Tuples Of Vectors Research Articles

A higher moment formula for the Siegel–Veech transform over quotients by Hecke triangle groups

We compute higher moments of the Siegel–Veech transform over quotients of SL (2,\mathbb{R}) by the Hecke triangle groups. After fixing a normalization of the Haar measure on SL (2,\mathbb{R}) we use geometric results and linear algebra to create explicit integration formulas which give information about densities of k -tuples of vectors in discrete subsets of \mathbb{R}^2 which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over SL (2,\mathbb{R})/ SL (2,\mathbb{Z}) .

The Weisfeiler–Leman algorithm and the diameter of Schreier graphs

We prove that the number of iterations taken by the Weisfeiler–Leman algorithm for configurations coming from Schreier graphs is closely linked to the diameter of the graphs themselves: an upper bound is found for general Schreier graphs, and a lower bound holds for particular cases, such as for Schreier graphs with G = SL_n (\mathbb F_q) (q > 2) acting on k -tuples of vectors in \mathbb F^n_q ; moreover, an exact expression is found in the case of Cayley graphs.

Generalizations of Buzano inequality for $n$-tuples of vectors in inner product spaces with applications

In this paper some generalizations of Buzano inequality for $n$-tuples of vectors in inner product spaces are given. Applications for norm and numerical radius inequalities for $n$-tuples of bounded linear operators and for functions of normal operators defined by power series with nonnegative coefficients are also provided.

Tuple lattice sieving

Lattice sieving is asymptotically the fastest approach for solving the shortest vector problem (SVP) on Euclidean lattices. All known sieving algorithms for solving the SVP require space which (heuristically) grows as$2^{0.2075n+o(n)}$, where$n$is the lattice dimension. In high dimensions, the memory requirement becomes a limiting factor for running these algorithms, making them uncompetitive with enumeration algorithms, despite their superior asymptotic time complexity.We generalize sieving algorithms to solve SVP with less memory. We consider reductions of tuples of vectors rather than pairs of vectors as existing sieve algorithms do. For triples, we estimate that the space requirement scales as$2^{0.1887n+o(n)}$. The naive algorithm for this triple sieve runs in time$2^{0.5661n+o(n)}$. With appropriate filtering of pairs, we reduce the time complexity to$2^{0.4812n+o(n)}$while keeping the same space complexity. We further analyze the effects of using larger tuples for reduction, and conjecture how this provides a continuous trade-off between the memory-intensive sieving and the asymptotically slower enumeration.

Degrees of convex dependence in recursively enumerable vector spaces

Let W be a recursively enumerable vector space over a recursive ordered field. We show the Turing equivalence of the following sets: the set of all tuples of vectors in W which are linearly dependent; the set of all tuples of vectors in W whose convex closures contain the zero vector; and the set of all pairs ( X, Y) of tuples in W such that the convex closure of X intersects the convex closure of Y. We also form the analogous sets consisting of tuples with given numbers of elements, and prove similar results on the Turing equivalence of these.