In a recent paper, Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general $H$-analogue of the Nakayama conjecture where $H$ is a set of positive integers.

Recently, J. Roe proved by a direct computation that the Baum–Connes assembly map can be expressed as a boundary map associated to a certain short exact sequence of C*-algebras. In this paper we deduce the same result as a corollary of the author's characterisation of the Baum–Connes assembly map.

The Borcherds isomorphism is proved to be Hecke equivariant if one considers multiplicative Hecke operators acting on the integral weight meromorphic modular forms. This answers a part of a question of Borcherds (see ‘Automorphic forms on ${\rm O}_{s+2,2}(R)$ and infinite products’, Invent. Math. 120 (1995) 161–213, 17.10), using his suggestion to define the multiplicative Hecke operators.

In the introduction to [3] it is mistakenly claimed that Mislin's proof uses Carlsson's proof of Segal's Burnside ring conjecture. In fact, Mislin uses instead the work of Carlsson, Miller and Lannes on Sullivan's fixed-point conjecture, and work of Dwyer and Zabrodski [1]. There is, however, a proof by Snaith [2] that depends on a version of the Segal conjecture.

This paper introduces the concept of orbit-homogeneity of permutation groups: a group $G$ is orbit-$t$-homogeneous if two sets of cardinality $t$ lie in the same orbit of $G$ whenever their intersections with each $G$-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of $t$-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan.

Further, this paper shows that any group generated by orbit-$t$-homogeneous subgroups is orbit-$t$-homogeneous, and that the condition becomes stronger as $t$ increases up to $\lfloor n/2\rfloor$, where $n$ is the degree. So any group $G$ has a unique maximal orbit-$t$-homogeneous subgroup $\Omega_t(G)$, and $\Omega_t(G)\le\Omega_{t-1}(G)$. Some structural results for orbit-$t$-homogeneous groups, and a number of examples, are also given.

Let $M\subset \mathbb{R}^n$ be a connected component of an algebraic set $\varphi^{-1}(0)$, where $\varphi$ is a polynomial of degree $d$. Assume that $M$ is contained in a ball of radius $r$. We prove that the geodesic diameter of $M$ is bounded by $2r\nu(n)d(4d-5)^{n-2}$, where $\nu(n)=2{\Gamma({1}/{2})\Gamma(({n+1})/{2})}{\Gamma({n}/{2})}^{-1}$. This estimate is based on the bound $r\nu(n)d(4d-5)^{n-2}$ for the length of the gradient trajectories of a linear projection restricted to $M$.

A proof is given to show that a positive function on the unit disk admits a harmonic majorant if and only if it has a certain explicit upper envelope that admits a superharmonic majorant. The (logarithmic) Lipschitz regularity of this superharmonic majorant is discussed.

A unipotent linear group over a skew field of characteristic 2 is nilpotent.

We prove that certain universality properties of the partial sums force a power series to have Ostrowski-gaps. This has interesting consequences for some classes of universal functions.

A finitary orbit equivalence mapping between the binary and ternary odometers is constructed.

We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one of P. Brinkmann that one can determine whether two cyclic words in a free group are mapped to each other by some power of a given automorphism. We also solve the power conjugacy problem, and give an algorithm to recognize whether two given elements of a finitely generated free group are twisted conjugated to each other with respect to a given automorphism.

We study deformations of functions on isolated singularities. A unified proof of the equality of Milnor and Tjurina numbers for functions on isolated complete intersections singularities and space curves is given. As a consequence, the base space of their miniversal deformations is endowed with the structure of an $F$-manifold, and we can prove a conjecture of V. Goryunov, stating that the critical values of the miniversal unfolding of a function on a space curve are generically local coordinates on the base space of the deformation.

Let $ S_n(x)=\sum_{k=1}^{n}({\sin(kx)})/{k} $ be Fejér's sine polynomial. We prove the following statements.

1. The inequality $ (S_n(x+y))^{\alpha} (x+y)^{\beta} \leq (S_n(x))^{\alpha}x^{\beta}+ (S_n(y))^{\alpha}y^{\beta} {(n\in \mathbb{N}; \; \alpha, \beta \in \mathbb{R})}$ holds for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\geq 0$ and $\alpha+\beta \leq 1$.

2. The converse of the above inequality is valid for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\leq 0$ and $\alpha+\beta \geq 1$.

3. For all $n\in\mathbb{N}$ and $x,y \in [0,\pi]$ we have $ 0\leq S_n(x)+S_n(y)- S_n(x+y)\leq \frac{3}{2}\sqrt{3}.$ Both bounds are best possible.

In this note it is shown that for $k$ a field, and for the four-dimensional algebra $\Lambda=k\langle x,y\rangle /\langle x^2,y^2,xy+qyx\rangle$ when $q^n\neq 1,0$ for all $n$, there exist a two-dimensional module $M$ and a family of two-dimensional modules $M_i$, $i=1,2,\ldots$, such that $\dim_k\Ext^i_\Lambda(M,M_j)=1$ for $i$ equal to 0, $j$ and $j+1$, and $\dim_k\Ext^i_\Lambda(M,M_j)=0$ otherwise. This is probably the most straightforward example giving a negative answer to a question raised by Maurice Auslander.

A compact subset of the unit circle is an interpolation set for conformal mappings if and only if it has logarithmic capacity zero.

There are many references showing that a classical solution to the Black–Scholes equation is a stochastic solution. However, it is the converse of this theorem that is most relevant in applications, and the converse is also more mathematically interesting. In this paper we establish such a converse. We find a Feynman–Kac-type theorem showing that the stochastic representation yields a classical solution to the corresponding Black–Scholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also obtain additional regularity results in the one-dimensional case.

Sufficient conditions are given on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$, the space of all bounded linear operators on $X$. A basic sequence $(e_n)$ is said to be quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, $(e_{k_n})$ dominates $(e_{\ell_n})$. If a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$.

The main result in this paper is a representation theorem for homogeneous orthogonally additive polynomials on Banach lattices. The representation theorem is used to study the linear span of the set of zeros of homogeneous real-valued orthogonally additive polynomials. It is shown that in certain lattices every element can be represented as the sum of two or three zeros or, at least, can be approximated by such sums. It is also indicated how these results can be used to study weak topologies induced by orthogonally additive polynomials on Banach lattices.

To form Riemann sums for generalized Riemann integrals, the domain of integration must be partitioned in a suitable manner. The existence of the required partitions is usually proved by a simple method of repeated bisection of the domain of integration. However, when the domain is the Cartesian product of infinitely many copies of the set of real numbers, this simple method of proof has frequently failed. A proof which works for infinite-dimensional spaces is provided here.

It is proved that if a Schur superalgebra is not semisimple, then it is neither cellular nor quasi-hereditary (Theorem 2), and it has infinite global dimension (Corollary 18). The algebra $S(m|n,r)$ with $m,n \ge 1$ is semisimple if and only if $p$, the characteristic of the ground field, is zero or greater than $r$, or when $m=n=1$ and $p$ does not divide $r$.