Scottish Book Research Articles

A Problem of S. Banach (Problem 106 from the ?Scottish Book?

A Problem of S. Banach (Problem 106 from the ?Scottish Book?

A problem of Orlicz in the Scottish book

A problem of Orlicz in the Scottish book

Determinacy of Banach games

For any A ⊂ R, the Banach game B(A) is the following infinite game on reals: Players I and II alternately play positive real numbers a1; a2, a3, a4,… such that for n > 1, an < an−1. Player I wins iff ai exists and is in A.This type of game was introduced by Banach in 1935 in the Scottish Book [15], Problem 43. The (rather vague) problem which Banach posed was to characterize those sets A for which I (II) has a winning strategy in B(A). (There are three parts to Problem 43. In the first, Mazur defined a game G**(A) for every set A ⊂ R and conjectured that II has a winning strategy in G**(A) iff A is meager and I has a winning strategy in G**(A) iff A is comeager in some neighborhood; this conjecture was proved by Banach. Presumably Banach had this result in mind when he asked the question about B(A), and hoped for a similar type of characterization.) Incidentally, Problem 43 of the Scottish Book appears to be the first time infinite games of any sort were studied by mathematicians.This paper will not provide the reader with any answer to Banach's question. I know of no nontrivial way to characterize when player I (or II) wins, and I suspect there is none. This paper is concerned with a different (also rather vague) question: For which sets A is the Banach game B(A) determined? To say that B(A) is determined means, of course, that one of the players has a winning strategy for B(A).

Problem 24 of the "Scottish book'' concerning additive functionals

Problem 24 of the "Scottish book'' concerning additive functionals

A Solution to a Problem of Eilenberg Concerning Dimension Lowering Mappings

This paper describes a map $f$ from a metric space $X$ (having the same dimension at each of its points) onto a space $Y$ such that $\dim X > \dim Y > 0$ with the property that $\dim K \geqslant \dim f(K)$ for every closed set $K$ contained in $X$. This answers a question posed by Eilenberg in 1936 in The Scottish Book. This question was answered with a more complicated example by Rubin, Schori, and Walsh.

A solution to a problem of Eilenberg concerning dimension lowering mappings

This paper describes a map f f from a metric space X X (having the same dimension at each of its points) onto a space Y Y such that dim ⁡ X > dim ⁡ Y > 0 \dim X > \dim Y > 0 with the property that dim ⁡ K ⩾ dim ⁡ f ( K ) \dim K \geqslant \dim f(K) for every closed set K K contained in X X . This answers a question posed by Eilenberg in 1936 in The Scottish Book. This question was answered with a more complicated example by Rubin, Schori, and Walsh.

Macghie's The Principles of Book-keeping'—The Second Scottish Book on Accounting

Macghie's The Principles of Book-keeping'—The Second Scottish Book on Accounting

The set of continuous nowhere differentiable functions

Let C be the space of all real-valued continuous functions defined on the unit interval provided with the uniform norm. In the Scottish Book, Banach raised the question of the descriptive class of the subset D of C consisting of all functions which are differentiable at each point of [0,1]. Banach pointed out that D forms a coanalytic subset of C and asked whether D is a Borel set. Later Mazurkiewicz showed that D is not a Borel set [3]. In this paper, we shall investigate the subset M of C consisting of all functions which do not have a finite derivative at any point of [0,1]. It is well known that M is residual in C [2]. We shall prove the following theorem.

The renaissance of public worship in the church of Scotland, 1865–1905

On his way to one of the early meetings of the church service society, Story met Charteris on the street, who asked what were their aims. He replied, ‘To restore the ring in marriage, the cross in baptism, and the denial of the cup to the laity’. Charteris fled in horror.Before discussing the renaissance of public worship in the church of Scotland it is necessary to look at the background. The lineage of reformed worship can be traced from Diebold Schwarz’s translation of the Hagenau Missal into German in 1524 through Bucer and Calvin, both of whom desired weekly communion, to the Scottish Book of Common Order (1564). When the civil authorities forbade weekly communion, Bucer and Calvin did not prepare an order of service for Sunday morning, but rubricated the order for communion as to how it should end when there was no celebration of the supper: that is, the eucharist was the norm for public worship.

On the Sum of Differentiable Functions

Ryll-Nardzewski has proposed the following problem (New Scottish Book, no. 119). If fn(x) are continuous, differentiable* functions in a closed finite interval, do there always exist constants cn (no cn = 0) (depending on the ), such that converges and is also a continuous differentiable function?

On a problem of Steinhaus about polygons

We solve a problem of Steinhaus (New Scottish Book, Problem 328; also Colloq. Math. 5 (1958), Problem 225); if p is a prime, does there exist a regular p-gon such that three of its diagonals cut in a point? The answer is in the negative, except in the case where the said diagonals are all parallel, which can clearly occur for any p ≥ 7.

A generalization of the Banach and Mazur game

satisfying (1.1) and such that s-= Z=0 x,eS whenever X2n, (n=O 1,2, ***) satisfy (1.2). If, on the other hand, B has a winning strategy on S, we say that S is avoidable (B can avoid S). In the sequel we shall also consider a game L{ 1km.} defined as a game G { km} satisfying the additional condition x0 < 1. C{ km} will be played exclusively on bounded sets SoC [0, I1. 12. Historical notes. Various variants of the game of Banach and Mazur are described in the so-called Scottish Book (see Colloq. Math. vol. 1 (1947) p. 57). One of them, which was defined by S. Mazur and later modified by S. Banach, is a special case of our game for km= 1, (m= 1, 2, ... ). This case was first considered by A. Turowicz [4], who proved that the set of all irrational numbers is unavoidable. His result was later generalized by S. Zubrzycki [5], who has shown that the complement of any countable set is unavoidable. A sufficient condition for avoidability of sets was given by S. Hartman [2 ]. Further, M. Reichbach [3 ] has constructed (for km = 1) a perfect unavoidable set of measure 0 and thus has given an answer to a question put by H. Steinhaus, concerning the existence of unavoidable sets of the first category.

The Prayer Book in Scotland, 1549–1949. By Gordon Donaldson. David Winter and Son, Dundee, 1949. pp. ii + 30. IS. - The Scottish Book of Common Prayer, 1929. Notes on its origin and growth, with illustrations from original documents. By Alan Campbell Don S.P.C.K. (1949). pp. 52. 6s. 6d.

The Prayer Book in Scotland, 1549–1949. By Gordon Donaldson. David Winter and Son, Dundee, 1949. pp. ii + 30. IS. - The Scottish Book of Common Prayer, 1929. Notes on its origin and growth, with illustrations from original documents. By Alan Campbell Don S.P.C.K. (1949). pp. 52. 6s. 6d. - Volume 2 Issue 4

The Scottish Book of Common Prayer

The Scottish Book of Common Prayer

Book Review: The Scottish Book of Common Prayer

Book Review: The Scottish Book of Common Prayer

The Scottish Book Clubs

THE history of the Scottish Book Clubs has still to be written. That no one has yet ventured upon it is partly due to the vastness of the subject, partly to the fact that in these degenerate times the student of history or literature is inclined to take his blessings for granted. Happily, the story is unfinished; for today the modern successors of the Bannatyne and Maitland go on from strength to strength. The Scottish History Society is now sixty years old, but as virile as in the days of its youth. The Scottish Text Society, which has given us monumental editions of Dunbar, Wyntoun, and Pitscottie, to name only a few of its triumphs, continues to maintain its high standards of editorship and production. In the North‐East, the Third Spalding Club proclaims to the world at large that Aberdeen has wealth still of scholars in the Joseph Robertson — Hill Burton tradition. In recent years one new name has been added to the roll of the printing fraternities. Some ten years ago, the Stair Society, established for the purpose of “encouraging the study, and advancing the knowledge of” Scots Law, began its labours. Though young, it has demonstrated beyond all doubt that it is worthy to rank with the great Clubs I have mentioned. True, its scope at first sight may appear limited: and those who have the layman's undefined distrust of legal affairs may pass with averted eye. The noble volumes of the Stair, however, bear eloquent witness to the close relationship of law and history, and afford illustrations of bygone life in Scotland which we would be sorry to lose. An extensive and distinguished membership, with regular publications of value, confirms that the work begun by men like David Laing and Cosmo Innes is still being carried on,—and right worthily too.

The psalter in the scottish prayer book

The psalter in the scottish prayer book

Scottish book of common prayer in 1662

Scottish book of common prayer in 1662