Limit-point (LP) criteria for real symmetric differential expressions of order 2n

1981 ◽  
Vol 88 (3-4) ◽  
pp. 203-217 ◽  
Author(s):  
H. Kurss ◽  
G. Meyer
Keyword(s):  
Limit Point ◽  
The Other ◽  
Special Cases ◽  
Interval Type ◽  
Image Position ◽  
Positive Coefficients

SynopsisAn interval-type LP criterion foris derived in which “positive” coefficients play a prominent role. When pn = 1 and all the other pi are zero this reduces to a result of Ismagilov (1962). Successive specializations are obtained with the growth of the pi constrained by monomials in x. Previous LP criteria of Everitt (1968) and Hinton (1972, 1974) are shown to be special cases.

Connection formulae for spectral functions associated with singular Dirac equations

2004 ◽  
Vol 134 (1) ◽  
pp. 215-223 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Dirac Equation ◽  
Spectral Function ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Dirac Equations ◽  
Special Cases ◽  
Limit Point Case ◽  
Image Position ◽  
Connection Formulae

We consider the Dirac equation given by with initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

scholarly journals XXVI.—On a Group of Linear Differential Equations of the 2nd Order, including Professor Chrystal's Seiche-equations

1906 ◽  
Vol 41 (3) ◽  
pp. 651-676 ◽  
Author(s):  
J. Halm
Keyword(s):  
Differential Equations ◽  
Stokes Equation ◽  
Mathematical Theory ◽  
General Type ◽  
The Other ◽  
Special Cases ◽  
Linear Differential ◽  
Image Position ◽  
Special Case ◽  
The Stokes Equation

It is readily seen that the two differential equationswhich play an important rôle in Professor Chrystal's mathematical theory of the Seiches, are special cases of the more general typeWith regard to the first, the Seiche-equation, this becomes at once apparent by writing a= − ½. Equation (2), on the other hand, which we may briefly call the Stokes equation [see Professor Chrystal's paper on “Some further Results in the Mathematical Theory of Seiches,” Proc. Roy. Soc. Edin., vol. xxv.] will be recognised as a special case (a = + 1) of the equationwhich is transformed into (3) by the substitution .

scholarly journals On inequalities for integral operators

1970 ◽  
Vol 11 (2) ◽  
pp. 126-133 ◽  
Author(s):  
G. O. Okikiolu
Keyword(s):  
General Setting ◽  
Integral Operators ◽  
The Other ◽  
Special Cases ◽  
Measure Spaces ◽  
Schur Inequality ◽  
Abstract Measure ◽  
Schur Theorem ◽  
Image Position ◽  
Similar Extension

In two papers [3] and [4], the author has extended the inequality of Schur (Theorem 319 of [2]) to cases involving kernels which satisfy identities of the formThe purpose of this paper is to prove a general inequality, which includes the above and also the inequality of Young (Theorem 281 of [2]) as special cases. We shall give the results a general setting by considering functions defined on abstract measure spaces. From this we shall deduce an extension to n dimensions of the results given in [3], which also generalises a similar extension of the Schur inequality given by Stein and Weiss. In fact some cases of the other results given in [5] will follow directly from our theorem.

scholarly journals Leader-chasing behavior in negative artificial triggered lightning flashes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Fukun Wang ◽  
Jianguo Wang ◽  
Li Cai ◽  
Rui Su ◽  
Wenhan Ding ◽  
...  
Keyword(s):  
High Speed ◽  
The Other ◽  
Lightning Flash ◽  
High Speed Camera ◽  
Return Stroke ◽  
Catching Up ◽  
Special Cases ◽  
Triggered Lightning

AbstractTwo special cases of dart leader propagation were observed by the high-speed camera in the leader/return stroke sequences of a classical triggered lightning flash and an altitude-triggered lightning flash, respectively. Different from most of the subsequent return strokes preceded by only one leader, the return stroke in each case was preceded by two leaders occurring successively and competing in the same channel, which herein is named leader-chasing behavior. In one case, the polarity of the latter leader was opposite to that of the former leader and these two combined together to form a new leader, which shared the same polarity with the former leader. In the other case, the latter leader shared the same polarity with the former leader and disappeared after catching up with the former leader. The propagation of the former leader in this case seems not to be significantly influenced by the existence of the latter leader.

scholarly journals The Dissociation of the Compound of Iodine and Thio-urea

1904 ◽  
Vol 24 ◽  
pp. 233-239 ◽  
Author(s):  
Hugh Marshall
Keyword(s):  
Nitric Acid ◽  
General Formula ◽  
Free Acid ◽  
The Other ◽  
Image Position

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

Extremely undecidable sentences

1982 ◽  
Vol 47 (1) ◽  
pp. 191-196 ◽  
Author(s):  
George Boolos
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Propositional Modal Logic ◽  
Image Position

Let ‘ϕ’, ‘χ’, and ‘ψ’ be variables ranging over functions from the sentence letters P0, P1, … Pn, … of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence A of modal logic, inductively define Aϕ by[and similarly for other nonmodal propositional connectives]; andwhere Bew(x) is the standard provability predicate for PA and ⌈F⌉ is the PA numeral for the Gödel number of the formula F of PA. Then for any ϕ, (−□⊥)ϕ = −Bew(⌈⊥⌉), which is the consistency assertion for PA; a sentence S is undecidable in PA iff both and , where ϕ(p0) = S. If ψ(p0) is the undecidable sentence constructed by Gödel, then ⊬PA (−□⊥→ −□p0 & − □ − p0)ψ and ⊢PA(P0 ↔ −□⊥)ψ. However, if ψ(p0) is the undecidable sentence constructed by Rosser, then the situation is the other way around: ⊬PA(P0 ↔ −□⊥)ψ and ⊢PA (−□⊥→ −□−p0 & −□−p0)ψ. We call a sentence S of PA extremely undecidable if for all modal sentences A containing no sentence letter other than p0, if for some ψ, ⊬PAAψ, then ⊬PAAϕ, where ϕ(p0) = S. (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

The Circular Cylinder With a Band of Uniform Pressure on a Finite Length of the Surface

1941 ◽  
Vol 8 (3) ◽  
pp. A97-A104 ◽  
Author(s):  
M. V. Barton
Keyword(s):  
Circular Cylinder ◽  
Finite Length ◽  
Infinite Series ◽  
Fundamental Problem ◽  
The Other ◽  
Complete Picture ◽  
Uniform Pressure ◽  
Uniform Tension ◽  
Special Cases ◽  
Stress And Deformation

Abstract The solution to the fundamental problem of a cylinder with a uniform pressure over one half its length and a uniform tension on the other half is found by using the Papcovitch-Neuber solution to the general equations. In this paper, the results, given analytically in terms of infinite-series expressions, are exhibited as curves giving a complete picture of the stress and deformation. The case of a cylinder with a band of uniform pressure of any length, with the exception of very small ones, is then solved by the method of superposition. The stresses and displacements are evaluated for the special cases of a cylinder with a uniform pressure load of 1 diam and 1/2 diam in length. The problem of a cylinder heated over one half its length is solved by the same means.

3. Laboratory Notes by Professor Tait (a.) Measurement of the Potential, required to produce Sparks of various lengths, in Air at different pressures, by a Holtz Machine

1878 ◽  
Vol 9 ◽  
pp. 332-333
Author(s):  
Messrs Macfarlane ◽  
Paton
Keyword(s):  
General Result ◽  
The Other ◽  
Considerable Distance ◽  
Image Position

The general result of these strictly preliminary experiments appears to show that for sparks not exceeding a decimetre in length (L), taken in air at different pressures (P), between two metal balls of 7mm·5 radius, the requisite potential (V), is expressed by the formulaThe Holtz machine employed is a double one, made by Ruhmkorff, and it was used with its small Leyden jars attached. The measurements had to be made with a divided-ring electrometer, so that two insulated balls, at a considerable distance from one another, were connected, one with the machine, the other with the electrometer.

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