E[n] ' { Z/nZ⊕ Z/nZ if p = 0 or p n, Z/nZ or {0} if p > 0 and n is a power of p. Proof. Assume p n and let ` be a prime dividing n. Then the multiplication-by-` map [`] is separable of degree `2, and therefore E[`] = ker[`] has order `2. Every nonzero element of E[`] has order `, and it follows that E[`] ' Z/`Z ⊗ Z/`Z. Thus the `-rank of E[`], and hence of E[n], is 2. If `e is the largest power...

The instructions under which raters quantify syllable prominence perception need to be simple in order to maintain immediate reactions. This leads to noise in the rating data that can be dealt with by normalization, e.g. setting central tendency = 0 and dispersion = 1 (as in Z-score normalization). Questions arise such as: Which parameter is adequate here to capture central tendency? Which refe...

The rank rk(R) of a ring R is the supremum of minimal cardinalities of generating sets of I as I ranges over ideals of R. Matson showed that every n ∈ Z+ occurs as the rank of some ring R. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension 0 or 1, we give four different constructions of rings of rank n (for all n ∈ Z+). Two constructions use one-dimension...

Background: Prior to applying genomic predictors to clinical samples, the genomic data must be properly normalized. The most effective normalization methods depend on the data from multiple patients. From a biomedical perspective this implies that predictions for a single patient may change depending on which other patient samples they are normalized with. This test set bias will occur when any...

Suppose that K and K' are knots inside the homology spheres Y Y', respectively. Let X = (K, K') be 3-manifold obtained by splicing complements of Z three-manifold 0 surgery on K. When Y' is an L-space, we use formula [1] to show rank (X ) bounded below (Y if τ(K 2 rank( (Z)) − (Y)) + 1 τ(K') ≠ 0.

The purpose of this study was to examine the results prediction breast cancer, which have been classified based on two types malignant and benign. method used in research is k-NN algorithm with normalization min-max Z-score, programming language R language. conclusion that highest k accuracy value = 5 21 an rate 98% using method. Whereas for Z-score at 15 97%. Thus considered better than Z-scor...

Three lectures on elliptic surfaces and curves of high rank Noam D. Elkies Over the past two years we have improved several of the (Mordell–Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P i ∈ E(Q) (the previous record was 24, by R. Martin and W. McMillen 2000), and the fi...

The problem of defining iteration for higher-order nested datatypes of arbitrary (finite) rank is solved within the framework of System F of higher-order parametric polymorphism. The proposed solution heavily relies on a general notion of monotonicity as opposed to a syntactic criterion on the shape of the type constructors such as positivity or even being polynomial. Its use is demonstrated fo...

Alpha is a functional language based on systems of aane recurrence equations over polyhedral domains. We present an extension of Alpha to deal with sparse polyhedral domains. Such domains are modeled by Z-polyhedra, namely the intersection of lattices and polyhedra. We summarize the mathematical closure properties of Z-polyhedra, and we show how the important features of Alpha, namely normaliza...