We derive formulas for HOMFLY polynomials of torus links using braid groups and linear recurrences.
problem Calculating HOMFLY polynomials for torus links.
method Using braid groups and linear recurrences, derived from the skein relation.
result Explicit formulas for HOMFLY polynomials of torus links T(3,n) and T(−3,n) are derived. Study provides explicit formula for complex 2D Kähler manifold quantization.
problem Quantization of complex 2D locally symmetric Kähler manifolds.
method Deformation quantization with separation of variables, solving recurrence relations.
result Explicit formula for star product on complex 2D locally symmetric Kähler manifolds.
SSA with a general recursive formula improves forecasting accuracy with structural breaks.
problem Forecasting time series with structural breaks.
method Singular Spectrum Analysis (SSA) with a general non-linear recurrence formula.
result SSA with the general recursive formula outperforms basic SSA and SSA bootstrap forecasting.
The simplest non-trivial solutions of WDVV equations are A_n and B_n-potentials, which describe metrics of K.Saito on spaces of versal deformation of A_n and B_n-singularities. These are some polynomials, which were known for n⩽ 4. We find some recurrence relations, which give a possibility to find all A_n an…
New invariant found for parabolic surfaces.
problem Identifying differential invariants of parabolic surfaces.
method Using Fels-Olver's recurrence formulas on parabolic jet bundles.
result Only one invariant, M, of order 5, is needed to generate all invariants. Researchers create a star product on a Grassmannian with separation of variables.
problem Constructing a star product with separation of variables on G2,4(C). method Solving recurrence relations using creation and annihilation operators on a Fock space.
result Explicit formula for a star product with separation of variables on G2,4(C). Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative re…
Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conw…
Paper develops formulas for shape derivatives in wave scattering.
problem Computing high order shape derivatives for wave scattering is challenging.
method Introduces elegant recurrence formulas using differential forms and Lie derivatives.
result Unified framework for computing high order shape perturbations in scattering problems.
New recurrence formula for stable twisted Betti numbers of configuration spaces and spaces of maximal tori.
problem Computing stable twisted Betti numbers of configuration spaces and spaces of maximal tori.
method Computing twisted point-counts on algebraic varieties and using Grothendieck-Lefschetz fixed point formula.
result Stable twisted Betti numbers are linearly recurrent in i.
New proofs for growth series of Coxeter groups using complex structures.
problem Proving new formulae for growth series of Coxeter groups.
method Using the structure of Coxeter complexes, Davis complexes, or Tits non-complexes.
result Several classical formulae for growth series are proved in a new way.
Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.
problem Finding optimal radii for packing circles in various plane regions.
method Deterministic analytic formulae and recurrence relations.
result Formulated analytic formulae for 2D circle packing on various plane shapes.
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.
We show that for a twist knot, the A-polynomial can be obtained from recurrences for the summand in Masbaum's formula of the colored Jones polynomial. Our result supports the AJ conjecture due to S.Garoufalidis.
The paper studies exponential functionals of processes with independent increments and their moments.
problem Analyzing the moments of exponential functionals of processes with independent increments.
method Deriving recurrent integral equations for Mellin transforms and applying them to calculate moments.
result Explicit formulas for the moments of It and I∞, and precise number of finite moments of I∞. We analyze generalization in deep learning models using random matrix theory.
problem Understanding the generalization error in deep learning models with random feature representations.
method Applying Random Matrix Theory to derive asymptotic generalization error formulas for various architectures.
result Linear ESNs are equivalent to ridge regression with exponentially time-weighted input covariance, revealing an inductive bias towards recent inputs.
Being able to forcast extreme volatility is a central issue in financial risk management. We present a large volatility predicting method based on the distribution of recurrence intervals between volatilities exceeding a certain threshold Q for a fixed expected recurrence time τQ. We find that the recurrence inter…
Develops a new method for solving equivalence problems in pseudo-groups.
problem Solving equivalence problems in pseudo-groups.
method Combining Cartan's equivalence method and equivariant moving frame for pseudo-groups.
result A hybrid equivalence method that extends and illuminates its two progenitors.
Solves PCR with fewer calls to ridge regression.
problem Principal component regression (PCR) with high accuracy.
method Reduces PCR to ridge regression calls and develops stable recurrence for matrix Chebyshev polynomials.
result Achieves PCR with multiplicative accuracy up to 1+γ using fewer calls. We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the as…
The study connects free subgroups in a specific group to hypermaps, deriving generating series and asymptotic formulas.
problem Counting and understanding free subgroups of a specific group and their conjugacy classes.
method Using a connection to hypermaps, deriving generating series and recurrence relations, and providing asymptotic formulas.
result The generating series for the numbers of free subgroups of finite index in the group is transcendental and provides non-linear recurrence relations.
It is well known that the compactifications of the canonical contact systems living on real jet spaces Jr(1,1), r≥2, are locally universal Goursat distributions, Δr, living on compact manifolds (called Goursat monsters) having open dense jet-like (Jr(1,1)-like) parts. By virtue of the results of …
Recursive KalmanNet combines neural networks with Kalman filters for precise state estimation.
problem State estimation in systems with noisy measurements and non-Gaussian noise.
method Recursive KalmanNet uses a recurrent neural network to estimate states with consistent error covariance, optimizing for Gaussian negative log-likelihood.
result Recursive KalmanNet outperforms conventional Kalman filters and deep learning-based estimators in non-Gaussian noise conditions.
Deep learning improves causal effect estimation from complex observational data.
problem Estimating causal effects from complex observational data with low bias.
method Unified deep learning framework using multitask recurrent neural networks.
result Deep learning estimator shows lower bias in causal effect estimates.
Universal Transformer improves sequence modeling accuracy.
problem Feed-forward sequence models fail on simple tasks.
method Parallel-in-time self-attentive recurrent model combining feed-forward and RNN properties.
result UTs outperform standard Transformers on various tasks.
Closed formulas for η-corrections in the once-punctured torus identified.
problem Identifying η-corrections in the Kauffman bracket skein algebra of the once-punctured torus.
method Explicit closed formulas for Chebyshev-threaded families and η-corrections.
result Explicit Chebyshev expansions and coefficients for η-corrections.
Explains RNN and LSTM fundamentals, derives formulas, and addresses training issues.
problem Lack of detailed formulas and unrolling techniques in LSTM and RNN literature.
method Derives canonical RNN and LSTM formulas from differential equations, proposes unrolling technique, addresses training difficulties.
result Provides a comprehensive understanding of RNN and LSTM, including detailed formulas and unrolling techniques.
LARNN improves LSTM's performance with attention and multi-head mechanism.
problem Improving LSTM's performance in sequence prediction tasks.
method Derived from LSTM and RNN, LARNN uses attention and multi-head mechanism to query past states.
result LARNN achieves 91.92% test accuracy, significantly better than vanilla LSTM.
Study eta invariant remainder on contact manifolds, improving previous results.
problem Eta invariant remainder in metric contact manifolds.
method Analyzes remainder term in semiclassical limit, using volumes of recurrence sets of Reeb flow.
result Improves remainder term for Anosov Reeb flows and certain elliptic flows.
Study on different types of recurrence in Finsler geometry.
problem Understanding various recurrence patterns in Finsler geometry.
method Adopted pullback approach to investigate three classes of recurrence: simple, Ricci, and concircular.
result Introduce and investigate four types of each class of recurrence, highlighting interrelationships and a new concept of generalized concircular recurrence.
Investigates new types of recurrence in Finsler geometry.
problem None explicitly stated in the abstract.
method Study of hyper-generalized recurrence and generalized conharmonic recurrence in Finsler geometry.
result Properties of hyper-generalized recurrence and generalized conharmonic recurrence are studied and their relations to other Finsler recurrences are explored.
Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.
problem Characterizing biharmonic hypersurfaces with recurrent operators.
method Analysis of various recurrent operators and their impact on biharmonic hypersurfaces.
result Some well-known recurrent operators play a significant role in making biharmonic hypersurfaces minimal.
Study on affine surfaces with specific algebraic properties.
problem Characterize homogeneous affine surfaces with Hessian rank 2.
method Investigate algebra of differential invariants under affine transformation group.
result Organize homogeneous models into inequivalent branches.
To generalize the notion of recurrent manifold, there are various recurrent like conditions in the literature. In this paper we present a recurrent like structure, namely, \textit{super generalized recurrent manifold}, which generalizes both the hyper generalized recurrent manifold and weakly generalized recurrent mani…
Simple conformally recurrent spaces are identified as pp-waves.
problem Characterizing conformally recurrent space-times.
method Analyzing dimension n>3 space-times.
result Simple conformally recurrent space-times are conformally recurrent pp-waves.
Two special Finsler spaces have been introduced and investigated, namely Rh-recurrent Finsler space and consircularly recurrent Finsler space. The defining properties of these spaces are formulated in terms of the first curvature tensor of Cartan connection. The following three results constitute the main object of …
The object of the present paper is to obtain the characterization of a warped product semi-Riemannian manifold with a special type of recurrent like structure, called super generalized recurrent. As consequence of this result we also find out the necessary and sufficient conditions for a warped product manifold to sati…
We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two o…
The present paper deals with the proper existence of a generalized class of recurrent manifolds, namely, hyper-generalized recurrent manifolds. We have established the proper existence of various generalized notions of recurrent manifolds. For this purpose we have presented a metric and computed its curvature propertie…
Study of infinite hyperbolic surfaces and their automorphisms.
problem Behavior of hyperbolic affine automorphisms on infinite translation surfaces.
method Analysis of dynamics, mixing properties, and polynomial decay.
result Hyperbolic affine automorphisms are not recurrent but have mixing properties.
We construct the generalized version of covariant Z_3-graded differential calculus introduced by one of us (R.K.), and then extended to the case of arbitrary Z_N grading. Here our main purpose is to establish the recurrence formulae for the N-th power of covariant q-differential D_q = d_q + A and to analyze more closel…
Stable recurrent models perform similarly to unstable ones, proving useful for sequence tasks.
problem The lack of stability in recurrent neural networks hinders their practical application.
method Theoretical analysis and empirical testing of stable recurrent models.
result Stable recurrent models can be well approximated by feed-forward networks, performing similarly to unstable ones.
Interneurons improve learning in neural networks by accelerating convergence.
problem Rapid adaptation to changing input statistics in neural networks.
method Two mathematically tractable recurrent linear neural networks were compared: one with direct recurrent connections and the other with interneurons that mediate recurrent communication.
result The network with interneurons converges more quickly than the network with direct recurrent connections, scaling logarithmically with initialization spectrum.
Unified recurrent networks reveal differences in complexity levels of grammars.
problem Understanding the complexity and behavior of recurrent networks.
method Connecting recurrent networks with deterministic finite automata and formal grammars.
result Unified recurrent networks improve performance and match grammars from different complexity levels.
Study predicts colorectal polyp recurrence using medical records and statistical models.
problem Identifying patient characteristics influencing colorectal polyp recurrence.
method Natural language processing for extracting polyp characteristics, Kaplan-Meier curves, Cox proportional hazards modeling, random survival forest models.
result Polyp size, number, location, and patient smoking status significantly influence recurrence risk.
Paper proves unique Finsler connections for scalar forms.
problem Existence and uniqueness of Finsler connections.
method Pullback approach to global Finsler geometry, study of horizontally recurrent connections.
result Existence and uniqueness of horizontally recurrent Finsler connections for scalar forms.
Enhanced speech emotion recognition using nonlinear recurrence dynamics.
problem Improving speech emotion recognition accuracy.
method Phase space reconstruction, Recurrence Plot, Recurrence Quantification Analysis, statistical functionals, feature fusion, Bidirectional Recurrent Neural Network.
result State-of-the-art performance on IEMOCAP with up to 10.7% improvement in accuracy.
It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold.