The boundary beta function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta function, expressing it as the gradient of the boundary entropy s at fixed nonzero temperature. The gradient formula implies that s dec...

We prove that the weak associativity for modules for vertex algebras are equivalent to a residue formula for iterates of vertex operators, obtained using the weak associativity and the lower truncation property of vertex operators, together with a known formula expressing products of components of vertex operators as linear combinations of iterates of components of vertex operators. By requirin...

We investigate the local structure factor S (z;q) at a free liquid-gas interface in systems with short-ranged intermolecular forces and determine the corrections to the leading-order, capillary-wave-like, Goldstone mode divergence of S (z;q) known to occur for parallel (i.e. measured along the interface) wavevectors [Formula: see text]. We show from explicit solution of the inhomogeneous Ornste...

In this article we get simple formulas for IE sups≤tX(s) where X is a spectrally positive or negative Lévy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain IP(sups≤t Zα(s) ≥ u) = α IP(Zα(t) ≥ u) for u ≥ 0 where Zα is a spectrally negative α-stable Lévy process with 1 < α ≤ 2 w...

This paper investigates the complexity of query problem for first-order formulas on quasi-unary signatures, that is, on vocabularies made of a single unary function and any number of monadic predicates. We first prove a form of quantifier elimination result: any query defined by a quasi-unary first-order formula can be equivalently defined, up to a suitable linear-time reduction, by a quantifie...

The class FORMULA[s]∘G consists of Boolean functions computable by size- s De Morgan formulas whose leaves are any from a G. We give lower bounds and (SAT, Learning, pseudorandom generators ( PRG )) algorithms for FORMULA[n 1.99 ]∘G, classes G with low communication complexity . Let R (k) be the maximum k -party number-on-forehead randomized function in Among other results, we show following: •...

We consider variants of Metric Temporal Logic (MTL) with the past operators S and SI . We show that these operators can be “eliminated” in the sense that for any formula in these logics containing the S and SI modalities, we can give a formula over an extended set of propositions which does not contain these past operators, and which is equivalent to the original formula modulo a projection ont...

The authors recently discovered some interesting relations between the Black-Scholes formula and last passage times of the Brownian exponential martingales, which invites one to seek analogous results for last passage times up to a …nite horizon. This is achieved in the present paper, where Yuri’s formula, as originally presented in Akahori, Imamura and Yano (2008), is also derived. We are most...

A short and simple derivation of the formula of Frobenius, which gives the dimensions of the irreducible representations of S(n), the symmetric group on any number, n, of symbols, is given. These dimensions are the characters of the identity element of the group, i.e., of the element all of whose cycles are unary. It is shown how a slight modification of Frobenius' formula yields, when n = 2p i...

(a) Consider a sequence of independent coin tosses with P[heads] = p and let Yn be the number of heads up to time n with Y0 = 0. Give the state space S and transition matrix for the process (Yn : n ∈ N0). Compute P[Yn = k] for all n, k ≥ 0 using the binomial formula. (b) Let (Zn : n ∈ N0) be a simple random walk on S = Z with transition matrix p(x, y) = pδy,x+1 + qδy,x−1. Use that Zn = 2Yn − n ...