Involutions with fixed point set RP(2m) ⊔ P(2m, 2n + 1)

Let (M, T) be a smooth closed manifold with a smooth involution T whose fixed point set is a disjoint union of an even-dimensional real projective space and a Dold manifold. In some cases, the equivariant bordism classes of (M, T) are determined.     

Methods for solving the matrix equation TB + CT = −A and its generalizations

A variety of physical problems give rise to special cases of the matrix equation

$\text{A + FRB}{}^2\text{+ FERB + CFRB + CFER = 0, (}{}^1\text{)}$

where it is required to determine the solution matrix R. One important special case, obtained by setting T = FRB and E=0 in (¹), is the matrix equation TB + CT =?A. In this article we suggest an algorithm for solving the latter matrix equation, and we extend the ideas behind this algorithm to solve (¹) for R. Some suggestions for the efficient implementation of the algorithms are also given.     

Soft set theory and uni–int decision making

We firstly redefine the operations of Molodtsov’s soft sets to make them more functional for improving several new results. We also define products of soft sets and uni–int decision function. By using these new definitions we then construct an uni–int decision making method which selects a set of optimum elements from the alternatives. We finally present an example which shows that the method can be successfully applied to many problems that contain uncertainties.     

On the nonlinear wave equation Utt−B(t,‖Ux‖)Uxx=f(x,t,U,Ux,Ut) associated with the mixed homogeneous conditions

In this paper we consider the following nonlinear wave equation

(1)     

Vust's Theorem and higher level Schur–Weyl duality for types B, C and D

Let G be a complex linear algebraic group, $\mathfrak{g}=\mathrm{Lie}(G)$ its Lie algebra and $e\in \mathfrak{g}$ a nilpotent element. Vust's Theorem says that in case of $G=\mathrm{GL}(V)$, the algebra ${\mathrm{End}}_{G_e}(V^{?d})$, where $G_e?G$ is the stabilizer of e under the adjoint action, is generated by the image of the natural action of d-th symmetric group ${\mathfrak{S}}_d$ and the linear maps $\{1^{?(i?1)}?e?1^{?(d?i)}|i=1,\dots ,d\}$. In this paper, we give an analogue of Vust's Theorem for $G=\mathrm{O}(V)$ and $\mathrm{SP}(V)$ when the nilpotent elements e satisfy that $\stackrel{￣}{G?e}$ is normal. As an application, we study the higher Schur–Weyl duality in the sense of [4] for types B, C and D, which establishes a relationship between W-algebras and degenerate affine braid algebras.     

Pruning method for optimal solutions of int–int decision making scheme

This paper aims to find a faster method for optimal solutions of Feng et al.’s int^m–intⁿ decision making scheme. We first give theoretical characterizations of optimal decision sets. Then we develop a pruning method which filters out those objects that cannot be elements of any optimal decision sets in the beginning. Experimental results have shown that our method has higher efficiency in computing the optimal solutions of this scheme, particularly when we are processing soft sets with a great quantity of data.     

Sur l'équation divu=fOn the equation divu=f

The main result is the following. Let $\text{Ω}$ be a bounded Lipschitz domain in $\text{R}{}^{\mathrm{d}}$, d?2. Then for every $\text{f∈}\text{L}{}^{\mathrm{d}}\text{(Ω)}$ with ∫f=0, there exists a solution $\text{u∈}\text{C}{}^0\text{(}\text{Ω}\text{)∩}\text{W}{}^{\mathrm{1,d}}\text{(Ω)}$ of the equation divu=f in $\text{Ω}$, satisfying in addition u=0 on $\text{?Ω}$ and the estimate

where C depends only on $\text{Ω}$. However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976.     

On simultaneous approximation to (α,α) with α+kα−1=0

We show that the modified Jacobi-Perron algorithm gives the best simultaneous approximation to (α,α²) with α³+kα−1=0. We claim the following facts:

(1)

the limit set of become an ellipse, where (p_n,q_n,r_n) is the nth convergent (p_n/q_n,r_n/q_n) of (α,α²) by the modified Jacobi-Perron algorithm,

(2)

the limit set of belongs to outside of the ellipse in (1).

     

基于l2/lq(q=2/3)最小化模型的块稀疏信号恢复

该文主要研究了块稀疏信号的恢复问题.利用q块限制等距性质(0<q≤1),通过极小化混合l₂/l_q(q=2/3)范数,建立了块稀疏信号恢复的一个充分条件,并且得到了在有噪声情形下信号恢复的误差界.通过数值实验,验证了该模型对于块稀疏信号的恢复有较高的成功率.     

Uniqueness of positive radial solutions for Δu−u+u=0 on an annulus

We prove uniqueness of positive radial solutions to the semilinear elliptic equation , subject to the Dirichlet boundary condition on an annulus in . As a by-product, our argument also provides a much simpler, if not the simplest, new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space .     

The plancherel formula for the line bundles on SL(n + 1, ℝ)/S(GL(1,ℝ) × GL(n, ℝ))

In this paper we obtain the Plancherel formula for the spaces of L²-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, ?) and H = S(GL(1, ?) × GL(n 1, ?)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χ_λ of the subgroup H. We follow the method of Faraut, Kosters and van Dijk.     

The hyperelliptic integrals and π

Text

A class of hyperelliptic integrals are expressed through hypergeometric functions, like those of Gauss, Lauricella and Appell, namely multiple power series. Whenever they can on their own be reduced to elliptic integrals through an algebraic transformation, we obtain a two-fold representation of the same mathematical object, and then several completely new π determinations through the above special functions and/or Euler integrals. All our π formulae have been successfully tested by means of convenient Mathematica_®'s packages and enter in a wide historical/sound context of π-formulae quite far from being exhausted. Due to their structure, the formulae's practical value does not lie in computing π, but in allowing, through π, a benchmark for computing the involved special functions, particularly those less elementary.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=UHZIHgAeCUc.     

Semiconductor final test scheduling with Sarsa(λ, k) algorithm

Semiconductor test scheduling problem is a variation of reentrant unrelated parallel machine problems considering multiple resource constraints, intricate {product, tester, kit, enabler assembly} eligibility constraints, sequence-dependant setup times, etc. A multi-step reinforcement learning (RL) algorithm called Sarsa(λ, k) is proposed and applied to deal with the scheduling problem with throughput related objective. Allowing enabler reconfiguration, the production capacity of the test facility is expanded and scheduling optimization is performed at the bottom level. Two forms of Sarsa(λ, k), i.e. forward view Sarsa(λ, k) and backward view Sarsa(λ, k), are constructed and proved equivalent in off-line updating. The upper bound of the error of the action-value function in tabular Sarsa(λ, k) is provided when solving deterministic problems. In order to apply Sarsa(λ, k), the scheduling problem is transformed into an RL problem by representing states, constructing actions, the reward function and the function approximator. Sarsa(λ, k) achieves smaller mean scheduling objective value than the Industrial Method (IM) by 68.59% and 76.89%, respectively for real industrial problems and randomly generated test problems. Computational experiments show that Sarsa(λ, k) outperforms IM and any individual action constructed with the heuristics derived from the existing heuristics or scheduling rules.     

Applying a non-equilibrium wall function in k–ε turbulent modelling of hydrodynamic circulating flow

Wall bounded flow with severe adverse pressure, separation, reattachment and stagnation has non-equilibrium (NE) exhibition. A wall function in turbulent flow is a remedy to avoid resolving near wall complex phenomena using predetermined functions as boundary conditions. The advantage of this case is permission to use a relatively coarse near wall cells and hence saving CPU time. Standard wall function (SWF) is a semi-empirical function that is just valid for constant shear near wall cell and local equilibrium flow. Popovac and Hanjalic introduced a non-equilibrium wall function as (PWF) with a blending method in v²–f model. To investigate PWF in circulating flow, standard k–ε model that has key role in complex and expensive industrial problems is used in this study. The approach derived by Popovac and Hanjalic retains the functional form of the SWF and can be easily implemented in existing code. Simulation results are validated against direct numerical simulation (DNS) on channel and experimental data on backward facing step (BS) and a sharp U bend flow. Prediction with PWF shows that use of this wall function in k–ε model has not any sensitive change in near equilibrium flow. However, produces an improvement in NE conditions like flow in circulation zones.     

Symmetric duality with (p, r) − ρ − (η, θ)-invexity

In this paper we introduce a new type of generalized invex function, called (p, r) − ρ − (η, θ)-invex function and study symmetric duality results under these assumptions. In our study the nonnegative orthants for the constraints are replaced by closed convex cones and their polars. We establish weak and strong duality theorems under (p, r) − ρ − (η, θ)-invexity assumptions for the symmetric dual problems. We also give many examples to justify our results.     

What did Gauss read in the Appendix?

In a clear analogy with spherical geometry, Lambert states that in an “imaginary sphere” the sum of the angles of a triangle would be less than π$\pi$. In this paper we analyze the role played by this imaginary sphere in the development of non-Euclidean geometry, and how it served Gauss as a guide. More precisely, we analyze Gauss’s reading of Bolyai’s Appendix in 1832, five years after the publication of Disquisitiones generales circa superficies curvas, on the assumption that his investigations into the foundations of geometry were aimed at finding, among the surfaces in space, Lambert’s hypothetical imaginary sphere. We also wish to show that the close relation between differential geometry and non-Euclidean geometry is already present in János Bolyai’s Appendix, that is, well before its appearance in Beltrami’s Saggio. From this point of view, one is able to answer certain natural questions about the history of non-Euclidean geometry; for instance, why Gauss decided not to write further on the subject after reading the Appendix.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

On an Infinite Family in πS

In this paper, a new family of homotopy elements in the stable homotopy groups of spheres represented by

n−2≥m≥5 and 3≤s<p.$n-2\ge m\ge 5\hspace{0.17em}\text{and}\hspace{0.17em}3\le s<p.$

in the Adams spectral sequence is detected, where n − 2 ≥ m ≥ 5 and 3 ≤ s < p.