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Published: Apr 14, 2021

Authors: Vladimir Drinfeld, Vladimir Drinfeld

Published: Apr 14, 2021

Authors: Vladimir Drinfeld, Vladimir Drinfeld

By a ring groupoid we mean an animated ring whose i-th homotopy groups are zero for all i>1. In this expository note we give an elementary treatment of the (2,1)-category of ring groupoids (i.e., without referring to general animated rings and without using n-categories for n>2). The note is motivated by the fact that ring stacks play a central role in the Bhatt-Lurie approach to prismatic cohomology.

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Published: Apr 15, 2021

Authors: William Gasarch, William Gasarch

Published: Apr 15, 2021

Authors: William Gasarch, William Gasarch

Hilbert's 10th problem, stated in modern terms, is: Find an algorithm that will, given $p \in \mathbb{Z}[x_1,\ldots,x_n]$ determine if there exists $a_1, a_2, \ldots, a_n \in \mathbb{Z}$ such that $p(a_1,\ldots,a_n)=0$. Davis, Putnam, Robinson, and Matijasevic showed that there is no such algorithm. We look at what happens (1) for fixed degree and number of variables, (2) for particular equations, and (3) for variants which reduce the number of variables needed for undecidability results.

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Published: Apr 15, 2021

Authors: Ivan Beschastnyi, Ivan Beschastnyi

Published: Apr 15, 2021

Authors: Ivan Beschastnyi, Ivan Beschastnyi

The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.

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Published: Apr 16, 2021

Authors: Duy Phan, Lassi Paunonen, Duy Phan

Published: Apr 16, 2021

Authors: Duy Phan, Lassi Paunonen, Duy Phan

We study output tracking and disturbance rejection for an Euler-Bernoulli beam with Kelvin-Voigt damping. The system has distributed control and pointwise observation. As our main result we design a finite-dimensional low-order internal model based controller that is based on a spectral Galerkin method and model reduction by Balanced Truncation. The performance of the designed controller is demonstrated with numerical simulations and compared to the performance of a low-gain internal model based controller.

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Published: Apr 16, 2021

Authors: Aravind Asok, Aravind Asok

Published: Apr 16, 2021

Authors: Aravind Asok, Aravind Asok

The quadric $\operatorname{Q}_{2n}$ is the ${\mathbb Z}$-scheme defined by the equation $\sum_{i=1}^n x_i y_i = z(1-z)$. We show that $\operatorname{Q}_{2n}$ is a homogeneous space for the split reductive group scheme $\operatorname{SO}_{2n+1}$ over ${\mathbb Z}$. The quadric $\operatorname{Q}_{2n}$ is know to have the ${\mathbb A}^1$-homotopy type of a motivic sphere and the identification as a homogeneous space allows us to give a characteristic independent affine representability statement for motivic spheres. This last observation allows us to give characteristic independent comparison results between Chow--Witt groups, motivic stable cohomotopy groups and Euler class groups.

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Published: Apr 15, 2021

Authors: Michele Carmassi, Michele Carmassi

Published: Apr 15, 2021

Authors: Michele Carmassi, Michele Carmassi

Let $G$ be a reductive linear algebraic group over an algebraically closed field $\mathbb{K}$ of characteristic $2$. Fix a parabolic subgroup $P$ such that the unipotent radical is abelian and a Levi subgroup $L\subseteq P$. We parametrize the orbits of a Borel $B\subseteq P$ over the Hermitian symmetric variety $G/L$ supposing the root system $\Phi$ is irreducible. For $\Phi$ simply laced we prove a combinatorial characterization of the Bruhat order over these orbits. We also prove a formula to compute the dimension of the orbits from combinatorial characteristics of their representatives.

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Published: Apr 15, 2021

Authors: Borys Kuca, Borys Kuca

Published: Apr 15, 2021

Authors: Borys Kuca, Borys Kuca

For a polynomial progression $$(x, x+P_1(y), ..., x+P_{t}(y)),$$ we define four notions of complexity: Host-Kra complexity, Weyl complexity, true complexity and algebraic complexity. The first two describe the smallest characteristic factor of the progression, the third one refers to the smallest-degree Gowers norm controlling the progression, and the fourth one concerns algebraic relations between terms of the progressions. We conjecture that these four notions are equivalent, which would give a purely algebraic criterion for determining the smallest Host-Kra factor or the smallest Gowers norm controlling a given progression. We prove this conjecture for all progressions whose terms only satisfy homogeneous algebraic relations and linear combinations thereof. This family of polynomial progressions includes, but is not limited to, arithmetic progressions, progressions with linearly independent polynomials $P_1, ..., P_t$ and progressions whose terms satisfy no quadratic relations. For progressions that satisfy only linear relations, such as $$(x, x+y^2, x+2y^2, x+y^3, x+2y^3),$$ we derive several combinatorial and dynamical corollaries: (1) an estimate for the count of such progressions in subsets of cyclic groups or totally ergodic dynamical systems; (2) a lower bound for multiple recurrence; (3) and a popular common difference result in cyclic groups. Lastly, we show that Weyl complexity and algebraic complexity always agree, which gives a straightforward algebraic description of Weyl complexity.

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Published: Apr 15, 2021

Authors: Pierrot, Adeline, et al

Published: Apr 15, 2021

Authors: Pierrot, Adeline, et al

This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit. Second, and unexpectedly given the above results, we show that for $\beta >0$ sufficiently small, the expected number of independent sets of size $\beta n$ in a uniform random cograph with $n$ vertices grows exponentially fast with $n$. We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions.

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Published: Apr 15, 2021

Authors: Mansi Suryawanshi, Narayan Rakshit, Jaydeb Sarkar, Mansi Suryawanshi

Published: Apr 15, 2021

Authors: Mansi Suryawanshi, Narayan Rakshit, Jaydeb Sarkar, Mansi Suryawanshi

Let $n > 1$. Let $\{U_{ij}\}_{1 \leq i < j \leq n}$ be $\binom{n}{2}$ commuting unitaries on some Hilbert space $\mathcal{H}$, and suppose $\mathcal{U}_n = \{U_{ij}\}_{i\neq j} \subseteq \mathcal{B}(\mathcal{H})$, where $U_{ji} := U_{ij}^*$, $1 \leq i < j \leq n$. An $n$-tuple of isometries $V = (V_1, \ldots ,V_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted isometry if $V_i$'s are in the commutator $\{U_{st}: s \neq t\}'$, and $V_i^*V_j=U_{ij}^*V_jV_i^*$, $i \neq j$. We prove that each $\mathcal{U}_n$-twisted isometry admits a von Neumann-Wold type orthogonal decomposition. We prove that the universal $C^*$-algebra generated by $\mathcal{U}_n$-twisted isometry is nuclear. The universal $C^*$-algebra generated by an $n$-tuple of $\mathcal{U}_n$-twisted unitaries is called the generalized noncommutative $n$-torus. We exhibit concrete analytic models of $\mathcal{U}_n$-twisted isometries, and establish connections between unitary equivalence classes of the irreducible representations of the $C^*$-algebras generated by $\mathcal{U}_n$-twisted isometries and the unitary equivalence classes of the non-zero irreducible representations of generalized noncommutative tori. Our motivation stems from the classical rotation $C^*$-algebras, Heisenberg group $C^*$-algebras, and a recent work of de Jeu and Pinto.

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Published: Apr 14, 2021

Authors: Jinniao Qiu, Cristina Cipriani, Hui Huang, Jinniao Qiu

Published: Apr 14, 2021

Authors: Jinniao Qiu, Cristina Cipriani, Hui Huang, Jinniao Qiu

Recently a continuous description of the particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in arXiv:2012.05613 where the authors formally showed the link between PSO and the consensus based optimization (CBO) through zero-inertia limit. This paper is devoted to solving this theoretical open problem proposed in arXiv:2012.05613 by providing a rigorous derivation of CBO from PSO through the limit of zero inertia, and a quantified convergence rate is obtained as well. The proofs are based on a probabilistic approach by investigating the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space and the results are illustrated with some numerical examples.

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