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If $K$ is a field of characteristic $p$ then the $p$torsion of the Brauer group, ${}_p{\rm Br\,}(K)$, is represented by a quotient of the group of $1$forms, $\Omega^1(K)$. Namely, we have a group isomorphism $$\alpha_p:\Omega^1(K)/\langle{\rm d}a,\, (a^pa){\rm dlog}b\, :\, a,b\in K,\, b\neq 0\rangle\to{}_p{\rm Br\,}(K),$$ given by $a{\rm d}b\mapsto [ab,b)_p$ $\forall a,b\in K$, $b\neq 0$. Here $[\cdot,\cdot )_p:K/\wp (K)\times K^\times/K^{\times p}\to{}_p{\rm Br\,}(K)$ denotes the ArtinSchreier symbol. In this paper we generalize this result. Namely, we prove that for every $n\geq 1$ we have a representation of ${}_{p^n}{\rm Br\,}(K)$ by a quotient of $\Omega^1(W_n(K))$, where $W_n(K)$ is the truncation of length $n$ of the ring of $p$typical Witt vectors, i.e. $W_{\{1,p,\ldots,p^{n1}\}}(K)$. Explicitly, we have a group isomorphism $$\alpha_{p^n}:\Omega^1(W_{p^n}(K))/\langle Fa{\rm d}ba{\rm d}Vb\, :\, a,b\in W_n(K),\, ([a^p][a]){\rm dlog}[b]\, :\, a,b\in K,\, b\neq 0\rangle\to{
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We construct a tower of fibrations approximating the derived mapping space between two simplicially enriched operads subject to mild conditions. The nth stage of the tower is obtained by neglecting operations with more than n inputs. The main theorem describes the layers of this tower.
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Let $(M,\textit{g},\sigma)$ be a compact Riemannian spin manifold of dimension $m\geq2$, let $\mathbb{S}(M)$ denote the spinor bundle on $M$, and let $D$ be the AtiyahSinger Dirac operator acting on spinors $\psi:M\to\mathbb{S}(M)$. We study the existence of solutions of the nonlinear Dirac equation with critical exponent \[ D\psi = \lambda\psi + f(\psi)\psi + \psi^{\frac2{m1}}\psi \tag{NLD} \] where $\lambda\in\mathbb{R}$ and $f(\psi)\psi$ is a subcritical nonlinearity in the sense that $f(s)=o\big(s^{\frac2{m1}}\big)$ as $s\to\infty$. A model nonlinearity is $f(s)=\alpha s^{p2}$ with $2<p<\frac{2m}{m1}$, $\alpha\in\mathbb{R}$. In particular we study the nonlinear Dirac equation \[ D\psi=\lambda\psi+\psi^{\frac2{m1}}\psi, \quad \lambda\in\mathbb{R}. \tag{BND} \] This equation is a spinorial analogue of the BrezisNirenberg problem. As corollary of our main results we obtain the existence of nontrivial solutions $(\lambda,\psi)$ of (BND) for every $\lambda>0$, ev
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We present an optimal algorithm for the threestage arbitrary polarization tracking using LithiumNiobatebased Polarization Controllers: device calibration, polarization state rotation, and stabilization. The theoretical model representing the lithiumniobatebased polarization controller is derived and the methodology is successfully applied. Results are numerically simulated in the MATLAB environment.
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We develop a model for an antiplane crack defect posed on a square lattice under an interatomic pairpotential with nearestneighbour interactions. In particular, we establish existence, local uniqueness and stability of solutions for small loading parameters and further prove qualitatively sharp farfield decay estimates. The latter requires establishing decay estimates for the corresponding lattice Green's function, which are of independent interest.
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In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a $1$Poincar\'e inequality. We show that at almost every point $x$ outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at $x$. We also show that, at codimension $1$ Hausdorff measure almost every measuretheoretic boundary point of a set $E$ of finite perimeter, there is an asymptotic limit set $(E)_\infty$ corresponding to the asymptotic expansion of $E$ and that every such asymptotic limit $(E)_\infty$ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of $(E)_\infty$ is Ahlfors codimension $1$ regular.
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We define automorphic vector bundles on the stack of $G$zips introduced by MoonenPinkWedhornZiegler and study their global sections. In particular, we give a combinatorial condition on the weight for the existence of nonzero mod $p$ automorphic forms on Shimura varieties of Hodgetype. We attach to the highest weight of the representation $V(\lambda)$ a mod $p$ automorphic form and we give a modular interpretation of this form in some cases.
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We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.
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Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higherorder logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We also address the relation of our axiom systems to alternative proposals from the literature, including an axiom set proposed by Freyd and Scedrov for which we reveal a technical issue (when encoded in free logic where free variables range over defined and undefined objects): either all operations, e.g. morphism composition, are total or their axiom system is inconsistent. The repair for this problem is quite straightforward, however.
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We define the mechanical complexity $C(P)$ of a convex polyhedron $P,$ interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria, and the mechanical complexity $C(S,U)$ of primary equilibrium classes $(S,U)^E$ with $S$ stable and $U$ unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class $(S,U)^E$ with $S, U > 1$ is the minimum of $2(f+vSU)$ over all polyhedral pairs $(f,v )$, where a pair of integers is called a polyhedral pair if there is a convex polyhedron with $f$ faces and $v$ vertices. In particular, we prove that the mechanical complexity of a class $(S,U)^E$ is zero if, and only if there exists a convex polyhedron with $S$ faces and $U$ vertices. We also discuss the mechanical complexity of the monostatic classes $(1,U)^E$ and $(S,1)^E$, and offer a complexitydependent prize for the compl
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Let G be a reductive padic group. Let $\Phi$ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that $\Phi$ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modiification of the arguments of J.F.Dat who proved our result in one direction.
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It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a constant magnetic field $b$ which generates a magnetic flux $\varphi$ per unit area, then any spectral island $\sigma_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $\mathcal{I}_b=M \varphi$. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any $2d$ BlochLandau operator $H_b$ which also has a bounded $\mathbb{Z}^2$periodic electric potential. Assume that $H_b$ has a spectral island $\sigma_b$ which remains isolated from the rest of the spectrum as long as $\varphi$ lies in a compact interval $[\varphi_1,\varphi_2]$. Then $\mathcal{I}_b=c_0+c_1\varphi$ on such intervals, where the constant $c_0\in \mathbb{Q}$ while $c_1\in \mathbb{Z}$
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The heavytailed distributions of corrupted outliers and singular values of all channels in lowlevel vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyperLaplacian. However, the use of such distributions generally leads to challenging nonconvex, nonsmooth and nonLipschitz problems, and makes existing algorithms very slow for largescale applications. Together with the analytic solutions to lpnorm minimization with two specific values of p, i.e., p=1/2 and p=2/3, we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis. We first define the double nuclear norm and Frobenius/nuclear hybrid norm penalties, and then prove that they are in essence the Schatten1/2 and 2/3 quasinorms, respectively, which lead to much more tractable and scalable Lipschitz optimization problems. Our experimental analysis shows that both our m
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We use bioriented incidence relations in order to construct a KempfLaksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated BottSamelson resolutions. The second process led to the introduction of singular flag varieties, algebrogeometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular singular flag variety produces an embedded resolution of the Schubert variety within the Grassmannian.
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On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the ChengYau solution to Fefferman's complex MongeAmpere equation up to the boundary is obstructed by a local CR invariant of the boundary. For a bounded strictly pseudoconvex domain $\Omega\subset \mathbb{C}^2$ diffeomorphic to the ball, we prove that the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball, subject to the existence of a holomorphic vector field satisfying a mild approximate tangency condition along the boundary. In particular, by considering the Euler vector field multiplied by $i$ the result applies to all domains in a large $C^1$ open neighborhood of the unit ball in $\mathbb{C}^2$. The proof rests on establishing an integral identity involving the CR curvature of $\partial \Omega$ for any holomorphic vector field defined in a neighborhood of the boundary. The notion of ambient holomorphic vector field along the CR boundary generalizes natur
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The rise of electric vehicles (EVs) is unstoppable due to factors such as the decreasing cost of batteries and various policy decisions. These vehicles need to be charged and will therefore cause congestion in local distribution grids in the future. Motivated by this, we consider a charging station with finitely many parking spaces, in which electric vehicles arrive in order to get charged. An EV has a random parking time and a random charging time. Both the charging rate per vehicle and the charging rate possible for the station are assumed to be limited. Thus, the charging rate of uncharged EVs depends on the number of cars charging simultaneously. This model leads to a layered queueing network in which parking spaces with EV chargers have a dual role, of a server (to cars) and customers (to the grid). We are interested in the performance of the aforementioned model, focusing on the fraction of vehicles that get fully charged. To do so, we develop several bounds and asymptotic (fluid
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We introduce the class of Cartan triples as a generalization of the notion of a Cartan MASA in a von Neumann algebra. We obtain a onetoone correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi's theorem to this setting. This context contains that of Fulman's generalization of Cartan MASAs and we discuss his generalization in an appendix.
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We introduce a categorical analogue of Saito's notion of primitive forms. Let $W$ denote the potential $\frac{1}{n+1} x^{n+1}$. For the category $MF(W)$ of matrix factorizations of $W$ we prove that there exists a unique, up to nonzero constant, categorical primitive form. The corresponding genus zero categorical GromovWitten invariants of $MF(W)$ are shown to match with the invariants defined through unfolding of singularities of $W$.
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In this paper, we prove a central limit theorem and a moderate deviation principle for a perturbed stochastic CahnHilliard equation defined on [0, T]x [0, \pi]^d, with d \in {1,2,3}. This equation is driven by a spacetime white noise. The weak convergence approach plays an important role.
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The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying simple graph, closing an open problem posed by Sopena. We find that such oriented graphs admit a forbidden subgraph characterization, and such graphs can be both identified and constructed in polynomial time. We study the analytic properties of this polynomial and show that there exist oriented graphs which have chromatic polynomials have roots, including negative real roots, that cannot be realized as the root of any chromatic polynomial of a simple graph.
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We develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first author's previous work. The efficiency is improved by a theoretical result showing that the sum of data variables appears in most coefficients of the generator polynomial of elimination ideal. Furthermore, applying the known structures of Newton polytopes of discriminants, we can also efficiently deduce discriminants of the elimination ideals. For instance, the discriminants of 3 by 3 matrix model and one JukesCantor model in phylogenetics (with sizes over 30 GB and 8 GB text files, respectively) can be computed by our methods.
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The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann, which reflects an unexpected symmetry among energyminimizing periodic configurations. This formal duality was later on translated into a purely combinatorial property by Cohn, Kumar, Reiher and Sch\"urmann, where the corresponding combinatorial object was called formally dual pair. Almost all known examples of primitive formally dual pairs satisfy that the two subsets have the same size. Indeed, prior to this work, there is only one known example having subsets with unequal sizes in $\mathbb{Z}_2 \times \mathbb{Z}_4^2$. Motivated by this example, we propose a lifting construction framework and a recursive construction framework, which generate new primitive formally dual pairs from known ones. As an application, for $m \ge 2$, we obtain $m+1$ pairwise inequivalent primitive formally dual pairs in $\mathbb{Z}_2 \times \mathbb{Z}_4^{2m}$, which have subsets with unequal sizes.
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This paper introduces assume/guarantee contracts on continuoustime control systems, hereby extending contract theories for discrete systems to certain new model classes and specifications. Contracts are regarded as formal characterizations of control specifications, providing an alternative to specifications in terms of dissipativity properties or setinvariance. The framework has the potential to capture a richer class of specifications more suitable for complex engineering systems. The proposed contracts are supported by results that enable the verification of contract implementation and the comparison of contracts. These results are illustrated by an example of a vehicle following system.
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We study certain structural properties of fine zonotopal tilings, or cubillages, on cyclic zonotopes $Z(n,d)$ of an arbitrary dimension $d$ and their relations to $(d1)$separated collections of subsets of a set $\{1,2,\ldots,n\}$. (Collections of this sort are well known as strongly separated ones when $d=2$, and as chord separated ones when $d=3$.)
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We examine the dynamics of cuttingandshuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cutandshuffled are relaxed to allow for nonorthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cuttingandshuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Nonorthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the nonorthogonalaxes parameter space. Arnold tongue intersections along a fundamental symmetry
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Endtoend learning of communication systems enables joint optimization of transmitter and receiver, implemented as deep neural networkbased autoencoders, over any type of channel and for an arbitrary performance metric. Recently, an alternating training procedure was proposed which eliminates the need for an explicit channel model. However, this approach requires feedback of realvalued losses from the receiver to the transmitter during training. In this paper, we first show that alternating training works even with a noisy feedback channel. Then, we design a system that learns to transmit real numbers over an unknown channel without a preexisting feedback link. Once trained, this feedback system can be used to communicate losses during alternating training of autoencoders. Evaluations over additive white Gaussian noise and Rayleigh blockfading channels show that endtoend communication systems trained using the proposed feedback system achieve the same performance as when trained
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We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual. Under this correspondence, which extends the previously known isoradial case, the urban renewal (local move for dimer models) is equivalent to the Miquel move (local move for circle patterns). As a consequence the Miquel dynamics on circle patterns is governed by the octahedron recurrence. As special cases of these circle pattern embeddings, we recover harmonic embeddings for resistor networks and sembeddings for the Ising model.
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In this paper, we present a distributionally robust optimization (DRO) approach for the transmission expansion planning (TEP) problem, considering both long and shortterm uncertainties on the system load and renewable generation. Longterm uncertainty is represented on two interrelated levels. At the first level, as is customary in industry applications, the deep uncertainty faced in economic, political, environmental, and technological development is addressed based on plausible visions of longterm future scenarios (trends), traced by current experts beliefs. Subsequently, uncertaintyrelated parameters defining the probability distributions of the uncertain factors are partially refined for each longterm scenario, thereby inducing an ambiguity set. Finally, for each longterm scenario and induced ambiguity set, the inherent risk model for the shortterm uncertainty is described by means of conditional probability distributions. The mathematical problem is formulated as a distribu
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We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that BenjaminiSchramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and formulate a Berrytype conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.
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We study the energy distribution of harmonic 1forms on a compact hyperbolic Riemann surface $S$ that has a small separating closed geodesic. The result is applied to the question how close the Jacobian torus of $S$ comes to a torus that splits. The aim is to answer this and related questions in terms of geometric data of $S$ such as its injectivity radius and the lengths of geodesics that form a homology basis. This is version 1 of an extended paper in which also non separating small geodesics are considered.
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Local and categorytheoretical entropies associated with an endomorphism of finite length (i.e., with zerodimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to categorytheoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic. Furthermore, given a flat morphism of CohenMacaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is "additive". Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.
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Adinkras are combinatorial objects developed to study supersymmetry representations. Gates et al. introduced the "gadget" as a function of pairs of adinkras, obtaining some mysterious results for $(n=4, k=1)$ adinkras with computeraided computation. Specifically, very few values of the gadget actually appear, suggesting a great deal of symmetry in these objects. In this paper, we compute gadgets symbolically and explain some of these observed phenomena with group theory and combinatorics. Guided by this work, we give some suggestions for generalizations of the gadget to other values of the $n$ and $k$ parameters.
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We consider a family of higherdimensional noncommutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori, and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz noncommutative solenoids are direct limits of the Toeplitz extensions of noncommutative tori. We consider natural dynamics on these Toeplitz algebras, and compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrised by the probability measures on an (ordinary) solenoid.
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In contrast with many other convex optimization classes, stateoftheart semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The proposed methodology, based on the primaldual hybrid gradient method, allows the presence of linear inequalities without the need of adding extra slack variables and avoids solving a linear system at each iteration. More importantly, it does simultaneously compute the dual variables associated with the linear constraints. The main contribution of this work is to achieve a substantial speedup by effectively adjusting the proposed algorithm in order to exploit the lowrank property inherent to several semidefinite programming problems. This proposed modification is the key element that allows the operator splitting method to efficiently scale to larger instances. Convergence guaran
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We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the onepoint compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this, we study the cardinal number given by the smallest possible character of remote and far sets of separable metrizable spaces. Finally, we prove that in some cases a countable space has far points.
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If $\mathfrak{g} \subseteq \mathfrak{h}$ is an extension of Lie algebras over a field $k$ such that ${\rm dim}_k (\mathfrak{g}) = n$ and ${\rm dim}_k (\mathfrak{h}) = n + m$, then the Galois group ${\rm Gal} \, (\mathfrak{h}/\mathfrak{g})$ is explicitly described as a subgroup of the canonical semidirect product of groups ${\rm GL} (m, \, k) \rtimes {\rm M}_{n\times m} (k)$. An Artin type theorem for Lie algebras is proved: if a group $G$ whose order isinvertible in $k$ acts as automorphisms on a Lie algebra $\mathfrak{h}$, then $\mathfrak{h}$ is isomorphic to a skew crossed product $\mathfrak{h}^G \, \#^{\bullet} \, V$, where $\mathfrak{h}^G$ is the subalgebra of invariants and $V$ is the kernel of the Reynolds operator. The Galois group ${\rm Gal} \,(\mathfrak{h}/\mathfrak{h}^G)$ is also computed, highlighting the difference from the classical Galois theory of fields where the corresponding group is $G$. The counterpart for Lie algebras of Hilbert's Theorem 90 is proved and based on
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D. Bennequin and P. Baudot introduced a cohomological construction adapted to information theory, called 'information cohomology', that characterizes Shannon entropy through a cocycle condition. This text develops the relation between information cohomology and topos theory. We also introduce several new constructions and results. First, we define generalized information structures, as categories of finite random variables related by a notion of extension or refinement; classical and quantum probability spaces appear as models (representations) for these general structures. Generalized information structures form a category with finite products and coproducts. We prove that information cohomology is invariant under isomorphisms of generalized structures. Secondly, we prove that the relativelyfree bar construction gives a projective object for the computation of cohomology. Thirdly, we provide detailed computations of $H^1$ for classical probabilities and describe the degenerate cases.
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For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.
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In this paper, we study gravitational instantons (i.e., complete hyperk\"aler 4manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known endALE, ALF, ALG or ALH. 2.In ALG and ALHnonsplitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a longstanding question of Yau in ALG and ALH cases. 3.In ALFD_k case, it must have an O(4)multiplet.
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We analyze the optimal harvesting problem for an ecosystem of species that experience environmental stochasticity. Our work generalizes the current literature significantly by taking into account nonlinear interactions between species, statedependent prices, and species injections. The key generalization is making it possible to not only harvest, but also `seed' individuals into the ecosystem. This is motivated by how fisheries and certain endangered species are controlled. The harvesting problem becomes finding the optimal harvestingseeding strategy that maximizes the expected total income from the harvest minus the lost income from the species injections. Our analysis shows that new phenomena emerge due to the possibility of species injections. It is wellknown that multidimensional harvesting problems are very hard to tackle. We are able to make progress, by characterizing the value function as a viscosity solution of the associated HamiltonJacobiBellman (HJB) equations. Moreov
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This paper deals with the MonteCarlo methods for evaluating expectations of functionals of solutions to McKeanVlasov Stochastic Differential Equations (MVSDE) with drifts of superlinear growth. We assume that the MVSDE is approximated in the standard manner by means of an interacting particle system and propose two importance sampling (IS) techniques to reduce the variance of the resulting Monte Carlo estimator. In the \emph{complete measure change} approach, the IS measure change is applied simultaneously in the coefficients and in the expectation to be evaluated. In the \emph{decoupling} approach we first estimate the law of the solution in a first set of simulations without measure change and then perform a second set of simulations under the importance sampling measure using the approximate solution law computed in the first step. For both approaches, we use large deviations techniques to identify an optimisation problem for the candidate measure change. The decoupling approac
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The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if they have equivalent independence polynomials. We extend previous work by showing that independence equivalence class of every odd path has size 1, while the class can contain arbitrarily many graphs for even paths. We also prove that the independence equivalence class of every even cycle consists of two graphs when $n\ge 2$ except the independence equivalence class of $C_6$ which consists of three graphs. The odd case remains open, although, using irreducibility results from algebra, we were able show that for a prime $p \geq 5$ and $n\ge 1$ the independence equivalence class of $C_{p^n}$ consists of only two graphs.
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In this paper, we provide an accessible introduction to the theory of locally convex supermanifolds in the categorical approach. In this setting, a supermanifold is a functor $\mathcal{M}\colon\mathbf{Gr}\to\mathbf{Man}$ from the category of Grassmann algebras to the category of locally convex manifolds that has certain local models, forming something akin to an atlas. We give a mostly selfcontained, concrete definition of supermanifolds along these lines, closing several gaps in the literature on the way. If $\Lambda_n\in\mathbf{Gr}$ is the Grassmann algebra with $n$ generators, we show that $\mathcal{M}_{\Lambda_n}$ has the structure of a so called multilinear bundle over the base manifold $\mathcal{M}_\mathbb{R}$. We use this fact to show that the projective limit $\varprojlim_n\mathcal{M}_{\Lambda_n}$ exists in the category of manifolds. In fact, this gives us a faithful functor $\varprojlim\colon\mathbf{SMan}\to\mathbf{Man}$ from the category of supermanifolds to the category of
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Infinitesimal bialgebras were introduced by Joni and Rota. An infinitesimal bialgebra is at the same time an algebra and coalgebra, in such a way that the comultiplication is a derivation. Twenty years after Joni and Rota, Aguiar introduced the concept of an infinitesimal (nonunitary) Hopf algebra. In this paper we study infinitesimal unitary bialgebras and infinitesimal unitary Hopf algebras, in contrary to Aguiar's approach. Using an infinitesimal version of the Hochschild 1cocycle condition, we prove respectively that a class of decorated planar rooted forests is the free cocycle infinitesimal unitary bialgebra and free cocycle infinitesimal unitary Hopf algebra on a set. As an application, we obtain that the planar rooted forests is the free cocycle infinitesimal unitary Hopf algebra on the empty set.
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Generalising a construction of Falconer, we consider classes of $G_\delta$subsets of $\mathbb{R}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes. As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsupsets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for $\alpha \geq 1$, \[ \mathrm{dim}_\mathrm{H}\, \{ \, y :  T_a^n (x)  y < n^{\alpha} \text{ infinitely often} \, \} = \frac{1}{\alpha}, \] for almost every $x \in [1a,1]$, where $T_a$ is a quadratic map with $a$ in a set of parameters described by Benedicks and Carleson.
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In this paper, we construct a theory of integration of Voevodsky motives over a perfect field $k$, and show that it circumvents some of the complications of motivic integration, leading to new arithmetic and geometric results concerning Kequivalent $k$varieties. One main application is that up to direct summing a common Chow motive, Kequivalent smooth projective $k$varieties have the same $\mathbb{Z}[1/p]$Chow motives ($p$ is the characteristic exponent of $k$), partially answering a conjecture of ChinLung Wang. In addition to generalizing a theorem of Kontsevich on the equality of Hodge numbers of Kequivalent smooth projective complex varieties, we show that such varieties have isomorphic \textit{integral} singular cohomology groups. On the arithmetic side, we show that Kequivalent smooth $k$varieties have isomorphic $\ell$adic Galois representations up to semisimplification. Furthermore, we connect this theory of integration of Voevodsky motives to the existence of motivic
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In this paper, we study an interference alignment (IA) scheme with finite time extension and beamformer selection method with low computational complexity for X channel. An IA scheme with a chain structure by the Latin square is proposed for Kx3 multipleinput multipleoutput (MIMO) X channel. Since the proposed scheme can have a larger set of possible beamformers than the conventional schemes, its performance is improved by the efficient beamformer selection for a given channel. Also, we propose a condition number (CN) based beamformer selection method with low computational complexity and its performance improvement is numerically verified.
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In this paper, we study a new notion of scaled minimaxity for sparse estimation in highdimensional linear regression model. We present more optimistic lower bounds than the one given by the classical minimax theory and hence improve on existing results. We recover sharp results for the global minimaxity as a consequence of our study. Fixing the scale of the signaltonoise ratio, we prove that the estimation error can be much smaller than the global minimax error. We construct a new optimal estimator for the scaled minimax sparse estimation. An optimal adaptive procedure is also described.
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In systems and synthetic biology, much research has focused on the behavior and design of single pathways, while, more recently, experimental efforts have focused on how crosstalk (coupling two or more pathways) or inhibiting molecular function (isolating one part of the pathway) affects systemslevel behavior. However, the theory for tackling these larger systems in general has lagged behind. Here, we analyze how joining networks (e.g., crosstalk) or decomposing networks (e.g., inhibition or knockouts) affects three properties that reaction networks may possessidentifiability (recoverability of parameter values from data), steadystate invariants (relationships among species concentrations at steady state, used in model selection), and multistationarity (capacity for multiple steady states, which correspond to multiple cell decisions). Specifically, we prove results that clarify, for a network obtained by joining two smaller networks, how properties of the smaller networks can b
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We prove a compactness theorem for full Booleanvalued models. As an application, we show that if $T$ is a complete countable theory and $\mathcal{B}$ is a complete Boolean algebra, then $\lambda^+$saturated $\mathcal{B}$valued models of $T$ exist. Moreover, if $\mathcal{U}$ is an ultrafilter on $T$ and $\mathbf{M}$ is a $\lambda^+$saturated $\mathcal{B}$valued model of $T$, then whether or not $\mathbf{M}/\mathcal{U}$ is $\lambda^+$saturated just depends on $\mathcal{U}$ and $T$; we say that $\mathcal{U}$ $\lambda^+$saturates $T$ in this case. We show that Keisler's order can be formulated as follows: $T_0 \trianglelefteq T_1$ if and only if for every cardinal $\lambda$, for every complete Boolean algebra $\mathcal{B}$ with the $\lambda^+$c.c., and for every ultrafilter $\mathcal{U}$ on $\mathcal{B}$, if $\mathcal{U}$ $\lambda^+$saturates $T_1$, then $\mathcal{U}$ $\lambda^+$saturates $T_0$.
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In this paper we consider the large genus asymptotics for two classes of SiegelVeech constants associated with an arbitrary connected stratum $\mathcal{H} (\alpha)$ of Abelian differentials. The first is the saddle connection SiegelVeech constant $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (\alpha) \big)$ counting saddle connections between two distinct, fixed zeros of prescribed orders $m_i$ and $m_j$, and the second is the area SiegelVeech constant $c_{\text{area}} \big( \mathcal{H}(\alpha) \big)$ counting maximal cylinders weighted by area. By combining a combinatorial analysis of explicit formulas of EskinMasurZorich that express these constants in terms of MasurVeech strata volumes, with a recent result for the large genus asymptotics of these volumes, we show that $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (\alpha) \big) = (m_i + 1) (m_j + 1) \big( 1 + o(1) \big)$ and $c_{\text{area}} \big( \mathcal{H}(\alpha) \big) = \frac{1}{2} + o(1)$, both as $\alpha = 2g  2$ tends
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Consider a chordal random curve model on a planar graph, in the scaling limit when a finemesh graph approximates a simplyconnected planar domain. The wellknown precompactness conditions of Kemppainen and Smirnov show that certain "crossing estimates" guarantee the subsequential weak convergence of the random curves in the topology of unparametrized curves, as well as in a topology inherited from curves on the unit disc via conformal maps. We complement this result by proving that proceeding to weak limit commutes with changing topology, i.e., limits of conformal images are conformal images of limits, without imposing any boundary regularity assumptions on the domains where the random curves lie. Treating such rough boundaries becomes necessary, e.g., in convergence proofs to multiple SLEs. The result in this generality has not been explicated before and is not trivial, which we demonstrate by giving warning examples and deducing strong consequences.
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FasterthanNyquist (FTN) signaling is a promising nonorthogonal physical layer transmission technique to improve the spectral efficiency of future communication systems but at the expense of intersymbolinterference (ISI). In this paper, we investigate the detection problem of FTN signaling and formulate the sequence estimation problem of any $M$ary phase shift keying (PSK) FTN signaling as an optimization problem that turns out to be nonconvex and nondeterministic polynomial time (NP)hard to solve. We propose a novel algorithm, based on concepts from semidefinite relaxation (SDR) and Gaussian randomization, to detect any $M$ary PSK FTN signaling in polynomial time complexity regardless of the constellation size $M$ or the ISI length. Simulation results show that the proposed algorithm strikes a balance between the achieved performance and the computational complexity. Additionally, results show the merits of the proposed algorithm in improving the spectral efficiency when compar
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Motivated by problems in percolation theory, we study the following 2player positional game. Let $\Lambda_{m \times n}$ be a rectangular gridgraph with $m$ vertices in each row and $n$ vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims $p$ (asyet unclaimed) edges of the board $\Lambda_{m \times n}$, while on each of his turns Breaker claims $q$ (asyet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the lefthand side of the board to its righthand side, otherwise Breaker wins. We call this game the $(p,q)$crossing game on $\Lambda_{m \times n}$. Given $m,n\in \mathbb{N}$, for which pairs $(p,q)$ does Maker have a winning strategy for the $(p,q)$crossing game on $\Lambda_{m \times n}$? The $(1,1)$case corresponds exactly to the popular game of Bridgit, which is well understood due to it being a special case of the older Shannon
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We obtain limit theorems for $\Phi(A^p)^{1/p}$ and $(A^p\sigma B)^{1/p}$ as $p\to\infty$ for positive matrices $A,B$, where $\Phi$ is a positive linear map between matrix algebras (in particular, $\Phi(A)=KAK^*$) and $\sigma$ is an operator mean (in particular, the weighted geometric mean), which are considered as certain reciprocal LieTrotter formulas and also a generalization of Kato's limit to the supremum $A\vee B$ with respect to the spectral order.
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We present a sample path dependent measure of causal influence between time series. The proposed causal measure is a random sequence, a realization of which enables identification of specific patterns that give rise to high levels of causal influence. We show that these patterns cannot be identified by existing measures such as directed information (DI). We demonstrate how sequential prediction theory may be leveraged to estimate the proposed causal measure and introduce a notion of regret for assessing the performance of such estimators. We prove a finite sample bound on this regret that is determined by the worst case regret of the sequential predictors used in the estimator. Justification for the proposed measure is provided through a series of examples, simulations, and application to stock market data. Within the context of estimating DI, we show that, because joint Markovicity of a pair of processes does not imply the marginal Markovicity of individual processes, commonly used pl
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Ptychography is a popular imaging technique that combines diffractive imaging with scanning microscopy. The technique consists of a coherent beam that is scanned across an object in a series of overlapping positions, leading to reliable and improved phase reconstruction. Computationally, ptychography is extremely expensive, as ptychographic microscopes allow for large fields to be imaged at high resolution. In this work, we propose a multigridbased optimization framework to reduce the computational burdens of largescale ptychographic phase retrieval. Our proposed method exploits the inherent hierarchical structures in ptychography through tailored restriction and prolongation operators for the object and data domains. Our numerical results show that our proposed scheme accelerates the convergence of its underlying solver and outperforms the stateoftheart method in the optics community.
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Nonconcave maximization has been the subject of much recent study in the optimization and machine learning communities, specifically in deep learning. Recent papers ((Ge \etal 2015, Lee \etal 2017) and references therein) indicate that first order methods work well and avoid saddles points. Results as in (Lee \etal 2017), however, are limited to the \textit{unconstrained} case or for cases where the critical points are in the interior of the feasibility set, which fail to capture some of the most interesting applications. In this paper we focus on \textit{constrained} nonconcave maximization. We analyze a variant of a wellestablished algorithm in machine learning called Multiplicative Weights Update (MWU) for the maximization problem $\max_{\mathbf{x} \in D} P(\mathbf{x})$, where $P$ is nonconcave, twice continuously differentiable and $D$ is a product of simplices. We show that MWU converges almost always for small enough stepsizes to critical points that satisfy the second order
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Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. First, we build on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain "critical function" changes sign. Here, we allow for more general parametrizations, which make it much easier to determine the existence of a sign change. This is particularly simple when the steadystate equations
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The contribution of this paper is twofold: First, we prove existence and uniqueness of the weighted maximum likelihood estimator of the multivariate Student$t$ distribution and propose an efficient algorithm for its computation that we call generalized multivariate myriad filter (GMMF). Second, we use the GMMF in a nonlocal framework for the denoising of images corrupted by different kinds of noise. The resulting method is very flexible and can handle very heavytailed noise such as Cauchy noise, but also also Gaussian or wrapped Cauchy noise.
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U.S. PIRG  a nonprofit that uses grassroots methods to advocate for political change  found that 90 percent of manufacturers it contacted claimed that a third party repair would void its warranty. "PIRG researched the warranty information of 50 companies in the Association of Home Appliance Manufacturers (AHAM)  an industry group of notorious for lobbying to protect is repair monopolies  and found that 45 of them claimed independent repair would void their warranty," Motherboard reports. From the report: PIRG poured over the documentation for 50 companies such as Bissell, Whirlpool, and Panasonic to document their warranty policies. When it couldn't find clear language about warranty and repair, it reached out to the companies via their customer service lines. The overwhelming majority of the companies told PIRG that independent repair would void the warranty. The 1975 MagnusonMoss Warranty Act states that no manufacturer who charges more than $5 for a product can put repair
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In a new video from robotics company Boston Dynamics, which Alphabet sold to SoftBank last year, a robot is shown hopping over a log and then up a series of blocks, an activity called parkour. From a report: In previous videos, the robot did a backflip  now it's leaping over obstacles and climbing up large, uneven stairs with fleetfooted ease. But Atlas wasn't always so graceful. In some of the first videos where Boston Dynamics' robots could walk upright, way back in 2015, Atlas lumbered through the woods, looking like it was narrowly avoiding falling with each step, rather than moving with any kind of purpose.
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