scholarly journals Computing trisections of 4-manifolds

2018 ◽  
Vol 115 (43) ◽  
pp. 10901-10907 ◽  
Author(s):  
Mark Bell ◽  
Joel Hass ◽  
Joachim Hyam Rubinstein ◽  
Stephan Tillmann
Keyword(s):  
Complexity Bounds ◽  
Four Manifolds

We describe an algorithm to compute trisections of orientable four-manifolds using arbitrary triangulations as input. This results in explicit complexity bounds for the trisection genus of a 4-manifold in terms of the number of pentachora (4-simplices) in a triangulation.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

Approximate Mining of Frequent -Subgraph Patterns in Evolving Graphs

2021 ◽  
Vol 15 (3) ◽  
pp. 1-35
Author(s):  
Muhammad Anis Uddin Nasir ◽  
Cigdem Aslay ◽  
Gianmarco De Francisci Morales ◽  
Matteo Riondato
Keyword(s):  
Sample Size ◽  
State Of The Art ◽  
Empirical Evaluation ◽  
Connected Subgraph ◽  
Dynamic Graphs ◽  
High Quality ◽  
Complexity Bounds ◽  
Approximation Quality ◽  
Uniform Sample ◽  
Waiting For Godot

“Perhaps he could dance first and think afterwards, if it isn’t too much to ask him.” S. Beckett, Waiting for Godot Given a labeled graph, the collection of -vertex induced connected subgraph patterns that appear in the graph more frequently than a user-specified minimum threshold provides a compact summary of the characteristics of the graph, and finds applications ranging from biology to network science. However, finding these patterns is challenging, even more so for dynamic graphs that evolve over time, due to the streaming nature of the input and the exponential time complexity of the problem. We study this task in both incremental and fully-dynamic streaming settings, where arbitrary edges can be added or removed from the graph. We present TipTap , a suite of algorithms to compute high-quality approximations of the frequent -vertex subgraphs w.r.t. a given threshold, at any time (i.e., point of the stream), with high probability. In contrast to existing state-of-the-art solutions that require iterating over the entire set of subgraphs in the vicinity of the updated edge, TipTap operates by efficiently maintaining a uniform sample of connected -vertex subgraphs, thanks to an optimized neighborhood-exploration procedure. We provide a theoretical analysis of the proposed algorithms in terms of their unbiasedness and of the sample size needed to obtain a desired approximation quality. Our analysis relies on sample-complexity bounds that use Vapnik–Chervonenkis dimension, a key concept from statistical learning theory, which allows us to derive a sufficient sample size that is independent from the size of the graph. The results of our empirical evaluation demonstrates that TipTap returns high-quality results more efficiently and accurately than existing baselines.

scholarly journals Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

2021 ◽  
Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff
Keyword(s):  
Computer Science ◽  
Affine Transformation ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Theoretical Computer Science ◽  
Sums Of Squares ◽  
Theoretical Computer ◽  
Complexity Bounds ◽  
Polynomial Optimization Problems ◽  
Transformation Of Variables

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

Quantum fluctuations of the metric, three-geometries and conformal embeddings in four-manifolds

2021 ◽  
Author(s):  
Simon Davis
Keyword(s):  
Path Integrals ◽  
Invariance Group ◽  
Two Dimensional ◽  
Four Dimensions ◽  
Physical Conditions ◽  
Exponential Expansion ◽  
Cotangent Bundles ◽  
Four Manifolds ◽  
Conformal Embeddings ◽  
Natural Relation

In this paper, connections between the path integrals for four-dimensional quantum gravity and string theory are emphasized. It is shown that there is a natural relation between these two path integrals based on the theorems on embeddings of two-dimensional surfaces in four dimensions and four-dimensional manifolds in ten dimensions. The isometry groups of the three-geometries that are spatial hypersurfaces confomally embedded in the four-manifolds are required to be subgroups of [Formula: see text], which is the invariance group of the Pfaffian differential system satisfied by one form in the cotangent bundles on the four-manifolds. Based on this and other physical conditions, the three-geometries are restricted to be [Formula: see text], [Formula: see text] and [Formula: see text] with a boundary, which may be included in the quantum gravitational path integral over four-manifolds which are closed at initial times followed by an exponential expansion compatible with supersymmetry.

scholarly journals A note on positive-definite, symplectic four-manifolds

2018 ◽  
Vol 21 (1) ◽  
pp. 257-270 ◽  
Author(s):  
Jennifer Hom ◽  
Tye Lidman
Keyword(s):  
Positive Definite ◽  
Four Manifolds
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