The article calculates a multiplying factor to convert rational Vassiliev invariants to integer-valued ones.
problem Converting rational valued Vassiliev invariants to integer-valued ones.
method Calculates the minimal multiplying factor λ needed for rational Vassiliev invariants to become integer-valued.
result Obtains a set of integer-valued Vassiliev invariants.
Paper proposes algorithms for BMF using integer programming.
problem Approximating binary input matrix as product of two smaller binary factors.
method Alternating optimization strategy using integer programming to solve subproblems and combine solutions.
result Proposed algorithms outperform state of the art on medium-scale problems.
Hardware-accelerated RBM solves large combinatorial problems and integer factorization.
problem Solving large combinatorial optimization and integer factorization problems.
method Logically synthesized RBM architecture, hardware acceleration, and efficient training methods.
result Hardware-accelerated RBM factorizes 16-bit numbers with 10000x speed and 32x power improvements.
The paper develops algorithms for Boolean matrix factorization using IP and heuristics.
problem Approximating binary input matrices as products of smaller binary factors.
method Alternating optimization with integer programming and greedy/local-search heuristics.
result Proposed methods improve scalability and performance compared to existing techniques.
We explicitly construct pseudo-Anosov maps on the closed surface of genus g with orientable foliations whose stretch factor λ is a Salem number with algebraic degree 2g. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d s…
Study on a new class of meanders with tangential intersections.
problem Enumerating and understanding meanders with transverse intersections.
method Developed a combinatorial framework, identified connections with other objects, and enumerated specific families.
result Completely enumerated several families of singular meanders.
IntSGD compresses SGD gradients without floats, converging as SGD.
problem Efficiently compressing stochastic gradients in distributed SGD.
method Adaptive integer compression of gradients, estimating scaling adaptively.
result IntSGD matches SGD's iteration complexity for convex and non-convex functions.
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.
The AJ conjecture is verified for certain connected sums of torus knots.
problem Verifying the AJ conjecture for specific connected sums of torus knots.
method Analyzing recurrence polynomials and their factorization properties.
result The AJ conjecture requires a modification for certain connected sums of torus knots.
This research connects quantum spectra of flag bundles to prime factorization of integers.
problem Understanding the quantum spectra of flag bundles and their relation to prime numbers.
method Functorial and inductive properties of vertical quantum cohomology, relating to analytic number theory.
result The degeneracy of the small vertical quantum spectrum of a Grassmann bundle is controlled by the prime factorization of ranks.
A new method computes Teichmüller polynomials from integer permutations.
problem Computing Teichmüller polynomials for fibered 3-manifolds.
method Using integer permutations to characterize pseudo-Anosov homeomorphisms and train tracks.
result Direct implementation of McMullen's algorithm for Teichmüller polynomials.
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
Proposes continuous convolution layers for flexible feature map resizing.
problem Fixed stride limitations in discrete convolution layers.
method Introduces Continuous Convolution (CC) layers that use learned continuous functions.
result Dynamic and consistent resizing of feature maps at any scale, non-integer and axis-dependent.
Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.
problem Computing stretch factors and foliations for pseudo-Anosov mapping classes efficiently.
method Quadratic-time algorithm using input word and length as complexity measure.
result First algorithm to compute stretch factors and foliations in sub-exponential time.
Paper introduces methods to create fair and accurate regression models.
problem Creating fair and accurate regression models.
method Mixed-integer optimization methods, exact formulations, branch-and-bound algorithm, coordinate descent algorithm.
result Developed methods produce fair and accurate models with reduced training times.
The GJMS operators of special Einstein products are factored into simpler operators and applied to solve the Q-Yamabe problem.
problem Factorization of GJMS operators in special Einstein products.
method Factorization of GJMS operators as a composition of second- and fourth-order differential operators.
result The Green's function for the GJMS operator of order 2k is positive for certain special Einstein products.
In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books…
Matrix factorization is a key tool in data analysis; its applications include recommender systems, correlation analysis, signal processing, among others. Binary matrices are a particular case which has received significant attention for over thirty years, especially within the field of data mining. Dictionary learning …
The study finds all trace field degrees for Torelli group mappings.
problem Identifying all possible trace field degrees for Torelli group mappings.
method Using Thurston-Veech construction of pseudo-Anosov maps, and providing examples of stretch factors with specific algebraic degrees.
result All integers 1≤d≤3g−3 are trace field degrees for g≥2. Skew parallelogram nets factorize, encompassing discrete differential geometry.
problem Factorization of polynomials in discrete differential geometry.
method Lax representation, Bäcklund transformations, factorization of polynomials.
result Skew parallelogram nets encompass all systems with polynomial representations.
The paper analyzes how behavioral investors make portfolio decisions using Markowitz Stochastic Dominance criteria.
problem Understanding how behavioral investors make portfolio decisions.
method Developed stochastic optimization problems and MILP models to capture subjective decision weights and probability weighting functions.
result The developed models can be used to formulate computationally tractable portfolio analysis problems.
Many tasks require finding groups of elements in a matrix of numbers, symbols or class likelihoods. One approach is to use efficient bi- or tri-linear factorization techniques including PCA, ICA, sparse matrix factorization and plaid analysis. These techniques are not appropriate when addition and multiplication of mat…
The analytic torsion is computed on fixed-point free and non fixed-point free factors (tessellations) of the three--sphere. We repeat the standard computation on spherical space forms (Clifford-Klein spaces) by an improved technique. The transformation to a simpler form of the spectral expression of the torsion on sphe…
Paper finds instantons for Kapustin-Witten equations on a specific manifold.
problem Existence of solutions to Kapustin-Witten equations on (0,∞)imesR2imesR. method Explains existence of solutions interpolating between two model solutions.
result Interpolation solutions exist with specific label constraints.
We use Floer's exact triangle to study the u-map (cup product with the 4-dimensional class) in the Floer cohomology groups of admissible SO(3) bundles over closed, oriented 3-manifolds. In the case of non-trivial bundles we show that (u^2-64)^n = 0 for some positive integer n. For homology 3-spheres Y the same holds fo…
We show that the Nielsen-Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum SU(n)-representations, for any fixed integer n≥2. In the Pseudo-Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) SU(2)-TQFT represen…
Let D be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/D is not compact, and that G has more than one noncompact simple factor. We show that D has no orientation-preserving actions on the real line. (In algebraic terms, this mean…
We study the fundamental group of an open n-manifold M of nonnegative Ricci curvature. We show that if there is an integer k such that any tangent cone at infinity of the Riemannian universal cover of M is a metric cone, whose maximal Euclidean factor has dimension k, then π1(M) is finitely generated. In p…
New algorithm for online convex minimization over integer lattice.
problem Online decision-making with nonlinear combinatorial objectives.
method Introduces online Latural-convex minimization and proposes efficient algorithms. result Tight regret bound for full information setting algorithm.
Identifying important components or factors in large amounts of noisy data is a key problem in machine learning and data mining. Motivated by a pattern decomposition problem in materials discovery, aimed at discovering new materials for renewable energy, e.g. for fuel and solar cells, we introduce CombiFD, a framework …
The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We descr…
If A is an abelian group and φ is an integer, let A(φ) be the subgroup of A consisting of elements a∈A such that φ⋅a=0. We prove that if D is a diagram of a classical link L and 0=φ0,φ1,…,φn−1 are the invariant factors of an adjusted Goeritz matrix of D, then the group $\mathcal{D}…
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.
We present a novel optimization strategy for training neural networks which we call "BitNet". The parameters of neural networks are usually unconstrained and have a dynamic range dispersed over all real values. Our key idea is to limit the expressive power of the network by dynamically controlling the range and set of …
Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain com…
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.
We present a new proof of Thurston's theorem that the unit ball of a seminorm on Rd taking integer values on Zd is a polyhedra defined by finitely many inequalities with integer coefficients.
Paper uses RL to optimize branching strategy in B&B algorithms.
problem Optimizing Branch and Bound algorithms for mixed integer linear programs.
method FMSTS, a Reinforcement Learning approach for variable selection.
result FMSTS outperforms commercial solvers in efficiency and generalization.
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 b2(W) for smooth cobordism between integer homology spheres. The paper shows deep connections between exotic smoothings of a small R^4 (the spacetime), the leaf space of codimension-1 foliations (related to noncommutative algebras) and quantization. At first we relate a small exotic R^4 to codimension-1 foliations of the 3-sphere unique up to foliated cobordisms and characterize…
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.
Erdős-Kac theorem applied to geodesics on modular surface.
problem Understanding the distribution of geodesics on modular surfaces.
method Analyzing the number of scattering geodesics with a fixed sojourn time.
result Gaussian behavior for the number of scattering geodesics on modular surface.
We use Nathanson's g-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets S to problems in additive number theory. If S consists of all powers of a fixed integer g, we find explicit formulas for the smallest positive intege…
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 p of the form 3k+1 is the norm of an Eisenstein integer. 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.
An elementary proof shows that quasi-isometric groups to integers are virtually integers.
problem Proving that quasi-isometric groups to integers are virtually integers.
method An elementary proof approach.
result Any finitely generated group quasi-isometric to the integers is virtually the integers.
Study area-minimizing subgraphs in integer lattices.
problem Finding the most efficient subgraphs in integer lattices.
method Formulated functions of bounded variations, classified subgraphs in 2D, proved properties in higher dimensions.
result Classified area-minimizing subgraphs in 2D integer lattice up to isomorphisms.