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Discrete linear Weingarten surfaces in space forms are characterized as special discrete 
$\unicode[STIX]{x1D6FA}$-nets, a discrete analogue of Demoulin’s 
$\unicode[STIX]{x1D6FA}$-surfaces. It is shown that the Lie-geometric deformation of 
$\unicode[STIX]{x1D6FA}$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.
We give an alternative proof of the theorem of Alikakos and Fusco concerning existence of heteroclinic solutions U : ℝ → ℝN to the system
Here a± are local minima of a potential W ∈ C2(ℝN) with W(a±) = 0. This system arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of Alikakos and Fusco and establishes existence by analysing the loss of compactness in minimizing sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.
A 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the 
q-Painlevé system.
In this paper, we consider an initial-value problem for the Korteweg–de Vries equation. The normalized Korteweg–de Vries equation considered is given by where 
$$\begin{equation*} u_{\tau }+u u_{x}+u_{xxx}=0, \quad -\infty <x<\infty ,\ \tau >0, \end{equation*}$$
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$ and 
$\tau $ represent dimensionless distance and time, respectively. In particular, we consider the case when the initial data has a discontinuous compressive step, where 
$u(x,0) =u_{0}>0$ for 
$x \le 0$ and 
$u(x,0)=0$ for 
$x>0$. The method of matched asymptotic coordinate expansions is used to obtain the detailed large-
$\tau $ asymptotic structure of the solution to this problem, which exhibits the formation of a dispersive shock wave.
Given two arbitrary sequences 
$({\lambda }_{j} )_{j\geq 1} $ and 
$({\mu }_{j} )_{j\geq 1} $ of real numbers satisfying we prove that there exists a unique sequence 
$$\begin{eqnarray*}\displaystyle \vert {\lambda }_{1} \vert \gt \vert {\mu }_{1} \vert \gt \vert {\lambda }_{2} \vert \gt \vert {\mu }_{2} \vert \gt \cdots \gt \vert {\lambda }_{j} \vert \gt \vert {\mu }_{j} \vert \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $, real valued, such that the Hankel operators 
${\Gamma }_{c} $ and 
${\Gamma }_{\tilde {c} } $ of symbols 
$c= ({c}_{n} )_{n\geq 0} $ and 
$\tilde {c} = ({c}_{n+ 1} )_{n\geq 0} $, respectively, are selfadjoint compact operators on 
${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have the sequences 
$({\lambda }_{j} )_{j\geq 1} $ and 
$({\mu }_{j} )_{j\geq 1} $, respectively, as non-zero eigenvalues. Moreover, we give an
explicit formula for 
$c$ and we describe the kernel of 
${\Gamma }_{c} $ and of 
${\Gamma }_{\tilde {c} } $ in terms of the sequences 
$({\lambda }_{j} )_{j\geq 1} $ and 
$({\mu }_{j} )_{j\geq 1} $. More generally, given two arbitrary sequences 
$({\rho }_{j} )_{j\geq 1} $ and 
$({\sigma }_{j} )_{j\geq 1} $ of positive numbers satisfying we describe the set of sequences 
$$\begin{eqnarray*}\displaystyle {\rho }_{1} \gt {\sigma }_{1} \gt {\rho }_{2} \gt {\sigma }_{2} \gt \cdots \gt {\rho }_{j} \gt {\sigma }_{j} \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $ of complex numbers such that the Hankel operators 
${\Gamma }_{c} $ and 
${\Gamma }_{\tilde {c} } $ are compact on 
${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have sequences 
$({\rho }_{j} )_{j\geq 1} $ and 
$({\sigma }_{j} )_{j\geq 1} $, respectively, as non-zero singular values.
