• Forthcoming articles

      Proceedings – Mathematical Sciences

    • The recollements induced by contravariantly finite subcategories

      Yonggang Hu Hailou Yao


      Let $\mathcal{X}$ be an admissible contravariantly finite subcategory of an abelian category $\mathscr{A}$. We show that $\mathscr{A}$ has finite global $\mathcal{X}$-resolution dimension if and only if there is a lower recollement of the homotopy category of bounded complexes over $\mathcal{X}$. We also give sufficient conditions such that the recollement ($\mathscr{A}$, $\mathscr{B}$, $\mathscr{C}$) of abelian categories can be lifted to a (lower or upper) recollement of relative derived categories with respect to balance pairs. Finally, we provide some applications.

    • Finite element method for hyperbolic heat conduction model with discontinuous coefficients in one dimension

      Tazuddin Ahmed Jogen Dutta


      In this article, a fitted finite element method is proposed and analyzed for non-Fourierbio heat transfer model in multi-layered media. Specifically, we employ the Maxwell--Cattaneo equation on the physical media which has a discontinuous coefficients.Convergence properties of the semidiscrete and fully discrete schemes are investigated in the $L^2$ norm. Optimal a priori error estimates for both the schemesare proved. Numerical experiment is conducted to confirm the theoretical findings.

    • Derivation modules for sum and gluing

      Joydip Saha Indranath Sengupta


      In this paper we explicitly compute the derivation module of quotients of polynomialrings by ideals formed by the sum or by some other gluing technique. We discusscases of monomial ideals and binomial ideals separately.

    • The universal ordinary deformation ring associated to a real quadratic field

      Haruzo Hida


      We study ring structure of the big ordinary Hecke algebra ${\mathbb T}$with the modular deformation $\rho_{\mathbb T}:\Gal(\bar{\mathbb Q}/{\mathbbQ})\rightarrow GL_2({\mathbb T})$ of an induced Artin representation $Ind_F^{\mathbb Q}\varphi$ from a real quadratic field $F$ with a fundamental unit $\varepsilon$, varying a prime $p>2$ split in $F$. Under mild assumptions (on the prime $p$), we prove that ${\mathbb T}$ is an integral domain free of even rank $e>0$ over the weight Iwasawa algebra $\Lambda$ \'etale outside$\Spec(\Lambda/p(\langle\varepsilon\rangle-1))$ for $\langle\varepsilon\rangle:=(1+T)^{log_p(\vep)/log_p(1+p)}\in{\mathbb Z}_p[[T]]\subset\Lambda$. If $p\nmid e$, ${\mathbb T}$ is shown to be a normal noetherian domain of dimension $2$ with ramification locus exactly given by $(\langle\varepsilon\rangle - 1)$. Moreover, only under $p$-distinguishedness, we prove that any modular specialization of weight $>1$ of $\rho_{\mathbb T}$ is indecomposable over the inertia group at $p$ (solving a conjecture of Greenberg without exception).

    • Herz-type Hardy spaces with variable exponents associated with operators satisfying Davies--Gaffney estimates

      Souad Ben Seghier Khedoudj Saibi


      Let $L$ be a one to one operator of type $w$, with $w\in[0,\pi/2]$, satisfying theDavies-Gaffney estimates. For $\alpha\in(0,\infty)$ and $p\in(0,\infty)$ and under the condition that $q(\cdot): {\R}^{n}\longrightarrow[1,\infty)$ satisfies the globally log-H\"{o}lder continuity condition, we introduce the Herz-type Hardy space with variable exponents associated to $L$ and establish its molecular decomposition. The atomic characterization and maximal function characterizations of the space are proved under the assumption that $L$ is a non-negative self-adjoint operator on $L^{2}({\R}^{n})$ whose heat kernels satisfy the Gaussian upper bound estimates. All the results are new even for the constant case.

    • Geometry of certain Brill--Noether locus on a very general sextic surface and Ulrich bundles

      Debojyoti Bhattacharya


      Let $X \subset \mathbb P^3$ be a very general sextic surface over complex numbers. In this paper we study certain Brill - Noether problems for moduli of rank $2$ stable bundles on $X$ and its relation with rank $2$ weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill - Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank $2$ weakly Ulrich bundles $\mathcal E$ on $X$ with $c_1(\mathcal E) =5H$ and $c_2 \geq 91$ and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill--Noether loci. Finally, using the existence of rank $2$ Ulrich bundles on $X$ we obtain some more non-empty Brill--Noether loci and investigate the possibility of existence of every even rank simple Ulrich bundles on $X$ .

    • The Dirichlet eigenvalues of the Laplacian on the full shift space

      Shrihari Sridharan Sharvari Neetin Tikekar


      The full one sided shift space over finite symbols is approximated by an increasingsequence of finite subsets of the space. The Laplacian on the space is then defined as a renormalised limit of the difference operators defined on these subsets. In this work, we determine the spectrum of these difference operators completely, using the method of spectral decimation. Further, we prove that under certain conditions, the renormalised eigenvalues of the difference operators converge to an eigenvalue of the Laplacian.

    • Study of nearly invariant subspaces with finite defect in Hilbert spaces

      Arup Chattopadhyay Soma Das


      In this article, we briefly describe nearly $T^{-1}$ invariant subspaces with finite defect for a shift operator $T$ having finite multiplicity acting on a separable Hilbert space $\mathcal{H}$ as a generalization of nearly $T^{-1}$ invariant subspaces introduced by Liang and Partington in \cite{YP}. In other words we characterize nearly $T^{-1}$ invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in \cite{CDP}. Furthermore, we also provide a concrete representation of the nearly $T_B^{-1}$ invariant subspaces with finite defect in a scale of Dirichlet-type spaces $\mathcal{D}_\alpha$ for $\alpha \in [-1,1]$ corresponding to any finite Blashcke product $B$, as was done recently by Liang and Partington for defect zero case (see Section 3, [16]).

    • On conjecture of Minkowski and Woods for $n=10$



      Let $\mathbb L$ be a lattice in $n$-dimensional Euclidean space $\mathbb R^n$ reduced in the sense of Korkine and Zolotareff and having a basis of the form $(A_1, 0, 0, \ldots , 0)$, $a_{2, 1}, A_2, 0, \ldots , 0), \ldots, (a_{n, 1}, a_{n, 2}, \ldots , a_{n, n-1}, A_n)$. A famous conjecture of Woods in Geometry of Numbers asserts that if $A_1 A_2\cdots A_n)=1$ and $A_i \leq A_1$ for each $i$ then any closed sphere in $\mathbb R^n$ of radius $\sqrt {n/4p}$ contains a point of $\mathbb L$. Together with a result of C. T. McMullen (2005), the truth of Woods' Conjecture for a fixed $n$, implies the long-standing classical conjecture of Minkowski on product of $n$ non-homogeneous linear forms for that value of $n$. In an earlier paper, {\em Proc. Indian Acad. Sci. (Math. Sci.)} {\bf 126} (2016) 501--548, we proved Woods' conjecture for $n = 9$. In this paper, we prove Woods' conjecture and hence Minkowski's conjecture for $n=10$.

    • Torsors on semistable curves and degenerations

      V Balaji


      In this paper, we answer two long-standing questions on the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first is the construction of a flat degeneration of the moduli of $G$-torsors on smooth projective curves when the smooth curve degenerates to an irreducible nodal curve and the second one is to give an intrinsic definition of (semi)stability for a $G$-torsor on an {\em irreducible projective nodal curve}. A generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools.

    • Affine monomial curves



      This article is an expository survey on affine monomial curves, where we discuss some research problems from the perspective of computation.

    • Splitting probability invariant for Cordes schemes

      Leena Jindal Anjana Khurana


      Let $S=(G,-1, D_G)$ be a Cordes scheme. In this paper, along with generalizing the results of splitting probability invariant of fields to Cordes schemes, it is also shown that $p_s(S)>1/2$ if and only if $S$ is a generalized Hilbert scheme except when $S$ is a scheme of a formally real field with $\lvert G/R(S) \rvert =4$, where $R(S)$ denotes the radical of $S$. We also prove that if $S_1, S_2$ are generalized Hilbert schemes, $P(S_i)=\{p_s(S_i^{T_n}) : \lvert T_n \rvert= 2^n \text{ and } n \in \mathbb{N}\}$; $i=1, 2$ and $\lvert P(S_1) \cap P(S_2) \rvert > 1$ then $P(S_1)=P(S_2)$.

    • An improved result in almost sure local central limit theorem for the partial sums

      Feng Xu Zhen Zeng


      The almost sure local central limit theorem is a general result which contains thealmost sure global central limit theorem. Let $\{X_k, k\geq 1$ be a sequence of independent and identically distributed random variables. Under a fairly general condition an universal result in almost sure local limit theorem for the partial sums $S_k =\Sigma^k_{i=1} X_i$ is established on the weight $d_k = k^{-1} \exp(\log^{\beta} k), 0 \leq \beta$ < 1/2:

      $$\lim_{n\rightarrow\infty}\frac{1}{D_n}\Sigma^n_{k=1}d_k\frac{I(a_k\leq S_k < b_k)}{P(a_k\leq S_k < b_k)}=1\ {\rm a.s.},$$

      where $D_n=\Sigma^n_{k=1}d_k, -\infty\leq a_k\leq 0\leq b_k\leq\infty$, $k=1, 2,\ldots$. This result extends previous results in the almost sure local central limit theorems from $d_k = 1/k$ to $d_k = k^{-1} \exp(\log ^\beta k)$, $0\leq\beta$ < 1/2.

    • Topological differences at infinity for nonlinear problems related to the fractional Laplacian



      In this paper we are interested in the fractional Yamabe-type equation $A_su = u^{\frac{n+2s}{n-2s}}$, $u > 0$ in $\Omega$ and $u = 0$ on $\partial\Omega$. Here $\Omega$ is a regular bounded domain of $\mathbb{R}^n$, $n\geq 2$ and $A_s$, $s \in (0, 1)$ represents the fractional Laplacian operator in $\Omega$ with zero Dirichlet boundary condition. Based on the theory of critical points at infinity of Bahri and the localization technique of Caffarelli and Silvestre. We compute the difference of topology induced by the critical points at infinity between the level sets of the variational functional associated to the problem. Our result can be seen as a nonlocal analog of the theorem of Bahri, Li and Rey [12] on the classical Yamabe-type equation.

    • Concentration behavior of solutions for quasilinear elliptic equations with steep potential well

      Jianhua Chen Xianjiu Huang Pingying Ling


      In this paper, we study the following quasilinear elliptic equation

      \begin{equation*} -\triangle u+\lambda V(x)u-[\triangle(1+u^{2})^{\frac{1}{2}}]\frac{u}{2(1+u^{2})^{\frac{1}{2}}}=|u|^{p-2}u,\,\,x\in\R^{N}, \end{equation*}

      where $N\geq 3$, $\lambda > 0$, $12-4\sqrt{6} < p < 2^{*}$, $V\in C(\R^{N},\R)$ and $V^{-1}(0)$ has nonempty interior. At first, we prove the existence of a nontrivial solution $u_{\lambda}$ via variational method. Then, the concentration behavior of $u_{\lambda}$ is also explored on the set $V^{-1}(0)$ as $\lambda\rightarrow\infty$.

    • On a conjecture of Kelly on (1, 3)-representation of Sylvester-Gallai designs

      C P Anil Kumar Anoop Singh


      We give an exact criterion of a conjecture of L. M. Kelly to hold true which is stated as follows. If there is a finite family $\Sigma$ of mutually skew lines in $\mathbb{R}^l,l\geq 4$ such that the 3-flat spanned by every two lines in $\Sigma$, contains at least one more line of $\Sigma$, then we have that all of the lines of $\Sigma$ are contained in a single 3-flat if and only if the arrangement of 3-flats is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, where we prove that, for $(2,5)$-representations of Sylvester--Gallai designs in $\mathbb{R}^6$, the analogous statement does not hold.

    • Brauer group of moduli of torsors under Bruhat--Tits group scheme $\mathcal{G}$ over a curve



      Let $X$ be a smooth projective curve over $\mathbb{C}$. We compute the Brauer group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of Bruhat--Tits group scheme $\mathcal{G}$-torsors. Let $M^{rs}_X (\mathcal{G})$ denote the regularly stable locus of the coarse moduli space of semi-stable $\mathcal{G}$-torsors. When $g(X)\geq 3$, we compute the kernel of $Br(M^{rs}_X (\mathcal{G}))\rightarrow Br(\mathcal{M}_X(\mathcal{G}))$.

    • 5-Rank of ambiguous class groups of quintic Kummer extensions

      Abdelmalek Azizi Fouad Elmouhib Mohamed Talbi


      Let $k =\mathbb{Q}\sqrt[5]{n}, \upzeta_5$, where $n$ is a positive integer 5th power-free, whose 5-class group denoted $C_{k,5}$ is isomorphic to $\mathbb{Z}/5\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$. Let $k_0 =\mathbb{Q}(\upzeta_5)$ be the cyclotomic field containing a primitive 5th root of unity 5. Let $C^{(\sigma)}_{k, 5}$ the group of the ambiguous classes under the action of $Gal(k/k_0) = \langle\sigma\rangle$. The aim of this paper is to determine all naturels $n$ such that the group of ambiguous classes $C^{(\sigma)}_{k, 5}$ has rank 1 or 2.

    • Interval maps where every point is eventually fixed

      V Kannan Pabitra Narayan Mandal


      An orbit pattern $\alpha$ is said to force an orbit pattern $\beta$, if any continuousinterval map which admits $\alpha$ also admits $\beta$. Among the orbit patterns that force only eventually fixed trajectories, we completely describe the forcing relation, by answering the question: which orbit patterns force which others? We provide two different ways to enlist them completely through formal languages. One is through constructed words and another is by derived words.

    • Swallowtails and cone-like singularities on a maxface



      When a connected component of the set of singular points of the maxface $X$ consists of only generalized cone-like singular point, we construct a sequence of maxfaces $X_n$, with an increasing number of swallowtails, converging to the maxface $X$. We include the general discussion toward this.

    • Strongly contracting geodesics in a tree of spaces



      Let $X$ be a tree of proper geodesic spaces with edge spaces strongly contracting and uniformly separated from each other by a number depending on the contraction function of edge spaces. Then we prove that the strongly contracting geodesics in vertex spaces are quasiconvex in $X$. We further prove that in $X$ if all the vertex spaces are uniformly hyperbolic metric spaces then $X$ is a hyperbolic metric space and vertex spaces are quasiconvex in $X$.

    • Transcendence of Hurwitz zeta type series and related questions

      Nabin Kumar Meher


      In this article, we prove the transcendence of special values of some Hurwitz zeta type series. Moreover, we find a linear independence criterion of these series under some mild conditions. We also show that, for any positive integer $k$ and for any $ a, b \in (0, 1) \cap \mathbb{Q}$ with $ a+b = 1$, at least one of the $\zeta(2k, a) $ or$\zeta(2k, b) $ must be transcendental.

    • Gabriel localizations with applications to flat epimorphisms of rings

      Abolfazl Tarizadeh


      In this article, new results on the Gabriel localizations are obtained. As an application of them, it is shown that a morphism of rings is a flat epimorphism of rings if and only if it corresponds to a kind of the Gabriel localizations. Using this result, then new progresses in the understanding the structure of flat epimorphisms of rings have been made. Especially among them, a set-theoretical gap in the structure of the ring$M(R)$, the maximal flat epimorphic extension of a ring $R$, has been fixed.

    • On a problem about the connectivity of the enhanced proper power graph of a finite group

      A Babai A Mahmoudifar


      Let $G$ be a group with identity element $e$. The proper power graph and properenhanced power graph of $G$, are denoted by $\Gamma^*_{P}(G)$ and$\Gamma^*_{EP}(G)$, respectively. Also, the prime graph of $G$, is denoted by$\Gamma_{GK}(G)$. In an article, Alipour and et al. asked which groups do have the property that $\Gamma^*_{P}(G)$ is connected? In this paper, we show that if$\Gamma_{GK}(G)$ is disconnected, then $\Gamma^*_{P}(G)$ and $\Gamma^*_{EP}(G)$ are disconnected. Moreover, we prove that if $G$ is a nilpotent group which is not a $p$-group, then $\Gamma^*_{EP}(G)$ is a connected graph.

    • Representations of the fundamental group and Higgs bundles on singular integral curves



      Let $Y$ be an integral projective complex curve. To representations of the topological fundamental group of $Y$ in the general linear group, we associate generalisedparabolic vector bundles and Hitchin pairs. We use this correspondence to study thevector bundles and Hitchin pairs on $Y$ associated to representations of the fundamental group in case the curve has singularities of type $A_{2s}; A_{2s-1} or ordinary $s$-points.

    • An elementary proof of Lelli Chiesa's theorem on constancy of second coordinate of gonality sequence



      Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article, we will give an alternative and elementary proof of Lelli Chiesa's Theorem in the case of $r = 2$. More precisely, we will prove that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $\mid L\mid$. Using Lelli Chiesa's theorem for $r \geq 3$, we also extend Lelli Chiesa's theorem in the case of $r = 2$ in weaker condition.

    • A weighted Erd\"{o}s--Ginzburg--Ziv constant for finite abelian groups with higher rank



      Let $G$ be a finite abelian group and $A \subset\mathbb{Z}$. The weighted zero-sum constant $s_A(G)$ (resp. $\eta_A(G))$ is defined as the least positive integer $t$, such that every sequence $S$ over $G$ with length $\geq t$ has an $A$-weighted zero-sum subsequence of length exp$(G)$ (resp. $\leq \exp(G))$. In this article, we investigate the value of $s_A(G)$ and $\eta_A(G)$ in the case $G = \mathbb{Z}_n \oplus \mathbb{Z}_n \oplus\cdots\oplus\mathbb{Z}_n$ where $n$ is a square free odd integer and $A$ is the set of integers co-prime to $n$. We also obtain certain properties about extremal zero-sum free sequences.

    • On the locus of higher order jets of entire curves in complex projective varieties



      For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green-Griffiths-Lang conjecture stipulates that entire curves drawn on a variety of general type should all be contained in a proper algebraic subvariety. We present here new results on the existence of differential equations that strongly restrain the locus of entire curves in the general context of foliated or directed varieties, under appropriate positivity conditions.

    • A Furstenberg--Zimmer structure theorm for $\sigma$-finite measur spaces

      Massoud Amini Jumah Swid


      We show that an analog of the Furstenberg--Zimmer structure theorem holds for σ -finite non atomic measure spaces and measure preserving strongly recurrent actionsof discrete groups. We adapt the idea of Tao in associating Hilbert modules to measure preserving extensions and show that for an isomorphic copy of the $L^2$-space, the tools of Zimmer structure theory could be applied.

    • A topological shadowing theorem



      We prove a topological version of the classical Shadowing Theorem in differentiable dynamics [10]. We use it to unify some stability results in the literature as the GH-topological, Hartman and Walters stability theorems [2], [13, 14], [17].

    • Approximation of nonlinear infinite delay equations in Banach spaces by finite delay equations

      G Divyabharathi T Sengadir


      In this paper, we show that the solution of an infinite delay equations in a Banach space given by a non-linear semigroups in a Fr\'{e}chet spaces, can be approximated by finite delay equations.

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