Whitney's Theorem Research Articles

Unstability problem of real analytic maps

AbstractAs well known, the stability of proper maps is characterized by the infinitesimal stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal stability does not imply stability; for instance, a Whitney umbrella is not stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.

Transformations establishing equivalence across neural networks: When have two networks learned the same task?

Transformations are a key tool in the qualitative study of dynamical systems: transformations to a normal form, for example, underpin the study of instabilities and bifurcations. In this work, we test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using diffusion maps with a Mahalanobis-like metric. If the construction succeeds, the two networks can be thought of as belonging to the same equivalence class. We first discuss transformation functions between only the outputs of the two networks; we then also consider transformations that take into account outputs (activations) of a number of internal neurons from each network. Whitney's theorem dictates the number of (generic) measurements from one of the networks required to reconstruct each and every feature of the second network. The construction of the transformation function relies on a consistent, intrinsic representation of the network input space. We illustrate our algorithm by matching neural network pairs trained to learn (a) observations of scalar functions, (b) observations of two-dimensional vector fields, and (c) representations of images of a moving three-dimensional object (a rotating horse). We also demonstrate reconstruction of a network's input (and output) from minimal partial observations of intermediate neuron activations. The construction of equivalences across different network instantiations clearly relates to transfer learning and will also be valuable in establishing equivalence between different machine learning-based tools.

A geometric representative for the fundamental class in KK-duality of Smale spaces

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module.

The list-coloring function of signed graphs

It is known that, for any k-list assignment L of an m-edge graph G, the number of L-list colorings of G is at least the number of the proper k-colorings of G when k>(m−1)/ln⁡(1+2). In this paper, we extend the Whitney's broken cycle theorem to L-colorings of signed graphs, by which we show that if k>(m3)+(m4)+m−1 then, for any k-assignment L, the number of L-colorings of a signed graph Σ with m edges is at least the number of the proper k-colorings of Σ. Further, if L is 0-free (resp., 0-included) and k is even (resp., odd), then the lower bound (m3)+(m4)+m−1 for k can be improved to (m−1)/ln⁡(1+2).

The characteristic polynomials of semigeneric threshold arrangements

<abstract><p>The semigeneric threshold arrangement is the hyperplane arrangement defined by $ x_i+x_j = a_{i} $, where $ 1 \leq i < j \leq n $ and $ a_i\text{'s} $ are generic elements. In this paper, we obtain a necessary and sufficient condition for subarrangements of the semigeneric threshold arrangement to be central from the perspective of simple graphs. Combining it with Whitney's theorem, we provide a formula for the characteristic polynomials of the semigeneric threshold arrangement and its subarrangements.</p></abstract>

Graph Structures and the Whitney Theorem

The article considers a method for constructing the structures of an inseparable undirected graph G based on the use of the structure of its edge graph L(G). The method is grounded on the mapping of the edge graph L(G) by the subgraphs of the graph G. It is shown that the construction of the set of isometric cycles of the edge graph L(G) generates subsets of the subgraphs of the graph G. In turn, the set of generated subgraphs of the graph G allows constructing various invariants of the graph G and its topological drawing. The metric properties of the system of isometric cycles of the edge graph L(G) make it possible to determine the weights of the edges and vertices. This allows to construct the numerical characteristics of the graph structure. In turn, the numerical characteristics of the digital invariant of the edge graph make it possible to apply Whitney's theorem to solve the problem of recognizing graph isomorphism. The algorithm for constructing a digital invariant of an inseparable graph belongs to the class of polynomial problems with a computational complexity of О(n6). The digital invariant of the edge graph does not depend on the numbering of the vertices and edges of the graph. It specifies the individual characteristics of the graph, which allows determining graph isomorphism by comparing the invariants.

A simple and elementary proof of Whitney's unique embedding theorem

In this note we give a short and elementary proof of a more general version of Whitney’s theorem that 3-connected planar graphs have a unique embedding in the plane. A consequence of the theorem is also that cubic plane graphs cannot be embedded in a higher genus with a simple dual. The aim of this paper is to promote a simple and elementary proof, which is especially well suited for lectures presenting Whitney’s theorem.

An elementary proof of the Loomis–Whitney theorem

An elementary proof of the Loomis–Whitney theorem

Watts Spaces and Smooth Maps

A new category is described, which generalizes a select variety of categories having smooth mappings as their class of morphisms, those like the C∞ manifolds and C∞ mappings between them. Categorical embeddings are produced to justify this claim of generalization. Theorems concerning the equivalence of smooth and continuous versions of different separation axioms are proved following the categorical discussion. These are followed by a generalization of Whitney's approximation theorem, a smooth version of the Tietze extension theorem, and a sufficient condition to guarantee that the connected components of these spaces are smoothly path-connected.

Clustering algorithm with strength of connectedness for m-polar fuzzy network models.

In this research study, we first define the strong degree of a vertex in an m-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete m-polar fuzzy graphs. Further, we introduce the concept of m-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete m-polar fuzzy graphs. We discuss connectivity parameters in m-polar fuzzy graphs with precise examples, and we investigate the m-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an m-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as ϵA-reachable vertices in m-polar fuzzy graphs, ϵA-connected m-polar fuzzy graphs, or ϵA-connected m-polar fuzzy subgraphs (in case the m-polar fuzzy graph itself is not ϵA-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.

Kernelization of Whitney Switches

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs $G$ and $H$ are 2-isomorphic, or equivalently, their cycle matroids are isomorphic if and only if $G$ can be transformed into $H$ by a series of operations called Whitney switches. In this paper we consider the quantitative question arising from Whitney's theorem: Given two 2-isomorphic graphs, can we transform one into another by applying at most $k$ Whitney switches? This problem is already \sf NP-complete for cycles, and we investigate its parameterized complexity. We show that the problem admits a kernel of size $\mathcal{O}(k)$ and thus is fixed-parameter tractable when parameterized by $k$.

The Noncrossing Bond Poset of a Graph

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$.
 Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.

A 2-isomorphism theorem for delta-matroids

Whitney's 2-Isomorphism Theorem characterises when two graphs have isomorphic cycle matroids. We present an analogue of this theorem for graphs embedded in surfaces by characterising when two graphs in surface have isomorphic delta-matroids.

Safe Controller Synthesis for Data-Driven Differential Inclusions

We consider the problem of designing finite-horizon safe controllers for a dynamical system for which no explicit analytical model exists and limited data only along a single trajectory of the system are available. Given samples of the states and inputs of the system, and additional side information in terms of regularity of the evolution of the states, we synthesize a controller such that the evolution of the states avoid some prespecified unsafe set over a given finite horizon. Motivated by recent results on Whitney's extension theorem, we use piecewise-polynomial approximations of the trajectories based on the data along with the regularity side information to formulate a data-driven differential inclusion model that can predict the evolution of the trajectories. For these classes of data-driven differential inclusions, we propose a safety analysis theorem based on barrier certificates. As a corollary of this theorem, we demonstrate that we can design controllers ensuring safety of the solutions to the data-driven differential inclusion over a finite horizon. From a computational standpoint, our results are cast into a set of sum-of-squares programs whenever the certificates are parametrized by polynomials of fixed degree and the sets are semialgebraic.

Flow polynomials of a Signed Graph

For a signed graph $G$ and non-negative integer $d$, it was shown by DeVos et al. that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $\Gamma$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $\Gamma$ of order $k$ with $\epsilon(\Gamma)=d$, where $\epsilon(\Gamma)$ is the largest integer $d$ for which $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We define a class of particular directed circuits in $G$, namely the fundamental directed circuits, and show that all $\Gamma$-flows (not necessarily nowhere-zero) in $G$ can be generated by these circuits. It turns out that all $\Gamma$-flows in $G$ can be evenly partitioned into $2^{\epsilon(\Gamma)}$ classes specified by the elements of order 2 in $\Gamma$, each class of which consists of the same number of flows depending only on the order of $\Gamma$. Using an extension of Whitney's broken circuit theorem of Dohmen and Trinks, we give a combinatorial interpretation of the coefficients in $F_d(G,x)$ for $d=0$ in terms of broken bonds. Finally, we show that the sets of edges in a signed graph that contain no broken bond form a homogeneous simplicial complex.

Whitney theorem for complex polynomial mappings

For given natural numbers d_1,d_2 we describe the topology of a generic polynomial mapping F=(f,g):Xrightarrow {{mathbb {C}}}^2, with deg fle d_1 and deg gle d_2. Here X is a complex plane or a complex sphere.

Smooth extension theorems for one variable maps

We characterize the real valued functions f defined on perfect subsets P of R which admit n-times differentiable extensions F:R→R. In this characterization no continuity of F(n) is imposed. In particular, it generalizes Jarník's Extension Theorem, according to which f admits differentiable extension F:R→R if, and only if, f is differentiable. The new characterization is also closely related to the Whitney's Extension Theorem, which characterizes partial maps f admitting n-times differentiable extensions F:R→R with continuous nth derivative F(n). We also provide an elegant description of a linear extension operator Tn:C(P)→C(R) such that Tn(f)∈Dn(R) for every Dn(R)-extendable f∈C(P) and Tn(f)∈Cn(R) whenever f∈C(P) is Cn(R)-extendable.

On the extension of Whitney ultrajets

We prove necessary and sufficient conditions for the validity of Whitney's extension theorem in the ultradifferentiable Roumieu setting with controlled loss of regularity.

Recursion relations for chromatic coefficients for graphs and hypergraphs

We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an explicit formula for the linear coefficient of the chromatic polynomial of the $r$-complete hypergraph in terms of roots of the Taylor polynomials for the exponential function.

Extensions and Smooth Approximations of Definable Functions in O-minimal Structures

In 1934, H. Whitney presented a series of papers which discussed how to determine whether a function or a jet of order m is the restriction of a C m -function on R n . In the first paper of the series, Whitney's Extension Theorem was proved. In the latter, Whitney answered special cases of the following question Question (Whitney's Extension Theorem, WEPn,m) Let f be a continuous function from a closed subset E of R n to R .How can we determine whether f is the restriction of a C m -function on R n ? In this dissertation, we work in o-minimal expansions of real closed ordered fields. Definable versions of Whitney's Extension Theorem and Whitney's Extension Problems will be discussed in this context. Definable set-valued maps are also studied; a definable version of Michael's Selection Theorem will be proved and used, in combination with a definable version of Whitney's Extension Theorem, to give a positive answer to a definable version of WEPn,1.In addition to the above problems, we also discuss smoothing problems. This is inspired by a series of papers by A. Fischer.In this series, a construction of a definable C m -approximation of a definable locally Lipschitz function is provided. Here, we also work in an o-minimal expansion of a real closed field and relax the condition further to just continuous.