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1. Tempered Homogeneous Spaces III

   Benoist, Y.; Kobayashi, T. (January 2021, Journal of Lie Theory)

   Abstract. Let 𝘎 be a real semisimple algebraic Lie group and 𝘏 a real reductive algebraic subgroup. We describe the pairs (𝘎,𝘏) for which the representation of 𝘎 in L²(𝘎/𝘏) is tempered. The proof gives the complete list of pairs (𝘎,𝘏) for which L²(𝘎/𝘏) is not tempered. When 𝘎 and 𝘏 are complex Lie groups, the temperedness condition is characterized by the fact that 𝘵𝘩𝘦 𝘴𝘵𝘢𝘣𝘪𝘭𝘪𝘻𝘦𝘳 𝘪𝘯 𝘏 𝘰𝘧 𝘢 𝘨𝘦𝘯𝘦𝘳𝘪𝘤 𝘱𝘰𝘪𝘯𝘵 𝘰𝘯 𝘎/𝘏 𝘪𝘴 𝘷𝘪𝘳𝘵𝘶𝘢𝘭𝘭𝘺 𝘢𝘣𝘦𝘭𝘪𝘢𝘯. 𝘔𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘤𝘴 𝘚𝘶𝘣𝘫𝘦𝘤𝘵 𝘊𝘭𝘢𝘴𝘴𝘪𝘧𝘪𝘤𝘢𝘵𝘪𝘰𝘯: Primary 22𝐄46; secondary 43𝐀85, 22𝐅30. 𝘒𝘦𝘺 𝘞𝘰𝘳𝘥𝘴: Lie groups, homogeneous spaces, tempered representations, unitary representations, matrix coefficients, symmetric spaces. 

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2. Uniformly Bounded Arrays and Mutually Algebraic Structures

   https://doi.org/doi:10.1215/00294527-2020-0004  

   Laskowski, Michael C.; Terry, Caroline A. (January 2020, Notre Dame journal of formal logic)

   We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T. 

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3. Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

   https://doi.org/10.15673/pigc.v16i2.2648  

   Abebaw, Tilahun; Gatsinzi, Jean-Baptiste; Yeruk, Smegnsh Demelash (June 2024, Proceedings of the International Geometry Center)

   The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models. 

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4. Asymptotics of singular values for quantum derivatives

   https://doi.org/10.1090/tran/8827  

   Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy (March 2023, Transactions of the American Mathematical Society)

   We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f f from the homgeneous Sobolev space W ˙ d 1 ( R d ) \dot {W}^1_d(\mathbb {R}^d) on R d . \mathbb {R}^d. The asymptotic coefficient ‖ ∇ f ‖ L d ( R d ) \|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal L d , ∞ , \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C ∗ C^{\ast } -algebraic notion of the principal symbol mapping on R d \mathbb {R}^d , as developed recently by the last two authors and collaborators. 

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5. Synthesizing Specifications

   https://doi.org/10.1145/3622861  

   Park, Kanghee; D'Antoni, Loris; Reps, Thomas (October 2023, Proceedings of the ACM on Programming Languages)

   Every program should be accompanied by a specification that describes important aspects of the code's behavior, but writing good specifications is often harder than writing the code itself. This paper addresses the problem of synthesizing specifications automatically, guided by user-supplied inputs of two kinds: i) a query posed about a set of function definitions, and ii) a domain-specific language L in which the extracted property is to be expressed (we call properties in the language L-properties). Each of the property is a best L-property for the query: there is no other L-property that is strictly more precise. Furthermore, the set of synthesized L-properties is exhaustive: no more L-properties can be added to it to make the conjunction more precise. We implemented our method in a tool, Spyro. The ability to modify both the query and L provides a Spyro user with ways to customize the kind of specification to be synthesized. We use this ability to show that Spyro can be used in a variety of applications, such as mining program specifications, performing abstract-domain operations, and synthesizing algebraic properties of program modules. 

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