Novel solutions for a model of wound healing angiogenesis

We prove the existence of novel, shock-fronted travelling wave solutions to a model of wound healing angiogenesis studied in Pettet et al (2000 IMA J. Math. App. Med. 17 395–413) assuming two conjectures hold. In the previous work, the authors showed that for certain parameter values, a heteroclinic orbit in the phase plane representing a smooth travelling wave solution exists. However, upon varying one of the parameters, the heteroclinic orbit was destroyed, or rather cut-off, by a wall of singularities in the phase plane. As a result, they concluded that under this parameter regime no travelling wave solutions existed. Using techniques from geometric singular perturbation theory and canard theory, we show that a travelling wave solution actually still exists for this parameter regime. We construct a heteroclinic orbit passing through the wall of singularities via a folded saddle canard point onto a repelling slow manifold. The orbit leaves this manifold via the fast dynamics and lands on the attracting slow manifold, finally connecting to its end state. This new travelling wave is no longer smooth but exhibits a sharp front or shock. Finally, we identify regions in parameter space where we expect that similar solutions exist. Moreover, we discuss the possibility of more exotic solutions.

A probabilistic framework for analysing the compositionality of conceptual combinations

Conceptual combination performs a fundamental role in creating the broad range of compound phrases utilised in everyday language. While the systematicity and productivity of language provide a strong argument in favour of assuming compositionality, this very assumption is still regularly questioned in both cognitive science and philosophy. This article provides a novel probabilistic framework for assessing whether the semantics of conceptual combinations are compositional, and so can be considered as a function of the semantics of the constituent concepts, or not. Rather than adjudicating between different grades of compositionality, the framework presented here contributes formal methods for determining a clear dividing line between compositional and non-compositional semantics. Compositionality is equated with a joint probability distribution modelling how the constituent concepts in the combination are interpreted. Marginal selectivity is emphasised as a pivotal probabilistic constraint for the application of the Bell/CH and CHSH systems of inequalities (referred to collectively as Bell-type). Non-compositionality is then equated with either a failure of marginal selectivity, or, in the presence of marginal selectivity, with a violation of Bell-type inequalities. In both non-compositional scenarios, the conceptual combination cannot be modelled using a joint probability distribution with variables corresponding to the interpretation of the individual concepts. The framework is demonstrated by applying it to an empirical scenario of twenty-four non-lexicalised conceptual combinations.

Commentary on the article by Swanenburg et al. ‘Validity and reliability of a German version of the Neck Disability Index (NDI-G)’

We commend Swanenburg et al. (2013) on translation, development, and clinimetric analysis of the NDI-G. However, the dual-factor structure with factor analysis and the high level of internal consistency (IC) highlighted in their discussion were not emphasized in the abstract or conclusion. These points may imply some inconsistencies with the final conclusions since determination of stable point estimates with the study's small sample are exceedingly difficult.

Secondary structure element alignment kernel method for prediction of protein structural classes

In this paper, we aim at predicting protein structural classes for low-homology data sets based on predicted secondary structures. We propose a new and simple kernel method, named as SSEAKSVM, to predict protein structural classes. The secondary structures of all protein sequences are obtained by using the tool PSIPRED and then a linear kernel on the basis of secondary structure element alignment scores is constructed for training a support vector machine classifier without parameter adjusting. Our method SSEAKSVM was evaluated on two low-homology datasets 25PDB and 1189 with sequence homology being 25% and 40%, respectively. The jackknife test is used to test and compare our method with other existing methods. The overall accuracies on these two data sets are 86.3% and 84.5%, respectively, which are higher than those obtained by other existing methods. Especially, our method achieves higher accuracies (88.1% and 88.5%) for differentiating the α + β class and the α/β class compared to other methods. This suggests that our method is valuable to predict protein structural classes particularly for low-homology protein sequences. The source code of the method in this paper can be downloaded at http://math.xtu.edu.cn/myphp/math/research/source/SSEAK_source_code.rar.

Energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

This work deals with the formulation and implementation of an energy-momentum conserving algorithm for conducting the nonlinear transient analysis of structures, within the framework of stress-based hybrid elements. Hybrid elements, which are based on a two-field variational formulation, are much less susceptible to locking than conventional displacement-based elements within the static framework. We show that this advantage carries over to the transient case, so that not only are the solutions obtained more accurate, but they are obtained in fewer iterations. We demonstrate the efficacy of the algorithm on a wide range of problems such as ones involving dynamic buckling, complicated three-dimensional motions, et cetera.

Application of commercially available, low-cost, miniaturised NIR spectrometers to the assessment of the sugar content of intact fruit

Recent decreases in costs, and improvements in performance, of silicon array detectors open a range of potential applications of relevance to plant physiologists, associated with spectral analysis in the visible and short-wave near infra-red (far-red) spectrum. The performance characteristics of three commercially available ‘miniature’ spectrometers based on silicon array detectors operating in the 650–1050-nm spectral region (MMS1 from Zeiss, S2000 from Ocean Optics, and FICS from Oriel, operated with a Larry detector) were compared with respect to the application of non-invasive prediction of sugar content of fruit using near infra-red spectroscopy (NIRS). The FICS–Larry gave the best wavelength resolution; however, the narrow slit and small pixel size of the charge-coupled device detector resulted in a very low sensitivity, and this instrumentation was not considered further. Wavelength resolution was poor with the MMS1 relative to the S2000 (e.g. full width at half maximum of the 912 nm Hg peak, 13 and 2 nm for the MMS1 and S2000, respectively), but the large pixel height of the array used in the MMS1 gave it sensitivity comparable to the S2000. The signal-to-signal standard error ratio of spectra was greater by an order of magnitude with the MMS1, relative to the S2000, at both near saturation and low light levels. Calibrations were developed using reflectance spectra of filter paper soaked in range of concentrations (0–20% w/v) of sucrose, using a modified partial least squares procedure. Calibrations developed with the MMS1 were superior to those developed using the S2000 (e.g. coefficient of correlation of 0.90 and 0.62, and standard error of cross-validation of 1.9 and 5.4%, respectively), indicating the importance of high signal to noise ratio over wavelength resolution to calibration accuracy. The design of a bench top assembly using the MMS1 for the non-invasive assessment of mesocarp sugar content of (intact) melon fruit is reported in terms of light source and angle between detector and light source, and optimisation of math treatment (derivative condition and smoothing function).

Generalized Burgers equations and Euler-Painleve transcendents. II

It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986)] that the Euler Painlevé equation yy[script `]+ay[script ']2+ f(x)yy[script ']+g(x) y2+by[script ']+c=0 represents the generalized Burgers equations (GBE's) in the same manner as Painlevé equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE ut+uaux+Ju/2t =(gd/2)uxx (the nonplanar Burgers equation) is considered. It is found that its self-similar form is again governed by the Euler Painlevé equation. The ranges of the parameter alpha for which solutions of the connection problem to the self-similar equation exist are obtained numerically and confirmed via some integral relations derived from the ODE's. Special exact analytic solutions for the nonplanar Burgers equation are also obtained. These generalize the well-known single hump solutions for the Burgers equation to other geometries J=1,2; the nonlinear convection term, however, is not quadratic in these cases. This study fortifies the conjecture regarding the importance of the Euler Painlevé equation with respect to GBE's. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Idealähtöistä koulualgebraa : IDEAA-opetusmallin kehittäminen algebran opetukseen peruskoulun 7. luokalla

This research is based on the problems in secondary school algebra I have noticed in my own work as a teacher of mathematics. Algebra does not touch the pupil, it remains knowledge that is not used or tested. Furthermore the performance level in algebra is quite low. This study presents a model for 7th grade algebra instruction in order to make algebra more natural and useful to students. I refer to the instruction model as the Idea-based Algebra (IDEAA). The basic ideas of this IDEAA model are 1) to combine children's own informal mathematics with scientific mathematics ("math math") and 2) to structure algebra content as a "map of big ideas", not as a traditional sequence of powers, polynomials, equations, and word problems. This research project is a kind of design process or design research. As such, this project has three, intertwined goals: research, design and pedagogical practice. I also assume three roles. As a researcher, I want to learn about learning and school algebra, its problems and possibilities. As a designer, I use research in the intervention to develop a shared artefact, the instruction model. In addition, I want to improve the practice through intervention and research. A design research like this is quite challenging. Its goals and means are intertwined and change in the research process. Theory emerges from the inquiry; it is not given a priori. The aim to improve instruction is normative, as one should take into account what "good" means in school algebra. An important part of my study is to work out these paradigmatic questions. The result of the study is threefold. The main result is the instruction model designed in the study. The second result is the theory that is developed of the teaching, learning and algebra. The third result is knowledge of the design process. The instruction model (IDEAA) is connected to four main features of good algebra education: 1) the situationality of learning, 2) learning as knowledge building, in which natural language and intuitive thinking work as "intermediaries", 3) the emergence and diversity of algebra, and 4) the development of high performance skills at any stage of instruction.

Water-wave scattering by two submerged plane vertical barriers-Abel integral-equation approach

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207-215, 1971) who employed a completely different approach involving a Schwarz-Christoffel transformation of complex-variable theory to solve the problem.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

The alpha method for solving differential algebraic inequality (DAI) systems

This paper describes an algorithm for ``direct numerical integration'' of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation's (DAE) Jacobian and this difficulty is addressed by a regularization property of the alpha method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

Mechanics of Deformation under Traction and Friction of a Micrometric Monolithic MoS2 Particle in Comparison with those of an Agglomerate of Nanometric MoS2 Particles

Tribology of small inorganic nanoparticles in suspension in a liquid lubricant is often impaired because these particles agglomerate even when organic dispersants are used. In this paper we use lateral force microscopy to study the deformation mechanism and dissipation under traction of two extreme configurations (1) a large MoS2 particle (similar to 20 mu m width) of about 1 mu m height and (2) an agglomerate (similar to 20 mu m width), constituting 50 nm MoS2 crystallites, of about 1 mu m height. The agglomerate records a friction coefficient which is about 5-7 times that of monolithic particle. The paper examines the mechanisms of material removal for both the particles using continuum modeling and microscopy and infers that while the agglomerate response to traction can be accounted for by the bulk mechanical properties of the material, intralayer and interlayer basal planar slips determine the friction and wear of monolithic particles. The results provide a rationale for selection of layered particles, for suspension in liquid lubricants.

A multistep transversal linearization (MTL) method in non-linear dynamics through a Magnus characterization

A new form of a multi-step transversal linearization (MTL) method is developed and numerically explored in this study for a numeric-analytical integration of non-linear dynamical systems under deterministic excitations. As with other transversal linearization methods, the present version also requires that the linearized solution manifold transversally intersects the non-linear solution manifold at a chosen set of points or cross-section in the state space. However, a major point of departure of the present method is that it has the flexibility of treating non-linear damping and stiffness terms of the original system as damping and stiffness terms in the transversally linearized system, even though these linearized terms become explicit functions of time. From this perspective, the present development is closely related to the popular practice of tangent-space linearization adopted in finite element (FE) based solutions of non-linear problems in structural dynamics. The only difference is that the MTL method would require construction of transversal system matrices in lieu of the tangent system matrices needed within an FE framework. The resulting time-varying linearized system matrix is then treated as a Lie element using Magnus’ characterization [W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., VII (1954) 649–673] and the associated fundamental solution matrix (FSM) is obtained through repeated Lie-bracket operations (or nested commutators). An advantage of this approach is that the underlying exponential transformation could preserve certain intrinsic structural properties of the solution of the non-linear problem. Yet another advantage of the transversal linearization lies in the non-unique representation of the linearized vector field – an aspect that has been specifically exploited in this study to enhance the spectral stability of the proposed family of methods and thus contain the temporal propagation of local errors. A simple analysis of the formal orders of accuracy is provided within a finite dimensional framework. Only a limited numerical exploration of the method is presently provided for a couple of popularly known non-linear oscillators, viz. a hardening Duffing oscillator, which has a non-linear stiffness term, and the van der Pol oscillator, which is self-excited and has a non-linear damping term.

Formal Verification of Full-Wave Rectifier: A Case Study

We present a case study of formal verification of full-wave rectifier for analog and mixed signal designs. We have used the Checkmate tool from CMU [1], which is a public domain formal verification tool for hybrid systems. Due to the restriction imposed by Checkmate it necessitates to make the changes in the Checkmate implementation to implement the complex and non-linear system. Full-wave rectifier has been implemented by using the Checkmate custom blocks and the Simulink blocks from MATLAB from Math works. After establishing the required changes in the Checkmate implementation we are able to efficiently verify, the safety properties of the full-wave rectifier.

Gauge-invariant reformulation of an anomalous gauge theory

An anomalous gauge theory can be reformulated in a gauge invariant way without any change in its physical content. This is demonstrated here for the exactly soluble chiral Schwinger model. Our gauge invariant version is very different from the Faddeev-Shatashvili proposal [L.D. Faddeev and S.L. Shatashvili, Theor. Math. Phys. 60 (1984) 206] and involves no additional gauge-group-valued fields. The status of the "gauge" A0=0 sometimes used in anomalous theories is also discussed and justified in our reformulation.