Paper finds coefficients of Catalan states using Θ_A-state expansion.
problem Finding coefficients of Catalan states of lattice crossings.
method Uses Θ_A-state expansion to express coefficients as a linear combination of other states.
result Shows that coefficients can be found using Θ_A-state expansion.
New formulas derived for lattice crossing coefficients, improving computation efficiency.
problem Computing coefficients of Catalan states in lattice crossings.
method Using plucking polynomial and Θ_A-state expansion, deriving new properties and formulas.
result Coefficients of Catalan states factor under specific conditions, leading to more efficient computation.
This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and an extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs pres…
We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollar…
New algebra counts components of arborescent knots and links.
problem Counting components of arborescent knots and links.
method Developed a new algebra called the crossing algebra.
result The crossing algebra counts the number of components for arborescent knots and links.
Paper applies theorem to find optimal investment boundary in stochastic capacity expansion.
problem Finding optimal investment boundary in a stochastic, time-inhomogeneous capacity expansion problem.
method Applies Bank and El Karoui Representation Theorem to solve first order conditions involving a non-integral term.
result Existence of base capacity ly⋆(t), showing optimal investment process becomes active at this level. Quantum trace map defines invariants for knots and links, confirming a length conjecture.
problem Defining invariants for knots and links in hyperbolic 3-manifolds.
method Introducing a quantum trace map for ideally triangulated knot complements, combining with state-integral models.
result Perturbative invariants determine an asymptotic expansion of the Jones polynomial, confirming the length conjecture.
A softmax operator applied to a set of values acts somewhat like the maximization function and somewhat like an average. In sequential decision making, softmax is often used in settings where it is necessary to maximize utility but also to hedge against problems that arise from putting all of one's weight behind a sing…
Agents learn state ambiguity from non-linear sensor data using Gaussian approximations.
problem Learning state representation from non-linear sensor data.
method Second-order Taylor approximation of Gaussian distribution for non-linear measurement functions.
result Induces a preference for states based on inferability from observations.
Study improves stock return prediction by switching between economic states, outperforming traditional methods.
problem Improving stock return prediction across economic regimes.
method State-switching specification using the slope of the yield curve, with an Aligned Economic Index.
result The Aligned Economic Index outperforms traditional predictors, especially during market turbulence.
New explanation of reservoir computing using random projections.
problem Understanding the randomness in reservoir computing.
method Constructing strongly universal reservoir systems as random projections of state-space systems.
result Approximation of any fading memory filters class by training a linear readout for each filter.
Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are probabilistic and non-parametric and as such are arguably more flexible, have a greater capacity to generalise, an…
We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.
F. Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n)-link invariant (a one…
New formula for implied volatility from Black-Scholes model.
problem Computing implied volatility from Black-Scholes model.
method Analytical solution using inverse Gaussian distribution.
result Explicit formulas for implied volatility with high precision.
We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori-Hecke algebras of type A.
Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
Paper describes a state sum formula for a graph coloring polynomial.
problem Counting n-face colorings of ribbon graphs for various n. method Combines topological quantum field theory and diagrammatic tensors.
result Describes a state sum formula for the total face color polynomial.
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Reinforcement learning algorithms can solve dynamic decision-making and optimal control problems. With continuous-valued state and input variables, reinforcement learning algorithms must rely on function approximators to represent the value function and policy mappings. Commonly used numerical approximators, such as ne…
Develops a martingale expansion for stochastic volatility models.
problem Approximating marginal distributions of stochastic volatility models.
method Martingale expansion framework for continuous stochastic volatility models.
result First-order perturbation expansions for small volatility-of-volatility and fast mean-reversion models.
We introduce the notion of a "state function" for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states fo…
Taylor expansions improve reinforcement learning policies.
problem Improving reinforcement learning policy optimization.
method Taylor expansion policy optimization.
result Taylor expansions enhance performance of distributed algorithms.
Analytic torsion expansions for symmetric and complex homogeneous spaces.
problem Calculating the full asymptotic expansion of analytic torsion for various spaces.
method Explicit calculation and comparison with existing results.
result Explicit full asymptotic expansions for symmetric and complex homogeneous spaces.
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
Study of hypersurfaces with specific expansion properties.
problem Existence and properties of hypersurfaces with prescribed null expansion.
method Adapted Eichmair's Perron approach for existence.
result Topology theorem for hypersurfaces with prescribed null expansion.
A new hypergraph expansion method treats vertices and hyperedges equally, improving node classification.
problem Information loss in hypergraph expansions on either vertex or hyperedge level.
method Proposes a new hypergraph formulation named line expansion (LE) that treats vertices and hyperedges symmetrically.
result The proposed line expansion method outperforms state-of-the-art baselines on five hypergraph datasets.
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.
Dropout increases the generalization of neural networks by expanding the weight space.
problem Understanding and improving the generalization of neural networks.
method Introducing weight expansion and showing that dropout leads to it.
result Dropout increases the generalization of neural networks by expanding the weight space.
New derivation of knot invariants from universal invariant.
problem Deriving knot invariants from universal invariant.
method Using Hopf algebra D and a Mathematica implementation to compute ZD(K). result Derivation of large-color expansion from universal invariant.
This work explores functional expansions to handle path dependence in various fields.
problem Path dependence and infinite-dimensional problems in non-Markovian systems.
method Generalizes Wiener series and functional Taylor expansion to handle static and dynamic functionals.
result Elegant separation of functionals from future trajectories in dynamic cases.
In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus correctio…
Paper calculates third coefficient in Kaehler-Einstein metric expansion.
problem Understanding Kaehler-Einstein metrics and their epsilon functions.
method Computes the third coefficient in the TYCZ-expansion of the epsilon function.
result Discovers the vanishing of the third coefficient's significance.
We describe the first known mean-field study of landing probabilities for random walks on hypergraphs. In particular, we examine clique-expansion and tensor methods and evaluate their mean-field characteristics over a class of random hypergraph models for the purpose of seed-set community expansion. We describe paramet…
Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.
problem Asymptotic expansion of heat trace for thermoelastic Dirichlet-to-Neumann map.
method Provided a method to obtain all coefficients of the asymptotic expansion.
result Explicitly gave the first two coefficients involving volume and total mean curvature of the boundary.
The paper proposes and proves asymptotic expansions for quantum invariants.
problem Quantum invariants and their expansions under varying metrics.
method Asymptotic expansion conjectures for relative Reshetikhin-Turaev, Turaev-Viro invariants and quantum 6j-symbols.
result Proved asymptotic expansions for special cases, showing geometric dependence on metrics.
New method models portfolios with leptokurtic risk factors using Gram-Charlier expansions.
problem Modeling portfolios with excess kurtosis.
method GC-like expansions of the hyperbolic-secant law to account for leptokurtosis.
result Portfolio distribution with risk factors modeled as GC-like expansions of the HS law.
Develops AMITE for analyzing neural network nonlinearities.
problem Addressing difficulties in verification, explainability, and security in neural network analysis.
method Analytically modified integral transform expansion (AMITE) for neural network nonlinearities.
result First to provide six mutually exclusive desired expansion properties.
We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.
Paper presents new expansions for option pricing with cash dividends.
problem No exact formula for European options with cash dividends.
method Uses Etore and Gobet's technique for piecewise lognormal process with jumps.
result Provides more robust first, second, and third-order expansions.
The validity of an approximation formula for European option prices under a general stochastic volatility model is proved in the light of the Edgeworth expansion for ergodic diffusions. The asymptotic expansion is around the Black-Scholes price and is uniform in bounded payoff func- tions. The result provides a validat…
The paper uses polyhedral expansions to capture the shape of compact metric spaces.
problem Capturing the shape of compact metric spaces using finite approximations.
method Inverse sequences of polyhedra based on finite approximations of a compact metric space.
result Proves the General Principle and computes inverse persistent homology groups.
We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kern…
We quantify predictive uncertainty using the posterior predictive variance.
problem Quantifying uncertainty in predictive models.
method Using the law of total variance, we generate expansions for the posterior predictive variance.
result Identify the main contributors to prediction intervals and quantify term-wise uncertainty.
The study improves volatility model pricing accuracy with new statistical expansions.
problem Improving option pricing accuracy in volatility models.
method Developed Edgeworth expansions for various volatility models.
result Enhanced statistical expansions for volatility models.
The notion of a symplectic expansion directly relates the topology of a surface to formal symplectic geometry. We give a method to construct a symplectic expansion by solving a recurrence formula given in terms of the Baker-Campbell-Hausdorff series.
For any strictly positive martingale S=exp(X) for which X has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log strike. We illustrate the versatility of our expansion by computing t…