In this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module.

AMS 2000 Mathematics subject classification: Primary 16D20

Let $G$ be a noncomplete $k$-connected graph such that the graphs obtained from contracting any edge in $G$ are not $k$-connected, and let $t(G)$ denote the number of triangles in $G$. Thomassen proved $t(G) \geq 1$, which was later improved by Mader to $t(G) \geq \frac{1}{3}|V(G)|$.

Here we show $t(G) \geq \frac{2}{3}|V(G)|$ (which is best possible in general).

Furthermore it is proved that, for $k \geq 4$, a $k$-connected graph without two disjoint triangles must contain an edge not contained in a triangle whose contraction yields a $k$-connected graph. As an application, for $k \geq 4$ every $k$-connected graph $G$ admits two disjoint induced cycles $C_1,C_2$ such that $G-V(C_1)$ and $G-V(C_2)$ are $(k-3)$-connected.

A new asymptotic approach is suggested for studying spectra of linear differential operators with periodic coefficients. The resulting formal recurrent procedure and its rigorous justification allow us to prove a classical theorem on the density of states in one dimension.

AMS 2000 Mathematics subject classification: Primary 34L20. Secondary 74J99; 78M35

Let $\mathcal{A}$ be a closed, point-separating sub-algebra of $C_0(X)$, where $X$ is a locally compact Hausdorff space. Assume that $X$ is the maximal ideal space of $\mathcal{A}$. If $f\in\mathcal{A}$, the set $f(X)\cup\{0\}$ is denoted by $\sigma(f)$. After characterizing the points of the Choquet boundary as strong boundary points, we use this equivalence to provide a natural extension of the theorem in [10], which, in turn, was inspired by the main result in [6], by proving the ‘Main Theorem’: if $\varPhi:\mathcal{A}\rightarrow\mathcal{A}$ is a surjective map with the property that $\sigma(fg)=\sigma(\varPhi(f)\varPhi(g))$ for every pair of functions $f,g\in\mathcal{A}$, then there is an onto homeomorphism $\varLambda:X\rightarrow X$ and a signum function $\epsilon(x)$ on $X$ such that

$$ \varPhi(f)(\varLambda(x))=\epsilon(x)f(x) $$

for all $x\in X$ and $f\in\mathcal{A}$.

AMS 2000 Mathematics subject classification: Primary 46J10; 46J20

We consider algorithms for group testing problems when nothing is known in advance about the number of defectives. Du and Hwang suggested measuring the quality of such algorithms by its so-called (first) competitive ratio (see the Introduction). Later, Du and Park suggested a second kind of competitive ratio. For each kind of competitiveness, we improve the best-known bounds: in the first case, from 1.65 to $1.5+\ep$, and in the second from 16 to 4.

In this paper we extract some conclusions about Newton non-degenerate ideals and the computation of Łojasiewicz exponents relative to this kind of ideal. This motivates us to study the Newton non-degeneracy condition on the Jacobian ideal of a given analytic function germ $f:(\mathbb{C}^n,0)\to(\mathbb{C},0)$. In particular, we establish a connection between Newton non-degenerate functions and functions whose Jacobian ideal is Newton non-degenerate.

AMS 2000 Mathematics subject classification: Primary 32S05. Secondary 57R45

We study a nonlinear second-order periodic problem driven by the scalar $p$-Laplacian with a non-smooth potential. We consider the so-called doubly resonant situation allowing complete interaction (resonance) with both ends of the spectral interval. Using variational methods based on the non-smooth critical-point theory for locally Lipschitz functions and an abstract minimax principle concerning linking sets we establish the solvability of the problem.

AMS 2000 Mathematics subject classification: Primary 34B15; 34C25

In this paper we study the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded. Apart from the standard operator norm on $B(H)$, we consider a rich variety of symmetric operator norms and spaces of operator-Lipschitz functions with respect to these norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of operator-Lipschitz functions.

AMS 2000 Mathematics subject classification: Primary 47A56. Secondary 47L20

This paper deals with a reaction–diffusion model with inner absorptions and coupled nonlinear boundary conditions of exponential type. The critical exponents are described via a pair of parameters that satisfy a certain matrix equation containing all the six nonlinear exponents of the system. Whether the solutions blow up or not is determined by the signs of the two parameters. A more precise analysis, depending on the geometry of $\varOmega$ and the absorption coefficients, is proposed for the critical sign of the parameters.

AMS 2000 Mathematics subject classification: Primary 35K55; 35B33

Let $L$ be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let $f:L\to\{\pm1\}$ be a homomorphism and, for $\alpha=\sum\alpha_\ell\ell\in\mathbb{Z}L$, define $\alpha^f=\sum f(\ell)\alpha_\ell\ell^{-1}$. Call $\alpha$ f-unitary if $\alpha^f=\alpha^{-1}$ or $\alpha^f=-\alpha^{-1}$. In this paper, we identify the RA loops $L$ with the property that all units in $\mathbb{Z}L$ are $f$-unitary. Along the way, we extend a famous theorem of Higman to a case still undecided in group rings.

AMS 2000 Mathematics subject classification: Primary 20N05. Secondary 17D05; 16S34; 16U60

The Ramsey Schur number $RS(s,t)$ is the smallest $n$ such that every 2-colouring of the edges of $K_n$ with vertices $1,2,\ldots,n$ contains a green $K_s$ or there are vertices $x_1,x_2,\ldots,x_t$ fulfilling the equation $x_1+x_2+\cdots+x_{t-1}=x_t$ and all edges $(x_i,x_j)$ are red. We prove $RS(3,3)=11, RS(3,t)=t^2-3$ for $t\equiv1\ (\mbox{mod}\ 6)$ and $t=8$, and $RS(3,t)\geq t^2-3$.

We consider elementary operators on centrally closed prime algebras that are Lie (or Jordan) homomorphisms or commutativity preservers.

AMS 2000 Mathematics subject classification: Primary 16N60. Secondary 16R50; 47B47

Complete disorder is impossible – this theme of Ramsey Theory, as stated by Theodore S. Motzkin, was a guiding theme throughout Walter Deuber's scientific life.

The singular boundary-value problem $(g(x'(t)))'=\mu f(t,x(t),x'(t))$, $x(0)=x(T)=0$ and $\max\{x(t):0\le t\le T\}=A$ is considered. Here $\mu$ is the parameter and the negative function $f(t,u,v)$ satisfying local Carathéodory conditions on $[0,T]\times(0,\infty)\times(\mathbb{R}\setminus\{0\})$ may be singular at the values $u=0$ and $v=0$ of the phase variables $u$ and $v$. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A>0$ such that the above problem with $\mu=\mu_A$ has a positive solution on $(0,T)$. The proofs are based on the regularization and sequential techniques and use the Leray–Schauder degree and Vitali’s convergence theorem.

AMS 2000 Mathematics subject classification: Primary 34B16; 34B18

For fixed positive integers $k,q,r$ with $q$ a prime power and large $m$, we investigate matrices with $m$ rows and a maximum number $N_q (m,k,r)$ of columns, such that each column contains at most $r$ nonzero entries from the finite field $GF(q)$ and any $k$ columns are linearly independent over $GF(q)$. For even integers $k \geq 2$ we obtain the lower bounds $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))})$, and $N_q(m,k,r) = \Omega (m^{((k-1)r)/(2(k-2))})$ for odd $k \geq 3$. For $k=2^i$ we show that $N_q(m,k,r) = \Theta ( m^{kr/(2(k-1))})$ if $\gcd(k-1,r) = k-1$, while for arbitrary even $k \geq 4$ with $\gcd(k-1,r) =1$ we have $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))} \cdot (\log m)^{1/(k-1)})$. Matrices which fulfil these lower bounds can be found in polynomial time. Moreover, for $\Char (GF(q)) > 2 $ we obtain $N_q(m,4,r) = \Theta (m^{\lceil 4r/3\rceil/2})$, while for $\Char (GF(q)) = 2$ we can only show that $N_q(m,4,r) = O (m^{\lceil 4r/3\rceil/2})$. Our results extend and complement earlier results from [7, 18], where the case $q=2$ was considered.

Some results are presented relating to questions raised in a recent paper by Anderson, Hayman and Pommerenke regarding the size of the set of boundary points of the unit disc at which a univalent function has a prescribed radial growth.

AMS 2000 Mathematics subject classification: Primary 30C55

A zigzag in a plane graph is a circuit of edges, such that any two, but not three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbours on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3-valent plane graph and, especially, of simple two-faced polyhedra, i.e., 3-valent 3-polytopes with only $a$-gonal and $b$-gonal faces, where $3 \leq a < b \leq 6$; the main cases are $(a,b)=(3,6), (4,6)$ and $(5,6)$ (the fullerenes).

We completely describe the zigzag structure for the case $(a,b)\,{=}\,(3,6)$. For the case $(a,b)\,{=}\,(4,6)$ we describe symmetry groups, classify all tight graphs with simple zigzags and give the upper bound 9 for the number of zigzags in general tight graphs. For the remaining case $(a,b)\,{=}\,(5,6)$ we give a construction realizing a prescribed zigzag structure.

This paper deals with local and global existence for the solutions of the heat equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied iswhere Ω ⊂ Rn (n ≥ 1) is a regular and bounded domain, ∂Ω = Γ0 ∪ Γ1, m > 1, 2 ≤ p < r, where r = 2(n − 1)/(n − 2) when n ≥ 3, r = ∞ when n = 1, 2 and u0 ∈ H1(Ω), u0 = 0 on Γ0. We prove local existence of the solutions in H1(Ω) when m > r/(r + 1−p) or n = 1, 2 and global existence when p ≤ m or the initial datum is inside the potential well associated to the stationary problem.

In this paper we study Mumford–Shah-type functionals associated with doubling metric measures or strong A∞ weights in the setting of the perimeter theory in the sense of Ambrosio and Miranda in metric spaces. We prove an existence theorem in a suitably defined class of special BV functions.