<abstract><p>With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that it converges to a log-periodic function of the form</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ A\sin \left( \log t\right) +B\cos \left( \log t\right) $\end{document} </tex-math></disp-formula></p>
<p>as$ \ t\rightarrow \infty, \ $where $ A, \ B $ are constants. Moreover, for any two numbers $ \alpha < \beta, \ $we are also able to construct a solution satisfying the oscillation limits</p>
<p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha,\ \ \ \limsup\limits _{t\rightarrow \infty}y\left( x,t\right) = \beta,\ \ \ x\in K $\end{document} </tex-math></disp-formula></p>
<p>on any compact subset$ \ K\subset \mathbb{R}. $</p></abstract>
We have established the estimation of deviation of continuous $2\pi$-periodic function $f(x)$ from the trigonometric polynomial of S.N. Bernstein's type that corresponds to it, by the modulus of continuity of the function $f(x)$.
A symmetric m-tilings model on the plane is assembled to be a phase portrait for a structurally stable Hamiltonian system. Integral of the system is the quasi-periodic function with m-fold rotational symmetry being result of the semi-dynamic system action on the unit interval. Some examples for pentagonal and heptagonal tilings has been built in detail. Some properties of an additive measure and order for tilings have been discussed.
We dealt with the nature of the points under the influence of periodic function chaotic functions associated functions chaotic and sufficient conditions to be a very chaotic functions Palace
AbstractWe investigate axion inflation where the gravitational Chern–Simons term is coupled to a periodic function of the inflaton. We find that tensor perturbations with different polarizations are amplified in different ways by the Chern–Simons coupling. Depending on the model parameters, the resonance amplification results in a parity-violating peak or a board plateau in the energy spectrum of gravitational waves, and the sharp cutoff in the infrared region constitutes a characteristic distinguishable from stochastic gravitational wave backgrounds produced by matter fields in Einstein gravity.
In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation ( − Δ ) α u + V ( x ) u = A ( ϵ x ) f ( u ) , x ∈ R N , where α ∈ ( 0 , 1 ), ϵ is a positive parameter, N > 2 α, ( − Δ ) α stands for the fractional Laplacian, f is a continuous function with subcritical growth, V ∈ C ( R N , R ) is a Z N -periodic function and A ∈ C ( R N , R ) satisfies some appropriate assumptions.
AbstractThe generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, $$f(z)^nf^{(k)}(z)$$
f
(
z
)
n
f
(
k
)
(
z
)
is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case $$k=1$$
k
=
1
. When $$k\ge 2$$
k
≥
2
the conjecture is shown to be true under certain conditions even if n is allowed to have negative integer values.
We present the generalization of M.I. Chernykh's results about the estimate of the best $L_2$-approximation of periodic function $f$ by trigonometric polynomials by its $L_2$-modulus of continuity, in the case of functions with values in Hilbert space.
On an asymptotically log-periodic solution to the graphical curve shortening flow equation
