• Forthcoming articles

      Proceedings – Mathematical Sciences

    • Positive Integer Solutions of Certain Diophantine Equations

      BIJAN KUMAR PATEL PRASANTA KUMAR RAY MANASI K SAHUKAR

      Abstract

      In this study, the Diophantine equations $x^{2}-32B_{n}xy-32y^{2} = \pm{32}^r, x^{4}-32B_{n}xy-32y^{2} = \pm{32}^r and x^{2}-32B_{n}xy-32y^4 = \pm{32}^r are considered and determine when these equations have positive integer solutions. Moreover, all positive integer solutions of these Diophantine equations in terms of balancing and Lucas-balancing numbers are also found out.

    • Codismantlability and projective dimension of the Stanley-Reisner ring of special hypergraphs

      Fahimeh Khosh Ahang Somayeh Moradi

      Abstract

      In this paper firstly, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley-Reisner ring of special hypergraphs in some sense.

    • Frobenius splitting of projective toric bundles

      He Xin

      Abstract

      We prove that the projectivization of the tangent bundle of a nonsingular toric variety is Frobenius split.

    • Properties of Singular Integral Operators 𝑆𝛼,𝛽

      Amit Samanta Santanu Sarkar

      Abstract

      For $\alpha, \beta \in L^{\infty}(S^{1})$, the singular integral operator 𝑆𝛼,𝛽 on 𝐿2(𝑆1) is defined by $S_{\alpha, \beta}f := \alpha Pf + \beta Qf$, where 𝑃 denotes the orthogonal projection of 𝐿2(𝑆1) onto the Hardy space 𝐻2(𝑆1), and 𝑄 denotes the orthogonal projection onto $H^{2}(S^{1})^{\perp}$. In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of 𝑆𝛼,𝛽. This work has shown that 𝑆𝛼,𝛽 may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of 𝑆𝛼,𝛽.

    • On 3-way combinatorial identities

      A K Agarwal Megha Goyal

      Abstract

      In this paper we provide combinatorial meanings to two generalized basic series with the aid of associated lattice paths. These results produce two new classes of infinite 3-way combinatorial identities. Five particular cases are also discussed. These particular cases provide new combinatorial versions of Göllnitz-Gordon identities and Göllnitz identity. Seven 𝑞-identities of Slater and five 𝑞-identities of Rogers are further explored using the same combinatorial object. These results extend the work of Goyal and Agarwal, Agarwal and Rana and Agarwal.

    • Sharp Trudinger-Moser type inequality invoking Hardy inequalities

      MOHAMED KHALIL ZGHAL

      Abstract

      We establish a sharp Trudinger-Moser type inequality invoking a Hardy inequality for any even dimension. This leads to a non compact Sobolev embedding in some Orlicz space. We also give a description of the lack of compactness of this embedding in the spirit of [8].

    • On Some U(n)-Symmetric Extremal K"ahler metrics of Non-Constant Scalar Curvature

      XIAOJUAN DUAN

      Abstract

      In this paper, we explicitly construct some rotationally symmetricextremal (pseudo-)KÄahler metrics of non-constant scalar curvature, which de-pend on some parameters, on some line bundles over projective spaces. We alsodiscuss the phase change phenomenon caused by the variation of parameters.

    • Alternating groups as a quotient of $PSL(2, \mathbb{Z}[i])$

      Qaiser Mushtaq Awais Yousaf

      Abstract

      In this study, we developed an algorithm to find the homomorphisms of the Picard group 𝑃𝑆𝐿(2,𝑍[𝑖]) into a finite group 𝐺. This algorithm is helpful to find a homomorphism (if it is possible) of the Picard group to any finite group of order less than 15! because of the limitations of the Gap and computer memory. Therefore, we obtain only five alternating groups 𝐴𝑛, where 𝑛 = 5, 6, 9, 13 and 14 as a quotient of Picard group. In order to extend the degree of the alternating groups, we use coset diagrams as a tool . In the end, we prove our main result with the help of three diagrams which are used as building blocks and proved that, for 𝑛 ≡ 1; 5; 6(mod 8), all but finitely many alternating groups An can be obtained as the quotients of the Picard group 𝑃𝑆𝐿(2,𝑍[𝑖]). A code in Groups Algorithms Programming (GAP) is developed to perform the calculation.

    • Global Weighted Estimates for Second-Order Nondivergence Elliptic and Parabolic Equations

      Fengping Yao

      Abstract

      In this paper we obtain the global weighted 𝐿𝑝 estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary we obtain 𝐿𝑝-type regularity estimates for such equations.

    • SOME ASPECTS OF SHIFT{LIKE AUTOMORPHISMS OF Ck

      SAYANI BERA KAUSHAL VERMA

      Abstract

      The goal of this article is two fold. First, using transcendental shift{like automorphisms of Ck; k \geq 3 we construct two examples of non{degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of Ck; k \geq 3 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon{Esterle in C2: The second example shows the existence of a Fatou{Bieberbach domain in Ck; k\geq3 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay{Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift{like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.

    • A GENERALIZATION OF ZERO DIVISOR GRAPHS ASSOCIATED TO COMMUTATIVE RINGS

      M AFKHAMI A ERFANIAN K KHASHYARMANESH N VAEZ MOOSAVI

      Abstract

      Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by \gamma^n_R, with R^n\{0}as the vertex set and two distinct vertices X and Y in Rn being adjacent if andonly if there exists an n \times n lower triangular matrix A over R whose entrieson the main diagonal are nonzero and one of the entries on the main diagonalis regular such that X^T AY = 0 or Y^T AX = 0, where, for a matrix B, B^T isthe matrix transpose of B. If n = 1, then \gamma^n_R is isomorphic to the zero divisor graph \gamma(R), and so \gamma^n_R is a generalization of \gamma(R) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of \gamma^n_R. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

    • A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

      M H Heydari M R Hooshmandasl C Cattani

      Abstract

      In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann-Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.

    • Cross-product of bessel functions: monotonicity patterns and functional inequalities

      Árpád Baricz Saminathan Ponnusamy Sanjeev Singh

      Abstract

      In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.

    • Positive solutions with single and multi-peak for semilinear elliptic equations with nonlinear boundary condition in the half-space

      Li Wang Peihao Zhao

      Abstract

      We consider the existence of single and multi-peak solutions of the following nonlinear elliptic Neumann problem

      $\begin{cases} - \Delta u + \lambda^2 u= Q (x) |u|^{p-2}u & ~\text{in}\quad \mathbb{R}_+^N,\\ \hspace{1.7cm}\frac{\partial u}{\partial n} = f(x, u) & ~\text{on}\quad \partial \mathbb{R}_+^N \end{cases} $

      where $\lambda$ is a large number, $p \in \left(2, \frac{2N}{N-2} \right)$ for $N \geq 3, f(x, u)$ is subcritical about $u$ and $Q$ is positive and has some non-degenerate critical points in $\mathbb{R}^{N}_{+}$ . For $\lambda$ large, we can get solutions which have peaks near the non-degenerate critical points of $Q$.

    • APPROXIMATE CONTROLLABILITY OF A NON-AUTONOMOUS DIFFERENTIAL EQUATION

      INDIRA MISHRA MADHUKANT SHARMA

      Abstract

      In this paper, we establish the approximate controllability results for a non-autonomous functional differential equation using the theory of linearevolution system, Schauder fixed point theorem, and by making use of resolventoperators. The obtained results in the paper, improve the existing ones in thisdirection, up to the considerable extent. An example is also given to illustratethe abstract results

    • A GENERALIZATION OF TOTAL GRAPHS

      M AFKHAMI K HAMIDIZADEH K KHASHYARMANESH

      Abstract

      Let R be a commutative ring with nonzero identity, L_n(R) be the set of all lower triangular n \times n matrices, and U be a triangular subset of R_n i.e. the product of any lower triangular matrix with the transpose of any element of U, belongs to U. The graph GT^n_U (R^n) is a simple graph whose vertices consists of all elements of R^n, and two distinct vertices (x_1; : : : ; x_n) and(y1; : : : ; y_n) are adjacent if and only if (x1 + y1; : : : ; x_n + y_n) \in U. The graph GT^n_U (Rn) is a generalization for total graphs. In this paper, we investigate the basic properties of GT^n_U (Rn). Moreover, we study the planarity of the graphs GT^n_U (U), GT^n_U (Rn\ U) and GT^n_U (R^n).

    • A NOTE ON GENERALIZED SKEW DERIVATIONS ON LIE IDEALS

      MOHAMMAD ASHRAF VINCENZO DE FILIPPIS

      Abstract

      Let R be a prime ring, Z(R) its center, C its extended centroid,L a Lie ideal of R, F a generalized skew derivation associated with a skewderivation d and automorphism α. Assume that there exist t ≥ 1 and m, n ≥ 0fixed integers such that vu = umF(uv)tun for all u, v 2 L. Then it is shownthat either L is central or char(R) = 2, R ⊆M2(C), the ring of 2×2 matricesover C, L is commutative and u2 2 Z(R), for all u 2 L. In particular, ifL = [R,R], then R is commutative.

    • Fourth Power Diophantine Equations in Gaussian Integers

      Farzali Izadi Rasool Naghdali Forooshani Amaneh Amiryousefi Varnousfaderani

      Abstract

      In this paper we examine a class of fourth power Diophantine equa-tions of the form x4 + kx2y2 + y4 = z2 and ax4 + by4 = cz2, in the GaussianIntegers, where a and b are prime integers.

    • 2-domination number of generalized Petersen graphs

      Davood Bakhshesh Mohammad Farshi Mohammad Reza Hooshmandasl

      Abstract

      Let $G = (V, E)$ be a graph. A subset $S \subseteq V$ is a $k$-dominating set ofG if each vertex in $V - S$ is adjacent to at least $k$ vertices in $S$. The $k$-domination number of $G$ is the cardinality of the smallest $k$-dominatingset of $G$. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs $P(5k+1, 2)$ and $P(5k+2, 2)$, for $k > 0$,is $4k + 2$ and $4k + 3$, respectively. This proves two conjectures dueto Yi-Jie Cheng (PhD thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs $P(2k, k)$ and $P(5k+4, 3)$. Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs $P(5k + 1, 3)$, $P(5k + 2, 3)$ and $P(5k + 3, 3)$.

    • Pair frames in Hilbert C^*-modules.

      AZANDARYANI M MIRZAEE FEREYDOONI A

      Abstract

      In this paper we introduce pair frames in Hilbert C*−modules and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and we get the necessary and sufficient conditions for a Hilbert C*−module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert C*−modules to pair frames andconsidering the stability of pair frames under invertible operators, we constructnew pair frames and we show that pair frames are stable under small perturbations.

    • Root and Critical Point Behaviors of Certain Sums of Polynomials

      Seon-Hong Kim Sung Yoon Kim Tae Hyung Kim Sangheon Lee

    • ON A GENERALIZATION OF SEMISIMPLE MODULES

      NAZIM AGAYEV CESIM CELIK TAHIRE OZEN

      Abstract

      Let R be a ring with identity. A module M_R is called an r-semisimple module if for any right ideal I of R, MI is a direct summand of M_R which is a generalization of semisimple and second modules. We investigate when an r-semisimple ring is semisimple and prove that a ring R with the number of nonzero proper ideals

      \leq 4 and J(R) = 0 is r-semisimple. Moreover, we prove that R is an r-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever R is an r-semisimple ring.

    • Uniformly locally univalent harmonic mappings

      Saminathan Ponnusamy Jinjing Qiao Xiantao Wang

      Abstract

      The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family BH(λ) of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of k-quasiconformal harmonic mappings in BH(λ) and the class BH(λ) are contained in the Hardy space of a specific exponent depending on the λ, respectively, and we also discuss the growth of coefficients for harmonic mappings in BH(λ).

    • Some Polynomials Associated with the r-Whitney Numbers

      CRISTINA B CORCINO ROBERTO B CORCINO ISTVAN MEZO JOSE L RAMIREZ

      Abstract

      In the present article we study three families of polynomials associated withthe r-Whitney numbers of the second kind. They are the $r$-Dowling polynomials, $r$-Whitney-Fubini polynomials and the $r$-Eulerian-Fubini polynomials. Then we derive several combinatorial results by using algebraic arguments (Rota's method), combinatorial arguments (set partitions) and asymptotic methods.

    • SOME INEQUALITIES FOR THE BELL NUMBERS

      FENG QI

      Abstract

      In the paper, the author presents derivatives of the generatingfunctions for the Bell numbers by induction and by the Fa`a di Bruno formula,recovers an explicit formula for the Bell numbers in terms of the Stirlingnumbers of the second kind, finds the (logarithmically) absolute and completemonotonicity of the generating functions for the Bell numbers, and constructssome inequalities for the Bell numbers. From these inequalities, the authorderives the logarithmic convexity of the sequence of the Bell numbers.

    • Character Average of Fourth Power of Dirichlet L-series at Unity

      VIVEK V RANE

    • Arithmetical Fourier and Limit values of elliptic modular functions

      NianLiang Wang

    • ANALYTIC SETS AND EXTENSION OF HOLOMORPHIC MAPS OF POSITIVE CODIMENSION

      MARYAM AL-TOWAILB OURIMI NABIL

    • SOME NEW ESTIMATES FOR THE HELGASON FOURIER TRANSFORM ON RANK 1 SYMMETRIC SPACES

      R DAHER S EL OUADIH

      Abstract

      New estimates are proved for the Helgason Fourier transform in thespace $L^2(X)$ on certain classes of functions characterized by the spherical modulus of continuity.

    • A note on the high power Diophantine equations

      Mehdi Baghalaghdam Farzali Izadi

    • $z$--CLASSES IN FINITE GROUPS OF CONJUGATE TYPE $(n, 1)$

      SHIVAM ARORA KRISHNENDU GONGOPADHYAY

    • Transcendence of some power series for liouville number arguments

      FATMA CALISKAN

      Abstract

      In this paper, we prove that some power series with rational coeffcients take either values of rational numbers or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. We then apply this result to an algebraic number field. In addition, we establish the $p$-adic analogues of these results and show that these results have analogues inthe field of $p$-adic numbers.

    • Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

      Cesar E Torres Ledesma

      Abstract

      In this article we consider the following fractional Hamiltonian systems$_{t}D^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u) +\lambdaL(t)u = \nablaW(t, u), t\in\mathbb{R}$,where $\alpha \in (1/2, 1), \lambda > 0$ is a parameter, $L \in C(\mathbb{R}, \mathbb{R}^{n\times n})$ and $W \in C^{1}(\mathbb {R}\times\mathbb{R}^n, \mathbb{R})$. Unlike most other papers on this problem, we require that $L(t)$ is a positive semi-definite symmetric matrix for all $t \in\mathbb{R}$, that is, $L(t) \equiv 0$ isallowed to occur in some finite interval $\mathbb{I}$ of $\mathbb{R}$. Under some mild assumptions on $W$, we establish the existence of nontrivial weak solution, which vanish on $\mathbb{R} \backslash \mathbb{I}$ as $\lambda\rightarrow\infty$, and converge to $\tilde{u}$ in $H^{\infty}(\mathbb{R})$; here $\tilde{u} \in E^{\infty}_{0}$ is nontrivialweak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $\mathbb{I}$. Furthermore we give a multiplicity results for (0.1).

    • MAXIMAL REGULARITY FOR NON-AUTONOMOUS STOCHASTIC EVOLUTION EQUATIONS IN UMD BANACH SPACES

      Viet Ton Ta Atsushi Yagi Yoshitaka Yamamoto

      Abstract

      A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partialdifferential equation.

    • Some functional inequalities on non-reversible Finsler manifolds

      Shin-ichi Ohta

      Abstract

      We continue our study of geometric analysis on (possibly non-reversible) Finslermanifolds, based on the Bochner inequality established by the author and Sturm.Following the approach of the $\Gamma$-calculus `a la Bakry et al, we show the dimensional versions of the Poincar´e–Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti–Mondino in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.

    • ON THE CONVERGENCE OF A NEW ITERATIVE ALGORITHM OF THREE INFINITE FAMILIES OF GENERALIZED NONEXPANSIVE MULTI-VALUED MAPPINGS

      DHIRENDRA BAHUGUNA ANUPAM SHARMA

      Abstract

      In this paper, we establish some weak and strong convergence theorems for a new iterative algorithm under some suitable conditions to approximate the common fixed point of three infinite families of multi-valued generalized non expansive mappings in a uniformly convex Banach spaces. Our results generalize and improve several previously known results of theexisting literature.

    • Augmentation Quotients for Real Representation Rings of Cyclic Groups

      Shan CHANG Hang LIU

    • On the Role of Associativity in Ramsey Algebras

      Andrew Rajah Wen Chean Teh Zu Yao Teoh

      Abstract

      It is known that semigroups are Ramsey algebras. This paper is an attempt to understand the role associativity plays in a binary system being a Ramsey algebra. Specifically, we show that the nonassociative Moufang loop of octonions is not a Ramsey algebra.

    • On the partition dimension of two-component graphs

      Debi Oktia Haryeni Edy Tri Baskoro Suhadi Wido Saputro Martin Baca Andrea Semanicova-Fenovcıkova

      Abstract

      In this paper, we continue investigating the partition dimension for disconnected graphs. We determine the partition dimension for some classes of disconnected graphs $G$ consisting of two components. If $G = G_{1}\cup G_{2}$, then we give the bounds of the partition dimension of $G$ for $G_{1} = P_{n}$ or $G_{1} = C_{n}$ and also for $pd(G_{1}) = pd(G_{2})$.

    • Allowable graphs of the nonlinear Schr$\"{o}$dinger equation and their applications

      NGUYEN BICH VAN

      Abstract

      We construct the definition of allowable graphs of the nonlinear Schr$\"{o}$dinger equation of arbitrary degree and use it to verify the separation and irreducibility (over the ring of integers) of the characteristic polynomials of all the possible graphs giving 3-dimensional blocks of the normal form of the nonlinear Schr$\"{o}$dinger equation. The method is purely algebraic and the obtained results will be useful in further studies of the nonlinear Schr$\"{o}$dinger equation

    • A CONSTRUCTIVE APPROACH TO THE FINITE WAVELET FRAMES OVER PRIME FIELDS

      ASGHAR RAHIMI NILOUFAR SEDDIGHI

      Abstract

      In this article, we present a constructive method for computingthe frame coefficients of finite wavelet frames over prime fieldsusing tools from computational harmonic analysis and group theory.

    • A CHARACTERIZATION OF FINITE $p$-GROUPS BY THEIR SCHUR MULTIPLIER

      SUMANA HATUI

    • The Fundamental Group and Extensions of Motives of Jacobians of Curves

      Subham Sarkar Ramesh Sreekantan

      Abstract

      In this paper we construct extensions of mixed Hodge structure coming from the mixed Hodge structure on the graded quotients of the group ring of the Fundamental group of a smooth projective pointed curve which correspond to the regulators of certain motivic cohomology cycles on the Jacobian of the curve essentially constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo [Col02] where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve.

    • Semifinite Bundles and the Chevalley-Weil Formula

      Shusuke Otabe

      Abstract

      In our previous paper, we studied the category of semifinite bundles on a proper variety defined over a field of characteristic 0. As a result, we obtained the fact that for a smooth projective curve defined over an algebraically closed field of characteristic 0 with genus g > 1, Nori fundamental group acts faithfully on the unipotent fundamental group of its universal covering. However, it was not mentioned about any explicit module structure. In this paper, we prove that the Chevalley-Weil formula gives a description of it

    • Comparison of graphs associated to ‎a‎ commutative ‎Artinian‎ ring

      MASOUD GHORAISHI KARIM SAMEI

    • polynomially peripheral range-preserving maps between Banach algebras

      M NAJAFI TAVANI

      Abstract

      Let $A$ and $B$ be two Banach function algebras and $p$ a two variable polynomial $p(z, w) = zw + az + bw + c$, $(a, b, c \in\mathbb{C})$. We characterize the general form of a surjection $T : A \rightarrow B$ which satisfies $Ran_{\pi}(p(T f, T g)) \cap Ran_{\pi}(p( f, g)) \neq\emptyset, ( f, g \in A and c \neq ab)$, where $Ran_{\pi}( f )$ is the peripheral range of $f$ .

    • HEISENBERG UNIQUENESS PAIRS CORRESPONDING TO FINITE NUMBER OF PARALLEL LINES

      SAYAN BAGCHI

      Abstract

      In this paper we study the Heisenberg uniqueness pairs correspondingto finite number of parallel lines $\Gamma$. We give a necessary condition and a sufficient condition for a subset $\Lambda$ of $\mathbb{R}^2$ so that ($\Gamma$, $\Lambda$) becomes a HUP.

    • Moduli space of Parabolic vector bundles over hyperelliptic curves

      SURATNO BASU SARBESWAR PAL

      Abstract

      Let $X$ be a smooth projective hyperelliptic curve of arbitrary genus $g$. Inthis short article we will classify the rank 2 stable vector bundles with parabolic structure along a reduced divisor of degree 4.

    • Nilpotent groups related to an automorphism

      Ahmad Erfanian Masoumeh Ganjali

      Abstract

      The aim of this paper is to state some results on an $\alpha$-nilpotent group, which was recently introduced by Barzegar and the first author, for any fixed automorphism $\alpha$ of a group $G$. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal sub-groups of a finite group.

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