Scattering matrices and Weyl functions

For a scattering system $\{A_\Theta,A_0\}$ consisting of selfadjoint extensions $A_\Theta$ and $A_0$ of a symmetric operator $A$ with finite deficiency indices, the scattering matrix \{S_\gT(\gl)\} and a spectral shift function $\xi_\Theta$ are calculated in terms of the Weyl function associated with the boundary triplet for $A^*$ and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schr\"odinger operators with point interactions.Comment: 39 page

Trace formulae for dissipative and coupled scattering systems

For scattering systems consisting of a (family of) maximal dissipative extension(s) and a selfadjoint extension of a symmetric operator with finite deficiency indices, the spectral shift function is expressed in terms of an abstract Titchmarsh-Weyl function and a variant of the Birman-Krein formula is proved.Comment: 38 page

Evaluation of machine learning algorithms for Health and Wellness applications: a tutorial

Research on decision support applications in healthcare, such as those related to diagnosis, prediction, treatment planning, etc., have seen enormously increased interest recently. This development is thanks to the increase in data availability as well as advances in artificial intelligence and machine learning research. Highly promising research examples are published daily. However, at the same time, there are some unrealistic expectations with regards to the requirements for reliable development and objective validation that is needed in healthcare settings. These expectations may lead to unmet schedules and disappointments (or non-uptake) at the end-user side. It is the aim of this tutorial to provide practical guidance on how to assess performance reliably and efficiently and avoid common traps. Instead of giving a list of do's and don't s, this tutorial tries to build a better understanding behind these do's and don't s and presents both the most relevant performance evaluation criteria as well as how to compute them. Along the way, we will indicate common mistakes and provide references discussing various topics more in-depth.Comment: To be published in Computers in Biology and Medicin

Trace formulae for dissipative and coupled scattering systems

For scattering systems consisting of a (family of) maximal dissipative extension(s) and a selfadjoint extension of a symmetric operator with finite deficiency indices, the spectral shift function is expressed in terms of an abstract Titchmarsh-Weyl function and a variant of the Birman-Krein formula is proved

Scattering matrices and Weyl functions

For a scattering system consisting of two selfadjoint extensions of a symmetric operator A with finite deficiency indices, the scattering matrix and the spectral shift function are calculated in terms of the Weyl function associated with the boundary triplet for A* and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar- and matrix-valued potentials, to Dirac operators and to Schroedinger operators with point interactions

Finite rank perturbations, scattering matrices and inverse problems : dedicated to the memory of our friend Peter Jonas (18.7.1941 - 18.7.2007)

In this paper the scattering matrix of a scattering system consisting of two selfadjoint operators with finite dimensional resolvent difference is expressed in terms of a matrix Nevanlinna function. The problem is embedded into an extension theoretic framework and the theory of boundary triplets and associated Weyl functions for (in general nondensely defined) symmetric operators is applied. The representation results are extended to dissipative scattering systems and an explicit solution of an inverse scattering problem for the Lax-Phillips scattering matrix is presented

Expressive Messaging on Mobile Platforms

We present a design for expressive multimodal messaging on mobile platforms. Strong context, simple text messages, and crude animations combine well to produce surprisingly expressive results

Alhaisen 3He/4He suhteen vaippalähde Karoon suuren magmaprovinssin Mg-rikkaimmille magmoille

The massive outpourings of Karoo and Ferrar continental flood basalts (CFBs) ~180 Ma ago mark the initial Jurassic rifting stages of the Gondwana supercontinent. The origin and sources of these eruptions have been debated for decades, largely due to difficulties in defining their parental melt and mantle source characteristics. Recent findings of Fe- and Mg-rich dikes (depleted ferropicrite suite) from Vestfjella, western Dronning Maud Land, Antarctica, have shed light on the composition of the deep sub-Gondwanan mantle: these magmas have been connected to upper mantle sources presently sampled by the Southwest Indian Ocean mid-ocean ridge basalts (SWIR MORBs) or to high 3He/4He plume-entrained non-chondritic primitive mantle sources formed early in Earth’s history. In an attempt to determine their He isotopic composition and relative contributions from magmatic, cosmogenic, and radiogenic He sources, we performed in-vacuo stepwise crushing and melting analyses of olivine mineral separates, some of which were abraded to remove the outer layer. The best estimate for the mantle isotopic composition is given by a sample with the highest amount of He released (> 50%) during the first crushing step of an abraded coarse fraction. It has a 3He/4He of 7.03 ± 0.23 (2σ) times the atmospheric ratio (Ra), which is indistinguishable from those measured from SWIR MORBs (6.3‒7.3 Ra; source 3He/4He ~6.4–7.6 Ra at 180 Ma) and notably lower than in the most primitive lavas from the North Atlantic Igneous Province (up to 50 Ra), considered to represent the epitome magmas from non-chondritic primitive mantle sources. Previously published trace element and isotopic (Sr, Nd, and Pb) compositions do not suggest a direct genetic link to any modern hotspot of Indian or southern Atlantic Oceans. Although influence of a mantle plume cannot be ruled out, the high magma temperatures and SWIR MORB-like geochemistry of the suite are best explained by supercontinent insulation of a precursory Indian Ocean upper mantle source. Such a model is also supported by the majority of the recent studies on the structure, geochronology, and petrology of the Karoo CFBs.The massive outpourings of Karoo and Ferrar continental flood basalts (CFBs) ~180 Ma ago mark the initial Jurassic rifting stages of the Gondwana supercontinent. The origin and sources of these eruptions have been debated for decades, largely due to difficulties in defining their parental melt and mantle source characteristics. Recent findings of Fe- and Mg-rich dikes (depleted ferropicrite suite) from Vestfjella, western Dronning Maud Land, Antarctica, have shed light on the composition of the deep sub-Gondwanan mantle: these magmas have been connected to upper mantle sources presently sampled by the Southwest Indian Ocean mid-ocean ridge basalts (SWIR MORBs) or to high 3He/4He plume-entrained non-chondritic primitive mantle sources formed early in Earth’s history. In an attempt to determine their He isotopic composition and relative contributions from magmatic, cosmogenic, and radiogenic He sources, we performed in-vacuo stepwise crushing and melting analyses of olivine mineral separates, some of which were abraded to remove the outer layer. The best estimate for the mantle isotopic composition is given by a sample with the highest amount of He released (> 50%) during the first crushing step of an abraded coarse fraction. It has a 3He/4He of 7.03 ± 0.23 (2σ) times the atmospheric ratio (Ra), which is indistinguishable from those measured from SWIR MORBs (6.3‒7.3 Ra; source 3He/4He ~6.4–7.6 Ra at 180 Ma) and notably lower than in the most primitive lavas from the North Atlantic Igneous Province (up to 50 Ra), considered to represent the epitome magmas from non-chondritic primitive mantle sources. Previously published trace element and isotopic (Sr, Nd, and Pb) compositions do not suggest a direct genetic link to any modern hotspot of Indian or southern Atlantic Oceans. Although influence of a mantle plume cannot be ruled out, the high magma temperatures and SWIR MORB-like geochemistry of the suite are best explained by supercontinent insulation of a precursory Indian Ocean upper mantle source. Such a model is also supported by the majority of the recent studies on the structure, geochronology, and petrology of the Karoo CFBs.Peer reviewe