scholarly journals On general curves lying on a quadric

1927 ◽  
Vol 1 (1) ◽  
pp. 19-30 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Present Note ◽  
Rational Curves ◽  
Stationary Points ◽  
Image Position

Introduction. The present note, though in continuation of the preceding one dealing with rational curves, is written so as to be independent of this. It is concerned to prove that if a curve of order n, and genus p, with k cusps, or stationary points, lying on a quadric, Ω, in space of any number of dimensions, is such that itself, its tangents, its osculating planes, … , and finally its osculating (h – 1)-folds, all lie on the quadric Ω, then the number of its osculating h-folds which lie on the quadric isTwo proofs of this result are given, in §§ 4 and 5.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

Sophocles, ‘Electra’ 1205–10

1973 ◽  
Vol 19 ◽  
pp. 45-46
Author(s):  
R. D. Dawe
Keyword(s):  
Present Note ◽  
Greek Tragedy ◽  
Image Position

The attribution of lines to different speakers in Greek tragedy is a matter on which MSS have notoriously little authority. As for Electra itself, there are at least three places where the name of the heroine has been incorrectly added in some or all MSS. In my Studies in the Text of Sophocles, I, 198, I list these places and suggest that the same error has happened at a fourth place, viz. 1323. The purpose of the present note is to suggest that at El. 1205–10 the same mistake has happened yet again.The situation is that Electra is holding the urn which she falsely believes to contain the ashes of her dead brother, Orestes. But Orestes is alive, and before her at this very moment. He is trying to persuade her to give up the urn. If the text before us had been preserved in a MS devoid of ascriptions to speakers, no one would have been so perverse as to do what all MSS and editors do in fact do, namely attribute the words οὔ φημ᾿ ἐάσειν to Orestes.

scholarly journals On the application of the operational calculus to the expansion of a function in a series of Legendre's functions of the second kind

1942 ◽  
Vol 7 (1) ◽  
pp. 1-2
Author(s):  
D. P. Banerjee
Keyword(s):  
Analytic Function ◽  
Present Note ◽  
Operational Calculus ◽  
Integral Function ◽  
Legendre Functions ◽  
Image Position

In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integralwhere f (x) is an analytic function of x, regular in the circlewhere an are constants and qn (ω) = in+1Qn (iω).

A System of Linear Equations with an Infinity of Unknowns

1924 ◽  
Vol 22 (3) ◽  
pp. 282-286
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Present Note ◽  
Linear Equations ◽  
The Other ◽  
System Of Linear Equations ◽  
Other Hand ◽  
Image Position ◽  
Monotonic Character

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations isthe numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

scholarly journals On stability and stationary points in nonlinear optimization

1986 ◽  
Vol 28 (1) ◽  
pp. 36-56 ◽  
Author(s):  
J. Guddat ◽  
H. Th. Jongen ◽  
J. Rueckmann
Keyword(s):  
Stability Condition ◽  
Minimization Problem ◽  
Level Sets ◽  
Constraint Qualification ◽  
Stationary Points ◽  
Feasible Set ◽  
Manifold With Boundary ◽  
Feasible Sets ◽  
Constrained Minimization Problem ◽  
Image Position

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:In summary, we provethat, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible setM[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible setM[H, G] is stable (perturbations ofHandGproduce homeomorphic feasible sets) if and only if MFCQ holds;under a stability condition, two lower level sets offwith a Kuhn-Tucker point between them are homotopically related by attachment of ak-cell (kbeing the stationary index in the sense of Kojima).

scholarly journals On the Number of Ways of Colouring a Map

1930 ◽  
Vol 2 (2) ◽  
pp. 83-91 ◽  
Author(s):  
George D. Birkhoff
Keyword(s):  
Present Note ◽  
The Third ◽  
Image Position

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the followingTheorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then(1)This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.

scholarly journals Some expansions of hypergeometric functions in series of hypergeometric functions

1976 ◽  
Vol 17 (1) ◽  
pp. 17-21 ◽  
Author(s):  
H. M. Srivastava ◽  
Rekha Panda
Keyword(s):  
Analytic Continuation ◽  
Hypergeometric Functions ◽  
Present Note ◽  
Expansion Formula ◽  
In Series ◽  
Image Position

Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula(1)which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r < q+s+l (or p+r = q+s+l and |zω| <1), p+t < q+u (or p + t = q + u and |z| < 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.

XI.—The Product of the Primary Minors of an n-by-(n + 1) Array.

1908 ◽  
Vol 28 ◽  
pp. 210-216
Author(s):  
Thomas Muir
Keyword(s):  
Present Note ◽  
General Theorem ◽  
Image Position

(1) IN a paper by A. Scholtz entitled “Sechs Punkte eines Kegelschnittes” (Archiv d. Math. u. Phys., lxii. pp. 317–324, year 1878) there appears a statement which, after correction of two misprints, runs thus:—where ξln=ylzn − ynzl. A year or so later there was published a paper by Hunyady with the title “Beitrag zur Theorie des Flächen zweiten Grades” (Crelle's Journ., lxxxix. pp. 47–69), in which it is asserted, again without proof, thatThe object of the present note is to formulate and prove a general theorem of which these are cases, and to draw certain deductions therefrom.

A Permanental Inequality—The Case of Equality

1966 ◽  
Vol 18 ◽  
pp. 1085-1090 ◽  
Author(s):  
Marvin Marcus ◽  
Henryk Minc
Keyword(s):  
Present Note ◽  
Hermitian Matrix ◽  
Normal Matrix ◽  
Characteristic Roots ◽  
Image Position

In (3) we proved that if A is a complex n-square normal matrix with characteristic roots α1 … , αn,then1If A is positive semi-definite hermitian, the inequality (1) becomes2This inequality partially answers the problem of determining the maximum permanent of a positive semi-definite hermitian matrix with prescribed characteristic roots (6). In (1), Brualdi and Newman proved that (2) also holds when A is an n-square circulant with non-negative entries. In a recent conversation Dr. Newman raised the question of determining the cases of equality in (1). In the present note we answer this question.

scholarly journals Two structure theorems for homeomorphism groups

1971 ◽  
Vol 4 (1) ◽  
pp. 63-68
Author(s):  
A. R. Vobach
Keyword(s):  
Metric Spaces ◽  
Present Note ◽  
Topological Properties ◽  
Compact Metric Space ◽  
Natural Structure ◽  
Compact Metric Spaces ◽  
Continuous Map ◽  
Structure Theorems ◽  
Image Position ◽  
Homeomorphism Groups

Let H(C) be the group of homeomorphisms of the cantor set, C onto itself. Let p: C → M be a (continuous) map of C onto a compact metric space M, and let G(p, M) be {h ∈ H(C) | ∀x ∈ C, p(x) = ph(x)}. G(p, M) is a group. The map p: C → M is standard, if for each (x, y) ∈ C × C such that p(x) = p(y), there is a sequence and a sequence such that xn → x and hn(xn) → y. Standard maps and their associated groups characterize compact metric spaces in the sense that: Two such spaces, M and N, are homeomorphic if and only if, given p standard from C onto M, there is a standard q from C onto N for which G(p, M) = h−1G(q, N)h, for some h ∈ H(C). That is, two compact metric spaces are homeomorphic if and only if they determine, via standard maps, the same classes of conjugate subgroups of H(C).The present note exhibits two natural structure theorems relating algebraic and topological properties: First, if M = H ∪ K (H, K ≠ π) , compact metric, and p : C → M are given, then G(p, M) is isomorphic to a subdirect product of G(p, M)/S(p, H\K) and G(p, M)/S(p, K\H) where, generally, S(p, N) is the normal subgroup of homeomorphisms supported on p−1M . Second, given M and N compact metric and p : M → N continuous and onto, let M ≠ M − CID*α ≠ 0 , where {Dα}α ∈ A is the collection of non-degenerate preimages of points in N Then there is a standard p : C → M such that fp : C → N is standard and there is a homomorphism.

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