For an algebraic K3 surface with complex multiplication (CM), algebraic fibres of the associated twistor space away from the equator are again of CM type. In this paper, we show that algebraic fibres corresponding to points at the same altitude of the twistor base ${S^2} \simeq \mathbb{P}_\mathbb{C}^1$ share the same CM endomorphism field. Moreover, we determine all the admissible Picard numbers of the twistor fibres.

Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.

We discuss a minimisation problem of the degree of the Chow–Mumford (CM) line bundle among all possible fillings of a polarised family with fixed general fibers, motivated by the study of the moduli space of K-stable Fano varieties. We show that such minimisation implies the slope semistability of the fiber if the central fiber is smooth.

The essential elements in any application of statistical theory are

(1) the set of hypothetical physical states θ which may give rise to the observed phenomena,

(2) the set of all possible observations or samples x,

(3) the set of actions α which may be taken when x is known.

The whole procedure in the common applications can be divided into two parts:

(A) the estimation of θ from the known observation x,

(B) action on the basis of this estimation.

Let be an open Riemann surface with finite genus and finite number of boundary components, and let be a closed Riemann surface. An open continuous function from to is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in either p or q times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all non-trivial functions having two valences between and .

The object of this paper is to study the geometry of (p, q)-maps and establish a generalized embedding theorem which asserts that the image surfaces of (p, q)-maps embed in p-fold closed coverings possibly having branch points off the image surfaces.

Attention has recently been directed to the measurement of the relative amplitudes of wireless echoes from the ionosphere, and some interesting methods of automatic registration have been devised (1, 2). For vertical incidence, and for wave-lengths shorter than 200 metres, visual observation shows that a single echo, even when separated from its accompanying oppositely polarized magneto-ionic component, shows considerable rapid fading (3). This fading is evident on the records published by White (1), and it is clear from these that it is not easy to determine the relative amplitudes of two echoes. This paper is an account of an apparatus which automatically integrates the amplitude of an echo over a time long compared with the period of the fading, and at the end of the period registers the integrated value on a recording galvanometer. The process is then repeated so that the final galvanometer trace represents the average value of the intensity throughout successive equal periods of time. It is further described how by using a double thread recording galvanometer, it is possible to record the averaged amplitudes of two different echoes (e.g. ordinary and extraordinary, or E and F) during alternate integrating periods.

In a recent paper (1), I considered the summability method (D, α) defined, for α > 0, by the sequence-to-sequence transformation

We note that, as is easily verified (and as was pointed out in (1)) a necessary and sufficient condition for the convergence of (1), and thus for the applicability of (D, α), is that

should converge. It was proved in (1) that, provided that (2) converges, a sequence summable (C, r) for any r > − 1 is necessarily summable (D, α). We now show that we can strengthen this result by replacing Cesàro by Abel summability. Moreover, we can omit the hypothesis that (2) converges provided that we interpret (1) as an Abel sum.

An action of a group, G, on a surface, F, consists of a homomorphism

ø: G → Homeo (F).

We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with ø(g) orientation-preserving for all g ε G. In addition we will consider only effective (ø is injective) free actions. Free means that ø(g) is fixed-point-free for all g ε G, g ≠ 1. This paper addresses the classification of such actions.

The problem stated in the title is studied for small values of the diffusivity ratio ε and the magnetic force coefficient N with the aim of finding the effect of rotation and the disc conductivity on the magnetic and velocity fields, currents, shear stresses, etc.

In the last two decades a family of closely related functions have been introduced for the solution of quantum-mechanical problems. In dealing with atomic and molecular systems Pauling (12) introduced in 1928 linear combinations of the eigenfunctions of a single atom, which were called hybrids. These have specific symmetry properties which were utilized by Kimball (6) to construct a group-theoretical method for their determination. More recently Lennard-Jones (8) defined linear combinations of molecular eigenfunctions which he called equivalent orbitals. These are formally similar to the hybrids and Lennard-Jones took over the Kimball method for their determination.

An approximate formula is found for the field produced on the boundary of an obstacle in an inhomogeneous medium by a source which is close to the boundary. Both harmonic and aperiodic waves are discussed.

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

1. Given a series

we will throughout write {sn} for the sequence of its partial sums; that is to say,

We will write p, q to denote the sequences {pn}, {qn}; we will write throughout

and will take P−1 = p−1 = 0.

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulae

for the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expression

for the characteristic projectors.

We study character varieties of symmetric knots and their reductions mod p. We observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry.

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.

Criteria for singularity and absolute continuity analogous to Zygmund's for Riesz products, are demonstrated for those generalized products introduced by Modica and Mortola.

In a recent paper [13], Kulkarni proves that, for $i\ges 0$, one has

$\Ext^{i}_{G}(M \otimes \delta(\mu), \nabla(\lambda))\cong \Ext^{i}{G}(M, \nabla(\lambda/\mu))$

– an isomorphism of spaces of extensions of rational modules. Here G is a general linear group scheme, $\lambda, \mu$ are partitions and M is a polynomial G-module. The modules $\Delta (\mu)$, $\nabla (\lambda)$ and $\nabla (\lambda/\mu)$ are respectively the Weyl module labelled by $\mu$, the induced module labelled by $\lambda$ and the skew module labelled by $\lambda /\mu$. A similar result is given in which the roles of $\Delta$ and $\nabla$ are interchanged.

Our purpose here is to set this result in the context of the representation theory of an arbitrary reductive group. Specifically, we prove a simple result which is valid for general reductive groups and derive Kulkarni's results from this. For convenience we work over an algebraically closed field k.

Envelopes of hypersurfaces arise in many different geometric situations. For example, the canal surfaces associated with a space curve can be obtained as the envelope of spheres of a fixed radius centred on that curve. (Other examples can be found in [11], chapter II and [4], chapter 5). In what follows we shall be entirely concerned with the local structure of such envelopes. So essentially we have a smooth map germ with 0 a regular value of the restriction of F to . For near 0 we have a smooth hypersurface near , where . The envelope of this family of hypersurfaces is obtained by eliminating t from the equations . Moreover, it can be identified with the discriminant of F viewed as an unfolding, by the x variables, of the function germ . This observation, together with standard results on versal unfoldings of functions, allows one to determine the local structure of many ‘generic’ geometric envelopes. See [2] for details.

We give a simple proof that a measure on a locally compact abelian group G is quasi-translation invariant with continuous translation densities if and only if it is equivalent to Haar measure on G and has a continuous positive density.