Hypergeometric τ-functions of the q-Painlevé system of type $E_{8}^{(1)}$

We present the τ-functions for the hypergeometric solutions to the q-Painlevé system of type E₈⁽¹⁾E_{8}^{(1)} in a determinant formula whose entries are given by Rahman’s q-hypergeometric integrals. By using the symmetry of the q-hypergeometric integral, we can construct 56 solutions and describe the action of W(E₇⁽¹⁾)W(E_{7}^{(1)}) on the solutions.     

Structure theorems of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-Azumaya algebras

Let k be a field and E(n) be the 2 ⁿ⁺¹-dimensional pointed Hopf algebra over k constructed by Beattie, Dăscălescu and Grünenfelder [J. Algebra, 2000, 225: 743–770]. E(n) is a triangular Hopf algebra with a family of triangular structures R _M parameterized by symmetric matrices M in M _n (k). In this paper, we study the Azumaya algebras in the braided monoidal category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } and obtain the structure theorems for Azumaya algebras in the category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } , where M is any symmetric n×n matrix over k.     

The Fermi-Walker Derivative on the Spherical Indicatrix of Spacelike Curve in Minkowski 3-Space

In this paper Fermi-Walker derivative, Fermi-Walker parallelism, non-rotating frame and Fermi-Walker terms Darboux vector concepts are given along the spherical indicatrix of a spacelike curve with a spacelike or timelike principal normal in \({E_{1}^{3}}\). First, we consider a spacelike curve in the Minkowski space and investigate the Fermi-Walker derivative along the tangent. The concepts with the Fermi-Walker derivative are analyzed along its tangent. Then, the Fermi-Walker derivative and its theorems are analyzed along the principal normal indicatrix and the binormal indicatrix of spacelike curve in \({E_{1}^{3}}\).     

On the facial structure of the unit ball in the dual space of a JB*-triple

It is shown that every proper weak* closed face of the closed unit ball E₁^*{E_1^*} in the dual space of a JB*-triple E coincides with set of all elements in the unit sphere of E* attaining their norm at a unique compact tripotent in E**. In particular every proper weak* closed face of the closed unit ball E₁^*{E_1^*} is weak*-semi-exposed. This result provides an affirmative answer to a conjecture posed over 20 years ago.     

On the Quaternionic B 2-Slant Helices in the Euclidean Space E 4

In this paper we give a new definition of harmonic curvature functions in terms of B ₂ and we define a new kind of slant helix which we call quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ by using the new harmonic curvature functions. Also we define a vector field D which we call Darboux quaternion of the real quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ and we give a new characterization such as: "a: I ì \mathbb R ? E⁴{``\alpha : I \subset {\mathbb R} \rightarrow E^4} is a quaternionic B ₂–slant helix ${\Leftrightarrow H^\prime_2 -KH_{1} = 0"}${\Leftrightarrow H^\prime_2 -KH_{1} = 0"} where H ₂, H ₁ are harmonic curvature functions and K is the principal curvature function of the curve α.     

The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz–Minkowski space $$\mathbb{L}^3$$

We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz–Minkowski space , with fundamental piece having a finite number (n + 1) of singularities, is a real analytic manifold of dimension 3n + 4. The underlying topology agrees with the topology of uniform convergence of graphs on compact subsets of {x ₃ = 0}.      

Local well-posedness for the space–time Monopole equation in Lorenz gauge

It is known from Czubak (Anal PDE 3(2):151–174, 2010) that the space–time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in H^s(\mathbbR²){H^s(\mathbb{R}^2)} for ${s>\frac{1}{4}}${s>\frac{1}{4}}. Here we prove local well-posedness for arbitrary initial data in H^s(\mathbbR²){H^s(\mathbb{R}^2)} with ${s>\frac{1}{4}}${s>\frac{1}{4}} in the Lorenz gauge.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

The upper bound of the $${{\rm L}_{\mu}^2}$$ spectrum

Let M be an n-dimensional complete non-compact Riemannian manifold, dμ = e ^h (x)dV(x) be the weighted measure and \triangle_m{\triangle_{\mu}} be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakry–émery curvature is bounded from below by Ric _m ≥ −(m − 1)K, K ≥ 0, then the bottom of the L_m²{{\rm L}_{\mu}^2} spectrum λ₁(M) is bounded by

Global existence of weak solutions for the Navier�CStokes equations with capillarity on the half-line

In this paper, we construct the global weak solutions to the initial-boundary problem for the Navier–Stokes system with capillarity in the half space \mathbbR₊¹{\mathbb{R}_+^1}. The result extends Eugene Tsyganov’s existence theorem which considered the problem in the finite region published in J. Differential Equaions 245:3936–3955, 2008.     

Discrete series characters for affine Hecke algebras and their formal degrees

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E_n⁽¹⁾{E_{n}^{(1)}} , n=6, 7, 8) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases).     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

Space curves of constant breadth in Minkowski 3-space

In this paper, we study spacelike and timelike curves of constant breadth in Minkowski 3-space. We show that in Minkowski 3-space spacelike and timelike curves of constant breadth are normal, helices, and spherical curves in some special cases. Furthermore, we give that the total torsion of a closed spacelike curve of constant breadth is zero while the total torsion of a simple closed timelike curve is equal to ${2\pi n, (n \in Z)}$ .     

Congruences modulo $$10^3$$ for Euler numbers

For a non-negative integer \(n\), let \(E_n\) be the \(n\) th Euler number. In this note, for any positive integer \(n\), we prove the following congruences:

$$\begin{aligned} {\left\{ \begin{array}{ll} E_{4n} \equiv 380n-375 \pmod {10^3}, \\ E_{4n+2} \equiv -460n+399 \pmod {10^3}. \end{array}\right. } \end{aligned}$$

Our proof is based on induction on \(n\) and elementary direct calculations.

     

Complete maximal spacelike hypersurfaces ofH 1 4 (c)*

In this paper, we prove that the hyperbolic cylinderH ¹(c ₁)×H ²(c ₂) is the only complete maximal spacelike hypersurfaces inH ₁ ⁴ (c) with nonzero constant Gauss-Kronecker curvature and give a characterization of complete maximal spacelike hypersurfaces ofH ₁ ⁴ (c) with constant scalar curvature. The project Supported by NNSFC, FECC and CPF     

Volume estimates in spaces of homogeneous polynomials

Given m we derive upper and lower estimates for the volume of the unit ball of the Banach space , all m-homogeneous polynomials defined on the nth section E _n of a symmetric Banach sequence space E (with constants depending on m only). We apply this result to obtain asymptotically correct orders of various important invariants of from local Banach space theory, as e.g. volume ratios, cotype and type 2 constants, projection constants or Banach–Mazur distances.     

Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures

Wintgen proved (C. R. Acad. Sci. Paris, 288:993–995, 1979) that the Gauss curvature K and the normal curvature K ^D of a surface in Euclidean 4-space \mathbb E⁴{\mathbb {E}^4} satisfy K + |K ^D | ≤ H ², where H ² is the squared mean curvature. A surface in \mathbb E⁴{\mathbb {E}^4} is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in \mathbb E⁴{\mathbb {E}^4} form an important family of surfaces, namely, surfaces with circular ellipse of curvature. In this article, we completely classify Wintgen ideal surfaces in \mathbb E⁴{\mathbb E^4} satisfying |K| = |K ^D | identically.     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.     

A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the w-ideals of E, we construct a fundamental regular semigroup WE called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that WE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to WE whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of WE. Moreover, when E is a biordered set of a semilattice Eo, WE is isomorphic to the Munn-semigroup TEo; and when E is the biordered set of a band B, WE is isomorphic to the Hall-semigroup WB.     

Integral formula of Minkowski type and new characterization of the Wulff shape

Given a positive function F on S^n which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in R^n＋1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in R^n＋1. We give some new characterizations of the Wulff shape by the use of our integral formulas of Minkowski type, in case F=1 which reduces to some well-known results.