Coherent wavelets form a unified basis of the multichannel perfect reconstruction analysis-synthesis filter bank of high resolution radar imaging and clinical magnetic resonance imaging (MRI). The filter bank construction is performed by the Kepplerian temporospatial phase detection strategy which allows for the stroboscopic and synchronous cross sectional quadrature filtering of phase histories in local frequency encoding multichannels with respect to the rotating coordinate frame of reference. The Kepplerian strategy and the associated filter bank construction take place in symplectic affine planes which are immersed as coadjoint orbits of the Heisenberg two-step nilpotent Lie group $G$ into the foliated three-dimensional real projective space $\mbox{ P}\bigl(\mbox{ R}\times\mbox{Lie}(G)^{\star}\bigr)$. Due to the factorization of transvections into affine dilations of opposite ratio, the Heisenberg group $G$ under its natural sub-Riemannian metric acts on the line bundle realizing the projective space $\mbox{ P}\bigl(\mbox{ R}\times \mbox{ Lie}(G)^{\star}\bigr)$. Its elliptic non-Euclidean geometry without absolute quadric, associated to the unitary dual $\hat{G}$, governs the design of the coils inside the bore of the MRI scanner system. It determines the distributional reproducing kernel of the tracial read-out process of quantum holograms excited and coexisting in the MRI scanner system. Thus the pathway of this paper leads from Keppler's approach to projective geometry to the Heisenberg approach to the sub-Riemannian geometry of quantum physics, and finally to the enormously appealing topic of ensemble quantum computing.

We show that a certain logarithmic growth condition on a bounded linear operator on a complex Banach space implies Bishop's property $(\beta )$, and discuss several applications of this result in local spectral theory.

By using the method of dierential inequalities we obtained some new and better results for certain classes of strongly starlike functions introduced recently in [3].

In this paper, we give a number theoretical application of a simple discrete map. For any prime $p\geq 5$, we show that the numbers in the set $G(1, p-2)=\{1, 2, 3, \cdots , p-2\}$ can be divided into disjoint triplets $\langle a, b, c\rangle $, such that $a\cdot b\cdot c\equiv 1$ (mod $p$). This is a generalization of the well known doublet--division, that is, the numbers in the set $\{1, 2, 3, \cdots , p-1\}$ can be divided into disjoint pairs $\langle a, a^*\rangle $, such that $a\cdot a^*\equiv 1$ (mod $p$). The reason that we can perform the triplet-division is because there exists a simple map $F$ defined on the set $G(1, p-2)$ such that any number $a\in G(1, p-2)$ is a period-3 point of the map. Furthermore, the triplet $\langle a,~F(a),~F^2(a)\rangle $ has the property that $a\cdot F(a)\cdot F^2(a)\equiv 1$ (mod $p$).

Consider the fixed-design nonparametric median regression model $Y_i=g(x_i)+e_i,\,i\geq 1$. For estimating the regression function $g(x)$ and its derivative, the nearest neighbor median estimators $\tilde{g}_{n,h}(x)$ are employed, where $h$ is the number of the nearest neighbors. Under mild regularity conditions, rates of convergence for the estimators are obtained.

In this paper, we prove a section theorem of Ky Fan type in interval space, and then, as its applications, some minimax inequalities and a xed point theorem are obtained.

In this work we consider the triviality of the solutions of the initial-boundary value problems for some semi-linear wave equations.

This paper is concerned with the pseudoconvex Riemann domain over the product space $\Bbb C^{\Bbb N}$. We investigate that the pseudoconvex Riemann domain is the product of $\Bbb C^{\Bbb N}-\{1,2,\cdots,n\}$ and a pseudoconvex Riemann domain over $\Bbb C^n$.

A path partition of a graph is a collection of vertex-disjoint paths that cover all vertices of the graph. The path-partition problem is to nd a path partition of minimum size. This paper gives a linear-time algorithm for the path-partition problem in bipartite distance-hereditary graphs.

A probabilistic approach to construct the singular monotone functions by using Markov chains is given, and the relation between the singular monotone functions and the strong law of Markov chains is revealed

Let $\gamma $ be the adiabatic index of self-gravitating, spherically symmetric motion of compressible viscous gas-star. When $\gamma \in (1,2]$, we prove the existence of nonisentropic equilibrium. Furthermore, at the adiabatic index $\gamma = \frac {4}{3}$, we found a family of particular solutions which corresponds to an expansive (contractive) gaseous star. Moreover, we find that there is a critical total mass $M_0$. If the total mass $M$ of star is less than $M_0$, then the star expands infinitely. However, if $M \geq M_0$, then there is an ``escape velocity'' $v_er$ associated with $M$ and the initial configuration of the star. If $v(0,r) \geq v_er$, then the star will expand infinitely. If $v(0,r) \lt v_er$, then it will collapse after a finite time.