Search Results (12)

The aim of this paper is to investigate the following non local p-Laplacian problem with data a bounded Radon measure $\vartheta \in M_b\left(\mathsf{\Omega }\right)$: ${(-\mathsf{\Delta })}_p^{s}u=\vartheta \mathrm{in}\mathsf{\Omega },$ with vanishing conditions outside $\mathsf{\Omega },$ and where $s\in (0,1),2-\frac{s}{N}<p\le N.$ An existence result is provided, and some sharp regularity has been investigated. More precisely, we prove by using some fractional isoperimetric inequalities the existence of weak solution u such that: 1. If $\vartheta \in M_b\left(\mathsf{\Omega }\right),$ then $u\in W_0^{s_1,q}\left(\mathsf{\Omega }\right)$ for all $s_1<s$ and $q<\frac{N(p-1)}{N-s}.$ 2. If $\vartheta$ belongs to the Zygmund space $LLo{g}^{\alpha }L\left(\mathsf{\Omega }\right),\alpha >\frac{N-s}{N},$ then the limiting regularity $u\in W_0^{s_1,\frac{N(p-1)}{N-s}}\left(\mathsf{\Omega }\right)$ (for all $s_1<s$). 3. If $\vartheta \in LLo{g}^{\alpha }L\left(\mathsf{\Omega }\right),$ and $\alpha =\frac{N-s}{N}$ with $p=N,$ then we reach the maximal regularity with respect to s and $N,u\in W_0^{s,N}\left(\mathsf{\Omega }\right).$ Full article

This paper comprehensively reviews the nonlinear dynamics given by some classes of constrained control problems which involve second-order partial derivatives. Specifically, necessary optimality conditions are formulated and proved for the considered variational control problems governed by integral functionals. In addition, the well-posedness and the associated variational inequalities are considered in the present review paper. Full article

The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University of Vigo and is devoted to presenting some Applications of Fubini’s theorem. In the first part, we present Brunn–Minkowski’s and Isoperimetric inequalities. The second part is devoted to the estimations of volumes of sections of balls in ${\mathbb{R}}^n$. Full article

In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called “cat states”, which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory, are also briefly discussed. Full article

Ben-Or and Linial, in a seminal work, introduced the full information model to study collective coin-tossing protocols. Collective coin-tossing is an elegant functionality providing uncluttered access to the primary bottlenecks to achieve security in a specific adversarial model. Additionally, the research outcomes for this versatile functionality has direct consequences on diverse topics in mathematics and computer science. This survey summarizes the current state-of-the-art of coin-tossing protocols in the full information model and recent advances in this field. In particular, it elaborates on a new proof technique that identifies the minimum insecurity incurred by any coin-tossing protocol and, simultaneously, constructs the coin-tossing protocol achieving that insecurity bound. The combinatorial perspective into this new proof-technique yields new coin-tossing protocols that are more secure than well-known existing coin-tossing protocols, leading to new isoperimetric inequalities over product spaces. Furthermore, this proof-technique’s algebraic reimagination resolves several long-standing fundamental hardness-of-computation problems in cryptography. This survey presents one representative application of each of these two perspectives. Full article

Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex bodies. Full article

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton–Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded. Full article

Comprehensive evaluation of forest state is the precondition and critical step for forest management. To solve the problem that the radar plot and unit circle only focus on the value of each the evaluation index, this paper proposes a novel method for comprehensively and simultaneously evaluating the functionality and inhomogeneity of forest state based on the modified unit circle method. We evaluated the forest state of the Quercus aliena BL. var. acuteserrata Maxim. ex Wenz. broad-leaved mixed forest in the Xiaolong Mountains Forest Area of Gansu Province and the Pinus koraiensis Sieb. et Zucc. broad-leaved mixed forest in Jilin Province in China. According to the principle of comprehensive, scientific and operability, 10 evaluation indices on forest structure and vitality were selected to construct the evaluation indicator system. Each index was normalized based on the assignment method and ensured to be strictly positive based on reciprocal transformation method. The areas and arc length of the closed graph, formed by connecting every two adjacent indicators, in the radar plot and unit circle were extracted. Based on the isoperimetric theorem (isoperimetric inequality), a comprehensive evaluation model was constructed. Compared with radar chart and unit circle method, each index in the newly proposed unit circle method is represented by an independent sector region, reflecting the contribution of the index to the overall evaluation result. Each index has the same relative importance weight, contributing to the estimation the relative sizes of each aspect of forest state. The unique area and arc length of the closed graph help summarize the overall performance with a global score. The expression effect of improved unit circle has been enhanced, and as an English proverb put it, “A picture is worth a thousand words.” The new proposed method simultaneously evaluates the functionality and inhomogeneity of the forest state and it is a powerful tool for the diagnosis of forest state problems and the decision-making of forest management. Full article

We study a class of isoperimetric problems on R^N with respect to weights that are powers of the distance to the origin. We consider different weights in the volume and in the perimeter. We investigate cases in which, among all smooth sets Ω in R^N with fixed weighted measure, the ball centered at the origin has minimum weighted perimeter. The results also imply a weighted Pólya- Szegö principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems. Full article

Searching for the dynamical foundations of Havrda-Charvát/Daróczy/ Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an N-Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a connection with the non-additive entropy under investigation. We present an isoperimetric interpretation of the non-extensive parameter and comment on further features of the system that can be probed through this tensor. Full article

In this review we summarize, expand, and set in context recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. We specifically consider the thermodynamics of higher-dimensional rotating asymptotically flat and AdS black holes and black rings in a canonical (fixed angular momentum) ensemble. We plot the associated thermodynamic potential—the Gibbs free energy—and study its behavior to uncover possible thermodynamic phase transitions in these black hole spacetimes. We show that the multiply-rotating Kerr-AdS black holes exhibit a rich set of interesting thermodynamic phenomena analogous to the “every day thermodynamics” of simple substances, such as reentrant phase transitions of multicomponent liquids, multiple first-order solid/liquid/gas phase transitions, and liquid/gas phase transitions of the van derWaals type. Furthermore, the reentrant phase transitions also occur for multiply-spinning asymptotically flat Myers–Perry black holes. These phenomena do not require a variable cosmological constant, though they are more naturally understood in the context of the extended phase space. The thermodynamic volume, a quantity conjugate to the thermodynamic pressure, is studied for AdS black rings and demonstrated to satisfy the reverse isoperimetric inequality; this provides a first example of calculation confirming the validity of isoperimetric inequality conjecture for a black hole with non-spherical horizon topology. The equation of state P = P(V,T) is studied for various black holes both numerically and analytically—in the ultraspinning and slow rotation regimes. Full article

We consider a pursuit-evasion problem where some lions have the task to clear a grid graph whose nodes are initially contaminated. The contamination spreads one step per time unit in each direction not blocked by a lion. A vertex is cleared from its contamination whenever a lion moves to it. Brass et al. [5] showed that n/2 lions are not enough to clear the n x n-grid. In this paper, we consider the same problem in dimension d > 2 and prove that Θ(n^d-1/√d) lions are necessary and sufficient to clear the n^d-grid. Furthermore, we analyze a problem variant where the lions are also allowed to jump from grid vertices to non-adjacent grid vertices. Full article