Communications in Contemporary Mathematics

We introduce the “skew apolarity lemma” and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in ${\wedge }^d{V}_{ℂ}$ with $d\le 3$ and $dim{V}_{ℂ}\le 8$. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.

We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass

In ${ℝ}^3$ we consider the vector fields

We consider the family of convex bodies obtained from an origin symmetric convex body $K$ by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call $K$-transform. In the special case, if $K$ is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector space generated by support functions of zonoids is dense in the family of support functions of origin symmetric convex bodies; and the cosine transform is injective. We show that these two properties are equivalent for general $K$. For $K$ being a generalized zonoid, we determine conditions that ensure the injectivity of the $K$-transform. Relations to mixed volumes and to a geometric description of one-sided stable laws are discussed. The later probabilistic application motivates our study of a family of convex bodies obtained as limits of sums of diagonally scaled ${\ell }_p$-balls.

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without inner boundary, we can obtain some inequalities involving the geometric quantities of the boundary and give some obstruction. We also discuss the 4-dimensional asymptotically flat extension of the 3-dimensional Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

Many earlier works were devoted to the study of the breakdown of superconductivity in type-II superconducting bounded planar domains, submitted to smooth magnetic fields. In the present contribution, we consider a new situation where the applied magnetic field is piecewise-constant, and the discontinuity jump occurs along a smooth curve meeting the boundary transversely. To handle this situation, we perform a detailed spectral analysis of a new effective model. Consequently, we establish the monotonicity of the transition from a superconducting to a normal state. Moreover, we determine the location of superconductivity in the sample just before it disappears completely. Interestingly, the study shows similarities with the case of corner domains subjected to constant fields.

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and interesting constructions such as time-slicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen’s universal algebra.

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities:

where $\mathrm{\Omega }\subset {ℝ}^n$ is a smooth bounded domain in ${ℝ}^n$ containing $0$ in its interior, and $f,g\in C(\overline{\mathrm{\Omega }})$ with $f^{+},g^{+}\not\equiv 0$ which may change sign in $\overline{\mathrm{\Omega }}$. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for $\lambda$ sufficiently small. The variational approach requires that $0<\alpha <2,$ $0<s<\alpha <n,$ $1<q<2<p\le 2_{\alpha }^{\ast }(s):=\frac{2(n-s)}{n-\alpha }$, and $\gamma <{\gamma }_H(\alpha )$, the latter being the best fractional Hardy constant on ${ℝ}^n$.

This paper studies the Cauchy problem of radial inhomogeneous Schrödinger maps (ISM) which arises from the integrable model of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system. Through a complex transformation the radial ISM is equivalent to an integro-differential Schrödinger equation. A new weighted Sobolev space ${{𝒲}}_{\beta }^{1,l}({ℝ}^{+})$ is introduced and the well-posedness of integro-differential Schrödinger equations, including the integral radial IMS, with small spherically symmetric initial data in one-dimensional energy space ${{𝒲}}_1^{1,2}({ℝ}^{+})$ is established. Furthermore, for $n\le 2$, we prove the existence of blow-up solutions for the integral radial ISM.

In this paper, we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form

The Brunn–Minkowski and Prékopa–Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn–Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of $log$-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa–Leindler inequality.

We prove that every 2-Segal space is unital.