P-adic L-functions in universal deformation families

We construct examples of p-adic L-functions over universal deformation spaces for $${{\,\mathrm{GL}\,}}_2$$ GL 2 . We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’.

Polynomials and number sets associated with the probability distribution of the hyperbolic cosine type for even values of the parameter

The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

Bribery and Control in Stable Marriage

We initiate the study of external manipulations in Stable Marriage by considering several manipulative actions as well as several manipulation goals. For instance, one goal is to make sure that a given pair of agents is matched in a stable solution, and this may be achieved by the manipulative action of reordering some agents' preference lists. We present a comprehensive study of the computational complexity of all problems arising in this way. We find several polynomial-time solvable cases as well as NP-hard ones. For the NP-hard cases, focusing on the natural parameter "budget" (that is, the number of manipulative actions one is allowed to perform), we also conduct a parameterized complexity analysis and encounter mostly parameterized hardness results.

Collider searches for scalar singlets across lifetimes

Spin-0 singlets arise in well-motivated extensions of the Standard Model. Their lifetime determines the best search strategies at hadron and lepton colliders. To cover a large range of singlet decay lengths, we investigate bounds from Higgs decays into a pair of singlets, considering signatures of invisible decays, displaced and delayed jets, and coupling fits of untagged decays. We examine the generic scalar singlet and the relaxion, and derive a matching as well as qualitative differences between them. For each model, we discuss its natural parameter space and the searches probing it.

Topic Modeling in Embedding Spaces

Topic modeling analyzes documents to learn meaningful patterns of words. However, existing topic models fail to learn interpretable topics when working with large and heavy-tailed vocabularies. To this end, we develop the embedded topic model (etm), a generative model of documents that marries traditional topic models with word embeddings. More specifically, the etm models each word with a categorical distribution whose natural parameter is the inner product between the word’s embedding and an embedding of its assigned topic. To fit the etm, we develop an efficient amortized variational inference algorithm. The etm discovers interpretable topics even with large vocabularies that include rare words and stop words. It outperforms existing document models, such as latent Dirichlet allocation, in terms of both topic quality and predictive performance.

Clumsy Packings of Graphs

Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains P. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is called clumsy if it is maximal of minimum size. Let $\mathrm{cl}(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $\mathrm{cl}(G,H)$, and in many cases asymptotically determine $\mathrm{cl}(G,H)$ for some generic classes of graphs G such as $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $\mathrm{cl}(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that $$\mathrm{cl}(K_n, H) = \frac{\binom{n}{2}- \mathrm{ex}(n,H)}{|E(H)|}+o(\mathrm{ex}(n,H)),$$where $ex(n,H)$ is the extremal number of $H$. A related natural parameter is $\mathrm{cov}(G,H)$, that is the smallest number of copies of $H$ in $G$ (not necessarily edge-disjoint) whose removal from $G$ results in an $H$-free graph. While clearly $\mathrm{cov}(G,H) \leqslant\mathrm{cl}(G,H)$, all of our lower bounds for $\mathrm{cl}(G,H)$ apply to $\mathrm{cov}(G,H)$ as well.