scholarly journals LIBERATION, FREE MUTUAL INFORMATION AND ORBITAL FREE ENTROPY

2018 ◽  
Vol 239 ◽  
pp. 205-231
Author(s):  
TAREK HAMDI
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Main Tool ◽  
Stochastic Calculus ◽  
Boundary Values ◽  
Spectral Measures ◽  
Orthogonal Projections ◽  
Free Entropy ◽  
Orbital Free ◽  
Subordinate Function

In this paper, we perform a detailed spectral study of the liberation process associated with two symmetries of arbitrary ranks: $(R,S)\mapsto (R,U_{t}SU_{t}^{\ast })_{t\geqslant 0}$, where $(U_{t})_{t\geqslant 0}$ is a free unitary Brownian motion freely independent from $\{R,S\}$. Our main tool is free stochastic calculus which allows to derive a partial differential equation (PDE) for the Herglotz transform of the unitary process defined by $Y_{t}:=RU_{t}SU_{t}^{\ast }$. It turns out that this is exactly the PDE governing the flow of an analytic function transform of the spectral measure of the operator $X_{t}:=PU_{t}QU_{t}^{\ast }P$ where $P,Q$ are the orthogonal projections associated to $R,S$. Next, we relate the two spectral measures of $RU_{t}SU_{t}^{\ast }$ and of $PU_{t}QU_{t}^{\ast }P$ via their moment sequences and use this relationship to develop a theory of subordination for the boundary values of the Herglotz transform. In particular, we explicitly compute the subordinate function and extend its inverse continuously to the unit circle. As an application, we prove the identity $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q))=-\unicode[STIX]{x1D712}_{\text{orb}}(P,Q)$.

Monotone methods and attractivity results for Volterra integro-partial differential equations

1981 ◽  
Vol 89 (1-2) ◽  
pp. 135-142 ◽  
Author(s):  
Andrea Schiaffino ◽  
Alberto Tesei
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Past History ◽  
Main Tool ◽  
Parabolic Type ◽  
Supremum Norm ◽  
Partial Differential ◽  
Diffusion Effects ◽  
Monotone Methods ◽  
Globally Attractive

SynopsisA Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

scholarly journals A New Proof of Central Limit Theorem for i.i.d. Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Zhaojun Zong ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Main Tool ◽  
Fundamental Result ◽  
Solution Theory ◽  
Independent Identically Distributed

Central limit theorem (CLT) has long and widely been known as a fundamental result in probability theory. In this note, we give a new proof of CLT for independent identically distributed (i.i.d.) random variables. Our main tool is the viscosity solution theory of partial differential equation (PDE).

scholarly journals Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates

2017 ◽  
Vol 27 (14) ◽  
pp. 2781-2802 ◽  
Author(s):  
Annalisa Buffa ◽  
Carlotta Giannelli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Convergence Rates ◽  
Main Tool ◽  
Adaptive Methods ◽  
Interpolation Operator ◽  
B Splines ◽  
Optimal Convergence Rates ◽  
Error Indicators ◽  
Suitable Approximation

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and continue the study of its numerical properties. We prove that our AIGM is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class. The main tool we use is the theory of adaptive methods, together with a local upper bound for the residual error indicators based on suitable properties of a well selected quasi-interpolation operator on hierarchical spline spaces.

scholarly journals Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1483
Author(s):  
Alexander Churkin ◽  
Stephanie Lewkiewicz ◽  
Vladimir Reinharz ◽  
Harel Dahari ◽  
Danny Barash
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Viral Hepatitis ◽  
Antiviral Treatment ◽  
Boundary Values ◽  
Partial Differential ◽  
Differential Equation Models ◽  
Age Structured

Parameter estimation in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation that addresses all cases of data points available presents a formidable challenge and efficiency considerations need to be employed in order for the method to become practical. In the case of age-structured models of viral hepatitis dynamics under antiviral treatment that deal with partial differential equations, a fully numerical parameter estimation method was developed that does not require an analytical approximation of the solution to the multiscale model equations, avoiding the necessity to derive the long-term approximation for each model. However, the method is considerably slow because of precision problems in estimating derivatives with respect to the parameters near their boundary values, making it almost impractical for general use. In order to overcome this limitation, two steps have been taken that significantly reduce the running time by orders of magnitude and thereby lead to a practical method. First, constrained optimization is used, letting the user add constraints relating to the boundary values of each parameter before the method is executed. Second, optimization is performed by derivative-free methods, eliminating the need to evaluate expensive numerical derivative approximations. The newly efficient methods that were developed as a result of the above approach are described for hepatitis C virus kinetic models during antiviral therapy. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for both the ordinary and partial differential equation models.

scholarly journals An Integral Transform Solution of the Differential Equation for the Transverse Motion of an Elastic Beam

1958 ◽  
Vol 11 (2) ◽  
pp. 87-93 ◽  
Author(s):  
J. Fulton
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Wide Class ◽  
Integral Transform ◽  
Classical Method ◽  
Elastic Beam ◽  
Integral Transforms ◽  
Transverse Motion ◽  
Boundary Values

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution.It is the purpose of this note to apply this technique to obtain the solution of the differential equation associated with the transverse motion of an elastic beam for a wide class of boundary conditions.

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