We propose a general theory for constructing functorial assignments Σ⟼ΩΣ∈E(Σ) for a large class of functors E from a certain category of bordered surfaces to a suitable target category of topological vector spaces. The construction proceeds by successive excisions of homotopy classes of embedded pai…
The Darboux-Egoroff system of PDEs with any number n≥3 of independent variables plays an essential role in the problems of describing n-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux-Egoroff system and des…
New 3D shapes can't be split into torus pieces.
problem 3D shapes without torus decompositions.
method Recursive definition from compact 3-manifolds.
result Examples of 3D shapes failing torus decomposition.
We derive a recursion relation for hyperbolic string vertices and apply it to string field theory.
problem Deriving a recursion relation for hyperbolic string vertices and its implications for string field theory.
method Using systolic volumes and a modified Mirzakhani's recursion, we construct a higher-order vertex determination for hyperbolic string field theory.
result The higher order vertices in hyperbolic string field theory are determined by the cubic vertex iteratively for any background.
In this article, we aim at improving the prediction of expert aggregation by using the underlying properties of the models that provide expert predictions. We restrict ourselves to the case where expert predictions come from Kalman recursions, fitting state-space models. By using exponential weights, we construct diffe…
This paper explores non-periodic folding of Spidron units, revealing nonlinear dynamics.
problem Understanding the kinematics and nonlinear phenomena of Spidron units.
method Analysis of single unit cell kinematics and recursive construction of multiple cells.
result Non-periodic folding restricts isotropic folding as the number of unit cells increases.
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.
Study torus knots in lens spaces using Gromov-Witten invariants and topological recursion.
problem Computing Gromov-Witten invariants for torus knots in lens spaces.
method Construct Lagrangian sub-manifolds and relate to topological recursion.
result Verify a conjecture in lens space for Gromov-Witten invariants.
Recurrent neural networks (RNNs) process input text sequentially and model the conditional transition between word tokens. In contrast, the advantages of recursive networks include that they explicitly model the compositionality and the recursive structure of natural language. However, the current recursive architectur…
Study dynamic Pareto-optimal allocations in multi-period economies with time-consistent risk measures.
problem Optimal allocation in multi-period pure-exchange economies with stochastic endowments and time-consistent risk measures.
method Introduced dynamic Pareto-optimal allocation processes and derived recursive and comonotone improvement theorems.
result Dynamic Pareto-optimal allocation processes can be constructed recursively and are comonotone.
Local classification of surfaces and hypersurfaces with radial mean curvature.
problem Classifying surfaces and hypersurfaces with specific curvature properties.
method Local classification and recursive construction method.
result Local classification of hypersurfaces with vanishing second mean curvature.
We prove the ADO invariants are a q-holonomic family and establish recursion relations.
problem Understanding the q-holonomic properties of ADO link invariants. method Proving the ADO invariants are a q-holonomic family and establishing recursion relations. result The ADO invariants for r≥2 are a q-holonomic family, satisfying independent recursion relations. Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on S3 and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on S3 and their invariants encoded in colored HOMFLY polynomials by means of topological recursion. In the context of the …
New method calculates super-hedging prices with transaction costs.
problem Super-hedging European contingent claims under proportional transaction costs.
method Explicit recursive scheme based on convex duality and Legendre-Fenchel transform.
result Computes super-hedging price and optimal strategy without martingale arguments.
Study finds root vertex in large networks with high probability.
problem Finding the root vertex in large growing networks.
method Constructs confidence sets for the root vertex in various random network models.
result Confidence sets of size independent of the number of vertices contain the root vertex with high probability.
New categorified homology expressions for torus knots and links.
problem Computing homology for colored torus knots and links.
method Recursive construction of categorified Young symmetrizers and comparison with row-colored homology.
result Verification of mirror symmetry conjectures for positive torus knots.
CEFOL uses deep learning for dynamic programming with recursive utility.
problem Challenges in solving dynamic programming problems with recursive utility.
method Introduces a separate neural network for certainty equivalent, uses first-order optimality conditions to learn value and policy functions.
result CEFOL achieves high accuracy in learning value and policy functions, matching VFI benchmarks.
Recursive neural networks have widely been used by researchers to handle applications with recursively or hierarchically structured data. However, embedded control flow deep learning frameworks such as TensorFlow, Theano, Caffe2, and MXNet fail to efficiently represent and execute such neural networks, due to lack of s…
Proposes categorification of Z-invariants for specific 3-manifolds.
problem Categorification of Z-invariants for negative definite plumbed 3-manifolds.
method Abelian categorification using 3d N=2 theory and log VOAs.
result Nested Weyl-type character formulas reconstruct Z^-invariants. The paper analyzes distances and volumes in lens spaces using recursion and formulas.
problem The problem of moments for distances between points on lens spaces.
method Derivation of recursion relations, formulas for moments and moment generating function, explicit formula for ball volumes.
result Explicit formulas for the volume of balls of all radii in lens spaces.
Paper defines Farey Recursive Functions and explores their properties.
problem Understanding recursive functions on rationals.
method Defined and studied Farey Recursive Functions using Farey graph.
result Farey Recursive Functions naturally connect to 2-bridge knots and links.
The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.
problem Computing volumes for physical gravity models.
method Topological recursion and physical two-dimensional gravity models.
result Derivation of Virasoro constraints and cut-and-join equations for generalized Mirzakhani's recursions.
We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds.
Causal trees struggle with accuracy in estimating treatment effects.
problem Estimating heterogeneous causal treatment effects using recursive decision trees.
method Adaptive recursive partitioning with and without sample splitting.
result Causal tree estimators can have uniform-norm errors decreasing more slowly than any power of the sample size.
Fractal Flow enhances normalizing flows with interpretable latent space and hierarchical modeling.
problem High-dimensional density estimation and generative modeling challenges.
method Integrates topic modeling (LDA) and fractal strategy into normalizing flows.
result Achieves latent clustering, controllable generation, and superior estimation accuracy.
This work proves lower bounds on a greedy teaching set construction algorithm.
problem Characterize the best-case teaching dimension of a concept class.
method A greedy algorithm that iteratively adds points to the teaching set to restrict the concept class the most.
result Lower bounds on the performance of the greedy approach for small k, extending up to k ≤ c*d for small constant c.
We study recursive-cube-of-rings (RCR), a class of scalable graphs that can potentially provide rich inter-connection network topology for the emerging distributed and parallel computing infrastructure. Through rigorous proof and validating examples, we have corrected previous misunderstandings on the topological prope…
New estimator reduces nested expectation estimation costs.
problem Estimating repeatedly nested expectations is computationally expensive.
method Recursive Estimator for Arbitrary Depth (READ) using randomized multilevel Monte Carlo.
result Optimal computational cost of O(ε^(-2)) for every fixed D.
Tab-TRM uses recursive model for insurance pricing on tabular data.
problem Insurance pricing on tabular data.
method Adapts recursive latent reasoning to insurance modeling using a compact, parameter-efficient network.
result Improves insurance pricing accuracy using iterative refinement of latent tokens.
In this paper, a new method is proposed for sparse PCA based on the recursive divide-and-conquer methodology. The main idea is to separate the original sparse PCA problem into a series of much simpler sub-problems, each having a closed-form solution. By recursively solving these sub-problems in an analytical way, an ef…
In a recurrent setting, conventional approaches to neural architecture search find and fix a general model for all data samples and time steps. We propose a novel algorithm that can dynamically search for the structure of cells in a recurrent neural network model. Based on a combination of recurrent and recursive neura…
New recursion formula for non-orientable surfaces resolves divergences.
problem Computing volumes of moduli spaces for non-orientable surfaces.
method Generalization of Mirzakhani's recursion to non-orientable surfaces, handling divergences with integral kernels.
result Regularized volumes can be computed with a cutoff on crosscap size.
Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is t…
Paper introduces v-CMC linking causality and utility.
problem Linking causality and utility for value theory.
method Developed a new causal independence principle (v-CMC) and proved its equivalence.
result Equivalence of local, global, and decomposition versions of v-CMC.
This paper studies recursive ensembles driven by Fibonacci updates, improving learning dynamics.
problem Improving learning dynamics in recursive ensemble learning.
method Develops second-order recursive architectures with Fibonacci-type update flows.
result Establishes global convergence conditions and generalization bounds for recursive ensembles.
Solves a recursion for Gromov-Witten invariants of the unknot.
problem Determining Gromov-Witten invariants for a specific Lagrangian brane.
method Uses a skein-theoretic recursion and geometric solutions.
result Solves the recursion to find the expected hook-content formula.
This study proposes an approach based on a perturbation technique to construct global solutions to dynamic stochastic general equilibrium models (DSGE). The main idea is to expand a solution in a series of powers of a small parameter scaling the uncertainty in the economy around a solution to the deterministic model, i…
New recursion found for hyperbolic sphere volumes.
problem Volume calculation of hyperbolic sphere moduli spaces.
method Proved a non-linear recursive relation.
result Generalized Zograf's result for conical points and geodesic boundaries.
This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.
problem Analyzing the relationship between Seiberg-Witten curves and topological recursion.
method Analytical approach using Seiberg-Witten family of curves.
result A generalized formula relating Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
LASER compresses recursive model activations by exploiting their low-dimensional structure.
problem Understanding and optimizing the geometric structure of recursive reasoning trajectories.
method Dynamic low-rank basis tracking via matrix-free subspace tracking with a fidelity-triggered reset mechanism.
result Recursive activations occupy a linear, low-dimensional subspace that can be compressed efficiently.
We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to Göttsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recu…
Topological recursion recovers a specific partition function for colored knots.
problem Recovering the extended Ooguri-Vafa partition function for colored HOMFLY-PT polynomials of torus knots.
method Applying topological recursion to the spectral curve of colored HOMFLY-PT polynomials of torus knots.
result Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.
We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brezín and Hikami. We express this recursion in special times in which all terms W1(g) of the genus expansion of the one-loop mean are polynomials. We find a generalization of th…
This paper concerns the recursive utility maximization problem under partial information. We first transform our problem under partial information into the one under full information. When the generator of the recursive utility is concave, we adopt the variational formulation of the recursive utility which leads to a s…
Unified framework for observables in n-plectic geometry.
problem Quantization of extended objects in higher geometric contexts.
method Develops a semi-simplicial set model for observables, using a Grassmann variable to encode submanifold codimensions.
result Establishes a categorified pre-n-Hilbert space and a quantization scheme matching multisymplectic geometry.
Machine learning finds new natural laws from noisy data.
problem Discovery of natural laws relies on human inspiration.
method Recursive-LASSO-based symbolic regression (RLS) method.
result Data-driven formulation of natural laws from noisy data.
We study an open problem of risk-sensitive portfolio allocation in a regime-switching credit market with default contagion. The state space of the Markovian regime-switching process is assumed to be a countably infinite set. To characterize the value function, we investigate the corresponding recursive infinite-dimensi…