We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ the unit cube $[0,1]^d$. If lengths their $y$ are evaluated in $\ell^p$-norm for any $p$ other than $1, 2, n-1$, and $\infty$, if dimension $d$ is large enough, then we show that numbers $\|y\|_{\ell^p}$ attain all values from an interval, length which depends only on $n$, $p$...

We show that under suitable non-degeneracy conditions, m points and n 2–dimensional algebraic surfaces in R satisfying certain “pseudoflat” requirements can have at most O ( mn + m + n ) incidences, provided that m ≤ n2− for any > 0 (where the implicit constant in the above bound depends on ), or m ≥ n. As a special case, we obtain the Szemerédi-Trotter theorem for 2–planes in R, again provided...

Econometricians typically make use of functional central limit theorems to prove results for I(1) processes. For example, to establish the limit distributions of unit root tests such as the Phillips–Perron and Dickey–Fuller tests, the functional central limit theorem plays a crucial role. In this paper, it is pointed out that for linear processes, minimal conditions that ensure that only a cent...

Since the time of ancient Greece and India, Pell’s equation, a2 − db2 = 1, has been studied and its solutions sought after. These solutions are significant, because they correspond to the units of the rings Z[ √ d]. Along with prime numbers, units are vital in understanding the structure and especially the factorization within different rings. Dirichlect’s Unit Theorem establishes the structure...

Residue Number System is a numerical system which arithmetic operations are performed parallelly. One of the main factors that affects the system’s performance is the complexity of reverse converter. It should be noted that the complexity of this part should not affect the earned speed of parallelly performed arithmetic unit. Therefore in this paper a high speed converter for moduli set {2n-1, ...

Residue Number System is a numerical system which arithmetic operations are performed parallelly. One of the main factors that affects the system’s performance is the complexity of reverse converter. It should be noted that the complexity of this part should not affect the earned speed of parallelly performed arithmetic unit. Therefore in this paper a high speed converter for moduli set {2n-1, ...

We prove that the multiplier algebra of the Drury-Arveson Hardy space H n on the unit ball in C n has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson in the unit disk. This result is obtained as a corollary of the Toeplitz Corona Theorem and a new Banach space result: the Besov-Sobolev space B p has the ”baby corona property” for all 0 ≤ σ < n p ...

In the unit on rings, I explained category theory and general rings at the same time. Then I talked mostly about commutative rings. In the unit on modules, I again mixed category theory into the basic notions and progressed to the structure theorem for finitely generated modules over PID’s. Jordan canonical forms were used as an application. The uniqueness part of the structure theorem was put ...