Conformal field theory complexity from Euler-Arnold equations

Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.

Fractional Differential Generalization of the Single Particle Model of a Lithium-Ion Cell

The effect of anomalous diffusion of lithium on the discharge curves and impedance spectra of lithium-ion batteries (LIB) is studied within the fractional differential generalization of the single-particle model. The distribution of lithium ions in electrolyte and electrode particles is expressed through the Mittag–Leffler function and the Lévy stable density. Using the new model, we generalize the equivalent circuit of LIB. The slope of the low-frequency rectilinear part of the Nyquist diagram does not always unambiguously determine the subdiffusion index and can be either larger or smaller than the slope corresponding to normal diffusion. The new aspect of capacity degradation related to a change in the type of ion diffusion in LIB components is discussed.

On time-fractional representation of an open system response

The article concerns dynamics of an open system (OS) considered as a subsystem of some closed Hamiltonian system. Description of such OS is carried out in terms of integro-differential generalization of the Liouville equation. The Mellin transform allows to present this equation in terms of fractional operators. In frame of these concepts the response problem is formulated and applied to the supercapacitor dynamics. A good agreement with experimental data is found.

Lexical Organization and Phonological Change in Treatment

Word frequency and neighborhood density are properties of lexical organization that differentially influence spoken-word recognition. This study examined whether these same properties also affect spoken-word production, particularly as related to children with functional phonological delays. The hypothesis was that differential generalization would be associated with a word's frequency and its neighborhood density when manipulated as input in phonological treatment. Using a multiple baseline across subjects design, 8 children (aged 3;10 to 5;4) were randomly enrolled in 1 of 4 experimental conditions targeting errored sounds in high-frequency, low-frequency, high-density, or low-density words. Dependent measures were generalization of treated sounds and untreated sounds within and across manner classes as measured during and following treatment. Results supported a hierarchy of phonological generalization by experimental condition. The clinical implications lie in planning for generalization through the input presented in treatment. Theoretically, the results demonstrate that lexical organization of words in the mental lexicon interacts with phonological structure in learning.

Syllable Onsets II

This study extends the application of the Sonority Sequencing Principle, as reported in J. A. Gierut (1999), to acquisition of word-initial 3-element clusters by children with functional phonological delays (ages in years;months: 3;4 to 6;3). The representational structure of 3-element clusters is complex and unusual because it consists of an s-adjunct plus a branching onset, which respectively violate and conform to the Sonority Sequencing Principle. Given the representational asymmetry, it is unclear how children might learn these clusters in treatment or whether such treatment may even be effective. Results of a single-subject staggered multiple-baseline experiment demonstrated that children learned the treated 3-element cluster in treatment but showed no further generalization to similar types of (asymmetric) onsets. Treatment of 3-element clusters did, however, result in widespread generalization to untreated singletons, including affricates. Moreover, there was differential generalization to untreated 2-element clusters, with individual differences being traced to the composition of children’s singleton inventories. Theoretically, the results suggest a segmental-syllabic interface that holds predictive potential for determining the effectiveness and effects of clinical treatment as based on the notion of linguistic complexity.

SU(N) LIE ALGEBRA, EXTENDED TODA LATTICE EQUATION AND THE EXACT SOLUTIONS

An integro-differential generalization of the Toda lattice equation is proposed via the zero curvature equation belonging to SU(N) Lie algebra. It is shown that the exact solutions for this equation can be constructed by the method of chiral vectors. Explicit results are given for SU(2) and SU(3). We also demonstrate that these equations are connected to the constrained WZW theory and hence Polyakov’s two-dimensional gravity.