Martins Kokainis

Approximation properties of higher degree F-transforms based on B-splines

The paper deals with the F-transform with polynomial components with respect to a generalized fuzzy partition given by B-splines. We investigate approximation properties of the inverse F-transform in this case and prove that using B-splines allows us to improve the quality of approximation of smooth functions.

Continuous and discrete higher-degree F-transforms based on B-splines

The paper deals with the continuous and discrete higher-degree fuzzy transforms (F-transforms with polynomial components) with respect to a generalized fuzzy partition given by B-splines. We investigate properties of the direct and inverse F-transforms in these cases and prove that using B-splines allows us to improve the quality of approximation of smooth functions.

Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games

We study quantum algorithms on search trees of unknown structure, in a model where the tree can be discovered by local exploration. That is, we are given the root of the tree and access to a black box which, given a vertex v, outputs the children of v. We construct a quantum algorithm which, given such access to a search tree of depth at most n, estimates the size of the tree T within a factor of 1± δ in Õ(√nT) steps. More generally, the same algorithm can be used to estimate size of directed acyclic graphs (DAGs) in a similar model. We then show two applications of this result: a) We show how to transform a classical backtracking search algorithm which examines T nodes of a search tree into an Õ(√Tn3/2) time quantum algorithm, improving over an earlier quantum backtracking algorithm of Montanaro (arXiv:1509.02374). b)We give a quantum algorithm for evaluating AND-OR formulas in a model where the formula can be discovered by local exploration (modeling position trees in 2-player games) which evaluates formulas of size T and depth To(1) in time O(T1/2+o(1)). Thus, the quantum speedup is essentially the same as in the case when the formula is known in advance.

Higher Degree F-transforms Based on B-splines of Two Variables

The paper deals with the higher degree fuzzy transforms (F-transforms with polynomial components) for functions of two variables in the case when two-dimensional generalized fuzzy partition is given by B-splines of two variables. We investigate properties of the direct and inverse F-transform in this case and prove that using B-splines as basic functions of fuzzy partition allows us to improve the quality of approximation.

Approximation by multivariate higher degree F-transform based on B-splines

The paper deals with the integral and discrete versions of the direct and inverse higher degree fuzzy transforms (F-transforms) of multivariate functions. The aim is to generalize to the multidimensional case the results known for the univariate F-transforms with respect to a generalized fuzzy partition given by B-splines. We prove that using multivariate B-splines as the generating functions of multidimensional fuzzy partition allows to improve the quality of approximation of multivariate functions and their derivatives.

All Classical Adversary Methods are Equivalent for Total Functions

We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity

Collocation Method for Linear BVPs via B-spline Based Fuzzy Transform

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Modified F-transform Based on B-splines

. That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. For partial functions, we show unbounded separations between

Nearly optimal separations between communication (or query) complexity and partitions

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Polynomials, quantum query complexity, and Grothendieck's inequality

and other adversary bounds, as well as between the relational and Kolmogorov complexity bounds. We also show that, for partial functions, fractional block sensitivity cannot give lower bounds larger than