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In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.  相似文献   

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The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature η–invariants for the family of Berger metrics on the odd dimensional spheres.  相似文献   

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The purpose of this paper is to bring a new light on the state-dependent Hamilton–Jacobi equation and its connection with the Hopf–Lax formula in the framework of a Carnot group $(\mathbf G ,\circ ).$ The equation we shall consider is of the form $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t}+ \Psi (X_{1}u, \ldots , X_{m}u)=0\qquad &{}(x,t)\in \mathbf G \times (0,\infty ) \\ {u}(x,0)=g(x)&{}x\in \mathbf G , \end{array} \right. \end{aligned}$$ where $X_{1},\ldots , X_{m}$ are a basis of the first layer of the Lie algebra of the group $\mathbf G ,$ and $\Psi : \mathbb{R }^{m} \rightarrow \mathbb{R }$ is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton–Jacobi equation can be given by the Hopf–Lax formula $$\begin{aligned} u(x,t) = \inf _{y\in \mathbf G }\left\{ t \Psi ^\mathbf{G }\left( \delta _{\frac{1}{t}}(y^{-1}\circ x)\right) + g(y) \right\} , \end{aligned}$$ where $\Psi ^\mathbf{G }:\mathbf G \rightarrow \mathbb{R }$ is the $\mathbf G $ -Legendre–Fenchel transform of $\Psi ,$ defined by a control theoretical approach. We recover, as special cases, some known results like the classical Hopf–Lax formula in the Euclidean spaces $\mathbb{R }^n,$ showing that $\Psi ^{\mathbb{R }^n}$ is the Legendre–Fenchel transform $\Psi ^*$ of $\Psi ;$ moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function $\Psi .$ In this paper we follow an optimal control problem approach and we obtain several properties for the value functions $u$ and $\Psi ^\mathbf G .$   相似文献   

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We present a new technique for analyzing the v0-Bockstein spectral sequence studied by Shimomura and Yabe. Employing this technique, we derive a conceptually simpler presentation of the homotopy groups of the E(2)-local sphere at primes p?5. We identify and correct some errors in the original Shimomura–Yabe calculation. We deduce the related K(2)-local homotopy groups, and discuss their manifestation of Gross–Hopkins duality.  相似文献   

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We investigate questions related to the minimal degree of invariants of finitely generated diagonalizable groups. These questions were raised in connection to security of a public key cryptosystem based on invariants of diagonalizable groups. We derive results for minimal degrees of invariants of finite groups, abelian groups and algebraic groups. For algebraic groups we relate the minimal degree of the group to the minimal degrees of its tori. Finally, we investigate invariants of certain supergroups that are superanalogs of tori. It is interesting to note that a basis of these invariants is not given by monomials.  相似文献   

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Let G be a compact, connected, simply-connected Lie group. We use the Fourier–Mukai transform in twisted K-theory to give a new proof of the ring structure of the K-theory of G.  相似文献   

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We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.  相似文献   

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In this paper Jeffery–Hamel flow has been studied and its nonlinear ordinary differential equation has been solved through homotopy analysis method (HAM). The obtained solution in comparison with the numerical ones represents a remarkable accuracy. The results also indicate that HAM can provide us with a convenient way to control and adjust the convergence region.  相似文献   

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In this paper, by applying the geometric criterion and time average property to Lotka–Volterra systems, some results for the global asymptotic stability of the systems are obtained. Furthermore, we consider Li–Wang Conjecture for a three-dimensional system which is transformed from a Lotka–Volterra system.  相似文献   

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The fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.  相似文献   

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The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s, Cayley, Sylvester, and the Irish mathematician, George Salmon, formulated the basic concepts, developed the key techniques, and set the research agenda for the field. As Cayley and Sylvester continued to extend the theory off and on through the 1880s, first Salmon in 1859 Salmon, George. 1859. Lessons introductory to the modern higher algebra, Dublin: Hodges & Smith.  [Google Scholar] and later Edwin Bailey Elliott in 1895 Elliott, Edwin Bailey. 1895. An introduction to the algebra of quantics, Oxford: Oxford University Press.  [Google Scholar] codified it in high-level textbooks. This paper sketches the development of nineteenth-century invariant theory in British hands against a backdrop of personal, nationalistic, and internationalistic mathematical goals.  相似文献   

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Given a homomorphically closed root class K of groups, we find a criterion for a Baumslag–Solitar group to be a residually K-group. In particular, we establish that all Baumslag–Solitar groups are residually soluble and a Baumslag–Solitar group is residually finite soluble if and only if it is residually finite.  相似文献   

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This expository essay is focused on the Shafarevich–Tate set of a group GG. Since its introduction for a finite group by Burnside, it has been rediscovered and redefined more than once. We discuss its various incarnations and properties as well as relationships (some of them conjectural) with other local–global invariants of groups.  相似文献   

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