We prove that the quantum SO(3)-invariant of an arbitrary 3-manifold is always an algebraic integer, if the order of the quantum parameter is co-prime with the order of the torsion part of $H_1(M,\BZ)$. An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-…
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Formula calculates instanton homology dimensions for knot surgeries over arbitrary fields.
problem Calculating instanton homology dimensions for knot surgeries over arbitrary fields.
method Established a dimension formula for framed instanton homology of knot surgeries over arbitrary fields.
result Formula generalizes instanton homology dimensions to arbitrary fields, including new results for and knots.
We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an…
Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on `Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an matrix whose entries are eleven mosaic tiles, represent…
Study shows alternating knots can't have similar crossings.
problem Cosmetic crossings in alternating knots.
method Examined homology groups of 4-manifolds.
result Subgroups of knot homology groups have co-prime orders.
The paper provides bounds for embedding manifolds into Euclidean spaces with group actions.
problem Finding explicit bounds for embedding manifolds into Euclidean spaces with group actions.
method The paper provides upper and lower bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
result Explicit bounds for the dimension of the Euclidean space required for equivariant embeddings of manifolds into Euclidean spaces for finite group actions.
Asymptotic Burnside lawsmath.GR
Novel groups exhibit contradictory behaviors with respect to Burnside laws.
problem Understanding probabilistic behaviors of groups under Burnside laws.
method Geometric analysis of relations, information-theoretic coding, combinatorial and probabilistic methods.
result Groups can satisfy Burnside laws with probability 1 for some generating sets and 0 for others.
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.
We present a new proof of Thurston's theorem that the unit ball of a seminorm on taking integer values on is a polyhedra defined by finitely many inequalities with integer coefficients.
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.
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.
We use Nathanson's -adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets to problems in additive number theory. If consists of all powers of a fixed integer , 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 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.
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.
New method finds lattice polygons that can be dissected into triangles with integer areas.
problem Finding lattice polygons that can be dissected into triangles with integer areas.
method A new version of Sperner's Lemma.
result Simple and complete description of lattice polygons that can be dissected into triangles with integer areas.
Neural networks with integer weights approximate continuous functions efficiently.
problem Approximating continuous functions using neural networks with integer weights.
method Integrates superexpressive activation functions and integer weights.
result Convergence rate of order for neural network regression.
New examples show non-integer Hausdorff dimensions in collapsing spaces.
problem Understanding Hausdorff dimensions in collapsing Ricci limit spaces.
method Provided examples of spaces with irregular Hausdorff dimensions.
result Hausdorff dimension of singular set exceeds regular set's dimension.
The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio …
Paper uses integer programming for non-convex boosting in classification.
problem Improving classification performance using non-convex optimization.
method Non-convex boosting via integer programming.
result Results comparable to or better than state-of-the-art.
We propose a simple yet powerful framework for modeling integer-valued data, such as counts, scores, and rounded data. The data-generating process is defined by Simultaneously Transforming and Rounding (STAR) a continuous-valued process, which produces a flexible family of integer-valued distributions capable of modeli…
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.
Improved RTM uses integer weights to reduce computation and increase interpretability.
problem Lack of interpretability in nonlinear regression models.
method Integer weighted RTM clauses, combined with a novel learning scheme.
result Significantly reduced computation cost with improved accuracy.
Paper develops machine learning algorithms to learn optimal integer weights for clinical risk scores.
problem Deriving optimal integer weights for clinical risk scores without computational burden.
method Flexible greedy optimization strategy to directly optimize a value function.
result Constructed an integer-weighted comorbidity score for measuring post-discharge mortality risk.
New symplectic embedding obstructions found for polydisks into half-integer ellipsoids.
problem Obstructing symplectic embeddings of polydisks into half-integer ellipsoids.
method Combinatorial criterion developed by Hutchings to obstruct symplectic embeddings.
result Optimal inclusion conditions for symplectic embeddings of polydisks into half-integer ellipsoids.
Horn(p,q)math.DG
The article describes Horn(p,q) with integer p and q.
problem Classical Horn's conjecture for Horn(p,q).
method Recursive description of Horn(p,q).
result Obtained a recursive description of Horn(p,q).
Infinite knots have non-integer trace values.
problem Finding knots with non-integer trace values.
method Proved existence of infinitely many non-homeomorphic hyperbolic knot complements with specific trace properties.
result Infinitely many non-homeomorphic hyperbolic knot complements with non-integer trace values.
Criteria found for graph drawings on surfaces.
problem Graph drawings on surfaces.
method Criteria for integer and modulo 2 embeddability.
result Found criteria for graph drawings on surfaces.
Proves surgery exact triangle for monopole Floer homology over integers.
problem Proving the surgery exact triangle for monopole Floer homology over integer coefficients.
method Modification of Kronheimer--Mrowka's local system and adaptation of Freeman's computation.
result Obtains a spectral sequence over integer coefficients for an oriented link in .
The paper uses clustering and integer programming to optimize stock selection for investment funds.
problem Maximizing profits and minimizing risk in stock markets.
method Data-oriented analysis and clustering techniques with integer programming.
result Reconstructed NASDAQ 100 index fund example demonstrates effectiveness.
The Arnold conjecture is proven for integers using Floer theory.
problem Proving the Arnold conjecture for symplectic manifolds over integers.
method Constructing a Hamiltonian Floer theory over the Novikov ring with integer coefficients.
result The number of 1-periodic orbits is bounded by the total Betti number over Z of the ambient space.
Optimizes intervention design for causal discovery using integer programming.
problem Identifying causal structures from observational data due to confounding variables.
method Uses integer programming to design minimal intervention sets for causal structure identifiability.
result Provides exact and modular solutions adaptable to various experimental settings and constraints.
Study of lambda lengths in figure eight knot complement using Eisenstein integers.
problem Determining lambda lengths in the figure eight knot complement.
method Using hyperbolic geometry and spinors, mapping lambda lengths to Eisenstein integers.
result Lambda lengths are precisely the Eisenstein integers, up to multiplication by a unit.
This paper explores how many positive integer surgeries on a knot produce a manifold rational homology cobordant to a lens space.
problem How many positive integer surgeries on a knot produce a manifold rational homology cobordant to a lens space?
method Uses Greene and McCoy's changemaker lattices from Heegaard Floer d-invariants and Aceto-Celoria-Park's rational cobordisms and integral homology.
result For a given knot, there are at most two positive integer surgeries that produce a manifold rational homology cobordant to a lens space.
New integer-valued functions for Legendrian knots.
problem Understanding Legendrian knots better.
method Using Legendrian fronts to derive integer-valued linear functions.
result Introduced new invariants similar to Arnold's basic invariant.
The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong -closure of vector fields with finitely many integer singularities.
method Characterization and decomposition of vector fields with finitely many integer singularities.
result Decomposition theorem for elements in , revealing information about mass-minimizing currents.
The study explores which sets of integers can be realized as the degrees of maps between manifolds.
problem Which sets of integers can be realized as the degrees of maps between manifolds?
method Analyzes the set of degrees of maps between closed oriented manifolds of the same dimension.
result Finite arithmetic progressions and geometric progressions starting from 1 can be realized as degrees of maps between manifolds.
Minimal spectral radii found for specific matrix types.
problem Finding smallest spectral radii for certain matrix classes.
method Analyzing skew-reciprocal integer matrices of fixed even dimensions.
result Most classes of matrices have smaller spectral radii than their reciprocal counterparts.
A short note on super-hedging an arbitrary number of European options with integer-valued strategiesq-fin.PR
This paper shows how to hedge financial risks with integer investments.
problem Evaluating the minimal super-hedging price with integer-valued strategies for arbitrary payoffs.
method Formulated a dynamic programming principle to evaluate the minimal super-hedging price with integer-valued strategies for continuous piecewise affine terminal claims.
result It is possible to evaluate the minimal super-hedging price with integer-valued strategies for discrete-time, arbitrary Ω.
We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.
In this paper we investigate discrete time trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices this has little effect on the core of the theory of no…
The paper proves smoothness of stationary varifolds.
problem Understanding the smoothness of stationary varifolds.
method Analyzing -dimensional integer rectifiable varifolds in open sets.
result The support of stationary varifolds is rectifiable.
Black-box Mixed-Variable Optimisation using a Surrogate Model that Satisfies Integer Constraintscs.LG
MVRSM optimizes expensive functions with mixed variables, outperforming state-of-the-art methods.
problem Minimizing expensive functions with mixed continuous and integer variables.
method Mixed-Variable ReLU-based Surrogate Modelling (MVRSM) using rectified linear units.
result MVRSM outperforms state-of-the-art methods on synthetic and real-life benchmarks.
We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of applications of the sp…
The study confirms conjectures about slopes of knots using knot Floer homology.
problem Verifying conjectures about non-integer characterizing slopes of knots.
method Using knot Floer homology, the study verifies conjectures for specific classes of knots.
result Almost all slopes are characterizing for many knots, and infinitely many for -space knots.