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In the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ -adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W -algebra of sp 2 N at level - N - 1 / 2 , following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function. [ABSTRACT FROM AUTHOR]

In the classical bargaining problem, we propose a very mild axiom of individual rationality, which we call possibility of utility gain. This requires that for at least one bargaining problem, there exists at least one player who reaches a higher utility level than their disagreement utility. This paper shows that the Nash solution (Nash in Econometrica 18(2):155–162, 1950) is characterized by possibility of utility gain and continuity with respect to feasible sets together with Nash's axioms except weak Pareto optimality. We also show that in Nash's theorem, weak Pareto optimality can be replaced by conflict-freeness (introduced by Rachmilevitch in Math Soc Sci 76(C):107–109, 2015). This demands that when the agreement most preferred by all players is feasible, this should be chosen. Furthermore, we provide alternative and unified proofs for other efficiency-free characterizations of the Nash solution. This clarifies the role of each axiom in the related results. [ABSTRACT FROM AUTHOR]

In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted K-theory of connected, simply-connected and simple compact Lie groups G, in both equivariant and nonequivariant settings. [ABSTRACT FROM AUTHOR]

This note corrects an error in our paper Anosov flows, growth rates on covers and group extensions of subshifts, Invent. Math. 223:445–483, 2021. This leaves our main results, Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 5.1, unchanged. We also fill a gap in the arguments presented in Sect. 9; this requires a small modification to the results in this section. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Avalanche Principle in [Ann. of. Math. (2), 2001, 154(1): 155–203]. [ABSTRACT FROM AUTHOR]

In this paper, we prove the convexity of trace functionals (A , B , C) ↦ Tr | B p A C q | s , for parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems. [ABSTRACT FROM AUTHOR]

Kovaleva and Spieksma (SIAM J Discrete Math 20(3):48–768, 2006) considered the problem of stabbing a given set of horizontal line segments with the smallest number of horizontal and vertical lines. The standard LP relaxation for this problem is easily shown to have an integrality gap of at most 2 by treating the horizontal and vertical lines separately. However, Kovaleva and Spieksma observed that threshold rounding can be used to obtain an integrality gap of e / (e - 1) ≈ 1.58 which is also shown to be tight. This is one of the rare known examples where the obvious upper bound of 2 on the integrality gap of the standard LP relaxation can be improved. Our goal in this paper is to extend their proof to two other problems where the goal is to stab a set R of objects with horizontal and vertical lines: in the first problem R is a set of horizontal and vertical line segments, and in the second problem R is a set of unit sized squares. The proof of Kovaleva and Spieksma essentially shows the existence of an appropriate threshold which yields the improved approximation factor. We begin by showing that a random threshold picked from an appropriate distribution works. This reduces the problem to finding an appropriate distribution for a desired approximation ratio. In the first problem, we show that the required distribution can be found by solving a linear program. In the second problem, while it seems harder to find the optimal distribution, we show that using the uniform distribution an improved approximation factor can still be obtained by solving a number of linear programs. [ABSTRACT FROM AUTHOR]

In the paper, we investigate the uniqueness problem of a meromorphic function and its difference operator to the most general setting via two shared set problems and thus improve a recent result of Chen–Chen (Bull Malays Math Sci Soc 35(3): 765-774, 2012). [ABSTRACT FROM AUTHOR]

This note tries to answer issues raised in Bhardwaj and Kumar (J Optim Theory Appl 163(2): 685-696, 2014). The research summarizes that the results obtained in Khan et al. (J Optim Theory Appl 159: 536-546, 2013) are sound and correct and it fulfills all the necessary requirements of its scope and objectives. [ABSTRACT FROM AUTHOR]

In the paper, we will present some important results from the theory of vector-valued invariant means. Discussing the results contained in the author's paper Badora (Ann Pol Math 58:147–159, 1993), we will add new facts that were previously not published. [ABSTRACT FROM AUTHOR]

In this paper, we consider sums related to the floor function. We improve previous results for some special arithmetic functions considered by O. Bordellès [On certain sums of number theory, Int. J. Number Theory, 18(9):2053– 2074, 2022], J. Stucky [The fractional sum of small arithmetic functions, J. Number Theory, 238:731–739, 2022], and J. Wu [Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski, Period. Math. Hung., 80(1):95–102, 2020]. It is worth emphasizing that we use much simpler methods to give much better results than previous. [ABSTRACT FROM AUTHOR]

Telescopers for a function are linear differential (resp. difference) operators annihilating the definite integral (resp. definite sum) of this function. They play a key role in Wilf–Zeilberger theory and algorithms for computing them have been extensively studied in the past 30 years. In this paper, we introduce the notion of telescopers for differential forms with D-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given. [ABSTRACT FROM AUTHOR]

Some global error bounds with undetermined parameters, which are not always valid, for the extended vertical linear complementarity problems (LCP) of CKV-type matrices and CKV-type B -matrices, are presented by Yan and Wang (Jpn. J. Ind. Appl. Math. 41:129–150, 2024). In this paper, new global error bounds for the extended vertical LCP of CKV-type matrices and CKV-type B -matrices are given, which depend only on the entries of the involved matrices. Numerical examples show that the new bounds are better than those provided in Zhang et al. (Comput. Optim. Appl. 42(3):335–352, 2009) and Wang et al. (Comput. Appl. Math. 40:148, 2021) in some cases. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring. [ABSTRACT FROM AUTHOR]

Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups O (b + , b -) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule Π , which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to Π as theta liftings of holomorphic Mp 2 (R) cusp forms by using the Borcherds' method (Invent Math 132:491–562, 1998). We propose a special choice of Schwartz functions to define the liftings, which yields precise descriptions of their Fourier expansions. [ABSTRACT FROM AUTHOR]

We derive Melnikov type conditions for the persistence of periodic solutions in perturbed slowly varying discontinuous differential equations. In contrast to Battelli and Fečkan (Mathematics 9(19):2449, 2021) we assume that the unperturbed (frozen) equation has a family of periodic solutions depending on some parameters. The result of this paper are motivated by and extend a result in Wiggins and Holmes (SIAM J Math Anal 18:592–611, 1987) where the authors considered a two dimensional hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation. [ABSTRACT FROM AUTHOR]

In Glickenstein (J Differ Geom 87: 201–237, 2011), Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers–Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces (Zhang et al. in Graph Models 76: 321–339, 2014) is a generalization of Chow–Luo's combinatorial Ricci flow for Thurston's circle packings (Chow and Luo in J Differ Geom 63: 97–129, 2003) and Luo's combinatorial Yamabe flow for vertex scalings (Luo in Commun Contemp Math 6: 765–780, 2004). We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half space. First, a narrow region principle for antisymmetric functions in unbounded domains is obtained, in which we remarkably weaken the decay condition u → 0 at infinity and only assume its growth rate does not exceed | x | γ (0 < γ < 2 s ) compared with (Adv. Math. 377:107463, 2021). Then we obtain asymptotic monotonicity of positive solutions of fractional parabolic equation on R + N × (0 , ∞) . [ABSTRACT FROM AUTHOR]

This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, ID ^ 1 i , over Heyting arithmetic (HA ), originally proved in full generality by Arai (Ann Pure Appl Log 162:807–815, 2011. https://doi.org/10.1016/j.apal.2011.03.002). The proof embeds ID ^ 1 i into the corresponding theory over Beeson's logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using a hierarchy of satisfaction predicates for almost negative formulae. It concludes by applying van den Berg and van Slooten's result (Indag Math 29:260–275, 2018. https://doi.org/10.1016/j.indag.2017.07.009) that Heyting arithmetic with partial terms plus the schema of self realizability for arithmetic formulae is conservative over HA . [ABSTRACT FROM AUTHOR]

In this paper, we describe an activity that involves economics and mathematics. It is included in the planning of orientation paths towards university studies within the Mathematical High School Project and is dedicated to students in the last years of high school. In particular, this research will deal with the issue of solving an economic problem using not only real analysis instruments but also geometrical topics concerning Euclidean geometry and topology. Mathematics becomes a language to understand and explain a real life problem, such as determining the optimal position of an airport, a nuclear reactor and so on. Some activities made use of dynamic geometry software and computer simulations. [ABSTRACT FROM AUTHOR]

This paper reports on students' conceptions of minima points. Written assignments and individual interviews uncovered salient, concept images, as well as erroneous mis-out examples that mistakenly regard examples as non-examples and mis-in examples that mistakenly grant non-examples the status of examples. We used Tall and Vinner's theoretical framework to analyze the students' errors that were rooted in mathematical and in real-life contexts. [ABSTRACT FROM AUTHOR]

In this paper we study the behaviour of solutions for a nonlocal hyperbolic equation. We use the Pohozaev manifold combined with a new technique to explicit two invariant regions in the space of initial data. On the first one the solution blows up (in finite or infinite time) and in the second one the solution exists globally. Additionally, we prove that the ground state solution of the elliptic problem associated to the original problem is unstable. The main goal of this paper is to present a new technique which allows us to consider nonlocal problems and to extend the classical result proved by Shatah (Trans Am Math Soc 290(2):701–710, 1985). [ABSTRACT FROM AUTHOR]

This paper is concerned with the two-species chemotaxis-competition model with degenerate diffusion, { u t = Δ u m 1 − χ 1 ∇ ⋅ (u ∇ w) + μ 1 u (1 − u − a 1 v) , x ∈ Ω , t > 0 , v t = Δ v m 2 − χ 2 ∇ ⋅ (v ∇ w) + μ 2 v (1 − a 2 u − v) , x ∈ Ω , t > 0 , 0 = Δ w + u + v − M ‾ (t) , x ∈ Ω , t > 0 , with ∫ Ω w (x , t) d x = 0 , t > 0 , where Ω : = B R (0) ⊂ R n (n ≥ 5) is a ball with some R > 0 ; m 1 , m 2 > 1 , χ 1 , χ 2 , μ 1 , μ 2 , a 1 , a 2 > 0 ; M ‾ (t) is the spatial average of u + v . In this paper, we show that if m 1 < 2 − 4 n , χ 1 > n (2 − m 1) n (2 − m 1) − 4 ⋅ max { 1 , a 1 } μ 1 and χ 2 > μ 2 a 2 or m 2 < 2 − 4 n , χ 1 > μ 1 a 1 and χ 2 > n (2 − m 2) n (2 − m 2) − 4 ⋅ max { 1 , a 2 } μ 2 , then there exist radially symmetric initial data such that the weak solution blows up in finite time in the sense that there is T ˜ max ∈ (0 , ∞) such that lim sup t ↗ T ˜ max (∥ u (t) ∥ L ∞ (Ω) + ∥ v (t) ∥ L ∞ (Ω)) = ∞. To obtain this result, we apply the method in the previous paper (Discrete Contin. Dyn. Syst., Ser. B 28(1):262–286, 2023) to derive an integral inequality for a moment-type functional, which was introduced by Winkler (Z. Angew. Math. Phys. 69(2):69, 2018). Moreover, before proving blow-up of solutions in the above model, we give a result on finite-time blow-up under the same conditions for m 1 , m 2 , χ 1 and χ 2 in the model with the terms Δ u m 1 , Δ v m 2 replaced with the nondegenerate diffusion terms Δ (u + ε) m 1 , Δ (v + ε) m 2 , where ε ∈ (0 , 1 ] . [ABSTRACT FROM AUTHOR]

The "Maker" movement is a cultural as well as educational phenomenon that has the potential to offer significant opportunities to students in conditions of social, economic and cultural disadvantage. The research reported in this paper, however, suggests that the simple provision of "Maker Spaces" for such activity is simplistic and not sufficient to realise this potential. The research involved a mixed methods study of a cohort of year 7 students (n = 26) in an Australian school located in a socio-economically disadvantaged outer-metropolitan region. The cohort undertook a range of Maker activities at a new "creativity centre" built at the school. Results indicate that the activities had positive impact on student attitudes towards science, technology, engineering and mathematics (STEM) overall, but that the impact was highly specific across attitudinal constructs. A strong ranging effect was also evident, suggesting that the impact of the experience was highly dependent on students' initial attitudes. Reflecting on these results, the paper also offers a reference framework that may help keep equity in mind when designing different kinds of Maker experience. [ABSTRACT FROM AUTHOR]

This paper concerns a distinction between circumstantial truths that hold because of the circumstances and acircumstantial truths that hold regardless of, or transcend, the circumstances. Previous discussions of the distinction tended to focus on its applications, such as to modality, logical truth, and essence. This paper focuses on developing the distinction largely, but not entirely, in abstraction from its potential applications. As such, the paper's main contribution is to further clarify the distinction itself. An indirect contribution is to help guide its future applications. [ABSTRACT FROM AUTHOR]

In this note I examine a number of statement made in Wadee and Yiatros (Z. Angew. Math. Phys. 61:565-577 ) in relation to my paper mentioned in the title. Most of the claims by Drs. Wadee and Yiatros are re-examined here in the proper context and it is shown that the results of the title paper are free of any errors. [ABSTRACT FROM AUTHOR]

In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n), 2 ∤ n, for a ∈ ℤ, ( a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

The article presents papers of the "Czech-Taiwanese Workshop on the Intermediate- and High-Energy Physics," that was held in Prague, Czech Republic from March 3-5, 2003, organized by Academy of Sciences of the Czech Republic, National Science Council of Taiwan, Czech Physical Society (Section of the Union of Czech Mathematicians and Physicists) and Physical Society of the Republic of China (Taiwan).

There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math. 106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman's work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research. [ABSTRACT FROM AUTHOR]

In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. 'Al.I. Cuza' Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above. [ABSTRACT FROM AUTHOR]

First of all, we indicate a severe error in the analysis of the main results of both Chakraborty [Ukr. Math. J., 72, No. 11, 1794–1806 (2021)] and Chakraborty–Chakraborty [Ukr. Math. J., 72, No. 7, 1164–1174 (2020)] to show that both these papers cease to be true. Further, pertinent to the results of these two papers, we consider the unique range set of a meromorphic function over a non-Archimedean field with the smallest possible weights 0 and 1 under the aegis of its most general form to improve the existing result. [ABSTRACT FROM AUTHOR]

The paper extends an earlier result of G.V. Kalachev et al. (Sb. Math. 210(8):1129–1147, 2019) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on R n with kernel from some L q , 1 < q < ∞ . On the other hand, E. Lieb (Ann. of Math. 118:(2):349–374, 1983) proved the existence of a maximizer for the Hardy-Littlewood-Sobolev inequality and remarked that in general a convolution maximizer for a kernel from weak L q may not exist. In this paper we axiomatize some properties used in the proof of the Kalachev-Sadov 2019 theorem and obtain a more general result. As a consequence, we prove that the convolution maximizer always exists for kernels from a slightly more narrow class than weak L q , which contains all Lorentz spaces L q , s with q ≤ s < ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we construct a symbol calculus yielding short exact sequences for the dual Toeplitz algebra generated by all bounded dual Toeplitz operators on the Hardy space associated with the polydisk D n in the unitary space C n , that have been introduced and well studied in our earlier paper (Benaissa and Guediri in Taiwan J Math 19: 31–49, 2015), as well as for the C*-subalgebra generated by dual Toeplitz operators with symbols continuous on the associated hypertorus T n . [ABSTRACT FROM AUTHOR]

Let X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map F : X → C C (Y) , for which the functionals f y ∗ (x) = inf y ∗ (F (x)) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001). [ABSTRACT FROM AUTHOR]

Let K and L be two convex bodies in R n , n ≥ 2 , with L ⊂ int K . We say that L is an equichordal body for K if every chord of K tangent to L has length equal to a given fixed value λ . Barker and Larman (Discrete Math. 241(1–3), 79–96 (2001)) proved that if L is a ball, then K is a ball concentric with L. In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. [ABSTRACT FROM AUTHOR]

In this paper we study a chemotaxis–haptotaxis model with p-Laplacian diffusion and non-flux boundary condition u t = ∇ · (| ∇ u | p - 2 ∇ u) - χ ∇ · (u ∇ v) - ξ ∇ · (u ∇ w) + μ u (1 - u - w) , (x , t) ∈ Ω × (0 , T) , v t = Δ v - v + u , (x , t) ∈ Ω × (0 , T) , w t = - v w , (x , t) ∈ Ω × (0 , T) , where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary ∂ Ω . We prove that when the diffusion parameter is appropriately large, i.e., p larger than some certain value, the global boundedness of the solutions does not depend on the relationship between the coefficient of logistic source term μ and the coefficient of chemotaxis term χ . And we find that the haptotaxis term does not affect the global boundedness of the solutions, in the other words, our result is consistent with the conclusion with no haptotaxis term (Zhuang et al. in Z Angew Math Phys 72:161, 2021). [ABSTRACT FROM AUTHOR]

In this paper, we establish local well-posedness for the Zakharov system on T d , d ≥ 3 in a low regularity setting. Our result improves the work of Kishimoto (J Anal Math 119:213–253, 2013). Moreover, the result is sharp up to ε -loss of regularity when d = 3 and d ≥ 5 as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain–Demeter's range of exponent for the linear decoupling inequality for paraboloid (Bourgain and Demeter in Ann Math 182:351–389) under the constraint that the input space-time function f satisfies supp f ^ ⊂ { (ξ , τ) ∈ R d + 1 : 1 - 1 N ≤ | ξ | ≤ 1 + 1 N , | τ - | ξ | 2 | ≤ 1 N 2 } for large N ≥ 1 . [ABSTRACT FROM AUTHOR]

In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill's equation based on L p -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill's equations of potential a (t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS. [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]

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 Nitsche's method for solving elliptic Dirichlet boundary control problems on curved domains with control constraints. By using Nitsche's method for the treatment of inhomogeneous Dirichlet boundary conditions, the L2 boundary control enters in the variational formulation in a natural sense. The idea was first used in Chang, et al. (Math. Anal. Appl.453, 529–557 2017) where the curved boundary was approximated by a broken line and a locally defined mapping was needed to obtain the numerical control on the curved boundary. In this paper, we develop a method defined on curved domains directly. We derive a priori estimates of quasi-optimal order for the control in the L2 norm, and quasi-optimal order for the state and adjoint state in energy norms. Numerical examples are provided to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract  We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) ⃛ ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions. [ABSTRACT FROM AUTHOR]

It is proved that the set of those natural numbers which cannot be written as n-Ω( n) is of positive lower density. Here Ω( n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca. [ABSTRACT FROM AUTHOR]

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

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Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of Criado and Santos (Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1641766), about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them. [ABSTRACT FROM AUTHOR]

Modern color science finds its birth in the middle of the nineteenth century. Among the chief architects of the new color theory, the name of the polymath Hermann von Helmholtz stands out. A keen experimenter and profound expert of the latest developments of the fields of physiological optics, psychophysics, and geometry, he exploited his transdisciplinary knowledge to define the first non-Euclidean line element in color space, i.e., a three-dimensional mathematical model used to describe color differences in terms of color distances. Considered as the first step toward a metrically significant model of color space, his work inaugurated researches on higher color metrics, which describes how distance in the color space translates into perceptual difference. This paper focuses on the development of Helmholtz's mathematical derivation of the line element. Starting from the first experimental evidence which opened the door to his reflections about the geometry of color space, it will be highlighted the pivotal role played by the studies conducted by his assistants in Berlin, which provided precious material for the elaboration of the final model proposed by Helmholtz in three papers published between 1891 and 1892. Although fallen into oblivion for about three decades, Helmholtz's masterful work was rediscovered by Schrödinger and, since the 1920s, it has provided the basis for all subsequent studies on the geometry of color spaces up to the present time. [ABSTRACT FROM AUTHOR]

This is a conceptual paper based on existing literature aiming to provide practical information on designing and implementing activities to promote children's computational thinking. Computational thinking is a relatively new term in early childhood education that refers to a specific problem-solving thinking process involving various logical and analytical thinking skills. Four foundational skills have been identified as core thinking skills of computational thinking: decomposition, abstraction, pattern recognition, and algorithm. We explain these four skills in this paper and their practical applications to teaching and learning in early childhood education. Early computational thinking skills are found in common early childhood activities. This paper identifies activities teachers can use in the classroom to explicitly promote children's computational thinking and provides a new perspective on how to adapt classroom activities to integrate computational thinking. In particular, we emphasize the need to vary the demands of the content in the activity and incorporate computational thinking based on children's needs and development to ensure that children progress through the thinking process. [ABSTRACT FROM AUTHOR]