On the Approximate Periodicity of Sequences Attached to Non-Crystallographic Root Systems

We study Fomin-Zelevinsky's mutation rule in the context of noncrystallographic root systems. In particular, we construct approximately periodic sequences of real numbers for the noncrystallographic root systems of rank 2 by adjusting the exchange relation for cluster algebras. Moreover, we describe matrix mutation classes for type H3 and H4

Time Quasilattices in Dissipative Dynamical Systems

We establish the existence of `time quasilattices' as stable trajectories in dissipative dynamical systems. These tilings of the time axis, with two unit cells of different durations, can be generated as cuts through a periodic lattice spanned by two orthogonal directions of time. We show that there are precisely two admissible time quasilattices, which we term the infinite Pell and Clapeyron words, reached by a generalization of the period-doubling cascade. Finite Pell and Clapeyron words of increasing length provide systematic periodic approximations to time quasilattices which can be verified experimentally. The results apply to all systems featuring the universal sequence of periodic windows. We provide examples of discrete-time maps, and periodically-driven continuous-time dynamical systems. We identify quantum many-body systems in which time quasilattices develop rigidity via the interaction of many degrees of freedom, thus constituting dissipative discrete `time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333. Submission to SciPos

Dualities and non-Abelian mechanics

Dualities are mathematical mappings that reveal unexpected links between apparently unrelated systems or quantities in virtually every branch of physics. Systems that are mapped onto themselves by a duality transformation are called self-dual and they often exhibit remarkable properties, as exemplified by an Ising magnet at the critical point. In this Letter, we unveil the role of dualities in mechanics by considering a family of so-called twisted Kagome lattices. These are reconfigurable structures that can change shape thanks to a collapse mechanism easily illustrated using LEGO. Surprisingly, pairs of distinct configurations along the mechanism exhibit the same spectrum of vibrational modes. We show that this puzzling property arises from the existence of a duality transformation between pairs of configurations on either side of a mechanical critical point. This critical point corresponds to a self-dual structure whose vibrational spectrum is two-fold degenerate over the entire Brillouin zone. The two-fold degeneracy originates from a general version of Kramers theorem that applies to classical waves in addition to quantum systems with fermionic time-reversal invariance. We show that the vibrational modes of the self-dual mechanical systems exhibit non-Abelian geometric phases that affect the semi-classical propagation of wave packets. Our results apply to linear systems beyond mechanics and illustrate how dualities can be harnessed to design metamaterials with anomalous symmetries and non-commuting responses.Comment: See http://home.uchicago.edu/~vitelli/videos.html for Supplementary Movi

Minkowski decompositions for generalized associahedra of acyclic type

We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the F-polynomials in terms of subword complexes as conjectured by S. Brodsky and the third author. We then show that the cluster complex is combinatorially isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by D. Speyer and L. Williams.Comment: 17 pages. v2: updated and extended examples, added footnote that Theorem 1.4 also follows from [AHL20, Theorems 4.1 & 4.2

Gitter und Anwendungen

The meeting focussed on lattices and their applications in mathematics and information technology. The research interests of the participants varied from engineering sciences, algebraic and analytic number theory, coding theory, algebraic geometry to name only a few

Crystallographic study on oligonucleotide coiled-coils

En la presente tesis doctoral se han realizado estudios estructurales de DNA. Estudios previos han demostrado que los coiled-coils de d(ATATATATATAT) y d(ATATATATAT) tienen unos parámetros geométricos muy diferentes. El objetivo de esta tesis es aclarar las propiedades de los coiled-coils.Con esta finalidad se han estudiado por cristalografía de Rayos X oligonucleótidos con diferentes secuencias y con extremos cohesivos que fijen la geometría de los coiled-coils. Se han utilizado oligonucleótidos con la secuencia d(CG)n(AT)m o (AT)m(CG)n y otros semejantes. En la mayor parte de ellos n=1 y m>1, con lo que el extremo cohesivo es normalmente la secuencia CG. La estructura (a una resolución de 3.1 Å) de los cristales generados por la secuencia d(CGATATATATAT) ha sido resuelta y publicada (De Luchi et al., ChemBiochem 2006, 7, 585-587). La estructura es isomorfa con la estructura de d(AT)6 y los enlaces a puente de hidrogeno entre las bases A y T son de tipo Hoogsteen, como en el caso de la secuencia d(ATATAT). Se han obtenido diferentes tipos de coiled-coils y se han estudiado sus características y propiedades.Hemos analizado las propiedades geométricas de los coiled coils: hemos visto que los parámetros que determinan el numero de oligonucleótidos por vuelta son el "kink angle" θ, y el ángulo de torsión τ. Ha sido estudiada la relación entre estos parámetro, en función del numero de oligonucleótidos por vuelta N y el ángulo de inclinación β del coiled-coil.Hemos intentado determinar si la formación de apareamientos de tipo Hoogsteen puede influir en la geometría de los coiled-coils, los resultados sugieren que los puentes de hidrogeno de tipo Hoosteeen favorecen la formación de coiled-coils, mientras los enlaces de tipo Watson-Crick generan mas fácilmente estructuras estándar de DNA pseudocontinuas. La secuencia d(CGATATGCATAT) genera columnas tradicionales de DNA, las bases centrales G y C, apareándose con enlaces Watson-Crick fuerzan las bases ATs a aparearse de la misma manera, generando así una hélice recta pseudocontinua.Como complemento de estos estudios se han obtenido las curvas de fusión de oligonucleótidos ricos en AT, los resultados, que incluyen una formula que permite un calculo aproximado de la temperatura de fusión de secuencias cortas de DNA, han sido publicados (De Luchi et al., Analytical Biochemistry, 2003, 322, 279-282).También he tenido la oportunidad de refinar la estructura de d(TAGG) en complejo con un derivado antraquinonico que ya había sido resuelta en nuestro laboratorio. Inesperadamente, encontramos que en esta estructura no se forman G-cuádruplex como fue descrito en solución por Kettani et al, las moléculas de fármaco, a través de interacciones de stacking forman un retículo de columnas perpendiculares una a la otra, estabilizado en los "crossing points" por las flexibles y cortas moléculas de DNA. La flexibilidad del DNA gracias a sus siete ángulos de torsión, le permite adoptar una conformación tal que se adapta a la estructura creada por las columnas de fármaco. La capacidad de formar diferentes tipos de enlaces a puente de hidrogeno estabiliza su conformación en este caso no canónica. En esta estructura hemos encontrado los siguientes tipos de enlaces a puente de hidrogeno: estándar Watson-Crick, reverse Watson-Crick, interacciones simétricas Guanina-Guanina

Atomic-level Characterization of Fe(001)/MgO(001)/Fe(001) Tunneling Magnetoresistance Structures and Spin-polarized Scanning Tunneling Microscopy

This thesis seeks to understand the Fe-MgO-Fe system through a series of atomic level studies of the topographic, electronic, and magnetic properties of these epitaxial films. This multilayer system is uniquely important because of its huge tunneling magnetoresistance (TMR) arising from spin coherence and strong spin filtering through the structure. MgO-based magnetic tunnel junctions have been actively investigated and are now successfully applied to commercial products such as non-volatile magnetic random access memories and read-write heads for hard disk. However, despite its popularity most work has been done on macroscopic samples and has focused on the device-level performance. Yet very little effort has been devoted towards the understanding at the atomic length scales including the effects of atomic steps and local variation in stoichiometry. The primary goal of this work is to elucidate the interplay between morphology, stoichiometry, local magnetism, and local electronic properties. To this end a multifaceted approach was used involving atomic/magnetic force microscopy (AFM/MFM), scanning tunneling microscopy (STM), scanning tunneling spectroscopy (STS), Auger electron spectroscopy, and low energy electron diffraction (LEED), which were operated in the cleanest possible conditions under an ultra-high vacuum. I linked the morphology directly to the formation of different magnetic domain configurations as a function of growth temperature and film thickness. I also correlated these atomic-level properties to the device-level performance. By investigating the topography and the surface electronic density of states with length scales in the nanometer regime, I found that the films had extremely inhomogeneous surface states. Because the structural defects such as surface steps, deep trenches and grain boundaries, as well as the existence of chemical impurities can perturb the spin-coherent tunneling, our observation of the electronic inhomogeneity can provide a direct clue for explaining the diminished TMR phenomenon on real systems compared to the theoretical expectation, which is one of longstanding problems to achieve high TMR in actual devices. In addition to the Fe/MgO/Fe work, I also demonstrated spin polarized STM which revealed the anti-ferromagnetic spin-structure of single crystal chromium and the magnetic domains structure of permalloy film on silicon oxide

Triplex formation as monitored by EPR spectroscopy and molecular dynamics studies of spin -probe -labeled DNAs

Molecular modeling has proven to be a powerful tool for studying structure and dynamics of biologically important molecules. Since the advent of nitroxide based spin-probes the electron paramagnetic resonance (EPR) study of spin-labeled macromolecules has been able to provide insight into structural and dynamics properties of DNAs, proteins, and related systems. Spin labels have been extensively used to study the dynamics of oligonucleotides. An example of this is 5-membered ring nitroxide 5-(2,2,5,5-tetramethyl-3-ethynylpyrrolidine-1-oxyl)-uridine, which has been previously used in our laboratory to monitor triplex formation. Because of the difficult synthetic steps involved in the synthesis of this particular probe a new spin labeled DNA base, 5-(2,2,6,6-tetramethyl-4-ethynylpiperidyl-3-ene-1-oxyl)-uridine (6sp-uridine) is introduced in the current study. This spin label, 6sp, is readily prepared, in half the number of steps required for the previous one, and yet behaves in a spectroscopically analogous manner to its counterpart. The 6sp has been used here to detect the formation of a triplex DNA and to examine the relative rigidity of triplex DNA as compared to double stranded DNA using circular dichroism and EPR spectroscopy. Their EPR spectra show larger changes in response to differences in the mobility of the oligonucleotides they are attached to.;Extending their use in application to DNAs we have conducted Molecular Dynamics (MD) studies on six different oligonucleotides (ONs) molecules using the suite of programs contained in AMBER 5.0 with the Cornell force field. Quantum mechanical calculations at B3LYP level with standard 6-31G* basis set using Gaussian98 were performed. Together with available crystallographic data for different types of nitroxide molecules (Barone et al., J. Am. Chem. Soc. 1998, 120, 7069--7078), new parameters for NO· and sp-hybridized carbon moieties have been developed for the Cornell force field. MD simulations on single-stranded (ss), double-stranded (ds) and triple-stranded or triplex (tx) spin-probe labeled DNAs along with unmodified analogues have been studied over the course of 1 ns. Structural and conformational properties of DNA molecules are described from the analysis of the trajectories. Dynamics of the spin-label was characterized by correlation time (tau c). Our results indicate slower nitroxide motion associated with tx-DNA rather than ds- and ss-DNAs. The presence of spin labels has a substantial effect on the conformation of ss DNA, while ds- and tx-DNA is not affected by the introduction of labels. We have also shown that the presence of the spin-label has small stabilizing effect on ds and tx DNAs