scholarly journals Explicit computation of some families of Hurwitz numbers, II

2020 ◽  
Vol 20 (4) ◽  
pp. 483-498
Author(s):  
Carlo Petronio
Keyword(s):  
Target Surface ◽  
Equivalence Classes ◽  
Weak Equivalence ◽  
Source Surface ◽  
Explicit Computation ◽  
Combinatorial Method ◽  
Branched Covers ◽  
Hurwitz Numbers ◽  
Branching Points

AbstractWe continue our computation, using a combinatorial method based on Gronthendieck’s dessins d’enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2k, and local degrees over the branching points of the form [2, …, 2], [2h + 1, 3, 2, …, 2], $\begin{array}{} \displaystyle \pi=[d_i]_{i=1}^\ell. \end{array}$ We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the di’s.

scholarly journals Monotone Hurwitz Numbers in Genus Zero

2013 ◽  
Vol 65 (5) ◽  
pp. 1020-1042 ◽  
Author(s):  
I. P. Goulden ◽  
Mathieu Guay-Paquet ◽  
Jonathan Novak
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Expansion ◽  
Symmetric Functions ◽  
Initial Conditions ◽  
Riemann Sphere ◽  
Branched Covers ◽  
Genus Zero ◽  
Hurwitz Numbers ◽  
Arbitrary Genus

AbstractHurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

scholarly journals Equivalence classes of functions on finite sets

1982 ◽  
Vol 5 (4) ◽  
pp. 745-762
Author(s):  
Chong-Yun Chao ◽  
Caroline I. Deisher
Keyword(s):  
Equivalence Class ◽  
Equivalence Classes ◽  
Permutation Groups ◽  
Weak Equivalence ◽  
Finite Sets ◽  
The Family ◽  
Polya's Theorem ◽  
Finite Set ◽  
Pólya’S Theorem

By using Pólya's theorem of enumeration and de Bruijn's generalization of Pólya's theorem, we obtain the numbers of various weak equivalence classes of functions inRDrelative to permutation groupsGandHwhereRDis the set of all functions from a finite setDto a finite setR,Gacts onDandHacts onR. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn's theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non-isomorphicfm-graphs relative to a given group.

The Symmetries of Genus One Handlebodies

1991 ◽  
Vol 43 (2) ◽  
pp. 371-404 ◽  
Author(s):  
John Kalliongis ◽  
Andy Miller
Keyword(s):  
Finite Groups ◽  
Focal Point ◽  
Complete Classification ◽  
Equivalence Classes ◽  
Weak Equivalence ◽  
Finite Group Actions ◽  
3 Dimensional ◽  
Low Dimensional Topology ◽  
Low Dimensional ◽  
Genus One

The symmetries of manifolds are a focal point of study in low-dimensional topology and yet, outside of some totally asymmetrical 3- and 4-manifolds, there are very few cases in which a complete classification has been attained. In this work we provide such a classification for symmetries of the orientable and nonorientable 3-dimensional handlebodies of genus one. Our classification includes a description, up to isomorphism, of all of the finite groups which can arise as symmetries on these manifolds, as well as an enumeration of the different ways in which they can arise. To be specific, we will classify the equivalence, weak equivalence and strong equivalence classes of (effective) finite group actions on the genus one handlebodies.

scholarly journals Topology and convexity in the space of actions modulo weak equivalence

2017 ◽  
Vol 38 (7) ◽  
pp. 2508-2536 ◽  
Author(s):  
PETER BURTON
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Convex Combination ◽  
Compact Convex ◽  
Type Theorem ◽  
Extreme Points ◽  
Equivalence Classes ◽  
Compact Convex Subset ◽  
Weak Equivalence ◽  
Convex Structure

We analyze the structure of the quotient $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We introduce a variant of an abstract construction of Fritz which encapsulates the convex combination operation on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$. This formalism allows us to define the geometric notion of an extreme point. We also discuss a topology on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ due to Abért and Elek in which it is Polish and compact, and show that this topology is equivalent to others defined in the literature. We show that the convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is compatible with the topology, and as a consequence deduce that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is path connected. Using ideas of Tucker-Drob, we are able to give a complete description of the topological and convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ for amenable $\unicode[STIX]{x1D6E4}$ by identifying it with the simplex of invariant random subgroups. In particular, we conclude that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ can be represented as a compact convex subset of a Banach space if and only if $\unicode[STIX]{x1D6E4}$ is amenable. In the case of general $\unicode[STIX]{x1D6E4}$ we prove a Krein–Milman-type theorem asserting that finite convex combinations of the extreme points of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ are dense in this space. We also consider the space $\text{A}_{{\sim}_{s}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of stable weak equivalence classes and show that it can always be represented as a compact convex subset of a Banach space. In the case of a free group $\mathbb{F}_{N}$, we show that if one restricts to the compact convex set $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})\subseteq \text{A}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$ consisting of the stable weak equivalence classes of free actions, then the extreme points are dense in $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$.

scholarly journals Weak containment of measure-preserving group actions

2019 ◽  
Vol 40 (10) ◽  
pp. 2681-2733 ◽  
Author(s):  
PETER J. BURTON ◽  
ALEXANDER S. KECHRIS
Keyword(s):  
Global Structure ◽  
Equivalence Classes ◽  
The Other ◽  
Unitary Representations ◽  
Weak Equivalence ◽  
Interesting Fact ◽  
Weak Containment ◽  
Probability Spaces ◽  
Measure Preserving Actions ◽  
Standard Probability

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

Evolution of Large-Scale Coronal Structures

1994 ◽  
Vol 144 ◽  
pp. 29-33
Author(s):  
P. Ambrož
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Large Scale ◽  
Coronal Magnetic Field ◽  
Source Surface ◽  
Solar Cycles ◽  
Characteristic Relationship ◽  
The Magnetic Field ◽  
Coronal Structures

AbstractThe large-scale coronal structures observed during the sporadically visible solar eclipses were compared with the numerically extrapolated field-line structures of coronal magnetic field. A characteristic relationship between the observed structures of coronal plasma and the magnetic field line configurations was determined. The long-term evolution of large scale coronal structures inferred from photospheric magnetic observations in the course of 11- and 22-year solar cycles is described.Some known parameters, such as the source surface radius, or coronal rotation rate are discussed and actually interpreted. A relation between the large-scale photospheric magnetic field evolution and the coronal structure rearrangement is demonstrated.

X-ray micro tomography in the scanning electron microscope

1978 ◽  
Vol 36 (1) ◽  
pp. 106-107 ◽  
Author(s):  
W. Brünger
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
Target Surface ◽  
The Body ◽  
X Rays ◽  
Cross Sectional ◽  
X Ray ◽  
Micro Tomography ◽  
Scanning Electron

Reconstructive tomography is a new technique in diagnostic radiology for imaging cross-sectional planes of the human body /1/. A collimated beam of X-rays is scanned through a thin slice of the body and the transmitted intensity is recorded by a detector giving a linear shadow graph or projection (see fig. 1). Many of these projections at different angles are used to reconstruct the body-layer, usually with the aid of a computer. The picture element size of present tomographic scanners is approximately 1.1 mm2.Micro tomography can be realized using the very fine X-ray source generated by the focused electron beam of a scanning electron microscope (see fig. 2). The translation of the X-ray source is done by a line scan of the electron beam on a polished target surface /2/. Projections at different angles are produced by rotating the object.During the registration of a single scan the electron beam is deflected in one direction only, while both deflections are operating in the display tube.

Sign in / Sign up

Export Citation Format

Share Document