text:
NULL
Countries | Regions
Countries | Regions
Article Types
Article Types
Year
Volume
Issue
Pages
IMR Press / FBL / Volume 27 / Issue 5 / DOI: 10.31083/j.fbl2705153
Open Access Original Research
Platelet TAU is Associated with Changes in Depression and Alzheimer's Disease
Show Less
1 Protein Core Facility, Institute of Medical Biochemistry, Biocenter, Medical University of Innsbruck, 6020 Innsbruck, Austria
2 Laboratory of Psychiatry and Experimental Alzheimer’s Research, Medical University of Innsbruck, 6020 Innsbruck, Austria
3 Department of Psychiatry and Psychotherapy A, Hall State Hospital, 6060 Hall in Tirol, Austria
*Correspondence: christian.humpel@i-med.ac.at (Christian Humpel)
Academic Editors: Fabio Moda and Giorgio Giaccone
Front. Biosci. (Landmark Ed) 2022 , 27(5), 153; https://doi.org/10.31083/j.fbl2705153
Submitted: 1 March 2022 | Revised: 4 April 2022 | Accepted: 22 April 2022 | Published: 12 May 2022
This is an open access article under the CC BY 4.0 license.
Abstract

Background: Platelets (thrombocytes) are small anuclear cells that play an important role in blood clotting. They are activated and dysfunctional in brain disorders, such as Alzheimer’s disease (AD) and depression. Platelets express the amyloid-precursor protein (APP) and release beta-amyloid${}_{40}$ into the blood. Recent evidence reports that platelets also express the microtubule-associated protein tau. In this study, we further characterized the molecular appearance of tau and examined its alterations in patients with neurocognitive impairment. Methods: Platelets were isolated from patients with AD, mild cognitive impairment (MCI) or depression and compared to healthy controls. Subsequently, FACS analysis was employed to characterize platelets for platelet surface P-selectin (CD62P). In order to enhance the detection levels, samples were pooled (15 samples per group) and analyzed by Lumipulse Assay, Western blots, and mass spectrometry. Results: Tau is expressed in human platelets and tau levels were decreased in platelets isolated from patients with AD and depression. Additionally, phospho-tau-181 was slightly increased in patients with depression. We show that tau is highly fragmented (20–40 kDa) in the platelet extracts using Western blot analysis. The mass spectrometry data did not show a clear identification of tau in the pooled platelet samples. Conclusions: Our data reveal that tau is found in platelets, possibly in a highly fragmented form. Tau levels may be used as a potential diagnostic approach to differentiate AD and depression from healthy controls.

Keywords
platelets
thrombocytes
tau
phospho-tau
mass spectrometry
Alzheimer
1. Introduction

The life expectancy of humans has markedly increased over the last 100 years. As age is the key risk factor for Alzheimer’s disease (AD), the number of patients suffering from AD and mixed forms of dementia will dramatically increase over the next 50 years. It is expected that there will be about 80 million AD patients worldwide in 2050. Sporadic AD is a progressive chronic neurodegenerative disorder (at least 97% of all cases are non-genetic) and is characterized by two major pathologies in the brain: the $\beta{}$-amyloid (A$\beta{}$) deposition (plaques) and tau pathology (hyperphosphorylated tau and neurofibrillary tangles). These pathologies are accompanied by cell death of cholinergic neurons, reactive astrogliosis, microglial activation, inflammation, and cerebrovascular damage, including blood-brain barrier (BBB) damage. The causes of sporadic AD are yet unknown, but so far the A$\beta{}$ cascade is the most prominent hypothesis, although it is now surrounded by more and more controversy [1, 2]. One explanation could be a dysfunctional clearance of A$\beta{}$ through the BBB [3]. Further, there is more and more evidence that cerebrovascular dysfunction might be an early event in the pathogenesis of AD [4, 5, 6, 7]. It is well established that A$\beta{}$ deposits are also found in brain vessels, the so-called cerebral amyloid angiopathy (CAA), but it is not known whether vessel damage is a primary event during AD progression or whether it is caused by brain damage [8, 9]. It has been hypothesized that a vessel damage causes activation of the blood-clotting system in the brain and that platelets become activated and migrate to lesion sites in the brain and use A$\beta{}$40 as a clotting substance [7, 10, 11].

Platelets are of particular interest in AD research because there is clear evidence that they become activated and dysfunctional in AD patients [11, 12, 13]. In an AD mouse model, we recently showed that platelets are early signs in vessels before A$\beta{}$ plaques develop [14, 15, 16]. We also provide clear evidence that platelets are found in blood vessels in postmortem brains of human AD patients [14]. Platelets contain high levels of APP. Both APP and A$\beta{}$ are stored in $\alpha{}$-granules of platelets and become released upon activation by agents like the physiological agonists thrombin and collagen. Several studies, including from our lab, showed that the platelet APP ratio (defined as the ratio between the upper 130 kDa and the lower 106–110 kDa isoforms) is significantly lower in AD patients compared to controls and patients with other forms of dementia [17, 18, 10, 19, 20]. While the pathology of APP and A$\beta{}$ seems to be well established, the role of the second important pathology, tau is nearly unknown. Recent novel evidence suggests that tau is found in the platelets, but the role is completely unknown. It has been shown that platelet tau may correlate with moderate cognitive decline since there was an increase in C-terminal platelet tau in cognitive impaired people with a mini-mental state examination scale (MMSE) of 24–27 [21, 22, 23, 24].

Tau is a microtubule-associated protein and physiologically stabilizes and regulates axonal transport [25, 26, 27, 28]. However, the physiological roles of tau seem to be more complex and by far not fully explored. Tau has more than 40 possible phosphorylation sites and the tau phosphorylation (pTau) is regulated by several kinases and phosphatases. It is extremely important to identify these critical phosphorylation sites in the tau protein, as they are either therapeutic or diagnostic targets [26, 27, 28, 29]. Tau and phospho-tau-181 are well-established biomarkers in cerebrospinal fluid to diagnose AD, with levels of $>$500 pg/mL and $>$60 pg/mL for tau and phospho-tau-181, respectively [30]. Tau is also markedly enhanced in Creutzfeldt-Jakob disease and can reach values of $>$3000 pg/mL [31]. In addition, tau is a general marker of neuronal degeneration and it is well known that stroke or multifocal cerebral microinfarcts (silent stroke) contributes to the progression of AD [32]. The function of human platelet tau is fully unknown as the detection levels of platelet tau are very low. To the best of our knowledge, the structure of tau is not established yet in human platelets using mass spectrometry. Thus, the aim of the present study is to characterize tau in human platelets. Moreover, we aimed to distinguish differences in platelets isolated from healthy controls and patients with mild cognitive impairment (MCI) and AD. In order to include an additional control we also aim to test platelets from patients with depression. This last group is also of special interest, as AD and MCI patients also complain from depressive symptoms. We hypothesize that tau is dysregulated in platelets of AD patients, as also the other AD specific biomarker (amyloid precursor protein) is altered.

2. Materials and Methods
2.1 Patients

Cognitively healthy subjects and patients suffering from AD, MCI or depression were recruited at the Landeskrankenhaus Hall/Tirol, Austria as reported in detail in several previous studies [19, 20]. The following groups were included in this study: healthy controls (group 1, n = 93), MCI patients (group 2, n = 111), AD patients (group 3, n = 233) and depressive patients without marked cognitive impairment (group 4, n = 76). All four groups were assessed by the same diagnostic procedure. Diagnosis of AD was established by a structured routine process including clinical assessment, extensive neuropsychological tests (including MMSE) and neuroimaging (magnetic resonance imaging, MRI) to exclude other brain pathologies. For each subject, the neuropsychological examination was conducted between 9:00 and 11:00 in the morning. Probable AD was diagnosed according to the current NINCDS-ADRDA (National Institute of Neurological and Communicative Disorders and Stroke and the AD and Related Disorders Association) criteria and confirmed for all participating patients. MCI was diagnosed according to previously published criteria [33]. The geriatric depression scale (GDS) was applied to each participant. A general blood examination was part of the routine diagnostic procedure. Exclusion criteria for healthy subjects, MCI and AD patients included (1) another primary neurological or mental disorder, (2) any kind of metabolic decompensation or any sign of peripheral inflammation (e.g., rheumatic disease), (3) long-term alcohol or drug abuse, (4) or any current, clinically significant cardiovascular disease. The study was approved by the ethics committee of Medical University of Innsbruck. All subjects and/or their caregivers enrolled in the study gave their informed consent.

2.2 Isolation of Human Platelets and Processing

Human platelets were isolated as previously described [20]. In short, 10 mL blood was collected in ethylenediaminetetraacetate (EDTA) tubes during normal clinical routine, centrifuged at 250 $\times{}$g for 10 min at RT to collect platelet-rich plasma (PRP). PRP was centrifuged for 10 minutes at RT (2300 $\times{}$g) and the pellet was dissolved in 1 mL Tyrode’s buffer and PGI2 was added in order to inhibit platelet activation. Next, the dissolved pellet was resuspended in 300 $\mathrm{\mu}$L PBS and an aliquot (20 $\mathrm{\mu}$L) used for FACS analysis, and the rest again centrifuged and the pellet stored at –80 °C until further use. In order to increase the detection levels, samples from 15 patients of the same group were pooled into 500 $\mathrm{\mu}$L PBS + protease inhibitor cocktail, then sonicated with a Branson Sonifier (10 strokes each 10 seconds on ice), then centrifuged at 14,000 $\times{}$g for 10 min, and the supernatant was collected. 400 $\mathrm{\mu}$L were analyzed by mass spectrometry. The rest was used for determination of total protein using Bradford assay, Western blot or Lumipulse tau analysis using the G600 II system (Fujirebio). As a control, platelets were isolated from adult 6 month old C57BL6 wildtype mice as reported previously by us [15, 16]. The animal experiment was approved by the Austrian Ministry of Science and Research (66.011/0055-WF/V/3b/2017) and all animal-related lab work was in line with the Austrian and international guidelines on animal welfare and experimentation.

2.3 FACS Analysis

FACS analysis was performed as described in detail by us [34]. Immediately after isolation, 2 $\mathrm{\mu}$L of mouse platelets were mixed with 46 $\mathrm{\mu}$L of FACs buffer (2 mM EDTA, 0.5% FCS ad 100 mL PBS, pH 7.1), and then the reaction started with addition of 2 $\mathrm{\mu}$L antibodies against FITC Mouse IgG1 (BD Biosciences, Cat: 555748, 1:25) or FITC Mouse Anti-Human CD62P (BD Biosciences, Cat: 555523, 1:25). The samples were vortexed, and incubated for 30 min at 4 °C in the dark, then centrifuged at 300 $\times{}$g for 10 min and the pellets were resuspended in 100 $\mathrm{\mu}$L of FACs flow (BD FACSFlow) and the FACs analysis was done with a BD FACScan.

2.4 Western Blot Analysis

Western blot analysis was performed for tau and phospho-tau396 as described by us in detail [35]. Platelet extracts (80 $\mathrm{\mu}$g per lane total protein) were denatured (70 °C for 10 min) and loaded onto 4–12% Bis-Tris polyacrylamide gels (Invitrogen Life Tech, Darmstadt, Germany) and separated by electrophoresis for 60 min at 200 V. Consecutively, samples were electrotransferred to nylon PVDF Immobilon-PSQ membranes (Millipore, Vienna, Austria) at 30 V for 90 min with 20% methanol transfer buffer (Invitrogen). Protein detection was performed with Western Breeze Chemiluminescent System (Invitrogen). Thus, membranes were blocked for 30 min with blocking solution at RT on the shaker, then incubated with the primary antibody tau-5 (Thermo AHB0042, 1:1000) or phospho-tau-396 (pTau396, BioLegend 807401, 1:10,000) or HT7 antibody (1:500, Thermo MN1000) overnight at 4 °C. Subsequently, blots were washed and incubated with anti-mouse (tau-5, HT7) or anti-rabbit (pTau396) antibodies for 30 min at room temperature (RT), again washed and incubated with CDP-Star chemiluminescent substrate solution (Invitrogen). Imaging was performed with a cooled CCD camera (SearchLight, ThermoScientific, Austria).

2.5 qRT-PCR

Collected platelets of 15 patients were pooled in a tube, homogenized by sonication and total RNA was extracted with QIAGEN RNeasy Mini Kit (Qiagen, Germany) according to manufacturer’s protocol. Total RNA concentrations were determined using Nanoquant Infinite M200 (Tecan, Switzerland). RNA integrity was checked in random samples with Fragment Analyzer (Advanced Analytical Technologies). In vitro reverse transcription was performed from 250 ng of total RNA in 20 $\mathrm{\mu}$L and carried out with Omniscript Reverse Transcription kit (Quiagen, Germany), including random hexamer primers (Promega) and RNAse inhibitor (Sigma) according to the manufacturer’s protocol. qRT-PCR was performed as described by us [35]. The relative abundance of human tau was assessed by TaqMan quantitative PCR (qRT-PCR) using a standard curve method based on normalization to housekeeping gene glyceraldehyde 3-phosphate dehydrogenase (GAPDH). TaqMan gene expression assay specific for human tau gene designed to span exon-exon boundary (Hs00902193_m1) and Gapdh assay (Hs02758991_g1) were used (Applied Biosystems). qRT-PCR was performed as 50 cycles (Ct = 19–30) in triplicates or duplicates, using 1.5 $\mathrm{\mu}$L total RNA equivalents of cDNA and the specific TaqMan gene expression assay for each 20 $\mathrm{\mu}$L reaction in TaqMan Universal PCR Master Mix (Applied Biosystems). Analysis was performed utilizing QuantStudio 6 (Applied Biosystems, Thermo Fisher Sci., Vienna, Austria) and Ct values for each gene expression assay were recorded for each individual reaction. All experiments were normalized to housekeeping Gapdh and relative expression was calculated by using the difference between Ct values of Gapdh and human tau.

2.6 WANGEL Analysis of tau

Platelet tau was characterized using our novel WANGEL (Western Blot-native Agarose gel elution) model developed by us recently (Humpel, 2022 submitted). Native Western Blots were performed as described above but directly after the gel run, different molecular weight bands were dissected with a scalpel with a size between 80 and 0 kDa related to the color molecular weight marker. For the elution, 2.5 mL sample vials (PE, 14 mm, Roth 5863.1) were used and 1 mL 4% agarose was pipetted at the bottom of the tubes and cooled. Then, the cut Western Blot bands were carefully placed on top of the hardened agarose. Next 1 mL 1% handwarm agarose was pipetted onto the extracted bands and cooled to harden. In the next step, the bottom of the vial was cut, and the tube closed with a Millicell culture insert (3 $\mathrm{\mu}$m, 12 mm, PITP01250, Merck). Next 1 mL of a MES/histidine puffer was carefully pipetted into the tube using a syringe and 10 $\mathrm{\mu}$L bromphenobuffer was pipetted to follow up the elution. These tubes were placed into the agarose gel chamber and proteins eluted for 20–60 min to the cathode. After the run, the buffer with the eluted proteins was collected and analyzed by Lumipulse assay or mass spectometry. Total tau levels were measured using automated Lumipulse technology (Fujirebio G600II); see https://www.fujirebio.com/en/products-solutions/lumipulse-g600ii.

2.7 Detection of Tau in Platelets Using Mass Spectrometry

Tau was analyzed by immunoprecipitation and mass spectrometry similar as reported earlier in detail for NT-proBNP [36, 37]. Briefly, a full length 2N4R tau standard (1000 pg/mL) or a CSF sample (1024 pg/mL) or a pooled platelet extract from 15 human patients was immunoprecipitated with a biotin-conjugated anti-Tau monoclonal antibody (20 $\mathrm{\mu}$L = 2 $\mathrm{\mu}$g HT7 antibody), which was coupled to streptavidin-coated magnetic microparticles (Thermo Scientific) overnight at 4 °C. The immunoprecipitate was washed, eluted with 2% formic acid and finally evaporated. The eluate was enzymatically digested with trypsin overnight and the resulting peptide fragments were analyzed using a nanoflow UHPLC instrument coupled to a Q Exactive HF or Orbitrap Eclipse FAIMS Pro mass spectrometer (Thermo Scientific) equipped with a Nanospray Flex ionization source. The identification of the peptides was performed using Proteome Discoverer Software 2.2 (Thermo Fisher Scientific, Vienna, Austria) with Sequest as search engine. The raw file was searched against a human database, to which the sequence of Tau441 and its various isoforms was added. Precursor and fragment mass tolerances were set to 10 ppm and 0.02 Da, respectively, and trypsin with up to 2 missed cleavages was chosen. Oxidation of methionine and phosphorylation of serine, threonine and tyrosine were set as dynamic, carbamidomethylation of cysteine as static modifications. The required false positive rate was set to 1% both at the peptide and protein level. In order to detect tau, the platelets of 15 patients were pooled for each group (G1: healthy controls, G2: MCI, G3: AD and G4: depression).

3. Results
3.1 Epidemiology

In the present study, 513 subjects were included, which were grouped in healthy controls (group 1, n = 93), MCI patients (group 2, n = 111), AD patients (group 3, n = 233) and patients with depression (group 4, n = 76) (Table 1). The age of the patients was 72–75 years, and only AD patients were slightly older (80.9 $\pm{}$ 0.9, p $<$ 0.05) (Table 1). In all groups, nearly 35–37% were male. Only in the depressive group, there were 22% male (Table 1). The MMSE score was 29.5 $\pm{}$ 0.1 for controls, which was significantly lower in the MCI and depression group and markedly lower in the AD group (Table 1). The geriatric depression scale (GDS) was 4.3 $\pm{}$ 0.4 in healthy controls, which was not different in the MCI and AD group, but significantly increased in the depression group (Table 1).

Table 1.Patients characteristics and platelet tau levels.
 Analyte Control MCI AD Depression n (before pooling) 93 111 233 76 Age [years] 74.0 $\pm$ 1.1 75.2 $\pm$ 0.7 ns 80.9 $\pm$ 0.9 * 72.2 $\pm$ 1.0 ns Male/% 35/37.6 39/35.1 87/37.3 17/22.3 MMSE 29.4 $\pm$ 0.1 27.6 $\pm$ 0.2 * 21.1 $\pm$ 0.3 *** 28.3 $\pm$ 0.2 * GDS 4.3 $\pm$ 0.4 5.1 $\pm$ 0.8 ns 4.0 $\pm$ 0.3 ns 8.2 $\pm$ 0.7 *** Pool of 15 (n) 6 7 15 5 Protein [mg/mL] 35.0 $\pm$ 1.8 42.9 $\pm$ 1.3 ns 45.8 $\pm$ 3.0 ns 35.0 $\pm$ 1.6 ns Tau [pg/mg] 818 $\pm$ 34 765 $\pm$ 18 ns 673 $\pm$ 52 * 658 $\pm$ 32 * pT181 [pg/mg] 3.2 $\pm$ 0.18 3.0 $\pm$ 0.03 3.3 $\pm$ 0.13 ns 4.0 $\pm$ 0.3 * Platelets were isolated from healthy controls or patients with mild cognitive impairment (MCI) or Alzheimer’s disease (AD) or with depression. Values are given as mean $\pm{}$ SEM; values in parenthesis give the number of samples. Statistical analysis was performed using One Way ANOVA with a subsequent Fisher LSD posthoc test (* p $<$ 0.05; *** p $<$ 0.001; ns, not significant). MMSE, Mini Mental State Examination; GDS, geriatric depression scale. Tau and phospho-tau-181 (pT181) were analyed by Lumipulse technology. Note: Protein and tau levels were determined in samples pooled from 15 patients per group.
3.2 Characterization of Human Platelets

Isolated human platelets were characterized for CD62P compared to a negative IgG control by FACS (Fig. 1A–C). The Lumipulse assay shows that platelets from healthy controls contain tau (approx. 820 pg/mg protein) and pTau-181 (approx. 3 pg/mg protein) (Table 1). Tau levels were significantly reduced in AD and depression but not in MCI (Table 1). Phospho-tau-181 levels were slightly increased in depression, but not in AD or MCI (Table 1).

Fig. 1.

Flow cytometry (FACS) analysis. (A) A representative sample of human platelets isolated from a healthy control shows a common population of cells in forward and sideward scatter and (C) markedly expressed CD62P (B) compared to an IgG control.

Western blot data shows a full-length tau standard with a size of approx. 60 kDa (Fig. 2) using the tau-5 antibody. No staining for platelet tau was found with the tau-5 antibody (Fig. 2). Using the HT-7 antibody, 2 clear bands were visible at approx. 45 and 30 kDa size, while a signal for phospho-tau-396 (approx. 50 kDa) was present in all samples (Fig. 2). A loading control GAPDH displayed a positive signal at 37 kDa (Fig. 2).

Fig. 2.

Western blot of platelet extracts. Human platelets were isolated, characterized by FACS and 15 samples per group pooled, extracted and 80 $\mathrm{\mu}$g protein was loaded on a gel and analyzed by Western Blots. As a control, a standard (St) of full-length tau (500 ng) was loaded which gives a strong band at approx. 50–60 kDa when stained with the tau-5 antibody. No staining for platelet tau was found with the tau-5 antibody. Using the HT-7 antibody, 2 clear bands were visible at approx. 45 and 30 kDa size, while a signal for phospho-tau-396 (approx. 50 kDa) was present in all samples. Note that the lanes were not always in line. This is because the gel was maximally loaded (80 $\mathrm{\mu}$g protein) in order to increase sensitivity. GAPDH served as a loading control and shows a band at approx. 37 kDa.

3.3 qRT-PCR

Platelets are anuclear cells, however, there are reports that mRNAs can be detected in platelets by qRT-PCR. RNA was isolated from human platelets using the Quiagen minikit and we achieved 1.1 $\pm{}$ 0.4 $\mathrm{\mu}$g/mL RNA (n = 6), which is far lower as isolated from control mouse cortex (209 $\pm{}$ 73 $\mathrm{\mu}$g/mL, n = 3). Using this RNA, we performed qRT-PCR for tau compared to gapd. Compared to control mouse cortex (3.71 $\pm{}$ 0.06 $\Delta{}$CT, n = 3), very low amounts of mRNA were found in wildtype mice (0.097 $\pm{}$ 0.076 $\Delta{}$CT, n = 3), while in human platelets 5 out of 6 samples were undetectable and one sample had 0.003 $\Delta{}$CT values.

3.4 WANGEL tau Detection

Using our novel WANGEL system, we show that full-length tau migrates to the cathode and elutes from the agarose gel as a 50–60 kDa protein (Fig. 3). However, when the platelet extract (a pool of 32 lanes) was eluted from the WANGEL system, several large and small molecular weight species could be detected, but the majority of the tau+ detection was seen as a 20–30 kDa fragment (Fig. 3).

Fig. 3.

Elution of tau from Western Blots linked to agarose gels. Tau protein (100 ng/lane) is separated on a Western Blot gel, then bands with between 0–100 kDa are cut, and tau eluted from agarose gels via the cathode. Note that tau elutes as a 60 kDa protein from Western-agarose gels (red triangles). In order to charcterize tau from platelet, an extract (89.5 ng/mL) was seperated on Western Blots, the lanes cut out and pooled from 32 lanes and analyzed by Lumipulse. Note that many large and small tau fragments were detected, but the majority of tau had a size of 20–30 kDa (blue circles). Values are given as pg/mL total tau measured by Lumipulse assay.

3.5 Detection of Tau in Platelets Using Mass Spectrometry

Applying an immunoprecipitation method with the HT7 antibody on the 2N4R full-length tau standard, 21 peptides could be identified by nanoLC ESI-MS/MS that covered 57% of the amino acid sequence of tau441. In a further experiment with cerebrospinal fluid of an AD patient (1024 pg/mL total tau), four tau peptides were identified with high confidence (data not shown). We used this method to investigate the different human platelet samples. Only a single peptide could be identified in a platelet extract pooled from 15 human patients of the same group (1577 pg/mg tau) with the HT7 antibody. The peptide with the sequence SGYSSPGSPGTPGSR corresponds to amino acid residues 203–217 of tau441, with an m/z = 697.32059 Da, charge state = 2, cross-correlation score (XCorr) = 3.74. The MS/MS spectrum and the assigned y- and b-series ions are shown in Fig. 4. The same peptide was found in the tau standard and in cerebrospinal fluid, respectively (data not shown). Tau could not be detected in all other platelet samples. Applying the tau-5 antibody instead of the HT7 antibody, no tau could be identified, neither in the standard sample nor in CSF or platelets.

Fig. 4.

Identification of a peptide from tau protein in platelets by mass spectrometry. Immunoprecipitation was performed to enrich tau from pooled platelets. Enriched proteins were analyzed with mass spectrometry and one unique peptide corresponding to amino acid residues 203–217 of tau441 was identified. (A) Shown is the MS/MS spectrum of peptide SGYSSPGSPGTPGSR (B) and the assigned y- and b-series ions.

4. Discussion

In the present study, we show that tau protein is expressed in human platelets. It occurs especially as a fragmented form. We demonstrate that tau protein levels in platelets of AD and depressive patients are significantly changed and reduced tau levels could distinguish AD patients and depressed people from healthy individuals.

Platelets are about 3 $\mathrm{\mu}$m small cells and are processed from megakaryocytes and do not have a nucleus. The isolation of human platelets from blood has been well described in our hands for several years [14, 38, 39]. For platelets characterization we used FACS analysis for CD62P, which is well established in our laboratory [34]. CD62P is located on the inner membrane of $\alpha{}$-granules and released on the outside upon activation where it acts as a receptor and is thus a platelet activation/degranulation marker. A negative control (IgG1) was incorporated in the protocol and the obtained values subtracted from the CD62P positive events. Platelets are sensitive cells and require careful handling during preparation. After decades of study, there is still a general need to establish concurrent methods for the different processing steps of platelets because the wide variation of reported protocols for platelet preparation leads to different biological responses and incomparable data. Several variables (such as, e.g., anticoagulants in collections tubes, time of processing, etc.) in the platelet preparation process need to be standardized to minimize artifactual stimulation during blood collection and sample handling. Thus, as blood was processed within 3 hours, and we are very experienced, we are confident that we analyzed healthy platelets. Blood was centrifuged to obtain platelet-rich plasma and further to give a high pure platelet pellet. In order to extract proteins from platelets, we extracted platelets by sonication and centrifugation, to give a pure extract of soluble proteins in the supernatant. In all cases, protease inhibitors were added to inhibit any degradation process.

Platelets modulate as mediators multiple cellular processes including angiogenesis and inflammation and are involved in a variety of pathologic conditions including AD. It is well known that both inflammatory and vascular processes accompany AD pathology and hence, it seems likely that platelets actively contribute to the progression of AD [9]. In murine models we have shown that platelets are first signs in AD as they concentrate in brain vessels and can also damage brain vessels and thus contribute to the AD pathology [15, 16]. Moreover, platelets contain proteins such as amyloid-precursor protein (APP) and produce beta-amyloid peptides, mainly the shorter 40 amino acid form [17, 18, 10, 19, 20, 40]. It is assumed that platelets use the beta-amyloid(40) peptide to clot the damaged vessel. Regarding the role of tau in platelets and AD, there is a strong mismatch in the literature. Although there are several reports that platelets contain tau (especially C-terminal tau), their role and structure is completely unknown [21, 22, 23, 24]. In the present study, we indeed verify that tau is found in platelets in a fragmented form and slightly decreased in late AD, however, we cannot link tau to any functional process.

Many studies now indicate that platelets in depressed people have a different level of activation [41, 42, 43, 44, 45]. Platelets contain a high concentration of serotonin, which is stored in vesicles and released by stimulation. Since serotonin plays a major role in depression, it could be that serotonin, which comes from platelets, plays a role in the pathogenesis of depression. In a previous study, we used platelets from depressive patients as an additional control, but also because AD and MCI patients complain from depressive episodes. Thus, a bio-profiling of platelets in severely depressed patients treated with medication was performed. We detected that the serotonin levels in blood platelets and different immunomodulatory compounds were significantly altered in these patients [46]. Interestingly, platelets not only contribute to the progression of depression, but also play a role in stress. In two previous papers we reported on differential changes in platelets reactivity induced by acute physical stress compared to persistent mental stress and also showed that stress enhances pro-inflammatory platelets activity [47, 48]. This present study shows that tau protein levels are decreased in depressed people followed by a slight enhanced phospho-tau-181 activation. Future studies are needed to further investigate this finding especially how and what the relationship is between serotonin and tau protein.

In order to characterize tau better, we performed a qRT-PCR, a Western Blot, a WANGEL detection and mass spectrometry. Platelets are anuclear cells; however, there are reports that mRNAs can be detected in platelets [49]. Although it seemed very unrealistic to detect tau mRNA in platelets, we performed a qRT-PCR. Our data supported our suggestion, that no tau mRNA was detectable in human platelets, but only in 1 sample out of 6 a very low tau expression was found by qRT-PCR, which could also be an artefact; this again shows that tau in platelets is expressed at very low concentrations. Thus, in order to catch tau in platelets we pooled up to 15 samples to concentrate tau. Using Western blots we did not see a full length tau protein (of approx. 60 kDa), but rather some smaller tau fragments of 40 and 30 kDa or smaller using the HT7 antibody. Only with an antibody recognizing the phospho-tau 396 form a large protein was seen, where we cannot really determine the specificity. In order to characterize the tau protein better, we performed a WANGEL analysis followed by the high sensitive Lumipulse assay. Here, a pool of 32 Western Blot lanes was extracted into agarose gels and eluted and analyzed and again we found that only very small tau fragments of 20–30 kDa were detectable. And finally applying immunoprecipitation and nanoLC MS, only a single peptide of tau was identified in one of the pooled platelet samples, while it was undetectable in all other platelet samples investigated. A comparison of different sample preparation protocols for the analysis of human platelets showed huge differences in the obtained protein concentrations [50]. A more efficient and replicable method may increase protein isolation and tau concentration, respectively. The fact that tau could not be detected clearly in platelets by mass spectrometry again suggests that it is of low abundance and/or fragmented, which results in a decreased binding to the HT7 antibody.

Taken together, our data highly suggest that tau is indeed present in human platelets, but at very low levels and only in a highly fragmented small (20–40 kDa) form. This was surprising to see but this result fully agrees with a prior study [51]. They report that multiple N-terminal and mid-domain fragments of tau were detected in pooled cerebrospinal fluid (CSF) with apparent sizes ranging from 20 kDa to approx. 40 kDa. The pattern of tau fragments in AD and control samples were similar. They also did not detect full-length tau and C-terminal fragments in their CSF samples. In order to concentrate the CSF, they used a RP-HPLC followed by a Western Blot and also used the same HT7 antibody (as we did) and found tau in fractions 3–7 representing small (20–40 kDa) tau fragments [51]. This is partly surprising, as tau is a well-established biomarker in CSF for AD with a pathological concentration of $>$500 pg/mL. The tau analysis is well established using different ELISAs or also our Lumipulse assay. The question now comes up, why is tau so highly fragmented, and why is no full length tau detectable in CSF (but also in platelets). More work is necessary to study the metabolism and stability of the tau protein and the functional role of smaller fragmented tau forms.

Due to a new generation of ultrasensitive immunoassays, phosphorylated tau epitopes are detectable in blood plasma. The currently best validated epitopes are phospho-tau-181 and phospho-tau-127. Plasma phosphorylated tau-181, for example shows 1.5 to 3.5 times higher concentrations in AD patients compared to controls [52]. The recently published plasma-phospho-tau-231 is also a promising new biomarker for incipient AD [53]. Furthermore, plasma phospho-tau-181 identified individuals who were A$\beta{}$-PET-positive regardless of the clinical diagnosis and correlated with cortical tau protein deposition measured by ${}^{18}$F-flortaucipir PET [52]. Furthermore, the altered tau in blood platelets very probably only represents a small part of the AD pathology. Similar to AD, this also applies to depression. However, for depression, further studies should show whether there is a connection between tau pathology and the serotonin system. Plasma phospho-tau-181 also allows individualised prediction of AD in patients with mild cognitive impairments [54]. Future longitudinal studies will show how tau protein changes in those MCI patients who convert to AD.

The question now arises, if platelet tau may become a good biomarker for dementia or depression. The answer is definitely no. As we have shown tau and phospho-tau-181 are expressed in platelets at very very low concentrations. This limits our tools to quantify tau. In order to overcome such a methodological problem, more sensitive assays must be used. We thus highly suggest to use SIMOA (single molecule array technology) to ultimately enhance the sensitivity for low abundance biomarkers up to the single molecule level. It will be highly interesting to use this technique, which is, however, very expensive and not useful for routine diagnosis. On the other hand, our data show that the differences between AD/depression and healthy controls are very low (1.3$\times$ reduction for tau) with a significance of p $\leq$ 0.05. Our experience in several clinical biomarker studies show, that at least a signifance of p $\leq$ 0.001 and a difference of at least 3–5 variances is necessary to establish a biomarker in the clinical routine. The same also holds for phospho-tau-181 and the increase in depression compared to controls (1.3$\times$ enhanced). Thus the present study will not discover a new biomarker but will definitely contribute to the idea that tau plays a role in platelet function in dementia and depression.

Definitely, this study has several limits. (1) The changes of tau in the brain of people suffering from severe neurocognitive disorder are well studied. However, in platelets, a functional role of tau is still largely unexplored. More work in humans, or in mouse models including tau transgenic will be necessary. (2) In platelets, the concentration of tau protein is very low. Therefore, future studies are necessary to examine whether it is possible to detect tau levels reliably at a specified level of confidence. The reliable reproducibility of the results are the prerequisite that the change in tau levels could establish as a future biomarker. More work is necessary to link tau to other vascular-related diseases. (3) As mentioned, tau exists as a small fragmented protein. This is really surprising, but it confirms work in human CSF and also questions why tau is so highly fragmented [51]. This fragmentation, however, does not limit the use as a biomarker in CSF for AD, as this method is well established internationally. It however opens the question, why tau is fragmented, and if is this only due to degradation and metabolism or has fragmented tau also a physiological role in the brain, e.g., to enter the nucleus. (4) In this regard, we suggest that full-length tau can be very instable and is metabolized to smaller fragmented peptides. This high fragmentation also limits our approach to detect tau by mass spectrometry.

5. Conclusions

Taken together we show that tau is expressed in human platelets, but exists mainly as a very highly small fragmented form. We show that tau in platelets is reduced in AD and depression and may differentiate them from healthy controls. We show that tau is very low in platelets and display a high heterogeneity. Thus, we conclude that tau in platelets may have a prominent function, but will not serve as a novel biomarker.

Author Contributions

CH designed the study, wrote the manuscript, and acquired funding. DSK performed the Western Blots. JM collected blood and diagnosed patients. BS performed the MS analysis. All authors contributed to editorial changes in the manuscript. All authors read and approved the final manuscript.

Ethics Approval and Consent to Participate

The study was performed according to local ethical approval protocol no. AN2015-0159 351/4.7 405/5.5 (4484a). Informed consent was obtained from all subjects enrolled in the study. The study was in accordance with the guidelines of the ethical commission of the Medical University Innsbruck.

Acknowledgment

We thank Karin Albrecht for excellent technical help.

Funding

This research received no external funding.

Conflict of Interest

The authors declare no conflict of interest.

Publisher’s Note: IMR Press stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Share


text:
# An alternative description of the first Stiefel-Whitney class

I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please?

The first Stiefel-Whitney class of a vector bundle is an element in the first cohomology group of the base space $H^1(B,Z_2)$. This element can be seen as a map $w_1 : \pi_1(B) \rightarrow Z_2$ (because of the universal coefficient theorem and the fact that $H_1$ is the abelianization of $\pi_1$), so it assigns to each loop based at a fixed point an element of $Z_2$. Now this number is $0$ if and only if that loop is orientation preserving (locally we have an orientation and when we go around a loop and come back to our first place, we can ask if orientation has changed or not)

-
Where did you hear this? –  Adam Saltz May 3 '13 at 1:27
@Adam from one of PhD students –  Mehdi May 3 '13 at 1:50
What definition(s) of $w_1$ do you know? If you have been studying Milnor-Stasheff, take a look at the obstruction theory chapter. –  Sam Lisi May 3 '13 at 9:39
@Sam I've read Milnor-Stasheff, but , the obstruction chapter was somewhat brief, so I haven't learned it well. should I use the fact that stiefel-whitney classes are mod 2 reduction of obstruction classes? –  Mehdi May 3 '13 at 16:06
@Mehdi : yes, that's what I was thinking about (to your first comment). I'll try to write something soon, but the short version of what I am thinking about is that the $\mathbb{Z}/2$ that shows up in the sequence $1 \to SO(n) \to O(n) \to \mathbb{Z}/2 \to 1$. –  Sam Lisi May 3 '13 at 16:21

The answer is more straightforward than what I was saying in the comments above, though the general principle of what I was saying holds. Let $n$ denote the dimension of $M$.

Recall that the Stiefel-Whitney classes are characterized by a set of four axioms. The normalization axiom says that $w_1$ of the unorientable line bundle over $S^1$ is the generator of $H^1(S^1, \mathbb{Z}/2)$ and that $w_1$ of the orientable line bundle is $0 \in H^1(S^1, \mathbb{Z}/2)$.

Fix now a loop $\gamma \colon S^1 \to M$. The question of whether $TM$ is orientable over $S^1$ corresponds to whether the determinant bundle $\Lambda^n TM$ is trivial or not when restricted to $\gamma$. Naturality of the Stiefel-Whitney class gives $< w_1( \gamma^* \Lambda^n TM), [S^1] > = < \gamma^* w_1(\Lambda^n TM), [S^1]>$ where this pairing is between cohomology/homology of $S^1$. Now, the LHS is equal to $1$ iff $\gamma^* w_1(\Lambda^n TM)$ is the non-trivial bundle, i.e. iff this loop is non-orientable. The RHS is equal to $< w_1(\Lambda^n TM), \gamma_*[S^1]>$. This shows that $w_1$ exhibits the property you claim.

-
This works beyond $w_1$ too. For a 4-manifold $M$ with 2nd cohomology represented by surfaces $\Sigma\subset M$, you can view $w_2(M)$ as a map $\Sigma\to \mathbb{Z}_2$ given by trivializability of $TM|_\Sigma$ (and so if this is 0 on all surfaces, then you've got a spin structure). –  Chris Gerig May 12 '13 at 21:07

If you have a triangulated manifold then it is orientable when its n simplices can be given signs (+/-1) so that the signed sum is an integer cycle.

If the manifold is not orientable then any signed sum of the n simplices is not a cycle and it turns out that what goes wrong is that some of the n-1 simplices in the boundary get counted twice rather than cancelling each other out.

This boundary which is a sum of doubles of n-1 simplices can be divided by two to give a 2 torsion Z homology class. This n-1 dimensional cycle represents the obstruction to orienting the manifold. I would strongly suspect that the first Stiefel Whitney class of the tangent bundle is the Poincare dual to this homology class. Check it out.

-
This seems true: the Bockstein $\beta:H_n(M;\mathbb{Z}_2)\to H_{n-1}(M)$ maps the $\mathbb{Z}_2$-fundamental class $[X]_2$ to that 2-torsion generator. Reducing mod-2 and taking the dual, we have a map $H^0(M;\mathbb{Z}_2)=\mathbb{Z}_2\to H^1(M;\mathbb{Z}_2)$ which I can believe sends $PD[X]_2$ to $w_1(M)$. –  Chris Gerig May 27 '13 at 21:49
+1. OK just worked it out and it is true; it follows from the definition of Stiefel-Whitney classes in terms of Steenrod squares of the Wu class (which involves the $\mathbb{Z}_2$-fundamental class). –  Chris Gerig May 28 '13 at 0:29

format: plain


text:
# Homotopy Type Theory algebraic formulation of dependent type theory (Rev #1)

## Idea

The goal of this line of work is to find algebraic formulations of dependent type theory. The syntax will then be the initial model.

## References

• Egbert Rijke, An algebraic formulation of dependent type theory, PDF


text:
Choose the correct alternative in the following:
Question:

Choose the correct alternative in the following:

If

A. $-\frac{2}{1+\mathrm{x}^{2}}$

B. $\frac{2}{1+x^{2}}$

C. $\frac{1}{2-x^{2}}$

D. $\frac{2}{2-x^{2}}$

Solution:

$y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$

Put $x=\tan \theta \Rightarrow \theta=\tan ^{-1} x$

$y=\sin ^{-1}\left(\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}\right)$

$y=\sin ^{-1}(\cos 2 \theta) \because \frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}=\cos 2 \theta$

$y=\sin ^{-1}\left(\sin \left(\frac{\pi}{2}-2 \theta\right)\right)$

$y=\frac{\pi}{2}-2 \theta$

Putting value of $\theta$ we get,

$y=\frac{\pi}{2}-2 \tan ^{-1} x$

Differentiating w.r.t $\mathrm{x}$ we get,

$\frac{\mathrm{dy}}{\mathrm{dx}}=0-2\left(\frac{1}{1+\mathrm{x}^{2}}\right)$

$\because \frac{\mathrm{d}}{\mathrm{dx}} \tan ^{-1} \mathrm{x}=\frac{1}{1+\mathrm{x}^{2}}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{2}{1+\mathrm{x}^{2}}=(\mathrm{A})$


text:
SERVING THE QUANTITATIVE FINANCE COMMUNITY

Topic Author
Posts: 11048
Joined: February 1st, 2005, 11:21 pm

There is absolutely no reason why the United States and China should be on a collision course. China is like a big boulder moving down the mountain. There is nothing you can do to stop it. The question is if you want to do something positive. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Every year China is constructing around 4,000 km of expressways, towards its target of connecting every city with a population of 200,000 or more to an 85,000 km national motorway network. Half the work is already done.
Last edited by TraderJoe on May 7th, 2006, 10:00 pm, edited 1 time in total.

DominicConnor
Posts: 11684
Joined: July 14th, 2002, 3:00 am

There is absolutely no reason why the United States and China should be on a collision course.I agree with you that there's no obvious reason for major tension between the two in the near future.But there is a huge swarm of relatively low probability issues, that combine to make me see it as almost intevitable.Taiwan is the first obvious one. China keeps trying to bully them, and they are under US protection. It's easy to forget that a respectable % ofAmerican military work is honourable and for the general good. S.Korea would be trashed within hours by Chinese backed North Korea if America left.China has this chip on it's shoulder about Japan, and sends thugs to beat up Japanese businessmen. That's hardly the stuff of wars, but it betrays an attitude that makes some rash act less than impossible.America also guarantees Japan from attack by China.China and India occasionally shoot at each other over the ownership of utterly useless and barren border areas, and it's not that long since China & Russia did the same. Neither are American allies, and there is no immediate threat of a real war, but it would be big, and big wars suck in 3rd parties.China also seems to regard any bit of water within 1,000 miles of anything they have a name for as Chinese. Mostly this is harmless but there are oil fields within China's reach, but not yet it's grasp.Oil is on course to be a source of conflict even move over the next 10-20 years, and again America backs the non-Chinese nations on this. (Currently).China isn't very nice. Historically, America has found the nastiest person in a region and backed them, and given that almost no one anywhere is nastier than China it can continue to expect America support.However, as we saw in Iran that really can turn out bad. If the evil old men fall in spite of America support, will the new regime hate America for helping their oppressors ?China sells nuclear technology to really bad people. America has never really taken action against France or Pakistan for their dodgy sales, but if the nukes were used against American interests that might change.China has a space program. OK, it's the only program in the world that makes NASA look good by comparison, but they're up there. For historical reasons, America has treated space as much like the high seas open to all.The idea of a common asset shared by all is utterly alien to China, if it feels it can occupy it, they will. The Clark orbit is getting more crowded, and is a finite resource. Also bits of crap are floating round at high velocity. China really really hates any form of communication it does not control. It might decide to claim ownership of the area of space that can broadcast down to China. Slight problem here of course is that China is too far north to actually be below a geostationary orbit. Means it will have to lean upon southern neighbours, and/or do the sort of things it already does at sea. Look up the incidence of piracy in that part of the world. And no, they are not one legged englishmen with loud laughs and a parrot. China is in on this as well. Historical precedent of course is that's how Britain's seaborne empire bootstrapped itself.There is nothing you can do to stop it. The question is if you want to do something positive.Suggestions ?

Posts: 1093
Joined: October 31st, 2003, 9:56 am

DCFC, I think you are a little over-paranoid towards China and North Korea.QuoteS.Korea would be trashed within hours by Chinese backed North Korea if America left.I think that this is very improbable. N. Korea would not try, and China would not back them either.S. Korea provides a lot of help to N. Korea. It is in the interest of the North to keep the South scared but not harmed.These days, China is only interested in economic growth, and would not embark into any war unless forced to, because it does not want to afford the cost. May be when they grow a lot stronger, this might change, but then a lot might have changed. I think the Chinese govt whatever you might of them is firstly and above all concerned about stability and growth.

DominicConnor
Posts: 11684
Joined: July 14th, 2002, 3:00 am

DCFC, I think you are a little over-paranoid towards China and North Korea.Right, that's why N.Korea has this huge offensive force within artillery range of Seoul ?I think that this is very improbable. N. Korea would not try, and China would not back them either.China backed it last time.The N.korean regime only has one friend, and that is China.S. Korea provides a lot of help to N. Korea. It is in the interest of the North to keep the South scared but not harmed.The problem with extortion is that every so often you need to break a few legs to show that you're serious. That can easily get out of hand.Also the victim may get it into his head to strike.If America suddenly pulled out, there would be a window of vulnerability before S.Korea built up it's capability. That would be the best time to strike.These days, China is only interested in economic growth, Chinese people certainly are, and good luck to them.China's leaders have a very different agenda.and would not embark into any war unless forced to, because it does not want to afford the cost.Clancy's law.Almost every nation that fired the first shot in war since the industrial revolution has lost. War is almost never a rational act.Even piss poor little wars like Iraq consume vast amounts of money, yet we observe they still happen.I doubt a direct war is likely any time soon. However, the boundary between selling arms to a country as a commercial transaction, and using it to build allies is often very blurred.What if Venezuela decided to swap some of the oil that China increasingly needs for a horde of cheap but respectable fighters, or perhaps missiles with the range to hit the USA ?China has full WMD capability, and a rapidly growing manufacturing base. I rather suspect that >5 years from now they will begin to own the low-mid range of international arms sales.Take Cuban missile crisis, and update it. China already sells lots of things to Iran. If a Chinese freighter carrying accurate long range ballistic missiles was spotted in the Gulf, what would the USA do ? What would Israel do ? Not saying this will lead to fights, but an expanding China means more of these points of friction.In any case the scenarios I described were semi-accidents, rather like pub fights. Someone bumps someone, they over-react, gets messy.May be when they grow a lot stronger, this might change, but then a lot might have changed.Things have already changed. I think the Chinese govt whatever you might of them is firstly and above all concerned about stability and growth."Stability" to China means very different things than to a civilised country.
Last edited by DominicConnor on September 13th, 2005, 10:00 pm, edited 1 time in total.

bhutes
Posts: 516
Joined: May 26th, 2005, 12:08 pm

Last edited by bhutes on November 9th, 2005, 11:00 pm, edited 1 time in total.

bhutes
Posts: 516
Joined: May 26th, 2005, 12:08 pm

Last edited by bhutes on November 9th, 2005, 11:00 pm, edited 1 time in total.

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

Chinese warships cruise near

Topic Author
Posts: 11048
Joined: February 1st, 2005, 11:21 pm

China's Success in Fighting PovertyAcross China, there were over 400 million fewer people living in extreme poverty in 2001 than 20 years previously. By 2001, China had met the foremost of the Millennium Development Goals — to reduce the 1990 incidence of poverty by half — and it had done so 14 years ahead of the 2015 target date for the developing world as a whole.

brontosaurus
Posts: 2035
Joined: May 10th, 2004, 8:33 pm

QuoteOriginally posted by: TraderJoeChina's Success in Fighting PovertyAcross China, there were over 400 million fewer people living in extreme poverty in 2001 than 20 years previously. By 2001, China had met the foremost of the Millennium Development Goals — to reduce the 1990 incidence of poverty by half — and it had done so 14 years ahead of the 2015 target date for the developing world as a whole.basically cheap labor is better than no labor
Last edited by brontosaurus on September 19th, 2005, 10:00 pm, edited 1 time in total.

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

some are more equal than others.....QuoteThe Groucho club, Soho House and now the Shanghai StarBy Jim Pickard in London Affluent business people will each be invited to pay $100,000 a year to belong to a Shanghai private members' club located inside the biggest Ferris wheel in the world.The club will be the most expensive in China, according to Xu Qun, chairman of Zhejiang Huamen, the property company behind the scheme. Work on the Shanghai Star, in a rundown area about 3km north of the Bund waterfront, is set to start in May and be completed in three years' time.The club, which will be a lava-lamp shaped building set within the wheel, will offer one of the best views in the city to its 300-400 members. Access to the restaurants, bars and meeting rooms of the five-storey club will be through an elevator. Mr Xu said he was in talks with Planet Hollywood, the leisure group, about operating the club.The entry charge is a sign of the growing wealth of China, which is rapidly emerging as the economic powerhouse of Asia. Merrill Lynch recently estimated that the country now had 300,000 dollar millionaires a figure which others believe may be “conservative”.The Shanghai Star is one of a number of giant Ferris wheels being planned across the world in cities such as Moscow and Singapore. It has been designed with the help of the technical team behind the London Eye, which has become one of the biggest tourist attractions in the UK since opening in January 2000.Mr Xu, in an interview at his house in Shanghai's French Concession, said the group was in talks with US groups to raise extra financing. But he insisted the company had enough money from six Chinese banks to proceed with the 2.8bn renminbi ($347m) project.The Star is one of a number of projects set to raise Shanghai's profile ahead of the 2010 Expo event, with engineers already at work on the world's tallest building and longest suspension bridge in the city.

Topic Author
Posts: 11048
Joined: February 1st, 2005, 11:21 pm

World's tallest building in Shanghai.

DominicConnor
Posts: 11684
Joined: July 14th, 2002, 3:00 am

Ppauper is the Ferris wheel for real ?If so, I have a mate from college who reckons who can build executive apartments in a hollowed out volcano.

brontosaurus
Posts: 2035
Joined: May 10th, 2004, 8:33 pm

QuoteOriginally posted by: TraderJoeWorld's tallest building in Shanghai.looks like a giant bottle opener

migalley
Posts: 3696
Joined: June 13th, 2005, 10:54 am

Apparently the straight section inside the hole was added as an afterthought because the building has some sort of Japanese connection and the Chinese authorities were concerned (rightly or wrongly) that the circular hole would be taken to represent a Japanese "Rising Sun". Of course, a Rising Sun in the middle of Shanghai would be deemed offensive by the general population given the long history of animosity betweeen China and Japan.

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

QuoteOriginally posted by: migalleyApparently the straight section inside the hole was added as an afterthoughtmaybe to allow arabs to fly through it ?


text:
# Proving that $\sqrt{pq} \ne (p + q)/2$ implies $p \ne q$ using the contrapositive

Prove by the contrapositive method, that if $p$ and $q$ are positive real numbers with the property that $\sqrt{pq}$ is not equal to $(p+q)/2$, then $p$ is not equal to $q$.

The proof is easy enough but I am not sure how to structure it logically. To use the contrapositive method instead of proving $A \implies B$, I try to prove $\neg B \implies \neg A$.

So I should work from the fact that $p = q$ and reach $\sqrt{pq} = (p+q)/2$.

However the only way I am able to do this is by algebraically manipulating the conclusion \begin{align*} \sqrt{pq} &= (p+q)/2\\ 4pq &= (p+q)^2\\ 4pq &= p^2 + 2pq+q^2\\ 0 &= (p-q)^2 \end{align*} therefore $p = q$.

But I should have been working from this fact and reached the conclusion, though in reality I worked from the conclusion and reached my assumption. My question is whether this approach is mathematically rigorous and if not how should I have proceeded. Thanks.

• In a proof by contraposition, you should not talk about "contradiction". – Frunobulax Aug 17 '14 at 12:23

We are trying to prove that

if $\sqrt{pq} \ne (p + q)/2$, then $p \ne q$ (where $p, q$ are positive)

by proving the contrapositive. This means that we must prove the equivalent statement

if $p = q$, then $\sqrt{pq} = (p + q)/2$.

So suppose that $p = q$. We then have \begin{align*} \sqrt{pq} &= \sqrt{p^2}\\ &= p\\ &= (p + p)/2\\ &= (p + q)/2 \end{align*} and the statement is proved.

Note that we did not presuppose that $\sqrt{pq} = (p+q)/2$. What you have proven is in fact the inverse of the original statement,

if $\sqrt{pq} = (p + q)/2$, then $p = q$.

Because $p=q$ we have $pq=q^2$ and thus $\sqrt{pq}=q$ as the numbers are positive. Likewise, we get $\sqrt{pq}=p$. If we add these two equations, we have $2\sqrt{pq}=p+q$. Now divide by $2$.

• Thanks, does that mean that my version would not be considered rigorous and has to be rewritten in your way? – pseudomarvin Aug 17 '14 at 12:31
• Sorry, forgot to answer that: If you explicitly say that all your manipulations above can be reversed, i.e. that the first and the last line are equivalent, then your proof is certainly correct. However, it is probably not what the person who asked the question had in mind when they specifically mentioned using the contrapositive method. – Frunobulax Aug 17 '14 at 12:33
• Or, do it the other way around: If $p=q$, then $p-q=0$. This implies $(p-q)^2=0$. Now work your way backwards through your equations above. Then you've proved that the conclusion you want to have follows from $p=q$. – Frunobulax Aug 17 '14 at 12:36
• Okay I think get it, thank you for your time. – pseudomarvin Aug 17 '14 at 12:38
• While I think the posted answers are more representative of how one is "supposed" to prove the stated fact, I think your sequence of equations (in the question) is quite clever, especially since they would enable to you write "if and only if" instead of merely "implies". – David K Aug 17 '14 at 13:10


text:
# Floer cohomology from mapping spaces of $\infty$ categories

There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at the nlab page on cohomology

I would like to understand if there's any non-contrived way that Floer-theoretic invariants fit into this picture.

I'm far from an expert. I'm aware of a number of works elucidating homotopy-theoretic aspects of Floer theory: for example the construction of Cohen-Jones-Segal, and the infinity category of Lagrangian cobordisms defined by Nadler-Tanaka. Do any of these constructions provide an answer to the question above?

• I know nothing about Floer homology, but the observation of Urs Schreiber is just a complicated way of stating that we like when our cohomology theories are (co)representable (be it by spectra, motives or whatever object you find suitable). – Denis Nardin Jul 12 '18 at 7:47
• Wikipedia mentions some recent work of Manolescu extending Cohen-Jones-Segal, seemingly in the direction of your question – მამუკა ჯიბლაძე Jul 12 '18 at 11:14
• You can also do worse than study the works of Thomas Kragh – Thomas Rot Jul 12 '18 at 11:49
• Thanks! I was aware of the Manolescu and Abouzaid-Kragh lines of work, but didn't want to turn the question into a glossary of vaguely relates things none of which I understand very well :( – zzz Jul 12 '18 at 13:20
• Cohomology \emph{theory} generally has a precise meaning i.e. there are certain axioms your cohomology groups satisfy(e.g. excision). Floer (co)homologies don't usually satisfy those literally. Though it is of course interesting to think about situations where Mayer-Vietoris, gluing formulas etc hold. – algebrachallenged Jul 12 '18 at 14:16


text:
summaryrefslogtreecommitdiffstats log msg author committer range
blob: 59078f0f877caf51aa7fb3fb99677fcf26081265 (plain)
 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 ``` `````` \$LastChangedDate: 2008-07-11 10:31:18 +0200 (ven., 11 juil. 2008) \$ roots roots of polynomials Calling Sequence [x]=roots(p) [x]=roots(p,'e') Arguments p polynomial with real or complex coefficients or vector of the polynomial coefficients in decreasing degree order (Matlab compatibility). Description x=roots(p) returns in the complex vector x the roots of the polynomial p. For real polynomials of degree <=100 the fast RPOLY algorithm (based on Jenkins-Traub method) is used. In the other cases the roots are computed as the eigenvalues of the associated companion matrix. Use x=roots(p,'e') to force this algorithm in any cases. Examples See Also poly spec companion Authors Serge Steer (INRIA) References The RPOLY algorithm is described in "Algorithm 493: Zeros of a Real Polynomial", ACM TOMS Volume 1, Issue 2 (June 1975), pp. 178-189 Jenkins, M. A. and Traub, J. F. (1970), A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration, SIAM J. Numer. Anal., 7(1970), 545-566. Jenkins, M. A. and Traub, J. F. (1970), Principles for Testing Polynomial Zerofinding Programs. ACM TOMS 1, 1 (March 1975), pp. 26-34 Used Functions The rpoly.f source codes can be found in the directory SCI/modules/polynomials/src/fortran of a Scilab source distribution. In the case where the companion matrix is used, the eigenvalue computation is perfomed using DGEEV and ZGEEV LAPACK codes. ``````


text:
# Solid rocket engine

Launch of a Scout solid rocket

A solid rocket engine or solid rocket drive is a rocket engine with a drive set made of solid material. Both reducing and oxidizing components are bound as solid substances. In contrast, there is the liquid rocket , in which both the oxidizer and the reducing fuel are carried along in liquid form, mixed and burned. In so-called hybrid rockets , solid fuel and liquid oxidizing agent are combined with one another.

The oldest rocket types are solid-based. For a long time only solid rockets were built.

## history

### Early use

Apollo rescue missile test

The oldest rockets were solid fuel rockets. They were probably built in the Byzantine Empire in the 7th century and consisted of bamboo as a rocket body and a mixture of saltpeter and sulfur as fuel.

Presumably independently of this, rockets propelled with black powder were developed in China in the 13th century . There they were used, among other things, for military purposes. They became known later in Europe, but only had their main meaning here as fireworks .

The British officer William Congreve developed a missile for military use in the early 19th century . It was used, for example, in the bombing of Copenhagen (1807) . Due to the progress in artillery , the missile experienced a rather shadowy existence. It was not until the end of the 19th century that research and development began to take place again in this area.

A rocket test (1931) by Karl Poggensee and Reinhold Tiling is named as the first successful launch of a solid rocket in Europe .

### Modern use

Solid rocket are used differently today, both military and civilian purposes, such as aviation and aerospace . They are due to their low price to help launch missiles ( " booster ") and aircraft ( RATO used) as well as in small school drives, as well as intercontinental ballistic missiles like the Trident be performed as a solid rockets. In addition, because of their high maximum acceleration, they are used as rescue missiles to quickly bring manned spaceships out of the danger zone of a failing launch vehicle.

In APCP , for example, ammonium perchlorate (NH 4 ClO 4 ) is used as an oxidizer, which results in, for example, 4 H 2 O + N 2  + 2 O 2  + Cl 2 when two molecules break down (in practice, HCl is also formed). Oxygen and chlorine react with aluminum to form Al 2 O 3 and AlCl 3 as well as a polymer binder to form H 2 O and CO 2 , which releases more energy. The mass fraction of aluminum is up to 30 percent.

Due to their simple structure, solid rockets can also be built in very small sizes, for example as small rockets for fireworks, signaling or as special rocket projectiles for the use of handguns . Such missiles have rather simple propellants like black powder .

### outlook

In the future there are plans to reduce the heavy weight of large solid rocket rockets by replacing steel with carbon fiber reinforced plastic. This could drastically reduce the empty mass of large solid rocket rockets. Calculations showed that this improvement in the full to empty mass ratio alone could increase the payload of the Ariane 5 with the aim of geostationary transfer orbit by 2  t .

These lightweight materials could also make pure solid rockets possible, which can transport large satellites economically into near-earth orbits.

## rating

With the exception of applications with swiveling nozzles, solid fuel rockets come from completely without moving parts or additional devices for fuel pumps or lines, which would specify a technical minimum mass. They also contain their fuel at all times, so that storage and refueling facilities are not required. This makes it easier to use, for example, for meteorological or ground-based scientific purposes; For military use, they can sometimes be stored in large numbers in magazines or carried and are ready for use in a short time.

The fuel itself is solid and therefore much easier to handle than liquid or gaseous fuels: it cannot escape in this form and can then be harmful to health or the environment. There are also no instabilities caused by sloshing liquid fuel with solid fuel. Due to the shape of the propellant, the so-called burn-off characteristics, i.e. the development of thrust over the burning time, can be seen and the burning time itself can be influenced very easily. In this way, thrust forces that are greater than those of liquid engines can also be achieved. In addition, most types of propellants mean that the center of gravity of the rocket changes relatively little during the burn, which is important for flight stability.

These advantages make solid rockets reliable in use, powerful and inexpensive to develop, manufacture, maintain and use.

Since solid rockets always contain their explosive fuel, they also pose a permanent increased risk. This also makes them heavier than comparably large liquid fuel rockets, which can be transported empty and only refueled when needed.

The products of combustion from solid fuel rockets are mostly ejected at a slower rate than the products of combustion from liquid propellant rockets. Because the thrust is according to the formula

${\ displaystyle {\ text {thrust}} = {\ frac {\ text {fuel mass}} {\ text {time}}} \ cdot {\ text {outflow speed}}}$

calculated, their advantage of high thrust has to be bought at the cost of high fuel consumption. This results in the short burning time compared to liquid rockets. A thrust control during the burn is not possible; and in the event of an incident, a solid fuel rocket cannot be switched off. Only during the manufacture of the booster can the thrust profile be influenced over the burning time, for example by filling different segments with differently reactive fuel mixtures or by shaping the fuel (see below).

The entire interior of a solid rocket is also its combustion chamber . When the fuel is burned, high pressures and temperatures occur. Therefore, the walls must be designed for relatively high loads. As the size of the rocket increases, the load on the combustion chamber wall increases with the same internal pressure, the walls have to become thicker and therefore heavier. For example, the empty mass of a solid rocket increases in comparison to the total mass with increasing size, while it continues to decrease in the case of liquid fuel rockets. Their maximum technical mass is therefore below that of other missile types.

Solid fuel rockets are often more polluting than other types of construction. When the fuel is burned, chlorine, hydrogen chloride, sulfur compounds or other toxic substances are produced, for example, depending on the filling. In the case of an Ariane 5 start with EAP P238 solid propellant engines, the total solid propellant mass is 476 tons. With a solids content of 86% by weight or 68% by weight of ammonium perchlorate (see Ariane 5 technology) in the solid propellant, the total mass of ammonium perchlorate is 324 tons. When an Ariane 5 is started, 100 tons of hydrogen chloride are produced . Hydrogen chloride then reacts with water to form hydrochloric acid (270 tons with 37 percent acidity).

## Fuel geometry and thrust curve

Simplified solid rocket.
1. Propellant with a cylindrical recess in the middle.
2. Detonator sets the surface of the propellant on fire.
3. Cylindrical recess acts as a combustion chamber.
4. Exhaust gas is throttled through a constriction to regulate thrust.
5. Exhaust gas emerges from the rocket.

In the simplest case, such as fireworks rockets, the entire interior of the solid rocket is filled with the fuel. This burns evenly from back to front. If the burning time is short, this is not a problem; However, if it is burned for a longer period of time, it leads to a very high thermal load on the rear, already “empty” part of the rocket through which the hot combustion gases flow.

To prevent this from happening, the fuel is shaped into a hollow tube that burns from the inside out. The rocket fuel that is still present acts as a heat insulator and protects the rocket hull from overheating. However, with increasing expansion of the cavity, the thrust of the engine also increases, since this is approximately proportional to the surface area of the burning fuel. In the case of rockets, on the other hand, the thrust requirement is usually highest in the launch phase, since the missile is the heaviest here and has to be accelerated first.

By appropriately shaping the cross section of the fuel pipe, however, the thrust course can be influenced in such a way that it corresponds to the requirements. Thus, the cavity can be made approximately star-shaped. When igniting, the surface of the burning fuel is then the largest. After the fuel spikes have burned away, the cross-section is approximately circular and the fuel surface and the thrust are therefore lower.


text:
# Dirichlet Conditions

What is meaning of a signal having a "finite number of maxima and minima during any single period of time"?

• Is the given solution clear enough? – Laurent Duval Jul 16 '17 at 15:00

$$f(t)=t^2 \cos (1/t)$$
with $f(0)=0$ by continuity. The ripples around $0$ get narrower and more numerous, and are infinitely many, in any non empty interval containing $0$.
The function is evidently continuous, even differentiable. On each interval defined by $$\left[\frac{1}{(k+1)\pi+\pi/2},\frac{1}{k\pi+\pi/2}\right]$$ the function $f$ vanishes at both ends, and since not identically $0$, it attains at least one extremum on this interval. Hence the function has an infinity of extrema on each interval $[0,a]$, $a>0$. The exact locations of the extremas can be computed with the derivative of $f$.


text:
# What is the position of the surviving mouse?

I have this question that I think it may be very interesting to all maths' lovers.

A cat caught $n$ (integer) mice and put them in line, numbered them from 1 to $n$, from left to right. He starts eating every other mouse, starting with the mouse at the 1st position, i.e. 1, 3, 5 ... (all mice at the odd positions will be gone). He then starts a new iteration, no matter if there is a surviving mouse at the end of the line, by going back to the left and eats every other mouse again, starting always with the first surviving mouse from the previous iteration.

Until there is one mouse left.

What is the position of the surviving mouse in the original sequence from 1 to n?

• Hint: after the first time eating through the line, what are the numbers of the surviving mice? From there, using the original numbers, what mice survive round 2? Can you see a pattern? – Arthur Apr 5 '17 at 19:09
• I can give two examples First example : n=7 From the sequence 1, 2, 3, 4, 5, 6, 7 1st iteration : 1, 3, 5, 7 will be gone, 2, 4, 6 remain 2nd iteration : 2, 6 will be gone Surviving mouse is the 4th. Second example : n=4 From the sequence 1, 2, 3, 4 1st iteration : 1, 3 will be gone, 2, 4 remain 2nd iteration : 2 will be gone Surviving mouse is the 4th – Jay Tee Apr 5 '17 at 19:13
• Josephus Numbers - not quite the same since this is a line, but almost. – Paul Sinclair Apr 5 '17 at 23:13

One can use recursion to solve such questions. Let $a_{n}$ be the number of the last mouse surviving when there are $n$ to begin with.

• If $n$ is odd, $n = 2k+1$, then after one round we are left with $k$ mice, $2, 4, \ldots, 2k$ and we do the same procedure. The mouse that is left is $2a_k$. Thus, $$a_{2k+1} = 2a_k.$$

• If $n$ is even, $n = 2k$, then after one round we are left with $k$ mice again, the same ones. Thus $$a_{2k} = 2a_k.$$

It follows by induction that $$a_n = 2^{\lfloor \log_2 n \rfloor}.$$

The other answers are correct, but let me offer an alternative way to see it. Suppose you had 7 mouses, for example. Number them in binary and go through the cat's iterations:

001 -> 001
010 -> 010 -> 010
011 -> 011
100 -> 100 -> 100
101 -> 101
110 -> 110 -> 110
111 -> 111

In the first round the cat eats all mice ending in 1, so that only the ones ending in 0 are left. On the second round the cat eats all mice ending in 10, so that only the ones ending in 00 are left. And so on. (Alternatively, you can drop one trailing zero from all mice after the first round and iterate with always eating mice ending in 1.)

That is, if a mouse has $k$ trailing zeroes, it will survive exactly $k$ rounds. The last survivor is the one with the largest amount of trailing zeroes.

The number of trailing zeroes in the set $\{1,\dots,n\}$ is always maximized by a power of two and the maximizer is unique. (Please ask if you want a proof for this statement. It might work as a separate question, too.) The largest power of two that does not exceed $n$ is indeed $a_n = 2^{\lfloor \log_2 n \rfloor}$ as others have pointed out.

If $n$ is even, then we have something like $[1, 2, 3, 4, 5, 6]$ becoming $[2, 4, 6]$, which is the same as playing a new game of size $n/2$ and multiplying the answer from $[1, 2, 3]$ by $2$.

If $n$ is odd, then we have something like $[1, 2, 3, 4, 5]$ becoming $[2, 4]$, which is the same as playing a new game of size $(n-1)/2$ and multiplying the answer from $[1, 2]$ by $2$.

We can simplify the two cases into one by saying $n$ becomes $\lfloor\frac{n}{2}\rfloor$ instead. Therefore:

$S(n) = 2S(\lfloor\frac{n}{2}\rfloor)$ where $S(1) = 1$

So $S(n) = 2^{\lfloor \log_2(n)\rfloor}$

This is also the same as finding the most significant bit of $n$ (i.e. the largest power of $2$ less than or equal to $n$). Write $n$ in binary and then set everything to $0$ except the leading bit.

Following Arthur's hint: after the $k$th round, the only mice that are left are the multiples of $2^{k+1}$. This is because in the $k$th round, the cat eats all the mice that are not divisible by $2^k$.

Let $2^k$ be the largest power of $2$ that is $\le n$, i.e. $k=\lfloor \log_2 n\rfloor$.

Then the last mouse is the largest mouse divisible by $2^k$, i.e. $m2^k$ where $m=\lfloor n/2^k\rfloor$.

So, the last mouse is $$m2^k = \lfloor n/2^k\rfloor 2^k = \lfloor 2^{\log_2 (n) - k}\rfloor 2^k = \lfloor 2^{\log_2 (n) - \lfloor \log_2 n\rfloor}\rfloor 2^{\lfloor \log_2 n\rfloor} = 2^{\lfloor \log_2 n\rfloor},$$ where we use the fact that $\log_2(n) - \lfloor \log_2 n \rfloor \in [0,1)$.


text:
## Intermediate Algebra (12th Edition)

$x=\left\{ -5,5 \right\}$
Using the properties of equality, the given equation, $|x|-1=4 ,$ is equivalent to \begin{array}{l}\require{cancel} |x|=4+1 \\\\ |x|=5 .\end{array} Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the equation above is equivalent to \begin{array}{l}\require{cancel} x=5 \text{ OR }x=-5 .\end{array} Hence, the solutions are $x=\left\{ -5,5 \right\} .$


text:
# Finding the velocity of a rock given its height as a function of time

I'm trying to learn calculus here, but I know I have to set the $h$ equal to 0 and find the time at when it's equal to 0, but I have no idea what to do after. Here is the question. How do I find out the velocity at that time?

If a rock is thrown upward on the planet Mars with a velocity of $10\;m/s$, its height (in meters) after $t$ seconds is given by $H = 10t − 1.86t^2$. Find the velocity of the rock when it hits the ground.

-
What are the steps i take primarily to find it after? Or if anyone could show me how its done hat would be awesome. –  soniccool Jun 17 '12 at 20:41
Can you find velocity as a function of $t$? –  user17794 Jun 17 '12 at 20:44

First we find the time(s) when the rock is at ground level. So set $10t-1.86t^2=0$ and solve for $t$. We get $t=0$ and $t=\frac{10}{1.86}$.

The velocity at time $t$ is the derivative of the displacement function $H(t)$. So the velocity at time $t$ is $10-(2)(1.86)t$. Substitute the value of $t$ we found above.

Remark: We can solve the problem instantly without calculus. The initial velocity is $10$. So by symmetry the velocity when it hits the ground on its return trip must be $-10$.

-
Alright i tried it out myself, it works. So after i solve it i dervice the original equation and just plugin the values! –  soniccool Jun 17 '12 at 21:00

These types of problems can all be solved by knowing the relationship between the position, velocity, and acceleration equations. In the following, by taking the derivative you can move from one equation to the next: $$\text{position} \to \text{velocity} \to \text{acceleration}$$ Similarly, to go from one equation to the next below, you need to integrate the relevant equation: $$\text{acceleration}\to\text{velocity}\to\text{position}$$ These equations make more sense when you know the general forms for each equation (assuming constant linear acceleration). The general position equation is given by $$s(t) = \frac{1}{2}at^2+v_0t+s_0$$ where $a$ is acceleration, $t$ is time, $v_0$ is initially velocity, and $s_0$ is the initial position. The general equation for velocity is $$v(t) = at+v_0$$ using the same variables. Finally, the equation for constant linear acceleration is simply $$a(t)=a$$

Here you're given the height, or position, equation and you want to get the velocity. From what I wrote above, it's clear that you want to take the derivative to get the velocity.

-

Find the positive root of $H(t)=10t-1.86t^2$. Let $t_1$ be that root. Then evaluate $H'(t_1)$.

Added: Plot of the height function $H(t)=10t-1.86t^2$ (in meters) vs. time $t$ (in seconds)

Added 2: Plot of $H(t)$ (black, in meters) and $H'(t)=10-3.72t$ (blue, in m/s) vs. time $t$ (in seconds)

-

You have $H(t)=10t-1.86 t^2$

The rock hits the floor when $H(x)=0$, this is, when $0=10t-1.86 t^2$.

It is clear the solution $t=0$ stands for the moment it is thrown, so we consider $0=10-1.86 t$, which leads to $t=\frac{1000}{186}$.

The speed of the rock is given by the magnitude of the velocity at the moment $t=\frac{1000}{186}$. We use the derivative, which is $H'(t)=10-3.72 t$. To get the value we evaluate $H'(\frac{1000}{186})=-10$

-
The $t$ you're looking for is actually $\frac{1000}{186}$ which gives the correct answer of $-10$ m/s. –  chris Jun 17 '12 at 21:00
@chris Sorry, I'll correct that, my fault. –  Pedro Tamaroff Jun 17 '12 at 21:10


text:
# Fuglede-Putnam-Rosenblum theorem

Let $A$ be a $C^{\ast}$-algebra with unit $e$.

The Fuglede-Putnam-Rosenblum theorem makes the assertion that for a normal element $a\in A$ the kernel of the commutator mapping $[a,-]\colon A\to A$ is a $\ast$-closed set.

The general formulation of the result is as follows:

Theorem. Let $A$ be a $C^{\ast}$-algebra with unit $e$. Let two normal elements $a,b\in A$ be given and $c\in A$ with $ac=cb$. Then it follows that $a^{\ast}c=cb^{\ast}$.

Lemma. For any $x\in A$ we have that $\exp(x-x^{\ast})$ is a element of $A$.

Proof. We have for $x\in A$ that $\exp(x-x^{\ast})^{\ast}\exp(x-x^{\ast})=\exp(x^{\ast}-x+x-x^{\ast})=\exp(0)=e$. And similarly $\exp(x-x^{\ast})\exp(x-x^{\ast})^{\ast}=e$. ∎

With this we can now give a proof the Theorem.

Proof. The condition $ac=cb$ implies by induction that $a^{k}c=cb^{k}$ holds for each $k\in\mathbb{N}$. Expanding in power series on both sides yields $\exp(a)c=c\exp(b)$. This is equivalent to $c=\exp(-a)c\exp(b)$. Set $U_{1}:=\exp(a^{\ast}-a),U_{2}:=\exp(b-b^{\ast})$. From the Lemma we obtain that $\|U_{1}\|_{A}=\|U_{2}\|_{A}=1$. Since $a$ commutes with $a^{\ast}$ und $b$ with $b^{\ast}$ we obtain that

 $\exp(a^{\ast})c\exp(-b^{\ast})=\exp(a^{\ast})\exp(-a)c\exp(b)\exp(b^{\ast})$

which equals $\exp(a^{\ast}-a)c\exp(b-b^{\ast})=U_{1}cU_{2}$.

Hence

 $\|\exp(a^{\ast})c\exp(-b^{\ast})\|\leq\|c\|.$

Define $f\colon\mathbb{C}\to A$ by $f(\lambda):=\exp(\lambda a^{\ast})c\exp(-\lambda b^{\ast})$. If we substitute $a\mapsto\lambda a,b\mapsto\lambda b$ in the last estimate we obtain

 $\|f(\lambda)\|\leq\|c\|,\ \lambda\in\mathbb{C}.$

But $f$ is clearly an entire function and therefore Liouville’s theorem implies that $f(\lambda)=f(0)=c$ for each $\lambda$.

This yields the equality

 $c\exp(\lambda b^{\ast})=\exp(\lambda a^{\ast})c.$

Comparing the terms of first order for $\lambda$ small finishes the proof. ∎

Title Fuglede-Putnam-Rosenblum theorem FugledePutnamRosenblumTheorem 2013-05-08 21:47:27 2013-05-08 21:47:27 karstenb (16623) karstenb (16623) 1 karstenb (16623) Theorem msc 47L30


text:
JShell 0.2
Jack Orenstein
Geophile, Inc.
jao@geophile.com

1. What is JShell?
2. Examples
3. Configuring and Running JShell
4. Files
5. Wildcards
6. Variables
7. Strings
8. Piping
9. Redirection
11. Command recall
12. Command usage
13. Extending JShell
14. Known problems
15. Helping out with JShell

# 1. What is JShell?

JShell is a UNIX-like shell written in Java, and intended primarily for working with Java programs. It was written to overcome the problems in working with Java on the Psion 5mx:

• eshell, the EPOC shell which can be used on the 5mx, has no environment variables.
• The command to start a Java program is quite lengthy, and this problem is exacerbated by the lack of environment variables. (Not to mention the 5mx's keyboard, which is good for a PDA but cramped nonetheless.)
• Every time a Java program is started, the Java Virtual Machine has to be loaded. On the 5mx this is time-consuming.
• If you want to compile on the 5mx, javac is your only choice. Because javac is written in Java, loading of the JVM is required. Also, the command for running javac is lengthy.
• Each Java program that executes simultaneously requires another JVM. Not only is there an additional delay as each JVM starts, the memory requirements of each JVM are considerable, especially on a machine with 16 meg RAM. This is a problem if you want to pipe the output from one Java program to another. (I'm not even sure whether eshell supports pipes.)

Because JShell is written in Java, running JShell starts a JVM. This is the only JVM you need. You can use JShell to run other Java programs (i.e. classes with a public static void main(String[]) method), and these run inside JShell's JVM as a thread. The JShell commands are Java classes which also run in the same JVM. You can extend the set of JShell commands. In fact, you are encouraged to do this and distribute your work to other JShell users.

JShell is designed to support development in Java. It is expected that you will be going through edit/compile/run/debug cycles. Java classes other than the JShell commands are reloaded, using a new class loader, each time the command is run. This permits the JShell JVM to work with different versions of the "same" class as you edit and recompile your Java source. You don't need to exit and restart JShell to pick up the latest versions of your Java classes.

While JShell was inspired by the Psion 5mx, it can be used on any Java platform. I've developed and used JShell on Windows NT, Linux, and EPOC.

# 2. Examples

The design of JShell was intended to be familiar to users of shells such as sh and csh. JShell is much simpler although somewhat less powerful. Here are some examples to give you an overview of JShell.

List the files in the current directory, with details on each file:

     5> ls :l

The 5 is a "job number". Each command that you run gets its own job number. The default prompt character is '>'. The ':' character is used for flags, by default. The traditional flag character, '-', is awkward to type on the 5mx while ':' is more accessible, (at least this is true on my 5mx; I've seen other keyboard layouts on other 5mxs). Both the prompt and flag characters can be changed.

List the java sources in the parent directory:

     6> ls ../*.java

List the files in the current directory, recursively, with details:

     7> ls :lR

You can examine the current character used for flags:

     8> env jshell.flag
:

And change it to something else:

     9> set jshell.flag -
jshell.flag=-

Then you can do this, for example:

     10> ls -lR

Here is how to compile all the java sources in this directory:

     11> javac *.java

Type all the .txt files in the current directory:

     12> cat *.txt

Make a new directory, foobar, and copy all the .txt files to it.

     13> mkdir foobar; cp *.txt foobar

Compile a bunch of Java sources in the background.

     14> javac *.java > errors.txt &

An & at the end of the line causes the command-line you entered (including all statements separated by semi-colons) to be executed in the background. You will get a new command prompt immediately after running the above command, but the file files.txt won't be complete until the command finishes executing.

You can view executing jobs using the jobs command, e.g.

        15> jobs
123: javac *.java > errors.txt &

The number before the colon is the "job" number. These numbers are unique within one execution of JShell. You can kill executing jobs by using the kill command. For example, the "javac *.java > errors.txt &" command can be kill like this:

        16> kill 123

# 3. Configuring and Running JShell

## EPOC

JShell.sis should be installed to your D drive. The batch file jshell.bat is written to assume that JShell is installed in D:\Java\JShell. If you don't have a D drive, it is doubtful that you will have enough memory for Java's classes.zip file, the JShell jar files and have enough memory left over to run JShell effectively. However, if you want to try this, you will need to modify jshell.bat and change all the references to the D drive to point to the C drive instead.

To run JShell, you will need to have installed eshell first. You can download eshell from the Symbian web site here.

Once you have eshell and JShell installed, go to the directory D:\Java\JShell. Run the command "jshell". In a few seconds, you should see a console with this prompt:

       0>

The first command will take a few seconds to run as the JShell class files get loaded. Subsequent commands should be faster.

## Windows and UNIX

Add jshell.jar and javacup.jar to your CLASSPATH. Then run jshell as follows:

        java jshell.JShell

# 4. Files

JShell uses UNIX-style file names with forward slashes and no concept of a logical drive (e.g. C: or D:). On DOS-like operating systems, (DOS, any version of Windows, EPOC), the drive letter is just the first part of the file name. So the DOS or EPOC path C:\foo\bar.txt becomes /C/foo/bar.txt. File names are case-sensitive only if the underlying operating system is, (so JShell on DOS and EPOC is case-insensitive, while JShell on UNIX is case-sensitive).

# 5. Wildcards

DOS has two wildcard characters, * which matches any number of characters, and ? which matches any one character. UNIX has two different kinds of wildcarding, "glob" patterns such as foo*.{cc,hh}, and regular expressions.

JShell takes an intermediate position. It uses UNIX-style glob patterns whenever a pattern is required. Regular expressions can be more powerful, but glob patterns should be sufficient for most purposes.

To summarize glob patterns:

 * matches zero or more characters. ? matches exactly one character. [abc] matches a, b or c. [a-zA-Z0-9] matches any letter or digit. {foo,bar} matches foo or bar. xyz matches xyz

For example, this command lists all the .java and .class files in the current directory:

       > ls *.{java,class}

# 6. Variables

A JShell variable begins with a letter or underscore and may consist of letters, underscores, digits and dots. Dots are permitted to provide for namespaces. All variables set by JShell begin with jshell. For example, jshell.prompt is the JShell prompt string.

Variables are set using the set command, e.g.

        100> set FOO xyz

and examined using the env command, e.g.

        101> env FOO
FOO=xyz

or the echo command:

        102> echo $FOO xyz  Note that variable names must be prefixed with$ to obtain the variable's value.

JShell's environment variables include the JVMs System properties. So if you type the "env" command, you'll see values for java.class.path, os.name, and all the other System properties. You'll also see JShell properties such as the current directory (jshell.dir), and whether output pauses after every page of output (jshell.page).

Here is a complete list of the environment variables defined by JShell. (Their values can be obtained by running the command "env 'jshell*'"):

 jshell.buffer Controls whether a buffer is used in handling console output. jshell.columns The number of columns of the output console. jshell.dir The current directory. The value of this variable can be observed and modified using the commands pwd, dirs, cd, pushd, and popd. jshell.flag The character used to denote a flag. jshell.history_size The number of previous commands tracked by JShell. jshell.lines If JShell pauses after each page of output (see jshell.page), then jshell.lines determines how many lines of output constitute a page. jshell.page Indicates whether jshell pauses after each page of output. jshell.prompt The prompt character.

# 7. Strings

Any sequence of characters (possibly including escaped characters) entered on the command line is a string. Strings may be unquoted, quoted with a single-quote character (') or quoted with a double-quote character ("). The rules for processing strings come from UNIX's sh shell:

 Unquoted string Single-quoted string Double-quoted string Escapes Processed Ignored Ignored Glob Patterns Expanded Not expanded Not expanded Environment variables Evaluated Not evaluated Evaluated

Examples:

1) These strings:

        a\bc    'a\bc'  "a\bc"

evaluate to abc, a\bc and a\bc respectively. (The last two strings contain four characters each, and the second character is the backslash.)

2) This command:

        ls *.java

lists all the java sources in the current directory. But these commands:

        ls '*.java'
ls "*.java"

list a file named "*.java" if it exists.

# 8. Piping

UNIX-style piping can be done using the ^ character. (This character was chosen instead of the more traditional | because | is not on the 5mx keyboard.)

# 9. Redirection

UNIX-style redirection can be done for standard output (not error output currently):

 < redirects input. > redirects output, overwriting the target file. >> redirects output, appending to the target file.

A command can be run in the background by typing & at the end of the command. For example, to get a recursive listing of your entire file system in files.txt as a background job, you can do this:

        200> ls :lR > files.txt &

If you enter multiple commands on a line, the entire set of commands runs in the background.

You can observe currently running jobs by using the jobs command, e.g.

        201> jobs
123: javac *.java > errors.txt &
157: ls :lR > files.txt &

You can kill a background job by using the kill command. E.g., here's how you would kill the recursive listing of the file system:

        202> kill 157

There is currently no support for putting a foreground job into the background.

# 11. Command recall

The history command can be used to find out about the last few commands you've executed. E.g.

        3> history
0: pwd
1: cd /foo/bar
2: javac *.java > errs.txt &

You can rerun the previous command by typing !!, or any other command by using it's number, e.g. !3.

By default, the last 100 commands are saved. You can change this limit by modifying the variable jshell.history_size. You can limit the amount of history shown to the n most recent commands by passing n to the history command, e.g.

        301> history 2
299: ls *.java
300: rm *.class

# 12. Command usage

Help on any command can be obtained by using the help command. help takes one argument, the name of the builtin JShell command, (i.e. the name of a class in the package jshell.command). Example:

        400> help cp
cp source ... destination
Copies source files to destination.
- If there are multiple sources, the destination
must be a directory.
- Directories are not copied by default.
- Use the -R flag to copy recursively.

Here is a complete list of the builtin commands:
• cat
• cd
• cp
• dirs
• echo
• env
• exit
• gc
• help
• history
• javac
• jobs
• kill
• ls
• mkdir
• popd
• pushd
• pwd
• rm
• save
• set

# 13. Extending JShell

JShell can run Java applications (i.e. Java classes that have a public static void main(String[] method), and JShell commands. A JShell command is a Java class with the following characteristics:

• It extends jshell.Command.
• It defines two methods:
•                         public void execute(String[] args)
public void usage()

• Input, output and error streams are accessed using the methods in(), out() and err() respectively.
• It calls checkForInterruption() periodically, (so that it can be killed from the command line in a timely manner).

A JShell command can be run by giving the fully qualified class name, followed by its arguments. For example, if you have written a class, com.mycompany.MyClass, which takes one integer argument, you could do this:

        123> com.mycompany.MyClass 419

Command-line arguments (from the JShell command line) are passed to execute's args argument. usage() prints out a usage message.

JShell commands can be connected using pipes, and input and output can be redirected.

Java applications run from JShell are problematic in a few ways. First, they can call System.exit which causes JShell to terminate immediately. Second, piping and redirection are not possible. JShell creates input and output streams for use by commands, but applications use System.in, System.err and System.out directly. It isn't clear how to, for example, pipe the output of one Java application to another when both use System.in and System.out. Another problem is that Java applications may not ever yield control, which means that they cannot be killed. (The kill commands uses Thread.interrupt, not Thread.stop which is deprecated in JDK 1.2 and can produce unpredictable results.)

It is sometimes convenient to use JShell to run Java commands, but in general, if you want to write a class to be run from JShell, you should write it as a command, following the guidelines described above. (Or look at any source in the jshell.command package, to see how JShell's builtin commands are coded.)

# 14. Known problems

• javac flags don't work. You can compile files (e.g. javac *.java), but don't try specifying any flags to the javac command.
• If you run the command "echo $foo" where foo is an undefined variable, echo will simply echo what you type (twice, in fact). This is actually documented behavior (see "help echo"), since "echo$foo" is equivalent to "echo" when foo is undefined. Unfortunately, I know of no way to regain useful control of JShell once this happens. You'll have to kill JShell, e.g. using Shift-Ctrl-Fn-K.

# 15. Helping out with JShell

JShell is an open-source project and contributions are welcome. Contributions can take the form of bug-fixes, new commands, suggestions, new features in the JShell core, etc. I'll act as coordinator and release updates periodically. If you want to help out but don't know what to work on, contact me at jao@geophile.com. If you send me your email address, I'll keep you posted on JShell developments.

Here are some projects that need to be done:

• Platform-specific documentation and help facilities (especially EPOC, but also Windows and Unix).
• Commands:
• mv: Put a general-purpose copy and delete implementation in jshell.OS (and called from jshell.command.mv). Allow OS-specific overrides, e.g. to spawn an mv command.
• grep: based on glob patterns, not regular expressions.
• chmod
• tail
• Get rid of all the CommandElementExpander$Transition* and LexerAutomaton$Transition classes. They greatly increase JShell's footprint. The functionality of these classes could be coded in a much more memory-efficient way, with only a slight loss in program readability.