About the convergence of the Fourier transform

The main result is the proof of the theorems, the results of which one can characterize as a weak form of the formula for the inversion of the bi-dimmensional Fourier transform. Sufficient conditions on a function are obtained for a weak (of degree $r$) convergence of bi-dimmensional Fourier transform for a function $f(x;y)$. These conditions have an integral form and describe the behavior of the function near the border of a rectangle. A similar theorem is proved, in which the Fourier transform of a function $f$ is replaced by the Fourier transform of another function $g$, the norm of the central difference of which does not exceed the norm of the central difference of $f$. The principal objective is to study the behavior of the Fourier transform of $g$ and $f$.

Lagrange Numbers Less Than Three

This chapter reflects the technical heart ofMarkoff’s theory. Themain theoremis that if the Lagrange number of a sequence is smaller than 3, then this sequence is the image under the substitution a->11, b->22 of a sequence satisfying theMarkoff property outlined inChapter 4. The proof goes through several technical lemmas: except for a finite number of terms, the sequence has only 1 and 2s; 121 and 212 are forbidden factors; the factors 1122 and 2211 are preceded and followed bywordswith a special lexicographical property; more forbidden factors; 1 and 2’s appear in pairs. The similar theorem for bi-infinite sequences is deduced.

Uniqueness of reduced alternating rational 3-tangle diagrams

Tangles were introduced by J. Conway. In 1970, he proved that every rational 2-tangle defines a rational number and two rational 2-tangles are isotopic if and only if they have the same rational number. So, from Conway's result we have a perfect classification for rational 2-tangles. However, there is no similar theorem to classify rational 3-tangles. In this paper, we introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.

A pair of characteristic subgroups for pushing-up. II

Many problems about local analysis in a finite group G reduce to a special case in which G has a large normal p-subgroup satisfying several restrictions. In 1983, R. Niles and G. Glauberman showed that every finite p-group S of nilpotence class at least 4 must have two characteristic subgroups S1 and S2 such that, whenever S is a Sylow p-subgroup of a group G as above, S1 or S2 is normal in G. In this paper, we prove a similar theorem with a more explicit choice of S1 and S2.

WHEN ARE TWO DEDEKIND SUMS EQUAL?

A natural question about Dedekind sums is to find conditions on the integers a1, a2, and b such that s(a1, b) = s(a2, b). We prove that if the former equality holds then b|(a1a2- 1)(a1- a2). Surprisingly, to the best of our knowledge such statements have not appeared in the literature. A similar theorem is proved for the more general Dedekind–Rademacher sums as well, namely that for any fixed non-negative integer n, a positive integer modulus b, and two integers a1and a2that are relatively prime to b, the hypothesis rn(a1, b) = rn(a2, b) implies that b|(6n2+ 1 - a1a2)(a2- a1).

Faithful representations of crossed products by actions of $\boldsymbol N^k$

We study a family of semigroup crossed products arising from actions of $\boldsymbol N^k$ by endomorphisms of groups. These include the Hecke algebra arising in the Bost-Connes analysis of phase transitions in number theory, and other Hecke algebras considered by Brenken. Our main theorem is a characterisation of the faithful representations of these crossed products, and generalises a similar theorem for the Bost-Connes algebra due to Laca and Raeburn.

Surgeries on periodic links and homology of periodic 3-manifolds

Fix a prime integer p. We show that a closed orientable 3-manifold M admits an action of Zp with the fixed point set S1 if and only if M can be obtained as the result of surgery on a p-periodic framed link L and Zp acts freely on the components of L. We prove a similar theorem for free Zp-actions. As an interesting application, we prove the following, rather unexpected result: for any M as above and for any odd prime p, H1(M, Zp) ≠ Zp. We also prove a similar criterion of 2-periodicity for rational homology 3-spheres.

Kaplansky's ternary quadratic form

This paper proves that ifNis a nonnegative eligible integer, coprime to 7, which is not of the formx2+y2+7z2, thenNis square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integernp2by two quadratic forms in the same genus, thepth coefficient of anL-function of a suitable elliptic curve, and the class number formula prove the theorem for large primes, leaving 3 cases which are easily numerically verified.

Lower Bound on VC-Dimension by Local Shattering

We show that the VC-dimension of a smoothly parameterized function class is not less than the dimension of any manifold in the parameter space, as long as distinct parameter values induce distinct decision boundaries. A similar theorem was published recently and used to introduce lower bounds on VC-dimension for several cases (Lee, Bartlett, & Williamson, 1995). This theorem is not correct, but our theorem could replace it for those cases and many other practical ones.

Universal and proximately universal limits

We present sufficient conditions on an approximate mapping F: X → Y of approximate inverse systems in order that the limit f: X → Y of F is a universal map in the sense of Holsztyński. A similar theorem holds for a more restrictive concept of a proximately universal map introduced recently by the second author. We get as corollaries some sufficient conditions on an approximate inverse system implying that the its limit has the (proximate) fixed point property. In particular, every chainable compact Hausdorif space has the proximate fixed point property.