DIMENSION FREE BOUNDEDNESS OF RIESZ TRANSFORMS FOR THE GRUSHIN OPERATOR

Let G = -Delta(xi) - vertical bar xi vertical bar(2) partial derivative(2)/partial derivative eta(2) be the Grushin operator on R-n x R. We prove that the Riesz transforms associated to this operator are bounded on L-p(Rn+1), 1 < p < infinity, and their norms are independent of dimension n.

Monopoles, Dirac operator, and index theory for fuzzy SU(3) / (U(1) x U(1))

The intersection of the ten-dimensional fuzzy conifold Y-F(10) with S-F(5) x S-F(5) is the compact eight-dimensional fuzzy space X-F(8). We show that X-F(8) is (the analogue of) a principal U(1) x U(1) bundle over fuzzy SU(3) / U(1) x U(1)) ( M-F(6)). We construct M-F(6) using the Gell-Mann matrices by adapting Schwinger's construction. The space M-F(6) is of relevance in higher dimensional quantum Hall effect and matrix models of D-branes. Further we show that the sections of the monopole bundle can be expressed in the basis of SU(3) eigenvectors. We construct the Dirac operator on M-F(6) from the Ginsparg-Wilson algebra on this space. Finally, we show that the index of the Dirac operator correctly reproduces the known results in the continuum.

An interior penalty method for distributed optimal control problems governed by the biharmonic operator

In this paper, a C-0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations. A priori and a posteriori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions. Numerical results justify the theoretical results obtained. The a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement. (C) 2014 Elsevier Ltd. All rights reserved.

L-p estimates for the wave equation associated to the Grushin operator

We prove that the solution of the wave equation associated to the Grushin operator G = -Delta -vertical bar x vertical bar(2)partial derivative(2)(t) is bounded on L-P (Rn+1), with 1 < p < infinity, when vertical bar 1/p - 1/2 vertical bar < 1/n+2.

A comparative study of a direct discretization and an operator-splitting solver for population balance systems

A direct discretization approach and an operator-splitting scheme are applied for the numerical simulation of a population balance system which models the synthesis of urea with a uni-variate population. The problem is formulated in axisymmetric form and the setup is chosen such that a steady state is reached. Both solvers are assessed with respect to the accuracy of the results, where experimental data are used for comparison, and the efficiency of the simulations. Depending on the goal of simulations, to track the evolution of the process accurately or to reach the steady state fast, recommendations for the choice of the solver are given. (C) 2015 Elsevier Ltd. All rights reserved.

Universality of the Stochastic Airy Operator

We introduce a new method for studying universality of random matrices. Let T-n be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, Tn converges to the stochastic Airy operator. In particular, the top edge of the Dyson beta ensemble and the corresponding eigenvectors are universal. As a byproduct, these ideas lead to conjectured operator limits for the entire family of soft edge distributions. (C) 2015 Wiley Periodicals, Inc.

Isolated singularities of polyharmonic operator in even dimension

We consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) <= Ce-gamma t, for some C, gamma > 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.

Trace Formulae in Operator Theory

In this article, we survey several kinds of trace formulas that one encounters in the theory of single and multi-variable operators. We give some sketches of the proofs, often based on the principle of finite-dimensional approximations to the objects at hand in the formulas.

Action of Hopf on the twisted modular Hecke operator

Let Gamma subset of SL2(Z) be a principal congruence subgroup. For each sigma is an element of SL2(Z), we introduce the collection A(sigma)(Gamma) of modular Hecke operators twisted by sigma. Then, A(sigma)(Gamma) is a right A(Gamma)-module, where A(Gamma) is the modular Hecke algebra introduced by Connes and Moscovici. Using the action of a Hopf algebra h(0) on A(sigma)(Gamma), we define reduced Rankin-Cohen brackets on A(sigma)(Gamma). Moreover A(sigma)(Gamma) carries an action of H 1, where H 1 is the Hopf algebra of foliations of codimension 1. Finally, we consider operators between the levels A(sigma)(Gamma), sigma is an element of SL2(Z). We show that the action of these operators can be expressed in terms of a Hopf algebra h(Z).

p Dissipative operator

In this paper the authors prove that the generalized positive p selfadjoint (GPpS) operators in Banach space satisfy the generalized Schwarz inequality, solve the maximal dissipative extension representation of p dissipative operators in Banach space by using the inequality and introducing the generalized indefinite inner product (GIIP) space, and apply the result to a certain type of Schrodinger operator.

Studies in nuclear reactor dynamics : I. The accuracy of point kinetics. II. The effect of delayed neutrons on the spectrum of the group-diffusion operator

This thesis is a theoretical work on the space-time dynamic behavior of a nuclear reactor without feedback. Diffusion theory with G-energy groups is used.

In the first part the accuracy of the point kinetics (lumped-parameter description) model is examined. The fundamental approximation of this model is the splitting of the neutron density into a product of a known function of space and an unknown function of time; then the properties of the system can be averaged in space through the use of appropriate weighting functions; as a result a set of ordinary differential equations is obtained for the description of time behavior. It is clear that changes of the shape of the neutron-density distribution due to space-dependent perturbations are neglected. This results to an error in the eigenvalues and it is to this error that bounds are derived. This is done by using the method of weighted residuals to reduce the original eigenvalue problem to that of a real asymmetric matrix. Then Gershgorin-type theorems .are used to find discs in the complex plane in which the eigenvalues are contained. The radii of the discs depend on the perturbation in a simple manner.

In the second part the effect of delayed neutrons on the eigenvalues of the group-diffusion operator is examined. The delayed neutrons cause a shifting of the prompt-neutron eigenvalue s and the appearance of the delayed eigenvalues. Using a simple perturbation method this shifting is calculated and the delayed eigenvalues are predicted with good accuracy.

Paranormal operator on a Hilbert space

In this thesis an extensive study is made of the set P of all paranormal operators in B(H), the set of all bounded endomorphisms on the complex Hilbert space H. T ϵ B(H) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. P contains the set N of normal operators and P contains the set of hyponormal operators. However, P is contained in L, the set of all T ϵ B(H) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, N≤P≤L.

If the uniform operator (norm) topology is placed on B(H), then the relative topological properties of N, P, L can be discussed. In Section IV, it is shown that: 1) N P and L are arc-wise connected and closed, 2) N, P, and L are nowhere dense subsets of B(H) when dim H ≥ 2, 3) N = P when dimH ˂ ∞ , 4) N is a nowhere dense subset of P when dimH ˂ ∞ , 5) P is not a nowhere dense subset of L when dimH ˂ ∞ , and 6) it is not known if P is a nowhere dense subset of L when dimH ˂ ∞.

The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ P to lie on a C²-smooth rectifiable Jordan curve G_o, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G_o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ P with σ(T) ≤ G_o, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ P with σ(T) ≤ G_o, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G_o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods.