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.

Using a variational approach we obtain the existence of at least three periodic solutions for discontinuous perturbations of the vector p-Laplacian operator .

We present here some general results of boundedness on LP for maximal operators of the form , where E is a subset of the positive real numbers and Tt is a dilation of a fixed multiplier operator. The range of values of p depends only on the decay at infinity of the multiplier and the Minkowski dimension of E. For the case being the maximal operator associated to a convex body, we prove that the norm of the operator is independent of the body.

In a forthcoming paper, N. M. Khan gives a condition for a variety of commutative semigroups V to be saturated in the sense of Howie and Isbell (1967) (i.e. epis are onto for each S ∈ V). We show the necessity of the condition by constructing a non-saturated semigroup which is a member of every commutative variety not satisfying Khan's condition. This determination of the saturated varieties of commutative semigroups enables us then to prove that these varieties form a sublattice of the lattice of varieties of all commutative semigroups.

In this work we derive by $\Gamma$-convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $\rho$ and study the limit as $\rho$ tends to $0$. The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x2x = xxx2 for every element; when A has a unitary central idempotent, it satisfies the identity (x2x2)x2 = x2(x2x2), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany.

We also introduce the class of e-quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e-quadratic algebras are shown, and new proofs of some well-known theorems are given.

In this paper, we will use optimal mass transport combining with suitable transforms to study the sharp constants and optimizers for a class of the Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. Moreover, we will investigate these inequalities with and without the monomial weights $x_{1}^{A_{1}} \cdots x_{N}^{A_{N}}$ on ℝN.

Given a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1

where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

We completely characterize the validity of the inequality $\| u \|_{Y(\mathbb R)} \leq C \| \nabla^{m} u \|_{X(\mathbb R)}$, where X and Y are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups: one for invariants mod Frobenius powers of the irrelevant ideal and one for cofixed spaces of polynomials.

A theorem of Rudin states that if B is the open unit ball in ℂN, N > 1, if 0<ρ < 1, if is the family of all complex lines in ℂN at a distance ρ from the origin and if f ∈ C(∂B) is such that for every Λ∈ the function f|Λ∂B has a continuous extension to Λ ∩ B which is holomorphic in Λ ∩ B, then f has a continuous extension to B which is holomorphic in B. In this paper we show that when N = 2, the theorem still holds if is replaced by a considerably smaller family.

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

\[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \]

$D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$

$\alpha _u,\alpha _w,\mu _u,\beta$

$\Omega \subset {\mathbb {R}}^{2}$

$\beta <1$

$(u,v,w,z)$

$(1, 0, 0, 0)$

$(L^{\infty }(\Omega ))^{4}$

$t\rightarrow \infty$

It is assumed that the glaciation of Northern Europe during the great Ice Age was due to a shift of the pole associated with appreciable variations of solar radiation.

The shift of the pole brought Europe into sufficiently high latitudes to permit of the formation of an ice-sheet; but the large variations of climate during the Ice Age, as shown by the interglacial epochs, were due to the oscillations of the solar energy.

If two complete cycles of solar radiation occurred during the Pleistocene Period, it is possible to account for four advances of the ice in the Alps as demonstrated by Penck and Brückner, but the interglacial epochs were not all warm. The Günz-Mindel and the Riss-Würm interglacial epochs occurred at the maximum of the solar radiation and were, therefore, warm interglacial epochs; but the Mindel-Riss interglacial epoch occurred at a minimum of solar radiation and was, therefore, a cold interglacial epoch.

At a maximum of solar radiation—that is, during a warm interglacial epoch—the climate of North-West Europe was warm and very wet, with a relatively small annual variation of temperature. As the intensity of solar radiation decreased, the mean temperature fell and the annual variation of temperature increased. At the same time the amount of precipitation decreased. The fall of temperature occurred sufficiently rapidly compared with the decrease in precipitation to cause the glaciers of the Alps to advance and for an ice-sheet to form over Scandinavia. As the solar radiation still further decreased, the lack of precipitation caused the glaciers of the Alps to retreat. At the minimum of solar radiation there was a cold interglacial epoch with low mean temperature, a large annual variation of temperature and very low precipitation; in fact, a truly continental climate.

With these changes of climate went a corresponding change in the flora, the sequence being: park land, forest, tundra, grass with sparse trees, and steppe. In this way it has been found possible to determine a sequence of climates and of fauna and flora for the whole Pleistocene Period, which is supported by the geological and archæological evidence available. In particular it is possible to arrange the sequence of human culture, the geological strata of East Anglia, and the history of the ice in the Alps into the scheme of climate change.

The two maxima of solar radiation were accompanied by increased precipitation in all parts of the world, so accounting for the two pluvial periods which are known to have occurred during the Pleistocene Period.

We show that a gap phenomenon occurs for general variational integrals for mappings from a domain Rn into a Riemannian manifold if has a non-trivial topology.

We study the existence of entropy solutions by assuming the right-hand side function f to be an integrable function for some elliptic nonlocal p-Laplacian type problems. Moreover, the existence of weak solutions for the corresponding parabolic cases is also established. The main aim of this paper is to provide some positive answers for the two questions proposed by Chipot and de Oliveira (Math. Ann., 2019, 375, 283-306).

Depuis Murray et Eenard deux travaux importants sont venus enrichir la bibliographie de la Glauconie: le mémoire de MM. Calderon et Cliaves, “Contribuciones al estudio de la Glauconita,” et l'œuvre magistrale de M. L. Cayeux, Etude micrographique des terrains sédimentaires.

Murray et Eenard avaient étudié spécialement la Glauconie actuelle; MM. Calderon et Cliaves baserent leur synthese de la Glauconie actuelle sur une analyse de Glauconie des roches sédimentaires; M. L. Cayeux limita strictement ses belles recherches à la Glauconie des roches sédimentaires et n'attaquapas le problème de sa genèse.

In a paper read in this Society on the 3rd of March last, “On the Law of Inertia,” &c., I had occasion to adduce for consideration a problem to the following effect:—

We investigate two settings of the Ginzburg—Landau system posed on a manifold where vortices are unstable. The first is an instability result for critical points with vortices of the Ginzburg—Landau energy posed on a simply connected, compact, closed 2-manifold. The second is a vortex annihilation result for the Ginzburg—Landau heat flow posed on certain surfaces of revolution with boundary.

This paper considers existence of periodic solutions for vector Liénard differential equations

In our main result we write

where Q(t, x) is a symmetric matrix and h(t, x) is sublinear. The key assumption relates the asymptotic behaviour as x →+ ∞ of the eigenvalues of Q(t, x) to the spectrum of the linear operator −d2/dt2 Several choices for Q(t, x) are considered which lead to known theorems and extend others. In the case of the Duffing equation

the assumptions are weakened.

Our approach is based on Leray-Schauder's degree theory and a priori estimates.

We construct the Conley index for maps. We do not assume any compactness of map or space. We prove the Ważewski property, additivity property and continuation property.