Mathematische Nachrichten最新文献

We investigate equivariant birational geometry of rational surfaces and threefolds from the perspective of derived categories.

We study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be characterized by any variant of the 290-Theorem of Bhargava–Hanke.

We study the set of intrinsic metrics on a given graph. This is a convex compact set and it carries a natural order. We investigate existence of largest elements with respect to this order. We show that the only locally finite graphs which admit a largest intrinsic metric are certain finite star graphs. In particular, all infinite locally finite graphs do not admit a largest intrinsic metric. For infinite graphs which are not locally finite the set of intrinsic metrics may be trivial as we show by an example. Moreover, we give a characterization for the existence of intrinsic metrics with finite balls for weakly spherically symmetric graphs.

Reaction–diffusion systems with mass dissipation are known to possess blow-up solutions in high dimensions when the nonlinearities have super quadratic growth rates. In dimension 1, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, that is, nonlinearities might have arbitrarily high growth rates, an additional entropy inequality had to be imposed. In this paper, we remove this extra entropy assumption completely and obtain global boundedness for reaction–diffusion systems with cubic intermediate sum condition. The novel idea is to show a nonconcentration phenomenon for mass dissipating systems, that is the mass dissipation implies a dissipation in a Morrey space $��M^{�1�,�\delta �}��\left(�\Omega �\right)��$ for some $��\delta �>�0�$. As far as we are concerned, it is the first time such a bound is derived for mass dissipating reaction–diffusion systems. The results are then applied to obtain global existence and boundedness of solutions to an oscillatory Belousov–Zhabotinsky system, which satisfies cubic intermediate sum condition but does not fulfill the entropy assumption. Extensions include global existence mass controlled systems with slightly super cubic intermediate sum condition.

In this paper, we present a description of invariant twisted Kähler–Einstein (tKE) metrics on flag varieties. Additionally, we delve into the applications of the concepts utilized in proving our main result, particularly concerning the existence of the invariant twisted constant scalar curvature Kähler metrics. Moreover, we provide a precise description of the greatest Ricci lower bound for arbitrary Kähler classes on flag varieties. From this description, we establish a sequence of inequalities linked to optimal upper bounds for the volume of Kähler metrics, relying solely on tools derived from the Lie theory. Further, we illustrate our main results through various examples, encompassing full flag varieties, the projectivization of the tangent bundle of $�P^{�n�+�1�}$, and families of flag varieties with a Picard number 2.

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain gave a nontrivial estimate for these and more general sums. In this work, we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.

We investigate a parabolic Monge–Ampère type equation on compact almost Hermitian manifolds and derive a priori gradient and second-order derivative estimates for solutions to this parabolic equation. These a priori estimates give us higher order estimates and a long-time solution. Then, we can observe its behavior as $��t�\to �\infty �$.

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem

In this paper, we derive the equations that constitute the nonlinear mathematical model of an extensible thermoelastic Timoshenko system. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations. The model takes account of the effects of extensibility, where the dissipations are entirely contributed by temperature. Based on the semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the considered problem under the condition (65). Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non-exponential stability of the same problem; but strongly stable by following a result due to Arendt–Batty. In the absence of additional mechanical dissipations, the system is often not highly stable. By adding a damping frictional function to the first equation of the nonlinear derived model with extensibility and using the multiplier method, we show that the solutions decay exponentially if Equation (85) holds.