A direct proof of uniqueness of square-root of a positive semi-definite tensor

Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.

COMPLETE SPACE-LIKE SUBMANIFOLDS IN LOCALLY SYMMETRIC SEMI-DEFINITE SPACES

The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized.

Semi-definite programming based pulse waveform design and its further analysis

In order to provide a judicious pulse waveform design required for ultra-wideband(UWB)communication to enable the UWB spectral mask compatible and coexistent with other existing wireless communication systems,a semi-definite programming(SDP)based pulse waveform design method for UWB radios is introduced and a further analysis is given in this paper.By using Sedumi and Yalmip toolboxes of Matlab,the procedure of solving the SDP problem is simplified.Simulation results show that this SDP based pulse waveform design method can be used to design pulses that fulfill the Federal Communications Commission(FCC)spectral mask strictly and optimize the power efficiency at the same time.This paper also analyzes the influences of the power efficiency duing to the changes of sampling interval and the number of combined pulses,and then the optimal sampling interval that maximizes the transmission power can be found.

A Large-update Interior-point Algorithm for Convex Quadratic Semi-definite Optimization Based on a New Kernel Function

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be O（√n（logn）2 log e/n）. This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and in optimization fields. Some computational results recent kernel functions introduced by some authors have been provided.

POSITIVE DEFINITE AND SEMI-DEFINITE SPLITTING METHODS FOR NON-HERMITIAN POSITIVE DEFINITE LINEAR SYSTEMS

In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting （PPS） methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

Characteristic finite element scheme and analysis the three-dimensional two-phase displacement semi-definite problem

NUMERICAL simulation of the two-phase (oil and water) displacement problem is the mathematical basis of energy sources. For two-dimensional positive problem, Douglas et al. put forward the well-known characteristic finite difference method and characteristic finite element method. However, for numerical analysis there exist some difficulties. They assumed that the problem is periodic and the diffusion matrix of the concentration equation is positive difinite

SEMI-DEFINITE RELAXATION ALGORITHM FOR SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES

The authors present a semi-definite relaxation algorithm for the scheduling problem with controllable times on a single machine. Their approach shows how to relate this problem with the maximum vertex-cover problem with kernel constraints (MKVC).The established relationship enables to transfer the approximate solutions of MKVCinto the approximate solutions for the scheduling problem. Then, they show how to obtain an integer approximate solution for MKVC based on the semi-definite relaxation and randomized rounding technique.

THE PRIMAL-DUAL POTENTIAL REDUCTION ALGORITHM FOR POSITIVE SEMI-DEFINITE PROGRAMMING

In this paper we introduce a primal-dual potential reduction algorithm for positive semi-definite programming. Using the symetric preserving scalings for both primal and dual interior matrices, we can construct an algorithm which is very similar to the primal-dual potential reduction algorithm of Huang and Kortanek [6] for linear programming. The complexity of the algorithm is either O(nlog(X0 · S0/ε) or O(nlog(X0· S0/ε) depends on the value of ρ in the primal-dual potential function, where X0 and S0 is the initial interior matrices of the positive semi-definite programming.

Fairness-Aware Harvested Energy Efficiency Algorithm for IRS-Aided Intelligent Sensor Networks with SWIPT

In this paper,a novel fairness-aware harvested energy efficiency-based green transmission scheme for wireless information and power transfer(SWIPT)aided sensor networks is developed for active beamforming of multiantenna transmitter and passive beamforming at intelligent reflecting surfaces(IRS).By optimizing the active beamformer assignment at the transmitter in conjunction with the passive beamformer assignment at the IRS,we aimtomaximize the minimumharvested energy efficiency among all the energy receivers(ER)where information receivers(IR)are bound to the signal-interference-noise-ratio(SINR)and the maximum transmitted power of the transmitter.To handle the non-convex problem,both semi-definite relaxation(SDR)and block coordinate descent technologies are exploited.Then,the original problem is transformed into two convex sub-problems which can be solved via semidefinite programming.Numerical simulation results demonstrate that the IRS and energy beamformer settings in this paper provide greater system gain than the traditional experimental setting,thereby improving the fairness-aware harvested energy efficiency of the ER.

Multiple phase detector of M-ary phase shift keying symbols in code division multiple access systems

A novel iterative technique, the phase descent search detection was proposed. This technique constrained the solution （PDS） algorithm, for M-ary phase shift keying （M-PSK） symbols to have a unit magnitude and it was based on coordinate descent iterations where coordinates were the unknown symbol phases. The PDS algorithm, together with a descent local search （also implemented as a version of the PDS algorithm）, was used multiple times with different initializations in a proposed multiple phase detector; the solution with the minimum cost was then chosen as the final solution. The simulation results show that for highly loaded multiuser scenarios, the proposed technique has a detection performance that is close to the single-user bound. The results also show that the multiple phase detector allows detection in highly overloaded scenarios and it exhibits near-far resistance. In particular, the detector has a performance that is significantly better, and complexity that is significantly lower, than that of the detector based on semi-definite relaxation.

Measurement of Incompatible Probability in Information Retrieval:A Case Study with User Clicks

The incompatible probability represents an important non-classical phenomenon, and it describes conflicting observed marginal probabilities, which cannot be satisfied with a joint probability. First, the incompatibility of random variables was defined and discussed via the non-positive semi-definiteness of their covariance matrixes. Then, a method was proposed to verify the existence of incompatible probability for variables. A hypothesis testing was also applied to reexamine the likelihood of the observed marginal probabilities being integrated into a joint probability space, thus showing the statistical significance of incompatible probability cases. A case study with user click-through data provided the initial evidence of the incompatible probability in information retrieval (IR), particularly in user interaction. The experiments indicate that both incompatible and compatible cases can be found in IR data, and informational queries are more likely to be compatible than navigational queries. The results inspire new theoretical perspectives of modeling the complex interactions and phenomena in IR.

Performance of magnetorheological elastomer based torsional vibration isolation system for dynamic loading conditions

Vibration isolation is an effective method to mitigate unwanted disturbances arising from dynamic loading conditions. With smart materials as suitable substitutes, the conventional passive isolators have attained attributes of semi-active as well as the active control system. In the present study, the non-homogenous field-dependent isolation capabilities of the magnetorheological elastomer are explored under torsional vibrations. Torsional natural frequency was measured using the serial arrangement of accelerometers. Novel methods are introduced to evaluate the torsional stiffness variations of the isolator for a semi-definite and a motor-coupled rotor system. For the semi-definite system, the isolation effect was studied using the frequency response functions from the modal analysis. The speed-dependent variations for motor-coupled rotor system were assessed using the shift in frequency amplitudes from torque transducers. Finite element method magnetics was used to study the variations in the non-homogenous magnetic field across the elastomer. The response functions for the semi-definite rotor system reveal a shift in the frequency in the effect of the magnetic field. Speed-dependent variations in the frequency domain indicate an increment of 9% in the resonant frequency of the system.

Robust MIMO Precoding for Cognitive Multiuser Relay Networks with Imperfect Channel State Information

In this paper, a novel robust precoder with imperfect channel state information(CSI)is proposed for multi-input multi-output(MIMO)cognitive multiuser networks equipped with relays. In the proposed model, the secondary users(SUs)are allowed to share the spectrum with the primary users(PUs)when the interference temperature(IT)is below a specific threshold. The transmitting strategy of relays is amplify-and-forward(AF), and the CSI error is characterized in terms of spherical uncertainty region. A minmax problem for the transmit power of the relays is considered when the mean square error(MSE)of SUs and the IT of PU meet their corresponding thresholds, and it is transformed into a semi-definite programming(SDP)problem to search for the solution. Numerical simulations demonstrate the effectiveness of the proposed precoder.

Performance analysis of ultra-wideband based wireless sensor networks in indoor office LOS environment

By analyzing the indoor office LOS channel model defined by IEEE 802.15.4a standard and deducing the variance for intra-symbol interference (IASI), inter-symbol interference (ISI) and multiuser interference (MUI), performance analysis model for ultra-wideband (UWB) based wireless sensor networks (WSN) is proposed, and the bit error rate (BER) formulation is also obtained. The comparison of performance analysis model with and without intra-symbol interference shows that the intra-symbol interference cannot be neglected and such interference will significantly decrease the system performance. In order to verify the proposed performance analysis model and BER formulation, the semi-definite programming (SDP) algorithm is used to obtain the pulses complying with the spectrum requirement of China and IEEE 802.15.4a standard, and by using such pulses, simulations and theoretical analysis are compared, the comparison results show the validity of the proposed performance analysis model and BER formulation.

Modulus-Based Matrix Splitting Iteration Methods for a Class of Stochastic Linear Complementarity Problem

For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.

Optimal Bounds for the Largest Eigenvalue of a 3 ×3 Correlation Matrix

A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more stringent than the optimal bounds by Wolkowicz and Styan in specific cases.

Inner Product Laplacian Embedding Based on Semidefinite Programming

This paper proposes an inner product Laplacian embedding algorithm based on semi-definite programming, named as IPLE algorithm. The new algorithm learns a geodesic distance-based kernel matrix by using semi-definite programming under the constraints of local contraction. The criterion function is to make the neighborhood points on manifold as close as possible while the geodesic distances between those distant points are preserved. The IPLE algorithm sufficiently integrates the advantages of LE, ISOMAP and MVU algorithms. The comparison experiments on two image datasets from COIL-20 images and USPS handwritten digit images are performed by applying LE, ISOMAP, MVU and the proposed IPLE. Experimental results show that the intrinsic low-dimensional coordinates obtained by our algorithm preserve more information according to the fraction of the dominant eigenvalues and can obtain the better comprehensive performance in clustering and manifold structure.

Fuzzy Multi-Objective Semi-Definition Programming

This paper first applies the fuzzy set theory to multi-objective semi-definite program-ming （MSDP）, and proposes the fuzzy multi-objective semi-definite programming （FMSDP） model whose optimal efficient solution is defined for the first time, too. By constructing a membership function, the FMSDP is translated to the MSDP. Then we prove that the optimal efficient solution of FMSDP is consistent with the efficient solution of MSDP and present the optimality condition about these programming. At last, we give an algorithm for FMSDP by introducing a new membership function and a series of transformation.

A potential reduction algorithm for an extended SDP problem

An extended semi-definite programming, the SDP with an additional quadratic term in the objective function, is studied. Our generalization is similar to the generalization from linear programming to quadratic programming. Optimal conditions for this new class of problems are discussed and a potential reduction algorithm for solving QSDP problems is presented. The convergence properties of this algorithm are also given.

On computing minimal H-eigenvalue of sign-structured tensors

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.