Estimation of population covariance matrices from samples of multivariate data is important. (1) Estimation of principle components and eigenvalues. (2) Construction of linear discriminant functions. (3) Establishing independence and conditional independence. (4) Setting confidence intervals on linear functions. Suppose we observed p dimensional multivariate samples X1, X2, · · · , Xn i.i.d. wi...

The administrative monopoly breaks the principle of justice, and has large harm to the society. The special chapter in Chinese Antimonopoly Law regulates the contents and corresponding legal responsibilities of administrative monopoly, but the law still has some deficiencies. The Chinese Antimonopoly Law should be perfected from increasing the operation property, confirming the comprehensive le...

The operator-theoretic renormalization group (RG) methods are powerful analytic tools to explore spectral properties of field-theoretical models such as quantum electrodynamics (QED) with non-relativistic matter. In this paper these methods are extended and simplified. In a companion paper, our variant of operator-theoretic RG methods is applied to establishing the limiting absorption principle...

It has long been appreciated that Shannon information can in principle be used to measure the informational content of sensory signals. But the lack of a general, context-sensitive method for modelling signal probability distributions has obstructed practical applications. Addressing this problem, the paper introduces an approach based on dependency models. Using Shannon’s concept of conditiona...

We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove existence of a global energy minimiser an...

In this paper we study potential-theoretic properties of the symmetric :-stable processes (0<:<2): establishing the boundary Harnack principle for ratios of :-harmonic functions on any open sets, identifying the Martin boundary with the Euclidean boundary for open sets with a certain interior fatness property, and extending earlier results on intrinsic ultracontractivity and the conditional gau...

The criterion of modulational instability the second order bright solitary wave is studied in this article. Principle consists initially seeking all solutions type which verify nonlinear partial differential equation governs dynamics propagation flattened optical fibers. When reference solution to be subjected a disturbance identified, next step establishing condition stability/instability.

In this paper, we prove a maximum principle for the general multi-term space-time-fractional transport equation and apply it establishing uniqueness of solution to an initial-boundary-value problem equation. We also derive some comparison principles solutions problems with different data. Finally, present Cauchy time-fractional on unbounded domain.

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

Orthogonal metric space is a considerable generalization of usual obtained by establishing perpendicular relation on set. Very recently, the notions orthogonality set and are described notable fixed point theorems given in orthogonal spaces. Some for generalizations contraction principle via altering distance functions spaces presented proved this paper. Furthermore, an example to clarify these...