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This chapter closes Part I presenting advanced topics, including dealing with variable length feature vectors, Feature Engineering and Deep Learning and automatic Feature Engineering (either supervised or unsupervised). It starts bridging the pure domain independent techniques to start drilling into problems of domain-specific importance. Variable length feature vectors has been a problem for fixed-size vector ML ever since. In general, techniques involve truncation, computing the most general tree and encoding paths on it or just destructive projection into a smaller plane. The chapter briefly delve into some Deep Learning concepts and what it entails for feature engineering. Automated Feature Learning using FeatureTools (the DataScience Machine) and genetic programming is covered, also Instance Engineering and Unsupervised Feature Engineering (in the form of autoencoders).
Design principles from the field of design engineering require that a product be designed as unambiguous, safe and simple as possible. Simplicity results on the one hand from an objective product-relevant side, on the other hand from the experience and knowledge of the user. A product that is perceived as simple by one person may seem complicated to another. From this, the questions arise, with which attributes simplicity can be described and how these are to be captured. In this paper, an evaluation system for the subjective attributes of simplicity is created using the fuzzy sets approach.
We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time
$\unicode[STIX]{x1D6FF}$
-dense (the orbit meets every ball of radius
$\unicode[STIX]{x1D6FF}$
), weakly dense and such that
$\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$
is dense for every
$\unicode[STIX]{x1D6E4}\subset \mathbb{C}$
that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.
We give a geometric interpretation of sheaf cohomology for higher degrees
$n\geq 1$
in terms of torsors on the member of degree
$d=n-1$
in hypercoverings of type
$r=n-2$
, endowed with an additional datum, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.
In this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.
We discuss how to define a basis for a general normed space (a ‘Schauder basis‘). We then consider orthonormal sets in inner-product spaces and orthonormal bases for separable Hilbert spaces. We give a number of conditions that ensure that a particular orthonormal sequence forms an orthonormal basis, and as an example, we discuss the L^2 convergence of Fourier series.
We recall the definition of a metric sapce, along with definitions of convergence, continuity, separability, and compactness. The treatment is intentionally brisk, but proofs are included.
The final chapter concerns two families of regular 5-polytopes, the second consisting of the double covers of the first. The starting point is a simple group, which is the automorphism group of a regular quotient of a 5-dimensional hyperbolic honeycomb. Armed with only modest information, it is first shown that the realization domain of this polytope is very simple. Two defining relations for the quotient were initially provided; subsequently, it is seen geometrically that one of them is redundant. As in the previous chapter, there is an extended family, among which there are polytopes whose facets and vertex-figures beong to the pentagonal 4-polytopes of Chapter 7 and the family of 4-polytopes described in Chapter 16. The initial quotients are non-orientable; the family of their double covers contains members that are universal as amalgamations. These families of polytopes correspond to actions of the automorphism groups on two of their maximal subgroups. The groups have another maximal subgroup, and though there are no nice related polytopes, nevertheless this gives rise to interesting symmetric sets. There is another quotient and a close relative which one might initially think belong to one of the families; that they do not, with a completely unrelated group, is perhaps surprising.
The bridging concept between the abstract and geometric is the theory of realizations. This chapter concentrates on symmetric sets, namely, finite sets on which a group of permutations acts transitively. After a discussion of their basic properties, the concept of their realizations is introduced, with operations on them (such as blending) showing that the family of their congruences classes has the structure of a convex cone. A key idea is that of the inner product and cosine vectors of realizations, which define them up to congruence. The theory up to this point is then illustrated by some examples. It is next shown that, corresponding to the tensor product of representations, there is a product of realizations. Another fundamental notion is that of orthogonality relations for cosine vectors. The different realizations derived from an irreducible representation of the abstract group may form a subcone of the realization cone that is more than 1-dimensional. These are looked at more closely, leading to a definition of cosine matrices for the general realization domain. There follows a discussion of cuts and their relationship with duality. Cosine vectors may have entries in some subfield of the real numbers, with implications for the corresponding realizations. The chapter ends with a brief account of how representations of groups are related to realizations.
We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.
This chapter introduces preference orderings and their representation by utility functions. Also, the consumer choice problem as a utility-maximization problem and its applications. Income and the substitution effect are discussed.
We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number
$p/q$
outside the Cantor middle-third set
$C$
to the set
$C$
, in terms of the denominator
$q$
. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing- digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.
Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of
$\sum _{p\in \pi (G)}d_p(G),$
where we are denoting by dp(G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.
The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set
$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
that is the infinite path space of the stationary
$k$
-Bratteli diagram
${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
, where
$\unicode[STIX]{x1D6EC}$
is a finite strongly connected
$k$
-graph. The Dirichlet form which we are interested in is induced by an even spectral triple
$(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$
and is given by
where
$\unicode[STIX]{x1D6EF}$
is the space of choice functions on
$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
. There are two ultrametrics,
$d^{(s)}$
and
$d_{w_{\unicode[STIX]{x1D6FF}}}$
, on
$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
which make the infinite path space
$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
an ultrametric Cantor set. The former
$d^{(s)}$
is associated to the eigenvalues of the Laplace–Beltrami operator
$\unicode[STIX]{x1D6E5}_{s}$
associated to
$Q_{s}$
, and the latter
$d_{w_{\unicode[STIX]{x1D6FF}}}$
is associated to a weight function
$w_{\unicode[STIX]{x1D6FF}}$
on
${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
, where
$\unicode[STIX]{x1D6FF}\in (0,1)$
. We show that the Perron–Frobenius measure
$\unicode[STIX]{x1D707}$
on
$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
has the volume-doubling property with respect to both
$d^{(s)}$
and
$d_{w_{\unicode[STIX]{x1D6FF}}}$
and we study the asymptotic behavior of the heat kernel associated to
$Q_{s}$
. Moreover, we show that the Dirichlet form
$Q_{s}$
coincides with a Dirichlet form
${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$
which is associated to a jump kernel
$J_{s}$
and the measure
$\unicode[STIX]{x1D707}$
on
$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$
, and we investigate the asymptotic behavior and moments of displacements of the process.
The Chebyshev conjecture posits that Chebyshev subsets of a real Hilbert space
$X$
are convex. Works by Asplund, Ficken and Klee have uncovered an equivalent formulation of the Chebyshev conjecture in terms of uniquely remotal subsets of
$X$
. In this tradition, we develop another equivalent formulation in terms of Chebyshev subsets of the unit sphere of
$X\times \mathbb{R}$
. We characterise such sets in terms of the image under stereographic projection. Such sets have superior structure to Chebyshev sets and uniquely remotal sets.
We motivate the definitions of sets of small doubling and approximate groups, and introduce their basic properties. We show that random sets of integers (suitably defined) have large expected doubling. We prove Freiman’s theorem that a subset of a group of doubling less than 2/3 is close to a finite subgroup. We prove the Plünnecke–Ruzsa inequalities, Ruzsa’s triangle inequality and Ruzsa’s covering lemma. We motivate in detail the definition of an approximate group, and reduce the study of sets of small doubling to the study of finite approximate groups. We show that the notions of small tripling and approximate group are stable under intersections and group homomorphisms. We introduce Freiman homomorphisms and present their basic properties.
In engineering design, surrogate models are often used instead of costly computer simulations. Typically, a single surrogate model is selected based on the previous experience. We observe, based on an analysis of the published literature, that fitting an ensemble of surrogates (EoS) based on cross-validation errors is more accurate but requires more computational time. In this paper, we propose a method to build an EoS that is both accurate and less computationally expensive. In the proposed method, the EoS is a weighted average surrogate of response surface models, kriging, and radial basis functions based on overall cross-validation error. We demonstrate that created EoS is accurate than individual surrogates even when fewer data points are used, so computationally efficient with relatively insensitive predictions. We demonstrate the use of an EoS using hot rod rolling as an example. Finally, we include a rule-based template which can be used for other problems with similar requirements, for example, the computational time, required accuracy, and the size of the data.
We consider closed orientable surfaces
$S$
of genus
$g>1$
and homeomorphisms
$f:S\rightarrow S$
isotopic to the identity. A set of hypotheses is presented, called a fully essential system of curves
$\mathscr{C}$
and it is shown that under these hypotheses, the natural lift of
$f$
to the universal cover of
$S$
(the Poincaré disk
$\mathbb{D}$
), denoted by
$\widetilde{f},$
has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, for
$C^{1+\unicode[STIX]{x1D716}}$
diffeomorphisms, we show the existence of rotational horseshoes having non-trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an
$f$
is a compact convex subset of
$\mathbb{R}^{2g}$
with maximal dimension and all points in its interior are realized by compact
$f$
-invariant sets and by periodic orbits in the rational case. Also,
$f$
has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where
$f$
is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.
For every integer
$k\geq 2$
and every
$A\subseteq \mathbb{N}$
, we define the
$k$
-directions sets of
$A$
as
$D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$
and
$D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$
, where
$\Vert \cdot \Vert$
is the Euclidean norm and
$A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$
. Via an appropriate homeomorphism,
$D^{k}(A)$
is a generalisation of the ratio set
$R(A):=\{a/b:a,b\in A\}$
. We study
$D^{k}(A)$
and
$D^{\text{}\underline{k}}(A)$
as subspaces of
$S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$
. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets
$X\subseteq S^{k-1}$
such that there exists
$A\subseteq \mathbb{N}$
satisfying
$D^{\text{}\underline{k}}(A)^{\prime }=X$
, where
$Y^{\prime }$
denotes the set of accumulation points of
$Y$
. Moreover, we provide a simple sufficient condition for
$D^{k}(A)$
to be dense in
$S^{k-1}$
. We conclude with questions for further research.
We define the notion of a completely determined Borel code in reverse mathematics, and consider the principle
$CD - PB$
, which states that every completely determined Borel set has the property of Baire. We show that this principle is strictly weaker than
$AT{R_0}$
. Any ω-model of
$CD - PB$
must be closed under hyperarithmetic reduction, but
$CD - PB$
is not a theory of hyperarithmetic analysis. We show that whenever
$M \subseteq {2^\omega }$
is the second-order part of an ω-model of
$CD - PB$
, then for every
$Z \in M$
, there is a
$G \in M$
such that G is
${\rm{\Delta }}_1^1$
-generic relative to Z.