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For K / F a finite Galois extension of number fields, the relative Pólya group Po (K / F) is the subgroup of the ideal class group of K generated by all the strongly ambiguous ideal classes in K / F. The notion of Ostrowski quotient Ost (K / F) , as the cokernel of the capitulation map into Po (K / F) , has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define Po (K / F) and Ost (K / F) which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For E an elliptic curve defined over F , we define the Ostrowski quotient Ost (E , K / F) and the coarse Ostrowski quotient Ost c (E , K / F) of E relative to K / F , for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group Ost c (E , K / F) and we analyze this theorem, in some details, for the class of curves E over quadratic extensions K / F. [ABSTRACT FROM AUTHOR]

In this paper, we analyze nonconforming virtual element methods on polytopal meshes with small faces for the second-order elliptic problem. We propose new stability forms for 2D and 3D nonconforming virtual element methods. For the 2D case, the stability form is defined by the sum of an inner product of approximate tangential derivatives and a weighed L 2 -inner product of certain projections on the mesh element boundaries. For the 3D case, the stability form is defined by a weighted L 2 -inner product on the mesh element boundaries. We prove the optimal convergence of the nonconforming virtual element methods equipped with such stability forms. Finally, several numerical experiments are presented to verify our analysis and compare the performance of the proposed stability forms with the standard stability form [B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016) 879–904]. [ABSTRACT FROM AUTHOR]

The Alperin weight conjecture is central to the modern representation theory of finite groups, and it is still open, despite many different approaches from different points of view. This paper surveys methods and results relating the Alperin weight conjecture, especially the recent developments in the inductive investigation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron Δ-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron Δ-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate various properties (e.g. nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear tri-harmonic equations in ℝn (n ≥ 2) and bi-harmonic equations in ℝ² with negative exponents. Such kind of equations arise from conformal geometry. [ABSTRACT FROM AUTHOR]

In this paper, using the q -Dunkl harmonic analysis introduced in Bettaibi and Bettaieb [q-Analog of the Dunkl transform on the real line, Tamsui Oxf. J. Math. Sci. 25(2) (2007) 117–205] and motivated by Wong's approach [M. W. Wong, Wavelet Transforms and Localization Operators, Vol. 136 (Springer Science & Business Media, Berlin, 2002)], we first define and study the q -wavelets, the continuous q -wavelet transforms. Next, we introduce the two-wavelet localization operators, the Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented. [ABSTRACT FROM AUTHOR]

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call Σ -Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Σ -Prikry. We showed that given a Σ -Prikry poset ℙ and a ℙ -name for a non-reflecting stationary set T , there exists a corresponding Σ -Prikry poset that projects to ℙ and kills the stationarity of T. In this paper, we develop a general scheme for iterating Σ -Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3) (2019) 479–500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer–Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs. [ABSTRACT FROM AUTHOR]

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. In this paper, we exhibit families of curly knots and links. [ABSTRACT FROM AUTHOR]

The way an author or a group of authors are cited tells more about the real impact of their work than authorship and collaborations. Indeed, the connections within the scientific community can be more accurately elicited from the co-citation graph than from the collaboration graph. We suggest some indices that can be drawn from the co-citation graph in order to capture the relevance of individual authors and the relevance of groups of authors. [ABSTRACT FROM AUTHOR]

The paper deals with rank, trace, eigenvalues and norms of the matrix C x = (x i x j) i , j = 1 n , where x i are ith components of any real sequence (x n). A result in this paper is that the Euclidean and spectral norms of the matrix C x is ∥ C x ∥ E = ∥ C x ∥ s = ∑ i = 1 n x i 2 . This is a generalization of the main result by Solak [Appl. Math. Comput.232 (2014) 919–921], with the proof based on a simple property of norms of real matrices. [ABSTRACT FROM AUTHOR]

A Carmichael number is a composite number such that divides for all integers coprime to . Korselt discovered that is a Carmichael number if and only if is square-free and for each prime divisor of . Let , a -number is defined to be a composite number , such that and for each prime . The set of all such that is a -number is called the Korselt set of and we denote this set by . In this paper, we investigate some properties of when is a power of a prime. [ABSTRACT FROM AUTHOR]

In this paper, we study the Gromov–Witten theory of the Hilbert scheme X [ 2 ] of two points on an elliptic surface X. Assume that | K X | contains an element supported on the smooth fibers of X. By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the 1 -point genus- 0 Gromov–Witten invariant 〈 w 〉 0 , d (β f − 2 β 2) X [ 2 ] up to some rational number m (d , X) depending only on d and X , where w ∈ H 4 (X [ 2 ] , ℂ) , d ≥ 1 , f is a smooth fiber of X , β f = x 0 + f ⊂ X [ 2 ] with x 0 ∈ X − f being a fixed point, and β 2 = { ξ ∈ X [ 2 ] | Supp (ξ) = { x 0 } }. Moreover, we propose a conjecture regarding m (d , X) , and prove that the conjecture is true for X = C × E where E is an elliptic curve and C is a smooth curve. [ABSTRACT FROM AUTHOR]

With the rapid change in the society, education must evolve to adapt to the digital native students of the 21st century. A change in the current educational paradigm is thus necessary. Information and Communication Technology (ICT) is a vehicle to achieve this objective of paradigm shift in education [2]. In this paper, we propose gamification approach [3] via a virtual reality (VR) through a report of a didactic experience. A mixed method research design was adopted for this study in which students' classification of polyhedral will be reported for the two different learning experiences using either wooden polyhedral or a VR tool within an immersive learning environment. The VR environment allows the learning of geometry in three-dimensional with a friendly interface that offers a multitude of options. It is worth highlighting the benefits of this VR tool in the teaching-learning process of geometry, making explicit the fundamental role played by ICT in meaningful learning of mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the boundedness of solutions for a class of fractional elliptic equations driven by the relativistic Schrôdinger operator (--Δ + I)s, with s ∈ (0, 1). The proof relies on a distributional Kato's inequality for (--Δ + I)s and on some properties of the Bessel kernel. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional p -Laplacian: (− Δ) p α u = f (x , u , ∇ u) , u > 0 in  Ω , u ≡ 0 in  ℝ n ∖ Ω , where Ω is a bounded or an unbounded domain which is convex in x 1 -direction, and (− Δ) p α is the fractional p -Laplacian operator defined by (− Δ) p α u (x) = C n , α , p P. V. ∫ ℝ n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + α p d y. Under some mild assumptions on the nonlinearity f (x , u , ∇ u) , we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional p -Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math.  335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math.  19(6) (2017) 1750018]. [ABSTRACT FROM AUTHOR]

In this paper, the idea of the Ricci flow is introduced and its significance and importance to related problems in mathematics had been discussed. Several functionals are defined and their behavior is studied under Ricci flow. A unique minimizer is shown to exist for one of the functionals. This functional evaluated at the minimizer is strictly increasing. The results for the first functional considered are extended to manifold with boundary. Finally, two physically motivated examples are presented. [ABSTRACT FROM AUTHOR]

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximum number of limit cycles for the unfolding of codimension-3 planar singularities with nilpotent linear parts. In [J. Math. Anal. Appl.499 (2017)], the authors proved that when parameter a ∈ (− 1 , 1) \ { 0 , − 2 / 3 } is rational, the corresponding problem could be transformed to solving semi-algebraic systems. At the same time, it is pointed out that when a = − 2 / 3 , the logarithmic function will appear according to the method, which makes it impossible to solve the problem. In this paper, we use some techniques to avoid the occurrence of logarithmic function, and get the corresponding system to produce at most two limit cycles. [ABSTRACT FROM AUTHOR]

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys.67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal.47 (2015) 4672–4684], where the initial condition 𝜃 0 ∈ W 3 + μ , 1 (μ > 0) plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like H s . As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 , Ω be a bounded (semi-)convex domain in ℝ n and the non-negative potential V belong to the reverse Hölder class RH n (ℝ n). Assume that p ∈ (1 , ∞) and ω ∈ A p (∂ Ω) , where A p (∂ Ω) denotes the Muckenhoupt weight class on ∂ Ω , the boundary of Ω. In this paper, the authors show that, for any p ∈ (1 , ∞) , the Neumann problem for the Schrödinger equation − Δ u + V u = 0 in Ω with boundary data in (weighted) L p is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J.43 (1994) 143–176] and Tao and Wang [Canad. J. Math.56 (2004) 655–672], via extending the range p ∈ (1 , 2 ] of p into p ∈ (1 , ∞). [ABSTRACT FROM AUTHOR]

This paper defines a new operation through extending the idea of the 0-dimensional crossing change and Shimizu's 2-dimensional region crossing change [A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Jpn. 66(3) (2014) 693–708, doi:10.2969/jmsj/06630693] to a 1-dimensional version called the arc crossing change. We will also prove that the arc crossing change is an unknotting operation with the help of Gauss diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we apply the preplay negotiation procedure proposed by Kalai [1981] [Preplay negotiations and the prisoner's dilemma, Math. Soc. Sci. 1, 375–379] to the n -person prisoners' dilemma with n ≥ 3 and examine whether it promotes cooperation. First, we demonstrate that every perfect equilibrium is a proper equilibrium for any extensive-form game in which players have only two alternatives in each information set. We show that if the preplay is carried out once, the preplay game has a proper equilibrium that realizes cooperation. We also show that if the preplay is executed twice, the game has a proper equilibrium that realizes cooperation regardless of the entering action profile (the starting point of the preplays). Finally, we demonstrate that the preplay game consisting of at least two preplays has a proper equilibrium that realizes cooperation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of commutative rings which is a generalization of ZD -rings and rings with Noetherian spectrum. A ring R is called strongly J -Noetherian whenever the ring R r is J -Noetherian for every non-nilpotent r ∈ R. We give some characterizations for strongly J -Noetherian rings and, among the other results, we show that if R [ [ x ] ] is strongly J -Noetherian, then R has Noetherian spectrum, which is a generalization of Theorem 2 in Gilmer and Heinzer [The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc.79 (1980) 13–16]. [ABSTRACT FROM AUTHOR]

Patrick Dehornoy's mathematical legacy is impressive and it has been celebrated in other papers of this volume. However, I think that there is another important aspect of Patrick's work that should be stressed, which is his involvement in scientific and managerial leadership in Mathematics. This text wants therefore to be a very personal memory of Patrick Dehornoy and of the impact that he had on my perspective on the relevance and importance of commitments to our community. [ABSTRACT FROM AUTHOR]

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]

The Laplace transform order of reversed residual life, introduced by [I. Elbatal, The Laplace order and ordering of reversed residual life, Appl. Math. Sci.36 (2007) 1773–1788.] is used in the stochastic comparison of two life distributions. In this paper, we aim to study new notions of stochastic comparisons based on bivariate Laplace transform order of reversed residual lives. We investigate relationships the new stochastic order has with other existing bivariate orders. The interpretation of the new orders and their applications in different contexts are also pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we study f-biharmonic Riemannian submersion, bi-f-harmonic Riemannian submersion and thus generalizing some results of [M. A. Akyol and Y.-L. Ou, Biharmonic Riemannian submersions, Ann. Math. Pura Appl.198 (2019) 559–570; Z. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z.269(3) (2011) 917–925]. Next, we obtain the conditions when the Riemannian submersion on its first factor is f-biharmonic as well as bi-f-harmonic. In the last section, we study f-biharmonic cylinders of a Riemannian submersion. [ABSTRACT FROM AUTHOR]

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 2 3 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not.1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of 2 3 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1). [ABSTRACT FROM AUTHOR]

The total irregularity index of a graph G is defined by Abdo et al. [H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014) 201–206] as irr t (G) = 1 2 ∑ u , v ∈ V (G) | deg G (u) − deg G (v) | , where deg G (u) denotes the degree of a vertex u ∈ V. In 2014, You et al. [L. H. You, J. S. Yang and Z. F. You, The maximal total irregularity of unicyclic graphs, Ars Comb. 114 (2014) 153–160.] characterized the graph having maximum i r r t value among all elements of the class U n (Unicyclic graphs) and Zhou et al. [L. H. You, J. S. Yang, Y. X. Zhu and Z. F. You, The maximal total irregularity of bicyclic graphs, J. Appl. Math. 2014 (2014) 785084, http://dx.doi.org/10.1155/2014/785084] characterized the graph having maximum irr t value among all elements of the class B n (Bicyclic graphs). In this paper, we characterize the aforementioned graphs with an alternative but comparatively simple approach. Also, we characterized the graphs having maximum irr t value among the classes T n (Tricyclic graphs), TET n (Tetracyclic graphs), PNT n (Pentacyclic graphs) and HEX n (Hexacyclic graphs). [ABSTRACT FROM AUTHOR]

We calculate the probability when a finite commutative ring is weakly nil-clean in terms of invariants associated only with the given whole ring. This continues the study of our recent paper concerned with the nil-clean case [P. Danchev and M. Samiei, The probability when a finite commutative ring is nil-clean, Trans. A. Razmadze Math. Inst. 176 (2022)]. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) and a set S ⊆ V (G) of size at least 2 , a path in G is said to be an S -path if it connects all vertices of S. Two S -paths P 1 and P 2 are said to be internally disjoint if E (P 1) ∩ E (P 2) = ∅ and V (P 1) ∩ V (P 2) = S. Let π G (S) denote the maximum number of internally disjoint S -paths in G. The k -path-connectivity π k (G) of G is then defined as the minimum π G (S) , where S ranges over all k -subsets of V (G). In [M. Hager, Path-connectivity in graphs, Discrete Math. 59 (1986) 53–59], the k -path-connectivity of the complete bipartite graph K a , b was calculated, where k ≥ 2. But, from his proof, only the case that 2 ≤ k ≤ min { a , b } was considered. In this paper, we calculate the situation that min { a , b } + 1 ≤ k ≤ a + b and complete the result. [ABSTRACT FROM AUTHOR]

It is well known that injective objects play a fundamental role in many branches of mathematics. The question whether a given category has enough injective objects has been investigated for many categories. Also, quasi-injective modules and acts have been studied by many categorists. In this paper, we study quasi-injectivity in the category of actions of an ordered monoid on ordered sets (P o s - S) with respect to embeddings. Also, we give the relation between injectivity, quasi-injectivity (with respect to embeddings), and poset completeness in the category P o s - S and some of its important subcategories. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of r circle diffeomorphisms with d breaks, r > 2 , with given size of breaks, converge to an invariant family of piecewise Möbius maps, of dimension 2 d. We prove that this invariant family identifies with a relative character variety χ (π 1 Σ , PSL (2 , ℝ) , h) where Σ is a d -holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group MCG (Σ). This action allows us to introduce the symplectic form which is preserved by renormalization. The invariant symplectic form is related to the symplectic form described by Guruprasad et al. [Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J.89(2) (1997) 377–412], and goes back to the earlier work by Goldman [The symplectic nature of fundamental groups of surfaces, Adv. Math.54(2) (1984) 200–225]. To the best of our knowledge the connection between renormalization in the nonlinear setting and symplectic dynamics had not been brought to light yet. [ABSTRACT FROM AUTHOR]

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the p -Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the p -Laplacian operator on closed orientate n -dimensional Lagrangian submanifolds in a complex space form n (4) with constant holomorphic sectional curvature 4 . As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525–533] to the p -Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for = 0 and = 1 , respectively. [ABSTRACT FROM AUTHOR]

In a recent paper, Sun posed six conjectures on the relations between T (a 1 , a 2 , ... , a k ; n) and N (a 1 , a 2 , ... , a k ; n) , where T (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 (x 1 + 1) 2 + a 2 x 2 (x 2 + 1) 2 + ⋯ + a k x k (x k + 1) 2 , where a 1 , a 2 , ... , a k are positive integers, n , x 1 , x 2 , ... , x k are arbitrary nonnegative integers, and N (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 2 + a 2 x 2 2 + ⋯ + a k x k 2 , where this time x 1 , x 2 , ... , x k are integers. In this paper, we prove Sun's six conjectures by using Ramanujan's theta function identities. [ABSTRACT FROM AUTHOR]

The classical Menon's identity [P. K. Menon, On the sum ∑ (a − 1 , n) [ (a , n) = 1 ] , J. Indian Math. Soc. (N.S.) 29 (1965) 155–163] states that ∑ a ∈ ℤ n ∗ gcd (a − 1 , n) = φ (n) σ 0 (n) , where for a positive integer n , ℤ n ∗ is the group of units of the ring ℤ n = ℤ / n ℤ , gcd (,) represents the greatest common divisor, φ (n) is the Euler's totient function and σ k (n) = ∑ d | n d k is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: ∑ a ∈ ℤ n ∗ b 1 , ... , b k ∈ ℤ n gcd (a − 1 , b 1 , ... , b k , n) χ (a) = φ (n) σ k n d , where k is a non-negative integer and χ is a Dirichlet character modulo n whose conductor is d. Our result can be viewed as an extension of Zhao and Cao's result [Another generalization of Menon's identity, Int. J. Number Theory13(9) (2017) 2373–2379] to k > 0. It can also be viewed as an extension of Sury's result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo58 (2009) 99–108] to Dirichlet characters. [ABSTRACT FROM AUTHOR]

This is a reply to Ramon Ferrer-I-Cancho's paper in this issue "Some Word Order Biases from Limited Brain Resources: A Mathematical Approach." In this reply, I challenge the Euclidean distance model proposed in that paper by proposing a simple alternative model based on linear ordering. [ABSTRACT FROM AUTHOR]

Given a spectral triple on a C ∗ -algebra together with a unital injective endomorphism α , the problem of defining a suitable crossed product C ∗ -algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of α () in can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on and α (). [ABSTRACT FROM AUTHOR]

Various ample divisors on smooth del Pezzo surfaces of degree 2 have been verified to allow polar cylinders in [I. Cheltsov, J. Park and J. Won, Cylinders in del Pezzo surfaces, Int. Math. Res. Not.2017(4) (2015) 1179–1230]. We show that affine cones over smooth del Pezzo surfaces of degree 2 polarized by such ample divisors are flexible in codimension one. All varieties in this paper are assumed to be defined over an algebraically closed field of characteristic zero. [ABSTRACT FROM AUTHOR]

The g -extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464–473] determined the 3 -extra edge-connectivity of balanced hypercubes B H n and conjectured that the g -extra edge-connectivity of B H n is λ g ( B H n) = 2 (g + 1) n − 4 g + 4 for 2 ≤ g ≤ 2 n − 1. In this paper, we confirm their conjecture for n ≥ 6 − 1 2 g + 1 and 2 ≤ g ≤ 8 , and disprove their conjecture for n ≥ 3 e g ( B H n) g + 1 and 9 ≤ g ≤ 2 n − 1 , where e g ( B H n) = max { | E ( B H n [ U ]) | | U ⊆ V (B H n) , | U | = g + 1 }. [ABSTRACT FROM AUTHOR]

A many-to-many k -disjoint path cover (k -DPC) of a graph G is a set of k vertex-disjoint paths joining k distinct pairs of source and sink in which each vertex of G is contained exactly once in a path. The balanced hypercube B H n , a variant of the hypercube, was introduced as a desired interconnection network topology. Let S = { s 1 , s 2 , ... , s 2 n − 2 } and T = { t 1 , t 2 , ... , t 2 n − 2 } be any two sets of vertices in different partite sets of B H n (n ≥ 2). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of B H n when | S | = | T | = 2. In this paper, we prove that there exists unpaired many-to-many (2 n − 2) -disjoint path cover of B H n (n ≥ 2) from S to T , which has improved some known results. The upper bound 2 n − 2 is best possible in terms of the number of disjoint paths in unpaired many-to-many k -DPC of B H n . [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the simplest first-order Hamilton–Jacobi equation in one space dimension, with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. Uniqueness of discontinuous viscosity solutions is proven, if the initial data function has a finite number of jump discontinuities. Main ingredients of the proof are the barrier effect of spatial discontinuities of a solution (which is linked to the boundedness of the Hamiltonian), and a comparison theorem for semicontinuous viscosity subsolution and supersolution. These are defined in the spirit of the paper [H. Ishii, Perron's method for Hamilton–Jacobi equations, Duke Math. J.55 (1987) 368–384], yet using essential limits to introduce semicontinuous envelopes. The definition is shown to be compatible with Perron's method for existence and is crucial in the uniqueness proof. We also describe some properties of the time evolution of spatial jump discontinuities of the solution, and obtain several results about singular Neumann problems which arise in connection with the above referred barrier effect. [ABSTRACT FROM AUTHOR]

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic Λ -metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic Λ -metric spaces, where Λ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case Λ = ℤ n taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ℤ n -metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy : (AMS- 2 0 9) , Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ℤ n -free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ℤ n , Trans. Amer. Math. Soc. 364 (2012) 2847–2882]. [ABSTRACT FROM AUTHOR]

The abstract should summarize the context, content and conclusions of the paper in less than 200 words. It should not contain any references or displayed equations. Typeset the abstract in 8 pt roman with baseline skip of 10 pt, making an indentation of 1.5 pica on the left and right margins. [ABSTRACT FROM AUTHOR]

The ancient Vedic mathematics is well known for quicker handy multiplications but its recognition as an integrated circuit core against existing hardware multipliers is not established. As optimized hardware implementation of binary multiplier is one of the prominent unsolved problems in computer architecture, this paper proposes efficient Urdhava Tiryakbhyam Vedic multiplier architecture and compares it with the set of hierarchical multiplication algorithms which generate multiplication result in a single clock cycle. Two innovative algorithms are proposed here, one with a compact structure and another for faster execution. Also, its optimized transistor level layout is designed and implemented. To maintain homogeneity for comparison, all the algorithms are programmed on a common HDL language platform and analyzed with the same tool and technology. Final results indicate that the proposed architecture delivers 15.5% less power delay product (PDP) compared to closest competitor algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct N -fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when N = 2. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer N which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a k -generalized Fibonacci numbers (F n (k)) n ∈ ℕ ∗ . Lastly, some applications to cryptography are given by using (F n (k)) n ∈ ℕ ∗ . [ABSTRACT FROM AUTHOR]

Dynamic (or stochastic) games are, in general, considerably more complicated to analyze than repeated games. This paper shows that for every deterministic dynamic game that is linear in the state, there exists a strategically equivalent representation as a repeated game. A dynamic game is said to be linear in the state if it holds for both the state transition function as well as for the one-period payoff function that (i) they are additively separable in action profiles and states and (ii) the state variables enter linearly. Strategic equivalence refers to the observation that the two sets of subgame perfect equilibria coincide, up to a natural projection of dynamic game strategy profiles on the much smaller set of repeated game histories. Furthermore, it is shown that the strategic equivalence result still holds for certain stochastic elements in the transition function if one allows for additional signals in the repeated game or in the presence of a public correlating device. [ABSTRACT FROM AUTHOR]

An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to 1 is an eigenform for suitable weights on the cells. [ABSTRACT FROM AUTHOR]