduction of an (a, η, μ) type condition, which serves as an important local condition of convex domains and better adapts to the study of boundary lower estimates; the other is the construction of super-solutions, which involves using the convex envelope of auxiliary functions over convex subdomains. As an application, we show the optimal boundary lower estimates of convex solutions to hyperbolic affine sphere equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a sliding method to investigate the monotonicity of solutions for nonlocal double phase equations. We first derive a narrow region principle in bounded domains. Then we illustrate how to utilize this new method of sliding to obtain monotonicity of solutions for nonlocal double phase equations in bounded domains and the whole space respectively. [ABSTRACT FROM AUTHOR]

Mathematicians frequently attend their peers' lectures to learn new mathematical content. The goal of this paper is to investigate what mathematicians learned from the lectures. Our research took place at a 2-week workshop on inner model theory, a topic of set theory, which was largely comprised of a series of lectures. We asked the six workshop organizers and seven conference attendees what could be learned from the lectures in the workshop, and from mathematics lectures in general. A key finding was that participants felt the motivation and road maps that were provided by the lecturers could facilitate the attendees' future individual studying of the material. We conclude by discussing how our findings inform the development of theory on how individuals can learn from lectures and suggest interesting directions for future research. [ABSTRACT FROM AUTHOR]

Mathematical Internet memes are examples of how the creative thrust characterising the Web 2.0 environment reaches the field of mathematics, translating mathematical statements into a new digital form endowed with an epistemic potential that is capable of initiating a process of mathematical argumentation. The research presented in this paper aims to shed light on the creative process of mathematical memes, contributing to building a body of knowledge on mathematical memes that, prospectively, could enable educators to profit from these objects in their teaching. Theoretically, this is based on a widened concept of creativity that focuses on the connection linking digital culture with mathematics, and on distinguishing and merging three perspectives to disclose the meanings of mathematical memes. Methodologically, the process of mathematical memes' creation is investigated through a reverse engineering approach on a dataset of about 2100 items collected in a 3-year-long ethnographic observation within online communities. The result is a heuristic action model of the creation process, that is validated by creating two new mathematical Internet memes that are shared online within the observed communities to explore if they retain the mathematical and epistemic characteristics of Web-found ones. [ABSTRACT FROM AUTHOR]

The purpose of this study was to identify and categorize in-the-moment teaching moves which support productive general social norms and sociomathematical norms during dialogic instruction by drawing on transcripts of classroom episodes. To understand how teachers construct in-the-moment teaching moves, this paper is grounded in a revised version of the Advancing Children's Thinking framework. In particular, the framework reveals three interconnected dimensions of teaching moves: (1) eliciting group strategies, (2) supporting conceptual understanding and group coordination, and (3) extending group mathematical thinking and coordination. Drawing on this framework, we use a novel self-study methodology to construct a sequence of in-the-moment teaching moves that support the development of productive social and sociomathematical norms in dialogic classrooms. These teaching moves include (1) inviting strategies and encouraging all members to actively listen; (2) exploring, clarifying, and questioning the mathematical details of all strategies as a group; (3) promoting understanding of differing strategies; (4) comparing and evaluating differing strategies as a group; and (5) connecting the group's thinking for making progress on the task/advancing current strategy to become viable. Implications for research and practice are offered. [ABSTRACT FROM AUTHOR]