Search

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series. [ABSTRACT FROM AUTHOR]

This study explored the unique connections between music and mathematics as perceived by four groups of experts: professional mathematicians and musicians, as well as teacher educators in these two fields. Using 2 × 2 study design, we studied four groups of participants, comprising theorists and educators from various Israeli universities. During semi-structured interviews, the study participants were asked about their views on the connections between mathematics and music. This study proposes a model of experts' conceptions of the connection between mathematics and music, which is of descriptive and explanatory power. that reveals differences between the four groups of experts. Theoreticians in both disciplines highlighted Mathematics as a key tool for music analysis and creation. Musical educators emphasized the role of music as a tool for learning mathematics. All the study participants, independently of the field of their expertise, value structure, beauty, sense of wonder, freedom and creative thinking as characteristics of both fields. Additionally, all the experts hold conceptions of the importance of integrating music and mathematics into various discipline. This study opened new doors for future research wherein utilization of experts' insights to craft integrated study modules of music and mathematics can be explored, a pursuit that carries substantial significance. [ABSTRACT FROM AUTHOR]

The home math environment is vital for early math development. Yet, there is limited understanding of how parents translated their attitudes towards helping children learn math (HCLM) into actions to influence their young children's math-related outcomes. Thus, the present study examined whether parents' perceived competence, value, and pressure about HCLM contributed to young children's math competence through parents' interest in HCLM, home math activities, and children's interest in math. One hundred forty-seven parents were surveyed about their attitudes, practices, and their children's math interest. Meanwhile, their children were individually tested on applied math problem solving. Results showed that parents' perceived competence and value about HCLM predicted their interest in HCLM, while parents' pressure about HCLM predicted children's math interest. Parents' interest in HCLM is positively linked to children's applied math problem solving via children's math interest. This study shed light on the importance of promoting parents' positive attitudes towards HCLM. [ABSTRACT FROM AUTHOR]

Understanding math and the ability to use math in various situations can greatly contribute to a successful life. Different math skills are of importance for understanding and supporting math ability. In addition, math interest may influence math ability. We investigated the relationship between math conceptual and procedural skills, their development over time, and the relation to math interest in Grades three and five. Participants in the study were Estonian students (N = 855), who were in Grade three (2019/2020) during the first testing and in Grade five (2021/2022) during the second testing. Students completed math competence testing which consisted of procedural and conceptual skill tasks and a questionnaire about math interest. Both person- and variable-oriented approaches were used in the analyses. Students' math conceptual and procedural skills in Grades three and five were moderately related to each other. Furthermore, based on latent profile analysis, three profiles of skills were identified in both grades, distinguishing between students with low, average, and high levels of skills in both skills, with the exception of an approximately average-skill group in Grade five that had low conceptual skills but average procedural skills. Students were likely to remain in similar-skill groups for both Grades three and five. Lastly, there was a significant difference in math interest in Grade three among the groups with stable low, stable average, and stable high skills from Grades three to five. Fostering interest in math may be beneficial for achieving a high level of math proficiency. Future research should prioritize the integration of person-oriented methods into studies aimed at enhancing understanding of math skill development. [ABSTRACT FROM AUTHOR]

The present study investigated multilevel models posited to predict student approaches to homework. Participants were 1,072 middle school students in China. Results revealed that deep and surface approaches were positively associated with performance-approach. Furthermore, deep approach to homework was associated negatively with homework cost, yet positively with mastery-approach, homework expectancy, and prior knowledge. Surface approach to homework was associated positively with homework cost and parent education, yet negatively with homework expectancy and mastery-approach. Females were less likely to use surface approach to homework than males. At the class level, surface approach to homework was negatively associated with parent education. Implications for homework practice and future investigation are discussed in terms of these results. [ABSTRACT FROM AUTHOR]

This study aimed to ascertain the contribution of children's effortful control (EC) to math achievement by testing the mediational involvement of math anxiety. Participants were 704 children (367 girls) aged between 7 and 12 years (M = 9.43, SD = 1.23). Children's EC was measured by parent's report, math anxiety was assessed through self-report, whereas math achievement was evaluated by performance on standard math tests. The models tested involved EC higher-order factor or specific EC components (activation control, attentional focusing, and inhibitory control). The results showed that activation control, attentional focusing, and EC higher-order factor contributed to math performance directly and through the mediation of math anxiety. Inhibitory control did not yield a significant effect on math performance. The findings suggest that children with higher dispositional effortful self-regulation tend to experience lower levels of math anxiety, with positive effects on math performance. [ABSTRACT FROM AUTHOR]

Early detection of and relevant information on children's mathematical difficulties is important to initiate targeted teaching and intervention. This study investigated the extent to which strategy use in single-digit addition provides additional predictive information about 61 grade one children's (6-year-old) mathematical achievement 3 years later that is not available from a standardised mathematics achievement test. Four predictors available in year one (arithmetic strategy use, mathematical achievement, non-verbal reasoning skills and sex) explained 54% of the variation in grade four mathematics achievement. Arithmetic strategy use was the most important single predictor of year four mathematics achievement (R2 = 30%) and explained an additional 12% variation if added to a model comprised by the three other year one predictors. This result suggests that systematically obtained measures of how young children solve single-digit arithmetic problems might provide useful information about their foundational number knowledge, which in turn may reveal how well they achieve later in school. [ABSTRACT FROM AUTHOR]

Interest in socioeconomic differences in academic motivation has been longstanding. However, previous research has often treated both low- and high-SES students as homogenous groups. This study aims to address this gap by investigating the developmental trajectory profiles of mathematics motivation during early adolescence, with a focus on variations within and across SES groups. Multigroup latent class growth analysis was used on a sample of 3718 early adolescents in China (initial Mage was 9.40 ± 0.52 years; 48.0% girls) across 2 years from grades 4 through 6. The analysis identified three distinct self-determined mathematics motivation trajectory profiles within each SES group: a good-quality profile (i.e., initially high autonomous but low controlled), a high-quantity profile (i.e., initially high both autonomous and controlled), and a low-quantity profile (i.e., initially low both autonomous and controlled). A greater proportion of low-SES students were observed within the low-quantity profile than within the good-quality profile. The study found that the failure-is-enhancing view was a protective factor against two relatively maladaptive motivational trajectory profiles (i.e., high-quantity profile and low-quantity profile), irrespective of socioeconomic background. These findings emphasize the importance of implementing motivational interventions for early adolescents that consider both structural factors (e.g., socioeconomic backgrounds) and psychological factors (e.g., failure beliefs), to foster students' academic development. [ABSTRACT FROM AUTHOR]

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Kollár (Publ. Res. Inst. Math. Sci. 40(3):689–708, 2004). [ABSTRACT FROM AUTHOR]

As a quantum affinization, the quantum toroidal algebra U q , q ¯ (gl ¨ n) is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations (Enriquez in Transform Groups 5(2):111–120, 2000; Feigin and Odesskii in Am Math Soc Transl Ser 2:185, 1998). In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation U q (gl ˙ n) ↷ C n (z) and its usual R-matrix R (z) ∈ End (C n ⊗ C n) (z) (see Faddeev et al. in Leningrad Math J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on U q , q ¯ (gl ¨ n) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra U q (gl ˙ n) ⊂ U q , q ¯ (gl ¨ n) . [ABSTRACT FROM AUTHOR]

We study infinite groups interpretable in three families of valued fields: V-minimal, power bounded T-convex, and p-adically closed fields. We show that every such group G has unbounded exponent and that if G is dp-minimal then it is abelian-by-finite. Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field K, its residue field k (when infinite), its value group Γ , or K / O , where O is the valuation ring. Our work uses and extends techniques developed in Halevi et al. (Adv Math 404:108408, 2022) to circumvent elimination of imaginaries. [ABSTRACT FROM AUTHOR]

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph K m , n , extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr (K 10 , n) ≥ 4.87057 n 2 - 10 n , cr (K 11 , n) ≥ 5.99939 n 2 - 12.5 n , cr (K 12 , n) ≥ 7.25579 n 2 - 15 n , cr (K 13 , n) ≥ 8.65675 n 2 - 18 n for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite. [ABSTRACT FROM AUTHOR]

We consider the problem of deterministically enumerating all minimum k-cut-sets in a given hypergraph for fixed constant k. The input here is a hypergraph G = (V , E) with non-negative hyperedge costs. A subset F ⊆ E of hyperedges is a k-cut-set if the number of connected components in G - F is at least k and it is a minimum k-cut-set if it has the least cost among all k-cut-sets. For fixed k, we call the problem of finding a minimum k-cut-set as Hypergraph- k -Cut and the problem of enumerating all minimum k-cut-sets as Enum-Hypergraph- k -Cut. The special cases of Hypergraph- k -Cut and Enum-Hypergraph- k -Cut restricted to graph inputs are well-known to be solvable in (randomized as well as deterministic) polynomial time (Goldschmidt and Hochbaum in Math Oper Res 19(1):24–37, 1994; Karger and Stein in J ACM 43(4):601–640, 1996; Kamidoi et al. in SIAM J Comput 36(5):1329–1341, 2007; Thorup, in: Proceedings of the 40th annual ACM symposium on theory of computing, STOC, 2008). In contrast, it is only recently that polynomial-time algorithms for Hypergraph- k -Cut were developed (Chandrasekaran et al. in Math Program 186:85–113, 2019; Fox et al., in: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, 2019; Chandrasekaran and Chekuri in Math Oper Res 47:3380–3399, 2022). The randomized polynomial-time algorithm for Hypergraph- k -Cut that was designed in 2018 (Chandrasekaran et al. 2019) showed that the number of minimum k-cut-sets in a hypergraph is O (n 2 k - 2) , where n is the number of vertices in the input hypergraph, and that they can all be enumerated in randomized polynomial time, thus resolving Enum-Hypergraph- k -Cut in randomized polynomial time. A deterministic polynomial-time algorithm for Hypergraph- k -Cut was subsequently designed in 2020 (Chandrasekaran and Chekuri 2022), but it is not guaranteed to enumerate all minimum k-cut-sets. In this work, we give the first deterministic polynomial-time algorithm to solve Enum-Hypergraph- k -Cut (this is non-trivial even for k = 2 ). Our algorithm is based on new structural results that allow for efficient recovery of all minimum k-cut-sets by solving minimum (S, T)-terminal cuts. Our techniques give new structural insights even for minimum cut-sets (i.e., minimum 2-cut-sets) in hypergraphs—we give a new proof showing that the number of minimum cut-sets in a n-vertex hypergraph is at most n 2 and they can all be enumerated in deterministic polynomial time for a given hypergraph. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading behavior of a two-species strong-weak competition system with two free boundaries. The model may describe how a strong competing species invades into the habitat of a native weak competing species. The asymptotic spreading speed of invading fronts has been determined by making use of semi-wave systems in Du et al. (J Math Pures Appl 107:253–287, 2017). Here we give a sharp estimate for the asymptotic spreading speed of invading fronts. Moreover, we prove that the solution of the free boundary problem evolves eventually into a semi-wave solution when the spreading happens, while the solution of the free boundary problem exponentially converges to a semi-trivial solution of such system when the vanishing happens. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In the present paper, we establish an infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with double normal frequencies. Applying it, we prove the existence of quasi-periodic solutions for one dimensional coupled nonlinear quantum harmonic oscillators (QHO) with a natural reversible structure. To compensate the lack of smoothing effect of perturbation, we introduce a class of vector fields with polynomial decay which extends the works of Grébert and Thomann (Commun Math Phys 307(2):383–427, 2011) for Hamiltonian QHO. To deal with the reversible, coupled perturbations in the equations, we also introduce a new class of generating vector fields during the KAM iteration. Moreover, the quasi-periodic solutions we obtain may not be linearly stable. This is obviously different from the result in Grébert and Thomann (2011) for Hamiltonian QHO. [ABSTRACT FROM AUTHOR]

In this paper, we explore the version of Hairer's regularity structures based on a greedier index set than trees, as introduced in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039) and algebraically characterized in (Linares et al. in Comm. Am. Math. Soc. 3:1–64, 2023). More precisely, we construct and stochastically estimate the renormalized model postulated in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039), avoiding the use of Feynman diagrams but still in a fully automated, i. e. inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range. We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. Our approach is analytic and top-down rather than combinatorial and bottom-up. [ABSTRACT FROM AUTHOR]

This is a continuation of our previous work (Ann. Sc. Norm. Super. Pisa Cl. Sci., 20, 1295–1324, 2020). Let (Σ, g) be a closed Riemann surface, where the metric g has conical singularities at finite points. Suppose G is a group whose elements are isometries acting on (Σ, g). Trudinger–Moser inequalities involving G are established via the method of blow-up analysis, and the corresponding extremals are also obtained. This extends previous results of Chen (Proc. Amer. Math. Soc., 1990), Iula–Manicini (Nonlinear Anal., 2017), and the authors (2020). [ABSTRACT FROM AUTHOR]

This article is a continuation of our investigations in the function space C(X) with respect to the topology τ B s of strong uniform convergence on B in line of (Chandra et al. in Indag Math 31:43-63, 2020; Das et al. in Topol Appl 310:108005, 2022) using the idea of strong uniform convergence (Beer and Levi in J Math Anal Appl 350:568-589, 2009) on a bornology. First we focus on the notion of the tightness property of (C (X) , τ B s) and some of its variations such as the supertightness, the Id-fan tightness and the T-tightness. Certain situations are discussed when C(X) is a k-space with respect to the topology τ B s . Next the notions of strong B -open game and γ B s -open game on X are introduced and some of its consequences are investigated. Finally, we consider discretely selective property and related games. On (C (X) , τ B s) several interactions between topological games related to discretely selective property, the Gruenhage game on (C (X) , τ B s) and certain games on X are presented. [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 research on the interpretation of mathematics as part of the sciences of mind, introduced as 'Geisteswissenschaften' in the German philosophy, reflects the relationship between mathematics and arts, including architecture. This study focuses on the early works of the philosopher Max Bense where he described the contours of an intellectual history of mathematics related to philosophy. He pointed out strong relationships between mathematics and all areas of human creation abilities. In particular, his investigations on the relationship between arts, including architecture, and mathematics in important historical phases reveal that the mathematics of an age is reflected aesthetically in the artistic styles or theories of art. His thesis posits that a high point in mathematics corresponds to a high point in art. Later on, Bense developed an aesthetic theory on the basis of mathematics, semiotics and information theory. [ABSTRACT FROM AUTHOR]

Quantitative stability for crystalline anisotropic perimeters, with control on the oscillation of the boundary with respect to the corresponding Wulff shape, is proven for n ≥ 3 . This extends a result of Neumayer (SIAM J Math Anal 48:172–1772, 2016) in n = 2 . [ABSTRACT FROM AUTHOR]

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg–Witten functional that in particular includes the Kapustin–Witten functional as a special case. We first demonstrate the smoothness of weak solutions to this generalized functional. We then establish the existence of weak solutions under the assumption that the structure group of the bundle is abelian, by verifying the Palais–Smale compactness. [ABSTRACT FROM AUTHOR]

We revisit the following nonlinear Schrödinger equation - ε 2 Δ u + V (x) u = u p , u > 0 , u ∈ H 1 (R N) , where ε > 0 is a small parameter, N ≥ 2 and 1 < p < 2 ∗ - 1 . It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive k-peak solutions to above problem when the critical points of V(x) are non-isolated and degenerate. We also give a specific formula for the Morse index of k-peak solutions when the critical point set of V(x) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential V(x). Our results generalize Grossi and Servadei's work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions. [ABSTRACT FROM AUTHOR]

In this study, we explore the valuation challenge posed by American options subject to regime switching, utilizing a model defined by a complex system of parabolic variational inequalities within an infinite domain. The initial pricing model is transformed into a linear complementarity problem (LCP) in a bounded rectangular domain, achieved through the application of a priori estimations and the introduction of an appropriate artificial boundary condition. To discretize the LCP, we employ a finite difference method (FDM), and address the resulting discretized system using a primal-dual active set (PDAS) strategy. The PDAS approach is particularly advantageous for its ability to concurrently determine the option's price and the optimal exercise boundary. This paper conducts an extensive convergence analysis, evaluating both the truncation error associated with the FDM and the iteration error of the PDAS. Comprehensive numerical simulations are performed to validate the method's accuracy and efficiency, underscoring its significant potential for application in the field of financial mathematics. [ABSTRACT FROM AUTHOR]

A topological index is a numerical property of a molecular graph that explains structural features of molecules. The potential of topological indices to discriminate between distinct structures is a significant topic to investigate. In this context, the exponential degree-based indices were put forward in the literature. The present work focuses on the exponential augmented Zagreb index (EAZ), which is defined for a graph G as E A Z (G) = ∑ v i v j ∈ E (G) e d i d j d i + d j - 2 3 , where d i represents the degree of the vertex v i and E(G) denotes the edge set of G. This work characterizes the maximal unicyclic graph for EAZ in terms of graph order, which was posed as an open problem in the recent article Cruz et al. (MATCH Commun Math Comput Chem 88:481-503, 2022). [ABSTRACT FROM AUTHOR]

Picard's theorem is among the most striking results in complex analysis and plays a decisive role in the development of the theory of entire and meromorphic functions. In this paper, we give an equivalence between Picard's theorem and entire solutions of (i) difference equation Δ η f + a (z) P (f) = 0 in C and (ii) differential equation D k f + h (z) g (f (z)) = 0 in terms of total derivatives in C n . We prove two Picard-type theorems that generalize the results of Lu (Kodai Math J 26:221–229, 2003). In addition, examples have been exhibited to show that certain assumptions in the main result cannot be improved. [ABSTRACT FROM AUTHOR]

The ratio of two magnitudes can take one of two values depending on the order they are operated on: a 'big' ratio of the larger to smaller magnitude, or a 'small' ratio of the smaller to larger. Although big and small ratio scales have different metric properties and carry divergent predictions for perceptual comparison tasks, no psychophysical studies have directly compared them. Two experiments are reported in which subjects implicitly learned to compare pairs of brightnesses and line lengths by non-symbolic feedback based on the scaled big ratio, small ratio or difference of the magnitudes presented. Results of Experiment 1 showed all three operations were learned quickly and estimated with a high degree of accuracy that did not significantly differ across groups or between intensive and extensive modalities, though regressions on individual data suggested an overall predisposition towards differences. Experiment 2 tested whether subjects learned to estimate the operation trained or to associate stimulus pairs with correct responses. For each operation, Gaussian noise was added to the feedback that was constant for repetitions of each pair. For all subjects, coefficients for the added noise component were negative when entered in a regression model alongside the trained differences or ratios, and were statistically significant in 80% of individual cases. Thus, subjects learned to estimate the comparative operations and effectively ignored or suppressed the added noise. These results suggest the perceptual system is highly flexible in its capacity for non-symbolic computation, which may reflect a deeper connection between perceptual structure and mathematics. [ABSTRACT FROM AUTHOR]

For 0 < α ≤ 1 and λ > 0, let G λ , α = f ∈ A : 1 - α + α z f ″ z / f ′ z z f ′ z / f z - 1 - α < λ , z ∈ D. (0.1) The general form of the Silverman class was introduced by Tuneski and Irmak [Int. J. Math. Math. Sci., 2006, Article ID 38089 (2006)]. Our differential-inequality formulation is based on several sufficient conditions for this class. Further, we consider a class Ω given by Ω = f ∈ A : z f ′ z - f z < 1 2 , z ∈ D. (0.2) For these two classes, we establish inclusion relations involving some well-known subclasses of S* and compute radius estimates featuring various pairings of these classes. [ABSTRACT FROM AUTHOR]

In [V. M. Abramov, Bull. Austral. Math. Soc., 104, 108 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present work, we provide an alternative proof of the existence of a positive solution in the general case. This proof is based on the application of a version of the M. A. Krasnosel'skii fixed-point theorem. The results are then extended to the equations with infinite matrices of the general type. [ABSTRACT FROM AUTHOR]

In this study, the theory of curves is reconstructed with fractional calculus. The condition of a naturally parametrized curve is described, and the orthonormal conformable frame of the naturally parametrized curve at any point is defined. Conformable helix and conformable slant helix curves are defined with the help of conformable frame elements at any point of the conformable curve. The characterizations of these curves are obtained in parallel with the conformable analysis Finally, examples are given for a better understanding of the theories and their drawings are given with the help of Mathematics. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associated with the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi (J Math Anal Appl 241:46–55, 2000). We establish that the trajectories x(t) of the continuous dynamical system (CDS) has an asymptotic behavior similar to the behavior of the sequences (x n) generated by the discrete viscosity approximation (DDS) [ABSTRACT FROM AUTHOR]

Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely P ∗ (n) , M (n) , T ∗ (n) , L (n) , K (n) , A (n) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo 2 a · 5 ℓ for P ∗ (n) , modulo 2 3 for M(n) and T ∗ (n) , where a = 3 , 4 and ℓ ≥ 1 . For instance, we prove that for n ≥ 0 , P ∗ (5 ℓ (4 n + 3) + 5 ℓ - 1) ≡ 0 (mod 2 3 · 5 ℓ). In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: M (5 n + 4) ≡ 0 (mod 5) , T ∗ (5 n + 3) ≡ 0 (mod 5). [ABSTRACT FROM AUTHOR]

Solving large-scale systems of nonlinear equations (SoNE) is a central task in mathematics that traverses different areas of applications. There are several derivative-free methods for finding SoNE solutions. However, most of the methods contributed to find SoNE solutions involve a monotone cost function. Methods dealing with pseudomonotone cost function remain rare. In this paper, we introduce two classes of derivative-free spectral three-term methods to solve large-scale continuous pseudomonotone SoNE. We combine the projection method of Solodov and Svaiter with the structure of the recently developed spectral three-term conjugate gradient method for unconstrained optimization by Amini and Faramarzi. We prove that the proposed methods possess sufficient descent property, trust region property, and global convergence without relying on Lipschitz continuity. Numerical experiments show that the proposed methods are efficient and competitive with existing methods. Finally, the proposed methods have been successfully applied to recover a sparse signal from incomplete and contaminated sampling measurements, yielding promising results. [ABSTRACT FROM AUTHOR]

In the context of claims problems, we formulate an axiom of invariance of a rule with respect to its "partial implementation": having applied the rule to a problem in its domain of definition, we imagine some claimants picking up their awards and we consider the problem of distributing what remains of the endowment among the other claimants: in this problem, the claims of the first group of claimants are set equal to zero and the endowment is adjusted down by the sum of their awards; we require of a rule that it assign to each remaining claimant the same amount as it did initially. We formulate several variants of this requirement of "partial-implementation invariance" and a "converse" of it. We investigate how the two requirements relate to known axioms and ask whether they are preserved by certain operators that have been defined on the space of rules. Our main result is a characterization in our fixed-population framework of a family of rules introduced and characterized by Young (Math Oper Res 12(3):398–414, 1987) in a variable-population framework. These rules are known under the name of "parametric rules". [ABSTRACT FROM AUTHOR]

We present various results about Euclidean preferences in the plane under ℓ 1 , ℓ 2 and ℓ ∞ norms. When there are four candidates, we show that the maximum size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in R 2 under norm ℓ 1 or ℓ ∞ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in the last position of a two-dimensional Euclidean preference profile under norm ℓ 1 or ℓ ∞ , which generalizes the case of one-dimensional Euclidean preferences (for which it is well known that at most two candidates can be ranked last). We generalize this result to 2 d (resp. 2d) for ℓ 1 (resp. ℓ ∞ ) for d-dimensional Euclidean preferences. We also establish that the maximum size of a two-dimensional Euclidean preference profile on m candidates under norm ℓ 1 is in Θ (m 4) , which is the same order of magnitude as the known maximum size under norm ℓ 2. Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm ℓ 2 for four candidates can be characterized by three inclusion-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. (Adv Appl Math 47(2):379–400, 2011). [ABSTRACT FROM AUTHOR]

The note is concerned with the functional equation λ 1 H 1 (f (x)) + λ 2 H 2 (f 2 (x)) + ⋯ + λ n H n (f n (x)) = F (x) , which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability. [ABSTRACT FROM AUTHOR]

We examine the association of structured and unstructured extracurricular activities on academic achievement for youth with a history of child welfare system involvement. Using longitudinal data from the National Survey of Child and Adolescent Well-being II (NSCAW II) and fixed effects ordinary least squares regression models we examine this association for 905 youth aged 11–17 years old. We find that even moderate levels of participation in structured extracurricular activities are associated with improvements in both math and reading achievement. Specifically, moving from non-participation to involvement in structured activities leads to a 4.46-point increase in applied problem solving and a 2.99-point increase in letter-word identification scores, net of the other controls. These findings suggest that children and youth with child welfare system involvement can benefit from participation in structured activities. This finding is significant in light of federal policy recognizing the importance of "normalizing" activities that enhance the well-being of children and youth in the child welfare system. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some versions of relative connectedness of subspaces of a topological space and we give some facts and relations among them. We prove that these relative versions satisfy some of the classical properties of connectedness. Additionally, we apply our results to the theory of hyperspaces, aiming to address a general problem posed by Arhangel'skii (Comment Math Univ Carolin 36:305–325, 1995, Problem 3). [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

We develop the asymptotic analysis as ε→0 for the natural gradient flow of the self-dual U(1)-Higgs energies on Hermitian line bundles over closed manifolds (Mn,g) of dimension n≥3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows—i.e., integral (n−2)-Brakke flows—generalizing results of (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (n−2)-cycle Γ0 in M, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the (n−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ0 with additional structure, similar to those produced via Ilmanen's elliptic regularization. [ABSTRACT FROM AUTHOR]

We prove an asymptotic for the second moment of quadratic twists of a modular L -function. This was previously known conditionally on GRH by the work of Soundararajan and Young (J. Eur. Math. Soc. 12(5):1097–1116, 2010). [ABSTRACT FROM AUTHOR]

Conditioned reinforcement for math, or math enjoyment, may be critical for students. If so, this should be established as early as pre-kindergarten, as conditioning the components of math as reinforcement arguably leads to more complex mathematical operations thus impacting individuals' future math ability and literacy (Engel et al., 2013), A multiple probe across dyads design with a nested multiple probe across dyads design was used to determine if establishing a preference for math influenced students' rate of learning math. The intervention tested the effect of the individualized reinforcement procedures, based on a sequence of conditioning procedures proposed in Buttigieg and Greer's (2023) study, on establishing conditioned reinforcement. Three participants required learn units, two participants required the stimulus–stimulus pairing procedure, and one participant required observational conditioning-by-denial to establish conditioned reinforcement for math. The dependent variable was each participant's rate of learning as measured by the number of learn units required to meet mastery criterion for four units of the Equivalence Based Functional Math Curriculum (EBF-Math; Weber et al., 2023). Results showed an educationally significant acceleration of learning following the establishment of conditioned reinforcement for math across all six participants. Results are discussed in terms of the significance of early math instruction and literacy. [ABSTRACT FROM AUTHOR]

Nearly 30 years after the end of apartheid in South Africa, the education system has yet to provide learners with an adequate school experience. With school closures during the pandemic, the urgency to provide a quality education in the townships became even more dire. In collaboration with the leadership of Charles Duna Primary School in Gqeberha, South Africa, Partnerships for Educational Excellence and Research (PEER) International reestablished a partnership to design a flexible implementation of methods and curricula to build numeracy repertoires. Professional development efforts leverage evidence-based best teaching and learning strategies included in the Morningside Model of Generative Instruction (Johnson et al., 2020). Weekly videoconferences target the development of basic computation skill acquisition and math problem solving skills. The dramatically different environment and available resources in township schools is a prevailing consideration in this transfer of instructional technology. Our project started early in 2022 and continues with optimism and determination. The purpose of this paper is to describe our constructional approach (Goldiamond, 1974/2002 Behavior and Social Issues, 11(2), 108-197) when invited to improve classroom conditions for teachers and learners. [ABSTRACT FROM AUTHOR]

The utilization of advanced intraoral scanners to acquire 3D dental models has gained significant popularity in the fields of dentistry and orthodontics. Accurate segmentation and labeling of teeth on digitized 3D dental surface models are crucial for computer-aided treatment planning. At the same time, manual labeling of these models is a time-consuming task. Recent advances in geometric deep learning have demonstrated remarkable efficiency in surface segmentation when applied to raw 3D models. However, segmentation of the dental surface remains challenging due to the atypical and diverse appearance of the patients' teeth. Numerous deep learning methods have been proposed to automate dental surface segmentation. Nevertheless, they still show limitations, particularly in cases where teeth are missing or severely misaligned. To overcome these challenges, we introduce a network operator called dilated edge convolution, which enhances the network's ability to learn additional, more distant features by expanding its receptive field. This leads to improved segmentation results, particularly in complex and challenging cases. To validate the effectiveness of our proposed method, we performed extensive evaluations on the recently published benchmark data set for dental model segmentation Teeth3DS. We compared our approach with several other state-of-the-art methods using a quantitative and qualitative analysis. Through these evaluations, we demonstrate the superiority of our proposed method, showcasing its ability to outperform existing approaches in dental surface segmentation. [ABSTRACT FROM AUTHOR]

Intelligent Tutoring Systems (ITS) have been widely used to enhance math learning, wherein teacher's involvement is prominent to achieve their full potential. Usually, ITSs depend on direct interaction between the students and a computer. Recently, researchers started exploring handwritten input (e.g., from paper sheets) aiming to provide equitable access to ITSs' benefits. However, research on math ITSs ability to handle handwritten input is limited and, to our best knowledge, no study has summarized its state of the art. This article fulfills that gap with a scoping review of handwritten recognition methods, characteristics, and applications of math ITSs compatible with handwritten input. Based on a search of 11 databases, we found eight primary studies that met our criteria. Mainly, we found that all ITSs depend on receiving handwritten input from a touchscreen interface, in contrast to recognizing solutions developed on paper. We also found that most ITSs focus on similar audiences (e.g., English speakers students), subjects (e.g., algebraic questions), and applications (e.g., in-class to understand student perceptions). Thus, towards enabling equitable access to ITSs, we propose ITS Unplugged (i.e., ITSs that i) run on low-cost, resource-restricted devices with little to no internet connection and ii) receive as well as return information in the format target users usually use) and contribute a research agenda concerning challenges of developing such ITSs. [ABSTRACT FROM AUTHOR]

Analogy has received attention as a form of inductive reasoning in the empirical sciences. Its role in mathematics has, instead, received less consideration. This paper provides a novel account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an incremental and a non-incremental form of confirmation by mathematical analogy. We offer an account of the former within the popular framework of Bayesian confirmation theory. As for the non-incremental notion, we defend its role in rationally informing the prior credences of mathematicians in those circumstances in which no new mathematical evidence is introduced. The resulting framework captures many important aspects of the use of analogical inference in the domain of pure mathematics. [ABSTRACT FROM AUTHOR]

This 5-year mathematics professional development project involves 27 elementary teachers prepared and supported as Elementary Mathematics Specialists (EMSs) in high-need, urban schools. Our inquiry centers on their experiences in the EMS preparation program and the development of important outcomes, specifically productive beliefs and teacher leadership. Data were gathered at the end of Year 1 via surveys of mathematical beliefs, coaching beliefs, and coaching practices, as well as individual and focus group interviews. The findings provide insights into participants' growth and work as a more knowledgeable other, illuminating specific emphases of their efforts and influences of the project. When it comes to mathematical beliefs, as participants shifted toward embracing cognitively-oriented pedagogical beliefs, they became more efficacious in this learner-centered pedagogy. The findings also show variability in specific mathematics coaching practices, including those most participants were and were not using, and areas for improvement. Several themes are evident in the interview data: growing confidence for the uncomfortable, creating and finding spaces for stepping up, advocating for learned mathematics pedagogy, and building teacher capacity through interactions. [ABSTRACT FROM AUTHOR]

The use of digital technology has the potential to support students' understanding in the mathematics classroom with the teacher playing a vital role. However, teaching with digital technology is not trivial, especially for teachers who are new to this. In this paper, we present an analysis of the enactment of a function lesson of a Sri Lankan mathematics teacher who used digital technology for the first time in her teaching. We combined the instrumental orchestration and ROG (resources, orientations and goals) frameworks into a conceptual framework to analyse her teaching. In particular, we used instrumental orchestration to identify how the teacher orchestrated the resources in her technology-rich classroom. This was combined with ROG theory to understand the reasons underpinning the decisions involved in moving from one orchestration to another. We demonstrate that this teacher showed diverse orchestrations and use the ROG framework to present these in the sequences in which they were used, formed into chains of orchestrations linked by goals. We propose that her didactical performance is a function of orchestration types over in-the-moment decision-making. [ABSTRACT FROM AUTHOR]

This study aims to analyze possible connections among the noticing skills of pre-service mathematics teachers, and specifically the skills of attending to students' thinking, interpreting students' mathematical understanding, and proposing teaching alternatives. We characterized these connections in terms of coherency, i.e., the extent to which pre-service teachers take into account (a) an identified critical event—including its mathematics and context—when interpreting students' statements, (b) their own interpretation of students' statements when interpreting the teacher's response, and (c) their own interpretation of students' statements when proposing alternative teaching strategies. The construct of coherency evolved during our attempts to analyze pre-service teachers' interpretations of the critical events they identified while observing mathematics lessons as part of their preparation program. In this paper, we elucidate the construct of coherency and its different levels and demonstrate how this construct relies upon and expands existing ideas from the literature on noticing. We propose a framework for identifying different coherency levels and the different patterns identified when using the framework to noticing among the pre-service teachers. Some of these patterns are supported by previous research, while others were unanticipated. We explore possible explanations for the emergence of these unanticipated patterns. [ABSTRACT FROM AUTHOR]