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1. The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

   https://doi.org/10.1007/s00526-021-01938-2  

   Banerjee, Agnid; Danielli, Donatella; Garofalo, Nicola; Petrosyan, Arshak (April 2021, Calculus of Variations and Partial Differential Equations)

   Abstract We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight$$|y|^a$$ ${\left|y\right|}^a$for$$a \in (-1,1)$$ $a\in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$(\partial _t - \Delta _x)^s$$ ${({\partial }_t-{\Delta }_x)}^s$for$$s \in (0,1)$$ $s\in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ $a=0$). 

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2. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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3. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa (April 2022, Mathematische Annalen)

   Abstract Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ $a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$ $a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$ $k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$ $a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$and that every$$m\times m$$ $m\times m$minor of the$$m\times k$$ $m\times k$matrix$$(a_{j,i})_{j,i}$$ ${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$ $\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$ $x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$ $\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. 

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4. Counting imaginary quadratic fields with an ideal class group of 5-rank at least 2

   https://doi.org/10.1007/s11139-025-01184-6  

   Bartz, Kollin; Levin, Aaron; Thamminana, Aman_Dhruva (August 2025, The Ramanujan Journal)

   Abstract We prove that there are$$\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}$$ $\gg \frac{X^{\frac{1}{3}}}{{(logX)}^2}$imaginary quadratic fieldskwith discriminant$$|d_k|\le X$$ $|d_k|\le X$and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound$$\gg X^{\frac{1}{4}}$$ $\gg X^{\frac{1}{4}}$in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curveCover$$\mathbb {Q}$$ $Q$such thatChas a rational Weierstrass point and the Jacobian ofChas a rational torsion subgroup of 5-rank 2. We deduce the main result from the existence of the curveCand a quantitative result of Kulkarni and the second author. 

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5. More on codes for combinatorial composite DNA

   https://doi.org/10.1007/s10623-025-01634-8  

   Ye, Zuo; Sabary, Omer; Gabrys, Ryan; Yaakobi, Eitan; Elishco, Ohad (May 2025, Designs, Codes and Cryptography)

   Abstract In this paper, we focus on constructing unique-decodable and list-decodable codes for the recently studied (t, e)-composite-asymmetric error-correcting codes ((t, e)-CAECCs). Let$$\mathcal {X}$$ $X$be an$$m \times n$$ $m\times n$binary matrix in which each row has Hamming weightw. If at mosttrows of$$\mathcal {X}$$ $X$contain errors, and in each erroneous row, there are at mosteoccurrences of$$1 \rightarrow 0$$ $1\to 0$errors, we say that a (t, e)-composite-asymmetric error occurs in$$\mathcal {X}$$ $X$. For general values ofm, n, w, t, ande, we propose new constructions of (t, e)-CAECCs with redundancy at most$$(t-1)\log (m) + O(1)$$ $(t-1)log\left(m\right)+O\left(1\right)$, whereO(1) is independent of the code lengthm. In particular, this yields a class of (2, e)-CAECCs that are optimal in terms of redundancy. Whenmis a prime power, the redundancy can be further reduced to$$(t-1)\log (m) - O(\log (m))$$ $(t-1)log\left(m\right)-O(log(m\left)\right)$. To further increase the code size, we introduce a combinatorial object called a weak$$B_e$$ $B_e$-set. When$$e = w$$ $e=w$, we present an efficient encoding and decoding method for our codes. Finally, we explore potential improvements by relaxing the requirement of unique decoding to list-decoding. We show that when the list size ist! or an exponential function oft, there exist list-decodable (t, e)-CAECCs with constant redundancy. When the list size is two, we construct list-decodable (3, 2)-CAECCs with redundancy$$\log (m) + O(1)$$ $log\left(m\right)+O\left(1\right)$. 

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