Abstract: We give an account of recent works on Jacquet-Rallis’ approach to the Gan-Gross-Prasad conjecture for unitary groups. We report on the present state of the Jacquet-Rallis relative trace formulae and on some current applications of it. We give also a precise computation of the constant that appears in the statement “Fourier transform and transfer commute up to a constant”. PubDate: 2019-03-07

Abstract: When \(A\in \mathscr{L}(\mathbb {X})\) and \(B\in \mathscr{L}(\mathbb {Y})\) are given, we denote by MC an operator acting on the Banach space \(\mathbb {X}\oplus \mathbb {Y}\) of the form \(M_{C}=\left (\begin {array}{cccccccc} A & C \\ 0 & B \\ \end {array}\right ) \) . In this paper, first we prove that σw(M0) = σw(MC) ∪{S(A∗) ∩ S(B)} and \(\mathbf {\sigma }_{aw}(M_{C})\subseteq \mathbf {\sigma }_{aw}(M_{0})\cup S_{+}^{*}(A)\cup S_{+}(B)\) . Also, we give the necessary and sufficient condition for MC to be obeys property (w). Moreover, we explore how property (w) survive for 2 × 2 upper triangular operator matrices MC. In fact, we prove that if A is polaroid on \(E^{0}(M_{C})=\{\lambda \in \text {iso}\sigma (M_{C}):0<\dim (M_{C}-\lambda )^{-1}\}\) , M0 satisfies property (w), and A and B satisfy either the hypotheses (i) A has SVEP at points \(\mathbf {\lambda }\in \mathbf {\sigma }_{aw}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)\) and A∗ has SVEP at points \(\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)\) , or (ii) A∗ has SVEP at points \(\mathbf {\lambda }\in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)\) and B∗ has SVEP at points \(\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(B)\) , then MC satisfies property (w). Here, the hypothesis that points λ ∈ E0(MC) are poles of A is essential. We prove also that if S(A∗) ∪ S(B∗), points \(\mathbf {\lambda }\in {E_{a}^{0}}(M_{C})\) are poles of A and points \(\mu \in {E_{a}^{0}}(B)\) are poles of B, then MC satisfies property (w). Also, we give an example to illustrate our results. PubDate: 2019-03-06

Abstract: This work establishes the nonuniform Berry-Esseen inequality for coordinate symmetric vectors. The nonuniform Lp (p ≥ 1) bound is also established. The main results are applied to projections of random vectors distributed according to a family of measures on the \({\ell _{r}^{n}}\) sphere and the \({\ell _{r}^{n}}\) ball, including cone measure and volume measure. PubDate: 2019-02-20

Abstract: We study the complex Monge-Ampère equation (ddcu)n = μ in a strictly pseudoconvex domain Ω with the boundary condition u = φ, where φ ∈ C(∂Ω). We provide a nontrivial sufficient condition for continuity of the solution u outside “small sets”. PubDate: 2019-02-19

Abstract: This paper investigates Nash games for stochastic singular systems with Markovian jumps. Based on the generalized Itô’s formula, the corresponding linear quadratic optimal control problem is studied for the first time. Then, we establish the existence of Nash strategies by means of generalized coupled Riccati algebraic equations. As an application, the stochastic H2/H∞ control with state, control, and external disturbance-dependent noise is discussed. PubDate: 2019-02-19

Abstract: Let R and S be polynomial rings of positive dimensions over a field k. Let I ⊆ R, J ⊆ S be non-zero homogeneous ideals none of which contains a linear form. Denote by F the fiber product of I and J in T = R ⊗kS. We compute homological invariants of the powers of F using the data of I and J. Under the assumption that either char k = 0 or I and J are monomial ideals, we provide explicit formulas for the depth and regularity of powers of F. In particular, we establish for all s ≥ 2 the intriguing formula depth(T/Fs) = 0. If moreover each of the ideals I and J is generated in a single degree, we show that for all s ≥ 1, reg Fs = maxi∈[1, s]{reg Ii + s − i, reg Ji + s − i}. Finally, we prove that the linearity defect of F is the maximum of the linearity defects of I and J, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product. PubDate: 2019-02-18

Abstract: It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley–Reisner ring (i.e., a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely generated as algebra over its degree zero piece, and that this fact can be read off in the corresponding simplicial complex; in the infinite case, we exhibit a 1–1 correspondence between potential new generators appearing on each graded piece and certain pairs of faces of such a simplicial complex, and we use it to provide an alternative proof of the fact that these Frobenius algebras can only be either principally generated or infinitely generated. PubDate: 2019-02-15

Abstract: In this paper, we give a new inequality for weighted Lebesgue spaces called Bohr-Nikol’skii inequality, which combines the inequality of Bohr-Favard and the Nikol’skii idea of inequality for functions in different metrics. PubDate: 2019-02-15

Abstract: Let \((A,\mathfrak {m})\) be a Cohen-Macaulay local ring of dimension d and let I be an \(\mathfrak {m}\) -primary ideal. Let G be the associated graded ring of A with respect to I and let \(\mathcal R = A[It,t^{-1}]\) be the extended Rees ring of A with respect to I. Notice t− 1 is a nonzero divisor on \(\mathcal R\) and \(\mathcal R/t^{-1}\mathcal R = G\) . So, we have Bockstein operators \(\beta ^{i} {\colon } {H}^{i}_{{G}_{+}}(G)(-1) \rightarrow {H}^{i + 1}_{{G}_{+}}(G)\) for i ≥ 0. Since βi+ 1(+ 1) ∘ βi = 0, we have Bockstein cohomology modules BHi(G) for i = 0,…,d. In this paper, we show that certain natural conditions on I implies vanishing of some Bockstein cohomology modules. PubDate: 2019-02-13

Abstract: We describe the equations and Gröbner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The ideals of these surfaces are nested in a simple way that allows us to analyze them inductively. We describe explicit Gröbner bases and syzygies for these objects over the integers and this lets us treat them in all characteristics simultaneously. PubDate: 2019-02-13

Abstract: We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This relationship leads to several new results and problems on the containments between these powers. PubDate: 2019-02-12

Abstract: Motivated by a conjecture of Huneke and Wiegand concerning torsion in tensor products of modules over local rings, we investigate the existence of ideals I in a one-dimensional Gorenstein local ring R satisfying \(\text {Ext}^{1}_{R}(I,I)= 0\) . PubDate: 2019-02-09

Abstract: We give a brief biography of Le Van Thiem and a survey on his contributions in mathematics and for the development of mathematics in Vietnam. PubDate: 2019-02-02

Abstract: In this paper, we use polarization to study the behavior of the depth and regularity of a monomial ideal I, locally at a variable xi, when we lower the degree of all the highest powers of the variable xi occurring in the minimal generating set of I, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If I is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of I have non-increasing depth and non-decreasing regularity. In particular, edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter. PubDate: 2019-01-31

Abstract: We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible. PubDate: 2019-01-30

Abstract: This paper derives several unicity results for a class of holomorphic mappings from the disc into compact Riemann surfaces as well as into the complex projective space ℙn(ℂ). This is done by using the Nevanlinna theory for holomorphic maps where the source is a disc developed by Ru-Sibony (to appear). PubDate: 2019-01-29

Abstract: It is known that when we apply a linear multistep method to differential-algebraic equations (DAEs), usually the strict stability of the second characteristic polynomial is required for the zero stability. In this paper, we revisit the use of linear multistep discretizations for a class of structured strangeness-free DAEs. Both explicit and implicit linear multistep schemes can be used as underlying methods. When being applied to an appropriately reformulated form of the DAEs, the methods have the same convergent order and the same stability property as applied to ordinary differential equations (ODEs). In addition, the strict stability of the second characteristic polynomial is no longer required. In particular, for a class of semi-linear DAEs, if the underlying linear multistep method is explicit, then the computational cost may be significantly reduced. Numerical experiments are given to confirm the advantages of the new discretization schemes. PubDate: 2019-01-29

Abstract: We introduce the concept of robust equilibrium in a multi-criteria transportation network and obtain a formula to compute the radius of robustness together with an algorithm to find robust equilibrium flows. PubDate: 2019-01-25

Abstract: In this paper, we study the common null point problem in Banach spaces. Then, using the shrinking projection method and ε-enlargement of maximal monotone operator, we prove two strong convergence theorems with nonsummable errors for solving this problem. PubDate: 2019-01-16