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1. Evaluation codes arising from symmetric polynomials

   https://doi.org/10.1007/s10623-025-01637-5  

   Gatti, Barbara; Korchmáros, Gábor; Nagy, Gábor P; Pallozzi_Lavorante, Vincenzo; Schulte, Gioia (May 2025, Designs, Codes and Cryptography)

   Abstract Datta and Johnsen (Des Codes Cryptogr 91:747–761, 2023) introduced a new family of evaluation codes in an affine space of dimension$$\ge 2$$ $\ge 2$over a finite field$${\mathbb {F}}_q$$ $F_q$where linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates. In this paper, we propose a generalization by taking low dimensional linear systems of symmetric polynomials. Computation for small values of$$q=7,9$$ $q=7,9$shows that carefully chosen generalized Datta–Johnsen codes$$\left[ \frac{1}{2}q(q-1),3,d\right] $$ $\left(\frac{1}{2},q,(q-1),,,3,,,d\right)$have minimum distancedequal to the optimal value minus 1. 

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2. Negative moments of the Riemann zeta-function

   https://doi.org/10.1515/crelle-2023-0091  

   Bui, Hung M; Florea, Alexandra (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $\zeta \left(s\right)${\zeta(s)}. For example, integrating ${\left|\zeta \left(\frac{1}{2}+\alpha +it\right)\right|}^{-2k}${|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}}with respect totfromTto $2T${2T}, we obtain an asymptotic formula when the shift α is roughly bigger than $\frac{1}{\log T}${\frac{1}{\log T}}and $k<\frac{1}{2}${k<\frac{1}{2}}. We also obtain non-trivial upper bounds for much smaller shifts, as long as $\log \frac{1}{\alpha }\ll \log \log T${\log\frac{1}{\alpha}\ll\log\log T}. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function. 

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3. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven J; Romero_Acosta, Francisco; et al (April 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$, let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$, where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$. Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$. 

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4. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian; Gusakova, Anna; Reitzner, Matthias; Schütt, Carsten; Thäle, Christoph; Werner, Elisabeth M (July 2024, Mathematische Annalen)

   Abstract Consider two half-spaces$$H_1^+$$ $H_1^{+}$and$$H_2^+$$ $H_2^{+}$in$${\mathbb {R}}^{d+1}$$ $R^{d+1}$whose bounding hyperplanes$$H_1$$ $H_1$and$$H_2$$ $H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ $S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$ $S^d$, which contains a great subsphere of dimension$$d-2$$ $d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$ $logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere. 

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5. A multiple-timing analysis of temporal ratcheting

   https://doi.org/10.1140/epje/s10189-024-00421-y  

   Hashemi, Aref; Gilman, Edward T; Khair, Aditya S (April 2024, The European Physical Journal E)

   AbstractWe develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies$$\omega $$ $\omega$and$$\alpha \omega $$ $\alpha \omega$, where$$\alpha $$ $\alpha$is a rational number. If$$\alpha $$ $\alpha$is a ratio of odd and even integers (e.g.,$$\tfrac{2}{1},\,\tfrac{3}{2},\,\tfrac{4}{3}$$ $\frac{2}{1},\frac{3}{2},\frac{4}{3}$), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for$$\alpha =2$$ $\alpha =2$. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for$$\alpha =\tfrac{3}{2}$$ $\alpha =\frac{3}{2}$and$$\tfrac{4}{3}$$ $\frac{4}{3}$appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these$$\alpha $$ $\alpha$values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems. Graphic abstract 

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