We study the appearance of $(k,r)$- integers in an arithmetic progression by using the exponent pair method. Moreover, we deduce the result to $r$-free integers in thearithmetic progression.

In this paper, we establish the boundary Harnack Principle for solutions to lineardegenerate elliptic equations in H\"{o}lder domains.

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.

In this note, for any number field, we define Dedekind harmonic numbers with respect to this number field. First, we show that they are not integers except finitely many of them. Then, we present a uniform and an explicit version of this result for quadratic number fields. Moreover, by assuming the Riemann Hypothesis for Dedekind zeta functions, we prove that the difference of two Dedekind harmonic numbers are not integers after a while if we have enough terms, and we prove the non-integrality of Dedekind harmonic numbers for quadratic number fields in another uniform way together with an asymptotic result.

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).

In this article, we study the deformations of Filippov algebroids. First, we define adifferential graded Lie algebra for a Filippov algebroid by introducing the notion ofFilippov multiderivations for a vector bundle. We then discuss deformations of aFilippov algebroid in terms of low-dimensional cohomology associated with thisdifferential graded Lie algebra. We define Nijenhuis operators on Filippov algebroids and characterize trivial deformations of Filippov algebroids in terms of these operators. Finally, we define finite order deformations and discuss the problem of extending a given finite order deformation to a deformation of a higher order.

In this paper, we investigate the frequent hypercyclicity of the Toeplitz operator $T_{\Phi}$ and their tensor products on the Hardy space with the symbol of the form $\Phi(z)=p\big(\frac{1}{z}\big)+\varphi(z)$, where $p$ is a polynomial and $\varphi\in H^{\infty}$. We also give some sufficient conditions for $T_{\Phi_{1}}\otimes T_{\Phi_{2}}$ to be hypercyclic and construct an example such that neither $T_{\Phi_{1}}$ nor $T_{\Phi_{2}}$ is hypercyclic, but $T_{\Phi_{1}}\otimes T_{\Phi_{2}}$ is frequently hypercyclic. Moreover, we also characterize the mixing property and chaoticity of $T_{\Phi}$ and $T_{\Phi_{1}}\otimes T_{\Phi_{2}}$.

An extension result for rectangular operator extreme states on operator spaces in ternary rings of operators is proved. It is established that for operator spaces in rectangular matrix spaces extreme states are conjugates of the inclusion mapimplemented by isometries or unitaries . Further, a characterisation of operator spaces of matrices for which the inclusion map is an extreme state is deduced. In the context of operator spaces, a version of Arveson's boundary theorem is proved. We also show that for any TRO extreme state on an operator space, the corresponding Paulsen map can be extended to a pure ucp map on the $C^*$ -algebra generated by the Paulsen system.

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 $X$ .

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be anirreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$,and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the$S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.

We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex.

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 $K_3/\mathbb{Q}$ be a non-normal cubic extension, which is given by anirreducible polynomial $g(x)=x^3+ax^2+bx+c$. In this paper, we study the mean value estimates related to the Dedekind zeta-function $\zeta_{K_3}(s)$. We first introduce a new ingredient to improve a previous result of L\"{u} \cite{lv1}. Moreover, we consider a generalized divisor problem and establish asymptotic formulas.

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.

In this paper, motivated by B. V. R. Bhat, J. Martin Lindsay, M. Mukherjee (Additiveunits of product systems, Trans. Amer. Math. Soc. 370 (2018)), we determine allcontinuous roots of the vacuum unit in the time ordered product system $\mathrm {I}\!\mathrm { \Gamma }(F)$, where $F$ is a two-sided Hilbert module over the $C*$-algebra $B$ of all bounded operators acting on a Hilbert space of finite dimension. Afterwards, we prove that the index of that product system and the Hilbert B--B module of all continuous roots of the vacuum unit are isomorphic as Hilbert two-sided modules.

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 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 $k =\mathbb{Q}\sqrt[5]{n}, \upzeta_5$, 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.

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.

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.