Major Theorems Research Articles

Minimal cover groups

Let F be a set of finite groups. A finite group G is called an F-cover if every group in F is isomorphic to a subgroup of G. An F-cover is called minimal if no proper subgroup of G is an F-cover, and minimum if its order is smallest among all F-covers. We prove several results about minimal and minimum F-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {Zq,Zr} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

A total variation version of Breuer–Major Central Limit Theorem under D1,2 assumption

A total variation version of Breuer–Major Central Limit Theorem under D1,2 assumption

A formally certified end-to-end implementation of Shor’s factorization algorithm

Quantum computing technology may soon deliver revolutionary improvements in algorithmic performance, but it is useful only if computed answers are correct. While hardware-level decoherence errors have garnered significant attention, a less recognized obstacle to correctness is that of human programming errors-"bugs." Techniques familiar to most programmers from the classical domain for avoiding, discovering, and diagnosing bugs do not easily transfer, at scale, to the quantum domain because of its unique characteristics. To address this problem, we have been working to adapt formal methods to quantum programming. With such methods, a programmer writes a mathematical specification alongside the program and semiautomatically proves the program correct with respect to it. The proof's validity is automatically confirmed-certified-by a "proof assistant." Formal methods have successfully yielded high-assurance classical software artifacts, and the underlying technology has produced certified proofs of major mathematical theorems. As a demonstration of the feasibility of applying formal methods to quantum programming, we present a formally certified end-to-end implementation of Shor's prime factorization algorithm, developed as part of a framework for applying the certified approach to general applications. By leveraging our framework, one can significantly reduce the effects of human errors and obtain a high-assurance implementation of large-scale quantum applications in a principled way.

A non-geodesic analogue of Reshetnyak’s majorization theorem

Abstract For any real number κ \kappa and any integer n ≥ 4 n\ge 4 , the Cycl n ( κ ) {{\rm{Cycl}}}_{n}\left(\kappa ) condition introduced by Gromov (CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), (Geom. i Topol. 7), 100–140, 299–300) is a necessary condition for a metric space to admit an isometric embedding into a CAT ( κ ) {\rm{CAT}}\left(\kappa ) space. For geodesic metric spaces, satisfying the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition is equivalent to being CAT ( κ ) {\rm{CAT}}\left(\kappa ) . In this article, we prove an analogue of Reshetnyak’s majorization theorem for (possibly non-geodesic) metric spaces that satisfy the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition. It follows from our result that for general metric spaces, the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition implies the Cycl n ( κ ) {{\rm{Cycl}}}_{n}\left(\kappa ) conditions for all integers n ≥ 5 n\ge 5 .

Limit theorems for the realised semicovariances of multivariate Brownian semistationary processes

In this article, we will introduce the realised semicovariance for Brownian semistationary (BSS) processes, which is obtained from the decomposition of the realised covariance matrix into components based on the signs of the returns and study its in-fill asymptotic properties. More precisely, weak convergence in the space of càdlàg functions endowed with the Skorohod topology for the realised semicovariance of a general Gaussian process with stationary increments is proved first. The proof is based on the Breuer–Major theorem and on a moment bound for sums of products of non-linearly transformed Gaussian vectors. Furthermore, we establish a corresponding stable convergence. Finally, a central limit theorem for the realised semicovariance of multivariate BSS processes is established. These results extend the limit theorems for the realised covariation to a result for non-linear functionals.

Orthogonal Generalized Symmetric Higher Bi-Left (Resp. Right) Centralizers on Semiprime Γ- Ring

Specialists in the field of algebra have highlighted many results related to the Γ-ring, and in my research, I will present some definitions and concepts and a major theorem parallel to the results of other researchers.

A fresh look at distribution theory for quadratic forms in jointly normally distributed random variables

Several major theorems pertaining to necessary and sufficient conditions under which quadratic forms in normal random vectors are chi-square distributed (both central and non central) are revisited and illuminated. Some of these results date back to the early 1950’s and are scattered among journals that are not all easily accessible. Moreover, some of these presentations are very terse, reflecting the tendency to reduce the amount of technical typesetting. Most of these results are not available in their entirety in current graduate level statistics texts. Accordingly, the purpose here is to collect these in one place and to offer complete detailed and clear proofs with some new perspectives. The general case in which the variance matrix of the normal random vector can be positive semidefinite is emphasized, since those results will subsume the positive definite case with appropriate reductions of conditions.

On parameter estimation of fractional Ornstein–Uhlenbeck process

Abstract We consider a problem of parameter estimation for the fractional Ornstein–Uhlenbeck model given by the stochastic differential equation d ⁢ X t = - θ ⁢ X t ⁢ d ⁢ t + d ⁢ B t H {dX_{t}=-\theta X_{t}dt+dB_{t}^{H}} , t ≥ 0 {t\geq 0} , where θ > 0 {\theta>0} is an unknown parameter to be estimated and B H {B^{H}} is a fractional Brownian motion with Hurst parameter H ∈ ( 0 , 1 ) {H\in(0,1)} . We provide an estimator for θ, and then we study its strong consistency and asymptotic normality. The main tool in our proofs is the paper [I. Nourdin, D. Nualart and G. Peccati, The Breuer–Major theorem in total variation: Improved rates under minimal regularity, Stochastic Process. Appl. 131 2021, 1–20].

Epistemic phase transitions in mathematical proofs

Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows exponentially as the argument expands. When a mathematician encounters a proof, how does she come to believe it? Here we show that, under a cognitively-plausible belief formation mechanism combining deductive and abductive reasoning, belief in mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with five hand-constructed cases including Euclid, Apollonius, Hernstein's Topics in Algebra, and Andrew Wiles's proof of Fermat's Last Theorem. Our results bear both on recent work in the history and philosophy of mathematics on how we understand proofs, and on a question, basic to cognitive science, of how we justify complex beliefs.

Comparison Geometry for an Extension of Ricci Tensor

For a complete Riemannian manifold M with a (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator \({L_A}(u): = div(A\nabla u)\) and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems such as mean curvature comparison theorem, Bishop–Gromov volume comparison theorem, Cheeger–Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applied to some Riemannian hypersurfaces of Riemannian or Lorentzian space forms.

The role of the algebraic structure in Wold-type decomposition

Abstract In recent works [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗ \ast -rings, Linear Algebra Appl. 539 2018, 117–133] and [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 2019, 102–118], the algebraic analogues of the three major decomposition theorems of Wold, Nagy–Foiaş–Langer and Halmos–Wallen were established in the larger category of Baer * {*} -rings. The results have their versions for commuting pairs in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. In this work, ignoring topological structures, we give an algebraic approach to obtain them in Baer * {*} -rings.

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

AbstractWe establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

Total variation estimates in the Breuer–Major theorem

This paper provides estimates for the convergence rate of the total variation distance in the framework of the Breuer–Major theorem, assuming some smoothness properties of the underlying function. The results are proved by applying new bounds for the total variation distance between a random variable expressed as a divergence and a standard Gaussian random variable, which are derived by a combination of techniques of Malliavin calculus and Stein’s method. The representation of a functional of a Gaussian sequence as a divergence is established by introducing a shift operator on the expansion in Hermite polynomials. Some applications to the asymptotic behavior of power variations of the fractional Brownian motions and to the estimation of the Hurst parameter using power variations are presented.

On Certain Generalizations of $$\mathcal {S}^*(\psi )$$

In this paper, we consider functions analytic in the unit disk that are subordinate to functions of the same type that are defined by certain differential subordinations. We prove several sharp majorization theorems and a product theorem.

How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?

The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer–Major theorem, asymptotic behavior of the third and fourth cumulants of fBm and the optimal fourth moment theorem.

Majorization and Minimal Energy on Spheres

In the present paper, we consider the majorization theorem (also known as Karamata's inequality) and the respective minima of the majorization (the so-called $M$-sets) for $f$-energy potentials of ...

The Breuer–Major theorem in total variation: Improved rates under minimal regularity

In this paper we prove an estimate for the total variation distance, in the framework of the Breuer–Major theorem, using the Malliavin–Stein method, assuming the underlying function g to be once weakly differentiable with g and g′ having finite moments of order four with respect to the standard Gaussian density. This result is proved by a combination of Gebelein’s inequality and some novel estimates involving Malliavin operators.

Unified extremal results for [formula omitted]-apex unicyclic graphs (trees)

A k-cone c-cyclic graph is the join of the complete graph Kk and a c-cyclic graph (if k=0, we get the usual connected graph). A k-apex tree (resp., k-apex unicyclic graph) is defined as a connected graph G with a k-subset Vk⊆V(G) such that G−Vk is a tree (resp., unicylic graph), but G−X is not a tree (resp., unicylic graph) for any X⊆V(G) with |X|<k. In this paper, we extend those extremal results and majorization theorems concerning connected graphs of Liu et al. (2019) to k-cone c-cyclic graphs. We also use a unified method to characterize the extremal maximum and minimum results of many topological indices in the class of k-apex trees and k-apex unicyclic graphs, respectively. The later results extend the main results of Javaid et al. (2019); Liu et al. (2020) and partially answer the open problem of Javaid et al. (2019). Except for the new majorization theorem, some new techniques are also established to deal with the minimum extremal results of this paper.

Difference equations related to majorization theorems via Montgomery identity and Green\u2019s functions with application to the Shannon entropy

In this paper we give generalized results of a majorization inequality by using extension of the Montgomery identity and newly defined Green’s functions (Mehmood et al. in J. Inequal. Appl. 2017(1):108, 2017). We obtain a generalized majorization theorem for the class of n-convex functions. We use Csiszár f-divergence and generalized majorization-type inequalities to obtain new generalized results. We further discuss our obtained generalized results in terms of the Shannon entropy and the Kullback–Leibler distance.

Majorization inequalities via Green functions and Fink\u2019s identity with applications to Shannon entropy

This paper is devoted to obtain generalized results related to majorization-type inequalities by using well-known Fink’s identity and new types of Green functions, introduced by Mehmood et al. (J. Inequal. Appl. 2017:108, 2017). We give a generalized majorization theorem for the class of n-convex functions. We utilize the Csiszár f-divergence and generalized majorization-type inequalities in providing the corresponding generalizations. As an application, we present the obtained results in terms of Shannon entropy and Kullback–Leibler distance.