The singular value decomposition is derived when the Radon transform is restricted to functions which are square integrable on the unit ball in Rn with respect to the weight W(x). It fulfilles mainly by means of the projection-slice theorem.The range of the Radon transform is spanned by products of Gegenbauer polynomials and spherical harmonics. The inverse transform of the those basis functions are given. This immediately
leads to an inversion formula by series expansion and range characterizations.

The solutions of the following bilinear stochastic differential equation are stud-
ied
dxt =
Xm
k=1
Ak
t xtdwk(t) + Btxtdt
where Ak
t , Bt are (deterministic) continuous matrix-valued functions of t and w1(t), · · ·,
wm(t) are m independent standard Brownian motions. Conditions are given such that the
solution is positive if the initial condition is positive. The equation the most probable path
must satisfy is also derived and applied to a mathematical finance problem.

The aim of this paper is to study the conditions by which a P-prime sub-
module can be expressed as a ¯nite intersection or union of P-prime submodules. Also
corresponding to dimension and rank of modules, some equivalent conditions for a ring to
be a Dedekind domain are given.

An operator T is said to be paranormal if ||T²x|| ≥ ||Tx||² holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], [14], respectively. Yamazaki and Yanagida [38] introduced the class of absolute-(p, r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈B(H) is called absolute-(p, r)-paranormal operator if |||T|^p|T*|^rx||^r ≥ |||T*|rx||^p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0. The famous result of Browder, that self adjoint operators satisfy Browder´s theorem, is extended to several classes of operators. In this paper we show that for any absolute-(p, r)-paranormal operator T, T satisfies Browder´s theorem and a-Browder´s theorem. It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p, r)-paranormal operator T, then E is self-adjoint if and only if the null space of T − μ, N(T − μ)  N(T* − μ).

t Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations
requires different techniques of unique character for each equation. By reducing these
differential equations of second order to a common solvable differential equation of first
order, a simple common solution is provided to cover all the existing standard solutions
of these named equations. It is easier than the method of generating functions and more
powerful than the Frobenius method of power series.

This is a survey paper on the study of compressible Navier-Stokes-Poisson equations. The emphasis is on the long time behavior of global solutions to multi-dimensional compressible Navier-Stokes-Poisson equations, and the optimal decay rates for both unipolar and bipolar compressible Navier-Stokes-Poisson equations are discussed.

new recursive vertex-deleting formula for the computation of the chromatic
polynomial of a graph is obtained in this paper. This algorithm is not only a good tool for
further studying chromatic polynomials but also the fastest among all the algorithms for
the computation of chromatic polynomials.

For any group G, denote by e(G) the set of orders of elements in G. Given
a finite group G, let h(e(G)) be the number of isomorphism classes of finite groups with
the same set e(G) of element orders. A group G is called k-recognizable if h(e(G)) =
k < 1, otherwise G is called non-recognizable. Also a 1-recognizable group is called a
recognizable (or characterizable) group. In this paper the authors show that the simple
groups PSL(3, q), where 3 < q  ±2 (mod 5) and (6, (q − 1)/2) = 1, are recognizable.

This article deals with some results concerning multiplication and comultiplication modules over a commutative ring.

In this paper, we will introduce how to apply Green's function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative  structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen's principle. Then, for other nonlinear
dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green's function of the linearized system and micro-local analysis, such as frequency decomposition and so on.

Numerical approximations of multi-dimensional shock waves sometimes exhibit an instability called the  carbuncle phenomenon.
Techniques for suppressing carbuncles are trial-and-error and lack in reliability and generality, partly because theoretical knowledge about carbuncles is equally unsatisfactory. It is not known which numerical schemes are affected in which circumstances, what causes carbuncles to appear and whether carbuncles are purely numerical artifacts or rather features of a continuum equation or model.

This article presents evidence towards the latter: we propose that carbuncles are a special class of entropy solutions which can be physically correct in some circumstances. Using "filaments'', we trigger a single carbuncle in a new and more reliable way, and compute the structure in detail in similarity coordinates. We argue that carbuncles can, in some circumstances, be valid vanishing viscosity limits.
Trying to suppress them is making a physical assumption that may be false.

In this article, we define almost prime submodules as a new generalization of prime and weakly prime submodules of unitary modules over a commutative ring with identity. We study some basic properties of almost prime submodules and give some char-acterizations of them, especially for (finitely generated faithful) multiplication modules.

In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid-ered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.