Baernstein’s star-function, maximum modulus points and a problem of Erdős

The paper is devoted to the development of Baernstein's method of \(T^{*}\)-function. We consider the relationship between the number of separated maximum modulus points of a meromorphic function and the \(T^{*}\)-function. The results of Bergweiler, Bock, Edrei, Goldberg, Heins, Ostrovskii, Petrenko, Wiman are generalized. We also give examples showing that the obtained estimates are sharp.

Effects of nanocrystalline calcium oxide particles on mechanical, thermal, and electrical properties of EPDM rubber

In this research, the effect of calcium oxide (CaO) nanocrystalline particles filled ethylene propylene diene monomer (EPDM) rubber composites is investigated, at different weight percentages (1.0, 2.0, 4.0, and 8 wt%) of CaO nanocrystalline particles using two methods of mixing. In one case conventional mixing on twin roll-mill was used, in the other case ultrasonic mixing as a pre-mixing was applied. CaO particles are synthesized by the precipitation method. The average crystallite size of CaO is 100 ± 20 nm. Adding CaO nanocrystalline particles increases the thermal stability of EPDM and the glass transition temperature. The hardness of EPDM rubber gradually increases with increasing the amount of CaO particles, the maximum hardness 64.2 observed in 8 wt% of CaO particles for both cases almost 26% higher than neat EPDM. Tensile strength decreases, while the maximum % modulus of the ultrasonic mixed sample was 1.48 MPa which is 24% higher than EPDM.

Some Inequalities for the Maximum Modulus of Rational Functions

For a polynomial p z of degree n , it follows from the maximum modulus theorem that max z = R ≥ 1 p z ≤ R n max z = 1 p z . It was shown by Ankeny and Rivlin that if p z ≠ 0 for z < 1 , then max z = R ≥ 1 p z ≤ R n + 1 / 2 max z = 1 p z . In 1998, Govil and Mohapatra extended the above two inequalities to rational functions, and in this paper, we study the refinements of these results of Govil and Mohapatra.

Computation of velocities and polarization vectors in weakly anisotropic media

To compute the phase velocities in the weakly anisotropic media, we propose to transform the Christoffel matrix K into an adapted coordinate system, and, then, apply the perturbation theory to the resulting matrix X. For a weakly anisotropic medium, the off-diagonal elements of the matrix X are small compared to the diagonal ones, and two of them are equal to 0. The diagonal elements of the matrix X are initial approximations of the phase velocities squared. To refine them, it is proposed to use either iterative schemes or Taylor series expansions. The initial terms of the series and the formulas of iterative schemes are expressed through the elements of the matrix X and have a compact analytical representation. The odd-order terms in the series are equal to 0. To approximate the phase velocities of the S1 and S2 waves, a stable method is proposed based on solving a quadratic equation with the coefficients being expressed in terms of the matrix elements and the precomputed value of the qP wave phase velocity squared. For all iterative schemes and series, the convergence conditions are derived. The polarization vector of the wave with the square of the phase velocity is defined as the column with maximum modulus of cofactor of the matrix K-I. The group velocities vectors are computed based on the known components of the polarization vector, the directional vector, and the density-normalized stiffness coefficients. The computational accuracy is demonstrated for the standard orthorhombic model. It is shown how the perturbation theory can be applied to media with strong anisotropy. To do this, first we need to apply several QR transforms or Jacobi rotations of the Christoffel matrix, and then use the perturbation theory. This method with four Jacobi rotations is applied to the calculation of the phase velocities squared for a triclinic medium with a maximum number (32) of singularity points. In this case, the phase velocities are computed with a relative error less than 0,004 %.

Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity

In this paper, we investigate spectral stability of traveling wave solutions to 1D quantum hydrodynamics system with nonlinear viscosity in the [Formula: see text], that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum stability.

On a Result of Hayman Concerning the Maximum Modulus Set

The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a “small” set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four.

Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

For a transcendental entire function f, the property that there exists $$r>0$$ r > 0 such that $$m^n(r)\rightarrow \infty $$ m n ( r ) → ∞ as $$n\rightarrow \infty $$ n → ∞ , where $$m(r)=\min \{|f(z)|:|z|=r\}$$ m ( r ) = min { | f ( z ) | : | z | = r } , is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).

On non-separated zero sequences of solutions of a linear differential equation

Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb {D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb {D}$ and such that possesses a solution having zeros precisely at the points $z_k$ , and the resulting function $a(z)$ has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ for any $p > 0$ . We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.

KINERJA SERAT KAWAT BENDRAT SEBAGAI BAHAN TAMBAH BETON FAS 0.4

Concrete is the most important part of a construction building. The purpose of this study was to examine how the comparison of physical and mechanical properties and optimum levels of the addition of straight tie wire as an added material with a water-cement ratio of 0.4. The percentage of addition of straight tie wire: 0%, 0.5%, 0.75%, 1.0%, of the total weight of the specimen with a tie-wire length of 8 cm. The test specimens for compressive strength, modulus of elasticity, and split tensile are in the form of a cylinder with a diameter of 15 cm and a height of 30 cm, and the specimen for flexural strength is a block with a length of 50 cm, a width of 10 cm and a height of 10 cm. The results show that the maximum compressive strength test on tie wire occurred at a percentage of 0.75% of 16.56 MPa. The maximum modulus of elasticity in tie wire occurred at a percentage of 0.75% of 15184.56 MPa. The maximum split tensile strength of tie wire occurred in a percentage of 0.75% of 1.165 MPa, and the maximum flexural strength of tie wire occurs at a percentage of 0.75% of 1.950 MPa. The research results concluded that the addition of a straight tie-wire to the concrete mixture could increase the compressive strength, split tensile strength, tensile strength, and elastic modulus of concrete.