Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Stability of a Difference Scheme for a Quasi-Linear Partial Differential Algebraic System of Equations of Index (k, 0)

Computational Mathematics and Mathematical Physics - A quasi-linear partial differential algebraic system of equations of index $$(k,0)$$ is considered. A spline collocation difference scheme with...     

A KP1 Scheme for Acceleration of Inner Iterations for the Transport Equation in 3D Geometry Consistent with Nodal Schemes: 2. Splitting Method for Solving the P1 System for Acceleration Corrections

Computational Mathematics and Mathematical Physics - An algorithm is proposed for solving the $${{P}_{1}}$$ system for acceleration corrections that arises in constructing a $$K{{P}_{1}}$$ scheme...     

The Chebyshev Spectral Method for Burgers-Like Equations

The Chebyshev polynomials have good approximation properties which are not affected by boundary values. They have higher resolution near the boundary than in the interior and are suitable for problems in which the solution changes rapidly near the boundary. Also, they can be calculated by FFT. Thus they are used mostly for initial-boundary value problems for P.D.E.'s (see [1, 3-4, 6, 8-11]). Maday and Quarterom discussed the convergence of Legendre and Chebyshev spectral approximations to the steady Burgers equation. In this paper we consider Burgers-like equations.$$\begin{cases}∂_iu+F(u)_x-vu_{zx}=0, & -1≤x≤1, 0＜t≤T \\ u (-1,t) =u (1,t) =0, & 0≤t≤T & (0.1)\\ u (x,0) =u_0(x), & -1≤x≤1\end{cases}$$ where $F\in C(R)$ and there exists a positive function $A\in C(R)$ and a constant $p＞1$ such that $$|F(z+y)-F(z)|\leq A(z)(|y|+|y|^p).$$ We develop a Chebyshev spectral scheme and a pseudospectral scheme for solving (0.1) and establish their generalized stability and convergence.     

Discrete second-order Ablowitz–Kaup–Newell–Segur equation and its modified form

Theoretical and Mathematical Physics - By introducing shift relations satisfied by a matrix $$\boldsymbol{r}$$ , we propose a generalized Cauchy matrix scheme and construct a discrete second-order...     

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

This paper presents a nonconforming finite element scheme for the planar biharmonic equation,which applies piecewise cubic polynomials(P_3) and possesses O(h~2) convergence rate for smooth solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H4 functions with O(h2) accuracy in the broken H~2 norm. Besides, a discrete strengthened Miranda-Talenti estimate(▽_h~2·,▽_h~2·) =(?h·, ?h·), which is usually not true for nonconforming finite element spaces, is proved.The finite element space does not correspond to a finite element defined with Ciarlet's triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.     

Stratification and $$\pi $$ π -cosupport: finite groups

Mathematische Zeitschrift - We introduce the notion of $$\pi $$ -cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give...     

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ P 4 of degree $$d = g+1$$ d = g + 1 and genus g

Archiv der Mathematik - We denote by $$\mathcal {H}_{d,g,r}$$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and...     

Class field theory for arithmetic schemes

Let X be a regular arithmetic scheme, i.e. a regular integral separated scheme flat and of finite type over Spec . Generalising classical class field theory for number fields, we define a class group C _X and show there is a natural surjective map whose kernel is the connected component of 0.      

Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system

An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson system in the two- dimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one, based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step, in the L ^∞ metric. The numerical solutions are proved to converge in L ^∞ towards the exact ones as ε and Δt tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs to . The rate of convergence is , which should be compared to the results of Besse who recently established in (SIAM J Numer Anal 42(1):350–382, 2004) similar rates for a uniform semi-Lagrangian scheme, but requiring that the initial data are in . Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations.     

On injective modules over Azumaya algebras over locally noetherian schemes

Let be an Azumaya algebra over a locally noetherian scheme X. We describe in this work quasi-coherent -bimodules which are injective in the category of sheaves of left -modules     

Existence and iteration of positive solution for a three-point boundary value problem with ap-Laplacian operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u(0) - B(u\prime (\eta )) = 0, u\prime (1) = 0 \hfill \\ \end{gathered} \right.$$ and $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u\prime (0) = 0, u(1) + B(u\prime (\eta )) = 0 \hfill \\ \end{gathered} \right.$$ The main tool is the monotone iterative technique. Here, the coefficientq(t) may be singular att = 0,1.     

Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration (q) on the set (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions ₊(q) and ₋(q) of (q) to the set ₊(q) of secant (hyperbolic) lines and to the set ₋(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme ₋(q) is pseudocyclic.We further show that the coherent configurations (q ²) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme ₊(q ²), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes ₊(q ²) and ₋(q ²). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.     

Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations

Applications of Mathematics - We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that $$\left| {{\omega...     

Some consequences of Wiesend’s higher dimensional class field theory

We use a result of G. Wiesend to establish the relation between the integral singular homology in degree zero and the abelianized tame fundamental group of a regular, connected scheme of finite type over Spec .      

Evidence for the charged θ-meson

A K-meson is found to decay at rest into a nearly relativistic secondary particle. The secondary particle produces a nuclear disintegration in flight and is identified as a π-meson of kinetic energy ～ 110 MeV. The event is interpreted as the decay of a θ^±-meson according to the scheme:

$$\theta ^ \pm \to \pi ^ \pm + \pi ^0 + (222 \pm 12 MeV.)$$     

Designs in Additive Codes over GF(4)

In this paper we discuss the security of digital signature schemes based on error-correcting codes. Several attacks to the Xinmei scheme are surveyed, and some reasons given to explain why the Xinmei scheme failed, such as the linearity of the signature and the redundancy of public keys. Another weakness is found in the Alabbadi-Wicker scheme, which results in a universal forgery attack against it. This attack shows that the Alabbadi-Wicker scheme fails to implement the necessary property of a digital signature scheme: it is infeasible to find a false signature algorithm D from the public verification algorithm E such that E(D ( )) = for all messages . Further analysis shows that this new weakness also applies to the Xinmei scheme.     

On conditional expectations in $$L^p(\mu ;L^q(\nu ;X))$$ L p ( μ ; L q ( ν ; X ) )

Positivity - Let $$(A,{\mathscr {A}},\mu )$$ and $$(B,{\mathscr {B}},\nu )$$ be probability spaces, let $${\mathscr {F}}$$ be a sub- $$\sigma $$ -algebra of the product $$\sigma $$ -algebra...