scholarly journals ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

2011 ◽  
Vol 85 (3) ◽  
pp. 463-475 ◽  
Author(s):  
MEI-RU CHEN ◽  
ZONG-XUAN CHEN
Keyword(s):  
Finite Order ◽  
Meromorphic Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
The Difference ◽  
Image Position

AbstractIn this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

Restrictions on meromorphic solutions of Fermat type equations

2020 ◽  
Vol 63 (3) ◽  
pp. 654-665
Author(s):  
Gary G. Gundersen ◽  
Katsuya Ishizaki ◽  
Naofumi Kimura
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Existence Theorems ◽  
Rational Functions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Nevanlinna Functions ◽  
Positive Integers

AbstractThe Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

scholarly journals On zeros and growth of solutions of complex difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min-Feng Chen ◽  
Ning Cui
Keyword(s):  
Entire Function ◽  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Meromorphic Solutions ◽  
Complex Difference ◽  
Difference Polynomial ◽  
Complex Difference Equations ◽  
The Difference ◽  
Growth Of Solutions

AbstractLet f be an entire function of finite order, let $n\geq 1$ n ≥ 1 , $m\geq 1$ m ≥ 1 , $L(z,f)\not \equiv 0$ L ( z , f ) ≢ 0 be a linear difference polynomial of f with small meromorphic coefficients, and $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 be a difference polynomial in f of degree $d\leq n-1$ d ≤ n − 1 with small meromorphic coefficients. We consider the growth and zeros of $f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$ f n ( z ) L m ( z , f ) + P d ( z , f ) . And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type $f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ f n ( z ) + P d ( z , f ) = p 1 e α 1 z + p 2 e α 2 z , where $n\geq 2$ n ≥ 2 , $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 is a difference polynomial in f of degree $d\leq n-2$ d ≤ n − 2 with small mromorphic coefficients, $p_{i}$ p i , $\alpha _{i}$ α i ($i=1,2$ i = 1 , 2 ) are nonzero constants such that $\alpha _{1}\neq \alpha _{2}$ α 1 ≠ α 2 . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

scholarly journals Meromorphic solutions of difference equations originated from Schwarzian differential equation

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 2003-2015
Author(s):  
Shuang-Ting Lan ◽  
Zhi-Bo Huang ◽  
Chuang-Xin Chen
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Difference Equation ◽  
Rational Function ◽  
Finite Order ◽  
Difference Equations ◽  
Meromorphic Functions ◽  
Meromorphic Solution ◽  
Meromorphic Solutions ◽  
The Difference

Let f (z) be a meromorphic functions with finite order , R(z) be a nonconstant rational function and k be a positive integer. In this paper, we consider the difference equation originated from Schwarzian differential equation, which is of form [?3f(z)?f(z)- 3/2(?2|(z))2]k = R(z)(?f (z))2k. We investigate the uniqueness of meromorphic solution f of difference Schwarzian equation if f shares three values with any meromrphic function. The exact forms of meromorphic solutions f of difference Schwarzian equation are also presented.

The role of differential phase contrast in electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 176-177
Author(s):  
E.M. Waddell ◽  
J.N. Chapman ◽  
R.P. Ferrier
Keyword(s):  
Phase Contrast ◽  
Practical Method ◽  
Position Vector ◽  
Differential Phase ◽  
Pure Phase ◽  
Differential Phase Contrast ◽  
The Difference ◽  
Image Position ◽  
First Moment

Dekkers and de Lang (1977) have discussed a practical method of realising differential phase contrast in a STEM. The method involves taking the difference signal from two semi-circular detectors placed symmetrically about the optic axis and subtending the same angle (2α) at the specimen as that of the cone of illumination. Such a system, or an obvious generalisation of it, namely a quadrant detector, has the characteristic of responding to the gradient of the phase of the specimen transmittance. In this paper we shall compare the performance of this type of system with that of a first moment detector (Waddell et al.1977).For a first moment detector the response function R(k) is of the form R(k) = ck where c is a constant, k is a position vector in the detector plane and the vector nature of R(k)indicates that two signals are produced. This type of system would produce an image signal given bywhere the specimen transmittance is given by a (r) exp (iϕ (r), r is a position vector in object space, ro the position of the probe, ⊛ represents a convolution integral and it has been assumed that we have a coherent probe, with a complex disturbance of the form b(r-ro) exp (iζ (r-ro)). Thus the image signal for a pure phase object imaged in a STEM using a first moment detector is b2 ⊛ ▽ø. Note that this puts no restrictions on the magnitude of the variation of the phase function, but does assume an infinite detector.

Kinematical and Dynamical CBED Pattern Models: Increasing the Accuracy of the Unit-Cell Parameter Measurements Based on CBED Patterns

1990 ◽  
Vol 48 (2) ◽  
pp. 540-541
Author(s):  
I.N. Yadhikov ◽  
S.K. Maksimov
Keyword(s):  
Reciprocal Lattice ◽  
Unit Cell ◽  
Reciprocal Lattice Vector ◽  
Unit Cell Parameters ◽  
Lattice Vector ◽  
Cell Parameters ◽  
Accuracy Of Measurements ◽  
The Difference ◽  
Image Position ◽  
Convergent Beam

Convergent beam electron diffraction (CBED) is widely used as a microanalysis tool. By the relative position of HOLZ-lines (Higher Order Laue Zone) in CBED-patterns one can determine the unit cell parameters with a high accuracy up to 0.1%. For this purpose, maps of HOLZ-lines are simulated with the help of a computer so that the best matching of maps with experimental CBED-pattern should be reached. In maps, HOLZ-lines are approximated, as a rule, by straight lines. The actual HOLZ-lines, however, are different from the straights. If we decrease accelerating voltage, the difference is increased and, thus, the accuracy of the unit cell parameters determination by the method becomes lower.To improve the accuracy of measurements it is necessary to give up the HOLZ-lines substitution by the straights. According to the kinematical theory a HOLZ-line is merely a fragment of ellipse arc described by the parametric equationwith arc corresponding to change of β parameter from -90° to +90°, wherevector, h - the distance between Laue zones, g - the value of the reciprocal lattice vector, g‖ - the value of the reciprocal lattice vector projection on zero Laue zone.

A quantitative evaluation of some factors affecting the non-fatty solids of cow's milk

1960 ◽  
Vol 27 (1) ◽  
pp. 19-32 ◽  
Author(s):  
W. H. Alexander ◽  
F. B. Leech
Keyword(s):  
Cell Count ◽  
Residual Effect ◽  
Clinical Mastitis ◽  
High Cell ◽  
Factors Affecting ◽  
Stage Of Lactation ◽  
Cell Counts ◽  
Satisfactory Analysis ◽  
The Difference ◽  
Image Position

SummaryTen farms in the county of Durham took part in a field study of the effects of feeding and of udder disease on the level of non-fatty solids (s.n.f.) in milk. Statistical analysis of the resulting data showed that age, pregnancy, season of the year, and total cell count affected the percentage of s.n.f. and that these effects were additive and independent of each other. No effect associated with nutritional changes could be demonstrated.The principal effects of the factors, each one freed from effects of other factors, were as follows:Herds in which s.n.f. had been consistently low over a period of years were compared with herds in which s.n.f. had been satisfactory. Analysis of the data showed that about 70% of the difference in s.n.f. between these groups could be accounted for by differences in age of cow, stage of lactation, cell count and breed.There was some evidence of a residual effect following clinical mastitis that could not be accounted for by residual high cell counts.The within-cow regression of s.n.f. on log cell count calculated from the Durham data and from van Rensburg's data was on both occasions negative.The implications of these findings are discussed, particularly in relation to advisory work.

Merciful heavens? A question in Aeschylus' Agamemnon

1974 ◽  
Vol 94 ◽  
pp. 100-113 ◽  
Author(s):  
Maurice Pope
Keyword(s):  
Oral Tradition ◽  
Correct Reading ◽  
The Difference ◽  
Image Position

In discussions of Aeschylus' theology one of the passages most often quoted is the so-called ‘hymn to Zeus’ in the first chorus of the Agamemnon (Ag. 160–83). Fraenkel in his commentary goes so far as to call it ‘the corner-stone not only of this play but of the whole trilogy’. The passage concludes with two lines which in all modern editions are read as a statement, though our oldest manuscript, the Medicean, writes them as a question. Textually the difference is merely one of accent, but the difference of accent carries with it a reversal of meaning. As a statement the lines mean that the gods are something to be grateful for, that there is some χάρις or kindness associated with them. Taken as a question they deny this. Clearly then it is of great importance for the interpretation of Aeschylus to decide which is the correct reading.The lines in question, written without accents, areOur oldest manuscript, M, as I have said, writes ποῦ with an accent. So does our next oldest, the manuscript 468 of the Biblioteca di San Marco, generally known as V. If this reading stems uncorrupted from the time when accents were first applied to the text of Aeschylus and if at that time the oral tradition of the poet's words was not yet dead, then it will not be destitute of authority. But the thread is far too tenuous to bear any weight of proof.Equally there can be no argument from authority on the side of reading the lines as a statement. For though Triclinius and the closely associated manuscript F write που without an accent as an enclitic, this is as likely as not to be due to simple conjecture.

The fractional parts of an additive form

1972 ◽  
Vol 72 (2) ◽  
pp. 209-212 ◽  
Author(s):  
R. J. Cook
Keyword(s):  
Quadratic Form ◽  
Analogous Result ◽  
The Difference ◽  
Image Position ◽  
Forms Of Higher Degree ◽  
Additive Form ◽  
Real Quadratic

Heilbronn (6) proved that for every ε ≥ 0 and N ≥ 1 and every real θ there is an integer x such that,where C(ε) depends only on ε and ∥α∥ is the difference between α and the nearest integer, taken positively. Danicic(1) obtained an analogous result for the fractional parts of nkθ, the proof of this is more readily accessible in Davenport(4). Danicic(2) also obtained an estimate for the fractional parts of a real quadratic form in n variables, and in order to extend this result to forms of higher degree it is desirable to first obtain results for additive forms.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals Degenerate Sklyanin algebras

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Andrey Smirnov
Keyword(s):  
Difference Operators ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Rational Solutions ◽  
R Matrix ◽  
Gauge Transformations ◽  
Sklyanin Algebras ◽  
Sklyanin Algebra ◽  
The Difference

AbstractNew trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

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