Global attractivity for a nonautonomous Nicholson’s equation with mixed monotonicities

We consider a Nicholson’s equation with multiple pairs of time-varying delays and nonlinear terms given by mixed monotone functions. Sufficient conditions for the permanence, local stability and global attractivity of its positive equilibrium K are established. The main novelty here is the construction of a suitable auxiliary difference equation x n+1 = h(x n ) with h having negative Schwarzian derivative, and its application to derive the attractivity of K for a model with one or more pairs of time-dependent delays. Our criteria depend on the size of some delays, improve results in recent literature and provide answers to open problems.

Universe as Klein–Gordon eigenstates

We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the$$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$ O β ( ρ , p ) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_\beta $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$\eta \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_\beta $$ t β ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.

The entropy of chaotic transitions of EEG phase growth in bipolar disorder with lithium carbonate

The application of chaos measures the association of EEG signals which allows for differentiating pre and post-medicated epochs for bipolar patients. We propose a new approach on chaos necessary for proof of EEG metastability. Shannon entropies of concealed patterns of Schwarzian derivatives from absolute instantaneous frequency transformations of EEG signals after Hilbert transform are compared and found significantly statistically different between pre and post-medication periods when fitted to von Bertalanffy’s functions. Schwarzian dynamics measures was compared at first baseline and then at the end of the first hour of one dose 300 mg lithium carbonate intake for the same subject in depressive patients. With an application of Schwarzian derivative on the prediction of von Bertalanffy’s models, integration and segregation of phase growth orbits of neural oscillations can be understood as an influence of chaos on the mixing of frequencies. A phase growth constant parameter was performed to determine the bifurcation parameter of von Bertalanffy’s model at each given non-overlapped EEG segment. Schwarzian derivative was sometimes very close positive near the origin but stayed negative for most of the number of segments. Lithium carbonate changed the chaotic invariants of the EEG Schwarzian dynamics and removed sharp boundaries in the bipolar spectrum.

Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

Let S H f be the Schwarzian derivative of a univalent harmonic function f in the unit disk D , compatible with a finitely generated Fuchsian group G of the second kind. We show that if S H f 2 1 − z 2 3 d x d y satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G , then S H f 2 1 − z 2 3 d x d y is a Carleson measure in D .

Entropy of Chaotic Transitions of EEG Phase Growth in Bipolar Disorder With Lithium Carbonate

The application of chaos theory measures in the association of EEG signals which allows for differentiating pre and postmedicated epochs for bipolar patients. We propose a new approach on positive Schwarzian chaos necessary for a proof of EEG power spectrum of metastability. Shannon entropies of those concealed patterns of positive Schwarzian derivatives from canonical angle transformations of EEG phases from Hilbert transform are compared and found significantly statistical different between pre and post medication periods. Entropy change of chaotic transition measures were compared at first baseline and then at the end of first hour of 300 mg lithium carbonate intake for the same subject in depressive patients. With an application of Schwarzian derivative on the prediction of the von Bertalanffy’s models, integration and segregation of phase growth orbits of neural oscillations can be understood as an influence of chaos on mixing of frequencies. A first order curve-fitting function was performed to determine bifurcation parameter of von Bertalanffy’s model at each given overlapped EEG segment. Schwarzian derivative was positive near the origin which revealed robust chaos. We founded that treatment with Lithium carbonate significantly altered Schwarzian spectrum of chaotic structure and entropy change in Schwarzian amplitudes even though it was not observed in classical EEG power spectrum. Lithium carbonate reduced the strong chaos spectrum of EEG Schwarzian dynamics and removed sharp boundaries in bipolar spectrum.