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.

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

The need for second-line antiretroviral therapy in adults in sub-Saharan Africa up to 2030: a mathematical modelling study.

BACKGROUND The number of patients in need of second-line antiretroviral drugs is increasing in sub-Saharan Africa. We aimed to project the need of second-line antiretroviral therapy in adults in sub-Saharan Africa up to 2030. METHODS We developed a simulation model for HIV and applied it to each sub-Saharan African country. We used the WHO country intelligence database to estimate the number of adult patients receiving antiretroviral therapy from 2005 to 2014. We fitted the number of adult patients receiving antiretroviral therapy to observed estimates, and predicted first-line and second-line needs between 2015 and 2030. We present results for sub-Saharan Africa, and eight selected countries. We present 18 scenarios, combining the availability of viral load monitoring, speed of antiretroviral scale-up, and rates of retention and switching to second-line. HIV transmission was not included. FINDINGS Depending on the scenario, 8·7-25·6 million people are expected to receive antiretroviral therapy in 2020, of whom 0·5-3·0 million will be receiving second-line antiretroviral therapy. The proportion of patients on treatment receiving second-line therapy was highest (15·6%) in the scenario with perfect retention and immediate switching, no further scale-up, and universal routine viral load monitoring. In 2030, the estimated range of patients receiving antiretroviral therapy will remain constant, but the number of patients receiving second-line antiretroviral therapy will increase to 0·8-4·6 million (6·6-19·6%). The need for second-line antiretroviral therapy was two to three times higher if routine viral load monitoring was implemented throughout the region, compared with a scenario of no further viral load monitoring scale-up. For each monitoring strategy, the future proportion of patients receiving second-line antiretroviral therapy differed only minimally between countries. INTERPRETATION Donors and countries in sub-Saharan Africa should prepare for a substantial increase in the need for second-line drugs during the next few years as access to viral load monitoring improves. An urgent need exists to decrease the costs of second-line drugs. FUNDING World Health Organization, Swiss National Science Foundation, National Institutes of Health.

MATHEMATICAL STUDIES ON THE EVOLUTIONARY CHANGE OF DNA SEQUENCES

With the aim of understanding the mechanism of molecular evolution, mathematical problems on the evolutionary change of DNA sequences are studied. The problems studied and the results obtained are as follows: (1) Estimation of evolutionary distance between nucleotide sequences. Studying the pattern of nucleotide substitution for the case of unequal substitution rates, a new mathematical formula for estimating the average number of nucleotide substitutions per site between two homologous DNA sequences is developed. It is shown that this formula has a wider applicability than currently available formulae. A statistical method for estimating the number of nucleotide changes due to deletion and insertion is also developed. (2) Biases of the estimates of nucleotide substitutions obtained by the restriction enzyme method. The deviation of the estimate of nucleotide substitutions obtained by the restriction enzyme method from the true value is investigated theoretically. It is shown that the amount of the deviation depends on the nucleotides in the recognition sequence of the restriction enzyme used, unequal rates of substitution among different nucleotides, and nucleotide frequences, but the primary factor is the unequal rates of nucleotide substitution. When many different kinds of enzymes are used, however, the amount of average deviation is generally small. (3) Distribution of restriction fragment lengths. To see the effect of undetectable restriction fragments and fragment differences on the estimate of nucleotide differences, the theoretical distribution of fragment lengths is studied. This distribution depends on the type of restriction enzymes used as well as on the relative frequencies of four nucleotides. It is shown that undetectability of small fragments or fragment differences gives a serious underestimate of nucleotide substitutions when the length-difference method of estimation is used, but the extent of underestimation is small when the site-difference method is used. (4) Evolutionary relationships of DNA sequences in finite populations. A mathematical theory on the expected evolutionary relationships among DNA sequences (nucleons) randomly chosen from the same or different populations is developed under the assumption that the evolutionary change of nucleons is determined solely by mutation and random genetic drift. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author). UMI ^