Fourier restriction theorems for degenerate curves

1987 ◽  
Vol 101 (3) ◽  
pp. 541-553 ◽  
Author(s):  
S. W. Drury ◽  
B. P. Marshall
Keyword(s):  
Finite Order ◽  
Smooth Manifold ◽  
Fourier Restriction ◽  
Restriction Theorems ◽  
Image Position

Fourier restriction theorems contain estimates of the formwhere σ is a measure on a smooth manifold M in ∝n. This paper is a continuation of [5], which considered this problem for certain degenerate curves in ∝n. Here estimates are obtained for all curves with degeneracies of finite order. References to previous work on this problem may be found in [5].

Fourier restriction theorems for curves with affine and Euclidean arclengths

1985 ◽  
Vol 97 (1) ◽  
pp. 111-125 ◽  
Author(s):  
S. W. Drury ◽  
B. P. Marshall
Keyword(s):  
Fourier Transform ◽  
Smooth Manifold ◽  
Fourier Restriction ◽  
Survey Article ◽  
Restriction Theorems ◽  
Schwartz Class ◽  
The Fourier Transform ◽  
Image Position

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequalityfor every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

scholarly journals ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

2011 ◽  
Vol 85 (3) ◽  
pp. 463-475 ◽  
Author(s):  
MEI-RU CHEN ◽  
ZONG-XUAN CHEN
Keyword(s):  
Finite Order ◽  
Meromorphic Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
The Difference ◽  
Image Position

AbstractIn this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

Congruence Relations Between the Traces of Matrix Powers

1949 ◽  
Vol 1 (3) ◽  
pp. 303-304 ◽  
Author(s):  
J. S Frame
Keyword(s):  
Finite Order ◽  
Rational Integer ◽  
Roots Of Unity ◽  
Finite Degree ◽  
Characteristic Roots ◽  
Congruence Relations ◽  
Image Position

Let A be a matrix of finite order n and finite degree d, whose characteristic roots are certain nth roots of unity a1, a2…, ad. We wish to prove a congruence (6) between the traces (tr) of certain powers of A, which is suggested by two somewhat simpler congruences (1) and (3). First, if tr (A) is a rational integer, it is easy to establish the familiar congruenceeven though tr(Ap) may not itself be rational.

An extension of a theorem of Kulakoff

1938 ◽  
Vol 34 (3) ◽  
pp. 316-320
Author(s):  
T. E. Easterfield
Keyword(s):  
Finite Order ◽  
Dihedral Group ◽  
Mixed Group ◽  
Quaternion Group ◽  
Sylow Subgroups ◽  
Cyclic Groups ◽  
Image Position

It has been shown by Kulakoff that if G is a group, not cyclic, of order pl, p being an odd prime, the number of subgroups of G of order pk, for 0 < k < l, is congruent to 1 + p (mod p2); and by Hall that if G is any group of finite order whose Sylow subgroups of G of order pk, p being odd, are not cyclic, then, for 0 < k < l, the number of subgroups of G of order pk is congruent to 1 + p (mod p2). No results were given for the case p = 2. In the present paper it is shown that analogous results hold for the case p = 2, but that the role of the cyclic groups is played by groups of four exceptional types: the cyclic groups themselves, and three non-Abelian types. These groups are defined as follows:(1) The dihedral group, of order 2k, generated by A and B, where(2) The quaternion group, of order 2k, generated by A and B, where(3) The "mixed" group, of order 2k, generated by A and B, where

scholarly journals Bilinear Fourier Restriction Theorems

2012 ◽  
Vol 18 (6) ◽  
pp. 1265-1290
Author(s):  
Ciprian Demeter ◽  
S. Zubin Gautam
Keyword(s):  
Fourier Restriction ◽  
Restriction Theorems

A note on integral functions

1932 ◽  
Vol 28 (3) ◽  
pp. 262-265 ◽  
Author(s):  
R. E. A. C. Paley
Keyword(s):  
Finite Order ◽  
Infinite Order ◽  
Integral Function ◽  
Zero Order ◽  
Image Position

1. Let f(z) denote an integral function of finite order ρ. We writeIt has been shown thatwhere hρ is a constant which depends only on ρ. We are naturally led to enquire whether some equation of the form (1.1) may be true with lim sup replaced by lim inf. In this note we show that the reverse is true. We construct an integral function of zero order for whichThe proof may easily be modified to construct a function of any finite order or of infinite order for which (1.2) is satisfied.

On groups of linear substitutions which contain irreducible metacyclical subgroups

1925 ◽  
Vol 22 (5) ◽  
pp. 788-792
Author(s):  
W. Burnside
Keyword(s):  
Finite Order ◽  
Prime Order ◽  
Primitive Root ◽  
Image Position

If G is a group of finite order which contains an operation P of prime order p, permutable only with its own powers, the order of G must, by Sylow's theorem, be of the form (1 + kp) pś, where s is a factor of p – 1. The greatest subgroup of G, which contains self-conjugately {P};, the subgroup generated by P, must be a metacyclical subgroup {S, P}, wherewhile g is a primitive root of the congruence gs = 1 (mod. p).

scholarly journals Transversally affine foliations

1976 ◽  
Vol 17 (2) ◽  
pp. 106-111 ◽  
Author(s):  
P. M. D. Furness ◽  
E. Fédida
Keyword(s):  
Coordinate Transformation ◽  
Smooth Manifold ◽  
Affine Structure ◽  
Coordinate Functions ◽  
Image Position

Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the formIf y′ is always affine in y, i.e.where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.

On elements of order four in certain free central extensions of groups

1989 ◽  
Vol 106 (1) ◽  
pp. 13-28 ◽  
Author(s):  
Ralph Stöhr
Keyword(s):  
Finite Order ◽  
Central Extension ◽  
Commutator Subgroup ◽  
Metabelian Group ◽  
The Other ◽  
Central Extensions ◽  
Image Position ◽  
Order Four ◽  
Torsion Free ◽  
Number Of Authors

Let F be a non-cyclic free group, R a normal subgroup of F and G = F/R, i.e.where π is the natural projection of F onto G, is a free presentation of G. Let R′ denote the commutator subgroup of R. The quotient F/[R′,F] is a free central extensionof the group F/R′, the latter being a free abelianized extension of G. While F/R′ is torsion-free (see, e.g. [2], p. 23), elements of finite order may occur in R′/[R′,F], the kernel of the free central extension (l·2). Since C. K. Gupta [1] discovered elements of order 2 in the free centre-by-metabelian group F/[F″,F] (i.e. (1·2) in the case R = F′), torsion in F/[R′,F] has been studied by a number of authors (see, e.g. [4–13]). Clearly the elements of finite order in F/[R′,F] form a subgroup T of the abelian group R′/[R′,F]. It will be convenient to write T additively. By a result of Kuz'min [5], any element of T has order 2 or 4. Moreover, it was pointed out in [5] that elements of order 4 may really occur. On the other hand, it has been shown in [11] that, if G has no 2-torsion, then T is an elementary abelian 2-group isomorphic to H4(G, ℤ2). So if T contains an element of order 4, then G must have 2-torsion. We also mention a result of Zerck [13], who proved that 2T is an invariant of G, i.e. it does not depend on the particular choice of the free presentation (1·1).

Sign in / Sign up

Export Citation Format

Share Document