The study of the ring of all formal series $a_{0}+a_{1}\binom{x}{1}+a_{2} \binom{x}{2}+\cdots$ with integer coefficients, denoted by $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$, or $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{n}]\hspace{-1.6pt}]_{n\geq0}$ for short, is motivated by the elementary number theoretical properties of the binomial coefficients. The binomial polynomials as well as the binomial coefficients and their generalizations can be found in different branches of mathematics, e.g. in algebra, analysis, combinatorics and in topology. The question of finding the remainder when dividing $\binom{n}{k}$ by a prime (Lucas's 1878 theorem) leads to base-$p$ expansions in the binomial coefficients and the consideration of integer-valued polynomials with rational coefficients. And although the study of these polynomials dates back to the seventeenth century, the study of this set as a ring began in 1936 with Skolem. More generally, the bijective correspondence between the set of functions defined on the set of non-negative integers and the series $a_{0}+a_{1}\binom{x}{1}+a_{2}\binom{x}{2}+\cdots$ is used by Mahler in the field of $p$-adic analysis and naturally leads to the expansion of Skolem's approach and the definition of $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$ or in fact of $R[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$ (with $R$ any ring). Iwasawa used such series in connection with the $p$-adic $L$-functions but without considering the ring.

Taking the argument used by De Giorgi to obtain the rectifiability of the reduced boundary of a set of finite perimeter in ℝN, we prove that a set E of finite perimeter in an open set Ω of ℝN may be approached, in the sense of the BV(Ω) norm, by sets whose boundary is included in a finite union of 1 hypersurfaces; more precisely, arbitrarily large parts (for the N−1 measure in Ω) of the essential boundaries of E and of the approximating set coincide and are included in a single 1 hypersurface.

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.

We consider a least-energy solution to a nonlinear elliptic equation arising in a nonlinear optics model. The equation is linear in a given bounded domain $D$ and nonlinear in $\mathbb{R}^N\setminus D$. We consider a singular limiting problem to obtain a relation between a location of a spiky solution and a geometry of $D$.

We prove the existence of the global smooth solutions to $p$-systems with damping under weaker conditions than those given by Lin and Zheng. The analysis is based on several key a priori estimates, which are obtained via the maximum principle.

We present a new necessary condition for similarity of indefinite Sturm–Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl–Titchmarsh $m$-functions. We also obtain necessary conditions for regularity of the critical points $0$ and $\infty$ of $J$-non-negative Sturm–Liouville operators. Using this result, we construct several examples of operators with the singular critical point $0$. In particular, it is shown that $0$ is a singular critical point of the operator

$$ -\frac{(\mathrm{sgn} x)}{(3|x|+1)^{-4/3}}\frac{\mathrm{d}^2}{\mathrm{d} x^2} $$

acting in the Hilbert space $L^2(\mathbb{R},(3|x|+1)^{-4/3}\,\mathrm{d} x)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a $J$-non-negative Sturm–Liouville operator of type $(\mathrm{sgn} x)(-\mathrm{d}^2/\mathrm{d} x^2+q(x))$ with the same properties.

We investigate the existence of solutions to systems of N differential equations representing connections between minima of potentials with several equal depths in ℝn. Using variational techniques and in particular a method introduced by Alikakos and Fusco we first prove such existence for N ≥ 2 and two minima. Dealing next with symmetric potentials corresponding to bulk free energies in crystals, we establish existence for N ≥ 2 in various cases of more than two minima. Finally, we obtain a sufficient condition establishing existence of connections to potentials which are not necessarily symmetric for arbitrary N and three minima.

Consider a space-like plane Π in Minkowski space. Under the presence of a uniform time-like potential directed towards Π, this paper analyses the configurations of shapes that show a space-like surface supported in Π with prescribed volume and show that it is a critical point of the energy of this system. Such a surface is called stationary and it is determined by the condition that the mean curvature is a linear function of the distance from Π and the fact that the angle of contact with the plate Π is constant. We prove that the surface must be rotational symmetric with respect to an axis orthogonal to Π. Next, we show existence and uniqueness of symmetric solutions for a prescribed angle of contact with Π. Finally, we study the shapes that a stationary surface can adopt in terms of its size. We thus derive estimates of its height and the enclosed volume by surface with the support plane.

We study an implicit second-order ordinary differential equation with complete integral. The class of second-order Clairaut-type equations is an important class of completely integrable equations. Indeed, the notion of second-order Clairaut-type equations is the generalization of second-order classical Clairaut equations. We give generic classifications of second-order Clairaut-type equations under the weak equivalence and the one-parameter strict equivalence relations.

We give a new and elementary proof of the known result: a non-constant mapping of finite distortion f : Ω ⊂ ℝn → ℝn is discrete and open, provided that its distortion function if n = 2 and that for some p > n − 1 if n ≥ 3.

We consider the Cauchy problem for quasilinear parabolic equations $u_t=\Delta\phi(u)+f(u)$, with the bounded non-negative initial data $u_0(x)$ ($u_0(x)\not\equiv0$), where $f(\xi)$ is a positive function in $\xi>0$ satisfying a blow-up condition $\int_1^{\infty}1/f(\xi)\,\mathrm{d}\xi<\infty$. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation $\mathrm{d} v/\mathrm{d} t=f(v)$ with the initial data $\|u_0\|_{L^{\infty}(\mathbb{R}^N)}>0$. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on $u_0$ for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on $u_0$ for blow-up with the least blow-up time, provided that $f(\xi)$ grows more rapidly than $\phi(\xi)$.

The Dirichlet problem

is considered, where B = {x ∈ ℝN : |x| < 1}, N ≥ 3, p > 1, K ∈ C2[0, 1] and K(r) > 0 for 0 ≤ r ≤ 1. A sufficient condition is derived for the uniqueness of radial solutions of (*) possessing exactly k − 1 nodes, where k ∈ ℕ. It is also shown that there exists K ∈ C∞[0, 1] such that (*) has at least three radial solutions possessing exactly k − 1 nodes, in the case 1 < p < (N + 2)/(N − 2).

In this paper, we study the multiplicity of positive solutions for the following semilinear elliptic equation:

\begin{alignat*}{2} -\Delta u+\lambda u&=f(x)u^{p-1}+h(x)u^{q-1} &\quad&\text{in }\mathbb{R}^{N}, \\ u&>0 &&\text{in }\mathbb{R}^{N}, \\ u&\in H^{1}(\mathbb{R}^{N}), \end{alignat*}

where $1\leq q<2<p<2^{\ast}$ ($2^{\ast}=2N/(N-2)$ if $N\geq3$ and $2^{\ast}=\infty$ if $N=1,2$), $\lambda>0,h\in L^{2/(2-q)}(\mathbb{R}^{N})\setminus\{0\}$ is non-negative and $f\in C(\mathbb{R}^{N})$. We will show how the shape of the graph of $f(x)$ affects the number of positive solutions.

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 study weak solutions for a class of free-boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.

The global entropy solution to the radial motion of isothermal self-gravitating gases is studied. A transformation of variables is applied. The global approximate solutions are constructed by a generalized shock-capturing scheme. The source terms include an integral which is non-local and is carefully estimated. The uniform estimate of the approximate solutions is obtained. The convergence of the approximate solutions is proved by the compensated compactness method. The global existence of weak entropy solution with large $L^{\infty}$ initial data is established. The solution allows a possible vacuum state.

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.)

We introduce the concept of causality into the framework of generalized pseudo-Riemannian geometry in the sense of Colombeau and establish the inverse Cauchy–Schwarz inequality in this context. As an application, we prove a dominant energy condition for some energy tensors as put forward by Hawking and Ellis. Our work is based on a new characterization of free elements in finite-dimensional modules over the ring of generalized numbers.

This paper is concerned with the existence of a global attractor for a semi-flowgoverned by the weak solutions to a nonlinear one-dimensional thermoviscoelasticsystem with clamped boundary conditions in shape memory alloys. The constitutiveassumptions for the Helmholtz free energy include the model for the study ofmartensitic phase transitions in shape memory alloys. To describe physicallyphase transitions between different configurations of crystal lattices, we workin a framework in which the strain u belongsto L∞. New approachesare introduced and more delicate estimates are derived to establish the crucialL∞ -estimate ofstrain u in the course of showing thecompactness of the orbit of the semi-flow and existence of an absorbing set.

A one-dimensional Ginzburg–Landau model that describes a superconducting closed thin wire with an arbitrary cross-section subject to a large applied magnetic field is derived from the three-dimensional Ginzburg–Landau energy in the spirit of Γ-convergence. Our result proves the validity of the formal result of Richardson and Rubinstein, which reveals the double limit of a large field and a thin domain. An additional magnetic potential related to the applied field is found in the limiting functional, which yields a parabolic background for the oscillatory phase transition curve between the normal and superconducting states.