Min-max Property Research Articles

A few characterizations of topological spaces with no infinite discrete subspace

We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible closed subsets. The full result extends well-known characterizations of posets with no infinite antichain. One of them is that FAC spaces are, equivalently, topological spaces in which every closed set contains a dense Noetherian subspace, or spaces in which every Hausdorff subspace is finite, or in which no subspace has any infinite relatively Hausdorff subset. The latter comes with a nice min-max property, extending an observation of Erdős and Tarski in the case of posets: on spaces with no infinite relatively Hausdorff subset, the cardinalities of relatively Hausdorff subsets are bounded, and the least upper bound is also the least cardinality of a family of closed irreducible subsets that cover the space.

Adversarial Learning With Cost-Sensitive Classes.

It is necessary to improve the performance of some special classes or to particularly protect them from attacks in adversarial learning. This article proposes a framework combining cost-sensitive classification and adversarial learning together to train a model that can distinguish between the protected and unprotected classes, such that the protected classes are less vulnerable to adversarial examples. We find in this framework an interesting phenomenon during the training of deep neural networks, called the Min-Max property, that is, the absolute values of most parameters in the convolutional layer approach 0 while the absolute values of a few parameters are significantly larger, becoming bigger. Based on this Min-Max property which is formulated and analyzed in a view of random distribution, we further build a new defense model against adversarial examples for adversarial robustness improvement. An advantage of the built model is that it performs better than the standard one and can combine with adversarial training to achieve improved performance. It is experimentally confirmed that, regarding the average accuracy of all classes, our model is almost as same as the existing models when an attack does not occur and is better than the existing models when an attack occurs. Specifically, regarding the accuracy of protected classes, the proposed model is much better than the existing models when an attack occurs.

The weak min-max property in Banach spaces

In this paper, we investigate the relationship between the weak min-max property and the diameter uniformity of domains in Banach spaces with dimension at least 2. As an application, we show that diameter uniform domains are invariant under relatively quasimöbius mappings.

A study on the uncertainty of convolutional layers in deep neural networks

This paper shows a Min–Max property existing in the connection weights of the convolutional layers in a neural network structure, i.e., the LeNet. Specifically, the Min–Max property means that, during the back propagation-based training for LeNet, the weights of the convolutional layers will become far away from their centers of intervals, i.e., decreasing to their minimum or increasing to their maximum. From the perspective of uncertainty, we demonstrate that the Min–Max property corresponds to minimizing the fuzziness of the model parameters through a simplified formulation of convolution. It is experimentally confirmed that the model with the Min–Max property has a stronger adversarial robustness, thus this property can be incorporated into the design of loss function. This paper points out a changing tendency of uncertainty in the convolutional layers of LeNet structure, and gives some insights to the interpretability of convolution.

Quasim

We study quasimöbius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as (diameter) uniformity, $\delta $-uniformity and the min-max property. Then we show

Min-max property in metric spaces with convex structure

In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings.

Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property

In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace. This effectively eliminates the search subspace growth and therewith the super-linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. The new restart technique is integrated into nonlinear iterative projection methods like the Nonlinear Arnoldi and Jacobi-Davidson methods. The efficiency of our new restart framework is demonstrated on a range of nonlinear eigenvalue problems: quadratic, rational and exponential including an industrial real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire. We also present an extension of the method to problems without minmax property but with eigenvalues which have a dominant either real or imaginary part and test it on two quadratic eigenvalue problems.

Analysis And Efficient Solution Of Stationary Schrödinger Equation Governing Electronic States Of Quantum Dots And Rings In Magnetic Field

AbstractIn this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number -L•i. We show, that each of those pair/quadruple of nonlinear problems allows for the min-max characterization of its eigenvalues under certain conditions, which are satisfied for our examples and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40% of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately.

Some properties of fractal interpolation functions on Sierpinski gasket

In this paper, we discuss some basic properties of uniform fractal interpolation functions (FIFs), which is a special class of FIFs, on Sierpinski gasket. We firstly study the min–max property of uniform FIFs. Then we present a necessary and sufficient condition such that uniform FIFs have finite energy. Normal derivative and Laplacian of uniform FIFs are also discussed.

A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.

LATTICED SIMULATION RELATIONS AND GAMES

Multi-valued Kripke structures are Kripke structures in which the atomic propositions and the transitions are not Boolean and can take values from some set. In particular, latticed Kripke structures, in which the elements in the set are partially ordered, are useful in abstraction, query checking, and reasoning about multiple view-points. The challenges that formal methods involve in the Boolean setting are carried over, and in fact increase, in the presence of multi-valued systems and logics. We lift to the latticed setting two basic notions that have been proven useful in the Boolean setting. We first define latticed simulation between latticed Kripke structures. The relation maps two structures M1 and M2 to a lattice element that essentially denotes the truth value of the statement "every behavior of M1 is also a behavior of M2". We show that latticed-simulation is logically characterized by the universal fragment of latticed µ-calculus, and can be calculated in polynomial time. We then proceed to defining latticed two-player games. Such games are played along graphs in which each transition have a value in the lattice. The value of the game essentially denotes the truth value of the statement "the ∨-player can force the game to computations that satisfy the winning condition". An earlier definition of such games involved a zig-zagged traversal of paths generated during the game. Our definition involves a forward traversal of the paths, and it leads to better understanding of multi-valued games. In particular, we prove a min-max property for such games, and relate latticed simulation with latticed games.

John disks, the Apollonian metric, and min-max properties

The main results of this paper are characterizations of John disks-the simply connected proper subdomains of the complex plane that satisfy a twisted double cone connectivity property. One of the characterizations of John disks is an analog of a result due to Gehring and Hag who found such a characterization for quasidisks. In both situations the geometric condition is an estimate for the domain’s hyperbolic metric in terms of its Apollonian metric. The other characterization is in terms of an arc min-max property.

Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting

In this work we derive a pair of nonlinear eigenvalue problems corresponding to the one-band effective Hamiltonian accounting for the spin-orbit interaction governing the electronic states of a quantum dot. We show that the pair of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions which are satisfied for our example of a cylindrical quantum dot and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise an efficient iterative projection method simultaneously handling the pair of nonlinear problems and thereby saving about 25% of the computation time as compared to the Nonlinear Arnoldi method applied to each of the problems separately.

A variational characterization of canonical angles between subspaces

Canonical angles between subspaces of a unitary space are characterized by a min-max property which involves inner products.

Partitioning chordal graphs into independent sets and cliques

We consider the following generalization of split graphs: A graph is said to be a ( k,ℓ)-graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k=ℓ=1.) Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by ( k,ℓ)-graphs in general. (For instance, being a ( k,0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal ( k,ℓ)-graphs. We also give an O( n( m+ n)) recognition algorithm for chordal ( k,ℓ)-graphs. When k=1, our algorithm runs in time O( m+ n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem—given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min–max property of chordal graphs: for each integer r⩾1, the maximum number of independent K r 's (i.e., of vertex disjoint subgraphs of G, each isomorphic to K r , with no edges joining two of the subgraphs) equals the minimum number of cliques of G that meet all the K r 's of G.

Orbitally stable standing waves for a system of coupled nonlinear Schrödinger equations

In this paper we consider the existence of standing waves for a system of coupled nonlinear 1-dimensional Schrodinger equations arising in several models of modern physics. By using P. L. Lions’ Concentration Compactness Principle, we prove the existence and the orbital stability of standing waves of minimal energy. In particular, under homogeneous nonlinearities we stabilish a min-max property which enables us to prove that the standing waves of minimal energy are orbitally stable ground states.

Variation on a theorem of König

A min-max property of bipartite graphs is stated; it is a variation on the theorem of König ‘maximum matching = minimum covering’; one shows that a certain inequality holds for any graph and the equality for bipartite graphs is derived from a simple network flow model.