The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Let ${\mathcal{D}}$ be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the ?? invariant ???= ??(M) for any orientable compact flat manifold M. After giving an explicit expression for ??(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining ??(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p??? 7, ??(s) can be expressed as a multiple of L _??(s), an L-function associated to a quadratic character mod p, while ??(0) is a (non-zero) integral multiple of the class number h _?p of the number field ${\mathbb Q(\sqrt {-p})}$ . In the case of metacyclic groups of odd order pq, with p, q primes, we show that ??(0) is a rational multiple of h _?p .     

On spherical twisted conjugacy classes

Let G be a simple algebraic group over an algebraically closed field of good odd characteristic, and let ?? be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical ??-twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. As a byproduct, we obtain a dimension formula for spherical twisted conjugacy classes that was originally obtained by J.-H. Lu in characteristic zero.     

Ensemble averages when β is a square integer

We give a hyperpfaffian formulation of partition functions and ensemble averages for Hermitian and circular ensembles when L is an arbitrary integer and ???=?L ² and when L is an odd integer and ???=?L ²?+?1.     

Moderate deviations and hypothesis testing for signal detection problem

We use moderate deviations to study the signal detection problem for a diffusion model.We establish a moderate deviation principle for the log-likelihood function of the diffusion model.Then applying the moderate deviation estimates to hypothesis testing for signal detection problem we give a decision region such that its error probability of the second kind tends to zero with faster speed than the error probability of the first kind when the error probability of the first kind is approximated by e-αr(T),where α>0,r(T)=o(T) and r(T) →∞ as the observation time T goes to infinity.     

A method to abstract a stochastic Petri net

We propose a method to abstract a given stochastic Petri net (SPN). We shall show that the reachability tree of the given SPN is isomorphic to a Markov renewal process. Then, the given SPN is transformed to a state transition system (STS) and the STS is reduced. The reduction of states on STS corresponds to a fusion of series transitions on the SPN. The reduced STS is again transformed to an abstract SPN. We show that it is helpful to use the notion of the conditional firstpassage time from a certain state to the others on the STS to reduce nonessential states, thus places and transitions on the given SPN. Mass functions, that is, the distribution functions of conditional first-passage time between preserved states on the reduced MRP, preserve firing probabilities of fused transitions. Firing probability of the preserved transition also preserves the stochastic properties of the fused transitions.     

Generating Functions for Alternating Descents and Alternating Major Index

In 2008, Chebikin introduced the alternating descent set, AltDes(??), of a permutation ?? =??? ₁ ··· ?? _n in the symmetric group S _n as the set of all i such that either i is odd and ?? _i >??? _i+1 or i is even and ?? _i <??? _i+1. We can then define altdes(??) =?|AltDes(??)| and ${{\rm altmaj}(\sigma) = \sum_{i \in AltDes(\sigma)}i}$ . In this paper, we compute a generating function for the joint distribution of altdes(??) and altmaj(??) over S _n . Our formula is similar to the formula for the joint distribution of des and maj over the symmetric group that was first proved by Gessel. We also compute similar generating functions for the groups B _n and D _n and for r-tuples of permutations in S _n . Finally we prove a general extension of these formulas in cases where we keep track of descents only at positions r, 2r, . . ..     

Criteria for Feller transition functions

This paper presents some conditions for the minimal Q-function to be a Feller transition function, for a given q-matrix Q. We derive a sufficient condition that is stated explicitly in terms of the transition rates. Furthermore, some necessary and sufficient conditions are derived of a more implicit nature, namely in terms of properties of a system of equations (or inequalities) and in terms of the operator induced by the q-matrix. The criteria lead to some perturbation results. These results are applied to birth-death processes with killing, yielding some sufficient and some necessary conditions for the Feller property directly in terms of the rates. An essential step in the analysis is the idea of associating the Feller property with individual states.     

Stochastic quantization and ergodic theorem for density of diffusions

For a given probability density function ρ(x) on Rd,we construct a(non-stationary) diffusion process xt,starting at any point x in Rd,such that 1/T∫T0 δ(xt-x)dt converges to ρ(x) almost surely.The rate of this convergence is also investigated.To find this rate,we mainly use the Clark-Ocone formula from Malliavin calculus and the Girsanov transformation technique.     

Balanced partitions

A famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. We begin by establishing a less well-known companion result, which states that both of these quantities are equal to the number of partitions of n into even parts along with exactly one triangular part. We then introduce the characteristic of a partition, which is determined in a simple way by the placement of odd parts within the list of all parts. This leads to a refinement of the aforementioned result in the form of a new type of partition identity involving characteristic, distinct parts, even parts, and triangular numbers. Our primary purpose is to present a bijective proof of the central instance of this new type of identity, which concerns balanced partitions—partitions in which odd parts occupy as many even as odd positions within the list of all parts. The bijection is accomplished by means of a construction that converts balanced partitions of 2n into unrestricted partitions of n via a pairing of the squares in the Young tableau.     

A geometric model for odd differential K-theory

Odd K-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential K-theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We characterise the odd Chern character and its transgression form in terms of a connection and Higgs field and discuss some applications. Our model can be seen as the odd counterpart to the Simons–Sullivan construction of even differential K-theory. We use this model to prove a conjecture of Tradler–Wilson–Zeinalian [16], which states that the model developed there also defines the unique differential extension of odd K-theory.     

On near optimal trajectories for a game associated with the ��-Laplacian

A two-player stochastic differential game representation has recently been obtained for solutions of the equation ???_?? u?=?h in a ${{\mathcal C}^2}$ domain with Dirichlet boundary condition, where h is continuous and takes values in ${{\mathbb R}{\setminus}\{0\}}$ . Under appropriate assumptions, including smoothness of u, we identify a family of diffusion processes that may arise as the vanishing ?? limit law of the state process, when both players play ??-optimally. We also identify the limit law of the state process under a sequence of near saddle points.     

Central limit theorem for a class of one-dimensional kinetic equations

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation??s solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ?? _??, then the limit is a scale mixture of ?? _??. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.     

On the dimension of Chowla–Milnor space

In a recent work, Gun, Murty and Rath defined the Chowla?CMilnor space and proved a non-trivial lower bound for these spaces. They also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of their lower bound will lead to irrationality of ??(k)/ ?? ^k for odd positive integers k?>?1. In this paper, we give an alternate proof of their theorem about the conditional lower bound.     

Quasi-stationary Distributions for a Class of Discrete-time Markov Chains

This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution—actually an infinite family of quasi-stationary distributions— if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step (killing) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth–death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.      

On the problem of maximizing the transition probability in an <Emphasis Type="Italic">n</Emphasis>-level quantum system using nonselective measurements

We consider the problem of maximizing the probability of transition from a given initial state to a given final state for an n-level quantum system using nonselective quantum measurements. We find an estimate from below for the maximum of the transition probability for any fixed number of measurements and find the measured observables on which this estimate is attained.     

Functional central limit theorems for sums of nearly nonstationary processes*

We study some Holderian functional central limit theorems for the polygonal partial-sum processes built on a first-order autoregressive process y _n,k ?=??? _n y _n,k?1?+??? _k with ? _n converging to 1 and i.i.d. centered square-integrable innovations. In the case where ? _n ?=?e ^??/n with a negative constant ??, we prove that the limiting process is an integrated Ornstein?CUhlenbeck one. In the case where ? _n ?=?1? ?? _n /n, with ?? _n tending to infinity slower than n, the convergence to Brownian motion is established in Holder space in terms of the rate of ?? _n and the integrability of the ?? _k s.     

On the Laplace Transform of the Lognormal Distribution

Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximation \(\widetilde {\mathcal {L}}(\theta )\) of the Laplace transform \(\mathcal {L}(\theta )\) which is obtained via a modified version of Laplace’s method. This approximation, given in terms of the Lambert W(?) function, is tractable enough for applications. We prove that ~(??) is asymptotically equivalent to ?(??) as ?? → ∞. We apply this result to construct a reliable Monte Carlo estimator of ?(??) and prove it to be logarithmically efficient in the rare event sense as ?? → ∞.     

View-based explicit knowledge

In the information-based definition of knowledge, an agent is said to know α at a state s if α is true in all states that look the same as s to that agent. However, in systems where an agent's view of the system is partial, even if a state s′ may be logically indistinguishable from a state s, s′ may not be visible from s. For instance, in a distributed system, all global states in which the agent's local state does not change look the same to that agent but this set of global states may not be accessible because the agent may not even be aware of the existence of many agents in the network. We propose a logic of explicit knowledge built on agents' views and show it to be decidable.     

On twin and anti-twin words in the support of the free Lie algebra

Let ${\mathcal{L}}_{K}(A)$ be the free Lie algebra on a finite alphabet A over a commutative ring K with unity. For a word u in the free monoid A ^? let $\tilde{u}$ denote its reversal. Two words in A ^? are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let l denote the left-normed Lie bracketing and ?? be its adjoint map with respect to the canonical scalar product on the free associative algebra K??A??. Studying the kernel of ?? and using several techniques from combinatorics on words and the shuffle algebra , we show that, when K is of characteristic zero, two words u and v of common length n that lie in the support of ${\mathcal{L}}_{K}(A)$ ??i.e., they are neither powers a ⁿ of letters a??A with exponent n>1 nor palindromes of even length??are twin (resp. anti-twin) if and only if u=v or $u = \tilde{v}$ and n is odd (resp. $u =\tilde{v}$ and n is even).