Study bundles over surfaces with specific fibers, determining characteristic numbers and obstructions.
problem Characterizing bundles over surfaces with highly connected fibers.
method Analyzing smooth and topological bundles, providing necessary and sufficient conditions, and computing characteristic numbers.
result Determine characteristic numbers and divisibility constraints on signatures and genera for bundles of this type.
New algorithms for fair item allocation with limited copies.
problem Fair division of numerous items with few copies.
method Modeling as a contextual bandit problem with sub-linear regret guarantees.
result Proposed algorithms achieve sub-linear regret in fair item allocation.
A new distributed algorithm for fitting sparse additive models with feature division and decorrelation.
problem Fitting high-dimensional sparse additive models efficiently and accurately.
method Divide, decorrelate, and conquer approach.
result Effective and efficient recovery of sparsity patterns and statistical inference for each component.
Lasso method applied to polynomial models with hierarchy constraints.
problem Estimating parameters in polynomial models with hierarchy constraints.
method Using lasso and standard quadratic programming techniques to estimate parameters.
result The proposed methodology outperforms existing techniques in terms of validation error and model size.
A new proof of an extension theorem with bounded generators.
problem Extension theorems in complex analysis.
method Skoda-type L2 division theorem with bounded generators. result The new division theorem allows α to be 1 in the norm of the datum. Solves division problem for L. Hörmander's systems.
problem Division problem for L. Hörmander's overdetermined systems.
method Formulates and proves divisibility criterion, coherence theorem.
result Establishes effective divisibility criterion and extends coherence theorem.
Proves divisibility relations for symplectic curve polynomials.
problem Divisibility relations for symplectic curve polynomials.
method New proofs of divisibility relations for Oka and Alexander polynomials of symplectic curves.
result Proves Libgober's divisibility relations for symplectic curves.
Uniform convexity in divisible domains leads to hyperbolic geometry.
problem Understanding the geometry of divisible convex sets in Finsler manifolds.
method Proving β-uniform convexity of a specific Finsler metric. result A strictly convex divisible domain induces a β-uniformly convex Finsler metric. Tropical division approximates polynomial division for neural networks.
problem Approximating polynomial division in max-plus semiring.
method Approximating Newton Polytope of dividend by divisor, then applying to neural networks.
result Minimizes a two-layer fully connected network for binary classification.
Paper tackles division difficulty, proposing new methods to improve accuracy.
problem Division is the most challenging arithmetic operation for both humans and computers.
method Proposes two novel approaches: Neural Reciprocal Unit (NRU) and Neural Multiplicative Reciprocal Unit (NMRU), and improves an existing division module.
result Improves division accuracy from 70.2% to 91.6%.
In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of p-planes in R2p when p>1. Moreover, this convex divisible domain is a model of the symmetric space associ…
Adopting a zonal structure of electricity market requires specification of zones' borders. In this paper we use social welfare as the measure to assess quality of various zonal divisions. The social welfare is calculated by Market Coupling algorithm. The analyzed divisions are found by the usage of extended Locational …
Division algorithm for surface group rings yields standard complexes and cohomological dimensions.
problem Understanding cohomological dimensions of surface group actions.
method Division algorithm for group rings of surface groups.
result Some 2-complexes with surface fundamental groups are standard.
The paper tackles hierarchical clustering with structural constraints, providing approximation guarantees and improving upon current techniques.
problem Exploiting prior information in hierarchical clustering for real-world applications.
method Top-down algorithms with provable approximation guarantees, using optimization viewpoint and constraint-based regularization.
result Improved solutions for hierarchical clustering with conflicting prior information.
Geometrically explains divisibility of Euler classes for spin modules.
problem Divisibility of KO-theoretical Euler classes for spin modules. method Geometric interpretation of divisibility using spin structures and Seiberg-Witten theory.
result Clarifies the role of reducibles in monopole equations.
New proof of divisibility property for certain algebraic varieties.
problem Divisibility property for LQEL varieties.
method Construction of Clifford algebra representations to Severi varieties.
result New proof of Russo's Divisibility Property for LQEL varieties.
The paper sets lower bounds on envy-free divisions in cake-cutting problems.
problem Finding the minimum number of envy-free divisions in cake-cutting problems.
method Analyzes two scenarios: classical and hybrid, with different constraints and allocations.
result Sharp bounds and examples for envy-free divisions in both scenarios.
The paper revisits Rokhlin's divisibility theorem and its significance.
problem Rokhlin's divisibility theorem on signatures of manifolds.
method Overview and retrace of Rokhlin's proof and further developments.
result Reaffirms the importance of Rokhlin's theorem in manifold theory.
Deep learning predicts downlink channel from uplink data, reducing signaling overhead.
problem Large signaling overhead for full DL CSI in FDD MIMO.
method Deep learning-based channel extrapolation (prediction).
result Deep learning can infer DL CSI from UL CSI without additional overhead.
Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.
problem Division Theorem in Skoda's context
method Degeneration approach inspired by B. Berndtsson and L. Lempert's L2 extension theorem result Simplified and extended proof of L2 extension theorem Under certain integrability and geometric conditions, we prove division theorems for the exact sequences of holomorphic vector bundles and improve the results in the case of Koszul complex. By introducing a singular Hermitian structure on the trivial bundle, our results recover Skoda's division theorem for holomorphic …
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
Constructs projective plane over octonions, proving no higher real division algebras.
problem Existence of higher-dimensional real division algebras.
method Using Adams' solution of the Hopf invariant 1 problem, constructs projective plane over octonions.
result No higher-dimensional real division algebras exist.
We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using this form…
Researchers infer gene activity in dividing cells, accounting for protein inheritance and division history.
problem Inferring protein production kinetics in dividing cells due to protein inheritance and division history.
method Adapted conditional normalizing flows to approximate intractable likelihoods from simulated data.
result Glc3 gene is mostly inactive under stress, with brief and transient expression.
A new method learns manifold-valued latents without an encoder.
problem Distorting data with intrinsic non-Euclidean structure.
method Riemannian generative decoder that learns latents directly.
result Learned representations respect the prescribed geometry and capture intrinsic non-Euclidean structure.
Geodesic connectedness proved for statistical manifolds with divisible cubic forms.
problem Geodesic connectedness of affine connections on statistical manifolds with divisible cubic forms.
method Analogy with Hopf-Rinow theorem in Riemannian geometry, establishing geodesic completeness.
result Geodesic connectedness established for statistical manifolds with divisible cubic forms.
Algorithm learns fair division from noisy feedback in uncertain markets.
problem Learning fair division in uncertain markets with noisy feedback.
method Wrapper algorithms using dual averaging to learn item and agent values from bandit feedback.
result Asymptotically achieves optimal Nash social welfare in linear Fisher markets.
Study of characteristic numbers in 24-dimensional String manifolds.
problem Characterizing and understanding characteristic numbers of 24-dimensional String manifolds.
method Using Pontryagin numbers, integral basis of String cobordism group, and divisibility results.
result Established 2- and 3-primary divisibilities of characteristic numbers.
An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have C1 boundary, and have word hyperbolic divid…
The Jones polynomial's divisibility is analyzed via local moves on virtual links.
problem Divisibility of the Jones polynomial under local moves on virtual links.
method Developed a general decomposition for the Jones polynomial of virtual links and analyzed divisibility conditions for various local moves.
result Succinct divisibility conditions on the Jones polynomial of virtual links differing via local moves.
In this note we introduce a construction which assigns to an arbitrary manifold bundle its fiberwise orientation covering. This is used to show that the zeta classes of unoriented surface bundles are not divisible in the stable range.
Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain tr…
Study shows non-symmetric convex sets have full boundary limits.
problem Understanding boundaries of non-symmetric convex sets.
method Proved using proximal limit set analysis.
result Proximal limit set equals full projective boundary for non-symmetric irreducible divisible convex sets.
Algorithm calculates polynomial Seiberg-Witten invariant from zeta-function of plumbed 3-manifolds.
problem Computing the Seiberg-Witten invariant for plumbed 3-manifolds.
method Némethi's division algorithm for zeta-functions.
result Explicit calculation of polynomial Seiberg-Witten invariant.
Study of congestion in negative curvature manifolds using fair-division algorithms.
problem Estimating and predicting the size and location of congestion core in negative curvature manifolds.
method Introducing a novel fair-division algorithm to estimate congestion core.
result Demonstrated the effectiveness of fair-division algorithms in estimating congestion core.
Research shows how certain flat structures behave in specific convex domains.
problem Understanding the behavior of codimension-1 simplices in divisible convex domains.
method Analyzes the set of codimension-1 flats and their images in quotient manifolds.
result The set of codimension-1 flats forms a finite collection of disjoint virtual tori, leading to cusped convex projective manifolds.
LLMs can collude in market divisions, maximizing profits.
problem Strategic collusion of LLM agents in multi-commodity markets.
method Examined LLMs in Cournot competition frameworks, analyzing pricing and resource allocation strategies.
result LLMs can monopolize specific commodities without direct human input or explicit collusion commands.
Constructs modular forms and proves divisibility results for odd-dimensional manifolds.
problem Constructing modular forms over specific groups and proving divisibility results.
method SL(2, Z) modular forms and Witten genus in odd dimensions.
result Obtained divisibility results of index of Toeplitz operators on spin and spin^c manifolds.
New invariants from divisibility of Lee classes for slice-torus.
problem Determining slice-torus knots using Lee class divisibility.
method Defined new invariants from divisibility of reduced Lee class invariants.
result New invariants coincide with Rasmussen invariant for certain cases.
In this article we consider a version of the geography question for simply-connected symplectic 4-manifolds that takes into account the divisibility of the canonical class as an additional parameter. We also find new examples of 4-manifolds admitting several symplectic structures, inequivalent under deformation and sel…
New 3D shape found that behaves oddly.
problem Finding a 3D shape that isn't hyperbolic.
method Created a specific 3D shape and showed it can't be hyperbolic.
result Discovered a 3D shape that is locally hyperbolic but not hyperbolic.
Study investigates asymptotic risk of overparameterized models, including deep neural networks.
problem Understanding the risk of overparameterized models, especially deep neural networks.
method Analyzes the upper bound of an asymptotic risk of an estimator with penalization, combining Fisher information matrix properties and extended Marchenko-Pastur law.
result Generalized results valid for models without linear-in-feature constraints, indicating small asymptotic risk for specific structures like divisibility.
HiPart offers an efficient, interactive tool for hierarchical clustering.
problem Efficient and interpretable hierarchical clustering for Big Data.
method Divisive hierarchical clustering algorithms with interactive visualizations.
result High computational efficiency and interpretability in Big Data applications.
We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al. We show that the message passing fixed-point equations obtained with this combination correspond to station…
A principle for specialized decision-making divides complex problems into manageable parts.
problem Complex decision-making problems beyond individual capabilities.
method An on-line learning rule that learns a partitioning of the problem space for specialized linear policies.
result The approach solves problems that exceed individual decision-makers' capabilities.
A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, w…
Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory bett…