Sentences over Integral Domains and Their Computational Complexities

LetRbe a Hilbertian domain and letKbe its fraction field. Letψ(x₁, …, x_n, y) be a quantifier free arithmetical formula overR. We may also takeψ(x₁, …, x_n, y) to be an arithmetical formula overK[x₁, …, x_n] and write it asψ(y). In this paper we show that ifRhas enough non-units and x₁…x_n y ψ(x₁, …, x_n, y), called an ⁿ  sentence, is true inR, then y ψ(y) is true inK[x₁, …, x_n]. Also, ifR=K[T], whereKis an infinite integral domain andx₁…x_n y ψ(x₁, …, x_n, y)is true inR, then y ψ(y) is true inR[x₁, …, x_n]. These results are applied to find the upper and lower bounds of the time complexities of various decision problems on diophantine equations with parameters and arithmetical sentences. Some of the results are: 1. The decision problem of sentences and diophantine equations with parameters over the ring of integers of a global field are co-NP-complete. 2. The decision problem of sentences over the ring of integers of a global field is NP-complete. 3. LetKbe an infinite domain, the time complexities of the decision problems of equations with parameters and sentences over the polynomial ringK[t] are polynomial time reducible to factoring polynomials overK. 4. The decision problem of sentences over all algebraic integer rings is in P. 5. The decision problem of sentences over all integral domains with characteristic 0 is in P. 6. The time complexity of the decision problem of sentences over all integral domains is polynomial time reducible to factoring integers overZand factoring polynomials over finite fields.     

Positive periodic solutions of second-order nonlinear differential systems with two parameters

By employing the Deimling fixed point index theory, we consider a class of second-order nonlinear differential systems with two parameters . We show that there exist three nonempty subsets of : Γ, Δ₁ and Δ₂ such that and the system has at least two positive periodic solutions for (λ,μ)Δ₁, one positive periodic solution for (λ,μ)Γ and no positive periodic solutions for (λ,μ)Δ₂. Meanwhile, we find two straight lines L₁ and L₂ such that Γ lies between L₁ and L₂.     

Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay

We regard the stochastic functional differential equation with infinite delay as the result of the effects of stochastic perturbation to the deterministic functional differential equation , where is defined by x_t(θ)=x(t+θ),θ(−∞,0]. We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.     

Long paths in hypercubes with a quadratic number of faults

A path between distinct vertices u and v of the n-dimensional hypercube Q_n avoiding a given set of f faulty vertices is called long if its length is at least 2ⁿ-2f-2. We present a function (n)=Θ(n²) such that if f(n) then there is a long fault-free path between every pair of distinct vertices of the largest fault-free block of Q_n. Moreover, the bound provided by (n) is asymptotically optimal. Furthermore, we show that assuming f(n), the existence of a long fault-free path between an arbitrary pair of vertices may be verified in polynomial time with respect to n and, if the path exists, its construction performed in linear time with respect to its length.     

On Some Schützenberger Conjectures

In this paper we consider factorizing codes C over A, i.e., codes verifying the factorization conjecture by Schützenberger. Let n be the positive integer such that aⁿC, we show how we can construct C starting with factorizing codes C′ with a^n′C′ and n′ < n, under the hypothesis that all words aⁱza^j in C, with z(A\a)A*(A\a) (A\a), satisfy i, j, > n. The operation involved, already introduced by Anselmo, is also used to show that all maximal codes C=P(A−1)S+1 with P, SZA and P or S in Za can be constructed by means of this operation starting with prefix and suffix codes. Old conjectures by Schützenberger have been revised.     

Approximating the Minimum Chain Completion problem

A bipartite graph G=(U,V,E) is a chain graph [M. Yannakakis, Computing the minimum fill-in is NP-complete, SIAM J. Algebraic Discrete Methods 2 (1) (1981) 77–79] if there is a bijection such that Γ(π(1))Γ(π(2))Γ(π(|U|)), where Γ is a function that maps a node to its neighbors.We give approximation algorithms for two variants of the Minimum Chain Completion problem, where we are given a bipartite graph G(U,V,E), and the goal is find the minimum set of edges F that need to be added to G such that the bipartite graph G^′=(U,V,E^′) (E^′=EF) is a chain graph.     

Bounded cost algorithms for multivalued consensus using binary consensus instances

In this paper, we present two bounded cost algorithms that solve multivalued consensus using binary consensus instances. Our first algorithm uses log₂n number of binary consensus instances where n is the number of processes, while our second algorithm uses at most binary consensus instances, where is the maximum length of the binary representation of all proposed values in the run. Both algorithms are significant improvements over the previous algorithm in [A. Mostefaoui, M. Raynal, F. Tronel, From binary consensus to multivalued consensus in asynchronous message-passing systems, Information Processing Letters 73 (5–6) (2000) 207–212], where the number of binary consensus instances needed to solve one multivalued consensus is unbounded.     

Wireless network design via 3-decompositions

We consider some network design problems with applications for wireless networks. The input for these problems is a metric space (X,d) and a finite subset UX of terminals. In the Steiner Tree with Minimum Number of Steiner Points (STMSP) problem, the goal is to find a minimum size set SX−U of points so that the unit-disc graph of S+U is connected. Let Δ be the smallest integer so that for any finite VX for which the unit-disc graph is connected, this graph contains a spanning tree with maximum degree Δ. The best known approximation ratio for STMSP was Δ−1 [I.I. Măndoiu, A.Z. Zelikovsky, A note on the MST heuristic for bounded edge-length Steiner trees with minimum number of Steiner points, Information Processing Letters 75 (4) (2000) 165–167]. We improve this ratio to (Δ+1)/2+1+ε.In the Minimum Power Spanning Tree (MPST) problem, V=X is finite, and the goal is to find a “range assignment” on the nodes so that the edge set contains a spanning tree, and ∑_vVp(v) is minimized. We consider a particular case {0,1}-MPST of MPST when the distances are in {0,1}; here the goal is to find a minimum size set SV of “active” nodes so that the graph (V,E₀+E₁(S)) is connected, where , and E₁(S) is the set the edges in with both endpoints in S. We will show that the (5/3+ε)-approximation scheme for MPST of [E. Althaus, G. Calinescu, I. Măndoiu, S. Prasad, N. Tchervenski, A. Zelikovsky, Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks, Wireless Networks 12 (3) (2006) 287–299] achieves a ratio 3/2 for {0,1}-distances. This answers an open question posed in [E. Lloyd, R. Liu, S. Ravi, Approximating the minimum number of maximum power users in ad hoc networks, Mobile Networks and Applications 11 (2006) 129–142].     

Reduction of Permutation-Invariant Polynomials: A Noncommutative Case Study

Let R be a commutative ring with 1, let RX₁,…,X_n/I be the polynomial algebra in the n≥4 noncommuting variables X₁,…,X_n over R modulo the set of commutator relations I={(X₁+···+X_n)*X_i=X_i*(X₁+···+X_n)|1≤i≤n}. Furthermore, let G be an arbitrary group of permutations operating on the indeterminates X₁,…,X_n, and let RX₁,…,X_n/I^G be the R-algebra of G-invariant polynomials in RX₁,…,X_n/I. The first part of this paper is about an algorithm, which computes a representation for any fRX₁,…,X_n/I^G as a polynomial in multilinear G-invariant polynomials, i.e., the maximal variable degree of the generators of RX₁,…,X_n/I^G is at most 1. The algorithm works for any ring R and for any permutation group G. In addition, we present a bound for the number of necessary generators for the representation of all G-invariant polynomials in RX₁,…,X_n/I^G with a total degree of at most d. The second part contains a first but promising analysis of G-invariant polynomials of solvable polynomial rings.     

A note on Rooted Survivable Networks

The (undirected) Rooted Survivable Network Design (Rooted SND) problem is: given a complete graph on node set V with edge-costs, a root sV, and (node-)connectivity requirements , find a minimum cost subgraph G that contains r(t) internally-disjoint st-paths for all tT. For large values of k=max_tTr(t) Rooted SND is at least as hard to approximate as Directed Steiner Tree [Y. Lando, Z. Nutov, Inapproximability of survivable networks, Theoret. Comput. Sci. 410 (21–23) (2009) 2122–2125]. For Rooted SND, [J. Chuzhoy, S. Khanna, Algorithms for single-source vertex-connectivity, in: FOCS, 2008, pp. 105–114] gave recently an approximation algorithm with ratio O(k²logn). Independently, and using different techniques, we obtained at the same time a simpler primal–dual algorithm with the same ratio.     

Nonnegative matrices with prescribed extremal singular values

We consider the problem of constructing nonnegative matrices with prescribed extremal singular values. In particular, given 2n−1 real numbers and , we construct an n×n nonnegative bidiagonal matrix B and an n×n nonnegative semi-bordered diagonal matrix C, such that and are, respectively, the minimal and the maximal singular values of certain submatrices B_j and C_j of B and C, respectively. By using a singular value perturbation result, we also construct an n×n nonnegative matrix with prescribed singular values σ₁≥≥σ_n.     

Drag and lift coefficients of inclined finite circular cylinders at moderate Reynolds numbers

The flow around two-dimensional cylinders at moderate Reynolds numbers has been much studied, both for cylinders perpendicular to the flow and for cylinders yawed to the flow. In contrast, yawed finite aspect ratio cylinders have received little attention. In this article we describe computer simulations of cylinders with aspect ratios 2  L/D  20 yawed at angles 0°  α  90° relative to a free stream. The simulations were carried out for Reynolds numbers in the range 1  Re  40. The simulations show that the Independence Principle [Zdravkovich MM. Flow around circular cylinders, vol. 2: applications. New York: Oxford University Press; 2003[1]] is not accurate for α  45°. We have also found that for all aspect ratios, the ratio of the lift to drag force reaches a maximum for 40° < α < 50°. Finally, we present C_L and C_D relationships as best curve fits to computational data.     

On systems of word equations with simple loop sets

Consider the infinite system S of word equations

For each , let T_k be the subsystem of S given by i{k,k+1,k+2}. We prove two properties of the above system.

(1) Let k≥1. If φ is a solution of T_k such that primitive roots of are of equal length, as well as primitive roots of , then φ is a solution of the whole S.

(2) If n=1 then, for any k≥2, a solution φ of T_k is also a solution of S.

Generating fuzzy rules from training instances for fuzzy classification systems

In recent years, many methods have been proposed to generate fuzzy rules from training instances for handling the Iris data classification problem. In this paper, we present a new method to generate fuzzy rules from training instances for dealing with the Iris data classification problem based on the attribute threshold value α, the classification threshold value β and the level threshold value γ, where α  [0, 1], β  [0, 1] and γ  [0, 1]. The proposed method gets a higher average classification accuracy rate than the existing methods.     

Liquid–solid equilibria in quinary system Na,Mg/Cl,SO4,NO3–H2O at 298.15 K

In order to develop the nitrate deposits found close to Lop Nur in the Xinjiang region in China, the solubilities of the system Na⁺,Mg²⁺/Cl⁻,SO₄²⁻, NO₃⁻–H₂O and its subsystems, the quaternary systems Na⁺,Mg²⁺/SO₄²⁻,NO₃⁻–H₂O and Mg²⁺/Cl⁻,SO₄²⁻,NO₃⁻–H₂O, were studied at 298.15 K. The phase diagrams were plotted according to the solubilities achieved. In the equilibrium phase diagram of Mg²⁺/Cl⁻,SO₄²⁻,NO₃⁻–H₂O, there are two invariant points, five univariant curves and four regions of crystallization: Mg(NO₃)₂6H₂O,MgCl₂6H₂O,MgSO₄7H₂O and MgSO₄(1–6)H₂O. In the equilibrium phase diagram of Na⁺,Mg²⁺/SO₄²⁻, NO₃⁻–H₂O, there are five invariant points, eleven univariant curves and seven regions of crystallization: Na₂SO₄,Na₂SO₄10H₂O,NaNO₃,MgSO₄Na₂SO₄4H₂O,NaNO₃Na₂SO₄2H₂O,Mg(NO₃)₂6H₂O and MgSO₄7H₂O. In the equilibrium phase diagram of the Na⁺, Mg²⁺/Cl⁻,SO₄²⁻,NO₃⁻–H₂O system, there are six invariant points, and ten regions of crystallization: NaCl, NaNO₃,Na₂SO₄,Na₂SO₄10H₂O,MgSO₄Na₂SO₄4H₂O, NaNO₃Na₂SO₄2H₂O,MgCl₂6H₂O,Mg(NO₃)₂6H₂O, MgSO₄(1–6)H₂O and MgSO₄7H₂O.     

Thermostable tensoresistors of Co doped GaSb–FeGa1.3 eutectic composites

The microstructure and tensoresistive properties of GaSb–FeGa_1.3 eutectic composites doped with 0.1% Co have been investigated. It was found that the Co impurity atoms mainly accumulate in the metallic inclusions. The length of the inclusions in GaSb–FeGa_1.3Co was measured to be about half of those in undoped GaSb–FeGa_1.3 eutectics. The tensometric characteristics of gauges based on GaSb–FeGa_1.3Co have been found to be more thermostable than undoped samples.     

Response time of nanostructured relative humidity sensors

Capacitive relative humidity (RH) sensors were fabricated by coating countersunk interdigitated electrode substrates with nanostructured TiO₂ films produced using glancing angle deposition. Areal capacitance increased from 1 nF cm⁻² to 800 nF cm⁻² as relative humidity was increased from 2% RH and 95% RH. For films deposited at 81° and with a thickness below 4 m, response time was (162±4) ms m⁻¹. Response times increased from 64 ms to 1440 ms as film thickness increased from 280 nm to 8.5 m. The linear dependence of response time with film thickness indicates that device response time is dominated by surface adsorption. Response time decreased with increasing deposition angle, with a slope of (−15.2±1.6) ms degree⁻¹ for the adsorption data, and (−17.3±2.5) ms degree⁻¹ for the desorption data. The optimum operating range of the sensors depends on deposition angle, and can be tuned to different ranges to match application needs.     

The query complexity of order-finding

We consider the problem where π is an unknown permutation on {0,1,…,2ⁿ−1}, y₀{0,1,…,2ⁿ−1}, and the goal is to determine the minimum r>0 such that π^r(y₀)=1. Information about π is available only via queries that yield π^x(y) from any x{0,1,…,2^m−1} and y{0,1,…,2ⁿ−1} (where m is polynomial in n). The main resource under consideration is the number of these queries. We show that the number of queries necessary to solve the problem in the classical probabilistic bounded-error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.     

A Logic for Reasoning about Generic Judgments

This paper presents an extension of a proof system for encoding generic judgments, the logic FOλ^Δ of Miller and Tiu, with an induction principle. The logic FOλ^Δ is itself an extension of intuitionistic logic with fixed points and a “generic quantifier”, , which is used to reason about the dynamics of bindings in object systems encoded in the logic. A previous attempt to extend FOλ^Δ with an induction principle has been unsuccessful in modeling some behaviours of bindings in inductive specifications. It turns out that this problem can be solved by relaxing some restrictions on , in particular by adding the axiom B≡x.B, where x is not free in B. We show that by adopting the equivariance principle, the presentation of the extended logic can be much simplified. Cut-elimination for the extended logic is stated, and some applications in reasoning about an object logic and a simply typed λ-calculus are illustrated.     

-Admissible control elements for diagonal semigroups on -spaces

For linear systems described by , where A is a diagonal operator on the state space l^r for some 1r<∞ and bl^r, we develop necessary and sufficient conditions for b to be p-admissible. This extends results by Ho, Russell and Weiss to the case r≠2.