Orthogonal Polynomial Representation of Imaginary-Time Green's Functions

We study the expansion of single-particle and two-particle imaginary-time Matsubara Green's functions of quantum impurity models in the basis of Legendre orthogonal polynomials. We discuss various applications within the dynamical mean-field theory (DMFT) framework. The method provides a more compact representation of the Green's functions than standard Matsubara frequencies and therefore significantly reduces the memory-storage size of these quantities. Moreover, it can be used as an efficient noise filter for various physical quantities within the continuous-time quantum Monte Carlo impurity solvers recently developed for DMFT and its extensions. In particular, we show how to use it for the computation of energies in the context of realistic DMFT calculations in combination with the local density approximation to the density functional theory (LDA+DMFT) and for the calculation of lattice susceptibilities from the local irreducible vertex function.Comment: 14 pages, 11 figure

Transformed Schatten-1 Iterative Thresholding Algorithms for Low Rank Matrix Completion

We study a non-convex low-rank promoting penalty function, the transformed Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a. We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-of-the-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions with ground truth matrix being multivariate Gaussian at varying covariance. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-q which is an iterative reweighted algorithm, non-convex in nature and best in performance

Breaking so(4) symmetry without degeneracy lift

We argue that in the quantum motion of a scalar particle of mass "m" on S^3_R perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, \ell, the 3D orbital angular momentum, and another, an external scale introducing continuous parameter, B, a loss of the geometric hyper-spherical so(4) symmetry of the free motion can occur that leaves intact the unperturbed {\mathcal N}^2-fold degeneracy patterns, with {\mathcal N}=(\ell +n+1) and n denoting the nodes number of the wave function. Our point is that although the number of degenerate states for any {\mathcal N} matches dimensionality of an irreducible so(4) representation space, the corresponding set of wave functions do not transform irreducibly under any so(4). Indeed, in expanding the Scarf I wave functions in the basis of properly identified so(4) representation functions, we find power series in the perturbation parameter, B, where 4D angular momenta K\in [\ell , {\mathcal N}-1] contribute up to the order \left(\frac{2mR^2B}{\hbar^2}\right)^{{\mathcal N}-1-K}. In this fashion, we work out an explicit example on a symmetry breakdown by external scales that retains the degeneracy. The scheme extends to so(d+2) for any d.Comment: Prepared for the proceedings of the conference "Lie Theory and Its Applications In Physics", June 17-23, 2013, Varna, Bulgari

Consistent maps and their associated representation theorems

A 2009 article of Allcock and Vaaler examined the vector space $\mathcal G := \overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ over $\mathbb Q$, describing its completion with respect to the Weil height as a certain $L^1$ space. By involving an object called a consistent map, the author began efforts to establish representation theorems for the duals of spaces related to $\mathcal G$. Specifically, we provided representation theorems for the algebraic and continuous duals of $\overline{\mathbb Q}^\times/{\overline{\mathbb Z}}^\times$. We explore further applications of consistent maps to provide representation theorems for duals of other spaces arising in the work of Allcock and Vaaler. In particular, we establish a connection between consistent maps and locally constant functions on the space of places of $\overline{\mathbb Q}$. We further apply our new results to recover, as a corollary, a main theorem of our previous work

Dynamics of the solutions of the water hammer equations

NOTICE: this is the author’s version of a work that was accepted for publication in Topology and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Topology and its Applications, [Volume 203, 15 April 2016, Pages 67-83] DOI10.1016/j.topol.2015.12.076¨[EN] In this note we provide a representation of the solution using an operator theoretical approach based on the theory of C-0-semigroups and cosine operator functions, when considering this phenomenon on a compressible fluid along an infinite pipe. We provide an integro-differential equation that represents this phenomenon and it only involves the discharge. In addition, the representation of the solution in terms of a specific C-0-semigroup lets us show that hypercyclicity and the topologically mixing property can occur when considering this phenomenon on certain weighted spaces of integrable and continuous functions on the real line. (C) 2016 Elsevier B.V. All rights reserved.The first and third authors are supported by MEC Projects MTM2010-14909 and MTM2013-47093-P. The first author is also supported by Programa de Investigación y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by CONICYT, under Fondecyt Grant number 1140258.Conejero, JA.; Lizama, C.; Ródenas Escribá, FDA. (2016). Dynamics of the solutions of the water hammer equations. Topology and its Applications. 203:67-83. https://doi.org/10.1016/j.topol.2015.12.076S678320

Normally preordered spaces and continuous multi-utilities

[EN] We study regular, normal and perfectly normal preorders by referring to suitable assumptions concerning the preorder and the topology of the space. We also present conditions for the existence of a countable continuous multi-utility representation, hence a Richter-Peleg multi-utility representation, by assuming the existence of a countable net weight.This research was carried out within the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni, Istituto Nazionale di Alta Matematica (Italy).Bosi, G.; Caterino, A.; Ceppitelli, R. (2016). Normally preordered spaces and continuous multi-utilities. Applied General Topology. 17(1):71-81. doi:10.4995/agt.2016.4561.SWORD7181171Alcantud, J. C. R., Bosi, G., & Zuanon, M. (2015). Richter–Peleg multi-utility representations of preorders. Theory and Decision, 80(3), 443-450. doi:10.1007/s11238-015-9506-zAlcantud, J. C. R., Bosi, G., Campión, M. J., Candeal, J. C., Induráin, E., & Rodríguez-Palmero, C. (2007). Continuous Utility Functions Through Scales. Theory and Decision, 64(4), 479-494. doi:10.1007/s11238-007-9025-7Bosi, G., & Herden, G. (2012). Continuous multi-utility representations of preorders. Journal of Mathematical Economics, 48(4), 212-218. doi:10.1016/j.jmateco.2012.05.001Bosi, G., & Herden, G. (2016). On continuous multi-utility representations of semi-closed and closed preorders. Mathematical Social Sciences, 79, 20-29. doi:10.1016/j.mathsocsci.2015.10.006Bosi, G., & Isler, R. (2000). Separation axioms in topological preordered spaces and the existence of continuous order-preserving functions. Applied General Topology, 1(1), 93. doi:10.4995/agt.2000.3026Bosi, G., & Mehta, G. B. (2002). Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. Journal of Mathematical Economics, 38(3), 311-328. doi:10.1016/s0304-4068(02)00058-7Caterino, A., & Ceppitelli, R. (2015). Jointly continuous utility functions on submetrizable kω-spaces. Topology and its Applications, 190, 109-118. doi:10.1016/j.topol.2015.04.012Caterino, A., Ceppitelli, R., & Holá, Ľ. (2013). Some generalizations of Backʼs Theorem. Topology and its Applications, 160(18), 2386-2395. doi:10.1016/j.topol.2013.07.033Caterino, A., Ceppitelli, R., & Maccarino, F. (2009). Continuous utility functions on submetrizable hemicompact k-spaces. Applied General Topology, 10(2), 187-195. doi:10.4995/agt.2009.1732Evren, Ö., & Ok, E. A. (2011). On the multi-utility representation of preference relations. Journal of Mathematical Economics, 47(4-5), 554-563. doi:10.1016/j.jmateco.2011.07.003K�nzi, H.-P. A. (1990). Completely regular ordered spaces. Order, 7(3), 283-293. doi:10.1007/bf00418656Minguzzi, E. (2011). Normally Preordered Spaces and Utilities. Order, 30(1), 137-150. doi:10.1007/s11083-011-9230-