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1. The Weil–Petersson gradient flow ofrenormalized volume and 3–dimensional convex cores

   https://doi.org/10.2140/gt.2023.27.3183  

   Bridgeman, Martin; Brock, Jeffrey; Bromberg, Kenneth (January 2023, Geometry topology)

   We use the Weil–Petersson gradient flow for renormalized volume to study the space CC(N;S,X) of convex cocompact hyperbolic structures on the relatively acylindrical 3-manifold (N;S). Among the cases of interest are the deformation space of an acylindrical manifold and the Bers slice of quasifuchsian space associated to a fixed surface. To treat the possibility of degeneration along flow-lines to peripherally cusped structures, we introduce a surgery procedure to yield a surgered gradient flow that limits to the unique structure M_geod in CC( N;S,X) with totally geodesic convex core boundary facing S. Analyzing the geometry of structures along a flow line, we show that if V_R(M) is the renormalized volume of M, then V_R(M)−V_R(M_geod) is bounded below by a linear function of the Weil Petersson distance d_WP(∂_c M,∂_cM_geod), with constants depending only on the topology of S. The surgered flow gives a unified approach to a number of problems in the study of hyperbolic 3-manifolds, providing new proofs and generalizations of well-known theorems such as Storm’s result that M geod has minimal volume for N acylindrical and the second author’s result comparing convex core volume and Weil–Petersson distance for quasifuchsian manifolds. 

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2. Algorithmic Aspects of Immersibility and Embeddability

   https://doi.org/10.1093/imrn/rnae170  

   Manin, Fedor; Weinberger, Shmuel (August 2024, International Mathematics Research Notices)

   Abstract We analyze an algorithmic question about immersion theory: for which $$m$$, $$n$$, and $$CAT=\textbf{Diff}$$ or $$\textbf{PL}$$ is the question of whether an $$m$$-dimensional $CAT$-manifold is immersible in $$\mathbb{R}^{n}$$ decidable? We show that PL immersibility is decidable in all cases except for codimension 2, whereas smooth immersibility is decidable in all odd codimensions and undecidable in many even codimensions. As a corollary, we show that the smooth embeddability of an $$m$$-manifold with boundary in $$\mathbb{R}^{n}$$ is undecidable when $n-m$ is even and $$11m \geq 10n+1$$. 

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3. The Choice and Agreement Problems of a Random Function

   Meir, Or; Tal, Avishay (March 2018, Information processing letters)

   The direct-sum question is a classical question that asks whether performing a task on m independent inputs is m times harder than performing it on a single input. In order to study this question, Beimel et al [BBKW14] introduced the following related problems: • The choice problem: Given m distinct instances, choose one of them and solve it. • The agreement problem: Given m distinct instances, output a solution that is correct for at least one of them. It is easy to see that these problems are no harder than performing the original task on a single instance, and it is natural to ask whether it is strictly easier or not. In particular, proving that the choice problem is not easier is necessary for proving a direct-sum theorem, and is also related to the KRW composition conjecture [KRW95]. In this note, we observe that in a variety of computational models, if f is a random function then with high probability its corresponding choice and agreement problem are not much easier than computing f on a single instance (as long as m is noticeably smaller than 2^n) 

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4. Group 14 Central Atoms and Halogen Bonding in Different Dielectric Environments: How Germanium Outperforms Silicon

   https://doi.org/10.1002/cplu.202100294  

   Donald, Kelling_J; Prasad, Supreeth; Wilson, Kaitlin (August 2021, ChemPlusChem)

   Abstract The nature of halogen bonding under different dielectric conditions remains underexplored, especially for inorganic systems. The structural and energetic properties of model halogen bonded complexes (R₃M−I—NH₃for R=H and F, and M=C, Si, and Ge) are examined computationally for relative permittivities between 1 and 109 using an implicit solvent model. We confirm and assess the exceptionally high maximum potentials at the sigma hole on I (V_s,max) in F₃Ge−I relative to cases where M=C or Si. In particular, Ge far outperforms Si in mediating inductive effects. Linear relationships, typically with R²>0.97, are identified betweenV_s,max, the full point charge on I in R₃M−I, and the interaction energy, and optimized I—N distance in the complexes. An anomalous trend is identified in which, for each M, F₃M−I—NH₃becomeslessstable as the optimized I—N distance getsshorterin different dielectric environments; it is explained using the F−I—NH₃complex as a reference. 

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5. The effect of adding randomly weighted edges

   https://doi.org/10.1137/20M1335418  

   Frieze, A. (January 2021, SIAM journal on discrete mathematics)

   We consider the following question. We have a dense regular graph $$G$$ with degree $$\a n$$, where $$\a>0$$ is a constant. We add $m=o(n^2)$ random edges. The edges of the augmented graph $G(m)$ are given independent edge weights $$X(e),e\in E(G(m))$$. We estimate the minimum weight of some specified combinatorial structures. We show that in certain cases, we can obtain the same estimate as is known for the complete graph, but scaled by a factor $$\a^{-1}$$. We consider spanning trees, shortest paths and perfect matchings in (pseudo-random) bipartite graphs. 

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