Lebesgue's Theorem Research Articles

The Blaschke–Lebesgue theorem revisited

The Blaschke–Lebesgue theorem revisited

Mixed volumes and the Blaschke–Lebesgue theorem

Mixed volumes and the Blaschke–Lebesgue theorem

Properties of the fractional Clifford–Fourier transform

ABSTRACT In this paper, we establish a Riemann–Lebesgue theorem and some real Paley–Wiener-type theorems for the fractional Clifford–Fourier transform (FrCFT). Furthermore, because of the non-commutativity of Clifford algebra, we study some basic properties from linearity to modulation of the FrCFT about the left-multiplied functions and the right-multiplied functions.

A Blaschke–Lebesgue theorem for the Cheeger constant

In this paper, we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the [Formula: see text]-Laplacian for any [Formula: see text] (this paper covers the case [Formula: see text] whereas the case [Formula: see text] was already known).

Tight description of faces of triangulations on the torus

The degree d(x) of a vertex or face x in a graph G is the number of incident edges. A face f=v1…vd(f) in a graph G on the plane or other orientable surface is of type (k1,k2,…), where k1≤k2≤…, if d(vi)≤ki for each i. By δ we denote the minimum vertex-degree of G.In 1989, Borodin confirmed Kotzig's conjecture of 1963 that every plane graph with minimum degree δ equal to 5 has a (5,5,7)-face or a (5,6,6)-face, where all parameters are tight. Recently, we proved that every torus triangulation with δ≥5 has a face of one of the types (5,5,8), (5,6,7), or (6,6,6), which is tight.It follows from the classical theorem by Lebesgue (1940) that every plane triangulation with δ≥4 has a 3-face of types (4,4,∞), (4,5,19), (4,6,11), (4,7,9), (5,5,9), or (5,6,7).In 1999, Jendrol' gave a similar description: “(4,4,∞), (4,5,13), (4,6,17), (4,7,8), (5,5,7), (5,6,6)” and conjectured that “(4,4,∞), (4,5,10), (4,6,15), (4,7,7), (5,5,7), (5,6,6)” holds.In 2002, Lebesgue's description was strengthened by Borodin to “(4,4,∞), (4,5,17), (4,6,11), (4,7,8), (5,5,8), (5,6,6)”.In 2014, Borodin and Ivanova obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': “(4,4,∞), (4,5,11), (4,6,10), (4,7,7), (5,5,7), (5,6,6)”, and recently proved another tight description of faces in plane triangulations with δ≥4: “(4,4,∞), (4,6,10), (4,7,7), (5,5,8), (5,6,7)”.It follows from Lebesgue's theorem of 1940 that every plane quadrangulation with δ≥3 has a face of one of the types (3,3,3,∞), (3,3,4,11), (3,3,5,7), (3,4,4,5). Recently, Borodin and Ivanova improved this description to “(3,3,3,∞), (3,3,4,9), (3,3,5,6), (3,4,4,5)”, where all parameters except possibly 9 are best possible and 9 cannot go down below 8.In 1995, Avgustinovich and Borodin proved the following tight description of the faces of torus quadrangulations with δ≥3: “(3,3,3,∞), (3,3,4,10), (3,3,5,7), (3,3,6,6), (3,4,4,6), (4,4,4,4)”, which also holds for each higher surface provided that its quadrangulation is large enough.Recently, Borodin and Ivanova proved that every triangulation with δ≥4 of the torus has a face of one of the types (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), which description is tight.The purpose of this paper is to prove that every triangulation with δ≥3 on the torus has a face of one of the types (3,6,24), (3,8,16), (3,12,12), (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), where all parameters are best possible.

Equivalence of Lebesgue's Theorem and Baire Characterization Theorem

Let $X$ be a complete separable metric space and $Y$ be a separable Banach space. We provide a proof of equivalence by linking explicitly the following statements:\\ \noindent \textbf{\textit{Lebesgue's Theorem.}} For every $\epsilon>0$ there exists a countable collection of closed sets $\left\lbrace C_n\right\rbrace $ of $X$ such that $$X=\bigcup_{n=1}^{\infty}C_n\;\;\text{and}\;\; \omega_f\left( C_n\right)<\epsilon\;\; \text{for each} \;\; n.$$ \textbf{\textit{Baire Characterization Theorem.}} For every nonempty perfect set $K\subset X$, the function $f|_K$ has at least one point of continuity in $K$. In fact, $C(f|_K)$ is dense in $K$.\\ \indent Moreover, replacing ``closed'' by ``open'' in the Lebesgue's Theorem, we obtain a characterization of continuous functions on space $X$.

Unrestricted Cesàro summability of $d$-dimensional Fourier series and Lebesgue points

We generalize the classical Lebesgue's theorem to multi-dimensional functions. We prove that the Cesàro means of the Fourier series of the multi-dimensional function $f\in L_1(\log L)^{d-1}(\mathbb{T}^d)\supset L_p(\mathbb{T}^d) (1<p<\infty)$ converge to $f$ at each strong Lebesgue point.

Sound source localization method by using a single moving microphone and inner product calculation

In this paper, a microphone is arranged to move along a line at a low speed while collecting sound pressure signals, then the mathematical relation is researched to locate the source and the localization processes are as follows. The collected continuous-time signals are segmented and each segment is transferred to the frequency domain by integral transformation. As a result, according to the Riemann Lebesgue theorem, the interference from the negative frequency term is eliminated, a correlation relationship between the real sound source and virtual sound source (a hypothetical sound source) is set up. By using the formed correlation, the position of the sound source is located by founding the maximum of the inner product module through optimization. Analyzing the details of the segmentation, it is found that the correlation will be enhanced when segment time length is integer multiples of signal's half period, and it benefits the localization accuracy. This method is also extended to more complicated multi-point source identification. Simulation examples and experiments show that the proposed method can locate the sound source. This method uses only one microphone, there is no demanding phase matching and calibration work like the of a microphone array, the cost of use is low.

Generalized convergence theorems for monotone measures

In this paper, we propose three types of absolute continuity for monotone measures and present some of their basic properties. By means of these three types of absolute continuity, we establish generalized Egoroff's theorem, generalized Riesz's theorem and generalized Lebesgue's theorem in the framework involving the ordered pair of monotone measures. The Egoroff theorem, the Riesz theorem and the Lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case. These three generalized convergence theorems include as special cases several previous versions of Egoroff-like theorem, Riesz-like theorem and Lebesgue-like theorem for monotone measures, respectively.

Fatou's Lemma in Its Classical Form and Lebesgue's Convergence Theorems for Varying Measures with Applications to Markov Decision Processes

The classical Fatou lemma states that the lower limit of a sequence of integrals of functions is greater than or equal to the integral of the lower limit. It is known that Fatou's lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with the integral of the lower limit in two parameters, where the second parameter is the argument of the functions. In the present paper, we provide sufficient conditions when Fatou's lemma holds in its classical form for a sequence of weakly converging measures. The functions can take both positive and negative values. Similar results for sequences of setwise converging measures are also proved. We also put forward analogies of Lebesgue's and the monotone convergence theorems for sequences of weakly and setwise converging measures. The results obtained are used to prove broad sufficient conditions for the validity of optimality equations for average-cost Markov decision processes.

Linearly continuous functions and $$F_\sigma $$-measurability

The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $\mathbb R^m$ this notion is near to the separate continuity for which it is required only the continuity on the straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately continuous function $f:\mathbb R^m\to\mathbb R$ is of the $(m-1)$-th Baire class. In this paper we prove that every linearly continuous function $f:\mathbb R^m\to\mathbb R$ is of the first Baire class. Moreover, we obtain the following result. If $X$ is a Baire cosmic topological vector space, $Y$ is a Tychonoff topological space and $f:X\to Y$ is a Borel-measurable (even BP-measurable) linearly continuous function, then $f$ is $F_\sigma$-measurable. Using this theorem we characterize the discontinuity point set of an arbitrary linearly continuous function on $\mathbb R^m$. In the final part of the article we prove that any $F_\sigma$-measurable function $f:\partial U\to \mathbb R$ defined on the boundary of a strictly convex open set $U\subset\mathbb R^m$ can be extended to a linearly continuous function $\bar f:X\to \mathbb R$. This fact shows that in the ``descriptive sense'' the linear continuity is not better than the $F_\sigma$-measurability.

Describing the neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 6 to 8

In 1940, Lebesgue proved that every 3-polytope with minimum degree 5 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (6,6,7,7,7),(6,6,6,7,9),(6,6,6,6,11),(5,6,7,7,8),(5,6,6,7,12),(5,6,6,8,10),(5,6,6,6,17),(5,5,7,7,13),(5,5,7,8,10),(5,5,6,7,27),(5,5,6,6,∞),(5,5,6,8,15),(5,5,6,9,11),(5,5,5,7,41),(5,5,5,8,23),(5,5,5,9,17),(5,5,5,10,14),(5,5,5,11,13).Recently, we proved that forbidding vertices of degree from 7 to 11 results in a tight description (5,5,6,6,∞), (5,6,6,6,15), (6,6,6,6,6).The purpose of this note is to prove that every 3-polytope with minimum degree 5 and no vertices of degrees 6, 7, and 8 has a 5-vertex whose neighborhood is majorized by one of the sequences (5,5,5,5,∞) and (5,5,5,10,12), which is tight and improves a corresponding description (5,5,5,5,∞), (5,5,9,5,17), (5,5,10,5,14), (5,5,11,5,13) that follows from the Lebesgue Theorem.

Colorful versions of the Lebesgue, KKM, and Hex theorem

Following and developing ideas of R. Karasev (2014) [10], we extend the Lebesgue theorem (on covers of cubes) and the Knaster–Kuratowski–Mazurkiewicz theorem (on covers of simplices) to different classes of convex polytopes (colored in the sense of M. Joswig). We also show that the n-dimensional Hex theorem admits a generalization where the n-dimensional cube is replaced by a n-colorable simple polytope. The use of specially designed quasitoric manifolds, with easily computable cohomology rings and the cohomological cup-length, offers a great flexibility and versatility in applying the general method.

Anti-Compacts and Their Applications to Analogs of Lyapunov and Lebesgue Theorems in Frechét Spaces

We introduce anti-compact sets (anti-compacts) in Frechet spaces. We thoroughly investigate the properties of anti-compacts and the scale of Banach spaces generated by anti-compacts. Special attention is paid to systems of anti-compact ellipsoids in Hilbert spaces. The existence of a system of anti-compacts is proved for any separable Frechet space E. Using the constructed theory, we obtain analogs of the Lyapunov theorem on the convexity and compactness of the range of vector measures in the class of separable Frechet spaces: We prove the convexity and compactness of the range of vector measure in a space \( {E}_{\overline{C}} \) generated by an anti-compact \( \overline{C} \). Also, the nondifferentiability problem with respect to the upper limit is investigated for the Pettis integral. We obtain differentiability conditions for the indefinite Pettis integrals in terms of the new weak integral boundedness and the σ-compact measurability. We prove an analog of the Lebesgue theorem on the differentiability of the indefinite Pettis integral for any strongly measurable integrand.

Coherent randomness tests and computing the $K$-trivial sets

We introduce Oberwolfach randomness, a notion within Demuth's framework of statistical tests with moving components; here the components' movement has to be coherent across levels. We show that a ML-random set computes all K -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a K -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis, such as the Lebesgue density theorem and Doob's martingale convergence theorem. We also show that random sets which are not Oberwolfach random satisfy highness properties (such as LR-hardness) which mean they are close to computing the halting problem. A consequence of these results is that a ML-random set failing the effective version of Lebesgue's density theorem for closed sets must compute all K -trivial sets. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem of algorithmic randomness. On the other hand these results settle stronger variants of the covering problem in the negative: no low ML-random set computes all K -trivial sets, and not every K -trivial set is computable from both halves of a random set.

The algebra of conditional sets and the concepts of conditional topology and compactness

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel–Lebesgue theorem, the Hahn–Banach theorem, the Banach–Alaoglu theorem and the Krein–Šmulian theorem are shown.

A Simple Proof of the Lebesgue Decomposition Theorem

Motivated by the notion of operator, a short and simple proof of Lebesgue's decomposition theorem is presented in this note.

The statistical convergence for sequences of fuzzy-number-valued functions

Based on the concept of statistical convergence of sequences of fuzzy numbers, the statistical convergence, uniformly statistical convergence and equi-statistical convergence of sequences of fuzzy-number-valued functions are defined and investigated in this paper. The relationship among statistical convergence, uniformly statistical convergence, equi-statistical convergence of sequences of fuzzy-number-valued functions and their representations of sequences of α-level cuts are discussed. In addition, the Egorov and Lebesgue theorems for the statistical convergence of sequences of fuzzy-number-valued functions are obtained in a finite measure set. Finally, the statistical convergence in measure for sequences of fuzzy-number-valued functions is investigated, and it is proved that the outer and inner statistical convergence in measure are equivalent in a finite measure set for a sequence of fuzzy-number-valued functions.

Measurable events indexed by products of trees

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ? 2, called the branching number of T, such that every t ? T has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T 1,...,T d ) of homogeneous trees and its level product ?T is the subset of the Cartesian product T 1×...×T d consisting of all finite sequences (t 1,...,t d ) of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product ?T of a vector homogeneous tree T. We show that, by refining the index set to the level product ?S of a vector strong subtree S of T, such families of events become highly correlated. An analogue of Lebesgue's density Theorem is also established which can be considered as the probabilistic version of the density Halpern-Lauchli Theorem.

A NOTE ON THE INTEGRAL REPRESENTATION OF THE VALUE FUNCTION

In this paper we propose an alternative assumption for the integral representation of the value function of Milgrom and Segal (2002). Instead of requiring that utility function has derivative almost everywhere, we impose that it has derivative in all its domain. The idea is to obtain conditions in order to apply the Lebesgue Theorem which provides at the same time an absolutely continuous value function and its integral representation. Our assumption is technically stronger than that of Milgrom and Segal (2002) but we argue that there is a substantial gain of economic interpretation in adding it. While it is difficult to interpret absolute continuity in terms of agent's preferences, the existence of the derivative everywhere means that all agent's choices are smooth.