Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0, 1}N . By an application of the Chen-Stein method, we show that U −2 log N/ log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we ...

In this paper we present alternate form of Radix-4/8 and split radix FFT’s based on DIF (decimation in frequency) version and discuss their implementation issues that further reduces the arithmetic complexity of power-of-two discrete Fourier Transform. This is achieved with circular shift operation on a subset of the output samples resulting from the decomposition in these FFT algorithms and a ...

In this paper we derive a deterministic fast discrete Fourier transform algorithm for a 2π-periodic function f whose Fourier coefficients with significantly large magnitude are contained inside a support interval of length B. The algorithm is based on a method for the efficient recovery of B-sparse 2π-periodic functions presented by Iwen in [15]. If a good bound B on the support length is known...

We study certain stationary and time-evolution problems of trapped Bose-Einstein condensates using the numerical solution of the Gross-Pitaevskii equation with both spherical and axial symmetries. We consider time-evolution problems initiated by changing the interatomic scattering length or harmonic trapping potential suddenly in a stationary condensate. These changes introduce oscillations in ...

Schmidt proved in 2014 that if ε > 0, almost all binary sequences of length n have peak sidelobe level between ( √ 2 − ε)√n log n and ( √ 2 + ε) √ n log n. Because of the small gap between his upper and lower bounds, it is difficult to find improved upper bounds that hold for almost all binary sequences. In this note, we prove that if ε > 0, then almost all binary sequences of length n have pea...

Given strings P of length m and T of length n over an alphabet of size σ, the string matching with k-mismatches problem is to find the positions of all the substrings in T that are at Hamming distance at most k from P . If T can be read only one character at the time the best known bounds are O(n √ k log k) and O(n+ n √ k/w log k) in the word-RAM model with word length w. In the RAM models (inc...

Well-preserved adult skulls (twenty-two male and 27 female) from White of Rasquera Goat breed were studied. The greatest length of the skull has been used to estimate head size. Horn length was measured with a tape along the whole length of the right sheath. The equations obtained were [log y = 3.231 – 1.417 log x] and [log y = 3.248 – 1.538 log x] for males and females respectively. It is poss...

We show that the number of t-ary trees with path length equal to p is exp ( h(t−1)t log t p log p (1 + o(1)) ) , where h(x)=−x log x−(1−x) log(1−x) is the binary entropy function. Besides its intrinsic combinatorial interest, the question recently arose in the context of information theory, where the number of t-ary trees with path length p estimates the number of universal types, or, equivalen...

In geometry, students are frequently required to split their attention across verbal and visual information. We measured the impact of the split-attention effect on geometry problem solving by analyzing log-file data generated by students who used two versions of Carnegie Learning’s intelligent tutoring system for geometry. The original version split students’ attention across a table and diagr...

This paper is a sequel to [B]. Our main result is an improvement of the density condition for a subset A ⊂ {1,. .. , N } to contain a nontrivial arithmetic progression of length 3. More specifically, we prove the following Theorem 1. (0.1) δ ≫ (log log N) 2 (log N) 2/3 (N assumed sufficiently large), then A contains nontrivial progressions of length 3.