We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb R^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right-hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the $L^1$-norm of perturbations in the data.

While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb {Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ have been published. In this paper we use solutions of certain symmetric diophantine systems to construct a number of families of elliptic curves whose coefficients are given in terms of several arbitrary parameters and whose generic rank ranges from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the elliptic curve $\leq 13$. It is, however, expected that the parametrized families of elliptic curves obtained in this paper would yield examples of elliptic curves of much higher rank.

Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental role in logic (Scott 1974) and in algebra (Lombardi and Quitte 2015). We call systems of ideals their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid $G$ and are equivariant with respect to its law, we call them equivariant systems of ideals for $G$: they describe all morphisms from $G$ to meet-semilattice-ordered monoids generated by (the image of) $G$. Taking a 1953 article by Lorenzen as a starting point, we also describe all morphisms from a commutative ordered group $G$ to lattice-ordered groups generated by $G$ through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen; we call them regular entailment relations. In particular, the free lattice-ordered group generated by $G$ is described through the finest regular entailment relation for $G$, and we provide an explicit description for it; it is order-reflecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen's research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain $R$, and specifically into Wolfgang Krull's ``Fundamentalsatz'' that $R$ may be represented as an intersection of valuation rings if and only if $R$ is integrally closed: his constructive substitute for this representation is the regularisation of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when one proceeds as if its elements are comparable.

Let $S$ be a polynomial ring over an algebraically closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height $4$ prime ideal. We give a finite characterization of the degree $2$ component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.

We consider intervals of the form $\left [ \frac {r}{s}, u \right ] $, where $r,s$ are positive integers with gcd($r,s$)=1 and $u$ is a real number, or of the form $[0, \smash {(\sqrt {a}{-}\sqrt {b})^2} ]$, where $a,b$ are positive integers. Thanks to a lemma of Chudnovsky, we give first a lower bound of the integer transfinite diameter of such intervals. Then, using the method of explicit auxiliary functions and our recursive algorithm, we explain how to get an upper bound for this quantity. We finish with some numerical examples. Secondly, we prove inequalities on the integer transfinite diameter of Farey intervals, i.e., intervals of the type $\bigl [ \frac {a}{q}, \frac {b}{s} \bigr ] $, where $|as-bq|=1$.

Fowler introduced the notion of a product system: a collection of Hilbert bimodules $\mathbf {X}=\{\mathbf {X}_p:p\in P\}$ indexed by a semigroup $P$, endowed with a multiplication implementing isomorphisms $\mathbf {X}_p\otimes _A \mathbf {X}_q\cong \mathbf {X}_{pq}$. When $P$ is quasi-lattice ordered, Fowler showed how to associate a $C^*$-algebra $\mathcal {NT}_\mathbf {X}$ to $\mathbf {X}$, generated by a universal representation satisfying some covariance condition. In this article we prove a uniqueness theorem for these so called Nica–Toeplitz algebras.

We propose and develop a new approach to prove the global existence of solutions for a class of $n$-dimensional Navier-Stokes equations.

We study the existence of traveling wave solutions in a higher-dimensional lattice cooperative system with stage structure. We establish the existence theorem of traveling wave solutions based on the upper and lower solutions method and Schauder's fixed point theorem. Then we construct a pair of upper and lower solutions to verify the existence of traveling wave solutions.

A nonnegative matrix $A$ is primitive if for some positive integer $m$ all entries in $A^m$ are positive. The smallest such $m$ is called the exponent of $A$ and written $\exp (A)$. For the class of primitive companion matrices $X$, we find $\exp (A)$ for certain $A \in X$. We find certain values of $m$ for which there is an $n \times n$ primitive companion matrix (for given $n$) with exponent $m$. We also propose open problems for further research.

We study separability and spectral properties of singular degenerate elliptic equations in vector-valued $L_{p}$ spaces. We prove that a realization operator according to this equation with some boundary conditions is separable and Fredholm in $L_{p}$. The leading part of the associated differential operator is not self-adjoint. The sharp estimate of the resolvent, discreteness of spectrum and completeness of root elements of this operator is obtained. Moreover, we show that this operator is positive and generates a holomorphic $C_{0}$-semigroups on $L_{p}$. In application, we examine the regularity properties of nonlocal boundary value problem for degenerate elliptic equation and for the system of degenerate elliptic equations of either finite or infinite number.

It is shown that Newton's inequalities and the related Maclaurin's inequalities provide several refinements of the fundamental arithmetic-geometric-harmonic mean inequality and Sierpinski's inequality in terms of the means and variance of positive real numbers. We also obtain some inequalities involving third and fourth central moments of real numbers.

We consider generators of a monoid consisting of effective divisor classes in Neron-Severi group for each isomorphism class of a projective plane bundle over an elliptic curve. Furthermore, we investigate the structure of the effective divisors which are algebraically equivalent to integral multiples of the generators of the monoid.

We investigate the initial-boundary value problem for a viscoelastic Petrovsky system of Kirchhoff-type with nonlinear source term. By applying the standard continuation principle, the existence of global solutions for this problem is proved and the exponential decay estimate of global solutions is obtained.

We study the existence of solutions for a class of Kirchhoff type problems with critical growth in $\mathbb {R}^N$:$$-\varepsilon ^2\biggl (a+b\int _{\mathbb {R}^N}|\nabla u|^2\,dx\biggr )\Delta u + V(x)u -\varepsilon ^2a\Delta (u^2)u = |u|^{22^\ast -2}u + h(x,u),$$ $(t, x) \in \mathbb {R} \times \mathbb {R}^N$. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We prove that it has at least one solution and for any $m \in \mathbb {N}$, it has at least $m$ pairs of solutions. The proofs are based on the variational methods and concentration-compactness principle.