The study classifies isotopy types of 3-periodic nets and their embeddings.
problem Classifying isotopy types of 3-periodic nets and their embeddings.
method Definition of entangled embedded periodic nets, classification methodology using linear graph knots.
result Enumeration and classification of isotopy classes for various 3-periodic nets.
Study finds the shortest triply periodic graph spanning a cubic lattice.
problem Finding the shortest periodic graph with a fixed volume.
method Analyzes the body centred cubic lattice and the gyroid surface.
result The shortest graph is the srs network with K4 quotient. Constructs graphs with singularities in a special space.
problem Creating graphs with specific singularities in a unique space.
method Using Weierstrass representation for minimal surfaces.
result Constructs entire singly periodic graphs with isolated cone-like singularities.
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
Complexity of periodic graphs linked to Mahler measure.
problem Understanding growth rates of graph complexity.
method Defining graph complexity and linking it to Mahler measure.
result Lehmer's question about polynomial roots is equivalent to complexity growth of certain graphs.
New graphs model periodic mapping classes and Seifert manifolds.
problem Modeling periodic mapping classes and Seifert manifolds.
method Introduced general tête-à-tête graphs and algorithms to model and reverse model Seifert manifolds and periodic mapping classes.
result General tête-à-tête graphs model all periodic mapping classes.
A graph G is said to be p-periodic, if the automorphism group Aut(G) contains an element of order p which preserves no edges. In this paper, we investigate the behavior of graph polynomials (Negmai and Tutte) with respect to graph periodicity. In particular, we prove that if p is a prime, then the coefficient…
Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.
problem Complexity of signed graphs and its relation to Alexander polynomials.
method Definition of graph complexity using Laplacian matrix and Mahler measure, linking to Alexander polynomials and Lehmer's question.
result Complexity growth of signed graphs is related to the growth rate of Alexander polynomials.
The study constructs periodic surfaces using graph theory and applies cyclically branched coverings to identify their conformal type.
problem Constructing and identifying the conformal type of periodic surfaces with a given geometric structure.
method Graph theory and cyclically branched coverings.
result Explicit cone metrics on compact Riemann surfaces can be realized as the quotient of triply periodic polyhedral surfaces.
The paper studies maps on graphs and their fibers, focusing on hyperelliptic graphs.
problem Analyzing maps on graphs and their fibers.
method Analyzes maps Phi and Phi* on Culler--Vogtmann's outer space CV_n and its quotient T_n, focusing on hyperelliptic graphs.
result Fibers of Phi in T_n are aspherical and pi_1-injective subspaces.
New method shows pseudo-Anosov flows on graph manifolds can be simplified.
problem Understanding pseudo-Anosov flows on graph manifolds.
method Constructing a partial Birkhoff section with genus one components that misses finitely many closed orbits.
result Every pseudo-Anosov flow on a graph manifold is almost equivalent to a totally periodic flow or a suspension Anosov flow.
Study homology of periodic cell complexes using quotient spaces and spectral sequences.
problem Quantifying homology in periodic cell complexes.
method Finite representation of periodic cell complexes, Mayer-Vietoris spectral sequence.
result Full recovery of homology generators for d-periodic graphs. We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period. The proof of the main result is based on computing the Yamada skein algebra of the solid torus then proving that this algebra injects into the Kauffman bracket …
We describe the family of minimal graphs on strips with boundary values ±∞ disposed alternately on edges of length one, and whose conjugate graphs are contained in horizontal slabs of width one in R3. We can obtain as limits of such graphs the helicoid, all the doubly periodic Scherk minimal surfac…
Study of discrete period matrices on embedded graphs, relating to Riemann surfaces.
problem Understanding discrete conformal structures on surfaces via period matrices.
method Combinatorial interpretation of period matrices, using homological quasi-trees and Laplacian determinants.
result Derived a combinatorial analogue of the Weil-Petersson potential and related it to homological quasi-trees.
New chiral minimal surfaces derived from quartz network.
problem Finding new triply-periodic minimal surfaces.
method Using dual graphs of quartz and its dual, generating area-minimizing meshes, and identifying flat point structures.
result Identified a new family of chiral triply-periodic minimal surfaces.
Lower bound on volumes of special mapping tori.
problem Calculating the minimum volume of compactified mapping tori.
method Using strongly irreducible end-periodic homeomorphisms and properties of pants graphs.
result Volume of compactified mapping tori is comparable to the translation length of the homeomorphism on pants graphs.
Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.
problem Counting periodic geodesics with specific commutator structure.
method Reduction to counting critical realizations of trivalent graphs.
result Asymptotic count of geodesics with bounded length and commutator structure.
The relation between time series irreversibility and entropy production has been recently investigated in thermodynamic systems operating away from equilibrium. In this work we explore this concept in the context of financial time series. We make use of visibility algorithms to quantify in graph-theoretical terms time …
We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of symmetries. The periodic mapping classes can in turn be used to construct sequences of pse…
We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorit…
New rigidity theorem for Scherk's surfaces and flat structures.
problem Characterizing and proving uniqueness of minimal surfaces and flat structures.
method Combining curvature estimates and geometric harmonic functions to construct fresh uniqueness results.
result Periodic minimal surfaces admit new uniqueness results.
The study finds infinitely many periodic orbits that can be used to modify Anosov flows.
problem Can surgeries on periodic orbits of Anosov flows produce equivalent flows?
method Analyzing suspension Anosov flows, the study identifies pairs of periodic orbits that can be used to modify the flow.
result For some suspension Anosov flows, there exist infinitely many pairs of periodic orbits that can be used to modify the flow.
Study mapping classes using curve singularities and pseudo-periodic homeomorphisms.
problem Characterize mapping classes representable by specific twists and homeomorphisms.
method Use pseudo-periodic homeomorphisms and curve singularities, introduce new graph and twist types.
result Characterize mapping classes that can be represented by tête-à-tête twists and generalize to boundary-free periodic classes.
Study infers volatility indicators from Bitcoin blockchain data.
problem Predicting extreme price volatility in Bitcoin.
method Non-negative decomposition of Bitcoin transaction graphs.
result EWI provides more predictive information than other methods.
Upper bound on 3-manifold volumes from surface homeomorphisms.
problem Bounding volumes of 3-manifolds from surface homeomorphisms.
method Using end-periodic homeomorphisms and pants graphs.
result Upper bound on infimal hyperbolic volume is asymptotically sharp.
We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …
Derives a formula for premium payments in multiple health states.
problem Valuation of premiums in complex health insurance models.
method Combines actuarial techniques with graph optimization.
result General matrix formula for net period premium paid.
For a closed oriented 3-manifold Y we define n(Y) to be the minimal non-negative number such that in each homotopy class of non-singular vector fields of Y there is a Morse-Smale vector field with less or equal to n(Y) periodic orbits. We combine the construction process of Morse-Smale flows given in [2] with h…
Mixed tête-à-tête twists model monodromies for isolated surface singularities.
problem Modeling monodromies for isolated complex surface singularities.
method Characterizing pseudo-periodic automorphisms as mixed tête-à-tête twists.
result Tête-à-tête twists coincide with monodromies for isolated surface singularities.
Formulae for Seiberg-Witten invariants derived from graph combinatorics.
problem Calculating Seiberg-Witten invariants for 3-manifolds.
method Combining Poincaré series and counting functions of a graph to derive surgery formulae.
result The periodic constant difference of invariants for surgeries on 3-manifolds.
Study on stable translation lengths of surface homeomorphisms and their approximations.
problem Understanding stable translation lengths of homeomorphisms and their finite approximations.
method Comparing stable translation lengths of homeomorphisms and their finite approximations on curve graphs.
result Stable translation length of homeomorphisms with dense periodic points equals the supremum of their approximations.
Let M be a rational homology sphere plumbed 3-manifold associated with a connected negative definite plumbing graph. We show that its Seiberg-Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial. For this, we construct the corresponding polytopes from the plumbing graphs toget…
In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by the period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H. Karcher. The technics that use is to solve a Jenkins-Serrin problem over a strip…
We consider the space of all representations of the commutator subgroup of a knot group into Z/p, p is prime. As proven by D. Silver and S. Williams, this space can be completely described by a finite oriented graph. We describe the lengths of cycles in this graph.
Nguyen's solutions converge to a grim reaper and plane.
problem Classifying semigraphical translators for mean curvature flow.
method Constructing a one-parameter family of translating solutions.
result Nearly complete classification of semigraphical translators.
From a graph G with constant valency v and a (non-compact) manifold C with v boundary components, we build a G-periodic manifold M. This process gives a class of topologically infinite manifolds which generalizes periodic manifolds and includes all riemannian coverings with finitely generated deck-group. Ou…
This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.
problem Constructing a discrete theory of real Riemann surfaces.
method Using quad-graphs and linear discretization of Cauchy-Riemann equations, constructing a symplectic homology basis.
result The discrete period matrix has the same canonical decomposition as in the smooth setting.
Model for material elasticity and plasticity using networks.
problem Understanding the elasticity and plasticity of materials.
method Developed a mathematical model based on networks, defining tension tensor for periodic graphs.
result The model explains elasticity and plasticity through local moves on graphs.
In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show …
NT-DCEG models dynamic processes with specific periodicity, proving useful for multivariate processes.
problem Modeling dynamic processes with specific periodicity and context-specific independences.
method Developed a method to distributively construct NT-DCEG models and used graph topology to infer context-specific independences.
result NT-DCEGs contain all discrete N time-slice Dynamic Bayesian Networks as special cases.
CHILI datasets tackle inorganic nanomaterials, advancing graph machine learning.
problem Challenges in modelling inorganic crystalline materials and nanomaterials with graph ML.
method Presented two large-scale datasets of inorganic nanomaterials, defined property and structure prediction tasks.
result Benchmarked performance of graph ML methods on inorganic nanomaterials, highlighting areas for future work.
We construct most symmetric Saddle towers in Heisenberg space i.e. periodic minimal surfaces that can be seen as the desingularization of vertical planes intersecting equiangularly. The key point is the construction of a suitable barrier to ensure the convergence of a family of bounded minimal disks. Such a barrier is …
We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz-Minkowski space $ł^3=(\r^3,dx_1^2+dx_2^2-dx_3^2),$ with fundamental piece having a finite number (n+1) of singularities, is a real analytic manifold of dimension 3n+4. The underlyin…
3D contact forms have supporting decompositions, leading to entropy results.
problem Existence of supporting decompositions for contact forms in 3D.
method Proving existence of broken book decompositions for nondegenerate contact forms.
result Nondegenerate Reeb vector fields on 3-manifolds have positive entropy or infinitely many periodic orbits.
We present a graph-based variational algorithm for multiclass classification of high-dimensional data, motivated by total variation techniques. The energy functional is based on a diffuse interface model with a periodic potential. We augment the model by introducing an alternative measure of smoothness that preserves s…
We show how to construct, for each r≥3, an ageometric, fully irreducible φ∈Out(Fr) whose ideal Whitehead graph is the complete graph on 2r−1 vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal …
Researchers create new triply periodic minimal surfaces by gluing saddle towers.
problem Creating triply periodic minimal surfaces without symmetry constraints.
method Gluing Karcher-Scherk saddle towers with phase differences and balancing under vertical interaction.
result Expands known triply periodic minimal surfaces into new 5-parameter families.