scholarly journals Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

10.37236/9422 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Christoph Goldner
Keyword(s):  
General Position ◽  
Tropical Geometry ◽  
Recursive Formula ◽  
Cross Ratio ◽  
Initial Values ◽  
Multiple Cross ◽  
Correspondence Theorem ◽  
The Given ◽  
Rational Degree

Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ points in general position. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio. This paper addresses the question whether there is a general Kontsevich's formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that, we use a correspondence theorem of Tyomkin that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet. We show that our recursive general Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.

scholarly journals Counting tropical rational space curves with cross-ratio constraints

2021 ◽  
Author(s):  
Christoph Goldner
Keyword(s):  
Tropical Geometry ◽  
Rational Curves ◽  
Cross Ratio ◽  
Current Paper ◽  
Higher Dimension ◽  
Rational Space ◽  
Space Curves ◽  
Correspondence Theorem ◽  
Floor Diagrams

AbstractThis is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

How many random digits are required until given sequences are obtained?

1982 ◽  
Vol 19 (03) ◽  
pp. 518-531 ◽  
Author(s):  
Gunnar Blom ◽  
Daniel Thorburn
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Probability Generating Function ◽  
Recursive Formula ◽  
Fixed Number ◽  
Digit Sequence ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

scholarly journals Enumeration of complex and real surfaces via tropical geometry

2018 ◽  
Vol 18 (1) ◽  
pp. 69-100
Author(s):  
Hannah Markwig ◽  
Thomas Markwig ◽  
Eugenii Shustin
Keyword(s):  
Tropical Geometry ◽  
Three Dimensional ◽  
Algebraic Surfaces ◽  
Lattice Path ◽  
Singular Surfaces ◽  
Dimensional Version ◽  
Correspondence Theorem

AbstractWe prove a correspondence theorem for singular tropical surfaces in ℝ3, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin’s lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in ℝ3. As application we show that there are pencils of real surfaces of degreedin ℙ3containing at least (3/2)d3+O(d2) singular surfaces, which is asymptotically comparable to the number 4(d− 1)3of all complex singular surfaces in the pencil. Our result relies on the classification of singular tropical surfaces [12].

scholarly journals Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

2021 ◽  
pp. 1-41
Author(s):  
RENZO CAVALIERI ◽  
PAUL JOHNSON ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Fock Space ◽  
Tropical Geometry ◽  
Geometric Interpretation ◽  
Toric Surfaces ◽  
Tropical Curves ◽  
Hirzebruch Surfaces ◽  
Witten Theory ◽  
Toric Degenerations ◽  
Correspondence Theorem ◽  
Floor Diagrams

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

scholarly journals Genus 0 characteristic numbers of the tropical projective plane

2013 ◽  
Vol 150 (1) ◽  
pp. 46-104 ◽  
Author(s):  
Benoît Bertrand ◽  
Erwan Brugallé ◽  
Grigory Mikhalkin
Keyword(s):  
Projective Plane ◽  
Tropical Geometry ◽  
Classical Problem ◽  
Integer Number ◽  
Hurwitz Numbers ◽  
Formula One ◽  
Characteristic Numbers ◽  
Pass Through ◽  
Finite Integer ◽  
Correspondence Theorem

AbstractFinding the so-called characteristic numbers of the complex projective plane$ \mathbb{C} {P}^{2} $is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given$d$and$g$one has to find the number of degree$d$genus$g$curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is$3d- 1+ g$so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when$g= 0$. Namely, we show that the tropical problem is well posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of$ \mathbb{C} {P}^{2} $in terms of open Hurwitz numbers.

How many random digits are required until given sequences are obtained?

1982 ◽  
Vol 19 (3) ◽  
pp. 518-531 ◽  
Author(s):  
Gunnar Blom ◽  
Daniel Thorburn
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Probability Generating Function ◽  
Recursive Formula ◽  
Fixed Number ◽  
Digit Sequence ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

RECALCULATION OF THE LOADS CURRENT OF ACTIVE MULTI-PORT NETWORKS ON THE BASIS OF PROJECTIVE GEOMETRY

2013 ◽  
Vol 22 (05) ◽  
pp. 1350031 ◽  
Author(s):  
A. A. PENIN
Keyword(s):  
Projective Geometry ◽  
Direct Current ◽  
Power Supply ◽  
Power Supply System ◽  
Traditional Approach ◽  
Cross Ratio ◽  
Supply System ◽  
The Cross ◽  
The Given

For an active multi-port network of a direct current, as a model of a power supply system, the problem of recalculation of the changeable loads currents is considered. The approach on the basis of projective geometry is used for interpretation of changes or "kinematics" of regime parameters of a circuit. The changes of the regime parameters are introduced otherwise, through the cross ratio of four points with use of projective coordinates. Easy-to-use formulas of the recalculation of the currents, which possess the group properties at change of conductivity of the loads, are obtained. It allows expressing the final values of the currents through the intermediate changes of the currents and conductivities. Disadvantages of the traditional approach, which uses the changes of resistance in the form of increments, are shown. The given approach is applicable to the analysis of "flowed" form processes of the various physical natures.

GREEN'S FUNCTIONS ON FRACTALS

Fractals ◽  
2000 ◽  
Vol 08 (04) ◽  
pp. 385-402 ◽  
Author(s):  
JUN KIGAMI ◽  
DANIEL R. SHELDON ◽  
ROBERT S. STRICHARTZ
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problem ◽  
Computer Experiments ◽  
Recursive Formula ◽  
Absolute Maximum ◽  
Initial Value ◽  
Sharp Estimates ◽  
Self Similar ◽  
The Given

For a regular harmonic structure on a post-critically finite (p.c.f.) self-similar fractal, the Dirichlet problem for the Laplacian can be solved by integrating against an explicitly given Green's function. We give a recursive formula for computing the values of the Green's function near the diagonal, and use it to give sharp estimates for the decay of the Green's function near the boundary. We present data from computer experiments searching for the absolute maximum of the Green's function for two different examples, and we formulate two radically different conjectures for where the maximum occurs. We also investigate a local Green's function that can be used to solve an initial value problem for the Laplacian, giving an explicit formula for the case of the Sierpinski gasket. The local Green's function turns out to be unbounded, and in fact not even integrable, but because of cancelation, it is still possible to form a singular integral to solve the initial value problem if the given function satisfies a Hölder condition.

Personal Problem Solving of Partners in Divorce Proceedings

2015 ◽  
Vol 36 (1) ◽  
pp. 82-103 ◽  
Author(s):  
Tereza Soukupova ◽  
Petr Goldmann
Keyword(s):  
Gender Differences ◽  
Problem Solving ◽  
Thematic Apperception Test ◽  
Scoring Systems ◽  
Personal Control ◽  
Personality Characteristics ◽  
Personal Problem ◽  
Thematic Apperception ◽  
And Gender ◽  
The Given

Abstract. The Thematic Apperception Test is one of the most frequently administered apperceptive techniques. Formal scoring systems are helpful in evaluating story responses. TAT stories, made by 20 males and 20 females in the situation of legal divorce proceedings, were coded for detection and comparison of their personal problem solving ability. The evaluating instrument utilized was the Personal Problem Solving System-Revised (PPSS-R) as developed by G. F. Ronan. The results indicate that in relation to card 1, men more often than women saw the cause of the problem as removable. With card 6GF, women were more motivated to resolve the given problem than were men, women had a higher personal control and their stories contained more optimism compared to men’s stories. In relation to card 6BM women, more often than men, used emotions generated from the problem to orient themselves within the problem. With card 13MF, the men’s level of stress was less compared to that of the women, and men were more able to plan within the context of problem-solving. Significant differences in the examined groups were found in those cards which depicted significant gender and parental potentials. The TAT can be used to help identify personality characteristics and gender differences.

Pattern Recognition in Histo-Pathology: Basic Considerations

1982 ◽  
Vol 21 (01) ◽  
pp. 15-22 ◽  
Author(s):  
W. Schlegel ◽  
K. Kayser
Keyword(s):  
Pattern Recognition ◽  
Graph Theory ◽  
Normal Tissue ◽  
Diagnostic Procedures ◽  
Minimal Distance ◽  
Malignant Growth ◽  
Colon Tissue ◽  
First Results ◽  
Tissue Structures ◽  
The Given

A basic concept for the automatic diagnosis of histo-pathological specimen is presented. The algorithm is based on tissue structures of the original organ. Low power magnification was used to inspect the specimens. The form of the given tissue structures, e. g. diameter, distance, shape factor and number of neighbours, is measured. Graph theory is applied by using the center of structures as vertices and the shortest connection of neighbours as edges. The algorithm leads to two independent sets of parameters which can be used for diagnostic procedures. First results with colon tissue show significant differences between normal tissue, benign and malignant growth. Polyps form glands that are twice as wide as normal and carcinomatous tissue. Carcinomas can be separated by the minimal distance of the glands formed. First results of pattern recognition using graph theory are discussed.

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