Existence of Solutions for S-L Singular Boundary Value Problems with p-Laplacian Operators

The existence of solutions for singular nonlinear two point boundary value problems subject to Sturm Liouville boundary conditions with p Laplacian operators is studied by the method of upper and lower solutions. The proof is based on an application of Schauder’s fixed point theorem to a modified problem whose solutions are that of the original one. At the same time, Arzela Ascoli theorem is used to prove that the defined operator N is a compact map.

UNIVERSAL BOUNDS FOR EIGENVALUES OF LAPLACIAN OPERATOR OF ANY ORDER

Let Ω be a connected bounded domain in R^n. Denote by λi the i-th eigenvalue of the Lapla^ian operator with any order p：{u=Эn→^-Эu=…=Эn→p-1^-Эp-1u=0 on ЭΩ （-△）pu=λu in Ω.In this article, we give some expressions for upper bound of the （k ＋ 1）-th eigenvalue ）λk＋l in terms of the first k eigenvalues.

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space ? n . Assume that $$0 < \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _k \leqslant \cdots $$ are the eigenvalues of the Dirichlet Laplacian operator with any order l: $$\left\{ \begin{gathered} ( - \vartriangle )^l u = \lambda u, in D \hfill \\ u = \frac{{\partial u}}{{\partial \vec n}} = \cdots = \frac{{\partial ^{l - 1} u}}{{\partial \vec n^{l - 1} }} = 0, on \partial D \hfill \\ \end{gathered} \right.$$ . Then we obtain an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues. $$\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )} \leqslant \frac{1}{n}[4l(n + 2l - 2)]^{\tfrac{1}{2}} \left\{ {\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{{l - 1}}{l}} \sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{1}{l}} } } } \right\}^{\tfrac{1}{2}} $$ . This ineguality is independent of the domain D. Furthermore, for any l ? 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.

Initial Value Problem of Singular n-Laplacian Equation

To show some theorems on the existence of singular initial value problem with n Laplacian operator, topology method and methods of analysis are employed. Some existence theorems for initial value problems with n Laplacian operators are established in three singular cases.

Multi Object Tracking Using Gradient-Based Learning Model in Video-Surveillance

On accomplishing an efficacious object tracking,the activity of an object concerned becomes notified in a forthright manner.An accurate form of object tracking task necessitates a robust object tracking procedures irrespective of hardware assistance.Such approaches inferred a vast computational complexity to track an object with high accuracy in a stipulated amount of processing time.On the other hand,the tracking gets affected owing to the existence of varied quality diminishing factors such as occlusion,illumination changes,shadows etc.,In order to rectify all these inadequacies in tracking an object,a novel background normalization procedure articulated on the basis of a textural pattern is proposed in this paper.After preprocessing an acquired image,employment of an Environmental Succession Prediction algorithm for discriminating disparate background environment by background clustering approach have been accomplished.Afterward,abstract textural characterizations through utilization of a Probability based Gradient Pattern(PGP)approach for recognizing the similarity between patterns obtained so far.Comparison between standardized frame obtained in prior and those processed patterns detects the motion exposed by an object and the object concerned gets identified within a blob.Hence,the system is resistant towards illumination variations.These illumination variation was interpreted in object tracking residing within a dynamic background.Devised approach certainly outperforms other object tracking methodologies like Group Target Tracking(GTT),Vi PER-GT,grabcut,snakes in terms of accuracy and average time.Proposed PGP-based pattern texture analysis is compared with Gamifying Video Object(GVO)approach and hence,it evidently outperforms in terms of precision,recall and F1 measure.

SEMILINEAR ELLIPTIC EQUATIONS ON FRACTAL SETS

In this article, the authors consider a class of semilinear elliptic equations on fractal sets under some new conditions, which are more weaker than those in usual cases. The authors get the non-trivial and non-negative solution of the zero boundary Dirichlet problem using Mountain Pass Lemma.

Resistance Distance and Kirchhoff Index of a Class of Join Graphs

For some complicated graphs obtained by graph operations,it is very difficult to compute resistance distance and Kirchhoff index.Define a new graph operation,and obtain a class of new join graphs:the subdivision-vertex-vertex join G_1* G_2.Then,describe the Laplacian matrix of the graph G_1 * G_2 and use generalized inverse of the Laplacian matrix to get formulas for resistance distance and Kirchhoff index.Through the obtained formulas,the resistance distance of any pairs of vertices and Kirchhoff index of the join graph can be computed.

A NEW LAPLACIAN COMPRISON THEOREM AND THE ESTIMATE OF EIGENVALUES

This paper establishes a new Laplacian comparison theorem which is specially useful tothe mathelds of nonpositive curvature. It leads naturally to the corresponding heat kernelcomparison and eigenvalue comparison theorems. Furthermore, a lower estimate of L2-spectrumof an n-dimensional non-compact complete Cartan-Hadamard manifold is given by (n-1)k/4,provided its mRicci curvature -(n -1)k (k= const. 0).

A UNIFORM GROWTH ESTIMATES OF SOLUTIONS OF ASYMPTOTICALLY ELLIPTIC OPERATORS

This paper introduces a generic eigenvalue flow of a parameter family of operators, where the corresponding eigenfunction is continuous in parameters. Then the author applies the result to the study of polynomial growth L-harmonic functions. Under the assumption that the operator has some weakly conic structures at infinity which is not necessarily unique, a Harnack type uniform growth estimate is obtained.

Weighted Marginal Fisher Analysis with Spatially Smooth for aircraft recognition

Due to limitations to extract invariant features for recognition when the aircraft presents various poses and lacks enough samples for training, a novel algorithm called Weighted Marginal Fisher Analysis with Spatially Smooth （WMFA-SS） for extracting invariant features in aircraft rec- ognition is proposed. According to the Graph Embedding （GE） framework, Heat Kernel function is firstly introduced to characterize the interclass separability when choosing the weights of penalty graph. Furthermore, Laplacian penalty is applied to constraining the coefficients to be spatially smooth in this algorithm. Laplacian penalty is able to incorporate the prior information that neigh- boring pixels are correlated. Besides, using a Laplacian penalty can also avoid the singularity of Laplacian matrix of intrinsic graph. Once compact representations of the images are obtained, it can be considered as invariant features and then be performed in classification to recognize different patterns of aircraft. Real aircraft recognition experiments show the superiority of our proposed WMFA-SS in comparison to other GE algorithms and the current aircraft recognition algorithm; the accuracy rate of our proposed method is 90.00% for dataset BH-AIR1.0 and 99.25% for dataset BH-AIR2.0.

Sublinear Elliptic Equation on Fractal Domains

This paper investigates sub-linear elliptic equations on self-similar fractal sets. With an appropriately defined Laplacian, we obtain the existence of nontrivial solutions of sub-linear elliptic equations -△u=λu- a（x）｜u｜q-1u-f（x,u）,with zero boundary Dirichlet conditions. The results are obtained by using Mountain Pass Lemma and Saddle Point Theorem.

On Triangular Lattice Boltzmann Schemes for Scalar Problems

We propose to extend the d’Humi`eres version of the lattice Boltzmann scheme to triangular meshes.We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant.On such meshes,it is possible to define the lattice Boltzmann scheme as a discrete particle method,without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice.We test this idea for the heat equation and perform an asymptotic analysis with the Taylor expansion method for two schemes named D2T4 and D2T7.The results show a convergence up to second order accuracy and set new questions concerning a possible super-convergence.

Bochner-Kodaira Techniques on Kahler Finsler Manifolds

A horizontal Hodge Laplacian operator □h is defined for Hermitian holomorphic vector bundles over PTM on Khler Finsler manifold,and the expression of □h is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection.The vanishing theorem is obtained by using the α_Hα_H-method on Kahler Finsler manifolds.

Fine edge detection in single-pixel imaging

Typical single-pixel imaging techniques for edge detection are mostly based on first-order differential edge detection operators.In this paper,we present a novel edge detection scheme combining Fourier single-pixel imaging with a second-order Laplacian of Gaussian(LoG)operator.This method utilizes the convolution results of an LoG operator and Fourier basis patterns as the modulated patterns to extract the edge detail of an unknown object without imaging it.The simulation and experimental results demonstrate that our scheme can ensure finer edge detail,especially under a noisy environment,and save half the processing time when compared with a traditional first-order Sobel operator.