Soft set theory applied to p-ideals of BCI-algebras related to fuzzy points

The notions of $(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})The notions of ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideals and fuzzy p-ideals with thresholds related to soft set theory are discussed. Relations between ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy ideals and ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideals are investigated. Characterizations of an ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideal and a fuzzy p-ideal with thresholds are displayed. Implication-based fuzzy p-ideals are discussed.     

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

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.     

Homogeneous Formulas and Symmetric Polynomials

We investigate the arithmetic formula complexity of the elementary symmetric polynomials S^k_n{S^k_n} . We show that every multilinear homogeneous formula computing S^k_n{S^k_n} has size at least k^W(logk)n{k^{\Omega(\log k)}n} , and that product-depth d multilinear homogeneous formulas for S^k_n{S^k_n} have size at least 2^W(k1/d)n{2^{\Omega(k^{1/d})}n} . Since Sⁿ_2n{S^{n}_{2n}} has a multilinear formula of size O(n ²), we obtain a superpolynomial separation between multilinear and multilinear homogeneous formulas. We also show that S^k_n{S^k_n} can be computed by homogeneous formulas of size k^O(logk)n{k^{O(\log k)}n} , answering a question of Nisan and Wigderson. Finally, we present a superpolynomial separation between monotone and non-monotone formulas in the noncommutative setting, answering a question of Nisan.     

Prediction of phosphorus content of electroless nickel–phosphorous coatings using artificial neural network modeling

A multilayer feedforward neural network with two hidden layers was designed and developed for prediction of the phosphorus content of electroless Ni–P coatings. The input parameters of the network were the pH, metal turnover, and loading of an electroless bath. The output parameter was the phosphorus content of the electroless Ni–P coatings. The temperature and molar rate of the bath were constant ( 91^° \textC, 0.4 \textNi^{\text + +} /\textH₂ \textPO₂ ^{- -} 91^\circ {\text{C}},\:0.4\,{\text{Ni}}^{{{\text{ + + }}}} /{\text{H}}_{2} {\text{PO}}_{2}^{{ - - }} ). The network was trained and tested using the data gathered from our own experiments. The goal of the study was to estimate the accuracy of this type of neural network in prediction of the phosphorus content. The study result shows that this type of network has high accuracy even when the number of hidden neurons is very low. Some comparison between the network’s predictions and own experimental data are given.     

On the efficiency of quantum algorithms for Hamiltonian simulation

We study algorithms simulating a system evolving with Hamiltonian H = ?_j=1^m H_j{H = \sum_{j=1}^m H_j} , where each of the H _j , j = 1, . . . ,m, can be simulated efficiently. We are interested in the cost for approximating e^-iHt, t ? \mathbbR{e^{-iHt}, t \in \mathbb{R}} , with error e{\varepsilon} . We consider algorithms based on high order splitting formulas that play an important role in quantum Hamiltonian simulation. These formulas approximate e ^−iHt by a product of exponentials involving the H _j , j = 1, . . . ,m. We obtain an upper bound for the number of required exponentials. Moreover, we derive the order of the optimal splitting method that minimizes our upper bound. We show significant speedups relative to previously known results.     

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:

Attractive energy contribution to nanoconfined fluids behavior: the normal pressure tensor

The aim of our research is to demonstrate the role of attractive intermolecular potential energy on normal pressure tensor of confined molecular fluids inside nanoslit pores of two structureless purely repulsive parallel walls in xy plane at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a uniform density. To achieve this we have derived the perturbation theory version of the normal pressure tensor of confined inhomogeneous fluids in nanoslit pores:

Complexity of Hard-Core Set Proofs

We study a fundamental result of Impagliazzo (FOCS’95) known as the hard-core set lemma. Consider any function f:{0,1}ⁿ?{0,1}{f:\{0,1\}^n\to\{0,1\}} which is “mildly hard”, in the sense that any circuit of size s must disagree with f on at least a δ fraction of inputs. Then, the hard-core set lemma says that f must have a hard-core set H of density δ on which it is “extremely hard”, in the sense that any circuit of size s￠=O(s/(\frac1e²log(\frac1ed))){s'=O(s/(\frac{1}{\epsilon^2}\log(\frac{1}{\epsilon\delta})))} must disagree with f on at least (1-e)/2{(1-\epsilon)/2} fraction of inputs from H.     

The Gaussian Surface Area and Noise Sensitivity of Degree-d Polynomial Threshold Functions

We prove asymptotically optimal bounds on the Gaussian noise sensitivity and Gaussian surface area of degree-d polynomial threshold functions. In particular, we show that for f a degree-d polynomial threshold function that the Gaussian noise sensitivity of f with parameter e{\epsilon} is at most \fracdarcsin(?{2e-e²})p{\frac{d\arcsin\left(\sqrt{2\epsilon-\epsilon^2}\right)}{\pi}} . This bound translates into an optimal bound on the Gaussian surface area of such functions, namely that the Gaussian surface area is at most \fracd?{2p}{\frac{d}{\sqrt{2\pi}}} . Finally, we note that the later result implies bounds on the runtime of agnostic learning algorithms for polynomial threshold functions.     

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.     

Entanglement and Berry phase in a 9 × 9 Yang–Baxter system

A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang–Baxterization approach, we obtain a unitary solution \breveR(q,j₁,j₂){\breve{R}(\theta,\varphi_{1},\varphi_{2})} of Yang–Baxter equation. It is shown that any pure two-qutrit entangled states can be generated via the universal \breveR{\breve{R}}-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the \breveR{\breve{R}}-matrix, and Berry phase of the Yang–Baxter system is investigated. Specifically, for j₁ = j₂{\varphi_{1}\,{=}\,\varphi_{2}}, the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.     

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.     

When Does Eddy Viscosity Damp Subfilter Scales Sufficiently?

Large eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence is that the LES-solution contains only scales of size ≥Δ, where Δ denotes some user-chosen length scale. This property enables us to perform a LES when it is not feasible to compute the full, turbulent solution of the Navier-Stokes equations. Therefore, in case the large eddy simulation is based on an eddy viscosity model we determine the eddy viscosity such that any scales of size <Δ are dynamically insignificant. In this paper, we address the following two questions: how much eddy diffusion is needed to (a) balance the production of scales of size smaller than Δ; and (b) damp any disturbances having a scale of size smaller than Δ initially. From this we deduce that the eddy viscosity ν _e has to depend on the invariants q = \frac12tr(S²)q = \frac{1}{2}\mathrm{tr}(S^{2}) and r = -\frac13tr(S³)r= -\frac{1}{3}\mathrm{tr}(S^{3}) of the (filtered) strain rate tensor S. The simplest model is then given by n_e = \frac32(D/p)² |r|/q\nu_{e} = \frac{3}{2}(\Delta/\pi)^{2} |r|/q. This model is successfully tested for a turbulent channel flow (Re  _τ =590).     

On the solution of the fuzzy Sylvester matrix equation

In this paper, we consider the fuzzy Sylvester matrix equation AX+XB=C,AX+XB=C, where A ? \mathbbR^{n ×n}A\in {\mathbb{R}}^{n \times n} and B ? \mathbbR^{m ×m}B\in {\mathbb{R}}^{m \times m} are crisp M-matrices, C is an n×mn\times m fuzzy matrix and X is unknown. We first transform this system to an (mn)×(mn)(mn)\times (mn) fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.     

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.     

Looking for the best constant in a Sobolev inequality: a numerical approach

A numerical method for the computation of the best constant in a Sobolev inequality involving the spaces H ²(Ω) and C⁰([`(W)])C^{0}(\overline{\Omega}) is presented. Green’s functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagrangian method, together with finite element approximations, and a Green’s functions based approach. Numerical experiments show the ability of the methods in computing this best constant for various two-dimensional domains, and the remarkable convergence properties of the augmented Lagrangian based iterative method.     

On the k-th root in circular arithmetic

The representation of thek-th root of a complex circular intervalZ={c;r} is considered in this paper. Thek-th root is defined by the circular intervals which include the exact regionZ ^1/k={z:z ^k ∈Z}. Two representations are given: (i) the centered inclusive disks \( \cup \{ c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} ; \mathop {\max }\limits_{z \in Z} |z^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} - c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} |\} \) and (ii) the diametrical inclusive disks with the diameter which is equal to the diameter of the regionZ ^1/k.     

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.