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The calculation of the probability of correct selection (PCS) shows how likely it is that the populations chosen as “best” truly are the top populations, according to a well-defined standard. PCS is useful for the researcher with limited resources or the statistician attempting to test the quality of two different statistics. This paper explores the theory behind two selection goals for PCS, -best and -best, and how they improve previous definitions of PCS for massive datasets. This paper also calculates PCS for two applications that have already been analyzed by multiple testing procedures in the literature. The two applications are in neuroimaging and econometrics. It is shown through these applications that PCS not only supports the multiple testing conclusions but also provides further information about the statistics used.
We prove that the map assigning to a given vector field the Lebesgue measure of the union of the basins of its attractors is lower semicontinuous in a residual subset of vector fields. Moreover, we prove that the Lebesgue measure of the union of the basins of attractors of a generic sectional axiom A vector field is total. For this, we also improve a result of Morales about sectional-hyperbolic sets. We also remark that homoclinic classes are topologically ergodic and that for a generic tame diffeomorphism, the union of the stable manifolds of the hyperbolic periodic orbits is dense in the manifold.
Let be a positive integer and let be an arbitrary, fixed positive integer. We define a generalized Fibonacci-type polynomial sequence by , , and for . Let represent the maximum real zero of . We prove that the sequence is decreasing and converges to a real number . Moreover, we prove that the sequence is increasing and converges to as well. We conclude by proving that is decreasing and converges to .
An iteration digraph generated by the function is a digraph on the set of vertices with the directed edge set . Focusing specifically on the function , we consider the structure of these graphs as it relates to the factors of . The cycle lengths and number of cycles are determined for various sets of integers including powers of 2 and multiples of 3.
We derive an effective quadrature scheme via a partitioned Duffy transformation for a class of Zienkiewicz-like rational bubble functions proposed by J. Guzmán and M. Neilan. This includes a detailed construction of the new quadrature scheme, followed by a proof of exponential error convergence. Briefly discussed is the functions application to the finite element method when used to solve Stokes flow and elasticity problems. Numerical experiments which support the theoretical results are also provided.
Let be a finite group. A subgroup of is called weakly -permutably embedded in if there is a subnormal subgroup of and an -permutably embedded subgroup of contained in such that and . The subgroup is called weakly -supplemented in if has a subgroup such that and , where is the largest -permutable subgroup of contained in . In this paper, we investigate the influence of weakly -permutably embedded and weakly -supplemented subgroups on the structure of finite groups. Some recent results are generalized.
A proposed measure of network cohesion for graphs arising from interrelated economic activity is studied. The measure is the largest singular value of a row-stochastic matrix derived from the adjacency matrix. It is shown here that among graphs on vertices, the star universally gives the (strictly) largest measure. Other universal comparisons among graphs with larger measures are difficult to make, but one is conjectured, and a selection of empirical evidence is given.
By directly considering Taylor coefficients and composite generating functions, we employ a generalized Faà di Bruno formula for higher partial derivatives using vector partitions to obtain identities that include explicit formulas for the Bernoulli and Euler numbers. The formulas we obtain are generalized analogs of the formulas obtained by D. C. Vella.
The crossing lemma holds in because a real line separates the plane into two disjoint regions. In removing a complex line keeps the remaining point-set connected. We investigate the crossing structure of affine line segment-like objects in by defining two notions of line segments between two points and give computational results on combinatorics of crossings of line segments induced by a set of points. One way we define the line segments motivates a related problem in , which we introduce and solve.
An edge ordering of a graph is an injection , where is the set of positive integers. A path in for which the edge ordering increases along its edge sequence is called an -ascent; an -ascent is maximal if it is not contained in a longer -ascent. The depression of is the smallest integer such that any edge ordering has a maximal -ascent of length at most . Applying the concept of ascents to edge colourings rather than edge orderings, we consider the problem of determining the minimum number of colours required to edge colour , , such that the length of a shortest maximal ascent is equal to . We obtain new upper and lower bounds for , which enable us to determine exactly for and and to bound by .
We study the envelope of the family of lines which bisect the interior region of a simple, closed curve in the plane. We determine this ‘bisection envelope’for polygons and show that polygons with no parallel pairs of sides are characterized by their bisection envelope. We show that the bisection envelope always has at least three and an odd number of cusps. We investigate the winding numbers of bisection envelopes, and use this to show that there are an infinite number of curves with any given bisection envelope and show how to generate them. We obtain results on the intersections of bisecting lines. Finally, we give a relationship between the ‘internal area’of a curve and that of its bisection envelope.
We study the 590 nonisomorphic degree 14 extensions of the 2-adic numbers by computing defining polynomials for each extension as well as basic invariant data for each polynomial, including the ramification index, residue degree, discriminant exponent, and Galois group. Our study of the Galois groups of these extensions shows that only 10 of the 63 transitive subgroups of occur as a Galois group. We end by describing our implementation for computing Galois groups in this setting, which is of interest since it uses subfield information, the discriminant, and only one other resolvent polynomial.
Mathematical tools from combinatorics and abstract algebra have been used to study a variety of musical structures. One question asked by mathematicians and musicians is: how many -note set classes exist in a -note chromatic universe? In the music theory literature, this question is answered with the use of Pólya’s enumeration theorem. We solve the problem using simpler techniques, including only Burnside’s lemma and basic results from combinatorics and abstract algebra. We use interval arrays that are associated with pitch class sets as a tool for counting.
We first describe how one associates a cubic curve to a given ternary trilinear form . We explore relations between the rank and border rank of the tensor and the geometry of the corresponding cubic curve. When the curve is smooth, we show there is no relation. When the curve is singular, normal forms are available, and we review the explicit correspondence between the normal forms, rank and border rank.
For any integer , there are infinitely many primes congruent to . In this note, the elementary argument of Thangadurai and Vatwani is modified to improve their upper estimate of the least such prime when itself is a prime greater than or equal to 5.