A system coupling and Donoghue classes of Herglotz-Nevanlinna functions.

We study the impedance functions of conservative L-systems with unbounded main operators. In addition to the generalized Donoghue class $\mathfrak M_\kappa$ of Herglotz-Nevanlinna functions considered earlier, we introduce ``inverse generalized Donoghue classes $\mathfrak M_\kappa^{-1}$ of functions satisfying a different normalization condition on the generating measure. We establish a connection between ``geometrical properties of two L-systems whose impedance functions belong to the classes $\mathfrak M_\kappa$ and $\mathfrak M_\kappa^{-1}$, respectively. After that we introduce a coupling of two L-system and show that if the impedance functions of two L-systems belong to the generalized Donoghue classes $\mathfrak M_{\kappa_1}$($\mathfrak M_{\kappa_1}^{-1}$) and $\mathfrak M_{\kappa_2}$($\mathfrak M_{\kappa_2}^{-1}$), then the impedance function of the coupling falls into the class $\mathfrak M_{\kappa_1\kappa_2}$. Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class $\mathfrak M=\mathfrak M_0$ is coupled with any other L-system, the impedance function of the coupling belongs to $\mathfrak M$ (the absorbtion property). Observing the result of coupling of $n$ L-systems as $n$ goes to infinity, we put forward the concept of a limit coupling which leads to the notion of the system attractor, two models of which (in the position and momentum representations) are presented. The talk is based on joint work with K.~A.~Makarov and E. Tsekanovski\u i (see the reference below). \begin{itemize} \item[{[1]}] S. Belyi, K. A.~ Makarov, and E.~Tsekanovski\u i: \textit{A system coupling and Donoghue classes of Herglotz-Nevanlinna functions}, Complex Analysis and Operator Theory, (2016), 10 (4), pp. 835-880. \end{itemize

Generalized Choquard equation with potential vanishing at infinity

In this paper we investigate the existence of solution for the following generalized Choquard equation $$-\Delta_{\Phi}u+V(x)\phi(|u|)u=\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^{\lambda}}dy\right)K(x)f(u(x)), \;\;x\in \mathbb{R}^{N}$$ where $N\geq 3$, $\lambda>0$ is a positive parameter, $V,K\in C(\mathbb R^N,[0,\infty))$ are nonnegative functions that may vanish at infinity, the function $f\in C(\mathbb{R}, \mathbb R)$ is quasicritical and $F(t)=\int_{0}^{t}f(s)ds$. This work incorporates the reflexive and non-reflexive cases taking into account from Orlicz-Sobolev framework. The non-reflexive case occurs when the $N$-function $\tilde{\Phi}$ does not verify the $\Delta_{2}$-condition. In order to prove our main results we employ variational methods and regularity results

A Vector-Valued Operational Calculus and Abstract Cauchy Problems.

Initial and boundary value problems for linear differential and integro-differential equations are at the heart of mathematical analysis. About 100 years ago, Oliver Heaviside promoted a set of formal, algebraic rules which allow a complete analysis of a large class of such problems. Although Heaviside\u27s operational calculus was entirely heuristic in nature, it almost always led to correct results. This encouraged many mathematicians to search for a solid mathematical foundation for Heaviside\u27s method, resulting in two competing mathematical theories: (a) Laplace transform theory for functions, distributions and other generalized functions, (b) J. Mikusinski\u27s field of convolution quotients of continuous functions. In this dissertation we will investigate a unifying approach to Heaviside\u27s operational calculus which allows us to extend the method to vector-valued functions. The main components are (a) a new approach to generalized functions, considering them not primarily as functionals on a space of test functions or as convolution quotients in Mikusinski\u27s quotient field, but as limits of continuous functions in appropriate norms, and (b) an asymptotic extension of the classical Laplace transform allowing the transform of functions and generalized functions of arbitrary growth at infinity. The mathematics are based on a careful analysis of the convolution transform $f \to k \star f.$ This is done via a new inversion formula for the Laplace transform, which enables us to extend Titchmarsh\u27s injectivity theorem and Foias\u27 dense range theorem for the convolution transform to Banach space valued functions. The abstract results are applied to abstract Cauchy problems. We indicate the manner in which the operational methods can be employed to obtain existence and uniqueness results for initial value problems for differential equations in Banach spaces

Non-ergodic phases in strongly disordered random regular graphs

We combine numerical diagonalization with a semi-analytical calculations to prove the existence of the intermediate non-ergodic but delocalized phase in the Anderson model on disordered hierarchical lattices. We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson and Thouless recursive scheme. This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered Random Regular Graphs (RRG) of N sites with the connectivity K=2. By extrapolation of the results of both approaches to N->infinity we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamic exponent D(W) with the accuracy sufficient to claim that they are non-trivial in the broad interval of disorder strength W_{E}<W<W_{c}. The thorough analysis of the exact diagonalization results for RRG with N>10^{5} reveals a singularity in D_{1,2}(W)-dependencies which provides a clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}\approx 10.0. We discuss the implications of these results for quantum and classical non-integrable and many-body systems.Comment: 4 pages paper with 5 figures + Supplementary Material with 5 figure

Two-point coordinate rings for GK-curves

Giulietti and Korchm\'aros presented new curves with the maximal number of points over a field of size q^6. Garcia, G\"uneri, and Stichtenoth extended the construction to curves that are maximal over fields of size q^2n, for odd n >= 3. The generalized GK-curves have affine equations x^q+x = y^{q+1} and y^{q^2}-y^q = z^r, for r=(q^n+1)/(q+1). We give a new proof for the maximality of the generalized GK-curves and we outline methods to efficiently obtain their two-point coordinate ring.Comment: 16 page