Ky Fan's Theorem Research Articles

A minimax approach to duality for linear distributional sensitivity testing

We consider the dual formulation of the problem of finding the maximum of E ν [ f ( X ) ] , where ν is allowed to vary over all the probability measures on a Polish space X for which d c ( μ , ν ) ≤ r , with d c an optimal transport distance, f a real-valued function on X satisfying some regularity, μ a ‘baseline’ measure and r ≥ 0 . Whereas some derivations of the dual rely on Fenchel duality, applied on a vector space of functions in duality with a vector space of measures, we impose compactness on X to allow the use of the minimax theorem of Ky Fan, which does not require vector space structure.

Variational Quantum Singular Value Decomposition

Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantum neural networks (or parameterized quantum circuits) could be trained to learn the singular vectors and output the corresponding singular values. Furthermore, we conduct numerical simulations of VQSVD for random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in recommendation systems and polar decomposition. Our work explores new avenues for quantum information processing beyond the conventional protocols that only works for Hermitian data, and reveals the capability of matrix decomposition on near-term quantum devices.

Some improvements on the Ky Fan theorem for tensors

The improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.

Optimal control of systems governed by fractional Laplacian in the minimax framework

ABSTRACT We consider some optimal control problem governed by a class of differential equations with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes are proved. The proof of the main result relies on the variational structure of the problem. To show that differential equations with the Dirichlet fractional Laplacian have a weak solution we employ the Ky Fan Theorem.

Some new inequalities for the minimum H-eigenvalue of nonsingular M-tensors

This paper is devoted to deriving some new upper and lower bounds for the minimum H-eigenvalue of irreducible nonsingular M-tensors. We demonstrate that the proposed bounds improve some existing ones and numerical examples are given to illustrate this fact. Additionally, we extend these bounds to general nonsingular M-tensors. The other main results in this paper is to provide sharper Ky Fan theorem and comparison theorem for nonsingular M-tensors which are better than the corresponding ones obtained by He and Huang (2014) [12].

Several New Estimates of the Minimum H-eigenvalue for Nonsingular $$\mathcal {M}$$ M -tensors

In this paper, several new estimates of the minimum H-eigenvalue for weakly irreducible nonsingular $$\mathcal {M}$$ -tensors, including the new Brauer-type estimates and the new S-type estimates, are derived. It is proved that the new estimates are tighter than some existing ones and numerical examples are given to verify this fact. The other main result of this paper is to provide a sharper Ky Fan-type theorem which is better than the original Ky Fan theorem for the nonsingular $$\mathcal {M}$$ -tensors.

On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi $-extreme points of a $\Phi $-convex compact metrizable space are replaced by the $\Phi $-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi$-extreme points of a $\Phi$-convex compact metrizable space are replaced by the $\Phi$-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

Some of Sion’s heirs and relatives

If one adds one extra assumption to the classical Knaster– Kuratowski–Mazurkiewicz (KKM) theorem, namely that the sets F i are convex, one gets the “Elementary” KKM theorem; the name is due to A. Granas and M. Lassonde (1995) who gave a simple proof of the Elementary KKM theorem and showed that despite being “elementary,” it is powerful and versatile. It is shown here that this Elementary KKM theorem is equivalent to Klee’s theorem, the Elementary Alexandroff– Pasynkov theorem, the Elementary Ky Fan theorem and the Sion–von Neumann minimax theorem, as well as a few other classical results with an extra convexity assumption; hence the adjective “elementary.” The Sion–von Neumann minimax theorem itself can be proved by simple topological arguments using connectedness instead of convexity. This work answers a question of Professor Granas regarding the logical relationship between the Elementary KKM theorem and the Sion–von Neumann minimax theorem.

Ky Fan theorem applied to Randić energy

Let G be a simple undirected graph of order n with vertex set V(G)={v1,v2,…,vn}. Let di be the degree of the vertex vi. The Randić matrix R=(ri,j) of G is the square matrix of order n whose (i,j)-entry is equal to 1/didj if the vertices vi and vj are adjacent, and zero otherwise. The Randić energy is the sum of the absolute values of the eigenvalues of R. Let X, Y, and Z be matrices, such that X+Y=Z. Ky Fan established an inequality between the sum of singular values of X, Y, and Z. We apply this inequality to obtain bounds on Randić energy. We also present results pertaining to the energy of a symmetric partitioned matrix, as well as an application to the coalescence of graphs.

Inequalities for M-tensors

In this paper, we establish some important properties of M-tensors. We derive upper and lower bounds for the minimum eigenvalue of M-tensors, bounds for eigenvalues of M-tensors except the minimum eigenvalue are also presented; finally, we give the Ky Fan theorem for M-tensors. MSC:15A18, 15A69, 65F15, 65F10.

Colorful Subhypergraphs in Kneser Hypergraphs

Using a $\mathbb{Z}_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a uniform hypergraph).

An extension of the Fitzpatrick theory

In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotone operator $\alpha: V\to {\mathcal P}(V')$ ($V$ being a real Banach space) there exists a lower semicontinuous, convex representative function $f_\alpha: V \times V'\to R\cup \{+\infty\}$ such that \begin{eqnarray} f_\alpha(v,v') \ge \langle v',v\rangle \quad\;\forall (v,v'), \qquad\quad f_\alpha(v,v') = \langle v',v\rangle \;\;\Leftrightarrow\;\;\; v'\in \alpha(v). \end{eqnarray} Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.

A class of quasilinear elliptic hemivariational inequality problems on unbounded domains

In this paper, we are concerned with the existence of solutions of a class of quasilinear elliptic hemivariational inequalities on unbounded domains. This kind of problems is more delicate due to the lack of compact embedding of the Sobolev spaces. By using the Clarke generalized directional derivatives for locally Lipschitz functions and some nonlinear function analysis techniques, such as the Ky Fan theorem for multivalued mappings, the theorem of finite intersection property, etc, we obtain the existence of solutions of the quasilinear elliptic hemivariational inequalities. Unlike those methods used in the references mentioned in this paper, we treat the case of unbounded domain by using the approximation of bounded domains.

From weierstrass to ky fan theorems and existence results on Optimization and Equilibrium Problems

In this work, we study coerciveness notions and their implications to existence conditions. We start with the presentation of classical ideas of coerciveness in the framework of Optimization Theory, and, then, using a classical technical result, introduced by Ky Fan in 1961, we extend these ideas first to Optimization Problems and then to Equilibrium Problems. We point out the importance of related conditions to the introduced coerciveness notion in order to obtain existence results for Equilibrium Problems, without using monotonicity or generalized monotonicity assumptions.

Borel-Carathéodory and Fan Type Inequalities for Hilbert Space Bicontractions

This paper deals with some operator versions of the classical Borel–Caratheodory and Ky Fan theorems for analytic functions of complex variable. Our results refer to the functional calculus with analytic functions on the unit bidisk induced by strict bicontractions on a complex Hilbert space.

Colourful Theorems and Indices of Homomorphism Complexes

We extend the colourful complete bipartite subgraph theorems of [G. Simonyi, G. Tardos, Local chromatic number, Ky Fan's theorem, and circular colorings, Combinatorica 26 (2006), 587--626] and [G. Simonyi, G. Tardos, Colorful subgraphs of Kneser-like graphs, European J. Combin. 28 (2007), 2188--2200] to more general topological settings. We give examples showing that the hypotheses are indeed more general. We use our results to show that the topological bounds on chromatic numbers of digraphs with tree duality are at most one better than the clique number. We investigate combinatorial and complexity-theoretic aspects of relevant order-theoretic maps.

Evolution of the 1984 KKM theorem of Ky Fan

Recently we established the Knaster, Kuratowski, and Mazurkiewicz (KKM) theory on abstract convex spaces. In our research, we noticed that there are quite a few generalizations and applications of the 1984 KKM theorem of Ky Fan compared with his celebrated 1961 KKM lemma. In a certain sense, the relationship between the 1984 theorem and hundreds of known generalizations of the original KKM theorem has not been recognized for a long period. There would be some reasons to explain this fact. Instead, in this paper, we give several generalizations of the 1984 theorem and some known applications in order to reveal the close relationship among them.MSC: 47H04, 47H10, 47J20, 47N10, 49J53, 52A99, 54C60, 54H25, 58E35, 90C47, 91A13, 91B50.

A saddle-point theorem for strongly and weakly convex functions

We prove a theorem on the existence, uniqueness, and continuous dependence on parameters for a saddle point in a type of minimax problem that arises, for example, in differential game theory. Our theorem on the existence of a saddle point does not follow from the well-known theorems of von Neumann, Ky Fan, Sion and others since the intersection of sublevel sets of the function considered may be disconnected and non-empty. The hypotheses of our theorem are stated in terms of the strong and weak convexity of functions defined on a Banach space. We study properties of strongly and weakly convex functions related to the operations of minimization and maximization. We obtain unimprovable estimates of convexity parameters for the infimal convolution (episum) and epidifference of functions. This results in the construction of a calculus of convexity parameters of functions with respect to epioperations. We give typical examples and show that the hypotheses of our theorems are essential.

Further Results for Perron–Frobenius Theorem for Nonnegative Tensors

For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.