• Volume 129, Issue 5

November 2019

• Fractional boundary value problem with $\psi$-Caputo fractional derivative

This paper is concerned with a boundary value problem for a nonlinearfractional differential equation involving a general form of Caputo fractional derivative operator with respect to new function $\psi$. The existence and uniqueness results of solutions are obtained. Our analysis relies on a variety of tools of fractional calculus together withfixed point theorems of Banach and Schaefer. The investigation of the results will be illustrated by providing a suitable example.

• Actions of some simple compact Lie groups on themselves

Let $G$ be a compact connected simple Lie group acting non-transitively, non-trivially on itself. Hsiang (Cohomology theory of topological transformation groups, (1975) (New York: Springer)) conjectured that the principal isotropy subgroup type must be the maximal torus and the action must be cohomologically similar to the adjoint action and the orbit space must be a simplex. But Bredon (Bull AMS 83(4) (1977) 711–718) found a simple counterexample, where the principal isotropy subgroup is not a maximal torus and which has no fixed point. In this work, we prove that if $SO(n), (n \geq 34)$ or $SU(3)$ acts smoothly (and nontrivially) on itself with non-empty fixed point set, then the principal isotropy subgroups are maximal tori.

• Fréchet phase space for nonlinear infinite delay equations in Banach spaces

In this paper, we prove the existence of solutions to nonlinear differential equations with infinite delay in Banach spaces. We construct a Fréchet space and the unique solutions are given by semigroups in this Fréchet space. Applications to partial differential equations with infinite delay are given.

• Infinitely many solutions for the stationary fractional ${p}$-Kirchhoff problems in $\mathbb{R}^{N}$

In the present paper, we investigate the existence of multiple solutions for the nonhomogeneous fractional $p$-Kirchhoff equation $$M\left(\int\int_\mathbb {R}^{2n}\frac{\mid u(x)-u(y)\mid^p}{\mid x-y\mid^{N+ps}}{\rm d}x{\rm d}y+\int_\mathbb{R}^{N}V(x)\mid u\mid^p {\rm d}x\right)$$ $$\times((-\delta)^s_p u+V(x)\mid u\mid^{p-2}u)=f(x, u)\ {\rm in}\ \mathbb {R}^N,$$ where $(-\Delta) _{p}^{s}$ is the fractional $p$-Laplacian operator, $0$ < $s$ < 1 < $p$ < $\infty$ with $sp$ < $N$, $M:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}_{0}^{+}$ is a nonnegative, continuous and increasing Kirchhoff function, the nonlinearity $f:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ is a $\rm{Carath\acute{e}odory}$ function that obeys some conditions which will be stated later and $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ is a non-negative potential function. We first establish the Bartsch--Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the variational method, $(S_{+})$ mapping theory and Krasnoselskii's genus theory.

• A note on the exponential diophantine equation $(a^{n} − 1)(b^{n} − 1) = x^{2}$

In 2002, Luca and Walsh (J. Number Theory 96 (2002) 152–173) solved the diophantine equation for all pairs $(a, b)$ such that $2\leq a$ < $b\leq 100$ with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation $(a^{n}−1)(b^{n}−1) = x^{2}$. It is also proved that this equation has no solutions if $a, b$ have opposite parity and $n$ >$4$ with $2|n$. Here, the equation is also solved for the pairs $(a, b) = (2, 50), (4, 49), (12, 45), (13, 76), (20, 77), (28, 49), (45, 100)$. Lastly, we show that when b is even, the equation $(a^{n} − 1)(b^{2n}a^{n} − 1) = x^{2}$ has no solutions $n, x$.

• A generalization of Grothendieck’s extension Panachées

We consider an extension problem for exact sequences in an abelian category and provide a criterion under which such a solution, if it exists is unique.

• Stable range one on non-zero divisors

In this paper, almost unit 1-stable range condition for a commutative ring with unity is defined and similar results which are already proved for the weakly unit 1-stable range are proved. Several properties of rings satisfying almost unit 1-stable range are discussed. The characterizations for rings such as presimplifiable semilocal rings and idealization $R(M)$ of a ring $R$ and an $R$-module $M$, satisfying almost unit 1-stable range is also given in this paper. Few examples are given to show that almost unit 1-stable range condition is different from other stable range conditions.

• Equivariant $K$-theory of quasitoric manifolds

Let $X(Q,\Lambda)$ be a quasitoric manifold associated to a simple convex polytope $Q$ and characteristic function $\Lambda$. Let $T\cong ({S}^1)^n$ denote the compact $n$-torus acting on $X=X(Q,\Lambda)$. The main aim of this article is to give a presentation of the $T$-equivariant $K$-ring of $X$, as a Stanley--Reisner ring over $K^{*}(pt)$. We also derive the presentation for the ordinary $K$-ring of $X$.

• A least-squares-based weak Galerkin finite element method for elliptic interface problems

This paper concerns the numerical approximation of elliptic interface problems via least-squares-based weak Galerkin (WG) finite element method. This method allows the use of totally discontinuous functions on finite element partitions consisting of arbitrary polygon shape. Further, the method is capable of solving the unknown and the flux simultaneously with optimal order convergence rates.

• BLO estimates for Marcinkiewicz integrals associated with Schrödinger operators

Let $L=-\Delta+V$ be a Schrödinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ , while the nonnegative potential $V$ belongs to the reverse Hölder class $B_{q}(q\geq1)$. In this paper, we will show that Marcinkiewicz integrals associated with Schrödinger operator are bounded from ${\rm BMO}_{L}$ to ${\rm BLO}_{L}$, when $V\in B_{d}$

• Transversal intersection of monomial ideals

In this paper, we prove conditions for transversal intersection of monomial ideals and derive a simplicial characterization of this phenomenon.

• Non-comaximal graph of ideals of a ring

Let R be a ring. The non-comaximal graph of R, denoted by $NC(R)$ is an undirected graph whose vertex set is the collection of all non-trivial (left) ideals of $R$ and any two distinct vertices $I$ and $J$ are adjacent if and only if $I + J \neq R$. The concepts of connectedness, independent set, clique and traversability of $NC(R)$ are discussed.

• Asymptotic series related to Ramanujan’s expansion for the harmonic number

In this paper, we present various asymptotic series for the harmonic number $H_{n}=\sum_{k=1}^{n}\frac{1}{k}$. For example, we give a pair of recurrence relations for determining the constants $a_{\ell}$ and $b_{\ell}$ such that $$H_{n}\sim\frac{1}{2}ln\left(2m+\frac{1}{3}\right)+\gamma+\sum_{\ell=1}^{\infty}\frac{a_{\ell}}{(2m+b_{\ell})^{2\ell}}\quad\text{as}\,n\to\infty,$$ where $m =\tfrac{1}{2}n(n+1)$ ($n\in\mathbb{N}:=\{1,2,\ldots\}$) is the $n$-th triangular number and $\gamma$ is the Euler-Mascheroni constant.

• Traveling salesman problem across well-connected cities and with location-dependent edge lengths

Consider $n$ nodes $\{Xi\}_{1\leq i \leq n}$ distributed independently across $N$ cities located in the unit square $S$, each according to a certain distribution $g_{N}(\cdot)$. Each city is modelled as an $r_{n} \times r_{n}$ square and $\rm{TSPC}_{n}$ denotes the weight of the minimum weighted length cycle containing all the $n$ nodes, where the edge length between nodes $X_{i}$ and $X_{j}$ is location-dependent and based on a metric $d$ that is equivalent to the Euclidean metric. We obtain variance estimates for $\rm{TSPC}_{n}$ and prove that if the cities are well connected in a certain sense, then $\rm{TSPC}_{n}$ appropriately centred and scaled converges to zero in probability. We also obtain large deviation type estimates for $\rm{TSPC}_{n}$. Using the proof techniques, we also study results $\rm{TSP}_{n}$ of the minimum length cycle in the unconstrained case, when the nodes are independently distributed throughout the unit square $S$ with location-dependent edge lengths. We obtain variance estimates and convergence in probability for $\rm{TSP}_{n}$ appropriately scaled and centred.

• Johnson graphs are panconnected

For any given $n,m \in \mathbb{N}$ with $m$ < $n$, the Johnson graph $J(n,m)$ is defined as the graph whose vertex set is $V=\{v\mid v\subseteq [n]=\{1,\ldots,n\}, |v|=m\}$, where two vertices $v$, $w$ are adjacent if and only if $|v\cap w|=m-1$. A graph $G$ of order $n$ > $2$ is panconnected if for every two vertices $u$ and $v$, there is a $u-v$ path of length $l$ for every integer $l$ with $d(u,v) \leq l \leq n-1$. In this paper, we prove that the Johnson graph $J(n,m)$ is a panconnected graph.

• Evolution equations with fractional Gross Laplacian and Caputo time fractional derivative

In this paper, we consider the evolution equations with fractional GrossLaplacian and generalized Caputo time fractional deravitive in infinite dimensional space of entire functions with growth condition. The convolution between a generalized function related to the Mittag–Leffler function and the initial condition has been given to demonstrate the explicit solutions. Moreover, we prove that the fundamental solution is related to the inverse stable subordinators and the symmetric $\alpha$-stable distribution.

• On solutions of the diophantine equation $F_{n} − F_{m} = 3^{a}$

In this paper, we find non-negative ($n, m, a$) integer solutions of thediophantine equation $F_{n} − F_{m} = 3^{a}$, where $Fn$ and $Fm$ are Fibonacci numbers. For proving our theorem, we use lower bounds in linear forms.

• On the rational closure of connected closed subgroups of connected simply connected nilpotent Lie groups

Let $G$ be a connected simply connected nilpotent Lie group with discrete uniform subgroup $\Gamma$. A connected closed subgroup $H$ of $G$ is called $\Gamma$-rational if $H\cap\Gamma$ is a discrete uniform subgroup of $H$. For a closed connected subgroup $H$ of $G$, let ${\frak I}(H, \Gamma)$ denote the identity component of the closure of the subgroup generated by $H$ and $\Gamma$. In this paper, we prove that ${\frak I}(H, \Gamma)$ is the smallest normal $\Gamma$-rational connected closed subgroup containing $H$. As an immediate consequence, we obtain that ${\frak I}(H, \Gamma)$ depends only on the commensurability class of $\Gamma$. As applications, we give two results. In the first, we determine explicitly the smallest $\Gamma$-rational connectedclosed subgroup containing $H$. The second is a characterization of ergodicity ofnilflow $(G/\Gamma, H)$ in terms of ${\frak I}(H, \Gamma)$. Furthermore, a characterization of the irreducible unitary representations of $G$ for which the restriction to $\Gamma$ remain irreducible is given.

• On the numerical solution of nonlinear integral equations on non-rectangular domains utilizing thin plate spline collocation method

The article investigates an approximate scheme to solve nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The integral equations considered in the current paper are considered together with either smooth or weakly singular kernels. The offered method utilizes thin plate splines as a basis in the discrete collocation method. We can regard thin plate splines as a type of the free shape parameter radial basis functions. These basis functions establish an accurate and stable technique to estimate an unknown function by using a set of scattered points on the solution domains. Since the thin plate splines have limited smoothness, the integrals appeared in the scheme cannot be estimated by classical integration rules. Therefore, we introduce a special precise quadrature formula on non-rectangular domains to compute these integrals. The proposed scheme does not require any mesh generations, so it is meshless and does not depend on the domain form. Error analysis is also provided for the method. The performance and convergence of the new approach are tested on four two-dimensional integral equations given on the wing, mushroom, pentagon and fish-like domains.

• Sparse subsets of the natural numbers and Euler’s totient function

In this article, we investigate sparse subsets of the natural numbers and study the sparseness of some sets associated to the Euler's totient function $\phi$ via the property of 'Banach density'. These sets related to the totient function are defined as follows: $V:=\phi(\mathbb{N})$ and $N_{i}:=\{N_{i}(m)\colon m\in V \}$ for $i = 1, 2, 3,$ where $N_{1}(m)=\max\{x\in \mathbb{N}\colon \phi(x)\leq m\}$, $N_{2}(m)=\max(\phi^{-1}(m))$ and $N_{3}(m)=\min(\phi^{-1}(m))$ for $m\in V$. Masser and Shiu (Pacific J. Math. 121(2) (1986) 407-426) called the elements of $N_{1}$ as sparsely totient numbers' and constructed an infinite family of these numbers. Here we construct several infinite families of numbers in $N_{2}\setminus N_{1}$ and an infinite family of composite numbers in $N_{3}$. We also study (i) the ratio $\frac{N_{2}(m)}{N_{3}(m)}$ which is linked to the Carmichael's conjecture, namely, $|\phi^{-1}(m)|\geq 2 ~\forall ~ m\in V$, and (ii) arithmetic and geometric progressions in $N_{2}$ and $N_{3}$. Finally, using the above sets associated to the totient function, we generate an infinite class of subsets of $\mathbb{N}$, each with asymptotic density zero and containing arbitrarily long arithmetic progressions.

• Automorphisms of relative Quot schemes

Let $X\rightarrow S$ be a smooth family of projective curves over an algebraically closed field $k$ of characteristic zero. Assume that both $X$ and $S$ are smooth projective varieties and let $E$ be a vector bundle of rank $r$ over $X$ and $\mathbb{P}(E)$ be its projectivization. Fix $d\geq 1$. Let $\mathcal{Q}(E,d)$ be the relative quot scheme of torsion quotients of $E$ of degree $d$. Then we show that if $r\geq 3$, then the identity component of the group of automorphisms of $\mathcal{Q}(E,d)$ over $S$ is isomorphic to the identity component of the group of automorphisms of $\mathbb{P}(E)$ over $S$. We also show that under additional hypotheses, the same statement is true in the case when $r=2$. As a corollary, the identity component of the automorphism group of flag scheme of filtrations of torsion quotients of $\mathcal{O}^{r}_{C}$, where $r\geq 3$ and $C$ a smooth projective curve of genus $\geq 2$ is computed.

• Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• Editorial Note on Continuous Article Publication

Posted on July 25, 2019