Marshall Hall Research Articles

Sales Revenue Maximization--A Note

IN the April I967 Journal of Industrial Economics, Marshall Hall tests the hypothesis that a firm maximizes sales revenue subject to its profit rate being above some critical level.' The hypothesis can be interpreted as saying that when profits are above (below) this minimal acceptable level, the firm will consciously attempt to increase (decrease) sales revenue. The test that Hall runs is elegant in its simplicity. If the effects on output decisions of all the factors other than profits are netted out, and if the change in profits is measured from a minimal profit constraint, then the hypothesis is verified by a positive relation between sales and the change in profits (cxij>to).2 The results of the test would appear to disprove the sales revenue maximization hypothesis. The axj, coefficients are not consistently positive. In fact, the coefficients alternate in sign as the lag increases. It is this author's contention that the test results are ambiguous for three reasons. Hall's procedure does not remove the effects on sales revenue of exogenous variables other than profits. It therefore

Marshall hall on crush injury and traumatic shock (1831)

Marshall hall on crush injury and traumatic shock (1831)

Historische Bemerkungen zum sogenannten Babinskischen Phänomen

A study of the neurological literature, particularly that of the 19th century, discloses that Babinski's credit for the phenomenon which carries his name, and on which he first reported on 22nd February 1896, is exaggerated when not only the introduction of this phenomenon to neurological diagnosis but also the priority of its discovery is attributed to him. This has been pointed out as early as 1952 in the second edition of Wartenberg's monograph Die Untersuchung der Reflexe. Prior to Babinski, investigators such as Marshall Hall (1842), R. D. Grainger (1837), A. Strümpell (1881), C. Westphal (1874) and E. Remak (1893) had described this sign, thus exerting a decisive influence on the entire further development. Strümpell (not Babinski, as is usually assumed) was the first to recognize the correlation between the occurrence of this phenomenon and a lesion of the pyramidal tract. Unlike current investigators, Babinski primarily sought the special diagnostic value of the sign in the differentiation between an organic neural disorder and a psychogenic disorder of behaviour. As component of a phylogenetically older programme of innervation, the sign is regarded by Walshe (1947) as a release phenomenon which ensures the individual's horizontal or vertical flight ( e.g. Wartenberg 1952). The phenomenon (component of a so-called mass flexor reflex) has of old been known to physicians as a pathognomonic sign, e.g. of diseases associated with hemiplegia. The so-called Babinski phenomenon can be expected, not only in the presence of a pyramidal tract lesion in the stricter sense, but in all cases in which, independent of certain pathways of transmission, there is within the cerebrospinal functional unit a disturbance of corticofugal information and, therefore, dissociation of the cerebrospinal ( i.e. disinhibition of the spinal) functional circuits.

A characterization of free projective planes

In a fundamental paper Marshall Hall defines a free plane to be a projective plane which either is degenerate or is generated as follows from a 'basis configuration/ ττ0, consisting of at least two points on a line together with two isolated points. For each pair of points not already joined in π0 create a distinct line that joins them and add it to 7Γ0. In the resulting configuration, πlf consider pairs of lines that do not intersect, and for each create a distinct intersection point and add it to πu thus forming π2. Continuing, construct 7Γ3, τr4, πδ, π6, etc. adding alternately lines and points as indicated above. Then π = U*^* (with the obvious incidence relation) is a projective plane. It is bydefinition a free plane. Hall proved that a free plane contains no confined configuration, that is, no finite configuration that, like the Desargues configuration, has ^ 3 points on each line and ^ 3 lines through each point. Further, using a complicated argument, he showed that, if a finitely generated plane contains no confined configuration, it is free. It follows that any finitely generated subplane of a free plane is free. L. I. Kopejkina [2] proved, shortly after, that an arbitrary subplane of a free plane is free. (Of interest is the analogy with free groups.) An exposition of Kopejkina's theorem appears in [3]. Because it suggests a more symmetrical definition of free plane that leads to very direct proofs of the above theorems, we introduce the notion of an extension process.

EPILEPTIFORM SEIZURES FROM CHRONIC ISOLATED CORTEX.

present experiments are concerned with further evidence of the epileptogenic properties of chronic partially isolated (denervated) monkey cerebral cortex. Marshall Hall 29 in 1841 remarked, The first effect of injury done to the nervous system is a diminution of its functions; whilst the second or ulterior effect is the augmentation of those functions. In 1880 Claude Bernard 3 stated, The excitability of all tissues seems to augment when they are separated from the nervous influences which dominate them. In subsequent years there followed a wide series of observations, largely on excitable peripheral structures, confirming and expanding the opinions of Hall and Bernard. In 1939 Cannon, 9 in his Hughlings Jackson lecture at the Montreal Neurological Institute, presented his Law of Denervation as follows: When in a series of efferent neurones a unit is destroyed an increased irritability to chemical agents develops in the isolated structure or structures, the effects

Marshall Hall

<h3>To the Editor:—</h3> The editorial on Marshall Hall, in the Sept 29 issue ofThe Journal(<b>181</b>:1132), presented his meritorious contributions to neurology, medicine, and physiology. We may note, too, that he made a mistake worth remembering. Mr. Ward, a surgeon, at Nottinghamshire in 1842, amputated the leg of a patient who was under hypnoanesthesia. ( He had been placed in a mesmeric trance.) When the case was reported, Marshall Hall noted that the patient did not evidence the anticipated reflex movements, and he suggested, unreasonably, that the patient was an impostor. An old adage asserts that an authority in one field is not necessarily an authority in another. This adage is often forgotten, but should be recalled more than ever today.

A New Kind of Free Extension for Projective Planes

Marshall Hall [l] shows how projective planes of very general structure may be constructed and at the same time exhibits an extensive class which are non-Desarguesian. Here we shall indicate how his method of free extension can be generalized to yield a class of planes which seem to be distinct from those which he obtains.A partial plane is a system consisting of two distinct sets of elements, a set of "points" P, Q,… and a set of "lines" l, m,…, and a relation between these two sets, called "incidence", such that for any two distinct points, there is at most one line incident with both (or, equivalently, for any two distinct lines, there is at most one point incident with both). A partial plane is complete if every two distinct points are joined by a line and every two distinct lines intersect in a point.

A natural linear ordering of basic commutators

The basic commutators derived from the generators of a free group were introduced by Philip Hall and studied extensively by Marshall Hall, Jr. There exists a natural linear ordering for these commutators. It is the purpose of this paper to show that this ordering is, in a certain sense, invariant under multiplication, i.e., under the process of forming the commutator with the same basic commutator on both sides of an inequality. We derive and state our results in the language of commutators in groups; obviously, they can be formulated also in terms of elements of a free Lie ring. Our investigation was motivated by a study of the smallest normal divisor in a free group containing a given basic commutator. We start out by giving some notation and definitions which will be used throughout this paper. G will be the free group on xi, x2, , x,. If a, bCG then (a, b) =a-1b-lab. The lower central series of G is the chain GiDG2D . . . GnDGn+lD . . . of subgroups defined by setting GC = G, Gn = (Gn-1, G), the group generated by all commutators of the form (an-1, b) with an_iCGn-i and bEG. We wish now to construct basic commutators and at the same time define a linear ordering on them. This is done by induction. The basic commutators of weight one with their linear order are x Cj and such that if C =(C8, Ce), then Cj> Ct. Let Cn,=(Cil, Cjl) and Cn2=(Ci2, Cj2) be of weight n. Then Cni > Cn2 if Cil > Ci2 or Cil = Ci2 but C,1 > Cj2. A basic commutator of weight n is greater than any of weight less than n. Thus, if r=2, let xi=x, x2=y so that the basic commutators of weight ? 3 in their order are

Book Reviews

Book Reviews

Some Aspects of Analysis and-Probability, by Irving Kaplansky, Marshall Hall Jr., Edwin Hewitt and Robert Fortet. Surveys in Applied Mathematics IV. John Wiley and Sons, New York, 1958, 243 pages. $9.00.

Some Aspects of Analysis and-Probability, by Irving Kaplansky, Marshall Hall Jr., Edwin Hewitt and Robert Fortet. Surveys in Applied Mathematics IV. John Wiley and Sons, New York, 1958, 243 pages. $9.00. - Volume 3 Issue 2

The Theory of Groups (Marshall Hall, Jr.)

The Theory of Groups (Marshall Hall, Jr.)

Difference sets in Abelian groups

1. Difference sets in finite groups have been studied at great lengtll in a paper by R. H. Bruck [I]. Marshall Hall, Jr.'s notion of a multiplier of a difference set has been extended by Bruck to difference sets in groups. The latter has also extended a theorem of Hall on multipliers [3] to abelian groups. Hall has later obtained a generalization of his theorem. The object of this paper is, in the first instance, to extend this generalized theorem of Hall on multipliers to difference sets in abelian groups. A new feature that has been introduced here is the application of the theory of characters to the discussion of the problem. By this method I have also been able to obtain a new class of different sets. The later part of the paper is concerned with the discussion of this new class. Let G be an abelian group of order v and let D be a subset of G containing k elements such that every element of G other than the identity can be expressed exactly X times in the form ab-' where a, b are elements of D. Then D is called a difference set of type (v, k, X). The numbers v, k, X satisfy the relation

Surveys in ‘Applied’ Mathematics

Some Aspects of Analysis and Probability By Irving Kaplansky, Marshall Hall Jr., Edwin Hewitt and Robert Fortet. (Surveys in Applied Mathematics, Vol. 4.) Pp. xi + 243. (New York: John Wiley and Sons, Inc.; London: Chapman and Hall, Ltd., 1958.) 72s. net.

Some Aspects of Analysis and Probability

Contributors: Irving Kaplansky, Marshall Hall, Jr., Edwin Hewitt, Robert Fortet London: Chapman and Hall Ltd. Pp. xi + 243. Price 72s. This is one of five Surveys in Applied Mathematics published as a joint project of the United States Office of Naval Research and Applied Mathematics Reviews. This project arose out of the increasing realization that there existed major areas of modern mathematics and theoretical mechanics the scope and trend of which were incompletely conveyed by the Western literature alone, quite apart from missing particular recent results.

On a result of Marshall Hall

MathematikaVolume 3, Issue 2 p. 109-110 Research Article On a result of Marshall Hall J. W. S. Cassels, J. W. S. Cassels Trinity College, CambridgeSearch for more papers by this author J. W. S. Cassels, J. W. S. Cassels Trinity College, CambridgeSearch for more papers by this author First published: 01 December 1956 https://doi.org/10.1112/S0025579300001765Citations: 13AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume3, Issue2December 1956Pages 109-110 RelatedInformation

Planar division neo-rings

Introduction. The notion of a division ring can be generalized to give a system whose addition is not necessarily associative, but which retains the property of coordinatizing an affine plane. Such a system will be called a planar division neo-ring (PDNR); examples of (infinite) PDNRs which are not division rings are known. If (R, +, *) is a finite power-associative PDNR, then (R, +) is shown to be commutative and to possess the inverse property. The center of an arbitrary PDNR, and the nucleus of a finite PDNR, are shown to be PDNRs. By means of these and similar properties it is demonstrated that all associative PDNRs of order ?250 are actually abelian. The main result is the following: if (R, +, *) is a finite associative and commutative PDNR of order n, and if p is any prime dividing n, then the mapping x-*xP is an automorphism of (R, +, .). Chiefly by means of this result, all associative and commutative PDNRs of order <250 are shown to have prime-power order. Chapter I contains results about the planar ternary rings developed by Marshall Hall [7], with a sketch of their connection with the complete sets of orthogonal latin squares associated with affine planes. Chapter II is devoted to strictly algebraic theory of PDNRs, mostly for the finite case. Chapter III contains the main theorem about automorphisms mentioned above, and examples of its application. In the Appendix will be found examples of infinite PDNRs which are not division rings. These results are from the author's doctoral dissertation at the University of Wisconsin; the author wishes to take this opportunity to express his gratitude to Professor R. H. Bruck for invaluable assistance in carrying out this research.

Note on the history of the Babinski reflex.

To the Editor:— In a previous note on the history of the Babinski reflex (A. M. A.Arch. Neurol. & Psychiat. 65: 713, 1951) it was pointed out that in 1893, three years before Babinski, a German neurologist, E. Remak reported the isolated dorsiflexion of the big toe on stroking the sole of the foot. The history of the Babinski reflex should contain a reference to the book of Marshall Hall (On the Diseases and Derangements of the Nervous System, London, H. Bailliere, 1841). There, on page 9, Hall writes: I have this day (January 3, 1841) seen... a case of the most complete hemiplegia of sensation and volition, of the left side, without loss of intellect, and with the perfect continuance of all the reflex actions: respiration, deglutition, the action of the sphincter ani, are unimpaired; and the following distinct reflex actions and phenomena were observed:... On tickling the

Finite Nets, I. Numerical Invariants

A finite net N of degree k, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line is simply a net of degree n+ 1, order n (Marshall Hall [1]). A loop of order n is essentially a net of degree 3, order n (Baer [1], Bates [1]). More generally, for , a set of k —2 mutually orthogonal n ⨯ n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose [1]) between affine planes and complete sets of orthogonal latin squares.

A Remark on a Theorem of Marshall Hall

A Remark on a Theorem of Marshall Hall

A remark on a theorem of Marshall Hall

characterizes quaternion algebras among associative but not commutative division rings. Our remark is that (1) characterizes CayleyDickson algebras among alternative but not associative division rings. This follows from Hall's proof and a result of A. A. Albert [1]. A theorem of R. D. Schafer [4, Theorem 4] permits us to conclude that (1) and Hall's Theorem L [2] (as universal) ensure that the coordinate ring of a projective plane is uniquely defined irrespective of the coordinate system. It is easy to verify (1) in a Cayley-Dickson algebra. We should like to sketch a proof of the converse, which is independent of Albert's result in [1]. Let A be an alternative division ring and let F be the set of all elements c CA which satisfy cx = xc for every x EA. Then F is a field2 and, when 3 is nonzero in A, F is the center of A [3; 5, Lemma 9]. If A satisfies (1), the proof of Hall's Lemma 1 [2, p. 262] yields x2=t(x)x-n(x) for every xzA not in F, where t(x), n(x)EF. Define t(c) = 2c, n(c) = c2 for cC F. When 3 is zero in A and A satisfies (1), F is still the center of A as we shall now show.3 Since 2 is a nonzero in A, for each xCA we have (x+c)2CF for some cCF. If, then, cl, c2 are in F, we have (cl, c2, (x+c)2)=0. By use of some identities of Zorn [6, (1.6) and (1.8); 5, (1)] we find that (Cl, C2, (X+C)2)=2(x+c)(c1, C2, x+c)=2(x+c) * (cl, c2, x). We infer that (cl, c2, x) = 0 for every cl, c2 in F and every x in A. Now let cEF and x, yEA. As before, (x+c')2CF for some c'EF and