Unstructured data are a promising new source of information that insurance companies may use to understand their risk portfolio better and improve the customer experience. However, these novel data sources are difficult to incorporate into existing ratemaking frameworks due to the size and format of the unstructured data. This paper proposes a framework to use street view imagery within a generalized linear model. To do so, we use representation learning to extract an embedding vector containing useful information from the image. This embedding is dense and low dimensional, making it appropriate to use within existing ratemaking models. We find that there is useful information included in street view imagery to predict the frequency of claims for certain types of perils. This model can be used as in a ratemaking framework but also opens the door to future empirical research on attempting to extract which characteristics within the image leads to increased or decreased predicted claim frequencies. Throughout, we discuss the practical difficulties (technical and social) of using this type of data for insurance pricing.

We obtain the following embedding theorem for symbolic dynamical systems. Let G be a countable amenable group with the comparison property. Let X be a strongly aperiodic subshift over G. Let Y be a strongly irreducible shift of finite type over G that has no global period, meaning that the shift action is faithful on Y. If the topological entropy of X is strictly less than that of Y and Y contains at least one factor of X, then X embeds into Y. This result partially extends the classical result of Krieger when $G = \mathbb {Z}$ and the results of Lightwood when $G = \mathbb {Z}^d$ for $d \geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.

We focus on modelling categorical features and improving predictive power of neural networks with mixed categorical and numerical features in supervised learning tasks. The goal of this paper is to challenge the current dominant approach in actuarial data science with a new architecture of a neural network and a new training algorithm. The key proposal is to use a joint embedding for all categorical features, instead of separate entity embeddings, to determine the numerical representation of the categorical features which is fed, together with all other numerical features, into hidden layers of a neural network with a target response. In addition, we postulate that we should initialize the numerical representation of the categorical features and other parameters of the hidden layers of the neural network with parameters trained with (denoising) autoencoders in unsupervised learning tasks, instead of using random initialization of parameters. Since autoencoders for categorical data play an important role in this research, they are investigated in more depth in the paper. We illustrate our ideas with experiments on a real data set with claim numbers, and we demonstrate that we can achieve a higher predictive power of the network.

Let $m\leqslant n\in \mathbb {N}$, and $G\leqslant \operatorname {Sym}(m)$ and $H\leqslant \operatorname {Sym}(n)$. In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$ and ${V_n(H)}$. When $n\equiv 1 \mod (m-1)$, and under some other technical conditions, we find an embedding of ${V_n(H)}$ into ${V_m(G)}$ via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$ and an embedding of ${V_m(G)}$ into ${V_n(H)}$.

Spatial data are a rich source of information for actuarial applications: knowledge of a risk’s location could improve an insurance company’s ratemaking, reserving or risk management processes. Relying on historical geolocated loss data is problematic for areas where it is limited or unavailable. In this paper, we construct spatial embeddings within a complex convolutional neural network representation model using external census data and use them as inputs to a simple predictive model. Compared to spatial interpolation models, our approach leads to smaller predictive bias and reduced variance in most situations. This method also enables us to generate rates in territories with no historical experience.

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; {\mathbb Z} )$. Our main result is a readily calculable classification of embeddings$N \to {\mathbb R}^7$up to isotopy, with an indeterminacy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9.

The group of knots $S^4\to {\mathbb R}^7$ acts on the set of embeddings $N\to {\mathbb R}^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1 \ne 0$, with an indeterminacy.

Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008.

For $N=S^1\times S^3$ we give a geometrically defined 1–1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set ${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$.

Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$ have the same homotopy $(2n-7)$ -type. In the four-dimensional case, this means that the spaces of smooth knots in homeomorphic $4$ -manifolds have sets $\pi _0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi _1$ . The result about $\pi _0$ is well-known and elementary, but the result about $\pi _1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie–Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie–Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on $\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$ . We use our model to show that for every choice of basepoint, each of the homotopy groups, $\pi _1$ and $\pi _2,$ of $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ contains an infinitely generated free abelian group.

We characterize the weight functions $u,v,w$ on $\left( 0,\infty \right)$ such that

$${{\left( \int\limits_{0}^{\infty }{{{f}^{*}}{{\left( t \right)}^{q}}w\left( t \right)}\,dt \right)}^{1/q}}\le C\,\,\underset{t\in \left( 0,\infty \right)}{\mathop{\sup }}\,{{f}_{u}}^{**}\left( t \right)v\left( t \right),$$

where

$${{f}_{u}}^{**}\left( t \right):={{\left( \int\limits_{0}^{t}{u\left( s \right)}\,ds \right)}^{-1}}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)u\left( s \right)\,ds.$$

As an application we present a new simple characterization of the associate space to the space ${{\Gamma }^{\infty }}\left( v \right)$, determined by the norm

$${{\left\| f \right\|}_{\Gamma \infty \left( v \right)}}=\,\underset{t\in \left( 0,\infty \right)}{\mathop{\sup }}\,{{f}^{**}}\left( t \right)v\left( t \right),$$

where

$${{f}^{**}}\left( t \right):=\frac{1}{t}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)\,ds.$$

We classify the generalized hexagons which are laxly embedded in projective space such that the embedding is flat and polarized. Besides the standard examples related to the hexagons defined over the algebraic groups of type ${{\text{G}}_{2}}$, ${}^{3}{{\text{D}}_{4}}$ and ${}^{6}{{\text{D}}_{\text{4}}}$ (and occurring in projective dimensions 5, 6, 7), we find new examples in unbounded dimension related to the mixed groups of type ${{\text{G}}_{2}}$ .

Let $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.

AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11

Let ${{\mathbf{P}}^{n}}$ be the $n$-dimensional projective space over some algebraically closed field $k$ of characteristic 0. For an integer $t\,\ge \,3$ consider the invertible sheaf $O\left( t \right)$ on ${{\mathbf{P}}^{n}}$ (Serre twist of the structure sheaf). Let $N\,=\,\left( \underset{n}{\mathop{t+n}}\, \right)$, the dimension of the space of global sections of $O\left( t \right)$, and let $k$ be an integer satisfying $0\,\le \,k\,\le \,N\,-\,\left( 2n\,+\,2 \right)$. Let ${{P}_{1}},\ldots ,{{P}_{k}}$ be general points on ${{\mathbf{P}}^{n}}$ and let $\pi :\,X\,\to \,{{\mathbf{P}}^{n}}$ be the blowing-up of ${{\mathbf{P}}^{n}}$ at those points. Let ${{E}_{i}}\,=\,{{\pi }^{-1}}\left( {{P}_{i}} \right)$ with $1\,\le \,i\,\le \,k$ be the exceptional divisor. Then $M={{\pi }^{*}}\left( O(t) \right)\,\otimes \,{{O}_{X}}\left( -{{E}_{1}}-\cdots -{{E}_{k}} \right)$ is a very ample invertible sheaf on $X$.

Call a category "sketchable" if it is the category of models in sets of some sketch. This paper explores the subtle boundary between sketchable and non-sketchable categories. We show that the category of small categories that have at least one initial object and functors that take an initial object to an initial object is sketchable. The same is true for weak initial objects, but is false for subinitial objects (that every object has at most one arrow to). Analogous results hold if we substitute finite limits for terminal object. We also show that the category of groups and center-preserving homomorphisms is not sketchable. We describe briefly how "higher-order" sketches can fill these gaps.