T ESTICULAR spermatid reserves have been quantified for bulls, buffalo-bulls, deer, boars, rabbits, guinea pigs, roosters and humans. Generally, a modification of the technique described by Amann and Almquist (1961) has been used. Daily sperm production can be calculated by dividing the value for testicular spermatid reserves by a time divisor which is the number of days of production these res...

We study the Cox ring and monoid of effective divisor classes ofM0,n ∼= BlPn−3, over an arbitrary ring R. We provide a bijection between elements of the Cox ring, not divisible by any exceptional divisor section, and pure-dimensional singular simplicial complexes on {1, . . . , n − 1} with weights in R\{0} satisfying a zero-tension condition. This leads to a combinatorial criterion for a diviso...

The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative rin...

We define a Deligne-Mumford stack XD,r which depends on a scheme X, an effective Cartier divisor D ⊂ X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into XD,r provides compactifications of the locally closed substacks of M̄g,n(X,β) corresponding to relative stable maps.

From a rational convex polytope of dimension r ≥ 2 J.P. Hansen constructed an error correcting code of length n = (q−1)r over the finite field Fq. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is obtained evaluating rational functions of the toric variety defined by the polytope at the algebraic torus, and it is an evaluation code in the ...

The Mumford class κ1 on Mg,0 was shown to be proportional to the cohomology class [ωWP ] of the Weil-Petersson form by Wolpert in [WO]. Furthermore he showed that the restriction of this class to any component of the compactyfying divisor coincides with the corresponding Weil-Petersson class. Arbarello and Cornalba introduced classes κ1 on Mg,n, proved a similar restriction property for these a...

Let k-l,ml,..~+k denote non-negative integers, and suppose the greatest common divisor of ml,...,mk is 1 . We show that if '1, "',sk are sufficiently long blocks of consecutive integers, then the set mlSl+ . ..+mkSk contains a sizable block of consecutive integers. For example; if m and n are relatively prime natural numbers, and U, U ? Vt V are integers with U-u 2 n-l , V-v 2 m-l 1 then the se...

This paper focuses on the interplay between the intersection theory and the Teichmüller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a non-simple zero, and verify a number of extremal effective divisors on the moduli spac...

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