Neeman's characterization of K(R-Proj) via Bousfield localization

Let R be an associative ring with unit and denote by $K(\mathrm{R}\text{-}\mathrm{Proj})$ the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that $K(\mathrm{R}\text{-}\mathrm{Proj})$ is ${\mathrm{?}}_1$-compactly generated, with the category $K^{+}(\mathrm{R}\text{-}\mathrm{proj})$ of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K(\mathrm{R}\text{-}\mathrm{Proj})$ vanishes in the Bousfield localization $K(\mathrm{R}\text{-}\mathrm{Flat})/〈K^{+}(\mathrm{R}\text{-}\mathrm{proj})〉$.     

Triangular C1-bialgebra defined as the direct sum of matrix algebras

Let $M_{*}(\mathbf{C})$ denote the ${\mathrm{C}}^1$-algebra defined as the direct sum of all matrix algebras $\{M_n(\mathbf{C}):n?1\}$. It is known that $M_{*}(\mathbf{C})$ has a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\phi }$. From a certain set of transformations of integers, we construct a universal R-matrix R of the ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi })$ such that the quasi-cocommutative ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi }\text{,}R)$ is triangular. Furthermore, it is shown that certain linear Diophantine equations are corresponded to the Yang–Baxter equations of R.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

A remark on the orbit structure of the complexification of a semisimple symmetric space

We consider the action of a real semisimple Lie group G on the complexification $G_{\mathbf{C}}/H_{\mathbf{C}}$ of a semisimple symmetric space $G/H$ and we present a refinement of Matsuki?s results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in $G_{\mathbf{C}}/H_{\mathbf{C}}$, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G-orbit structure of $G_{\mathbf{C}}/H_{\mathbf{C}}$. Every such point $\overline{p}$ lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair $(\mathfrak{g},\mathfrak{h})$. The slice representation at $\overline{p}$ is equivalent to the isotropy representation of a real reductive symmetric space, namely $Z_G(p^4)/G_{\overline{p}}$. In principle, this gives the possibility to explicitly parametrize all G-orbits in $G_{\mathbf{C}}/H_{\mathbf{C}}$.     

A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

In this paper, we apply the variational method with Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for the planar four-body problem with two pairs of equal masses $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ on $[0,T]$ satisfies the SPBC if the boundaries $q(0)\in \mathbf{A}$ and $q(T)\in \mathbf{B}$, where A and B are two structural configuration spaces in ${({\mathbf{R}}^2)}^4$ and they depend on a rotation angle $\theta \in (0,2\pi )$ and the mass ratio $\mu =\frac{m_2}{m_1}\in {\mathbf{R}}^{+}$.We show that there is a region $\mathrm{\Omega }?(0,2\pi )\times R^{+}$ such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying the SPBC $\{q(t)\in H^1([0,T],{({\mathbf{R}}^2)}^4)|q(0)\in \mathbf{A},q(T)\in \mathbf{B}\}$ for any $(\theta ,\mu )\in \mathrm{\Omega }$. The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if θ is commensurable with π or a quasi-periodic solution if θ is not commensurable with π. In the variational method with the SPBC, we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries.The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when $(\theta ,\mu )=(\frac{4\pi }{5},1)$. Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the equal mass choreographies.     

A topological nullstellensatz for tensor-triangulated categories

Let $\mathrm{Spec}(\mathbb{T})$ be the spectrum of a tensor-triangulated category $(\mathbb{T},?,\mathbf{1})$. We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in $(\mathbb{T},?,\mathbf{1})$ and the collection of open subsets of $\mathrm{Spec}(\mathbb{T})$ in inverse topology. In fact, we prove a more general result in terms of supports on $(\mathbb{T},?,\mathbf{1})$ and work by combining methods from commutative algebra, topology and tensor triangular geometry.     

Ideals in deformation quantizations over Z/pnZ

Let k be a perfect field of characteristic $p>2$. Let $A_1$ be an Azumaya algebra over a smooth symplectic affine variety over k. Let $A_n$ be a deformation quantization of $A_1$ over $W_n(\mathbf{k})$. We prove that all $W_n(\mathbf{k})$-flat two-sided ideals of $A_n$ are generated by central elements.     

Beyond abstract elementary classes: On the model theory of geometric lattices

Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, $({\mathbf{K}}^3,?)$, is ω-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in $({\mathbf{K}}^3,?)$ the Smoothness Axiom fails, and so $({\mathbf{K}}^3,?)$ is not an AEC.