A new look at the Jordan-Hölder theorem for semimodular lattices

We show that in a semimodular lattice L of finite length, from any prime interval we can reach any maximal chain C by an up- and a down-perspectivity. Therefore, C is a congruence-determining sublattice of L.     

Vector-Valued John–Nirenberg Inequalities and Vector-Valued Mean Oscillations Characterization of BMO

We establish a vector-valued John–Nirenberg inequality for oscillations measured in a Banach function space (B.f.s.) norm. This inequality generalizes several existing results on John–Nirenberg inequalities on function spaces such as rearrangement-invariant B.f.s. and Lebesgue spaces with variable exponents. Moreover, this inequality also offers a new characterization of BMO in terms of the weighted vector-valued mean oscillation.     

Generalised Gagliardo–Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO

Using elementary arguments based on the Fourier transform we prove that for ${1 \leq q < p < \infty}$ and ${s \geq 0}$ with s > n(1/2 ? 1/p), if ${f \in L^{q,\infty} (\mathbb{R}^n) \cap \dot{H}^s (\mathbb{R}^n)}$ , then ${f \in L^p(\mathbb{R}^n)}$ and there exists a constant c _p,q,s such that $$\| f \|_{L^{p}} \leq c_{p,q,s} \| f \|^\theta _{L^{q,\infty}} \| f \|^{1-\theta}_{\dot{H}^s},$$ where 1/p = θ/q + (1?θ)(1/2?s/n). In particular, in ${\mathbb{R}^2}$ we obtain the generalised Ladyzhenskaya inequality ${\| f \| _{L^4} \leq c \| f \|^{1/2}_{L^{2,\infty}} \| f \|^{1/2}_{\dot{H}^1}}$ .We also show that for s = n/2 and q > 1 the norm in ${\| f \|_{\dot{H}^{n/2}}}$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon–Zygmund decompositions.     

On the Brézis Nirenberg Stampacchia-type theorems and their applications

In this paper, we obtain several new generalized KKM-type theorems under a new coercivity condition and the condition of intersectionally closedness which improves condition of transfer closedness. As applications, we obtain new versions of equilibrium problem, minimax inequality, coincidence theorem, fixed point theorem and an existence theorem for an 1-person game.     

A variant of Hardy's and Miyachi's theorems for the Bessel–Struve transform

In this paper, we establish new versions of Hardy's and Miyachi's theorems for the Bessel–Struve transform.     

Liouville theorems for the Navier–Stokes equations and applications

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R ⁿ × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].     

Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C _n depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L ^∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.     

A new look at school algebra—Past,present, and future

The articles in this issue arose from the keynote presentations and discussions of the Algebra Working Group of the Seventh International Congress on Mathematical Education (ICME-7) held in Québec City, Canada, in August 1992 (official title of the Working Group: The Place of Algebra in Secondary and Tertiary Education). Before presenting an overview of the articles and how they all fit together, let me first say a few words about the origins and structure of the Working Group.     

On the constants for some fractional Gagliardo–Nirenberg and Sobolev inequalities

We consider the inequalities of Gagliardo–Nirenberg and Sobolev in ${\mathbb{R}}^d$, formulated in terms of the Laplacian $\Delta$ and of the fractional powers $D^n?{\sqrt{?\Delta }}^n$ with real $n?0$; we review known facts and present novel, complementary results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the ${?}^2$ case where, for all sufficiently regular $f:{\mathbb{R}}^d\to \mathbb{C}$, the norm $6D^{j}f{6}_{{?}^r}$ is bounded in terms of $6f{6}_{{?}^2}$ and $6D^{n}f{6}_{{?}^2}$, for $1∕r=1∕2?(?n?j)∕d$, and suitable values of $j,n,?$ (with $j,n$ possibly noninteger). In the special cases $?=1$ and $?=j∕n+d∕2n$ (i.e., $r=+\infty$), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general ${?}^2$ case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the sharp constants are confined to quite narrow intervals. Several examples are given, including the numerical values of the previously mentioned bounds.     

A look at the past and present of optimization – An editorial

This special issue of the European Journal of Operational Research is devoted to the EURO XXIV Conference, that was held at the facilities of the University of Lisbon (Portugal) from July 11 to July 14, 2010. With over 700 sessions for a total of approximately 2350 presentations, and with 2700 participants (delegates and accompanying persons) coming from 69 countries, this was the largest EURO conference ever.     

A remark on the Brauer–Fowler theorems

Archiv der Mathematik - Let G be a finite group of even order that has no 2-rank 1. We will prove, using only elementary methods, that there is an involution $$t in G$$ such that $$|G| &lt;...     

A new foundation for the mean–variance analysis

In this paper, I re-examine how the mean–variance analysis is consistent with its traditional theoretical foundations, namely, stochastic dominance and the expected utility theory. Then I propose a simplified version of the coarse utility theory as a new foundation. I prove that, by assuming risk aversion and the normality of asset variables, the simplified model is well behaved; indifference curves are convex and the opportunity set is concave. Therefore, there exist global optimal portfolios in the market. Finally, I prove that decision-making in accordance with the simplified model is consistent with the mean–variance analysis.     

A new look at some basic concepts in arbitrage pricing theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we establish a version of the Kramkov's optional decomposition theorem in the setting of equivalent martingale measures. Based on this theorem, we have a new look at some basic concepts in arbitrage pricing theory: superhedging, fair price, attainable contingent claims, complete markets and etc.     

Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.     

Regularity and vanishing theorems for Yang–Mills–Higgs pairs

Archiv der Mathematik - We obtain a vanishing theorem for Yang–Mills–Higgs pairs on Euclidean and hyperbolic spaces in dimensions greater than 4, as well as a regularity theorem more...     

Variation Operators for Semigroups and Riesz Transforms on BMO in the Schrödinger Setting

In this paper we prove that the variation operators of the heat semigroup and the truncations of Riesz transforms associated to the Schrödinger operator are bounded on a suitable BMO type space.