We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)action. This improves upon the classification theorem of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.Mathematics Subject Classification: 55P42, 55P91 and 55U35
a b s t r a c tWe prove that the Goodwillie tower of a weak equivalence preserving functor from spaces to spectra can be expressed in terms of the tower for stable mapping spaces. Our proof is motivated by interpreting the functors P n and D n as pseudo-differential operators which suggests certain 'integral' presentations based on a derived Yoneda embedding. These models allow one to extend computational tools available for the tower of stable mapping spaces. As an application we give a classical expression for the derivative over the basepoint.
In this paper, we prove that the family of binomials $x_1^{a_1}
\cdots x_m^{a_m}-y_1^{b_1}\cdots y_n^{b_n}$ with $\gcd(a_1,
\ldots, a_m, b_1, \ldots, b_n)=1$ is irreducible by identifying
the connection between the irreducibility of a binomial in
${\mathbb C}[x_1, \ldots, x_m, y_1, \ldots, y_n]$ and ${\mathbb
C}(x_2, \ldots, x_m, y_1, \ldots, y_n)[x_1]$. Then we show that
the necessary and sufficient conditions for the irreducibility of
this family of binomials is equivalent to the existence of a
unimodular matrix $U_i$ with integer entries such that $(a_1,
\ldots, a_m, b_1, \ldots, b_n)^T=U_i \be_i$ for $i\in \{1, \ldots,
m+n\}$, where $\be_i$ is the standard basis vector.
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