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Character expansion is introduced and explicitly constructed for the (non-colored) HOMFLY polynomials of the simplest knots. Expansion coefficients are not the knot invariants and can depend on the choice of the braid realization. However,…
We compute the closest convex piecewise linear-quadratic (PLQ) function with minimal number of pieces to a given univariate piecewise linear-quadratic function. The Euclidean norm is used to measure the distance between functions. First, we…
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…
This paper presents a comprehensive analysis of a broad range of variations of the stochastic proximal point method (SPPM). Proximal point methods have attracted considerable interest owing to their numerical stability and robustness…
In this paper, we are concerned with stationarity conditions and qualification conditions for optimization problems with disjunctive constraints. This class covers, among others, optimization problems with complementarity, vanishing, or…
Existing works on "black-box" model interpretation use local-linear approximations to explain the predictions made for each data instance in terms of the importance assigned to the different features for arriving at the prediction. These…
Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a…
In the present paper, we construct a simple invariant which provides a sliceness obstruction for {\em free knots}. This obstruction provides a new point of view to the problem of studying cobordisms of curves immersed in 2-surfaces, a…
Convex integration and the holonomic approximation theorem are two well-known pillars of flexibility in differential topology and geometry. They may each seem to have their own flavor and scope. The goal of this paper is to bring some new…
While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due to the curse of…
Regression splines are smooth, flexible, and parsimonious nonparametric function estimators. They are known to be sensitive to knot number and placement, but if assumptions such as monotonicity or convexity may be imposed on the regression…
In the era of big data, an ever-growing volume of information is recorded, either continuously over time or sporadically, at distinct time intervals. Functional Data Analysis (FDA) stands at the cutting edge of this data revolution,…
For a product of i.i.d. random maps or a memoryless stochastic flow on a compact space $X$, we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies…
Some properties of generalized convexity for sets and for functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is…
We consider the approximation of a continuous function, defined on a compact set of the $d$-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best…
We derive a variant of the nonsmooth maximum principle for problems with pure state constraints. The interest of our result resides on the nonsmoothness itself since, when applied to smooth problems, it coincides with known results.…
We discuss technical results on learning function approximations using piecewise-linear basis functions, and analyze their stability and convergence using nonlinear contraction theory.
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
In this paper, we present a general framework for constructively proving the existence and of stationary localized solutions, spatially periodic solutions, and branches of spatially periodic solutions in the 1D Thomas model. Specifically,…
We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise…