Determinantal representations and Bézoutians

A major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral, i.e. the solution sets of linear matrix inequalities. We will use sum-of-squares decompositions of certain bilinear forms called Bézoutians to approach this problem. More precisely, we show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that \(q \cdot h\) has a definite determinantal representation. Besides commutative algebra, the proof relies on results from real algebraic geometry.     

Vanishing of Rabinowitz Floer homology on negative line bundles

Following Frauenfelder (Rabinowitz action functional on very negative line bundles, Habilitationsschrift, Munich/München, 2008), Albers and Frauenfelder (Bubbles and onis, 2014. arXiv:1412.4360) we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. Ritter (Adv Math 262:1035–1106, 2014) showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem \(\mathrm {SH}=0\Leftrightarrow \mathrm {RFH}=0\) (Ritter in J Topol 6(2):391–489, 2013), does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak–Frauenfelder–Oancea long exact sequence Cieliebak et al. (Ann Sci Éc Norm Supér (4) 43(6):957–1015, 2010).     

Directional marginal productivity: a foundation of meta-data envelopment analysis

Differential characteristics of the production function represent elasticity measures and marginal rates of production technologies; in particular, marginal productivity (MP) plays an important role in economic theory and applications. This study provides a theoretical foundation of directional marginal productivity (DMP) supporting the meta-data envelopment analysis (meta-DEA) which measures the efficiency via marginal-profit-maximized orientation. In addition, the segmented marginal rate of technical substitution is developed based on DMP. In fact, DMP is developed to address finding the improving direction of the efficient firm on the frontier towards the marginal profit maximization. This approach, which emphasizes “planning” over “efficiency evaluation”, forms the basis for transforming a typical “ex-post” DEA into an “ex-ante” DEA study. Two case studies show that the DMP provides an explicit span of directions for productivity improvement via a trade-off between these distinct directions.     

Productivity development of Norwegian institutions of higher education 2004–2013

Productivity growth of institutions of higher education is of interest for two main reasons: education is an important factor for productivity growth of the economy, and in countries where higher education is funded by the public sector, accountability of resource use is of key interest. Educational services consist of teaching, research and the “third mission” of dissemination of knowledge to the society at large. A bootstrapped Malmquist productivity change index is used to calculate productivity development for Norwegian institutions of higher education over the 10-year period 2004–2013. The confidence intervals from bootstrapping allow part of the uncertainty of point estimates stemming from sample variation to be revealed. The main result is that the majority of institutions have had a positive productivity growth over the total period. However, when comparing with growth in labour input, the impact on productivity varies a lot.     

A short note on the derivation of the elastic von Kármán shell theory

We derive the Γ-limit of scaled elastic energies h^?4E ^h (u ^h ) associated with deformations u ^h of a family of thin shells \({S^h} = \left\{ {z = x + t\vec n\left( x \right);x \in S, - g_1^h\left( x \right) < t < g_2^h\left( x \right)} \right\}\). The obtained von Kármán theory is valid for a general sequence of boundaries g ₁ ^h , g ₂ ^h converging to 0 in an appropriate manner as h vanishes. Our analysis relies on the techniques and extends the results in [10] and [11].     

Existence of three solutions for quasilinear elliptic equations: an Orlicz-Sobolev space setting

In this paper, we establish the existence of three weak solutions for quasilinear elliptic equations in an Orlicz-Sobolev space via an abstract result recently obtained by Ricceri in [13].     

Numerical algorithm for solving multi-pantograph delay equations on the half-line using Jacobi rational functions with convergence analysis

A new spectral Jacobi rational-Gauss collocation (JRC) method is proposed for solving the multi-pantograph delay differential equations on the half-line. The method is based on Jacobi rational functions and Gauss quadrature integration formula. The main idea for obtaining a semi-analytical solution for these equations is essentially developed by reducing the pantograph equations with their initial conditions to systems of algebraic equations in the unknown expansion coefficients. The convergence analysis of the method is analyzed. The method possesses the spectral accuracy. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Indeed, the present method is compared favorably with other methods.     

Some inequalities and limit theorems under sublinear expectations

In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob’s inequality for submartingale and Kolmogrov’s inequality. By Kolmogrov’s inequality, we obtain a special version of Kolmogrov’s law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.     

Asymptotic behaviour of hessian equation with boundary blow up

We consider the boundary blow up problem for k-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their perturbed problem has a unique positive solution, which satisfies some asymptotic behaviors to unperturbed problems under appropriate structure hypotheses for perturbed terms.