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In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research.
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In this paper we prove the concavity of the $k$-trace functions, $Amapsto (text{Tr}_k[exp(H+ln A)])^{1/k}$, on the convex cone of all positive definite matrices. $text{Tr}_k[A]$ denotes the $k_{mathrm{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. As an application, we use the concavity of these $k$-trace functions to derive tail bounds and expectation estimates on the sum of the $k$ largest (or smallest) eigenvalues of a sum of random matrices.
The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedguts junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p = o(1)$, there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general $p$ that applies to `global functions, i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgains sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgains theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a $p$-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Turan number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Turan number, answering a question of Mubayi and Verstraete. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the Furedi--Jiang--Seiver conjecture on path expansions.
Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, this paper establishes a new central limit theorem (CLT) for a linear spectral statistic (LSS) of high dimensional sample correlation matrices for the case where the dimension p and the sample size $n$ are comparable. This result is of independent interest in large dimensional random matrix theory. Meanwhile, we apply the linear spectral statistic to an independence test for $p$ random variables, and then an equivalence test for p factor loadings and $n$ factors in a factor model. The finite sample performance of the proposed test shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from different cities in China is also conducted.
In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of $n$ disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the $n$th isoperimetric constant and the number obtained by taking the minimum over all $n$-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the ${bf NP}$-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J.~Malik and J.~Shi ($2000$) we study the close relationships to the well-known $k$-means algorithm and normalized cuts method.
We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyals category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.
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