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1. Inequalities for the Radon Transform on Convex Sets

   https://doi.org/10.1093/imrn/rnab122  

   Giannopoulos, Apostolos; Koldobsky, Alexander; Zvavitch, Artem (May 2021, International Mathematics Research Notices)

   Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $$K$$ and $$L$$ be star bodies in $${\mathbb R}^n,$$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $$K$$ and $$L$$, respectively, so that $$\|g\|_\infty =g(0)=1.$$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $$C$$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of $${\mathbb R}^n,$$ and $$d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$$ is the outer volume ratio distance from $$K$$ to the class of generalized $$k$$-intersection bodies in $${\mathbb R}^n.$$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem. 

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2. A New Family of Algebraically Defined Graphs with Small Automorphism Group

   https://doi.org/10.37236/10707  

   Lazebnik, Felix; Taranchuk, Vladislav (January 2022, The Electronic Journal of Combinatorics)

   Let $$p$$ be an odd  prime, $q=p^e$, $$e \geq 1$$, and $$\mathbb{F} = \mathbb{F}_q$$ denote the finite field of $$q$$ elements.  Let $$f: \mathbb{F}^2\to \mathbb{F}$$ and  $$g: \mathbb{F}^3\to \mathbb{F}$$  be functions, and  let $$P$$ and $$L$$ be two copies of the 3-dimensional vector space $$\mathbb{F}^3$$. Consider a bipartite graph $$\Gamma_\mathbb{F} (f, g)$$ with vertex partitions $$P$$ and $$L$$ and with edges defined as follows: for every $$(p)=(p_1,p_2,p_3)\in P$$ and every $$[l]= [l_1,l_2,l_3]\in L$$, $$\{(p), [l]\} = (p)[l]$$ is an edge in $$\Gamma_\mathbb{F} (f, g)$$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question  appeared in Nassau: Given $$\Gamma_\mathbb{F} (f, g)$$,  is it always possible to find a function $$h:\mathbb{F}^2\to \mathbb{F}$$ such that the graph $$\Gamma_\mathbb{F} (f, h)$$  with the same vertex set as $$\Gamma_\mathbb{F} (f, g)$$ and with edges $(p)[l]$  defined in a similar way  by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $$\Gamma_\mathbb{F} (f, g)$$ for infinitely many $$q$$?  In this paper we show that the  answer to the question is negative and the graphs $$\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$$ provide such an example for $$p \equiv 1 \pmod{3}$$. Our argument is based on proving that the automorphism group of these graphs has order $$p$$, which is the smallest possible order of the automorphism group of graphs of the form $$\Gamma_{\mathbb{F}}(f, g)$$. 

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3. Negative Moments of L –Functions With Small Shifts Over Function Fields

   https://doi.org/10.1093/imrn/rnad118  

   Florea, Alexandra (June 2023, International Mathematics Research Notices)

   Abstract We consider negative moments of quadratic Dirichlet $$L$$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $$\mathbb{F}_{q}[x]$$, we obtain an asymptotic formula for the $$k^{\textrm{th}}$$ shifted negative moment of $$L(1/2+\beta ,\chi _{D})$$, in certain ranges of $$\beta $$ (e.g., when roughly $$\beta \gg \log g/g $$ and $k<1$). We also obtain non-trivial upper bounds for the $$k^{\textrm{th}}$$ shifted negative moment when $$\log (1/\beta ) \ll \log g$$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $$\beta \gg g^{-\frac{1}{2k}+\epsilon }$$. 

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4. Twisted Mahler Discrete Residues

   https://doi.org/10.1093/imrn/rnae238  

   Arreche, Carlos_E; Zhang, Yi (October 2024, International Mathematics Research Notices)

   Abstract Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p> 1$. Here we develop a notion of $$\lambda $$-twisted Mahler discrete residues for $$\lambda \in \mathbb{Z}$$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$p^{\lambda } g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations. 

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5. Maximizing Submodular Functions under Submodular Constraints

   Padmanabhan, Madhavan R; Zhu, Yanhui; Basu, Samik; Pavan A. (July 2023, Uncertainty in Artificial Intelligence, {UAI} 2023)

   Evans, Robin; Shpitser, Ilya (Ed.)

   We consider the problem of maximizing submodular functions under submodular constraints by formulating the problem in two ways: \SCSKC and \DiffC. Given two submodular functions $$f$$ and $$g$$ where $$f$$ is monotone, the objective of \SCSKC problem is to find a set $$S$$ of size at most $$k$$ that maximizes $f(S)$ under the constraint that $$g(S)\leq \theta$$, for a given value of $$\theta$$. The problem of \DiffC focuses on finding a set $$S$$ of size at most $$k$$ such that $h(S) = f(S)-g(S)$$ is maximized. It is known that these problems are highly inapproximable and do not admit any constant factor multiplicative approximation algorithms unless NP is easy. Known approximation algorithms involve data-dependent approximation factors that are not efficiently computable. We initiate a study of the design of approximation algorithms where the approximation factors are efficiently computable. For the problem of \SCSKC, we prove that the greedy algorithm produces a solution whose value is at least $$(1-1/e)f(\OPT) - A$, where $$A$$ is the data-dependent additive error. For the \DiffC problem, we design an algorithm that uses the \SCSKC greedy algorithm as a subroutine. This algorithm produces a solution whose value is at least $$(1-1/e)h(\OPT)-B$, where $$B$$ is also a data-dependent additive error. A salient feature of our approach is that the additive error terms can be computed efficiently, thus enabling us to ascertain the quality of the solutions produced. 

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