Evgeniy Timofeev

• Yaroslavl State University

Let \(\Omega = A^N\) - be a space of right-sided infinite sequences drawn from a finite alphabet \(A = \{0,1\}\), \(N = {1,2,\dots} \), \[\rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} \] - a metric on \(\Omega\), and \(\mu\) - a probability measure on \(\Omega\). Let \(\boldsymbol{\xi_0}, \boldsymbol{\xi_1}, \dots,...

Let Ω = AN be a space of right-sided infinite sequences drawn from a finite alphabet A = {0,1}, N = {1,2,…}. Let ρ(x, y)Σk=1∞|x k − y k |2−k be a metric on Ω = AN, and μ the Bernoulli measure on Ω with probabilities p0, p1 > 0, p0 + p1 = 1. Denote by B(x,ω) an open ball of radius r centered at ω. The main result of this paper \(\mu (B(\omega ,r))r...

The Takagi function is a simple example of a continuous yet nowhere differentiable function and is given as T(x) = Σk=0∞ 2−n ρ(2nx), where \(\rho (x) = \mathop {\min }\limits_{k \in \mathbb{Z}} |x - k|\). The moments of the Takagi function are given as M n = ∫01xnT(x)dx. The estimate \({M_n} = \frac{{1nn - \Gamma '(1) - 1n\pi }}{{{n^2}1n2}} + \frac...

Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t − 1), where p, q > 0, q = 1 − p, p ≠ q. The variables M n = ∫01tndL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), whe...

Let \(\Omega = A^{N}\) be a space of right-sided innite sequences drawn from a nite alphabet \(A = \{0,1\}\), \(N = \{1,2,\dots\}\). Let $$\label{rho} \rho(\boldsymbol{x},\boldsymbol{y}) =\sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k}$$- be a metric on \(\Omega = A^{N}\), and \(\mu\) - the Bernoulli measure on \(\Omega\) with probabilities \(p_0,p_1>0\),...

Let \(\Omega = A^{N}\) be a space of right-sided infinite sequences drawn from a finite alphabet \(A = \{0,1\}\), \(N = \{1,2,\dots \}\), \[\label{rho} \rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} \] a metric on \(\Omega = A^{N}\), and \(\mu\) is a probability measure on \(\Omega\). Let \(\boldsymbol{\xi_0}, \bolds...

Asymptotic Formula for the Moments of Bernoulli Convolutions Timofeev E. A. Received February 8, 2016 For each λ, 0 < λ < 1, we define a random variable ∞ Yλ =(1−λ)ξnλn, n=0 where ξn are independent random variables with P{ξn =0}=P{ξn =1}= 1. 2 The distribution of Yλ is called a symmetric Bernoulli convolution. The main result of this paper is Mn =...

Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by T(x) = ∞ ᢘ k=0 2−nρ(2nx), where ρ(x) = min k∈Z |x − k|. The moments of Takagi function are defined as Mn = ᝈ 1 0 xnT(x) dx. The main result of this paper is the following: Mn = lnn − Γ᝘(1) − lnπ n2 ln 2 + 1 2n2 + 2 n2 ln 2 φ(n) + O(n−2.99), where...

Recall the Lebesgue's singular function. We define a Lebesgue's singular function \(L(t)\) as the unique continuous solution of the functional equation$$L(t) = qL(2t) +pL(2t-1),$$where \(p,q>0\), \(q=1-p\), \(p\ne q\).The moments of Lebesque' singular function are defined as$$M_n = \int_0^1t^n dL(t), \quad n = 0, 1, \dots$$The main result of this p...

Recall Lebesgue’s singular function. Imagine flipping a biased coin with probability p of heads and probability q = 1 − p of tails. Let the binary expansion of ξ ∈ [0, 1]: ξ = ∑∞ k=1 ck2−k be determined by flipping the coin infinitely many times, that is, ck = 1 if the k-th toss is heads and ck = 0 if it is tails. We define Lebesgue’s singular function...

We consider the problem of the nonparametric entropy estimation of a stationary ergodic process. Our approach is based on the nearest-neighbor distances. We propose a broad class of metrics on the space Ω = AN of right-sided infinite sequences drawn from a finite alphabet A. The new metric has a parameter which is a non-increasing function. We appl...

A new class of metrics on a space of right-sided infinite sequences drawn from a binary alphabet was introduced.

We introduce a new metric on a space of right-sided infinite sequences drawn from a finite alphabet. Emerging from a problem of entropy estimation of a discrete stationary ergodic process, the metric is important on its own part and exhibits some interesting properties. For example, the measure of a ball is discontinuous at every binary rational va...

We consider the problem of improving the efficiency of the nonparametric entropy estimation for a stationary ergodic process. Our approach is based on the nearest-neighbor distances. We propose a broad class of metrics on the space of right-sided infinite sequences drawn from a finite alphabet. The new metric has a parameter that is a nonincreasing...

A problem of improving the efficiency of nonparametric entropy estimation for discrete stationary ergodic processes is considered. The estimation depends on selection of underlying metric on the space of right-sided infinite sequences. Proposed is a new family of metrics which depend on a set of parameters. The estimator is linearly dependent on th...

A problem of nonparametric entropy estimation for discrete stationary ergodic processes is considered. The estimation is based on so-called ”nearest-neighbor method”. It is shown that, for Bernoulli measures, the estimator is unbiased, i.e. converges to the (inverse) entropy of the process. Moreover, for symmetric Bernoulli measures, the unbiased e...

A new metric on a space of right-sided infinite sequences drawn from a finite alphabet is proposed. This metric was introduced in the problem of estimating the entropy of discrete stationary processes. It has a number of interesting properties. For example, the measure of a ball is discontinuous at any binary rational value of logr, where r is the...

Proposed is a new fast algorithm for entropy estimation of a given input word. The algorithm utilizes k-nearest neighbor search of a given dictionary. The time complexity of the search is independent of the dictionary size.

We introduce a new metric on a space of right-sided infinite sequences drawn from a finite alphabet. Emerging from a problem of entropy estimation of a discrete stationary ergodic process, the metric is important on its own part and exhibits some interesting properties. Notably, the number of distinct metric values for a set of sequences of length...

A problem of improving the accuracy of nonparametric entropy estimation for a stationary ergodic process is considered. New weak metrics are introduced and relations between metrics, measures, and entropy are discussed. Based on weak metrics, a new nearest-neighbor entropy estimator is constructed and has a parameter with which the estimator is opt...

We consider the problem of nonparametric entropy (entropy rate) estimation. We study the technique of nonparametric entropy estimation based on the so-called “nearest neighbor distances” and obtain a closed-form expression of the bias for Markov measures. This bias is a discontinuous function of transition probabilities. Bibliography: 20 titles.

The exact asymptotic form for the bias of entropy estimator [8] for Bernoulli measures is found. Keywordsentropy–nonparametric estimate–moments–Bernoulli measure

New invariants of measures, called the β-statentropy, are described. They are similar to the entropy and the HP-spectrum for dimensions. The β- statentropy admits construction of a statistical estimator calculated by n independent points distributed in accordance with a given measure. The accuracy of this estimator is O(n−c), where c is some consta...

In this work we consider a problem of optimizing the data transmission mechanism of Internet transport protocols. We use a priority discipline without time measurements by a receiver, which has the property of universality. The possibility of an application of this discipline for improving performances of the original Transmission Control Protocol...

In the paper, a new invariant of measures and dynamical systems, called statentropy, is described. A statistical estimator for statentropy, computed without using auxiliary estimates of measures, is constructed. It is proved that the proposed statistical estimator is consistent under fairly general restrictions. We show that for exact dimensional m...

An experiment on fluid temperature measurement at the constant depth of 3000 m is carried out to explain the time variations, caused by influence of tidal potential. It is ascertained that the time variations of fluid temperature have the small (0.001-0.005°C) f(t) component with the day period. This component has the significant correlation with t...

New characteristics of functions called unsmoothness indicators and calculated from a few thousand measurements are proposed. A characteristic feature of the unsmoothness indicators is that they change only slightly if a sufficiently smooth systematic error is introduced into the functions or if the argument is subjected to smooth small variations....

A statistical estimate for generalized dimensions of a set A Ì \mathbbRmA \subset \mathbb{R}^m based on the computation of average distances to the closest points in a sample of elements of A is given. For smooth manifolds with Lebesgue measures and for self-similar fractals with self-similar measures, the estimate is proved to be consistent.

A priority discipline without time measurements is described. The universality of this discipline is proved. The application of this discipline for the performance of TCP is shown.

An estimate is given for the dimension of a set M ⊂ ℝn based on the calculation of averaged distances to the nearest point for all elements of a sampling subset of M. For smooth manifolds, it is shown that this estimate is a consistent estimate for the dimension of the manifold and is independent of the distribution of the points of the sampling se...

In this work, the circuit and mathematical models of a pulse-stream neuron are developed. The number of the random bit generators used is on the order of logN, where N is the number of the neuron inputs. The output of such a neuron is a random variable whose expectation varies linearly with the synaptic weights provided that the stream density of i...

The problem of finding the queueing discipline that minimizes a linear function of mean queue lengths has been considered for a multiserver queueing system in [G. P. Klimov, Teor. Veroyatn. Primen. 19, 558-576 (1974; Zbl 0378.60102)] and for a more general system with branching streams of secondary customers of n species in [M. Yu. Kitaev and V. V....

A mathematical model of a data-flow neuron by pulse stream density is constructed. The stationary distribution of the states of such a neuron is found to be approximately normal.

An algorithm is designed, which computes the probabilistic allocation of customers to servers in an M n /G n /m queue system so that a given function of mean waiting time attains its (local) minimum. For a linear function, the allocation problem is reduced to minimizing a nonlinear function on a polyhedron such that no more than one extremum exists...

Neurons that represent numbers as pulse-stream densities are considered. A circuit model of such neurons and their operation are discussed.

Neurons, using the flow density of unit amplitude pulses as an information storage medium, are considered. A scheme realization of the flow neuron and devices necessary for the chosen technique of information representation is given. The circuit is based on the digital element base. Modeling results of a full-connective network consisting of the st...

There is solved a problem of obtaining discipline when the minimum value of a given function depending on mean queue lengths is realized in the service system with one device, Poisson input and splitting secondary flows of requests of several types with arbitrary service duration. An algorithm is presented for parameters determination of the optima...

For a queue with one server, Poisson input, and branching secondary flows of demands of various types and arbitrary service durations, the problem of finding a discipline providing the minimum to a given function of mean queue lengths has been solved. An algorithm for determining the parameters of the optimal discipline is described.

We discuss neurons that use the density (number of pulses per unit time) of a stream of unit-amplitude pulses (bits) as a carrier of information. We describe the devices needed to implement this method of representing information using a purely digital (binary) element basis, along with a circuit realization of such a 'bit-stream neuron.' To illust...

A new queueing discipline is proposed, which achieves any prescribed mean waiting time under stationary conditions in the GI|Gn|1 queue. Mean waiting times for customers of each type are obtained for the HM|Gn|1 and GI|HMn|1 queues. A polynomial-time algorithm is described to determine the parameters of the queueing discipline given the mean waitin...

A system, where requests of n classes are serviced without interruptions, is considered. The problem of searching for the service discipline is solved, when the average waiting times for requests of each class in a queue under stationary conditions have the preset values. For this purpose a new service discipline and n polynomial algorithm for dete...

An optimization problem of mean waiting time of requests in a queue is considered for GI∥Mn∥1 systems. A new approach to arbitrary function optimization of n variables is proposed. Conventional results are generalized to exponentially distributed service times with various parameters. It is shown that the possible values of function arguments form...

A polynomial algorithm based on reduction to a polyhedron minimization problem is suggested for finding the minimum of the prescribed F(Wl, Wn) function depending on mean queuing times in the queuing GI|Gn|1 system. The algorithm determines the function minimum for some special polyhedron, for which the polynomial algorithm is proposed to determine...

Optimization of the average waiting time function F(W 1 ,···,W n ) of calls is examined for a GI n /M/1 queueing system. The set of all sorts of values of (W 1 ,···,W n ) for various queue disciplines forms a polyhedron which is the face of a polymatroid. An algorithm is described that minimizes the function F after a polynomial number of operation...

The following problem is considered: given an undirected graph G with n vertices (and possible multiple edges), find all minimum edge cuts. An algorithm to solve this problem in time O(\lambda n^2) is constructed, where \lambda is the minimum number of edges of a cut.