Real hypersurfaces of quaternionic projective space satisfying\nabla _{U_i } A = 0

We classify real hypersurfaces of quaternionic projective space satisfying , i=1,2,3.Dedicated to Prof. Nikolaus Stephanidis on his 65th birthday.Research partially supported by DGICYT Grant PS87-0115-CO3-02.     

Classification of multiple unilateral weighted shifts by $$\mathbb{A}_{\aleph _0 }$$

In this paper, it is characterized when a multiple unilateral weighted shift belongs to the classes \mathbbA_n ( 1 \leqslant n \leqslant à₀ )\mathbb{A}_n \left( {1 \leqslant n \leqslant \aleph _0 } \right). As a result, we perfect and generalize the previous conclusions given by H. Bercovici, C. Foias, and C. Pearcy. Moreover, we remark that Question 21 posed by Shields has been negatively answered.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

On integral graphs which belong to the class\overline {\alpha K_{a,a} \cup \beta {\rm K}_{b,b} }

LetG be a simple graph and letG denote its complement. We say thatG is integral if its spectrum consists of integral values. In this work we establish a characterization of integral graphs which belong to the class $\overline {\alpha K_{a,a} \cup \beta {\rm K}_{b,b} } $ wheremG denotes them-fold union of the graphG.     

Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.     

A Version of the {\cal G} \rm{-conditional Bipolar Theorem in} {L^0 (\mathbb{R}^{d}_{+};\Omega}, {\cal F},\mathbb{P})

Motivated by applications in financial mathematics, Ref. 3 showed that, although fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of : if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extended by Ref. 1 in the multidimensional case, replacing by a closed convex cone K of [0, )^d, and by Ref. 12 who provided a conditional version in the unidimensional case. In this paper, we show that the conditional bipolar theorem of Ref. 12 can be extended to the multidimensional case. Using a decomposition result obtained in Ref. 3 and Ref. 1, we also remove the boundedness assumption of Ref. 12 in the one dimensional case and provide less restrictive assumptions in the multidimensional case. These assumptions are completely removed in the case of polyhedral cones K.     

On finite 2-groups of the type $ {\mathbb{Z}_2} \wr G $

A wreath product of the type \mathbbZ₂ \wr G {\mathbb{Z}_2} \wr G is considered for any finite 2-group G. The monomorphism of such a group in the well-known Kaloujnine group P _2,m is studied for a suitable natural m.     

An analogue of the Moutard transformation for the Goursat equation\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y }

We present a new Bäcklund-type transformation for the nonlinear equation $\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } $ studied by É. Goursat. Goursat found a linearization transformation and some properties of this equation, which make it similar to the Moutard equation u_xy=M(x, y)u. However, this Goursat transformation does not provide proper superposition formulas. We give the necessary extended superposition formulas.     

Effect of a random real on\kappa \to \left( {\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right)} \right)^2

It is consistent that $\kappa \to (\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right))^2 $ holds in the random extension.     

A lemma on binomial coefficients and applications to Lee weights modulo 2 e of codes over {\mathbb{Z}_4}

In this work, we prove that if C is a free ${\mathbb{Z}_4}$ -module of rank k in ${\mathbb{Z}_4^n}$ , and ${j\in \mathbb{Z}}$ and e ≥ 1, then the number of codewords in C with Lee weight congruent to j modulo 2 ^e is divisible by ${2^{\left \lfloor \large{\frac{k-2^{e-2}}{2^{e-2}}} \right \rfloor}}$ . We prove this result by introducing a lemma and applying the lemma in one of the theorems proved by Wilson. The method used is different than the one used in our previous work on Lee weight enumerators in which more general results were obtained. Moreover, Wilson’s methods are used to prove that the results obtained are sharp by calculating the power of 2 that divides the number of codewords in the trivial code ${\mathbb{Z}_{4}^k}$ with Lee weight congruent to j modulo 2 ^e .     

On the\left| {\bar N,p_n } \right|_k summability factors for infinite seriessummability factors for infinite series

In this paper a theorem on \(\left| {\bar N,p_n } \right|_k \) summability factors of infinite series, which generalizes a theorem of Bor [2], has been proved.     

Moduli Spaces of Stable Polygons and Symplectic Structures on\overline {\mathcal{M}} _{0,n}

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to , brings up a new tool to study the Kähler topology of . A wild but precise conjecture on the shape of the Kähler cone of is given in the end.     

The Isometric Extension of an Into Mapping from the Unit Sphere S（t（2）^∞） to S（L^1（μ））

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.     

Antibasis theorems for {\Pi^0_1} classes and the jump hierarchy

We prove two antibasis theorems for ${\Pi^0_1}$ classes. The first is a jump inversion theorem for ${\Pi^0_1}$ classes with respect to the global structure of the Turing degrees. For any ${P\subseteq 2^\omega}$ , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists ${A \in P}$ of degree a. For any degree ${{\bf a \geq 0'}}$ , let ${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$ . We prove that, for any ${{\bf a \geq 0'}}$ and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$ then P contains a member of every degree. For any degree ${{\bf a \geq 0'}}$ such that a is recursively enumerable (r.e.) in 0', let ${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$ . The second theorem concerns the degrees below 0'. We prove that for any ${{\bf a\geq 0'}}$ which is recursively enumerable in 0' and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$ then P contains a member of every degree.     

The exact order of\sum\nolimits_{k = 1}^{\text{n}} {|{\text{x - x}}_{\text{k}} } |^8 |{\text{l}}_{\text{k}} ({\text{x)|}}^{\text{t}}

In this paper we give the exact order of \(\sum\nolimits_{k = 1}^{\text{n}} {|{\text{x - x}}_{\text{k}} } |^5 .\) for any fixed nonnegative integers s and t, which is n^?s, n^?s lnn and n^1?t for s≤t?2, s=t?1 and s≥t, respectively.