bluuebunny/crossref_metadata_2025_split huggingface dataset:
Integrals allied to Airy's integrals
Authors: Macrobert T. M. 
Published: May 2009

Airy's integralscan be expressed ([1], [2]) in terms of Bessel functions. In this paper integrals of the typesare discussed. Various subsidiary formulae are given in § 2, some integrals of the typeare evaluated in § 3, and from these the integrals of the Airy type are derived in § 4.
A note on Mathieu functions
Authors: Bell M. 
Published: May 2009

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equationThe eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce toaN = bN = N2when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namelySuppose first that N is an odd integer. Then there is an expansionwhereThese functions π satisfyandOn Substituting (3) in (1), one obtains the algebraic equationwhereExplicitly,{11} = q{lm} = 0 otherwise.
On rearrangements of infinite series
Authors: Robertson A. P. 
Published: May 2009

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.(i) What is the condition on a given series for every rearrangement to converge?(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).
Integrals involving hypergeometric functions and E-functions
Authors: Macrobert T. M. 
Published: May 2009

In § 2 a number of integrals in which the integrand contains a product of a hypergeometric function and an E-function will be evaluated. The following formulae will be employed in the proofs.If ρ +σ = α + β + γ + 1, and if α, β or γ is zero or a negative integer,this is Sallschütz's theorem [1].If R(γ - ½α - ½β)< - ½,This theorem was given by Wastson [2] for negative integral values of α and later by Whipple [3] for general values α.R(γ) > 0,This formula was given by Whipple [3]l is a positive integer,
A Note on Some Inequalities
Authors: Flett T. M. 
Published: May 2009

In the course of some recent work on Fourier series [5, 6] I had occasion to use a number of integral inequalities which were generalizations or limiting cases of known results. These inequalities may perhaps have other applications, and it seems worth while to collect them together in a separate note with one or two further results of a similar nature.For any number k, used as an index (exponent), and such that K > 1, we write k' = k′(k–1), so that k and k′ are conjugate indices in the sense of Hölder's inequality.
The Packing of Spheres in the Space lp
Authors: Burlak Jane A. C. , Rankin R. A. , Robertson A. P. 
Published: May 2009

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we writeThe unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if¶ y- z ¶≥ 2aThe statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.
Multiplication of Strongly Summable Series
Authors: Boyd A. V. 
Published: May 2009

Given a series we define , by the relationsThe series is said to be summable (C, k), where k > - 1, to the sum s ifto be summable (C, - 1) to s if it converges to s and nan = o(l); to be absolutely summable (C, k), or summable | C, k, to s if it is summable (C, k) to s andand to be strongly Cesàro summable to s with order k > 0 and index p or summable [C; k, p] to s, if
Invariant Measures on Coset Spaces
Authors: Świerczkowski S. 
Published: May 2009

In this note we consider measures on a left coset space G/H, where G is a locally compact group and H is a closed subgroup. We assume the natural topology in G/H and we denote the generic element of this space by xH (x∈G). Every element t∈G defines a homeomorphism of G/H given by t(xH) = (tx)H. A. Weil showed that a Baire measure on G/H invariant under all these homeomorphisms can exist only ifΔ(ξ) = δ(ξ) for each ξ ∈ H,where Δ(x), δ(ξ) denote the modular functions in G, H [6, pp. 42–45]. We shall devote our investigations to inherited measures on G/H (cf. [3] and the definition below) invariant under homeomorphisms belonging to a normal and closed subgroup T ⊂ G.
On P. Hall's Generalisation of a Theorem of Frobenius
Authors: Sehgal S. K. 
Published: May 2009

1. It is a well known theorem due to Frobenius that the number of solutions of the equationXn = 1in a finite group G, is a multiple of the greatest common divisor (n, g) of n and the order g of G. Frobenius himself proved later that the number of solutions of the equationXn = awhere a is a fixed element of G, is a multiple of (n, ga), ga being the order of the centralizer Z(a) of ain G.
A Relation between Laplace and Hankel Transforms
Authors: Bhonsle B. R. 
Published: May 2009

The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equationand its Hankel transform of order v is defined by the equationThe object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.
Dual integral equations with trigonometrical kernels
Authors: Sneddon Ian N. 
Published: May 2009

In the analysis of mixed boundary value problems in the plane, we encounter dual integral equations of the typeIf we make the substitutions cos we obtain a pair of dual integral equations of the Titchmarsh type [1, p. 334] with α = − 1, v = − ½ (in Titchmarsh's notation). This is a particular case which is not covered by Busbridge's general solution [2], so that special methods have to be derived for the solution.
A characterization of inaccessible cardinals
Authors: Preston G. B. 
Published: May 2009

A cardinal number which is too large to be reached by some process is generally said to be inaccessible by that process. Many kinds of inaccessible cardinals have been discussed and for a general survey the book of H. Bachmann [1, Chapter 7] may be consulted. We consider here two inaccessibility properties. We shall denote the cardinal of a set X by |X|. The first inaccessibility property will be called regularity: the cardinal| X| will be said to be regular if there does not exist a disjoint cover {X1: i ε I} of X such that(i)|X1|<|X|, for each i in I, and(ii)|I|<|X|.
An embedding theorem with amalgamation for cancellative semigroups
Authors: Howie J. M. 
Published: May 2009

Let {Si; i ε I} be a finite or infinite family of cancellative semigroups. Let U be a cancellative semigroup, and suppose that there exists, for each i in I, a monomorphism φi: u→ Si. We are interested in finding a semigroup T with the following properties.(a) For each i in I, there is a monomorphism λi: Si → T such that uφiλi = uøjλi for all u ɛ U and all i, j in I. That is to say, there exists a monomorphism λ: U → T which equals øiλi for all i in I.Siλi∩Sjλj = Uλ (i, j ε I; i ≠ j).
An arithmetic characterization of the parabolic points of G(2cos π/5)
Authors: Rosen David 
Published: May 2009

By the group G(2 cos π/q) we mean the group of linear fractional transformations of the complex plane onto itself, generated by V(z)= — 1/z and U(z) = z+λq, where λq = 2 cos (π/q), qbeing a positive integer greater than 2. In this paper we shall be concerned only with the group given by q = 5, and we shall therefore omit the subscript 5 on the λ. We note that λ = λ5 satisfies the equationx2–x–l=0;(1)henceλ = (l + 5½)/2.
On the asymptotic expansion of Airy's Integral
Authors: Copson E. T. 
Published: May 2009

The integral functionis known as Airy's Integral since, when z is real, it is equal to the integralwhich first arose in Airy's researches on optics. It is readily seen that w= Ai(z) satisfies the differential equation d2w/dz2 = zw, an equation which also has solutions Ai(ωz), Ai(ω2z), where ω is the complex cube root of unity, exp 2/3πi. The three solutions are connected by the relation.
A lemma about the Epstein zeta-function
Authors: Ennola Veikko 
Published: May 2009

Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shapewith y > 0. We write (for s > 1)The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.
Generalizations of a well-known result in matrix theory
Authors: Thompson R. C. 
Published: May 2009

Let A and C be m × m matrices and let B and D be n × n matrices, all with elements in a field F. Let AT denote the transpose of A. A well-known theorem states that, if every m × m matrix X for which AX = XA also satisfies CX = XC, then C = φ(A) for some polynomial φ(λ). In this note we establish the following simple generalizations.Theorem 1. Let A and B have the same minimal polynomial m(λ). If each m × n matrix X over F for which AX = XB also satisfies CX = XD, then C = φ(A) and D = φ(B) for a polynomial φ(λ) over F.
An extension of a theorem of Gordon
Authors: Wright E. M. 
Published: May 2009

In what follows all small Latin letters denote non-negative integers or functions whose values are non-negative integers. Let N = (n1, …, nj) be a j-dimensional vector and let q = q (k; N) = q(k; n1, …, nj) be the number of partitions of N into just k parts, each part being a vector whose components are non-negative integers. We writefor the generating function of q. We have.
A Note on the Hardy-Hille and Mehler formulas
Authors: Al-Salam W. A. , Carlitz L. 
Published: May 2009

Let and Hn(x) be the nth Laguerre and Hermite polynomials, respectively. Two well-known bilinear generating formulas are the Hardy-Hille formula [1, p. 101]and the mehler formula [1, p. 377]This suggests the following problem. Consider the equationwhere fa(x) is a polynomial in x of degree n with highest coefficient equal to 1,A0 = B0 = 1. We shall also assume that ak = 1 and y0y1y2 … yk–1 ≠ 0. We seek all sets of polynomials {fn(x)} which satisfy (1.3), (1.4) and (1.5).
Affine semigroups over an arbitrary field
Authors: Clark W. Edwin 
Published: May 2009

Let ℒ V denote the algebra of all linear transformations on an n-dimensional vector space V over a field Φ. A subsemigroup S of the multiplicative semigroup of ℒ V will be said to be an affine semigroup over Φ if S is a linear variety, i.e., a translate of a linear subspace of ℒ V.This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.
Integrals involving E-functions
Authors: Ragab F. M. , Simary M. A. 
Published: May 2009

In this paper two integrals involving E-functions are evaluated in terms of E-functions. The formulae to be established are:where n is a positive integer, andwhere n is a positive integer, andthe prime and the asterisk denoting that the factor sin {(s–s)π/2n} and the parameter βq+s–βq+s + 1 are omitted. The definitions and properties of MacRobert's E-function can be found in [1, pp. 348–352] and [3, pp. 203–206].
Congruences on a bisimple ω-semigroup
Authors: Munn W. D. , Reilly N. R. 
Published: May 2009

In a semigroup S the set E of idempotents is partially ordered by the rule that e≦ƒ if and only if eƒ=e=ƒe. We say that S is an ω-semigroup if E={ei: i=0, 1, 2, …}, whereBisimple ω-semigroups have been classified in [10]. From a group G and an endomorphism α of G a bisimple ω-semigroup S(G, α) can be constructed by a process described below in § 1: moreover, any bisimple ω-semigroup is isomorphic to one of this type.
A remark on the distributive law for an ideal in a commutative ring
Authors: Jensen C. U. 
Published: May 2009

Let R be a commutative ring, with an identity element. It is the purpose of this note to establish conditions for an arbitrary but fixed ideal a of R to satisfy the distributive lawfor all ideals b and c of R. In particular, in the Noetherian case, this will be related to the decomposition of a into prime ideals. We start withProposition 1. For a fixed ideal a in a commutative ring R with an identity element, the following conditions are equivalent.
An Infinite Integral involving a Product of Two Modified Bessel Functions of the Second Kind
Authors: Macrobert T. M. 
Published: May 2009

The formula to be established iswhere l, m, n. are any numbers real or complex and R(b)>0. A similar result, involving Bessel Functions of the First Kind, was obtained by Hanumanta Rao [Mess, of Maths., XLVII. (1918), pp. 134–137].
Integrals of E-Functions expressed in Terms of E-Functions
Authors: Ragab F. M. 
Published: May 2009

In § 2 the following two formulae will be established.If, when p≧q + l, R(ar + k)>0, r = 1, 2,…, p and | ampz |<π,If p≧q, the result holds if the integral is convergent.
A novel antibiotic, linezolid, reduces intraperitoneal adhesion formation in the rat uterine horn model
Authors: AYTAN HAKAN , CALISKAN AHMET C. , YENER TAMER , DEMIRTURK FAZLI , AYTAN PELIN , YENISEHIRLI AYDAN 
Published: May 2009

AbstractObjective. To examine the effects of linezolid in prevention of adhesion formation in a rat uterine horn model. Design. Prospective randomized study. Setting. University Experimental Animal Research Laboratory Center. Sample. Ninety female Wistar albino rats. Methods. A dose response study was first conducted with 60 Wistar albino rats that were randomly assigned to six equal groups by administering vehicle (control), 5 mg/kg (Group 1), 15 mg/kg (Group 2), 50 mg/kg (Group 3), 100 mg/kg (Group 4), and 150 mg/kg (Group 5) linezolid starting three days before the adhesion inducing operation and continuing for 14 days postoperatively. Adhesion was scored clinically with a scoring system. The minimum effective dose was found to be 100 mg/kg/day. With this dose time response (starting three days before the operation and continuing for seven days), only preoperative and postoperative administration studies were conducted. Main outcome measures. Extent and severity of adhesions. Results. Total adhesion scores in the control and Groups 1 and 2 were significantly higher when compared with Groups 4 and 5, but not with Group 3. There were no significant differences in the adhesion scores between Groups 3, 4, and 5. In time response arm a total of 10 days treatment was not as effective as 17 days treatment. In postoperative and preoperative arms of the study, it was found that administration of linezolid only postoperatively or preoperatively did not affect adhesion formation significantly when compared with the control group. Conclusion. Linezolid was found to reduce intraperitoneal adhesion formation.
Comparison of aortic valve area measured by magnetic resonance imaging and dual-source computed tomography
Authors: Bruder O. , Jochims M. , Hunold P. , Jensen C. , Forsting M. , Sabin G.V. , Barkhausen J. , Schlosser T. 
Published: May 2009

Background: Aortic valve stenosis is the most common type of valve lesion in Europe and North America. Patient treatment is based on disease severity, which is classified by determining the aortic valve area (AVA). Purpose: To compare dual-source computed tomography (DSCT) with magnetic resonance (MR) imaging for quantifying AVA. Material and Methods: Thirty-two patients, 28 with normal aortic valve function and four with aortic valve stenosis, who underwent DSCT coronary angiography (Somatom Definition; Siemens, Erlangen, Germany), were included in this study. Retrospective ECG-gated contrast-enhanced DSCT scans with dose-reducing tube current modulation were performed, and data sets were reconstructed in 3% steps of the R–R interval (slice thickness 0.75 mm, increment 0.6 mm). Planimetry of the AVA in systole was assessed on cross-sectional images by multiplanar reformations. Within 48 hours, MR was performed with a 1.5T scanner (Magnetom Sonata; Siemens, Erlangen, Germany) using a balanced steady-state free-precession cine sequence (repetition/echo time 3/1.5 ms, flip angle 60°, spatial resolution 1.4x1.4 mm2). Cine sequences of the left ventricular outflow tract (LVOT) were obtained in two orthogonal planes, and MR planimetry was performed on cross-sectional images of the aortic valve perpendicular to the LVOT images. Results: AVA assessment by DSCT and MR was feasible in all 32 patients. Mean AVA values determined by DSCT and MR were 4.73±1.5 cm2 and 4.69±1.4 cm2, respectively. A strong positive correlation was found between both imaging modalities ( R=0.98, P<0.001). Bland-Altman analysis demonstrated an excellent intermodality agreement, with a slight underestimation of AVA by DSCT. The mean difference was -0.04 cm2, with a standard deviation of 0.32 cm2. Conclusion: Retrospective ECG-gated contrast-enhanced DSCT with tube current modulation provides an accurate imaging technique for the assessment of the AVA. Further studies are required to determine whether DSCT also allows for AVA quantification in patients with aortic valve stenosis.
The Cognitive Neuroscience of Memory
Authors: Eichenbaum Howard 
Published: May 2009

AbstractThis book details the brain's remarkable capacity for memory. The book is organized into sections corresponding to its four major themes: Connection considers how memory is based on alterations to the communication between nerve cells. Cognition discusses the fundamental psychological structure of memory. Compartmentalization involved the notion that the different forms of memory are accomplished by distinct brain systems. Consolidation refers to processes by which memories are transformed from a labile trace into a permanent store. The book provides insights into how memory works and how it is fundamental to all aspects of cognition, behavior, and emotion.
Amnesia—Learning about Memory from Memory Loss
Authors: Eichenbaum Howard 
Published: May 2009

AbstractThis chapter provides a detailed overview of a patient called H.M., in order to provide a closer perspective on the nature of his amnesia. It explores the distinction between declarative and procedural memory using several examples from the experimental literature on amnesia. H.M. had been severely epileptic for several years. In an effort to alleviate his disorder, the medial temporal lobe area was removed, and the surgery did reduce the frequency of his seizures considerably. However, following the surgery this patient became severely amnesic, and yet showed hardly any other neurological deficits.
Two Distinct Stages of Memory Consolidation
Authors: Eichenbaum Howard 
Published: May 2009

AbstractThis chapter considers two aspects of memory consolidation, one that involves a fixation of memory within synapses over a period of minutes or hours, and another that involves a reorganization of memories that occurs over weeks to years. Topics discussed include cellular events that mediate memory fixation, modulation of memory fixation, influencing memory by hormonal activation associated with emotional arousal, the role of the amygdala in enhancing memory in humans, and brain systems that mediate prolonged processes in memory reorganization.
Evolution
Authors: Iwaniuk Andrew N. 
Published: May 2009

AbstractThis chapter discusses the evolution of the laboratory rat from the origins of rodents in general to the speciation of the genus Rattus. It provides a summary of the evolutionary events leading to R. norvegicus. Because palaeontology, taxonomy, and phylogenetics are intimately related to one another, the chapter is organized in terms of the taxonomy of R. norvegicus. Evolutionary relationships and palaeontological history are discussed with reference to other groups of the same taxonomic rank. For example, the order Rodentia is placed in the context of other mammalian orders. Summarizing the evolutionary history of R. norvegicus provides a basic understanding of how the species has evolved that may be instructive in interpreting the results of behavioral experimentation and/or comparative analyses.
Somatosensation
Authors: Hermer-Vazquez Linda , Hermer-Vazquez Raymond , Chapin John K. 
Published: May 2009

AbstractThis chapter discusses five principles for how sensorimotor behaviors are learned and performed by rats, based on new findings from neuroscience. It focuses on nonvibrissal somatosensory processing but also considers examples from the whisker-tactile system or from other sensory modalities. These principles are: (i) analyzed somatosensory feedback information is constantly influencing the ascending somatosensory data stream in rats; (ii) rats are constantly evaluating information across multiple timescales to more accurately predict what will happen in their world; (iii) information from multiple spatial scales is processed simultaneously in the rat; (iv) rat sensory and motor processing are constantly influencing one another; and (v) rat behaviors appear to be organized into survival-related repertoires that can be adapted to novel circumstances.
Immune System
Authors: Anisman Hymie , Kusnecov Alexander W. 
Published: May 2009

AbstractThis chapter presents a brief overview of the functioning of the immune system, followed by a description of how immune alterations can affect central nervous system (CNS) processes and behavior in the rat. It also describes how factors that have an impact on psychological processes, most notably stressors, may come to affect immune functioning. In so doing, it introduces numerous caveats concerning the conditions and limitations that determine the nature of the effects observed, making it clear that disentangling the impact of various manipulations on immune activity is complex.
Maternal Behavior
Authors: Rees Stephanie L. , Lovic Vedran , Fleming Alison S. 
Published: May 2009

AbstractThis chapter describes maternal behavior of the laboratory rat and outlines various methods of observing and quantifying this behavior. Although in some rodent biparental species males also show parental behavior, this is not the case for most rodents, including R. norvegicus. However, under certain experimental conditions, males also show many of the components of behavior normally shown by the mother rat. The general and specific methods for the testing of maternal behavior are described. Several environmental and situational factors that affect the expression of maternal behavior are considered.
Social Learning
Authors: Galef, Jr. Bennett G. 
Published: May 2009

AbstractThis chapter begins with a brief review of the literature on social influences on food choices of Norway rats. It describes fieldwork strongly suggesting that interactions between adult free-living rats and their young can determine which foods the young come to eat. It then describes several behavioral processes that have been shown in the laboratory to be sufficient to influence food choice in young rats. Finally, it discusses a type of social influence on rats' food preferences that has already proved to be useful in studies of the physical substrates of learning and memory.