bluuebunny/crossref_metadata_2025 huggingface dataset:
Definition, size of the problem, screening and diagnostic criteria: Who should be screened, cost‐effectiveness, and feasibility of screening
Authors: Jovanovic Lois
Published: January 2009

The epidemic of obesity and increased prevalence of diabetes necessitates reconsideration of its definition and screening by separating type 2 diabetic women from women with gestational diabetes to provide a better model of care. The unique problems concerning an undiagnosed type 2 diabetic woman are not major issues in the gestational diabetic woman. This review underscores the need for a unique diagnosis for women with moderate to severe hyperglycemia and/or other evidence of long‐standing diabetes complications, and the urgent need for more intensive surveillance and treatment than is otherwise available for gestational diabetic women.
Perinatal mortality in pregestational diabetes
Authors: Melamed Nir , Hod Moshe
Published: January 2009

Recent studies have found that the perinatal mortality (PNM) rate in pregnancies complicated by pregestational diabetes remains significantly higher than in the general population. These observations are probably related to lack of optimal glycemic control, and a significant proportion of unplanned pregnancies. The prevalence of pregnancies complicated by type 2 diabetes is increasing. Although these pregnancies are often perceived as a less serious condition, type 2 diabetes appears to be associated with PNM rates that are at least as high as in type 1 diabetic pregnancies. This article reviews the epidemiology, etiology, and recommendations for prevention of PNM in pregnancies complicated by pregestational diabetes.
Women's health in the 21st century
Authors: Wolfson Elaine M. , DeKalb Adriane , Rojhani Ariella
Published: January 2009

Although women's health has been under‐attended for most of the 20th century, it has gained international attention in recent decades. Medical and social research on heart disease, lung cancer, HIV/AIDS, and trachoma indicate that bio‐socio variables affect women's health differently from men's. With regard to diabetes, data on pregnancy, diabetic ketoacidosis (DKA), depression, and heart disease corroborate the differentials between men and women. Data also indicate that social factors place diabetic women at a disadvantage regarding access to treatment and outcomes. Ascertaining the precise interactions that cause these differences and applying this information to policies and programs are imperative in the 21st century.
Social and economic consequences of diabetes in women from low‐income countries: A case study from Bangladesh
Authors: Mahtab Hajera , Habib Samira H.
Published: January 2009

Diabetes poses a serious threat to low‐income countries such as Bangladesh. It is one of the leading causes of premature morbidity and mortality, and requires life‐long healthcare services. Women with diabetes are affected in all stages of their lives. Uniquely, diabetes affects the health of mothers and their unborn children. Poverty, ignorance, and gender discrimination adversely affect women with diabetes. However, the Diabetic Association of Bangladesh has played an increasingly effective role in providing comprehensive socio‐medicare and life‐long follow‐up of diabetic women, free of charge. This is reflected by the increasing attendance of women with diabetes, especially from rural areas.
Obstetric outcomes of women with intracranial neoplasms
Authors: Johnson Nadine , Sermer Mathew , Lausman Andrea , Maxwell Cynthia
Published: January 2009

Objective

To determine the obstetric outcomes of women diagnosed with a primary intracranial neoplasm prior to or during pregnancy and the puerperium.

Methods

Demographic data, neurologic, obstetric, and neonatal outcomes were extracted retrospectively from charts identified from two hospital databases.

Results

Twenty‐two patients with 25 pregnancies were identified. Thirteen patients were diagnosed prior to pregnancy (group 1); tumor growth or recurrence during pregnancy occurred in 3 patients. Nine patients were diagnosed during pregnancy and the puerperium (group 2); 7 underwent neurosurgery at a mean gestational age of 27 ± 5.7 weeks; 2 experienced permanent visual loss. Significantly more gliomas occurred in group 1, but there were significantly more meningiomas in group 2. Numbers of cesarean delivery, preterm delivery at less than 34 weeks, and NICU admission were higher in group 2 than group 1, but this was not statistically significant.

Conclusion

Although not statistically significant, neurologic deterioration during pregnancy may be associated with cesarean delivery, preterm delivery, and NICU admission.
Solutions of Euclid's Problems, with a rule and one fixed aperture of the compasses, by the Italian geometers of the sixteenth century
Authors: Mackay J. S.
Published: January 2009

Chasles in his Aperçu, Historique sur l'origine et le développement des Méthodes en Géométrie (seconde édition, 1875, pp. 214–215) makes the following statement:

“Essays of the same kind as the geometry of the rule and that of the compasses, and which hold, so to speak, the mean between the two, had long previously engaged the attention of famous mathematicians. Cardan first of all in his book De Subtilitate had resolved several of Euclid's problems by the straight line and a single aperture of the compasses, as if one had in practice only a rule and invariable compasses. Tartalea was not long in following his rival on this field, and extended this mode of treatment to some new problems. (General trattato di numeri et misure; 5ta parte, libra terzo; in-fol. Venise, 1560). Finally, a learned Piedmontese geometer, J.–B. de Benedictis, made it the object of a treatise entitled: Resolutio omnium Euclidis problematum, aliorumque ad hoc necessario inventorum, una tantummodo circini data apertura; in-4°. Venise, 1553.”
Composition de Mathématiques Élémentaires proposée au Concours d' Agrégation de 1886
Authors: Aubert M. Paul
Published: January 2009

On donne un cercle et deux points P et Q situés sur un diamètre, on joint les points P et Q aux extrémités A et B d'ux diamètre du cercle par les droites I'A et QB qui se coupent au point M. On fait tourner le diamètre AB et on demande

I. D' étudier les variations du rappoet de construire la figure quand le rapport a unedonnée.

II. D' étudier les variations de l' angle AMB, et de construire la figure quand cet angle a une valeur donnée.

III. A′ et B′ étant les seconds points d' intersection des droites MA, MB avec la circonférence donnée, trouver le, lieu du centre du cercle circonscrit au triangle MA′B′.
On Vortices
Authors: Chree C.
Published: January 2009

Helmholtz and most, if not all, subsequent writers on vortex motions have, except in obtaining the fundamental equations, confined themselves to fluid of invariable density.

In the following paper are considered some simple systems of vortices in a compressible fluid. To show that such systems are of considerable importance it is sufficient to refer to the phenomena of cyclonic storms. It may be as well, however, to state that though the vortices are here treated as compressible, the circumstances are still so different from those found in nature that the results obtained could bear only a general resemblance at most to the phenomena of storms.
Historical Notes on a Geometrical Theorem and its Developments. [18th century.]
Authors: Mackay J. S.
Published: January 2009

The theorem is—

The distance between the circumscribed and the inscribed centres of a triangle it a mean proportional between the circumscribed radius and its excess above the inscribed diameter.
Note on the Kinematics of a Quadrilateral
Authors: Dallas R. J.
Published: January 2009

I send a note on the following problem, a solution of which was requested of me by one of the tutors at King's College, Cambridge.

We are given a quadrilateral of four jointed bars ABCD (fig. 83). The bar CD being held fast, find the tangent to the locus of P, the intersection of DA, CB in any position; and verify the following construction for the radius of curvature of the path of P:—

Let PQ be the third diagonal, draw through P a perpendicular to PQ meeting BA, CD in L and L′; through L and L′ draw parallels to PQ meeting AD in M and M′; through M and M′ draw perpendiculars to AD meeting the normal at P in O and O′; then will
An Exercise on Logarithmic Tables
Authors: Professor Tait
Published: January 2009

In reducing some experiments, I noticed that the logarithm of 237 is about 2.37 …. Hence it occurred to me to find in what cases the figures of a number and of its common logarithm are identical:—i.e., to solve the equation

where m is any positive integer.
Geometrical Proof of the Tangency of the Inscribed and Nine-Point Circles
Authors: Harvey William
Published: January 2009

S (fig. 85) is the circumscribed centre, and O the orthocentre of the triangle ABC; AX the perpendicular from A on BC, and P the middle point of BC.

SP produced bisects the arc BC in V, and I, the centre of the inscribed circle, lies on AV, and is so situated that AI.IV = 2Rr. (See Note). Also the angle XAV = angle AVS = angle SAV
On a class of Differential Equations whose solutions satisfy Integral Equations
Authors: Whittaker E. T.
Published: January 2009

The science of the solution of Differential Equations has been in great measure systematized by the aid of ideas borrowed from the Theory of Functions, the equations being classified according to the singularities possessed by their solutions. In the case of linear Differential Equations of the second order

the solutions can have no singularities except at the singularities of the functions q(x) and r(x) (and possibly also at x = ∞ ): these equations may therefore be classified simply according to the number and nature of these singularities.
Taylor's Cubics associated with a Triangle in Non-Euclidean Geometry
Authors: Sommerville D. M. Y.
Published: January 2009

§ 1. The famous theorem of the pedal line of a triangle in ordinary geometry can be stated as follows:—“Given a triangle ABC and a point P such that the feet of the perpendiculars X, Y, Z, dropped from P on the sides of the triangle, are collinear, then the locus of P is the circumcircle.” In noneuclidean geometry this locus is not a circle or even a curve of the second degree, but a cubic; and in both cases the envelope of the line XYZ is a curve of the third class. The explanation of the inconsistency in ordinary geometry is that the complete locus consists of the circumcircle together with the straight line at infinity.
On the Roots of a Derivative of a Rational Function
Authors: Ford L. R.
Published: January 2009

This paper extends some familiar theorems concerning the relations between the roots of a polynomial and those of its first derivative to the more general case of the rational function with a pole at a single point.
On Spheroidal Harmonics and Allied Functions
Authors: Jeffery G. B.
Published: January 2009

In a paper recently read before this Society, Mr E. Blades obtained a general formula for spheroidal harmonics in the form of the general solution of Laplace's equation given by Professor Whittaker,

If spheroidal coordinates r, θ, φ are defined by

the result obtained is
On the Oscillation Functions derived from a Discontinuous Function
Authors: Ford L. R.
Published: January 2009

In this paper are introduced what we shall term “successive oscillation functions.” These functions are derived from functions of a real variable. The word “function” as here used has its widest meaning. We say y is a function of x in an interval of the the x-axis, if given any value of x, in the interval one or more values of y are thereby determined. The values of the function may be determined by any arbitrary law whatsoever. We shall deal with discontinuous functions; the theorems will be true for continuous functions, but will be trivial, except in the case of functions which are discontinuous and whose points of discontinuity are infinite in number. We shall assume in what follows that the values of the function lie between finite limits.
On the Pfaffian Invariant of a Skew-Symmetric Matrix
Authors: Schwerdtfeger H.
Published: January 2009

Let P be an n-rowed skew-symmetric matrix of rank 2r with elements out of an infinite field F. Denoting by x, y columns of n variables (indeterminates over F) xv, yv (v = 1, …., n), and by x′, y′ the corresponding row matrices, we consider the skew-symmetric bilinear form y′Px. It is well known that for every P a regular homogeneous substitution over F can be found so that
An Inversion Formula for the Generalised Laplace Transform
Authors: Varma R. S.
Published: January 2009

Recently I have given a generalisation of the Laplace integral

in the form

where Wk, m (x) stands for Whittaker Functions.
A Derivation of Certain Variational Principles for Mixed Boundary Value Problems in Potential Theory
Authors: Bartlett C. C. , Noble B.
Published: January 2009

We consider the following problem: A potential function φ satisfies Laplace's equation ∇2φ = φxx + φyy = 0 in a region R bounded by a closed curve C on which mixed boundary conditions are specified, i.e. φ = f(s) on a part A of the boundary and ∂φ/∂n = g(s) on a part B, where C = A + B and distance along C is denoted by s. Electrostatic problems of this type have been solved approximately in (1) and (2) by formulating them in terms of integral equations and then applying variational principles to the integral equations. In that approach, attention is concentrated on integrals over the boundary of the region R. The most common type of variational principle for potential problems involves integrals over the region R rather than integrals over the boundary of R. An example is given by the Rayleigh-Ritz method which depends on the stationary character of Dirichlet's integral

In this paper we show that the variational principles used in (1), (2), are closely connected with the more usual type of variational principles, by deriving the principles used in (1), (2) from inequalities deduced by considering integrals of type (1) over the region R.
Some Finite Analogues of the Poisson Summation Formula
Authors: Carlitz L.
Published: January 2009

1. Guinand (2) has obtained finite identities of the type

where m, n, N are positive integers and either

or

where γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).
The Spectral Sequence of a Covering
Authors: Simms D. J.
Published: January 2009

Let be a covering of a topological space X and ℱ a sheaf of abelian groups over X. By a well known result of Leray, (3) II theorems 5.2.4. and 5.4.1., if is open, or closed and locally finite, there exists a spectral sequence {Er} satisfying isomorphisms and for some filtration of the graded group H*(X, ℱ). ℋq(ℱ) denotes the system of coefficients over : s→Hq(| s |, ℱ).
On the non-vanishing of Poincaré series
Authors: Lehner J.
Published: January 2009

Let M = SL(2, Z) be the classical modular matrix group. One form of the Poincaré series on M is

here z ∈ H={z = x + iy: y >0}, q ≧ 2 and m ≧ 1 are integers, and the summation is over a complete system of matrices (ab: cd) in M with different lower row. The problem of the identical vanishing of the Poincaré series for different values of m and q goes back to Poincaré.
Analogues of the braid group whose graphs are stars
Authors: Napthine Andrew K.
Published: January 2009

Let G be a group having presentation

a braid relator and a commutator respectively, and define the graph of such a group as having vertices labelled x1,…, xn + 1 and such that xi and xj are joined if and only if rij = (xi, xj).
On the (J, pn, qn) method of summation
Authors: Kwee B.
Published: January 2009

In the following discussion we shall assume that pn≧0, qn≧0 for all n and that qn + 1 > qn → ∞. The (J, pn, qn) method of summation is defined as follows.

The series with the partial sum sn, is called summable (J, pn, qn) to s, and we write if the series

and converge to the sum functions p*(x) and p(s)(x) respectively for 0<x<1 and if τ(x) = p(s)(x)/p*(x)→s as x→1–0.
The hypercore of a semigroup
Authors: Hall T. E. , Munn W. D.
Published: January 2009

In this paper the “hypercore” of a semigroup S is defined to be the subsemigroup generated by the union of all the subsemigroups of S without non-universal cancellative congruences, provided that at least one such subsemigroup exists: otherwise it is taken to be the empty set. It is shown first that if the hypercore of S is nonempty (which holds, for example, when S contains an idempotent) then it is the largest subsemigroup of S with no non-universal cancellative congruence, is full and unitary in S, and is contained in the identity class of every group congruence on S (Theorem 1).
Limits and colimits in categories of d.g. near-rings
Authors: Mahmood Suraiya J.
Published: January 2009

In this paper, using the idea of upper and lower faithful d.g. near-rings, introduced in (6), we show that the category of all faithful d.g. near-rings is a reflective as well as coreflective subcategory of the category of all d.g. near-rings. We also prove that both and are complete and cocomplete categories.
On the endomorphism near-ring of a free group
Authors: Zeamer R. Warwick
Published: January 2009

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.
The natural partial order on a regular semigroup
Authors: Nambooripad K. S. Subramonian
Published: January 2009

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.
Tame near-rings and N-groups
Authors: Scott S. D.
Published: January 2009

Throughout this paper any near-ring N will be left distributive, zero symmetric and have an identity. Furthermore, all N-groups will be unitary. All groups considered will be written additively. This does not imply commutativity.
Spectral measures on spaces not containing l∞
Authors: Gillespie T. A.
Published: January 2009

The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then

(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that

(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that

(iii) every prespectral operator on X is spectral.

(See also (1, Theorem 6.11, p. 165) for (iii).)
Limit circle criteria for 2nth order differential operators
Authors: Becker Ronald I.
Published: January 2009

A formally self-adjoint differential operatorLis said to be of limit circle type at infinity if its highest order coefficient is zero-free and all solutionsxofL(x) = 0 are square-integrable on [a, ∞). (We will drop reference to “at infinity” in what follows.)

For the second-order case

Dunford and Schwartz (3) p. 1409 prove thatgiven

thenLis of limit circle type if and only if
A theorem on compatible N-groups
Authors: Scott S. D. , Lyons C. G.
Published: January 2009

A near-ring N is a set N with binary operations + and · satisfying the conditions (1) (N, +) is a group, (2) (N, ·) is a semigroup, and (3) · satisfies one of the distributive laws over +. (N, +) need not be an abelian group and if the left distributive law holds, i.e. a · (b + c) = a · b + a · c for all a, b, c ∈ N, then N is called a left near-ring. Similarly, the notion of a right near-ring may be defined.
Generators for the sporadic group Co3 as a (2, 3, 7) group
Authors: Worboys M. F.
Published: January 2009

A (2, 3, 7)-group is a group generated by two elements, one an involution and the other of order 3, whose product has order 7. Known finite simple examples of such groups are PSL(2, 7), PSL(2, p) where p is prime and p ≡ ±1 (mod 7), PSL(2, p3) where p is prime and p≢0, ±1 (mod 7), groups of Ree type of order q3(q3 + 1)(q − 1) where q = 32n+1 and n > 0, the sporadic group of order 23 · 3 · 5 · 7 · 11 · 19 discovered by Janko, and the Hall–Janko–Wales group of order 27 · 33 · 52 · 7 [4, 2]. G. Higman in an unpublished paper has shown that every sufficiently large alternating group is a (2, 3, 7)-group. Here we show that the sporadic group Co3 discovered by Conway [1] is a (2, 3, 7)-group.
The number of Cayley integers of given norm
Authors: Lamont P. J. C.
Published: January 2009

Using results obtained by J. W. L. Glaisher [1, 2] for the number of representations Rr,s(n) of n as a sum of r odd and s even squares, formulae are derived for the number of Cayley integers of given norm n in certain orders ℴ. When computer generating order elements of given norm, the formulae can be used to verify that all the required elements have been obtained.
On compact action in JB-algebras
Authors: Bunce L. J.
Published: January 2009

A real Jordan algebra which is also a Banach space with a norm which satisfies

for each pair a, b of elements, is said to be a JB-algebra. A JB-algebra which is also a Banach dual space is said to be a JBW-algebra.
Semigroups under a sandwich operation
Authors: Hickey J. B.
Published: January 2009

For any element a of a semigroup (S,·), we may define a “sandwich” operation ∘ on the set S by x ∘ y = xay (x, y ∈ S). Under this operation the set S is again a semigroup; we denote this semigroup by (S, a) and call it a variant of S. Variants of semigroups of binary relations have been studied by Chase [6, 7]. In this paper we consider variants of arbitrary semigroups.
Primary decomposition in enveloping algebras
Authors: Brown K. A. , Lenagan T. H.
Published: January 2009

Recently, the first author and, independently, A. V. Jategaonkar have shown that every factor ring of U(g), the universal enveloping algebra of a finite dimensional complex Lie algebra, has a primary decomposition if g is solvable and almost algebraic. On the other hand, a suitable factor ring of U(SL(2, ℂ) fails to have a primary decomposition (1).
On complex projective hypersurfaces
Authors: Bruce J. W.
Published: January 2009

In this paper we prove various results concerning monodromy groups associated with nonsingular complex projective hypersurfaces. Most of these results are already known but proofs are either unavailable or are algebraic and require a lot of machinery. The groups in question are those obtained from the second Lefschetz theorem (see (1)) applied to (a) the general Veronese variety, (b) a nonsingular projective hypersurface. By embedding the monodromy group of an extraordinary local isolated singularity (discovered by Libgober (8)) in these global monodromy groups we obtain necessary and sufficient conditions for the global groups to be finite. For case (a) we also obtain information on the structure of the dual to the Veronese variety which is of use when considering the monodromy group. The author gratefully acknowledges the financial support of the Stiftung Volkswagenwerk for a vist to the IHES during which this paper was written.
Remarks on module-finite pairs
Authors: Huckaba James A. , Papick Ira J.
Published: January 2009

Let R ⊊ T be an extension of commutative rings having the same identity. A. Wadsworth (10) studies the situation when R and T are integral domains, and all rings between R and T are Noetherian. In this case (R, T) is called a Noetherian pair. In a similar vein, E. Davis (4) studies normal pairs and I. Papick (8) shows when coherent pairs are Noetherian pairs.
On general curves lying on a quadric
Authors: Baker H. F.
Published: January 2009

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 is

Two proofs of this result are given, in §§ 4 and 5.
On the uninodal quartic curve
Authors: Richmond H. W.
Published: January 2009

In comparison with the general plane quartic on the one hand, and the curves having either two or three nodes on the other, the uninodal curve has been neglected. Many of its properties may of course be deduced from those of the general quartic in the limiting case when an oval shrinks to a point or when two branches approach and ultimately unite. The modifications of properties of the bitangents are shewn more clearly by Geiser's method, in which these lines are obtained by projecting the lines of a cubic surface from a point on the surface. As the point moves up to and crosses a line on the surface, the quartic acquires a node and certain pairs of bitangents obviously coincide, viz. those obtained by projecting two lines coplanar with that on which the point lies. A nodal quartic curve and its double tangents may also be obtained by projecting a cubic surface which has a conical point from an arbitrary point on the surface. Each of these three methods leads us to the conclusion that, when a quartic acquires a node, twelve of the double tangents coincide two and two and become six tangents from the node, and the other sixteen remain as genuine bitangents: the twelve which coincide are six pairs of a Steiner complex.
On the meaning of an equation in dual coordinates
Authors: Richmond H. W.
Published: January 2009

If L, M, N denote Prof. Study's Dual Coordinates of a straight line (see Proc. Edinburgh Math. Soc., 44 (1926), 90–97), any (homogeneous) equation F (L, M, N) = 0 must define a certain system of lines. By the nature of dual numbers we must have

where U and V are functions of l, m, n, λ, μ, ν, the ordinary (Pluckerian) coordinates. Since F = 0 implies U = 0 and V = 0 the system of lines is a congruence. But it is a congruence of a very special kind, whose nature will now be considered.
A note on the correlation of classes
Authors: Whittaker J. M.
Published: January 2009

Let R be a (1 — 1) relation between the members of two similar classes A, B1. It correlates the members of a subclass X of A to the members of a certain subclass Y of B1 and thus defines a relation ρ connecting X and Y. It is clear that ρ is a (1 – 1) relation and that it has the property (M). If X1ρ Y1, X2ρY2, then implies
Proof of the Conditions for a Turning Value of f (x, y) without the use of Taylor's Theorem
Authors: McWhan John
Published: January 2009

The standard method of establishing rigidly the tests for a maximum or minimum value of a function of two independent variables, depending as it does on the use of Taylor's Theorem and on a very critical consideration of the Remainder in that theorem, presents difficulties so considerable that it is not surprising that most text-books on the Calculus frankly decline to enter on the discussion, and assume the necessity and sufficiency of the well known Lagrange's Condition. It is the object of this paper to show that a satisfactory proof of the tests may be given, from the purely geometrical standpoint, without recourse to Taylor's Theorem. The method requires only an elementary knowledge of the process of changing the independent variables in partial derivatives, and may therefore be introduced comparatively early in the Calculus course.
The solution of Cauchy's Problem for linear partial differential equations, with constant coefficients, by means of integrals involving complex variables
Authors: Stewart C. A.
Published: January 2009

The object of this paper is to show how the theory of integrals involving complex variables may be applied to the integration of linear partial differential equations, possessing real, distinct characteristics and constant coefficients. The problem considered is a Cauchy problem (with analytic data)—typical of the equation of real characteristics and the method taken is that of Riemann. For simplicity of exposition, the second order hyperbolic equation is considered, but the results are given in such a form as to indicate an obvious generalisation to equations of higher order.
A special Pencil of Binary Quartics
Authors: Vaidyanathaswamy R.
Published: January 2009

I. If the Jacobian sextic of a pencil of binary quartics is known, the pencil itself is determined in five ways. The explicit determination of the pencil in terms of the irrational invariants of has been effected by Stephanos.

There are two known cases in which a pencil of quartics admits of rational determination from its Jacobian. In these, the rationally determinable pencil differentiates itself algebraically from its remaining four co-Jacobian pencils.
On the Latent Roots of Certain Matrices
Authors: Aitken A. C.
Published: January 2009

In the present note certain known theorems on the latent roots of matrices are deduced from the fundamental theorem that a matrix A can be expressed in the form PQP-1, where P is non-singular and Q has zero elements everywhere to the left of the principal diagonal, and the latent roots of A in the diagonal. [The presence or absence of non-zero elements to the right of the diagonal is known to depend on the nature of the “elementary divisors” of the “characteristic determinant” of A, but in what follows these will not concern us.]
On some Legendre Function Formulae
Authors: MacRobert T. M.
Published: January 2009

The various formulae for the Legendre Functions, and the relations between these formulae, have been studied by Kummer,Riemann, Olbricht, Hobson, Barnes, Whipple, and others. Hobson obtained some of the relations directly, by expres ing the functions as Pochhammer integrals, and expanding in a number of series each with its own region of convergence. To obtain some of the other formulae, such as (i) below, he transformed the differential equation, and then expressed the functions in terms of the solutions of the transformed equation. Barnes succeeded, by means of his wellknown integrals involving Gamma Functions, in deducing all the formulae directly from the formulae which define the functions.Notes on the history of the subject and references to previous work will be found in the papers by Hobson and Barnes.
On an Integral due to Ramanujan, and some ideas suggested by it
Authors: MacRobert T. M.
Published: January 2009

By applying Fourier's Integral Theorem to a well-known formula, due to Cauchy, expressed in the form

where R (μ + v) > 1, Ramanujan has shown that

where t is any real number.
A Matrix Form of Taylor's Theorem
Authors: Turnbull H. W.
Published: January 2009

The following pages continue a line of enquiry begun in a work On Differentiating a Matrix, (Proceedings of the Edinburgh Mathematical Society (2) 1 (1927), 111-128), which arose out of the Cayley operator , where xij is the ijth element of a square matrix [xij] of order n, and all n2 elements are taken as independent variables. The present work follows up the implications of Theorem III in the original, which stated that

where s (Xr) is the sum of the principal diagonal elements in the matrix Xr. This is now written ΩsXr = rXr – 1 and Ωs is taken as a fundamental operator analogous to ordinary differentiation, but applicable to matrices of any finite order n.
A geometrical theorem and the resultant of two binary forms
Authors: Baker H. F.
Published: January 2009

It was proved by Salmon (Geom. of three dimensions (1882), p. 331) that the chords of the curve of intersection of two algebraic surfaces of order m and n. which can be drawn from an arbitrary point,meet the curve upon a surface of order (m — 1) (n — 1); it was proved by Valentiner (Acta Math. 2 (1883), p. 191), and by Noether (Berlin. Abh. (1882), Zur Grundlegung u.s.w., p. 27), that the surface of order (m — 1) (n — 1) is a cone, with vertex at the point from which the chords are drawn; and a converse theorem was given by Halphen (J. de l' école Polyt. 52 (1882), p. 106). But the proofs given by Valentiner and Noether have not the elementary character that seems desirable, Noether's proof in particular depending on the theory of the canonical series upon the curve.
On The Uniformisation of Algebraic Curves of Genus 3
Authors: Mursi M.
Published: January 2009

An algebraic equation

determines, in general, s as a many valued function of z. If s and z can be expressed as one valued functions of a third variable t, then t is called the uniformising variable. As Poincaré showed, s and z are automorphic functions of t.
Solution with Axial Symmetry of Einstein's Equations of Teleparallelism
Authors: McVittie G. C.
Published: January 2009

Einstein has recently adopted a new set of field-equations in his Unified Field-Theory of Gravitation and Electricity, the so-called theory of parallelism at a distance or Teleparallelism, and has given a solution of these equations with spherical symmetry, corresponding to the field of a charged mass-particle. In the present paper we discuss the solution of these equations with axial symmetry, which corresponds to a statical field whose field-variables depend on a single coordinate only, viz. the coordinate which is measured along the axis of symmetry.
Quadratic Equations in a Cyclic Number System
Authors: Wilson R.
Published: January 2009

Mr D. E. Littlewood has recently discussed the properties of the quadratic equation over the real quaternions and shown that the solutions correspond to the common intersections of four quadrics in four-space. Although complex quaternion solutions may arise, the system of real quaternions to which the coefficients belong is a division algebra. It is of interest, therefore, to discuss the solution of the quadratic when the coefficients are drawn from a system containing divisors of zero.
A “Cubical” Universe
Authors: McCrea W. H.
Published: January 2009

Einstein has studied a universe in which the time coordinate t is uncurved and the spatial section is the surface of a sphere

in four dimensions. Some interest attaches to the case where this surface is replaced by

where n is a positive integer.
On the Solution of Differential Equations by Definite Integrals
Authors: Whittaker E. T.
Published: January 2009

It is well known that in many cases the solutions of a linear differential equation can be expressed as definite integrals, different solutions of the same equation being represented by integrals which have the same integrand, but different paths of integration. Thus, the various solutions of the hypergeometric differential equation

can be represented by integrals of the type

the path of integration being (for one particular solution) a closed circuit encircling the point t = 0 in the positive direction, then the point t = 1 in the positive direction, then the point t = 0 in the negative direction, and lastly the point t = 1 in the negative direction; or (for another particular solution) an arc in the t-plane joining the points t = 1 and t = ∞.
On Absolute Summability for any Positive Order
Authors: Winn C. E.
Published: January 2009

Absolute summability according to Cesàro's method has been defined by Fekete for positive integral orders, as follows:—

Denoting the rth partial sum of a series Σun by and its rth mean, namely2, by we can regard as the sum of the series

.
Supplement to a Note on Recurrent Sequences
Authors: Walsh C. E.
Published: January 2009

A lemma used in the above paper can be extended to cover the case of a sequence Sn determined by a recurrence relation of the form

r being any positive integer.

For convenience denote everywhere. Let us suppose that an inequality of the form

holds for p = 2, 3, … ., n — 1. The constant K can be determined from the initial values of Sn, when kn and ln have been found.
Note on the “sum” of an integral function
Authors: Whittaker J. M
Published: January 2009

Theorem 1 of a recent paper “On the asymptotic periods of integral functions” can be replaced by the following more precise result.

If f(z) is an integral function of order ρ there is an integral function g(z), of order ρ, such that

The improvement consists in showing that, if ρ < 1, g (z) can be chosen to be of order ρ, not merely of order less than or equal to one.
On a Simple Type of Integro-differential Equation
Authors: Gillespie R. P.
Published: January 2009

In a previous paper in these Proceedings the author discussed conditions for a maximum or minimum of functions of integrals of the type

using the methods of the Calculus of Variations. In the effort to establish a third necessary condition for a minimum—the analogue of Jacobi's condition in the ordinary variational problem—it was found that the analogue of Jacobi's Equation was an integrodifferential equation of the form
On certain Quadric Hypersurfaces in Riemannian Space
Authors: Weatherburn C. E.
Published: January 2009

The use of geodesic polar coordinates in the intrinsic geometry of a surface leads to the concept of a geodesic circle, i.e. the locus of points at a constant distance from the pole 0 along the geodesics through 0. A geodesic hypersphere is the obvious generalisation of this for a Riemannian Vn. We propose to consider more general central quadric hypersurfaces of Vn, which we define as follows. Let xi (i = 1, 2, …, n) be a system of coordinates in Vn, whose metric is gijdxidxj, and let aij be the components in the x's of a symmetric covariant tensor of the second order, evaluated at the point 0, which is taken as pole.
On the Topology of Certain Algebraic Threefold Loci
Authors: Todd J. A.
Published: January 2009

The study of the topological properties of algebraic surfaces, considered as continua of four real dimensions, has thrown much light on the theory of the birational invariants of such loci. The results obtained for surfaces have been generalised to varieties of higher dimension by Hodge, and, particularly, by Lefschetz. Apart from this, little seems to be known about the general topological properties of algebraic loci of three (or more) dimensions, the detailed study of which seems to present considerable difficulty. In particular, apart from the general theorems of Lefschetz, nothing seems to be known about the cycles of three dimensions of an algebraic V3. The object of the present paper is to study these cycles on certain quite special V3, in the hope that some insight may be gained into the general theory.
Polynomial expansions in the Borel region
Authors: Hurt J. T. , Ford L. R.
Published: January 2009

This paper deals with certain expansions of analytic functions in series of polynomials. Explicit forms for the polynomials are given in terms of the coefficients of the Taylor's expansion and of prescribed positive constants hk. Under suitable conditions, to be presently discussed, the series converge to the function in the Borel region.
Birational transformations with isolated fundamental points
Authors: Todd J. A.
Published: January 2009

It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, then

where E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.
On extensions of Pascal's theorem (Second paper) Paul Serret's theorem
Authors: Richmond H. W.
Published: January 2009

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.
Congruence properties of self-contained balanced sets
Authors: Gupta Hansraj
Published: January 2009

Let the set b1b2, b3, …., bk; of k non-negative integers be denoted by Bk. Let ξ;Bk denote the set ξb1 ξb2, ξb3, …. ξbk; ξ being any integer > 1. Without loss of generality, we can suppose that b ≦bi+1.
Principal directions in a gravitational field
Authors: Watson A. G. D. , Turnbull H. W.
Published: January 2009

In a general metric space of four dimensions, with an interval given by , where – ds2 = gμνdxμdxν, where g = ║gμν║<0, we can choose locally galilean coordinates at any point. The initial directions of the axes can be fixed in an absolute fashion as the directions of the principal axes of the quadric Gμνdxμdxν = const., is the contracted Riemann-Christoffel tensor.
A note on the zeros of the Bessel function
Authors: Wilson R.
Published: January 2009

This note contains an elementary proof of the well-known result that every solution of Besse's equation

with a real parameter n has an infinite number of real positive zeros.
On certain expansions involving Bessel functions and Whittaker's M-functions
Authors: Mitra S. C.
Published: January 2009

Adopting the notation of Barnes1 and Fox, let us write

If q > p — l, the series on the right of (1) represents an integral function, while if q = p — 1, the series converges only inside or on the circle | x |=1.
On certain infinite integrals involving Struve functions and parabolic cylinder functions
Authors: Mitra S. C.
Published: January 2009

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 that

From (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.
A Plane Quartic with Eight Undulations
Authors: Edge W. L.
Published: January 2009

1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curve

where (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points

The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.
The Semi-Algebra Generated by a Compact Linear Operator
Authors: Bonsall F. F. , Tomiuk B. J.
Published: January 2009

We prove that if t is a compact linear operator that is not quasi-nilpotent and is appropriately normalised, then the closed semi-algebra A(t) generated by t is locally compact. The theory of locally compact semi-algebras (2) is therefore applicable to A(t), and we show that it can be used to obtain spectral properties of t.
Mean Value Theorems for Vector Valued Functions
Authors: McLeod Robert M.
Published: January 2009

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we have

for some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to be

where cka, b, 0 k, and .
A Note on Double Transformations of Certain Hypergeometric Functions
Authors: Singh R. P.
Published: January 2009

Rainville (6) has discussed the following Euler transformation of the hypergeometric function
Green's functions and Riemann's method
Authors: Mackie A. G.
Published: January 2009

Methods for solving boundary value problems in linear, second order, partial differential equations in two variables tend to be somewhat rigidly partitioned in some of the standard text-books. Problems for elliptic equations are sometimes solved by finding the fundamental solution which is defined as a solution with a given singularity at a certain point. Another approach is by way of Green's functions which are usually defined as solutions of the original homogeneous equations now made inhomogeneous by the introduction of adelta function on the right hand side. The Green's function coincides with the fundamental solution for elliptic equations but exhibits a totally different type of singularity for parabolic or hyperbolic equations. Boundary value problems for hyperbolic equations can often by solved by Riemann's method which depends on the existence of an auxiliary function called the Riemann or sometimes the Riemann-Green function. The main object of this paper is to show the close relationship between Riemann's method and the method of Green's functions. This not only serves to unify different methods of solution of boundary value problems but also provides an additional method of determining Riemann functions for given hyperbolic equations. Before establishing these relationships we shall survey the general approach to boundary value problems through the use of the Green's function.
A criterion for absolute Cesàro summability of negative order of a Fourier series
Authors: Chen Yung-Ming
Published: January 2009

Let f(x) be integrable L(0, 2π) and periodic with period 2π, and let ψ(t) be the conjugate function of with respect to the variable t, where x is onsidered as an arbitrary constant. The following theorems are due to K. K. Chen (1), (2), pp. 111–124.