Abstract We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices Λ ⊂ R 2 $\Lambda \subset \mathbb {R}^2$ normal upper Lamda subset of double struck upper R squared with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be seen as the inhomogeneous counterpart to Montgomery’s 1988 result, which proved that the hexagonal lattice minimizes the maximum.
Suppose J=(f1,…,fn) $J = (f_1, \dots , f_n)$ upper J equals left parenthesis f 1 comma ellipsis comma f Subscript n Baseline right parenthesis is an n-generated ideal in any ring R. We prove a general Briançon-Skoda-type containment relating the integral closure Jn+k−1― $\overline {J^{n+k-1}}$ upper J Superscript n plus k minus 1 Baseline overbar with ordinary powers Jk $J^k$ upper J Superscript k . We prove that our result implies the full Briançon-Skoda containment Jn+k−1―⊆Jk $\overline {J^{n+k-1}} \subseteq J^k$ upper J Superscript n plus k minus 1 Baseline overbar subset of or equal to upper J Superscript k for pseudo-rational singularities (for instance regular rings), and even for the weaker condition of birational derived splinters. Our methods also yield the containment Jn+k―⊆Jk $\overline {J^{n+k}} \subseteq J^k$ upper J Superscript n plus k Baseline overbar subset of or equal to upper J Superscript k for Du Bois singularities and even for a characteristic-free generalization. Our Briançon-Skoda-type theorem also implies well-known closure-based Briançon-Skoda results Jn+k−1―⊆(Jk)cl $\overline {J^{n+k-1}} \subseteq (J^k)^{\mathrm {cl}}$ upper J Superscript n plus k minus 1 Baseline overbar subset of or equal to left parenthesis upper J Superscript k Baseline right parenthesis Superscript cl where, for instance, cl $\mathrm {cl}$ cl is tight or plus closure in characteristic p>0 $p> 0$ p greater than 0 , or ep $\mathrm {ep}$ ep closure or extension and contraction from R+^ $\widehat {R^+}$ ModifyingAbove upper R Superscript plus Baseline With caret in mixed characteristic. Our proof relies on a study of the tensor product of the derived image of the structure sheaf of a partially normalized blowup of J with the Buchsbaum-Eisenbud complex (equivalently the Eagon-Northcott complex) associated to (f1,…,fn)k $(f_1,\dots ,f_n)^k$ left parenthesis f 1 comma ellipsis comma f Subscript n Baseline right parenthesis Superscript k .
We study deformation theory of mod p Galois representations of p-adic fields with values in generalised reductive group schemes, such as L-groups and C-groups. We show that the corresponding deformation rings are complete intersections of expected dimension. We determine their irreducible components in many cases and show that they and their special fibres are normal and complete intersection.
We construct integral models of Shimura varieties of abelian type with parahoric level structure over odd primes. These models are & eacute;tale locally isomorphic to corresponding local models.
In this work, we establish the convergence of 2D, stationary Navier-Stokes flows with viscosity epsilon > 0, (u(epsilon), v (epsilon)) to to the classical Prandtl boundary layer, ((-)u(p), (-)v(p)), posed on the domain (0, infinity)(0, infinity): ||u(epsilon)-(-)u(p)||(infinity)(Ly)less than or similar to root epsilon < x > (- 1 /4 + delta), ||v(epsilon-)root epsilon v p||(infinity)(Ly) less than or similar to root epsilon < x >(- 1 /2) This validates Prandtl's boundary layer theory globally in the x-variable for a large class of boundary layers, including the entire one parameter family of the classical Blasius profiles, with sharp decay rates. The result demonstrates asymptotic stability in two senses simultaneously: (1) asymptotic as epsilon -> 0 and (2) asymptotic as x -> infinity. In particular, our result provides the first rigorous confirmation for the Navier-Stokes equations that the boundary layer cannot "separate" in these stable regimes, which is very important for physical and engineering applications.
