Major Theorems Research Articles

The maximum [formula omitted]-spectral radius and the majorization theorem of [formula omitted]-uniform supertrees

Let α be a real number with 0≤α<1, G be a uniform hypergraph, and Aα(G)=αD(G)+(1−α)A(G), where D(G) and A(G) are the diagonal degree tensor and the adjacency tensor of G, respectively. The spectral radius of Aα(G) is called the α-spectral radius of G. In this paper, some properties for the α-spectral radius of connected k-uniform hypergraphs are investigated, the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with a given degree sequence is characterized, and the majorization theorem for k-uniform supertrees is obtained. By applying the majorization theorem, we determine the k-uniform supertrees obtaining the three largest α-spectral radius among all k-uniform supertrees with n vertices, give a new proof ordering the eight largest spectral radius among all k-uniform supertrees with n vertices, propose an open problem for further research, and characterize the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with given different parameters, e.g., the maximum degree △, or the number of pendent vertices.

Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion

In this letter, we construct cusum change-point tests for the Hurst exponent and the volatility of a discretely observed fractional Brownian motion. As a statistical application of the functional Breuer–Major theorems by Bégyn (2007) and Nourdin and Nualart (2019), we show under infill asymptotics consistency of the tests and weak convergence to the Kolmogorov–Smirnov law under the no-change hypothesis. The test is feasible and pivotal in the sense that it is based on a statistic and critical values which do not require knowledge of any parameter values. Consistent estimation of the break date under the alternative hypothesis is established. We demonstrate the finite-sample properties in simulations and a data example.

Continuous Breuer–Major theorem: Tightness and nonstationarity

Let $Y=(Y(t))_{t\geq 0}$ be a zero-mean Gaussian stationary process with covariance function $\rho :\mathbb{R}\to \mathbb{R}$ satisfying $\rho (0)=1$. Let $f:\mathbb{R}\to \mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_{\mathbb{R}}|\rho (s)|^{d}\,ds<\infty $, then the celebrated Breuer–Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_{\varepsilon }:=\sqrt{\varepsilon }\int_{0}^{\cdot /\varepsilon }f(Y(s))\,ds$ converge to those of $\sigma W$ as $\varepsilon \to 0$, where $W$ is a standard Brownian motion and $\sigma $ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes. The goal of this paper is twofold. First, we investigate the tightness in the Breuer–Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [J. Multivariate Anal. 80 (2002) 191–216] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that $|f|^{p}$ must be integrable with respect to the standard Gaussian measure for some $p$ strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities. Second, and motivated by a problem of geometrical nature, we extend the continuous Breuer–Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Brownian motion concludes the paper.

Structure theorems for operators associated with two domains related to μ-synthesis

A commuting tuple of n operators (S1,…,Sn−1,P) defined on a Hilbert space H, for which the closed symmetrized polydiscΓn={(∑i=1nzi,∑1≤i<j≤nzizj,…,∏i=1nzi):|zi|≤1,i=1,…,n} is a spectral set is called a Γn-contraction. Also a triple of commuting operators (A,B,P) for which the closed tetrablock E‾ is a spectral set is called an E-contraction, whereE={(x1,x2,x3)∈C3:1−zx1−wx2+zwx3≠0∀z,w∈D‾}. There are several decomposition theorems for contraction operators in the literature due to Sz. Nagy, Foias, Levan, Kubrusly, Foguel and few others which reveal structural information of a contraction. In this article, we obtain analogues of six such major theorems for both Γn-contractions and E-contractions. In each of these decomposition theorems, the underlying Hilbert space admits a unique orthogonal decomposition which is provided by the last component P. The central role in determining the structure of a Γn-contraction or an E-contraction is played by positivity of some certain operator pencils and the existence of a unique operator tuple associated with a Γn-contraction or an E-contraction.

The role of algebraic structure in the invariant subspace theory

Following a recent work of authors [1], we establish the algebraic analogs of three major decomposition theorems of Nagy-Foias-Langer, Halmos-Wallen and Fishel in Baer ⁎-rings. This approach not only provides purely algebraic and shorter proofs of these decomposition theorems, but also suggests the possibility of eliminating the role of analytic structure in results related to the invariant subspace theory.

Unified extremal results of topological indices and spectral invariants of graphs

Identifying graphs with extremal properties is an extensively studied topic in both topological graph theory and spectral graph theory. As observed in the literature, for many graph categories the extremal graphs with respect to some prescribed topological index or spectral invariant are the same. Therefore, it is an interesting problem to find a unified method to identify the extremal graphs for a set of topological or spectral invariants. Here we consider the majorization theorem to obtain extremal graphs in the class of trees, unicyclic graphs and bicyclic graphs with given order and fixed maximum degree, independence number, matching number, domination number, and/or number of pendant vertices, respectively.

The functional Breuer–Major theorem

Let $$X=\{ X_n\}_{n\in \mathbb {Z}}$$ be zero-mean stationary Gaussian sequence of random variables with covariance function $$\rho $$ satisfying $$\rho (0)=1$$. Let $$\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}$$ be a function such that $$\mathbb {E}[\varphi (X_0)^2] 2$$.

Majorization theorems for strongly convex functions

In the article, we present several majorization theorems for strongly convex functions and give their applications in inequality theory. The given results are the improvement and generalization of the earlier results.

An Improved Second-Order Poincaré Inequality for Functionals of Gaussian Fields

We present an improved version of the second-order Gaussian Poincare inequality, first introduced in Chatterjee (Probab Theory Relat Fields 143(1):1–40, 2009) and Nourdin et al. (J Funct Anal 257(2):593–609, 2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative central limit theorems for nonlinear functionals of stationary Gaussian fields related to the Breuer–Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.

Inequalities for the eigenvectors associated to extremal eigenvalues in rank one perturbations of symmetric matrices

This paper considers the eigenvectors involved in rank one perturbations of symmetric matrices. A doubly stochastic matrix based on the inner products between initial and perturbed eigenvectors is introduced to derive several relations for the latter ones. A majorization theorem used together with this doubly stochastic matrix provides the main results of the paper, which deal with the eigenvectors associated to the largest and smallest non-zero eigenvalues. Further developments are also suggested for infinitesimal perturbations using convergent power expansions of both eigenvalues and eigenvectors.

Annihilators and extensions of idempotent-generated ideals

We define a ring R to be right -Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. This class of rings generalizes the class of right p.q.-Baer rings. We also define a ring R to be right I-extending if each ideal generated by an idempotent is essential, as a right R-module, in a right ideal generated by an idempotent. It is shown that the two conditions are equivalent in a semiprime ring. From the definition of a right -Baer ring, we define a generalization of a prime ring which is equivalent to the condition that all nonzero cyclic projective right R-modules are faithful. Such a ring is called right I-prime. For a semiprime ring, we show the existence of a -Baer hull. We also provide some results about the p.q.-Baer hull and when it is equal to the -Baer hull. Polynomial and formal power series rings are studied with respect to the right -Baer condition. In general, a formal power series ring over one indeterminate in which its base ring is right p.q.-Baer ring is not necessarily right p.q.-Baer. However, if the base ring is right -Baer then the formal power series ring over one indeterminate is right -Baer. The last section is devoted to matrix extensions of right -Baer rings. A characterization of when a 2-by-2 generalized upper triangular matrix ring is right -Baer is given. The last major theorem is a decomposition of a -Baer ring R, satisfying a mild finiteness condition, into a direct sum of rings where A is a direct sum of right I-prime rings; and B is a generalized triangular matrix ring with right I-prime rings down the main diagonal. Examples illustrating and delimiting our results are provided.

Oscillatory Breuer–Major theorem with application to the random corrector problem

In this paper, we present an oscillatory version of the celebrated Breuer–Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures.

Inductive groupoids and cross-connections of regular semigroups

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann--Schein--Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

The Majorization Theorems of Single-Cone Trees and Single-Cone Unicyclic Graphs

A single-cone tree (unicyclic graph) is the join of a complete graph $$K_1$$ and a tree (unicyclic graph). Suppose $$\pi =(d_1, d_2, \ldots , d_n)$$ and $$\pi ^{\,\prime }=(d_1^{\,\prime }, d_2^{\,\prime }, \ldots , d_n^{\,\prime })$$ are two non-increasing degree sequences. We say $$\pi $$ is majorizated by $$\pi ^{\,\prime }$$, denoted by $$\pi \lhd \pi ^{\,\prime }$$, if and only if $$\pi \ne \pi ^{\,\prime }$$, $$\sum \nolimits _{i=1}^{n} d_i=\sum \nolimits _{i=1}^{n} d_i^{\,^{\,\prime }}$$, and $$\sum \nolimits _{i=1}^j d_i\le \sum \nolimits _{i=1}^j d_i^{\,^{\,\prime }}$$ for all $$j=1, 2, \ldots , n-1$$. We use $$J_{\pi }$$ to denote the class of single-cone trees (unicyclic graphs) with degree sequence $$\pi $$. Suppose that $$\pi $$ and $$\pi ^{\,\prime }$$ are two different non-increasing degree sequences of single-cone trees (unicyclic graphs). Let $$\rho $$ and $$\rho ^{\,\prime }$$ be the largest spectral radius of the graphs in $$J_{\pi }$$ and $$J_{\pi ^{\,\prime }}$$, respectively, $$\mu $$ and $$\mu ^{\,\prime }$$ be the largest signless Laplacian spectral radius of the graphs in $$J_{\pi }$$ and $$J_{\pi ^{\,\prime }}$$, respectively. In this paper, we prove that if $$\pi \lhd \pi ^{\,\prime }$$, then $$\rho <\rho ^{\,\prime }$$ and $$\mu <\mu ^{\,\prime }$$.

Extremal graphs for vertex-degree-based invariants with given degree sequences

For a symmetric bivariable function f(x,y), let the connectivity function of a connected graph G be Mf(G)=∑uv∈E(G)f(d(u),d(v)), where d(u) is the degree of a vertex u. In this paper, we prove that for an escalating (de-escalating) function f(x,y), there exists a BFS-graph with the maximum (minimum) connectivity function Mf(G) among all graphs with a c-cyclic degree sequence π=(d1,d2,…,dn) and dn=1, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invariants based on degree are included.

Some majorization integral inequalities for functions defined on rectangles

In this paper, we first prove an integral majorization theorem related to integral inequalities for functions defined on rectangles. We then apply the result to establish some new integral inequalities for functions defined on rectangles. The results obtained are generalizations of weighted Favard’s inequality, which also provide a generalization of the results given by Maligranda et al. (J. Math. Anal. Appl. 190:248–262, 1995) in an earlier paper.

Functions generating (m,M,varvec{Psi })-Schur-convex sums

The notion of (m,M,Psi )-Schur-convexity is introduced and functions generating (m,M,Psi )-Schur-convex sums are investigated. An extension of the Hardy–Littlewood–Pólya majorization theorem is obtained. A counterpart of the result of Ng stating that a function generates (m,M,Psi )-Schur-convex sums if and only if it is (m,M,psi )-Wright-convex is proved and a characterization of (m,M,psi )-Wright-convex functions is given.

Fixed points of rational type contractions in G-metric spaces

We establish three major fixed-point theorems for functions satisfying generalized rational type almost contraction conditions. Firstly we consider the case of a single mapping, secondly we look at the case of a triplet of mappings and we conclude by the case of a family of mappings. The theorems we present generalize similar results already obtained by Abbas, Rhoades, Gaba, and others. The operators we consider are all of the weakly Picard type.

On the ordering of distance-based invariants of graphs

Let d(u, v) be the distance between u and v of graph G, and let Wf(G) be the sum of f(d(u, v)) over all unordered pairs {u, v} of vertices of G, where f(x) is a function of x. In some literatures, Wf(G) is also called the Q-index of G. In this paper, some unified properties to Q-indices are given, and the majorization theorem is illustrated to be a good tool to deal with the ordering problem of Q-index among trees with n vertices. With the application of our new results, we determine the four largest and three smallest (resp. four smallest and three largest) Q-indices of trees with n vertices for strictly decreasing (resp. increasing) nonnegative function f(x), and we also identify the twelve largest (resp. eighteen smallest) Harary indices of trees of order n ≥ 22 (resp. n ≥ 38) and the ten smallest hyper-Wiener indices of trees of order n ≥ 18, which improve the corresponding main results of Xu (2012) and Liu and Liu (2010), respectively. Furthermore, we obtain some new relations involving Wiener index, hyper-Wiener index and Harary index, which gives partial answers to some problems raised in Xu (2012).

Weighted majorization inequalities for n-convex functions via extension of Montgomery identity using Green function

New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Čebyšev type inequalities. Mean value theorems are also discussed for functional related to new results.