Let α be a polynomial Poisson bivector on a finite-dimensional vector space V over C. Then Kontsevich [K97] gives a formula for a quantization f ⋆ g of the algebra S(V )∗. We give a construction of an algebra with the PBW property defined from α by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra T (V ∗) by relations xi ⊗ xj − xj ⊗ xi = Rij(~) wh...

We study a Markov random process describing a muscle molecular motor behavior. Every motor is either bound up with a thin filament or unbound. In the bound state the motor creates a force proportional to its displacement from the neutral position. In both states the motor spend an exponential time depending on the state. The thin filament moves at its velocity proportional to average of all dis...

We consider vortices in the nonlocal two-dimensional Gross-Pitaevskii equation with the interaction potential having the Lorentz-shaped dependence on the relative momentum. It is shown that in the Fourier series expansion with respect to the polar angle the unstable modes of the axial n-fold vortex have orbital numbers l satisfying 0 < |l| < 2|n|, similar as in the local model. Numerical simula...

The napkin problem was first posed by John H. Conway, and written up as a 'toughie' in " Mathematical Puzzles: A Connoisseur's Collection, " by Peter Winkler. To paraphrase Winkler's book, there is a banquet dinner to be served at a mathematics conference. At a particular table, n men are to be seated around a circular table. There are n napkins, exactly one between each of the place settings. ...

For each natural number n, poset T , and |T |–tuple of scalars Q, we introduce the ramified partition algebra P (T) n (Q), which is a physically motivated and natural generalization of the partition algebra [24, 25] (the partition algebra coincides with case |T | = 1). For fixed n and T these algebras, like the partition algebra, have a basis independent of Q. We investigate their representatio...

We consider the one-person game of peg solitaire on a triangular board of arbitrary size. The basic game begins from a full board with one peg missing and finishes with one peg at a specified board location. We develop necessary and sufficient conditions for this game to be solvable. For all solvable problems, we give an explicit solution algorithm. On the 15-hole board, we compare three simple...

We construct a sequence of functions that uniformly converge (on compact sets) to the price of Asian option, which is written on a stock whose dynamics follows a jump diffusion, exponentially fast. Each of the element in this sequence solves a parabolic partial differential equation (not an integro-differential equation). As a result we obtain a fast numerical approximation scheme whose accurac...

Let X be a locally compact Hausdorff space with n proper continuous self maps τ i : X → X for 1 ≤ i ≤ n. To this we associate various topological conjugacy algebras; and two emerge as the natural candidates for the universal algebra of the system, the tensor algebra A(X, τ) and the semicrossed product C 0 (X) × τ F + n. The C*-envelope of A(X, τ) is the Cuntz-Pimsner algebra C * (X, τ) as defin...

We study the BGG Category O over a skew group ring, involving a finite group acting on a regular triangular algebra. We relate the representation theory of the algebra to Clifford theory for the skew group ring, and obtain results on block decomposition, semisimplicity, and enough projectives. O is also shown to be a highest weight category; the BGG Reciprocity formula is slightly different bec...

X iv :m at h/ 04 05 42 8v 8 [ m at h. G T ] 1 7 Ju n 20 05 Virtual Knot Theory – Unsolved Problems Roger Fenn University of Sussex Department of Mathematics Sussex BN1 9RF, England [email protected] and Louis H. Kauffman Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 South Morgan St., Chicago IL 60607-7045, USA [email protected] and Vassily O. ...