Clifford-algebraic random walks on the hypercube

The n-dimensional hypercube is a simple graph on 2ⁿ vertices labeled by binary strings, or words, of length n. Pairs of vertices are adjacent if and only if they differ in exactly one position as binary words; i.e., the Hamming distance between the words is one. A discrete-time random walk is easily defined on the hypercube by “flipping” a randomly selected digit from 0 to 1 or vice-versa at each time step. By associating the words as blades in a Clifford algebra of particular signature, combinatorial properties of the geometric product can be used to represent this random walk as a sequence within the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k ≥ 0 by a formal power series expansion of elements in the algebra. Furthermore, by inducing a walk on a larger Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Moreover, because the Clifford algebras used in the current work are canonically isomorphic to fermion algebras, everything appearing here can be rewritten using fermion creation/annihilation operators, making the discussion relevant to quantum mechanics and/or quantum computing.     

Weighted Inequalities for the Generalized Maximal Operator in Martingale Spaces

The generalized maximal operator M in martingale spaces is considered. For 1 < p ≤ q < ∞, the authors give a necessary and sufficient condition on the pair ([^(m)]\hat \mu , v) for M to be a bounded operator from martingale space L ^p (v) into L ^q ([^(m)]\hat \mu ) or weak-L ^q ([^(m)]\hat \mu ), where [^(m)]\hat \mu is a measure on Ω × ℕ and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.     

Unitary Spaces on Clifford Algebras

For the complex Clifford algebra (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These representations take into account the structure of unitary space on Clifford algebra. The work of N.M. is supported in part by the Russian President’s grant NSh-6705.2006.1.     

Higher order commutators for a class of rough operators

In this paper we study the (L ^p (u ^p ),L ^q (v ^q )) boundedness of the higher order commutatorsT _Ω,α,b ^m andM _Ω,α,b ^m formed by the fractional integral operatorT _Ω,α , the fractional maximal operatorM _Ω,α , and a functionb(x) in BMO(v), respectively. Our results imporve and extend the corresponding results obtained by Segovia and Torrea in 1993 [9]. The project was supported by NNSF of China.     

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> Estimates of solution for <Emphasis Type="Italic">m</Emphasis>-Laplacian parabolic equation with a nonlocal term

In this paper, we consider the global existence, uniqueness and L ^∞ estimates of weak solutions to quasilinear parabolic equation of m-Laplacian type u _t − div(|∇u| ^m−2∇u) = u|u| ^β−1 ∫_Ω |u| ^α dx in Ω × (0,∞) with zero Dirichlet boundary condition in tdΩ. Further, we obtain the L ^∞ estimate of the solution u(t) and ∇u(t) for t > 0 with the initial data u ₀ ∈ L ^q (Ω) (q > 1), and the case α + β < m − 1.     

Bochner algebras and their compact multipliers

Let \mathfrakA\mathfrak{A} be a normed algebra with identity, Ω be a locally compact Hausdorf space and λ be a positive Radon measure on Ω with supp(λ) = Ω. In this paper, we establish a necessary and sufficient condition for L ¹(Ω, \mathfrakA\mathfrak{A}) to be an algebra with pointwise multiplication. Under this condition, we then characterize compact and weakly compact left multipliers on L ¹(Ω, \mathfrakA\mathfrak{A}).     

Clifford algebra and the Lorentz transformation with (p, q) form

In this paper, the concepts of Lorentz inner product with (p, q) form, the Lorentz space and the Lorentz transformation with (p, q) form are given by using Clifford algebra. It is shown that L_m^p,q is the Lorentz transformation with (p, q) form, and the matrix equality relation of Minkowski space with (n − 1, 1) form is given. The examples are given to illustrate the corresponding results.     

On generalized energy equality of the Navier-Stokes equations

We show that for every initial dataa εL ²(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL ^s (0,T;L ^q (Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL ^s (ε, T; L ^q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.     

H ∞ -calculus for the Stokes operator on L q -spaces

 It is proved that the Stokes operator on a bounded domain, an exterior domain, or a perturbed half-space Ω admits a bounded H ^∞ -calculus on L _q (Ω) if q(1,∞). Received: 25 January 2002; in final form: 2 October 2002 / Published online: 16 May 2003     

The embedding ofU q(sl (2)) and sine algebras in generalized Clifford algebras

In this note, we establish the connection between certain quantum algebras and generalized Clifford algebras (GCA). Precisely, we embed the quantum tori Lie algebra andU _q(sl (2)) in GCA.     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

---

Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

For a locally compact group G, L^1 (G) is its group algebra and L^∞(G) is the dual of L^1 (G).Lau has studied the bounded linear operators T:L^∞(G)→L^∞(G) which commute with convolutions and translations. For a subspace H of L^∞(G), we know that M(L^∞(G),H), the Banach algebra of all bounded linear operators on L^∞(G) into H which commute with convolutions, has been studied by Pyre and Lau. In this paper, we generalize these problems to L(K)^*, the dual of a hypergroup algebra L(K) in a very general setting, i.e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L^1(G) but also most of the semigroup algebras.Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however,we succeed in getting some interesting results.     

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

The non commuting matrix elements of matrices from quantum groupGL _q (2;C) withq≡ω being then-th root of unity are given a representation as operators in Hilbert space with help ofC ₄ ⁽ⁿ⁾ generalized Clifford algebra generators appropriately tensored with unit 2×2 matrix infinitely many times. Specific properties of such a representation are presented. Relevance of generalized Pauli algebra to azimuthal quantization of angular momentum alà Lévy-Leblond [10] and to polar decomposition ofSU _q (2;C) quantum algebra alà Chaichian and Ellinas [6] is also commented. The case ofq∈C, |q|=1 may be treated parallely.     

Exponential actions and maximal ?-invariant ideals

Let V be an exponential ?-module, ? being an exponential Lie algebra. Put ? = exp ?. Then every orbit of V under the action of ? admits a closed orbit in its closure. If G= exp ? is a nilpotent Lie group and ? an exponential algebra of derivations of ?, then ? = exp ? acts on G, L ¹(G), (?) and the maximal ?-invariant ideals of L ¹(G), resp. of (?) coincide with the kernels Ker Ω, resp. Ker Ω∩ (?), where Ω is a closed orbit of ?^*. Received: 6 December 1996 / Revised version: 7 December 1997     

Laplacian Operators and Radon Transforms on Grassmann Graphs

Let Ω be a vector space over a finite field with q elements. Let G denote the general linear group of automorphisms of Ω and let us consider the left regular representation associated with the natural action of G on the set X of linear subspaces of Ω. In this paper we study a natural basis B of the algebra End_G(L ₂(X)) of intertwining maps on L ₂(X). By using a Laplacian operator on Grassmann graphs, we identify the kernels in B as solutions of a basic hypergeometric difference equation. This provides two expressions for these kernels. One in terms of the q-Hahn polynomials and the other by means of a Rodrigues type formula. Finally, we obtain a useful product formula for the mappings in B. We give two different proofs. One uses the theory of classical hypergeometric polynomials and the other is supported by a characterization of spherical functions in finite symmetric spaces. Both proofs require the use of certain associated Radon transforms.     

Clifford-Hermite-monogenic operators

In this paper we consider operators acting on a subspace ℳ of the space L ₂ (ℝ^m; ℂ_m) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ_s,k of specific Clifford basis functions of L ₂(ℝ^m; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳ_s,k into a similar space ℳ_s′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.     

Clifford algebras and geometric algebra

The geometric algebra as defined by D. Hestenes is compared with a constructive definition of Clifford algebras. Both approaches are discussed and the equivalence between a finite geometric algebra and the universal Clifford algebra R _{p, q} is shown. Also an intermediate way to construct Clifford algebras is sketched. This attempt to conciliate two separated approaches may be useful taking into account the recognized importance of Clifford algebras in theoretical and applied physics.     

A bilinear estimate for biharmonic functions in Lipschitz domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in _boxclose^d{\mathbb{R}^{d}} and 1 < q < ∞, the solvability of the L ^q Dirichlet problem for Δ ² u = 0 in Ω with boundary data in WA ^1,q (∂Ω) is equivalent to that of the L ^p regularity problem for Δ ² u = 0 in Ω with boundary data in WA ^2,p (∂Ω), where \frac1p + \frac1q=1{\frac{1}{p} + \frac{1}{q}=1}. This duality relation, together with known results on the Dirichlet problem, allows us to solve the L ^p regularity problem for d ≥ 4 and p in certain ranges.     

Exponential decay for distributed bilinear control systems with damping

Let Ω be a bounded open domain in ℝ ^N ,A an unbounded, selfadjoint, positive and coercive linear operator onL ² (Ω). We consider feedback stabilization for the distributed bilinear control systemy″(t)+Ay(t)+Dy′(t)+u(t)By(t)=0, whereD andB are linear bounded operators fromL ²(Ω) toL ²(Ω). Under suitable assumptions onB andD, a nonlinear feedback ensuring uniform exponential decay of solutions is given. Various applications to vibrating processes are presented.     

The Ornstein-Uhlenbeck semigroup in exterior domains

Let Ω be an exterior domain in It is shown that Ornstein-Uhlenbeck operators L generate C₀-semigroups on L^p(Ω) for p ∈ (1, ∞) provided ∂Ω is smooth. The method presented also allows to determine the domain D(L) of L and to prove L^p − L^q smoothing properties of e^tL. If ∂Ω is only Lipschitz, results of this type are shown to be true for p close to 2. Received: 16 December 2004; revised: 4 February 2005