The problem of local feedback equivalence for 1-dimensional control systems of the 1-st order is considered. The algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.
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In this paper we work toward the Homflypt skein module of the lens spaces , , using braids. In particular, we establish the connection between , the Homflypt skein module of the solid torus ST, and and arrive at an infinite system, whose solution…
In this paper we present recent results toward the computation of the HOMFLYPT skein module of the lens spaces , , via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, , for knots and links in ST, the universal analogue of th…
Researchers develop a new basis to simplify solving infinite systems for HOMFLYPT skein module of lens spaces.
problem Computing the HOMFLYPT skein module of lens spaces via braids.
method Using a new basis of the HOMFLYPT skein module of the solid torus, relating infinite systems of equations obtained by performing braid band moves on elements in and via a map .
result Reduced complexity in solving the infinite system of equations for using braids.
Let denote the -punctured disk in the complex plane, where the punctures are on the real axis. An -braid is said to be \emph{reducible} if there exists an essential curve system $\C$ in , called a \emph{reduction system} of , such that $α*\C=\C$ where $α*\C$ denotes the action of the braid o…
In building intelligent transportation systems such as taxi or rideshare services, accurate prediction of travel time and distance is crucial for customer experience and resource management. Using the NYC taxi dataset, which contains taxi trips data collected from GPS-enabled taxis [23], this paper investigates the use…
An important step for the computation of the HOMFLYPT skein module of the lens spaces via braidsmath.GT
We prove that, in order to derive the HOMFLYPT skein module of the lens spaces from the HOMFLYPT skein module of the solid torus, , it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant for knots and links in the solid torus, braid ba…
Self-training improves generalization by fitting to reliable pseudo-labels or gradually improving the classification plane.
problem Understanding how self-training improves generalization in high-dimensional Gaussian mixtures.
method Analyzing iterative self-training on binary Gaussian mixtures in the asymptotic limit.
result ST improves generalization by fitting to reliable pseudo-labels or gradually improving the classification plane.
Spatio-temporal (ST) data for urban applications, such as taxi demand, traffic flow, regional rainfall is inherently stochastic and unpredictable. Recently, deep learning based ST prediction models are proposed to learn the ST characteristics of data. However, it is still very challenging (1) to adequately learn the co…
In this paper, we introduce an equivalence relation on the set of local moves and classify local moves, called the extended -moves, up to the equivalence. Moreover, by inducing a binary relation on the set of equivalence classes of local moves, we show that an extended -move realizes the crossing change or the …
We show that two Alexander biquandles M and M' are isomorphic iff there is an isomorphism of Z[s,1/s,t,1/t]-modules h:(1-st)M --> (1-st)M' and a bijection g:O_s(A) --> O_s(A') between the s-orbits of sets of coset representatives of M/(1-st)M and M'/(1-st)M' respectively satisfying certain compatibility conditions.
We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `compl…
In this paper, we develop a reinforcement learning (RL) based system to learn an effective policy for carpooling that maximizes transportation efficiency so that fewer cars are required to fulfill the given amount of trip demand. For this purpose, first, we develop a deep neural network model, called ST-NN (Spatio-Temp…
New model infers causal relationships from spatio-temporal data, even with unobserved confounders.
problem Challenges in inferring causal relationships from spatio-temporal data due to unobserved confounders.
method Spatio-Temporal Hierarchical Causal Models (ST-HCMs) that extend hierarchical causal modeling to the spatio-temporal domain, using the Spatio-Temporal Collapse Theorem.
result Validated the effectiveness of ST-HCMs on both synthetic and real-world datasets, demonstrating robust causal inference in complex dynamic systems.
New method for computing Kauffman bracket skein module of lens spaces using unoriented braids.
problem Computing Kauffman bracket skein module of lens spaces for .
method Developed a braid theoretic approach via unoriented braids, introducing a new algebra and invariant.
result Computed the Kauffman bracket skein module of lens spaces and extended to .
Let be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure determined by a pair of opposite Borel subgroups . We prove that for each in the Weyl group of , the double Bruhat cell in , together with the …
Spatio-temporal (ST) data, which represent multiple time series data corresponding to different spatial locations, are ubiquitous in real-world dynamic systems, such as air quality readings. Forecasting over ST data is of great importance but challenging as it is affected by many complex factors, including spatial char…
Arnold introduced invariants , and for generic planar curves. It is known that both and are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves for uknotting.…
This paper proposes a geometry-aware active learning framework for spatiotemporal dynamic systems.
problem Challenges in modeling complex dynamic systems with 3D geometries and time evolution.
method Geometry-aware spatiotemporal Gaussian Process (G-ST-GP) and adaptive active learning strategy.
result The proposed framework outperforms traditional methods in predicting high-dimensional dynamic behaviors.
ST-MAML tackles task ambiguity in meta-learning by encoding tasks with stochastic representations.
problem Handling tasks from multiple distributions is challenging for meta-learning due to task ambiguity.
method ST-MAML uses a stochastic neural network module to encode tasks and propagate task representations to revise input variable encoding.
result ST-MAML matches or outperforms state-of-the-art methods on various tasks.
The Straight-Through (ST) estimator is a widely used technique for back-propagating gradients through discrete random variables. However, this effective method lacks theoretical justification. In this paper, we show that ST can be interpreted as the simulation of the projected Wasserstein gradient flow (pWGF). Based on…
This paper presents a decomposition of the 3-strand Singular Braid Group.
problem Decomposing the 3-strand Singular Braid Group.
method Investigation of the group through presentations and isomorphisms.
result Proves is isomorphic to the singular pure braid group and is a semi-direct product.
QC-ST and CoCo methods correct batch effects in metabolomics data.
problem Batch effects in metabolomics data obscure biological variations.
method QC-ST for simultaneous detection of QC samples' mean vectors and covariance matrices, CoCo for covariance correction.
result QC-ST and CoCo improve batch effect correction in metabolomics datasets.
In this paper we give an alternative basis, , for the Kauffman bracket skein module of the solid torus, . The basis is obtained with the use of the Tempereley--Lieb algebra of type B and it is appropriate for computing the Kauffman bracket sk…
Surface mount technology (SMT) is a process for producing printed circuit boards. Solder paste printer (SPP), package mounter, and solder reflow oven are used for SMT. The board on which the solder paste is deposited from the SPP is monitored by solder paste inspector (SPI). If SPP malfunctions due to the printer defec…
Menger 1934 revisitedq-fin.RM
Karl Menger's 1934 paper on the St. Petersburg paradox contains mathematical errors that invalidate his conclusion that unbounded utility functions, specifically Bernoulli's logarithmic utility, fail to resolve modified versions of the St. Petersburg paradox.
We introduce and study a notion of `Sasaki with torsion structure' (ST) as an odd-dimensional analogue of Kähler with torsion geometry (KT). These are normal almost contact metric manifolds that admit a unique compatible connection with 3-form torsion. Any odd-dimensional compact Lie group is shown to admit such a stru…
New pricing theory solves St. Petersburg paradox.
problem St. Petersburg Paradox unresolved for 280 years.
method Proposes new pricing theory with fair pricing rules.
result New pricing theory resolves the paradox.
STS clarifies chaos and stochastic dynamics, linking algebraic topology and physics.
problem Chaos and stochastic dynamics in arbitrary form SDEs.
method Supersymmetric theory of stochastic dynamics (STS) using generalized transfer operator (GTO) and topological field theories (TFT).
result Positive 'pressure' in GTOs corresponds to spontaneous breakdown of topological supersymmetry, explaining 1/f noise.
Explicitly found generators of cohomology for SL_n(Z) using sharbly cycles and cosharbly cocycles.
problem Finding explicit generators for the cohomology of SL_n(Z).
method Using sharbly cycles and cosharbly cocycles, and applying Borel-Serre duality.
result Explicitly found generators of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles.
Researchers compute the skein module of a solid torus using braids.
problem Computing the HOMFLYPT skein module of .
method Braid-theoretic techniques to solve an infinite system of equations.
result The free part of the skein module is generated by the empty link, with all other elements being torsion.
New framework for 3D spatial topology enumeration and identification.
problem Efficient navigation through complex engineering system topologies.
method Mathematical spatial graph theory to represent, enumerate, and identify unique topological classes.
result Identification of distinctive 3D topological classes for engineering systems.
Proposes FairDRL-ST for fair spatio-temporal mobility prediction.
problem Fairness concerns in spatio-temporal AI applications.
method Disentangled representation learning, adversarial learning.
result Achieves fairness in spatio-temporal mobility prediction without performance loss.
Proposes a method to forecast spatial-temporal data with limited training data.
problem Forecasting with nodes having no temporal training data.
method Temporal data augmentation and spatial graph topology learning.
result Improves forecasting performance on nodes without training data.
The paper studies pseudo and singular links in a solid torus, developing invariants and algebraic structures.
problem Understanding and classifying links with missing crossing information in a solid torus.
method Introducing pseudo and singular links, constructing invariants, and developing algebraic structures.
result Formulated and proved the Alexander and Markov theorems for pseudo and singular links in a solid torus.
We prove a Chekanov-type theorem for the spherization of the cotangent bundle of a closed manifold . It claims that for Legendrian submanifolds in the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies.
The structure set $\ST^{TOP}(M)$ of an -dimensional topological manifold for has a homotopy invariant functorial abelian group structure, by the algebraic version of the Browder-Novikov-Sullivan-Wall surgery theory. An element $(N,f) \in \ST^{TOP}(M)$ is an equivalence class of -dimensional ma…
StreamEnsemble dynamically selects ML models for ST data streams to improve predictive accuracy.
problem Predictive queries over spatiotemporal data streams are challenging due to varying distributions and patterns.
method Dynamic selection and allocation of ML models based on time series distributions and characteristics.
result Significantly outperforms traditional ensemble and single model approaches, reducing prediction error by over 10 times.
Contact round surgeries on help in constructing and understanding contact 3-manifolds.
problem Constructing contact 3-manifolds using Legendrian surgeries.
method Introducing contact round surgeries of indices 1 and 2, and associating them with surgery diagrams.
result Every closed connected contact 3-manifold can be obtained by a sequence of contact round surgeries on Legendrian knots in .
ST-MTM: Masked Time Series Modeling with Seasonal-Trend Decomposition for Time Series Forecastingcs.LG
ST-MTM models complex time series by decomposing and masking seasonal and trend components.
problem Forecasting complex time series with intricate temporal variations.
method Seasonal-Trend Decomposition with Masking and Contrastive Learning.
result ST-MTM achieves superior forecasting performance compared to existing methods.
Improved Thompson Sampling outperforms existing Bayesian optimization methods.
problem Thompson Sampling's performance in Bayesian optimization is suboptimal compared to other methods.
method Developed Stagger Thompson Sampler (STS), which more precisely samples the optimal arm with less computation.
result STS outperforms TS, PSS, and other acquisition methods in various optimization tasks.
Paper proposes a new method to assess default risk using CEV process in KMV model.
problem Assessing default risk with constant volatility assumption.
method Apply CEV process to KMV model to estimate firm asset value dynamics and equivalent volatility.
result CEV-KMV model fits market better for credit risk forecasting.
Reintroducing Straight-Through Estimators as Principled Methods for Stochastic Binary Networksstat.ML
Reintroduces straight-through estimators for binary neural networks.
problem Training neural networks with binary weights and activations is challenging due to gradient issues and discrete weight optimization.
method Derives ST methods as estimators in the SBN model, analyzes properties and estimation accuracy, explains latent weights and mirror descent method.
result Reintroduces ST methods as sound approximations and provides clearer application and improvements.
For an -dimensional space-time define a mapped null hypersurface to be a smooth map (that is not necessarily an immersion) such that there exists a smooth field of null lines along that are both tangent and -orthogonal to We study relations between mapped null hyp…
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
We present a generic framework for spatio-temporal (ST) data modeling, analysis, and forecasting, with a special focus on data that is sparse in both space and time. Our multi-scaled framework is a seamless coupling of two major components: a self-exciting point process that models the macroscale statistical behaviors …
In this paper, by modifying the arguments in \cite{WY}, we get some rigidity theorems on compact manifolds with nonempty boundary. The results in this paper are similar with those in \cite{ST} and \cite{WY}. Like \cite{ST} and \cite{WY}, we still use quasi-spherical metrics introduced by \cite{Ba} to get monotonicity o…
Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.
problem Study of Arnold-type invariants of immersed curves and surfaces.
method Framework on dual complexes, locally normalized maps, finite-difference structures, and Shumakovitch-type identities.
result Unified evaluation of Arnold-type invariants and on dual skeleta.