Practical Criteria for H-Tensors and Their Application

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

Spontaneous polarization effects on solid high harmonic generation in ferroelectric lithium niobate crystals

High-order harmonic generation (HHG) in ferroelectric lithium niobate (LiNbO$_{3}$) is investigated theoretically by solving the semi-conductor Bloch equations. Because of the spontaneous polarization, even-order harmonics are produced in the HHG spectra of the LiNbO$_{3}$ crystal driven by a monochromatic multi-cycle 3300-nm laser. Our numerical calculation shows that they are originated from the suppression of one half-optical cycle HHG process in each cycle of the driving field due to the spontaneous polarization. We also illustrate that the spontaneous polarization will increase the harmonic yield and extend the maximally attainable cutoff energy at the same time. We further report that the carrier-envelope phase dependence of HHG spectra changes from a minimum period of $\pi$ rad to 2$\pi$ rad when the laser polarization direction is parallel/anti-parallel to the spontaneous polarization direction in LiNbO$_{3}$ crystal. This is promising to be utilized as an isolated attosecond pulse (IAP) gating mechanism. Moreover, the two-color relative phase dependence of HHG in LiNbO$_{3}$ is also investigated and shows broken inversion-symmetry.

Asymptotic and Oscillatory Properties of Noncanonical Delay Differential Equations

In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the significance of our results, we apply them to delay differential equation of Euler-type.

Some Oscillation Results for Even-Order Differential Equations with Neutral Term

The objective of this work is to study some new oscillation criteria for even-order differential equation with neutral term rxzn−1xγ′+qxyγζx=0. By using the Riccati substitution and comparison technique, several new oscillation criteria are obtained for the studied equation. Our results generalize and improve some known results in the literature. We offer some examples to illustrate the feasibility of our conditions.

Maximum principle for higher order operators in general domains

We first prove De Giorgi type level estimates for functions in W 1,t (Ω), Ω ⊂ R N $ \Omega\subset{\mathbb R}^N $ , with t > N ≥ 2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

New Oscillation Results of Even-Order Emden–Fowler Neutral Differential Equations

The objective of this study is to establish new sufficient criteria for oscillation of solutions of even-order delay Emden-Fowler differential equations with neutral term rıyı+mıygın−1γ′+∑i=1jqiıyγμiı=0. We use Riccati transformation and the comparison with first-order differential inequalities to obtain theses criteria. Moreover, the presented oscillation conditions essentially simplify and extend known criteria in the literature. To show the importance of our results, we provide some examples. Symmetry plays an essential role in determining the correct methods for solutions to differential equations.

Mathematical solutions to the problem of creating basic decenter optical mirror models from the coma and astigmatism correction conditions

A serious drawback of reflective optics is a center without central screening that degrades the image quality. To eliminate it, rotations or displacements of mirrors are introduced, but there appear even-order non-elementary aberrations that must be corrected. The creation of compositions with decentered catoptric elements requires further development of the calculation and methodological base. The exact formulas are obtained for calculation of real rays from the astigmatism and coma correction conditions for the given angles of incidence of the main ray on the mirror surfaces and the “oblique” thickness d, that determines their mutual position. Based on the proposed formulas, a new method for parametric calculation of decentered mirror systems has been created, which allows one to compose algorithms and to design both basic models and complex mirror systems from off-axis mirrors. The development of new algorithms for two- and three-mirror off-center lenses will increase the accumulated potential of computational optics. The scope of the proposed technique can be expanded in terms of the number of components.

Heron’s cubic root iteration method with a new approach

Heron’s cubic root iteration formula conjectured by Wertheim is proved and extended for any odd order roots. Some possible proofs are suggested for the roots of even order. An alternative proof of Heron’s general cubic root iterative method is explained. Further, Lagrange’s interpolation formula for nth root of a number is studied and found that Al-Samawal’s and Lagrange’s method are equivalent. Again, counterexamples are discussed to justify the effectiveness of the present investigations.

On the Oscillation of Solutions of Differential Equations with Neutral Term

In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples.