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1. Resistance scaling on 4 N -carpets

   https://doi.org/10.1515/forum-2020-0330  

   Canner, Claire; Hayes, Christopher; Huang, Robin; Orwin, Michael; Rogers, Luke G. (December 2021, Forum Mathematicum)

   Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F , let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F . 

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2. Crystal structures of three copper(II)–2,2′-bipyridine (bpy) compounds, [Cu(bpy) ₂ (H ₂ O)][SiF ₆ ]·4H ₂ O, [Cu(bpy) ₂ (TaF ₆ ) ₂ ] and [Cu(bpy) ₃ ][TaF ₆ ] ₂ and a related coordination polymer, [Cu(bpy)(H ₂ O) ₂ SnF ₆ ] _n

   https://doi.org/10.1107/S2056989021000633  

   Nisbet, Matthew L.; Hiralal, Emily; Poeppelmeier, Kenneth R. (February 2021, Acta Crystallographica Section E Crystallographic Communications)

   null (Ed.)

   We report the hydrothermal syntheses and crystal structures of aquabis(2,2′-bipyridine-κ 2 N , N ′)copper(II) hexafluoridosilicate tetrahydrate, [Cu(bpy) 2 (H 2 O)][SiF 6 ]·4H 2 O (bpy is 2,2′-bipyridine, C 10 H 8 N 2 ), (I), bis(2,2′-bipyridine-3κ 2 N , N ′)-di-μ-fluorido-1:3κ 2 F : F ;2:3κ 2 F : F -decafluorido-1κ 5 F ,2κ 5 F -ditantalum(V)copper(II), [Cu(bpy) 2 (TaF 6 ) 2 ], (II), tris(2,2′-bipyridine-κ 2 N , N ′)copper(II) bis[hexafluoridotantalate(V)], [Cu(bpy) 3 ][TaF 6 ] 2 , (III), and catena -poly[[diaqua(2,2′-bipyridine-κ 2 N , N ′)copper(II)]-μ-fluorido-tetrafluoridotin-μ-fluorido], [Cu(bpy)(H 2 O) 2 SnF 6 ] n , (IV). Compounds (I), (II) and (III) contain locally chiral copper coordination complexes with C 2 , D 2 , and D 3 symmetry, respectively. The extended structures of (I) and (IV) are consolidated by O—H...F and O—H...O hydrogen bonds. The structure of (III) was found to be a merohedral (racemic) twin. 

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3. Averages Along the Primes: Improving and Sparse Bounds

   https://doi.org/10.1515/conop-2020-0003  

   Han, Rui; Krause, Ben; Lacey, Michael T.; Yang, Fan (February 2020, Concrete Operators)

   Abstract Consider averages along the prime integers ℙ given by {\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓ p -improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds {N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}. The maximal function 𝒜 * f = sup N |𝒜 N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known. 

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4. Counting Real Roots in Polynomial-Time via Diophantine Approximation

   https://doi.org/10.1007/s10208-022-09599-z  

   Rojas, J. Maurice (November 2022, Foundations of Computational Mathematics)

   Peter Burgisser (Ed.)

   Suppose A = {a 1 , . . . , a n+2 } ⊂ Z n has cardinality n + 2, with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = ( f 1 , . . . , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log(dH ). We also discuss a number- theoretic hypothesis that would imply a further speed-up to time polynomial in n as well. 

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5. Quantum algorithms and approximating polynomials for composed functions with shared inputs

   https://doi.org/10.22331/q-2021-09-16-543  

   Bun, Mark; Kothari, Robin; Thaler, Justin (September 2021, Quantum)

   We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q ( f ) , we give an algorithm for evaluating F using O ~ ( Q ( f ) ⋅ n ) quantum queries. This improves on the bound of O ( Q ( f ) ⋅ n ) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O ~ ( n 1 − 2 − d ) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0 ∘ ⊕ circuits of depth d + 1 require size Ω ~ ( n 1 / ( 1 − 2 − d ) ) ≥ ω ( n 1 + 2 − d ) to compute the Inner Product function even on average. The previous best size lower bound was Ω ( n 1 + 4 − ( d + 1 ) ) and only held in the worst case (Cheraghchi et al., JCSS 2018). 

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