An interesting result of Ivanov implies that a non-aspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture. In this point of view, we prove the asphericity of the length-6 relative presentation $\langle H,x: xh_1xh_2xh_3xh_4xh_5xh_6\rangle$, provided that each coefficient is torsion free.

For $a\in(0,\tfrac12]$ and $r\in(0,1)$, let $\mu_a(r)$ be the so-called generalized Grötzsch function which appears in Ramanujan's generalized modular equations. In this paper, several sharp inequalities for $\mu_a(r)$ are obtained and a conjecture on $\mu_a(r)$, which was presented by Qiu and Vuorinen in 1999, is proved.

The concepts of d- and nd-Frattini chief factors of a finite group are introduced. Their ingrainment into that of the extended Frattini dual subgroup becomes the natural dual to Frattini and supplemented chief factors. Not only does a dual of the strengthened form of the Jordan–Hölder theorem arise, but also the $p$-nilpotent radical becomes the intersection of the centralizers of the nd-Frattini chief factors. As a result, a class $\mathfrak{F}$ of groups is a full integrated local formation $\mathrm{LF}(f)$ if and only if each nd-Frattini chief factor in $G\in\mathfrak{F}$ is $f$-central.

Let $\mu$ be a positive Radon measure on $\mathbb{R}^d$ which satisfies $\mu(B(x,r))\le Cr^{n}$ for any $x\in\mathbb{R}^d$ and $r>0$ and some fixed constants $C>0$ and $n\in(0,d]$. In this paper, a new characterization of the space $\rbmo(\mu)$, which was introduced by Tolsa, is given. As an application, it is proved that the $L^p(\mu)$-boundedness with $p\in(1,\infty)$ of Calderón–Zygmund operators is equivalent to various endpoint estimates.

We prove the following two new optimal immersion results for complex projective space. First, if $n\equiv3\,\Mod 8$ but $n\not\equiv3\,\Mod 64$, and $\alpha(n)=7$, then $CP^{n}$ can be immersed in $\mathbb{R}^{4n-14}$. Second, if $n$ is even and $\alpha(n)=3$, then $CP^n$ can be immersed in $\mathbb{R}^{4n-4}$. Here $\alpha(n)$ denotes the number of 1s in the binary expansion of $n$. The first contradicts a result of Crabb, which said that such an immersion does not exist, apparently due to an arithmetical mistake. We combine Crabb's method with that developed by the author and Mahowald.

In this paper we state and prove a new general Hilbert-type inequality in $\mathbb{R}^{n}$ with $k\geq2$ non-conjugate exponents. Using Selberg's integral formula, this result is then applied to obtain explicit upper bounds for the doubly weighted Hardy–Littlewood–Sobolev inequality and some further Hilbert-type inequalities for $k$ non-negative functions and non-conjugate exponents.

Let $H_n$, $n\ge3$, be the space of all $n\times n$ Hermitian matrices. Assume that a map $\phi:H_n\to H_n$ preserves commutativity in both directions (no linearity or bijectivity of $\phi$ is assumed). Then $\phi$ is a unitary similarity transformation composed with a locally polynomial map possibly composed by the transposition. The same result holds for injective continuous maps on $H_n$ preserving commutativity in one direction only. We give counter-examples showing that these two theorems cannot be improved or extended to the infinite-dimensional case.

In this paper, we are concerned with the existence of unbounded orbits of the mapping

\begin{align*}\theta_1&=\theta+2\pi+\frac{1}{\rho}\mu(\theta)+o(\rho^{-1}), \\ \rho_1&=\rho+c-\mu'(\theta)+o(1),\quad\rho\to\infty,\end{align*}

where $c$ is a constant and $\mu(\theta)$ is $2\pi$-periodic. Assume that $c\not=0$, that $\mu(\theta)$ is non-negative (or non-positive) and that $\mu(\theta)$ has finitely many degenerate zeros in $[0,2\pi]$. We prove that every orbit of the given mapping tends to infinity in the future or in the past for sufficiently large $\rho$. On the basis of this conclusion, we further prove that the equation $x''+f(x)x'+V'(x)+\phi(x)=p(t)$ has unbounded solutions provided that $V$ is an isochronous potential at resonance and $F(x)$ ($F(x)=\int_0^xf(s)\,\mathrm{d} s$) and $\phi(x)$ satisfy some limit conditions. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this equation.

This work is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form

$$\mathcal{L}u+|u|^{q-1}u \leq\mathcal{L}v+|v|^{q-1}v,$$

where $q>0$ is a given real number and $\mathcal{L}$ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form given by the relation

$$\mathcal{L}=\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\bigg[a_{ij}(x)\frac{\partial}{\partial x_j}\bigg].$$

We assume that $n\geq2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots,n$, are measurable bounded functions on $\mathbb{R}^n$ such that $a_{ij}(x)=a_{ji}(x)$ and that the corresponding quadratic form is non-negative. The results obtained in this work were announced by the author in 2005.

The usual asymptotic expansion for the solutions of an elliptic linear problem with oscillatory periodic coefficients is known to not be accurate near the boundary. In order to obtain a better approximation it is necessary to add to this expansion a boundary-layer term. This term has been obtained by other authors in the case of a plane boundary, such that its normal is proportional to some period. We consider the case where the normal is arbitrary.špace{-8pt}

In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like $k^{-7/6}$ as $k\to0$. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function $k^{-7/6}$ that we obtained in an earlier paper.

The essential spectrum of the singular matrix differential operator of mixed order determined by the operator matrix

$$\begin{pmatrix} -\dfrac{\mathrm{d}}{\mathrm{d} x}\rho(x) \dfrac{\mathrm{d}}{\mathrm{d} x}+q(x) & \dfrac{\mathrm{d}}{\mathrm{d} x}\dfrac{\beta(x)}{x} \\[12pt] -\dfrac{\beta(x)}{x}\dfrac{\mathrm{d}}{\mathrm{d} x} & \dfrac{m(x)}{x^2} \end{pmatrix}$$

is studied. Investigation of the essential spectrum of the corresponding self-adjoint operator is continued but now without assuming that the quasi-regularity conditions are satisfied. New conditions that guarantee that the operator is semi-bounded from below are derived. It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential operator is given by the range $\text{range}((m\rho-\beta^2)/\rho x^2)$ in the case where the quasi-regularity conditions are not satisfied.

Let $\varOmega$ be a domain in $\mathbb{R}^N$ (possibly unbounded), $N\geq2$, $1<p<\infty$, and let $V\in L_{\mathrm{loc}}^\infty(\varOmega)$. Consider the energy functional $\mathcal{Q}_V$ on $C_{\mathrm{c}}^\infty(\varOmega)$ and its Gâteaux derivative $\mathcal{Q}_V^\prime$, respectively, given by

$$\mathcal{Q}_V(u)\eqdef\frac{1}{p}\int_\varOmega(|\nabla u|^p+V|u|^p)\,\mathrm{d} x,\qquad\mathcal{Q}_V^\prime(u)= \mathrm{div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u,$$

for $u\in C_{\mathrm{c}}^\infty(\varOmega)$. Assume that $\mathcal{Q}_V>0$ on $C_{\mathrm{c}}^{\infty}(\varOmega)\setminus\{0\}$ and that $\mathcal{Q}_V$ does not have a ground state (in the sense of a null sequence for $\mathcal{Q}_V$ that converges in $L_{\mathrm{loc}}^p(\varOmega)$ to a positive function $\varphi\in C_{\mathrm{loc}}^1(\varOmega)$, a ground state). Finally, let $f\in\mathcal{D}^\prime(\varOmega)$ be such that the functional $u\mapsto \mathcal{Q}_V(u)-\langle u,f\rangle:C_{\mathrm{c}}^\infty(\varOmega)\to\mathbb{R}$ is bounded from below. Then the equation

$$\mathcal{Q}_V^\prime(u)=f$$

has a solution $u_0\in W^{1,p}_{\mathrm{loc}}(\varOmega)$ in the sense of distributions. This solution also minimizes the functional $u\mapsto\mathcal{Q}_V^{**}(u)-\langle u,f\rangle:C_{\mathrm{c}}^\infty(\varOmega)\to \mathbb{R}$, where $\mathcal{Q}_V^{**}$ denotes the bipolar ($\varGamma$-regularization) of $\mathcal{Q}_V$ and $\mathcal{Q}_V^{**}$ is the largest convex, weakly lower semicontinuous functional on $C_{\mathrm{c}}^\infty(\varOmega)$ that satisfies $\mathcal{Q}_V^{**}\leq\mathcal{Q}_V$. (The original energy functional $\mathcal{Q}_V$ is not necessarily convex.)

The oldest competition for an optimal (area-maximizing) shape was won by the circle. But if the fixed perimeter is measured by the line integral of $|\mathrm{d} x|+|\mathrm{d} y|$, a square would win. Or if the boundary integral of $\max(|\mathrm{d} x|,|\mathrm{d} y|)$ is given, a diamond has maximum area. For any norm in $\mathbb{R}^2$, we show that when the integral of $\|(\mathrm{d} x,\mathrm{d} y)\|$ around the boundary is prescribed, the area inside is maximized by a ball in the dual norm (rotated by $\pi/2$).

This ‘isoperimetrix' was found by Busemann. For polyhedra it was described by Wulff in the theory of crystals. In our approach, the Euler–Lagrange equation for the support function of $S$ has a particularly nice form. This has application to computing minimum cuts and maximum flows in a plane domain.

We prove the convergence in certain weighted spaces in momentum space of eigenfunctions of $H=T-\lambda V$ as the energy goes to an energy threshold. We do this for three choices of kinetic energy $T$, namely the non-relativistic Schrödinger operator, the pseudorelativistc operator $\sqrt{-\Delta+m^2}-m$, and the Dirac operator.

This work is devoted to the identification of parameters in a problem of pollution modeled by a semi-linear parabolic equation. We use the notion of sentinels introduced by J. L. Lions, (Lions, J. L. 1992 Sentinelles pour les systèmes distribués à données incomplètes. Masson, Paris.) re-visited in a more general framework. We prove the existence of such sentinels by solving a problem of null controllability with constraint on the control. The key of our results is an observability inequality of Carleman type adapted to the constraint.

Consider the two-dimensional autonomous systems of differential equations

$$\dot{x}=-y+\lambda x+P(x,y),\qquad\dot{y}=x+\lambda y+Q(x,y),$$

where $\lambda$ is a real constant and $P$ and $Q$ are $\mathcal{C}^{\infty}$-functions of order greater than or equal to two. These systems, so-called centre-focus-type systems, have either a centre or a focus at the origin. In this work, we give necessary and sufficient conditions of isochronicity using normal forms. We characterize the systems which have either an isochronous centre or an isochronous focus at the origin by means of the existence of a commutator of the field. Moreover, we prove that the maximum order of a weak isochronous focus for quadratic systems is two, and that for systems with cubic nonlinearities is three.

A mathematical model is developed for mud cake growth and fluid invasion around a horizontal wellbore during drilling. The non-axisymmetry of the invasion front is addressed for the case of water based drilling mud and isotropic rock formation. It is shown that the invasion profile may lose convexity due to gravitation even though the filtrate density is the same as that of the pore fluid in the rock.

In this paper I consider a system of equations describing two stratified fluids flowing in closed, slightly inclined ducts. In the framework of the shallow water approximation with turbulent friction acting on the wall and at the interface, I investigate a special class of periodic travelling waves containing stable moving jumps, namely, roll waves and periodic slug flows. The modulation equations of roll waves and slugs are derived and a nonlinear stability criterion is obtained. As for plug flow, only a gas–liquid system is considered. The stability criterion is expressed in terms of an integro-differential relation. For self-similar cross-section, this criterion is simplified to a relation for a function of one variable only.

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.