Fractional Vector Calculus and Fractional Maxwell's Equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered.Comment: 42 pages, LaTe

Computation of Gr\"obner Bases for Two-Loop Propagator Type Integrals

The Gr\"obner basis technique for calculating Feynman diagrams proposed in [O.V. Tarasov, Acta Physica Polonica, v. B29 (1998) 2655] is applied to the two-loop propagator type integrals with arbitrary masses and momentum. We describe the derivation of Gr\"obner bases for all integrals with 1PI topologies and present elements of the Gr\"obner bases.Comment: 4 pages, LaTeX, to appear in the Proceedings of ACAT-03, Tsukuba, Japa

Stationary Solutions of Liouville Equations for Non-Hamiltonian Systems

We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems.Comment: 22 page

Fractional Generalization of Liouville Equations

In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouvile equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered considered as a normalization condition for systems in fractional phase space. The interpretation of the fractional space is discussed.Comment: 9 pages, LaTe

Dynamical differential equations compatible with rational qKZ equations

For the Lie algebra $gl_N$ we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the $gl_N$ rational quantized Knizhnik-Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the $gl_N$ Weyl group.Comment: 7 pages, AmsLaTe

Possible Experimental Test of Continuous Medium Model for Fractal Media

We use the fractional integrals to describe fractal media. We consider the fractal media as special ("fractional") continuous media. We discuss the possible experimental testing of the continuous medium model for fractal media that is suggested in Phys. Lett. A. 336 (2005) 167-174. This test is connected with measure of period of the Maxwell pendulum with fractal medium cylinder.Comment: 9 page

Why We Need Corpus Linguistics in Intuition-Based Semantics

The following method is popular in some areas of philosophy and linguistics when trying to describe the semantics of a given sentence Φ. Present ordinary speakers with scenarios that involve an utterance of Φ, ask them whether these utterances are felicitous or infelicitous and then construct a semantics that assigns the truth-value True to felicitous utterances of Φ and the truth-value False to infelicitous utterances of Φ. The author makes five observations about this intuition-based approach to semantics; their upshot is that it should be revised in favour of a more nuanced method. The author suggests that this method should be based on corpus linguistics and makes some tentative remarks about what it might look like and which questions we need to address in order to develop it

Trade Liberalization and Welfare Inequality: a Demand-Based Approach

There is strong evidence that different income groups consume di¤erent bundles of goods. This evidence suggests that trade liberalization can a¤ect welfare inequality within a country via changes in the relative prices of goods consumed by di¤erent income groups (the price effect). In this paper, I develop a framework that enables us to explore the role of the price effect in determining welfare inequality. There are two core elements in the model. First, I assume that heterogenous in income consumers share identical but nonhomothetic preferences. Secondly, I consider a monopolistic competition environment that leads to variable markups a¤ected by trade and trade costs. I �nd that trade liberalization does affect the prices of different goods differently and, as a result, can bene�fit some income classes more than others. In particular, I show that the relative welfare of the rich with respect to that of the poor has a hump shape as a function of trade costs