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1. 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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2. 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. 

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3. Lieb–Thirring Inequality for the 2D Pauli Operator

   https://doi.org/10.1007/s00220-024-05177-2  

   Frank, Rupert_L; Kovařík, Hynek (January 2025, Communications in Mathematical Physics)

   Abstract By the Aharonov–Casher theorem, the Pauli operatorPhas no zero eigenvalue when the normalized magnetic flux$$\alpha $$ $\alpha$satisfies$$|\alpha |<1$$ $\left|\alpha \right|<1$, but it does have a zero energy resonance. We prove that in this case a Lieb–Thirring inequality for the$$\gamma $$ $\gamma$-th moment of the eigenvalues of$$P+V$$ $P+V$is valid under the optimal restrictions$$\gamma \ge |\alpha |$$ $\gamma \ge \left|\alpha \right|$and$$\gamma >0$$ $\gamma >0$. Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order$$\gamma \ge 1$$ $\gamma \ge 1$. 

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4. Higher-group symmetry of (3+1)D fermionic $$\mathbb{Z}_2$$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry

   https://doi.org/10.21468/SciPostPhys.16.5.122  

   Barkeshli, Maissam; Hsin, Po-Shen; Kobayashi, Ryohei (January 2024, SciPost Physics)

   It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D\mathbb{Z}_2 ${\mathbb{Z}}_2$gauge theory with an emergent fermion, and point out that pumping chiralp+ip $p+ip$topological states gives rise to a\mathbb{Z}_{8} ${\mathbb{Z}}_8$0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization ofT^3 $T^3$(3-torus) andT^2 \rtimes_{C_2} S^1 $T^2{⋊}_{C_2}{S}^1$(2-torus bundle over the circle) respectively, and pumpingp+ip $p+ip$states. Our considerations also imply the possibility of a logicalT $T$gate by placing the code on\mathbb{RP}^3 ${ℝ\mathbb{P}}^3$and pumping ap+ip $p+ip$topological state. 

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5. An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

   https://doi.org/10.1007/s10240-024-00144-y  

   De Lellis, Camillo; Nardulli, Stefano; Steinbrüchel, Simone (February 2024, Publications mathématiques de l'IHÉS)

   Abstract We consider integral area-minimizing 2-dimensional currents$$T$$ $T$in$$U\subset \mathbf {R}^{2+n}$$ $U\subset R^{2+n}$with$$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$, where$$Q\in \mathbf {N} \setminus \{0\}$$ $Q\in N\setminus \left\{0\right\}$and$$\Gamma $$ $\Gamma$is sufficiently smooth. We prove that, if$$q\in \Gamma $$ $q\in \Gamma$is a point where the density of$$T$$ $T$is strictly below$$\frac{Q+1}{2}$$ $\frac{Q+1}{2}$, then the current is regular at$$q$$ $q$. The regularity is understood in the following sense: there is a neighborhood of$$q$$ $q$in which$$T$$ $T$consists of a finite number of regular minimal submanifolds meeting transversally at$$\Gamma $$ $\Gamma$(and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$$Q=1$$ $Q=1$. As a corollary, if$$\Omega \subset \mathbf {R}^{2+n}$$ $\Omega \subset R^{2+n}$is a bounded uniformly convex set and$$\Gamma \subset \partial \Omega $$ $\Gamma \subset \partial \Omega$a smooth 1-dimensional closed submanifold, then any area-minimizing current$$T$$ $T$with$$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$is regular in a neighborhood of $$\Gamma $$ $\Gamma$. 

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