Neural networks struggle with periodic functions, a new activation fixes this.
problem Neural networks fail to learn simple periodic functions.
method Proposed a new activation function, x+sin2(x), to learn periodic functions. result The new activation function successfully learns and predicts periodic functions.
Binary encoding enables neural networks to extrapolate periodic functions.
problem Extrapolating periodic functions without prior knowledge of their form.
method Normalized Base-2 Encoding (NB2E) for continuous numerical values.
result MLPs using NB2E can successfully extrapolate diverse periodic signals.
Periodic activation functions improve neural network reliability and interpretability.
problem Neural networks reinforce hidden biases, making them unreliable and hard to interpret.
method Introduce periodic activation functions in Bayesian neural networks to establish a connection with stationary Gaussian process priors.
result Periodic activation functions, including sinusoidal, triangular, and ReLU, improve model performance and sensitivity to perturbations.
Model for multi-period carbon market pricing with allowances.
problem Carbon market pricing with multiple trading periods and compliance times.
method Singular forward-backward stochastic differential equations (SDEs).
result Value function convergence to infinite period model under certain conditions.
FNN approximates functions and solves PDEs with periodic BCs.
problem Approximating and solving periodic functions and PDEs.
method Fourier neural network architecture with activation and loss functions.
result FNN can solve PDEs with periodic BCs and is interpretable.
Study shows mean action of periodic orbits in annuli is bounded by their Calabi invariant.
problem Understanding the average distortion of periodic orbits in area-preserving annuli.
method Analyzes action functions and Calabi invariants of diffeomorphisms near annulus boundaries.
result Infimum of mean action of periodic orbits is bounded by their Calabi invariant.
Quantum neural networks approximate periodic functions more efficiently.
problem Approximating periodic functions with quantum neural networks.
method Using Jackson's inequality to construct a QNN that approximates a trigonometric polynomial of the function.
result Quantum neural networks can achieve better approximation results with fewer parameters for smoother functions.
In this paper we show how to bypass the usual difficulties in the analysis of elliptic integrals that arise when solving period problems for minimal surfaces. The method consists of replacing period problems with ordinary Sturm-Liouville problems involving the support function. We give a practical application by provin…
The study counts periodic orbits on smooth manifolds, adding ghost orbits for completeness.
problem Counting periodic orbits of vector fields on smooth closed manifolds.
method Enlarging the space of orbits to include ghost orbits, defining weight functions, and showing constancy under deformation.
result The weight function remains constant as the vector field moves and Γ deforms. In the present paper, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. All results are derived under weak conditions on the asset returns. No assumption on the correlation structure between different time points is needed a…
A novel approach to quantizing neural networks using periodic functions as regularizers.
problem Quantization of neural network parameters to reduce memory usage and computational cost.
method Using periodic functions (sine, cosine, hat) as regularizers during training to push weights into discrete points.
result Quantized models achieve the same accuracy as original models on CIFAR-10 and ImageNet datasets.
Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.
problem Constructing semi-discrete and discrete surfaces explicitly.
method Using Jacobi elliptic functions and τ-functions.
result Explicit constructions and periodicities of semi-discrete and discrete surfaces.
Trefoil knots can be inscribed using periodic functions.
problem Proving the existence of a trefoil knot inscribed using a specific function.
method Analytic function with periodicity and non-vanishing derivative.
result A sequence of points parameterized by a function forms a trefoil knot.
The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided th…
The paper classifies periodic solitons in curve flows on the light-cone.
problem Investigating periodic solitons in curve flows on the light-cone.
method Deriving Harnack inequality for heat flow, classifying space-periodic solitons for a third-order curvature flow.
result Closed soliton solutions form a family of transcendental curves with specific rotation indices.
The presence of log-periodic structures before and after stock market crashes is considered to be an imprint of an intrinsic discrete scale invariance (DSI) in this complex system. The fractal framework of the theory leaves open the possibility of observing self-similar log-periodic structures at different time scales.…
New connection between DNNs and Sharkovsky's Theorem for depth-width trade-offs.
problem Understanding why some functions are hard to represent by shallow ReLU networks.
method Connection to Sharkovsky's Theorem and analysis of dynamical systems.
result Lower bounds for width needed to represent periodic functions as a function of depth.
Study transcendence of abelian differential periods from bi-algebraic perspective.
problem Arithmetic and functional transcendence of periods of abelian differentials.
method Bi-algebraic structure on strata of abelian differentials.
result Characterization of arithmetic points and proof of linear bi-algebraic curves.
We show that for any n real periodic functions f_1,..., f_n with the same period, such that f_i>0 for i<n, and a real number e >0, there is a closed curve in R^{n+1} with curvatures k_1, ..., k_n such that |k_i(t)-f_i(t)| < e for all i and t. This neither holds for closed curves in the hyperbolic space H^{n+1}, nor for…
Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circ…
Combines kernels to create flexible priors in BNNs for seasonal and trend data.
problem Creating flexible priors in Bayesian neural networks for complex data.
method Derives BNN architectures from kernel combinations and periodic functions.
result BNNs can produce periodic kernels useful for capturing seasonal and trend data.
Study shows no periodic geodesics in jet space.
problem Existence of periodic geodesics in jet space.
method Characterization and classification of subRiemannian geodesics.
result No periodic geodesics found in the space of k-jets.
We introduce a new class of forward performance processes that are endogenous and predictable with regards to an underlying market information set and, furthermore, are updated at discrete times. We analyze in detail a binomial model whose parameters are random and updated dynamically as the market evolves. We show tha…
Paper proposes a robust framework for detecting multiple periodic components in time series.
problem Detecting multiple periodic components in time series with interlaced patterns and external noise.
method Applying maximal overlap discrete wavelet transform to isolate periodic components, ranking them by wavelet variance, and detecting single periodicity robustly.
result The proposed algorithm outperforms other methods for both single and multiple periodicity detection.
Study optimal periodic dividend strategies for risky businesses with transaction costs.
problem Optimal periodic dividend strategies for spectrally positive Lévy risk processes with fixed transaction costs.
method Investigates periodic (bu,bl) strategies for a Poisson arrival process of decision times. result A periodic (bu,bl) strategy is optimal with lump sum dividends net of transaction costs. The aim of this paper is to compare statistical properties of stock price indices in periods of booms with those in periods of stagnations. We use the daily data of the four stock price indices in the major stock markets in the world: (i) the Nikkei 225 index (Nikkei 225) from January 4, 1975 to August 18, 2004, of (ii…
Study optimizes inventory restocking for demand processes with exponential replenishment.
problem Optimizing inventory restocking for demand processes with exponential replenishment.
method Developed periodic barrier replenishment policies for spectrally positive Lévy demand processes.
result Optimal policies and value functions are concisely written in terms of scale functions.
This paper optimizes periodic dividend strategies for Lévy processes with transaction costs.
problem Maximizing dividends for spectrally negative Lévy processes with fixed transaction costs.
method Using periodic strategies and fixed transaction costs, the paper calculates the value function and shows optimality conditions.
result A sufficient condition for optimality is that the Lévy measure is completely monotonic.
Study optimal portfolio strategies with periodic evaluation under short-selling prohibition.
problem Optimal portfolio strategies with periodic evaluation under short-selling prohibition.
method Reformulate the original problem into an auxiliary one-period optimization problem and introduce dual control problem.
result Derive and verify the value function and optimal constrained portfolio for the original problem.
We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. A key role is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the utility function of a representative market maker.
This paper optimizes portfolio management in incomplete markets with stochastic factors, considering periodic wealth evaluations.
problem Optimizing portfolio performance in an incomplete market model with stochastic factors and periodic wealth evaluations.
method Developed a martingale duality approach to find optimal portfolio processes and dual minimizers.
result Established the existence of optimal portfolio processes and identified dual minimizers as the 'least favorable' market completion.
Proves a theorem for mechanical systems with reflections.
problem Action functionals on paths with reflections.
method Proves a Morse index theorem for action functionals on paths that can reflect.
result Action functionals on paths with reflections have a well-defined Morse index.
We prove the Livšic Theorem for arbitrary GL(m,R) cocycles. We consider a hyperbolic dynamical system f:X→X and a Hölder continuous function A:X→GL(m,R). We show that if A has trivial periodic data, i.e. A(fn−1p)...A(fp)A(p)=Id for each periodic point p=fnp, then there …
DEPTS learns to forecast periodic time series with improved accuracy.
problem Forecasting periodic time series is challenging due to complex dependencies and diverse periods.
method DEPTS uses a decoupled formulation with an expansion module and a periodicity module to handle these challenges.
result DEPTS significantly improves forecasting accuracy, reducing errors by up to 20%.
New method for approximating periodic kernels on high-dimensional data.
problem Inefficient modelling of periodicity in higher-dimensional problems.
method Index Set Fourier Series Features
result Significantly less predictive error compared to alternative methods.
Analyzes compound interest with constant payments and interest rate.
problem Examines the properties of compound interest balance and payment functions.
method Analyzes the outstanding balance and payment functions for constant payments and interest rate.
result The outstanding balance function is not generally concave in the interest rate.
BEKAN uses RBFs and evolutionary methods to solve PDEs with boundary conditions.
problem Enforcing boundary conditions in neural networks for PDE solutions.
method Boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (BEKAN) with radial basis functions (RBFs). Incorporates Dirichlet, periodic, and Neumann conditions.
result BEKAN outperforms MLP and B-splines KAN in solving PDEs with boundary conditions.
We will show that the period T of a closed orbit of the planar circular restricted three-body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, 2T=kπ+∫Ωg where k is an integer, Ω is the region enclosed by the periodic orbit and $g:\mathbb{R}^2\to \m…
Study optimizes dividend strategies for risk processes with Lévy jumps.
problem Optimizing dividend payments in risk processes with Lévy jumps.
method Analyzes spectrally positive and negative Lévy processes, using scale functions.
result Periodic barrier strategy is optimal for spectrally negative Lévy processes with completely monotone Lévy density.
We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …
We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in R3. These new examples form branches issuing from the H-family, the rPD-family, the tP-family, and the tD-family, that converge to some degenerate embedding of the families. As to nonde…
In this paper we derive the exact solution of the multi-period portfolio choice problem for an exponential utility function under return predictability. It is assumed that the asset returns depend on predictable variables and that the joint random process of the asset returns and the predictable variables follow a vect…
Avanzi et al. (2016) recently studied an optimal dividend problem where dividends are paid both periodically and continuously with different transaction costs. In the Brownian model with Poissonian periodic dividend payment opportunities, they showed that the optimal strategy is either of the pure-continuous, pure-peri…
For a real valued periodic smooth function u on R, n≥0, one defines the osculating polynomial φs (of order 2n+1) at a point s∈R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex …
Mixed tête-à-tête twists model monodromies for isolated surface singularities.
problem Modeling monodromies for isolated complex surface singularities.
method Characterizing pseudo-periodic automorphisms as mixed tête-à-tête twists.
result Tête-à-tête twists coincide with monodromies for isolated surface singularities.
Paper defines untangling number to measure entanglement complexity in 3-periodic networks.
problem Measuring the complexity of entanglement in 3-periodic networks.
method Defining ground states through knot-theoretic crossing diagrams and measuring untangling number.
result Introduced untangling number as a measure of entanglement complexity.
The paper explores the geometry of level lines of quasiperiodic functions with many periods.
problem Describing the geometry of level lines of quasi-periodic functions with a large number of periods.
method Generalizes the Novikov problem to the multidimensional case of quasiperiodic functions.
result Arises of open or closed level lines of arbitrarily large sizes.
Reduces learning periodic neural networks to lattice problems, proving hardness under cryptographic assumptions.
problem Learning single periodic neurons in noisy environments.
method Reduction to worst-case lattice problems, using LLL algorithm.
result Polynomial-time algorithms for learning these functions are hard under cryptographic assumptions.