Showing papers in "Journal of The Korean Mathematical Society in 2021"
TL;DR: In this paper, the Peterson hit problem was studied in the context of finding the minimal set of generators for a polynomial algebra over the Steenrod algebra, which is equivalent to determining a base for the space of hits.
Abstract: Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the polynomial algebra $\mathbb Z/2[x_1, x_2, \ldots, x_d]$ as a connected unstable $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ It is equivalent to determining a $\mathbb Z/2$-basis for the space of "cohits" $Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d.$ This $Q\mathscr P_d$ is also a representation of $GL_d$ over $\mathbb Z/2.$ The problem for $d= 5$ is not yet completely solved, and unknown in general. In this work, we give an explicit solution to the hit problem of five variables in the generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5,\ s =8$ and $t$ an arbitrary non-negative integer. An application of this study to the cases $t = 0$ and $t = 1$ shows that the Singer algebraic transfer of rank $5$ is an isomorphism in the bidegrees $(5, 5+(13.2^{0} - 5))$ and $(5, 5+(13.2^{1} - 5)).$ Moreover, the result when $t\geq 2$ was also discussed. Here, the Singer transfer of rank $d$ is a $\mathbb Z/2$-algebra homomorphism from $GL_d$-coinvariants of certain subspaces of $Q\mathscr P_d$ to the cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A_2}^{d, d+*}(\mathbb Z/2, \mathbb Z/2).$ It is one of the useful tools for studying mysterious Ext groups and the Kervaire invariant one problem.
11 citations
TL;DR: In this paper, it was shown that if a Kenmotsu metric satisfies Lg(λ) = 0 on a (2n+ 1)-dimensional manifold M 2n+1, then either ξλ = −λ or M 2 n+1 is Einstein.
Abstract: The present paper deals with the study of Fischer-Marsden conjecture on a Kenmotsu manifold. It is proved that if a Kenmotsu metric satisfies Lg(λ) = 0 on a (2n+ 1)-dimensional Kenmotsu manifold M2n+1, then either ξλ = −λ or M2n+1 is Einstein. If n = 1, M3 is locally isometric to the hyperbolic space H3(−1).
6 citations
TL;DR: In this paper, it was shown that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blowdown and symplectic antiflips.
Abstract: We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic varieties.
3 citations
3 citations
TL;DR: In this paper, the equivalence between different notions of unique range sets, namely, unique range set, weighted unique set, and weak-weighted unique set under certain conditions is shown.
Abstract: In this paper, we exhibit the equivalence between different notions of unique range sets, namely, unique range sets, weighted unique range sets and weak-weighted unique range sets under certain conditions.\par Also, we present some uniqueness theorems which show how two meromorphic functions are uniquely determined by their two finite shared sets. Moreover, in the last section, we make some observations that help us to construct other new classes of unique range sets.
2 citations
TL;DR: In this article, the strong law of large numbers and complete convergence for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables, by using the Marcinkiewicz-Zygmund type moment inequality and Roenthal type moment inequalities for AANA random variables.
Abstract: In this paper, we main study the strong law of large numbers and complete convergence for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables, by using the Marcinkiewicz-Zygmund type moment inequality and Roenthal type moment inequality for AANA random variables. As an application, the complete consistency for the weighted linear estimator of nonparametric regression models based on AANA errors is obtained. Finally, some numerical simulations are carried out to verify the validity of our theoretical result.
2 citations
TL;DR: In this paper, the authors define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurface in (C √ n)n and prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes.
Abstract: First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in (C∗)n. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin’s decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.
2 citations
TL;DR: In this paper, the authors prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces, and they also classify real hyperspaces with Killing with respect to the k-th generalized Tanaka-Webster connection.
Abstract: We prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces. We also classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Killing with respect to the k-th generalized Tanaka-Webster connection.
2 citations
TL;DR: In this paper, the Klein-Gordon-Maxwell equations arising from a semilocal gauge field model were studied, and it was shown that there exist infinitely many standing wave solutions for p ∈ (2, 6) which are radially symmetric.
Abstract: In this article, we study the Klein-Gordon-Maxwell equations arising from a semilocal gauge field model. This model describes the interaction of two complex scalar fields and one gauge field, and generalizes the classical Klein-Gordon equation coupled with the Maxwell electrodynamics. We prove that there exist infinitely many standing wave solutions for p ∈ (2, 6) which are radially symmetric. Here, p comes from the exponent of the potential of scalar fields. We also prove the nonexistence of nontrivial solutions for the critical case p = 6.
2 citations
2 citations
TL;DR: In this article, the authors introduced the notion of strong Galois H-progenerator object for a finite cocommutative Hopf quasigroup H in a symmetric monoidal category C.
Abstract: In this paper we introduce the notion of strong Galois Hprogenerator object for a finite cocommutative Hopf quasigroup H in a symmetric monoidal category C. We prove that the set of isomorphism classes of strong Galois H-progenerator objects is a subgroup of the group of strong Galois H-objects introduced in [3]. Moreover, we show that strong Galois H-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if H is finite, we find exact sequences of Picard groups related with invertible left H-(quasi)modules and an isomorphism Pic(HMod) ∼= Pic(C)⊕G(H∗) where Pic(HMod) is the Picard group of the category of left H-modules, Pic(C) the Picard group of C, and G(H∗) the group of group-like morphisms of the dual of H.
TL;DR: In this article, the authors analyze whether the completion map of a ring homomorphism is Ohm-Rush and show that the answer is typically "yes" in dimension one, but "no" in higher dimension, and in any case it coincides with the content map having good algebraic properties.
Abstract: One says that a ring homomorphism $R \rightarrow S$ is Ohm-Rush if extension commutes with arbitrary intersection of ideals, or equivalently if for any element $f\in S$, there is a unique smallest ideal of $R$ whose extension to $S$ contains $f$, called the content of $f$. For Noetherian local rings, we analyze whether the completion map is Ohm-Rush. We show that the answer is typically `yes' in dimension one, but `no' in higher dimension, and in any case it coincides with the content map having good algebraic properties. We then analyze the question of when the Ohm-Rush property globalizes in faithfully flat modules and algebras over a 1-dimensional Noetherian domain, culminating both in a positive result and a counterexample. Finally, we introduce a notion that we show is strictly between the Ohm-Rush property and the weak content algebra property.
TL;DR: The MacWilliams identity for complete weight enumerators of ZpZp[u]/〈u〉-cyclic codes is established and the generator polynomials and minimal spanning sets of these codes in both the cases are determined.
Abstract: In this paper we study the algebraic structure of ZpZp[u]/ 〈uk〉-cyclic codes, where uk = 0 and p is a prime. A ZpZp[u]/〈u〉-linear code of length (r + s) is an Rk-submodule of Zp × Rs k with respect to a suitable scalar multiplication, where Rk = Zp[u]/〈u〉. Such a code can also be viewed as an Rk-submodule of Zp[x]/〈x−1〉×Rk[x]/〈x−1〉. A new Gray map has been defined on Zp[u]/〈u〉. We have considered two cases for studying the algebraic structure of ZpZp[u]/〈u〉-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) 6= 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ZpZp[u]/〈u〉-linear codes. Examples have been given to construct ZpZp[u]/〈u〉-cyclic codes, through which we get codes over Zp using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.
TL;DR: In this paper, the existence of ground state sign-changing solutions for the following Schrödinger-Poisson-Kirchhoff type system was studied. But the ground state solutions were not considered.
Abstract: In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following SchrödingerPoisson-Kirchhoff system − (1+b ∫ R3 |∇u|dx)4u+V (x)u+k(x)φu=λf(x)u+|u|u, in R, −4φ=k(x)u, in R, where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ1 and λ1 is the first eigenvalue of the problem −4u + V (x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ1. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.
TL;DR: In this article, a set of sufficient conditions for the normalized form of the Mittag-Leffler and Wright functions to have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit disk are presented.
Abstract: The main focus of the present paper is to present new set of sufficient conditions so that the normalized form of the Mittag-Leffler and Wright functions have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit disk. Interesting consequences and examples are derived to support that these results are better than the existing ones and improve several results available in the literature.
TL;DR: In this paper, the notions of expansiveness, shadowing property and topological stability for group actions on metric spaces were introduced and a version of Walters's stability theorem was given for groups actions on locally compact metric spaces.
Abstract: In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for group actions on metric spaces and give a version of Walters’s stability theorem for group actions on locally compact metric spaces. Moreover, we show that if G is a finitely generated virtually nilpotent group and there exists g ∈ G such that if Tg is expansive and has the shadowing property, then T is topologically stable.
TL;DR: In this article, two general iterative algorithms (one implicit algorithm and one explicit algorithm) were introduced for finding a common element of the solution set of the variational inequality problems for a continuous monotone mapping, the zero point set of a set-valued maximal Monotone operator, and the fixed point sets of a continuous pseudocontractive mapping in a Hilbert space.
Abstract: In this paper, we introduce two general iterative algorithms (one implicit algorithm and one explicit algorithm) for finding a common element of the solution set of the variational inequality problems for a continuous monotone mapping, the zero point set of a set-valued maximal monotone operator, and the fixed point set of a continuous pseudocontractive mapping in a Hilbert space. Then we establish strong convergence of the proposed iterative algorithms to a common point of three sets, which is a solution of a certain variational inequality. Further, we find the minimum-norm element in common set of three sets.
TL;DR: For complete manifolds with α-Bach tensor (which is defined by (1.2) flat, the authors provided some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature.
Abstract: For complete manifolds with α-Bach tensor (which is defined by (1.2)) flat, we provide some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moveover, some Einstein metrics have also been characterized by some L n 2 -integral inequalities. Furthermore, we also give some rigidity characterizations for constant sectional curvature.
TL;DR: This paper studies the parameter estimation problem for damped shallow arches and presents the Parameter Estimation Algorithm that identifies the unknown parameters sequentially, showing numerically that the algorithm achieves a successful parameter estimation for models defined by arbitrary parameters, including the critical ones.
Abstract: Design and maintenance of large span roof structures require an analysis of their static and dynamic behavior depending on the physical parameters defining the structures. Therefore, it is highly desirable to estimate the parameters from observations of the system. In this paper we study the parameter estimation problem for damped shallow arches. We discuss both symmetric and non-symmetric shapes and loads, and provide theoretical and numerical studies of the model behavior. Our study of the behavior of such structures shows that it is greatly affected by the existence of critical parameters. A small change in such parameters causes a significant change in the model behavior. The presence of the critical parameters makes it challenging to obtain good estimation. We overcome this difficulty by presenting the Parameter Estimation Algorithm that identifies the unknown parameters sequentially. It is shown numerically that the algorithm achieves a successful parameter estimation for models defined by arbitrary parameters, including the critical ones.