Tilings of nonorientable surfaces by Steiner triple systems

A Steiner triple system of order n (for short, STS(n)) is a system of three-element blocks (triples) of elements of an n-set such that each unordered pair of elements occurs in precisely one triple. Assign to each triple (i,j,k) ? STS(n) a topological triangle with vertices i, j, and k. Gluing together like sides of the triangles that correspond to a pair of disjoint STS(n) of a special form yields a black-and-white tiling of some closed surface. For each n ≡ 3 (mod 6) we prove that there exist nonisomorphic tilings of nonorientable surfaces by pairs of Steiner triple systems of order n. We also show that for half of the values n ≡ 1 (mod 6) there are nonisomorphic tilings of nonorientable closed surfaces.     

Star reduction among minimal length addition chains

An addition chain for a natural number n is a sequence \({1=a_0 < a_1 < \cdots < a_r=n}\) of numbers such that for each 0 < i ≤ r, a _i  = a _j  + a _k for some 0 ≤ k ≤ j < i. The minimal length of an addition chain for n is denoted by ?(n). If j = i ? 1, then step i is called a star step. We show that there is a minimal length addition chain for n such that the last four steps are stars. Then we conjecture that there is a minimal length addition chain for n such that the last \({\lfloor\frac{\ell(n)}{2}\rfloor}\)-steps are stars. We verify that the conjecture is true for all numbers up to 2¹⁸. An application of the result and the conjecture to generate a minimal length addition chain reduce the average CPU time by 23–29% and 38–58% respectively, and memory storage by 16–18% and 26–45% respectively for m-bit numbers with 14 ≤ m ≤ 22.     

On transitive partitions of an <Emphasis Type="Italic">n</Emphasis>-cube into codes

We present methods to construct transitive partitions of the set E ⁿ of all binary vectors of length n into codes. In particular, we show that for all n = 2 ^k ? 1, k ≥ 3, there exist transitive partitions of E ⁿ into perfect transitive codes of length n.     

On the Construction of Transitive Codes

Application of some known methods of code construction (such as the Vasil'ev, Plotkin, and Mollard methods) to transitive codes satisfying certain auxiliary conditions yields infinite classes of large-length transitive codes, in particular, at least ?k/2?² nonequivalent perfect transitive codes of length n = 2 ^k ? 1, k > 4. A similar result is valid for extended perfect transitive codes.     

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.     

Shadow Prices in Territory Division

We consider a geographic optimization problem in which we are given a region R, a probability density function f(?) defined on R, and a collection of n utility density functions u _i (?) defined on R. Our objective is to divide R into n sub-regions R _i so as to “balance” the overall utilities on the regions, which are given by the integrals \(\iint _{R_{i}}f(x)u_{i}(x)\, dA\). Using a simple complementary slackness argument, we show that (depending on what we mean precisely by “balancing” the utility functions) the boundary curves between optimal sub-regions are level curves of either the difference function u _i (x) ? u _j (x) or the ratio u _i (x)/u _j (x). This allows us to solve the problem of optimally partitioning the region efficiently by reducing it to a low-dimensional convex optimization problem. This result generalizes, and gives very short and constructive proofs of, several existing results in the literature on equitable partitioning for particular forms of f(?) and u _i (?). We next give two economic applications of our results in which we show how to compute a market-clearing price vector in an aggregate demand system or a variation of the classical Fisher exchange market. Finally, we consider a dynamic problem in which the density function f(?) varies over time (simulating population migration or transport of a resource, for example) and derive a set of partial differential equations that describe the evolution of the optimal sub-regions over time. Numerical simulations for both static and dynamic problems confirm that such partitioning problems become tractable when using our methods.     

Making Doubling Metrics Geodesic

The starting point of our research is the following problem: given a doubling metric ?=(V,d), can one (efficiently) find an unweighted graph G′=(V′,E′) with V?V′ whose shortest-path metric d′ is still doubling, and which agrees with d on V×V? While it is simple to show that the answer to the above question is negative if distances must be preserved exactly. However, allowing a (1+ε) distortion between d and d′ enables us bypass this hurdle, and obtain an unweighted graph G′ with doubling dimension at most a factor O(log?ε ^?1) times the doubling dimension of G.More generally, this paper gives algorithms that construct graphs G′ whose convex (or geodesic) closure has doubling dimension close to that of ?, and the shortest-path distances in G′ closely approximate those of ? when restricted to V×V. Similar results are shown when the metric ? is an additive (tree) metric and the graph G′ is restricted to be a tree.     

Stability of properties of Kolmogorov complexity under relativization

Assume that a tuple of binary strings \(\bar a\) = 〈a ₁ ..., a _n 〉 has negligible mutual information with another string b. Does this mean that properties of the Kolmogorov complexity of \(\bar a\) do not change significantly if we relativize them to b? This question becomes very nontrivial when we try to formalize it. In this paper we investigate this problem for a special class of properties (for properties that can be expressed by an ?-formula). In particular, we show that a random (conditional on \(\bar a\)) oracle b does not help to extract common information from the strings a _i .     

Guaranteed strategies for escaping encirclement

This paper considers a conflict situation on the plane as follows. A fast evader E has to break out the encirclement of slow pursuers P _{j1,...,j n} = {P _j1,..., P _jn }, n ≥ 3, with a miss distance not smaller than r ≥ 0. First, we estimate the minimum guaranteed miss distance from E to a pursuer P _a , a ∈ {j ₁,..., j _n }, when the former moves along a given straight line. Then the obtained results are used to calculate the guaranteed estimates to a group of two pursuers P _b,c = {P _b , P _c }, b, c ∈ {j ₁,..., j _n }, b ≠ c, when E maneuvers by crossing the rectilinear segment P _b P _c , and the state passes to the domain of the game space where E applies a strategy under which the miss distance to any of the pursuers is not decreased. In addition, we describe an approach to the games with a group of pursuers P _{j1,... jn} , n ≥ 3, in which E seeks to break out the encirclement by passing between two pursuers P _b and P _c , entering the domain of the game space where E can increase the miss distance to all pursuers by straight motion. By comparing the guaranteed miss distances with r for all alternatives b, c ∈ {j ₁,..., j _n }, b ≠ c, and a ? {b, c}, it is possible to choose the best alternative and also to extract the histories of the game in which the designed evasion strategies guarantee a safe break out from the encirclement.     

Motion of spin-half particles in the axially symmetric field of naked singularities of the static <Emphasis Type="Italic">q</Emphasis>-metric

Quantum-mechanical motion of a spin-half particle is examined in the axially symmetric fields of static naked singularities formed by a mass distribution with a quadrupole moment (q-metric). The analysis is performed by means of the method of effective potentials of the Dirac equation generalized to the case where radial and angular variables are not separated. If ?1 < q < qlim, |qlim| ? 1, where q is the quadrupolemoment in proper units, the naked singularities do not exclude the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For an oblate mass distribution, the naked singularities of the q-metric are separated from a Dirac particle by infinitely large repulsive barriers followed by a potential well which deepens while moving apart from the equator (from θ = θ _min or θ = π ? θ _min) toward the poles. The poles make an exception, and at 0 < q < q*, there are some points θ _i for particle states with j ≥ 3/2.     

General Independence Sets in Random Strongly Sparse Hypergraphs

We analyze the asymptotic behavior of the j-independence number of a random k-uniform hypergraph H(n, k, p) in the binomial model. We prove that in the strongly sparse case, i.e., where \(p = c/\left( \begin{gathered} n - 1 \hfill \\ k - 1 \hfill \\ \end{gathered} \right)\) for a positive constant 0 < c ≤ 1/(k ? 1), there exists a constant γ(k, j, c) > 0 such that the j-independence number α _j (H(n, k, p)) obeys the law of large numbers \(\frac{{{\alpha _j}\left( {H\left( {n,k,p} \right)} \right)}}{n}\xrightarrow{P}\gamma \left( {k,j,c} \right)asn \to + \infty \) Moreover, we explicitly present γ(k, j, c) as a function of a solution of some transcendental equation.     

Redshift in the model of embedded spaces

The non-configurational geometrization of the electromagnetic field can be realized using the Model of Embedded Spaces (MES). This model assumes the existence of proper 4D space-time manifolds of particles with a nonzero rest mass and declares that physical space-time is the metric result of the dynamic embedding of these manifolds: the value of the partial contribution of the element manifold is determined by element interactions. The space of the model is provided with a Riemann-like geometry, whose differential formalism is described by a generalization of the gradient operator ?/?x ⁱ → ?/?x ⁱ + 2u ^k ?²/?x[ ⁱ ?u ^k ], where u ⁱ = dx ⁱ /ds is a matter velocity. In the paper, the redshift effect existing in the space of MES is considered, and its electromagnetic component is analyzed. It is shown that for cold matter of the modern Universe this component reduces to a shift in electric fields and is described by the expression \(\Delta {\omega _e}/\omega \simeq \mp \sqrt k \Delta {\varphi _e}/{c^2} = \mp 0.861 \cdot {10^{ - 21}}\Delta {\varphi _e}\left( V \right)\), where the potential is measured in volts and the sign must be determined experimentally. Testing of the effect is the “experimentum crusis” for MES.     

A new strategy for generating shortest addition sequences

An addition sequence problem is given a set of numbers X = {n ₁, n ₂, . . . , n _m }, what is the minimal number of additions needed to compute all m numbers starting from 1? This problem is NP-complete. In this paper, we present a branch and bound algorithm to generate an addition sequence with a minimal number of elements for a set X by using a new strategy. Then we improve the generation by generalizing some results on addition chains (m = 1) to addition sequences and finding what we will call a presumed upper bound for each n _j , 1 ≤ j ≤ m, in the search tree.     

Bimagic Vertex Labelings

The notion of the equivalence of vertex labelings on a given graph is introduced. The equivalence of three bimagic labelings for regular graphs is proved. A particular solution is obtained for the problem of the existence of a 1-vertex bimagic vertex labeling of multipartite graphs, namely, for graphs isomorphic with K_{n, n, m}. It is proved that the sequence of bi-regular graphs K_n(ij)?=?((K_n???1???M)?+?K₁)???(u_nu_i)???(u_nu_j) admits 1-vertex bimagic vertex labeling, where u_i, u_j is any pair of non-adjacent vertices in the graph K_n???1???M, u_n is a vertex of K₁, M is perfect matching of the complete graph K_n???1. It is established that if an r-regular graph G of order n is distance magic, then graph G + G has a 1-vertex bimagic vertex labeling with magic constants (n?+?1)(n?+?r)/2?+?n² and (n?+?1)(n?+?r)/2?+?nr. Two new types of graphs that do not admit 1-vertex bimagic vertex labelings are defined.     

Similarity of binary relations based on <Emphasis Type="Italic">L</Emphasis>-fuzzy topologies

Binary relations play an important role in rough set theory. This paper investigates the similarity of binary relations based on L-fuzzy topologies, where L is a boolean algebra. First, rough approximations based on a boolean algebra are proposed through successor neighborhoods on binary relations. Next, L-fuzzy topologies induced by binary relations are investigated. Finally, similarity of binary relations is introduced by using the L-fuzzy topologies and the fact that every binary relation is solely similar to some preorder relation is proved. It is worth mentioning that similarity of binary relations are both originated in the L-fuzzy topology and independent of the L-fuzzy topology.     

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.     

Propelinear codes related to some classes of optimal codes

A code is said to be propelinear if its automorphism group contains a subgroup that acts regularly on codewords. We show propelinearity of complements of cyclic codes C _1,i , (i, 2 ^m ? 1) = 1, of length n = 2 ^m ? 1, including the primitive two-error-correcting BCH code, to the Hamming code; the Preparata code to the Hamming code; the Goethals code to the Preparata code; and the Z₄-linear Preparata code to the Z₄-linear perfect code.     

Chain equilibria in secure strategies

In this paper we introduce a modification of the concept of Equilibrium in Secure Strategies (EinSS), which takes into account the non-uniform attitudes of players to security in non-cooperative games. In particular, we examine an asymmetric attitude of players to mutual threats in the simplest case, when all players are strictly ordered by their relation to security. Namely, we assume that the players can be reindexed so that each player i in his behavior takes into account the threats posed by players j > i but ignores the threats of players j < i provided that these threats are effectively contained by some counterthreats. A corresponding equilibrium will be called a Chain EinSS. The conceptual meaning of this equilibrium is illustrated by two continuous games that have no pure Nash equilibrium or (conventional) EinSS. The Colonel Blotto two-player game (Borel 1953; Owen 1968) for two battlefields with different price always admits a Chain EinSS with intuitive interpretation. The product competition of many players on a segment (Eaton, Lipsey 1975; Shaked 1975) with the linear distribution of consumer preferences always admits a unique Chain EinSS solution (up to a permutation of players). Finally, we compare Chain EinSS with Stackelberg equilibrium.     

Minimization of the maximal lateness for a single machine

Consideration was given to the classical NP-hard problem 1|r_j|L_max of the scheduling theory. An algorithm to determine the optimal schedule of processing n jobs where the job parameters satisfy a system of linear constraints was presented. The polynomially solvable area of the problem 1|r_j|L_max was expanded. An algorithm was described to construct a Pareto-optimal set of schedules by the criteria L_max and C_max for complexity of O(n³logn) operations.     

One-Nonterminal Conjunctive Grammars over a Unary Alphabet

Systems of equations of the form X _i =φ _i (X ₁,…,X _n ) (1≤ i ≤ n) are considered, in which the unknowns are sets of natural numbers. Expressions φ _i may contain the operations of union, intersection and elementwise addition \(S+T=\{m+n\mid m\in S\), n∈T}. A system with an EXPTIME-complete least solution is constructed in the paper through a complete arithmetization of EXPTIME-completeness. At the same time, it is established that least solutions of all such systems are in EXPTIME. The general membership problem for these equations is proved to be EXPTIME-complete. Among the consequences of the result is EXPTIME-completeness of the compressed membership problem for conjunctive grammars.