Few galaxies have been found between the redshift ranges z < ∼ 1 probed by magnitude-limited surveys and z > ∼ 3 probed by Lyman-break surveys. Comparison of galaxy samples at lower and higher redshift suggests that large numbers of stars were born and the Hubble sequence began to take shape at the intermediate redshifts 1 < z < 3, but observational challenges have prevented us from observing t...

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable; so is the theory of multiplication in all those domains. By Gödel’s Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q; though Tarski proved that this theory is decidable in the domains of R and C. The theory of...

It is shown that Dirichlet's theorem on primes in an arithmetic progression is equivalent to the statement that every unit of a certain quotient ring Z of the nonstandard integers is the image of an infinite prime. The ring Z is the completion of Z relative to the "natural" topology on Z. 1. Notation. Throughout this note A shall denote the natural numbers, Z the rational integers, and P the po...

lim h→0 f(z + h)− f(z) h if this limit exists. If the derivative of f exists at z, we denote its value by f ′(z), and we say f isholomorphic at z. If f is holomorphic at every point of an open set U we say that f is holomorphic on U . This definition naturally leads to several basic remarks. First, the definition formally looks identical to the limit definition of a derivative of a function of ...

Let P (x) be a unimodal polynomial of degree m with nonnegative coefficients and a mode n for nonnegative integers n m. We study the unimodality of P (x+ z) for real numbers z = 1 or z 2 and show that: if z = 1, P (x + z) is unimodal provided that m − n 4; if z 2, then P (x+ z) is unimodal provided that m− n 2z + 1; and we also show that the given conditions are best possible. Additionally, we ...

One of the most important challenges when building road infrastructure is selection appropriate mechanization, on which efficiency construction and life exploitation depends largely. As machinery, pavers occupy a significant place in civil engineering projects, so their selection, depending category, very activity. The objective this paper to develop an intelligent Fuzzy MCDM (Multi-Criteria De...

Let Z=(pq) be the integer residue ring modulo pq with odd prime numbers p and q. This paper studies the distinctness problem of modulo 2 reductions of two primitive sequences over Z=(pq), which has been studied by H.J. Chen and W.F. Qi in 2009. First, it is shown that almost every element in Z=(pq) occurs in a primitive sequence of order n > 2 over Z=(pq). Then based on this element distributio...

We discuss the case of simultaneous mixing of gauge bosons and mixing of fermions in a model independent way and for a variety of extra-fermion representations. In this context we analyze a class of lepton family violating processes, namely Z→ eτ̄ , Z→ μτ̄ , Z→ eμ̄, μ→ eeē, τ → eeē, τ → μμμ̄, τ → eμμ̄ and τ → μeē in the presence of one extra neutral gauge boson, Z′, with universal, non-universal or ...

We consider the families of finite Abelian groups Z/pZ × Z/pZ, Z/pZ and Z/pZ×Z/qZ for p, q two distinct prime numbers. For the two first families we give a simple characterization of all functions whose support has cardinality k while the size of the spectrum satisfies a minimality condition. We do it for a large number of values of k in the third case. Such equality cases were previously known...

In this work, we present two sets of full asymptotic expansions for the modified Bessel functions Iν(z) and Kν(z) and a full asymptotic expansion for Iν(z)Kν(z) as ν → ∞ and z is fixed with | arg z| < π. In particular, we show that Iν(z) ∼ (12z) ν Γ(ν + 1) ∞ ∑ m=0 bm(z) νm as ν → ∞, | arg ν| ≤ π − δ, and Kν(z) ∼ 1 2 Γ(ν) (12z) ν ∞ ∑ m=0 (−1) bm(z) νm as ν → ∞, | arg ν| ≤ 12π − δ, where, for eac...