We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants and behave under the generalized connected sums.
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Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is ag…
Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Can the Minkowski sum of two compact convex bodies be made smoother by rotating one of them? We construct two infinitely differentiable strictly convex plane bodies such that after any generic rotation (in the Baire category sense) of one of the summands the Minkowski sum is not five times differentiable. On the other …
Gradient methods converge exponentially in concave network games.
problem Finding Nash equilibria in concave network zero-sum games.
method Gradient Ascent and Optimistic Gradient Ascent analyses.
result Exponential convergence rates in various game settings.
Generalized Dedekind sumsmath.GT
Classical Dedekind sums are connected to the modular group through the construction of a (Dedekind) symbol on the cusp set of the modular group. In this paper we study generalizations of Dedekind symbols and sums that can be associated to certain Fuchsian groups uniformizing 1-punctured tori.
Just as war is sometimes fallaciously represented as a zero sum game -- when in fact war is a negative sum game - stock market trading, a positive sum game over time, is often erroneously represented as a zero sum game. This is called the "zero sum fallacy" -- the erroneous belief that one trader in a stock market exch…
Summing over 3-manifolds using TQFT partition functions.
problem Summing over all 3-manifolds with fixed boundary.
method Rewriting the sum over 3-manifolds as a sum over homology groups, using TQFT partition functions and topological boundary conditions.
result Existence of a distribution of 2d TQFTs whose ensemble average equals the sum over 3-manifolds.
We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result.
Closed-Form Parametric Equation for the Minkowski Sum of Ellipsoids in and Associated Volume Boundsmath.MG
Derives exact formula for Minkowski sum of ellipsoids in N-space.
problem Finding volume bounds for Minkowski sum of ellipsoids.
method Closed-form parametric equation derivation and volume bounds calculation.
result Upper and lower volume bounds for Minkowski sum of ellipsoids.
Gradient Descent Ascent converges to von-Neumann solution in hidden zero-sum games.
problem Understanding dynamics of zero-sum games with hidden structure.
method Gradient Descent Ascent applied to hidden zero-sum games with specific convex-concave structure.
result Gradient Descent Ascent converges to von-Neumann solution in strictly convex-concave hidden games.
The end sum of surfacesmath.GT
Study end sum for surfaces and prove uniqueness results.
problem Uniqueness of end sum for surfaces and related manifolds.
method Analyzing end sum and adding a 1-handle at infinity for surfaces.
result The end sum of two surfaces with compact boundary is uniquely determined by the chosen ends.
Study Nash equilibrium in non-zero-sum game with Bermudan strategies.
problem Optimizing pay-offs in non-linear non-zero-sum games.
method Recursive construction to find Nash equilibrium.
result Existence of Nash equilibrium in non-zero-sum game.
New methods optimize sums of bivariate functions on finite domains.
problem Optimizing functions with multiple arguments that are sums of bivariate functions.
method Measure-valued extensions, -approximation, entropy-regularization, linear programming, coordinate ascent.
result Tractable problem formulations solvable with various methods.
We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.
Study symmetry groups and curves from sums of exponentials.
problem Understanding the geometry and symmetry of curves from sums of exponentials.
method Analysis of symmetry groups, winding numbers, and parametrization of the unit circle.
result Unified method for constructing curves with specific properties.
Researchers found multiple ways to end-sum 4-manifolds, contradicting a previous conjecture.
problem Nonuniqueness of end-sums in 4-manifolds.
method Explicit examples and detailed discussion of end-cohomology algebra.
result Uncountably many distinct proper homotopy types from end-sums.
ELBO converges to a sum of entropies for many generative models.
problem Understanding the convergence of variational lower bounds in unsupervised learning.
method Analyzing the ELBO for a broad class of generative models, showing it equals a sum of entropies.
result The ELBO is equal to a sum of entropies at stationary points for many generative models.
Proves a formula for a special invariant of 4-manifolds.
problem Calculating the Bauer-Furuta invariant for connected sums of 4-manifolds.
method Uses a finite dimensional approximation of the Seiberg-Witten monopole map to derive a formula for the families Bauer-Furuta invariant of a fibrewise connected sum.
result Derives a general connected sum formula for the families Bauer-Furuta invariant.
The study allows for connected sums in manifolds with positive intermediate Ricci curvature.
problem Performing connected sums in manifolds with positive intermediate Ricci curvature.
method Introducing and utilizing -core metrics to show the possibility of connected sums.
result Connected sums are possible under certain conditions involving -core metrics.
Characterizes Stein surfaces with finite homotopy rank-sum.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …
The paper studies Gauss sums and their applications in algebra and topology.
problem Computing signatures of bilinear forms and Poincaré spaces.
method Investigates properties of Gauss sums and applies them to algebraic and topological problems.
result Reproves and generalizes results on signatures mod 8 of bilinear and Poincaré spaces.
High dimensional superposition models characterize observations using parameters which can be written as a sum of multiple component parameters, each with its own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated sparse vectors, etc. In this paper, we consider general superposition models …
Generative models' ELBOs converge to entropy sums, proving for various models.
problem Proving convergence of ELBOs to entropy sums for various generative models.
method Proofs for individual models under realistic conditions.
result ELBOs of various generative models converge to entropy sums at all stationary points.
New algorithms learn graph structures privately, matching best results.
problem Private learning of graph structures with multiple blocks.
method Sum-of-squares relaxation and exponential mechanism for score function.
result Matches statistical utility of previous best non-private methods.
Sum formula for relative Seiberg-Witten invariants in 4-manifolds.
problem Defining and studying relative Seiberg-Witten invariants for 4-manifolds with submanifolds.
method Defined relative Seiberg-Witten invariants and proved a sum formula.
result A sum formula relating relative SW invariants of 4-manifolds and their submanifolds.
Urysohn Width and Surgeriesmath.MG
Study bounds Urysohn width of manifolds under surgeries.
problem Bounding Urysohn width of manifolds after surgeries.
method Analyzes connected sums and universal covers, applies to general surgeries.
result Optimal constants in estimates of width bounds are shown.
M-theory on compact eight-manifolds with -holonomy is a framework for geometric engineering of 3d gauge theories coupled to gravity. We propose a new construction of such -manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four…
Unified framework for stability and generalization of Push-Sum in decentralized learning over directed graphs.
problem Understanding stability and generalization of Push-Sum in decentralized learning over directed networks.
method Developed a unified uniform-stability framework for SGP algorithm, incorporating imbalance-aware consistency bounds.
result Established finite-iteration stability and optimization guarantees for convex and non-convex objectives.
Estimation is the computational task of recovering a hidden parameter associated with a distribution , given a measurement sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems ca…
New method for sequential probability assignment reduces regret using contextual Shtarkov sums.
problem Minimizing regret in sequential probability assignment with arbitrary hypothesis classes.
method Introducing contextual Shtarkov sum and contextual Normalized Maximum Likelihood (cNML) algorithm.
result The contextual Shtarkov sum characterizes minimax regret and provides a minimax optimal strategy.
Let be an invertible matrix and be the inverse of . In this paper, we consider the generalized Liouville system: \label{abeq1} Δ_g u_i+\sum_{j=1}^n a_{ij}ρ_j(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}-1)=0\quad\text{in \,}M, where and $ρ_j\in \mathb…
In this paper we determine the integral homology and cohomology groups of a closed 4-manifold X obtained as the generalized fibre sum of two closed 4-manifolds M and N along embedded surfaces of genus g and self-intersection zero. If the homologies of the 4-manifolds are torsion free and the surfaces represent indivisi…
This paper tackles learning Stackelberg equilibrium in asymmetric games efficiently from noisy samples.
problem Learning Stackelberg equilibrium in asymmetric, general-sum games efficiently from noisy samples.
method The paper initiates the theoretical study of sample-efficient learning of the Stackelberg equilibrium in bandit feedback setting.
result Sharp positive results on sample-efficient learning of Stackelberg equilibrium with value optimal up to a fundamental gap identified.
Sharp concentration results for sums of heavy-tailed random variables.
problem Analyzing sums of independent heavy-tailed random variables.
method Using concentration inequalities and large deviation principles for distributions satisfying specific tail bounds.
result Sharp concentration inequalities and large deviation results for sums of heavy-tailed random variables.
Abstract: Study gyrations of sphere products and connected sums, generalizing Fico's Lemmata.
problem Understanding the homotopy type of gyrations of sphere products and connected sums.
method Recasting Fico's Lemmata into modern homotopy theoretic setting.
result Generalization of Fico's Lemmata to sphere products and connected sums.
Let M denote the total space of a Lefschetz fibration, obtained by blowing up a Lefschetz pencil on an algebraic surface. We consider the n-fold fibre sum M(n), generalizing the construction of the elliptic surfaces E(n). For a Lefschetz pencil on a simply-connected minimal surface of general type we partially calculat…
Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yie…
Study connects knot polynomials with number theory sums.
problem Alexander polynomials and Dedekind sums of torus knots.
method No specific method mentioned; connects known concepts.
result Established relationship between knot theory and number theory.
A universal state summath.QA
Defines a universal state sum construction for various TQFTs.
problem No specific problem stated; universal construction for TQFTs.
method Defines a universal state sum construction using n-categories with specific conditions.
result Produces state sums from n-categories and handle decompositions of n+1-manifolds.
Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.
problem Proving the existence of smooth solutions for Demailly's system.
method Used Demailly's system and Leray-Schauder degree theory to reduce the problem.
result Proved existence of smooth solutions for direct sums of ample line bundles.
Sharp bounds for Dirichlet sums lead to improved Bayesian algorithm analysis.
problem Improving Bayesian algorithm performance through precise deviation bounds.
method Novel integral representation of Dirichlet sum density, Gaussian approximation, complex analysis.
result Significantly sharpened regret bounds for Multinomial Thompson Sampling.
In this paper we will show that the generalized connected sum construction for constant scalar curvature metrics can be extended to the zero scalar curvature case. In particular we want to construct solutions to the Yamabe equation on the generalized connected sum M = M_1 (\sharp_K) M_2 of two compact Riemannian manifo…
Proposes a new training algorithm for zero-sum games to avoid convergence issues.
problem Gradient-based training leads to weak convergence and cyclic dynamics in zero-sum architectures.
method Follow the perturbed leader algorithm with neural mediating agent.
result Guarantees convergence to mixed Nash equilibrium without cyclic behaviors.