scholarly journals Monodromy Filtrations and the Topology of Tropical Varieties

2012 ◽  
Vol 64 (4) ◽  
pp. 845-868 ◽  
Author(s):  
David Helm ◽  
Eric Katz
Keyword(s):  
Topological Space ◽  
Betti Numbers ◽  
Geometric Interpretation ◽  
Monodromy Operator ◽  
Natural Class ◽  
Multiplicity Free ◽  
Tropical Varieties

AbstractWe study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, “multiplicity-free” parameterization of Trop(X) by a topological space ГXand give a geometric interpretation of the cohomology of ГXin terms of the action of a monodromy operator on the cohomology ofX. This gives bounds on the Betti numbers of ГXin terms of the Betti numbers ofXwhich constrain the topology of Trop(X). We also obtain a description of the top power of the monodromy operator acting on middle cohomology ofXin terms of the volume pairing on ГX.

scholarly journals A note on the characteristic rank and related numbers

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Ľudovít Balko ◽  
Július Korbaš
Keyword(s):  
Vector Bundle ◽  
Topological Space ◽  
Betti Numbers ◽  
Sharp Inequality ◽  
Grassmann Manifolds ◽  
The Real ◽  
Partitions Of Integers

AbstractThis note quantifies, via a sharp inequality, an interplay between (a)the characteristic rank of a vector bundle over a topological space X (b)the ℤ2-Betti numbers of X, and(c)sums of the numbers of certain partitions of integers.In a particular context, (c) is transformed into a sum of the readily calculable Betti numbers of the real Grassmann manifolds.

Mathematical simulation of error functions of decision-making at tolerance control of measuring techniques availability

Metrologiya ◽  
2020 ◽  
pp. 3-15
Author(s):  
Rustam Z. Khayrullin ◽  
Alexey S. Kornev ◽  
Andrew A. Kostoglotov ◽  
Sergey V. Lazarenko
Keyword(s):  
Measurement Error ◽  
Geometric Interpretation ◽  
Measuring Instruments ◽  
Measured Quantity ◽  
Metrological Examination ◽  
Technical Documentation ◽  
Error Functions ◽  
Measuring Techniques ◽  
Tolerance Control ◽  
Laws Of Distribution

Analytical and computer models of false failure and undetected failure (error functions) were developed with tolerance control of the parameters of the components of the measuring technique. A geometric interpretation of the error functions as two-dimensional surfaces is given, which depend on the tolerance on the controlled parameter and the measurement error. The developed models are applicable both to theoretical laws of distribution, and to arbitrary laws of distribution of the measured quantity and measurement error. The results can be used in the development of metrological support of measuring equipment, the verification of measuring instruments, the metrological examination of technical documentation and the certification of measurement methods.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

The Spatial Dimensions of Social Networks

2020 ◽  
pp. 367-383
Author(s):  
Zachary P. Neal
Keyword(s):  
Social Networks ◽  
Topological Space ◽  
Network Science ◽  
Spatial Dimensions

The first law of geography holds that everything is related to everything else, but near things are more related than distant things, where distance refers to topographical space. If a first law of network science exists, it would similarly hold that everything is related to everything else, but near things are more related than distant things, but where distance refers to topological space. Frequently these two laws collide, together holding that everything is related to everything else, but topographically and topologically near things are more related than topographically and topologically distant things. The focus of the spatial study of social networks lies in exploring a series of questions embedded in this combined law of geography and networks. This chapter explores the questions that have been asked and the answers that have been offered at the intersection of geography and networks.

Extremal Betti numbers of graded ideals

2003 ◽  
Vol 43 (3-4) ◽  
pp. 235-244 ◽  
Author(s):  
Marilena Crupi ◽  
Rosanna Utano
Keyword(s):  
Betti Numbers

scholarly journals A Geometric Perspective on Information Plane Analysis

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 711
Author(s):  
Mina Basirat ◽  
Bernhard C. Geiger ◽  
Peter M. Roth
Keyword(s):  
Neural Network ◽  
Mutual Information ◽  
Geometric Interpretation ◽  
Neural Network Training ◽  
Neural Network Learning ◽  
Network Learning ◽  
Plane Analysis ◽  
Network Training ◽  
Hidden Layer ◽  
The Impact

Information plane analysis, describing the mutual information between the input and a hidden layer and between a hidden layer and the target over time, has recently been proposed to analyze the training of neural networks. Since the activations of a hidden layer are typically continuous-valued, this mutual information cannot be computed analytically and must thus be estimated, resulting in apparently inconsistent or even contradicting results in the literature. The goal of this paper is to demonstrate how information plane analysis can still be a valuable tool for analyzing neural network training. To this end, we complement the prevailing binning estimator for mutual information with a geometric interpretation. With this geometric interpretation in mind, we evaluate the impact of regularization and interpret phenomena such as underfitting and overfitting. In addition, we investigate neural network learning in the presence of noisy data and noisy labels.

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