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.     

Observational constraints on a hyperbolic potential in brane-world inflation

We focus on the large field of a hyperbolic potential form, which is characterized by a parameter f, in the framework of the brane-world inflation in Randall-Sundrum-II model. From the observed form of the power spectrum P _R (k), the parameter f should be of order 0.1m _p to 0.001m _p , the brane tension must be in the range λ ~ (1?10)×10⁵⁷ GeV⁴, and the energy scale is around V₀ ^1/4 ~ 10¹⁵ GeV. We find that the inflationary parameters (n _s , r, and dn _s /d(ln k) depend only on the number of e-folds N. The compatibility of these parameters with the last Planck measurements is realized with large values of N.     

On Supersymmetric M-brane configurations with an R<Stack><Subscript>*</Subscript><Superscript>1,1</Superscript></Stack> /Z<Subscript>2</Subscript> submanifold

We obtain new examples of partly supersymmetric M-brane solutions defined on products of Ricci-flat manifolds, which contain a two-dimensional Lorentzian submanifold R _* ^1,1 /Z ₂ with one parallel spinor. The examples belong to the following configurations: M2, M5, M2 ∩M5 and M5 ∩M5. Among them, an M2 solution with N = 1/32 fractional number of preserved supersymmetries is presented.     

Guaranteed approach with the farthest of the runaways

Games of the family {Λ _N } _N?2 are formulated and studied with the application of generalized Isaacs’s approach. The game Λ _N is a simplest model of the counteraction of one persecutor P and coalition N of E ^N runaways for the case when the payoff is the distance up to the coalition of E ^N equal to the Euclidean distance between P and the farthest from the runaways; P is in command of the termination moment. Moreover, an approach within the limits of which in games with a smooth terminal payoff are generated strategies prescribing players’ motions in the directions of local gradients of the payoff is described. The approach is used for constructing pursuit strategies in games in which smooth approximations of the maximum of Euclidean distances up to the runaways are in place of payoffs. Pursuit strategies prescribing the motion in the direction of the farthest of the runaways are studied. A numerical simulation of the development of the games Λ₂ and Λ₃ is conducted in using different strategies by the players.     

Two-armed bandit problem for parallel data processing systems

We consider application of the two-armed bandit problem to processing a large number N of data where two alternative processing methods can be used. We propose a strategy which at the first stages, whose number is at most r ? 1, compares the methods, and at the final stage applies only the best one obtained from the comparison. We find asymptotically optimal parameters of the strategy and observe that the minimax risk is of the order of N ^α , where α = 2 ^r?1/(2 ^r ? 1). Under parallel processing, the total operation time is determined by the number r of stages but not by the number N of data.     

A novel constant degree and constant congestion DHT scheme for peer-to-peer networks

1 Introduction and related work In recent years, peer-to-peer computing has attracted significant attention from both industry field and academic field[1-3]. The core component of many proposed peer-to- peer systems is the distributed hash table (DHT) schemes[4,5] that use a hash table-like interface to publish and look up data objects. Many proposed DHT schemes[6-15] are based on some traditional interconnection to- pology: Chord[6], Tapestry[7,8], Pastry[9] are based on hypercube topolog…     

On the Smallest Size of an Almost Complete Subset of a Conic in PG(2, <Emphasis Type="Italic">q</Emphasis>) and Extendability of Reed–Solomon Codes

Abstract—In the projective plane PG(2, q), a subset S of a conic C is said to be almost complete if it can be extended to a larger arc in PG(2, q) only by the points of C \ S and by the nucleus of C when q is even. We obtain new upper bounds on the smallest size t(q) of an almost complete subset of a conic, in particular,

$$t(q) < \sqrt {q(3lnq + lnlnq + ln3)} + \sqrt {\frac{q}{{3\ln q}}} + 4 \sim \sqrt {3q\ln q} ,t(q) < 1.835\sqrt {q\ln q.} $$

The new bounds are used to extend the set of pairs (N, q) for which it is proved that every normal rational curve in the projective space PG(N, q) is a complete (q+1)-arc, or equivalently, that no [q+1,N+1, q?N+1]_q generalized doubly-extended Reed–Solomon code can be extended to a [q + 2,N + 1, q ? N + 2]_q maximum distance separable code.

     

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.     

On s-uniform property of compressing sequences derived from primitive sequences modulo odd prime powers

Let Z/(p^e) be the integer residue ring modulo p^e with p an odd prime and e ≥ 2. We consider the suniform property of compressing sequences derived from primitive sequences over Z/(p^e). We give necessary and sufficient conditions for two compressing sequences to be s-uniform with α provided that the compressing map is of the form ?(x₀, x₁,...,x_e?1) = g(x_e?1) + η(x₀, x₁,..., x_e?2), where g(x_e?1) is a permutation polynomial over Z/(p) and η is an (e ? 1)-variable polynomial over Z/(p).     

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.     

The bounded core for games with restricted cooperation

A game with restricted (incomplete) cooperation is a triple (N, v, Ω), where N represents a finite set of players, Ω ? 2^N is a set of feasible coalitions such that N ∈ Ω, and v: Ω → R denotes a characteristic function. Unlike the classical TU games, the core of a game with restricted cooperation can be unbounded. Recently Grabisch and Sudhölter [9] proposed a new solution concept—the bounded core—that associates a game (N, v,Ω) with the union of all bounded faces of the core. The bounded core can be empty even if the core is nonempty. This paper gives two axiomatizations of the bounded core. The first axiomatization characterizes the bounded core for the class G^r of all games with restricted cooperation, whereas the second one for the subclass G_bc^r ? G^r of the games with nonempty bounded cores.     

On the Advice Complexity of the k-server Problem Under Sparse Metrics

We consider the k-Server problem under the advice model of computation when the underlying metric space is sparse. On one side, we introduce Θ(1)-competitive algorithms for a wide range of sparse graphs. These algorithms require advice of (almost) linear size. We show that for graphs of size N and treewidth α, there is an online algorithm that receives O (n(log α + log log N))^* bits of advice and optimally serves any sequence of length n. We also prove that if a graph admits a system of μ collective tree (q, r)-spanners, then there is a (q + r)-competitive algorithm which requires O (n(log μ + log log N)) bits of advice. Among other results, this gives a 3-competitive algorithm for planar graphs, when provided with O (n log log N) bits of advice. On the other side, we prove that advice of size Ω(n) is required to obtain a 1-competitive algorithm for sequences of length n even for the 2-server problem on a path metric of size N ≥ 3. Through another lower bound argument, we show that at least \(\frac {n}{2}(\log \alpha - 1.22)\) bits of advice is required to obtain an optimal solution for metric spaces of treewidth α, where 4 ≤ α < 2k.     

The pursuit-evasion game on the 1-skeleton graph of a regular polyhedron. I

We consider a game between a group of n pursuers and one evader moving with the same maximum velocity along the 1-skeleton graph of a regular polyhedron. The goal of the paper is finding, for each regular polyhedron M, a number N(M) with the following properties: if n ≥ N(M), the group of pursuers wins, while if n < N(M), the evader wins. Part I of the paper is devoted to the case of polyhedra in ?³; Part II will be devoted to the case of ? ^d , d ≥ 5; and Part III, to the case of ?⁴.     

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.     

On conditional diagnosability and reliability of the BC networks

An n-dimensional bijective connection network (in brief, BC network), denoted by X _n , is an n-regular graph with 2 ⁿ nodes and n2 ^n?1 edges. Hypercubes, crossed cubes, twisted cubes, and Möbius cubes all belong to the class of BC networks (Fan and He in Chin. J. Comput. 26(1):84–90, [2003]). We prove that the super connectivity of X _n is 2n?2 for n≥3 and the conditional diagnosability of X _n is 4n?7 for n≥5. As a corollary of this result, we obtain the super connectivity and conditional diagnosability of the hypercubes, twisted cubes, crossed cubes, and Möbius cubes.     

Generalized cryptanalysis of RSA with small public exponent针对公钥e小于等于N的0.5次幂的RSA算法的广义密码分析

In this paper, we demonstrate that there exist weak keys in the RSA public-key cryptosystem with the public exponent e = N^α ≤ N^0.5. In 1999, Boneh and Durfee showed that when α ≈ 1 and the private exponent d = N^β < N^0.292, the system is insecure. Moreover, their attack is still effective for 0.5 < α < 1.875. We propose a generalized cryptanalytic method to attack the RSA cryptosystem with α ≤ 0.5. For \(c = \left\lfloor {\frac{{1 - \alpha }}{\alpha }} \right\rfloor \) and e^γc ≡ d (mod e^c), when γ, β satisfy \(\gamma < 1 + \frac{1}{c} - \frac{1}{{2\alpha c}}and\beta < \alpha c + \frac{7}{6} - \alpha \gamma c - \frac{1}{3}\sqrt {6\alpha + 6\alpha c + 1 - 6\alpha \gamma c} \), we can perform cryptanalytic attacks based on the LLL algorithm. The basic idea is an application of Coppersmith’s techniques and we further adapt the technique of unravelled linearization, which leads to an optimized lattice. Our advantage is that we achieve new attacks on RSA with α ≤ 0.5 and consequently, there exist weak keys in RSA for most α.     

Minimizing the maximal weighted lateness of delivering orders between two railroad stations

We consider the planning problem for freight transportation between two railroad stations. We are required to fulfill orders (transport cars by trains) that arrive at arbitrary time moments and have different value (weight). The speed of trains moving between stations may be different. We consider problem settings with both fixed and undefined departure times for the trains. For the problem with fixed train departure times we propose an algorithm for minimizing the weighted lateness of orders with time complexity O(qn ² log n) operations, where q is the number of trains and n is the number of orders. For the problem with undefined train departure and arrival times we construct a Pareto optimal set of schedules optimal with respect to criteria wL _max and C _max in O(n ² max{n log n, q log v}) operations, where v is the number of time windows during which the trains can depart. The proposed algorithm allows to minimize both weighted lateness wL _max and total time of fulfilling freight delivery orders C _max.     

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.     

<Emphasis Type="Italic">α</Emphasis>-Systems of differential inclusions and their unification

This paper introduces α-systems of differential inclusions on a bounded time interval [t₀, ?] and defines α-weakly invariant sets in [t₀, ?] × ?_n, where ?_n is a phase space of the differential inclusions. We study the problems connected with bringing the motions (trajectories) of the differential inclusions from an α-system to a given compact set M ? ?_n at the moment ? (the approach problems). The issues of extracting the solvability set W ? [t₀, ?] × ?ⁿ in the problem of bringing the motions of an α-system to M and the issues of calculating the maximal α-weakly invariant set W^c ? [t₀, ?] × ?ⁿ are also discussed. The notion of the quasi-Hamiltonian of an α-system (α-Hamiltonian) is proposed, which seems important for the problems of bringing the motions of the α-system to M.     

Analysis of Multi-Sort Algorithm on Multi-Mesh of Trees (MMT) architecture

Various sorting algorithms using parallel architectures have been proposed in the search for more efficient results. This paper introduces the Multi-Sort Algorithm for Multi-Mesh of Trees (MMT) Architecture for N=n ⁴ elements with more efficient time complexity compared to previous architectures. The shear sort algorithm on Single Instruction Multiple Data (SIMD) mesh model requires \(4\sqrt{N}+O\sqrt{N}\) time for sorting N elements, arranged on a \(\sqrt{N}\times \sqrt{N}\) mesh, whereas Multi-Sort algorithm on the SIMD Multi-Mesh (MM) Architecture takes O(N ^1/4) time for sorting the same N elements, which proves that Multi-Sort is a better sorting approach. We have improved the time complexity of intrablock Sort. The Communication time complexity for 2D Sort in MM is O(n), whereas this time in MMT is O(log?n). The time complexity of compare–exchange step in MMT is same as that in MM, i.e., O(n). It has been found that the time complexity of the Multi-Sort on MMT has been improved as on Multi-Mesh architecture.