Abstract Let $$\Gamma $$ Γ be a compact patch of a well-curved $$C^{n+1}$$ C n + 1 curve in $$\mathbb {R}^n$$ R n with induced Lebesgue measure $$\textrm{d} \lambda $$ d λ , and let $$g \mapsto \widehat{g \,\textrm{d}\lambda }$$ g ↦ g d λ ^ be the Fourier extension operator for $$\Gamma $$ Γ . Then we have, for arbitrary non-negative weights w , $$\begin{aligned} \int _{B_R} |\widehat{g \,\textrm{d}\lambda }|^2w \le C_{n,a} R^{a} \sup _S \left( \int _S w\right) \int _\Gamma |g|^2 \, \textrm{d} \lambda \end{aligned}$$ ∫ B R | g d λ ^ | 2 w ≤ C n , a R a sup S ∫ S w ∫ Γ | g | 2 d λ for any $$a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$$ a > n - 3 2 + 2 n - 2 n 2 ( n + 1 ) , where the $$\sup $$ sup is over all 1-neighbourhoods S of hyperplanes whose normals are parallel to the tangent at some point of $$\Gamma $$ Γ . This represents partial progress on the Mizohata–Takeuchi conjecture for curves in dimensions $$n \ge 3$$ n ≥ 3 , improving upon the exponent $$a=n-1$$ a = n - 1 which can be obtained as a consequence of the Agmon–Hörmander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves.

Let C ⊆ [ 0 , 1 ] \mathcal {C}\subseteq [0,1] be a Cantor set. In the classical C ± C \mathcal {C}\pm \mathcal {C} problems, modifying the “size” of C \mathcal {C} has a magnified effect on C ± C \mathcal {C}\pm \mathcal {C} . However, any gain in C \mathcal {C} necessarily results in a loss in C c \mathcal {C}^c , and vice versa. This interplay between C \mathcal {C} and its complement C c \mathcal {C}^c raises interesting questions about the delicate balance between the two, particularly in how it influences the “size” of C c − C \mathcal {C}^c-\mathcal {C} . One of our main results indicates that the Lebesgue measure of C c − C \mathcal {C}^c-\mathcal {C} has a greatest lower bound of 3 2 \frac {3}{2} .

We investigate a class of n -dimensional ( n\geq 2 ) free boundary elliptic problems, which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its Hausdorff dimension being less than n , which in turn leads to its Lebesgue measure being zero. As a corollary, we obtain the uniqueness of the so-called reservoirs-connected solution of the dam problem in higher dimension. Our proof of the porosity relies on the Lipschitz continuity of the solution and an appropriately constructed barrier function.

Let [Formula: see text], [Formula: see text], be a bounded connected open set with Lipschitz boundary. We consider the weighted eigenvalue problem [Formula: see text] in [Formula: see text] with [Formula: see text], [Formula: see text] and with homogeneous Dirichlet and Robin boundary conditions. First, we study weak* continuity, convexity and Gâteaux differentiability of the map [Formula: see text], where [Formula: see text] is the principal eigenvalue. Then, denoting by [Formula: see text] the class of rearrangements of a fixed weight [Formula: see text] and assuming that [Formula: see text] is positive on a set of positive Lebesgue measure, we investigate the minimization and maximization of [Formula: see text] over [Formula: see text]. The minimization problem has been already discussed in some papers; here we give an alternative treatment of some known results about the existence and characterization of minimizers of [Formula: see text]. We underline that our approach allows us to deal with Dirichlet and Robin boundary conditions together. Instead, to our best knowledge, the maximization problem has been only partially addressed in the literature; it turns out that the maximization of [Formula: see text] is more intricate than its minimization. In our work we discuss existence, uniqueness and characterization of maximizers both in [Formula: see text] and in its weak* closure [Formula: see text]. In particular, we provide an original full description of the unique maximizer in the case of Dirichlet boundary conditions. To prove this result we give a generalization of a lemma of Burton [Rearrangements of functions, maximization of convex functionals and vortex rings, Math. Ann. 276 (1987) 225–253, doi:10.1007/bf01450739] about rearrangement of functions, which we did not find in literature. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favorable and unfavorable habitats in order to increase the chances of survival or extinction of a population.

Abstract We prove that a partially hyperbolic attracting set for a $C^2$ vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical/SRB measures, which are also Gibbs states along the central-unstable direction. This extends to continuous time systems a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set. In codimension two the same result holds, assuming only the trajectories on the trapping region admit a sequence of times with asymptotical sectional expansion, on a positive volume subset. We present several examples of application, including the existence of physical measures for asymptotically sectional hyperbolic attracting sets, and obtain physical measures in an alternative unified way for many known examples: Lorenz-like and Rovella attractors, and sectional-hyperbolic attracting sets (including the multidimensional Lorenz attractor).

Abstract The aim of this article is to study a variant of the Poincaré inequality in Sobolev spaces $$W^{k,p(\cdot )}(\Omega ) $$ W k , p ( · ) ( Ω ) where $$\Omega $$ Ω is a given open set of $$\mathbb {R}^{n}$$ R n with finite Lebesgue measure and, consequently, the Friedrichs type inequality.

Differential calculus constitutes a fundamental mathematical tool in contemporary architectural design and analysis. This documentary research examines the practical applications of differential calculus in architecture, demonstrating how abstract mathematical concepts transform into tangible constructive solutions. The study analyzes the application of mathematical limits to establish design restrictions, guarantee structural safety, and optimize the behavior of high-rise buildings and complex geometries. Derivatives emerge as crucial analytical instruments for calculating curvatures, optimizing organic structures, and analyzing energy efficiency in emblematic works such as the Guggenheim Museum Bilbao. Differential geometry enables the modeling of complex surfaces and non-Euclidean spaces, while integrals facilitate the quantification of geometric properties, structural load analysis, and optimization of thermal behavior in sustainable buildings. The document also explores advanced applications such as multiple integrals, Gaussian series, partial derivatives, and Lebesgue measure theory, evidencing their relevance in parametric design and computational architecture. The results demonstrate that mastery of differential calculus not only improves the technical precision of architectural design but also expands the frontiers of creativity, enabling architects to create habitable, safe, and aesthetically innovative spaces that respond with scientific rigor to contemporary social and environmental needs.

Let μ be a translation invariant measure on ( R d , B ( R d ) ) and let λ denote the Lebesgue measure on R d . If there exists a set U ⊂ R d that is either a box or an open set, such that 0 < μ ( U ) = λ ( U ) < ∞ , it is a simple exercise to show that μ = λ | B ( R d ) . Is the same conclusion true if U is merely a Borel set? The main purpose of this short note is to construct a measure that provides a negative answer to this question. Incidentally, this construction provides a new example of a translation invariant measure with a rich domain and range that is not Hausdorff, a problem previously studied by K. E. Hirst.

The work is devoted to the study of complex-valued continuous symmetric polynomials on Cartesian products of complex Banach spaces of Lebesgue integrable functions. Let $L_p$, where $p\in [1;+\infty)$, be the complex Banach space of all complex-valued functions on $[0;1]$, the $p$th powers of absolute values of which are Lebesgue integrable. Let $\Xi_{[0;1]}$ be the set of all bijections $\sigma:[0;1] \to [0;1]$ such that both $\sigma$ and $\sigma^{-1}$ are measurable and preserve Lebesgue measure, i.e. $\mu(\sigma(E)) = \mu(\sigma^{-1}(E)) = \mu(E)$ for every Lebesgue measurable set $E\subset [0;1]$, where $\mu$ is Lebesgue measure. A function $f$ on the Cartesian product $L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$, where $p_1,p_2, \ldots, p_n \in [1;+\infty)$, is called symmetric if $f((x_1\circ\sigma;x_2\circ\sigma;\ldots;x_n\circ\sigma))=f((x_1;x_2;\ldots;x_n))$ for every $\sigma\in \Xi_{[0;1]}$ and $(x_1;x_2;\ldots;x_n)\in L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$. We construct an algebraic basis of the algebra of all complex-valued continuous symmetric polynomials on $L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$. Also we construct some isomorphisms of Fréchet algebras of complex-valued entire symmetric functions of bounded type on $L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$.

Abstract In this paper we study the class of optimal entropy-transport problems introduced by Liero, Mielke and Savaré in Inventiones Mathematicae 211 in 2018. This class of unbalanced transport metrics allows for transport between measures of different total mass, unlike classical optimal transport where both measures must have the same total mass. In particular, we develop the theory for the important subclass of semi-discrete unbalanced transport problems, where one of the measures is diffuse (absolutely continuous with respect to the Lebesgue measure) and the other is discrete (a sum of Dirac masses). We characterize the optimal solutions and show they can be written in terms of generalized Laguerre diagrams. We use this to develop an efficient method for solving the semi-discrete unbalanced transport problem numerically. As an application, we study the unbalanced quantization problem, where one looks for the best approximation of a diffuse measure by a discrete measure with respect to an unbalanced transport metric. We prove a type of crystallization result in two dimensions – optimality of a locally triangular lattice with spatially varying density – and compute the asymptotic quantization error as the number of Dirac masses tends to infinity.

Let X be a real or complex Banach space. Let S(X) denote the unit sphere of X. For x∈S(X), let Sx={x∗∈S(X∗):x∗(x)=1}. A lot of Banach space geometry can be determined by the ‘quantum’ of the state space Sx. In this paper, we mainly study the norm compactness and weak compactness of the state space in the space of Bochner integrable functions and c0-direct sums of Banach spaces. Suppose X is such that X∗ is separable and let μ be the Lebesgue measure on [0,1]. For f∈L1(μ,X), we demonstrate that if Sf is norm compact, then f is a smooth point. When μ is the discrete measure, we show that if (xi)∈S(ℓ1(X)) and ∥xi∥≠0 for all i∈N, then S(xi) is weakly compact in ℓ∞(X∗) if and only if Sxi/∥xi∥ is weakly compact in X∗ for each i∈N and diam(Sxi/∥xi∥)→0. For discrete c0-sums, we show that for (xi)∈c0(X), S(xi) is weakly compact if and only if for each i0∈N such that ∥xi0∥=1, the state space Sxi0 is weakly compact.

We prove the existence of response quasi-periodic solutions for the forced inviscid Burgers-Hilbert equation, close to any fixed, nonzero constant velocity, provided that the forced term is small enough and the frequency belongs to a Borel set with asymptotically full Lebesgue measure. The proof is based on a reduction of transport operators and Nash-Moser iteration.

We prove a Hardy-type inequalities in Dunkl setting, integrated with an HI-potential. Our approach utilizes the h-harmonic expansion of functions and integrating techniques such as integral transformations, spherical coordinate formulas, and separation of variables, we derive the main result presented in Theorem 1. These outcomes build upon and extend the foundational work of Ghoussoub and Moradifam (2013), which addressed Hardy-type inequalities involving the Laplace operator and the Lebesgue measure in conjunction with an HI-potential. Consequently, our findings advance the generalization of Hardy inequality within broader context of Dunkl theory. Moreover, this research carries substantial implications for analyzing differential equations and partial differential equations that exhibit singularities, thereby providing enhanced understanding of the qualitative properties and behaviors of solutions in these equation classes. This extension not only refines existing inequalities but also opens avenues for applications in mathematical physics and functional analysis.

ABSTRACTSeveral notions of concentration function have been proposed in the statistical literature. Here we will refer to the definition by Cifarelli and Regazzini (1987), used for the comparison of probability measures in very general probability spaces. We will consider the concentration function in more restricted settings, typically those of interest to practitioners, that is, discrete or absolutely continuous (with respect to the Lebesgue measure) probability measures. We will show how a sophisticated mathematical tool can be used to provide graphs and indices that are easily interpreted by practitioners. Some examples will support this statement.This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences &gt; Exploratory Data Analysis Statistical and Graphical Methods of Data Analysis &gt; Statistical Graphics and Visualization

It is well-known that if a, b are irrational numbers, then a b need not be an irrational number. Let M be a set of real numbers. In this note it is proved that if M is any of (i) the set of all irrational real numbers, (ii) the set of all transcendental real numbers, (iii) the set of all non-computable real numbers, (iv) the set of all real normal numbers, (v) the set of all real numbers of irrationality exponent equal to 2, (vi) the set of all real Mahler S-numbers, (vii) or indeed any subset of R of full Lebesgue measure, then, for each positive real number s ≠ 1 , there exist a , b ∈ M such that s = a b . The analogous result for complex numbers is also proved. These results are proved using measure theory.

Abstract In this paper, we investigate the Hausdorff measure of shrinking target sets in ‐dynamical systems. These sets are dynamically defined in analogy to the classical theory of weighted and multiplicative Diophantine approximation. While the Lebesgue measure and Hausdorff dimension theories for these sets are well‐understood, much remains unknown about the Hausdorff measure theory. We show that the Hausdorff measure of these sets is either zero or full depending upon the convergence or divergence of a certain series, thus providing a rather complete measure theoretic description of these sets.

In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman–DeWitt–Faddeev–Popov ghost fields. Such fields result from the fiber-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock–Kurzweil integral is first introduced, because it makes it possible to integrate functions like [Formula: see text]. The Lebesgue measure on [Formula: see text] is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space [Formula: see text], which contains [Formula: see text] as a continuous dense embedding, and contains both the test functions [Formula: see text] and their dual [Formula: see text], the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to [Formula: see text], where [Formula: see text] is any separable Hilbert space. Last, the existence of a unique universal definition of time, [Formula: see text], that we call historical time, is proven. We use [Formula: see text] as the order parameter for our construction of Feynman’s time ordered operator calculus, which in turn is used to extend the path integral in order to include all time-dependent groups and semigroups with a reproducing kernel representation.

Abstract We study the distribution $$\mu $$ μ in the central limit theorem for V-monotone independence. Using its Cauchy–Stieltjes transform, we prove that $$\mu $$ μ is absolutely continuouswith respect to the Lebesgue measure on the real line and we give its density $$\rho $$ ρ in an implicit form. We present a computer-generated graph of $$\rho $$ ρ .

Against the backdrop of the deep integration of the global digital economy and cross-border e-commerce, and addressing the lack of measure theory and quantitative analysis challenges for high-dimensional dynamic data in this field, this study constructs a measure theory system for cross-border e-commerce markets that combines mathematical rigor and economic interpretability based on Carathéodory’s extension theorem. Based on functional analysis and measure theory, the study defines the market fundamental set as the topological product space of a time index set and a multi-dimensional transaction state space. By constructing a combined structure of a left-open right-closed interval semiring and a power set semiring that satisfies the closure of Boolean algebra operations, an algebraic framework is established for the unified measurement of continuous and discrete variables. On the semiring structure, a σ-finite premeasure integrating Lebesgue measure and counting measure is defined. With the help of the countable covering mechanism generated by outer measure and the measure screening rules of Carathéodory’s measurability condition, the axiomatic extension from premeasure to complete measure on the σ-algebra is completed. Through the verification of Carathéodory’s condition for subsets of null sets and the transmission of outer measure monotonicity, the completeness of the measure space is strictly proved, and the core property that “subsets of null sets must be measurable” is established, providing a solid measure-theoretic foundation for mathematical modeling of cross-border e-commerce markets. At the empirical analysis level, the study uses micro-panel data on global cross-border e-commerce transactions from 2018 to 2024. Through the Kolmogorov-Smirnov test in non-parametric hypothesis testing, the distribution isomorphism between the theoretical measure and empirical data is verified. Based on the measure space theory, a Generalized Method of Moments (GMM) panel regression model is constructed. System GMM and Difference GMM estimation methods are used to handle endogeneity issues. Combined with instrumental variable methods and lag variable techniques, key parameters such as the logarithmic elasticity of economic scale between importing and exporting countries, the spatial decay effect of geographical distance, and the asymmetric inhibitory effect of tariff policies are quantitatively analyzed. A graph neural network model integrating measure theory is innovatively designed. By introducing a completeness regular term, the measure constraints on null sets and their subsets are achieved. Combined with the SHAP value interpretability analysis method, the marginal contribution of each characteristic variable in model decision-making is revealed. The study finds that the constructed measure space not only satisfies the axiomatic requirements of modern measure theory such as completeness and σ-finiteness, but also through the empirical tests of the GMM model and graph neural network, it is confirmed that it can effectively characterize the economic scale effect, spatial distance decay law, and policy sensitivity characteristics in cross-border e-commerce transactions, providing a methodological innovation paradigm based on measure theory for quantitative analysis in the field of international business in the digital economy era.

In this paper, the existence of positive strong solutions to a Dirichlet [Formula: see text]-Laplacian problem with reaction both singular at zero and highly discontinuous is investigated. In particular, it is only required that the set of discontinuity points has Lebesgue measure zero.