Order-unit quantum Gromov-Hausdorff distance

We introduce a new distance dist_oq between compact quantum metric spaces. We show that dist_oq is Lipschitz equivalent to Rieffel's distance dist_q, and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to dist_oq. As applications, we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a fixed compact group is determined by the multiplicities of the actions, generalizing Rieffel's work on noncommutative tori and integral coadjoint orbits of semisimple compact connected Lie groups; we also show that the θ-deformations of Connes and Landi are continuous in the parameter θ.     

Variational problem with degeneracy at the boundary of the interval

The weighted L _p-norms of derivatives are estimated in terms of the weighted L _p-norm of the highest derivative and the traces of the function and its derivatives at the given points of closure of the bounded interval; weights are powers of the distance to the nearest endpoint of the interval. For functions with zero traces, sharper estimates are established. For the integral quadratic functional with degenerate coefficients, we prove the existence and uniqueness of the solution to the problem of minimization of a functional on a function class with zero traces.     

On the inverse problem of the product of a semi-classical form by a polynomial

A form (linear functional) u is called regular if there exists a sequence of polynomials {P_n}_n⩾0, deg P_n=n which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist polynomials Φ and Ψ such that D(Φu) + Ψu = 0, where D designs the derivative operator.On certain regularity conditions, the product of a semi-classical form by a polynomial, gives a semi-classical form. In this paper, we consider the inverse problem: given a semi-classical form v, find all regular forms u which satisfy the relation x²u = −λv, $\text{λ ∈}\text{C}{}^1$. We give the structure relation (differential-recurrence relation) of the orthogonal polynomial sequence relatively to u. An example is treated with a nonsymmetric form v.     

The fractional quantum derivative and its integral representations

The quantum fractional derivative is defined using formulations analogue to the common Grünwald–Letnikov derivatives. While these use a linear variable scale, the quantum derivative uses an exponential scale and is defined in R⁺ or R⁻. Two integral formulations similar to the usual Liouville derivatives are deduced with the help of the Mellin transform.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β ₁∈(0,1) and β ₂∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

Modèles linéaires généralisés fonctionnels avec dérivée

We consider the functional generalized linear model whose response function is a linear operator depending on an explanatory variable X belonging to a functional space. It has been studied, among others, by Cardot and Sarda [4]. In this paper, we consider the functional generalized linear model with derivative component, denoted MLGFD in the following, whose response function depends on a linear operator of X and on its derivative. We propose estimators for the unknown functional parameters and provide convergence rates.     

Persistent currents in open spin chains

Nonparallel boundary magnetic fields can induce a longitudinal (spin) current in a quantum spin chain. We use functional approaches to formulate the spectral problem for the spin-1/2 Heisenberg model subject to a class of integrable boundary conditions in terms of an infinite hierarchy of nonlinear integral equations. From these equations, we compute finite-size corrections to the ground-state energy of the antiferromagnetic chain and the induced spin current for a certain range of boundary parameters.     

The Haar measure on finite quantum groups

By a finite quantum group, we will mean in this paper a finite-dimensional Hopf algebra. A left Haar measure on such a quantum group is a linear functional satisfying a certain invariance property. In the theory of Hopf algebras, this is usually called an integral. It is well-known that, for a finite quantum group, there always exists a unique left Haar measure. This result can be found in standard works on Hopf algebras. In this paper we give a direct proof of the existence and uniqueness of the left Haar measure on a finite quantum group. We introduce the notion of a faithful functional and we show that the Haar measure is faithful. We consider the special case where the underlying algebra is a -algebra with a faithful positive linear functional. Then the left and right Haar measures coincide. Finally, we treat an example of a root of unity algebra. It is an example of a finite quantum group where the left and right Haar measures are different. This note does not contain many new results but the treatment of the finite-dimensional case is very concise and instructive.

     

Nonclassical analog of the Goursat problem for a three-dimensional equation with highest derivative

In the present paper, we study the Goursat problem for a three-dimensional equation with highest derivative of fifth order with L _p -coefficients and establish a homeomorphism between certain pairs of Banach spaces by reducing this problem to the equivalent Volterra integral equation.     

Construction of an analytic solution for one class of Langevin-type equations with orthogonal random actions

We find an analytic representation of a solution of the Itô-Langevin equations in R ³ with orthogonal random actions with respect to the vector of the solution. We construct a stochastic process to which the integral of the solution weakly converges as a small positive parameter with the derivative in the equation tends to zero.     

Algebraic aspects of the Hofstadter problem in graphene

We consider the Hofstadter problem on a honeycomb lattice. ita relevance to the quantum group U _q (sl ₂) is explicitly shown. We point out the reducibility of the corresponding characteristic polynomials and conjecture its relation to supersymmetric quantum mechanics.     

Universal integrability objects

We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group U_q(L(sl ₂ )). We give a complete set of the functional relations correcting inexactitudes in the previous considerations. We especially attend to the interrelation of the representations used to construct the universal transfer operators and Q-operators.     

The theorem on convergence with a functional for integral functionals with p(x)- and p(x, u)-growth

We continue studying weak convergence for the integral functionals satisfying p(x)- and p(x, u)-growth conditions. We obtain the theorem on convergence with a functional and some results on the relation between integral functionals and their abstract lower semicontinuous extensions.     

Quantum sℓ(2) action on a divided-power quantum plane at even roots of unity

We describe a nonstandard version of the quantum plane in which the basis is given by divided powers at an even root of unity q = e^iπ/p. It can be regarded as an extension of the “nearly commutative” algebra ?[X, Y] with XY = (?1)^pYX by nilpotents. For this quantum plane, we construct a Wess-Zumino-type de Rham complex and find its decomposition into representations of the 2p ³ -dimensional quantum group $ \bar {\mathcal{U}} $ _q s?(2) and its Lusztig extension U _q s?(2); we also define the quantum group action on the algebra of quantum differential operators on the quantum plane.     

Lipschitzian Properties of Integral Functionals on Lebesgue Spaces $\bf{L_p, 1 \leqslant {p} < \infty}$

We give complete characterizations of integral functionals which are Lipschitzian on a Lebesgue space L _p with p ≠ ∞. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces. In every cases, the Lipchitzian properties of the integral functional can be described by growth conditions on the subdifferentials of the integrand which are equivalent to Lipschitzian properties of the integrand.     

Chiral solitons in a quantum potential

We study chiral solitons in a quantum potential using a dimensional reduction of the problem for (2+1)-dimensional anyons. We show that the integrable version of the model is described by a family of the resonant derivative nonlinear Schrödinger equations. For a quantum potential strength s > 1, we show that the chiral soliton interaction has a resonance. We investigate the semiclassical quantization procedure for solitons.