Nash multiplicities and isolated points of maximum multiplicity

Let X be an algebraic variety defined over a field of characteristic zero, and let $\xi \in \underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$ be a point in the closed subset of maximum multiplicity of X. We provide a criterion, given in terms of arcs, to determine whether ξ is isolated in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$. More precisely, we use invariants of arcs derived from the Nash multiplicity sequence to characterize when ξ is an isolated point in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$.     

A sharper uncertainty principle

The work strengthens the result established by L. Cohen on uncertainty principle involving phase derivative. We propose stronger uncertainty principles not only in the classical setting for Fourier transform, but also for self-adjoint operators. We also deduce the conditions that give rise to the equal relation of the uncertainty principle. Examples are provided to show that the new uncertainty principle is truly sharper than the existing ones in literature.     

Euler–Goursat-like formula via Laplace–Borel duality

The Goursat formula for the hypergeometric function extends the Euler–Gauss relation to the case of logarithmic singularities.     

3D Uranyl Organic Frameworks Supported by Rigid Octadentate Carboxylate Ligand: Synthesis,Structure Diversity,and Luminescence Properties

Seven three dimensional (3D) uranyl organic frameworks (UOFs), formulated as [NH₄][(UO₂)₃(HTTDS)(H₂O)] ( 1 ), [(UO₂)₄(HTTDS)₂](HIM)₆ ( 2 , IM=imidazole), [(UO₂)₄(TTDS)(H₂O)₂(Phen)₂] ( 3 , Phen=1,10-phenanthroline), [Zn(H₂O)₄]_0.5[(UO₂)₃(HTTDS)(H₂O)₄] ( 4 ), and {(UO₂)₂[Zn(H₂O)₃]₂(TTDS)} ( 5 ), {Zn(UO₂)₂(H₂O)(Dib)_0.5(HDib)(HTTDS)} ( 6 , Dib=1,4-di(1H-imidazol-1-yl)benzene) and [Na]{(UO₂)₄[Cu₃(u₃-OH)(H₂O)₇](TTDS)₂} ( 7 ) have been hydrothermally prepared using a rigid octadentate carboxylate ligand, tetrakis(3,5-dicarboxyphenyl)silicon(H₈TTDS). These UOFs have different 3D self-assembled structures as a function of co-ligands, structure-directing agents and transition metals. The structure of 1 has an infinite ribbon formed by the UO₇ pentagonal bipyramid bridged by carboxylate groups. With further introduction of auxiliary N-donor ligands, different structure of 2 and 3 are formed, in 2 the imidazole serves as space filler, while in 3 the Phen are bound to [UO₂]²⁺ units as co-ligands. The second metal centers were introduced in the syntheses of 4–7 , and in all cases, they are part of the final structures, either as a counterion ( 4 ) or as a component of framework ( 5 − 7 ). Interesting, in 7 , a rare polyoxometalate [Cu₃(μ₃-OH)O₇(O₂CR)₄] cluster was found in the structure. It acts as an inorganic building unit together with the dimer [(UO₂)₂(O₂CR)₄] unit. Those uranyl carboxylates were sufficiently determined by single crystal X-ray diffraction, and their topological structures and luminescence properties were analyzed in detail.     

Influence of pressure on electron spectra of metals and semiconductors

Abstract

We have already reported the results of direct observations of electron-topological phase transition (ETT) in cadmium'. The appearance of new dHvA-frequencies corresponding to the Fermi surface (FS) change, i.e. restoring of folding of hole “monster” and electron “needle” appearance is observed under pressure. In t h i s report we are going to enlarge on the ETT consequences study in cadmium-on the advent of anomalous electronic features in transverse magnetoresistance and thermoelectric power.     

Taylor Expansions of Distributions

Classical Taylor expansions of holomorphic functions in the complex plane are extended to distributions in Rⁿand in domains     

The lifespan of solutions to the inviscid 3D Boussinesq system

In this paper, we consider the inviscid 3D Boussinesq system in Besov spaces and obtain the lower bound for the lifespan of solutions.     

Interior C 1, γ-Regularity for Weak Solutions of Nonlinear Second Order Parabolic Systems

The interior C ^{γ, γ/2}-regularity for the first gradient with respect to the space variables of a weak solution to a class of nonlinear second order parabolic systems is proven under the assumption that oscillations of coefficients are controlled by the ellipticity constant.     

Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions

In this paper we study the family of embeddings ${\mathrm{\Phi }}_t$ of a compact ${\mathrm{RCD}}^{?}(K,N)$ space $(X,\mathsf{d},\mathfrak{m})$ into $L^2(X,\mathfrak{m})$ via eigenmaps. Extending part of the classical results [10], [11] known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaled pull-back metrics ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ in $L^2(X,\mathfrak{m})$ induced by ${\mathrm{\Phi }}_t$. Moreover we discuss the behavior of ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.