Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

Let 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by

$$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$

$(A_{n}(\alpha))_{n=0}^{\infty}$

$$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$

In this paper, we extend and complement previous works about propagation in kinetic reaction–transport equations. The model we study describes particles moving according to a velocity-jump process, and proliferating according to a reaction term of monostable type. We focus on the case of bounded velocities, having dimension higher than one. We extend previous results obtained by the first author with Calvez and Nadin in dimension one. We study the large time/large-scale hyperbolic limit via an Hamilton–Jacobi framework together with the half-relaxed limits method. We deduce spreading results and the existence of travelling wave solutions. A crucial difference with the mono-dimensional case is the resolution of the spectral problem at the edge of the front, that yields potential singular velocity distributions. As a consequence, the minimal speed of propagation may not be determined by a first-order condition.

We show that if A is a compact C*-algebra without identity that has a faithful *-representation in the C*-algebra of all compact operators on a separable Hilbert space and its multiplier algebra admits a minimal central projection p such that pA is infinite-dimensional, then there exists a Hilbert A1-module admitting no frames, where A1 is the unitization of A. In particular, there exists a frame-less Hilbert C*-module over the C*-algebra $K(\ell^2) \dotplus \mathbb{C}I_{\ell^2}$.

Given a Banach space X and a real number α ≥ 1, we write: (1) D(X) ≤ α if, for any locally finite metric space A, all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C, the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C; (2) D(X) = α+ if, for every ϵ > 0, the condition D(X) ≤ α + ϵ holds, while D(X) ≤ α does not; (3) D(X) ≤ α+ if D(X) = α+ or D(X) ≤ α. It is known that D(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D((⊕n=1∞Xn)p) ≤ 1+ for every nested family of finite-dimensional Banach spaces {Xn}n=1∞ and every 1 ≤ p ≤ ∞. (2) D((⊕n=1∞ ℓ∞n)p) = 1+ for 1 < p < ∞. (3) D(X) ≤ 4+ for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).

We consider a non-stationary incompressible non-Newtonian Stokes system in a porous medium with characteristic size of the pores ϵ and containing a thin fissure of width ηϵ. The viscosity is supposed to obey the power law with flow index $\frac{5}{3}\leq q\leq 2$. The limit when size of the pores tends to zero gives the homogenized behaviour of the flow. We obtain three different models depending on the magnitude ηϵ with respect to ϵ: if ηϵ ≪ $\varepsilon^{q\over 2q-1}$ the homogenized fluid flow is governed by a time-dependent non-linear Darcy law, while if ηϵ ≫ $\varepsilon^{q\over 2q-1}$ is governed by a time-dependent non-linear Reynolds problem. In the critical case, ηϵ ≈ $\varepsilon^{q\over 2q-1}$, the flow is described by a time-dependent non-linear Darcy law coupled with a time-dependent non-linear Reynolds problem.

A natural composition ⊙ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for the p-lower central series spectral sequence of a simplicial group. It is proved that the rth differential satisfies a ‘Leibniz rule with suspension’: d r (a ⊙ σ b) = ±d r a ⊙ b + a ⊙ d r σ b, where σ is the suspension homomorphism.

We add to the list of Banach spaces X for which it is known that the space of bounded linear operators on X has a unique maximal ideal. In particular, the result holds if X is a subsymmetric direct sum of ℓp or of the Schlumprecht space S. We also show that two recently identified ideals in L(J p ), where J p is the pth James space, each contains a unique maximal ideal.

Many fundamental properties of graph C*-algebras may be determined directly from the structure of the underlying graph, and because of this, they have been celebrated as C*-algebras that can be seen. This paper shows how permutative endomorphisms of graph C*-algebras can be represented by labelled directed multigraphs that give visual representations of the endomorphisms and facilitate computations. This formalism provides a useful calculus for permutative automorphisms and allows efficient exhaustive construction of such automorphisms.

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.

In this paper, we have shown that the quantum isometry group of Cr*(ℤ), denoted by ℚ(ℤ, S) as in Goswami and Mandal, Rev. Math. Phys.29(3) (2017), 1750008, with respect to a symmetric generating set S does not depend on the generating set S. Moreover, we have proved that the result is no longer true if the group ℤ is replaced by $\underbrace{\mathbb{Z} \times \mathbb{Z} \times\cdots \times \mathbb{Z}}_{n \ copies} \ \forall \ n>1$.

We prove that a quotient singularity ℂn/G by a finite subgroup G ⊂ SLn(ℂ) has a crepant resolution only if G is generated by junior elements. This is a generalization of the result of Verbitsky (Asian J. Math. 4(3) (2000), 553–563). We also give a procedure to compute the Cox ring of a minimal model of a given ℂn/G explicitly from information of G. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities that admit projective symplectic resolutions.

Let Mn, n ≥ 3, be a complete hypersurface in $\mathbb{S}$n+1. When Mn is compact, we show that Mn is a homology sphere if the squared norm of its traceless second fundamental form is less than $\frac{2(n-1)}{n}$. When Mn is non-compact, we show that there are no non-trivial L2 harmonic p-forms, 1 ≤ p ≤ n − 1, on Mn under pointwise condition. We also show the non-existence of L2 harmonic 1-forms on Mn provided that Mn is minimal and $\frac{n-1}{n}$-stable. This implies that Mn has only one end. Finally, we prove that there exists an explicit positive constant C such that if the total curvature of Mn is less than C, then there are no non-trivial L2 harmonic p-forms on Mn for all 1 ≤ p ≤ n − 1.

We characterize those non-negative, measurable functions ψ on [0, 1] and positive, continuous functions ω1 and ω2 on ℝ+ for which the generalized Hardy–Cesàro operator

$$\begin{equation*}(U_{\psi}f)(x)=\int_0^1 f(tx)\psi(t)\,dt\end{equation*}$$

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, $\mathbb S$n × $\mathbb S$2n is a space of type (0, 1) and the one-point union $\mathbb S$n ∨ $\mathbb S$2n ∨ $\mathbb S$3n is a space of type (0, 0)). It is known that a finite group G that contains ℤp ⊕ ℤp ⊕ ℤp, p a prime, cannot act freely on $\mathbb S$n × $\mathbb S$2n. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G cannot contain ℤp ⊕ ℤp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that ℤ2 is the only group that can act freely on X.