Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Scalable 2D Convex Hull and Triangulation Algorithms for Coarse Grained Multicomputers

In this paper we describe scalable parallel algorithms for building the convex hull and a triangulation ofncoplanar points. These algorithms are designed for thecoarse grained multicomputermodel:pprocessors withO(n/p)⪢O(1) local memory each, connected to some arbitrary interconnection network. They scale over a large range of values ofnandp, assuming only thatn⩾p^1+ε(ε>0) and require timeO((T_sequential/p)+T_s(n, p)), whereT_s(n, p) refers to the time of a global sort ofndata on approcessor machine. Furthermore, they involve only a constant number of global communication rounds. Since computing either 2D convex hull or triangulation requires timeT_sequential=Θ(n log n) these algorithms either run in optimal time,Θ((n log n)/p), or in sort time,T_s(n, p), for the interconnection network in question. These results become optimal whenT_sequential/pdominatesT_s(n, p) or for interconnection networks like the mesh for which optimal sorting algorithms exist.     

Taylor approximation for hybrid systems

We propose a new approximation technique for Hybrid Automata. Given any Hybrid Automaton H, we call Approx(H, k) the Polynomial Hybrid Automaton obtained by approximating each formula ϕ in H with the formulae ϕ_k obtained by replacing the functions in ϕ with their Taylor polynomial of degree k. We prove that Approx(H, k) is an over-approximation of H. We study the conditions ensuring that, given any ϵ > 0, some k₀ exists such that, for all k > k₀, the “distance” between any vector satisfying ϕ_k and at least one vector satisfying ϕ is less than ϵ. We study also conditions ensuring that, given any ϵ > 0, some k₀ exists such that, for all k > k₀, the “distance” between any configuration reached by Approx(H, k) in n steps and at least one configuration reached by H in n steps is less than ϵ.     

On identifying codes that are robust against edge changes

Assume that G = (V, E) is an undirected graph, and C ⊆ V. For every v ∈ V, denote I_r(G; v) = {u ∈ C: d(u,v) ≤ r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets I_r(G; v) for v ∈ V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r = 1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide.     

Fault diagnosis in loop-connected systems

The paper considers fault diagnosis in a large system comprising a collection of small subsystems or units which can test one another for the existence of a faulty condition. If subsystem α is not faulty and tests subsystem β, a correct indication of the status of β is obtained; if α is faulty, the test outcome contains meaningless information. A particular form of interconnection is examined. For a system with n units u_o,u₁,…,u_{n ? 1}, for each i unit u_i tests u_{i + 1},u_{i + 2},…,u_{i + A} (modulo n arithmetic being understood), where A is a preselected integer. If t is the maximum number of faulty units, we show that when t ? A, all faults are immediately diagnosable if n ? 2t + 1; we also show that when t ? A, at least A faults can be diagnosed if and only if n ? s(t ? As) + t + A + 1, where s is the integer which maximizes the quadratic function f(x) = x(t ? Ax) of the integer variable x.     

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.     

Fast Constructive Recognition of a Black Box Group Isomorphic toSnorAnusing Goldbach’s Conjecture

The well-known Goldbach Conjecture (GC) states that any sufficiently large even number can be represented as a sum of two odd primes. Although not yet demonstrated, it has been checked for integers up to 10¹⁴. Using two stronger versions of the conjecture, we offer a simple and fast method for recognition of a gray box group G known to be isomorphic to S_n(or A_n) with knownn ≥  20, i.e. for construction1of an isomorphism from G toS_n (or A_n). Correctness and rigorous worst case complexity estimates rely heavily on the conjectures, and yield times of O([ρ + ν + μ ] n log²n) or O([ ρ + ν + μ ] n logn / loglog n) depending on which of the stronger versions of the GC is assumed to hold. Here,ρ is the complexity of generating a uniform random element of G, ν is the complexity of finding the order of a group element in G, and μ is the time necessary for group multiplication in G. Rigorous lower bound and probabilistic approach to the time complexity of the algorithm are discussed in the Appendix.     

Gröbner Bases Applied to Finitely Generated Field Extensions

Using a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensionsk (x) / k(g), resp. how to determine the (separable) degree if k(x) / k(g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k(x) / k(g) and a minimal primary decomposition of a suitable ideal is described. For Galois extensions the field-ideal correspondence can also be used to determine the elements of the Galois group.     

Investigation of pollution flashover on high voltage insulators using artificial neural network

High voltage insulators form an essential part of the high voltage electric power transmission systems. Any failure in the satisfactory performance of high voltage insulators will result in considerable loss of capital, as there are numerous industries that depend upon the availability of an uninterrupted power supply. The importance of the research on insulator pollution has been increased considerably with the rise of the voltage of transmission lines. In order to determine the flashover behavior of polluted high voltage insulators and to identify to physical mechanisms that govern this phenomenon, the researchers have been brought to establish a modeling. Artificial neural networks (ANN) have been used by various researches for modeling and predictions in the field of energy engineering systems. In this study, model of V_C = f (H, D, L, σ, n, d) based on ANN which compute flashover voltage of the insulators were performed. This model consider height (H), diameter (D), total leakage length (L), surface conductivity (σ) and number of shed (d) of an insulator and number of chain (n) on the insulator.     

Bin packing problems with rejection penalties and their dual problems

In this paper we consider the following problems: we are given a set of n items {u₁, … , u_n} and a number of unit-capacity bins. Each item u_i has a size w_i ∈ (0, 1] and a penalty p_i ⩾ 0. An item can be either rejected, in which case we pay its penalty, or put into one bin under the constraint that the total size of the items in the bin is no greater than 1. No item can be spread into more than one bin. The objective is to minimize the total number of used bins plus the total penalty paid for the rejected items. We call the problem bin packing with rejection penalties, and denote it as BPR. For the on-line BPR problem, we present an algorithm with an absolute competitive ratio of 2.618 while the lower bound is 2.343, and an algorithm with an asymptotic competitive ratio arbitrarily close to 1.75 while the lower bound is 1.540. For the off-line BPR problem, we present an algorithm with an absolute worst-case ratio of 2 while the lower bound is 1.5, and an algorithm with an asymptotic worst-case ratio of 1.5. We also study a closely related bin covering version of the problem. In this case p_i means some amount of profit. If an item is rejected, we get its profit, or it can be put into a bin in such a way that the total size of the items in the bin is no smaller than 1. The objective is to maximize the number of covered bins plus the total profit of all rejected items. We call this problem bin covering with rejection (BCR). For the on-line BCR problem, we show that no algorithm can have absolute competitive ratio greater than 0, and present an algorithm with asymptotic competitive ratio 1/2, which is the best possible. For the off-line BCR problem, we also present an algorithm with an absolute worst-case ratio of 1/2 which matches the lower bound.     

A generalization of the Schützenberger product of finite monoids

For each n?1, an n-ary product ? on finite monoids is constructed. This product has the following property: Let Σ be a finite alphabet and Σ¹ the free monoid generated by Σ. For i = 1, …,n, let A_i be a recognizable subset of Σ¹, M(A_i) the syntactic monoid of A_n and M(A₁?A_n) the syntactic monoid of the concatenation product A₁?A_n. Then M(A₁?A_n)< ? (M(A₁),…,M(A_n)). The case n = 2 was studied by Schützenberger. As an application of the generalized product, I prove the theorem of Brzozowski and Knast that the dot-depth hierarchy of star-free sets is infinite.     

Computing Modular Invariants of p-groups

Let V be a finite dimensional representation of a p -group, G, over a field,k , of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k [ V ]^G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k [ V ] G. We use these methods to analyse k [2V₃ ]^U₃where U₃is the p -Sylow subgroup ofGL₃ (F_p) and 2 V₃is the sum of two copies of the canonical representation. We give a generating set for k [2 V₃]^U₃forp =  3 and prove that the invariants fail to be Cohen–Macaulay forp >  2. We also give a minimal generating set for k [mV₂ ]^Z / pwere V₂is the two-dimensional indecomposable representation of the cyclic group Z / p.     

Evaluation of response characteristics of resistive oxygen sensors based on porous cerium oxide thick film using pressure modulation method

In order to reduce the response time of resistive oxygen sensors using porous cerium oxide thick film, it is important to ascertain the factors controlling response. Pressure modulation method (PMM) was used to find the rate-limiting step of sensor response. This useful method measures the amplitude of sensor output (H(f)) for the sine wave modulation of oxygen partial pressure at constant frequency (f). In PMM, “break” response time, which is minimum period in which the sensor responds precisely, can be measured. Three points were examined: (1) simulated calculations of PMM were carried out using a model of porous thick film in which spherical particles are connected in a three-dimensional network; (2) sensor response speed was experimentally measured using PMM; and (3) the diffusion coefficient and surface reaction coefficient were estimated by comparison between experiment and calculation. The plot of log f versus log H(f) in the high f region was found to have a slope of approximately −0.5 for both porous thick film and non-porous thin film, when the rate-limiting step was diffusion. Calculations showed the response time of porous thick film was 1/20 that of non-porous thin film when the grain diameter of the porous thick film was the same as the thickness of non-porous thin film. At 973 K, “break” response time (t_b) of the resistive oxygen sensor was found by experiment to be 109 ms. It was concluded that the response of the resistive oxygen sensor prepared in this study was strongly controlled by diffusion at 923–1023 K, since the experiment revealed that the slope of plot of log f versus log H(f) in the high f region was approximately −0.5. At 923–1023 K, the diffusion coefficient of oxygen vacancy in porous ceria (D_V) was expressed as follows: D_V (m²s⁻¹) = 5.78 × 10⁻⁴ exp(−1.94 eV/kT). At 1023 K, the surface reaction coefficient (K) was found to exceed 10⁻⁴ m/s.     

Modal analysis and modified cascade neural networks in identification of geometrical parameters of circular arches

A hybrid computational system, composed of the finite element method (FEM) and cascade neural network system (CNNs), is applied to the identification of three geometrical parameters of elastic arches, i.e. span l, height f and cross-sectional thickness h. FEM is used in the direct (forward) analysis, which corresponds to the mapping α = {l, f, h} → {ω_j}, where: α – vector of control parameters, ω_j – arch eigenfrequencies. The reverse analysis is related to the identification procedure in which the reverse mapping is performed {ω_j} → {α_i}. For the identification purposes a recurrent, three level CNNs of structure (D^k-H^k-1)_s was formulated, where: k – recurrence step, s = I, II, III-levels of cascade system. The Semi-Bayesian approach is introduced for the design of CNNs applying the MML Maximum Marginal Likelihood) criterion. The computation of hyperparameters is performed by means of the Bayesian procedure evidence. The numerical analysis proves a great numerical efficiency of the proposed hybrid approach for both the perfect (noiseless) values of eigenfrequencies and noisy ones simulated by an added artificial noise.     

Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing

An instance of the k -Steiner forest problem consists of an undirected graph G=(V,E), the edges of which are associated with non-negative costs, and a collection $\mathcal{D}=\{(s_{1},t_{1}),\ldots,(s_{d},t_{d})\}An instance of the k -Steiner forest problem consists of an undirected graph G=(V,E), the edges of which are associated with non-negative costs, and a collection D={(s₁,t₁),?,(s_d,t_d)}\mathcal{D}=\{(s_{1},t_{1}),\ldots,(s_{d},t_{d})\} of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest ℱ⊆G connects a demand (s _i ,t _i ) when it contains an s _i -t _i path. Given a profit k _i for each demand (s _i ,t _i ) and a requirement parameter k, the goal is to find a minimum cost forest that connects a subset of demands whose combined profit is at least k. This problem has recently been studied by Hajiaghayi and Jain (SODA ’06), whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k -subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research. In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of O(min{n^2/3,?d}·logd)O(\min \{n^{2/3},\sqrt{d}\}\cdot \log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands. We believe that the approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.     

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

This paper presents an algorithm for the Quillen–Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k [ x₀, . . . ,x_n ] / I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis forP .     

Transition-aware DVS algorithm for real-time systems using tree structure analysis

Dynamic voltage scaling (DVS) is a key technique for embedded real-time systems to reduce energy consumption by lowering the supply voltage and operating frequency. Many existing DVS algorithms have to generate the canonical schedules or estimate the lengths of slack time in advance for generating the voltage scaling decisions. Therefore, these methods have to compute the schedules with exponential time complexities in general. In this paper, we consider a set of jitter-controlled, independent, periodic, hard real-time tasks scheduled according to preemptive pinwheel model. Our approach constructs a tree structure corresponding to a schedule and maintains the data structure at each early-completion point. Our approach consists of off-line and on-line algorithms which consider the effects of transition time and energy. The off-line and on-line algorithm takes O(k + n log n) and O(k + (p_max/p_min)) time complexity, respectively, where n, k, p_max and p_min denotes the number of tasks, jobs, longest and shortest task period, respectively. Experimental results show that the proposed approach is effective in reducing computational complexity, transition time and energy overhead.