Archiv der Mathematik最新文献

We revisit the group structure on elliptic curves and give a simple and elementary proof of the associativity of the addition. We do this by providing an explicit formula for the sum of three points, only using the explicit definition of the group structure. In the process, we find a nice geometric interpretation of the sum of three points on the elliptic curve

For a chain of cycles (Gamma ), we prove that ({{,textrm{Cliff},}}(Gamma )={{,textrm{gon},}}(Gamma )-2).

It is shown that the restriction of a polynomial to a sphere satisfies a Logvinenko–Sereda–Kovrijkine type inequality (a specific type of uncertainty relation). This implies a spectral inequality for the Laplace–Beltrami operator, which, in turn, yields observability and null-controllability with explicit estimates on the control costs for the spherical heat equation that are sharp in the large and in the small time regime.

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof, we use realisations of characters in (ell )-adic cohomology groups of Deligne–Lusztig varieties as well as block theoretic considerations.

We compute the canonical trace of generic determinantal rings and provide a sufficient condition for the trace to specialize. As an application, we determine the canonical trace (tr (omega _R)) of a Cohen–Macaulay ring R of codimension two, which is generically Gorenstein. It is shown that if the defining ideal I of R is generated by n elements, then (tr (omega _R)) is generated by the ((n-2))-minors of the Hilbert–Burch matrix of I.

There is a well-known class of algebras called Igusa–Todorov algebras which were introduced in relation to the finitistic dimension conjecture. As a generalization of Igusa–Todorov algebras, the new notion of (m, n)-Igusa–Todorov algebras provides a wider framework for studying derived dimensions. In this paper, we give methods for constructing (m, n)-Igusa–Todorov algebras. As an application, we present for general Artin algebras a relationship between the derived dimension and the representation distance. Moreover, we end this paper to show that the main result can be used to give a better upper bound for the derived dimension for some classes of algebras.

In this paper, we investigate the existence and stability of almost periodic mild solutions to the non-autonomous Oseen–Navier–Stokes equations (ONSE) in the exterior domain (Omega subset mathbb {R}^3) of a rigid body under the actions of almost periodic external forces. Our method is based on the (L^p-L^q) smoothness of the evolution family corresponding to linearized equations in combination with interpolation spaces and fixed point theorems.

The need to control the residual of a potentially nonlinear function (mathcal {F}) arises in several situations in mathematics. For example, computing the zeros of a given map, or the reduction of some cost function during an optimization process are such situations. In this note, we discuss the existence of a curve (tmapsto x(t)) in the domain of the nonlinear map (mathcal {F}) leading from some initial value (x_0) to a value u such that we are able to control the residual (mathcal {F}(x(t))) based on the value (mathcal {F}(x_0)). More precisely, we slightly extend an existing result from J.W. Neuberger by proving the existence of such a curve, assuming that the directional derivative of (mathcal {F}) can be represented by (x mapsto mathcal {A}(x)mathcal {F}(x_0)), where (mathcal {A}) is a suitable defined operator. The presented approach covers, in case of (mathcal {A}(x) = -textsf{Id}), some well known results from the theory of so-called continuous Newton methods. Moreover, based on the presented results, we discover an approximate inverse function result.

This paper gives a second way to solve the one-dimensional minimization problem of the form : 

for scalar-valued functions f on the half line, where (f') and (f'') are its derivatives and (varepsilon ) and (mu ) are positive parameters with (varepsilon < frac{mu ^2}{4}.) This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).

We show that the first two k-invariants of (textrm{Top}/textrm{O}) vanish and give some applications.