Let $\mathfrak {g}$ be a complex semisimple Lie algebra with associated Yangian $Y_{\hbar }\mathfrak {g}$. In the mid-1990s, Khoroshkin and Tolstoy formulated a conjecture which asserts that the algebra $\mathrm {D}Y_{\hbar }\mathfrak {g}$ obtained by doubling the generators of $Y_{\hbar }\mathfrak {g}$, called the Yangian double, provides a realization of the quantum double of the Yangian. We provide a uniform proof of this conjecture over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ which is compatible with the theory of quantized enveloping algebras. As a by-product, we identify the universal R-matrix of the Yangian with the canonical element defined by the pairing between the Yangian and its restricted dual.

We observe a fundamental relationship between Steenrod operations and the Artin–Schreier morphism. We use Steenrod's construction, together with some new geometry related to the affine Grassmannian, to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the $K$-theoretic version of the quantum Coulomb branch. At the end of the paper, we investigate what our ideas produce on the categorical level. We find that they yield, after a little fiddling, a construction which corresponds, under the geometric Satake equivalence, to the Frobenius twist functor for representations of the Langlands dual group. We also describe the unfiddled answer, conditional on a conjectural ‘modular derived Satake’, and, though it is more complicated to state, it is in our opinion just as neat and even more compelling.

We prove that every symplectic manifold is a coadjoint orbit of the group of automorphisms of its integration bundle, acting linearly on its space of momenta, for any group of periods of the symplectic form. This result generalizes the Kirilov–Kostant–Souriau theorem when the symplectic manifold is homogeneous under the action of a Lie group and the symplectic form is integral.

The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimension d=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimension d≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.

In this paper we study categories ${\mathcal{O}}$ over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden et al. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories ${\mathcal{O}}$ . We use these structures to study shuffling functors of Braden et al. (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.

Visual retrieval and classification are of growing importance for a number of applications, including surveillance, automotive, as well as web and mobile search. To facilitate these processes, features are often computed from images to extract discriminative aspects of the scene, such as structure, texture or color information. Ideally, these features would be robust to changes in perspective, illumination, and other transformations. This paper examines two approaches that employ dimensionality reduction for fast and accurate matching of visual features while also being bandwidth-efficient, scalable, and parallelizable. We focus on two classes of techniques to illustrate the benefits of dimensionality reduction in the context of various industrial applications. The first method is referred to as quantized embeddings, which generates a distance-preserving feature vector with low rate. The second method is a low-rank matrix factorization applied to a sequence of visual features, which exploits the temporal redundancy among feature vectors associated with each frame in a video. Both methods discussed in this paper are also universal in that they do not require prior assumptions about the statistical properties of the signals in the database or the query. Furthermore, they enable the system designer to navigate a rate versus performance trade-off similar to the rate-distortion trade-off in conventional compression.

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.

TanDEM-X (TDX; TerraSAR-X add-on for digital elevation measurement) is an innovative spaceborne X-band earth observation mission that will be launched in June 2010. This paper gives an overview of the TDX mission concept, summarizes the basic products and illustrates the achievable performance. In more detail the effect of quantization on the interferometric performance is discussed. Finally, new imaging techniques are outlined.

Let M be a smooth manifold, ${\cal S}$ the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on ${\cal S}$. This cohomology space is closely related to the Vect(M)-modules, ${\cal D}$λ(M), of linear differential operators on the space of tensor densities on M of degree λ.