scholarly journals Poincaré and Log–Sobolev Inequalities for Mixtures

Entropy ◽  
2019 ◽  
Vol 21 (1) ◽  
pp. 89 ◽  
Author(s):  
André Schlichting
Keyword(s):  
Blow Up ◽  
Functional Inequality ◽  
Multidimensional Case ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Mixture Ratio ◽  
Mixture Parameter ◽  
Sobolev Constants ◽  
The One ◽  
Poincaré Constant

This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the χ 2 -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

scholarly journals CONCENTRATION INEQUALITIES FOR GIBBS MEASURES

2011 ◽  
Vol 14 (01) ◽  
pp. 79-104
Author(s):  
IOANNIS PAPAGEORGIOU
Keyword(s):  
Gibbs Measures ◽  
Type Inequality ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Infinite Dimensional ◽  
Unbounded Spin Systems ◽  
The One ◽  
Concentration Properties ◽  
Weaker Conditions ◽  
Unbounded Spin

We are interested in Sobolev type inequalities and their relationship with concentration properties on higher dimensions. We consider unbounded spin systems on the d-dimensional lattice with interactions that increase slower than a quadratic. At first we assume that the one-site measure satisfies a modified log-Sobolev inequality with a constant uniformly on the boundary conditions and we determine conditions so that the infinite-dimensional Gibbs measure satisfies a concentration as well as a Talagrand type inequality, similar to the ones obtained by Barthe and Roberto6 for the product measure. Then a modified log-Sobolev type concentration property is obtained under weaker conditions referring to the log-Sobolev inequalities for the boundary free measure.

scholarly journals Gauge theories on compact toric manifolds

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Francesco Fucito ◽  
Jose Francisco Morales ◽  
Massimiliano Ronzani ◽  
Ekaterina Sysoeva ◽  
...  
Keyword(s):  
Partition Function ◽  
Gauge Theories ◽  
Blow Up ◽  
Gauge Coupling ◽  
Piecewise Constant Function ◽  
Partition Functions ◽  
Piecewise Constant ◽  
Toric Manifolds ◽  
Holomorphic Anomaly ◽  
The One

AbstractWe compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

scholarly journals Sobolev inequalities for fractional Neumann Laplacians on half spaces

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Roberta Musina ◽  
Alexander I. Nazarov
Keyword(s):  
Half Space ◽  
Sobolev Inequalities ◽  
Neumann Laplacian ◽  
Sobolev Constants

Abstract We consider different fractional Neumann Laplacians of order {s\in(0,1)} on domains {\Omega\subset\mathbb{R}^{n}} , namely, the restricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{R}}}} , the semirestricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sr}}}} and the spectral Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sp}}}} . In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.

Variability and repeatability of olfactometric results of n-butanol, pig odour and a synthetic gas mixture

2004 ◽  
Vol 50 (4) ◽  
pp. 65-73 ◽  
Author(s):  
N. Defoer ◽  
H. Van Langenhove
Keyword(s):  
Synthetic Mixture ◽  
Gas Mixture ◽  
Mixture Ratio ◽  
Air Samples ◽  
Complete Dataset ◽  
Synthetic Gas ◽  
Pig Farms ◽  
The Mean ◽  
The One ◽  
Mean Variance

For the purposes of a research project for the Flemish authorities, olfactometric measurements were carried out at six closed pig farms and six fattener farms. The results of these olfactometric measurements were compared with the olfactometric results of n-butanol samples and samples of a synthetic gas mixture of ethanethiol, methylacetate and 2-propanol in nitrogen, both analysed on the same days as the air samples from the pig farms. The results of the n-butanol tests for all panellists showed that nobody was qualified according to the CEN criteria, and that, consequently, these criteria are rather stringent. Comparing the variability of the results for the three different odours showed that the mean and standard deviation of the mean variance were not significantly different for the three odour types, which means that the repeatability of the panellist results was equal for the examined odour types. The principle of traceability was checked by comparing the variance of the n-butanol, pig odour and synthetic mixture ratio. For the complete dataset, the principle of traceability could not been proven for n-butanol. For the restricted dataset, the principle of traceability was more valid for n-butanol than for the mixture, but differences were small. Finally, normalization was looked for with regard to olfactometric measurements of air samples from pig farms based either on n-butanol or on the synthetic mixture. Both models had low determination coefficients, but the model based on the synthetic mixture gave better results than the one based on n-butanol.

scholarly journals New Sharp Gagliardo–Nirenberg–Sobolev Inequalities and an Improved Borell–Brascamp–Lieb Inequality

2018 ◽  
Vol 2020 (10) ◽  
pp. 3042-3083 ◽  
Author(s):  
François Bolley ◽  
Dario Cordero-Erausquin ◽  
Yasuhiro Fujita ◽  
Ivan Gentil ◽  
Arnaud Guillin
Keyword(s):  
Half Space ◽  
Minkowski Inequality ◽  
Point Of View ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Geometric Point ◽  
Functional Point

Abstract We propose a new Borell–Brascamp–Lieb inequality that leads to novel sharp Euclidean inequalities such as Gagliardo–Nirenberg–Sobolev inequalities in $ {\mathbb{R}}^n$ and in the half-space $ {\mathbb{R}}^n_+$. This gives a new bridge between the geometric point of view of the Brunn–Minkowski inequality and the functional point of view of the Sobolev-type inequalities. In this way we unify, simplify, and generalize results by S. Bobkov–M. Ledoux, M. del Pino–J. Dolbeault, and B. Nazaret.

scholarly journals Existence and nonexistence of entire solutions to the logistic differential equation

2003 ◽  
Vol 2003 (17) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Vicentiu Radulescu
Keyword(s):  
Blow Up ◽  
Entire Solutions ◽  
One Dimensional ◽  
Nonexistence Result ◽  
Nonnegative Solutions ◽  
Existence And Nonexistence ◽  
Logistic Problem ◽  
Whole Space ◽  
The Mean ◽  
The One

We consider the one-dimensional logistic problem(rαA(|u′|)u′)′=rαp(r)f(u)on(0,∞),u(0)>0,u′(0)=0, whereαis a positive constant andAis a continuous function such that the mappingtA(|t|)is increasing on(0,∞). The framework includes the case wherefandpare continuous and positive on(0,∞),f(0)=0, andfis nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results depending on the growth ofpandA. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.

FREE-BOUNDARY PROBLEM OF THE ONE-DIMENSIONAL EQUATIONS FOR A VISCOUS AND HEAT-CONDUCTIVE GASEOUS FLOW UNDER THE SELF-GRAVITATION

2013 ◽  
Vol 23 (08) ◽  
pp. 1377-1419 ◽  
Author(s):  
MORIMICHI UMEHARA ◽  
ATUSI TANI
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Global Solvability ◽  
The Self ◽  
Fundamental Result ◽  
System Of Equations ◽  
One Dimensional ◽  
Unique Existence ◽  
The One

In this paper we consider a system of equations describing the one-dimensional motion of a viscous and heat-conductive gas bounded by the free-surface. The motion is driven by the self-gravitation of the gas. This system of equations, originally formulated in the Eulerian coordinate, is reduced to the one in a fixed domain by the Lagrangian-mass transformation. For smooth initial data we first establish the temporally global solvability of the problem based on both the fundamental result for local in time and unique existence of the classical solution and a priori estimates of its solution. Second it is proved that some estimates of the global solution are independent of time under a certain restricted, but physically plausible situation. This gives the fact that the solution does not blow up even if time goes to infinity under such a situation. Simultaneously, a temporally asymptotic behavior of the solution is established.

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