bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Pair of Dual Integral Equations Involving Bessel Functions of the First and the Second Kind
Authors: Srivastav R. P. 
Published: January 2009

In this paper we give a method for the solution of the dual integral equationswhere Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation forsubject to the usual regularity conditions and the following boundary conditions:
Explicit solutions for a system of coupled Lyapunov differential matrix equations
Authors: Jodar L. , Mariton M. 
Published: January 2009

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the typewhere Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].
Integrals of dynamical systems linear in the velocities
Authors: Collinson C. D. 
Published: January 2009

Kilmister (1) has discussed the existence of linear integrals of a dynamical system specified by generalized coordinates qα(α = 1, 2, …, n) and a Lagrangianrepeated indices being summed from 1 to n. He derived covariant conditions for the existence of such an integral, conditions which do not imply the existence of an ignorable coordinate. Boyer (2) discussed the conditions and found the most general Lagrangian satisfying the conditions for the case of two degrees of freedom (n = 2).
On the Fourier coefficients of a discontinuous function
Authors: Bhatnagar S. P. 
Published: January 2009

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).We writeand suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).
Sets of Semi-Commutative Matrices: Part II
Authors: Williamson J. 
Published: January 2009

In this section we extend the definition of an E-set, so that it includes sets of the typewhere the only restriction on the Ei is that they be non-singular. We now consider matrices of the typewhere each ei takes independently the values 0, 1, …, n − 1, while the a(ei) are either complex numbers or else matrices of order r, the product a(ei) E(ei), in the latter case, being interpreted as the direct product of the two matrices a(ei) and E(ei).
On the Summability of Series by a Method of Valiron
Authors: Hyslop J. M. 
Published: January 2009

The method of summability with which I shall be concerned here is denoted by (V, α ) and is defined as follows:—The series Σαn is said to be summable (V, α ) to the sum s ifThis is a particular case of a method due to Valiron in which μ–2α is replaced by a function of μ.
A Theorem on Alternants
Authors: Littlewood D. E. 
Published: January 2009

The following is a direct proof of a theorem by Zia-ud-Din.Let {ν} ≡ {ν1, ν2, …, νp)} be any S-function of weight r + s such thatin the alternant denoted in the theorem by A(αβγ….). Let {μ} be an S-function of weight s, equal to and let {λ} be an S-function of weight r, such that {λ}δ(x1, …,xp) is obtained with coefficient gλμν by diminishing the indices in the alternant {ν}δ(x1, …,xp) according to the theorem.
Two Multiplier Theorems for H1(U2)
Authors: Oberlin Daniel M. 
Published: January 2009

Let H1(U2) be the Hardy space of the bidisc as described in (3). Each function f ∈ H1(U2) has a Taylor expansion of the form . For 0<p<∞, a doubly-indexed sequence is said to be a multiplier of H1(U2) into lp ifThis paper is concerned with the cases p = 2 and p = 1. Theorem 1 characterises the multipliers of into l2 and is an analogue in two variables of an old result of Hardy and Littlewood. Theorem 2 characterises the sequences (an)n≥0 such that (an+m)n,m≥0 is a multiplier of H1(U) into l1
Functorial radicals and non-abelian torsion, II
Authors: Feigelstock Shalom , Klein Aaron 
Published: January 2009

The object of this paper is to complete and continue some matters in [1].In [1], Section 2, the torsion and torsion-free functors, whose operation on the category of abelian groups are well known, were extended to the category of all groups as follows. For a group A, put t0(A)= 0 and t1(A) = the subgroup of A generated by the torsion elements of A. Inductively define tn+1(A)/tn(A)=t1(A)/tn(A)), for every positive integer n. Then T(A)=∪ntn(A) is the smallest subgroup H of A such that A/H is torsion-free, [1], Th. 2.2. A group A satisfying T(A) = A was called a pre-torsion group. In [1], 2.12 an example was constructed of a group A satisfying t1(A)≠t2(A)=A. The question was posed whether for every positive integer n there exist groups A, satisfying tn–1(A)≠tn(A)=A. Here we give an affirmative answer. In fact, such groups will be constructed, as well as pre-torsion groups A with tk(A)≠A for every positive integer k, see Section 1.
Geometrical aspects of the system and applications to the nonlinear wave equation
Authors: Cieciura G. , Grundland A. M. 
Published: January 2009

Let E be n-dimensional (n≧2) real vector space with a nondegenerate symmetric scalar product (.|.):E × E → R1 with an arbitrary signature (p, n–p). Let us consider a second order partial differential equation (P.D.E.) of the form:where φ is a given function of two variables, v is an unknown function (defined on an open subset 0 ⊂E), |∇ν|2: =(∇ν|∇ν) is the square of the gradient ∇ν of the function ν and ∇2, denotes the Laplace-Beltrami operator.
Corrigenda: Inverse multi-parameter eigenvalue problems for matrices II
Authors: Browne Patrick J. , Sleeman B. D. 
Published: January 2009

In the paper referred to above (see [1]), the following corrections should be made.(1) Hypothesis 2.2 should read:(2) The set E⊂ℝn should be defined as(3) F0(υ)=υ+x should read F(υ)=υ–x.(4) The unique solution of F0(υ)=0 is υ=x (not υ= –x as printed).(5) Page 345, line 14:
On the Latent Roots of Compound Determinants and Brill's Determinants
Authors: Whittaker E. T. 
Published: January 2009

The roots of the equation in λwere named by Sylvester the latent roots of the determinant | a11a22 … ann |. So early as 1852, Sylvester showed that if any determinant D is given, we can at once write down a determinant whose latent roots are the squares of the latent roots of D: this determinant is in fact the square of D, the process of squaring being performed by multiplying rows into columns : so that, e.g. if the latent roots of are λ1 and λ2, then the roots of are and . Spottiswoode had also shown in 1851 that the latent roots of the reciprocal of a determinant are the reciprocals of the latent roots of the determinant itself. Both these theorems were soon found to be particular cases of a general theorem which was enunciated by Sylvester thus: The latent roots of any function of a matrix are respectively the same functions of the latent roots of the matrix itself.
The Apolar Locus of Two Tetrads of Points
Authors: Milne William P. 
Published: January 2009

In the present paper I propose to investigate the fundamental geometrical properties of the Apolar Locus of two tetrads of points in a plane.The Apolar Locus of two tetrads of points K, L, M, N and P, Q, R, S is defined in the locus of the point X, moving so that the pencils X[K, L, M, N] and X[P, Q, R, S] are apolar.
On a Class of Continued Fractions
Authors: Ford Lester R. 
Published: January 2009

The continued fractions treated in this paper are of the general formwhere s0, s1, s2, … are real integers (positive, negative, or zero). An arbitrary real number can, of course, be developed in such a fraction in an infinite variety of ways. The continued fractions discussed here have a number of striking properties and present numerous contrasts with the ordinary continued fraction usually employed.
Projective Geometry of the Sphere
Authors: Allardice R. E. 
Published: January 2009

AbstractIf A and B be two fixed points on a great circular arc and P a variable point on the arc, there are two and only two possible positions of the point P corresponding to a given valae of the ratio sinAP/sinBP, provided arcs measured in one direction from A or B be considered positive, and in the opposite direction negative; and these two points are antipodal.
On the Theory of Contours, and its Applications in Physical Science
Authors: Peddie W. 
Published: January 2009

1. In the first part of this paper we have considered merely the contours of curves, that is, contour points, and the method of obtaining the various physical diagrams. In this part we shall consider chiefly the contours of surfaces; that is, contour lines.If any curve be cut by planes parallel to that of (x, y) and if the various points of intersection be projected on any one of these planes, say z = 0, the contour points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break in continuity by projecting every point of the curve upon the plane z = 0. But such a line may be regarded as the intersection, by the plane z = 0, (see fig. 48) of a cylindrical surface whose generating lines are parallel to the z-axis and are drawn from the given curve to meet that plane. We have here then the intersection of a given surface by a surface over which z is constant. But this satisfies our definition of a contour line. This case of a cylindrical surface supplies the simplest system of contour lines by giving z different values. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is that in which the same values of the x and y co-ordinates of a point on the curve correspond to different values of the z-co-ordinate.
Question proposée an Concours Général pour la classe de Mathématiques Spéciales, Juin, 1886.
Authors: Aubert M. Paul 
Published: January 2009

Étant donnés une surface du second ordre S et deux points A et B, on mène par le point B une sécante qui rencontre la surface aux points C, C′, et le plan polaire du point A au point D. Soient M et M′ les points of la droite AD rencontre les plans qui touchent la surface aux points C et C′. La sécante BD tournant autour du point B, on demands1°. Le lieu décrit par les points M et M′.2°. Ce lieu se compose de deux surfaces du second ordre, dont l' une est inépendante de la position du point B, et l' autre Σ dépend de la position de ce point. Chercher ce que devient la surface Σ quand, dans la construction qui donne les points de cette surface, on fait jouer au point A le rôle du point B, et inversement.3°. Le point A restant fixe, déterminer les positions occupées par le point B quand la surface Σ n' α pas un centre unique à distance finie.
Note on a Formula in Quaternions
Authors: Allardice R. E. 
Published: January 2009

The formula referred is the condition for the coplanarity of the extremities of four coinitial vectors; namely, if α, β, γ, δ, are the vectors, then(See Kelland and Tait's Quaternions, p. 62.)In the first place, it may be pointed out that these equations do involve one condition on the four vectors α, β, γ, δ. For, since the vanishing of a vector involves three conditions, and only the three ratios of the four quantities a, b, c, d, are involved, it is always possible to determine these four quantities so that the condition aα + bβ + cγ +dδ = 0 shall be satisfied. But, in general, the equation a + b + c + d = 0 will not be satisfied; and thus one condition is involved.
On the condition that the straight line lx+my+nz=0 should be a normal to the conic (a, b, c, f, g, h)(x, y, z)2=0 the co-ordinates being trilinear
Authors: Pinkerton R. H. 
Published: January 2009

The condition in question may be found by using the following theorem:-If the equation in trilinear co-ordinates
Some Geometrical Theorems
Authors: Allardice R. E. 
Published: January 2009

The greatest line joining two points in the perimeter of a polygon is a side or a diagonal of the polygon.For (fig. 2) PS is obviously less than one or other of the lines PQ, PR. Hence if AB, CD (fig. 3) are any two finite lines, L and M any points, one in each of these lines, LM is less than LC, or LD and LC is less than CA or CB. From this the theorem follows at once. The theorem and the above proof apply to crossed and gauche polygons.
The solution of the equation √{(x−a)(x−b)}(x−c)(x−d)}=e
Authors: Dickson J. D. Hamilton 
Published: January 2009

If P=a+b−c−d, and K=cd−ab, the solution of the equationmay be most readily found by putting (x−a)(x−b)=z2; whence (x−c)(x−d)=z2+Px+K, and therefore =z2−2ez+e2, and finally x is given by the equation
Some relations between the orthic and the median triangles
Authors: Pressland A. J. 
Published: January 2009

Let ABC be the triangle, X, Y, Z the feet of the altitudes, H the orthocentre, A′,B′,C′ the mid points of the sides.Let ZY meet A′B′ in D, A′C′ in D′, B′C′ in R:ZX meet B′C′ in E, A′B′ in E′, A′C′ in S:XY meet C′A′ in F, B′C′ in F′, A′B′ in T.
On the history and degree of certain geometrical approximations
Authors: Pressland A.J. 
Published: January 2009

According to Luther's translation 1 Kings vi. 31 should read— “At the entrance of the choir he made two doors with pentagonal door-posts.” This is probably a wrong translation, for nowhere on Asiatic monuments of this time has a pentagon been found. Prof. A. Merx in Cantor's Vorlesungen über Geschichte der Mathemalik, Vol. i., p. 91.It is probable that the construction of the regular pentagon is due to the Pythagorean school as a consequence of I. 47. (Ibid, p. 151.)
On the history and degree of certain geometrical approximations
Authors: Pressland A.J. 
Published: January 2009

§1. Since the former paper on this subject was read, Prof. Cantor has published the second volume of his history of Mathematics. This has necessitated various additions to the paper, which can perhaps be best given as an appendix.On page 413 Prof. Cantor says that the construction of Dürer's pentagon is found in a book called Geometria deutsch, which was lately discovered in the town library at Nürnberg, and gives 1487 as the upper limit to its date. The construction is said to be “mitunverrücktem Zirckel,” the same expression that Schwenter applies to Dürer's solution.
On a surface of the third order
Authors: Allardice R.E. 
Published: January 2009

In the second part of Professor Chrystal's Algebra (Exer. V., No. 4) the following exercise is given:-If 2xyz-x2-y2-z2+1=0, and x,y,z are all real, then all, or none, of the quantities x,y,z lie between-1 and + 1.
Integrals of the form and allied Integrals
Authors: Gibson George A. 
Published: January 2009

1. If θ be the eccentric angle of a point P on an ellipse whose semi-axes are a and b and if x, y be the rectangular co-ordinates of a point Q, then PQ2 = (x − acosθ)2 + (y − bsinθ)2. When a = b and x2 + y2 = f2, PQ2 = f2 2afcosθ + a2, if the line from which θ is measured passes through Q. The integralis a well-known one. I propose in the present paper to consider the more general form which the integral takes when P lies on an ellipse.
A Problem of Lewis Carroll's, and the rational solutions of a Diophantine Cubic
Authors: Tweedie C. 
Published: January 2009

§ 1. In the Life and Letters of Lewis Carroll occurs the following extract from his Diary:—“Dec. 19 (Sun).—Sat up last night till 4 a.m., over a tempting problem, sent me from New York, ‘to find 3 equal rational-sided rt.-angled ∆&s.’ I found two, whose sides are 20, 21, 29; 12, 35, 37; but could not find three.' (v. page 343.)
The determination of the centre of gravity of a circular arc by reference to a principle of Dynamics
Authors: Crawford G. E. 
Published: January 2009

Mr H. Poincaré, in a recent work (la Valeur de la Science) speaks of the interest attaching to proofs of theorems in pure geometry by arguments derived from mechanical or physical considerations. The following is a case in point.To find the centre of gravity of a circular arc by reference to a principle of dynamics.
A method of finding (i) the double points of a unicursal curve, (ii) unicursal quartics with three given double points
Authors: Bell R. J. T. 
Published: January 2009

The equationswhere f1(t), f2(t)φ(t) are polynomials in t, determine a unicursal curve. If α and β can be found so thatthen the point (α, β) is on the curve and is given by two values of t, t1 and t2, which are the roots of the equation at2 + 2bt + c =0.
Integral Equations and the Determination of Green's Functions in the Theory of Potential
Authors: Carslaw H. S. 
Published: January 2009

In the Theory of Potential the term Green's Function, used in a slightly different sense by Maxwell, now denotes a function associated with a closed surface S, with the following properties:—(i) In the interior of S, it satisfies ∇2V = 0.(ii) At the boundary of S, it vanishes.(iii) In the interior of S, it is finite and continuous, as also its first and second derivatives, except at the point (x1, y1,z1).
Local Electromagnetic Fields
Authors: Bateman H. 
Published: January 2009

§ 1. The radiation of electromagnetic charges from a moving point.Let the electric charge associated with a moving point Q at time s be f(s). This charge is supposed to vary on account of the radiation of electric charges from Q in a variable direction, which, at time s, has direction cosines l (s), m (s), n (s) Using ξ(s), η(s), ζ(s) to denote the coordinates of Q at time s, and defining the effective time τ by means of the equationthe electromagnetic field may be specified by means of the potentialswhere
On the stationary eccentricity of a system of four-tangent conics
Authors: Edwards F. E. 
Published: January 2009

Let the convex quadrilateral formed by the four given tangents be ABA′B′, and O the intersection of the diagonals. Let OA and OB be taken as axes of x and y. Denote OA, OA′, OB and OB′ by a, a′, b and b′, a and b being positive, and a′ and b′ negative. The tangential equation of the system is thenwhere k is a variable parameter; for the equation is satisfied when the straight line lx + my + 1 = 0 passes through any two adjacent angular points of the quadrilateral.
The Factors of (a, b, c, f, g, h)(x, y, z)2–λ(x2+y2+z2)
Authors: Bell Robert J. T. 
Published: January 2009

f λS is the product of two factors of the form αx + βy + γz if λ is a root of a discriminating cubicA well-known proof of the reality of the roots of the cubic is as follows :—Write
On Determinants whose elements are Determinants
Authors: Whittaker E. T. 
Published: January 2009

The present paper is concerned with determinants whose elements are themselves determinants. The best-known determinants of this kind are those whose elements are minors of a given determinant; these are called the “adjugate” and the “compounds” of the given determinant. Determinants whose elements are themselves determinants also occur frequently in Muir's theory of Extensionals.The determinants considered in the present paper are of a somewhat more general type than adjugates, compounds, and extensionals; the principal result obtained is “Theorem A” (at the end of § 2), which relates to determinants whose elements are formed from any number of arrays. It is shown in § 3 that many other formulae, both new and old, may be obtained by specialising the arrays in “Theorem A.”
Determinantal Systems of Apolar Triads in the Twisted Cubic
Authors: Marr W. L. 
Published: January 2009

It has already been shown that nine points can be found on a conic to form six triads all apolar to a given triad (ABC), the hessian lines of these six triads and of ABC being concurrent.The construction of the corresponding system in the twisted cubic is easy. In the first place, the poles of the six triads P1Q2R3, etc., will lie on the plane of the triad ABC to which they are all apolar. Secondly, the hessian lines of these triads, which in the conic were concurrent, will now be generating lines of a quadric circumscribing the twisted cubic. Now the pole of a triad is in the osculating plane at any of the points of the triad, hence the osculating plane at Pl will contain the poles of the triads P1Q2R3 and P1Q2R3
The Modified Bessel Function Kn(z)
Authors: MacRobert T. M. 
Published: January 2009

Gray and Mathews, in their treatise on Bessel Functions, define the function Kn(z) to beWe shall denote this function by Vn(z). This definition only holds when z is real, and R(n)≧0. The asymptotic expansion of the function is also given; but the proof, which is said to be troublesome and not very satisfactory, is omitted. Basset (Proc. Camb. Phil. Soc., Vol. 6) gives a similar definition of the function.
Some Extensions of Pincherle's Polynomials
Authors: Humbert Pierre 
Published: January 2009

The polynomials which satisfy linear differential equations of the second order and of the hypergeometric type have been the object of extensive work, and very few properties of them remain now hidden; the student who seeks in that direction a subject for research is compelled to look, not after these functions themselves but after generalisations of them. Among these may be set in first place the polynomials connected with a differential equation of the third order and of the extended hypergeometric type, of which a general theory has been given by Goursat. The number of such polynomials of which properties have been studied in particular is rather small; in fact, Appell's polynomialsand Pincherle's polynomials, arising from the expansionsare, so far as I know, the only well-known ones. To show what can be done in these ways, I shall briefly give the definition and principal properties of some polynomials analogous to Pincherle's and of some allied functions.
On the Relation between Pincherle's Polynomials and the Hypergeometric Function
Authors: Baker Bevan B. 
Published: January 2009

1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equationand Pn(x) satisfies the differential equation
Geometrical Interpretation of a few concomitants of the cubic in the Argand Plane
Authors: Saddler W. 
Published: January 2009

be the binary cubic whose coefficients a1, are complex numbers represented on the Argand Plane. Then if its roots are z1, z2, z3, the three corresponding points form the vertices of a triangle A1 A2 A3. Let this triad of points be said to represent the cubic. Then its Hessianis represented by a certain pair of other points; likewise every first polarassociates a definite pair of points (z) with any given point (y).
Note on the Geometry of two Binary quadratics in the complex plane
Authors: Saddler W. 
Published: January 2009

Ifrepresent the points A, B and C, D respectively and ifthen A, B, C, D are four harmonic points on a circle. If 0 is the mid point of AB then OA2 = OC. OD and ∠COB = ∠DOB.