Partial Moduli Research Articles

Conjugate function and the modulus of continuity of k-th order

The estimates of the partial moduli of continuity of k-th order (k≧3) of the conjugate functions of several variables are obtained in the space ℂ(Tn). The exactness of these estimates is established by proper examples.

Optimal Riser Design by STL Arbitrarily Slitting Method and Genetic Algorithm

It is essential to consider the proper position and volume of riser during the design of feeding system. In this paper, we are presenting a new method of designing optimal riser in the steel casting processes. A technique of slitting arbitrarily 3D entity model by STL (Stereolithography) was used to obtain accurate values of the partial modulus of casting, and then a mathematical model of the process of the riser design was optimized by a genetic algorithm (GA); with the help of the CAE system, which has an ability to calculate automatically and verify the validity of the optimized results, we will pursue the goal of obtaining the desired riser with optimal size and distribution but without causing any defect in casting. Thus, by combining the numerical optimization with the traditional riser design, our method proposed here will be more practical and reliable. In the end, we give an example to demonstrate the feasibility of our optimized approach in the riser design. Keywords：Riser design, Genetic Algorithm, casting optimization, STL

Two-side estimates for sums of absolute values of Fourier coefficients of functions from H ω (T m ) with the use of partial moduli of smoothness

Two-side estimates of sums of absolute values of Fourier coefficients for functions of many variables from the classes H ω (T m ) are established in the paper with the help of partial moduli of smoothness.

Inequalities for Partial Moduli of Continuity and Partial Derivatives

We obtain pointwise and integral type estimates of higher-order partial moduli of continuity in C via partial derivatives. Also, a Gagliardo–Nirenberg type inequality for partial derivatives in a fixed direction is proved. Our methods enable us to study the case when different partial derivatives belong to different spaces, including the space L1.

Lp -moduli of continuity of sections of functions

Let ƒ(x, y) be a function in Lp(ℝ2) (1 ≤ p < ∞) and let ω1(ƒ; δ)p be the partial modulus of continuity of ƒ in Lp with respect to the variable x. For almost all y ∈ ℝ the sections ƒy(x) = ƒ(x, y) belong to Lp(ℝ). We find sharp estimates of the Lp-modulus of continuity of ƒy in terms of ω1(ƒ; δ)p.

Non-Abelian walls in supersymmetric gauge theories

The Bogomol'nyi-Prasad-Sommerfield (BPS) multiwall solutions are constructed in supersymmetric $\mathrm{U}({N}_{\mathrm{C}})$ gauge theories in five dimensions with ${N}_{\mathrm{F}}(g{N}_{\mathrm{C}})$ hypermultiplets in the fundamental representation. Exact solutions are obtained with full generic moduli for infinite gauge coupling and with partial moduli for finite gauge coupling. The generic wall solutions require nontrivial configurations for either gauge fields or off diagonal components of adjoint scalars depending on the gauge. Effective theories of moduli fields are constructed as world volume gauge theories. Nambu-Goldstone and quasi-Nambu-Goldstone scalars are distinguished and worked out. Total moduli space of the BPS non-Abelian walls including all topological sectors is found to be the complex Grassmann manifold $\mathrm{S}\mathrm{U}({N}_{\mathrm{F}})/[\mathrm{S}\mathrm{U}({N}_{\mathrm{C}})\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}({N}_{\mathrm{F}}\ensuremath{-}{N}_{\mathrm{C}})\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]$ endowed with a deformed metric.

Quantitative Inheritance Properties for Simultaneous Approximation by Tensor Product Operators

Our results describe how quantitative properties of univariate operators are inherited by the tensor product of their parametric extensions. This includes statements concerning simultaneous approximation. The estimates are in terms of partial and total moduli of smoothness of higher order. Applications are given for cubic interpolatory splines and Bernstein operators. Further applications are possible.

The Correction Theorem for Anisotropic Spaces

The following old problem is solved. Given an e> 0, a function f:[0,1]n→ ℝ, and the partial moduli of continuity of this function evaluated in a symmetric space X, find a set I(e) of measure larger than 1-e such that the partial uniform moduli of continuity of f determined for the points of this set admit an unimprovable (with respect to all restrictions to sets of measure larger than 1- e) estimate of partial uniform moduli of continuity and write out this estimate of the uniform partial moduli of continuity.

Linearization as a method for determining parameters of relaxation spectra

A new method for determining parameters of a relaxation spectrum of visco-elastic bodies is proposed. The basic idea of the method consists of presenting measured frequency dependencies of the dynamic modules components by the power series. The coefficients of these series are found by comparing varying values with experimental data by linear functional of errors. The parameters of a spectrum are then calculated from the coefficients of the series. This procedure allows us to overcome the main problem of finding relaxation times, i.e. to pass from non-linear to linear functional of errors. It gives us the unique possibility of unambiguously estimating weights (partial moduli) of lines in a discrete relaxation spectrum and relaxation times because there is only a single minimum in the functional of errors.

A multidimensional analog of a theorem due to zygmund

Zygmund proved an inequality describing the dependence of the modulus of continuity of the adjoint function on that of the original function lying in the space of 2π-periodic continuous functions. The present article contains estimates of partial moduli of continuity of the adjoint function of several variables in the spaceC. Examples show that these estimates are sharp.

Multidimensional analogue of a theorem of Privalov

A criterion is established for the continuity of functions that are conjugate in the sense of Cesari to a given function in the class where , , () are the partial moduli of continuity of and () are moduli of continuity. Best possible estimates of the partial modulus of continuity of a function conjugate to are obtained in the case when the () satisfy two specific conditions. These conditions on the modulus of continuity are shown to be necessary and sufficient in order that the conjugation operator violate the invariance of the class in the same way as it violates that of ().

Об одной теореме влож ения для функций мног их переменных

In the paper the best possible conditions are found which should be imposed on the orders of partial and mixed integral moduli of continuity of a function fromLp(TN) (1<p<∞,T≡[0,2π],N=2, 3, ...), in order that the imbedding $$f \in \exp (\sqrt[\sigma ]{{L(T^N )}}),2 \leqq \sigma< \infty ,$$ , is valid.

Effect of polydispersity in polystyrene on stress relaxation

AbstractStress relaxation master curves are presented for twenty‐three polystyrene samples whose weight‐average molecular weights Mw varied from 20,800 to 310,000 and whose heterogeneity indices varied from 1.0 to 15.92. The maximum relaxation times τm, the associated partial modulus Em, and the flow viscosity η were determined for these samples at 115°C. It is shown that at 115°C. the relation log τm (minutes) = −15.10 + 3.4 Mw is obeyed. It is also shown that the quantity αB = τmEm/η correlates with the heterogeneity index of the sample. It is concluded that stress relaxation studies can be used to obtain weight‐average molecular weights and also to provide a qualitative measure of the heterogeneity index.

Relaxation Time Spectrum, Elasticity, and Viscosity of Rubber. II

Abstract In Part I of this work, the occurrence of a relaxation time spectrum in high-polymer materials was described, and a quantitative expression for the relaxation time spectrum for rubber and rubberlike materials was found, based on the creep curve observed for these materials, i.e., the distribution density was given with which the partial elastic moduli are distributed among the relaxation times. In the present paper, conclusions have been derived regarding the elastic and viscous behavior of the materials on the basis of the relaxation time spectrum reported. It has been found that the expected creep curve in such a material, i.e., the curve of the change of length after a definite time at a constantly held stress, must be practically identical with the reciprocal value of the elastic modulus E, which can be determined as a time function from the decrease of stress after rapid deformation. The E modulus observed at time t after rapid deformation is a function of all portions of the relaxation time spectrum in such a way, however, that E(t) is mainly determined by those portions of the spectrum, for which the relaxation time τ is greater than t. The accuracy with which the distribution of the partial elastic moduli can be computed from the available experimental data is not equally large for all regions of the relaxation time spectrum. The possible errors lie both in the region of very large relaxation times τ and very small times. But it appears that the error in the E modulus and in the viscosity, due to the contributions of the inaccurately known portions of the spectrum, is small in all cases and that these inaccuracies represent only a small constant added to the E modulus and the viscosity in the field of practical interest. The dynamic elastic modulus, in the frequency range 10−2 to 104 per second, on the basis of the relaxation time spectrum, is found experimentally to be almost independent of the period. On the other hand, the dynamic viscosity increases proportionally to the period. For a periodTs, those restoring force mechanisms whose relaxation time is somewhat but not much smaller than Ts contribute almost entirely to the magnitude of the dynamic viscosity. The amount of heat developed in a test-sample per cc. and per second by periodic displacement increases proportional to the frequency of the applied deformation. The proportionality factor can be calculated from the deformation-time curve observed at constant load, i.e., from the creep curve. It is evident that the distribution density in the relaxation time spectrum in the region τ&amp;lt;10−4 second increases somewhat more rapidly than the extrapolation of the formula valid for the region 10−2 to 104 second would indicate. It was shown that those restoring force mechanisms for which the relaxation time is greater than 10−2 second are probability mechanisms, whereas in the case of shorter relaxation times, energy mechanisms occur in increasing proportions in addition to, or in place of, the probability mechanisms. The occurrence of probability mechanisms having the relaxation time t* is to be interpreted in such a way that linear sections of molecular weight M*, which must be smaller than the molecular weight Mf of the lattice link or of the total molecule, need a time t* to change their configuration or orientation noticeably in the interior of the mass in which they are embedded. Accordingly, a relationship between t* and M* can be given t* depends on the viscosity η* by which the embedding medium opposes a Brownian movement by the linear section of molecular weight M*. Conclusions can be drawn thatη* increases rapidly with M*, e.g., exponentially. Furthermore, as is shown, the relaxation time of a restoring force mechanism considered not only through its contribution to the viscosity determines the relaxation times of all the other restoring force mechanisms, but also by their contributions to the viscosity is itself determined by the total of all the other relaxation times.