Hyperbolic torsion polynomials of pretzel knots

In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].

Extrapolating body masses in large terrestrial vertebrates

Despite more than a century of interest, body-mass estimation in the fossil record remains contentious, particularly when estimating the body mass of taxa outside the size scope of living animals. One estimation approach uses humeral and femoral (stylopodial) circumferences collected from extant (living) terrestrial vertebrates to infer the body masses of extinct tetrapods through scaling models. When applied to very large extinct taxa, extant-based scaling approaches incur obvious methodological extrapolations leading some to suggest that they may overestimate the body masses of large terrestrial vertebrates. Here, I test the implicit assumption of such assertions: that a quadratic model provides a better fit to the combined humeral and femoral circumferences-to-body mass relationship. I then examine the extrapolation potential of these models through a series of subsetting exercises in which lower body-mass sets are used to estimate larger sets. Model fitting recovered greater support for the original linear model, and a nonsignificant second-degree term indicates that the quadratic relationship is statistically linear. Nevertheless, some statistical support was obtained for the quadratic model, and application of the quadratic model to a series of dinosaurs provides lower mass estimates at larger sizes that are more consistent with recent estimates using a minimum convex-hull (MCH) approach. Given this consistency, a quadratic model may be preferred at this time. Still, caution is advised; extrapolations of quadratic functions are unpredictable compared with linear functions. Further research testing the MCH approach (e.g., the use of a universal upscaling factor) may shed light on the linear versus quadratic nature of the relationship between the combined femoral and humeral circumferences and body mass.

Una “buena” manera de hablar acerca de grados: bien con adjetivos en español

This article provides a detailed description of the syntactic and semantic properties of the degree term

K-theory Schubert calculus of the affine Grassmannian

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

On partitions of n into k summands

In his recent paper on partitions (1), Jakub Intrator proved that the number p(n, k) of partitions of n into exactly k summands, 1 < k ≦ n, is given by a polynomial of degree exactly k − 1 in n, the first [(k+1)/2] coefficients of which (starting with the coefficient of the highest degree term), are independent of n and the rest depend on the residue of n modulo the least common multiple of the integers 1, 2, 3, …, k. He even showed (ignoring the case k = 3) that the [(k+3)/2]-th coefficient in the polynomial depends only on the parity of n and is not the same for n even and n odd.

High Common Factor

As the colleges now are requiring highest common factor by factoring methods only, any plan whereby the number of factors to be tried can be lessened is certainly worth while. For in general we must regard as possible any binomial factor whose first-degree term is an integral divisor of the highest term of the given expression and whose independent term is an integral divisor of the independent term of the expression. Thus in such a problem as, “Find the H.C.F. of 5x3 − 21x2 + 5x − 4 and 5x3 − 19x2 + 5x + 4” (McCurdy’s Exercise Book, page 40, example 4) the possible factors that a student might try and must try, unless he were very lucky in those he chose to try first, are x ± 1, x ± 2, x ± 4, 5x ± 1, 5x ± 2, 5x ± 4. And further if the given expressions were, say, cubics having no common binomial factor at all but with a quadratic one instead (e. g., 2x3 + 5x2 + x − 3 and 2x3 − x2 − 5x + 3) I doubt if the ordinary first-year student would get any result unless it were unity.