How do we read pie charts? This seems like a straightforward question to answer, but it turns out that most of what you’ve probably heard is wrong. We don’t actually know whether we use angle, area, or arc length. In a short paper at the VIS conference this week I’m presenting a study I ran […]

We all use data all the time, but what exactly is data? How do different programs know what to do with our data? How is visualizing data different from other uses of data? And isn’t everything inside a computer data in the end? The latest episode of eagereyesTV looks at what data is and what […]

This book from 1945 contains a very interesting mix of different charts made by the ISOTYPE Institute, some classic and some quite unusual. As a book about labor and unemployment, it also makes extensive use of Gerd Arntz’s famous unemployed man icon. Michael Young and Theodor Prager’s There’s Work for All is part of a […]

Visualization turns data into images, but are images themselves data? There are often claims that they are, but then you mostly see the images themselves without much additional data. In this video, I look at image browsers, a project classifying selfies along a number of criteria, and the additional information stored in HEIC that makes […]

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Corners Picks *** Sunday *** 17 September 2017

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Double Chance Picks *** Monday *** 18 September 2017

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We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.

We study two notions of topological entropy of correspondences introduced by Friedland and Dinh-Sibony. Upper bounds are known for both. We identify a class of holomorphic correspondences whose entropy in the sense of Dinh-Sibony equals the known upper bound. This provides an exact computation of the entropy for rational semigroups. We also explore a connection between these two notions of entropy.

We prove a Marstrand type slicing theorem for the subsets of the integer square lattice. This problem is the dual of the corresponding projection theorem, which was considered by Glasscock, and Lima and Moreira, with the mass and counting dimensions applied to subsets of $mathbb{Z}^{d}$. In this paper, more generally we deal with a subset of the plane that is $1$ separated, and the result for subsets of the integer lattice follow as a special case. We show that the natural slicing question in this setting is true with the mass dimension.

We give a complete (and surprisingly simple) description of the affine Hecke category for $ ilde{A}_2$ in characteristic zero. More precisely, we calculate the Kazhdan-Lusztig polynomials, give a recursive formula for the projectors defining indecomposable objects and, for each coefficient of a Kazhdan-Lusztig polynomial, we produce a set of morphisms with such a cardinality.

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.

A topological dynamical system $(X,f)$ induces two natural systems, one is on the probability measure spaces and other one is on the hyperspace.

We introduce a concept for these two spaces, which is called entropy order, and prove that it coincides with topological entropy of $(X,f)$. We also consider the entropy order of an invariant measure and a variational principle is established.

We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Delta + V - iepsilon x^2$, $|V(x)|leq e^{-2gamma |x|}$ converge, as $ epsilon o 0+ $, to the poles of the meromorphic continuation of $ ( -Delta + V -lambda^2 )^{-1} $ uniformly on compact subsets of $ extrm{Re},lambda>0$, $ extrm{Im},lambda>-gamma$, $arglambda > pi/8$.

In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution, and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases

We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic.

In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--R"{o}hrle, the usual notion of $G$-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of $G$. We show that other variations of this notion, such as relative complete reducibility and $sigma$-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M. This allows us to express the h-vector of the broken circuit complex of M in terms of the intersection theory of the conormal fan of M. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincar'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. The Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M, when combined with the Hodge-Riemann relations for the conormal fan of M, implies Brylawski's and Dawson's conjectures that the h-vectors of the broken circuit complex and the independence complex of M are log-concave sequences.

In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

Applying the equivariant moving frames method, we construct convergent normal forms for real-analytic 5-dimensional totally nondegenerate CR submanifolds of C^4. These CR manifolds are divided into several biholomorphically inequivalent subclasses, each of which has its own complete normal form. Moreover it is shown that, biholomorphically, Beloshapka's cubic model is the unique member of this class with the maximum possible dimension seven of the corresponding algebra of infinitesimal CR automorphisms. Our results are also useful in the study of biholomorphic equivalence problem between CR manifolds, in question.

In this article, we prove that the completeness of the Hamilton flow and essential self-dajointness are equivalent for real principal type operators on the circle. Moreover, we study spectral properties of these operators.

Consider that $u_0$ nodes are aware of some piece of data $d_0$. This note derives the expected time required for the data $d_0$ to be disseminated through-out a network of $n$ nodes, when communication between nodes evolves according to a graphical Markov model $overline{ mathcal{G}}_{n,hat{p}}$ with probability parameter $hat{p}$. In this model, an edge between two nodes exists at discrete time $k in mathbb{N}^+$ with probability $hat{p}$ if this edge existed at $k-1$, and with probability $(1-hat{p})$ if this edge did not exist at $k-1$. Each edge is interpreted as a bidirectional communication link over which data between neighbors is shared. The initial communication graph is assumed to be an Erdos-Renyi random graph with parameters $(n,hat{p})$, hence we consider a emph{stationary} Markov model $overline{mathcal{G}}_{n,hat{p}}$. The asymptotic "$u_0$-expected flooding time" of $overline{mathcal{G}}_{n,hat{p}}$ is defined as the expected number of iterations required to transmit the data $d_0$ from $u_0$ nodes to $n$ nodes, in the limit as $n$ approaches infinity. Although most previous results on the asymptotic flooding time in graphical Markov models are either emph{almost sure} or emph{with high probability}, the bounds obtained here are emph{in expectation}. However, our bounds are tighter and can be more complete than previous results.

We relate the $L^p$-mapping properties of the Bergman projections on two domains in $mathbb{C}^n$, one of which is the quotient of the other under the action of a finite group of biholomorphic automorphisms. We use this relation to deduce the sharp ranges of $L^p$-boundedness of the Bergman projection on certain $n$-dimensional model domains generalizing the Hartogs triangle.

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $omega$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $omega$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.

Connections between set theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set theoretic $R$-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for $R$-matrices being Baxterized solutions of the $A$-type Hecke algebra ${cal H}_N(q=1)$. We show in the case of the reflection algebra that there exists a "boundary" finite sub-algebra for some special choice of "boundary" elements of the $B$-type Hecke algebra ${cal B}_N(q=1, Q)$. We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the $B$-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the $B$-type Hecke algebra. These are universal statements that largely generalize previous relevant findings, and also allow the investigation of the symmetries of the double row transfer matrix.

In this work we consider the initial-value problem associated with a coupled system of generalized Korteweg-de Vries equations. We present a relationship between the best constant for a Gagliardo-Nirenberg type inequality and a criterion for the existence of global solutions in the energy space. We prove that such a constant is directly related to the existence problem of solitary-wave solutions with minimal mass, the so called ground state solutions. To guarantee the existence of ground states we use a variational method.

We compute non-extremal three-point functions of scalar operators in $mathcal{N}=4$ super Yang-Mills at tree-level in $g_{YM}$ and at finite $N_c$, using the operator basis of the restricted Schur characters. We make use of the diagrammatic methods called quiver calculus to simplify the three-point functions. The results involve an invariant product of the generalized Racah-Wigner tensors ($6j$ symbols). Assuming that the invariant product is written by the Littlewood-Richardson coefficients, we show that the non-extremal three-point functions satisfy the large $N_c$ background independence; correspondence between the string excitations on $AdS_5 imes S^5$ and those in the LLM geometry.

This paper deals with an optimal control problem associated to the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.

Willems et al.'s fundamental lemma asserts that all trajectories of a linear system can be obtained from a single given one, assuming that a persistency of excitation condition holds. This result has profound implications for system identification and data-driven control, and has seen a revival over the last few years. The purpose of this paper is to extend Willems' lemma to the situation where multiple (possibly short) system trajectories are given instead of a single long one. To this end, we introduce a notion of collective persistency of excitation. We will then show that all trajectories of a linear system can be obtained from a given finite number of trajectories, as long as these are collectively persistently exciting. We will demonstrate that this result enables the identification of linear systems from data sets with missing data samples. Additionally, we show that the result is of practical significance in data-driven control of unstable systems.

Miller and Sheffield introduced a notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $mathbb{C}$ and Gaussian free fields (GFFs) on them under conformal equivalence. They considered the situation in which the conformal transformations are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct processes of GFF on $mathbb{H}$ (the upper half-plane) and $mathbb{O}$ (the first orthant of $mathbb{C}$) by coupling zero-boundary GFFs on these domains with stochastic log-gases defined on parts of boundaries of the domains, $mathbb{R}$ and $mathbb{R}_+$, called the Dyson model and the Bru--Wishart process, respectively, using multiple SLEs evolving in time. We prove that the obtained processes of GFF are stationary. The stationarity defines an equivalence relation between GFFs, and the pairs of time-evolutionary domains and stationary processes of GFF will be regarded as generalizations of the imaginary surfaces studied by Miller and Sheffield.

Inserting renewable energy in the electric grid in a decentralized manneris a key challenge of the energy transition. However, at local scale, both production and demand display erratic behavior, which makes it delicate to match them. It is the goal of Energy Management Systems (EMS) to achieve such balance at least cost. We present EMSx, a numerical benchmark for testing control algorithms for the management of electric microgrids equipped with a photovoltaic unit and an energy storage system. EMSx is made of three key components: the EMSx dataset, provided by Schneider Electric, contains a diverse pool of realistic microgrids with a rich collection of historical observations and forecasts; the EMSx mathematical framework is an explicit description of the assessment of electric microgrid control techniques and algorithms; the EMSx software EMSx.jl is a package, implemented in the Julia language, which enables to easily implement a microgrid controller and to test it. All components of the benchmark are publicly available, so that other researchers willing to test controllers on EMSx may reproduce experiments easily. Eventually, we showcase the results of standard microgrid control methods, including Model Predictive Control, Open Loop Feedback Control and Stochastic Dynamic Programming.

We investigate quasistatic evolution in finite plasticity under the assumption that the plastic strain is compatible. This assumption is well-suited to describe the special case of dislocation-free plasticity and entails that the plastic strain is the gradient of a plastic deformation map. The total deformation can be then seen as the composition of a plastic and an elastic deformation. This opens the way to an existence theory for the quasistatic evolution problem featuring both Lagrangian and Eulerian variables. A remarkable trait of the result is that it does not require second-order gradients.