We show examples that mutants may not be visible as such from minimal crossing diagrams, and of knots with the same polynomial invariants and same hyperbolic volume, which are not mutants.

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are sharp in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.

In Part I of the paper, a simple theory of infinite systems of linear equations of the type L2 is presented. In Part II, these results are applied to the elastostatic problem of determining the stress distribution in an axially loaded bar, held in position by a centrally placed circular bolt. Some numerical results are obtained and these are compared with experimental observation.

In an improved method of strip-theory calculation of the performance of an airscrew a factor K occurs in the formula

which, for an airscrew with N blades, connects the total interference velocity w1 and the circulation Γ at radius r. In this formula ø is the inclination to the plane of rotation of the airscrew of the velocity of the air relative to the airscrew blade; for infinite pitch ø=½π For large but finite pitch the calculation of K is difficult; in computing tables of K it is therefore useful to have the values for infinite pitch, which can be calculated by a much simpler theory than that needed for the general case†.

1. This paper is a sequel to (2). The notation is the same and we assume the reader to be familiar with that paper. We there proved various theorems which established conditions under which ‘almost all’ labelled (n, q) graphs contained a k-cycle, i.e. under which the proportion of all labelled (n, q) graphs which contained a k-cycle tended to 1 as n → ∞. We here consider a different but related problem. If a graph contains a cycle of length k for every k such that 3 ≤ k ≤ k0, we say that the graph is pancyclic up to k0. We find conditions on k0, n and q such that almost all (n, q) graphs are pancyclic up to k0. If k0 = 0(1), the result follows trivially from Theorem 3a of (1) (Theorem 1 of (2)), but if k0 → ∞with n the matter is not so simple. We shall prove the following theorem.

1. Introduction. Let Ω be a non-singular quadric in an n-dimensional projective space PG(n, K) whose coordinate field is K. With respect to Ω the linear subspaces of PG(n, K) fall into various types: the subspaces of a given type each have the same dimension and the same geometrical kind of quadric section with Ω. Each element of the collineation group Γ preserving Ω takes a subspace into one of the same type, but Γ may divide the subspaces of a given type into several transitivity classes or orbits.

We study the four-genus of linear combinations of torus knots: g4(aT(p, q) #-bT(p′, q′)). Fixing positive p, q, p′, and q′, our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram–Levine signature function; for the second, the recently defined Upsilon function of Ozsváth–Stipsicz–Szabó determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.

We compute certain twists of the classical modular curve X(8). Searching for rational points on these twists enables us to find non-trivial pairs non-isogenous elliptic curves over ℚ whose 8-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over ℚ.

In this paper, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the 𝔪-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the F-jumping numbers and test ideals associated to an arbitrary polynomial over an F-finite field.

An alternative procedure is developed for examining the solutions of the Low equation. The ambiguities of those solutions which have a finite number of zeros are shown to satisfy the conditions derived earlier by Castillejo, Dalitz and Dyson. The method presented here has the advantage that it makes no special use of the positiveness of the cut-off function, and so can be used when this function changes sign. It is shown that the arbitrariness of the solution increases with the number of these sign changes, but that a completely negative cut-off function still satisfies the CDD conditions.

Some apparent anomalies in oscillograms obtained in the course of electrical impulse tests of a material used in the construction of lightning arresters have led to a fuller investigation of the behaviour of the test circuit as applied to the study of a material with a non-linear voltage-current characteristic.

This investigation has been carried out by evaluation of solutions of the equations of a simplified form of the circuit numerically and by means of the differential analyser, and by further oscillograms taken under conditions suggested by the behaviour of these solutions.

The apparent anomalies which originally gave rise to the investigation have been shown to be a real feature of the response of the circuit, and not a spurious effect due to the recording apparatus, and the study of the initial response of the circuit to the impulse leads to the interpretation of another feature of the oscillographic records which had been previously regarded as an instrumental error.

Suppose throughout that

and that

is an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

The 3-dimensional potential due to a charged circle can be derived in various ways which lead to the same expression in terms of the toroidal coordinate σ through Legendre functions of s (= cosh σ). Applying the same methods to the magnetic potential of a circular current we find various results: (a) Legendre's equation yields no solution (§4·1). (b) The appropriate transformation from cylindrical to toroidal coordinates gives a Fourier series (equation (48)) the coefficients of which, though not Legendre functions, are related to them by the identity (§4·2) By this method we also derive expressions in elliptic integrals (equations (56), (58)). (c) Direct integration of the solid angle gives an alternative expression in elliptic integrals (equation (59)).

Whenever a covariant (resp. contravariant) functor F from a category of groups is given, it is natural to ask the following question: if a homomorphismof groups induces the isomorphism (resp., ), is f itself an isomorphism?

Many of the basic concepts in this paper are defined in (2), and it will be useful to the reader if he is familiar with that paper. In it the concept of a loop transversal to a subgroup S of a group G is defined and discussed, where if G is a transitive permutation group and no explicit S is mentioned it is assumed that S is a point stabilizer. In (3) there appeared the following problem (P):

Let G be a transitive permutation group which is 2-closed.

(i) Is there a fixed-point-free transversal in G (i.e. to a point stabilizer)?

(ii) Is there a loop transversal in G?

Unfortunately a note was added in which it was falsely stated that the answer to both (i) and (ii) is yes.

The theory of Fourier transforms

can be developed from the functional equation K(s) K(1 – s) = 1, where K(s) is the Mellin transform of the kernel k(x).

In this paper I show that reciprocities can be obtained which are analogous to the Fourier transforms above but which develop from the much more general functional equation

The reciprocities are obtained by using fractional integration. In addition to the reciprocities we have analogues of the Parseval theorem and of the discontinuous integrals usually associated with Fourier transforms.

In order to simplify the analysis I confine myself to the case n = 1 and to L2 space.

Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q + 1, q + 1)/S[U(q + 1, q) × U(1)], and SU(q + 1, q + 1)/SL(n, ℂ) ⋊ H(n) (p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

We give a closed formula for the multivariable Conway potential function of any graph link in a homology sphere. As corollaries, we answer three questions by Walter Neumann [8] about graph links.

We consider a random walk defined in the following way. We have a set of states indexed by n where n takes on all negative and positive integral values and zero. When we are at state n, there is a probability per unit time λ of going to n + 1, and a probability per unit time λ of going to n − l. Let us start out at n = 0, and study Wn(t), the probability of being at n at time t. Continuity of probability requires that whence since G(s, 0) = 1, we have It follows from the well-known result .

We will construct differential forms on the embedding spaces Emb(j, n) for n − j ≥ 2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(j, n) by Emb(j, n), the homotopy fiber of the inclusion Emb(j, n) ↪ Imm(j, n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(j, n) and Emb(j, n). In particular we obtain nontrivial cohomology classes (for example, in H3(Emb(2, 5))) of higher degrees than those of the first nonvanishing homotopy groups.