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.     

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.     

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

Direction-based surrounder queries for mobile recommendations

Location-based recommendation services recommend objects to the user based on the user’s preferences. In general, the nearest objects are good choices considering their spatial proximity to the user. However, not only the distance of an object to the user but also their directional relationship are important. Motivated by these, we propose a new spatial query, namely a direction-based surrounder (DBS) query, which retrieves the nearest objects around the user from different directions. We define the DBS query not only in a two-dimensional Euclidean space \mathbbE{\mathbb{E}} but also in a road network \mathbbR{\mathbb{R}} . In the Euclidean space \mathbbE{\mathbb{E}} , we consider two objects a and b are directional close w.r.t. a query point q iff the included angle Daqb{\angle aqb} is bounded by a threshold specified by the user at the query time. In a road network \mathbbR{\mathbb{R}} , we consider two objects a and b are directional close iff their shortest paths to q overlap. We say object a dominates object b iff they are directional close and meanwhile a is closer to q than b. All the objects that are not dominated by others based on the above dominance relationship constitute direction-based surrounders (DBSs). In this paper, we formalize the DBS query, study it in both the snapshot and continuous settings, and conduct extensive experiments with both real and synthetic datasets to evaluate our proposed algorithms. The experimental results demonstrate that the proposed algorithms can answer DBS queries efficiently.     

Quadratic Lower Bound for Permanent Vs. Determinant in any Characteristic

In Valiant’s theory of arithmetic complexity, the classes VP and VNP are analogs of P and NP. A fundamental problem concerning these classes is the Permanent and Determinant Problem: Given a field \mathbbF{\mathbb{F}} of characteristic ≠ 2, and an integer n, what is the minimum m such that the permanent of an n × n matrix X = (x_ij) can be expressed as a determinant of an m × m matrix, where the entries of the determinant matrix are affine linear functions of x_ij ’s, and the equality is in \mathbbF[X]{\mathbb{F}}[{\bf X}]. Mignon and Ressayre (2004) proved a quadratic lower bound m = W(n²)m = \Omega(n^{2}) for fields of characteristic 0. We extend the Mignon–Ressayre quadratic lower bound to all fields of characteristic ≠ 2.     

Characterizing sets of jobs that admit optimal greedy-like algorithms

The “Priority Algorithm” is a model of computation introduced by Borodin, Nielsen and Rackoff ((Incremental) Priority algorithms, Algorithmica 37(4):295–326, 2003) which formulates a wide class of greedy algorithms. For an arbitrary set \mathbbS\mathbb{S} of jobs, we are interested in whether or not there exists a priority algorithm that gains optimal profit on every subset of \mathbbS\mathbb{S} . In the case where the jobs are all intervals, we characterize such sets \mathbbS\mathbb{S} and give an efficient algorithm (when \mathbbS\mathbb{S} is finite) for determining this. We show that in general, however, the problem is NP-hard.     

Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, ${\int_t^{t+T}\alpha(s){\rm d}s \geq \mu}Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, . In particular, when α(t) becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback u = Kx, with K only depending on (A, B) and possibly on μ, T, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when A is neutrally stable and when the system is the double integrator. Notation  A continuous function is of class , if it is strictly increasing and is of class if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function is said to be a class -function if, for any t ≥ 0, and for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L ₂-norm of matrices.     

VPSPACE and a Transfer Theorem over the Reals

We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class \sf PAR_{\mathbb R}\sf {PAR}_{\mathbb {R}} of decision problems that can be solved in parallel polynomial time over the real numbers collapses to \sfP_{\mathbb R}\sf{P}_{\mathbb {R}}. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate \sfP_{\mathbb R}\sf{P}_{\mathbb {R}} from \sfNP_{\mathbb R}\sf{NP}_{\mathbb {R}}, or even from \sfPAR_{\mathbb R}\sf{PAR}_{\mathbb {R}}.     

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.     

Approximating Minimum-Power Degree and Connectivity Problems

Power optimization is a central issue in wireless network design. Given a graph with costs on the edges, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected network design problems under the power minimization criteria. Given a graph G=(V,E)\mathcal{G}=(V,\mathcal{E}) with edge costs {c(e):e∈ℰ} and degree requirements {r(v):v∈V}, the Minimum-Power Edge-Multi-Cover\textsf{Minimum-Power Edge-Multi-Cover} (MPEMC\textsf{MPEMC} ) problem is to find a minimum-power subgraph G of G\mathcal{G} so that the degree of every node v in G is at least r(v). We give an O(log n)-approximation algorithms for MPEMC\textsf{MPEMC} , improving the previous ratio O(log ⁴ n). This is used to derive an O(log n+α)-approximation algorithm for the undirected $\textsf{Minimum-Power $\textsf{Minimum-Power ($\textsf{MP$\textsf{MP ) problem, where α is the best known ratio for the min-cost variant of the problem. Currently, _boxclosen-k)\alpha=O(\log k\cdot \log\frac{n}{n-k}) which is O(log k) unless k=n−o(n), and is O(log ² k)=O(log ² n) for k=n−o(n). Our result shows that the min-power and the min-cost versions of the $\textsf{$\textsf{ problem are equivalent with respect to approximation, unless the min-cost variant admits an o(log n)-approximation, which seems to be out of reach at the moment.     

Computing Diameter in the Streaming and Sliding-Window Models

We investigate the diameter problem in the streaming and sliding-window models. We show that, for a stream of nn points or a sliding window of size nn, any exact algorithm for diameter requires W(n)\Omega(n) bits of space. We present a simple e\epsilon-approximation algorithm for computing the diameter in the streaming model. Our main result is an e\epsilon-approximation algorithm that maintains the diameter in two dimensions in the sliding-window model using O((1/e^3/2) log³n(logR+loglogn + log(1/e)))O(({1}/{\epsilon^{3/2}}) \log^{3}n(\log R+\log\log n + \log ({1}/{\epsilon}))) bits of space, where RR is the maximum, over all windows, of the ratio of the diameter to the minimum non-zero distance between any two points in the window.     

On-line graph coloring of {\mathbb{P}_5}-free graphs

Kierstead et al. (SIAM J Discret Math 8:485–498, 1995) have shown 1 that the competitive function of on-line coloring for -free graphs (i.e., graphs without induced path on 5 vertices) is bounded from above by the exponential function . No nontrivial lower bound was known. In this paper we show the quadratic lower bound . More precisely, we prove that is the exact competitive function for ()-free graphs. In this paper we also prove that 2 - 1 is the competitive function of the best clique covering on-line algorithm for ()-free graphs.     

Free binary decision diagrams for the computation of EAR n

Free binary decision diagrams (FBDDs) are graph-based data structures representing Boolean functions with the constraint (additional to binary decision diagram) that each variable is tested at most once during the computation. The function EAR_n is the following Boolean function defined for n × n Boolean matrices: EAR_n(M) = 1 iff the matrix M contains two equal adjacent rows. We prove that each FBDD computing EAR_n has size at least and we present a construction of such diagrams of size approximately .     

On metric rigidity for some classes of codes

A code C in the n-dimensional metric space $ \mathbb{F}_q^n $ \mathbb{F}_q^n over the Galois field GF(q) is said to be metrically rigid if any isometry I: C → $ \mathbb{F}_q^n $ \mathbb{F}_q^n can be extended to an isometry (automorphism) of $ \mathbb{F}_q^n $ \mathbb{F}_q^n . We prove metric rigidity for some classes of codes, including certain classes of equidistant codes and codes corresponding to one class of affine resolvable designs.     

Lower estimation of approximation rate for neural networks

Let SFd and Πψ,n,d = { nj=1bjψ(ωj·x+θj) :bj,θj∈R,ωj∈Rd} be the set of periodic and Lebesgue’s square-integrable functions and the set of feedforward neural network (FNN) functions, respectively. Denote by dist (SF d, Πψ,n,d) the deviation of the set SF d from the set Πψ,n,d. A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist (SFd, Πψ,n,d) is proved. That is, dist(SF d, Πψ,n,d) (nlogC2n)1/2 . T...     

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^d$$

We study mutually unbiased maximally entangled bases (MUMEB’s) in bipartite system \(\mathbb {C}^d\otimes \mathbb {C}^d (d \ge 3)\). We generalize the method to construct MUMEB’s given in Tao et al. (Quantum Inf Process 14:2291–2300, 2015), by using any commutative ring R with d elements and generic character of \((R,+)\) instead of \(\mathbb {Z}_d=\mathbb {Z}/d\mathbb {Z}\). Particularly, if \(d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}\) where \(p_1, \ldots , p_s\) are distinct primes and \(3\le p_1^{a_1}\le \cdots \le p_s^{a_s}\), we present \(p_1^{a_1}-1\) MUMEB’s in \(\mathbb {C}^d\otimes \mathbb {C}^d\) by taking \(R=\mathbb {F}_{p_1^{a_1}}\oplus \cdots \oplus \mathbb {F}_{p_s^{a_s}}\), direct sum of finite fields (Theorem 3.3).     

Cardinal interpolation with periodic polysplines on strips

Abstract  We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in , polyharmonic of order p on each strip , , and periodic in its last n variables, whose restriction to the parallel hyperplanes , , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal -splines. Keywords: cardinal -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63     

Unconditionally Stable Finite Difference,Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations

We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the L_s^￥(0,T;H_h¹)L_{s}^{\infty}(0,T;H_{h}^{1}) norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the L_s²(0,T;H_h²)L_{s}^{2}(0,T;H_{h}^{2}) norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.     

A Comparison of Asymptotically Scalable Superscalar Processors

The poor scalability of existing superscalar processors has been of great concern to the computer engineering community. In particular, the critical-path lengths of many components in existing implementations grow as Θ(n ² ) where n is the fetch width, the issue width, or the window size. This paper describes two scalable processor architectures, Ultrascalar I and Ultrascalar II, and compares their VLSI complexities (gate delays, wire-length delays, and area.) Both processors are implemented by a large collection of ALUs with controllers (together called execution stations ) connected together by a network of parallel-prefix tree circuits. A fat-tree network connects an interleaved cache to the execution stations. These networks provide the full functionality of superscalar processors including renaming, out-of-order execution, and speculative execution. The difference between the processors is in the mechanism used to transmit register values from one execution station to another. Both architectures use a parallel-prefix tree to communicate the register values between the execution stations. Ultrascalar I transmits an entire copy of the register file to each station, and the station chooses which register values it needs based on the instruction. Ultrascalar I uses an H-tree layout. Ultrascalar II uses a mesh-of-trees and carefully sends only the register values that will actually be needed by each subtree to reduce the number of wires required on the chip. The complexity results are as follows: The complexity is described for a processor which has an instruction-set architecture containing L logical registers and can execute n instructions in parallel. The chip provides enough memory bandwidth to execute up to M(n) memory operations per cycle. (M is assumed to have a certain regularity property.) In all the processors, the VLSI area is the square of the wire delay. Ultrascalar I has gate delay O(log n) and wire delay \tau_wires = \Theta(\sqrt{n}L) if $M(n)$ is $O(n^{1/2-\varepsilon})$, \tau_wires = \Theta(\sqrt{n}(L+\log n)) if $M(n)$ is $\Theta(n^{1/2})$, \tau_wires = \Theta(\sqrt{n}L+M(n)) if $M(n)$ is $\Omega(n^{1/2+\varepsilon})$ for ɛ>0 . Ultrascalar II has gate delay Θ(log L+log n) . The wire delay is Θ(n) , which is optimal for n=O(L) . Thus, Ultrascalar II dominates Ultrascalar I for n=O(L ² ) , otherwise Ultrascalar I dominates Ultrascalar II. We introduce a hybrid ultrascalar that uses a two-level layout scheme: Clusters of execution stations are layed out using the Ultrascalar II mesh-of-trees layout, and then the clusters are connected together using the H-tree layout of Ultrascalar I. For the hybrid (in which n&ge; L ), the wire delay is Θ(\sqrt nL+M(n)) , which is optimal. For n&ge; L , the hybrid dominates both Ultrascalar I and Ultrascalar II. We also present an empirical comparison of Ultrascalar I and the hybrid, both layed out using the Magic VLSI editor. For a processor that has 32 32-bit registers and a simple integer ALU, the hybrid requires about 11 times less area. Received June 11, 2000, and in revised form March 20, 2001, and in final form August 19, 2001. Online publication April 5, 2002.     

A note on balanced immunity

For a finite alphabet ∑ we define a binary relation on \(2^{\Sigma *} \times 2^{2^{\Sigma ^* } } \) , called balanced immunity. A setB ? ∑* is said to be balancedC-immune (with respect to a classC ? 2^Σ* of sets) iff, for all infiniteL εC, $$\mathop {\lim }\limits_{n \to \infty } \left| {L^{ \leqslant n} \cap B} \right|/\left| {L^{ \leqslant n} } \right| = \tfrac{1}{2}$$ Balanced immunity implies bi-immunity and in natural cases randomness. We give a general method to find a balanced immune set'B for any countable classC and prove that, fors(n) =o(t(n)) andt(n) >n, there is aB εSPACE(t(n)), which is balanced immune forSPACE(s(n)), both in the deterministic and nondeterministic case.