In this article we introduce combinatorial multi-color discrepancies and generalize several classical results from 2–color discrepancy theory to c colors (c ≥ 2). We give a recursive method that constructs c–colorings from approximations of 2–color discrepancies. This method works for a large class of theorems like the ‘six standard deviations’ theorem of Spencer, the Beck–Fiala theorem, the re...

We investigate the computable content of the Uniform Boundedness Theorem and of the closely related Banach-Steinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It turns out that the answer depends on how the seque...

In this short note we show that the bound provided by Theorem 3.8 in 4] is not correct by exhibiting a counter example. We identify the aw in the proof of Theorem 3.8 in 4] and then we prove a correct bound on P S .

This paper is an expository piece on edge-chromatic graph theory. The central theorem in this subject is that of Vizing. We shall then explore the properties of graphs where Vizing’s upper bound on the chromatic index is tight, and graphs where the lower bound is tight. Finally, we will look at a few generalizations of Vizing’s Theorem, as well as some related conjectures.

We derive an isoperimetric theorem for sequences genenerated by feedback and consider block codes for the binary broadcast channel with two receivers and noiseless feedback. We get an outer bound by applying the isoperimetric theorem on the achievable rates for special cases of these codes with unequal error protection. We get a lower bound with a generalized Varshamov-Gilbert construction.

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.

In this paper, stability of slopes with a shallow foundation located on top including the shape of the failure mechanism is investigated in static and seismic states. This is a three dimensional problem which is analyzed two dimensionally because 3D analyses are complex and appropriate softwares are not available. The conventional method of bearing capacity evaluation of foundations on top of ...

A general moment bound for sums of products of Gaussian vector’s functions extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general central limit theorem for triangular arrays of nonlinear functionals of multidimensional non-stationary Gaussian sequences is proved. This theorem extends the previous results of Breuer and Major (1981), Arcones (1994) and others. A Berry-Ess...

This note complements an earlier paper of the author by providing a lower bound theorem for Minkowski sums of polytopes. In [AS16], we showed an analogue of McMullen’s Upper Bound theorem for Minkowski sums of polytopes, estimating the maximal complexity of such a sum of polytopes. A common question in reaction to that research was the question for an analogue for the Barnette’s Lower Bound The...

We make explicit the exponential bound on degrees of polynomials appearing in Effective Quillen-Suslin Theorem, and apply it jointly with Hilbert-Burch Theorem to show that syzygy module a sequence m n variables defining complete intersection ideal grade two is free, basis can be computed bounded degrees. In known cases, these bounds improve previous results.