Partial integral operators of Fredholm type on Kaplansky-Hilbert module over $L_0$

The article studies some characteristic properties of self-adjoint partially integral operators of Fredholm type in the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $L_{0}\left(\Omega_{2}\right)$. Some mathematical tools from the theory of Kaplansky-Hilbert module are used. In the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $ L_{0} \left (\Omega _ {2} \right)$ we consider the partially integral operator of Fredholm type $T_{1}$ ($ \Omega_{1} $ and $\Omega_{2} $ are closed bounded sets in $ {\mathbb R}^{\nu_{1}}$ and $ {\mathbb R}^{\nu_{2}},$ $\nu_{1}, \nu_{2} \in {\mathbb N} $, respectively). The existence of $ L_{0} \left (\Omega _ {2} \right) $ nonzero eigenvalues for any self-adjoint partially integral operator $T_{1}$ is proved; moreover, it is shown that $T_{1}$ has finite and countable number of real $L_{0}(\Omega_{2})$-eigenvalues. In the latter case, the sequence $ L_{0}(\Omega_{2})$-eigenvalues is order convergent to the zero function. It is also established that the operator $T_{1}$ admits an expansion into a series of $\nabla_{1}$-one-dimensional operators.

Positive Solutions for BVP of Fractional Differential Equation with Integral Boundary Conditions

In this paper, we consider a class of boundary value problems of nonlinear fractional differential equation with integral boundary conditions. By applying the monotone iterative method and some inequalities associated with Green’s function, we obtain the existence of minimal and maximal positive solutions and establish two iterative sequences for approximating the solutions to the above problem. It is worth mentioning that these iterative sequences start off with zero function or linear function, which is useful and feasible for computational purpose. An example is also included to illustrate the main result of this paper.

Holomorphic functions with distinguished properties on infinite dimensional spaces

In this paper, we develop a method to construct holomorphic functions that exist only on infinite dimensional spaces. The following types of holomorphic functions f:U→ℂ on some open subsets U of an infinite dimensional complex Banach space are constructed: (1) f is bounded holomorphic on U and is continuously, but not uniformly continuously extended to U¯; (2) f is continuous on U¯ and holomorphic of bounded type on U, but f is unbounded on U; (3) f is holomorphic of bounded type on U and f cannot be continuously extended to U¯. The technique we develop is powerful enough to provide, in the cases (2) and (3) above, large algebraic structures formed by such functions (up to the zero function, of course).

Polynomials inducing the zero function on chain rings

We provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that map the maximal ideal [Formula: see text] into one of its powers [Formula: see text], where [Formula: see text] is a discrete valuation ring with a finite residue field. We use this to provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that send [Formula: see text] to zero, where [Formula: see text] is a finite commutative local principal ideal ring.

Quarks in a Polar Model

Current-quark masses are compared to the rest masses allowed by the Helmholtz equation in a polar model. Within the uncertainty of the current u quark mass determination, the current quark mass coincides with the rest mass allowed by the Helmholtz equation in the polar model in accordance with the second root of the zero Neumann function. Current d quark mass coincides with the rest mass calculated in accordance with the third root of the Bessel zero function. On the basis of a comparison of these results with the results obtained earlier for ordinary real particles u and d quarks stability is discussed.

On Fε2-planar mappings with function ε of (pseudo-)Riemannian manifolds

In this paper we study special mappings between n-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced PQ?-projectivity of Riemannian metrics, with constant ? ? 0,1 + n. These mappings were studied later by Matveev and Rosemann and they found that for ? = 0 they are projective. These mappings could be generalized for case, when ? will be a function on manifold. We show that PQ?- projective equivalence with ? is a function corresponds to a special case of F-planar mapping, studied by Mikes and Sinyukov (1983) with F = Q. Moreover, the tensor P is derived from the tensor Q and non-zero function ?. We assume that studied mappings will be also F2-planar (Mikes 1994). This is the reason, why we suggest to rename PQ? mapping as F?2. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.

A generalization of the m-topology on C(X) finer than the m-topology

It is well known that the component of the zero function in C(X) with the m-topology is the ideal C?(X). Given any ideal I ? C?(X), we are going to define a topology on C(X) namely the mI-topology, finer than the m-topology in which the component of 0 is exactly the ideal I and C(X) with this topology becomes a topological ring. We show that compact sets in C(X) with the mI-topology have empty interior if and only if X n T Z[I] is infinite. We also show that nonzero ideals are never compact, the ideal I may be locally compact in C(X) with the mI-topology and every Lindel?f ideal in this space is contained in C?(X). Finally, we give some relations between topological properties of the spaces X and Cm(X). For instance, we show that the set of units is dense in Cm(X) if and only if X is strongly zero-dimensional and we characterize the space X for which the set r(X) of regular elements of C(X) is dense in Cm(X).

Large Function Algebras with Certain Topological Properties

LetFbe a family of continuous functions defined on a compact interval. We give a sufficient condition so thatF∪{0}contains a densec-generated free algebra; in other words,Fis denselyc-strongly algebrable. As an application we obtain densec-strong algebrability of families of nowhere Hölder functions, Bruckner-Garg functions, functions with a dense set of local maxima and local minima, and nowhere monotonous functions differentiable at all but finitely many points. We also study the problem of the existence of large closed algebras withinF∪{0}whereF⊂RXorF⊂CX. We prove that the set of perfectly everywhere surjective functions together with the zero function contains a2c-generated algebra closed in the topology of uniform convergence while it does not contain a nontrivial algebra closed in the pointwise convergence topology. We prove that an infinitely generated algebra which is closed in the pointwise convergence topology needs to contain two valued functions and infinitely valued functions. We give an example of such an algebra; namely, it was shown that there is a subalgebra ofRRwith2cgenerators which is closed in the pointwise topology and, for any functionfin this algebra, there is an open setUsuch thatf-1(U)is a Bernstein set.

Monotone Iterative Methods of Positive Solutions for Fractional Differential Equations Involving Derivatives

This paper studies the existence and computing method of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions. By applying monotone iterative methods, the existence results of positive solutions and two iterative schemes approximating the solutions are established. The interesting point of our method is that the iterative scheme starts off with a known simple function or the zero function and the nonlinear term in the fractional differential equation is allowed to depend on the unknown function together with derivative terms. Two explicit numerical examples are given to illustrate the results.