Methods of applied mathematics with a software overview

This textbook, now in its second edition, provides students with a firm grasp of the fundamental notions and techniques of applied mathematics as well as the software skills to implement them. The text emphasizes the computational aspects of problem solving as well as the limitations and implicit assumptions inherent in the formal methods. Readers are also given a sense of the wide variety of problems in which the presented techniques are useful. Broadly organized around the theme of applied Fourier analysis, the treatment covers classical applications in partial differential equations and boundary value problems, and a substantial number of topics associated with Laplace, Fourier, and discrete transform theories. Some advanced topics are explored in the final chapters such as short-time Fourier analysis and geometrically based transforms applicable to boundary value problems. The topics covered are useful in a variety of applied fields such as continuum mechanics, mathematical physics, control theory, and signal processing. Replete with helpful examples, illustrations, and exercises of varying difficulty, this text can be used for a one- or two-semester course and is ideal for students in pure and applied mathematics, physics, and engineering. Key features of the software overview: Now relies solely on the free software tools Octave, Maxima, and Python. Appendix introduces all of these tools at a level suitable to those with some programming experience Provides references to sources of further learning. Code snippets incorporated throughout the text. All graphics and illustrations generated using these tools. Praise for the first edition: “The author mixed in a remarkable way theoretical results and applications illustrating the results. Flexibility of presentation (increasing and decreasing level of rigor, accessibility) is a key feature...The book contains extensive examples, presented in an intuitive way with high quality figures (some of them quite spectacular)…” – Mathematica “...Davis's book has many novel features being quite different from most other textbooks on applied mathematics.... Mainly it has a clear and consistent exposition with a strong focus on mathematical fundamentals and useful techniques. It has numerous extensive examples, illustrations, comments, and a very modern graphical presentation of results. “…The book has style. Every theorem and mathematical result has a wonderful appealing comment.” – Studies in Informatics and Control

Hypercholesterolemia induced cerebral small vessel disease

<div><p>Background</p><p>While hypercholesterolemia plays a causative role for the development of ischemic stroke in large vessels, its significance for cerebral small vessel disease (CSVD) remains unclear. We thus aimed to understand the detailed relationship between hypercholesterolemia and CSVD using the well described <i>Ldlr</i>^<i>-/-</i> mouse model.</p><p>Methods</p><p>We used <i>Ldlr</i>^<i>-/-</i> mice (n = 16) and wild-type (WT) mice (n = 15) at the age of 6 and 12 months. <i>Ldlr</i>^<i>-/-</i> mice develop high plasma cholesterol levels following a high fat diet. We analyzed cerebral capillaries and arterioles for intravascular erythrocyte accumulations, thrombotic vessel occlusions, blood-brain barrier (BBB) dysfunction and microbleeds.</p><p>Results</p><p>We found a significant increase in the number of erythrocyte stases in 6 months old <i>Ldlr</i>^<i>-/-</i> mice compared to all other groups (<i>P</i> < 0.05). <i>Ldlr</i>^<i>-/-</i> animals aged 12 months showed the highest number of thrombotic occlusions while in WT animals hardly any occlusions could be observed (<i>P</i> < 0.001). Compared to WT mice, <i>Ldlr</i>^<i>-/-</i> mice did not display significant gray matter BBB breakdown. Microhemorrhages were observed in one <i>Ldlr</i>^<i>-/-</i> mouse that was 6 months old. Results did not differ when considering subcortical and cortical regions.</p><p>Conclusions</p><p>In <i>Ldlr</i>^<i>-/-</i> mice, hypercholesterolemia is related to a thrombotic CSVD phenotype, which is different from hypertension-related CSVD that associates with a hemorrhagic CSVD phenotype. Our data demonstrate a relationship between hypercholesterolemia and the development of CSVD. <i>Ldlr</i>^<i>-/-</i> mice appear to be an adequate animal model for research into CSVD.</p></div