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

Altered collagen-induced adhesion and aggregate formation of <i>5Htt</i>^<i>-/-</i> platelets under flow.

<p>(A) Whole blood from <i>Wt</i> or <i>5Htt</i>^<i>-/-</i> mice was perfused over a collagen coated surface (0.2 mg/mL) at a shear rate of 1000 s^-1. Representative brightfield (BF) and fluorescence (DyLight 488 conjugated anti-GPIX) images of aggregate formation on collagen after 4 minutes of perfusion time are shown. Mean surface coverage (%) ± SD and relative thrombus volume expressed as integrated fluorescence intensity (IFI) ± SD in both <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> mice. (B) Impaired procoagulant activity of blood samples from <i>5Htt</i>^<i>-/-</i> mice. PS exposure was detected by Annexin-V-Dylight 488 under similar flow conditions as described above. (C) Restored adhesion and aggregate formation of <i>5Htt</i>^<i>-/-</i> platelets by co-infused 5-HT on a collagen coated surface under flow. Heparinized whole blood of either <i>Wt</i> or <i>5Htt</i>^<i>-/-</i> mice was perfused with 10 μM 5-HT over a collagen coated surface (0.2 mg/mL) at a shear rate of 1000 s^-1.</p

Abolished 5-HT uptake in <i>5Htt</i>^<i>-/-</i> platelets.

<p>Defective (hem)ITAM induced integrin activation, α-granule release and aggregation responses in <i>5Htt</i>^{<b><i>-/-</i></b>} platelets. (A) Platelet count and (B) size measured by FACS analysis or with a hematology analyzer (Sysmex). (C) Measurement of released platelet 5-HT before and after agonist dependent activation. 5-HT ELISA was performed with washed platelets of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> mice (n.d.: not detectable). Total 5-HT concentration was quantified in platelet lysates. (D) 5-HT concentration was measured in blood plasma with a 5-HT ELISA performed with platelet poor plasma (PPP) of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> mice. (E) 5-HIAA and (F) melatonin concentration was measured in urine and plasma samples of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i>, respectively with a 5-HIAA and melatonin ELISA. (G) Spreading of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> platelets on a fibrinogen coated surface in the presence of thrombin. Washed platelets of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> mice were allowed to spread for 5, 15 and 30 min after stimulation with 0.01 U/mL thrombin. Statistical evaluation of the percentage of spread platelets at different spreading stages. 1: roundish; 2: only filopodia; 3: filopodia and lamellipodia; 4: fully spread. (H) Flow cytometric analysis of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> platelets. Integrin αIIbβ3 activation (upper panel) and degranulation (lower panel) in response to the indicated agonists were measured on a FACSCalibur. Results presented as MFI ± SD (I) Aggregation responses of <i>5Htt</i>^<i>-/-</i> platelets (grey line) compared to <i>Wt</i> platelets (black line) measured by change in light transmission upon activation with indicated agonists. ADP measurements were performed in platelet rich plasma (PRP), whereas all others were performed in washed platelets (Thr: thrombin; U46.: stable thromboxane A₂ analogue U46619; CRP: collagen-related peptide; Coll: HORM collagen; Rhd: rhodocytin; CVX: convulxin).</p

Store operated Ca²⁺ entry is reduced in <i>5Htt</i>^<i>-/-</i> platelets.

<p>(A) Normal Ca²⁺ store release (upper panel), but reduced Ca²⁺ response to platelet agonists in the presence of extracellular CaCl₂ (lower panel) in <i>5Htt</i>^<i>-/-</i> platelets. Fura-2-loaded <i>Wt</i> (black bars) or <i>5Htt</i>^<i>-/-</i> (grey bars) platelets were stimulated with the indicated agonists in calcium-free medium or in the presence of extracellular 1 mM CaCl₂, and [Ca²⁺]_i was monitored by fluorimetry. Representative measurements and maximal Δ[Ca²⁺]_i ± standard deviation (SD) are shown. (B) Platelets were labeled and stimulated under similar conditions as Fig 4A in the presence of 10 μM 5-HT. Representative measurements and maximal Δ[Ca²⁺]_i ± standard deviation (SD) are shown. (C) Unaltered cytoplasmic Ca²⁺ level in resting platelets and TG induced Ca²⁺ store release in <i>5Htt</i>^<i>-/-</i> platelets. (D) Reduced TG induced SOCE in <i>5Htt</i>^<i>-/-</i> platelets. SOCE was measured in fura-2-loaded platelets stimulated with 0.1 or 0.05 μM TG for 5 min followed by the addition of 1 mM extracellular CaCl₂. Representative Ca²⁺ curves were indicated (black line: <i>Wt</i>; gray line: <i>5Htt</i>^<i>-/-</i>). Maximal Δ[Ca²⁺]_i of SOCE were quantified. Data are mean ± SD.</p

Altered hemostasis and thrombus formation in <i>5Htt</i>^<i>-/-</i> mice.

<p>(A) Prolonged tail bleeding times of <i>5Htt</i>^<i>-/-</i> mice. Each symbol represents one animal. (B-C) Impaired occlusion time of <i>5Htt</i>^<i>-/-</i> mice in models of arterial thrombosis. The abdominal aorta was injured by firm compression with forceps and blood flow was monitored by an ultrasonic flow probe until complete vessel occlusion occurred or for 30 min (B). Time to stable vessel occlusion of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> mice was determined. Each symbol represents one animal.(C) The right carotid artery was injured by the application of a FeCl₃ soaked filter paper and blood flow was monitored for 30 min using a Doppler flow probe. Each symbol represents one animal. (D) Ischemic stroke development in <i>5Htt</i>^<i>-/-</i> mice using the transient middle cerebral artery occlusion (tMCAO) model with 60 min. Representative images of coronal brain sections (left) stained with 2,3,5-triphenyltetrazolium chloride after 24 hours are shown. Infarct volume was measured by planimetry 24 h after (right panel). (E) Bederson score and (F) grip test were determined 24 h after tMCAO. Each symbol represents one animal. Results represent mean ± SD. (G) Number of infiltrated leukocytes in the ischemic brain of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> mice. Representative pictures of Ly6B.2 immunostaining show similar number of leukocytes in both <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> ischemic brains. (H) Proposed regulatory role of SOCE in serotonergic system in platelets. During platelet agonist dependent Ca²⁺ store depletion, STIM1 binds Ora1 and modulates SOCE. Enhanced Orai1 activity supports degranulation and 5-HT secretion. Secreted platelet 5-HT further amplifies Ca²⁺ store depletion through 5HTR2A-Gq-PLCβ signaling. At the same time, 5-HT uptake is inhibited by SOCE to keep secreted platelet 5-HT in the extracellular milieu. The functional link between Orai1 mediated SOCE and 5HTT is so far unknown. Abbreviations: phospholipase C (PLC), inositol trisphosphate receptor (IP₃R), inositol trisphosphate (IP₃), 5-HT transporter (5HTT), serotonin (5-HT), serotonin receptor (5HTR2A).</p

Defective (hem)ITAM signaling is rescued by addition of extracellular 5-HT.

<p>(A) Flow cytometric analysis of <i>Wt</i> and <i>5Htt</i>^<i>-/-</i> platelets in the presence of 10 μM 5-HT. Agonists and concentrations are indicated. Integrin αIIbβ3 activation was detected by JON/A-PE (upper panel) and degranulation was detected by anti-P-selectin-FITC as a marker of α-granule secretion (lower panel). Results are MFI ± SD. (B) Aggregation responses of <i>5Htt</i>^<i>-/-</i> platelets (grey line) compared to <i>Wt</i> platelets (black line) in the presence of 10 μM 5-HT upon activation with indicated agonists. Measurements were performed in washed platelets. Light transmission was recorded on a Fibrintimer 4-channel aggregometer and representative aggregation traces of at least 3 individual experiments are depicted. (Rest: resting platelets, Thr: thrombin, U46: U46619, a stable thromboxane A₂ analogue, CRP: collagen-related peptide; CVX: convulxin, Rhd: rhodocytin, Coll.: HORM collagen).</p