More Like this

1. Regularity of area minimizing currents mod p

   https://doi.org/10.1007/s00039-020-00546-0  

   De Lellis, Camillo; Hirsch, Jonas; Marchese, Andrea; Stuvard, Salvatore (October 2020, Geometric and Functional Analysis)

   Abstract We establish a first general partial regularity theorem for area minimizing currents$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$, for everyp, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of anm-dimensional area minimizing current$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$cannot be larger than$$m-1$$ $m-1$. Additionally, we show that, whenpis odd, the interior singular set is$$(m-1)$$ $(m-1)$-rectifiable with locally finite$$(m-1)$$ $(m-1)$-dimensional measure. 

   more » « less
2. 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. 

   more » « less
3. Photo-response of the $$N=Z$$ nucleus $$^{24}$$Mg

   https://doi.org/10.1140/epja/s10050-023-01111-7  

   Deary, J; Scheck, M; Schwengner, R; O’Donnell, D; Bemmerer, D; Beyer, R; Hensel, Th; Junghans, A R; Kögler, T; Müller, S E; et al (September 2023, The European Physical Journal A)

   Abstract The electricE1 and magneticM1 dipole responses of the$$N=Z$$ $N=Z$nucleus$$^{24}$$ ${}^{24}$Mg were investigated in an inelastic photon scattering experiment. The 13.0 MeV electrons, which were used to produce the unpolarised bremsstrahlung in the entrance channel of the$$^{24}$$ ${}^{24}$Mg($$\gamma ,\gamma ^{\prime }$$ $\gamma ,{\gamma }^{\prime }$) reaction, were delivered by the ELBE accelerator of the Helmholtz-Zentrum Dresden-Rossendorf. The collimated bremsstrahlung photons excited one$$J^{\pi }=1^-$$ $J^{\pi }=1^{-}$, four$$J^{\pi }=1^+$$ $J^{\pi }=1^{+}$, and six$$J^{\pi }=2^+$$ $J^{\pi }=2^{+}$states in$$^{24}$$ ${}^{24}$Mg. De-excitation$$\gamma $$ $\gamma$rays were detected using the four high-purity germanium detectors of the$$\gamma $$ $\gamma$ELBE setup, which is dedicated to nuclear resonance fluorescence experiments. In the energy region up to 13.0 MeV a total$$B(M1)\uparrow = 2.7(3)~\mu _N^2$$ $B\left(M1\right)\uparrow =2.7\left(3\right){\mu }_N^2$is observed, but this$$N=Z$$ $N=Z$nucleus exhibits only marginalE1 strength of less than$$\sum B(E1)\uparrow \le 0.61 \times 10^{-3}$$ $\sum B\left(E1\right)\uparrow \le 0.61\times {10}^{-3}$ e$$^2 \, $$ ${}^2$fm$$^2$$ ${}^2$. The$$B(\varPi 1, 1^{\pi }_i \rightarrow 2^+_1)/B(\varPi 1, 1^{\pi }_i \rightarrow 0^+_{gs})$$ $B(\Pi 1,1_i^{\pi }\to 2_1^{+})/B(\Pi 1,1_i^{\pi }\to 0_{\mathrm{gs}}^{+})$branching ratios in combination with the expected results from the Alaga rules demonstrate thatKis a good approximative quantum number for$$^{24}$$ ${}^{24}$Mg. The use of the known$$\rho ^2(E0, 0^+_2 \rightarrow 0^+_{gs})$$ ${\rho }^2(E0,0_2^{+}\to 0_{\mathrm{gs}}^{+})$strength and the measured$$B(M1, 1^+ \rightarrow 0^+_2)/B(M1, 1^+ \rightarrow 0^+_{gs})$$ $B(M1,1^{+}\to 0_2^{+})/B(M1,1^{+}\to 0_{\mathrm{gs}}^{+})$branching ratio of the 10.712 MeV$$1^+$$ $1^{+}$level allows, in a two-state mixing model, an extraction of the difference$$\varDelta \beta _2^2$$ $\Delta {\beta }_2^2$between the prolate ground-state structure and shape-coexisting superdeformed structure built upon the 6432-keV$$0^+_2$$ $0_2^{+}$level. 

   more » « less
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 (August 2023, 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. 

   more » « less
5. Enumeration of Rooted Binary Unlabeled Galled Trees

   https://doi.org/10.1007/s11538-024-01270-8  

   Agranat-Tamir, Lily; Mathur, Shaili; Rosenberg, Noah A. (March 2024, Bulletin of Mathematical Biology)

   Abstract Rooted binarygalledtrees generalize rooted binary trees to allow a restricted class of cycles, known asgalls. We build upon the Wedderburn-Etherington enumeration of rooted binary unlabeled trees withnleaves to enumerate rooted binary unlabeled galled trees withnleaves, also enumerating rooted binary unlabeled galled trees withnleaves andggalls,$$0 \leqslant g \leqslant \lfloor \frac{n-1}{2} \rfloor $$ $0⩽g⩽\lfloor \frac{n-1}{2}\rfloor$. The enumerations rely on a recursive decomposition that considers subtrees descended from the nodes of a gall, adopting a restriction on galls that amounts to considering only the rooted binarynormalunlabeled galled trees in our enumeration. We write an implicit expression for the generating function encoding the numbers of trees for alln. We show that the number of rooted binary unlabeled galled trees grows with$$0.0779(4.8230^n)n^{-\frac{3}{2}}$$ $0.0779(4.{8230}^n)n^{-\frac{3}{2}}$, exceeding the growth$$0.3188(2.4833^n)n^{-\frac{3}{2}}$$ $0.3188(2.{4833}^n)n^{-\frac{3}{2}}$of the number of rooted binary unlabeled trees without galls. However, the growth of the number of galled trees with only one gall has the same exponential order 2.4833 as the number with no galls, exceeding it only in the subexponential term,$$0.3910n^{\frac{1}{2}}$$ $0.3910n^{\frac{1}{2}}$compared to$$0.3188n^{-\frac{3}{2}}$$ $0.3188n^{-\frac{3}{2}}$. For a fixed number of leavesn, the number of gallsgthat produces the largest number of rooted binary unlabeled galled trees lies intermediate between the minimum of$$g=0$$ $g=0$and the maximum of$$g=\lfloor \frac{n-1}{2} \rfloor $$ $g=\lfloor \frac{n-1}{2}\rfloor$. We discuss implications in mathematical phylogenetics. 

   more » « less