We strengthen the nondeterministic hierarchy theorem for non-deterministic polynomial time to show that the lower bound holds against sub-linear advice. More formally, we show that for any constants d and d′ such that 1 6 d < d′, and for any time-constructible bound t = o(n), there is a language in NTIME(n) which is not in NTIME(t)/n1/d . The best known earlier separation of Fortnow, Santhanam ...

We study combinatorial bounds for geometric permutations of balls with bounded size disparity in d-space. Our main contribution is the following theorem: given a set S of n disjoint balls in R , if n is sufficiently large and the radius ratio between the largest and smallest balls of S is γ , then the maximum number of geometric permutations of S is O(γ logγ ). When d = 2, we are able to prove ...

The previous best estimates are due to Chang [Cha02] who showed the above result (up to logarithmic factors) with 2 in place of 7/4. Note that one cannot hope to improve the dimension bound past ⌊K − 1⌋, or the exponent of K in the size bound below 1; at the end of [Cha02] Chang (using arguments of Bilu [Bil99]) actually shows how to bootstrap the dimension bound to ⌊K−1⌋ for a small cost in th...

We investigate the number of limit cycles of a planar quadratic vector field with a perturbed center-like singular point. An upper bound is obtained on the number of δ-good limit cycles of such a vector field (Theorem 1). Here δ is a parameter characterizing the limit cycles: it shows how far those cycles are from the singular points of the vector field and from the infinite points. The bound a...

Main result of this paper is the following theorem: given δ, 0 < δ < 1/3 and n ∈ N there exists an (n + 1) × n inner matrix function F ∈ H∞ (n+1)×n such that I ≥ F ∗(z)F (z) ≥ δI ∀z ∈ D, but the norm of any left inverse for F is at least [δ/(1−δ)]−n ≥ (2δ). This gives a lower bound for the solution of the Matrix Corona Problem, which is pretty close to the best known upper b bound C · δ−n−1 log...

For a given visual cryptography scheme, it is possible to present a basis matrices for it and most of constructions are based on basis matrices. In this paper we introduce a lower bound for the pixel expansion of visual cryptography schemes with basis matrices. To make the main theorem more flexible, we will introduce a lower bound based on induced matchings of hypergraph of qualified sets. As ...

The complexity class ZPP (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. r For starters, we provide a new characterization: ZPP equals the restriction of BPP where the algorithm is only allowed t...

Hirst investigated a slight variant of Hindman's Finite Sums Theorem called Hilbert's Theorem and proved it equivalent over RCA0 to the In nite Pigeonhole Principle for all colors. This gave the rst example of a natural restriction of Hindman's Theorem provably much weaker than Hindman's Theorem itself. We here introduce another natural variant of Hindman's Theorem which we name the Adjacent Hi...

We prove that a locally compact space with an upper curvature bound is topological manifold if and only all of its spaces directions are homotopy equivalent not contractible. discuss applications to homology manifolds, limits Riemannian manifolds deduce sphere theorem.

We prove that the error in prime number theorem can be quantitatively improved beyond Riemann Hypothesis bound by using versions of Montgomery’s conjecture for pair correlation zeros zeta-function which are uniform long ranges and with suitable terms.