New proof shows integer norms on integer lattices are polyhedra.
problem Characterizing norms on integer lattices.
method New proof of Thurston's theorem.
result Unit ball of integer norms on integer lattices is a polyhedron.
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48 results for “integer inequalities”
The paper proves inequalities for lattice eigenvalues, answering a question.
problem Finding bounds for Dirichlet Laplace eigenvalues on integer lattices.
method Proving analogues of existing inequalities for eigenvalues.
result Answers a question posed by Chung and Oden about eigenvalues on integer lattices.
Study inequalities between knot invariants and compute new bounds.
problem Understanding inequalities and bounds between knot invariants.
method Directed graph of inequalities, database construction, systematic transitivity criterion.
result 139 new exact values for unknotting number and doubly slice genus.
We give a very short and rather elementary proof of Gromov's filling volume inequality for n-dimensional Lipschitz cycles (with integer and Z_2-coefficients) in -spaces. This inequality is used in the proof of Gromov's systolic inequality for closed aspherical Riemannian manifolds and is often regarded as the…
We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the -skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…
Efficient algorithm for cell detection and segmentation in crowded images.
problem Instance segmentation in biological images with overlapping cells.
method Column generation approach with exact optimization and odd set inequalities.
result Rapid and accurate instance segmentation on multiple datasets.
The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.
problem Failure of combinatorial isoperimetric inequality in the free factor complex.
method Construction of a coarsely Lipschitz function from the upward link of a free factor to integers.
result A loop in the free factor complex requires linearly growing number of 2-simplices to fill.
Researchers find the minimal genus for a specific type of surface bundle.
problem Determining the minimal genus of bundles.
method Constructing embedded surfaces to find exact minimal genus values.
result Exact values of minimal genus functions for bundles are determined.
The paper establishes a new inequality for knot invariants and applies it to show differences between two invariants can be arbitrarily large.
problem Defining and analyzing a new knot invariant and proving its properties.
method Using the positive -twisted double of a knot, the paper defines and proves an inequality involving it.
result The paper proves an inequality relating different knots and shows the difference between two invariants can be arbitrarily large.
The study proves a new positive energy theorem for manifolds with specific curvature properties.
problem Proving a new positive energy theorem for manifolds with specific curvature properties.
method Establishing a systolic inequality relating boundary mean curvature to the systole of the boundary.
result Obtaining a new positive energy theorem, with equality for Horowitz-Myers metrics.
New homology from 3D cobordism to integers.
problem Understanding 4-manifolds with boundary and their homology.
method Instanton homology with coefficients.
result Found a non-linear homomorphism with bounds on 4-manifolds.
Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.
problem Proving properties of graphs with specific curvature conditions.
method Graph-theoretic modified nonlinear heat-flow method, including point-mass consequences and diffusive exit-time control.
result Volume doubling and Poincaré inequalities for graphs with nonnegative Bakry-Émery curvature.
Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the…
The study finds knots with Stein structures beyond the typical upper bound.
problem Finding knots with Stein structures beyond the known upper bound.
method Constructing specific knots and framings to demonstrate Stein structures.
result The largest framing for which a knot admits a Stein structure can be arbitrarily larger than the usual upper bound.
The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.
problem Improving bounds on skein tree depth and delta-crossing numbers for knots and links.
method Theoretical and computational analysis of skein trees and knot invariants.
result New upper and lower bounds on skein tree depth and delta-crossing numbers are derived.
This note quantifies, via a sharp inequality, an interplay between (a) the characteristic rank of a vector bundle over a topological space X, (b) the Z/2Z-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 B…
New MIP model and algorithms for Bayesian network learning.
problem Optimization problems with acyclic constraints on directed graphs.
method Mixed integer programming model and iterative algorithms based on topological orders.
result Significantly lower number of constraints and efficient search of topological orders.
The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.
problem Establishing inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.
method Applying effective very ampleness of adjoint bundles, log-concavity, and Khovanskii-Teissier inequalities.
result For any projective manifold X and ample line bundle L, there exists a universal bivariate polynomial Q_λ(x, y) with deg Q ≤ d, such that the inequality holds.
Simple geodesics on spherical tetrahedra identified for specific angles.
problem Identifying simple closed geodesics on regular tetrahedra in spherical space.
method Analyzing pairs of coprime integers (p,q) to find angles α1 and α2.
result Existence and non-existence of simple closed geodesics for specific angles.
It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…
New distances defined between surfaces in 4-manifolds, with a new inequality proved.
problem Defining and comparing distances between surfaces in 4-manifolds.
method Using techniques similar to Gabai's proof, proving an inequality between two distance notions.
result Proved that stabilisation distance is at most one more than singularity distance.
The study characterizes Clifford hypersurfaces in terms of curvature constants.
problem Characterizing Clifford hypersurfaces in terms of curvature constants.
method Defined constants σ_k and used integral inequalities to show bounds on curvature.
result For specific conditions, σ_k ≥ n^k, with equality for Clifford hypersurfaces.
New statistical Minkowski distances for Gaussian mixtures with closed-form formulas.
problem Computing distances for Gaussian mixture models efficiently.
method Proposed novel statistical distances based on Minkowski's inequality for Gaussian mixtures.
result Closed-form formula for Gaussian mixture models with integer exponents.
The paper proves inequalities for isometries in loxodromic Kleinian groups.
problem Discreteness criteria for subgroups of PSL.
method Generalization of discreteness criteria, using trace inequalities and optimization problems.
result Inequalities involving traces and hyperbolic displacements for loxodromic Kleinian groups.
Constructs 4-manifolds with positive Euler characteristic proving a conjecture.
problem Finding aspherical 4-manifolds with positive Euler characteristic.
method Explicit construction of 4-manifolds with specified properties.
result Proves a conjecture by constructing a 4-manifold with Euler characteristic 1.
Study Chern number inequalities for negative curvature Kähler manifolds.
problem Chern number inequalities for compact Kähler manifolds with negative sectional curvature.
method Study -harmonic forms on lifting bundle over universal covering space, observe relationship between Laplace-Beltrami eigenvalues and Euler characteristic.
result Euler characteristic inequality involving sectional curvature and Laplace-Beltrami eigenvalues.
The paper classifies periodic solitons in curve flows on the light-cone.
problem Investigating periodic solitons in curve flows on the light-cone.
method Deriving Harnack inequality for heat flow, classifying space-periodic solitons for a third-order curvature flow.
result Closed soliton solutions form a family of transcendental curves with specific rotation indices.
Maximizes Laplace eigenvalues on a sphere, proving a 2002 conjecture.
problem Maximizing Laplace eigenvalues on a sphere.
method Using properties of harmonic maps and bounds on harmonic degree.
result Sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere.
The paper analyzes the spectrum of Morse-Smale flows on manifolds.
problem Understanding the dynamics and topology of Morse-Smale flows on manifolds.
method Spectral analysis of the generator of Morse-Smale flows on anisotropic spaces of currents.
result The Morse complex is the image of the de Rham complex under a spectral projector.
The paper constructs corks and exotic 4-manifolds with controlled shadow-complexity.
problem Constructing exotic 4-manifolds with controlled shadow-complexity.
method Using corks of Mazur type, the authors construct exotic pairs of 4-manifolds with controlled shadow-complexity.
result An infinite family of exotic pairs of 4-manifolds with controlled shadow-complexity.
Researchers found a quadratic estimate for embedding higher-dimensional simplices into sphere-connected sums.
problem Estimating the number of handles required for embedding higher-dimensional simplices into sphere-connected sums.
method Combining geometric topology, combinatorics, and linear algebra.
result Presented a quadratic estimate for embedding -faces of -simplex.
We explore a natural generalization of systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry of numbers. In particular, we show that if an optical hypersurface of contact type in the cotangent bundle of the 2-dimensional torus encl…
Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.
problem Proving explicit formulas and inequalities for fractional operators on hyperbolic spaces.
method Scattering theory on hyperbolic space, Helgason-Fourier analysis, and special function analysis.
result Sharp constants in fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities coincide with Euclidean space constants.
The paper studies extremal hypersurfaces in ellipsoids using centro-affine geometry.
problem Characterizing extremal hypersurfaces in centro-affine geometry.
method Analyzing invariant submanifold flows and deriving variational formulas.
result Circles on with radius are equi-centro-affine maximal.
Let be a Morse-Bott function on a compact smooth finite dimensional manifold . The polynomial Morse inequalities and an explicit perturbation of defined using Morse functions on the critical submanifolds of show immediately that , where …
The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…
Let M and N be two closed (not necessarily orientable) surfaces, and f a continuous map from M to N. By definition, the minimal multiplicity MMR[f] of the map f denotes the minimal integer k having the following property: f can be deformed into a map g such that the number |g^{-1}(c)| of preimages of any point c in N u…
The paper examines stochastic inequalities involving minimum and maximum claim amounts.
problem Investigating stochastic inequalities for claim amounts with random number of claims.
method Analyzing stochastic order and reversed hazard rate order for minimum and maximum claim amounts.
result Strengthening and generalizing existing results in the literature.
New q-deformed integers help compute Jones polynomials efficiently.
problem Computing Jones polynomials of rational links efficiently.
method Defining q-deformed integers from pairs of coprime integers and using them to compute Jones polynomials.
result Efficient algorithm for computing Jones polynomials of rational links.
Generalized Steinberg module presentation for Gaussian and Eisenstein integers.
problem Presenting Steinberg modules for specific number rings.
method Generalization of Bykovskii's presentation to Gaussian and Eisenstein integers.
result Generalization does not yield a presentation for all Euclidean number rings.
Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.
problem Obstructing knots in integer homology spheres using surgery.
method Extending Heegaard Floer homology obstructions to all integer homology spheres for both positive and negative surgeries.
result Deduced a lower bound on for smooth cobordism between integer homology spheres.
New algorithm clusters Gaussian mixtures with unknown covariance efficiently.
problem Clustering data from a mixture of Gaussians with unknown covariance.
method Developed an efficient spectral algorithm based on a Max-Cut integer program.
result Achieves optimal misclassification rate with quadratic sample size.
IDF++ improves integer discrete flows for lossless compression.
problem Theoretical limitations of integer discrete flows for lossless compression.
method Investigated and improved integer discrete flows, addressing gradient bias and architecture modifications.
result Different architecture modifications improve integer discrete flows for lossless compression.
New links split by integer homology spheres but not by others.
problem Characterizing links split by integer homology spheres.
method Constructing specific links and homology spheres.
result Infinite families of links and homology spheres split by specific ones but not by others.
Investigates trading with integer constraints in discrete time.
problem Trading with discrete, integer quantities under integer constraints.
method Establishes a novel theory of integer arbitrage-free pricing and hedging for non-rational price processes.
result The set of prices of a contingent claim is either empty or dense in an interval.
Geometric proof shows primes of form 3k+1 are norms of Eisenstein integers.
problem Geometric proof of primes of form 3k+1 being norms of Eisenstein integers.
method Geometric proof using Penner's λ-length and norms of Eisenstein integers.
result Every prime of the form is the norm of an Eisenstein integer.
From Sets to Multisets: Provable Variational Inference for Probabilistic Integer Submodular Modelscs.LG
New method for probabilistic modeling of integer submodular functions.
problem Lack of probabilistic modeling for integer submodular functions.
method Proposed Generalized Multilinear Extension and block-coordinate ascent algorithm.
result Demonstrated effectiveness and viability on real-world datasets.
A new STAR framework models integer-valued data with flexible distributions.
problem Modeling integer-valued data with flexibility and accuracy.
method Simultaneously Transforming and Rounding (STAR) a continuous-valued process.
result STAR framework designs a new BART model for integer-valued data with impressive predictive accuracy.