text:
## SFB-Seminar on Supersymmetries

Next seminars:

Speaker Date Room Title
Dominik Ostermayr (Universität zu Köln) Monday, 13.07.2015,
14:15 - 16:00
NA 4/64 Automorphism Supergroups of Supermanifolds
Abstract.

A classical theorem states that the group of automorphisms of a manifold M preserving a G-structure of finite type is a Lie group. We show how to generalize this statement to the category of cs manifolds and give some examples, some of which being generalizations of classical notions, others being particular to the super case. Notably, we have to introduce a new notion of supermanifolds which we call mixed supermanifolds.

## Archive

Speaker Date Room Title
Tilmann Wurzbacher (RUB) Wednesday, 04.03.2015,
10:15 - 12:00
NA 4/24 On orbits of Lie supergroups
Abstract.

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points'. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy supergroups and orbits at general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov-Kostant- Souriau type. Applying a generalisation of Kirillov's orbit method to the parity changed variants of the Heisenberg group, the Clifford supergroup and the odd Heisenberg supergroup, we obtain universal' families of representations parametrised by a supermanifold.

Speaker Date Room Title
Ricardo Kennedy (Universität zu Köln) Friday, 14.11.2014,
10:15 - 11:45
NA 4/64 Homotopy theory of topological insulators
Speaker Date Room Title
Hermann Schulz-Baldes (FAU Erlangen-Nürnberg) Thursday, 26.06.2014,
13:00 - 14:30
NA 4/64 Topological insulators and index theory
Abstract.

According to a widely accepted terminology, a topological insulator is a (independent) Fermion system which has surface modes that are not exposed to Anderson localization. This stability results from topological constraints given by non-trivial invariants like non-commutative Chern numbers and higher winding numbers, but sometimes also more subtle $Z_2$ invariants associated to adequate Fredholm operators with symmetries. Prime examples are quantum Hall systems, but the talk also considers chiral and BdG systems as well as time-reversal symmetric systems with $Z_2$ invariants.

Speaker Date Room Title
David Kyed (Odense) Friday, 25.04.2014,
10:15
NA 4/64 L2-Betti numbers for locally compact groups
Abstract.

In a recent joint work with Petersen and Vaes, we prove that the L2-Betti numbers of a locally compact group coincide, up to a natural scaling constant, with the L2-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of it. As a consequence, we obtain that the L2-Betti numbers of any lattice equal the ones for the ambient group scaled with the covolume of the lattice in question. Moreover, several vanishing results are derived from this, including the vanishing of the reduced L2-cohomology for amenable locally compact groups. In my talk I will elaborate on these results and survey the theory of L2-Betti numbers.

Speaker Date Room Title
Michel Egeileh (American Univ. Beirut/Lebanon) Friday, 10.01.2014,
10:15
NA 4/64 Smoothness of maps and of spaces in infinite dimensions
Abstract.

In his thesis, Adrien Douady introduced the notion of (infinite-dimensional) Banach analytic spaces using the language of "functored spaces". On the other hand, Pierre Mazet defined a notion of analytic spaces in the context of locally convex spaces. We discuss the second approach and show, by a theorem of Douady, that the structure of infinite-dimensional analytic space is very weak, allowing for pathological phenomena. In particular, there exists maps defined on analytic spaces which are "weakly analytic" but not analytic. Finally, we present a new result showing that, for an appropriate category of locally convex spaces (including Fréchet spaces), smoothness of maps is equivalent to their "scalarwise smoothness" and to their "weak smoothness". This allows the definition of a large class of infinite-dimensional manifolds via locally-ringed spaces.

Speaker Date Room Title
Mauro Spera (Brescia) Friday, 20.09.2013,
10:15
NA 4/64 On geometric and physical aspects of some unitary Riemann surface braid group representations
Abstract.

In this talk we discuss the simplest unitary Riemann surface braid group representations, which are constructed geometrically via representations of the Weyl-Heisenberg group- by means of stable bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail with the help of noncommutative geometric tools, and, time permitting, an application of Fourier-Mukai techniques is given as well.

Speaker Date Room Title
Christian Becker (Universität Potsdam) Tuesday, 16.07.2013,
10:15 h
NA 2/24 Introduction to differential cohomology
Abstract.

Motivated by the construction of the Chern-Simons invariants of closed 3- manifolds, Cheeger and Simons in the late 1970’s introduced the notion of differential characters. From a topological point of view, the ring of differen- tial characters is a refinement of integral cohomology by differential forms. Differential characters provide a particular model for what is now called differential cohomology. From a geometric point of view, differential char- acters are higher degree analogues of (isomorphism classes of) hermitean line bundles with connection. They are used in mathematical physics to describe higher degree analogues of gauge fields and field strengths for the gauge group U(1). Elementary concepts available for complex line bun- dles with connection, like holonomy along closed paths, parallel transport, (nonvanishing) sections and their covariant derivatives, have counterparts for higher degree differential cohomology classes. In the talk I will consider differential cohomology from a topological and a geometric perspective. I will focus on differential characters as a model for differential cohomology. I will also explain the similarities and differences between differential cohomology and singular cohomology.

Speaker Date Room Title
Christian Becker (Universität Potsdam) Tuesday, 16.07.2013,
16:15 h
NA 1/58 Differential cohomology, stratifolds and the Cheeger-Chern-Simons construction
Abstract.

Differential extensions of cohomology theories have appeared in several con- texts in topology, geometry and mathematical physics. For example, com- plex line bundles with connection are classified by the differential cohomol- ogy group of degree 2. This defines a U(1) gauge theory wich classically models electromagnetic potentials and fields. Complex line bundles are also the starting point of geometric quantization. Higher differential cohomology groups classify gerbes and higher form fields used in String theory. In the same way as line bundles allow to compute holonomy along closed curves, higher differential cohomology classes have holonomies along higher dimensional closed oriented manifolds. More generally, we may re- place smooth manifolds by stratifolds. This allows a clear geometric ap- proach to basic constructions for differential cohomology like (uniqueness of) the ring structure and fiber integration. In the talk I will introduce differential cohomology, mainly by the spe- cific model of differential characters. I will explain the identification of low degree differential cohomology groups with well-known objects from geometry, topology and mathematical physics, including the relation to Chern-Simons forms and Chern-Weil theory. Finally I will explain the use of stratifolds to geometrically construct the ring structure, fiber integration and transgression to loop space.

Am 28.06.2013 findet das "SFB-Seminar on symmetries" nicht statt, aufgrund des am gleichen Tage in Wuppertal stattfindenden Seminares "Complex algebraic geometry and complex analysis" (gemeinsam mit der Universität Wuppertal).

Am 21.06.2013 findet das "SFB-Seminar on symmetries" nicht statt, aufgrund eins SFB TR 12 Workshops in Bad Honnef (20.06.2013 bis 23.06.2013).

Speaker Date Room Title
Wolfgang Palzer (Universität Köln) Friday, 14.06.2013,
10:15 h
NA 4/64 The Fourier Transformation on Symmetric Superspaces – The Rank One Case
Abstract.

In order to have a useful Fourier transform on symmetric superspaces, we need to understand the asymptotic behaviour of certain spherical functions, which are eigenfunctions of the Laplacian. In studying their asymptotic behaviour, we are led to the so-called Harish-Chandra c-function - a necessary ingredient to define the inverse Fourier transform. The proof of the inversion formula involves coordinate transformations resulting in boundary terms, which have to be dealt with appropriately.

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Speaker Date Room Title
Zain Shaikh (Universität Köln) Friday, 17.05.2013,
10:15 h
NA 4/64 Superbosonization and the Laplace transform of conical superfunctions
Abstract.

The superbosonisation formula of Littelmann-Sommers-Zirnbauer relates a flat Berezin integral over a linear superspace (LHS) to an integral over a symmetric superspace Omega (RHS). The LHS can be interpreted as averages of observables in condensed matter physics, and the importance of the formula stems from the fact that the RHS is well-suited to calculate asymptotics of such averages. Alexander Alldridge and I have given a short conceptual proof of this formula by developing a theory of Laplace transforms on the relevant superspaces. The Laplace transform of the LHS is straightforward, whereas the RHS is more intricate. The real work is to define and calculate the Laplace transform of conical superfunctions over Omega. This leads to a function that is analogous to the Gindikin Gamma-function in the purely even case, which will have consequences for analytic representations of the Lie supergroups in the picture. In this talk, I will discuss this proof, define the conical superfunctions, give some indication of how to calculate their Laplace transform, and (if time permits) talk about the representation theory. Am 26.04.2013 fand das "SFB-Seminar on symmetries" nicht statt, aufgrund des am gleichen Tage in Bochum stattfindenden Seminares "Complex algebraic geometry and complex analysis" (gemeinsam mit der Universität Wuppertal).

Speaker Date Room Title
Natalie Schieferstein (Ruhr-Universität Bochum) Friday, 03.05.2013,
10:15 h
NA 4/64 Elliptische Integrale, der elliptische Sinus und meromorphe Funktionen auf komplex-eindimensionalen Tori (Präsentation BSc-Arbeit).
Speaker Date Room Title
Andreas Osterloh (Universität Duisburg-Essen) Friday, 19.04.2013,
10:15 h
NA 4/64 Invariant theory and entanglement: a connection of the polynomial degree of an invariant and the length of the state
Abstract.

Entanglement is a phenomenon that is invariant with respect to local unitaries, or even more, the local SL transformations. The theory of multipartite entanglement for qubits is therefore well described by the Cayley-Hilbert omega-process and singlet contractions. Here I show how it is correlated to the antilinear expectation value of an SL invariant operator. Consequences of this is a balancedness condition that points towards the length of the state, i.e. the way it is decomposed into product states of some local basis.

Speaker Date Room Title
Benjamin Schwarz (Universität Paderborn) Friday, 12.04.2013,
10:15 h
NA 4/64 Nearly holomorphic sections on Kähler manifolds
Abstract.

We describe the notion of nearly holomorphic functions on Kähler manifolds which was first introduced by G. Shimura. This notion extends to vector bundles and corresponding nearly holomorphic sections. We show that if $X$ is a compact Hermitian symmetric manifold, and $E \to X$ is anirreducible homogeneous Hermitian vector bundle, then the space of nearly holomorphic sections is dense in the space of $L^2$-sections. Finally, we give some representation theoretic applications of this result.

Speaker Date Room Title
Nathan Grieve (Queen's University, Canada) Tuesday, 09.04.2013,
14:15 h
NA 2/24 Index conditions, cup-products and number fields
Abstract.

We will discuss a cohomological condition which we place on a pair of vector bundles on an abelian variety. This condition is motivated on several fronts, notably D. Mumford's index theorem, as well as my own results concerning cup-products. I will explain these results, consider some examples, and show that this condition can be satisfied, in all possible instances, by a large class of abelian varieties. To see why, we will consider this condition as it applies to abelian varieties with real multiplication.

Speaker Date Room Title
Hannah Bergner (Ruhr-Universität Bochum) Tuesday, 12.02.2013,
14:15 h
NA 4/64 Lie superalgebras of vector fields and local actions
Speaker Date Room Title
Björn Villa (Ruhr-Universität Bochum) Friday, 01.02.2013,
12:15 h
NA 4/64 Kählerian structures on coadjoint orbits
Speaker Date Room Title
Florian Hanisch (Universität Potsdam) Friday, 18.01.2013,
10:15 h
NA 4/64 Variational problems on supermanifolds and applications to supergeodesics
Speaker Date Room Title
Christopher Graw (Ruhr-Universität Bochum) Friday, 11.01.2013,
10:15 h
NA 4/64 Real supergroup orbits in complex Supergrassmannians
Speaker Date Room Title
Alexander Kahle (MPIM Bonn) Tuesday, 18.12.2012,
16:15 h
NA 1/58 Supersymmetric field theories and differential K-theory
Speaker Date Room Title
Majdi BEN HALIMA (Sfax, Tunisia) Friday, 07.12.2012,
10:15 h
NA 4/64 On sums of coadjoint orbits
Abstract.

For compact connected Lie groups, a connection between the restriction of irreducible representations and the projection of coadjoint orbits has been discovered by Heckman, and then generalized by Guillemin and Sternberg in the framework of symplectic geometry. This provides in particular a connection between sums of coadjoint orbits of a compact connected Lie group G and tensor products of irreducible representations of G. In the first half of this talk, we explain applications of the latter result to certain matrix spectral problems. In the second half, using elementary representation-theoretic arguments, we prove an extension of the above-mentioned result of Heckman to all irreducible finite-dimensional representations of a compact semisimple Lie group.

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Speaker Date Room Title
Prof. Dr. Alexander Isaev (Canberra/Australia) Tuesday, 27.11.2012,
14:15 h
NA 1/58 Explicit characterization of isolated hypersurface singularities by their moduli algebras
Abstract.

The Mather-Yau theorem states that a germ of a hypersurface in complex space is completely characterized by a certain commutative algebra, the Tjurina algebra. The proof of the Mather-Yau theorem does not provide an explicit procedure for recovering the germ from the algebra, and finding a way of doing it is an interesting open problem. In this talk I will concentrate on the case of homogeneous singularities. One way to make the Mather-Yau theorem explicit in this case is to utilize classical invariant theory. This approach leads to an intriguing conjecture concerning classical invariants, and I will discuss progress on the conjecture to date. The talk will mainly involve elementary concepts from commutative algebra and classical invariant theory and should be suitable for a broad mathematical audience.

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Speaker Date Room Title
Roman Avdeev (Moscow) Friday, 23.11.2012,
10:15 h
NA 4/64 Spherical homogeneous spaces and harmonic analysis on them (II)
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Speaker Date Room Title
Roman Avdeev (Moscow) Friday, 16.11.2012,
10:15 h
NA 4/64 Spherical homogeneous spaces and harmonic analysis on them (I)
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Speaker Date Room Title
Prof. Dr. Bruce Gilligan (Regina/Canada) Friday, 09.11.2012,
10:15 h
NA 4/64 Holomorphic reductions of pseudoconvex homogeneous manifolds
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Speaker Date Room Title
Stéphane Garnier (Ruhr-Universität) Friday, 02.11.2012,
10:15 h
NA 4/64 Vector fields and metrics on supermanifolds
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Speaker Date Room Title
Stéphane Garnier (Ruhr-Universität) Friday, 19.10.2012,
14:15 h
NA 4/64 Generalized cohomology theories and spectra (Intro)
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Speaker Date Room Title
Aprameyan Parthasarathy (Universität Marburg) Friday, 29.06.2012,
10:15 h
NA 4/64 A fixed point formula on the Oshima compactification of a Riemannian symmetric space
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Speaker Date Room Title
Ricardo Kennedy Friday, 18.05.2012,
10:15 h
NA 4/64 Homotopy classification of topological insulators II
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Speaker Date Room Title
Hannah Bergner Friday, 11.05.2012,
10:15 h
NA 4/64 Palais' Konstruktion lokaler Lie Gruppenwirkungen
Abstract.

In dem Vortag werde ich die in "Global formulation of the Lie theory of transformation groups" von Palais beschriebene Konstruktion erkären, welche zeigt, dass zu jeder endlich-dimensionalen Lie Unteralgebra der Vektorfelder auf einer Mannigfaltigkeit eine lokale Lie Gruppenwirkung existiert, deren infinitesimale Wirkung genau der Wirkung der Lie Unteralgebra auf der Mannigfaltigkeit entspricht.

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Speaker Date Room Title
Arturo Prat-Waldron Friday, 04.05.2012,
10:15 h
NA 4/64 On the Construction of the Dirac Operator of a Spin Manifold via Geometric Quantization
Abstract.

In this talk we revisit the geometric quantization of a certain symplectic supermanifold obtained from considering a one-dimensional supersymmetric sigma model on a Riemannian manifold. We study the relation between the metaplectic correction in the super-setting and spin structures and we construct the heat kernel of the Dirac operator of a spin manifold by quantizing an appropriate Hamiltonian function using a supersymmetric version of the Blattner-Kostant-Stenberg kernel method. We explain how this point of view fits into the Stolz-Teichner program of studying cohomology theories via super-symmetric field theories.

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Speaker Date Room Title
Ricardo Kennedy Thursday, 03.05.2012,
10:15 h
NA 4/64 Homotopy classification of topological insulators
Abstract.

After their theoretical prediction, several examples of topological insulators have been realised experimentally. Despite this success, the question of the very definition of a 'topological insulator' is not settled. In this talk, I will motivate a mathematical definition of a topological insulator as a homotopy equivalence class of maps from a sphere or a torus into a complex Grassmannian, subject to contraints due to symmetries. Furthermore, I will present some methods of determining these equivalence classes in special cases.

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Speaker Date Room Title
Dr. Christian Miebach Friday, 27.04.2012,
10:15 h
NA 4/64 Holomorphe Reduktion von komplex-homogenen Mannigfaltigkeiten

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Speaker Date Room Title
Wolfgang Palzer (University of Cologne) Friday, 13.04.2012,
10:15 h
NA 4/64 The Stokes theorem on supermanifolds
Wolfgang Palzer (University of Cologne) Friday, 13.04.2012,
14:15 h
NA 4/64 Discussion with Wolfgang Palzer
Abstract.

We investigate integration over non-compact supermanifolds. Since the Berezin integral on superdomains is not invariant under changes of coordinates, a global definition of this integral is only possible after fixing a retraction. Once this is done and a reasonable analog of differential forms are defined — integral forms in the case of supermanifolds — we will deduce a super version of Stokes' Theorem. We conclude by investigating the behavior of ‘boundary terms’, which appear under the aforementioned change of coordinates/retractions.

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Speaker Date Room Title
Prof. Dr. Arkadij Onishchik (Yaroslavl/Russia) Wednesday, 22.02.2012,
14:00 h
NA 4/24 Non-Abelian Cohomology and Supermanifolds
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Speaker Date Room Title
Dr. Valdemar Tsanov (Bochum) Friday, 10.02.2012,
10:15 h
NA 4/64 Branching of representations, semistability and Schubert calculus
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Speaker Date Room Title
Miriam Kümmel (Bochum) Friday, 03.02.2012,
10:15 h
NA 4/64 Anwendungen des Spektralsatz fuer selbstadjungierte Operatoren in der Darstellungstheorie
Abstract.

Wir wiederholen den Spektralsatz für unbeschränkte selbstadjungierte Operatoren in drei Versionen und wenden ihn auf grundlegende Probleme der Darstellungstheorie an (Lemma von Schur bzw. Satz von Stone). Es handelt sich um einen Vortrag im Niveau für fortgeschrittene Studenten/junge Postdoktoranden und NICHT um einen Vortrag über rezente Forschung!

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Speaker Date Room Title
Dr. Valdemar Tsanov (Bochum) Monday, 30.01.2012,
12:00 h
NA 1/58 The Belkale-Kumar product on cohomology of flag manifolds
Abstract.

Let X be a complex flag manifold. Belkale and Kumar have de fined a new product on the cohomology space H*(X,C), which can be obtained as a deformation of the usual cup product. Another way to de fined the new product is by interpreting H*(X,C) as a suitable Lie algebra cohomology space, and taking the natural product there. An insight into the geometric meaning of the new product is gained through the notion of Levi-movability.
The Belkale-Kumar product is a useful technical tool in the study of semistability and branching rules for representations.

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Speaker Date Room Title
Prof. Dr. Gerald Schwarz (Brandeis University) Friday, 27.01.2012,
10:15 h
NA 4/64 Real geometric invariant theory and double coset spaces
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Speaker Date Room Title
Dr. Florin Dumitrescu (Hamburg) Friday, 20.01.2012,
10:15 h
NA 4/64 Superconnections and supergeometry
Abstract.

We use the language of supermanifolds to interpret geometrically superconnections (a la Quillen) as parallel transport along superpaths. This generalizes the usual parallel transport interpretation of connections.

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Speaker Date Room Title
Guillaume Tomasini Wednesday, 18.01.2012,
15:00 h
NA 4/64 The unitary dual of the universal cover of SU(1,1) and tensor products
Abstract.

In this talk, we shall derive by algebraic methods the classification of all simple unitary representation of the universal cover of SU(1,1), recovering a result first due to Pukanszky. This approach does (almost) not require any knowledge from representation theory. In a second part of the talk, we shall use this algebraic situation to understand the (discrete spectrum of the) decomposition of tensor products of two simple unitary representations of the universal cover of SU(1,1). The talk is based on a joint work with Bent Orsted.

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Speaker Date Room Title
Dr. Mathieu Molitor Monday, 19.12.2011,
12:15 h
NA 1/58 Information geometry, Kähler geometry and quantum mechanics
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Speaker Date Room Title
Prof. Dr. Ivan Dimitrov Friday, 16.12.2011,
10:15 h
NA 4/64 Lagrangian subalgebras of classical Lie superalgebras
Abstract.

I will first explain the theorems describing the Lagrangian subalgebras of complex simple Lie algebras (due to Belavin and Drinfeld) and of complex reductive Lie algebras (due to Delorme). I will then define the classical Lie superalgebras and will describe their Lagrangian subalgebras. This is a joint work with Milen Yakimov, LSU.

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Speaker Date Room Title
Dr. Will Kirwin Monday, 28.11.2011,
12:00 h
NA 1/64 Unitarity in quanization commutes with reduction I
Dr. Will Kirwin Monday, 05.12.2011,
12:00 h
NA 1/58 Unitarity in quanization commutes with reduction II
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Speaker Date Room Title
Prof. Dr. Frédéric Campana Friday, 25.11.2011,
10:15 h
NA 4/64 An orbifold version of Miyaoka's generic semi-positivity theorem
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Speaker Date Room Title
Christoph Wockel Friday, 28.10.2011,
10:15 h
NA 4/64 The diffeomorphism supergroup of a finite-dimensional supermanifold
Abstract.

Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism su- pergroup of a finite-dimensional supermanifold. The reduced part of this supergroup is the automorphism group of this supermanifold, which can be endowed with the structure of a Frechet-Lie group in case the supermanifold is compact. We will expain this con- struction and how it can be used to endow the dif- feomorphism supergroup with the structure of a Lie supergroup.


text:
Natural language understanding system

# Montague

An experimental natural-language understanding system.

Unlike conventional, machine-learning NLU systems, Montague uses formal semantics to interpret English sentences.

When Montague reads a sentence, it looks up each word in its lexicon to assign it a meaning in its logical language (an extension of first-order logic). Then, it recursively combines the words into phrases and computes the meaning of each phrase. When the sentence has been translated into a logical formula, Montague evaluates the formula against its model of the world to determine its truth value.

## Installation

$pip3 install montague-nlu  Alternatively, you can install a development version with setup.py: $ git clone https://github.com/iafisher/montague.git
$python3 setup.py develop --user  Once you do so, you can play with Montague's command-line interface: $ montague

## Limitations

The Montague system is still in early beta and suffers from many limitations.

• Montague has no knowledge of syntax. Its only criterion for grouping two phrases is whether they are linearly adjacent and whether their types are compatible. This leads Montague to interpret nonsense sentences like "Every good is child."
• Montague will fail to interpret a sentence if it contains a word that is not in its lexicon.
• Important modules of formal semantics, like plurality, tense, aspect, theta roles, intensionality, and indexicals, have yet to be implemented.

## Project details

Uploaded source
Uploaded py3

format: plain


text:
# Does division algorithm have a meaning?

Emphasis added: I'm trying to find a real world interpretation of division algorithm

Division in $$\mathbb{Z}$$ has a well defined analogue in our physical world. Since, in general, division algorithm $$a=bq+r$$ is not unique, does it have an easy-to-understand meaning? Let's say $$a,b,q,r \in \mathbb{Z}[\sqrt{11}]$$, $$b \ne 0$$. Suppose $$a=1$$ and $$b=2$$, we have,

\begin{align} 1 + \sqrt{11} &= (6-2\sqrt{11})\ (-3-\sqrt{11}) + (-3+\sqrt{11}) \\ &= (6-2\sqrt{11})\ (-4-\sqrt{11}) + ( 3-\sqrt{11}) \\ &= (6-2\sqrt{11})\ (-103-31\sqrt{11}) + ( -63- 19\sqrt{11}) \\ &= (6-2\sqrt{11})\ ( 96-31\sqrt{11}) + (-1257+379\sqrt{11}) \\ &= \dots \text{and many more, although I don't know if it is finite or not.} \end{align}

And note that: \begin{align} N(\alpha) = N(1 + \sqrt{11}) = 10 &\text{ and } N(\beta) = N(6-2\sqrt{11}) = 8 \\ N(q_{1}) = N(-3-\sqrt{11}) = 2 &\text{ and } N(r_{1}) = N(-3+\sqrt{11}) = 2 \\ N(q_{2}) = N(-4-\sqrt{11}) = 5 &\text{ and } N(r_{2}) = N(3-\sqrt{11}) = 2 \\ N(q_{3}) = N(-103-31\sqrt{11}) = 38 &\text{ and } N(r_{3}) = N(-63- 19\sqrt{11}) = 2 \\ N(q_{4}) = N(96-31\sqrt{11}) = 1355 &\text{ and } N(r_{4}) = N(-1257+379\sqrt{11}) = 2. \\ \end{align}

Background added: I do aware of the follows

A norm is well-defined if $$N(\alpha\beta) \ge N(\alpha)$$. Note that $$N(\alpha\beta) = N(\alpha)N(\beta)$$ and $$N(\alpha)=1 \iff \alpha\ \text{is a unit}$$.

There is no doubt that $\mathbb{Z}[\sqrt{11}]$ is norm-euclidean. With $$a,b \in \mathbb{Z}[\sqrt{11}]$$, $$b \ne 0$$, suppose the nearest lattice point to $$a/b \in \mathbb{Q}[\sqrt{11}]$$ is $$m+n\sqrt{11} \in \mathbb{Z}[\sqrt{11}]$$, one possible division algorithm can be defined by considering six lattice points as quotient $$q=(m+x)+(n+y)\sqrt{11}$$ with $$(x,y) \in \{ (0,0),\ (1,0),\ (-1,0), (2,0),\ (2,1),\ (-5,2) \}$$, which satisfies $$N(r) if $$r \ne 0$$.

I pick $$\mathbb{Z}[\sqrt{11}]$$, since it is the first quadratic integer ring, among $$\mathbb{Z}[\sqrt{d}]$$ with $$d \in \{ -2, -1,\ 2,\ 3,\ 6,\ 7,\ 11,\ 14,\ 19 \}$$, that it is not trivial to prove it is a Euclidean domain. Although $$\mathbb{Z}[\sqrt{14}]$$ is Euclidean but it is not norm-Euclidean.

Let me rephrase my question (19th February)

Perhaps a more fundamental question is that do different definitions of norm have geometric meaning. Let $$R=\mathbb{Z}[\sqrt{d}] = \{ s+t\sqrt{d} : s,t \in \mathbb{Z} \}$$ be a quadratic integer ring. The most usual definition of norm is $$N: R \setminus \{ 0 \} \rightarrow \mathbb{Z}_{\ge 0}$$ by setting $$s+t\sqrt{d} \mapsto |s^2 - dt^2 |$$. For negative values of $$d$$, $$N$$ is the modulus squared for any given element in $$R$$. For $$d=-1$$ and $$d=-2$$, elements in $$\mathbb{Z}[i]$$ and $$\mathbb{Z}[\sqrt{-2}]$$ are nicely ordered (along $$\mathbb{Z}_{\ge 0}$$) in concentric circles and ellipses respectively by their moduli squared. For positive values of $$d$$, an element (not $$R$$) has a division algorithm, if $$\alpha/\beta$$ falls within a starfish shape, enclosed by two pairs of hyperbolae, with a good choice of quotient $$q$$. Elements of the same (or close enough by magnitude) norms are no longer nicely ordered (geometrically) but all over the places.

Thanks @BillDubuque for the Lenstra papers. They are very useful to a non maths person like me.

• If I'm not mistaken, I think $\left|\operatorname N(a+b\sqrt{11})\right|=\left|a^2-11b^2\right|$ provides a norm to do Euclidean division with. Feb 18 at 23:03
• In order to use the Euclidean Algorithm in a domain $R,$ you must have a Euclidean function $\upsilon : R \setminus \{0_R\} \to \mathbb Z_{\geq 0}.$ As @DeriveFoiler has suggested, the map defined by $\upsilon(a + b \sqrt{11}) = a^2 - 11b^2$ is a Euclidean function on $\mathbb Z[\sqrt{11}].$ Check this link for other examples of quadratic integer rings that possess at least one Euclidean function. Feb 18 at 23:39
• See Hendrik Lenstra's Math. Intelligencer expositions on Euclidean number rings for a good introduction to this and related topics.. Feb 18 at 23:52
• @ChanTaiMan could you elaborate on "a well-defined analogue in our physical world?" Feb 19 at 0:02
• @BadamBaplan: I don't know, hence my question. In $\mathbb{Z}$, I have eleven apples (a); and I want groups of four (b). I have two groups (quotient q) and three leftover (reminder r). Feb 19 at 1:55


text:
# a question related to two competing patterns in coin tossing

If I have a two-sided coin with probability $p$ showing head. I repeatedly toss it until either HTHTH or HTHH appears. Can you calculate

1) the probability when I got HTHTH, and

2) the expected value of the number of tosses before I stop?

Thanks.

-
So the answers in the previous question: math.stackexchange.com/questions/18093/… didn't help? I believe the approach in my answer there tackles this one too. You just need two accepting states in the DFA, one for HTHTH and one for HTTH. –  Aryabhata Jan 21 '11 at 6:41
@Moron: to be honest, I am not familiar with DFA. Without a concrete worked example, I do not know how to carry out the calculation. Can you please give a detailed solution to this problem, some numbers/calculational steps please? –  Qiang Li Jan 21 '11 at 6:54
@QIang: THat would require time which I don't have right now. Perhaps this link might help in the meantime: docs.google.com/… . As to what a DFA is, check the wiki for deterministic finite automaton. –  Aryabhata Jan 21 '11 at 7:00
Maybe this will help too: cs.brown.edu/research/ai/dynamics/tutorial/Documents/… –  Aryabhata Jan 21 '11 at 7:01
You also ought to be able to use a Markov chain approach similar to the accepted answer on the question to which Moron linked. –  Isaac Jan 21 '11 at 9:51

Let $q$ be the probability of tails. Then the answers are

$$1) \frac{q}{1+q}$$

$$2) \frac{1+pq+p^2q+p^2q^2}{p^3q(1+q)}.$$

The ideas and notation in the following derivation are courtesy of Section 8.4 ("Flipping Coins") of Concrete Mathematics. See that section for more details and examples.

Imagine this as a contest in which Player A wins if HTHTH appears first and Player B wins if HTHH appears first. Let $S_A$ and $S_B$ denote the sum of the winning sequences of tosses for HTHTH and HTHH, respectively. (This is overloading the addition operator, but it works.) Let $N$ denote the sum of the sequences in which neither HTHTH nor HTHH has occurred yet. ($N$ includes the empty sequence.) Thus $$S_A = \text{HTHTH + THTHTH + HHTHTH + TTHTHTH + THHTHTH + HHHTHTH} + \cdots,$$ $$S_B = \text{HTHH + THTHH + HHTHH + TTHTHH + THHTHH + HHHTHH} + \cdots,$$ $$N = \text{1 + H + T + HH + HT + TH + TT} + \cdots.$$

Then, by substituting $p$ for H and $q$ for T, $S_A$, $S_B$, and $N$ become the probability that $A$ wins, the probability that $B$ wins, and the expected number of tosses until the game ends, respectively. The first two claims are straightforward. The third follows from the fact that if $X$ is a nonnegative random variable then $E[X] = \sum_{k=0}^{\infty} P(X > k).$

As sums of sequences, we also have $$N \text{ HTHTH} = S_A + S_A \text{ TH} + S_A \text{ THTH} + S_B \text{ THTH},$$ $$N \text{ HTHH} = S_A \text{ H} + S_A \text{ THH} + S_B + S_B \text{ THH}.$$

The idea behind the first equation is that the sequences of tosses obtained by appending HTHTH to $N$ must be a sequence of winning tosses for $A$ or $B$ plus possibly some extra tosses. In particular, each sequence in $N$ HTHTH must be exactly one of 1) a winning sequence for $A$, 2) a winning sequence for $A$ followed by TH, 3) a winning sequence for $A$ followed by THTH, or 4) a winning sequence for $B$ followed by THTH. These are all possibilities because they represent all ways in which part of the sequence HTHTH can begin (i.e., overlap) another occurrence of HTHTH or (in 4) in which HTHH can begin an occurrence of HTHTH. The idea behind the second equation is similar.

Then, substituting $p$ for H and $q$ for T we have $$N p^3q^2 = S_A + S_A pq + S_A p^2q^2 + S_B p^2q^2,$$ $$N p^3q = S_A p + S_A p^2 q + S_B + S_B p^2q.$$

Simultaneously solving these two equations with $S_A + S_B = 1$ yields $$S_A = \frac{q}{1+q},$$ $$S_B = \frac{1}{1+q},$$ $$N = \frac{1+pq+p^2q+p^2q^2}{p^3q(1+q)}.$$

Update, in response to OP's question in the comments: Why does substituting $p$ for H and $q$ for T and the fact that $E[X] = \sum_{k=0}^{\infty} P(X > k)$ for a nonnegative random variable all mean that $N$ is the expected number of tosses until the game ends?

Let $X$ be the number of tosses until the game ends. $P(X > 0)$ is just 1. $P(X > 1)$ is the probability that we haven't seen one of the patterns yet with a single toss; i.e., H + T, with $p$ subbed in for H and $q$ for T. $P(X > 2)$ is the probability that we haven't seen one of the patterns yet with two tosses, which is HH + HT + TH + TT, with $p$ subbed in for H and $q$ for T. $P(X > 3)$ is..., etc. This doesn't get interesting until we look at $P(X > 4)$, which would include all four-length sequences of H and T except for HTHH. Adding all those sequences in which neither HTHH nor HTHTH has occurred yet is exactly $N$, and subbing in $p$ for H and $q$ for T in $N$ thus gives $E[X]$.

See also the solution to Problem 8.21 in Concrete Mathematics, p. 578 (second edition).

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I came back to this problem today. I am still wondering why "The third follows from the fact that if $X$ is a nonnegative random variable then $E[X]=\sum_{k=0}^\infty{P(X>k)}$." I know this identity, but I don't see how it can be applied to the situation described here. We have "$N$ denote the sum of the sequences in which neither HTHTH nor HTHH has occurred yet", and we need to get "the expected number of tosses until the game ends". How to bridge these two using the identity above? May thanks. –  Qiang Li Feb 2 '11 at 4:42
@Qiang Li: I've updated my answer to address the question in your comment. –  Mike Spivey Feb 2 '11 at 5:44

There is a direct, and rather automatic, way to compute the probability to hit A=HTHTH first rather than B=HTHH.

Both motives begin by HTH hence one can wait until HTH first appears. Then, either (1) the next letter is H, or (2) the two next letters are TH, or (3) the two next letters are TT. If (1) happens, B won. If (2) happens, A won. If (3) happens, one has to wait for more letters to know who won. The important fact in case (3) is that, since the last letters are TT, A or B must be entirely produced again.

Hence, $p_B=p_1+p_3p_B$ and $p_A=p_2+p_3p_A$, where $p_i$ for $i=$ 1, 2 and 3, is a shorthand for the conditional probability that ($i$) happens starting from the word HTH. Since $p_1=p$, $p_2=qp$ and $p_3=q^2$, one gets $p_B=p_1/(1-p_3)=p/(1-q^2)$, hence $$p_B=1/(1+q),\quad p_A=q/(1+q).$$ Similarly, a standard, and rather automatic, way to compute the mean number of tosses before this happens is to consider a Markov chain on the state space made of the prefixes of the words one wishes to complete.

Here, the states are 0 (for the empty prefix), 1=H, 2=HT, 3=HTH, B=HTHH, 4=HTHT and A=HTHTH. The transitions are from 0 to 1 and 0, from 1 to 2 and 1, from 2 to 3 and 0, from 3 to B and 4 and from 4 to A and 0. The transitions from B and from A are irrelevant. The next step is to compute $n_s$ the number of tosses needed to produce A or B starting from any state $s$ amongst 0, 1, 2, 3 and 4, knowing that one is in fact only interested in $n_0$.

The $n_s$ are solutions of a Cramér system which reflects the structure of the underlying Markov chain: $$n_0=1+pn_1+qn_0,\quad n_1=1+pn_1+qn_2,\quad n_2=1+pn_3+qn_0,$$ $$n_3=1+qn_4,\quad n_4=1+qn_0.$$ Solving this system of equations backwards, that is, going from the last equation back to the first one, yields $n_3$ in terms of $n_0$, then $n_2$ in terms of $n_0$, then $n_1$ in terms of $n_0$, and finally an equation for $n_0$ alone, which yields Mike's formula for $n_0$, namely: $$n_0=\frac{1+pq+p^2q+p^2q^2}{p^3q(1+q)}.$$ An accessible reference for these techniques (in the context of genomic sequence analysis) is the book DNA, Words and Models by Robin, Rodolphe and Schbath, at Cambridge UP.

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+1: Just for the record, though, in Concrete Mathematics it is claimed that their method (the one in my answer) for calculating $n_0$ is simpler than solving the system of equations resulting from the Markov chain. The Markov chain approach is also the approach indicated by Moron in the first comment on the OP. –  Mike Spivey Feb 2 '11 at 9:47
Mike: CM invokes simplicity for one motif. First, even in this case, I do not find them very convincing. Second, no such claim is made for two motives (see for instance equation (8.79) at the bottom of page 409 in CM second edition). Third, and more importantly to me, I must think to apply CM method correctly (for instance, should it be $S_AT$ or $S_A$ or something else in the RHS of the third equation of (8.78)?) and the Markov chain method is entirely automatic. But, as they say: Des goûts et des couleurs... :-) [Did you elucidate the $+p$ problem in the numerator?] –  Did Feb 2 '11 at 10:28
@Didier: Fair enough. :) Unfortunately, I don't see the mistake in either of our solutions, yet; they both look right to me! Maybe I'll try again in the morning when my brain is less foggy. :) –  Mike Spivey Feb 2 '11 at 11:02
Mike: Got it! When at prefix 1=H, either one moves to prefix 2=HT or one stays at prefix 1=H, right? Hence $n_1=1+pn_0+qn_1$ (instead of $n_1=1+pn_0+qn_2$ as I first wrote, erroneously). This yields your formula. I corrected my post. –  Did Feb 2 '11 at 13:47
@Didier: That's it! I'm glad you found the resolution; it was bothering me that two correct approaches were yielding different answers. :) (FYI: If you preface your comments with @ then the addressee gets a notification that there is a new comment for them the next time he/she checks the site.) –  Mike Spivey Feb 2 '11 at 19:59

Assume that the coin tossing is infinite. Define the state at time $n$ to be the most recent $5$ outcomes when $n \geq 5$ and the $n$ most recent outcomes when $n <5$. This is a Markov chain. Now $\pi_{HTHTH}$(the limiting probability) is equal to the inverse of the mean time to go from state HTHTH to HTHTH. Thus $\pi_{HTHTH} = p^{3}q^2$ (probability of getting HTHTH) where $q=1-p$. So $1/(p^{3}q^2)$ is the mean time to go from HTHTH to HTHTH. Thus $$E(\text{time to get} \ HTHTH) = E(\text{time to get} \ HT)+ \frac{1}{p^{3}q^{2}}$$ $$= E(\text{time to get} \ H)+ \frac{1}{pq}+\frac{1}{p^{3}q^{2}} = \frac{1}{p}+\frac{1}{pq}+\frac{1}{p^{3}q^{2}}$$

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Sorry, but does this actually answer either of the part of the problem? I'm (not very familiar with Markov chains and am) confused about what the mean time tells you. (And I'm assuming $q = 1 - p$; is that right?) –  Ben Alpert Jan 22 '11 at 2:55
@Ben Alpert: It answers the probability of getting HTHTH which is $p^{3}q^2$ where $q=1-p$. –  PEV Jan 22 '11 at 2:58
@PEV: So if that's a fair coin, you're saying the probability is $1/32$. Of course, that's the probability that a random flip sequence of length 5 is equal to HTHTH but doesn't seem to answer whether HTHTH or HTHH is more likely to appear first. –  Ben Alpert Jan 22 '11 at 3:01
@Ben Alpert: The probability of obtaining HTHH is $p^{3}q$. –  PEV Jan 22 '11 at 3:03
@PEV: I can't tell if I'm just being really stupid here, but I'm still unsure how that helps. For example, if with a fair coin you're looking for TH vs HH, it's much more likely ($3/4$ vs. $1/4$) that you'll see TH first because once you see a single T, it's impossible to get HH before seeing TH. How do we get the same answer using your method? (Presumably that would be $pq$ vs $p^2$.) –  Ben Alpert Jan 22 '11 at 3:06


text:
# How to prove this property in a Poisson process?

For a Poisson process show, for $s < t$, that $P(N(s)=k|N(t)=n) = \binom{n}{k} (\frac{s}{t})^k (1-\frac{s}{t})^{n-k}$

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The joint probability mass function for $(N(s), N(t))$ for $s<t$ is, using the property of independent increments, i.e. $N(t) \sim N(s) + N(t-s)$: $$\mathbb{P}(N(s) = k, N(t) = n) = \mathbb{P}(N(s) = k, N(t) - N(s) = n - k) = \\\frac{(\mu s)^k}{k!} \mathrm{e}^{-\mu s} \cdot \frac{(\mu (t-s))^{n-k}}{(n-k)!} \mathrm{e}^{-\mu(t-s)} \cdot [ n-k \geqslant 0, k \geqslant 0]$$ Then, assuming $n \geqslant k \geqslant 0$: $$\mathbb{P}(N(s) =k | N(t) = n) = \frac{\mathbb{P}(N(s)=k,N(t)=n)}{\mathbb{P}(N(t)=n)} = \\ \frac{\frac{\mu^k s^k}{k!} \cdot \frac{\mu^{n-k} (t-s)^{n-k}}{(n-k)!} \cdot \mathrm{e}^{-\mu t} }{ \frac{\mu^n t^n}{n!} \mathrm{e}^{-\mu t}} = \left( \frac{s}{t} \right)^k \left( 1- \frac{s}{t} \right)^{n-k} \binom{n}{k}$$

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+1. Neat'n easy. –  Did Mar 16 '12 at 0:20

Let

$\ \ \ \ A$ be the number of events occurring in the interval $[0,s]$,

$\ \ \ \ B$ be the number of events occurring in the interval $[s,t]$,

and

$\ \ \ \ C$ be the number of events occurring in the interval $[0,t]$.

Using the fact that in a Poisson process, events occurring in one time frame are independent of events occurring in a different time frame, we may write the sought after probability as \tag{1} \eqalign{P ( A =k \mid C= n) &={P\bigl((A=k)\cap ( C=n) \bigr)\over P(C=n)}\cr &={P\bigl((A=k)\cap ( B=n-k) \bigr)\over P(C=n)}\cr &={P (A=k) P ( B=n-k) \over P(C=n)} .}

Now, if the Poisson process has parameter $\lambda$, then

$\ \ \ \ A$ has Poisson distribution with parameter $s\lambda$,

$\ \ \ \ B$ has Poisson distribution with parameter $(t-s)\lambda$,

and

$\ \ \ \ C$ has Poisson distribution with parameter $t\lambda$.

So, using equation $(1)$: \eqalign{ P( A =k \mid C= n) &={ {(s\lambda)^k e^{-s\lambda}\over k!} {( (t-s)\lambda)^{n-k} e^{-(t-s)\lambda}\over (n-k)!} \over {(t\lambda)^n e^{-t\lambda}\over n!}}\cr &={n\choose k} {s^k(t-s)^{n-k} \lambda^k \lambda^{n-k} e^{-t\lambda}\over t^n\lambda^ne^{-t\lambda}}\cr &={n\choose k} {s^k(t-s)^{n-k}\over t^kt^{n-k}}\cr &={n\choose k} {\Bigl({s\over t}\Bigr)^k \Bigl(1-{s\over t}\Bigr)^{n-k}}.\cr }

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text:
• ### Towards QCD-assisted hydrodynamics for heavy-ion collision phenomenology(1805.02985)

May 8, 2018 nucl-th
Heavy-ion collisions are well described by a dynamical evolution with a long hydrodynamical phase. In this phase the properties of the strongly coupled quark-gluon plasma are reflected in the equation of state (EoS) and the transport coefficients, most prominently by the shear and bulk viscosity over entropy density ratios $\eta$/s(T) and $\zeta$/s(T), respectively. While the EoS is by now known to a high accuracy, the transport coefficients and in particular their temperature and density dependence are not well known from first-principle computations yet, as well as the possible influence they can have once used in hydrodynamical simulations. In this work, the most recent QCD-based parameters are provided as input to the MUSIC framework. A ratio $\eta$/s(T) computed with a QCD based approach is used for the first time \cite{Haas:2013hpa,Christiansen:2014ypa}. The IP-Glasma model is used to describe the initial energy density distribution, and UrQMD for the dilute hadronic phase. Simulations are performed for Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV, for different centrality intervals. The resulting kinematic distributions of the particles produced in the collisions are compared to data from the LHC, for several experimental observables. The high precision of the experimental results and the broad variety of observables considered allow to critically verify the quality of the description based on first-principle input to the hydrodynamic evolution.
• ### Predictions for Cold Nuclear Matter Effects in $p+$Pb Collisions at $\sqrt{s_{_{NN}}} = 8.16$ TeV(1707.09973)

Dec. 8, 2017 hep-ph
Predictions for cold nuclear matter effects on charged hadrons, identified light hadrons, quarkonium and heavy flavor hadrons, Drell-Yan dileptons, jets, photons, gauge bosons and top quarks produced in $p+$Pb collisions at $\sqrt{s_{_{NN}}} = 8.16$ TeV are compiled and, where possible, compared to each other. Predictions of the normalized ratios of $p+$Pb to $p+p$ cross sections are also presented for most of the observables, providing new insights into the expected role of cold nuclear matter effects. In particular, the role of nuclear parton distribution functions on particle production can now be probed over a wider range of phase space than ever before.
• ### The Electron-Ion Collider: Assessing the Energy Dependence of Key Measurements(1708.01527)

Sept. 8, 2017 hep-ph, hep-ex, nucl-ex, nucl-th
We provide an assessment of the energy dependence of key measurements within the scope of the machine parameters for a U.S. based Electron-Ion Collider (EIC) outlined in the EIC White Paper. We first examine the importance of the physics underlying these measurements in the context of the outstanding questions in nuclear science. We then demonstrate, through detailed simulations of the measurements, that the likelihood of transformational scientific insights is greatly enhanced by making the energy range and reach of the EIC as large as practically feasible.
• ### Momentum broadening in unstable quark-gluon plasma(1607.02359)

Jan. 24, 2017 hep-th, hep-ph, nucl-th
Quark-gluon plasma produced at the early stage of ultrarelativistic heavy ion collisions is unstable, if weakly coupled, due to the anisotropy of its momentum distribution. Chromomagnetic fields are spontaneously generated and can reach magnitudes much exceeding typical values of the fields in equilibrated plasma. We consider a high energy test parton traversing an unstable plasma that is populated with strong fields. We study the momentum broadening parameter $\hat q$ which determines the radiative energy loss of the test parton. We develop a formalism which gives $\hat q$ as the solution of an initial value problem, and we focus on extremely oblate plasmas which are physically relevant for relativistic heavy ion collisions. The parameter $\hat q$ is found to be strongly dependent on time. For short times it is of the order of the equilibrium value, but at later times $\hat q$ grows exponentially due to the interaction of the test parton with unstable modes and becomes much bigger than the value in equilibrium. The momentum broadening is also strongly directionally dependent and is largest when the test parton velocity is transverse to the beam axis. Consequences of our findings for the phenomenology of jet quenching in relativistic heavy ion collisions are briefly discussed.
• ### Constraints for proton structure fluctuations from exclusive scattering(1612.00041)

Nov. 30, 2016 hep-ph, nucl-th
We constrain the average density profile of the proton and the amount of event-by-event fluctuations by simultaneously calculating the coherent and incoherent exclusive diffractive vector meson production cross section in deep inelastic scattering. Working within the Color Glass Condensate picture, we find that the gluonic density of the proton must have large geometric fluctuations in order to describe the experimentally measured large incoherent cross section.
• ### Tracing the origin of azimuthal gluon correlations in the color glass condensate(1509.03499)

Dec. 22, 2015 hep-ph, nucl-th
We examine the origins of azimuthal correlations observed in high energy proton-nucleus collisions by considering the simple example of the scattering of uncorrelated partons off color fields in a large nucleus. We demonstrate how the physics of fluctuating color fields in the color glass condensate (CGC) effective theory generates these azimuthal multiparticle correlations and compute the corresponding Fourier coefficients v_n within different CGC approximation schemes. We discuss in detail the qualitative and quantitative differences between the different schemes. We will show how a recently introduced color field domain model that captures key features of the observed azimuthal correlations can be understood in the CGC effective theory as a model of non-Gaussian correlations in the target nucleus.
• ### The importance of the bulk viscosity of QCD in ultrarelativistic heavy-ion collisions(1502.01675)

Sept. 12, 2015 hep-ph, nucl-th
We investigate the consequences of a nonzero bulk viscosity coefficient on the transverse momentum spectra, azimuthal momentum anisotropy, and multiplicity of charged hadrons produced in heavy ion collisions at LHC energies. The agreement between a realistic 3D hybrid simulation and the experimentally measured data considerably improves with the addition of a bulk viscosity coefficient for strongly interacting matter. This paves the way for an eventual quantitative determination of several QCD transport coefficients from the experimental heavy ion and hadron-nucleus collision programs.
• ### Extracting the bulk viscosity of the quark-gluon plasma(1408.0024)

July 31, 2014 hep-ph, nucl-th
We investigate the implications of a nonzero bulk viscosity coefficient on the azimuthal momentum anisotropy of ultracentral relativistic heavy ion collisions at the Large Hadron Collider. We find that, with IP-Glasma initial conditions, a finite bulk viscosity coefficient leads to a better description of the flow harmonics in ultracentral collisions. We then extract optimal values of bulk and shear viscosity coefficients that provide the best agreement with flow harmonic coefficients data in this centrality class.
• ### Thermally Fluctuating Second-Order Viscous Hydrodynamics and Heavy-Ion Collisions(1407.1077)

July 3, 2014 nucl-th
The fluctuation-dissipation theorem requires the presence of thermal noise in viscous fluids. The time and length scales of heavy ion collisions are small enough so that the thermal noise can have a measurable effect on observables. Thermal noise is included in numerical simulations of high energy lead-lead collisions, increasing average values of the momentum eccentricity and contributing to its event by event fluctuations.
• ### Effect of initial-state nucleon-nucleon correlations on collective flow in ultra-central heavy-ion collisions(1406.7792)

June 30, 2014 hep-ph, nucl-th
We investigate the effect of nucleon-nucleon correlations on the initial condition of ultra-central heavy ion collisions at LHC energies. We calculate the eccentricities of the MC-Glauber and IP-Glasma models in the 0--1% centrality class and show that they are considerably affected by the inclusion of such type of correlations. For an IP-Glasma initial condition, we further demonstrate that this effect survives the fluid-dynamical evolution of the system and can be observed in its final state azimuthal momentum anisotropy.
• ### Dilepton production in high energy heavy ion collisions with 3+1D relativistic viscous hydrodynamics(1211.0022)

Oct. 31, 2012 hep-ph, nucl-th
We present a first calculation of the dilepton yield and elliptic flow done with 3+1D viscous hydrodynamical simulations of relativistic heavy ion collisions at the top RHIC energy. A comparison with recent experimental data from the STAR collaboration is made.
• ### Finite lifetime effects on the photon production from a quark-gluon plasma(0906.1734)

June 9, 2009 hep-ph, nucl-th
We use the real-time Keldysh formalism to investigate finite lifetime effects on the photon emission from a quark-gluon plasma (QGP). We provide an ansatz which eliminates the divergent contribution from the vacuum polarization and renders the photon spectrum UV-finite if the time evolution of the QGP is described in a suitable manner.
• ### Dileptons from vector mesons with dropping masses in a nonequilibrium quantum field theoretical framework(hep-ph/0606028)

June 2, 2006 hep-ph
The influence of time dependent medium modifications of low mass vector mesons on dilepton production is investigated using nonequilibrium quantum field theory. By working in the two-time representation memory effects are fully taken into account. For different scenarios the resulting dilepton yields are compared to quasi-equilibrium calculations and remarkable differences are found leading to the conclusion that memory effects can not be neglected when calculating dilepton yields from heavy ion collisions. Emphasis is put on a dropping mass scenario since it was recently claimed to be ruled out by the NA60 data. Here the memory effects turn out to be particularly important.
• ### Dilepton production from hot hadronic matter in nonequilibrium(hep-ph/0509026)

Jan. 11, 2006 hep-ph, nucl-th
The influence of time dependent medium modifications of low mass vector mesons on dilepton yields is investigated within a nonequilibrium quantum field theoretical description on the basis of the Kadanoff-Baym equations. Time scales for the adaption of the spectral properties to changing self energies are given and, under use of a model for the fireball evolution, nonequilibrium dilepton yields from the decay of rho- and omega-mesons are calculated. In a comparison of these yields with those from calculations that assume instantaneous (Markovian) adaption to the changing medium, quantum mechanical memory effects turn out to be important.
• ### Dilepton production from non-equilibrium hot hadronic matter(hep-ph/0504278)

April 28, 2005 hep-ph
It is investigated under which conditions an adiabatic adaption of the dynamic and spectral information of vector mesons to the changing medium in heavy ion collisions, as assumed in schematic model calculations and microscopic transport simulations, is a valid assumption. Therefore time dependent medium modifications of low mass vector mesons are studied within a non-equilibrium quantum field theoretical description. Timescales for the adaption of the spectral properties are given and non-equilibrium dilepton yields are calculated, leading to the result that memory effects are not negligible for most scenarios.


text:
# Data¶

To find out how to fetch data in bulk, check out the documentation about data access.

## Elements¶

The followig data are currently available:

Name

Type

Comment

Unit

Data Source

abundance_crust

float

Abundance in the Earth’s crust

mg/kg

[22]

abundance_sea

float

Abundance in the seas

mg/L

[22]

annotation

str

Annotations regarding the data

atomic_number

int

Atomic number

float

pm

[49]

float

Atomic radius by Rahm et al.

pm

[41, 42]

atomic_volume

float

Atomic volume

cm3/mol

atomic_weight

float

Atomic weight1

[32, 60]

atomic_weight_uncertainty

float

Atomic weight uncertainty1

[32, 60]

block

str

Block in periodic table

boiling_point

float

Boiling temperature

K

c6

float

C_6 dispersion coefficient in a.u.

a.u.

[13, 52]

c6_gb

float

C_6 dispersion coefficient in a.u. (Gould & Bučko)

a.u.

[21]

cas

str

Chemical Abstracts Serice identifier

float

pm

[10]

float

Covalent radius by Cerdero et al.2

pm

[16]

float

Single bond covalent radius by Pyykko et al.

pm

[39]

float

Double bond covalent radius by Pyykko et al.

pm

[38]

float

Triple bond covalent radius by Pyykko et al.

pm

[40]

cpk_color

str

Element color in CPK convention

HEX

[57]

density

float

Density at 295K10

g/cm3

[22, 63]

description

str

Short description of the element

dipole_polarizability

float

Dipole polarizability

a.u.

[47]

dipole_polarizability_unc

float

Dipole polarizability uncertainty

a.u.

[47]

discoverers

str

The discoverers of the element

discovery_location

str

The location where the element was discovered

dipole_year

int

The year the element was discovered

electron_affinity

float

Electron affinity3

eV

[6, 22]

electrons

int

Number of electrons

electrophilicity

float

Electrophilicity index

eV

[35]

en_allen

float

Allen’s scale of electronegativity4

eV

[28, 29]

en_ghosh

float

Ghosh’s scale of electronegativity

[18]

en_mulliken

float

Mulliken’s scale of electronegativity

eV

[33]

en_pauling

float

Pauling’s scale of electronegativity

[22]

econf

str

Ground state electron configuration

evaporation_heat

float

Evaporation heat

kJ/mol

fusion_heat

float

Fusion heat

kJ/mol

gas_basicity

float

Gas basicity

kJ/mol

[22]

geochemical_class

str

Geochemical classification

[55]

glawe_number

int

Glawe’s number (scale)

[19]

goldschmidt_class

str

Goldschmidt classification

[55, 56]

group

int

Group in periodic table

heat_of_formation

float

Heat of formation

kJ/mol

[22]

ionenergy

tuple

Ionization energies

eV

[23]

list

Ionic and crystal radii in pm9

pm

[26, 48]

is_monoisotopic

bool

Is the element monoisotopic

bool

isotopes

list

Isotopes

jmol_color

str

Element color in Jmol convention

HEX

[61]

lattice_constant

float

Lattice constant

Angstrom

lattice_structure

str

Lattice structure code

mass_number

int

Mass number (most abundant isotope)

melting_point

float

Melting temperature

K

mendeleev_number

int

Mendeleev’s number5

[37, 53]

float

pm

[1]

float

Metallic radius with 12 nearest neighbors

pm

[1]

molar_heat_capacity

float

Molar heat capacity @ 25 C, 1 bar

J/(mol K)

[22]

molcas_gv_color

str

Element color in MOCAS GV convention

HEX

[62]

name

str

Name in English

name_origin

str

Origin of the name

neutrons

int

Number of neutrons (most abundant isotope)

oxistates

list

Oxidation states

period

int

Period in periodic table

pettifor_number

float

Pettifor scale

[37]

proton_affinity

float

Proton affinity

kJ/mol

[22]

protons

int

Number of protons

sconst

float

Nuclear charge screening constants6

[14, 15]

series

int

Index to chemical series

sources

str

Sources of the element

specific_heat_capacity

float

Specific heat capacity @ 25 C, 1 bar

J/(g K)

[22]

symbol

str

Chemical symbol

thermal_conductivity

float

Thermal conductivity @25 C

W/(m K)

uses

str

Applications of the element

float

pm

[22]

float

Van der Waals radius according to Alvarez7

pm

[5, 54]

float

Van der Waals radius according to Batsanov

pm

[8]

float

Van der Waals radius according to Bondi

pm

[9]

float

Van der Waals radius from the DREIDING FF

pm

[31]

float

Van der Waals radius from the MM3 FF

pm

[3]

float

Van der Waals radius according to Rowland and Taylor

pm

[44]

float

Van der Waals radius according to Truhlar

pm

[30]

float

Van der Waals radius from the UFF

pm

[43]

## Isotopes¶

Name

Type

Comment

Unit

Data Source

abundance

float

Relative Abundance

[59]

g_factor

float

Nuclear g-factor8

[51]

half_life

float

Half life of the isotope

[32]

half_life_unit

str

Unit in which the half life is given

[32]

bool

[58]

mass

float

Atomic mass

Da

[58]

mass_number

int

Mass number of the isotope

[58]

mass_uncertainty

float

Uncertainty of the atomic mass

[58]

spin

float

Nuclear spin quantum number

float

b [100 fm2]

[50]

Data Footnotes

1(1,2)

Atomic Weights

Atomic weights and their uncertainties were retrieved mainly from ref. [60]. For elements whose values were given as ranges the conventional atomic weights from Table 3 in ref. [32] were taken. For radioactive elements the standard approach was adopted where the weight is taken as the mass number of the most stable isotope. The data was obtained from CIAAW page on radioactive elements. In cases where two isotopes were specified the one with the smaller standard deviation was chosen. In case of Tc and Pm relative weights of their isotopes were used, for Tc isotope 98, and for Pm isotope 145 were taken from CIAAW.

2

Covalent Radius by Cordero et al.

In order to have a more homogeneous data for covalent radii taken from ref. [16] the values for 3 different valences for C, also the low and high spin values for Mn, Fe Co, were respectively averaged.

3

Electron affinity

Electron affinities were taken from [22] for the elements for which the data was available. For He, Be, N, Ar and Xe affinities were taken from [6] where they were specified for metastable ions and therefore the values are negative.

• Electron affinity of niobium was taken from [27].

• Electron affinity of cobalt was taken from [11].

• Electron affinity of lead was taken from [12].

4

Allen’s configuration energies

The values of configurational energies from refs. [28] and [29] were taken as reported in eV without converting to Pauling units.

5

Mendeleev numbers

Mendeleev numbers were mostly taken from [53] but the range was extended to cover the whole periodic table following the prescription in the article of increasing the numbers going from top to bottom in each group and group by group from left to right in the periodic table.

6

Nuclear charge screening constants

The screening constants were calculated according to the following formula

$\sigma_{n,l,m} = Z - n\cdot\zeta_{n,l,m}$

where $$n$$ is the principal quantum number, $$Z$$ is the atomic number, $$\sigma_{n,l,m}$$ is the screening constant, $$\zeta_{n,l,m}$$ is the optimized exponent from [14, 15].

For elements Nb, Mo, Ru, Rh, Pd and Ag the exponent values corresponding to the ground state electronic configuration were taken (entries with superscript a in Table II in [15]).

For elements La, Pr, Nd and Pm two exponent were reported for 4f shell denoted 4f and 4f’ in [15]. The value corresponding to 4f were used since according to the authors these are the dominant ones.

7

van der Waals radii according to Alvarez

The bulk of the radii data was taken from Ref. [5], but the radii for noble gasses were update according to the values in Ref. [54].

8(1,2)

The data regarding g-factors and electric quadrupole moments was parsed from easyspin webpage (accessed 25.01.2017) where additional notes are mentioned:

• Typo for Rh-103: Moment is factor of 10 too large

• 237Np, 239Pu, 243Am magnetic moment data from [22], section 11-2

• In quadrupole moment data - a typo for Ac-227: sign should be +

9

Ionic radii for Actinoid (III) ions

Ionic radii values for 3+ Actinoids were with coordination number 9 were taken from [26]. In addition crystal_radius values were computed by adding 14 pm to the ionic_radius values according to [48].

10

Densities

Density values for solids and liquids are always in units of grams per cubic centimeter and can be assumed to refer to temperatures near room temperature unless otherwise stated. Values for gases are the calculated ideal gas densities at 25°C and 101.325 kPa.

Original values for gasses are converted from g/L to g/cm3.

For elements where several allotropes exist, the density corresponding to the most abundand are reported (for full list refer to [22]), namely:

• Antimony (gray)

• Berkelium (α form)

• Carbon (graphite)

• Phosphorus (white)

• Selenium (gray)

• Sulfur (rhombic)

• Tin (white)

For elements where experimental data is not available, theoretical estimates taken from [63] are used, namely for:

• Astatine

• Francium

• Einsteinium

• Fermium

• Mendelevium

• Nobelium

• Lawrencium

• Rutherfordium

• Dubnium

• Seaborgium

• Bohrium

• Hassium

• Meitnerium

• Roentgenium

• Copernicium

• Nihonium

• Flerovium

• Moscovium

• Livermorium

• Tennessine

• Oganesson