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1. Efficient Computation of a Semi-Algebraic Basis of the First Homology Group of a Semi-Algebraic Set

   https://doi.org/10.1007/s00454-024-00626-0  

   Basu, Saugata; Percival, Sarah (September 2024, Discrete & Computational Geometry)

   Let $$\R$$ be a real closed field and $$\C$$ the algebraic closure of $$\R$$. We give an algorithm for computing a semi-algebraic basis for the first homology group, $$\HH_1(S,\mathbb{F})$$, with coefficients in a field $$\FF$$, of any given semi-algebraic set $$S \subset \R^k$$ defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. More precisely, if the given quantifier-free formula involves $$s$$ polynomials whose degrees are bounded by $$d$$, the complexity of the algorithm is bounded by $$(s d)^{k^{O(1)}}$$. This algorithm generalizes well known algorithms having singly exponential complexity for computing a semi-algebraic basis of the zero-th homology group of semi-algebraic sets, which is equivalent to the problem of computing a set of points meeting every semi-algebraically connected component of the given semi-algebraic set at a unique point. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity. As an intermediate step in our algorithm we construct a semi-algebraic subset $$\Gamma$$ of the given semi-algebraic set $$S$$, such that $$\HH_q(S,\Gamma) = 0$$ for $q=0,1$. We relate this construction to a basic theorem in complex algebraic geometry stating that for any affine variety $$X$$ of dimension $$n$$, there exists Zariski closed subsets \[ Z^{(n-1)} \supset \cdots \supset Z^{(1)} \supset Z^{(0)} \] with $$\dim_\C Z^{(i)} \leq i$, and $$\HH_q(X,Z^{(i)}) = 0$$ for $$0 \leq q \leq i$$. We conjecture a quantitative version of this result in the semi-algebraic category, with $$X$$ and $$Z^{(i)}$$ replaced by closed semi-algebraic sets. We make initial progress on this conjecture by proving the existence of $$Z^{(0)}$$ and $$Z^{(1)}$$ with complexity bounded singly exponentially (previously, such an algorithm was known only for constructing $$Z^{(0)}$$). 

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2. Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation

   https://doi.org/10.1353/ajm.2023.a913295  

   Farah, Luiz Gustavo; Holmer, Justin; Roudenko, Svetlana; Yang, Kai (December 2023, American Journal of Mathematics)

   Abstract: We consider the quadratic Zakharov-Kuznetsov equation $$\partial_t u + \partial_x \Delta u + \partial_x u^2=0$$ on $$\Bbb{R}^3$$. A solitary wave solution is given by $Q(x-t,y,z)$, where $$Q$$ is the ground state solution to $$-Q+\Delta Q+Q^2=0$$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $$Q$$ in the energy space, evolves to a solution that, as $$t\to\infty$$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $$x>\delta t-\tan\theta\sqrt{y^2+z^2}$$ for $$0\leq\theta\leq{\pi\over 3}-\delta$$. 

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3. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia; Love, Jonathan (March 2024, Research in Number Theory)

   Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$of two curves over$$\mathbb {Q} $$ $Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$ $J_i$is the Jacobian variety of$$C_i$$ $C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ $E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$ $\epsilon$has finite image. 

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4. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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5. Hadronic effects in Møller scattering at NNLO

   https://doi.org/10.1007/JHEP08(2022)183  

   Erler, Jens; Ferro-Hernández, Rodolfo; Freitas, Ayres (August 2022, Journal of High Energy Physics)

   A bstract Two-loop electroweak corrections to polarized Møller scattering are studied in two different schemes at low energies. We find the finite Q 2 corrections to be well under control. The hadronic and perturbative QCD corrections to the γZ two-point function are incorporated through the weak mixing angle at low energies, which introduce an error of 0 . 08 × 10 − 3 in the weak charge of the electron $$ {Q}_W^e $$ Q W e . Furthermore, by studying the scheme dependence, we obtain an estimate of the current perturbative electroweak uncertainty, $$ \delta {Q}_W^e $$ δ Q W e ≈ 0 . 23 × 10 − 3 , which is five times smaller than the precision estimated for the MOLLER experiment ( $$ \delta {Q}_W^e $$ δ Q W e = 1 . 1 × 10 − 3 ). Future work is possible to reduce the theory error further. 

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