Approximations for a Three Dimensional Scan Statistic

Let X _ijk ,1 ≤ i ≤ N ₁,1 ≤ j ≤ N ₂, 1 ≤ k ≤ N ₃ be a sequence of independent and identically distributed 0 − 1 Bernoulli trials. X _ijk  = 1 if an event has occurred at the i,j,k ^th location in a three dimensional rectangular region and X _ijk  = 0, otherwise. For 2 ≤ m _j  ≤ N _j  − 1,1 ≤ j ≤ 3, a three dimensional discrete scan statistic is defined as the maximum number of events in any m ₁×m ₂×m ₃ rectangular sub-region in the entire N ₁×N ₂×N ₃ rectangular region. In this article, a product-type approximation and three Poisson approximations are derived for the distribution of this three dimensional scan statistic. Numerical results are presented to evaluate the accuracy of these approximations and their use in testing for randomness.     

Approximations of the exponential of an operator with periodic coefficients

We consider the operator exponential e ^−tA , t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in \mathbbRⁿ {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm ||   ·   ||_{L²( \mathbbRⁿ ) ? L²( \mathbbRⁿ )} {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order O( t ^{- \fracm2} ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N _α (x), x ? \mathbbRⁿ x \in {\mathbb{R}^n} , with multi-indices α of length | a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε ^m ) in the L ²-norm as ε → 0. Bibliography: 14 titles.     

On the Representation of Compact Nilpotent Rings by Triangular Matrices

We prove that every compact nilpotent ring R of characteristic p > 0 can be embedded in a ring of upper triangular matrices over a compact commutative ring. Furthermore, we prove that every compact topologically nilpotent ring R of characteristic p > 0, is embedded in a ring of infinite triangular matrices over \mathbbF_p^w(R)\mathbb{F}_{p}^{w(R)}.     

Frobenius Extensions and Tilting Complexes

Let n ≥ 1 be an integer and π a permutation of I = {1, ⋯ ,n}. For any ring R, we provide a systematic construction of rings A which contain R as a subring and enjoy the following properties: (a) 1 = ∑  _i ∈ I e _i with the e _i orthogonal idempotents; (b) e _i x = xe _i for all i ∈ I and x ∈ R; (c) e _i A e _j  ≠ 0 for all i, j ∈ I; (d) e _i A _A  ≇ e _j A _A unless i = j; (e) every e _i Ae _i is a local ring whenever R is; (f) e _i A _A  ≅ Hom _R (Ae _π(i),R _R ) and _A Ae _π(i) ≅  _A Hom _R (e _i A, _R R) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(e _i ) = e _π(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map is a tilting complex. Dedicated to Takeshi Sumioka on the occasion of his 60th birthday.     

Exact tail asymptotics in bivariate scale mixture models

Let (X, Y) = (RU ₁, RU ₂) be a given bivariate scale mixture random vector, with R > 0 independent of the bivariate random vector (U ₁, U ₂). In this paper we derive exact asymptotic expansions of the joint survivor probability of (X, Y) assuming that R has distribution function in the Gumbel max-domain of attraction, and (U ₁, U ₂) has a specific local asymptotic behaviour around some absorbing point. We apply our results to investigate the asymptotic behaviour of joint conditional excess distribution and the asymptotic independence for two models of bivariate scale mixture distributions.     

The ratio of the extreme to the sum in a random sequence

For X ₁ , X ₂ , ..., X _n a sequence of non-negative independent random variables with common distribution function F(t), X _(n) denotes the maximum and S _n denotes the sum. The ratio variate R _n  = X _(n) / S _n is a quantity arising in the analysis of process speedup and the performance of scheduling. O’Brien (J. Appl. Prob. 17:539–545, 1980) showed that as n → ∞, R _n →0 almost surely iff is finite. Here we show that, provided either (1) is finite, or (2) 1 − F (t) is a regularly varying function with index ρ < − 1, then . An integral representation for the expected ratio is derived, and lower and upper asymptotic bounds are developed to obtain the result. Since is often known or estimated asymptotically, this result quantifies the rate of convergence of the ratio’s expected value. The result is applied to the performance of multiprocessor scheduling.      

Annihilating power values of co-commutators with generalized derivations

Let R be a prime ring with its Utumi ring of quotient U, H and G be two generalized derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(H(u)u − uG(u)) ⁿ  = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then there exist b′,c′ ∈ U such that H(x) = b′x + xc′, G(x) = c′x for all x ∈ R with ab′ = 0, unless R satisfies s ₄, the standard identity in four variables.     

A(α)-acceptability of pade approximations to function exp (Q)

In this paper, a necessary and sufficient condition of A(a)-acceptability for rational approximations to the functionexp(q) and some sufficient conditions to guarantee A(α)-acceptability of Padé approximations[^(R)]_mⁿ (q)\hat R_m^n (q) to the functionexp(q) are given, where a ∈(0,π/2), n≤m,m≥2. Furthermore, it is proved that the condition of A(α)-acceptability of rational approximations toexp(q) is equivalent to the nonnegatively of a real polynomial in interval (−∞,0). Finally, we prove that[^(R)]₄¹ (q)\hat R_4^1 (q) is A(π/3)-acceptable. Based on this conclusion, two A(π/3)-stable multiderivative (hybrid) one-step methods are constructed.     

A distortion theorem for the class of typically real functions

The paper continues the studies of the well-known class T of typically real functions f(z) in the disk U = {z:|z| < 1}. The region of values of the system {f(z ₀), f(z ₀), f(r ₁), f(r ₂),…, f(r _n )} in the class T is investigated. Here, z ₀ ∈ U, Im z ₀ ≠ 0, 0 < r _j  < 1 for j = 1,…, n, n ≥ 2. As a corollary, the region of values of f′(z ₀) in the class of functions f ∈ T with fixed values f(z ₀) and f(r _j ) (j = 1,…, n) is determined. The proof is based on the criterion of solvability of the power problem of moments. Bibliography: 10 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 33–45.     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows. Received August 10, 2001; in revised form June 25, 2002     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows.     

Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and _R M _S is an S-R-bimodule. In the main theorem of this paper we show that if T _S is a tilting S-module, then under certain homological conditions on the S-module M _S , one can extend T _S to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and T _S , and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M _S is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M _S has a finite projective resolution and Ext _S ⁿ (M _S , S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, Hom _S (M, S)).     

Transcendence of Certain Reciprocal Sums of Linear Recurrences

 Suppose that {R _n } _n ⩾ 0 is a sequence of integers satisfying a binary linear recurrence relation with suitable conditions. We prove the transcendency of the numbers

Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

We consider amalgamated free product II₁ factors M = M _1*B M _2*B … and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitarily conjugated into one of the M _i ’s. We apply this to the case where the M _i ’s are w-rigid II₁ factors, with B equal to either C, to a Cartan subalgebra A in M _i , or to a regular hyperfinite II₁ subfactor R in M _i , to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N _1 * CN_2*C …) ^t , for some t > 0 and some other similar inclusions of algebras C ⊂ N _i then, after a permutation of indices, (B ⊂ M _i ) is inner conjugate to (C ⊂ N _i ) ^t , for all i. Taking B = C and , with {t _i } _i⩾1 = S a given countable subgroup of R ₊ ^*, we obtain continuously many non-stably isomorphic factors M with fundamental group equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying and Out(M) abelian and calculable. Taking B = R, we get examples of factors with , Out(M) = K, for any given separable compact abelian group K.     

Milnor K-theory of rings,higher Chow groups and applications

If R is a smooth semi-local algebra of geometric type over an infinite field, we prove that the Milnor K-group K ^M _n (R) surjects onto the higher Chow group CH ⁿ (R , n) for all n≥0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CH ⁿ (R , n) modulo equivalence as a linear combination of “symbolic elements” defined as graphs of units in R. As a byproduct we get a new and entirely geometric proof of results of Gabber, Kato and Rost, related to the Gersten resolution for the Milnor K-sheaf. Furthermore it is also shown that in the semi-local PID case we have, under some mild assumptions, an isomorphism. Some applications are also given. Oblatum 17-XII-1998 & 1-X-2001?Published online: 18 January 2002     

Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H ₀ : f≡ 0, (the signal f is zero) against the nonparametric alternative H ₁ : f∈Λ_ɛ where Λ_ɛ is a set of functions on R ¹ of the form Λ_ɛ = {f : f∈?, ϕ(f) ≥ Cψ_ɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψ_ɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C ^* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C ^* and is not possible for C < C ^*. We propose asymptotically minimax test statistics. Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000     

On the second derivative of the spherical contact distribution function of smooth grain models

We consider a stationary grain model Ξ in ℝ ^d with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density. Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001     

On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process

It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential equation dX(t) = μ(X(t))dt + σ(X(t)) dB _t , X(0) = x ₀, through b + Y(t), where b > x ₀ and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B _t . In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B _t , for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately the FPT density; some examples and numerical results are also reported.     

Almost sure convergence of sample range

Let X ₁, X ₂, ... be i.i.d. random variables. The sample range is R _n = max {X _i , 1 ≤ i ≤ n} − min {X _i , 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α _k ), (β _k ) then we have