Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Gröbner basis for the multiview ideal of $n$ generic cameras. As the cameras move, the multiview varieties vary in a family of dimension $11n\,-\,15$. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme.

Let $\left( \text{ }\!\!\chi\!\!\text{ ,}\,d,\,\mu \right)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu \left( \left\{ x \right\} \right)\,=\,0$ for all $x\,\in \,\text{ }\!\!\chi\!\!\text{ }$. In this paper, we show that the boundedness of a Calderón–Zygmund operator $T$ on ${{L}^{2}}\left( \mu \right)$ is equivalent to that of $T$ on ${{L}^{p}}\left( \mu \right)$ for some $p\,\in \,\left( 1,\,\infty \right)$, and that of $T$ from ${{L}^{1}}\left( \mu \right)$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. As an application, we prove that if $T$ is a Calderón–Zygmund operator bounded on ${{L}^{2}}\left( \mu \right)$, then its maximal operator is bounded on ${{L}^{p}}\left( \mu \right)$ for all $p\,\in \,\left( 1,\,\infty \right)$ and from the space of all complex-valued Borel measures on $\text{ }\!\!\chi\!\!\text{ }$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

We prove a homogenization formula for a sub-Laplacian are left invariant Hörmander vector fields) on a connected Lie group Gof polynomial growth. Then using a rescaling argument inspired from M. Avellanedaand F. H. Lin [2], we prove Harnack inequalities for the positive solutions of the equation (∂/∂t+ L)u= 0. Using these inequalities and further exploiting the algebraic structure of Gwe prove that the Riesz transforms , are bounded on Lq,1 < q <+∞ and from L1to weak-L1.

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.

Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.

Let $\ell $ be a length function on a group $G$, and let ${{M}_{\ell }}$ denote the operator of pointwise multiplication by $\ell $ on ${{\ell }^{2}}\left( G \right)$. Following Connes, ${{M}_{\ell }}$ can be used as a “Dirac” operator for $C_{r}^{*}\left( G \right)$. It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$, which defines a metric on the state space of $C_{r}^{*}\left( G \right)$. We show that if $G$ is a hyperbolic group and if $\ell $ is a word-length function on $G$, then the topology from this metric coincides with the weak-$*$ topology (our definition of a “compact quantum metric space”). We show that a convenient framework is that of filtered ${{C}^{*}}$-algebras which satisfy a suitable “Haagerup-type” condition. We also use this framework to prove an analogous fact for certain reduced free products of ${{C}^{*}}$-algebras.

In this paper, the authors first establish the Hardy-Littlewood-Sobolev theorems of fractional integration on the Herz spaces and Herz-type Hardy spaces. Then the authors give some applications of these theorems to the Laplacian and wave equations.

Let the function of two real variablesf(x, y) be given over the square

A finite lattice L of order n is dismantlable [6] if there is a chain L1 ⊂ L2 ⊂ . . . ⊂ Ln = L of sublattices of L such that |Li| = i for every i = 1, 2, . . . , n. In [1] it was shown that every finite planar lattice is dismantlable. Furthermore, every lattice L with |L| ≦ 7 is dismantlable [6]; in fact, every large enough lattice contains a dismantlable sublattice with precisely n elements [4]. As well, such lattices are closed under the formation of sublattices and homomorphic images [6].

Let π be the Desarguesian affine plane of order n = pr, for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags, of π. Further, G is doubly transitive on π if G is doubly transitive on the points of π. Clearly, G is flag transitive if G is doubly transitive on π.

The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).

Given n points with c1 of one colour, c2 of another colour, up to k colours, linear graphs are formed with the restriction that no line connects points of the same colour. Following fairly standard terminology, coloured graphs with this restriction will be called point chromatic graphs. Giving the points numerical labels running from 1 to ci for points of the ith colour (i = 1, 2, … , k) forms point labelled chromatic graphs. Note that this description is slightly different from assigning a label and a colour independently to each point.

The problems considered here arose in connection with the interesting use by Loeb [8] and Anderson [1], [2] of Loeb's measure construction [7] to define measures on certain topological spaces. The original problem, from which the results given here developed, was to identify precisely the family of sets on which these measures are defined.

To be precise, let be a set theoretical structure and * a nonstandard extension of , as in the usual framework for nonstandard analysis (see [10]). Let X be a Hausdorff space in and stx the standard part map for X, defined on the set of nearstandard points in *X. Suppose, for example, µ is an internal, finitely additive probability measure defined on the internal subsets of *X.

In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums

1

have a constant value σ(α) for k = 1, 2, … , n. We noted that K2 is magic and showed that K3 and K4 are not magic.

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is a free group.

Our main result is the following monotonicity property for moment sequences μ. Let p be fixed, 1 ≤ p < ∞: then

is an increasing function of r(r = 1,2,…). From this we derive a sharp lower bound for an arbitrary Hausdorff matrix acting on ℓp.The corresponding upper bound problem was solved by Hardy.

In (2), Tutte has shown that the number, Bn, of rooted non-separable planar maps having n edges is [2(3n — 3)!]/[n! (2n — 1)!]. Rooting was accomplished by designating one edge as the root, orienting it, and distinguishing between its sides as left and right. We shall here compute the number, Bn,m, of rooted non-separable planar maps having n edges and such that the face to the left of the root is incident with exactly m edges, which maps will be said to be of type [n, m].

It is known that every sublattice A of a free lattice satisfies the following conditions:

(W) For all a, b, c, d ∈ A, if ab ≤ c + d, then ab ≤ c or ab ≤ d or a ≤ c+dor b ≤ c + d.

(SD) For all u, a, b, c ∈ A, if u = a + b = a + c, then u = a + bc.

(SD′) For all u, a, b, c ∈ A, if u = ab = ac, then u = a(b + c).

In fact, (W) is one of the four conditions used in Whitman (4) to characterize free lattices, and in Jónsson (3) it was shown that (SD) and (SD′) follow from Whitman's canonical representations of elements of a free lattice.

This note is concerned with lattices that satisfy one or more of the above conditions, and especially with finite lattices that satisfy all three conditions.

Let Γn be the representation group or spin group (9; 4) of the symmetric group Sn. Then the irreducible representations of Γn can be allocated into two classes which we shall call (i) ordinary representations, which are the irreducible representations of the symmetric group, and (ii) spin or projective representations.

As is well known (3; 5), there is an ordinary irreducible representation [λ] corresponding to every partition (λ) = (λ1, λ2, . . . , λm) of n with

λ1 ≥ λ2 ≥ . . . ≥ λm > 0.

A graph G consists of a finite set of p points and q lines joining pairs of these points. Each line joins two distinct points and no pair of points is joined by more than one line. A subgraph of G is a graph whose points and lines are also in G. If every pair of points of a graph is joined by a line, the graph is called complete and is denoted by Kp. A planar graph can be embedded in the plane, that is, drawn in the plane in such a way that none of its lines intersect.

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p2n for every positive integer n and every odd prime p.

A set D of k-letter words is called a comma-free dictionary (2), if whenever (a1a2 . . . ak) and (b1b2 . . . bk) are in D, the "overlaps" (a2a3 . . . akb1), (a3a4 . . . akb1b2), . . . , (akb1 . . . bk-1) are not in D. We say that two k-letter words are in the same equivalence class if one is a cyclic permutation of the other. An equivalence class is called complete if it contains k distinct members. Comma-freedom is violated if we choose words from incomplete equivalence classes, or if more than one word is chosen from the same complete class.