Generalized fuzzy interior ideals of semigroups

The concept of $(\overline{\in},\overline{\in} \vee \overline{q})The concept of ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups is introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy interior ideals, (∈, ∈ ∨ q)-fuzzy interior ideals and ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups. Finally, we give some characterization of [F] _t by means of (∈, ∈ ∨ q)-fuzzy interior ideals.     

New types of fuzzy ideals of near-rings

In this paper, we consider the $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy and $(\overline{\in}_{\gamma},\overline{\in}_{\gamma} \vee \; \overline{\hbox{q}}_{\delta})$ -fuzzy subnear-rings (ideals) of a near-ring. Some new characterizations are also given. In particular, we introduce the concepts of (strong) prime $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy ideals of near-rings and discuss the relationship between strong prime $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy ideals and prime $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy ideals of near-rings.     

General types of ({\in,}\,{\in}\,{\vee}\,{\rm q})-fuzzy filters in BL-algebras

As a generalization of an ( ? ,  ?  ú q)({\in,}\,{\in}\,{\vee}\, \hbox{q})-fuzzy filter in a BL-algebra, the notion of an ( ? ,  ?  ú q_k)({\in,}\,{\in}\,{\vee}\,\hbox{q}_k)-fuzzy filter in a BL-algebra is introduced, and related properties are investigated. Characterizations of an ( ? ,  ?  ú q_k)({\in,}\,{\in\,\vee}\,\hbox{q}_k)-fuzzy filter are considered. The implication-based fuzzy filters of a BL-algebra are discussed.     

On (\overline{\in},\overline{\in} \vee \overline{q})-fuzzy ideals of BCI-algebras

The concepts of $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy (p-, q- and a-) ideals of BCI-algebras are introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy (p-, q- and a-) ideals, (??,?????? q)-fuzzy (p-, q- and a-) ideals and $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy (p-,q- and a-) ideals of BCI-algebras. Moreover, we prove that a fuzzy set??? of a BCI-algebra X is an $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy a-ideal of X if and only if it is both an $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy p-ideal and an $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy q-ideal. Finally, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals.     

Generalized fuzzy bi-ideals of semigroups

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of (∈, ∈ ∨q)-fuzzy subgroups introduced by Bhakat is one among them. In this paper, using the relations between fuzzy points and fuzzy sets, the concept of a fuzzy bi-ideal with thresholds is introduced and some interesting properties are investigated. The acceptable nontrivial concepts obtained in this manner are the (∈, ∈ ∨q)-fuzzy bi-ideals and -fuzzy bi-ideals, which are extension of the concept of a fuzzy bi-ideal in semigroup.     

Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

Fuzzy n-ary hypergroups related to fuzzy points

In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called $(\in,\in \vee q)In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup. By using this new idea, we consider the ( ? , ? úq)(\in,\in\vee q)-fuzzy n-ary sub-hypergroup of a n-ary hypergroup. This newly defined ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup is a generalization of the usual fuzzy n-ary sub-hypergroup. Finally, we consider the concept of implication-based fuzzy n-ary sub-hypergroup in an n-ary hypergroup and discuss the relations between them, in particular, the implication operators in £\poundsukasiewicz system of continuous-valued logic are discussed.     

Process control using VSI cause selecting control charts

The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean, process variance, or both, ultimately affecting the quality of products. We construct the variable sampling interval (VSI) Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub>{Z_{\overline{X}}-Z_{S_X^2}} and Z<sub>[`(e)]}-Z_Se²{Z_{\bar{{e}}}-Z_{S_e^2}} control charts to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment, respectively. The performance of the proposed VSI control charts is measured by the adjusted average time to signal derived using a Markov chain approach. An example of the cotton yarn producing system shows the application and performance of the proposed joint VSI Z_{[`(X)]</sub> -Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}} -Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts in detecting shifts in mean and variance for the two dependent process steps with incorrect adjustment. Furthermore, the performance of the VSI Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}}-Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts and the fixed sampling interval Z_{[`(X)]</sub> -Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}} -Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts are compared by numerical analysis results. These demonstrate that the former is much faster in detecting small and median shifts in mean and variance. When quality engineers cannot specify the values of variable sampling intervals, the optimum VSI Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}}-Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts are also proposed by using the Quasi-Newton optimization technique.     

Lower Bounds for Agnostic Learning via Approximate Rank

We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose that there exist functions f₁, ?, f_r\phi_{1}, \ldots , \phi_{r} : {− 1, 1}ⁿ → \mathbbR{\mathbb{R}} with the property that every disjunction f on n variables has $\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3$\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3 for some reals a₁, ?, a_r\alpha_{1}, \ldots , \alpha_{r}. We prove that then $r \geq exp \{\Omega(\sqrt{n})\}$r \geq exp \{\Omega(\sqrt{n})\}, which is tight. We prove an incomparable lower bound for the concept class of decision lists. For the concept class of majority functions, we obtain a lower bound of W(2ⁿ/n)\Omega(2^{n}/n) , which almost meets the trivial upper bound of 2ⁿ for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi (1992) and show that the regression-based agnostic learning algorithm of Kalai et al. (2005) is optimal.     

Continuous point projection to planar freeform curves using spiral curves

We present an efficient algorithm for projecting a continuously moving query point to a family of planar freeform curves. The algorithm is based on the one-sided Hausdorff distance from the trajectory curve (of the query point) to the planar curves. Using a bounding volume hierarchy (BVH) of the planar curves, we estimate an upper bound [`(h)]\overline{h} of the one-sided Hausdorff distance and eliminate redundant curve segments when they are more than distance [`(h)]\overline{h} away from the trajectory curve. Recursively subdividing the trajectory curve and repeating the same elimination procedure to the BVH of the remaining curves, we can efficiently determine where to project the moving query point. The explicit continuous point projection is then interpreted as a curve reparameterization problem, for which we propose a few simple approximation techniques. Using several experimental results, we demonstrate the effectiveness of the proposed approach.     

Spreading of Messages in Random Graphs

Consider the following model on the spreading of messages. A message initially convinces a set of vertices, called the seeds, of the Erdős-Rényi random graph G(n,p). Whenever more than a ρ∈(0,1) fraction of a vertex v’s neighbors are convinced of the message, v will be convinced. The spreading proceeds asynchronously until no more vertices can be convinced. This paper derives lower bounds on the minimum number of initial seeds, min-seed(n,p,d,r)\mathrm{min\hbox{-}seed}(n,p,\delta,\rho), needed to convince a δ∈{1/n,…,n/n} fraction of vertices at the end. In particular, we show that (1) there is a constant β>0 such that min-seed(n,p,d,r)=W(min{d,r}n)\mathrm{min\hbox{-}seed}(n,p,\delta,\rho)=\Omega(\min\{\delta,\rho\}n) with probability 1−n ^−Ω(1) for p≥β (ln (e/min {δ,ρ}))/(ρ n) and (2) min-seed(n,p,d,1/2)=W(dn/ln(e/d))\mathrm{min\hbox{-}seed}(n,p,\delta,1/2)=\Omega(\delta n/\ln(e/\delta)) with probability 1−exp (−Ω(δ n))−n ^−Ω(1) for all p∈[ 0,1 ]. The hidden constants in the Ω notations are independent of p.     

The 1-Versus-2 Queries Problem Revisited

The 1-versus-2 queries problem, which has been extensively studied in computational complexity theory, asks in its generality whether every efficient algorithm that makes at most 2 queries to a Σ _k ^p -complete language L _k has an efficient simulation that makes at most 1 query to L _k . We obtain solutions to this problem for hypotheses weaker than previously considered. We prove that:

Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems

We present in this paper an analysis of a semi-Lagrangian second order Backward Difference Formula combined with hp-finite element method to calculate the numerical solution of convection diffusion equations in ℝ². Using mesh dependent norms, we prove that the a priori error estimate has two components: one corresponds to the approximation of the exact solution along the characteristic curves, which is O(Dt²+h^m+1(1+\frac\mathopen|logh|Dt))O(\Delta t^{2}+h^{m+1}(1+\frac{\mathopen{|}\log h|}{\Delta t})); and the second, which is O(Dt^p+|| [(u)\vec]-[(u)\vec]_h||_L^￥)O(\Delta t^{p}+\| \vec{u}-\vec{u}_{h}\|_{L^{\infty}}), represents the error committed in the calculation of the characteristic curves. Here, m is the degree of the polynomials in the finite element space, [(u)\vec]\vec{u} is the velocity vector, [(u)\vec]_h\vec{u}_{h} is the finite element approximation of [(u)\vec]\vec{u} and p denotes the order of the method employed to calculate the characteristics curves. Numerical examples support the validity of our estimates.     

Geometric Spanners for Weighted Point Sets

Let (S,d) be a finite metric space, where each element p∈S has a non-negative weight w (p). We study spanners for the set S with respect to the following weighted distance function:

Communication Complexity Under Product and Nonproduct Distributions

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R_∈(f) and $D^\mu_\in (f)$D^\mu_\in (f) denote the randomized and μ-distributional communication complexities of f, respectively (∈ a small constant). Yao’s well-known minimax principle states that $R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}$R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}. Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function f : {0, 1}ⁿ ×{0, 1}ⁿ ? {0, 1}f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\} for which max_{μ product} {D^m _? (f)} = Q(1)  but R _? (f) = Q(n)\{D^\mu_\in (f)\} = \Theta(1) \,{\textrm but}\, R_{\in} (f) = \Theta(n). We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).     

Quadrilateral meshes with provable angle bounds

In this paper, we present an algorithm that utilizes a quadtree data structure to construct a quadrilateral mesh for a simple polygonal region in which no newly created angle is smaller than 18.43^° (=arctan(\frac13)){{18.43}}^{\circ} ({=}\hbox{arctan}(\frac{1}{3})) or greater than 171.86^° (=135^° + 2arctan(\frac13)){{171.86}}^{\circ} ({=}{{135}}^{\circ} + 2\hbox{arctan}(\frac{1}{3})). This is the first known result, to the best of our knowledge, on a direct quadrilateral mesh generation algorithm with a provable guarantee on the angles.     

An improved algorithm on the content of realizable fuzzy matrices

An n×nn{\times}n fuzzy matrix A is called realizable if there exists an n×tn{\times}t fuzzy matrix B such that A=B\odot B^T,A=B\odot B^{T}, where \odot\odot is the max–min composition. Let r(A)=min{p:A=B\odot B^T, B ? L^n×p}.r(A)={min}\{p:A=B\odot B^{T}, B\in L^{n\times p}\}. Then r(A)r(A) is called the content of A. Since 1982, how to calculate r(A) for a given n×nn{\times}n realizable fuzzy matrix A was a focus problem, many researchers have made a lot of research work. X. P. Wang in 1999 gave an algorithm to find the fuzzy matrix B and calculate r(A) within [r(A)]ⁿ²[r(A)]^{n^{2}} steps. Therefore, to find a simpler algorithm is a problem what we have to consider. This paper makes use of the symmetry of the realizable fuzzy matrix A to simplify the algorithm of content r(A)r(A) based on the work of Wang (Chin Ann Math A 6: 701–706, 1999).     

Computational Divided Differencing and Divided-Difference Arithmetics

Tools for computational differentiation transform a program that computes a numerical function F(x) into a related program that computes F(x) (the derivative of F). This paper describes how techniques similar to those used in computational-differentiation tools can be used to implement other program transformations—in particular, a variety of transformations for computational divided differencing. The specific technical contributions of the paper are as follows:– It presents a program transformation that, given a numerical function F(x) defined by a program, creates a program that computes F[x ₀, x ₁], the first divided difference of F(x), where – It shows how computational first divided differencing generalizes computational differentiation.– It presents a second program transformation that permits the creation of higher-order divided differences of a numerical function defined by a program.– It shows how to extend these techniques to handle functions of several variables.The paper also discusses how computational divided-differencing techniques could lead to faster and/or more robust programs in scientific and graphics applications.Finally, the paper describes how computational divided differencing relates to the numerical-finite-differencing techniques that motivated Robert Paige's work on finite differencing of set-valued expressions in SETL programs.     

Distributed discovery of large near-cliques

Given an undirected graph and 0 ￡ e ￡ 1{0\le\epsilon\le1}, a set of nodes is called an e{\epsilon}-near clique if all but an e{\epsilon} fraction of the pairs of nodes in the set have a link between them. In this paper we present a fast synchronous network algorithm that uses small messages and finds a near-clique. Specifically, we present a constant-time algorithm that finds, with constant probability of success, a linear size e{\epsilon}-near clique if there exists an e³{\epsilon^3}-near clique of linear size in the graph. The algorithm uses messages of O(log n) bits. The failure probability can be reduced to n ^−Ω(1) by increasing the time complexity by a logarithmic factor, and the algorithm also works if the graph contains a clique of size Ω(n/(log log n) ^α ) for some a ? (0,1){\alpha \in (0,1)}. Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.