Rank Theorem Research Articles

Level sets of nonsmooth functions, Part 2: Lipschitz and piecewise-differentiable manifolds

This paper introduces piecewise-differentiable (PCr) manifolds according to a unified general framework that also applies to nonsmooth Lipschitz manifolds and smooth manifolds. We present definitions of nonsmooth manifolds and embedded submanifolds for abstract sets as well as for subsets of Rn, and explore the relationships between them. The PCr and Lipschitz Rank Theorems from Part 1 of this series, in terms of the Clarke Jacobian and B-subdifferential generalized derivative sets, are used to characterize level sets of functions between nonsmooth manifolds as embedded submanifolds. We illustrate how the Level Set Theorems developed in this paper can be applied to functions on Euclidean space, including a piecewise-differentiable process model for distillation columns.

A maximum rank theorem for solutions to the homogenous complex Monge–Ampère equation in a $$\mathbb {C}$$-convex ring

A maximum rank theorem for solutions to the homogenous complex Monge–Ampère equation in a $$\mathbb {C}$$-convex ring

Level sets of nonsmooth functions, Part 1: Lipschitz and piecewise-differentiable Rank Theorems

We present a piecewise-differentiable (PCr) Rank Theorem and extend a previously stated Lipschitz Rank Theorem, with the goal of characterizing the level sets of nonsmooth functions f:Rn→Rm. When the appropriate conditions are satisfied by the generalized derivatives of f, the Rank Theorems allow us to express a given level set f−1(c)⊂Rn locally as the graph of a nonsmooth function, within a homeomorphic transformation of the same class as f. We define PCr and Lipschitz submersions, immersions and maps of constant rank in terms of the most general conditions under which the corresponding Rank Theorems are applicable. Moreover, we develop sufficient conditions that are more easily verifiable for practical applications and relate them to existing full-rank conditions from the literature.

Constraint Qualification with Schauder Basis for Infinite Programming Problems

We consider infinite programming problems with constraint sets defined by systems of infinite number of inequalities and equations given by continuously differentiable functions defined on Banach spaces. In the approach proposed here we represent these systems with the help of coefficients in a given Schauder basis. We prove Abadie constraint qualification under the new infinite-dimensional Relaxed Constant Rank Constraint Qualification Plus and we discuss the existence of Lagrange multipliers via Hurwicz set. The main tools are: Rank Theorem and Lyusternik–Graves theorem.

Borel’s rank theorem for Artin 𝐿-functions

Borel’s rank theorem identifies the ranks of algebraic K K -groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin L L -functions by considering equivariant algebraic K K -groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic K K -theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in L L -functions.

Autoencoders for a manifold learning problem with a jacobian rank constraint

We formulate the manifold learning problem as the problem of finding an operator that maps any point to a close neighbor that lies on a “hidden” k-dimensional manifold. We call this operator the correcting function. Under this formulation, autoencoders can be viewed as a tool to approximate the correcting function. Given an autoencoder whose Jacobian has rank k, we deduce from the classical Constant Rank Theorem that its range has a structure of a k-dimensional manifold. A k-dimensionality of the range can be forced by the architecture of an autoencoder (by fixing the dimension of the code space), or alternatively, by an additional constraint that the rank of the autoencoder mapping is not greater than k. This constraint is included in the objective function as a new term, namely a squared Ky-Fan k-antinorm of the Jacobian function. We claim that this constraint is a factor that effectively reduces the dimension of the range of an autoencoder, additionally to the reduction defined by the architecture. We also add a new curvature term into the objective. To conclude, we experimentally compare our approach with the CAE+H method on synthetic and real-world datasets.

About systems of vectors and subspaces in finite dimensional space recovering vectorsignal

The subject of this paper are the systems of vectors and subspaces in finite dimensional spaces admitting the recovery of an unknown vector-signal by modules of measurements. We analyze the relationship between the properties of recovery by modules of measurements and recovery by norms of projections and the properties of alternative completeness in Euclidean and unitary spaces. The theorem on ranks of one linear operator is considered, the result of which in some cases can be regarded as another criterion for the possibility to restore a vector-signal. As a result of this work, the equivalence of the alternative completeness property and the statement of the rank theorem for Euclidean space is proved. It is shown that the rank theorem in the real case can be extended to the systems of subspaces.
 The questions about the minimum number of vectors admissible for reconstruction by modules of measurements are considered. The results available at the moment are presented, which are summarized in the form of a table for spaces of dimension less than 10. Also the known results to the question of the minimum number of subspaces admitting reconstruction by the norms of projections are briefly given.

Constant rank theorems for curvature problems via a viscosity approach

An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map Brendle et al. (Acta Math 219:1–16, 2017) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument.

Weak Harnack inequalities for eigenvalues and a constant rank theorem for level sets

We consider convex level sets of solutions of a class of quasilinear elliptic equations. We prove a weak Harnack inequality for the eigenvalues of the Weingarten tensor of level sets and then obtain a quantitative version of constant rank theorem.

The Lp Minkowski type problem for a class of mixed Hessian quotient equations

Existence and uniqueness of the admissible solutions to the Lp Minkowski type problem for mixed Hessian operators are proved. By using the new phenomenon of “inverse convexity” of these operators, we show the existence of the geometric convex solutions by the Full Rank Theorem.

A new geometric approach to the calculation of transmission zeros in the signal processing theory

In the paper, a new method concerning the estimation of transmission zeros is introduced. Following the notion related to the Smith–McMillan-like technique allowing us to determine the crucial zeros, a new solution covering the geometric approach is effectively presented and confirmed in the manuscript. Indeed, the existing non-intuitive method providing non-unique computational effort even for the well-conditioned environmental matrices can successfully be replaced without loss of generality. Therefore, in order to overcome the indicated difficulties, the rank-nullity theorem is adapted to determine the transmissions zeros, for which the matrix depending on variable coefficients loses its rank. The benefits of the new scheme in modeling of non-square MIMO signal processing systems are clearly visible. It should be emphasized that currently only the Smith–McMillan factorization allows us to calculate the transmission zeros. However, in the proposed method, it can be done faster, more intuitively and effectively.

Constant rank theorems for Li-Yau-Hamilton type matrices of heat equations

In this paper, we prove constant rank theorems for Li-Yau-Hamilton type matrices of heat equations on Riemannian manifolds as well as on Kähler manifolds under some curvature conditions.

Secure and Efficient Distributed Relay-Based Rekeying Algorithm for Group Communication in Mobile Multihop Relay Network

In mobile multihop relay (MMR) networks, Relay multicast rekeying algorithm (RMRA) is meant to ensure secure multicast communication and selective updating of keys in MMR networks. However, in RMRA, the rekeying is carried out after a specific interval of time, which cannot ensure the security for multicast communication on joining the member. Secondly, the rekeying scheme generates a huge communication overhead on the serving multihop relay base station (MR-BS) on frequent joining of members. Lastly, there is nothing about when a member left the group communication. Thus, the rekeying scheme of RMRA fails to provide forward and backward secrecy and also is not scalable. To solve this problem, an improved rekeying scheme based on broadcasting a new seed value on joining and leaving of a member for updating the ongoing key management is proposed. The proposed scheme solves the issue of forward and backward secrecy and the scalability in a very simplified way. The forward and backward secrecy of the proposed scheme has been extensively validated by formal method using rank theorem. Furthermore, mathematical derivation showed that the proposed scheme out-performed the RMRA in terms of communication cost and complexity.

A quantitative constant rank theorem for quasiconcave solutions to fully nonlinear elliptic equations

We investigate the fully nonlinear elliptic equations F(D2u,Du,u,x)=0, which satisfy the structural condition previously posed by Bianchini-Longinetti-Salani in 2009. By establishing a novel differential inequality, we prove a weak Harnack inequality for the principal curvatures of the level surfaces of the solutions. This result is indeed a quantitative version of the constant rank theorem showed by Guan-Xu in 2013.

Some applications of transversality for infinite dimensional manifolds

We present some transversality results for a category of Frechet manifolds, the so-called MCk - Frechet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem.

A proof of A. Gabrielov’s rank theorem

This article contains a complete proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung theorem. We include, furthermore, new extensions of the rank theorem (concerning the Zariski main theorem and elimination theory) to commutative algebra.

Local forms of morphisms of colored supermanifolds

In [13], [17] and [29], we introduced the category of colored supermanifolds (Z2n-supermanifolds or just Z2n-manifolds (Z2n=Z2×…×Z2 (n times))), explicitly described the corresponding Z2n-Berezinian and gave first insights into Z2n-integration theory. The present paper contains a detailed account of parts of the Z2n-differential calculus and of the Z2n-variants of the trilogy of local theorems, which consists of the inverse function theorem, the implicit function theorem and the constant rank theorem.

Weak Harnack inequalities for eigenvalues and constant rank theorems

We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian and Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.

Interplay of non-convex quadratically constrained problems with adjustable robust optimization

In this paper we explore convex reformulation strategies for non-convex quadratically constrained optimization problems (QCQPs). First we investigate such reformulations using Pataki’s rank theorem iteratively. We show that the result can be used in conjunction with conic optimization duality in order to obtain a geometric condition for the S-procedure to be exact. Based upon known results on the S-procedure, this approach allows for some insight into the geometry of the joint numerical range of the quadratic forms. Then we investigate a reformulation strategy introduced in recent literature for bilinear optimization problems which is based on adjustable robust optimization theory. We show that, via a similar strategy, one can leverage exact reformulation results of QCQPs in order to derive lower bounds for more complicated quadratic optimization problems. Finally, we investigate the use of reformulation strategies in order to derive characterizations of set-copositive matrix cones. Empirical evidence based upon first numerical experiments shows encouraging results.

Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p < 1

Lp-Christoffel-Minkowski problem arises naturally in the Lp-Brunn-Minkowski theory. It connects both curvature measures and area measures of convex bodies and is a fundamental problem in convex geometric analysis. Since the lack of Firey's extension of Brunn-Minkowski inequality and constant rank theorem for p<1, the existence and uniqueness of Lp-Brunn-Minkowski problem are difficult problems. In this paper, we prove a uniqueness theorem for solutions to Lp-Christoffel-Minkowski problem with p<1 and constant prescribed data. Our proof is motivated by the idea of Brendle-Choi-Daskaspoulos's work on asymptotic behavior of flows by powers of the Gaussian curvature. One of the highlights of our arguments is that we introduce a new auxiliary function Z which is the key to our proof.