New cohomology theory for planar graphs with perfect matchings.
problem Understanding cohomology of planar trivalent graphs with perfect matchings.
method Introducing a cohomology theory and defining new polynomials.
result 2-factor polynomial can indicate 4-face colorability.
The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.
problem Developing a stable homotopy type for planar trivalent graphs with perfect matchings.
method Defining a cover functor from the 2-factor flow category to the cube flow category, realizing the 2-factor spectrum, and showing it's an invariant.
result The stable homotopy type of the 2-factor spectrum is an invariant of planar trivalent graphs with perfect matchings.
Paper proposes an analytical pricing model for puttable bonds with credit risk.
problem Analytical pricing of puttable bonds with credit risk.
method Developed a 2-factor structural PDE model and derived analytical pricing formula under specific conditions.
result Derived analytical pricing formula for puttable bonds with credit risk.
Conditions of Stability for explicit finite difference scheme and some results of numerical analysis for a unified 2 factor model of structural and reduced form types for corporate bonds with fixed discrete coupon are provided. It seems to be difficult to get solution formula for PDE model which generalizes Agliardi's …
Paper finds instantons for Kapustin-Witten equations on a specific manifold.
problem Existence of solutions to Kapustin-Witten equations on (0,∞)imesR2imesR. method Explains existence of solutions interpolating between two model solutions.
result Interpolation solutions exist with specific label constraints.
Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M)…
Same genus-2 fibration structures for specific types found by different researchers.
problem Identifying equivalent genus-2 fibrations of type (4, 3).
method Comparing Lefschetz fibration structures of genus-2 fibrations of type (4, 3).
result Lefschetz fibration structures are the same for genus-2 fibrations of type (4, 3).
In this article, we consider a 2 factors-model for pricing defaultable bond with discrete default intensity and barrier where the 2 factors are stochastic risk free short rate process and firm value process. We assume that the default event occurs in an expected manner when the firm value reaches a given default barrie…
No Lagrangian Klein bottles found in S2imesS2.
problem Existence of Lagrangian Klein bottles in S2imesS2. method Luttinger surgery to show non-existence in a specific homology class.
result No Lagrangian Klein bottles in S2imesS2. In this paper, we mainly prove a theorem with a corollary establishing two characterizations of the Calabi composition of hyperbolic hyperspheres, where the second characterization (i.e., the corollary) has been given via a dual correspondence theorem earlier but now we would like to use a very direct method. Note that…
Algorithm finds small confidence sets for arbitrary distributions.
problem Learning high-density regions in arbitrary distributions.
method Competitive with sets from a concept class with bounded VC-dimension.
result Algorithm finds a confidence set with volume exp(ildeO(d1/2)) competitive with optimal ball. The universal order 1 invariant f^U of immersions of a closed orientable surface into R^3, whose existence has been established in [N3], takes values in the group G_U = K \oplus Z/2 \oplus Z/2 where K is a countably generated free Abelian group. The projections of f^U to K and to the first and second Z/2 factors are de…
Improved private learning of halfspaces with reduced sample complexity.
problem Private learning of halfspaces with reduced sample complexity.
method Iterative algorithm for solving linear feasibility problem, improving state-of-the-art results.
result Sample complexity reduced to d2.5⋅2log∗∣G∣, improving d2 factor. This work shows how penalising bias terms in norm regularisation leads to sparse solutions.
problem Understanding the relation between parameter norm regularization and the sparsity of neural network solutions.
method Analyzes one hidden ReLU layer networks with unidimensional data, showing the norm required for function representation and the importance of the bias term's norm.
result Penalising the bias terms in regularisation leads to sparse solutions, enforcing the uniqueness and sparsity of the minimal norm interpolator.
Hikami observed a discontinuity in a WRT invariant at roots of unity.
problem Discontinuity of a WRT invariant at roots of unity.
method Using Bailey's lemma and false theta functions.
result Generalized Hikami's observation about the discontinuity.
We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball B4(c)⊂R4 into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form Mλ=(S2×S2,μω0⊕ω0) where ω0…
CPPO learns policies from partial offline data in MDPs with structural assumptions.
problem Offline Reinforcement Learning with partial coverage assumption.
method Constrained Pessimistic Policy Optimization (CPPO) using a function class and model class constraint.
result CPPO achieves PAC guarantee with partial coverage, learning competitive policies.
We present three families of exact, cohomogeneity-one Einstein metrics in (2n+2) dimensions, which are generalizations of the Stenzel construction of Ricci-flat metrics to those with a positive cosmological constant. The first family of solutions are Fubini-Study metrics on the complex projective spaces CPn+1, w…
New method selects best offline RL policies from logged data.
problem Hyperparameter selection challenges offline RL.
method Offline hyperparameter selection for RL algorithms.
result Reliable ranking and selection of policies across hyperparameters.
Study impacts of feeding cost risk on aquaculture valuation and decision making.
problem Impact of stochastic feeding costs on aquaculture valuation and decision making.
method Using Schwartz-2-factor model and deep neural networks to infer decision boundary.
result Accounting for stochastic feeding costs leads to superior performance in decision rules.
New uniformity tester ensures consistent results across different samples.
problem Non-replicable behavior of uniformity testing algorithms.
method Develops a replicable uniformity tester with improved sample complexity.
result Achieves nearly linear dependence on replicability factor ρ. A new DRL model optimizes hedging with market impact for low-liquidity stocks.
problem Optimizing hedging strategies for stocks with limited liquidity.
method Integrates Deep Reinforcement Learning with realistic market impact features.
result Optimal hedging policies learned from DRL model perform better in low-liquidity scenarios.
New findings connect shaped and unshaped neural networks using differential equations.
problem Understanding the behavior of neural networks with different activation scaling methods.
method Deriving differential equation-based asymptotic characterizations for shaped and unshaped neural networks.
result Two types of unshaped networks converge to the same infinite-depth-and-width limit at initialization.
The paper presents efficient methods for identifying causal graphs with latent variables.
problem Recovering causal graphs with latent variables while minimizing intervention costs.
method Two intervention cost models (linear and identity) are considered. Algorithms are provided for both models.
result Upper bounds on the number of interventions needed for recovery, and approximation factors for the linear cost model.
This paper develops dimension-agnostic inference methods for high-dimensional data.
problem Understanding how classical inference methods behave in high-dimensional settings.
method Using variational representations, sample splitting, and self-normalization to create a refined test statistic.
result The resulting statistic has a Gaussian limiting distribution regardless of how dimensionality scales with sample size.
The paper proposes a new method to estimate interest rates consistently under both risk-neutral and real-world measures.
problem Consistent estimation of interest rates under both risk-neutral and real-world measures.
method Proposes a framework using progressive and square-integrable functions to specify the change of measure, and introduces two time-dependent candidates: step and linear functions.
result The proposed methods produce more stable and realistic long-term interest rate forecasts compared to using a constant function.
New F-polynomial distinguishes knotoid diagrams not previously possible.
problem Polynomial invariants for knotoids.
method Introduced F-polynomial and constructed distinguishing examples. result New invariant F distinguishes knotoids not previously possible. The paper studies polynomials and ideals from colored Jones polynomials for links.
problem Understanding the structure of colored Jones polynomials for links.
method Investigates commutative and noncommutative ideals derived from colored Jones polynomials.
result Formulates the link version of the AJ conjecture.
Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.
problem Examining relationships between Affine Index Polynomial and Sawollek Polynomial.
method New approach to extract Affine Index Polynomial from Sawollek Polynomial.
result Constructs a concise proof of Mellor's Theorem.
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.
problem Understanding and characterizing knot polynomials from Gaussian calculus.
method Gaussian calculus of generating series for noncommutative algebras, connected sum of knots.
result Half of the polynomials vanish and three polynomials are explicitly given.
The paper defines and classifies Cappell-Shaneson polynomials.
problem Characterizing Cappell-Shaneson polynomials.
method Algebraic conditions on polynomials, reduction modulo primes, and construction of infinite series.
result Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5, and several infinite series of degree 6.
Developed algorithms to compute three polynomial invariants of veering triangulations.
problem Computing polynomial invariants of veering triangulations.
method Introduced and used algorithms for taut, veering, and Teichmüller polynomials based on upper and lower tracks of veering triangulations.
result Proved that the lower and upper taut polynomials are equal but the veering polynomials can differ.
A new kernel test avoids permutations for independence testing.
problem Intractable null distributions of kernel statistics.
method Developed xHSIC and xdCov, avoiding permutations.
result New tests have limiting Gaussian distributions under null.
We introduce the problem of hidden Hamiltonian cycle recovery, where there is an unknown Hamiltonian cycle in an n-vertex complete graph that needs to be inferred from noisy edge measurements. The measurements are independent and distributed according to $\calP_n$ for edges in the cycle and $\calQ_n$ otherwise. This …
SAM optimizer struggles to converge to global minima or stationary points in practical settings.
problem Limited convergence of SAM optimizer to global minima or stationary points in practical scenarios.
method Deterministic and stochastic versions of SAM with constant perturbation size and gradient normalization were studied.
result SAM has limited capability to converge to global minima or stationary points in many scenarios.
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.
Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.
problem Understanding polynomial invariants of pretzel links.
method Revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links P(1,1,n). result Reveals properties of Alexander-Conway and Kauffman bracket polynomials for P(1,1,n). Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between A-polynomial and colored Jones polynomial. method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between A-polynomial and colored Jones polynomial potential function. Algorithm estimates principal eigenvector with adaptive sensing, improving over non-adaptive methods.
problem Estimating principal eigenvector with limited scalar measurements.
method Compressed variant of Oja's algorithm using two adaptive measurements per sample.
result Convergence rate of O(λ1λ2d2/(Δ2t)) after t iterations, matching information-theoretic lower bound. Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…
This paper investigates the equivalence between Yamada polynomial and Jones polynomial of associated links for brunnian θ-curves.
problem Understanding the relationship between Yamada polynomial and Jones polynomial for θ-curves.
method Investigates the equivalence between the normalized Yamada polynomial of θ-curves and the Jones polynomial of their associated links.
result Shows that the two polynomials are equivalent for brunnian θ-curves.
The taut polynomial equals a twisted Alexander polynomial.
problem Understanding the relationship between taut polynomials and Alexander polynomials.
method Defined taut polynomial of veering triangulations and proved it equals a twisted Alexander polynomial.
result The taut polynomial equals a twisted Alexander polynomial of the underlying manifold.
Quantum polynomials are derived from a specific tribracket structure.
problem Quantum enhancement polynomials for oriented links.
method Defined using a canonical two-element tribracket, proving polynomials can be derived from five specific ones.
result Universal quantum enhancement polynomials are strictly stronger than the Jones polynomial.
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomial…
New polynomials show all knots of a certain type have simple structure.
problem Classifying knots with specific properties.
method Introduced new L-polynomials and F-polynomials for virtual knots, and defined totally flat-trivial knots.
result All Akimova-Matveev knots are totally flat-trivial and their affine index polynomials can be calculated.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
Innovates polynomial invariant for tribrackets.
problem Counting and distinguishing knots and links.
method Introduces subtribracket polynomials and uses them to enhance counting.
result Enhanced counting invariant for knots and links.