In this note we discuss some recent results on extreme steady waves under gravity. They include the existence and regularity theorems for highest waves on finite depth with and without vorticity. Furthermore, we state new results concerning the asymptotic behavior of surface profiles near stagnation points. In particular, we find that the wave profile of an extreme wave is concave near each crest, provided that the vorticity is negative near the surface.

Orthogonal polynomials $$P_{n}(\lambda)$$ are oscillating functions of $$n$$ as $$n\to\infty$$ for $$\lambda$$ in the absolutely continuous spectrum of the corresponding Jacobi operator $$J$$ . We show that, irrespective of any specific assumptions on the coefficients of the operator $$J$$ , the amplitude and phase factors in asymptotic formulas for $$P_{n}(\lambda)$$ are linked by certain universal relations found in the paper. Our proofs rely on the study of a time-dependent evolution generated by suitable functions of the operator $$J$$ .

V. I. Arnold classified simple (i.e., having no moduli for classification) singularities (function germs) and also simple boundary singularities, that is, function germs invariant with respect to the action $$\sigma(x_1; y_1,\dots, y_n)=(-x_1; y_1,\dots, y_n)$$ of the group $${\mathbb Z}_2$$ . In particular, he showed that a function germ (a germ of a boundary singularity) is simple if and only if the intersection form (respectively, the restriction of the intersection form to the subspace of anti-invariant cycles) of a germ in $$3+4s$$ variables stably equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding space is finite. In a previous paper the authors obtained analogues of the latter statements for function germs invariant with respect to an arbitrary action of the group $${\mathbb Z}_2$$ and also for corner singularities. This paper presents an analogue of the simplicity criterion in terms of the intersection form for functions invariant with respect to a number of actions (representations) of the group $${\mathbb Z}_3$$ .

Let $$K$$ be a compact Hausdorff space, $$C(K)$$ be the real Banach space of all continuous functions on $$K$$ endowed with the supremum norm, and $$C(K)^+$$ be the positive cone of $$C(K)$$ . A weak stability result for the symmetrization $$\Theta=(f(\,\boldsymbol\cdot\,)-f(-\;\boldsymbol\cdot\,)/2$$ of a general $$\varepsilon$$ -isometry $$f$$ from $$C(K)^+\cup-C(K)^+$$ to a Banach space $$Y$$ is obtained: For any element $$k\in K$$ , there exists a $$\phi\in S_{Y^\ast}$$ such that $$|\langle\delta_k,x\rangle-\langle\phi,\Theta(x)\rangle|\le3\varepsilon/2\quad\text{for all }\,x\in C(K)^+\cup-C(K)^+.$$ This result is used to prove new stability theorems for the symmetrization $$\Theta$$ of $$f$$ .

This paper studies spaces of distributions on a dyadic half-line, which is the positive half-line equipped with bitwise binary addition and Lebesgue measure. We prove the nonexistence of a space of dyadic distributions which satisfies a number of natural requirements (for instance, the property of being invariant with respect to the Walsh–Fourier transform) and, in addition, is invariant with respect to multiplication by linear functions. This, in particular, is evidence that the space of dyadic distributions suggested by S. Volosivets in 2009 is optimal. We also show applications of dyadic distributions to the theory of refinement equations and wavelets on the dyadic half-line.

We describe the location of the essential spectrum of one-dimensional Dirac operators with singular potentials supported on discrete sets in ℝ in terms of limit operators.

Basic results for nonlinear operators are given. These results include nonlinear versions of the classical uniform boundedness theorem and Hahn-Banach theorem. Further, mappings of a metrizable space into another normed space can belong to some normed spaces if one defines suitable norms.

For any small e > 0, a two-dimensional elastic waveguide is constructed such that λe = e4 is the only eigenvalue in the vicinity of the lower bound λ† = 0 of the continuous spectrum. This result is rather unexpected, because an acoustic waveguide (the Neumann problem for the Laplace operator) with an arbitrary small localized perturbation cannot support a trapped mode at a low frequency.

Let Ω be a C1,s bounded domain (s > 1/2) in ℝd, and let $${{\cal A}^\varepsilon } = - {\rm{div}}\;A(x,x/\varepsilon )\nabla $$ be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that e is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of $${{\cal A}^\varepsilon }$$ are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as e → 0, for $${({{\cal A}^\varepsilon } - \mu {\rho ^\varepsilon })^{ - 1}}$$ and $$\nabla {({{\cal A}^\varepsilon } - \mu {\rho ^\varepsilon })^{ - 1}}$$ in the operator topology on L2. Here ρe(x)= ρ(x,x/e) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both e and μ, we then proceed to similar questions for the solution to the initial boundary-value problem $${\rho ^\varepsilon }{\partial _t}{v_\varepsilon } = - {{\cal A}^\varepsilon }{v_\varepsilon }$$ .

The paper considers the Schrodinger equation with an additional linear potential on the entire real line. A transformation operator with a condition at −∞ is constructed. The Gelfand-Levitan integral equation is obtained on the half-line (−∞, x).