Solvability of subprincipal type operators

In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of order $k \ge 2$ at a nonradial involutive manifold $\Sigma_2$. We shall assume that the operator is of subprincipal type, which means that the $k$:th inhomogeneous blowup at $\Sigma_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $\Sigma_2$, but transversal to the symplectic leaves of $\Sigma_2$ at the characteristics. When $k = \infty$ this blowup reduces to the subprincipal symbol. We also assume that the blowup is essentially constant on the leaves of $\Sigma_2$, and does not satisfying the Nirenberg-Treves condition (${\Psi}$). We also have conditions on the vanishing of the normal gradient and the Hessian of the blowup at the characteristics. Under these conditions, we show that $P$ is not solvable.Comment: Changed the formulation of Theorem 2.15, added an assuption. Corrected errors and clarified the arguments. Added reference

Differences Between Seriously Emotionally Disturbed and Socially Maladjusted Boys on Measures of Academic Achievement and Ratings of Emotional and Behavioral Problems

Applied Behavioral Studie

Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.Comment: 60 pages, 3 figures. J. Pseudo-Differ. Oper. Appl., to appear. Revised according to referee report, including minor changes to Corollary 1.8. The final publication will be available at link.springer.co

Den økonomiske Grænse for de forskjellige Bevægkræfters Anvendelse i Landbrugets Tjeneste.

Den økonomiske Grænse for de forskjellige Bevægkræfters Anvendelse i Landbrugets Tjeneste

Beretning om Arbejdsprøve med Kværne, Knusemaskiner og Oliekagebrydere i Randers, samt en Efterprøve med Kværne og Knusemaskiner i Nakskov.

Beretning om Arbejdsprøve med Kværne, Knusemaskiner og Oliekagebrydere i Randers, samt en Efterprøve med Kværne og Knusemaskiner i Nakskov

Arbejdsprøve med Radsaamaskiner, Geduingsspredemaskiner og Godningsmoller.

Arbejdsprøve med Radsaamaskiner, Geduingsspredemaskiner og Godningsmoller

Landbrugets Redskaber og Maskiner 1892.

Landbrugets Redskaber og Maskiner 1892

Landbrugets Bedskaber og Maskiner 1895.

Landbrugets Bedskaber og Maskiner 1895

Beretning om den paa Borreby i Marts 1893 afholdte Prøve med Petroleums lokomobiler til Tærskning

Beretning om den paa Borreby i Marts 1893 afholdte Prøve med Petroleums lokomobiler til Tærsknin

Residual Stresses from Incremental Hole Drilling Using Directly Deposited Thin Film Strain Gauges

Background: Commonly, polymer foil-based strain gauges are used for the incremental hole drilling method to obtain residual stress depth profiles. These polymer foil-based strain gauges are prone to errors due to application by glue. For example zero depth setting is thus often erroneous due to necessary removal of polymer foil and glue. This is resulting in wrong use of the calibration coefficients and depth resolution and thus leading to wrong calculations of the obtained residual stress depth profiles. Additionally common polymer foil-based sensors are limited in their application regarding e.g. exposure to high temperatures. Objective: This paper aims at a first step into the qualification of directly deposited thin film strain gauges for use with the incremental hole drilling method. With the directly deposited sensors, uncertainties regarding the determination of calibration coefficients and zero depth setting due to the absence of glue can be reduced to a minimum. Additionally, new areas of interest such as the investigation of thermally sprayed metallic layers can be addressed by the sensors due to their higher temperature resilience and their component inherent minimal thickness. Methods: For the first time, different layouts of directly deposited thin film strain gauges for residual stress measurements were manufactured on a stainless steel specimen. Strain measurements during incremental hole drilling using a bespoke hole drilling device were conducted. Residual stress depth profiles were calculated using the Integral method of the ASTM E837 standard. Afterwards, strain measurements with conventional polymer foil-based strain gauges during incremental hole drilling were conducted and residual stress depth profiles were calculated accordingly. Finally the obtained profiles were compared regarding characteristic values. Results: The residual stress depth profiles obtained from directly deposited strain gauges generally match the ones obtained from conventional polymer foil based strain gauges. With the novel strain gauges, zero depth setting is simplified due to the absence of glue and polymer foil. With the direct deposition, a wide variety of rosette designs is possible, enabling a more detailed evaluation of the strain field around the drilled hole. Conclusions: The comparative analysis of the obtained residual stress depth profiles shows the general feasibility of directly deposited strain gauges for residual stress measurements. Detailed investigations on uncertainty sources are still necessary. © 2022, The Author(s)