Spherical Derivatives Research Articles

Goufo-Caputo fractional viscoelastic photothermal model of an unbounded semiconductor material with a cylindrical cavity

In order to describe the photothermal dynamic behavior of viscoelastic semiconductor materials, the authors of this work propose a fractional modification of the Kelvin-Voigt type. The generalized Mittag-Leffler function method with two parameters was utilized as the non-local, nonsingular kernel of the Goufo-Caputo fractional operator, which is similar to the AB fractional derivative. Also, a new concept of photovoltaic heat transfer has been developed to describe the process of photothermal heating in semiconductors, in which light energy is converted into thermal energy by absorption. This unique model is grounded in the theoretical framework of the Moore-Gibson-Thompson (MGT) equation, and it offers additional insights into the underlying process of light sensitive heat transfer, as well as the intricate interaction between heating signals, elasticity, and plasma. As a direct application of the proposed model, the photothermal interactions that occur when an infinitely rotating semiconductor material with a cylindrical hole is subjected to a laser thermal flux and immersed in a constant axial magnetic field have been investigated. These findings may have implications for developing optically absorbent nanostructures and their use in photothermal energy generation. For this reason, the comparison of modified photoelasticity models with spherical derivatives and classical models applied to semiconducting materials seems interesting.

Revisiting the Rellich inequality

We revisit the Rellich inequality from the viewpoint of isolating the contributions from radial and spherical derivatives. This naturally leads to a comparison of the norms of the radial Laplacian and Laplace–Beltrami operators with the standard Laplacian. In the case of the Laplace–Beltrami operator, the three-dimensional case is the most subtle and here we improve a result of Evans and Lewis by identifying the best constant. Our arguments build on certain identities recently established by Wadade and the second and third authors, along with use of spherical harmonics.

Boundary characteristics of meromorphic functions with summable spherical derivation and annular functions. Consideration

In this paper we formulate classical theorems Plesner and Meyer on the boundary behavior of meromorphic functions and their refinement and strengthening - Gavrilov's and Kanatnikov's theorems. An application of these theorems to classes of meromorphic functions with integrable spherical derivative and annular holomorphic functions is presented. Collingwood's theorem on boundary singularities of the Tsuji function as well as Kanatnikov's theorems are formulated. Kanatnikov's theorems strengthen and generalize Collingwood's theorem to broader classes of meromorphic functions with summable spherical derivatives. Special attention is paid to the boundary properties of annular holomorphic functions. The behavior of annular holomorphic functions on the boundary of the unit circle is considered. It is shown that Gavrilov's P-sequences play an important role in the study of the boundary properties of holomorphic and meromorphic functions.

Normality and Product of Spherical Derivatives

Normality and Product of Spherical Derivatives

On Some Classes with Norms of Meromorphic Function Spaces Defined by General Spherical Derivatives

The main concerned target of this article is to define and study some concerned classes of meromorphic function spaces using the general spherical derivatives. The general Besov-type classes of meromorphic functions as well as the general normal functions are considered intensively and both are compared deeply with each other. Specifically, multiple results concerning general meromorphic-type classes as well as non-normal classes are obtained by the help of general spherical derivatives. The concerned results are proved by constructing some specific mild conditions on the sequences of points belonging to the concerned meromorphic-type classes. The obtained results generalize and improve the corresponding previous results in some concerned respects. The concerned proofs and methods are simply presented.

Improved Hardy inequalities and weighted Hardy type inequalities with spherical derivatives

The first purpose of this paper is to set up several enhancements of the Hardy type inequalities on certain subspaces of the Sobolev spaces. The second aim is to explore the role of the spherical derivative in such improvements and to prove some weighted versions of the Hardy inequality with spherical derivative. As applications of our results, we establish several Hardy’s inequalities and the Hardy inequalities with spherical derivatives on half-spaces.

Hardy type inequalities with spherical derivatives

A Hardy type inequality is presented with spherical derivatives in $${\mathbb {R}}^{n}$$ with $$n\ge 2$$ in the framework of equalities. This clarifies the difference between contribution by radial and spherical derivatives in the improved Hardy inequality as well as nonexistence of nontrivial extremizers without compactness arguments.

Ultrafast Melting of Metal–Organic Frameworks for Advanced Nanophotonics

AbstractThe conversion of metal–organic frameworks (MOFs) into derivatives with a well‐defined shape and composition is considered a reliable way to produce efficient catalysts and energy capacitors at the nanometer scale. Yet, approaches based on conventional melting of MOFs provide the derivatives such as amorphous carbon, metal oxides, or metallic nanoclusters with an appropriate morphology. Here ultrafast melting of MOFs is utilized by femtosecond laser pulses to produce a new generation of derivatives with complex morphology and enhanced nonlinear optical response. It is revealed that such a nonequilibrium process allows conversion of interpenetrated 3D MOFs comprising flexible ligands into well‐organized spheres with a metal oxide dendrite core and amorphous organic shell. The ability to produce such derivatives with a complex morphology is directly dependent on the electronic structure, crystal density, ligand flexibility, and morphology of initial MOFs. An enhanced second harmonic generation and three‐photon luminescence are also demonstrated due to the resonant interaction of 100–1000 nm spherical derivatives with light. The results obtained are in the favor of new approaches for melting special types of MOFs for nonlinear nanophotonics.

Fabrication and Characterization of Nanoenergetic Hollow Spherical Hexanitrostibene (HNS) Derivatives.

The spherization of nanoenergetic materials is the best way to improve the sensitivity and increase loading densities and detonation properties for weapons and ammunition, but the preparation of spherical nanoenergetic materials with high regularization, uniform size and monodispersity is still a challenge. In this paper, nanoenergetic hollow spherical hexanitrostibene (HNS) derivatives were fabricated via a one-pot copolymerization strategy, which is based on the reaction of HNS and piperazine in acetonitrile solution. Characterization results indicated the as-prepared reaction nanoenergetic products were HNS-derived oligomers, where a free radical copolymerization reaction process was inferred. The hollow sphere structure of the HNS derivatives was characterized by scanning electron microscopy (SEM), transmission electron microscope (TEM), and synchrotron radiation X-ray imaging technology. The properties of the nanoenergetic hollow spherical derivatives, including thermal decomposition and sensitivity are discussed in detail. Sensitivity studies showed that the nanoenergetic derivatives exhibited lower impact, friction and spark sensitivity than raw HNS. Thermogravimetric-differential scanning calorimeter (TG-DSC) results showed that continuous exothermic decomposition occurred in the whole temperature range, which indicated that nanoenergetic derivatives have a unique role in thermal applications. Therefore, nanoenergetic hollow spherical HNS derivatives could provide a new way to modify the properties of certain energetic compounds and fabricate spherical nanomaterials to improve the charge configuration.

On the real differential of a slice regular function

Abstract In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called spherical expansion), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.

Normal functions and products of spherical derivatives

In this paper, we obtain a characterization of normal functions in terms of the products of certain spherical derivatives, which improves a related result due to Chen and Lappan.

On the exceptional set in a conditional theorem of Littlewood

In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, “most” preimages of almost all points are concentrated in a small subset of the plane. In 1988, Lewis and Wu proved Littlewood’s conjecture. Using techniques from complex dynamics, we construct entire functions of finite order with a bounded set of singular values for which the set of exceptional preimages is infinite, with logarithmically growing cardinality.

Three-Dimensional Wave Gradiometry for Polarized Seismic Waves

Abstract We expand the theoretical development of seismic‐wave gradiometry to 3D using the polarized wavefield. First, we develop a map that relates the Cartesian spatial derivatives to the spatial derivatives in spherical coordinates. We then develop a set of partial differential equations that relate the spherical derivatives of the observed wavefield to the body‐wave radiation patterns. Because the spherical derivatives of the wavefield depend on the direction of wave propagation, we also develop relationships to find the propagation direction using simple ratios of the Cartesian‐based wave derivatives. Finally, we test the robustness of estimating the wave‐propagation direction, and its sensitivity to noise.

Spherical derivatives and normal families

We show that a family of functions meromorphic in a plane domain D whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each f meromorphic in the unit disk D, infz∈Df#(z) ≤ 1/2, where f# denotes the spherical derivative of f.

Synthesis of Multiwall Carbon Nanotubes in the Pore System and/or on the Outer Surface of Mesoporous MCM-41 Structures of Various Morphology

Multiwall carbon nanotubes were synthesized either on the outer surface of iron containing mesoporous silicates using catalytic chemical vapor deposition (CCVD) or in the pore system of morphologically different mesoporous materials (hexagonal or spherical shapes) with graphitization of the template molecules. Transmission electron microscopy (TEM) study shows that the CCVD method resulted in long, bent and well graphitized carbon nanotubes on impregnated samples irrespective to the morphology of the silicate. Isomorphously substituted spherical MCM-41 with low Si/Fe ratio was found to be active catalysts for carbon nanotube production in CCVD as well. Synthesis of MWNTs with graphitization of template molecules in the pores of MCM-41 was successful in hexagonal MCM-41 samples irrespective that they contain or not iron in the silicate framework. Carbon nanotube formation was not observed in spherical derivatives of these samples during the graphitization process.

A New Approach to Support Point Theory for the Class $$\mathcal{S}$$

In this paper, we introduce a new technique for measuring the hypothetical non-monotonicity of the argument of the vector tracing the omitted arc of a support point of the class \({\cal S}\). It has been previously shown that the number of sign changes of (arg w(t))′ on the omitted arc is finite. Here, we derive upper bounds for that number in terms of both the spherical and Schwarzian derivatives. Thus, our innovative approach identifies an inherently interesting connection between support point theory and these derivatives.

Synthesis and Complexation Properties of Some Novel Lariat-Crown Ethers.

Several new lariat-crown ethers bearing either bridged bisdioxine or tetraoxaadamantane units as chiral substituents are prepared by reacting the corresponding amino-crown ether derivatives with the dimeric alpha-oxoketene, the latter obtained by flash vacuum pyrolysis of a furan-2,3-dione precursor. Complexation properties towards differently charged metal ions are investigated by H-1 NMR titration to obtain complexation constants (K-c-values for potassium/ sodium rhodanides: 480-1100 mol dm(-3)), as well as extraction experiments to explore the metal ion transportation abilities of the new lariat crown derivatives. In particular, a significantly increased ability to transport metal ions from water into chloroform was found with spherical tetraoxaadamantyl derivatives when compared with the free amino-benzocrown ethers.

Normal families of holomorphic functions and mappings on a Banach space

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented.

Value distribution of the Painlevé Transcendents

We consider the solutions of the First Painleve Differential Equationω″=z+6w2, commonly known as First Painleve Transcendents. Our main results are the sharp order estimate λ(w)≤5/2, actually an equality, and sharp estimates for the spherical derivatives ofw andf(z)=z−1w(z2), respectively:w#(z)=O(|z|3/4) andf#(z)=O(|z|3/2). We also determine in some detail the local asymptotic distribution of poles, zeros anda-points. The methods also apply to Painleve’s Equations II and IV.

Products of spherical derivatives and normal functions

AbstractChen and Gu [1] have given some results relating to normal families, and, in this paper, we give versions of these results valid for normal functions. In the process, we improve some of our previous results involving products of certain spherical derivatives as they relate to normal functions. Some examples are given to show the sharpness of our results.