A generalization of the finiteness problem of the local cohomology modules

Let R be a commutative Noetherian ring and \(\mathfrak{a}\) an ideal of R. We introduce the concept of \(\mathfrak{a}\) -weakly Laskerian R-modules, and we show that if M is an \(\mathfrak{a}\) -weakly Laskerian R-module and s is a non-negative integer such that Ext _R ^j \((R/\mathfrak{a},H_\mathfrak{a}^i (M))\) is \(\mathfrak{a}\) -weakly Laskerian for all i < s and all j, then for any \(\mathfrak{a}\) -weakly Laskerian submodule X of \(H_\mathfrak{a}^s (M)\) , the R-module \(Hom_R (R/\mathfrak{a},H_\mathfrak{a}^s (M)/X)\) is \(\mathfrak{a}\) -weakly Laskerian. In particular, the set of associated primes of \(H_\mathfrak{a}^s (M)/X\) is finite. As a consequence, it follows that if M is a finitely generated R-module and N is an \(\mathfrak{a}\) -weakly Laskerian R-module such that \(H_\mathfrak{a}^i (N)\) (N) is \(\mathfrak{a}\) -weakly Laskerian for all i < s, then the set of associated primes of \(H_\mathfrak{a}^s (M,N)\) (M,N) is finite. This generalizes the main result of S. Sohrabi Laleh, M.Y. Sadeghi, and M.Hanifi Mostaghim (2012).     

On Bipartite Distance-Regular Graphs with Intersection Numbers c i = (q i ? 1)/(q ? 1)

Let q denote an integer at least two. Let ?? denote a bipartite distance-regular graph with diameter D ?? 3 and intersection numbers c _i = (q ⁱ ? 1)/(q ? 1), 1 ?? i ?? D. Let X denote the vertex set of ?? and let ${V = \mathbb{C}^X}$ denote the vector space over ${\mathbb{C}}$ consisting of column vectors whose coordinates are indexed by X and whose entries are in ${\mathbb{C}}$ . For ${z \in X}$ , let ${{\hat z}}$ denote the vector in V with a 1 in the z-coordinate and 0 in all other coordinates. Fix ${x, y \in X}$ such that ?(x, y) = 2, where ? denotes the path-length distance function. For 0 ?? i, j ?? D define ${w_{ij} = \sum {\hat z}}$ , where the sum is over all ${z \in X}$ such that ?(x, z) = i and ?(y, z) = j. We define W?=?span{w _ij | 0 ?? i, j ?? D}. In this paper we consider the space ${MW={\rm span} \{mw \mid m \in M, w \in W\}}$ , where M is the Bose?CMesner algebra of ??. We observe that MW is the minimal A-invariant subspace of V which contains W, where A is the adjacency matrix of ??. We give a basis for MW that is orthogonal with respect to the Hermitean dot product. We compute the square-norm of each basis vector. We compute the action of A on the basis. For the case in which ?? is the dual polar graph D _D (q) we show that the basis consists of the characteristic vectors of the orbits of the stabilizer of x and y in the automorphism group of ??.     

A Note on Drazin Invertibility for Upper Triangular Block Operators

A bounded linear operator A acting on a Banach space X is said to be an upper triangular block operators of order n, and we write ${A \in \mathcal{UT}_{n}(X)}$ , if there exists a decomposition of ${X = X_{1} \oplus . . . \oplus X_{n}}$ and an n × n matrix operator ${(A_{i,j})_{\rm 1 \leq i, j \leq n}}$ such that ${A = (A_{i, j})_{1 \leq i, j \leq n}, A_{i, j} = 0}$ for i > j. In this note we characterize a large set of entries A _{i, j} with j > i such that ${\sigma_{\rm D} (A) = {\bigcup\limits_{i = 1}^{n}} \sigma_{\rm D} (A_{i, i})}$ ; where σ_D(.) is the Drazin spectrum. Some applications concerning the Fredholm theory and meromorphic operators are given.     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .     

A remark on Nesterenko’s theorem for Ramanujan functions

We study the algebraic independence of values of the Ramanujan q-series $A_{2j+1}(q)=\sum_{n=1}^{\infty}n^{2j+1}q^{2n}/(1-q^{2n})$ or S _2j+1(q) (j≥0). It is proved that, for any distinct positive integers i, j satisfying $(i,j)\not=(1,3)$ and for any $q\in \overline{ \mathbb{Q}}$ with 0<|q|<1, the numbers A ₁(q), A _2i+1(q), A _2j+1(q) are algebraically independent over $\overline{ \mathbb{Q}}$ . Furthermore, the q-series A _2i+1(q) and A _2j+1(q) are algebraically dependent over $\overline{ \mathbb{Q}}(q)$ if and only if (i,j)=(1,3).     

On the annihilators and attached primes of top local cohomology modules

Let ${\mathfrak{a}}$ be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}$ , where ${T_R(\mathfrak{a}, M)}$ is the largest submodule of M such that ${{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}$ . Several applications of this result are given. Among other things, it is shown that there exists an ideal ${\mathfrak{b}}$ of R such that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}$ . Using this, we show that if ${ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}$ , then ${{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}$ These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]).     

Supercritical Holes for the Doubling Map

For a map \({S : X \to X}\) and an open connected set (= a hole) \({H \subset X}\) we define \({\mathcal{J}_H(S)}\) to be the set of points in X whose S-orbit avoids H. We say that a hole H ₀ is supercritical if

1. for any hole H such that \({\overline{H}_0 \subset H}\) the set \({\mathcal{J}_H(S)}\) is either empty or contains only fixed points of S;

1. for any hole H such that \({\overline{H} \subset H_0}\) the Hausdorff dimension of \({\mathcal{J}_H(S)}\) is positive.

The purpose of this note is to completely characterize all supercritical holes for the doubling map Tx =  2x mod 1.     

Bulk universality for generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν _ij with a subexponential decay. Let ${\sigma_{ij}^2}$ be the variance for the probability measure ν _ij with the normalization property that ${\sum_{i} \sigma^2_{ij} = 1}$ for all j. Under essentially the only condition that ${c\le N \sigma_{ij}^2 \le c^{-1}}$ for some constant c?>?0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M ^?1.     

Existence of nontrivial solutions of linear functional equations

In this paper we study the functional equation $$\sum_{i=1}^n a_i f(b_i x+c_i h)=0 \quad (x, h \in \mathbb{C})$$ where a _i , b _i , c _i are fixed complex numbers and \({f \colon \mathbb{C} \to \mathbb{C}}\) is the unknown function. We show, that if there is i such that \({b_i / c_i \neq b_j /c_j}\) holds for any \({1 \leq j \leq n,\ j \neq i}\) , the functional equation has a nonconstant solution if and only if there are field automorphisms \({\phi_1, \ldots, \phi_k}\) of \({\mathbb{C}}\) such that \({\phi_1 \cdots \phi_k}\) is a solution of the equation.     

Equilibrium Problems for Infinite Dimensional Vector Potentials with External Fields

We consider a minimal energy problem with an external field over noncompact classes of infinite dimensional vector measures $(\mu^i)_{i\in I}$ on a locally compact space. The components μ ⁱ are positive measures normalized by $\int g_i\,d\mu^i=a_i$ (where a _i and g _i are given) and supported by closed sets A _i with the sign +?1 or ??1 prescribed such that A _i ?∩?A _j ?=?? whenever ${\rm sign}\,A_i\ne{\rm sign}\,A_j$ , and the law of interaction between μ ⁱ , i?∈?I, is determined by the matrix $\bigl({\rm sign}\,A_i\,{\rm sign}\,A_j\bigr)_{i,j\in I}$ . For positive definite kernels satisfying Fuglede’s condition of consistency, sufficient conditions for the existence of equilibrium measures are established and properties of their uniqueness, vague compactness, and strong and vague continuity are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also obtain variational inequalities for the weighted equilibrium potentials, single out their characteristic properties, and analyze continuity of the equilibrium constants. The results hold, e.g., for classical kernels in $\mathbb R^n$ , $n\geqslant 2$ , which is important in applications.     

Finite groups with \mathbb{P }-subnormal subgroups

A subgroup $H$ of a group $G$ is called $\mathbb{P }$ -subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset \cdots \subset H_n=G$ such that $|H_i:H_{i-1}|$  is a prime for all $i$ . In this paper we study groups with $\mathbb{P }$ -subnormal 2-maximal subgroups, and groups with $\mathbb{P }$ -subnormal primary cyclic subgroups.     

Null curves in {\mathbb{C}^3} and Calabi–Yau conjectures

For any open orientable surface M and convex domain ${\Omega\subset \mathbb{C}^3,}$ there exist a Riemann surface N homeomorphic to M and a complete proper null curve F : N → Ω. This result follows from a general existence theorem with many applications. Among them, the followings:

For any convex domain Ω in ${\mathbb{C}^2}$ there exist a Riemann surface N homeomorphic to M and a complete proper holomorphic immersion F : N → Ω. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and Ω is the solid right cylinder ${\{x \in \mathbb{C}^2 \,|\, \mbox{Re}(x) \in D\},}$ then F can be chosen so that Re(F) : N → D is proper.

There exist a Riemann surface N homeomorphic to M and a complete bounded holomorphic null immersion ${F:N \to {\rm SL}(2, \mathbb{C}).}$

There exists a complete bounded CMC-1 immersion ${X:M \to \mathbb{H}^3.}$

For any convex domain ${\Omega \subset \mathbb{R}^3}$ there exists a complete proper minimal immersion (X _j ) _j=1,2,3 : M → Ω with vanishing flux. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and ${\Omega=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 \,|\, (x_1,x_2) \in D\},}$ then X can be chosen so that (X ₁, X ₂) : M → D is proper.

Any of the above surfaces can be chosen with hyperbolic conformal structure.     

Connectivity of Strong Products of Graphs

The strong product ${G\boxtimes H}$ of graphs G = (V ₁, E ₁) and H = (V ₂, E ₂) is the graph with vertex set ${V(G \boxtimes H)=V_1\times V_2}$ , where two distinct vertices ${(x_1, x_2), (y_1, y_2)\in V_1\times V_2}$ are adjacent in ${G\boxtimes H}$ if and only if x _i  = y _i or ${x_i y_i\in E_i}$ for i = 1, 2. We introduce so called I-sets and L-sets in the strong product ${G\boxtimes H}$ and prove that every minimum separating set in ${G\boxtimes H}$ is either an I-set or an L-set in ${G\boxtimes H}$ . Some bounds and exact results for connectivity of strong products follow from this characterization. The result is then generalized to an arbitrary number of factors in the strong product.     

Structural and geometric characteristics of sets of convergence and divergence of multiple Fourier series with J k -lacunary sequence of rectangular partial sums

We study multiple trigonometric Fourier series of functions f in the classes $L_p \left( {\mathbb{T}^N } \right)$ , p > 1, which equal zero on some set $\mathfrak{A}, \mathfrak{A} \subset \mathbb{T}^N , \mu \mathfrak{A} > 0$ (µ is the Lebesgue measure), $\mathbb{T}^N = \left[ { - \pi ,\pi } \right]^N$ , N ≥ 3. We consider the case when rectangular partial sums of the indicated Fourier series S _n (x; f) have index n = (n ₁, ..., n _N ) ∈ ? ^N , in which k (k ≥ 1) components on the places {j ₁, ..., j _k } = J _k ? {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with J _k -lacunary sequence of partial sums). A correlation is found of the number k and location (the “sample” J _k ) of lacunary sequences in the index n with the structural and geometric characteristics of the set $\mathfrak{A}$ , which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure $\mathfrak{A}_1$ of the set $\mathfrak{A}$ .     

On the images of horodisks under holomorphic self-maps of the unit disk

Suppose that f is a holomorphic self map of the unit disk ${\mathbb{D}}$ . Recently several monotonicity results related to the image of smaller disks under f have been proved. These results extend the classical Schwarz lemma in various ways. We prove analogous monotonicity results in the context of Julia’s boundary Schwarz lemma. A horodisk is a disk internally tangent to the unit circle. For positive ${\lambda}$ , we denote by ${H_{\lambda}}$ the disk of radius ${\lambda/(1\,+\,\lambda)}$ centered at the point ${1/(1\,+\,\lambda)}$ . This is a horodisk that touches the unit circle at the point 1. Suppose that f(1) = 1 (in the sense of radial limit) and denote by ${f^{\prime}(1)}$ the angular derivative. By Julia’s lemma ${f(H_{\lambda})\,\subset H_{{\lambda}f^{\prime}(1)}}$ . Let ${\Psi_f(\lambda)\,=\,\inf\,\{\rho > 0 : f(H_{\lambda}) \subset H_\rho\}}$ . We show that the function ${\Psi_f(\lambda)/\lambda}$ is a decreasing function of ${\lambda}$ and that ${\lim_{\lambda\,\to\,0+} \Psi_f(\lambda)/\lambda = f^\prime(1)}$ . This result implies that the constant ${f^\prime(1)}$ in Julia’s lemma is the best possible.     

A spinorial characterization of hyperspheres

Let M be a compact orientable n-dimensional hypersurface, with nowhere vanishing mean curvature H, immersed in a Riemannian spin manifold ${\overline{M}}$ admitting a non trivial parallel spinor field. Then the first eigenvalue ${\lambda_1(D_{M}^{H})}$ (with the lowest absolute value) of the Dirac operator ${D_{M}^{H}}$ corresponding to the conformal metric ${\langle\;,\;\rangle^{H}=H^{2}\,\langle\;,\;\rangle}$ , where ${\langle\;,\;\rangle}$ is the induced metric on M, satisfies ${\left|\lambda_1(D_{M}^{H})\right|\le \frac{n}{2}}$ . By applying the Bourguignon-Gauduchon first variational formula, we obtain a necessary condition for ${\left|\lambda_1(D_{M}^{H})\right|=\frac{n}{2}}$ . As a consequence, we prove that round hyperspheres are the only hypersurfaces of the Euclidean space satisfying the equality in the Bär inequality $$\lambda_1(D_{M})^{2}\le \frac{n^{2}}{4{vol}(M)}\int_{M} H^{2}\, dV,$$ where D _M stands now for the Dirac operator of the induced metric.     

Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness

In this paper we study viscosity solutions to the system $$\begin{array}{ll} \min \{ -\mathcal{H}u_i(x,t)-\psi _i(x,t),u_i(x,t) - \max_{j \neq i} (-c_{i ,j} (x,t) + u_j (x,t)) \} = 0,\\ u_i(x,T)=g_i (x), \, i \in \{1,\ldots , d \},\end{array}$$ where \({(x,t)\in{\mathbb{R}}^{N} \times [0,T]}\) . Concerning \({{\mathcal{H}}}\) , we assume that \({{\mathcal{H}}={\mathcal{L}}+{\mathcal{I}}}\) where \({{\mathcal{L}}}\) is a linear, possibly degenerate, parabolic operator of second order and \({{\mathcal{I}}}\) is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when the dynamics of the underlying state variables is described by an N-dimensional Levy process. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the data, i.e., on the operator \({{\mathcal{H}}}\) and on the continuous functions \({\psi_i}\) , c _i,j , and g _i . Using the comparison principle we prove the existence of a unique viscosity solution (u ₁, . . . , u _d ) to the system by Perron’s method. Our main contribution is that we establish existence and uniqueness of viscosity solutions, in the setting of Levy processes and non-local operators, with no sign assumption on the switching costs {c _{i, j} } and allowing c _{i, j} to depend on x as well as t.     

Some new classes of Kadison-Singer lattices in Hilbert spaces

Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).     

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

Given an f-structure ${\varphi}$ on a manifold M, together with a compatible metric g and connection ${\nabla}$ on M, we construct an odd firstorder differential operator D whose principal symbol is of the type considered in [13]. In the special case of a CR-integrable almost ${\mathcal {S}}$ -structure, we show that when ${\nabla}$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator D is given by D = ${{\sqrt {2} (\overline {\partial}_b + \overline{\partial}_{b}^{\ast})}}$ , where ${\overline {\partial}_b}$ is the tangential Cauchy-Riemann operator. We then describe two types of “quantization” of manifolds with f-structure that reduce to familiar methods in symplectic geometry in the case that ${\varphi}$ is a compatible almost complex structure, and to the contact quantizations defined in [16] when ${\varphi}$ comes from a contact metric structure.