Search results for "LYN"

Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials

2001

Abstract No infinite dimensional Banach space X is known which has the property that for m ⩾2 the Banach space of all continuous m -homogeneous polynomials on X has an unconditional basis. Following a program originally initiated by Gordon and Lewis we study unconditionality in spaces of m -homogeneous polynomials and symmetric tensor products of order m in Banach spaces. We show that for each Banach space X which has a dual with an unconditional basis ( x * i ), the approximable (nuclear) m -homogeneous polynomials on X have an unconditional basis if and only if the monomial basis with respect to ( x * i ) is unconditional. Moreover, we determine an asymptotically correct estimate for the …

Shadow trees of Mandelbrot sets

2003

A multilinear Lindenstrauss theorem

2006

Abstract We show that the set of N -linear mappings on a product of N Banach spaces such that all their Arens extensions attain their norms (at the same element) is norm dense in the space of all bounded N -linear mappings.

Nonlinear systems solver in floating-point arithmetic using LP reduction

2009

This paper presents a new solver for systems of nonlinear equations. Such systems occur in Geometric Constraint Solving, e.g., when dimensioning parts in CAD-CAM, or when computing the topology of sets defined by nonlinear inequalities. The paper does not consider the problem of decomposing the system and assembling solutions of subsystems. It focuses on the numerical resolution of well-constrained systems. Instead of computing an exponential number of coefficients in the tensorial Bernstein basis, we resort to linear programming for computing range bounds of system equations or domain reductions of system variables. Linear programming is performed on a so called Bernstein polytope: though,…

Interpolation and approximation in L2(γ)

2007

Assume a standard Brownian motion W=(W"t)"t"@?"["0","1"], a Borel function f:R->R such that f(W"1)@?L"2, and the standard Gaussian measure @c on the real line. We characterize that f belongs to the Besov space B"2","q^@q(@c)@?(L"2(@c),D"1","2(@c))"@q","q, obtained via the real interpolation method, by the behavior of a"X(f(X"1);@t)@[email protected]?f(W"1)-P"X^@tf(W"1)@?"L"""2, where @t=(t"i)"i"="0^n is a deterministic time net and P"X^@t:L"2->L"2 the orthogonal projection onto a subspace of 'discrete' stochastic integrals x"[email protected]?"i"="1^nv"i"-"1(X"t"""i-X"t"""i"""-"""1) with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the…

On the ∗-cocharacter sequence of 3×3 matrices

2000

Abstract Let M 3 (F) be the algebra of 3×3 matrices with involution * over a field F of characteristic zero. We study the ∗ -polynomial identities of M 3 (F) , where ∗=t is the transpose involution, through the representation theory of the hyperoctahedral group B n . After decomposing the space of multilinear ∗ -polynomial identities of degree n under the B n -action, we determine which irreducible B n -modules appear with non-zero multiplicity. In symbols, we write the nth ∗ -cocharacter χ n (M 3 (F),*)=∑ r=0 n ∑ λ⊢r,h(λ)⩽6 μ⊢n−r,h(μ)⩽3 m λ,μ χ λ,μ , where λ and μ are partitions of r and n−r , respectively, χ λ,μ is the irreducible B n -character associated to the pair (λ,μ) and m λ,μ ⩾0 i…

Ordinary and graded cocharacter of the Jordan algebra of 2x2 upper triangular matrices

2014

Abstract Let F be a field of characteristic zero and U J 2 ( F ) be the Jordan algebra of 2 × 2 upper triangular matrices over F . In this paper we give a complete description of the space of multilinear graded and ordinary identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . For every Z 2 -grading of U J 2 ( F ) we compute the multiplicities in the graded cocharacter sequence and furthermore we compute the ordinary cocharacter.

Polynomial method to study the entanglement of pure N-qubit states

2009

We present a mapping which associates pure N-qubit states with a polynomial. The roots of the polynomial characterize the state completely. Using the properties of the polynomial we construct a way to determine the separability and the number of unentangled qubits of pure N-qubit states.

A Polynomial Quantum Query Lower Bound for the Set Equality Problem

2004

The set equality problem is to tell whether two sets A and B are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any ω(1) query lower bound when sets A and B are given by quantum oracles. We will show that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles \(\Omega(\sqrt[5]{\frac{n}{\ln n}})\) times, where n=|A|=|B|.

The Monadic Quantifier Alternation Hierarchy over Grids and Graphs

2002

AbstractThe monadic second-order quantifier alternation hierarchy over the class of finite graphs is shown to be strict. The proof is based on automata theoretic ideas and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. Considering grids where the width is a function of the height, we prove that the difference between the levels k+1 and k of the monadic hierarchy is witnessed by a set of grids where this function is (k+1)-fold exponential. We then transfer the hierarchy result to the class of directed (or undirected) graphs, using an encoding technique called strong reduction. It is notable that one can obtain sets of graphs which occur arbitrar…