Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

Action of braid groups on determinantal ideals, compact spaces and a stratification of Teichmüller space

We show that for n≥3 there is an action of the braid group B _n on the determinantal ideals of a certain n×n symmetric matrix with algebraically independent entries off the diagonal and 2s on the diagonal. We show how this action gives rise to an action of B _n on certain compact subspaces of some Euclidean spaces of dimension . These subspaces are real semi-algebraic varieties and include spheres of dimension -1 on which the kernel of the action of B _n is the centre of B _n . We investigate the action of B _n on these subspaces. We also show how a finite number of disjoint copies of the Teichmüller space for the n-punctured disc is naturally a subset of this ℝ and how this cover (in the broad sense) of Teichmüller space is a union of non-trivial B _n -invariant subspaces. The action of B _n on this cover of Teichmüller space is via polynomial automorphisms. For the case n=3 we show how to define modular forms on the 3-dimensional Teichmüller space relative to the action of B ₃. Oblatum 29-V-2000 & 7-XI-2000?Published online: 5 March 2001     

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL₂(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝⁿ or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.     

Weighted Inequalities for Multilinear Potential Operators and their Commutators

We prove weighted strong inequalities for the multilinear potential operator T_f{\cal T}_{\phi} and its commutator, where the kernel ϕ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. Moreover we prove weighted weak type inequalities for the multilinear maximal operator M_j,L^B\mathcal{M}_{\varphi,L^{B}} associated to an essentially nondecreasing function φ and to the Orlicz space L ^B for a given Young function B. This result allows us to obtain a weighted weak type inequality for the operator T_f{\cal T}_{\phi}.     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

Fluctuations,higher anharmonisms,and the Ginzburg-Landau-Devonshire expansion for barium titanate

We show that the strong temperature dependence of the coefficients B ₁ and B ₂ of terms of the form P_x ⁴ and P_x ² P _y ² in the Ginzburg-Landau-Devonshire expansion for barium titanate can result from the action of thermal polarization fluctuations if the fourth-order anharmonic constants are anomalously small, i.e., if the nonlinearity of the form P ⁴ and higher anharmonisms play comparable roles. We calculate the regular (noncritical) fluctuation contributions to B ₁ and B ₂ in the first approximation in the sixth- and eighth-order anharmonic constants. Both contributions increase monotonically with temperature, which agrees with the forms of the experimentally observed temperature dependences of these coefficients. The theory allows finding the ratio of the fluctuation components B ₁ and B ₂ without additional assumptions. This ratio also turns out to be very close to the experimental value.     

The Heat Kernel Expansion for the Electromagnetic Field in a Cavity

We derive the first six coefficients of the heat kernel expansion for the electromagnetic field in a cavity by relating it to the expansion for the Laplace operator acting on forms. As an application we verify that the electromagnetic Casimir energy is finite. Communicated by Vincent Rivasseau submitted 17/02/03, accepted: 04/07/03     

A generating function for all semi-magic squares and the volume of the Birkhoff polytope

We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B _n of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B _n and any of its faces.     

Generalized unitaries and the Picard group

After discussing some basic facts about generalized module maps, we use the representation theory of the algebra ℬ^a(E) of adjointable operators on a HilbertB-moduleE to show that the quotient of the group of generalized unitaries onE and its normal subgroup of unitaries onE is a subgroup of the group of automorphisms of the range idealB _E ofE inB. We determine the kernel of the canonical mapping into the Picard group ofB _E in terms of the group of quasi inner automorphisms ofB _E . As a by-product we identify the group of bistrict automorphisms of the algebra of adjointable operators onE modulo inner automorphisms as a subgroup of the (opposite of the) Picard group.     

Generalised Kostka–Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles

Let G be a simple algebraic group and B a Borel subgroup. We consider generalisations of Lusztig’s q-analogues of weight multiplicity, where the set of positive roots is replaced with the multiset of weights of a B-submodule N of an arbitrary finite-dimensional G-module V. The corresponding polynomials in q are called generalised Kostka–Foulkes polynomials (gKF). We prove vanishing theorems for the cohomology of line bundles on G ×  _B N and derive from this a sufficient condition for the non-negativity of the coefficients of gKF. We also consider in detail the case in which V is the simple G-module whose highest weight is the short dominant root and N is the B-submodule whose weights are all short positive roots.     

Obtaining a banded system of equations in complete spline interpolation problem via B-spline basis

We study the problem of interpolation by a complete spline of 2n − 1 degree given in B-spline representation. Explicit formulas for the first nand the last ncoefficients of B-spline decomposition are found. It is shown that other B-spline coefficients can be computed as a solution of a banded system of an equitype linear equations.     

Are the coefficients of a polynomial well-conditioned functions of its roots?

Summary We show that the coefficients of a polynomial can be rather illconditioned functions of its given roots. In certain cases, however, the coefficients are well conditioned, e.g. if the roots lie symmetrically in a closed angular sector for an angle less than centered at the origin of the complex plane. We introduce and study condition numbersB _l(k) for the sensitivity of the polynomial coefficienta _k to changes in the rootx _l and develop an algorithm to compute the condition numbers and to decide whether a given set of roots will generate the polynomial coefficients in an ill way. This algorithm is an essential preliminary for certain numerical pole placement algorithms in linear control theory.Dedicated to the memory of Prof. Dr. Fritz Reutter (1911–1990)     

The Fejér-Riesz inequality for the Besov spaces

A Fejér-Riesz inequality for the weighted Besov spaces B _p,q is given. Some characterizations of functions in B _p,q in terms of their Taylor coefficients are obtained.     

Second-order subelliptic operators on Lie groups III: Hölder continuous coefficients

Let G be a connected Lie group with Lie algebra and an algebraic basis of . Further let denote the generators of left translations, acting on the -spaces formed with left Haar measure dg, in the directions . We consider second-order operators corresponding to a quadratic form with complex coefficients , , , . The principal coefficients are assumed to be H?lder continuous and the matrix is assumed to satisfy the (sub)ellipticity condition uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators in nondivergence form for which the principal coefficients are at least once differentiable. Received January 24, 1997 / Accepted June 5, 1998     

Reflecting the pascal matrix about its main antidiagonal

Let B denote either of two varieties of order n Pascal matrix, i.e., one whose entries are the binomial coefficients. Let B_R denote the reflection of B about its main antidiagonal. The matrix B is always invertible modulo n; our main result asserts that B^-1 ≡ B^R mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of the difficulties with the matrix exponential under modular arithmetic. We then use our main result to extend the "Fibonacci diagonal" property of Pascal matrices.     

Some bounds for coefficients of schlicht pseudo-conformal mappings

Summary The author considers a schlicht pseudo-conformal mapping of a domainB in the (z ₁ , z ₂ )-space onto the Reinhardt circular domainC in the (ξ ₁ , ξ ₂ )-space by a pair of functions (see (1)§ 1). The domainB is assumed to include the bicylinder ((2)§ 2) and to omit four planes z_k=a_k, z_k=b_k, k=1, 2. Upper bounds for a sequence of the coefficients _μv ^(k) of the developments (1) are given, see p. 304. The upper bounds depend only on a_k, b_k, k=1, 2, and on the radii r_k of the bicylinder ((2)§ 2). The bounds are obtained by using the method of the kernel function. The result can be considered as an analogue to the inequalities of Grunsky in the theory of functions of one complex variable. To Enrico Bompiani on his scientific Jubiles This paper was prepared under the sponsorship of the N. S. F.     

Obtaining upper bounds of heat kernels from lower bounds

We show that a near‐diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on‐diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off‐diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near‐diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. © 2007 Wiley Periodicals, Inc.     

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B _H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ? ^d , like, for example, (B ₁(|C(t)|),…, B _d (|C(t)|)) and (C ₁(|C(t)|),…, C _d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B ₁(|B ₂(…|B _n+1(t)|…)|)|) are governed by fractional diffusion equations.     

Estimation of the parameters for unstable AR models

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Solvability of Boundary Value Problems for Operator-Differential Equations of Mixed Type: The Degenerate Case

We obtain some results on solvability of boundary value problems for the equation B u _t –L u =f (t(0,)), where B and L are selfadjoint and dissipative operators defined in a Hilbert space E. The kernel of B may be nontrivial and L is uniformly dissipative on a subspace M of finite codimension.