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The well-known notion of an extending module is closely linked to that of a Baer module. A right R -module M is called extending if every submodule of M is essential in a direct summand. On the other hand, a right R -module M is called Baer if for all N ≤ M , l S (N) ≤ ⊕ S S where S = End R (M). In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc.  21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: "a module M is -nonsingular extending if and only if M is -cononsingular Baer". M R is called -nonsingular if ∀ φ ∈ S such that Ker φ ≤ e M , φ = 0. Moreover, M R is called -cononsingular if for any N ≤ M with φ N ≠ 0 for all 0 ≠ φ ∈ S , implies N ≤ e M. In view of this result, every Baer module which happens to be -cononsingular will automatically become an extending module. In this paper, our main focus is the study of -cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of -cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of -cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman's result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an L 2 -almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)]. [ABSTRACT FROM AUTHOR]

In this paper, we study a discrete momentum consensus-based optimization (Momentum-CBO) algorithm which corresponds to a second-order generalization of the discrete first-order CBO [S.-Y. Ha, S. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci. 30 (2020) 2417–2444]. The proposed algorithm can be understood as the modification of ADAM-CBO, replacing the normalization term by unity. For the proposed Momentum-CBO, we provide a sufficient framework which guarantees the convergence of algorithm toward a global minimum of the objective function. Moreover, we present several experimental results showing that Momentum-CBO has an improved success rate of finding the global minimum compared to vanilla-CBO and show the stability of Momentum-CBO under different initialization schemes. We also show that Momentum-CBO can be used as the alternative of ADAM-CBO which does not have a proper convergence analysis. Finally, we give an application of Momentum-CBO for Lyapunov function approximation using symbolic regression techniques. [ABSTRACT FROM AUTHOR]

This editorial paper reviews the articles published in a special issue devoted to the application of active particle methods applied to the study of the collective dynamics of large systems of interacting entities in science and society. The applications presented in this special issue focus on the study of financial markets, cell dynamics in the context of cancer modeling, vehicle and crowd vehicle and crowd dynamics, and classical problems in the kinetic theory of active particles and swarm theory. A critical analysis is proposed to look forward to research. Perspectives with emphasis on the interaction of multiple social dynamics. [ABSTRACT FROM AUTHOR]

We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f (λ (D 2 u)) = g (x) with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli et al. [The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301], Trudinger [On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on Perron's method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure conditions as in [L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301; N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] but without requiring the concavity of f. The equations in our setting may embrace the well-known Monge–Ampère equations, Hessian equations and Hessian quotient equations as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field. [ABSTRACT FROM AUTHOR]

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]

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]

[M. Ma, The connectivity of exchanged hypercubes, Discrete Math., Algorithms Appl. 2(2) (2010) 213–220.] proved that the vertex connectivity and the edge connectivity of the exchanged hypercube are each equal to the minimum degree of the graph. This paper presents a proof of that result, which is substantially simpler and more intuitive. [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]

This is a corrigendum to the authors paper [A reconstruction theorem for complex polynomials, Int. J. Math. 26(9) (2015) 1550073]. Here we present a completely rewritten and self-contained Sec. 4 of the aforementioned paper and provide the reader with the corrected proof of the main result that briefly states: "Given a generic polynomial P of degree at least 2 , there exist positive integers N , M such that Re (P k (z)) = Re (P k (w)) for all 0 ≤ k ≤ N implies P M (z) = P M (w) ". [ABSTRACT FROM AUTHOR]

In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with Ric ≥ (n − 1) κ , κ = 0 or 1 , which extends the results in [A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math. 76(3) (2023) 585–603]. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel–Daners inequality for Robin Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

The following inequality ∫ 0 c | ψ (z) ψ ′ (z) | d z ≤ c 4 ∫ 0 c (ψ ′ (z)) 2 d z is well known in the literature as Opial's inequality [Z. Opial, Sur une inégalité, Polon. Math. 8 (1960) 29–32], it had/has been studied extensively since its appearance. This paper investigates some of its generalizations in the prospect of generalized integral operators. From huge literature on this inequality, the results for superquadratic functions are considered and further analyzed for some integral operators. Several inequalities have been deduced for fractional integral operators by analyzing particular kernels. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053]. [ABSTRACT FROM AUTHOR]

Algebraic quantum groupoids have been developed by two of the authors of this note (AVD and SHW) in a series of papers [A. Van Daele and S. Wang, Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples, Banach Center Publ. 98 (2012) 367–415; A. Van Daele and S. Wang, Weak multiplier Hopf algebras I. The main theory, J. Reine Angew. Math. 705 (2015) 155–209; A. Van Daele and S. Wang, Weak multiplier Hopf algebras II. The source and target algebras, preprint (2014), arXiv: 1403.7906v2 [math.RA]; A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], see also [A. Van Daele, Algebraic quantum groupoids. An example, preprint (2017), arXiv: 1702.04903 [math.RA]]. By an algebraic quantum groupoid, we understand a regular weak multiplier Hopf algebra with enough integrals. Regular multiplier Hopf algebroids are obtained also by two authors of this note (TT and AVD) in [T. Timmermann and A. Van Daele, Regular multiplier Hopf algebroids. Basic theory and examples, Commun. Algebra 46 (2017) 1926–1958]. Integral theory and duality for those have been studied by one author here (TT) in [T. Timmermann, Integration on algebraic quantum groupoids, Int. J. Math. 27 (2016) 1650014, arXiv: 1507.00660 [QA]; T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]. In these papers, the term algebraic quantum groupoid is used for a regular multiplier Hopf algebroid with a single faithful integral. Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. In the paper, Weak multiplier Hopf algebras III. Integrals and duality [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], one of the main results is that the dual of an algebraic quantum groupoid, admits a dual of the same type. In the paper, On duality of algebraic quantum groupoids [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]], a result of the same nature is obtained for regular multiplier Hopf algebroids with a single faithful integral. The duality of regular weak multiplier Hopf algebras with a single integral can be obtained from the duality of regular multiplier Hopf algebroids (see [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]). That is however not the obvious way to obtain this result. It is more difficult and less natural than the direct way followed in [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]]. We will discuss this statement further in the paper. Nevertheless, it is interesting to investigate the relation between the two approaches to duality in greater detail. This is what we do in this paper. We build further on the intimate relation between weak multiplier Hopf algebras and multiplier Hopf algebroids as studied in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. We now add the presence of integrals. That seems to be done best in a framework of dual pairs. It is in fact more general than the duality of these objects coming with integrals. We are convinced that the material we present in this paper will provide a deeper understanding of the duality of algebraic quantum groupoids, both within the framework of weak multiplier Hopf algebras, as well as more generally for multiplier Hopf algebroids. Finally, we feel it is also appropriate to include some historical comments on the development of these duality theories. [ABSTRACT FROM AUTHOR]

As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar N = 4 SYM amplitudes, J. High Energy Phys. 2021 (2021) 65]. [ABSTRACT FROM AUTHOR]

A question dating to Mardešić and Prasolov's 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits lim n   (n > 0) of a certain inverse system A indexed by ω ω to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all n -coherent families of functions indexed by ω ω to be trivial. In this paper, we prove that, in any forcing extension given by adjoining ℶ ω -many Cohen reals, lim n A vanishes for all n > 0. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional Δ -system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of lim n A for all n > 0. [ABSTRACT FROM AUTHOR]

For a graph G, ω (G) , χ (G) represent the clique number and the chromatic number of G, respectively. A hereditary family of graphs is called χ -bounded with χ -binding function f if χ (G) ≤ f (ω (G)) for all G ∈ . A result of Schiermeyer shows that the class of (P 5 , d a r t) -free graphs has a χ -binding function f (ω) = ω 2 [L. Esperet, L. Lemoine, F. Maffray and G. Morel, The chromatic number of (P 5 , K 4) -free graphs, Discrete Math. 313 (2013) 743–754]. In this paper, we prove that the class of (P 5 , d a r t) -free graphs has a χ -binding function f (ω) = 3 4 ω 2 . For the class of (P 5 , C 5 , d a r t) -free graphs, we give a χ -binding function f (ω) = ω + 1 2 . [ABSTRACT FROM AUTHOR]

In secure domination [A. P. Burger, M. A. Henning and J. H. Van Vuuren, Vertex covers and secure domination in graphs, Quaest Math. 31 (2008) 163–171; E. J. Cockayne, Irredundance, secure domination and maximum degree in trees, Discrete Math. 307(1) (2007) 12–17; E. J. Cockayne, O. Favaron and C. M. Mynhardt, Secure domination, weak Roman domination and forbidden subgraph, Bull. Inst. Combin. Appl. 39 (2003) 87–100; C. M. Mynhardt, H. C. Swart and E. Ungerer, Excellent trees and secure domination, Util. Math. 67 (2005) 255–267], a vertex outside has the chance of coming inside the dominating set by replacing an element of the set without affecting domination. This idea is combined with equitability. Secure equitable dominating set is introduced and studied. Other concepts like independence and rigid security are also studied in this paper. [ABSTRACT FROM AUTHOR]

In 1978, Doro, in his paper [S. Doro, Simple moufang loops, Math. Proc. Camb. Philos. Soc. 83 (1978) 377–392] published the following conjecture: If the nucleus of a Moufang loop is trivial, then the commutant is a normal subloop. By working in the multiplication group of the loop we prove that in case of finite Moufang loops with trivial nucleus, the commutant is normal if and only if it is trivial. [ABSTRACT FROM AUTHOR]

This study investigates the axisymmetric postbuckling of functionally graded graphene platelets reinforced composite (FG-GPLRC) annular plates resting on nonlinear elastic medium in thermal environment. Five kinds of graphene platelets (GPLs) distribution patterns including U -pattern, X -pattern, O -pattern, A -pattern, and V -pattern have been considered. The nonlinear equilibrium equations and associated boundary conditions are obtained based upon the Mindlin plate theory. The governing equations are solved via the generalized differential quadrature method (GDQM). Afterwards, the direct iterative method is implemented to accomplish postbuckling loads using the buckling mode deflection. In order to confirm the accuracy of the present model, comparisons between our data with those published in the available literature are put forth. Eventually, this paper emphasizes the impact of diverse parameters such as geometrical parameters of the structure, GPLs patterns and their geometric, GPLs weight fraction, boundary conditions, elastic medium's parameters and temperature change on the buckling and postbuckling response of nanocomposite annular plates. It can be found that elastic medium overshadows the applicability of distribution patterns and weight fraction of GPLs. [ABSTRACT FROM AUTHOR]

This editorial paper presents the papers published in a special issue devoted to the modeling and simulation of mutating virus pandemics in a globally connected world. The presentation is proposed in three parts. First, motivations and objectives are presented according to the idea that mathematical models should go beyond deterministic population dynamics by considering the multiscale, heterogeneous features of the complex system under consideration. Subsequently, the contents of the papers in this issue are presented referring to the aforementioned complexity features. Finally, a critical analysis of the overall contents of the issue is proposed, with the aim of providing a forward look to research perspectives. [ABSTRACT FROM AUTHOR]

Let K be a number field. The Pólya field concept is used to know when the module of integer-valued polynomials over the ring of integers K of K has a regular basis. In [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398], the author used cohomological results from [C. B. Setzer, Units over totally real C 2 × C 2 fields, J. Number Theory  12 (1980) 160–175] to answer questions raised in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71] on biquadratic Pólya fields. Here we first prove that number fields were omitted from the list of exceptional fields cited in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71]. We therefore identify new biquadratic Pólya fields, where the prime number 2 is totally ramified. This result corrects and completes some others on the literature. On the other hand, we show that the main results of [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398] and this paper can be proved with a single method using Kuroda/Kubota's unit formula without cohomological results of [C. B. Setzer, Units over totally real C 2 × C 2 fields, J. Number Theory  12 (1980) 160–175]. [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]

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]

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]

For a homogeneous space G / H of reductive type, we consider the tangential homogeneous space G / H . In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino's condition by direct calculations. In this paper, some easy-to-check necessary conditions (= obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister's theorem (the existence problem of common zero points of polynomials of odd degree). [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 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]

This paper defines the Euler–Poincaré characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [Thanh and Viet, Mixed multiplicities of maximal degrees, J. Korean Math. Soc.55(3) (2018) 605–622] seem to show a natural and nice picture of mixed multiplicities of maximal degrees. [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]

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]

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]

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: − div (A (x , ∇ u)) = μ in Ω , u = 0 on ∂ Ω , in Lorentz–Morrey spaces, where Ω ⊂ ℝ n (n ≥ 2), μ is a finite Radon measure, A is a monotone Carathéodory vector-valued function defined on W 0 1 , p (Ω) and the p -capacity uniform thickness condition is imposed on the complement of our domain Ω. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case 2 ≤ p ≤ n , where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case p > 2 − 1 n . Here in this study, we particularly restrict ourselves to the singular case 3 n − 2 2 n − 1 < p ≤ 2 − 1 n . The results are central to generalize our technique of good- λ type bounds in the previous work [M.-P. Tran, Good- λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results. [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]

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]

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]

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]

It is much more practical to use the t -norm than the minimum in defining the fuzzy graph. Hence, the concept of t -fuzzy graph was introduced by Mordeson et al. in [J. N. Mordeson and S. Mathew, t -norm Fuzzy Graphs, New Mathematics and Natural Computation14(1) (2018) 129–143]. In this paper, we concentrate on the relatively new concept of fuzzy adjacency matrix of that t -fuzzy graphs. The purpose of this paper is to further develop a product of this fuzzy adjacency matrix based on t -fuzzy graphs. Moreover, two kinds of Boolean matrix are defined and relation between Normal Boolean matrix, product of adjacency matrix and 1-level cut Boolean matrix is obtained. [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]

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]

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]