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I've been trying to find a vector function where $A_x(x,y)$ and $A_y(x,y)$ are periodic functions with period $\ell_q$, of the form $A(x,y)=(By+A_x(x,y),A_y(x,y))$ such that the discretised version of $$F(x_1,x_2):=(x_1,x_2)\cdot\int_0^1A\left(q_1-sx_1,q_2-sx_2\right)ds,$$ i.e., $x_1=\frac{\ell_q m_1}{N}$, $x_2=\frac{\ell_q m_2}{N}$, $q_1=\frac{\ell_q l_1}{N}$ $q_2=\frac{\ell_q l_2}{N}$, is $2Np$ periodic in the $m$ argument, i.e.,

$$F(m_1+2Np_1, m_2+2Np_2)=F(m_1,m_2)$$ and when $p_1,p_2\in\mathbb{Z}$

The linear term in $2Np$ periodic when $B=2\pi n$ when $n\in\mathbb{Z}$, but I can see to find a condition for the rest of the vector function.

Does anybody know a way that would help?

Thanks.

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