Those $(2m-1)$-edge-colourings of a spanning subgraph of $K_{2m}$, consisting of $K_{r}$ and independent edges, that can be embedded in a $(2m-1)$-edge-colouring of $K_{2m}$ are characterised.

Waldspurger's involution for the genuine irreducible supercuspidal representations of $\widetilde{\mathrm{SL}_{2}(F)}$ is parametrized in terms of types in the case $F$$p$-adic with $p$ odd. In particular, it is shown that the in-volution is given by conjugating by an element of $\widetilde{\mathrm{GL}_{2}(F)}$ and twisting one of the defining parameters of an associated type by a quadratic character, the relevant parameter being a character on the norm one elements of a quadratic extension.

Let $E$ be an elliptic curve defined over a number field $F$. The paper concerns the structure of the $p^{\backslash}\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_{\backslash}\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division points of an abelian variety $A$ defined over $F$. The main focus of the paper is the calculation of the Gal$(F_{\backslash}\infty/F)$-Euler characteristic of the $p^{\backslash}\infty$-Selmer group of $E$. The main theory is illustrated with the example of an elliptic curve of conductor 294.

Patchworking of singular hypersurfaces is used to construct projective hypersurfaces with prescribed singularities. For all $n\,{\geq}\, 2$, an asymptotically proper existence result is deduced for hypersurfaces in $\P^n$ with singularities of corank at most 2 prescribed up to analytical or topological equivalence. In the case of $T$-smooth hypersurfaces with only simple singularities, the result is even asymptotically optimal, that is, the leading coefficient in the sufficient existence condition cannot be improved, which is new even in the case of plane curves. Furthermore, an asymptotically proper existence result is proved for hypersurfaces in $\P^n$ with quasihomogeneous singularities. The estimates substantially improve all known (general) existence results for hypersurfaces with these singularities.

The theory of order $p$ linear differential equations with univalued solutions can be presented in terms of the Schubert calculus in the Grassmannian of $p$-dimensional subspaces of the vector space of complex polynomials in one variable. The regular singular points and the exponents of an equation determine an intersection of Schubert varieties.

Heine and Stieltjes in their studies of order $p = 2$ linear differential equations with polynomial coefficients having a polynomial solution of a prescribed degree discovered that such a solution was given by a critical point of a certain remarkable symmetric function. In terms of the Schubert calculus, the critical points determine the elements of the intersection of Schubert varieties.

The case $p > 2$ is studied. A function whose critical points determine the non-degenerate elements in the intersection of Schubert varieties is presented. The number of Fuchsian equations with univalued solutions and prescribed exponents at singular points is estimated from above by the intersection number of the corresponding Schubert classes. Conjecturally, for generic disposition of singular points, these numbers coincide, that is, the intersection of Schubert varieties is transversal and consists of non-degenerate elements only. For $p = 2$ the conjecture was proved earlier by the author.

Graded versions of the principal series modules of the category $\cO$ of a semisimple complex Lie algebra $\mg$ are defined. Their combinatorial descriptions are given by some Kazhdan–Lusztig polynomials. A graded version of the Duflo–Zhelobenko four-term exact sequence is proved. This gives results about composition factors of quotients of the universal enveloping algebra of $\mg$ by primitive ideals; in particular an upper bound is obtained for the multiplicities of such composition factors. Explicit descriptions are given of principal series modules for Lie algebras of rank $2$. It can be seen that these graded versions of principal series representations are neither rigid nor Koszul modules.

A functorial and categorical defined cyclotomic trace is given, extending the usual one for rings to ring spectra. There are two ingredients to this: first a cyclotomic trace is needed that accepts a categorical input with few restrictive assumptions. This is important in its own right, since this allows one to transport rich structures through the cyclotomic trace. Secondly, a sufficiently nice model is needed for the category of finitely generated free modules which is functorial in the ring spectrum.

The normalized positive definite class functions $\C G\,{\longrightarrow}\,\C$ are determined for all those direct limits $G$ of finite alternating groups $\Alt(\Omega_i)$ for which the embeddings $\Alt(\Omega_i)\,{\longrightarrow}\,\Alt(\Omega_j)$ are natural in the sense that every non-trivial $\Alt(\Omega_i)$-orbit in $\Omega_j$ is natural.

A version of interpolation inequalities for derivatives in logarithmic Orlicz spaces is obtained where the first gradient of $u$ is estimated in terms of $u$ and its second gradient. One of the Orlicz functions considered is supposed to be ${\backslash\lambda^p}$. The motivation, examples and applications are discussed.

A numerical quasi-isometry invariant $R(\Gamma)$ of a finitely generated group $\Gamma$ is defined whose values parametrize the difference between $\Gamma$ being uniformly embeddable in a Hilbert space and $C^{*}_{r}(\Gamma)$ being exact.

The paper investigates the vectorial Dirichlet problem defined by $$\begin{cases}\sigma_j(\nabla u(x))=1, \backslash, x\in\Omega \text{a.e.},\, j=1,\ldots, n u(x)=\varphi(x),\,x\in\partial\Omega. \end{cases}$$ Here $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with boundary $\partial\Omega$, and $\sigma_j(A)$ ($j=1, \ldots , n$) denote the singular values of the gradient $\nabla u(x)$. The existence of solutions is established under one of the following assumptions: $\varphi: \overline{\Omega} \longrightarrow \mathbb{R}^n$ is continuous on $\overline{\Omega}$ and locally contractive on $\Omega$, or $\varphi: \partial\Omega \longrightarrow \mathbb{R}^n$ is contractive on $\partial\Omega$. This extends a result due to Dacorogna and Marcellini. The approach is based on the Baire category method developed earlier by the authors.

It is proved that there are two minimal $\Pi_1^0$ classes that are not automorphic.

Let $T$ be a sublinear operator such that $(T\chi_E)^*(t)\,{\le}\, h(t, |E|)$ for some positive function $h(t,s)$ and every measurable set $E$. Then it is shown that under some conditions on the operator $T$, this restricted weak type estimate can be extended to the set of all functions $f\,{\in}\, L^1$ such that $\n f.\infty\,{\le}\, 1$, in the sense that $(Tf)^*(t)\,{\le}\, h(t, \n f.1)$. This inequality allows strong type estimates for $T$ to be obtained on several classes of spaces, such as logarithmic spaces and Lorentz spaces. A similar problem for operators $T$ acting on radial functions is also studied.

It is shown that the matrix spectral factorization mapping is sequentially continuous from $\LL^p$ to $\HH^{2p}$ (where $1\,{\le}\, p\,{<}\,\infty$), under the additional assumption of uniform integrability of the logarithms of the spectral densities to be factorized. It is shown, moreover, that this condition is necessary for continuity, as well as sufficient.

Hoh's calculus for pseudo-differential operators with negative-definite symbols is extended to the case of parameter-dependent symbols. Then this parameter-dependent calculus is applied to the study of subordinate sub-Markovian semigroups.

The paper considers new components of a key space of a moduli space of minimal surfaces in flat 4-tori and calculates their dimensions. An example of minimal surfaces in 4-tori is constructed, and an element of the moduli is obtained. In the process of construction, an example is given of minimal surfaces with a good property in a 3-torus that is distinct from classical examples.