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1. Computing the Caratheodory Number of a Point

   Bereg, Sergey; Haghpanah, Mohammadreza (August 2020, Canadian Conference on Computational Geometry)

   Caratheodory’s theorem says that any point in the convex hull of a set $$P$$ in $R^d$ is in the convex hull of a subset $P'$ of $$P$$ such that $$|P'| \le d + 1$$. For some sets P, the upper bound d + 1 can be improved. The best upper bound for P is known as the Caratheodory number [2, 15, 17]. In this paper, we study a computational problem of finding the smallest set $P'$ for a given set $$P$$ and a point $$p$$. We call the size of this set $P'$, the Caratheodory number of a point p or CNP. We show that the problem of deciding the Caratheodory number of a point is NP-hard. Furthermore, we show that the problem is k-LDT-hard. We present two algorithms for computing a smallest set $P'$, if CNP= 2,3. Bárány [1] generalized Caratheodory’s theorem by using d+1 sets (colored sets) such that their convex hulls intersect. We introduce a Colorful Caratheodory number of a point or CCNP which can be smaller than d+1. Then we extend our results for CNP to CCNP. 

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2. 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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3. A dip-and-read impedimetric electrochemical sensor for orthophosphate monitoring

   https://doi.org/10.21203/rs.3.rs-4691772/v1  

   Moreira, Geisianny; Shaw, Alex B; Amin, Nafisa; Gao, Wei; McLamore, Eric (July 2024, Research Square)

   Abstract Phosphorus (P) is an essential element for all life forms and a finite resource. P cycle plays a vital role in regulating primary productivity, making it a limiting nutrient for agricultural production and increasing the development of fertilizers through extractive mining. However, excessive P may cause detrimental environmental effects on aquatic and agricultural ecosystems. As a result, there is a pressing need for conservation and management of P loads through analytical techniques to measure P and precisely determine P speciation. Here, we explore a new 2D sorbent structure (GO-PDDA) for sensing orthophosphate in aqueous samples. The sorbent mimics a group of phosphate-binding proteins in nature and is expected to bind orthophosphate in solution. Laser-induced graphene (LIG) was coated with GO-PDDA using a drop-cast method. Electrochemical impedance spectroscopy was used as a transduction technique for electrochemical sensing of orthophosphate (HPO₄²⁻) and selectivity assay for chloride, sulfate and nitrate in buffer at pH 8. The analytical sensitivity was estimated to be 347 ± 90.2 Ω/ppm with a limit of detection of 0.32 ± 0.04 ppm. Selectivity assays demonstrate that LIG-GO-PDDA is 95% more selective for ortho-P over sulfate and 80% more selective over chloride and nitrate. The developed sensor can be reused after surface regeneration with an acidic buffer (pH 5), with slight changes in sensor performance. Our results show that the sorbent structure is a promising candidate for developing electrochemical sensors for environmental monitoring of orthophosphate and may provide reliable data to support sustainable P management. 

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4. Affine Stresses, Inverse Systems, and Reconstruction Problems

   https://doi.org/10.1093/imrn/rnad294  

   Murai, Satoshi; Novik, Isabella; Zheng, Hailun (December 2023, International Mathematics Research Notices)

   Abstract A conjecture of Kalai asserts that for $$d\geq 4$$, the affine type of a prime simplicial $$d$$-polytope $$P$$ can be reconstructed from the space of affine $$2$$-stresses of $$P$$. We prove this conjecture for all $$d\geq 5$$. We also prove the following generalization: for all pairs $(i,d)$ with $$2\leq i\leq \lceil \frac d 2\rceil -1$$, the affine type of a simplicial $$d$$-polytope $$P$$ that has no missing faces of dimension $$\geq d-i+1$$ can be reconstructed from the space of affine $$i$$-stresses of $$P$$. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: it was proved by Nagel that for any simplicial $(d-1)$-sphere $$\Delta $$ and $$1\leq k\leq \lceil \frac {d}{2}\rceil -1$$, $$g_{k}(\Delta )$$ is at least as large as the number of missing $(d-k)$-faces of $$\Delta $$; here we show that, for $$1\leq k\leq \lfloor \frac {d}{2}\rfloor -1$$, equality holds if and only if $$\Delta $$ is $$k$$-stacked. Finally, we show that for $$d\geq 4$$, any simplicial $$d$$-polytope $$P$$ that has no missing faces of dimension $$\geq d-1$$ is redundantly rigid, that is, for each edge $$e$$ of $$P$$, there exists an affine $$2$$-stress on $$P$$ with a non-zero value on $$e$$. 

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5. Regularity of weak solution of variational problems modeling the Cosserat micropolar elasticity.

   https://doi.org/10.1093/imrn/rnaa202  

   Li, Yimei; Wang, Changyou (January 2022, International mathematics research notices)

   In this paper, we consider weak solutions of the Euler–Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $$p$$-harmonic maps (⁠#2\le p\le 3$$⁠). We show that if a weak solution is stationary, then its singular set is discrete for $2<3$ and has zero one-dimensional Hausdorff measure for $p=2$⁠. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $$p\in [2, 32/15]$$. 

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