Moreover, the above upper bound 1/(qN) is best possible apart from a scalar multiple when one considers the golden ratio θ = ( √ 5 − 1)/2 (for a proof, one may use Theorem 181 and bottom of page 162 in [2] on certain properties of continued fractions). During his work in [1], the author accidentally stumbled across the following analogous question: Question 1. For any real θ and any integer N ≥...

In [J. Graph Theory 13 (1989) 749–762], McCuaig and Shepherd gave an upper bound of the domination number for connected graphs with minimum degree at least two. In this paper, we propose a simple strategy which, together with the McCuaig-Shepherd theorem, gives a sharp upper bound of the domination number via the number of leaves. We also apply the same strategy to other domination-like invaria...

We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n log n upper bound on the ordered graph extremal function of F = ({1, 3}, {1, 5}, {2, 3}, {2, 4}) due to Füredi to the n(log n)2(log log n)3 upper bound in the hypergraph case. We use Davenp...

this paper considers the economic lot scheduling problem, that is, the problem of scheduling several products on a single facility so as to minimize holding and setup costs. combination of frequency and timing as well as production quantity make this problem np-hard. a heuristic is developed to obtain a good solution to elsp. the proposed heuristic makes use of the simulated annealing technique...

The main purpose of this note is to prove an upper bound on the number of lattice points of a centrally symmetric convex body in terms of the successive minima of the body. This bound improves on former bounds and narrows the gap towards a lattice point analogue of Minkowski’s second theorem on successive minima. Minkowski’s proof of his second theorem is rather lengthy and it was also criticis...

Monsky’s theorem from 1970 states that a square cannot be dissected into an odd number n of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monsky’s theorem to “constrained framed maps”; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for th...

Abstract. A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i > 0 interior lattice points. We give a new proof for this theorem and classify polygons with maximal volume. Then we show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimension. From a theorem of Victor Batyrev the finiteness of ...

We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis smoothing argument. The upper bound applies to more general families samples, including dependent variables, interest study rates convergence for empirical measures. Following recent pde approach by L. Ambrosio, F. Stra D. Trevisan, we also adapt simple lower bo...

We prove a new upper bound on the second moment of Maass form symmetric square L -functions defined over Gaussian integers. Combining this estimate with recent result Balog–Biro–Cherubini–Laaksonen, we improve error term in prime geodesic theorem for Picard manifold.

Let q denote a large prime number, a be any integer. Let N(q, a) denote the smallest positive integer n for which φ(n) ≡ a (mod q). The number N(q, a) exists. Indeed, if a+1 ≡ 0 (mod q) then one can take n = q. Otherwise, one can take n to be a prime from the arithmetical progression a + 1 (mod q). The problem of upper bound estimates for N(q, a) has been a subject of study of the work of Fried...