Sequence X1 Research Articles

Simulating events of unknown probabilities via reverse time martingales

AbstractLet s∈(0,1) be uniquely determined but only its approximations can be obtained with a finite computational effort. Assume one aims to simulate an event of probability s. Such settings are often encountered in statistical simulations. We consider two specific examples. First, the exact simulation of non‐linear diffusions ([3]). Second, the celebrated Bernoulli factory problem ([10, 13]) of generating an f(p)‐coin given a sequence X1,X2,… of independent tosses of a p‐coin (with known f and unknown p). We describe a general framework and provide algorithms where this kind of problems can be fitted and solved. The algorithms are straightforward to implement and thus allow for effective simulation of desired events of probability s. Our methodology links the simulation problem to existence and construction of unbiased estimators. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 441–452, 2011

FINITENESS PROPERTIES OF LOCAL COHOMOLOGY MODULES AND GENERALIZED REGULAR SEQUENCES

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. The purpose of this paper is to show that if M is finitely generated and dim M/𝔞M > 1, then the R-module ∪{N|N is a submodule of [Formula: see text] and dim N ≤ 1} is 𝔞-cominimax and for some x ∈ R is Rx + 𝔞-cofinite, where t ≔ gdepth (𝔞, M). For any nonnegative integer l, it is also shown that if R is semi-local and M is weakly Laskerian, then for any submodule N of [Formula: see text] with dim N ≤ 1 the associated primes of [Formula: see text] are finite, whenever [Formula: see text] for all i < l. Finally, we show that if (R, 𝔪) is local, M is finitely generated, [Formula: see text] for all i < l, and [Formula: see text] then there exists a generalized regular sequence x1, …, xl ∈ 𝔞 on M such that [Formula: see text].

On a Variant of Van Der Waerden's Theorem

AbstractGiven positive integers

Linear systems and Multiplicity of ideals

A result of P. Samuel ([17] p. 186, Chap.II, Theoreme 5) says that in a local noetherian ring (O,M) of Krull dimension d in which the residual field k is infinite, the multiplicity of a M-primary ideal I is equal to the multiplicity of an ideal (x1, . . . , xd) generated by some parameter sequence x1, . . . , xd contained in I. By a theorem of Rees ([16] p.142 Theorem 9.44), this implies that the ideals I and (x1, . . . , xd) have the same integral closure in the ring O.

A Characterization of Gorenstein Rings and Grothendieck's Conjecture

Given a commutative Noetherian local ring (R, 𝔪), it is shown that R is Gorenstein if and only if there exists a system of parameters x1,…,xd of R which generates an irreducible ideal and [Formula: see text] for all t > 0. Let n be an arbitrary non-negative integer. It is also shown that for an arbitrary ideal 𝔞 of a commutative Noetherian (not necessarily local) ring R and a finitely generated R-module M, [Formula: see text] is finitely generated if and only if there exists an 𝔞-filter regular sequence x1,…,xn∈ 𝔞 such that [Formula: see text] for all t > 0.

On the boundedness of one recurrent sequence in a banach space

We establish necessary and sufficient conditions under which a sequence x 0 = y 0 , x n+1 = Ax n + y n+1 , n ≥ 0, is bounded for each bounded sequence $\left\{ {y_n :n \geqslant 0} \right\} \subset \left\{ {\left. {x \in \bigcup\nolimits_{n = 1}^\infty {D\left( {A^n } \right)} } \right|\sup _{n \geqslant 0} \left\| {A^n x} \right\| < \infty } \right\}$ , where A is a closed operator in a complex Banach space with domain of definition D(A) .

Optimization of 6061T6 CNC Boring Process Using the Taguchi Method and Grey Relational Analysis

Optimization of 6061T6 CNC Boring Process Using the Taguchi Method and Grey Relational Analysis

Notes on the nuclearity of random subalgebras of $${\mathcal {O}_2}$$

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x 1, x 2, ... we have $${\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0$$ . Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra $${\mathcal {O}_2}$$ such that for an independent sequence x 1, x 2, ... of μ-distributed random elements x i we have $${\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0$$ . We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] d ) is d.

Extreme values for Benedicks–Carleson quadratic maps

AbstractWe consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0,X1,… , given by Xn=fan, for every integer n≥0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of fa. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of Mn=max {X0,…,Xn−1} is the same as that which would apply if the sequence X0,X1,… was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of Mn is of type III (Weibull).

Adaptive Coding and Prediction of Sources With Large and Infinite Alphabets

The problem of predicting a sequence x1,x2,. . . generated by a discrete source with unknown statistics is considered. Each letter xt+1 is predicted using the information on the word x1x2 hellip xt only. This problem is of great importance for data compression, because of its use to estimate probability distributions for PPM algorithms and other adaptive codes. On the other hand, such prediction is a classical problem which has received much attention. Its history can be traced back to Laplace. We address the problem where the sequence is generated by an independent and identically distributed (i.i.d.) source with some large (or even infinite) alphabet and suggest a class of new methods of prediction.

Tot(M) and modules of generalized fractions

Throughout this paper, R is a commutative Notherian ring with non-zero identity and M is a finitely generated R-module. This paper is concerned with a certain complex C(U(x),M) of R-modules and R-homomorphism which involves modules of generalized fractions derived from M and the sequence x1, x2, . . . , xn. The complex is described as follows: for each i ∈ N (throughout we use N to denote the set of positive integers). Set U(x)i = {(x1 1 , x2 2 , . . . , xi i ) | ∃j, 0 ≤ j ≤ i s.t. α1, . . . , αj ∈ N, and αj+1 = · · · = αi = 0}, where xr is interpreted as xn whenever r > n. Then, for each i ∈ N, U(x)i is a triangular subset [4,2.1] of R. We can [2, page 420] from the complex 0 → M e0 −→ U(x)−1 1 M e 1 −→ · · · ei −→ U(x)−i−1 i+1 M e i+1 −→ · · · of R-modouls and R-homomorphisms, which we denote by C(U(x),M). Here U(x)−i i M is modoule of generalized fractions of M with respect to the triangular subset U(x)i of R , and the homomorphisms e (i ≥ 0) are given by the following formulas. e(b) = b (1) for all b ∈ M and, for each i ∈ N,

An Algorithm for Universal Lossless Compression With Side Information

This paper proposes a new algorithm based on the Context-Tree Weighting (CTW) method for universal compression of a finite-alphabet sequence x1 n with side information y1 n available to both the encoder and decoder. We prove that with probability one the compression ratio converges to the conditional entropy rate for jointly stationary ergodic sources. Experimental results with Markov chains and English texts show the effectiveness of the algorithm

Divisibility properties of certain recurrent sequences

Let g and m be two positive integers, and let F be a polynomial with integer coefficients. We show that the recurrent sequence x0 = g, xn = x n−1 n + F(n), n = 1, 2, 3,…, is periodic modulo m. Then a special case, with F(z) = 1 and with m = p > 2 being a prime number, is considered. We show, for instance, that the sequence x0 = 2, xn = x n−1 n + 1, n = 1, 2, 3, …, has infinitely many elements divisible by every prime number p which is less than or equal to 211 except for three prime numbers p = 23, 47, 167 that do not divide xn. These recurrent sequences are related to the construction of transcendental numbers ζ for which the sequences [ζn!], n = 1, 2, 3, …, have some nice divisibility properties. Bibliography: 18 titles.

Identification and Functional Expression of a Family of Nicotinic Acetylcholine Receptor Subunits in the Central Nervous System of the Mollusc Lymnaea stagnalis

We described a family of nicotinic acetylcholine receptor (nAChR) subunits underlying cholinergic transmission in the central nervous system (CNS) of the mollusc Lymnaea stagnalis. By using degenerate PCR cloning, we identified 12 subunits that display a high sequence similarity to nAChR subunits, of which 10 are of the alpha-type, 1 is of the beta-type, and 1 was not classified because of insufficient sequence information. Heterologous expression of identified subunits confirms their capacity to form functional receptors responding to acetylcholine. The alpha-type subunits can be divided into groups that appear to underlie cation-conducting (excitatory) and anion-conducting (inhibitory) channels involved in synaptic cholinergic transmission. The expression of the Lymnaea nAChR subunits, assessed by real time quantitative PCR and in situ hybridization, indicates that it is localized to neurons and widespread in the CNS, with the number and localization of expressing neurons differing considerably between subunit types. At least 10% of the CNS neurons showed detectable nAChR subunit expression. In addition, cholinergic neurons, as indicated by the expression of the vesicular ACh transporter, comprise approximately 10% of the neurons in all ganglia. Together, our data suggested a prominent role for fast cholinergic transmission in the Lymnaea CNS by using a number of neuronal nAChR subtypes comparable with vertebrate species but with a functional complexity that may be much higher.

A remark on proper sequences of modules

A bound for the depth of a quotient of the symmetric algebra, S(E), of a finitely generated module E, over a C.M. ring by an ideal of S(E) generated by a subsequence of x1, . . . , xn is obtained in the case when E satisfies the sliding depth condition, with maximal irrelevant ideal generated by a proper sequence x1, . . . , xn in E

Some decompositions of Bernoulli sets and codes

A decomposition of a set X of words over a d-letter alphabet A = {a1,...,ad} is any sequence X1,...,Xd,Y1,...,Yd of subsets of A* such that the sets Xi, i = 1,...,d, are pairwise disjoint, their union is X, and for all i, 1 ≤ i ≤ d, Xi ~ aiYi, where ~ denotes the commutative equivalence relation. We introduce some suitable decompositions that we call good, admissible, and normal. A normal decomposition is admissible and an admissible decomposition is good. We prove that a set is commutatively prefix if and only if it has a normal decomposition. In particular, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition.

Sequential selection of random vectors under a sum constraint

We observe a sequence X1, X2,…, Xn of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the Xi we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1]d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {x ∈ Q | 〈x, θ〉 ≤ 1}, where θ ∈ ℝd, θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Sequential selection of random vectors under a sum constraint

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

THE DISTRIBUTION OF FIRST OCCURRENCE OF PATTERNS OF OUTCOMES OF BERNOULLI EXPERIMENTS

Let X₁, X₂, .... be a sequence of independent Bernoulli random variables. For any fixed integer K ≥ 1, let $\underline x _K = (x_1 ,x_2 ,...,x_K )$ be any one of the 2K vectors each of whose components is 0 or 1. We study the first occurrence of the pattern X̱K in X₁, X₂ ,..., and characterise the factors that explain why some sequences X̱K are more likely to occur before other sequences of the same length. Explicit formulae and a simple recursion are derived for the distribution of the first occurrence of the vector x̱K in the sequence X₁, X₂, ....

Expected Sums of General Parking Functions

A (u 1, u 2, . . . )-parking function of length n is a sequence (x 1, x 2, . . . , x n) whose order statistics (the sequence (x (1), x (2), . . . , x (n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x (i)≤ u (i). The Goncarov polynomials g n (x; a 0, a 1, . . . , a n-1) are polynomials biorthogonal to the linear functionals e(a i ) D i , where e(a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Goncarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u i =a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u i+1 - u i . Finally, we give analogues of our results for real-valued parking functions.