Examples and Counterexamples最新文献

In the present paper, we have solved the equation $x^2-(p^2{q}^2\pm 3p)y^2=k^t,x^2-(p^2{q}^2\pm 5p)y^2=k^t$ and expressed its positive integer solutions in terms of generalized Fibonacci, generalized Lucas and generalized Pell, generalized Pell–Lucas sequences. With the help of this equation, we have found units of $Z\left[\sqrt{885}\right]$ and $Z\left[\sqrt{915}\right]$ in terms of generalized Fibonacci, generalized Lucas, generalized Pell and generalized Pell–Lucas numbers.

We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator $B$ in terms of existence and maximal monotonicity of the proto-derivative of $B$.

We continue the investigation on the spectrum of operators arising from the discretization of partial differential equations. In this paper we consider a three field formulation recently introduced for the finite element least-squares approximation of linear elasticity. We discuss in particular the distribution of the discrete eigenvalues in the complex plane and how they approximate the positive real eigenvalues of the continuous problem. The dependence of the spectrum on the Lamé parameters is considered as well and its behavior when approaching the incompressible limit.

Numerical investigations of mechanical stresses for phase transforming battery electrode materials on the particle scale are computationally highly demanding. The limitations are mainly induced by the strongly varying spatial and temporal scales of the underlying phase field model, which require an ultra fine mesh and time resolution, however, solely at specific stages in space and time. To overcome these numerical difficulties we present a general-purpose space and time adaptive solution algorithm based on an $hp$-adaptive finite element method and a variable-step, variable-order time integrator. At the example of a chemo-mechanical electrode particle model we demonstrate the computational savings gained by the $hp$-adaptivity. In particular, we compare the results to an $h$-adaptive finite element method and show the reduction of computational complexity.

The evolutionary dynamics of zero-sum and non zero-sum games under replicator equations could be drastically different from each other. In zero-sum games, heteroclinic cycles naturally occur whenever the species of the population supersede each other in cyclic fashion (like for the Rock–Paper–Scissors game). In this case, the highly erratic oscillations may cause the divergence of the time averages. In contrast, it is a common belief that all “reasonable” replicator equations of non-zero sum games satisfy “The Folk Theorem of Evolutionary Game Theory” which asserts that $\left(i\right)$ a Nash equilibrium is a rest point; $\left(ii\right)$ a stable rest point is a Nash equilibrium; $\left(iii\right)$ a strictly Nash equilibrium is asymptotically stable; $\left(iv\right)$ any interior convergent orbit evolves to a Nash equilibrium. In this paper, we propose two distinct classes of replicator equations generated by Schur-convex potential functions which exhibit two opposing phenomena: stable/predictable and historic/unpredictable behavior. In the latter case, the time averages of the orbit will slowly oscillate during the evolution of the system and do not converge to any limit. This will eventually cause the divergence of higher-order repeated time averages.

A code $C\subseteq {\{0,1,2\}}^n$ is said to be trifferent with length $n$ when for any three distinct elements of $C$ there exists a coordinate in which they all differ. Defining $T\left(n\right)$ as the maximum cardinality of trifferent codes with length $n$, $T\left(n\right)$ is unknown for $n\ge 5$. In this note, we use an optimized search algorithm to show that $T\left(5\right)=10$ and $T\left(6\right)=13$.

The following model $p\left(t\right)u^{\prime }\left(t\right)-p\left(0\right)\beta -{\int }_0^t\left(\left(gu\right)\left(s\right)+\sum _{i=1}^m\left({\xi }_i\left(s\right)x\left({\nu }_i\left(s\right)\right)-{\psi }_i\left(s\right)x\left({\mu }_i\left(s\right)\right)\right)\right)dQ\left(s\right)=-F\left(t\right)$of the Stieltjes string on the segment $[0,l]$ with a nonlocal initial value condition $u\left(0\right)=\alpha$ is considered. $D$-stability conditions of the unique solution of the mentioned problem are established. Also, example is presented.

We extend Stein’s lemma for averages that explicitly contain the Gaussian random variable at a power. We present two proofs for this extension of Stein’s lemma, with the first being a rigorous proof by mathematical induction. The alternative, second proof is a constructive formal derivation in which we express the average not as an integral, but as the action of a pseudodifferential operator defined via the Gaussian moment-generating function. In extended Stein’s lemma, the absolute values of the coefficients of the probabilist’s Hermite polynomials appear, revealing yet another link between Hermite polynomials and normal distribution.