Archiv der Mathematik最新文献

The Ma–Qiu index of a group is the minimum number of normal generators of the commutator subgroup. We show that the Ma–Qiu index gives a lower bound of the presentation distance of two groups, the minimum number of relator replacements to change one group to the other. Since many local moves in knot theory induce relator replacements in knot groups, this shows that the Ma–Qiu index of knot groups gives a lower bound of the Gordian distance based on various local moves. In particular, this gives a unified and simple proof of the Nakanishi index bounds of various unknotting numbers, including virtual or welded knot cases.

We prove that the weak Hilbert property ascends along a morphism of varieties over an arbitrary field of characteristic zero, under suitable assumptions.

Based on the notions of conciseness and semiconciseness, we show that these properties are not equivalent by proving that a word originally presented by Ol’shanskii is semiconcise but not concise. We further establish that every (1/m)-concise word is semiconcise by proving that when the group-word w takes finitely many values in G, the iterated commutator subgroup ([w(G), G, {mathop {dots }limits ^{(m)}}, G]) is finite for some (m in mathbb {N}) if and only if ([w(G), G]) is finite.

In this paper, a strengthened Bonnesen inequality for closed convex plane curves will be given by strengthening its right-hand side into a nonnegative function S(r). For constant width curves, S(r) can be further written in a clearer form which can give us a strengthened Bonnesen-type inequality and an enhanced Gage inequality.

If G is p-solvable, we prove that there exists a McKay bijection that respects the decomposition numbers (d_{chi varphi }), whenever (varphi ) is linear.

In this paper, we strengthen a result of Seager regarding the number of orbits of a solvable primitive linear group.

Classically, Gohberg-type lemmas provide lower bounds for the distance of suitable pseudodifferential operators acting in a Hilbert space to the ideal of compact operators, in terms of “the behavior of the symbol at infinity”. In this article, the pseudodifferential operators are associated to a compact Abelian group (textsf{X}) and an important role is played by its Pontryagin dual ({widehat{textsf{X}}}) . Hörmander-type classes of symbols are not always available; they will be replaced by crossed product (C^*)-algebras involving a vanishing oscillation condition, which anyway is more general even in the particular cases allowing a full pseudodifferential calculus. In addition, the distance to a large class of operator ideals is controlled; the compact operators only form a particular case. This involves invariant closed subsets of certain compactifications of the dual group or, equivalently, invariant ideals of (ell ^infty ({widehat{textsf{X}}})) .

In this work, we investigate and characterize linear functionals (L: mathcal {V}subsetneq mathbb {R}[x_1,dots ,x_n]rightarrow mathbb {R}) with finite-dimensional (mathcal {V}) and absolutely continuous representing measures (mu ), i.e., (textrm{d}mu (x) = g(x),textrm{d}x) for some density g. We focus on the regularity of the density g.

In this paper, we explicitly compute two kinds of algebraic Belyi functions on Bring’s curve. One is related to a congruence subgroup of (textrm{SL}_2({mathbb {Z}})) and the other is related to a congruence subgroup of the triangle group (Delta (2,4,5)subset textrm{SL}_2({mathbb {R}}).) To carry out the computation, we use elliptic cusp forms of weight 2 for the former case and the automorphism group of Bring’s curve for the latter case. We also discuss a suitable base field (a number field) for describing isomorphisms between Hulek–Craig’s curve, Bring’s curve, and another algebraic model obtained as a modular curve.

In this paper, we obtain the (L^p) boundedness of the Berezin transform on fat Hartogs triangles for a restricted range of p, which is proved to be sharp.