Professor Rudolf Gorenflo and his Contribution to Fractional Calculus

MSC 2010: 26A33 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary

Fractional Fokker-Planck-Kolmogorov type Equations and their Associated Stochastic Differential Equations

MSC 2010: 26A33, 35R11, 35R60, 35Q84, 60H10 Dedicated to 80-th anniversary of Professor Rudolf Gorenflo

Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary

Matrix-Variate Statistical Distributions and Fractional Calculus

MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthday

Solutions of Fractional Diffusion-Wave Equations in Terms of H-functions

MSC 2010: 35R11, 42A38, 26A33, 33E12

Fractional variational problems depending on indefinite integrals and with delay

The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example, where we nd analytically the minimizer.

A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

Structural Connectivity Of The Default Mode Network And Cognition In Alzheimer׳s Disease.

Disconnectivity between the Default Mode Network (DMN) nodes can cause clinical symptoms and cognitive deficits in Alzheimer׳s disease (AD). We aimed to examine the structural connectivity between DMN nodes, to verify the extent in which white matter disconnection affects cognitive performance. MRI data of 76 subjects (25 mild AD, 21 amnestic Mild Cognitive Impairment subjects and 30 controls) were acquired on a 3.0T scanner. ExploreDTI software (fractional Anisotropy threshold=0.25 and the angular threshold=60°) calculated axial, radial, and mean diffusivities, fractional anisotropy and streamline count. AD patients showed lower fractional anisotropy (P=0.01) and streamline count (P=0.029), and higher radial diffusivity (P=0.014) than controls in the cingulum. After correction for white matter atrophy, only fractional anisotropy and radial diffusivity remained significantly lower in AD compared to controls (P=0.003 and P=0.05). In the parahippocampal bundle, AD patients had lower mean and radial diffusivities (P=0.048 and P=0.013) compared to controls, from which only radial diffusivity survived for white matter adjustment (P=0.05). Regression models revealed that cognitive performance is also accounted for by white matter microstructural values. Structural connectivity within the DMN is important to the execution of high-complexity tasks, probably due to its relevant role in the integration of the network.

Exercise training and caloric restriction prevent reduction in cardiac Ca(2+)-handling protein profile in obese rats

Previous studies show that exercise training and caloric restriction improve cardiac function in obesity. However, the molecular mechanisms underlying this effect on cardiac function remain unknown. Thus, we studied the effect of exercise training and/or caloric restriction on cardiac function and Ca(2+) handling protein expression in obese rats. To accomplish this goal, male rats fed with a high-fat and sucrose diet for 25 weeks were randomly assigned into 4 groups: high-fat and sucrose diet, high-fat and sucrose diet and exercise training, caloric restriction, and exercise training and caloric restriction. An additional lean group was studied. The study was conducted for 10 weeks. Cardiac function was evaluated by echocardiography and Ca(2+) handling protein expression by Western blotting. Our results showed that visceral fat mass, circulating leptin, epinephrine, and norepinephrine levels were higher in rats on the high-fat and sucrose diet compared with the lean rats. Cardiac nitrate levels, reduced/oxidized glutathione, left ventricular fractional shortening, and protein expression of phosphorylated Ser(2808)-ryanodine receptor and Thr(17-)phospholamban were lower in rats on the high-fat and sucrose diet compared with lean rats. Exercise training and/or caloric restriction prevented increases in visceral fat mass, circulating leptin, epinephrine, and norepinephrine levels and prevented reduction in cardiac nitrate levels and reduced: oxidized glutathione ratio. Exercise training and/or caloric restriction prevented reduction in left ventricular fractional shortening and in phosphorylation of the Ser(2808)-ryanodine receptor and Thr(17)-phospholamban. These findings show that exercise training and/or caloric restriction prevent cardiac dysfunction in high-fat and sucrose diet rats, which seems to be attributed to decreased circulating neurohormone levels. In addition, this nonpharmacological paradigm prevents a reduction in the Ser(2808)-ryanodine receptor and Thr(17-)phospholamban phosphorylation and redox status. (Hypertension. 2010;56:629-635.)

Exercise training reduces cardiac angiotensin II levels and prevents cardiac dysfunction in a genetic model of sympathetic hyperactivity-induced heart failure in mice

The role of exercise training (ET) on cardiac renin-angiotensin system (RAS) was investigated in 3-5 month-old mice lacking alpha(2A-) and alpha(2C-)adrenoceptors (alpha(2A)/alpha(2C)ARKO) that present heart failure (HF) and wild type control (WT). ET consisted of 8-week running sessions of 60 min, 5 days/week. In addition, exercise tolerance, cardiac structural and function analysis were made. At 3 months, fractional shortening and exercise tolerance were similar between groups. At 5 months, alpha(2A)/alpha(2C)ARKO mice displayed ventricular dysfunction and fibrosis associated with increased cardiac angiotensin (Ang) II levels (2.9-fold) and increased local angiotensin-converting enzyme activity (ACE 18%). ET decreased alpha(2A)/alpha(2C)ARKO cardiac Ang II levels and ACE activity to age-matched untrained WT mice levels while increased ACE2 expression and prevented exercise intolerance and ventricular dysfunction with little impact on cardiac remodeling. Altogether, these data provide evidence that reduced cardiac RAS explains, at least in part, the beneficial effects of ET on cardiac function in a genetic model of HF.

Quantum symmetries and the Weyl–Wigner product of group representations

In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase space. Each such phase-space representation is a Weyl–Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by 'factorizing' the Weyl–Wigner product. However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized. The conditions under which this is possible are examined. Examples are presented.

Group theory and quasiprobability integrals of Wigner functions

The integral of the Wigner function of a quantum-mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su(1, 1) approximate to so(2, 1). The explicit relation between the spectra of operators associated with discs and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials.

Field quantization, photons and non-Hermitean modes

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.

gl(2 vertical bar 2) current superalgebra and non-unitary conformal field theory

Motivated by application of current superalgebras in the study of disordered systems such as the random XY and Dirac models, we investigate gl(2\2) current superalgebra at general level k. We construct its free field representation and corresponding Sugawara energy-momentum tensor in the non-standard basis. Three screen currents of the first kind are also presented. (C) 2003 Elsevier B.V. All rights reserved.