We establish a general sharp inequality for warped products in real space form. As applications, we show that if the warping function $f$ of a warped product $N_1\times_fN_2$ is a harmonic function, then

(1) every isometric minimal immersion of $N_1\times_fN_2$ into a Euclidean space is locally a warped-product immersion, and

(2) there are no isometric minimal immersions from $N_1\times_f N_2$ into hyperbolic spaces.

Moreover, we prove that if either $N_1$ is compact or the warping function $f$ is an eigenfunction of the Laplacian with positive eigenvalue, then $N_1\times_f N_2$ admits no isometric minimal immersion into a Euclidean space or a hyperbolic space for any codimension. We also provide examples to show that our results are sharp.

AMS 2000 Mathematics subject classification: Primary 53C40; 53C42; 53B25

In this paper we give a relation between the Futaki invariant for a compact complex manifold $M$ and the holonomy of a determinant line bundle over a loop in the base space of any principal $G$-bundle, where $G$ is the identity component of the maximal compact subgroup of the complex Lie group consisting of all biholomorphic automorphisms of $M$. Using the property of the Futaki invariant, we show that the holonomy is an obstruction to the existence of the Einstein–Kähler metrics on $M$. Our main result is Theorem 2.1.

AMS 2000 Mathematics subject classification: Primary 32Q20. Secondary 58J28; 58J52

We prove the following.

(1) The inequalities

$$ \biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!a}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!a}\leq 2\leq\biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!b}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!b} $$

hold for all $x>0$ if and only if

$$ -1.204\,64\ldots=2+\frac{1}{\gamma}-\frac{1}{6}\biggl(\frac{\pi}{\gamma}\biggr)^{\!2}\leq a\leq0\leq b. $$

(2) For all real numbers $x\in(0,1]$ we have

$$ x^{\alpha}\leq\frac{1}{2}\biggl(\frac{1}{\varGamma(x)}+\frac{1}{\varGamma(1/x)}\biggr)\leq x^{\beta}, $$

with the best possible constants

$$ \alpha=1.321\,76\dots\link{and}\beta=0. $$

These theorems extend and complement a result of Gautschi (from 1974), who proved that for all $x>0$ the harmonic mean of $\varGamma(x)$ and $\varGamma(1/x)$ is greater than or equal to $1$.

AMS 2000 Mathematics subject classification: Primary 33B15; 26D15

In this note we show that two conjectures of George Weiss on admissible and weakly admissible observation operators fail to hold in general for the right shift semigroup in the case that the output space is infinite dimensional.

AMS 2000 Mathematics subject classification: Primary 47D06; 32A37; 93B28

We study a three parameter deformation $\mathcal{U}_{abc}$ of $\mathcal{U}(\mathfrak{sl}_2)$ introduced by Le Bruyn in 1995. Working over an arbitrary algebraically closed field of characteristic zero, we determine the centres, the finite-dimensional irreducible representations, and, when the parameter $a$ is not a non-trivial root of unity, the prime ideals of those $\mathcal{U}_{abc}$, with $ac\neq0$, which are conformal as ambiskew polynomial rings.

AMS 2000 Mathematics subject classification: Primary 16W35; 17B37. Secondary 16S36; 16S80

The main result establishes that a weak solution of degenerate elliptic equations can be approximated by a sequence of solutions of non-degenerate elliptic equations. To this end we prove an approximation theorem for $A_p$-weights.

AMS 2000 Mathematics subject classification: Primary 35J50. Secondary 35D05

The system $\dot{X}=TX+Q(X)$ (in $\mathbb{R}^{n}$), where $T$ is linear and $Q$ is quadratic, is considered via commutative algebras. The case of the linearized system having a centre manifold spanned on vectors $E_{1}$, $E_{2}$ (and $TE_{1}=\omega E_{2}$, $TE_{2}=-\omega E_{1}$) is studied. It is shown that for $\Span(E_{1},E_{2})$ being a subalgebra (of the algebra corresponding to the form $Q(X)$), the system is stable. Necessary and sufficient conditions are given for stability of the system in the case where $\mathrm{span}(E_{1},E_{2})$ is not a subalgebra.

AMS 2000 Mathematics subject classification: Primary 34A34; 34C15; 34C27; 34C35. Secondary 13M99

Let $\tau$ be a partition of the positive integer $n$. A partition of the set $\{1,2,\dots,n\}$ is said to be of type $\tau$ if the sizes of its classes form the partition $\tau$ of $n$. It is known that the semigroup $S(\tau)$, generated by all the transformations with kernels of type $\tau$, is idempotent generated. When $\tau$ has a unique non-singleton class of size $d$, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of $S(\tau)$. We further develop existing techniques, associating with a subset $U$ of the set of all idempotents of $S(\tau)$ with kernels of type $\tau$ a directed graph $D(U)$; the directed graph $D(U)$ is strongly connected if and only if $U$ is a generating set for $S(\tau)$, a result which leads to a proof if the fact that the rank and the idempotent rank of $S(\tau)$ are both equal to

$$ \max\biggl\{\binom{n}{d},\binom{n}{d+1}\biggr\}. $$

AMS 2000 Mathematics subject classification: Primary 20M20; 05A18; 05A17; 05C20

We prove that spaces with an uncountable $\omega$-independent family fail the Kunen–Shelah property. Actually, if $\{x_i\}_{i\in I}$ is an uncountable $\omega$-independent family, there exists an uncountable subset $J\subset I$ such that $x_j\notin\overline{\co}(\{x_i\}_{i\in J\setminus\{j\}})$ for every $j\in J$. This improves a previous result due to Sersouri, namely that every uncountable $\omega$-independent family contains a convex right-separated subfamily.

AMS 2000 Mathematics subject classification: Primary 46B20; 46B26

We solve a joint similarity problem for pairs of operators of Foias–Williams/Peller type on weighted Bergman spaces. We show that for the single operator, the Hardy space theory established by Bourgain and Aleksandrov–Peller carries over to weighted Bergman spaces, by establishing the relevant weak factorizations. We then use this fact, together with a recent dilation result due to the first author and Rochberg, to show that a commuting pair of such operators is jointly polynomially bounded if and only if it is jointly completely polynomially bounded. In this case, the pair is jointly similar to a pair of contractions by Paulsen’s similarity theorem.

AMS 2000 Mathematics subject classification: Primary 47B35; 47B47

In this paper, we establish the existence of infinitely many solutions to a Neumann problem involving the p-Laplacian and with discontinuous nonlinearities. The technical approach is mainly based on a very recent result on critical points for possibly non-smooth functionals in a Banach space due to Marano and Motreanu, namely Theorem 1.1 in a paper that is to appear in the journal J. Diff. Eqns (see Theorem 2.3 in the body of this paper). Some applications are presented.

AMS 2000 Mathematics subject classification: Primary 35A15; 35J65; 35R05

We consider the Sturm–Liouville equation

$$ -y''+qy=\lambda y\quad\text{on }[0,1], $$

subject to the boundary conditions

$$ y(0)\cos\alpha=y'(0)\sin\alpha,\quad\alpha\in[0,\pi), $$

and

$$\frac{y'}{y}(1)=a\lambda+b-\sum_{k=1}^N\frac{b_k}{\lambda-c_k}. $$

Topics treated include existence and asymptotics of eigenvalues, oscillation of eigenfunctions, and transformations between such problems.

AMS 2000 Mathematics subject classification: Primary 34B24; 34L20

The explicit defining equations of a new family of curves whose members have a trivial automorphism group are given. Each member is defined for characteristic zero and all but a finite number of characteristics greater than zero. This family, in conjunction with a previously appearing family of the author’s, provides explicit examples of algebraic curves which possess only the trivial automorphism for each genus greater than three. The family is then used to construct Riemann surfaces without anticonformal automorphisms and Klein surfaces with no non-trivial automorphisms.

AMS 2000 Mathematics subject classification: Primary 14H37; 30F50; 30F99

A cotorsion theory is defined as a pair of classes Ext-orthogonal to each other. We give a hereditary condition (HC) which is satisfied by the (flat, cotorsion) cotorsion theory and give properties satisfied by arbitrary cotorsion theories with an HC. Given a cotorsion theory with an HC, we consider the class of all modules having a special precover with respect to the first class in the cotorsion theory and show that this class is closed under extensions. We then raise the question of whether this class is resolving or coresolving.

AMS 2000 Mathematics subject classification: Primary 18G15. Secondary 18E40

The well-behaved Sylow theory for soluble groups is exploited to prove an Euler product for zeta functions counting certain subgroups in pro-soluble groups. This generalizes a result of Grunewald, Segal and Smith for nilpotent groups.

AMS 2000 Mathematics subject classification: Primary 20F16; 11M99

In this paper we give a short, direct proof, using only properties of the Haagerup tensor product, that if an operator algebra $A$ possesses a diagonal in the Haagerup tensor product of $A$ with itself, then $A$ must be isomorphic to a finite-dimensional $C^*$-algebra. Consequently, for operator algebras, the first Hochschild cohomology group $H^1(A,X)=0$ for every bounded, Banach $A$-bimodule $X$, if and only if $A$ is isomorphic to a finite-dimensional $C^*$-algebra.

AMS 2000 Mathematics subject classification: Primary 46L06. Secondary 46L05

Precise bounds are given for the quantity

$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$

where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.

If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.

If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.

AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

In 1950 Erdös proved that if $x\equiv2\,036\,812\ (\mo5\,592\,405)$ and $x\equiv3\ (\mo62)$, then $x$ is not of the form $2^n+p$, where $n$ is a non-negative integer and $p$ is a prime. In this note we present a theorem on integers of the form $2^n+cp$, in particular we completely determine all those integers $c$ relatively prime to $5\,592\,405$ such that the residue class $2\,036\,812(\mo5\,592\,405)$ contains integers of the form $2^n+cp$.

AMS 2000 Mathematics subject classification: Primary 11P32. Secondary 11A07; 11B25; 11B75

Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincaré duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which $K_0$ contains torsion, and construct non-free stably free modules for the group algebras of certain virtually free groups.

AMS 2000 Mathematics subject classification: Primary 19A31. Secondary 16S34; 20J05; 55N15; 57P10

Felix and Murillo introduced the group $\mathrm{Aut}_\varOmega(X)$ of self-maps $f$ of $X$, which satisfy $\sOm f=1_{\varOmega X}$, and proved that the group is nilpotent with the order of nilpotency bounded by the Lusternik–Schnirelmann category of $X$. In this paper we construct a spectral sequence converging to the group $\mathrm{Aut}_\varOmega(X)$ and derive several interesting consequences.

AMS 2000 Mathematics subject classification: Primary 55P42