Some identities on derangement and degenerate derangement polynomials

In combinatorics, a derangement is a permutation that has no fixed points. The number of derangements of an n-element set is called the n-th derangement number. In this paper, as natural companions to derangement numbers and degenerate versions of the companions we introduce derangement polynomials and degenerate derangement polynomials. We give some of their properties, recurrence relations and identities for those polynomials which are related to some special numbers and polynomials.Comment: 12 page

Stochastic Differential Systems with Memory: Theory, Examples and Applications

The purpose of this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise , and the classical ``heat-bath model of R. Kubo , modeling the motion of a ``large molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde\u27s). We then establish pathwise existence and uniqueness of solutions to these classes of sfde\u27s under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde\u27s is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from zero. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel $\sigma$-algebra of the state space. This chapter also contains a derivation of a formula for the weak infinitesimal generator of the semigroup for sufficiently regular functions, and for a large class of quasitame functions. In Chapter III, we study pathwise regularity of the trajectory random field in the time variable and in the initial path. Of note here is the non-existence of the stochastic flow for the singular sdde $dx(t)= x(t-r) dW(t)$ and a breakdown of linearity and local boundedness. This phenomenon is peculiar to stochastic delay equations. It leads naturally to a classification of sfde\u27s into regular and singular types. Necessary and sufficient conditions for regularity are not known. The rest of Chapter III is devoted to results on sufficient conditions for regularity of linear systems driven by white noise or semimartingales, and Sussman-Doss type nonlinear sfde\u27s. Building on the existence of a compacting stochastic flow, we develop a multiplicative ergodic theory for regular linear sfde\u27s driven by white noise, or general helix semimartingales (Chapter IV). In particular, we prove a Stable Manifold Theorem for such systems. In Chapter V, we seek asymptotic stability for various examples of one-dimensional linear sfde\u27s. Our approach is to obtain upper and lower estimates for the top Lyapunov exponent. Several topics are discussed in Chapter VI. These include the existence of smooth densities for solutions of sfde\u27s using the Malliavin calculus, an approximation technique for multidimensional diffusions using sdde\u27s with small delays, and affine sfde\u27s

Notes on Macdonald polynomials and the geometry of Hilbert schemes

These notes are based on a series of seven lectures given in the combinatorics seminar at U.C. San Diego in February and March, 2001. My lectures at the workshop which is the subject of this proceedings volume covered a portion of the same material in a more abbreviated form

On the Redundancy of D-Ary Fano Codes

We study the redundancy of D-ary Fano source codes. We show that a novel splitting criterion allows to prove a bound on the redundancy of the resulting code which sharpens the guarantee provided by Shannon’s classical result for the case of an optimal code. In particular we show that, for any D≥ 2 and for every source distribution p= p1, ⋯, pn, there is a D-ary Fano code that satisfies the redundancy bound 1$$\begin{aligned} \overline{L} - H_D(\mathbf{p}) \le 1- p_{\min }, \end{aligned}$$L¯-HD(p)≤1-pmin, where, L¯ denotes the average codeword length, pmin= min ipi, and HD(p)=-∑i=1npilogD(pi) is the D-ary entropy of the source. The existence of D-ary Fano codes achieving such a bound had been conjectured in [ISIT2015], where, however, the construction proposed achieves the bound only for D= 2, 3, 4. In [ISIT2020], a novel construction was proposed leading to the proof that the redundancy bound in (1) above also holds for D= 5 (and some other special cases). This result was attained by a dynamic programming based algorithm with time complexity O(Dn) (per node of the codetree). Here, besides proving that the redundancy bound in (1) can be achieved, unconditionally, for every D> 3, we also significantly improve the time complexity of the algorithm building a D-ary Fano code tree achieving such a bound: We show that, for every D≥ 4, a D-ary Fano code tree satisfying (1) can be constructed by an efficient greedy procedure that has complexity O(Dlog 2n) per node of the codetree (i.e., improving from linear time to logarithmic time in n)