scholarly journals Directed Rooted Forests in Higher Dimension

10.37236/5819 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Olivier Bernardi ◽  
Caroline J. Klivans
Keyword(s):  
Generating Function ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Graph Laplacian ◽  
Connected Components ◽  
Higher Dimension ◽  
Cell Complexes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Rooted Forest

For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

scholarly journals Rigidity and trace properties of divergence-measure vector fields

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Gian Paolo Leonardi ◽  
Giorgio Saracco
Keyword(s):  
Vector Fields ◽  
Extremal Solution ◽  
Divergence Measure ◽  
Higher Dimension ◽  
Regular Domain ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Divergence Free Vector ◽  
Prescribed Mean Curvature Equation ◽  
Rigidity Property

AbstractWe consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where {\varphi(t)} is a non-negative convex function vanishing only at {t=0}. We show that this property is always satisfied in dimension {n=2}, while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when {\varphi(t)=ct^{2}}) in dimension {n\geq 4}. The validity of the quadratic rigidity, which we prove in dimension {n=2}, implies the existence of the trace of a divergence-measure vector field ξ on an {\mathcal{H}^{1}}-rectifiable set S, as soon as its weak normal trace {[\xi\cdot\nu_{S}]} is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

scholarly journals Connected components of the space of proper gradient vector fields

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Maciej Starostka
Keyword(s):  
Vector Fields ◽  
Connected Components ◽  
Gradient Vector ◽  
Proper Maps

AbstractWe show that there exist two proper gradient vector fields on $$\mathbb {R}^n$$ R n which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

scholarly journals Detection of the homotopy type of an object using differential invariants of an approximating map

2019 ◽  
Vol 43 (4) ◽  
pp. 611-617
Author(s):  
S.V. Kurochkin
Keyword(s):  
Neural Network ◽  
Data Analysis ◽  
Homotopy Type ◽  
Morse Theory ◽  
Object Representation ◽  
Differential Invariants ◽  
Topological Data Analysis ◽  
Higher Dimension ◽  
Differential Topology ◽  
Open Questions

A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

GENERATING FUNCTION FOR CUBIC INTERACTION VERTICES OF HIGHER SPIN FIELDS IN ANY DIMENSION

1993 ◽  
Vol 08 (25) ◽  
pp. 2413-2426 ◽  
Author(s):  
R. R. METSAEV
Keyword(s):  
Generating Function ◽  
Wide Class ◽  
Light Cone ◽  
Arbitrary Dimension ◽  
Space Time ◽  
Relativistic Dynamics ◽  
Higher Spin ◽  
Spin Fields ◽  
Higher Spin Fields

Using the light-cone formulation of relativistic dynamics we present a wide class of cubic interaction vertices for higher spin fields of any symmetry in arbitrary dimension of space-time. The solution is obtained in terms of generating function for interaction vertices.

scholarly journals Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes

2019 ◽  
Vol 29 (06) ◽  
pp. 951-1007
Author(s):  
Nithi Rungtanapirom ◽  
Jakob Stix ◽  
Alina Vdovina
Keyword(s):  
Arbitrary Dimension ◽  
Congruence Subgroups ◽  
Quaternion Algebras ◽  
Ramanujan Graphs ◽  
Dimension Formula ◽  
Cubical Sets ◽  
Higher Dimensional ◽  
Residually Finite Groups ◽  
Doubling Construction ◽  
Effective Use

We construct vertex transitive lattices on products of trees of arbitrary dimension [Formula: see text] based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher-dimensional cubical version of Ramanujan graphs (optimal expanders).

STRONG COUPLING FROM TAU LEPTON DECAYS

2013 ◽  
Vol 28 (24) ◽  
pp. 1360006 ◽  
Author(s):  
MATTHIAS JAMIN
Keyword(s):  
Perturbative Qcd ◽  
Strong Coupling ◽  
Theoretical Description ◽  
Operator Product ◽  
Hadronic Decays ◽  
Open Questions ◽  
Higher Dimensional ◽  
Low Energies ◽  
Consistent Manner ◽  
Lepton Decays

A determination of the strong coupling αs at rather low energies is possible through the analysis of hadronic decays of the τ lepton. In turn, the low energy necessitates sufficient control over perturbative QCD corrections, the nonperturbative condensate contributions in the framework of the operator product expansion (OPE), as well as corrections going beyond the OPE, the duality violations (DVs). Perturbative QCD uncertainties arise from open questions regarding the renormalization group resummation of the series. The fit quantities are moment integrals of the τ spectral function data in a certain energy window and care should be taken to have good perturbative behavior of those moments as well as control over higher-dimensional operator corrections. Furthermore, all parameters occurring in the theoretical description should be extracted from fits to the data in a self-consistent manner.

scholarly journals Mixing in High-Dimensional Expanders

2017 ◽  
Vol 26 (5) ◽  
pp. 746-761 ◽  
Author(s):  
ORI PARZANCHEVSKI
Keyword(s):  
Arbitrary Dimension ◽  
Simplicial Complexes ◽  
Similar Type ◽  
High Dimensional ◽  
Hodge Laplacian ◽  
Laplace Spectrum ◽  
Open Questions

We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to asmixing, orpseudo-randomness). Recently, an analogue of this lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of pseudo-randomness as in graphs. In this paper we remove the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex. We discuss various applications and present some open questions.

The Structure of the Polytope of Hereditary Information

2019 ◽  
Vol 8 (2) ◽  
pp. 7-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Nucleic Acid ◽  
Hydrogen Bonds ◽  
Phosphoric Acid ◽  
High Intensity ◽  
Information Exchange ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Nitrogen Bases ◽  
Higher Dimensional ◽  
Low Dimensional

The representations of the sugar molecule and the residue of phosphoric acid in the form of polytopes of higher dimension are used. Based on these ideas and their simplified three-dimensional images, a three-dimensional image of nucleic acids is constructed. The geometry of the neighborhood of the compound of two nucleic acid helices with nitrogen bases has been investigated in detail. It is proved that this neighborhood is a cross-polytope of dimension 13 (polytope of hereditary information), in the coordinate planes of which there are complementary hydrogen bonds of nitrogenous bases. The structure of this polytope is defined, and its image is given. The total incident flows from the low-dimensional elements to the higher-dimensional elements and vice versa of the hereditary information polytope are calculated equal to each other. High values of these flows indicate a high intensity of information exchange in the polytope of hereditary information that ensures the transfer of this information.

Automatic First Arrival Picking Workflow by Global Path Tracing

Geophysics ◽  
2021 ◽  
pp. 1-46
Author(s):  
Dongliang Zhang ◽  
Tong W. Fei ◽  
Song Han ◽  
Constantine Tsingas ◽  
Yi Luo ◽  
...  
Keyword(s):  
Field Data ◽  
Amplitude Ratio ◽  
Optimal Path ◽  
Two Dimensions ◽  
Higher Dimension ◽  
Path Tracing ◽  
Higher Dimensional ◽  
First Arrival ◽  
The Stability ◽  
The One

It can be challenging to pick high quality first arrivals on noisy seismic datasets. The stability and smoothness criteria of the picked first arrival are not satisfied for datasets with shingles and interferences from unexpected and backscattered events. To improve first arrival picking, we propose an automatic first arrival picking workflow using global path tracing to find a global solution for first arrival picking with the condition of smoothness of the traced path. The proposed methodology is composed of data preconditioning, global path tracing, and final addition of traced and piloted travel times to compute the total picked travel time. We propose several ways to precondition the dataset, including the use of amplitude and amplitude ratio with and without a pilot. 2D global path tracing is comprised of two steps, namely, accumulation of energy on the potential path and backtracking of the optimal path with a strain factor for smoothness. For higher dimensional datasets, two strategies were adopted. One was to split the higher-dimension data into sub-domains of two dimensions to which 2D global path tracing was applied. The alternative method was to smooth the preconditioned dataset in directions except for the one used to trace the path before applying 2D global path tracing. Next, we discussed the importance of choosing proper parameters in both data preconditioning and constraining global path tracing. We demonstrated the robustness and stability of the proposed automatic first arrival picking via global path tracing using synthetic and field data examples.

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