MathSciDoc: An Archive for Mathematician ∫

In this paper, we compare some deterministic and probabilistic techniques in the study of upper bounds in problems related to certain mean square discrepancie with respect to balls in the$d$-dimensional unit torus, and show that the quality of these techniques depends in an intricate way on the dimension$d$under consideration.

We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A. Tsikh for the case of a reduced complete intersection.

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ℂ^{$N$}. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter,$Math. Ann.$$337$(2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ℂ^{$N$}. Our results extend to certain weights and measures defined on cones in ℝ^{$N$}.

We prove that spherical spectral analysis and synthesis hold in Damek–Ricci spaces and derive two-radius theorems.

We study the tropical lines contained in smooth tropical surfaces in ℝ^{3}. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines.

We construct a 2-dimensional complex manifold$X$which is the increasing union of proper subdomains that are biholomorphic to ℂ^{2}, but$X$is not Stein.

The main facts about Hausdorff and packing measures and dimensions of a Borel set$E$are revisited, using$determining$set functions $\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)$ , where $\mathcal{B}_E$ is the family of all balls centred on$E$and α is a real parameter. With mild assumptions on φ_{α}, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set$V$in ℝ^{$D$}, these notions are used to introduce the$interior$and$exterior$measures and dimensions of any Borel subset of ∂$V$. We stress that these dimensions depend on the choice of φ_{α}. Two determining functions are considered, φ_{α}($B$)=Vol_{$D$}($B$∩$V$)diam($B$)^{α-$D$}and φ_{α}($B$)=Vol_{$D$}($B$∩$V$)^{α/$D$}, where Vol_{$D$}denotes the$D$-dimensional volume.

We show that the transition probability of the Markov chain ($G$($i$,1),...,$G$($i$,$n$))_{$i$≥1}, where the$G$($i$,$j$)’s are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for$G$($m$,$n$). We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.

We describe$T$-equivariant Schubert calculus on$G$($k$,$n$),$T$being an$n$-dimensional torus, through derivations on the exterior algebra of a free$A$-module of rank$n$, where$A$is the$T$-equivariant cohomology of a point. In particular,$T$-equivariant Pieri’s formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Giambelli and determinantal restriction formulas for the Grassmannian,$Pure Appl. Math. Quart.$$2$(2006), 699–717).

A Łojasiewicz-type estimate is a powerful tool in studying the rigidity properties of the harmonic map heat flow. Topping proved such an estimate using the Riesz potential method, and established various uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to $\mathbb{S}^{2}$ ($J. Differential Geom.$$45$(1997), 593–610). In this note, using an inequality due to Sobolev, we will derive the same estimate for maps from $\mathbb{S}^{2}$ to a compact Kähler manifold$N$with nonnegative holomorphic bisectional curvature, and use it to establish the uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to$N$, which generalizes Topping’s result.