Paper recovers lattice signal partitions efficiently.
problem Estimating lattice partition from noisy data.
method Uses dyadic CART for computationally-efficient partition recovery.
result Consistently estimates partition with optimal error rate.
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48 results for “partition lattice”
Researchers prove conjecture about contractible subcomplexes in noncrossing partition link.
problem Understanding contractibility of subcomplexes in the noncrossing partition link.
method Combining contractibility of flag complexes' stars with noncrossing hypertrees theory.
result Proved conjecture about contractible subcomplexes in the noncrossing partition link.
New spectral sequence helps determine when cohomology of configuration spaces is finite.
problem Determining when cohomology of configuration spaces is finitely generated.
method Derived a spectral sequence from the homological algebra of the partition lattice.
result Criterion for finitely generated module .
GPU-accelerated particle methods outperform neural samplers in LFT benchmarks.
problem High-dimensional multimodal sampling problems in lattice field theory.
method GPU-accelerated particle Monte Carlo methods (Sequential Monte Carlo and nested sampling).
result These methods match or outperform neural samplers in sample quality and wall-clock time.
We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ``ultra-local'' Poisson bracket algebra defined on di…
Ehrenborg and Jung recently related the order complex for the lattice of d-divisible partitions with the simplicial complex of pointed ordered set partitions via a homotopy equivalence. The latter has top homology naturally identified as a Specht module. Their work unifies that of Calderbank, Hanlon, Robinson, and Wach…
For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain.
There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the arrangement still holds. An application of the theory shows that all minimal cellular …
Characterizes elements in braid conjugacy classes using noncrossing partitions.
problem Conjugacy problem for periodic braids.
method Characterization of elements in super summit set via noncrossing partitions.
result Directly provides a conjugating element and determines the size of the super summit set.
This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group on the cohomology of the configuration space of ordered points in . This cohomology is known to vanish outside of dimensions divisible by ; it is shown here that the -…
The paper proves actions of lattices in higher rank groups have cost one.
problem Fixed price question for higher rank semisimple Lie groups.
method Low intensity Poisson point processes and geometry of Voronoi tessellations.
result Proves all probability measure preserving actions of lattices in higher rank groups have cost one.
A new discretisation of a doubled, i.e. BF, version of the pure abelian Chern-Simons theory is presented. It reproduces the continuum expressions for the topological quantities of interest in the theory, namely the partition function and correlation function of Wilson loops. Similarities with free spinor field theory a…
New framework for detecting complex interactions in multivariate data.
problem Insufficient pairwise measures fail to capture multivariate data complexities.
method Lattice theory and operator functions to derive higher-order information-theoretic measures.
result Streitberg Information fully characterizes all interactions among variables.
New measures and tests for high-order interactions in complex data.
problem Challenges in capturing high-order interactions in multivariate data.
method Hierarchy of -order interaction measures and kernel-based tests.
result Established statistical significance of high-order interactions.
New combinatorial method connects knot invariants to reflection groups.
problem Computing knot invariants using combinatorial techniques.
method Relating dual braid group generators, Hecke images of pure braids, and reflection groups.
result The -HOMFLYPT polynomial can be computed as a solution to factorization problems.
Let G be a torus acting linearly on a complex vector space M, and let X be the list of weights of G in M. We determine the equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit di…
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.
Meanders in a Cayley graphmath.CO
A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph of the symmetric group S_n as generated by all (n choose 2) transpositions. Let …
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
New geometric object for polynomials simplifies complex data.
problem Understanding the combinatorial and geometric properties of polynomials.
method Introducing a compact planar 2-complex for polynomials with distinct roots.
result Extracts combinatorial data from a geometric structure of polynomials.
Approximate inference in high-dimensional, discrete probabilistic models is a central problem in computational statistics and machine learning. This paper describes discrete particle variational inference (DPVI), a new approach that combines key strengths of Monte Carlo, variational and search-based techniques. DPVI is…
Hybrid subgroups found in non-arithmetic PU(2,1) lattices.
problem Exploring hybrid subgroups in non-arithmetic PU(2,1) lattices.
method Exploring hybrid subgroups of certain non-arithmetic lattices in PU(2,1). Showing that Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
result Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
New non-arithmetic lattice found in PU(3,1)
problem Arithmeticity of Couwenberg-Heckman-Looijenga lattices
method Study of arithmeticity and non-arithmetic lattices in PU(n,1)
result Found a non-arithmetic lattice in PU(3,1) not commensurable to Deligne-Mostow lattice
New property identifies arithmetic lattices from nonuniform lattices.
problem Characterizing arithmetic lattices among nonuniform lattices.
method Introduced Bounded Clustering (B-C) property.
result B-C property uniquely identifies arithmetic lattices.
New lattice stick knot condition identified.
problem Determining if 2D lattice knots project to 3D lattice sticks.
method Provided a necessary and sufficient condition.
result Identified a condition for 2D lattice knots to project to 3D lattice sticks.
Simplified proof for lattice link projections.
problem Necessary and sufficient condition for lattice link projections.
method Shorter and simpler proof of existing result.
result Simplified proof for lattice link projections.
We study 4d superconformal indices for a large class of N=1 superconformal quiver gauge theories realized combinatorially as a bipartite graph or a set of "zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we call a "double Yang-Baxter move", gives the Seiberg duality of the gauge theory, and t…
Comparison of Lattice-Free and Lattice-Based Sequence Discriminative Training Criteria for LVCSReess.AS
This work compares lattice-free and lattice-based training criteria for LVCSR.
problem Improving acoustic model performance in speech recognition.
method Direct comparison of lattice-free and lattice-based sequence discriminative training criteria using GPU.
result Lattice-free MMI performance is comparable to lattice-based criteria, while lattice-based sMBR remains superior.
Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.
problem Determining the stick number and edge length of knots in a hexagonal lattice.
method Introducing a linear transformation between lattices to prove strict inequalities and classifying knots.
result Only trefoil and figure-eight knots are 11-stick knots in the hexagonal lattice.
Extended dual Coxeter and Artin groups theory to rank-three systems.
problem Extend dual Coxeter and Artin groups theory to rank-three systems.
method Geometric, combinatorial, and topological techniques.
result Proved the conjecture, triviality of the center, and solubility of the word problem for rank-three Artin groups.
We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology o…
New lattices from 4-manifolds show manifold properties.
problem Identifying distinct 4-manifolds from their lattices.
method Examined smooth, simply-connected 4-manifolds bounded by fixed homology 3-spheres.
result Found two distinct sets of Niemeier lattices.
Arithmetic groupsmath.GT
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
Upper bound for lattice stick number of spatial graphs.
problem Finding the minimum number of sticks in a cubic lattice to represent spatial graphs.
method Defined lattice stick number for spatial graphs and presented an upper bound in terms of crossing number.
result An upper bound for the lattice stick number of spatial graphs.
Triangulates permutahedra for Coxeter groups, revealing braid group connections.
problem Triangulating permutahedra for Coxeter groups.
method Constructs triangulations using total linear stability and height functions.
result Explicitly relates two braid group presentations.
This article classifies reflective hyperbolic lattices of rank 4.
problem Classifying reflective hyperbolic lattices of a specific rank.
method Using automorphism groups and reflections to classify lattices.
result Complete classification of 1.2-reflective maximal anisotropic lattices of rank 4.
New thin subgroups found in special linear groups via bending techniques.
problem Finding thin subgroups of lattices in special linear groups.
method Techniques from convex projective geometry.
result Infinitely many non-commensurable lattices with thin subgroups.
New rigidity theorem for product of lattices.
problem Understanding quasi-isometry of product lattices.
method Demonstrated rigidity for product of non-uniform rank one lattice and nilpotent lattice.
result Any quasi-isometric group is an extension of a non-uniform rank one lattice by a nilpotent lattice.
This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…
This paper proves the trefoil and figure-8 knots have the smallest lattice stick numbers.
problem Determining the minimum number of straight line segments for knot constructions in a cubic lattice.
method Mathematical proof for specific knot types (trefoil and figure-8) with lattice stick numbers less than 15.
result The trefoil and figure-8 knots are the only knot types with lattice stick numbers less than 15.
The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.
problem Locally finite 2-complexes and their automorphism groups contain incommensurable lattices.
method Constructing lattices in combinatorial models of Baumslag-Solitar groups and analyzing their properties.
result The constructed lattices are incommensurable and have specific properties like isomorphic Cayley graphs.
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a -theoretic formula for an index-type invariant.
result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
Vertex distortion measures how far lattice knots deviate from straight lines.
problem Measuring how much lattice knots deviate from straight paths.
method Analogous to smooth knots, study vertex distortion in lattice knots.
result Vertex distortion is 1 only for the unknot and can be arbitrarily high.
Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…
The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.
problem Residual finiteness of lattices in and existence of smooth projective surfaces.
method Proved residual finiteness of certain lattices and constructed surfaces using central extensions.
result First examples of residually finite lattices in and construction of surfaces with specific fundamental groups.
Proves properties of arithmetic lattices and hyperbolic manifolds.
problem Properties of arithmetic lattices and hyperbolic manifolds.
method Study of normalizers of lattices and subgroup growth theory.
result Every arithmetic lattice has the property of being the normalizer of many sublattices.
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…
Quantitative CT predicts ILD patterns and prognosis.
problem Diagnosing and predicting prognosis of fibrosing ILD patterns.
method High-resolution CT texture features, TM model for classification and survival analysis.
result TM model outperforms histogram-based model in distinguishing UIP from non-UIP patterns and allows for survival group partitioning.