We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev–Slobodeckiĭ seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.

In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.

We study $L^p$-Sobolev regularity estimates for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb {R}^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb {R}^3$.

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated with the primitive Maass cusp form over $\Gamma _0(N)$.

We consider the existence and stability of constrained solitary wave solutions to the generalized Ostrovsky equation

\begin{align*}\partial_x\left(\partial_t u+ \alpha\partial_x u+\partial_x(f(u))+\beta \partial_x^3u\right)=\gamma u,\quad \|u\|_{L^2}^2=\lambda \gt 0,\end{align*}

$f(s)=\alpha_0|s|^p+\alpha_1|s|^{p-1}s$

$\alpha_0,\alpha_1 \gt 0$

$\alpha\in\mathbb{R}$

$\beta\gamma=-1$

$1 \lt p \lt 5$

$\omega \gt \alpha-2$

Since the communication of the first of this series of articles to the Eoyal Society of Edinburgh in April, and its publication in the Philosophical Magazine in May and June, the stability or instability of the steady motion of a viscous fluid has been proposed as subject for the Adams Prize of the University of Cambridge for 1888.

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\operatorname K$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\operatorname K$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of $\operatorname K$-theory.

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$-analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$, the $d$-th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$-Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$-Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$-Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

Let Aij, l≦j≦k, be bounded Hermitean operators on Hilbert spaces Hi, 1≦i≦k, and let be the induced operators on . An important operator for multiparameter theory is δ: H →H denned by δ = det the determinant being expanded formally. Various definiteness properties of δ are critical for multiparameter spectral theory.

We use the operators Aij to construct a numerical matrix δ(δ) upon which we use Geršgorin theory to investigate the non-singularity and definiteness of δ. Diagonal dominance properties of the array [Aij] are also discussed.

Let F be a real analytic function on a real analytic manifold X. Let P be a linear differential operator on X such that , where Q is an ordinary differential operator with analytic coefficients whose singular points are all regular. For each (isolated) critical value z of F, we construct locally an F-invariant solution u of the equation Pu - v, v being an arbitrary F-invariant distribution supported in F−1(z). The solution u is constructed explicitly in the form of a series of F-invariant distributions.

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

We consider two questions on the geometry of Lipschitz-free $p$-spaces $\mathcal {F}_p$, where $0< p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal {M}, \rho )$ is an infinite doubling metric space (e.g. an infinite subset of an Euclidean space), then $\mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$ for every $\alpha \in (0,\,1)$ and $0< p\leq 1$. An upper bound on the Banach–Mazur distance between the spaces $\mathcal {F}_p ([0, 1]^d, |\cdot |^\alpha )$ and $\ell _p$ is given. Moreover, we tackle a question due to Albiac et al. [4] and expound the role of $p$, $d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from $(K, |\cdot |_1)$, where $K=\cup _{Q\in \mathcal {R}} Q$ is a union of a collection $\emptyset \neq \mathcal {R} \subseteq \{ Rw + R[0,\,1]^d: w\in \mathbb {Z}^d\}$ of cubes in $\mathbb {R}^d$ with side length $R>0$, into the Lipschitz-free $p$-space $\mathcal {F}_p (V, |\cdot |_1)$ over their vertices.

In this paper, we study divergence properties of the Fourier series on Cantor-type fractal measure, also called the mock Fourier series. We give a sufficient condition under which the mock Fourier series for doubling spectral measure is divergent on a set of strictly positive measure. In particular, there exists an example of the quarter Cantor measure whose mock Fourier sums are not almost everywhere convergent.

Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $\Omega$. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.

In the present memoir the author enters very fully into the figure-of-8 wave movements, described by the wing in space, to which he first directed attention in March 1867. He has adduced the experiments with natural and artificial wings, on which his description was originally based, and has shown, by the aid of original models and a large number of diagrams and drawings, that artificial wings can be made to approach indefinitely near to natural ones, not only in their structure, but also in their movements.

(a) The oblate ellipsoid of revolution is proved in Thomson and Tait's Natural Philosophy (first edition, § 776, and the Table of § 772) to be stable, if the condition of being an ellipsoid of revolution be imposed. It is obviously not stable for very great eccentricities without this double condition of being both a figure of revolution and ellipsoidal.

(b) If the condition of being a figure of revolution is imposed, without the condition of being an ellipsoid, there is, for large enough moment of momentum, an annular figure of equilibrium which is stable, and an ellipsoidal figure which is unstable. It is probable, that for moment of momentum greater than one definite limit and less than another, there is just one annular figure of equilibrium, consisting of a single ring.

These experiments were begun in January last at the request of Professor Tait, and have been carried on since then in his laboratory. The subject suggested itself to Professor Tait during his experiments on the effects of great pressures on thermometers carried out for the “Challenger” Expedition Commission, and it was indeed by part of the apparatus purchased by him for this purpose that the experiments were made. The method adopted was of an indirect nature, and consisted in determining first the change of temperature produced by a sudden change of pressure to which the water was subjected.