Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

Extraction of Tangential Momentum and Normal Energy Accommodation Coefficients by Comparing Variational Solutions of the Boltzmann Equation with Experiments on Thermal Creep Gas Flow in Microchannels

In the present paper, we provide an analytical expression for the first- and second-order thermal slip coefficients, σ1,T and σ2,T, by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules. The Cercignani-Lampis scattering kernel of the gas-surface interaction has been considered in order to take into account the influence of the accommodation coefficients (αt, αn) on the slip parameters. Comparing our theoretical results with recent experimental data on the mass flow rate and the slip coefficient for five noble gases (helium, neon, argon, krypton, and xenon), we found out that there is a continuous set of values for the pair (αt, αn) which leads to the same thermal slip parameters. To uniquely determine the accommodation coefficients, we took into account a further series of measurements carried out with the same experimental apparatus, where the thermal molecular pressure exponent γ has been also evaluated. Therefore, the new method proposed in the present work for extracting the accommodation coefficients relies on two steps. First of all, since γ mainly depends on αt, we fix the tangential momentum accommodation coefficient in such a way as to obtain a fair agreement between theoretical and experimental results. Then, among the multiple pairs of variational solutions for (αt, αn), giving the same thermal slip coefficients (chosen to closely approximate the measurements), we select the unique pair with the previously determined value of αt. The analysis carried out in the present work confirms that both accommodation coefficients increase by increasing the molecular weight of the considered gases, as already highlighted in the literature.

Variational Solutions for Resonances by a Finite-Difference Grid Method

We demonstrate that the finite difference grid method (FDM) can be simply modified to satisfy the variational principle and enable calculations of both real and complex poles of the scattering matrix. These complex poles are known as resonances and provide the energies and inverse lifetimes of the system under study (e.g., molecules) in metastable states. This approach allows incorporating finite grid methods in the study of resonance phenomena in chemistry. Possible applications include the calculation of electronic autoionization resonances which occur when ionization takes place as the bond lengths of the molecule are varied. Alternatively, the method can be applied to calculate nuclear predissociation resonances which are associated with activated complexes with finite lifetimes.

Global continuity of variational solutions weakening the one-sided bounded slope condition

We study regularity properties of variational solutions to a class of Cauchy–Dirichlet problems of the form { ∂ t ⁡ u - div x ⁡ ( D ξ ⁢ f ⁢ ( D ⁢ u ) ) = 0 in ⁢ Ω T , u = u 0 on ⁢ ∂ 𝒫 ⁡ Ω T . \left\{\begin{aligned} \displaystyle\partial_{t}u-\operatorname{div}_{x}(D_{% \xi}f(Du))&\displaystyle=0&&\displaystyle\phantom{}\text{in }\Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on }\partial% _{\mathcal{P}}\Omega_{T}.\end{aligned}\right. We do not impose any growth conditions from above on f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\to\mathbb{R}} , but only require it to be convex and coercive. The domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is mainly supposed to be bounded and convex, and for the time-independent boundary datum u 0 : Ω ¯ → ℝ {u_{0}\colon\overline{\Omega}\to\mathbb{R}} we only require continuity. These requirements are weaker than a one-sided bounded slope condition. We prove global continuity of the unique variational solution u : Ω T → ℝ {u\colon\Omega_{T}\to\mathbb{R}} . If the boundary datum is Lipschitz continuous, we obtain global Hölder continuity of the solution.

$L^{p}-$Variational solutions of multivalued backward stochastic differential equations

We prove the existence and uniqueness of the $L^{p}-$variational solution, with $p>1,$ of the fo\-llo\-wing multivalued backward stochastic differential equation with $p-$integrable data: \[ \left\{ \begin{array}[c]{l} -dY_{t}+\partial_{y}\Psi(t,Y_{t})dQ_{t}\ni H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<\tau,\\[0.2cm] Y_{\tau}=\eta, \end{array} \right. \] where $\tau$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and $\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous function $y\mapsto\Psi(t,y).$

Expectancy-based rhythmic entrainment as continuous Bayesian inference

When presented with complex rhythmic auditory stimuli, humans are able to track underlying temporal structure (e.g., a “beat”), both covertly and with their movements. This capacity goes far beyond that of a simple entrained oscillator, drawing on contextual and enculturated timing expectations and adjusting rapidly to perturbations in event timing, phase, and tempo. Previous modeling work has described how entrainment to rhythms may be shaped by event timing expectations, but sheds little light on any underlying computational principles that could unify the phenomenon of expectation-based entrainment with other brain processes. Inspired by the predictive processing framework, we propose that the problem of rhythm tracking is naturally characterized as a problem of continuously estimating an underlying phase and tempo based on precise event times and their correspondence to timing expectations. We present two inference problems formalizing this insight: PIPPET (Phase Inference from Point Process Event Timing) and PATIPPET (Phase and Tempo Inference). Variational solutions to these inference problems resemble previous “Dynamic Attending” models of perceptual entrainment, but introduce new terms representing the dynamics of uncertainty and the influence of expectations in the absence of sensory events. These terms allow us to model multiple characteristics of covert and motor human rhythm tracking not addressed by other models, including sensitivity of error corrections to inter-event interval and perceived tempo changes induced by event omissions. We show that positing these novel influences in human entrainment yields a range of testable behavioral predictions. Guided by recent neurophysiological observations, we attempt to align the phase inference framework with a specific brain implementation. We also explore the potential of this normative framework to guide the interpretation of experimental data and serve as building blocks for even richer predictive processing and active inference models of timing.

An evolutionary Haar-Rado type theorem

In this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ f ( D u ) ) + D u g ( x , u ) = 0 , where u attains time-independent boundary values $$u_0$$ u 0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values $$u_0$$ u 0 admit a modulus of continuity $$\omega $$ ω and the estimate $$|u(x,t)-u_0(\gamma )| \le \omega (|x-\gamma |)$$ | u ( x , t ) - u 0 ( γ ) | ≤ ω ( | x - γ | ) holds, then u admits the same modulus of continuity in the spatial variable.

Soluções variacionais e numéricas da Equação de Schrödinger 1D submetida ao potencial de Pöschl-Teller

<p>This paper presents the numerical and variational solutions of the 1D Schrödinger Equation submitted to the Pöschl-Teller potential. The methods used were the Variational Method and the Finite Difference Method. They were presented in a didactic and detailed way with the purpose of instructing both undergraduate and graduate students, about the applicability and effectiveness of the aforementioned methods. We use the Pöschl-Teller potential due to the fact that it is little explored in the books of Quantum Mechanics used in undergraduation courses and also because of its diverse applications, such as in Bose-Einstein condensates, waveguides, topological defects in field theory and so on. We conclude this paper comparing the variational and numerical solutions with the analytical solution and present the advantages of each method.</p>