Occurrence indices of elements in linear recurrence sequences over primary residue rings

We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring \(\mathbb{Z}_{p^n } \). We give conditions on the characteristic polynomial F(x) of a linear recurrence sequence u which guarantee that all elements of the ring occur in u. For the case where F(x) is a reversible Galois polynomial over \(\mathbb{Z}_{p^n } \), we give upper bounds for occurrence indices of elements in a linear recurrence sequence u. A situation where the characteristic polynomial F(x) of a linear recurrence sequence u is a trinomial of a special form over ?₄ is considered separately. In this case we give tight upper bounds for occurrence indices of elements of u.     

Expansion of Self-Similar Functions in the Faber–Schauder System

Let Ω = A^N be a space of right-sided infinite sequences drawn from a finite alphabet A = {0,1}, N = {1,2,…}. Let ρ(x, y)Σ_k=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 \(\mu (B(\omega ,r))r + \sum\nolimits_{n = 0}^\infty {\sum\nolimits_{j = 0}^{{2^n} - 1} {{\mu _{n,j}}} } (\omega )\tau ({2^n}r - j)\), where τ(x) = 2min {x,1 ? x}, 0 ≤ x ≤ 1, (τ(x) = 0, if x < 0 or x > 1 ), \({\mu _{n,j}}(\omega ) = (1 - {p_{{\omega _{n + 1}}}})\prod _{k = 1}^n{p_{{\omega _k}}} \oplus {j_k}\), \(j = {j_1}{2^{n - 1}} + {j_2}{2^{n - 2}} + ... + {j_n}\). The family of functions 1, x, τ(2 ⁿ r ? j), j = 0,1,…, 2 ⁿ ? 1, n = 0,1,…, is the Faber–Schauder system for the space C([0,1]) of continuous functions on [0, 1]. We also obtain the Faber–Schauder expansion for Lebesgue’s singular function, Cezaro curves, and Koch–Peano curves. Article is published in the author’s wording.     

Polar Codes with Higher-Order Memory

We introduce a construction of a set of code sequences {C_n^(m) : n ≥ 1, m ≥ 1} with memory order m and code length N(n). {C_n^(m)} is a generalization of polar codes presented by Ar?kan in [1], where the encoder mapping with length N(n) is obtained recursively from the encoder mappings with lengths N(n ? 1) and N(n ? m), and {C_n^(m)} coincides with the original polar codes when m = 1. We show that {C_n^(m)} achieves the symmetric capacity I(W) of an arbitrary binary-input, discrete-output memoryless channel W for any fixed m. We also obtain an upper bound on the probability of block-decoding error P_e of {C_n^(m)} and show that \({P_e} = O({2^{ - {N^\beta }}})\) is achievable for β < 1/[1+m(? ? 1)], where ? ∈ (1, 2] is the largest real root of the polynomial F(m, ρ) = ρ^m ? ρ^{m ? 1} ? 1. The encoding and decoding complexities of {C_n^(m)} decrease with increasing m, which proves the existence of new polar coding schemes that have lower complexity than Ar?kan’s construction.     

On the <Emphasis Type="Italic">k</Emphasis>-accessibility of cores of <Emphasis Type="Italic">TU</Emphasis>-cooperative games

This paper proposes a strengthening of the author’s core-accessibility theorem for balanced TU-cooperative games. The obtained strengthening relaxes the influence of the nontransitivity of classical domination αv on the quality of the sequential improvement of dominated imputations in a game v. More specifically, we establish the k-accessibility of the core C(α _v ) of any balanced TU-cooperative game v for all natural numbers k: for each dominated imputation x, there exists a converging sequence of imputations x₀, x₁,..., such that x₀ = x, lim x _r ∈ C(α _v ) and x_r?m is dominated by any successive imputation x _r with m ∈ [1, k] and r ≥ m. For showing that the TU-property is essential to provide the k-accessibility of the core, we give an example of an NTU-cooperative game G with a ”black hole” representing a nonempty closed subset B ? G(N) of dominated imputations that contains all the α _G -monotonic sequential improvement trajectories originating at any point x ∈ B.     

On derivative-based global sensitivity criteria

A mathematical model f(x) given in the unit n-dimensional cube, where x = (x ₁, ..., x _n ), is considered. How can one estimate the global sensitivity of f(x) with respect to x _i ? If f(x) ∈ L ₂, global sensitivity indices help answer this question. Being less reliable, derivative-based sensitivity criteria are sometimes easier-to-compute. In this work, a new derivative-based global sensitivity criterion is compared to the respective global sensitivity index. These estimates are proven to coincide in the particular case when f(x) linearly depends on x _i . However, Monte Carlo approximations to the derivative-based criterion converge faster. Thus, the global derivative-based sensitivity criterion can prove useful when f(x) depends on x _i almost linearly. It can be also used to find nonessential variables x _i .     

The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.     

Negation-Limited Complexity of Parity and Inverters

In negation-limited complexity, one considers circuits with a limited number of NOT gates, being motivated by the gap in our understanding of monotone versus general circuit complexity, and hoping to better understand the power of NOT gates. We give improved lower bounds for the size (the number of AND/OR/NOT) of negation-limited circuits computing Parity and for the size of negation-limited inverters. An inverter is a circuit with inputs x ₁,…,x _n and outputs ¬ x ₁,…,¬ x _n . We show that: (a) for n=2 ^r ?1, circuits computing Parity with r?1 NOT gates have size at least 6n?log?₂(n+1)?O(1), and (b) for n=2 ^r ?1, inverters with r NOT gates have size at least 8n?log?₂(n+1)?O(1). We derive our bounds above by considering the minimum size of a circuit with at most r NOT gates that computes Parity for sorted inputs x ₁≥???≥x _n . For an arbitrary r, we completely determine the minimum size. It is 2n?r?2 for odd n and 2n?r?1 for even n for ?log?₂(n+1)??1≤r≤n/2, and it is ?3n/2??1 for r≥n/2. We also determine the minimum size of an inverter for sorted inputs with at most r NOT gates. It is 4n?3r for ?log?₂(n+1)?≤r≤n. In particular, the negation-limited inverter for sorted inputs due to Fischer, which is a core component in all the known constructions of negation-limited inverters, is shown to have the minimum possible size. Our fairly simple lower bound proofs use gate elimination arguments in a somewhat novel way.     

On ruin probability minimization under excess reinsurance

The problem of ruin probability minimization in the Cramer-Lundberg risk model under excess reinsurance is studied. Together with traditional maximization of the Lundberg characteristic coefficient R is considered the problem of direct calculation of insurer’s ruin probability ? _r (x) as an initial-capital function x under the prescribed level of net-retention r. To solve this problem, we propose the excess variant of the Cramer integral equation which is an equivalent to the Hamilton-Jacobi-Bellman equation. The continuation method is used for solving this equation; by means of it is found the analytical solution to the Markov risk model. We demonstrated on a series of standard examples that with any admissible value of x the ruin probability ? _x (r): = ? _r (x) is usually a unimodal function r. A comparison of the analytic representation of ruin probability ? _r(x) with its asymptotic approximation with x → ∞ was conducted.     

Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function

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 = ∫₀¹t ⁿ dL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), where the function τ(x) is periodic in log₂x with the period 1 and is given as \(\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}\), \({z_k} = \frac{{2\pi ik}}{{1n2}}\), k ≠ 0. The proof is based on poissonization and the Mellin transform.     

The Sum 2KM(x)−K(x) Over All Prefixes x of Some Binary Sequence Can be Infinite

We consider two quantities that measure complexity of binary strings: KM(x) is defined as the negative logarithm of continuous a priori probability on the binary tree, and K(x) denotes prefix complexity of a binary string x. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence ω such that the sum of 2^KM(x)?K(x) over all prefixes x of ω is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.     

Linear complexity problems of level sequences of Euler quotients and their related binary sequences欧拉商的权位序列及其二元序列的线性复杂度问题

The Euler quotient modulo an odd-prime power p^r (r > 1) can be uniquely decomposed as a p-adic number of the form \(\frac{{u^{(p - 1)p^{r - 1} } - 1}}{{p^r }} \equiv a_0 (u) + a_1 (u)p + \cdots + a_{r - 1} (u)p^{r - 1} (\bmod p^r ), \gcd (u,p) = 1,\) where 0 ? a_j (u) < p for 0 ? j ? r?1 and we set all a_j (u) = 0 if gcd(u, p) > 1. We firstly study certain arithmetic properties of the level sequences (a_j (u))_u?0 over \(\mathbb{F}_p \) via introducing a new quotient. Then we determine the exact values of linear complexity of (a_j (u))_u?0 and values of k-error linear complexity for binary sequences defined by (a_j (u))_u?0.     

On finite groups with large degrees of irreducible character

Let G be a finite nontrivial group with an irreducible complex character χ of degree d = χ(1). According to the orthogonality relation, the sum of the squared degrees of irreducible characters of G is the order of G. N. Snyder proved that, if G = d(d + e), then the order of the group G is bounded in terms of e for e > 1. Y. Berkovich demonstrated that, in the case e = 1, the group G is Frobenius with the complement of order d. This paper studies a finite nontrivial group G with an irreducible complex character Θ such that G ≤ 2Θ(1)² and Θ(1) = pq where p and q are different primes. In this case, we have shown that G is a solvable group with an Abelian normal subgroup K of index pq. Using the classification of finite simple groups, we have established that the simple non-Abelian group, the order of which is divisible by the prime p and not greater than 2p ⁴ is isomorphic to one of the following groups: L ₂(q), L ₃(q), U ₃(q), S _z(8), A ₇, M ₁₁, and J ₁.     

On polynomial solutions of the linear stationary control system

It is known that the controllable system x′ = Bx + Du, where the x is the n-dimensional vector, can be transferred from an arbitrary initial state x(0) = x ⁰ to an arbitrary finite state x(T) = x ^T by the control function u(t) in the form of the polynomial in degrees t. In this work, the minimum degree of the polynomial is revised: it is equal to 2p + 1, where the number (p ? 1) is a minimum number of matrices in the controllability matrix (Kalman criterion), whose rank is equal to n. A simpler and a more natural algorithm is obtained, which first brings to the discovery of coefficients of a certain polynomial from the system of algebraic equations with the Wronskian and then, with the aid of differentiation, to the construction of functions of state and control.     

On risk concentration for convex combinations of linear estimators

We consider the estimation problem for an unknown vector β ∈ Rp in a linear model Y = Xβ + σξ, where ξ ∈ Rⁿ is a standard discrete white Gaussian noise and X is a known n × p matrix with n ≥ p. It is assumed that p is large and X is an ill-conditioned matrix. To estimate β in this situation, we use a family of spectral regularizations of the maximum likelihood method β^α(Y) = H ^α(X ^T X) β ^?(Y), α ∈ R⁺, where β ^?(Y) is the maximum likelihood estimate for β and {H ^α(·): R⁺ → [0, 1], α ∈ R⁺} is a given ordered family of functions indexed by a regularization parameter α. The final estimate for β is constructed as a convex combination (in α) of the estimates β^α(Y) with weights chosen based on the observations Y. We present inequalities for large deviations of the norm of the prediction error of this method.     

Indefinite summation of rational functions with additional minimization of the summable part

An algorithm of indefinite summation of rational functions is proposed. For a given function f(x), it constructs a pair of rational functions g(x) and r(x) such that f(x) = g(x + 1) ? g(x) + r(x), where the degree of the denominator of r(x) is minimal, and, when this condition is satisfied, the degree of the denominator of g(x) is also minimal.     

On horizons and wormholes in k-essence theories

We study the properties of possible static, spherically symmetric configurations in k-essence theories with the Lagrangian functions of the form F(X), X ≡ ?,_α ?,^α. A no-go theorem has been proved, claiming that a possible black-hole-like Killing horizon of finite radius cannot exist if the function F(X) is required to have a finite derivative dF/dX. Two exact solutions are obtained for special cases of kessence: one for F(X) = F ₀ X ^1/3, another for F(X) = F ₀|X|^1/2 ? 2Λ, where F ₀ and Λ are constants. Both solutions contain horizons, are not asymptotically flat, and provide illustrations for the obtained nogo theorem. The first solution may be interpreted as describing a black hole in an asymptotically singular space-time, while in the second solution two horizons of infinite area are connected by a wormhole.     

On weak isometries of Preparata codes

Two codes C ₁ and C ₂ are said to be weakly isometric if there exists a mapping J: C ₁ → C ₂ such that for all x, y in C ₁ the equality d(x, y) = d holds if and only if d(J(x), J(y)) = d, where d is the code distance of C ₁. We prove that Preparata codes of length n ≥ 2¹² are weakly isometric if and only if the codes are equivalent. A similar result is proved for punctured Preparata codes of length at least 2¹⁰ ? 1.     

Central limit theorem and large deviations of the fading Wyner cellular model via product of random matrices theory

We apply the theory of products of random matrices to the analysis of multi-user communication channels similar to the Wyner model, which are characterized by short-range intra-cell broadcasting. We study fluctuations of the per-cell sum-rate capacity in the non-ergodic regime and provide results of the type of the central limit theorem (CLT) and large deviations (LD). Our results show that CLT fluctuations of the per-cell sum-rate C _m are of order \(1/\sqrt m \), where m is the number of cells, whereas they are of order 1/m in classical random matrix theory. We also show an LD regime of the form P(|C _m ? C| > ?) ≤ e ^?mα with α = α(?) > 0 and C = \(\mathop {\lim }\limits_{m \to \infty } \) C _m , as opposed to the rate \(e^{ - m^2 \alpha } \) in classical random matrix theory.     

Linear Time Algorithms for Generalizations of the Longest Common Substring Problem

In its simplest form, the longest common substring problem is to find a longest substring common to two or multiple strings. Using (generalized) suffix trees, this problem can be solved in linear time and space. A first generalization is the k -common substring problem: Given m strings of total length n, for all k with 2≤k≤m simultaneously find a longest substring common to at least k of the strings. It is known that the k-common substring problem can also be solved in O(n) time (Hui in Proc. 3rd Annual Symposium on Combinatorial Pattern Matching, volume 644 of Lecture Notes in Computer Science, pp. 230–243, Springer, Berlin, 1992). A further generalization is the k -common repeated substring problem: Given m strings T ⁽¹⁾,T ⁽²⁾,…,T ^(m) of total length n and m positive integers x ₁,…,x _m , for all k with 1≤k≤m simultaneously find a longest string ω for which there are at least k strings \(T^{(i_{1})},T^{(i_{2})},\ldots,T^{(i_{k})}\) (1≤i ₁<i ₂<???<i _k ≤m) such that ω occurs at least \(x_{i_{j}}\) times in \(T^{(i_{j})}\) for each j with 1≤j≤k. (For x ₁=???=x _m =1, we have the k-common substring problem.) In this paper, we present the first O(n) time algorithm for the k-common repeated substring problem. Our solution is based on a new linear time algorithm for the k-common substring problem.     

On the optimal choice of quality metric in image compression: a soft computing approach

Images take lot of computer space; in many practical situations, we cannot store all original images, we have to use compression. Moreover, in many such situations, compression ratio provided by even the best lossless compression is not sufficient, so we have to use lossy compression. In a lossy compression, the reconstructed image ? is, in general, different from the original image I. There exist many different lossy compression methods, and most of these methods have several tunable parameters. In different situations, different methods lead to different quality reconstruction, so it is important to select, in each situation, the best compression method. A natural idea is to select the compression method for which the average value of some metric d(I,?) is the smallest possible. The question is then: which quality metric should we choose? In this paper, we show that under certain reasonable symmetry conditions, L ^p metrics d(I,?)=∫|I(x)??(x)| ^p dx are the best, and that the optimal value of p can be selected depending on the expected relative size r of the informative part of the image.