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.
Introduces q-transpose for q-deformed modular group matrices.
problem Understanding q-deformed rational numbers and their properties. method Introduces q-transpose and applies it to refine q-deformed modular group actions. result New proof and refinement of Leclere and Morier-Genoud's trace palindromicity theorem.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups. Compactifies stability space for A2 category, introducing q-deformed rational numbers.
problem Stability conditions in triangulated categories and their compactifications.
method Embedding into an infinite-dimensional projective space, using B3 braid group action. result Two orbits in the boundary correspond to q-deformed rational numbers. Link between braid groups and q-deformed rationals solves a classification problem.
problem Classifying faithful complex specializations of the Burau representation of braid group B3.
method Established a link between Burau representation and q-deformed rational numbers.
result Proved faithfulness of Burau representation outside a specific annulus.
Finite specializations of a q-deformed modular group at roots of unity.
problem Understanding the finiteness of specializations of a q-deformed modular group at roots of unity.
method Introduced a q-deformed modular group and studied its specializations at roots of unity.
result For ζn being a primitive nth root of unity, PSLq(2,Z)∣q=ζn is finite if and only if Gq(ζn) is finite. Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
Unified treatment of gauge theories and Yang-Mills theory duality.
problem Unified treatment of gauge theories and Yang-Mills theory duality.
method Cohomological localization techniques and Atiyah-Singer index theorem.
result Unified framework and simplified derivations of localization formulas.
We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq(2) is treated in some detail.
Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.
problem Understanding the skein-valued cluster transformation in Legendrian surfaces.
method Geometric considerations of moduli of holomorphic curves.
result Skein-valued cluster transformation of Legendrian surfaces related by surgery.
Study of q-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.
problem Geometry of q-rationals and their properties. method Construction and analysis of deformed Farey triangulation and deformed modular surface; definition and study of Springborn operations.
result Derivation of a formula for the q-deformed midpoint and new q-deformation of Markov numbers. We describe a natural q-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type A. We then describe an extension of this construction involving a cluster variety called the symplectic double.
The paper constructs TQFTs and Schrödinger representations for Heisenberg group.
problem Constructing TQFTs and Schrödinger representations for Heisenberg group.
method Using Lagrangian correspondences and q-deformation of U(1).
result Normalization of Schrödinger bimodule action reproduces abelian TQFT.
We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R-mat…
The theoretical basis for a candidate variational principle for the information bottleneck (IB) method is formulated within the ambit of the generalized nonadditive statistics of Tsallis. Given a nonadditivity parameter q, the role of the \textit{additive duality} of nonadditive statistics (q∗=2−q) in relating…
Study tangle equations linking enzyme actions to knot theory.
problem Proving the Jones Unknot conjecture and understanding tangle solutions.
method Analyzing framed tangle equations and introducing Kauffman bracket ratios.
result Unique rational solutions for tangle equations imply the Jones Unknot conjecture.
We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also impl…
We developed a strategic of optimal portfolio based on information theory and Tsallis statistics. The growth rate of a stock market is defined by using q-deformed functions and we find that the wealth after n days with the optimal portfolio is given by a q-exponential function. In this context, the asymptotic optim…
We show that the A2 clasps in the Karoubi envelope of A2 spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type A2. The A2 spider is a diagrammatic description of the representation category for Uq(sl3) and the $…
We conjecture formulae of the colored superpolynomials for a class of twist knots Kp where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomial for the twist k…
Holomorphic structures on quantum flag manifolds uniquely defined.
problem Defining unique holomorphic structures on quantum flag manifolds.
method Constructing covariant q-deformed holomorphic structures. result Holomorphic structures are unique for simple relative Hopf modules.
Extends quantum trace map to SL3(C) for 3D surfaces.
problem Generalizing quantum trace map to higher dimensions.
method Definition of SL3(C) quantum trace invariant.
result Construction of SL3(C) quantum trace map.
The notion of a Kähler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kähler structure has a canonically associated triple satisf…
We compute the noncommutative de Rham cohomology for the finite-dimensional q-deformed coordinate ring Cq[SL2] at odd roots of unity and with its standard 4-dimensional differential structure. We find that H1 and H3 have three additional modes beyond the generic q-case where they are 1-dimensional, while $H…
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.
We review a construction of a new class of algebraic curves, called super-A-polynomials, and their quantum generalizations. The super-A-polynomial is a two-parameter deformation of the A-polynomial known from knot theory or Chern-Simons theory with SL(2,C) gauge group. The two parameters of the super-A-polynomial encod…
We extend neural networks with fractional and mixed activation functions for better function approximation.
problem Limitations in approximating higher-order smooth functions in complex spaces.
method Incorporating fractional exponents in activation functions and defining new density functions.
result Improved accuracy and broader applicability of neural network approximation theory.
Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…
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. 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.
Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
problem Integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
method Constructs a bihamiltonian integrable hierarchy of hydrodynamic type.
result Integrable hierarchy possesses Virasoro symmetries and a tau structure.
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.
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.
Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.
problem Describing and constructing nontrivial cycles in Habiro cohomology.
method Using either the Picard-Fuchs equation or push-forward of elements of the Habiro ring.
result Explicit classes for 1-parameter Calabi-Yau families and q-holonomic modules.
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 n2β+d−2βlog2n 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.