Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

Lepton Flavor Violation in the Standard Model with general Dimension-Six Operators

We study lepton flavor observables in the Standard Model (SM) extended with all dimension-6 operators which are invariant under the SM gauge group. We calculate the complete one-loop predictions to the radiative lepton decays μ → eγ, τ → μγ and τ → eγ as well as to the closely related anomalous magnetic moments and electric dipole moments of charged leptons, taking into account all dimension-6 operators which can generate lepton flavor violation. Also the 3-body flavor violating charged lepton decays τ ± → μ ± μ + μ −, τ ± → e ± e + e −, τ ± → e ± μ + μ −, τ ± → μ ± e + e −, τ ± → e ∓ μ ± μ ±, τ ± → μ ∓ e ± e ± and μ ± → e ± e + e − and the Z 0 decays Z 0 → ℓ+iℓ−j are considered, taking into account all tree-level contributions.

Beyond questionnaires – Exploring adult education teachers' mathematical beliefs with pictures and interviews

Because of the impact that mathematical beliefs have on an individual’s behaviour, they are generally well researched. However, little mathematical belief research has taken place in the field of adult education. This paper presents preliminary results from a study conducted in this field in Switzerland. It is based on Ernest’s (1989) description of mathematics as an instrumental, Platonist or problem solving construct. The analysis uses pictures drawn by the participants and interviews conducted with them as data. Using a categorising scheme developed by Rolka and Halverscheid (2011), the author argues that adults’ mathematical beliefs are complex and especially personal aspects are difficult to capture with said scheme. Particularly the analysis of visual data requires a more refined method of analysis.

Full and hat inductive definitions are equivalent in NBG

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA 0, those in second order set theory extending NBG (as well as those in n + 3-th order number theory extending the so-called Bernays−Gödel expansion of full n + 2-order number theory etc.) are. In this article, we establish the equivalence between Δ10\bf-LFP and Δ10\bf-FP, which assert the existence of a least and of a (not necessarily least) fixed point, respectively, for positive elementary operators (or between Δn+20\bf-LFP and Δn+20\bf-FP). Our proof also shows the equivalence between ID 1 and ^ID1, both of which are defined in the standard way but with the starting theory PA replaced by ZFC (or full n + 2-th order number theory with global well-ordering).

Delay effects in the response of low-grade gliomas to radiotherapy: a mathematical model and its therapeutical implications

Low-grade gliomas (LGGs) are a group of primary brain tumours usually encountered in young patient populations. These tumours represent a difficult challenge because many patients survive a decade or more and may be at a higher risk for treatment-related complications. Specifically, radiation therapy is known to have a relevant effect on survival but in many cases it can be deferred to avoid side effects while maintaining its beneficial effect. However, a subset of LGGs manifests more aggressive clinical behaviour and requires earlier intervention. Moreover, the effectiveness of radiotherapy depends on the tumour characteristics. Recently Pallud et al. (2012. Neuro-Oncology, 14: , 1-10) studied patients with LGGs treated with radiation therapy as a first-line therapy and obtained the counterintuitive result that tumours with a fast response to the therapy had a worse prognosis than those responding late. In this paper, we construct a mathematical model describing the basic facts of glioma progression and response to radiotherapy. The model provides also an explanation to the observations of Pallud et al. Using the model, we propose radiation fractionation schemes that might be therapeutically useful by helping to evaluate tumour malignancy while at the same time reducing the toxicity associated to the treatment.

Pushing forward in embodied cognition: May we mouse the mathematical mind?

Freely available software has popularized “mousetracking” to study cognitive processing; this involves the on-line recording of cursor positions while participants move a computer mouse to indicate their choice. Movement trajectories of the cursor can then be reconstructed off-line to assess the efficiency of responding in time and across space. Here we focus on the process of selecting among alternative numerical responses. Several studies have recently measured the mathematical mind with cursor movements while people decided about number magnitude or parity, computed sums or differences, or simply located numbers on a number line. After some general methodological considerations about mouse tracking we discuss several conceptual concerns that become particularly evident when “mousing” the mathematical mind.

Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L1-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

Sample size considerations using mathematical models: an example with Chlamydia trachomatis infection and its sequelae pelvic inflammatory disease.

BACKGROUND The success of an intervention to prevent the complications of an infection is influenced by the natural history of the infection. Assumptions about the temporal relationship between infection and the development of sequelae can affect the predicted effect size of an intervention and the sample size calculation. This study investigates how a mathematical model can be used to inform sample size calculations for a randomised controlled trial (RCT) using the example of Chlamydia trachomatis infection and pelvic inflammatory disease (PID). METHODS We used a compartmental model to imitate the structure of a published RCT. We considered three different processes for the timing of PID development, in relation to the initial C. trachomatis infection: immediate, constant throughout, or at the end of the infectious period. For each process we assumed that, of all women infected, the same fraction would develop PID in the absence of an intervention. We examined two sets of assumptions used to calculate the sample size in a published RCT that investigated the effect of chlamydia screening on PID incidence. We also investigated the influence of the natural history parameters of chlamydia on the required sample size. RESULTS The assumed event rates and effect sizes used for the sample size calculation implicitly determined the temporal relationship between chlamydia infection and PID in the model. Even small changes in the assumed PID incidence and relative risk (RR) led to considerable differences in the hypothesised mechanism of PID development. The RR and the sample size needed per group also depend on the natural history parameters of chlamydia. CONCLUSIONS Mathematical modelling helps to understand the temporal relationship between an infection and its sequelae and can show how uncertainties about natural history parameters affect sample size calculations when planning a RCT.

Non-self-adjoint graphs

On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.

Pseudospectra in non-Hermitian quantum mechanics

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT -symmetric quantum mechanics.

How Entrepreneurs Become Skilled Cultural Operators

Cultural entrepreneurship and symbolic management perspectives portray entrepreneurs as skilled cultural operators and often assume them to be capable from the outset to purposefully use ‘cultural resources' in order to motivate resource-holding audiences to support their new ventures. We problematize this premise and develop a model of how entrepreneurs become skilful cultural operators and develop the cultural competences necessary for creating and growing their ventures. The model is grounded in a case study of an entrepreneur who set up shop and sought to acquire resources in a culturally unfamiliar setting. Our model proposes that two adaptive sensemaking processes - approval-driven sensemaking and autonomy-driven sensemaking - jointly facilitate the gradual development of cultural competences. These processes jointly enable entrepreneurs to gain cultural awareness and calibrate their symbolic enactments. Specifically, while approval-driven sensemaking facilitates recognizing cultural resources to symbolically couple a venture's identity claims more tightly with the cultural frames of targeted audiences and gain legitimate distinctiveness, autonomy-driven sensemaking enables recognizing cultural constraints and more effective symbolic decoupling to shield the venture from constraining cultural frames and defend the venture's autonomy and resources. We conclude the paper with a discussion of the theoretical implications of our study for cultural entrepreneurship and symbolic management research.