Search

For rational functions , where is a polynomial of degree at the most and , with we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions . In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000) 1797–1807]. [ABSTRACT FROM AUTHOR]

We consider the following problem stated in 1993 by Buttazzo and Kawohl (Math Intell 15:7–12, 1993): minimize the functional ∫ ∫ Ω (1 + | ∇ u (x , y) | 2 ) - 1 d x d y in the class of concave functions u : Ω → [0,M], where Ω ⊂ R 2 is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets. [ABSTRACT FROM AUTHOR]

We consider the energy-critical (corotational) 1-equivariant wave maps into the two-sphere. By the seminal work (Raphaël and Rodnianski in Publ Math Inst Hautes Études Sci 115:1–122, 2012) of Raphaël and Rodnianski, there is an open set of initial data whose forward-in-time development blows up in finite time with the blow-up rate λ (t) = (T - t) e - | log (T - t) | + O (1) . In this paper, we show that this e O (1) -factor in fact converges to the universal constant 2 e - 1 , and hence these solutions contract at the universal rate λ (t) = 2 e - 1 (T - t) e - | log (T - t) | (1 + o t → T (1)) . Our proof is inspired by recent works on type-II blow-up dynamics for parabolic equations. The key improvement is in the construction of an explicit invariant subspace decomposition for the linearized operator perturbed by the scaling generator in the dispersive case, from which we obtain a more precise ODE system determining λ (t) . [ABSTRACT FROM AUTHOR]

Some of the key challenges for STEM education in Turkey include the problems encountered by teachers when integrating diverse disciplines and technologies into STEM activities, the need for teachers to acquire the skills necessary to prepare and implement appropriate STEM activities for the educational level of their students and their school context, the difficulties faced by teachers when following the Engineering Design (ED) problem-solving process in the implementation of STEM activities, and teachers' heavy workloads, which impede the collaboration required to develop and implement STEM activities. A field study was carried out in Samsun, Turkey between 2017 and 2018 to develop a Design Thinking (DT) approach customized to support STEM teachers. The field study adopted an action research methodology and involved two main studies with fifth-grade teachers and students, facilitated by a researcher-designer. Qualitative methods including co-design workshops, focus groups, interviews, and observation were employed to gather data. The paper presents a seven-step DT approach, customized to support pre-service and in-service secondary school teachers in collaboratively designing and implementing classroom STEM activities. The results indicate that structuring the STEM activity design as a specialized DT process provides a complementary framework. Collaboration among teachers in a structured STEM activity design process supports the integration of disciplines and the development of STEM activities that correspond to the students' level of knowledge and ability. Empathy building, which facilitates familiarization with other teachers and students, enhances collaboration among teachers and aids the development of STEM activities relevant to the students' level. [ABSTRACT FROM AUTHOR]

In this paper we develop a Nash-Moser iteration type reducibility approach to prove the (inverse) localization for some d-dimensional discrete almost-periodic operators with power-law long-range hopping. We also provide a quantitative lower bound on the regularity of the hopping. As an application, some results of Sarnak (Comm Math Phys 84(3):377–401, 1982), Pöschel (Comm Math Phys 88(4):447–463, 1983), Craig (Comm Math Phys 88(1):113–131, 1983) and Bellissard et al. (Comm Math Phys 88(4):465–477, 1983) are generalized to the power-law hopping case. [ABSTRACT FROM AUTHOR]

The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu (in Commun Math Phys 350(2):699–747, 2017) via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland to allow for the singularity at the bifurcation point. For both the corotating and the counter-rotating pairs, along the global curve of solutions either the angular fluid velocity vanishes or the two patches self-intersect. [ABSTRACT FROM AUTHOR]

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Γ -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Γ -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004). [ABSTRACT FROM AUTHOR]

This paper finds new quasi-polynomials over Z for the number p k (n) of partitions of n with parts at most k. Methods throughout are elementary. We derive a small number of polynomials (e.g., one for k = 3 , two for k = 4 or 5, six for k = 6 ) that, after addition of appropriate constant terms, take the value p k (n) . For example, for 0 ≤ r < 6 and for all q ≥ 0 , p 3 (6 q + r) = p 3 (r) + π 0 (q , r) , a polynomial of total degree 2 in q and r. In general there are M ⌊ k / 2 ⌋ = lcm { 1 , 2 , ... , ⌊ k / 2 ⌋ } such polynomials. In two variables q and s, they take the form ∑ a i , j q i s j with a i , j ∈ Z , which we call the proper form for an integer-valued polynomial. They constitute a quasi-polynomial of period M ⌊ k / 2 ⌋ for the sequence (p k (n) - p k (r)) with n ≡ r (mod M k) . For each k the terms of highest total degree are the same in all the polynomials and have coefficients dependent only on k. A second theorem, combining partial fractions and the above approach, finds hybrid polynomials over Q for p k (n) that are easier to determine than those above. We compare our results to those of Cayley, MacMahon, and Arkin, whose classical results, as recast here, stand up well. We also discuss recent results of Munagi and conclude that circulators in some form are inevitable. At k = 6 we find serious errors in Sylvester's calculation of his "waves." Sylvester JJ (Q J Pure Appl Math 1:141–152, 1855). The results are generalized to the (not very different) problem called "making change," where significant improvements to existing approaches are found. We find an infinitude of new congruences for p k (n) for k = 3 , 4 , and one new one for k = 5 . Reduced modulo m the periodic sequence (p k (n)) is investigated for periodicity and zeros: we find, from scratch, a simple proof of a known result in a special case. [ABSTRACT FROM AUTHOR]

In the paper Targino (Outer Lipschitz Geometry of Complex Algebraic Plane Curves. International Mathematics Research Notices, rnac202. https://doi.org/10.1093/imrn/rnac202, 2022) Renato Targino shows that bi-Lipschitz type of a plane curve is determined by the local ambient topological properties of curves. Here we show that it is no longer true in higher dimensions. However we show that the bi-Lipschitz type of space curves is determined by the number of singular points and by the local ambient topological type of a generic projection of such curves into the affine plane. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some sharp upper and lower bounds for the spectral radii of nonnegative k-uniform tensors (resp., nonnegative tensors) by using the i-th q-times-average slice sum which was introduced by C. Lü, L. H. You and Y. F. Huang [AIMS Math., 2020, 5(3): 1799–1819]. When k = 2, q =1, our results can deduce the main results of D. P. Huang and L. H. You [J. Appl. Math., 2016, 3: 1–7]. We also find that the upper bounds of nonnegative k-uniform tensors we obtained can not be compared with the upper bounds presented by [AIMS Math., 2020, 5(3): 1799–1819] by numerical examples. As applications, we obtain some sharp upper bounds for the α-spectral radii of k-uniform hypergraphs (resp., k-uniform directed hypergraphs). [ABSTRACT FROM AUTHOR]

The notion of approximate fixed point sequence, emphasized in Chidume (Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer-Verlag London, Ltd., London, 2009), is a very useful tool in proving convergence theorems for fixed point iterative schemes in the class of nonexpansive-type mappings. In the present paper, our aim is to present simple and unified alternative proofs of some classical fixed point theorems emerging from Banach contraction principle, by using a technique based on the concepts of approximate fixed point sequence and graphic contraction. [ABSTRACT FROM AUTHOR]

We consider the sine-Gordon (SG) equation in 1 + 1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H 1 × L 2 . It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in Hoffman and Wayne (Differ Int Equ 26(3–4):303–320, 2013), Muñoz and Palacios (Ann. IHP C Analyse Nonlinéaire 36(4):977–1034, 2019) and Virial estimates around odd perturbations of the vacuum solution, in the spirit of Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and ϕ 4 at the level of linear Bäcklund transformations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constructive proof of the termination of the normal form (NF) algorithm for multivariate polynomials, as well as the related concept of admissible ordering < on monomials. In classical mathematics, the well-quasiorder property of relation < is derived from Dickson's lemma, and this is sufficient to justify the termination of the NF algorithm. In provable programming based on constructive type theory (Coq and Agda), a somewhat stronger condition (in constructive mathematics) of the well-foundedness of the ordering (in its constructive version) is required. We propose a constructive proof of this theorem (T) for < , which is based on a known method that we refer to here as the "pattern method." This theorem on the well-foundedness of an arbitrary admissible ordering is also important in itself, independently of the NF algorithm. We are not aware of any other works on constructive proof of this theorem. However, it turns out that it follows, not very difficultly, from the results achieved by other researchers in 2003. We program this proof in the Agda language in the form of our library AdmissiblePPO-wellFounded of provable computational algebra programs. This development also uses the theorem to prove termination of the NF algorithm for polynomials. Thus, the library also contains a set of provable programs for polynomial algebra, which is significantly larger than that needed to prove Theorem T. [ABSTRACT FROM AUTHOR]

Creating sustained, transformative change within and across organizations is challenging, particularly when those undertaking change act as individuals. COMmunities of Practice (CoPs) are organically created collaborations among like-minded participants, working toward a common set of goals (Lave & Wenger, 1991; Wenger-Trayner & Wenger-Trayner, 2014). CoPs offer an avenue for members to connect individuals across various boundaries. In this paper, we investigate the ways in which regional CoP leaders experience value participating in their community, using the Communities for Mathematics Inquiry in Teaching (COMMIT) Network as our unit of study. The COMMIT Network is a grant funded project aimed at engaging mathematics faculty at institutions of higher education in regional CoPs around teaching with inquiry. In this study we examine the experiences of CoP leaders nested within this network setting. We interviewed 19 leaders from eight United States regions to understand their perceptions of individual and collective value participating in the regional CoP and COMMIT Network structures. We framed our study on Wenger et al. (2011). Promoting and assessing value creation in communities and networks: A conceptual framework. Open University of the Netherlands.) Value Framework. Our findings show that leaders found Immediate Value as individuals participating in a collaborative, supportive CoP environment and they found Realized Value in terms of the impact their CoP could make on instructional practices, both in their region and the network. An unexpected finding examines how future opportunities for value creation may influence long-term sustainability and transformation of college mathematics instruction. We provide implications for the ways that regional CoPs, along with CoP networks, can provide value for members through such communities. [ABSTRACT FROM AUTHOR]

Despite the documented negative outcomes accompanying sexual harassment, the experience of sexual harassment among STEM faculty members remains underexamined. In this paper, we explore how two sexual harassment variables—gender harassment and sexual harassment climate—are linked to four facets of faculty well-being: job burnout, turnover intentions, psychological distress, and self-rated physical health. Using data from STEM faculty at a mid-sized university located in the upper Midwest (N = 117 faculty members), we find gender harassment is associated with lower self-rated physical health and higher turnover intentions among women STEM faculty. In contrast, gender harassment was not significant in predicting well-being outcomes among men STEM faculty. Instead, the sexual harassment climate features more prominently in the experiences of STEM men faculty, with the perception that sexual harassment charges will be treated seriously (the sexual harassment climate) being negatively related to men's job burnout and psychological distress and positively related to men's self-rated physical health. Taken together, our findings extend the existing literature by documenting that outcomes for STEM men faculty are more strongly shaped by the perceived institutional climate surrounding sexual harassment, whereas women's outcomes are more intricately linked to experiencing gender harassment. [ABSTRACT FROM AUTHOR]

We investigate the functional equation f (p (x)) = q (f (x)) where p and q are given real functions. In the paper "On solvability of f (p (x)) = q (f (x)) for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494", we solved the problem of the solvability of f (p (x)) = q (f (x)) under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output 'solvable/not solvable'. [ABSTRACT FROM AUTHOR]

For the Helmholtz equation Δu + k2u = 0 in 2D domain S, there exists a unique solution if k2 is not exactly equal to an eigenvalue λ of the Laplace eigenvalue problem Δu + λu = 0 in S. One important criterion for numerical methods is that there must exist no spurious (i.e., superfluous) eigenvalues so that the above unique solution can be obtained correctly. For exterior problems, the method of fundamental solutions (MFS) using Hankel functions suffers from spurious eigenvalues. New modified Hankel functions have been proposed in Zhang et al. (Appl. Numer. Math. 145, 236–260, 2019) to eliminate all spurious eigenvalues. In this paper, we study bounded multiply connected domains. The MFS algorithms without spurious eigenvalues and their strict analysis are our goals. First, we study bounded simply connected domains by the MFS. The algorithms using Hankel functions are free from spurious eigenvalues. A brief error analysis is provided. Next, we focus on bounded multiply connected domains and choose an annular domain for analysis. The Hankel functions and the modified Hankel functions in [47] are chosen as the exterior fundamental solutions (FS) and the interior FS, respectively. Such combined FS eliminate all spurious eigenvalues, and new error and stability analyses are explored. Bounds of errors involve degeneracy, and those of the condition number involve a gap to eigenvalues. Numerical experiments are carried out to support the analysis made, and better pseudo-boundaries of source nodes are also investigated numerically. The new error and stability analyses in this paper are new and essential, thus providing a solid theoretical basis of the MFS for the Helmholtz equation in 2D bounded simply connected and multiply connected domains. [ABSTRACT FROM AUTHOR]

The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m (z) (m ∈ N , κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by J κ 1 , κ 2 κ 3 , m (z) : = Γ m (κ 1 + 1) Γ (κ 1 + κ 2 + 1) 2 2 κ 1 + κ 2 z 1 − (κ 2 / 2) − κ 1 J κ 1 , κ 2 κ 3 , m (z) , where J κ 1 , κ 2 κ 3 , m (z) : = (1 − 2 κ 1 − κ 2) J κ 1 , κ 2 κ 3 , m (z) + z (J κ 1 , κ 2 κ 3 , m (z)) ′ and J κ 1 , κ 2 κ 3 , m (z) = (z 2) 2 κ 1 + κ 2 ∑ n = 0 ∞ (− 1) n Γ m (n + κ 1 + 1) Γ (n κ 3 + κ 1 + κ 2 + 1) (z 2) 2 n , was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for J κ 1 , κ 2 κ 3 , m (z) to be starlike and convex of order α, 0 ≤ α < 1 , inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function J κ 1 , κ 2 κ 3 , m (z) defined by I κ 1 , κ 2 κ 3 , m (z) = J κ 1 , κ 2 κ 3 , m (z) ∗ z / (1 + z) is in the class of starlike and convex functions, respectively. Further work regarding the function J κ 1 , κ 2 κ 3 , m (z) is underway and will be presented in a forthcoming paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) A = (A α β x α β) , where A α β is a 2 × 2 matrix over a field F and x α β is an indeterminate for α = 1 , 2 , ⋯ , μ and β = 1 , 2 , ⋯ , ν . This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (SIAM J Matrix Anal Appl 16(3):719–734, 1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos et al. (Comput Complex 27:561–593, 2018) and Garg et al. (Found. Comput. Math. 20:223–290, 2020), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial O ((μ ν) 2 min { μ , ν }) -time algorithm for computing the symbolic rank of a (2 × 2) -type generic partitioned matrix of size 2 μ × 2 ν . Our algorithm is inspired by the Wong sequence algorithm by Ivanyos et al. for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of A for an arbitrary field F . [ABSTRACT FROM AUTHOR]

This paper is concerned with a quasilinear degenerate Keller–Segel system of parabolic–parabolic type. It was proved in Ishida and Yokota (J. Differ. Equ. 252:2469–2491, 2012) that if q > m + 2 n , then the system has a global weak solution under smallness conditions for initial data, where m describes the intensity of diffusion, q shows the nonlinearity, and n denotes the dimension. The smallness conditions were relaxed in Wang et al. (Z. Angew. Math. Phys. 70:18 pp., 2019) when q = 2 . The purpose of this paper is to obtain global existence and boundedness under more general conditions for initial data in the case m + 2 n < q < m + 4 n + 2 and to relax the conditions assumed in Ishida and Yokota (J. Differ. Equ. 252:2469–2491, 2012). [ABSTRACT FROM AUTHOR]

In a paper titled, “The Unreasonable Effectiveness of Mathematics”, published 20 years after Wigner’s seminal paper, the mathematician Richard W. Hamming discussed what he took to be Wigner’s problem of Unreasonable Effectiveness and offered some partial explanations for this phenomenon. Whether Hamming succeeds in his explanations as answers to Wigner’s puzzle is addressed by other scholars in recent years (Azeri 2020) I, on the other hand, raise a more fundamental question: does Hamming succeed in raising the same question as Wigner? The answer is no. My goal is to show that Hamming’s reading misses Wigner’s highly original formulation of the problem.Through a close and contextual reading of Wigner’s work, as I will show, we are led in new directions in addressing and solving the applicability problem. [ABSTRACT FROM AUTHOR]

Nexus Network Journal guest editor Roberta Spallone introduces papers selected from the Nexus Conference 2023 in Turin for detailed review, expansion and development. These papers identify close links between architecture, mathematics, and specific areas of interest related to historical periods, architectural cultures, typological elements, and analysis tools. [ABSTRACT FROM AUTHOR]

The Weyl law of the Laplacian on the flat torus T n is concerning the number of eigenvalues ≤ λ 2 , which is equivalent to counting the lattice points inside the ball of radius λ in R n . The leading term in the Weyl law is c n λ n , while the sharp error term O (λ n - 2) is only known in dimension n ≥ 5 . Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. This result establishes the sharpness of the general theorems for the Schrödinger operators H V = - Δ g + V in the previous work (Huang and Zhang (Adv Math, arXiv:2103.05531)) of the authors, and extends the 3-dimensional results of Frank and Sabin (Sharp Weyl laws with singular potentials. arXiv:2007.04284) to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li–Yau's heat kernel estimates, Blair–Sire–Sogge's eigenfunction estimates, and Duhamel's principle for the wave equation. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

In 2010, Gábor Czédli and E. Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices. Advances in Mathematics, 225, 2455–2463 (2010)]. That is to say: What are the geometric lattices G such that a given finite semimodular lattice L has a cover-preserving embedding into G with the smallest ∣G∣? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices G of a given semimodular lattice L and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice G equal the length of L and the number of non-zero join-irreducible elements of L, respectively. Therefore, we solve the problem on the best cover-preserving embedding of a given semimodular lattice raised by Gábor Czédli and E. Tamás Schmidt. [ABSTRACT FROM AUTHOR]

To reconcile the standard possible worlds model of knowledge with the intuition that ordinary agents fall far short of logical omniscience, a Stalnakerian strategy appeals to two components. The first is the idea that mathematical and logical knowledge is at bottom metalinguistic knowledge. The second is the idea that non-ideal minds are often fragmented. In this paper, we investigate this Stalnakerian reconciliation strategy and argue, ultimately, that it fails. We are not the first to complain about the Stalnakerian strategy. But in contrast to existing complaints, we want to cause trouble for the strategy directly on its home turf. That is, we will advance our objection while granting both the plausibility of the fragmentation component—save for an extreme version of it—and that of the metalinguistic component. Once our central objection to the Stalnakerian strategy is in place, we will show how it negatively affects Adam Elga and Augustín Rayo's recent attempt to apply the Stalnakerian strategy in the context of Bayesian decision theory. [ABSTRACT FROM AUTHOR]

Mathematical education, both mass education, and university education of non-mathematicians, are in an abominable state, and rapidly degrading. We argue that the instruction of non-mathematicians should be dramatically reformed both as substance and style. With traditional approach, such a transformation would take decades, with unclear results. But we do not have this time. The advent of Computer Algebra Systems gives the mathematics community a chance to reverse the trend. We should make a serious attempt to seize this opportunity. In the present paper we describe one such project of reform implemented at the St Petersburg State University. [ABSTRACT FROM AUTHOR]

The paper discusses an example of an educational project in modern mathematics in which school students create mathematics that is new to them. The mathematical results produced by the students in the theory of definability also have an "absolute" novelty, i.e., are the basis for professional publications. The described course was based on recent results of this article's authors in the theory of definability. The new results were obtained by a group of 10 schoolchildren from different regions of Russia. [ABSTRACT FROM AUTHOR]

The paper describes in detail a class of educational problems from an elementary school course of mathematics and computer science. This course has been implemented over the past decades by a team led by Academician of the RAS A.L. Semenov. In the problems, it is necessary to find, build, or list all objects that satisfy a certain system of conditions. The student conducts these activities in a visual world of basic objects of discrete mathematics and computer science: strings (finite sequences of symbols), bags (multisets), tables, trees, and statements containing quantifiers. The connections of these problems with problems in computational combinatorics (counting the number of options), with search problems in the theory of computational complexity, with "big ideas," and general cognitive strategies for their formation in education are considered. The content of education in our approach is more adequate to the context of the modern world. [ABSTRACT FROM AUTHOR]

Over the last three decades, educational researchers and policymakers have increasingly promoted instructional strategies that centralize group work in mathematics. One difficulty teachers face in implementing group-based instruction in mathematics involves facilitating meaningful group interaction amongst students. In this paper, we explore student noticing as a novel strategy for supporting college students to collaborate effectively during group work in mathematics. First, we construct a noticing framework named student noticing of collaborative practices which provides a lens for "seeing" how students notice their collaborative practices. Then, we use the framework to explore how 25 college students noticed their collaborative practices in mathematics. After working on a novel mathematics task in groups, the college students listened to audiorecordings of their group interactions and responded to reflection questions about the effectiveness of their collaboration. We identified themes regarding how and what students noticed related to their collaborative practices. The findings reveal that students attended to many aspects of their collaboration, including their talking turns and propensity to listen to others. Students demonstrated a desire to change their collaborative practices in the future. The findings imply that teachers and researchers might leverage student noticing as a tool for improvement in mathematics group-based classrooms. [ABSTRACT FROM AUTHOR]

Under mild assumptions, we prove that any random multifunction can be represented as the set of minimizers of an infinitely many differentiable normal integrand, which preserves the convexity of the random multifunction. This result is an extended random version of work done by Azagra and Ferrera (Proc Am Math Soc 130(12):3687–3692, 2002). We provide several applications of this result to the approximation of random multifunctions and integrands. The paper ends with a characterization of the set of integrable selections of a measurable multifunction as the set of minimizers of an infinitely many differentiable integral function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for d ≥ 1 , there exist C 0 , C 1 > 0 , depending on ‖ m ‖ L 1 only, such that C 0 d ≤ supremum of total population ≤ C 1 d. However, when replaced by random diffusion, a conjecture, proposed by Wei-Ming Ni and justified in Bai et al. (Proc. Am. Math. Soc. 144:2161–2170, 2016), indicates that in the one-dimensional case, supremum of total population = 3 ‖ m ‖ L 1. This reflects serious discrepancies between models with local and nonlocal dispersal strategies. Furthermore, we provide an equivalent characterization about the combination of resource distribution and diffusion rate such that the corresponding total population could reach the optimal order d as d goes to infinity. [ABSTRACT FROM AUTHOR]

We prove a theorem that generalizes Lemma 4 in the paper of A. S. Mamontov [Sib. Math. J., 54, No. 1, 114-118 (2013)] concerning the validity of the Baer–Suzuki theorem in groups of period 12. The results of the present work can be used in studying groups with a given set of element orders, also called a spectrum. [ABSTRACT FROM AUTHOR]

In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve X were investigated and the sets of minimal generators were determined for all points in X (F q 2) and X (F q 6) \ X (F q 2) . This paper completes their work by settling the remaining cases, that is, for points in X (F ¯ q) \ X (F q 6) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in X (F q 7) \ X (F q) and we give a bound on the Feng–Rao minimum distance d ORD . For q = 3 we provide a table that also reports the exact values of d ORD . As a further application we construct quantum codes from F q 7 -rational points of the GK-curve. [ABSTRACT FROM AUTHOR]

Deflationists hold that it does not seem to us, in experience, as though time robustly passes. There is some recent empirical evidence that appears to support this contention. Equally, empirical evidence suggests that we naïvely represent time as dynamical. Thus deflationists are faced with an explanatory burden. If, as they maintain, the world seems to us in experience as though it is non-dynamical, then why do we represent time as dynamical? This paper takes up the challenge of investigating, on the part of the deflationist, one candidate explanation. We hypothesise that people's belief that the future is what we call deliberatively open partly explains why they represent time as dynamical. In a series of two experiments we test this hypothesis. We find no evidence that beliefs about deliberative openness explain why people represent time as dynamical. Hence there remains an explanatory burden for the deflationist to discharge. [ABSTRACT FROM AUTHOR]

Tarski's hierarchical solution to the Liar paradox is widely viewed as ad hoc. In this paper I show that, on the contrary, Tarski's solution is justified by a sound philosophical principle that concerns the inner structure of truth. This principle provides a common philosophical basis to a number of solutions to the Liar paradox, including Tarski's and Kripke's. Tarski himself may not have been aware of this principle, but by providing a philosophical basis to his hierarchical solution to the paradox, it undermines the ad-hocness objection to this solution. Indeed, it contributes to the defense of Tarski's theory against other objections as well. [ABSTRACT FROM AUTHOR]

The paper is devoted to the introduction of a new class of generalized set-valued nonlinear variational-like inequality problems in the setting of Banach spaces. By means of the notion of P- η -proximal mapping, we prove its equivalence with a class of generalized implicit Wiener–Hopf equations and employ the obtained equivalence relationship and Nadler's technique to suggest a new iterative algorithm for finding an approximate solution of the considered problem. The existence of solution and the strong convergence of the sequences generated by our proposed iterative algorithm to the solution of our considered problem are verified. The problem of finding a common element of the set of solutions of a generalized nonlinear variational-like inequality problem and the set of fixed points of a total asymptotically nonexpansive mapping is also investigated. The final section deals with the investigation and analysis of the main results appeared in Kazmi and Bhat (Appl Math Comput 166:164–180, 2005) and some comments relating to them are given. The results presented in this article extend and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

Let (M, g) be a compact Riemann surface with unit area, h ∈ C ∞ (M) a function which is positive somewhere, ρ > 0 , p i ∈ M and α i ∈ (- 1 , + ∞) for i = 1 , ⋯ , ℓ , we consider the mean field equation Δ v + 4 π ∑ i = 1 ℓ α i 1 - δ p i = ρ 1 - h e v ∫ M h e v d μ , on M, where Δ and d μ are the Laplace–Beltrami operator and the area element of (M, g) respectively. Using variational method and blowup analysis, we prove some existence results in the critical case ρ = ρ ¯ : = 8 π (1 + min { 0 , min 1 ≤ i ≤ ℓ α i }) . These results can be seen as partial generalizations of works of Chen and Li (J Geom Anal 1:359–372, 1991), Ding et al. (Asian J Math 1:230–248, 1997), Mancini (J Geom Anal 26:1202–1230, 2016), Yang and Zhu (Proc Amer Math Soc 145:3953–3959, 2017), Sun and Zhu (arXiv:2012.12840) and Zhu (arXiv:2212.09943). Among other things, we prove that the blowup (if happens) must be at the point where the conical angle is the smallest one and h is positive, this is the most important contribution of our paper. [ABSTRACT FROM AUTHOR]

Ampliation quasisimilarity was applied as a tool in Foias and Pearcy (J Funct Anal 219:134–142, 2005) to reduce the hyperinvariant subspace problem to a particular class of operators. The seemingly weaker pluquasisimilarity relation was introduced in Bercovici et al. (Acta Sci Math Szeged 85:681–691, 2019) and studied also in Kérchy (Acta Sci Math Szeged 86:503–520, 2020). The problem whether these two relations are actually equivalent is addressed in the present paper. The following more general, related question is studied in details: under what conditions is the operator A a quasiaffine transform of B, whenever A can be injected into B and A can be also densely mapped into B. [ABSTRACT FROM AUTHOR]

Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let P 2 , 2 → be the orientation of C 4 which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of P 2 , 2 → , and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let P k → and C k → denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of C k → -free digraphs of order n for all n , k ∈ N ∗ , as well as the extremal digraphs attaining this maximum size. Similar result is obtained for P k → where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of C 4 except P 2 , 2 → . In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global existence of the axisymmetric strong solutions to the vacuum free boundary problem of isentropic compressible Navier–Stokes equations with small initial data and self-gravitation. Moreover, motivated by the results of Lane-Emden solutions for spherically symmetric flows in Luo (Adv Math 291:90–182, 2016), the nonlinear asymptotic stability of the axisymmetric solution is proved with detailed convergence rates. We show that the velocity components u ϕ and u z have better decay rates (1 + t) - 1 than u r due to the special structures of the equations. The solutions capture the precise physical behavior that the sound speed is C 1 / 2 -Hölder continuous across the vacuum boundary provided that the adiabatic exponent γ ∈ (5 / 4 , 2) . The main difficulties of this problem lie in the singularity at the symmetry axis, the degeneracy of the system near the free boundary and the strong coupling between the three components of velocity, especially for the lower-order estimate of angular velocity v ϕ in Lagrangian coordinates. We overcome these obstacles by making a new bounded assumption on | x - 1 / 2 v x ϕ | + | x - 3 / 2 v ϕ | , introducing a new functional E 2 (t) : = μ 2 ‖ x 1 2 v x ϕ - v ϕ / x , v x z (· , t) ‖ L 2 (see (2.8)), which helps to obtain some estimates of time derivatives of v ϕ and v z (see (4.9)), and using a truncation function method. [ABSTRACT FROM AUTHOR]

In this paper, we present a new ellipsoid-type algorithm for solving nonsmooth problems with convex structure. Examples of such problems include nonsmooth convex minimization problems, convex-concave saddle-point problems and variational inequalities with monotone operator. Our algorithm can be seen as a combination of the standard Subgradient and Ellipsoid methods. However, in contrast to the latter one, the proposed method has a reasonable convergence rate even when the dimensionality of the problem is sufficiently large. For generating accuracy certificates in our algorithm, we propose an efficient technique, which ameliorates the previously known recipes (Nemirovski in Math Oper Res 35(1):52–78, 2010). [ABSTRACT FROM AUTHOR]

The lace expansion for the Ising two-point function was successfully derived in (Sakai in Commun Math Phys 272:283–344, 2007, Proposition 1.1). It is an identity that involves an alternating series of lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster x-space decay (as the two-point function cubed) above the critical dimension d c ( = 4 for finite-variance models) if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of (Sakai in Commun Math Phys 272:283–344, 2007, Lemma 4.2), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic (Sakai 2007, Lemma 4.2), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in (Sakai 2007, Proposition 4.1) but nonetheless obey the same fast decay above the critical dimension d c . Consequently, the lace-expansion results for the Ising and φ 4 models in the literature are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients. [ABSTRACT FROM AUTHOR]

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo (Commun Contemp Math 6: 765–780, 2004) is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial α -curvature for vertex scaling of piecewise linear metrics on surfaces is a discretization of Gaussian curvature on surfaces. In this paper, we investigate the prescribing combinatorial α -curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory (J. Differ. Geom. 109: 223–256, 2018) for piecewise linear metrics on surfaces and variational principles with constraints, we prove some Kazdan-Warner type theorems for prescribing combinatorial α -curvature problem, which generalize the results obtained in Gu-Luo-Sun-Wu (J. Differ. Geom. 109: 223–256, 2018), Xu (Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I. arXiv:1806.04516v2) on prescribing combinatorial curvatures on surfaces. Gu-Luo-Sun-Wu (J. Differ. Geom. 109: 223–256, 2018) conjectured that one can prove Kazdan-Warner's theorems in Kazdan (Ann Math 99: 14–47, 1974), Kazdan (Ann Math 101: 317-331, 1975) via approximating smooth surfaces by polyhedral surfaces. This paper takes the first step in this direction. [ABSTRACT FROM AUTHOR]

The ergodic properties of the randomly forced Navier–Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier–Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers Kuksin et al. (Geom Funct Anal 30(1):126–187, 2020) and Shirikyan (J Eur Math Soc, 2020) and using the asymptotic compactness of the dynamics. [ABSTRACT FROM AUTHOR]

In this paper we investigate different questions related to bent–negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form x · π (y) ⊕ h (y) [Maiorana–McFarland (M )] is bent–negabent, and more generally, we study the non-existence of bent–negabent functions in the M class. We also identify some functions that are bent–negabent. Next, we continue the recent work by Mandal et al. (Discrete Appl Math 236:1–6, 2018) on rotation symmetric bent–negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent–negabent function in 4 p k variables, where p is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results. [ABSTRACT FROM AUTHOR]

In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré's works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré's works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré's stated motivations for topology and the Poincaré conjecture. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up phenomena connected to the following porous-medium problem with gradient terms under Robin boundary conditions: { u t = Δ u m + k 1 u p − k 2 | ∇ u | q in Ω × (0 , t ∗) , ∂ u ∂ ν + γ u = 0 on ∂ Ω × (0 , t ∗) , u (x , 0) = u 0 (x) ≥ 0 in Ω ‾ , where Ω ⊂ R n (n ≥ 3 ) is a bounded and convex domain with smooth boundary ∂Ω. The constants p, q, m are positive, and p > q > m > 1 , q > 2 . By making use of the Sobolev inequality and the differential inequality technique, we obtain a lower bound for the blow-up time of the solution. In addition, an example is given as an application of the abstract results obtained in this paper. Our results can be regarded as an answer to the open question raised by Li et al. in (Z. Angew. Math. Phys. 70:1–18, 2019). [ABSTRACT FROM AUTHOR]

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form f (x) - f (y) . [ABSTRACT FROM AUTHOR]

The complete forcing number of a graph G is the cardinality of a minimum subset of E(G) to which the restriction of every perfect matching M is a forcing set of M. In a previous paper (He et al., J Math Chem 59:1767–1784, 2021), we presented a complete forcing set of a hexagonal system in terms of elementary edge-cut cover, and a lower bound for the complete forcing number of a normal hexagonal system by matching numbers of some subgraphs of its inner dual graph. In this paper, we show that the complete forcing number of a normal hexagonal system without 2 × 3 subsystems attains the above lower bound. As an example, we present an expression of the complete forcing numbers of pyrene systems. Besides, for a parallelogram hexagonal system, we obtain that its complete forcing number is larger than the lower bound by at most 1. [ABSTRACT FROM AUTHOR]