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The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points…
Free coherent states for a system with p degrees of freedom are defined. An isomorphism of the space of free coherent states to the space of distributions on p-adic disk is constructed.
Entropies associated with spatial subsystems in conventional local quantum field theories are typically divergent when the spatial regions have boundaries. However, in certain linear combinations of the entropies for various subsystems,…
We develop an approach to investigate the non-perturbative dynamics of quantum field theories, in which specific vacuum field fluctuations are treated as the low-energy dynamical degrees of freedom, while all other vacuum field…
Recently, a couple of investigations related to symmetry breaking phenomena, 'spontaneous stochasticity' and 'ergodicity breaking' have led to significant impacts in a variety of fields related to the stochastic processes such as economics…
We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces, give several techniques for computing lower bounds for it, and show that it is equal to a limit of…
We consider coupling an ordinary quantum field theory with an infinite number of degrees of freedom to a topological field theory. On R^d the new theory differs from the original one by the spectrum of operators. Sometimes the local…
A challenging problem is to find an algorithm to decide whether a morphism is k-power-free. We provide such an algorithm when k >= 3 for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite…
Understanding the dependence of entanglement entropy on the renormalized mass in quantum field theories can provide insight into phenomena such as quantum phase transitions, since the mass varies in a singular way near the transition. Here…
We pedagogically present the information theory as originally established, explaining its essential ideas and paying attention to the expression employed to measure the amount of information. Also we discussed relationships between…
The role of symmetries in formation of quantum dynamics is discussed. A quantum version of the d'Alambert's principle is proposed to take into account symmetry constrains for quantum case. It is noted that in this approach one can find, in…
Using a one-dimensional macromolecule in aqueous solution as an illustration, we demonstrate that the relative entropy from information theory, $\sum_k p_k\ln(p_k/p_k^*)$, has a natural role in the energetics of equilibrium and…
We derive and classify properties of the power-spectrum tensor for systems in general steady-states, including stationary states not necessarily corresponding to equilibrium configurations. We establish a rigorous connection between the…
We study the correlation dynamics of a system composed of arbitrary numbers of qutrits interacting with a common environment. Initially, the system is assumed to be in a low dimensional subspace of the Hamiltonian called "decoherence-free…
We study the entanglement entropy within a spherical region for a free scalar field in a squeezed state in 3+1 dimensions. We show that, even for small squeezing, a volume term appears, whose coefficient is essentially independent of the…
Standard entropy calculations in quantum field theory, when applied to a subsystem of definite volume, exhibit area-dependent UV divergences that make a thermodynamic interpretation troublesome. In this paper we define a renormalized…
Dynamical systems theory is especially well-suited for determining the possible asymptotic states (at both early and late times) of cosmological models, particularly when the governing equations are a finite system of autonomous ordinary…
Inspired by Katok's intermediate entropy property [Inst. Hautes \'Etudes Sci. Publ. Math. 51 (1980), 137-173], we introduce and study the notion of entropy flexibility for discrete-time and continuous-time dynamical systems. By using…
Notions of topological free entropy and of free capacity are introduced in the $C^*$-algebra context. Basic properties, basic problems and connections to potential theory and random matrix theory are discussed.
We propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define…