S-Lemma with Equality and Its Applications

Let $f(x)=x^TAx+2a^Tx+c$ and $h(x)=x^TBx+2b^Tx+d$ be two quadratic functions having symmetric matrices $A$ and $B$. The S-lemma with equality asks when the unsolvability of the system $f(x)<0, h(x)=0$ implies the existence of a real number $\mu$ such that $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$. The problem is much harder than the inequality version which asserts that, under Slater condition, $f(x)<0, h(x)\le0$ is unsolvable if and only if $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$ for some $\mu\ge0$. In this paper, we show that the S-lemma with equality does not hold only when the matrix $A$ has exactly one negative eigenvalue and $h(x)$ is a non-constant linear function ($B=0, b\not=0$). As an application, we can globally solve $\inf\{f(x)\vert h(x)=0\}$ as well as the two-sided generalized trust region subproblem $\inf\{f(x)\vert l\le h(x)\le u\}$ without any condition. Moreover, the convexity of the joint numerical range $\{(f(x), h_1(x),\ldots, h_p(x)):~x\in\Bbb R^n\}$ where $f$ is a (possibly non-convex) quadratic function and $h_1(x),\ldots,h_p(x)$ are affine functions can be characterized using the newly developed S-lemma with equality.Comment: 34 page

Rigidity controllable as-rigid-as-possible shape deformations

Shape deformation is one of the fundamental techniques in geometric processing. One principle of deformation is to preserve the geometric details while distributing the necessary distortions uniformly. To achieve this, state-of-the-art techniques deform shapes in a locally as-rigid-as-possible (ARAP) manner. Existing ARAP deformation methods optimize rigid transformations in the 1-ring neighborhoods and maintain the consistency between adjacent pairs of rigid transformations by single overlapping edges. In this paper, we make one step further and propose to use larger local neighborhoods to enhance the consistency of adjacent rigid transformations. This is helpful to keep the geometric details better and distribute the distortions more uniformly. Moreover, the size of the expanded local neighborhoods provides an intuitive parameter to adjust physical stiffness. The larger the neighborhood is, the more rigid the material is. Based on these, we propose a novel rigidity controllable mesh deformation method where shape rigidity can be flexibly adjusted. The size of the local neighborhoods can be learned from datasets of deforming objects automatically or specified by the user, and may vary over the surface to simulate shapes composed of mixed materials. Various examples are provided to demonstrate the effectiveness of our method

An Anti-Eavesdropping Strategy for Precoding-Aided Spatial Modulation With Rough CSI of Eve

In this paper, an anti-eavesdropping strategy is proposed for secure precoding-aided spatial modulation networks, under the assumption that the rough channel state information of eavesdropper can be obtained at the transmitter. Traditionally, artificial noise (AN) can be always projected into the null-space of the legitimate channel, however it may lead to some security loss since this strategy dispenses with a holistic consideration for secure transmissions. To reduce the computational complexity of our optimization problem, we derive a closed-form expression that is a loose bound of the approximate rate over the illegitimate channel. Then a concave maximization problem is formulated for optimizing the covariance matrix of AN. Simulation results show that our proposed low-complexity scheme performs closely to the method which directly maximizes the approximate secrecy rate expression, and harvests significant secrecy rate gains compared with the traditional null-space projection benchmark

The risk factors and preventive measures for the recurrence of endometrial polyps

Endometrial polyps (EPs) are a frequently encountered gynecologic disease with abnormal uterine bleeding and infertility being the two common presenting problems, and hysteroscopic polypectomy is an effective method to remove them. The postoperative polyp recurrence might result in reappearance of abnormal uterine bleeding or infertility, whereas factors influencing the postoperative recurrence potential have limited data. Endometrial polyp recurrence remains a concern with recurrence rates of 2.5% to 43.6%. As such, it is critical to identify the risk factors and the preventive measures for endometrial recurrence, especially in reproductive-age women desiring future conception, to aid in clinical counselling and decision making. The recurrence of EPs is related to estrogen stimulation and endometrial hyperplasia. The progesterone-containing drugs are currently the most commonly used method to prevent the recurrence of EPs. In this article, authors aim to discuss the high-risk factors of EPs recurrence and the preventive measures for EPs recurrence. The preventive measures will focus on the combined oral contraceptives (COCs) and the levonorgestrel-releasing intrauterine system (LNG-IUS)

Photoluminescence pressure coefficients of InAs/GaAs quantum dots

We have investigated the band-gap pressure coefficients of self-assembled InAs/GaAs quantum dots by calculating 17 systems with different quantum dot shape, size, and alloying profile using atomistic empirical pseudopotential method within the ``strained linear combination of bulk bands'' approach. Our results confirm the experimentally observed significant reductions of the band gap pressure coefficients from the bulk values. We show that the nonlinear pressure coefficients of the bulk InAs and GaAs are responsible for these reductions. We also find a rough universal pressure coefficient versus band gap relationship which agrees quantitatively with the experimental results. We find linear relationships between the percentage of electron wavefunction on the GaAs and the quantum dot band gaps and pressure coefficients. These linear relationships can be used to get the information of the electron wavefunctions.Comment: 8 pages, 2 tables, 4 figure

Hollow Titanium Silicalite Zeolite: From Fundamental Research to Commercial Application in Environmental-Friendly Catalytic Oxidation Processes

The systematical investigation on the synthesis, characterization, formation mechanism, and catalytic application of hollow titanium silicalite (HTS) zeolite has been reviewed. HTS is prepared through a “dissolution–recrystallization” post-treatment in the presence of template under hydrothermal conditions. Compared with TS-1, HTS is of unique hollow voids and with high framework Ti content, which significantly increase the mass diffusion and the amount of active sites, respectively. Thus, HTS zeolite displays high catalytic activity and stability in many oxidation processes with H2O2 oxidant, that is, cyclohexanone ammoximation, phenol hydroxylation, propylene epoxidation, Baeyer-Villiger oxidation of cyclohexanone, and selective oxidation of aromatics and cycloalkanes. The former three ones have been commercialized and run smoothly, which have promising economic and environmental significance