bluuebunny/crossref_metadata_2025 huggingface dataset:
Special train algebras arising in genetics II
Authors: Gonshor Harry
Published: January 2009

We shall extend some of the results of (7) to the case of multiple alleles, our primary concern being that of polyploidy combined with multiple alleles. Generalisations often tend to make the computations more involved as is expected. Fortunately here, the attempt to generalise has led to a new method which not only handles the case of multiple alleles, but is an improvement over the method used in (7) for the special case of polyploidy with two alleles. This method which consists essentially of expressing certain elements of the algebra in a so-called “ factored ” form, gives greater insight into the structure of a polyploidy algebra, and avoids a great deal of the computation with binomial coefficients, e.g. see (7), p. 46.
The enumeration of tree-like polyhexes
Authors: Harary Frank , Read Ronald C.
Published: January 2009

A problem of considerable interest in combinatorial analysis is that of determining the number of ways in which a connected figure can be constructed in the plane by assemblingnregular hexagons in such a way that two hexagons abut on each other, if at all, along the whole of a common edge. Examples of these constructions can be seen in the various figures in this paper.
The interaction of a contact discontinuity with sound waves according to the linearised Navier-Stokes equations
Authors: Anderson W. M.
Published: January 2009

The resolution of a small initial discontinuity in a gas is examined using the linearised Navier-Stokes equations. The smoothing of the resultant contact surface and sound waves due to dissipation results in small flows which interact. The problem is solved for arbitrary Prandtl number by using a Fourier transform in space and a Laplace transform in time. The Fourier transform is inverted exactly and the density perturbation is found as two asymptotic series valid for small dissipation near the contact surface and the sound waves respectively. The modifications to the structures of the contact surface and the sound waves are exhibited.
Endomorphism near-rings
Authors: Malone J. J. , Lyons C. G.
Published: January 2009

The study of near-rings is motivated by consideration of the system generated by the endomorphisms of a (not necessarily commutative) group. Such endomorphism near-rings also furnish the motivation for the concept of a distributively generated (d.g.) near-ring. Although d.g. near-rings have been extensively studied, little is known about the structure of endomorphism near-rings. In this paper results are presented which enable one to give the elements of the endomorphism near-ring of a given group. Also, some results relating to the right ideal structure of an endomorphism near-ring are presented. These concepts are applied to present a detailed picture of the properties of the endomorphism near-ring of (S3, +).
Genetic algebras associated with sex linkage
Authors: Holgate P.
Published: January 2009

Most work in genetic algebras has been concerned with inheritance which is symmetric with respect to sex, in that the characters studied are determined by genes located at autosomal loci, and it is assumed that the segregation pattern is the same in males and females. When asymmetric situations are studied, the development of the theory is complicated by the higher dimensions of the algebras, and by a feature to which Etherington (3, p. 40) drew attention, namely the fact that the passage from the gametic to the zygotic algebra no longer quite corresponds to the process of duplication, as it does in the symmetric case. Etherington gave some results for the gametic and zygotic algebras of a single sex linked diallelic locus, and its properties were discussed further by Gonshor (4, p. 44). In a second paper (5, p. 334) Gonshor studied sex linkage in the case of multiple alleles, choosing a canonical basis which exhibited very clearly the multiplication table and ideal structure of the algebra. His treatment from the statement of the multiplication table in terms of the natural basis to its expression in terms of a canonical basis, is repeated in the displayed relations (4)–(8) below, for completeness and to establish the present notation.
Embedding topological semigroups in topological groups
Authors: McKilligan Sheila A.
Published: January 2009

If we consider a semigroup, its algebraic structure may be such that it is isomorphic to a subsemigroup of a group, or is algebraically embeddable in a group. This problem was investigated in 1931 by Ore who obtained in (4) a set of necessary conditions for this embedding. A necessary condition is that the semigroup should be cancellative: for any a, x, y in the semigroup either xa = ya or ax = ay implies that x = y. Malcev in (3) showed that this was not sufficient. It is enough to note that his example was a non-commutative semigroup: a commutative cancellative semigroup is embeddable algebraically in a group.
Contributions to genetic algebras
Authors: Gonshor H.
Published: January 2009

Etherington introduced certain algebraic methods into the study of population genetics (6). It was noted that algebras arising in genetic systems tend to have certain abstract properties and that these can be used to give elegant proofs of some classical stability theorems in population genetics (4, 5, 9, 10).
On the integral representation of the endomorphisms of Mikuśinski's operator field
Authors: Bleyer András
Published: January 2009

We proved in (1) that every continuous endomorphism can be generated on a subring of the fieldM. More precisely, the ringHof piecewise polynomial functions has the property that every isomorphism fromHintoM, continuous in the sequential topology ofH, can be extended to a continuous endomorphism ofMwhere the notion of continuity inMis the usual sequential one.
Generating varieties of topological groups
Authors: Brooks M. S. , Morris Sidney. A. , Saxon Stephen. A.
Published: January 2009

Recently several papers on varieties of topological groups have appeared. In this note we investigate the question: if Ω is a class of topological groups, what topological groups are in the variety V(Ω) generated by Ω that is, what topological groups can be “manufactured” from Ω using repeatedly the operations of taking subgroups, quotient groups and arbitrary cartesian products? We seeka general theorem which will be useful for investigating V(Ω) for well-known classesΩ.
Near-rings with identities on dihedral groups
Authors: Johnson Marjory J.
Published: January 2009

Normal right submodules and right ideals need not coincide in an arbitrary near-ring. Berman and Silverman (1) have shown that in a near-ring (N, +, ·) with a two-sided zero (i.e. x · 0 = 0 · x = 0, for all x ∈ N) a right ideal is also a right submodule. If (N, +, ·) is in fact a distributively generated near-ring, then all normal right submodules are also right ideals. (See (5).)
A problem on cyclic subgroups of finite groups
Authors: Laffey Thomas J.
Published: January 2009

Let G be a finite group and let S be a subgroup of G with
Primary and coprimary decompositions
Authors: Moore D. J.
Published: January 2009

Let R be an associative, commutative ring with identity, and let A be a (unitary) R-module. It is well known that if A is a Noetherian R-module then every submodule of A has a primary decomposition in A. The object of the present paper is to dualise this result; that is, to show that if A is an Artinian R-module then every submodule of A can be expressed as the sum of a finite number of coprimary submodules of A.
Generalisations of nilpotency of rings
Authors: Robertson Edmund F.
Published: January 2009

In (5) and (6) we studied certain subgroups of infinite dimensional linear groups over rings. In particular we investigated how the structure of the subgroups was related to the structure of the rings over which the linear groups were defined. It became clear that it might prove useful to study generalised nilpotent properties of rings analogous to Baer nilgroups and Gruenberg groups. We look briefly at some classes of generalised nilpotent rings in this paper and obtain a lattice diagram exhibiting all the strict inclusions between the classes.
On the existence of positive solutions of Au = λBu
Authors: Kershaw D.
Published: January 2009

It is well known that sufficient conditions for the existence of a positive vector u which satisfies the matrix equation Au = λu are that A should be non-negative and irreducible. This result, the qualitative part of the Perron-Frobenius theorem, has been proved in a variety of ways, one of the most attractive of which is that given by Alexandroff and Hopf in their treatise “ Topologie ”. The aim of this note is to show how their method can be adapted to deal with the generalised eigenvalue problem defined by Au = λBu where A and B are square matrices.
On non-negative spectrum in Banach algebras
Authors: Yood Bertram
Published: January 2009

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).
Invariant convolution algebras
Authors: Baker J. W.
Published: January 2009

Let A be a commutative, semi-simple, convolution measure algebra in the sense of Taylor (6), and let S denote its structure semigroup. In (2) we initiated a study of some of the relationships between the topological structure of A^ (the spectrum of A), the algebraic properties of S, and the way that A lies in M(S). In particular, we asked when it is true that A is invariant in M(S) or an ideal of M(S) and also whether it is possible to characterise those measures on S which are elements of A. It appeared from (2) that if A is invariant in M(S) then S must be a union of groups and that A^ must be a space which is in some sense “ very disconnected ”. In (3) we showed that if A^ is discrete then A is “ approximately ” an ideal of M(S). (What is meant by “ approximately ” is explained in (3); it is the best one can expect since algebras which are approximately equal have identical structure semigroups and spectra.) In this paper we round off some of the results of (2) and (3). We show that if A is invariant in M(S) then A^ is totally disconnected, and that if A^ is totally disconnected then S is an inverse semigroup (union of groups). From these two crucial facts it is fairly straight-forward to obtain a complete characterisation of algebras A (and their structure semigroups) for which (i) A^ is totally disconnected, (ii) A is invariant in M(S), or (iii) A is an ideal of M(S).
Some cyclic inequalities
Authors: Baston V. J.
Published: January 2009

In this note we prove some cyclic inequalities which are generalisations of known results. We shall assume throughout that ai+n = ai ≧ 0 for all i, that no denominator in the statement of a result vanishes and finally that p, m and q are positive integers. We shall also use A(i, m) to denote with the convention that A(i, 0) = 0. The most interesting of our results is probably Theorem 2 since, in the special case p = 1, m = 2, r = 0, it gives a lower bound of ⅓n for the Shapiro sum . Although it is by no means best possible, see (2), our method implicitly gives a really simple way of obtaining this lower bound which, incidentally, is an improvement on Rankin's original result (5).
l-simple lattice-ordered groups
Authors: Glass A. M. W.
Published: January 2009

Let G be a lattice-ordered group (l-group) and H a subgroup of G. H is said to be an l-subgroup of G if it is a sublattice of G. H is said to be convex if h1, h2 ∈ H and h2 ≦ g ≦ h2 imply g ∈ H. The normal convex l-subgroups (l-ideals) of an l-group play the same role in the study of lattice-ordered groups as do normal subgroups in the investigation of groups. For this reason, an l-group is said to be l-simple if it has no non-trivial l-ideals. As in group theory, a central task in the examination of lattice-ordered groups is to characterise those l-groups which are l-simple.
A completeness theorem for a nonlinear problem
Authors: Brown K. J.
Published: January 2009

As is well known, the linear Sturm-Liouville eigenvalue problem on a bounded real interval [a, b] possesses a family of eigenfunctions which is a complete orthonormal system for the real Hilbert space L2[a, b], i.e. there exists a sequence of eigenfunctions un such that (ui, uj) = δij (Kronecker delta) for i, j ∈ N (the set of positive integers) and, if u ∈ L2[a, b], u = where cj = (u, uj). Pimbley (4, p. 113), raises the question as to whether similar completeness results hold for nonlinear problems. In this note we show that certain nonlinear Sturm-Liouville eigenvalue problems possess eigenfunctions which form a basis for L2[a, b], i.e. there exists a sequence of eigenfunctions {vn} for the nonlinear problem such that every u ∈ L2[a, b] can be expressed in the form u = by means of a unique sequence {cn} of real numbers.
On the mean convergence of Markov operators
Authors: Atalla Robert E.
Published: January 2009

Throughout the paper, T will be a Markov operator on C(X) (X compact T2), i.e. a continuous positive operator such that Te = e (e the unit function). P will be the set of Borel probability measures on X, which we shall often think of as linear functionals on C(X), and , where T' is the adjoint of T. Let
Automorphisms of transformation semigroups
Authors: Fitzpatrick S. P. , Symons J. S. V.
Published: January 2009

It is common property in the theory of transformation semigroups that the presence of all the constant maps ensures that automorphisms are induced by a permutation of the underlying set. Essentially, this goes back to Malcev (2); it has been extensively generalised by Sullivan in (4). For semigroups which do not contain the constants (for example, all surjective transformations of a set, or all injections) there is, as yet, no similar result. The purpose of this note is to provide one.
On a result of M. Heins
Authors: Jenkins James A.
Published: January 2009

Some years ago Heins (1) proved that a Riemann surface which can be conformally imbedded in every closed Riemann surface of a fixed positive genus g is conformally equivalent to a bounded plane domain. In the proof the main effort is required to prove that a surface satisfying this condition is schlichtartig. Heins gave quite a simple proof of the remaining portion (1; Lemma 1). The main part of the proof depended on exhibiting a family of surfaces of genus g such that a surface which could be conformally imbedded in all of them was necessarily schlichtartig. Another proof using a different construction was recently given by Rochberg (2). We will give here a further proof based on the method of the extremal metric and using a further construction which is in some ways more direct than those previously given.
Perfect semigroups
Authors: Fountain John
Published: January 2009

A semigroup S with identity is (left) perfect if every unitary left S-system has a projective cover. This is the semigroup analogue of the definition of left perfect rings introduced in (1). The investigation of perfect semigroups was initiated by Isbell (4), who proved that a semigroup is perfect if and only if it satisfies two conditions referred to as conditions A and D.
Perfect categories
Authors: Isbell John
Published: January 2009

This note extends to categories Fountain's theorem (2) that for a perfect monoid S, every flat S-set is projective. (The converse is known (4).)
A note on pseudo-umbilical surfaces
Authors: Houh Chorng-Shi
Published: January 2009

We follow the notations and basic equations of Chen (2). Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. We choose a local field of orthonormal frames e1, …, em in Rm(c) such that, restricted to M, the vectors e1, e2 are tangent to M. Let ω1, …, ωm be the field of dual frames. Then the structure equations of Rm(c) are given by
Commuting well-bounded operators on Hilbert spaces
Authors: Gillespie T. A.
Published: January 2009

A bounded linear operator T on a complex reflexive Banach space is said to be well-bounded if it is possible to choose a compact interval J = [a, b] and a positive constant M such that

for every complex polynomial p, where ‖p‖J denotes sup {|p(t)|:t ∈ J}. Such operators were introduced and first studied by Smart (4). They are of interest principally because they admit (and in fact are characterised by) an integral representation similar to, but in general weaker than, the integral representation of a self-adjoint operator on a Hilbert space. (See (2) and (4) for details.) It is easily seen, by verifying (1) directly, that T is well-bounded if it is a scalar-type spectral operator with real spectrum.
Convergence in Sequence Spaces
Authors: Green H. F.
Published: January 2009

In a perfect sequence space α, on which a norm is defined, we can consider three types of convergence, namely projective convergence, strong projective convergence and distance convergence. In the space σ∞, when distance is defined in the usual way, the last two types of convergence coincide and are distinct from projective convergence ((2), p. 316). In the space σ1 all three types of convergence coincide ((2), p. 316). It will be shown in this paper that, if distance convergence and projective convergence coincide, then all three types of convergence coincide. It will not be assumed that the limit under one convergence is also the limit under the other convergence.
Relatively Complete Fields
Authors: Rayner F. J.
Published: January 2009

Let K be a valued field, let v denote its valuation and B its valuation ring. Let P denote the valuation ideal. For each a in B, let ā denote the residue class a + P in the field B/P; for f(x)=∑arxr in B[x], let f(x) denote ∑ārxr in B/P[x]. Let Λp denote the leading coefficient of a polynomial p, and ∂p the degree of a non-zero polynomial.
On the Zeros of a Class of Canonical Products of Integral Order
Authors: Bowen N. A.
Published: January 2009

In (1) I obtained † an asymptotic formula for the number of zeros of an arbitrary canonical product II(z) of integral order but not of mean type, all of whose zeros lie on a single radius, from a knowledge of the asymptotic behaviour of (i)log | П(z)| as | z | = r→ ∞ along another radius l, with certain side conditions. After proving the analogous theorem in which log | П(z)| in (i) is replaced by , I show in this note that, at a cost of replacing l by two radii l1 and l2, both of these theorems may be generalised to include a class of canonical products of integral order whose zeros lie along a whole line. In one of the resulting theorems ‡ (Theorem II) I find the asymptotic number of zeros on each half of the line of zeros; another theorem (Theorem III) includes a previous result of mine.§
Long Waves on a Rotating Earth in the Presence of a Semi-Infinite Barrier
Authors: Chambers LL. G.
Published: January 2009

The problem considered is that of long gravity waves approaching, from an arbitrary direction, a semi-infinite barrier, the whole system being in rotation. It is shown that the rotation gives rise to a wave in the shadow region whose amplitude depends upon the angle of incidence, but whose form is independent of it and which travels along the barrier without attenuation in that direction. The work is an extension and simplification of previous work by Crease, involving the use of methods previously developed by the author.
A Class of Integral Transforms
Authors: Wimp Jet
Published: January 2009

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))

and the generalised Mehler transform pair (7)

These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).
Some Inequalities involving (r!)1/r
Authors: Minc Henryk , Sathre Leroy
Published: January 2009

In a recent investigation of a conjecture on an upper bound for permanents of (0, 1)-matrices (2) we obtained some inequalities involving the function (r!)1/r which are of interest in themselves. Probably the most interesting of them, and certainly the hardest to prove, is the inequality

where ø(r = (r!)1/r. In the present paper we prove (1) and other inequalities involving the function ø(r.
Primitive Rings of Infinite Matrices
Authors: Sands A. D.
Published: January 2009

E. C. Posner (5) has shown that a ring R is primitive if and only if the corresponding matrix ring Mn(R) is primitive. From this result he is able to deduce that the primitive ideals in Mn(R) are precisely those ideals of the form Mn(P), where P is a primitive ideal in R. This affords an alternative proof that the Jacobson radical of Mn(R) is Mn(J), where J is the Jacobson radical of R. But Patterson (3, 4) has shown that this last result does not hold in general for rings of infinite matrices and thus that the above result concerning primitive ideals cannot be extended to the infinite case. Nevertheless in this paper we are able to show that Posner's result on primitive rings does extend to infinite matrix rings. Patterson's result depends on showing that if the Jacobson radical J of R is not right vanishing then a certain matrix with entries from J does not lie in the Jacobson radical of the infinite matrix ring. In the final section of this paper we consider a ring R with this property and exhibit a primitive ideal in the infinite matrix ring, which does not arise, as above, from a primitive ideal in R. Finally the Jacobson radical of this ring is determined.
Commutation Problems Involving Rings of Infinite Matrices
Authors: Patterson E. M.
Published: January 2009

Let R be a ring and let J be the set of all integers. In the set M(R) of all mappings A: J×J→R, let addition and multiplication be defined by

Here aij denotes the image of (i, j) under A and bij, cij, dij are similarly defined for the mappings B, C, D. In (2) we require A, B to be such that the sum is defined and is in R. Thus, in general, M(R) is not closed with respect to multiplication.
Ancestral Rings
Authors: Peinado Rolando E.
Published: January 2009

A ring R is said to be a P-ancestral ring if all proper non-zero sub-rings of R have property P. If p is the property that every proper non-zero sub-ring of R is a (two-sided) ideal then the ring Z of rational integers furnishes an example of a P-ancestral ring.
Total Graphs and Traversability
Authors: Behzad Mehdi , Chartrand Gary
Published: January 2009

With every graph G (finite and undirected with no loops or multiple lines) there is associated a graph L(G), called the line-graph of G, whose points correspond in a one-to-one manner with the lines of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of Gare adjacent. This concept was originated by Whitney (3). In a similar way one can associate with G another graph which we call its total graph and denote by T(G). This new graph has the property that a one-to-one correspondence can be established between its points and the elements (the set of points and lines) of G such that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent (if both elements are points or both are lines) or theyare incident
A Note on Griffith Cracks
Authors: Lowengrub M.
Published: January 2009

The problem of determining the distribution of stress in the neighbourhood of a Griffith crack, denned by ∣x∣ ≦ 1, y = 0, which is subject to an internal pressure varying along the length of the crack has been solved by Sneddon and Elliott (3) and by England and Green (1). In the former paper a Fourier cosine method is used to arrive at a solution while in the latter paper the problem is reduced to an Abel integral equation by making use of integral representations of the complex potentials given in Green and Zerna (2). Neither paper deals with the calculation of the stress intensity factor

which is extremely important to workers in fracture mechanics.
Some Fractional q-Integrals and q-Derivatives
Authors: Al-Salam Waleed A.
Published: January 2009

A q-analogue of the integral ∣f(t)dt is defined by means of

which is an inverse of the q–derivative

The present author (2) has recently obtained a q–nalogue of a formula of Cauchy, namely,

where, for real or complex α and N a positive integer,
On the Elementary Proof of the Prime Number Theorem
Authors: Levinson N.
Published: January 2009

Perhaps the simplest elementary proof of the prime number theorem, see Erdös (2) and Selberg (5), is Wright's modification (8), (3, p. 362) of Selberg's original proof (5). Another variant is due to V. Nevanlinna (4). Wright's proof uses Selberg's idea of smoothing the weighting process which occurs in the Selberg inequality, (1.2) below, by iterating this inequality. Here it will be shown that the proof requires less ingenuity if use is made of a further smoothing operation, namely first integrating the Selberg inequality itself. Integration has been used on a related inequality by Breusch (1 to obtain a remainder term. This method also makes proof by contradiction unnecessary.
On Extending Semigroup Characters
Authors: Fulp Ronald
Published: January 2009

The principal result of this paper is a characterisation of those commutative semigroups S which have the property that each character of each subsemigroup of S can be extended to a character of S. This work was partially inspired by the discovery that Theorem 5.65 of (1) is incorrect; it is related to that of Hill,who has obtained a different solution of the problem (2). Warne and Williams proved in (4) that any bounded character defined on an inverse subsemigroup of an inverse semigroup can be extended to the semigroup.
Useful Theorems on Commutative Non-associative Algebras
Authors: Losey Nora
Published: January 2009

Recently J. M. Osborn has investigated the structure of a simple commutative non-associative algebra with unity element satisfying a polynomial identity, (4), (5) and (6). From his work it seems likely that if such an algebra is of degree three or more it is necessarily power-associative. In (4) he establishes a hierarchy of identities with the property that each identity is satisfied by an algebra satisfying no preceding identity. Following (5), (6),the next identity to consider is

where a, c and h are elements of the ground field.
On the Sum
Authors: Mcdonagh Sean
Published: January 2009

Erdős (1) has proved the following result:

Theorem A.Every integral polynomial g(n) of degree k ≧ 3, represents for infinitely many integers n a(k-1)th power-free integer provided, in the case where k is a power of 2, there exists an integer n such that g(n)≢0 (mod 2k-1).
An estimate for trigonometrical sums over square-free integers with a constant number of prime factors
Authors: Mc Donagh Sean
Published: January 2009

1. In deriving an expression for the number of representations of a sufficiently large integer N in the form

where k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the type

where ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sum

where α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the form

where s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.
Generation of an acoustic pulse by a baffled circular piston
Authors: Chadwick P. , Tupholme G. E.
Published: January 2009

The generation of acoustic disturbances in a fluid of semi-infinite extent by the motion of a circular piston surrounded by a plane rigid baffle has been studied quite extensively (see (1), (2), (3), (4) and further references given in these papers). Attention has been devoted mainly to the case in which the piston executes a harmonic oscillation of small amplitude, and only comparatively recently has Oberhettinger (2) demonstrated how the time-harmonic solution can be used to solve the more general problem in which the normal velocity of the piston is an arbitrary function of time. The purpose of the present paper is two-fold. Firstly, we point out that for arbitrary normal motion of the piston the " baffled piston problem " can be solved directly, and in a particularly simple manner, by means of a technique involving integral transforms which has been applied by Mitra (5) and Eason (6) to the study of shear wave propagation in an elastic half-space. Secondly, we give a more detailed account than appears to be available in the literature of the structure of the sound pulse generated by the arbitrary normal motion of a baffled piston.
On the absolute summability by Borel's integral method of the derived Fourier series and its conjugate series
Authors: Sharma R. M.
Published: January 2009

Mohanty (1) and (3) considered the absolute summability of conjugate series and Fourier series by Borel's integral method by proving the following theorems.
The number of partitions of a set of N points in k dimensions induced by hyperplanes
Authors: Harding E. F.
Published: January 2009

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.
Generalised Dirichlet series and Hecke's functional equation
Authors: Berndt Bruce C.
Published: January 2009

The generalised zeta-function ζ(s, α) is defined by

where α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).
On functions and equations in distributive lattices
Authors: Rudeanu Sergiu
Published: January 2009

In [1], R. L. Goodstein has extended some well-known theorems on functions and equations in a Boolean algebra to the case of a distributive lattice L with 0 and 1. The purpose of this paper is to prove that most of Goodstein's theorems, as well as some additional results, are still valid in the case when L is not required to have least and greatest elements.
On the absolute Cesaro summability factors of trigonometric series
Authors: Singh Niranjan
Published: January 2009

Let be any given infinite series with sn as its n-th partial sum.

We write

and

where
Circular and transverse wave solutions of non-linear hyperbolic systems of equations
Authors: Waterston Robin J.
Published: January 2009

Circularly and transversely polarised (henceforth called circular and transverse) waves have been shown to occur as solutions of non-linear equations governing a wide range of physical phenomena, including finite elasticity (1), magnetohydrodynamics (2), and gyromagnetism (3), but only when the material properties of the medium are isotropic with respect to the direction of wave propagation. This paper is an attempt to unify and generalise these results.
Nearly directed subspaces of partially ordered linear spaces
Authors: Jameson G. J. O.
Published: January 2009

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that
On the Fredholm integral equation associated with pairs of dual integral equations
Authors: Hutson V.
Published: January 2009

Consider the Fredholm equation of the second kind

where

and Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.
On radicals of infinite matrix rings
Authors: Sands A. D.
Published: January 2009

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.
Continuous functions of bounded nth variation
Authors: Brown Gavin
Published: January 2009

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).
On Riesz operators
Authors: Alexander J. C.
Published: January 2009

In (4) Vala proves a generalization of Schauder's theorem (3) on the compactness of the adjoint of a compact linear operator. The particular case of Vala's result that we shall be concerned with is as follows. Let t1 and t2 be non-zero bounded linear operators on the Banach spaces Y and X respectively, and denote by 1T2 the operator on B(X, Y) defined by
On the greatest prime Factor of a polynomial
Authors: Keates M.
Published: January 2009

Let ƒ(x) be a non-linear polynomial with rational integer coefficients, and for integral x let P(x) denote the greatest (positive) prime factor of ƒ(x). Pólya (1) has proved that if ƒ(x) is of degree 2 and has distinct roots then P(x)→∞ as x→∞. It is probably well-known that, provided ƒ(x) has distinct roots, this is true whatever the degree of ƒ(x). There does not appear to be a proof of this in the literature, but it is easily deducible from a result of Siegel (2). These results, however, are non-effective, although effective results have been obtained for a number of special polynomials. Chowla (3) has proved that, if ƒ(x) = x2 +1, then P(x)>C log log x, where C is an absolute positive constant. Analogous results have been proved for some polynomials of the form ax2+b and for some of the form ax3 + b by Mahler and Nagell respectively (4).
A procedure for summing asymptotic series
Authors: Scraton R. E.
Published: January 2009

When a function ƒ(x) possesses an asymptotic series

this series provides a useful means of evaluating ƒ(x) for large values of x. The usual procedure is to sum all the terms in S(x) up to, but excluding, the term of smallest magnitude. The degree of accuracy obtained by this method cannot normally be improved by direct summation of S(x), but sometimes better accuracy can be obtained by using one of the familiar devices for accelerating the convergence of series. Simple δ2-extrapolation may be successful, and Rosser (1) and others have used the Euler transformation to some effect. The method given here provides, in suitable cases, a more effective means of evaluating ƒ(x) from the series for a wide range of values of x.
A Choquet-Deny theorem for affine functions on a Choquet simplex
Authors: Priestley H. A.
Published: January 2009

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).

I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.
On functions of bounded boundary rotation I
Authors: Brannan D. A.
Published: January 2009

Let Vk denote the class of functions

which map conformally onto an image domain ƒ(U) of boundary rotation at most kπ (see (7) for the definition and basic properties of the class kπ). In this note we discuss the valency of functions in Vk, and also their Maclaurin coefficients.

In (8) it was shown that functions in Vk are close-to-convex in . Here we show that Vk is a subclass of the class K(α) of close-to-convex functions of order α (10) for , and we give an upper bound for the valency of functions in Vk for K>4.
Operators with a Single Spectrum
Authors: West T. T.
Published: January 2009

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to be

and the spectrum of T is the complement of ρ(T) in Z.
Characterisations of Generalised Uniserial Algebras. III
Authors: Wall Drury W.
Published: January 2009

Let A be a finite dimensional algebra with identity element over a field. A is generalised uniserial if every primitive left ideal and every primitive right ideal of A has only one compositions series. In the previous papers in this series (6, 7) generalised uniserial algebras have been characterised as algebras all of whose residue class algebras are of certain types. The purpose of this paper is to extend the earlier results by showing that in order that A be generalised uniserial it is sufficient to require weaker conditions on merely a finite sequence of residue class algebras of A.
A Note on the Asymptotic Expansion of a Ratio of Gamma Functions
Authors: Fields Jerry L.
Published: January 2009

Many problems in mathematical analysis require a knowledge of the asymptotic behaviour of Γ(z + α)/Γ(z + β) for large values of |z|, where α and β are bounded quantities. Tricomi and Erdélyi in (1), gave the asymptotic expansion

where the are the generalised Bernoulli polynomials, see (2), defined by

In this note, we show that if, instead of considering z to be the large variable, we consider a related large variable, (1) can be improved from a computational viewpoint.
A Note on Logarithmic Summability (L)
Authors: Sitaraman Y.
Published: January 2009

It is well known that two mutually related summability methods for a sequence sn(n = 0, 1, 2, …) are any Cesaro method (C, γ) of order γ > 0 and the Abel method (A). The notation used in this statement is that of Hardy ((1), pp. 96-7, 71) and the statement itself can be amplified as follows. Summability (C, γ), γ > 0, of sn implies (i) sn = o(nγ), (ii) summability (A) of sn. Also, as a conditional converse of this result, we have Offord's result ((3), first part of Theorem 2), that hypothesis (i), and hypothesis (ii) suitably strengthened, together imply summability (C, γ), γ > 0, of sn. It is the object of this note to bring to light a second pair of summability methods mutually related like the methods (C, γ) and (A), by following an argument which is essentially similar to Offord's but differs sufficiently from Offord's in details to justify a separate and self-contained treatment of the second pair of methods.
On a Splitting Theorem of Gaschütz
Authors: Rose John S.
Published: January 2009

LetGbe any finite group, andpany prime number. (All groups to be considered here are finite, and we assume this without further comment.) We denote byKp(G) the unique smallest normal subgroup ofGfor which the quotientG/Kp(G) is ap-group.G/Kp(G) is called thep-residualofG. W. Gaschütz (2, Satz 7) has proved the following

Theorem.Set K = Kp(G).If the Sylow p-subgroups of K are abelian, then G splits over K.
Dual Integral Equations with Trigonometric Kernels
Authors: Lowndes J. S.
Published: January 2009

Consider the dual equations

where
The “Fourier” Theory of the Cardinal Function
Authors: Whittaker J. M.
Published: January 2009

The generalised Riesz-Fischer theorem states that if

is convergent, with 1 < p ≤ 2, then

is the Fourier series of a function of class . When p > 2 the series (2) is not necessarily a Fourier series; neither is it necessarily a Fourier D-series. It will be shown below that it must however be what may be called a “Fourier Stieltjes” series. That is to say, the condition (1) with (p > 1) implies that there is a continuous function F (x) such that
A General Formula of Polynomial Interpolation
Authors: Aitken A. C.
Published: January 2009

The operations of differentiation, ordinary and central differencing, divided differencing, and forming linear combinations of the results of these, have in common the distributive, associative and commutative properties, and the further property that when performed on a general polynomial of the nth degree they produce a polynomial of the n−1th degree, while when performed on a constant they yield zero.
Remarks on the Algebra of the 4-nodal Cubic Surface
Authors: Saddler W.
Published: January 2009

The derivation of the 4-nodal Cubic Surface and the Quartic Surface, of which it is a particular case, are well known: certain new results of interest from the point of view of symmetry, and extension to n-fold space, are provided by the symbolic algebra. In particular a simple proof is given, in § 2, of the theorem that a symmetry exists between the four vertices of the tetrahedron and the fifth point whose locus is the cubic surface, and this property can be extended to the case of n + 1 points in n-fold space with one additional point.
Interpolation by means of Given Differences (Central-Difference Notation)
Authors: Sheppard W. F.
Published: January 2009

Formulae of interpolation in terms of given central differences might be regarded as falling into two groups, A and B. In group A, the simplest cases are those in which each given difference is one of the two which in the difference table lie nearest to the preceding given difference; the differences are all natural differences (i.e., are not mean differences), and are all expressed in the centraldifference notation. Any such formula can be a central-difference formula for a certain range of the variable: but that is a matter with which we are only incidentally concerned. What I have to do is to examine the formula as determined by the series of differences given. I have then to see how the formula is affected when an ordinary difference is replaced by a mean difference. This brings us to group B, which comprises two formulae only: the Newton-Stirling formula, which expresses the required quantity in terms of a tabulated value and its central differences; and the Newton-Bessel formula, which expresses it in terms of the mean of two tabulated values and the central differences of this mean.
On Appell's Function P (θ, φ)
Authors: Humbert Pierre
Published: January 2009

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansion

where j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as
On Small Deformation of Sub-Spaces of a Flat Space
Authors: Walker A. G.
Published: January 2009

The object of this paper is to introduce the differential operator, ▽, generalised for a Riemannian space Vn immersed in a flat space Vp, and then to discuss the general small deformation of Vn.
On Functions of Integrals
Authors: Gillespie R. P.
Published: January 2009

The object of this paper is to discuss conditions for a maximum or minimum of functions of integrals of the type

employing the methods of the Calculus of Variations.
Certain Expansions in the Algebra of Quantum Mechanics
Authors: McCoy Neal H.
Published: January 2009

§ 1. Introduction. The algebra of quantum mechanics is characterized by the fact that the variables p, q obey all the laws of ordinary algebra except that multiplication is non-commutative and instead there exists a relation of the form

where c is a real or complex scalar constant and is thus commutative with both p and q.
On the Composition of Simultaneous Differential Systems of the First Order
Authors: Mursi-Ahmed M.
Published: January 2009

§ 1. Consider the system of n first order linear differential equations:

together with the n boundary conditions

where aij, bij are constants and where we assume for simplicity of notation that gii = 0.
On the Canonical Form of a Rational Integral Function of a Matrix
Authors: Rutherford D. E.
Published: January 2009

It is well known that the square matrix, of rank n−k + 1,

which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.
A Theorem on the Complete Integral
Authors: Daniels H. E.
Published: January 2009

The object of the present note is to show that a well-known theorem in the theory of non-linear partial differential equations, which is usually proved analytically, admits of a geometrical proof which exhibits the relations concerned in a more intuitive manner.
A Generalisation of Jacobi's Fundamental Formulae
Authors: Rushforth J. M.
Published: January 2009

1. Jacobi obtained his well-known formulae by a purely algebraic method, but it was not until H. J. S. Smith had obtained them similarly, but by the use of a more symmetrical notation, that they were put into the form by which they are known today.
Interpolated Derivatives
Authors: Spain B.
Published: January 2009

In a previous paper [Spain, Proc. Roy. Soc. Edinburgh, Vol. LX (1940), 134], I have shown that the application of the cardinal function to the problem of interpolating the derivatives yields the result

This formula is valid for x > a (the constant of integration), and R(n) < 0. The analytical continuation for R(n) ≥ 0. is indicated in the paper just quoted. The first term is the familiar expression for a fractional derivative, but the second term is not Riemann's complementary function. Furthermore, this result is unsatisfactory because it is impossible to perform the repeated operation of a fractional derivative of a fractional derivative.