The structure of a stationary Gaussian process near a local maximum with a prescribed height$u$has been explored in several papers by the present author, see [5]–[7], which include results for moderate$u$as well as for$u$→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ$(t) t ∈ R$^{n}}, with mean zero and the covariance function$r$($t$). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.

In this paper we use a homological approach to obtain upper bounds for a few homological invariants of FI_G-modules V. These upper bounds are expressed in terms of the generating degree and torsion degree, which measure the top and socle of V under actions of non-invertible morphisms in the category respectively.

This essay mainly involves a method to count the number of permutations of size n with length of the longest increasing subsequence equal to 2. The question is raised as the research project of Algebraic Combinatorics in 2016 Tsinghua Math Camp. This essay will show the relationshi between a permutation's longest increasing subsequence andits correspondingRobinson-Schensted map. Moreover, theessay will develop a general formula to count thenumberwith the help of the hook length formula.

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Abstract Let T: V W be a surjective real linear isometry between full real Hilbert C-modules over real C-algebras A and B, respectively. We show that the following conditions are equivalent:(a) T is a 2-isometry;(b) T is a complete isometry;(c) T preserves ternary products;(d) T preserves inner products;(e) T is a module map. When A and B are commutative, we give a full description of the structure of T.