Further results on integral generalized inverses of integral matrices

This paper further investigates integral generalized inverses of integral matrices.     

Existence results for Urysohn integral inclusions involving the Henstock integral

This paper presents two existence results for Urysohn integral inclusions in Banach spaces, the set-valued integral involved being the Henstock integral. In particular, the existence of continuous solutions for integral inclusions of Volterra and of Hammerstein type is obtained. Our results extend the existence theorems given in literature in the single- or set-valued case, under Bochner or Pettis integrability hypothesis. The compactness of solutions set is also studied.     

A constructive integral equivalent to the integral of Kurzweil

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.     

Liouville type theorems for integral equations and integral systems

In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in R ^N . The main technique we use is the method of moving planes in an integral form.     

A Daniell integral approach to nonstandard Kurzweil-Henstock integral

A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from [9]. The theory is recovered together with a few new results.     

Gronwall-Bellman type integral inequalities for abstract Lebesgue integral

In this paper, we consider Gronwall-Bellman type integral inequalities and the corresponding integral equations for scalar functions of several variables involving abstract Lebesque integrals. Some delay and advance effects are also included. Since the inequalities and the equations are in a general form, the results may have different applications.     

Generalized Lebesgue integral

A new definition of integral-like functionals exploiting the ideas of the Lebesgue integral construction and extending the idea of pan-integrals is given. Some convergence theorems for sequence of measurable functions are discussed. As a result, a theoretical basis for applications of the generalized Lebesgue integral is provided. Several types of integrals known from the literature are shown to be special cases of generalized Lebesgue integral.     

Constructably Laplacian integral graphs

A graph is Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. We consider the class of constructably Laplacian integral graphs - those graphs that be constructed from an empty graph by adding a sequence of edges in such a way that each time a new edge is added, the resulting graph is Laplacian integral. We characterize the constructably Laplacian integral graphs in terms of certain forbidden vertex-induced subgraphs, and consider the number of nonisomorphic Laplacian integral graphs that can be constructed by adding a suitable edge to a constructably Laplacian integral graph. We also discuss the eigenvalues of constructably Laplacian integral graphs, and identify families of isospectral nonisomorphic graphs within the class.     

Volterra integral equations and evolution equations with an integral condition

We suggest a simple method for reducing problems with an integral condition for evolution equations to a Volterra integral equation of the first kind. For Volterra equations of the convolution type, we indicate necessary and sufficient solvability conditions for the case in which the right-hand side lies in some classes of functions of finite smoothness. We use these conditions to construct examples of nonexistence of a local solution for the heat equation with an integral condition.     

The widest continuous integral

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock-Kurzweil integrals, is considered. We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V. The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given.     

Packing and covering with integral feasible flows in integral supply-demand networks

Polynomial-time algorithms are presented for solving combinatorial packing and covering problems defined from the integral feasible flows in an integral supply-demand network. These algorithms are also shown to apply to packing and covering problems defined by the minimal integral solutions to general totally unimodular systems. Analogous problems arising from matroid bases are also discussed and it is demonstrated that a means for solving such problems is provided by recent work of Cunningham.     

Gaussian integral circulant digraphs

The spectrum of a digraph in general contains real and complex eigenvalues. A digraph is called a Gaussian integral digraph if it has a Gaussian integral spectrum that is all eigenvalues are Gaussian integers. In this paper, we consider Gaussian integral digraphs among circulant digraphs.     

On integral groups

Summary An algorithm for the determination of all integral classes of reducible integral matrix groups of given dimension from those of lower dimension is described. For dimensionn=4 there are 567 such classes.     

The fuzzy integral

In this paper we define the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure. We show that the monotone convergence theorem and Fatou's lemma are still true in this new setting. We study some of the properties of this integral, and show that it coincides with another fuzzy integral defined in the literature. Our main result is a convergence theorem, that is in a way stronger than the Lebesgue-dominated convergence theorem. This holds when the fuzzy measure is also assumed to be subadditive.     

A simplified multidimensional integral

We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.     

Behavior of integral curves in the neighborhood of optimal integral manifolds

We describe an explicit construction of optimal integral manifolds [1] for a quasilinear system of differential equations that uses the method of successive approximations. We study the behavior of integral curves in the neighborhood of optimal integral manifolds. We cite a numerical method of synthesis of optimal control and prove its justification.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1049–1060, August, 1992.     

整图的构造

In this paper, some new classes of integral graphs are given in two new ways. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving diophantine equations. Some classes are infinite. The discovery of these classes is a new contribution to the search of such integral graphs.     

Denjoy integral and Henstock-Kurzweil integral in vector lattices,II

In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.      

Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results

For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.     

Cauchy-type LG* integral

We examine a Cauchy-type LG^* integral. We give the characteristics of the class of functions representable by such an integral and prove that this class coincides with the class of functions representable by the Cauchy LG^* integral [3].Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 645–653, November, 1976.