We present a comprehensive and updated Python-based open software to calculate continuous symmetry measures (CSMs) their related chirality measure (CCM) of molecules across chemistry. These descriptors are used quantify distortion levels molecular structures on scale were proven insightful in numerous studies. The input information includes the coordinates geometry desired cyclic point group (

We examine the number of cycles length k in a permutation, as function on symmetric group. write it explicitly combination characters irreducible representations. This allows to study formation long interchange process, including precise formula for probability that permutation is one cycle at given time t, and estimates cases shorter cycles.

Abstract Quantifying imperfect symmetry of molecules can help explore the sources, roles and extent structural distortion. Based on established methodology continuous chirality measures, we develop a set three-dimensional molecular descriptors to estimate distortion large structures. These geometrical quantify gap between desirable (or chirality) actual one. They are global parameters geometry, intuitively defined, have ability detect even minute changes given molecule across chemistry,...

Les inégalités d'opérateurs de nature géométrique ont été très utiles pour étudier le mélange des marches aléatoires et la mécanique quantique. Nous suggérons une nouvelle approche exhiber ce type en utilisant l'inégalité « pieuvre » (octopus inequality) Caputo, Liggett Richthammer.

Symmetry of proteins, an important source their elegant structure and unique functions, is not as perfect it may seem. In the framework continuous symmetry, in which symmetry no longer a binary yes/no property, such imperfections can be quantified used global descriptor three-dimensional structure. We present improved algorithm for calculating measure proteins that takes into account complete set atoms including all side chains. Our method advantage protein sequence division peptides order...

Soit H un corps gauche de dimension finie sur son centre k. Nous résolvons le Problème Inverse Galois des fractions H(X) l’anneau fonctions polynomiales en la variable X et à coefficients dans H, si k contient ample.

Abstract Inspired by Aldous' conjecture for the spectral gap of interchange process and its recent resolution Caputo, Liggett, Richthammer, we define an associated order on irreducible representations Sn. is equivalent to certain being comparable in this order, hence determining “Aldous order” completely a generalized question. We show few additional entries order.

We use representation theory to write a formula for the magnetisation of quantum Heisenberg ferromagnet. The core new result is spectral decomposition function $\alpha_k 2^{\alpha_1+\dotsb+\alpha_n}$ where $\alpha_k$ number cycles length k permutation. In mean-field case, we simplify further, arriving at closed-form expression magnetisation, which allows analyse phase transition.

We study the stopping time of Collatz map for a polynomial $f \in \mathbb{F}_2[x]$, and bound it by $O({\rm deg} (f)^{1.5})$, improving upon quadratic proven Hicks, Mullen, Yucas Zavislak. also prove existence arithmetic sequences unbounded length in times certain polynomials, phenomenon observed classical map.

Let R be the ring of polynomials in n central variables over real quaternion algebra H, and let I a left ideal R. We prove that if polynomial p vanishes at all common zeros H^n with commuting coordinates, then as slice regular quaternionic function, H^n. This confirms conjecture Gori, Sarfatti Vlacci, who settled two dimensional case.

Following the work of first and last authors [2], we further analyze structure a zero set left ideal in ring central polynomials over quaternion algebra H. We describe "algebraic hull" point H^n prove it is product spheres. Using this description give new proof to conjecture Gori, Sarfatti Vlacci. also show that main result [2] does not extend general division algebras.

We define the notion of a semicharacter group G: A function from to ℂ*, whose restriction any abelian subgroup is homomorphism. conjecture that for finite group, order semicharacters divisible by group. prove holds some important families groups, including Symmetric groups and GL(2, q).

À l'aide de la théorie des représentations, nous donnons une formule pour magnétisation du ferro-aimant Heisenberg quantique. La nouveauté-clé est décomposition spectrale fonction αk2α1+⋯+αn où αk le nombre cycles longueur k d'une permutation. Dans cas à champ moyen, simplifions encore formule, ce qui donne expression fermée permet d'analyser transition phase.

Inspired by Aldous' conjecture for the spectral gap of interchange process and its recent resolution Caputo, Liggett Richthammer, we define an associated order on irreducible representations S_n. is equivalent to certain being comparable in this order, hence determining "Aldous order" completely a generalized question. We show few additional entries order.