Let Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the boundfor any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds:We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, thenFinally we show that if g is a generator of F*p then for any M < p the number of solutions of the equationis less than . This implies that if p1/2 < M < p, then

We study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.

We prove a domination theorem for operators on Köthe function spaces by probability measures which includes both the Maurey–Rosenthal domination theorem and the Pisier domination theorem as special cases.

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

A notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m > 2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m < 6 in general, and for all choices of m when the satellite is based on a cable knot pattern.

We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants.

We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

We will give an explicit formula of Ozsváth–Szabó's correction terms of lens spaces. Applying the formula to a restriction studied by P. Ozsváth and Z. Szabó in [12] and [13], we obtain several constraints of lens spaces which are constructed by a positive Dehn surgery in 3-sphere. Some of the constraints are results which are analogous to results which were known in [6] and [20] before. The constraints completely determine knots yielding L(p, 1) by positive Dehn surgery.

We introduce a geometric invariant of knots in S3, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots.

The purpose of this paper is to provide a simple relation between the Witten–Reshetikhin-Turaev SO(3) invariant and the Hennings invariant associated to quantum .

We discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruence classes of maximal surfaces in 3. It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2. In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.

A Hermitian form q on the dual space, *, of the Lie algebra, , of a simply connected complex Lie group, G, determines a sub-Laplacian, Δ, on G. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, on G associated to etΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions in L2(G, ρt) onto Jt0 (a subspace of) the dual of the universal enveloping algebra of . Here we give a very different proof of the surjectivity of the Taylor map under the assumption that G is nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense in Jt0 when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions in L2(G, ρt), for appropriate t, when G is the complex Heisenberg group.

For differential equations P(y(k), y)=0, where P is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.

Let [a1(x), a2(x), . . .] be the continued fraction expansion of x ∈ [0,1). Write Tn(x)=max{ak(x):1 ≤ k ≤ n}. Philipp [6] proved thatOkano [5] showed that for any k ≥ 2, there exists x ∈ [0, 1) such that T(x)=1/log k. In this paper we show that, for any α ≥ 0, the setis of Hausdorff dimension 1.

Let k be the set of all k-monotone functions on (−1, 1), i.e., those functions f for which the kth divided differences [x0,. . ., xk; f] are nonnegative for all choices of (k+1) distinct points x0,. . .,xk in (−1, 1). We obtain estimates (which are exact in a certain sense) of kth Ditzian–Totik q-moduli of smoothness of functions in k∩p(−1, 1), where 1 ≤ q < p ≤ ∞, and discuss several applications of these estimates.

Functional data analysis, or FDA, is a relatively new and rapidly growing area of statistics. A substantial part of the interest in the field derives from new types of data that are generated through the application of new technologies. Statistical methodologies, such as linear regression, which are effectively finite-dimensional in conventional statistical settings, become infinite-dimensional in the context of functional data. As a result, the convergence rates of estimators based on functional data can be relatively slow, and so there is substantial interest in methods for dimension reduction, such as principal components analysis (PCA). However, although the statistical development of PCA for FDA has been underway for approximately two decades, relatively high-order theoretical arguments have been largely absent. This makes it difficult to assess the impact that, for example, eigenvalue spacings have on properties of eigenvalue estimators, or to develop concise first-order limit theory for linear functional regression. This paper shows how to overcome these hurdles. It develops rigorous arguments that underpin stochastic expansions of estimators of eigenvalues and eigenfunctions, and shows how to use them to answer statistical questions. The theory is based on arguments from operator theory, made more challenging by the requirement of statisticians that closeness of functions be measured in the L∞, rather than L2, metric. The statistical implications of the properties we develop have been discussed elsewhere, but the theoretical arguments that lie behind them have not been presented before.