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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 this paper, for d ≥ 1 and s ∈ (0 , d 2) , we study the Bianchi–Egnell quotient Q (f) = inf f ∈ H ˙ s (R d) \ B ‖ (- Δ) s / 2 f ‖ L 2 (R d) 2 - S d , s ‖ f ‖ L 2 d d - 2 s (R d) 2 dist H ˙ s (R d) (f , B) 2 , f ∈ H ˙ s (R d) \ B , where S d , s is the best Sobolev constant and B is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when d = 1 , there is a neighborhood of B on which the quotient Q (f) is larger than the lowest value attainable by sequences converging to B . This behavior is surprising because it is contrary to the situation in dimension d ≥ 2 described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for d = 1 , Q (f) has no minimizer on H ˙ s (R d) \ B , which again would be contrary to the situation in d ≥ 2 . As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every d ≥ 1 . For d ≥ 2 , this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For d = 1 we make some numerical observations which support the conjecture stated above. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space (Xia and Wang in Math Ann 385(1–2):863–879, 2023), and Szegö–Weinberger inequalities in rank-1 symmetric spaces obtained by Aithal and Santhanam (Trans Am Math Soc 348(10):3955–3965, 1996). [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, 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]

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, we establish some reversed dynamic inequalities of Hilbert type on time scales nabla calculus by applying reversed Hölder's inequality, chain rule on time scales, and the mean inequality. As particular cases of our results (when T = N and T = R ), we get the reversed form of discrete and continuous inequalities proved by Chang-Jian, Lian-Ying and Cheung (Math. Slovaca 61(1):15–28, 2011). [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 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]

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]

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]