Study Markov cubature rules for polynomial processes.
problem Tractability of path-dependent tasks in polynomial process models.
method Discretizations using finite state Markov processes with moment matching conditions.
result Markov cubature rules aid American option pricing.
Generalizes ruling polynomials to Legendrian tangles and proves their composition property.
problem Computing augmentation numbers for Legendrian tangles.
method Generalization of ruling polynomials to Legendrian tangles and proving their composition property.
result Ruling polynomials for Legendrian tangles satisfy the composition axiom.
Study Legendrian graph invariants via augmentation and ruling polynomials.
problem Equivalence of Legendrian isotopy invariants.
method Use augmentation number and ruling polynomial for front projection.
result Show equivalence between augmentation number and ruling polynomial.
New invariants for Legendrian graphs defined and proven.
problem Defining and proving invariants for Legendrian graphs.
method Defined ruling invariants and proved their properties.
result Ruling invariants are compatible with vertex-identifying operations and vertical cuts/gluings.
For Legendrian links in the 1-jet space of S1 we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kau…
We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar met…
New formulas link knot invariants from DGA and satellite polynomials.
problem Establishing relationships between knot invariants.
method Introducing new polynomials and formulas linking DGA and satellite invariants.
result Arbitrary m-graded satellite ruling polynomials are determined by DGA of K.
Legendrian knot representations linked to colored Kauffman polynomial.
problem Relating Legendrian knot representations to colored Kauffman polynomial.
method Introducing ungraded n-colored ruling polynomial and relating it to the n-colored Kauffman polynomial. result Ungraded representation numbers of Legendrian knot DG-algebra agree with n-colored Kauffman polynomial specialization. We show that for any Legendrian link L in the 1-jet space of S1 the 2-graded ruling polynomial, RL2(z), is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover RL2(z) as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, …
For any Legendrian link, L, in (\R^3, \ker(dz-y\,dx)) we define invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L. Generalizing a result of Ng and Sabloff for the…
Homotopy cardinality counts augmentations of Legendrian knots.
problem Counting augmentations of Legendrian knots.
method Assigning ruling polynomials and proving homotopy cardinality results.
result Homotopy cardinality of representation categories is a multiple of ruling polynomials.
Study of knot polynomials for twist satellites, generalizing cabling.
problem Lifting decomposition rule to superpolynomials for positive and negative twistings.
method General decomposition of satellite's colored HOMFLY polynomial in terms of original knot's contributions.
result Knot polynomials for twist satellites are related to Vogel's universality.
For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. This work is motivated by Khovanov's work on categorification of the…
Invariant Causal Set Covering Machines avoid spurious associations.
problem Learning algorithms for rule-based models are vulnerable to spurious associations.
method Building on invariant causal prediction, propose Invariant Causal Set Covering Machines for conjunctions/disjunctions of binary-valued rules.
result The method can identify causal parents of a variable of interest in polynomial time.
Machine learning improves searching for polynomial proofs.
problem Automatically searching for proofs of polynomial inequalities.
method Deep reinforcement learning guiding inference rules in semi-algebraic proof systems.
result Reduces the size of linear programs by several orders of magnitude.
Study geometric bases for A-polynomials in SU(3) using arcade formalism.
problem Understanding relations between different knots and their representations.
method Use arcade formalism and Kuberberg rule to simplify calculations.
result Equations for knot polynomials become independent of representation.
This study analyzes adversarial training on linearly separable data and finds that gradient updates can achieve large margins in polynomial iterations.
problem Ensuring robustness in machine learning models trained on linearly separable data.
method Analysis of adversarial training with gradient updates on linearly separable data.
result Gradient updates in adversarial training can achieve large margins in polynomial iterations, whereas non-smooth methods require exponentially many iterations.
The Berglund-Hübsch rule connects Calabi-Yau orbifolds to Sasakian manifolds.
problem Connecting Calabi-Yau orbifolds to Sasakian manifolds.
method Applying the Berglund-Hübsch transpose rule to associate Sasaki manifolds.
result Four seven-dimensional Sasakian manifolds of positive Ricci curvature are associated with a K3 orbifold.
A graph G is said to be p-periodic, if the automorphism group Aut(G) contains an element of order p which preserves no edges. In this paper, we investigate the behavior of graph polynomials (Negmai and Tutte) with respect to graph periodicity. In particular, we prove that if p is a prime, then the coefficient…
The colored HOMFLY polynomial is the quantum invariant of oriented links in S3 associated with irreducible representations of the quantum group Uq(slN). In this paper, using an approach to calculate quantum invariants of links via cabling-projection rule, we derive a formula for the colored HOMFLY polyn…
New examples show Sasaki manifolds without extremal metrics.
problem Finding Sasaki manifolds without extremal metrics.
method Using Berglund-Hübsch transpose rule and relative K-stability.
result Examples of Sasaki manifolds without extremal metrics.
This note is a write-up of a talk given by the author at the Meeting of the Sociedade Portuguesa de Matematica in July 2012. We describe Jaeger's HOMFLY-PT expansion of the Kauffman polynomial and how to generalize it to other quantum invariants using the so-called "branching rules" for Lie algebra representations. We …
We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively n…
It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in R3 vanishes on a real analytically ruled two-dimensional surface S⊂R3 then S is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial…
We present an easy example of mutant links with different Khovanov homology. The existence of such an example is important because it shows that Khovanov homology cannot be defined with a skein rule similar to the skein relation for the Jones polynomial.
Develops a new fuzzy model using QPs and ewl2 regularization to improve local region behavior.
problem Inability of constant and linear functions to accurately describe local regions in fuzzy models.
method Applied Fuzzy C-Means for structure identification, used QPs as consequents, introduced ewl2 regularization.
result Improved model's ability to describe local regions without overfitting.
It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describi…
Develops a generalized version of Chung's Lemma for stochastic optimization methods.
problem Establishing asymptotic convergence rates for stochastic optimization methods under various step size rules.
method Generalized version of Chung's Lemma for a broader family of step size rules.
result Demonstrates tight non-asymptotic convergence rates for various stochastic methods.
In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian ci…
Paper shows domain recursion is more powerful than previously thought, enabling faster inference.
problem Intractable probabilistic inference in relational models.
method Study of domain recursion rule and its impact on lifted inference.
result Domain recursion extends the range of models for which lifted inference is polynomial-time.
New algorithm solves complex stopping problems with robust optimization.
problem Solving complex stochastic optimal stopping problems.
method Simulation-based robust optimization with exact reformulation as a zero-one bilinear program.
result Developed polynomial-time heuristics and algorithms for practical solution.
In a physical neural system, where storage and processing are intimately intertwined, the rules for adjusting the synaptic weights can only depend on variables that are available locally, such as the activity of the pre- and post-synaptic neurons, resulting in local learning rules. A systematic framework for studying t…
Skein theory classifies UFCs with specific fusion rules.
problem Classifying unitary fusion categories with specific fusion rules.
method Graphical calculus and rotation operator action on a canonical basis.
result Explicit formulae for Fqqqq when k=2 and C is ribbon. Study disproves conjecture about low-degree polynomials in hypothesis testing.
problem Conjecture about limitations of polynomial-time algorithms in hypothesis testing.
method Used counterexamples to refute the conjecture and modified the conjecture to rule out the counterexample.
result Disproved conjecture about limitations of low-degree polynomials in hypothesis testing.
Slice-polynomial functions help compute twistor discriminant loci of cubic scrolls.
problem Computing the twistor discriminant locus of cubic scrolls in CP3. method Introduced slice-polynomial functions and their companions/extensions, used twistor theory.
result Generically constant cardinality of pre-images for slice-polynomial functions.
Algorithm maximizes wealth from best pairs rebalancing rule in hindsight.
problem Maximizing wealth from best pairs rebalancing rule in hindsight.
method Extends Ordentlich and Cover's max-min universal portfolio to achieve a percentage of the hindsight-optimized wealth.
result Achieves a compound-annual growth rate arbitrarily close to the best pairs rebalancing rule in hindsight.
Study local moduli of Sasaki-Einstein metrics on specific polynomial links.
problem Understanding the local moduli of Sasaki-Einstein metrics on links of invertible polynomials.
method Analyzing Sasaki-Einstein metrics on links of invertible polynomials of cycle type and Thom-Sebastiani sums.
result For polynomials of cycle type, local moduli spaces are zero-dimensional. For Thom-Sebastiani sums, dimensions are positive.
New Sasaki-Einstein 7-spheres found via Berglund-Hübsch transpose.
problem Finding new Sasaki-Einstein 7-spheres.
method Berglund-Hübsch transpose rule applied to Kähler-Einstein 3-folds.
result 75 new Sasaki-Einstein rational homology 7-spheres discovered.
The paper connects knot homology, quantum 6j-symbols, and complements of knots.
problem Investigating the relationship between knot homology, quantum 6j-symbols, and knot complements.
method Developed a grading rule for HOMFLY-PT and Kauffman homology, found relationships between A-polynomials, and conjectured closed-form expressions for quantum 6j-symbols and knot complements.
result Closed-form expressions for SO(N) quantum 6j-symbols and conjectured expressions for (a,t)-deformed F_K for knot complements.
A new method for sampling on manifolds reduces density estimation errors.
problem Sampling on implicitly defined manifolds in various applications.
method Polynomial-Maximization Moment (PMM) estimator replacing local k-nearest-neighbour density estimate.
result Reduces density estimation errors by 22--36% on asymmetric gamma and boundary-spacing regimes.
The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling, therefore its Thurston-Bennequin number is maximal among Legendrian …
Paper proposes a method for early stopping in regression using reproducing kernels.
problem Early stopping for iterative learning algorithms in nonparametric regression.
method Data-driven rule based on minimum discrepancy principle, validated by fixed-point analysis of localized Rademacher complexities.
result The proposed rule is minimax-optimal and performs comparably to cross-validation.
A new sequencing rule prevents miners from front-running transactions in decentralized exchanges.
problem Miners exploit their privileged position to front-run transactions, leading to unfair profits.
method Introduce verifiable sequencing rules that constrain transaction execution order and are verifiable.
result A verifiable sequencing rule ensures users receive at least fair execution prices, preventing front-running.
We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We prese…
Clarifies interest rate cap rules for loans with unconventional cash flows.
problem Ambiguity in applying interest rate caps to loans with non-conventional internal rate of return (IRR).
method Clarified conventional IRR definition, axiomatized, and extended to all loans.
result Unique extension of interest rate cap rule for all loans, based on net present value test.
KnotMosaics package simplifies knot theory computations in SageMath.
problem Efficiently computing knot mosaic diagrams and their properties.
method Developed a SageMath package for knot mosaic diagrams, implementing validation, strand tracing, and computation algorithms.
result Enabled easy computation of knot mosaic diagrams and their properties.
Developable ruled surfaces generated by curvature axes of curves.
problem Creating simple and understandable ruled surfaces for practical design.
method Investigating a straightforward method to generate developable ruled surfaces using curvature axes of curves.
result Developable ruled surfaces are generated by the curvature axes of curves, and these surfaces are developable.
Diffusion models learn hierarchical composition rules from data.
problem How many samples do generative models need to learn hierarchical composition rules?
method Theoretical and empirical investigation of diffusion models on probabilistic context-free grammars.
result Diffusion models learn hierarchical composition rules with sample complexity scaling polynomially with context size.