Parameterized Intractability of Even Set and Shortest Vector Problem

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W [1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP) , in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W [1]-hard to approximate (under randomized reductions) to some constant factor.

A note on discrete wave packet frames in ℂN

We show that for every nonzero vector [Formula: see text] in [Formula: see text], the discrete wave packet system [Formula: see text] constitutes a frame for the unitary space [Formula: see text]. An application of this result is given, where frame conditions cannot be derived from discrete wavelet systems in [Formula: see text]. The canonical dual of discrete wave packet frame is also discussed.

Reiterative Distributional Chaos on Banach Spaces

If we change the upper and lower densities in the definition of distributional chaos of a continuous linear operator on a Banach space [Formula: see text] by the Banach upper and Banach lower densities, respectively, we obtain Li–Yorke chaos. Motivated by this, we introduce the notions of reiterative distributional chaos of types [Formula: see text], [Formula: see text] and [Formula: see text] for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and present the conditions for the existence of a vector subspace [Formula: see text] of [Formula: see text], such that every nonzero vector in [Formula: see text] is both irregular for [Formula: see text] and distributionally near zero for [Formula: see text].

Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector

In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.

The almost semimonotone matrices

A (strictly) semimonotone matrix A ∈ ℝn×n is such that for every nonzero vector x ∈ ℝn with nonnegative entries, there is an index k such that xk > 0 and (Ax)k is nonnegative (positive). A matrix which is (strictly) semimonotone has the property that every principal submatrix is also (strictly) semimonotone. Thus, it becomes natural to examine the almost (strictly) semimonotone matrices which are those matrices which are not (strictly) semimonotone but whose proper principal submatrices are (strictly) semimonotone. We characterize the 2 × 2 and 3 × 3 almost (strictly) semimonotone matrices and describe many of their properties. Then we explore general almost (strictly) semimonotone matrices, including the problem of detection and construction. Finally, we relate (strict) central matrices to semimonotone matrices.

Space-efficient classical and quantum algorithms for the shortest vector problem

A lattice is the integer span of some linearly independent vectors. Lattice problems have many significant applications in coding theory and cryptographic systems for their conjectured hardness. The Shortest Vector Problem (SVP), which asks to find a shortest nonzero vector in a lattice, is one of the well-known problems that are believed to be hard to solve, even with a quantum computer. In this paper we propose space-efficient classical and quantum algorithms for solving SVP. Currently the best time-efficient algorithm for solving SVP takes 2^{n+o(n)} time and 2^{n+o(n)} space. Our classical algorithm takes 2^{2.05n+o(n)} time to solve SVP and it requires only 2^{0.5n+o(n)} space. We then adapt our classical algorithm to a quantum version, which can solve SVP in time 2^{1.2553n+o(n)} with 2^{0.5n+o(n)} classical space and only poly(n) qubits.