Nonnegative Integrable Function Research Articles

Integral means of solutions of the one-dimensional Poisson equation with Robin boundary conditions

Consider the one-dimensional Poisson equation −u″=f on the interval [−π,π], where f is an non-negative integrable function, with Robin boundary conditions −u′(−π)+αu(−π)=u′(π)+αu(π)=0, where α>0 is a constant. In this paper we prove inequalities for the convex integral means of solutions of this problem, using the polarization of functions and its properties. We find solutions with maximal convex integral means and prove their uniqueness.

On a nonlinear Robin problem with an absorption term on the boundary and L 1 data

Abstract We deal with existence and uniqueness of nonnegative solutions to: − Δ u = f ( x ) , in Ω , ∂ u ∂ ν + λ ( x ) u = g ( x ) u η , on ∂ Ω , \left\{\begin{array}{ll}-\Delta u=f\left(x),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ \frac{\partial u}{\partial \nu }+\lambda \left(x)u=\frac{g\left(x)}{{u}^{\eta }},\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where η ≥ 0 \eta \ge 0 and f , λ f,\lambda , and g g are the nonnegative integrable functions. The set Ω ⊂ R N ( N > 2 ) \Omega \subset {{\mathbb{R}}}^{N}\left(N\gt 2) is open and bounded with smooth boundary, and ν \nu denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of p p -Laplacian type jointly with nonlinear boundary conditions. We prove the existence of an entropy solution and check that, under natural assumptions, this solution is unique. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

On a geometric combination of functions related to Prékopa–Leindler inequality

AbstractWe introduce a new operation between nonnegative integrable functions on , that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prékopa–Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the 0‐sum, we get an alternative proof of the log‐Brunn–Minkowski inequality for unconditional convex bodies and for convex bodies with n symmetries.

Packing Feedback Arc Sets in Tournaments Exactly

Let [Formula: see text] be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set (FAS) if [Formula: see text] contains no cycles (directed). A collection [Formula: see text] of FASs (with repetition allowed) is called an FAS packing if each arc e is used at most w(e) times by the members of [Formula: see text]. The purpose of this paper is to give a characterization of all tournaments [Formula: see text] with the property that, for every nonnegative integral weight function w defined on A, the minimum total weight of a cycle is equal to the maximum size of an FAS packing. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11801266 and 11971228]; the Chinese Academy of Sciences [Grants XDA27000000 and ZDBS-LY-7008]; the Ministry of Science and Technology of China [Grant 2018AAA0101002]; the Research Grants Council of Hong Kong; the Fundamental Research Funds for the Central Universities [Grant 020314380035].

Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

Approximating connected safe sets in weighted trees

For a graph G and a non-negative integral weight function w on the vertex set of G, a set S of vertices of G is w-safe if w(C)≥w(D) for every component C of the subgraph of G induced by S and every component D of the subgraph of G induced by the complement of S such that some vertex in C is adjacent to some vertex of D. The minimum weight w(S) of a w-safe set S is the safe number s(G,w) of the weighted graph (G,w), and the minimum weight of a w-safe set that induces a connected subgraph of G is its connected safe number cs(G,w). Bapat et al. showed that computing cs(G,w) is NP-hard even when G is a star. For a given weighted tree (T,w), they described a polynomial time 2-approximation algorithm for cs(T,w) as well as a polynomial time 4-approximation algorithm for s(T,w). Addressing a problem they posed, we present a PTAS for the connected safe number of a weighted tree. Our PTAS partly relies on an exact pseudopolynomial time algorithm, which also allows to derive an asymptotic FPTAS for restricted instances. Finally, we extend a bound due to Fujita et al. from trees to block graphs.

Ranking tournaments with no errors II: Minimax relation

Abstract A tournament T = ( V , A ) is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Mobius ladders as a subgraph; such a tournament is referred to as Mobius-free. In the first paper we have given a structural description of all Mobius-free tournaments, and have proved that every CM tournament is Mobius-free. In this second paper we establish the converse by using our structural theorems and linear programming approach.

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments.

Formula omitted]-widths of certain function classes defined by the modulus of continuity

Several exact inequalities are found between the best approximation by trigonometric polynomials in L2 of differentiable periodic functions, in the sense of Weyl, averaged with the weight of the modulus of continuity of arbitrary fractional order. For some classes of functions, defined by specific moduli of continuity, the exact values of the various n-widths in L2 are calculated. In particular the problem of minimizing the constants in inequalities of Jackson–Stechkin type over all subspaces of dimension N is solved. It is proved that this value is equal to the exact value of the different widths of the class L2(α)(β,p,h,φ), where φ is a non-negative integrable weight function on (0,h](0<h≤π/n).

Integer round-up property for the chromatic number of some h-perfect graphs

We prove that every h-perfect line graph and every t-perfect claw-free graph G has the integer round-up property for the chromatic number: for every non-negative integral weight function c on the vertices of G, the weighted chromatic number of (G, c) can be obtained by rounding up its fractional relaxation. As a corollary, we obtain that the weighted chromatic number can be computed in polynomial-time for these graphs. Finally, we show a new example of a graph operation which preserves the integer round-up property for the chromatic number, and use it to provide a first example of a t-perfect 3-colorable graph which does not have this property.

Quantitative results on continuity of the spectral factorization mapping in the scalar case

In the scalar case, the spectral factorization mapping $f\to f^+$ puts a nonnegative integrable function $f$ having an integrable logarithm in correspondence with an outer analytic function $f^+$ such that $f = |f^+|^2$ almost everywhere. The main question addressed here is to what extent $\|f^+ - g^+\|_{H_2}$ is controlled by $\|f-g\|_{L_1}$ and $\|\log f - \log g\|_{L_1}$.

Example of a Monotonic, Everywhere Differentiable Function on ℝ Whose Derivative Is not Continuous

We construct an example of a monotonic function which is differentiable everywhere, but the derivative is not continuous. This is done using a nonnegative discontinuous integrable function whose every point is a Lebesgue point.

On an Integral Transform of a Class of Analytic Functions

For α, γ ≥ 0 and β &lt; 1, let 𝒲β(α, γ) denote the class of all normalized analytic functions f in the open unit disc E = {z:|z | &lt; 1} such that ℜeiϕ((1 − α + 2γ)(f(z)/z) + (α − 2γ)f′(z)+γzf′′(z) − β) &gt; 0, z ∈ E for some ϕ ∈ ℝ. It is known (Noshiro (1934) and Warschawski (1935)) that functions in 𝒲β(1,0) are close‐to‐convex and hence univalent for 0 ≤ β &lt; 1. For f ∈ 𝒲β(α, γ), we consider the integral transform , where λ is a nonnegative real‐valued integrable function satisfying the condition . The aim of present paper is, for given δ &lt; 1, to find sharp values of β such that (i) Vλ(f) ∈ 𝒲δ(1,0) whenever f ∈ 𝒲β(α, γ) and (ii) Vλ(f) ∈ 𝒲δ(α, γ) whenever f ∈ 𝒲β(α, γ).

STABILITY OF THE PRÉKOPA–LEINDLER INEQUALITY

We prove a stability version of the Pr\'ekopa-Leindler inequality.

Comparison and regularity results for degenerate elliptic equations related to Gauss measure

This article deals with a Dirichlet problem relative to a linear elliptic equation with zero-order terms, whose ellipticity condition is given in terms of the function ν(x)ϕ(x), where is the density of Gauss measure, ν(x) is a nonnegative integrable function. Using the notion of rearrangement with respect to Gauss measure, we prove a comparison result with a problem of the same type defined in a half space, with data depending only on the first variable. Also, some regularity results for the solutions in Lorentz–Zygmund spaces are obtained by us.

Chebyshev inequality for Sugeno integrals

In this paper we study Chebyshev inequality for Sugeno integrals on an arbitrary fuzzy measure space. Particularly, we obtain other Chebyshev type inequalities which relate the measure of a level set of the maximum and the sum of two non-negative integrable functions and their integrals. Finally, we apply our results to the case of comonotone functions.

Про додатні розв'язки рівняння реакції-дифузії з правою частиною типу Каратеодорі

In the paper for reaction-diffusion equation with Caratheodory nonlinear term under conditions, which do not guarantee uniqueness of Cauchy problem solution, we prove the global resolvability in the class of nonnegative integrable functions.

A Min-Max Theorem on Tournaments

We present a structural characterization of all tournaments $T=(V,A)$ such that, for any nonnegative integral weight function defined on V, the maximum size of a feedback vertex set packing is equal to the minimum weight of a triangle in T. We also answer a question of Frank by showing that it is $NP$-complete to decide whether the vertex set of a given tournament can be partitioned into two feedback vertex sets. In addition, we give exact and approximation algorithms for the feedback vertex set packing problem on tournaments.

A Min-Max Relation on Packing Feedback Vertex Sets

Let G be a graph with a nonnegative integral function w defined on V(G). A collection ℱ of subsets of V(G) (repetition is allowed) is called a feedback vertex set packing in G if the removal of any member of ℱ from G leaves a forest, and every vertex v ∈ V(G) is contained in at most w(v) members of ℱ. The weight of a cycle C in G is the sum of w(v), over all vertices v of C. The purpose of this paper is to characterize all graphs with the property that, for any nonnegative integral function w, the maximum cardinality of a feedback vertex set packing is equal to the minimum weight of a cycle.

A prediction problem in $L^2 (w)$

For a nonnegative integrable weight function w on the unit circle T, we provide an expression for p = 2, in terms of the series coefficients of the outer function of w, for the weighted L P distance inf f ∫ T |1 - f[Pwdμ, where μ is the normalized Lebesgue measure and f ranges over trigonometric polynomials with frequencies in [{..., -3, -2, -1}\{-n}]Ufm}, m > 0, n ≥ 2. The problem is open for p # 2.