scholarly journals Birational transformations with isolated fundamental points

1938 ◽  
Vol 5 (3) ◽  
pp. 117-124 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Algebraic Surface ◽  
Canonical System ◽  
The Other ◽  
Canonical Systems ◽  
Birational Transformations ◽  
Birational Transformation ◽  
Image Position ◽  
Simple Points

It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, thenwhere E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.

On the canonical systems of certain algebraic surfaces

1937 ◽  
Vol 33 (3) ◽  
pp. 311-314
Author(s):  
D. Pedoe
Keyword(s):  
Algebraic Surface ◽  
Base Point ◽  
Canonical System ◽  
Algebraic Surfaces ◽  
Simple Point ◽  
Canonical Systems ◽  
Base Points ◽  
Complete Linear System ◽  
Pass Through ◽  
Simple Points

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

Birational transformations possessing fundamental curves

1938 ◽  
Vol 34 (2) ◽  
pp. 144-155 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Algebraic Variety ◽  
General Character ◽  
Topological Methods ◽  
Canonical Systems ◽  
Birational Transformations ◽  
Birational Transformation ◽  
Relative Invariants

On any algebraic variety of d dimensions there exist certain systems of equivalence which are relative invariants under birational transformation. These systems include the canonical systems (Todd(3,4)) which can be defined for each dimension from zero to d − 1, and the systems defined by the intersections of these canonical systems among themselves. It is natural to enquire what is the precise behaviour of these invariant systems under birational transformations. Relatively few results of this kind are known. For threefolds, Segre has recently (2) investigated the transformations undergone by the invariant systems in any algebraic (not necessarily birational) transformation whose fundamental points and curves are of general character. More recently, A. Bassi (1) has obtained by topological methods the relations between the Zeuthen-Segre invariants of two Vd in an (α, α′) correspondence. I have recently (6) discussed the transformation of the invariant systems on a Vd for birational transformations on the assumption that the fundamental points are isolated and of general character.

On a group of 1440 birational transformations of four variables that arises in considering the projective equivalence of double sixes

1926 ◽  
Vol 23 (2) ◽  
pp. 103-108
Author(s):  
W. Burnside
Keyword(s):  
The Other ◽  
Birational Transformations ◽  
Actual Position ◽  
Projective Equivalence ◽  
Image Position ◽  
Usual Notation

The lines of a double-six will here be represented by the usual notationwhere two lines whose symbols are in the same line or same column of this scheme are non-intersectors and all other pairs of lines intersect. Any six of the lines, no two of whose symbols are in the same column, and just three are in the same row, are generators of a quadric, and the actual position in space of each of the other six is determined by the two points in which it intersects this quadric.

scholarly journals On the stability of linear canonical systems with periodic coefficients

1965 ◽  
Vol 5 (2) ◽  
pp. 169-195 ◽  
Author(s):  
W. A. Coppel ◽  
A. Howe
Keyword(s):  
Differential Equations ◽  
Hamiltonian Systems ◽  
Hermitian Matrix ◽  
Coefficient Matrix ◽  
Canonical System ◽  
Linear Differential Equations ◽  
Canonical Systems ◽  
Linear Differential ◽  
The Stability ◽  
Image Position

By a linear canonical system we mean a system of linear differential equations of the formwhereJis an invertible skew-Hermitian matrix andH(t) is a continuous Hermitian matrix valued function. We reserve the name Hami1tonia for real canonical systems withwhereIkdenotes thek×kunit matrix. In recent years the stability properties of Hamiltonian systems whose coefficient matrixH(t) is periodic have been deeply investigated, mainly by Russian authors ([2], [3], [5], [7]). An excellent survey of the literature is given in [6]. The purpose of the present paper is to extend this theory to canonical systems. The only work which we know of in this direction is a paper by Yakubovič [9].

Canonical systems on a reducible variety

1939 ◽  
Vol 35 (3) ◽  
pp. 389-393 ◽  
Author(s):  
R. E. Macpherson
Keyword(s):  
Linear System ◽  
Canonical System ◽  
General Linear ◽  
Singular Variety ◽  
Canonical Systems ◽  
Image Position

The object of this note is to extend the proof, for the general canonical varieties of any dimension k < d on a Vd whose existence was established by Dr Todd in a recent paper, of the property of adjunction given by the relationwhere S is any non-singular variety of dimension d−1 lying on Vd, and Xk[S] denotes the canonical system of dimension k on S. This formula was proved by Todd in the case when Vd is a general non-singular variety, and S is non-singular and belongs to a sufficiently general linear system. Using this result, I show how to remove the latter restriction. (1·1) cannot be taken, without further proof, as defining canonical systems on virtual or isolated varieties, as is done by B. Segre, since the canonical systems on an effective isolated variety are already well defined.

The fixed part of the canonical system on an algebraic surface

1938 ◽  
Vol 34 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Patrick Du Val
Keyword(s):  
Algebraic Surface ◽  
Canonical System ◽  
Rational Curve ◽  
Variable Part ◽  
Exceptional Curve ◽  
Base Points ◽  
Fixed Part ◽  
The Given ◽  
Simple Points

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

Base conditions and covariant systems in an algebraic threefold

1939 ◽  
Vol 35 (2) ◽  
pp. 166-179
Author(s):  
J. G. Semple
Keyword(s):  
Linear System ◽  
Canonical System ◽  
Canonical Systems ◽  
Adjoint Systems ◽  
Canonical Curves ◽  
Image Position

If F is a free linear system of surfaces in an algebraic threefold V which is either non-singular or possesses only normal singularities, then F has Jacobian and adjoint surfaces, J2(F) and A2(F), and Jacobian and adjoint curve systems, J1(F) and A1(F), such thatwhere X2, X2 are the canonical systems of surfaces and curves on V, and X1(F) is the canonical system of curves of F. The imposition of base elements (points or curves) Ei, of assigned multiplicities λi, on F defines a system F1 which we may represent formally by the equationand it is natural to enquire how the Jacobian systems of F1 differ from those of F, and how we may define adjoint systems A2(F1) and A1(F1) which cut on F1 its canonical curves and sets respectively.

scholarly journals The Dissociation of the Compound of Iodine and Thio-urea

1904 ◽  
Vol 24 ◽  
pp. 233-239 ◽  
Author(s):  
Hugh Marshall
Keyword(s):  
Nitric Acid ◽  
General Formula ◽  
Free Acid ◽  
The Other ◽  
Image Position

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

Extremely undecidable sentences

1982 ◽  
Vol 47 (1) ◽  
pp. 191-196 ◽  
Author(s):  
George Boolos
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Propositional Modal Logic ◽  
Image Position

Let ‘ϕ’, ‘χ’, and ‘ψ’ be variables ranging over functions from the sentence letters P0, P1, … Pn, … of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence A of modal logic, inductively define Aϕ by[and similarly for other nonmodal propositional connectives]; andwhere Bew(x) is the standard provability predicate for PA and ⌈F⌉ is the PA numeral for the Gödel number of the formula F of PA. Then for any ϕ, (−□⊥)ϕ = −Bew(⌈⊥⌉), which is the consistency assertion for PA; a sentence S is undecidable in PA iff both and , where ϕ(p0) = S. If ψ(p0) is the undecidable sentence constructed by Gödel, then ⊬PA (−□⊥→ −□p0 & − □ − p0)ψ and ⊢PA(P0 ↔ −□⊥)ψ. However, if ψ(p0) is the undecidable sentence constructed by Rosser, then the situation is the other way around: ⊬PA(P0 ↔ −□⊥)ψ and ⊢PA (−□⊥→ −□−p0 & −□−p0)ψ. We call a sentence S of PA extremely undecidable if for all modal sentences A containing no sentence letter other than p0, if for some ψ, ⊬PAAψ, then ⊬PAAϕ, where ϕ(p0) = S. (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

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