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.

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.

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.

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).

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.

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 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.

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]).

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 $n=10$.

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.

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

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 $P(S_1)=P(S_2)$.

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.

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.

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 $\lambda\rightarrow\infty$.

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.

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 $Br(M^{rs}_X (\mathcal{G}))\rightarrow Br(\mathcal{M}_X(\mathcal{G}))$.

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.

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.

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.

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 $X$.

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.

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.

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.

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.

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.

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.

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.

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.

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].

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.