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# Product Sigma Algebra

## Homework Statement

I should mention beforehand that I do not come from a math background so I may ask some trivial questions.

I am reading the book "Real Analysis" by Folland for a course I am taking and am attempting to understand a definition of product sigma algebra. It is stated in the book that if we have an indexed collection of non-empty sets, $$\{ X_{\alpha} \}_{\alpha \in A}$$, and we have

$$X = \prod_{x \in \alpha}^{} X_{\alpha}$$

and

$$\pi_{\alpha} : X \to X_{\alpha}$$

the coordinate maps. If $$M_{\alpha}$$ is a $$\sigma$$-algebra on $$X_{\alpha}$$ for each $$\alpha$$, the product $$\sigma$$-algebra on $$X$$ is the $$\sigma$$-algebra generated by

{$$\pi_{\alpha}^{-1}(E_{\alpha}) : E_{\alpha} \in M_{\alpha}, \alpha \in A$$}.

Now the first time I read this definition(by first time I mean the thousandth time), I thought that one has to just pick one $$E_{\alpha}$$ from some $$M_{\alpha}$$ and take it's inverse map(pre image?) to get some collection of sets. Then generating a sigma algebra from that collection would yield the product sigma algebra. However, I was told that we have to take the inverse map of every $$E_{\alpha}$$ inside every $$M_{\alpha}$$. I don't know how this is implied by the definition stated in the book but it makes more sense. Maybe I am not familiar with the notation because I would have expected some "for all" symbols in the definition somewhere.

Secondly, I wanted to think about what $$\pi_{\alpha}^{-1}(E_{\alpha})$$ would look like just for a single $$E_{\alpha}$$.

## The Attempt at a Solution

I tried to take an example like this, Consider:

$$\{ R_{1}, R_{2} \}$$, the real numbers. Then $$X = R_{1} \times R_{2}$$. Suppose I have sigma algebras $$M_{1}$$ and $$M_{2}$$. Now, What is $$\pi_{\alpha}^{-1} (E_{\alpha})$$ for some element in say $$M_{1}$$?

Is it $$E_{\alpha} \times R_{2}$$?

I don't know. Even this is just a guess. I'm hoping someone can give me some hints on understanding this since I am not able to easily find references on product sigma algebras.

Maybe I am not familiar with the notation because I would have expected some "for all" symbols in the definition somewhere.

Well, the "for all" symbol is implied by the notation. Let P be a certain property (I'll give examples later), then

$$\{x\in A~\vert~P(x)\}$$

means that you take all elements in A which satisfy P. An example:

$$\{n^2~\vert~n\in \mathbb{N}\}$$

By definition this means that you take all the squares in $$\mathbb{N}$$. Thus, you take all the elements in $$\mathbb{N}$$, and you square it. Another example:

$$\{\{x\}~\vert~x\in \mathbb{R}\}$$

means that you take all the $$x\in \mathbb{R}$$, and you take the singleton set of it.

$$\{\pi^{-1}(E_\alpha)~\vert~E_\alpha\in M_\alpha, \alpha\in A\}$$

means that you take all the $$\alpha\in A$$ and all the $$E_\alpha\in M_\alpha$$, and you calculate $$\pi^{-1}(E_\alpha)$$.

Is it $$E_{\alpha} \times R_{2}$$?

This is correct! $$\pi^{-1}(E_\alpha)=E_\alpha\times R_2$$...

Thanks a lot. I guess now I will move on to the next page in the book. I don't know if I should open a new thread or not but here goes.

This is Proposition 1.3 in Folland. I am interested more in the notation than in the proof given in the book. The proposition is

If $$A$$ is countable, then the product sigma-algebra is the sigma-algebra generated by $$\{ \prod_{\alpha \in A} E_{\alpha} : E_{\alpha} \in M_{\alpha} \}$$.

The proof given is this:

If $$E_{\alpha} \in M_{\alpha}$$, then $$\pi_{\alpha}^{-1} (E_{\alpha}) = \prod_{\beta \in A} E_{\beta}$$ where $$E_{\beta} = X_{\beta}$$ for $$\beta \notin \alpha$$.

On the other hand, $$\prod_{\alpha \in A} E_{\alpha} = \cap_{\alpha \in A} \pi_{\alpha}^{-1} (E_{\alpha})$$

Next he uses a lemma to prove this. I am more interested in the two equalities he uses.

Before anything I will ask the stupid question. Shouldn't one of the elements in $$M_{\alpha}$$ be the empty set? That is there should be some $$E_{j}$$ which will just be the empty set. Then the cartesian product of all elements in $$M_{\alpha}$$ should be the empty set. Where am I goofing up?

Anyway again I consider my example where I have $$\{ R_{1}, R_{2} \}$$, the reals and then have $$M_{1}$$ for $$R_{1}$$ and $$M_{2}$$ for $$R_{2}$$.

Now suppose I consider $$E_{1}$$ in $$M_{1}$$. Then the first equality should be $$\pi_{1}^{-1} E_{1} = E_{1} \times R_{2}$$, which I can understand.

Next, in the second equality, the LHS I already asked about before. The RHS looks like it should be = $$E_{1} \times R_{1} \cap E_{2} \times R_{1} \cap E_{3} \times R_{1} \cap ........$$, but I don't see how this condenses to $$E_{1} \times E_{2} \times .....$$ which I am assuming is the RHS.


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# A core-shell Au@Cu2-xSe heterogeneous steel nanocomposite for photoacoustic and computed tomography dual-imaging-guided photothermal boosted chemodynamic remedy | Journal of Nanobiotechnology

### Supplies

Hydrogen Tetrachloroaurate(III) Trihydrate (HAuCl4·3H2O) and methylene blue (MB) have been obtained from Shanghai Titan Expertise Co., Ltd. Copper(II) sulfate pentahydrate (CuSO4·5H2O), selenium dioxide (SeO2), sodium citrate dihydrate, ascorbic acid, and hydrogen peroxide (H2O2) have been bought from Sinopharm Chemical Reagent Co., Ltd. Polyvinylpyrrolidone (PVP, Mw ~55,000), 2′,7′- dichlorofluorescein diacetate (DCFH-DA), and Calcein AM have been supplied by Sigma Aldrich. Roswell Park Memorial Institute 1640 (RPMI-1640) was supported by Corning Included. Fetal bovine serum (FBS), trypsin, and penicillin-streptomycin resolution have been provided from Gibco Firm. Cell Counting Equipment-8 (CCK-8) resolution was bought from Dojindo Laboratories. Hoechst 33,342, 4′,6-diamidino-2-phenylindole (DAPI), and propidium iodide (PI) have been acquired from Beyotime Biotechnology. Mitochondrial hydroxyl radical detection assay, anti-gamma H2AX (phospho S139) antibody [EP854(2)Y], and goat polyclonal secondary antibody to rabbit IgG – H&L (Alexa Fluor® 647) have been bought from Abcam Firm. Accutase™ Cell Detachment Answer and FITC Annexin V Apoptosis Detection Equipment have been provided from BD Bioscience. All chemical substances and reagents have been used as acquired with out additional purification.

### Synthesis of Au NPs

HAuCl4 aqueous resolution ready by mixing 1.215 mL of 10 mM HAuCl4 resolution and 50 mL of deionized water was heated to slight boiling. A condenser was utilized to forestall the evaporation of the water. Then 50 mL of 10 mg mL−1 sodium citrate resolution was added into the boiling HAuCl4 resolution. After conserving boiling for 15 min below steady stirring, the combination was allowed to chill to room temperature. The resulted Au NPs have been collected by centrifuging the suspension at 15,000 rpm for 15 min.

### Synthesis of Au@Cu2-xSe NPs

5.5 mL of Au NPs redispersed in deionized water was blended with 1.6 mL of 10 mg mL−1 PVP resolution and stored stirring at 30 °C for 1 h, adopted by the addition of 0.1 mL of 0.1 M SeO2 and 0.3 mL of 0.2 M ascorbic acid resolution. After response for 30 min, a blended resolution of 0.1 mL of 0.2 M CuSO4 and 0.4 mL of 0.2 M ascorbic acid was added. After stirring the suspension at 30 °C for an additional 10 h, the Au@Cu2-xSe NPs have been collected by centrifuging the suspension at 15,000 rpm for 15 min and washed 3 times with deionized water.

### Synthesis of Cu2-xSe NPs

1.6 mL of 10 mg mL−1 PVP resolution was added into 5.5 mL of deionized water and stored stirring at 30 °C, adopted by the addition of 0.1 mL of 0.1 M SeO2 and 0.3 mL of 0.2 M ascorbic acid resolution. After response for 30 min, a blended resolution of 0.1 mL of 0.2 M CuSO4 and 0.4 mL of 0.2 M ascorbic acid was added. After stirring the suspension at 30 °C for an additional 10 h, the Cu2-xSe NPs have been collected by centrifuging the suspension at 15,000 rpm for 15 min and washed 3 times with deionized water.

### Characterization

The morphology and construction have been characterised by transmission electron microscopy (TEM, JEOL JEM-2100 F). The crystal construction was decided by X-ray diffraction (XRD, Rigaku D/max2500). The chemical state was analyzed by X-ray photoelectron spectroscopy (XPS, Thermo Escalab 250), and the binding vitality of C 1 s peak at 284.8 eV was taken as an inside customary. Fourier remodel infrared (FTIR) spectrum was recorded on Bruker Tensor-27 spectrometer. Ultraviolet-visible-infrared (UV-vis-NIR) absorption spectrum was scanned on Persee TU-1901 spectrometer. The steel component content material was decided by inductively coupled plasma optical emission spectroscopy (ICP-OES, Varian 710ES). Hydrodynamic dimension and zeta potential have been characterised by a particle dimension zeta potential analyzer (Malvern ZEN3690).

### Photothermal efficiency

An 808 NIR laser was used to estimate the photothermal efficiency of Au@Cu2-xSe NPs. The NIR energy density was maintained at 1.0 W cm−2 in all exams. The amount of Au@Cu2-xSe aqueous dispersions in every take a look at was 1.5 mL. Temperature adjustments have been recorded by a thermal digicam. For concentration-dependent photothermal efficiency, numerous Au@Cu2-xSe aqueous dispersions in a quartz cuvette with concentrations of 0, 25, 50, 100, and 200 µg mL−1 have been uncovered to laser for 10 min. To judge the photothermal stability, Au@Cu2-xSe aqueous dispersion with a focus of fifty µg mL−1 was irradiated by laser for 10 min (laser on) adopted by pure cooling with out irradiation for an additional 10 min (laser off). Such heating/cooling processes have been repeated 6 instances. Heating-cooling curve was obtained by constantly irradiating 50 µg mL−1 dispersion with laser for 15 min, after which turning off the laser to permit the dispersion to naturally cool to room temperature. Following beforehand reported methodology [8, 20, 28], the photothermal conversion effectivity was calculated based on Eqs. 14.

$$eta =frac{{hS({T_{hbox{max} }} – {T_{hbox{max} ,{{textual content{H}}_{textual content{2}}}{textual content{O}}}})}}{{I(1 – {{10}^{ – {A_lambda }}})}}$$

(1)

$$hS=frac{{m{C_{textual content{p}}}}}{{{tau _s}}}$$

(2)

$$t= – {tau _{textual content{s}}}ln theta$$

(3)

$$theta =frac{{T – {T_{{textual content{sur}}}}}}{{{T_{hbox{max} }} – {T_{{textual content{sur}}}}}}$$

(4)

the place η is the photothermal conversion effectivity, h is the heat-transfer coefficient, S is the floor space of container, T is the temperature of Au@Cu2-xSe aqueous dispersion, Tmax is the utmost temperature of Au@Cu2-xSe aqueous dispersion, Tmax,H2O is the utmost temperature of pure water, Tsur is the ambient temperature of environment, I is the laser energy, Aλ is the absorbance of Au@Cu2-xSe aqueous dispersion on the wavelength of 808 nm, m and Cp are the mass (1.5 g) and warmth capability (4.2 J g−1) of solvent (water), t is the cooling time, τs is the time fixed of pattern system, θ is the driving pressure temperature. τs may be obtained by making use of the linear time knowledge from the cooling interval vs. –lnθ (Fig. 2D, E).

### Photothermal enhanced Fenton-like catalytic efficiency

An MB degradation assay was used to watch the ·OH era. First, blended aqueous options of Au@Cu2-xSe and MB remained static state at the hours of darkness for 30 min at 25, 37, and 42 °C, respectively. Second, after including H2O2, the mixtures have been then remained static state at the hours of darkness for an additional 30 min at 25, 37, or 42 °C, respectively. The ultimate concentrations of Au@Cu2-xSe NPs, MB, and H2O2 in mixtures have been set at 100 µg mL−1, 10 µg mL−1, and 10 mM, respectively. Lastly, after eradicating Au@Cu2-xSe NPs through centrifugation at 15,000 rpm for 15 min, the absorbance of every supernatant at 664 nm was measured. In these experiments, the pure MB resolution (10 µg mL−1), and the combination of MB (10 µg mL−1) and H2O2 (10 mM) resolution, have been chosen as management.

Subsequent, ESR was used to substantiate the era of ·OH in Fenton-like response. DMPO was used as a spin entice agent for ·OH. After conserving the blended aqueous options of Au@Cu2-xSe, H2O2, and DMPO for five min at 25, 37, and 42 °C, the mixtures have been used to report ESR spectra by a spectrometer (Bruker ELEXSYS E500, CW X-band), respectively. The ultimate concentrations of Au@Cu2-xSe, H2O2, and DMPO have been set at 50 µg mL−1, 100 µM, and 50 mM, respectively. The combination of Au@Cu2-xSe (50 µg mL−1) and DMPO (50 mM), and the combination of H2O2 (100 µM) and DMPO (50 mM), have been chosen as management.

To show the photothermal accelerated ·OH era in Fenton-like response, 1.5 mL of blended aqueous resolution of Au@Cu2-xSe NPs, MB, and H2O2 was irradiated with or with out 808 nm NIR laser at 1.0 W cm−2 for 10 min. The ultimate concentrations of Au@Cu2-xSe NPs, MB, and H2O2 in combination have been set at 100 µg mL−1, 10 µg mL−1, and 10 mM, respectively. The Temperature rise was recorded by thermal digicam. Lastly, after eradicating Au@Cu2-xSe NPs through centrifugation at 15,000 rpm for 15 min, the absorbance of supernatant at 664 nm was measured.

### Detection of intracellular ROS

CLSM (Zesis LSM800) was adopted to detect the intracellular ROS by utilizing DCFH-DA (Sigma-Aldrich) as ROS probe. Cells have been divided into six teams, together with (1) management, (2) H2O2, (3) Au@Cu2-xSe, (4) Au@Cu2-xSe + H2O2, (5) Au@Cu2-xSe + NIR, and (6) Au@Cu2-xSe + H2O2 + NIR. First, 2 × 105 4T1 tumor cells have been seeded into the CLSM-specific tradition disk after which cultured at 37 °C below 5% CO2 for twenty-four h. Second, after changing the tradition medium with 1 mL of FBS-free RPMI-1640 medium containing Au@Cu2-xSe (50 µg mL−1) or H2O2 (100 µM), the cells in group (5) and (6) have been irradiated with NIR laser (1.0 W cm−2) for five min, respectively. After irradiation, the cells continued to incubate for 30 min. Third, after washing the cells 3 times with PBS, the cells have been incubated with 1 mL of FBS-free RPMI-1640 medium containing DCFH-DA (10 µM) and Hoechst 33,342 (5 µg mL−1) for 20 min. Fourth, the cells have been washed 3 times with PBS adopted by the addition of 1 mL of FBS-free RPMI-1640 medium. Then the fluorescence photos of cells have been collected by CLSM.

### Detection of intracellular ·OH radicals

Mitochondrial hydroxyl radical detection assay package (Abcam) was used to detect the intracellular ·OH. First, 2 × 105 4T1 tumor cells have been seeded into the CLSM-specific tradition disk after which cultured for twenty-four h. Second, after changing the tradition medium with 200 µL of the ready OH580 stain working resolution, the cells continued to incubate for 1 h with out mild. Third, after washing the cells one time with PBS, the cells have been incubated with 1 mL of RPMI-1640 full medium containing Au@Cu2-xSe (50 µg mL−1) or H2O2 (100 µM), after which irradiated with or with out NIR laser (1.0 W cm−2) for five min. After irradiation, the cells continued to incubate for 30 min. Fourth, the cells have been incubated with 1 mL of PBS containing Hoechst 33,342 (5 µg mL−1) for 10 min. Fifth, the cells have been washed 3 times with PBS adopted by the addition of 1 mL assay buffer. Then the fluorescence photos of cells have been collected by CLSM.

### In vitro cytotoxicity assay

The cytotoxicity of Au@Cu2-xSe towards HEK293 regular cells and 4T1 tumor cells have been examined through a CCK-8 assay. HEK293 regular cells or 4T1 tumor cells have been seeded into 96-well plate at 1 × 104 cells/effectively and incubated for twenty-four h. Then the cells have been incubated in 100 µL of the entire medium containing Au@Cu2-xSe NPs at numerous concentrations of 0, 12.5, 25, 50, 100, and 200 µg mL−1. After 24 h, the tradition medium was eliminated and the cells have been washed 3 times with PBS. Then the cells have been incubated in 100 µL of full medium containing 10 µL CCK-8. After one other 1–2 h, the optical densities of every effectively have been recorded at 450 nm on a microplate reader (BioTek Cytation 3).

### In vitro mobile uptake

First, 4T1 tumor cells have been seeded into 6-well plate at 1.5 × 105 cells/effectively after which cultured at 37 °C below 5% CO2 for twenty-four h. Second, after changing the tradition medium with 1 mL of the entire medium containing Au@Cu2-xSe NPs (50 µg mL−1), the cells have been then incubated for 0.5, 1, 2, 4, and eight h, respectively (three wells at every time level). Third, the tradition medium was eliminated and the cells have been washed 3 times with PBS, digested with 0.25% trypsin (Gibco), and centrifuged at 1500 rpm for 3 min. When the collected cells have been redispersed in 1 mL PBS, the cells have been counted. Lastly, the content material of Au in PBS was analyzed by ICP-MS.

### In vitro PTT and CDT

4T1 tumor cells have been divided into eight teams, together with (1) management, (2) H2O2, (3) NIR, (4) H2O2 + NIR, (5) Au@Cu2-xSe, (6) Au@Cu2-xSe + H2O2, (7) Au@Cu2-xSe + NIR, and (8) Au@Cu2-xSe + H2O2 + NIR. First, 4T1 tumor cells have been seeded into 96-well plate at 1 × 104 cells/effectively and incubated for twenty-four h. Second, after changing the tradition medium with 100 µL of the entire medium containing Au@Cu2-xSe (50 µg mL−1) or H2O2 (100 µM), the cells have been irradiated with or with out NIR laser (1.0 W cm−2) for five min. After irradiation, the cells continued to incubate for twenty-four h. Third, the tradition medium was eliminated and the cells have been washed 3 times with PBS. Then the cells have been incubated in 100 µL of full medium containing 10 µL CCK-8 for an additional 1 h. Lastly, the optical densities of every effectively have been recorded at 450 nm on a microplate reader.

CLSM was then employed to watch the cell dying induced by hyperthermia and ·OH radicals. First, 2 × 105 4T1 tumor cells have been seeded into the CLSM-specific tradition disk after which cultured at 37 °C below 5% CO2 for twenty-four h. Second, after changing the tradition medium with 1 mL of the entire medium containing Au@Cu2-xSe (50 µg mL−1) or H2O2 (100 µM), the cells have been irradiated with or with out NIR laser (1.0 W cm−2) for five min. After irradiation, the cells continued to incubate for 4 h. Third, the tradition medium was changed by 1 mL of PBS containing Calcein AM (2 µM) and PI (4 µM) and co-incubated with the cells for an additional 20 min. Lastly, the fluorescence photos of cells have been collected instantly by CLSM.

### In vitro apoptosis and necrosis evaluation by move cytometry

First, 4T1 tumor cells have been seeded into 12-well plate at 7 × 104 cells/effectively after which cultured at 37 °C below 5% CO2 for twenty-four h. Second, after changing the tradition medium with 1 mL of the entire medium containing Au@Cu2-xSe (50 µg mL−1) or H2O2 (100 µM), the cells have been irradiated with or with out NIR laser (1.0 W cm−2) for five min. After irradiation, the cells continued to incubate for twenty-four h. Third, the cells have been digested with Accutase™ Cell Detachment Answer (BD Biosciences) and re-suspended in PBS. Then, the cells have been stained based on the FITC Annexin V Apoptosis Detection Equipment (BD Biosciences) and quantified by move cytometry (Beckman Cytoflex).

### In vitro DNA harm evaluation

First, 2 × 105 4T1 tumor cells have been seeded into the CLSM-specific tradition disk after which cultured for twenty-four h. Second, after changing the tradition medium with 1 mL of RPMI-1640 full medium containing Au@Cu2-xSe (50 µg mL−1) or H2O2 (100 µM), the cells have been irradiated with or with out NIR laser (1.0 W cm−2) for five min. After irradiation, the cells continued to incubate for six h. Third, these cells have been mounted with 4% paraformaldehyde for 10 min, washed a number of instances with PBS, permeabilized with immunostaining permeabilization buffer with triton X-100 (Beyotime Biotechnology) for 10 min, and blocked with QuickBlock blocking buffer (Beyotime Biotechnology) for 10 min at room temperature. Forth, after washing 3 times with PBS, the cells have been stained with anti-gamma H2AX (phospho S139) antibody [EP854(2)Y] (Abcam, dilution 1:400) at 4 °C in a single day. Fifth, after washing 3 times with PBS, the cells have been incubated with the goat polyclonal secondary antibody to rabbit IgG-H&L (Alexa Fluor® 647) (Abcam, dilution 1:400) for 1 h. Sixth, after washing 3 times with PBS, the cells have been stained with DAPI (Beyotime Biotechnology) for five min. Lastly, the cells have been washed 3 times with PBS adopted by the addition of 1 mL PBS. Then the fluorescence photos of cells have been collected by CLSM.

### In vitro hemolysis assay

1 mL of contemporary blood was taken from BALB/c mouse and saved in a tube containing EDTA. 2 mL of saline was added and blended. Crimson blood cells (RBC) have been collected by centrifuging the blood at 5000 rpm for 3 min. Then the RBC was washed 3 times with saline till the supernatant grew to become colorless. After discarding the supernatant, 100 µL of RBC was taken and redispersed into 5 mL saline to organize 2% quantity fraction RBC suspension. Then a sequence of mixtures have been ready by mixing 500 µL RBC suspension and 500 µL saline containing Au@Cu2-xSe with totally different concentrations. The ultimate concentrations of Au@Cu2-xSe within the mixtures have been set at 12.5, 25, 50, 100, and 200 µg mL−1, respectively. The combination of 500 µL RBC suspension and 500 µL saline, and the combination of 500 µL RBC suspension and 500 µL pure water, have been used as detrimental and optimistic controls, respectively. After remaining steady at room temperature for five h, these mixtures have been centrifuged at 15,000 rpm for 15 min. Lastly, the absorbance (A) of supernatant in numerous mixtures was recorded at 570 nm on microplate reader for calculating the hemolysis charge of RBC. Hemolysis charge was calculated based on Eq. 5.

$${textual content{Hemolysis charge}}=frac{{{{textual content{A}}_{{textual content{pattern}}}} – {{textual content{A}}_{{textual content{saline}}}}}}{{{{textual content{A}}_{{{textual content{H}}_{textual content{2}}}{textual content{O}}}}{textual content{-}}{{textual content{A}}_{{textual content{saline}}}}}} instances 100%$$

(5)

### In vivo toxicity

After 200 µL of saline (management group) or 200 µL of Au@Cu2-xSe suspended in saline (2 mg mL−1) have been injected into mice through intravenous injection (n = 3, per group), mice have been sacrificed at 14 d to gather blood for hematological analyses and tissues (together with coronary heart, liver, spleen, lung, and kidney) for H&E stain. In these 14 days, weights of mice have been measured each 2 days.

### Tumor mannequin

100 µL PBS containing 4T1.2 cells (2 × 106 cells mL−1) was subcutaneously injected beside the foreleg of a BALB/c feminine mouse (6-weeks outdated). When tumor grows to ~80 cm3, the tumor-bearing mouse was utilized in subsequent in vivo experiments.

### In vitro and in vivo PA imaging

PA imaging was carried out on an ultrasound-photoacoustic dual-mode imaging system (VEVO LAZR-X, Fujifilm VisualSonics). The excitation wavelength for PA imaging was 808 nm. For in vitro PA imaging, numerous Au@Cu2-xSe aqueous dispersions with concentrations of 0, 25, 50, 100, and 200 µg mL−1 have been injected into polyethylene tubes for PA testing, respectively. For in vivo PA imaging, 200 µL saline containing Au@Cu2-xSe NPs (2 mg mL−1) have been intravenously injected into tumor-bearing mice (n = 3). Then the ultrasound and photoacoustic indicators at tumor web site have been recorded at 1, 2, 4, 6, and eight h post-injection, respectively. The indicators earlier than injection have been used as management.

### In vitro and in vivo CT imaging

CT imaging was collected from the Siemens Inveon PET/CT imaging system. For in vitro CT imaging, numerous Au@Cu2-xSe aqueous dispersions with concentrations of 0, 0.625, 1.25, 2.5, and 5 mg mL−1 have been positioned in small tubes for CT imaging. In vivo CT imaging was performed with tumor-bearing mice (n = 3, per group) after intravenous or intratumoral injection of Au@Cu2-xSe NPs. For intravenous injection, the CT photos have been collected at 1, 2, 4, 6, and eight h post-injection of 200 µL saline containing Au@Cu2-xSe NPs (2 mg mL−1). The CT photos earlier than injection have been used as management. For intratumoral injection, the CT photos have been collected after injection of 150 µL saline containing Au@Cu2-xSe NPs with 0, 6, and eight mg mL−1, respectively.

### In vivo PTT and CDT

Tumor-bearing mice have been divided randomly into 5 teams (n = 5, per group): (1) saline; (2) saline + NIR (1.5 W cm−2); (3) Au@Cu2-xSe; (4) Au@Cu2-xSe + NIR (1.0 W cm−2); (5) Au@Cu2-xSe + NIR (1.5 W cm−2). Mice in group (1) and (2) have been intravenously injected with 200 µL pure saline, whereas mice in group (3), (4), and (5) have been intravenously injected with 200 µL saline containing Au@Cu2-xSe NPs (2 mg mL−1). At 4 h post-injection, tumors in group (2) and (5) have been irradiated with 808 nm NIR laser for five min at 1.5 W cm−2, whereas tumors in group (4) have been irradiated with 808 nm NIR laser for five min at 1.0 W cm−2. The temperature adjustments at tumor websites have been recorded with a thermal digicam. The size and width of tumors and weight of mice have been measured each 2 days. The tumor quantity (V) was calculated based on equation V = (width2 × size)/2. The relative tumor quantity (V/V0) was obtained by normalizing the tumor quantity to the preliminary tumor quantity (V0). Moreover, the tumors at 20 h post-treatment have been collected for H&E and TUNEL analyses. For in vivo ROS detection in tumor, 200 µL saline containing DCFH-DA (100 µM) was intratumorally injected into the tumor-bearing mice at 3.5 h post-injection of saline or Au@Cu2-xSe NPs. 30 min later, NIR irradiation was administrated. Then, tumors in every group have been collected for fluorescence evaluation.

Intratumoral injection of Au@Cu2-xSe NPs was additionally carried out to judge synergistic remedy of PTT + CDT. Tumor-bearing mice have been divided randomly into 4 teams (n = 5, per group): (1) saline; (2) saline + NIR; (3) Au@Cu2-xSe; (4) Au@Cu2-xSe + NIR. After intratumoral injection of 100 µL pure saline or 100 µL saline containing Au@Cu2-xSe NPs (2 mg mL−1), tumors in group (2) and (4) have been then irradiated with 808 nm NIR laser for five min at 0.5 W cm−2. Tumors at 20 h post-treatment have been collected for H&E and TUNEL analyses.

### Statistical evaluation

Bars show imply ± s.d., and statistical evaluation was carried out utilizing Pupil’s t-test and the P values have been supplied (***P < 0.001; **P < 0.01; *P < 0.05).

format: plain


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# How does fluid viscosity determine viscous friction between fluid and the moving plate?

When fluid is placed between two parallel plates such that one plate is moving and other is stationary, fluid will start flowing. Between plate and the fluid there is viscous friction given by equation: $$F = -\eta A \frac{\text{d}v}{\text{d}y}$$ where $$\eta$$ is fluid viscosity, $$A$$ is area of a plate and $$\text{d}v/\text{d}y$$ is a velocity gradient.

Since fluid viscosity is a measure of intermolecular or cohesive forces in a fluid, how can its value determine viscous friction between fluid and the plate since interactions between fluid and the plate aren't the same like between fluid molecules?


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# Import of titles with math symbols from APS doesn't work

Hi,

Consider the following item
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.107006

When I import it with Zotero Connector addon in Firefox the math gets broken. So I see in the Zotero UI this

$({\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{K}}_{x}){\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$

Then I export it to bibtex and I see this

\$(\{\textbackslash mathrm\{\vphantom{\}\}}{{Ba}}\vphantom\{\}\vphantom\{\}\_\{1\textbackslash ensuremath\{-\}x\}\{\textbackslash mathrm\{\vphantom{\}\}}{{K}}\vphantom\{\}\vphantom\{\}\_\{x\})\{\textbackslash mathrm\{\vphantom{\}\}}{{Fe}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash mathrm\{\vphantom{\}\}}{{As}}\vphantom\{\}\vphantom\{\}\_\{2\}\$

So basically the math environment is escaped, and I see those special commands instead of the equation.

However if I download the bib file and then import it to zotero it works correctly. So I can see this in the UI

(Ba₁₋ₓKₓ)Fe₂As₂

and then also correct exported bibtex format

({{Ba}}{$_{1-}$}{{\textsubscript{x}K\textsubscript{x}}}){{Fe}}{$_{2}$}{{As}}{$_{2}$}

************************
Zotero version: 5.0.96.2
Firefox version: 90.0
Zotero connector version: 5.0.86

Regards,
• Zotero doesn't support math, so as far as Zotero is concerned, this is just text. You can sort of work around this using BBT, either by wrapping that title in <script>...</script> or by having that applied automatically.

If you import $({\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{K}}_{x}){\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$ by hand with BBT installed it will actually translate that to (Ba₁₋ₓKₓ)Fe₂As₂ which would be exported back to bibtex (but as text, not math) and would also work with Word bibliographies. This has some obvious limits, not everything mathy has a decent text representation.
• Thanks for the answer, but this doesn't solve my problem.
What I need is to be able to import bibliography with firefox addon and then export it to bibtex. Currently it is broken, so Zotero is unfortunately not suitable for me, because of this math bug.
• @AbeJellinek -- APS currently pulls RIS -- might make more sense to use BibTeX given the prevalence of formulas in the field (and the fact that it looks like they just put pseudo-MathJax into RIS titles)


text:
# C-shapes in custom blocks: improved "editing behavior"

Introduction
Recently I've been playing with custom blocks having at least one C-shape command among its inputs. What I've experienced is that using such C-shaped custom block makes for rather clunky editing.

Description of the issue
When a variable or a reporter is dragged anywhere near the C-shape a semi-transparent white rectangular veil appears, covering the C-shape and its contents; if one accidentally releases the variable or reporter block - even if only for a fraction of a second - it settles into the C-shape and causes its original contents to be ejected, which may be many lines of code. (I don't know how to shoot all this in Snap!, but I'm sure others will recognize what I've described).
It even applies to "official" library blocks, like (from "List utilities").

A solution exists! (or so I suppose)
Snap! primitives with C-shape commands, such as:
, , and
... won't be associated with this kind of havoc. So I guess it must be solvable.

Proposal
My proposal is that the "editing behavior" (for lack of a better word) of custom blocks with C-shape commands as inputs will be aligned with that of their cousins amongst Snap! primitives.

You should have an option,instead of the behaviour getting deleted
Variables with rings inside are expected to go into c slots.

I may not understand what you mean. I wasn't saying no variables etc. may be substituted into slots of blocks contained inside the C-shape. All I've proposed is to do away with custom blocks behaving differently, while one is editing, from otherwise similar primitive blocks. Basically I'm asking for to behave just like .

i agree with this, in any of the primitive blocks youd use the RUN block, not drop it in the C. dropping a variable in the C is confusing because it's not very clear from the visuals that you can do it or what it will actually do.
i think most people's first experience with this is doing it by accident, which i think says a lot

thx 4 the 1st vote! (assuming it was yours)

Yes! This has definitely been a significant annoyance when creating large projects with custom blocks.
Personally, my solution to this problem would be to add an option in the input settings menu, similar to read-only, that would make it so nothing could overwrite what is inside the block input if something is already in it.

Concept art explaining my idea

Being able to drop reporters - specifically variable getters - into custom command blocks' c-slots is an important pedagogical design to foster the understanding of "procedure as data", one of the truly powerful ideas in computing. Consider this recursive definition of while:

That's what we mean when we talk about "Beauty and Joy of Computing"!

[edit: ] I realize most of you aren't teaching / learning this in your CS classes. But that's what we made Snap! for, and that's totally what we believe kids have a right to be taught.

That's what I was thinking when making my suggestion above, as having variables containing code, or other such things is very useful.

OK, now I understand the clunky editing of C-shape custom blocks is a direct consequence of a deliberate design decision, itself based on Snap’s educational character. That I hadn’t expected, but it makes sense now, thank you for explaining! (perhaps @d4s_over_dt4 tried to convey the same principle in post #2). So I’ll gladly rest my case (thx for the votes, people).

BTW the recursive WHILE block is a true brain teaser; it really only works the way you wrote it, with “test” being a predicate (not a Boolean), “test” being CALL-ed, and the WHILE tail recursion being an independent block (not RUN). Makes me wonder how many BJoC-students do grasp it .

i think the main issue is that variables can be dropped in even when the C already has code in it. in most other cases, dropping a variable into a filled slot doesn't work (at least when other slots are around it), and it seems odd that the least used case would be the easiest to do.

I don't think this is true; when you drag an expression over any filled slot you get a halo around the current expression and if you let go, the new expression replaces the old one. (But the old one doesn't just disappear; it stays in the scripting area, not attached to anything.) So I'm surprised that y'all are surprised that this is how it works even if it's a C-slot.

The fact that you can't put variables in the C-slot of (some) primitives is a misfeature, inherited from Scratch.

if you stack many reporters (such as operators), it's often impossible to replace any specific filled slot, i think there's a priority system.
i know there's a halo around the C, but i don't lock my mouse in place, check the halo, then release the mouse button. it would take years to make projects at that rate. the issue is that misdrops happen, and when they do it's easy to lose track of where things were, especially because it's now difficult to find where the C slot is, the popped out script is almost certainly overlapping other scripts, and after dealing with that you've probably lost track of where you were in the code in the beginning

That’s not a misfeature, it’s a misbug

I hadda look that up; I knew the word existed but had forgotten what it meant. It's an unintended behavior that turns out useful. But the thing about C slots of primitives isn't unintended; it's just wrong.

My statement was actually meant as a pun on INABIAF - I wasn’t even aware of misbug being a known (tongue-in-cheek) expression. Nonetheless, it was at least half-serious.

I fully understand Snap! developers sometimes choosing to prioritize pedagogical objectives over ease of use. However, if application of common (primitive) constructions such as FOR, REPEAT / WHILE and IF ...ELSE is, deliberately, going to be made more difficult out of principle, IMO the baby is being thrown out with the bathwater … students may be annoyed instead of encouraged to further explore Snap!’s many instructive features.

Although of course I'm familiar with the underlying sentence, this is the first time I've seen it acronymized. You learn something every day. (Strictly speaking it should be INBIF, but I get that that'd be less funny.)

It's not the case that we always put principle over convenience. Maybe the biggest counterexample is sprite inheritance: In principle, every attribute of the parent sprite should be shared with its clones, not copied, when the clone is created. But in fact some attributes are shared and others are copied, based entirely on how people ordinarily use each attribute. That sort of bothers me, but Jens talked me into it by citing the example of

which people use in a gazillion projects.

Even when it comes to dropping expressions into slots, we pay attention to convenience. Back a million years ago, in BYOB, if you had overlapping scripts (overlapping in their actual placement in the scripting area) and dropped something, it filled the topmost slot, iirc, which was a simple rule, but I convinced Jens that it should prefer empty slots even if the topmost one is filled. I'm probably misremembering the details, but the point is, we've tried to make it easy to use.

But yeah, in this case, you want to make it impossible to do something uncommon, so that's a big ask.


text:
# Differentiating the $W$ lambert function to find optimum angle for maximum horizontal range of a projectile with air resistance

Here is the equation I obtained, representing the horizontal range of a projectile in air resistance as a function of its initial release angle-

$$x=\frac{V_0\cos(\theta)}{B}-\frac{V_0\cos(\theta)}{B}\,\exp\left(\frac{BV_0\sin(\theta)}{g}\right)\exp\left(W\left(\frac{BV_0\sin(\theta)\exp\left(\frac{BV_0\sin(\theta)}{-g}\right)}{-g}\right)\right),$$

where:

$$x=$$ horizontal distance travelled by projectile

$$V_0=$$ initial velocity of projectile

$$g=$$ acceleration due to gravity

$$\theta=$$ initial release angle

I have assumed that the force of air resistance acting on the projectile is equal to a constant k multiplied by the velocity of the projectile.

So, $$F=kv.$$

In my equation, $$B= k/m,$$ where $$m$$ is the mass of the projectile.

Also, $$W(\cdot)$$ is the $$W$$ Lambert function.

I have been trying to differentiate the expression with respect to $$\theta$$ in order find an expression for the value of $$\theta$$ that would result in maximum horizontal range, assuming all other terms in the equation are constant.

How can this be done?

• According to this diagram, $X=0$. Can you show how you found the equation for $X$? – Peter Foreman Jul 8 at 14:18
• Since the equation for $X$ is not too long, can you type it up? – automaticallyGenerated Jul 8 at 14:21
• Curious about $F=kv.$ Is that the magnitude of the force equal to a constant times the magnitude of the velocity? – Adrian Keister Jul 8 at 14:41
• Yes. k is a constant, which depends on the properties of the fluid that the projectile is travelling through. F is the force of air resistance, the magnitude of which is equal to the constant k times the magnitude of the velocity of the projectile. – LZR789 Jul 8 at 14:45
• So what's the vector equation? I'm guessing $\mathbf{F}=-k\mathbf{v};$ is that correct? – Adrian Keister Jul 8 at 15:07

Starting from $$\mathbf{F}=-k\,\mathbf{v}-mg\,\hat{\mathbf{j}}=m\dot{\mathbf{v}},$$ and initial conditions $$\mathbf{v}_0=v_0\langle\cos(\theta),\sin(\theta)\rangle$$ and $$\mathbf{x}_0=\langle 0,0\rangle,$$ we obtain \begin{align*} m\ddot{x}&=-k\dot{x} \\ \ddot{x}&=-B\dot{x} \\ \dot{x}&=Ce^{-Bt} \\ \dot{x}&=v_0\cos(\theta)\,e^{-Bt} \\ x&=-\frac{v_0\cos(\theta)}{B}\,e^{-Bt}+C\\ x(t)&=\frac{v_0\cos(\theta)}{B}\,\left(1-e^{-Bt}\right),\;\text{and} \\ m\ddot{y}&=-k\dot{y}-mg \\ \ddot{y}&=-B\dot{y}-g \\ \int\frac{d \dot{y}}{B\dot{y}+g}&=-\int dt\\ \frac1B\int\frac{d \dot{y}}{\dot{y}+g/B}&=-t+C\\ \frac1B\,\ln|\dot{y}+g/B|&=-t+C\\ \ln|\dot{y}+g/B|&=-Bt+C, \quad\text{absorbing into constant}\\ \dot{y}+g/B&=Ce^{-Bt},\quad\text{switching to multiplicative constant}\\ v_0\sin(\theta)+\frac{g}{B}&=C\\ \dot{y}&=\left(v_0\sin(\theta)+\frac{g}{B}\right)e^{-Bt}-\frac{g}{B}\\ y&=-\left(\frac{v_0\sin(\theta)+\frac{g}{B}}{B}\right)e^{-Bt}-\frac{gt}{B}+C\\ y(t)&=\frac{v_0\sin(\theta)+mg/k}{B}\left(1-e^{-Bt}\right)-\frac{mgt}{k}. \end{align*} Solving the second equation $$y(t)=0$$ for $$t$$ yields $$t_{\text{range}}=\frac{mg W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+mg+k v_0 \sin (\theta )}{B mg}.$$ Plugging this back into $$x$$ yields $$x_{\text{range}}(\theta)=\frac{v_0 \cos (\theta )}{B} \left(1-\exp \left(- W\left(-e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (1+(k v_0/(mg)) \sin (\theta ))\right)+1+(k v_0/(mg)) \sin (\theta )\right)\right).$$ Now it is a fact that $$\dot{W}(t)=\frac{W(t)}{t(1+W(t))}.$$ Using this, we can differentiate this expression (it's quite a monster) to obtain $$x_{\text{range}}'(\theta)=-\frac{v_0 \left(mg W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+mg+k v_0 \sin (\theta )\right) \left((mg \sin (\theta )+k v_0) W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+\sin (\theta ) (mg+k v_0 \sin (\theta ))\right)}{B (mg+k v_0 \sin (\theta ))^2 \left(W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+1\right)}.$$ Solving $$x_{\text{range}}'(\theta)=0$$ for $$\theta$$ will be tricky. We can eliminate the denominator and just solve $$-v_0 \left(mg W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+mg+k v_0 \sin (\theta )\right) \left((mg \sin (\theta )+k v_0) W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+\sin (\theta ) (mg+k v_0 \sin (\theta ))\right) =0.$$ Mathematica does not immediately find a solution, but we note that this expression is factored. That is, we can solve either of \begin{align*} mg W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+mg+k v_0 \sin (\theta )&=0,\quad\text{or}\\ (mg \sin (\theta )+k v_0) W\left(-\frac{e^{-\frac{k v_0 \sin (\theta )}{mg}-1} (mg+k v_0 \sin (\theta ))}{mg}\right)+\sin (\theta ) (mg+k v_0 \sin (\theta ))&=0. \end{align*} The first one initially looks a bit more promising. Let $$s=\sin(\theta),$$ so that we have \begin{align*} mg W\left(-\frac{e^{-\frac{k v_0 s}{mg}-1} (mg+k v_0 s)}{mg}\right)+mg+k v_0 s&=0 \\ W\left(-\frac{e^{-\frac{k v_0 s}{mg}-1} (mg+k v_0 s)}{mg}\right)+1+(k v_0/(mg)) s&=0 \\ W\left(-\frac{e^{-\frac{k v_0 s}{mg}-1} (mg+k v_0 s)}{mg}\right)&=-1-\frac{kv_0s}{mg} \\ -\frac{e^{-\frac{k v_0 s}{mg}-1} (mg+k v_0 s)}{mg}&=-\left(\frac{mg+kv_0s}{mg}\right)\exp\left(-1-\frac{kv_0s}{mg}\right), \end{align*} which you can see is a tautology! That is, it conveys no information. Therefore, any solution must be from the other factor. Let us work on that (again, $$s=\sin(\theta)$$) \begin{align*} (mgs+k v_0) W\left(-\frac{e^{-\frac{k v_0 s}{mg}-1} (mg+k v_0 s)}{mg}\right)+s (mg+k v_0 s)&=0 \\ (mg s+k v_0) W\left(-\frac{e^{-\frac{k v_0 s}{mg}-1} (mg+k v_0 s)}{mg}\right)&=-s (mg+k v_0 s) \\ W\left(-\frac{e^{-\frac{k v_0 s}{mg}-1} (mg+k v_0 s)}{mg}\right)&=-s\,\frac{mg+kv_0s}{mgs+kv_0} \\ -\frac{e^{-\frac{k v_0 s-mg}{mg}} (mg+k v_0 s)}{mg}&=-s\,\frac{mg+kv_0s}{mgs+kv_0}\,\exp\left(-s\,\frac{mg+kv_0s}{mgs+kv_0}\right) \\ \exp\left(-\frac{k v_0 s-mg}{mg}\right) &=\frac{mgs}{mgs+kv_0}\,\exp\left(-s\,\frac{mg+kv_0s}{mgs+kv_0}\right). \end{align*} In canceling $$mg+kv_0s,$$ we are eliminating the solution $$s=-mg/(kv_0),$$ corresponding to $$\theta=\arcsin\left(-\frac{mg}{kv_0}\right).$$ A negative answer seems physically impossible, so we rule it out. We continue: \begin{align*} \exp\left(\frac{mg-k v_0 s}{mg}\right) &=\frac{mgs}{mgs+kv_0}\,\exp\left(-s\,\frac{mg+kv_0s}{mgs+kv_0}\right)\\ \frac{mg-k v_0 s}{mg}&=\ln\left(\frac{mgs}{mgs+kv_0}\right)-s\,\frac{mg+kv_0s}{mgs+kv_0}\\ \frac{g-B v_0 s}{g}&=\ln\left(\frac{gs}{gs+Bv_0}\right)-s\,\frac{g+Bv_0s}{gs+Bv_0}. \end{align*} Unfortunately, this is pretty much as far as we can go. $$s$$ appears in far too many places in this equation to think that there is an analytical solution. My recommendation is to take this equation, plug in the values you need for $$m, g, k,$$ and $$v_0,$$ and then solve it numerically.
• I think you lost an $m$ somewhere: \begin{align*}F_y&=-kv_y-g \\ m\ddot{y}&=-k\dot{y}-g \\ \ddot{y}&=-\frac{k}{m}\,\dot{y}-\frac{g}{m} \\ \ddot{y}+B\dot{y}&=-\frac{g}{m}. \end{align*} The $k$ is bound up in $B,$ and depending on how you manipulate things, you might get something slightly different. – Adrian Keister Jul 9 at 14:46
• Oh, you're right, my mistake. I'll need to go through and replace all $g$'s with $mg$'s to fix it. – Adrian Keister Jul 9 at 15:45


text:
$$\require{cancel}$$

# VI: Some Relativistic Quantum Fields

Thumbnail: In this Feynman diagram, an electron and a positron annihilate, producing a photon (represented by the blue sine wave) that becomes a quark–antiquark pair, after which the antiquark radiates a gluon (represented by the green helix). Image used with permission (CC BY 2.5; Joel Holdsworth).


text:
# Freudenthal (1981)

Major Problems of Mathematics Education

## Abstract

The article does not have an abstract.

## Summary

In this article, Hans Freudenthal articulates 11 major problems of mathematics education. (Most of Freudenthal's spellings are preserved below.)

1. Why can Jennifer not do arithmetic?
In this problem, Freudenthal critiques educators from trying to diagnose students' difficulties as a doctor would diagnose an illness, claiming educators too often diagnose for what is wrong, not why something is wrong. Instead, Freudenthal suggests we rely on the "vast resources of human experience" (p. 135) to build "paradigmatic cases, paradigms of diagnosis and prescription, for the benefit of practitioners and as bricks for theory builders" (p. 135).
2. How should children learn?
Freudenthal extends this to "How should people learn?" and how we learn to observe learning processes. He suggests answers lie in "observing learning processes, analysing them and reporting paradigms — learning processes within the total educational system, learning processes of pupils, groups, classes, teachers, school teams, councillors, teacher students, teacher trainers, and of the observer himself" (pp. 136-137).
3. How to use progressive schematisation and formalisation in teaching any mathematical subject whatever?
This question receives Freudenthal's longest reply, including illustrations of what he means by schematisation. Freudenthal claims that the "history of mathematics has been a learning process of progressive schematising, (p. 140) but claims that "Youngsters need not repeat the history of mankind but they should not be expected either to start at the very point where they preceding generation stopped" (p. 140). He clarifies that schematising should be seen as a psychological progression or network of possible progressions.
4. How to keep open the sources of insight during the training process, and how to stimulate retention of insight, in particular in the process of schematising?
This problem tackles the problem of learning math conceptually versus procedurally, which Freudenthal contrasts with the terms "insight" versus "training." He feels that more mathematics has been learned by insight than we typically recognize, but it becomes a problem when the insights of the learner is superseded by those of the teacher and textbook author. Freudenthal feared that too much training, particularly premature training, endangers the ability for the learner to retain their first insights and actively use them when schematising.
5. How to stimulate reflecting on one's own physical, mental and mathematical activities?
Freudenthal believed that reflection could help learners retain their insights, but purposeful reflection is lost in the learners' efforts to master rules that seem disconnected to those insights.
6. How to develop a mathematical attitude?
Freudenthal claims attitude can be described by example, but we lacked the language to describe problems, strategies for changing perspectives, understanding precision, knowing when mathematics might not be applicable, and "dealing with one's own mathematical activity as a subject matter in order to reach a higher level" (p. 143). Freudenthal uses these examples to warn against what he says is the usual mistake: "testing a mathematical attitude by asking questions about an attitude towards mathematics (p. 143).
7. How is mathematical learning structured according to levels and can this structure be used in attempts at differentiation?
In this problem Freudenthal addresses the fact that learners learn differently and at different rates, yet need to work together.
8. How to create suitable contexts in order to teach mathematising?
Freudenthal focuses this problem more squarely on mathematical subject matter and the need for contexts which to mathematize. He briefly describes one of the foundations of Realistic Mathematics Education, that the "'real world' is represented by a meaningful context involving a mathematical problem" (p. 144) with 'meaningful' meaning a context that's meaningful to the learners.
9. Can you teach geometry by having the learner reflect on his spatial intuitions?
Freudenthal asks that learners become more aware of their intuitive grasp of space as a way of understanding geometry, but questions how this leads to formal subject matter knowledge.
10. How can calculators and computers be used to arouse and increase mathematical understanding?
This problem is less about using technology to calculate, but rather how technology can be a tool that helps learners explore mathematical relationships and fundamental concepts.
11. How to design educational development as a strategy for change?
Freudenthal claims that educational problems need to be solved in the educational process, not in the laboratory, and that learning will be a slow and social process. Curriculum development, says Freudenthal, is not enough. Our problems need to be addressed longitudinally, across levels, and in context.

Freudenthal continues the article by commenting on the influence of textbooks and teacher training. Freudenthal saw teachers relying on textbooks because they've been inadequately trained, but believed good textbooks — ones that mathematized the environment and led both teachers and learners along paths of retention and insight — were worth developing. Teacher training, on the other hand, was inadequate because while it exposes student teachers to short term learning processes, it's not possible to give preservice teachers deep understanding of long term learning processes.

Lastly, Freudenthal comments on educational research, much of which he did not find useful. However, he believed the quality was improving, even if the sheer quantity of research made the quality difficult to find. He hoped for less research of education, and more research in education.

## Also

APA
Freudenthal, H. (1981). Major problems of mathematics education. Educational Studies in Mathematics, 12(2), 133–150. doi:10.1007/BF00305618
BibTeX
@article{Freudenthal1981,
author = {Freudenthal, Hans},
doi = {10.1007/BF00305618},
journal = {Educational Studies in Mathematics},
number = {2},
pages = {133--150},
title = {{Major problems of mathematics education}},


text:
Posted on Categories:Generalized linear model, 广义线性模型, 数据科学代写, 统计代写, 统计代考

统计代写|广义线性模型代写Generalized linear model代考|PSQF6270 No specified distribution

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统计代写|广义线性模型代写Generalized linear model代考|No specified distribution

There are occasions when the distribution of the response variable is unknown, but the experimenter believes that the link function $g(\cdot)$ and variance function $V(\mu)$ can be specified. Without knowledge of the distribution, the $\log$ likelihood function $\ell(\boldsymbol{\beta})$ cannot be constructed. However, a quasi-likelihood may be formulated. The mathematical details are provided in McCullagh \& Nelder (1989, Chapter 9 ). To perform an analysis in $\mathrm{R}$, one needs to make some modifications to the arguments of the $g l m$ function in R. An example is given in Faraway (2006, Section 7.4).
Faraway (2006, Section 3.1) contains the analysis of data from a dataset called gala (for Galapagos Islands) under the assumption that the response variable has a Poisson distribution. Running the commands
data(gala)
gala <- gala $[,-2]$
modp <- glm(Species $.$, family $=$ poisson, data $=$ gala $)$
summary (modp)
gives the output
Deviance Residuals:

$$\begin{array}{rrrrr} \text { Min } & 1 Q & \text { Median } & 3 Q & \text { Max } \ -8.2752 & -4.4966 & -0.9443 & 1.9168 & 10.1849 \end{array}$$
Coefficients:

统计代写|广义线性模型代写Generalized linear model代考|Bayesian Experimental Design

Bayesian analysis is a large and rapidly growing field of statistics. An oversimplified description is to say that, when estimating some parameter(s), one begins with an expression of prior knowledge or belief about the value of the parameters. This is combined with empirical knowledge about the value of the parameters that is gained by conducting an experiment. The result is a posterior description of the statistical behaviour of the parameters. Many books, including Carlin \& Louis (2009), provide a coverage of Bayesian analysis.

Estimation of parameter values does not occur in the determination of an optimal design. However, the expression Bayesian experimental design is often used for the procedure about to be described because we use prior belief about the value of the vector of parameters, $\boldsymbol{\beta}$, to assist us in the determination of the design.

Of course, we have been expressing prior belief about the value of $\boldsymbol{\beta}$ in previous chapters, when using the nominated value of $\boldsymbol{\beta}$ as though we are certain that it is correct. In what follows, we recognise some uncertainty about our knowledge of $\boldsymbol{\beta}$. Many people who feel uncomfortable specifying the value of the parameter vector $\boldsymbol{\beta}$ in the methods considered in previous chapters may feel happier using Bayesian design methods.
An approach that is fully Bayesian, often referred to as a decisiontheoretic approach, was reviewed by Chaloner \& Verdinelli (1995), and is addressed by Overstall \& Woods (2017). Consult either of these references for more information. The aim of this approach is to maximise the experimenter’s gain from using one of a collection of designs if given one of a collection of vectors of response variables when the parameter vertor is one from a collection of such vectors. It requires the maximisation of an expected utility function (Overstall \& Woods, 2017, eq. 1)
$$U(\boldsymbol{\delta})=\iint_{\Psi, \mathcal{Y}} u(\boldsymbol{\delta}, \boldsymbol{\psi}, \boldsymbol{y}) \pi(\boldsymbol{y}, \boldsymbol{\psi} \mid \boldsymbol{\delta}) d \boldsymbol{y} d \boldsymbol{\psi}$$
where the symbols are defined in Overstall \& Woods $(2017$, p. 458).

统计代写|义线性模型代写Generalized linear model代考|No specified distribution

Faraway (2006 年，第 $3.1$ 节) 在假设响应变量具有泊松分布的情况下，对来自名为 gala (加拉帕戈斯群岛) 的数据集的数据 进行了分析。运行命令 data(gala)
gala<- gala[, $-2]$
modp <-glm(物种.， 家庭=泊松，数据=节日)
summary (modp)

$\begin{array}{lllllllll}\text { Min } 1 Q & \text { Median } & 3 Q \quad \operatorname{Max}-8.2752 & -4.4966 & -0.9443 & 1.9168 & 10.1849\end{array}$

统计代写|广义线性模型代写Generalized linear model代考|Bayesian Experimental Design

Louis (2009)，都隄供了贝叶斯分析的内容。

Chaloner $\backslash \&$ \& Verdinelli (1995) 回顾了完全贝叶斯的方法，通常称为决策理论方法，并由 Overstall $\ \&$ Woods (2017) 解决。

2017, eq. 1)
$$U(\boldsymbol{\delta})=\iint_{\Psi, \mathcal{Y}} u(\boldsymbol{\delta}, \boldsymbol{\psi}, \boldsymbol{y}) \pi(\boldsymbol{y}, \boldsymbol{\psi} \mid \boldsymbol{\delta}) d \boldsymbol{y} d \boldsymbol{\psi}$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


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# Math Help - Regression line and correlation coefficient help

1. ## Regression line and correlation coefficient help

 year Value (\$) 0 1,500 1 1,1200 2 1,100 3 1,00 4 800 5 400 6 200

b. given that the equation of the regression line between value and year is y= -207.14x + 1507.14 with an R^2 value of 0.9622, how reliable do you think the regression equation is at predicting the future value of a computer in 5 years time?

c. calculate the value of the correlation coefficient between year and value

d. estimate the value of the computer in year 7, using the regression equation

2. ## Re: Regression line and correlation coefficient help

year = 0 as it is lower than after a year on, do you think a computer can appreciate?

year = 1 is the comma in the correct place? Should it be 1,200?

year = 3, should this be 1,000?

3. ## Re: Regression line and correlation coefficient help

Originally Posted by hollymxox
b. given that the equation of the regression line between value and year is y= -207.14x + 1507.14 with an R^2 value of 0.9622, how reliable do you think the regression equation is at predicting the future value of a computer in 5 years time?
Did you find any t-stats or p-values?

Originally Posted by hollymxox
[COLOR=#333333]

c. calculate the value of the correlation coefficient between year and value
$r =\sqrt{0.9622}$

Originally Posted by hollymxox
d. estimate the value of the computer in year 7, using the regression equation
substitute x=7 into the equation

4. ## Re: Regression line and correlation coefficient help

Yes it is suppose to be going down past 0 and yes the numbers should be 1,200 and 1,000.

Is there any way can you show me the work or explain it a little better?

5. ## Re: Regression line and correlation coefficient help

and I dont even know what t-stats or p-values are...


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 The infinite growth of solutions of second order linear complex differential equations with completely regular growth coefficient Bull. Korean Math. Soc. 2021 Vol. 58, No. 2, 419-431 https://doi.org/10.4134/BKMS.b200321Published online September 7, 2020Printed March 31, 2021 Guowei Zhang Anyang Normal University Abstract : In this paper we discuss the classical problem of finding conditions on the entire coefficients $A(z)$ and $B(z)$ guaranteeing that all nontrivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order. We assume $A(z)$ is an entire function of completely regular growth and $B(z)$ satisfies three different conditions, then we obtain three results respectively. The three conditions are (1) $B(z)$ has a dynamical property with a multiply connected Fatou component, (2) $B(z)$ satisfies $T(r,B)\sim \log M(r,B)$ outside a set of finite logarithmic measure, (3) $B(z)$ is extremal for Denjoy's conjecture. Keywords : Entire function, infinite order, complex differential equation MSC numbers : 30D35, 34M10, 37F10 Supported by : This work was supported by NSFC(no.11426035), the key scientific research project for higher education institutions of Henan Province, China (no. 18A110002) and training program for young backbone teachers of colleges and universities in Henan Province, China (no. 2017GGJS126) Downloads: Full-text PDF   Full-text HTML


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# Homework Help: Green's function in acoustics,method of descent

1. Jan 1, 2018

### davon806

1. The problem statement, all variables and given/known data
Hi there,
I was reading a book discussing on the topic of compact Green's function in 2D. However,I have been stuck for a while on some mathematical manipulations depicted below.

2. Relevant equations

3. The attempt at a solution
In 2nd box,I guess the author was trying to pull out the differentiation sign and make use of the fact that the delta function is an even function.However,the delta function in the first box is differentiating with respect to the single term within the bracket rather than t?

2. Jan 1, 2018

### Delta²

if $f(x)$ is a function and $x=g(t)$ is any linear function of t (such that $\frac{\partial g}{\partial{t}}=c_1$) then from the chain rule it follows that $\frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}c_1$.

To state it more clearly, the operators $\frac{\partial}{\partial t}, \frac{\partial}{\partial Y}$ have the same effect (up to a constant) because the argument $x=g(t,Y)$ of the delta function is linear both in t and Y.

Last edited: Jan 1, 2018
3. Jan 1, 2018

### davon806

For the primed delta in 2nd box, what is delta differentiating with respect to?

4. Jan 2, 2018

### Delta²

I don't see any primed delta in the 2nd box, if you mean at first box, there it is like saying $\delta'(x)$ that is differentiate w.r.t x and then replace $x=t-\tau-...$

5. Jan 2, 2018

### Delta²

I see now Y is a vector $(y_1,y_2)$ if we want to do this in detail we ll have to use taylor's theorem with two variables but again the key point is that the term $\hat x \cdot Y=x_1y_1+x_2y_2$ is linear in $y_1 , y_2$

Last edited: Jan 2, 2018


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# Exact value (trig)

• September 14th 2010, 04:23 AM
TwoPlusTwo
Exact value (trig)
How do I find the exact value of an expression like this one?

u=sin(x+pi/4)

when sin(x)= 1/3 and x is in the range 0-90 degrees.

Thanks.
• September 14th 2010, 04:33 AM
Prove It
$\sin{(\alpha \pm \beta)} = \sin{\alpha}\cos{\beta} \pm \cos{\alpha}\sin{\beta}$.
• September 14th 2010, 05:00 AM
TwoPlusTwo
Quote:

Originally Posted by Prove It
$\sin{(\alpha \pm \beta)} = \sin{\alpha}\cos{\beta} \pm \cos{\alpha}\sin{\beta}$.

Yeah, that’s what I thought, but what should I do when I have this?

u=(√2/2)sin(x) + (√2/2)cos(x)
• September 14th 2010, 05:14 AM
Prove It
You know that $\sin{x} = \frac{1}{3}$.

You should also know that $\cos{x} = \sqrt{1 - \sin^2{x}}$ from the Pythagorean Theorem.

That means $\cos{x} = \sqrt{1 - \left(\frac{1}{3}\right)^2}$

$= \sqrt{1 - \frac{1}{9}}$

$= \sqrt{\frac{8}{9}}$

$= \frac{2\sqrt{2}}{3}$.
• September 14th 2010, 05:38 AM
TwoPlusTwo
Allright! Got it now. I assume you mean:

cos(x)=√1-cos^2(x)

Which makes:

cos(x)=√8/3 not √10/3

That gave me the right answer. Thanks!


text:
# publicmoritz/libtommath forked from libtom/libtommath

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LibTomMath is written entire in portable ISO C source code and designed to have an applicationinterface much like that of MPI from Michael Fromberger. LibTomMath was written from scratch by Tom St Denis but designed to be drop in replacement for the MPI package. The algorithms within the library are derived from descriptions as provided in the Handbook of Applied Cryptography and Knuth'sThe Art of Computer Programming''. The library has been extensively optimized and should provide quite comparable timings as compared to many free and commercial libraries.LibTomMath was designed with the following goals in mind:\begin{enumerate}\item Be a drop in replacement for MPI.\item Be much faster than MPI.\item Be written entirely in portable C.\end{enumerate}All three goals have been achieved. Particularly the speed increase goal. For example, a 512-bit modular exponentiation isfour times faster\footnote{On an Athlon XP with GCC 3.2} with LibTomMath compared to MPI.Being compatible with MPI means that applications that already use it can be ported fairly quickly. Currently there are a few differences but there are many similarities. In fact the average MPI based application can be ported in under 15minutes. Thanks goes to Michael Fromberger for answering a couple questions and Colin Percival for having the patience and courtesy tohelp debug and suggest optimizations. They were both of great help!\section{Building Against LibTomMath}Building against LibTomMath is very simple because there is only one source file. Simply add bn.c'' to your project and copy both bn.c'' and bn.h'' into your project directory. There is no configuration nor building required before hand.If you are porting an MPI application to LibTomMath the first step will be to remove all references to MPI and replace them with references to LibTomMath. For example, substitute \begin{verbatim}#include "mpi.h"\end{verbatim}with \begin{verbatim}#include "bn.h"\end{verbatim}Remove mpi.c'' from your project and replace it with bn.c''. Note that currently MPI has a few more functions thanLibTomMath has (e.g. no square-root code and a few others). Those are planned for future releases. In the interim work arounds can be sought. Note that LibTomMath doesn't lack any functions required to build a cryptosystem.\section{Programming with LibTomMath}\subsection{The mp\_int Structure}All multiple precision integers are stored in a structure called \textbf{mp\_int}. A multiple precision integer isessentially an array of \textbf{mp\_digit}. mp\_digit is defined at the top of bn.h. Its type can be changed to suita particular platform. For example, when \textbf{MP\_8BIT} is defined\footnote{When building bn.c.} a mp\_digit is a unsigned char and holds seven bits. Similarly when \textbf{MP\_16BIT} is defined a mp\_digit is a unsigned short and holds 15 bits. By default a mp\_digit is a unsigned long and holds 28 bits. The choice of digit is particular to the platform at hand and what available multipliers are provided. For MP\_8BIT either a $8 \times 8 \Rightarrow 16$ or $16 \times 16 \Rightarrow 16$ multiplier is optimal. When MP\_16BIT is defined either a $16 \times 16 \Rightarrow 32$ or $32 \times 32 \Rightarrow 32$ multiplier is optimal. Bydefault a $32 \times 32 \Rightarrow 64$ or $64 \times 64 \Rightarrow 64$ multiplier is optimal. This gives the library some flexibility. For example, a i8051 has a $8 \times 8 \Rightarrow 16$ multiplier. The 16-bit x86 instruction set has a $16 \times 16 \Rightarrow 32$ multiplier. In practice this library is not particularlydesigned for small devices like an i8051 due to the size. It is possible to strip out functions which are not required to drop the code size. More realistically the library is well suited to 32 and 64-bit processors that have decentinteger multipliers. The AMD Athlon XP and Intel Pentium 4 processors are examples of well suited processors.Throughout the discussions there will be references to a \textbf{used} and \textbf{alloc} members of an integer. Theused member refers to how many digits are actually used in the representation of the integer. The alloc member refersto how many digits have been allocated off the heap. There is also the $\beta$ quantity which is equal to $2^W$ where $W$ is the number of bits in a digit (default is 28). \subsection{Basic Functionality}Essentially all LibTomMath functions return one of three values to indicate if the function worked as desired. A function will return \textbf{MP\_OKAY} if the function was successful. A function will return \textbf{MP\_MEM} ifit ran out of memory and \textbf{MP\_VAL} if the input was invalid. Before an mp\_int can be used it must be initialized with \begin{verbatim}int mp_init(mp_int *a);\end{verbatim}For example, consider the following.\begin{verbatim}#include "bn.h"int main(void){   mp_int num;   if (mp_init(&num) != MP_OKAY) {      printf("Error initializing a mp_int.\n");   }   return 0;} \end{verbatim}A mp\_int can be freed from memory with\begin{verbatim}void mp_clear(mp_int *a);\end{verbatim}This will zero the memory and free the allocated data. There are a set of trivial functions to manipulate the value of an mp\_int. \begin{verbatim}/* set to zero */void mp_zero(mp_int *a);/* set to a digit */void mp_set(mp_int *a, mp_digit b);/* set a 32-bit const */int mp_set_int(mp_int *a, unsigned long b);/* init to a given number of digits */int mp_init_size(mp_int *a, int size);/* copy, b = a */int mp_copy(mp_int *a, mp_int *b);/* inits and copies, a = b */int mp_init_copy(mp_int *a, mp_int *b);\end{verbatim}The \textbf{mp\_zero} function will clear the contents of a mp\_int and set it to positive. The \textbf{mp\_set} function will zero the integer and set the first digit to a value specified. The \textbf{mp\_set\_int} function will zero the integer and set the first 32-bits to a given value. It is important to note that using mp\_set can have unintended side effects when either the MP\_8BIT or MP\_16BIT defines are enabled. By default the library will accept the ranges of values MPI will (and more).The \textbf{mp\_init\_size} function will initialize the integer and set the allocated size to a given value. The allocated digits are zero'ed by default but not marked as used. The \textbf{mp\_copy} function will copy the digits(and sign) of the first parameter into the integer specified by the second parameter. The \textbf{mp\_init\_copy} willinitialize the first integer specified and copy the second one into it. Note that the order is reversed from that ofmp\_copy. This odd bug'' was kept to maintain compatibility with MPI.\subsection{Digit Manipulations}There are a class of functions that provide simple digit manipulations such as shifting and modulo reduction of powersof two. \begin{verbatim}/* right shift by "b" digits */void mp_rshd(mp_int *a, int b);/* left shift by "b" digits */int mp_lshd(mp_int *a, int b);/* c = a / 2^b */int mp_div_2d(mp_int *a, int b, mp_int *c);/* b = a/2 */int mp_div_2(mp_int *a, mp_int *b);/* c = a * 2^b */int mp_mul_2d(mp_int *a, int b, mp_int *c);/* b = a*2 */int mp_mul_2(mp_int *a, mp_int *b);/* c = a mod 2^d */int mp_mod_2d(mp_int *a, int b, mp_int *c);\end{verbatim}Both the \textbf{mp\_rshd} and \textbf{mp\_lshd} functions provide shifting by whole digits. For example, mp\_rshd($x$, $n$) is the same as $x \leftarrow \lfloor x / \beta^n \rfloor$ while mp\_lshd($x$, $n$) is equivalentto $x \leftarrow x \cdot \beta^n$. Both functions are extremely fast as they merely copy digits within the array. Similarly the \textbf{mp\_div\_2d} and \textbf{mp\_mul\_2d} functions provide shifting but allow any bit count to be specified. For example, mp\_div\_2d($x$, $n$, $y$) is the same as $y =\lfloor x / 2^n \rfloor$ while mp\_mul\_2d($x$, $n$, $y$) is the same as $y = x \cdot 2^n$. The \textbf{mp\_div\_2} and \textbf{mp\_mul\_2} functions are legacy functions that merely shift right or left one bit respectively. The \textbf{mp\_mod\_2d} functionreduces an integer mod a power of two. For example, mp\_mod\_2d($x$, $n$, $y$) is the same as $y \equiv x \mbox{ (mod }2^n\mbox{)}$.\subsection{Basic Arithmetic}Next are the class of functions which provide basic arithmetic.\begin{verbatim}/* compare a to b */int mp_cmp(mp_int *a, mp_int *b);/* c = a + b */int mp_add(mp_int *a, mp_int *b, mp_int *c);/* c = a - b */int mp_sub(mp_int *a, mp_int *b, mp_int *c);/* c = a * b */int mp_mul(mp_int *a, mp_int *b, mp_int *c);/* b = a^2 */int mp_sqr(mp_int *a, mp_int *b);/* a/b => cb + d == a */int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d);/* c == a mod b */#define mp_mod(a, b, c) mp_div(a, b, NULL, c)\end{verbatim}The \textbf{mp\_cmp} will compare two integers. It will return \textbf{MP\_LT} if the first parameter is less thanthe second, \textbf{MP\_GT} if it is greater or \textbf{MP\_EQ} if they are equal. These constants are the same as fromMPI.The \textbf{mp\_add}, \textbf{mp\_sub}, \textbf{mp\_mul}, \textbf{mp\_div}, \textbf{mp\_sqr} and \textbf{mp\_mod} are allfairly straight forward to understand. Note that in mp\_div either $c$ (the quotient) or $d$ (the remainder) can be passed as NULL to ignore it. For example, if you only want the quotient $z = \lfloor x/y \rfloor$ then a call such as mp\_div(\&x, \&y, \&z, NULL) is acceptable.There is a related class of single digit'' functions that are like the above except they use a digit as the secondoperand.\begin{verbatim}/* compare against a single digit */int mp_cmp_d(mp_int *a, mp_digit b);/* c = a + b */int mp_add_d(mp_int *a, mp_digit b, mp_int *c);/* c = a - b */int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);/* c = a * b */int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);/* a/b => cb + d == a */int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);/* c = a mod b */#define mp_mod_d(a,b,c) mp_div_d(a, b, NULL, c)\end{verbatim}Note that care should be taken for the value of the digit passed. By default, any 28-bit integer is a valid digit that canbe passed into the function. However, if MP\_8BIT or MP\_16BIT is defined only 7 or 15-bit (respectively) integers can be passed into it.\subsection{Modular Arithmetic}There are some trivial modular arithmetic functions.\begin{verbatim}/* d = a + b (mod c) */int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);/* d = a - b (mod c) */int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);/* d = a * b (mod c) */int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);/* c = a * a (mod b) */int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c);/* c = 1/a (mod b) */int mp_invmod(mp_int *a, mp_int *b, mp_int *c);/* c = (a, b) */int mp_gcd(mp_int *a, mp_int *b, mp_int *c);/* c = [a, b] or (a*b)/(a, b) */int mp_lcm(mp_int *a, mp_int *b, mp_int *c);/* d = a^b (mod c) */int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);\end{verbatim}These are all fairly simple to understand. The \textbf{mp\_invmod} is a modular multiplicative inverse. That is itstores in the third parameter an integer such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$ provided such integer exists. Ifthere is no such integer the function returns \textbf{MP\_VAL}.\subsection{Radix Conversions}To read or store integers in other formats there are the following functions.\begin{verbatim}int mp_unsigned_bin_size(mp_int *a);int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);int mp_to_unsigned_bin(mp_int *a, unsigned char *b);int mp_signed_bin_size(mp_int *a);int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);int mp_to_signed_bin(mp_int *a, unsigned char *b);int mp_read_radix(mp_int *a, unsigned char *str, int radix);int mp_toradix(mp_int *a, unsigned char *str, int radix);\end{verbatim}The integers are stored in big endian format as most libraries (and MPI) expect. The \textbf{mp\_read\_radix} and \textbf{mp\_toradix} functions read and write (respectively) null terminated ASCII strings in a given radix. Valid valuesfor the radix are between 2 and 64 (inclusively). \section{Timing Analysis}\subsection{Observed Timings}A simple test program demo.c'' was developed which builds with either MPI or LibTomMath (without modification). Thetest was conducted on an AMD Athlon XP processor with 266Mhz DDR memory and the GCC 3.2 compiler\footnote{With buildoptions -O3 -fomit-frame-pointer -funroll-loops''}. The multiplications and squarings were repeated 10,000 times each while the modular exponentiation (exptmod) were performed 10 times each. The RDTSC (Read Time Stamp Counter) instructionwas used to measure the time the entire iterations took and was divided by the number of iterations to get anaverage. The following results were observed.\begin{small}\begin{center}\begin{tabular}{c|c|c|c}\hline \textbf{Operation} & \textbf{Size (bits)} & \textbf{Time with MPI (cycles)} & \textbf{Time with LibTomMath (cycles)} \\\hlineMultiply & 128 & 1,394 & 915 \\Multiply & 256 & 2,559 & 1,893 \\Multiply & 512 & 7,919 & 3,770 \\Multiply & 1024 & 28,460 & 9,970 \\Multiply & 2048 & 109,637 & 32,264 \\Multiply & 4096 & 467,226 & 129,645 \\\hline Square & 128 & 1,288 & 1,147 \\Square & 256 & 1,705 & 2,129 \\Square & 512 & 5,365 & 3,755 \\Square & 1024 & 18,836 & 9,267 \\Square & 2048 & 72,334 & 28,387 \\Square & 4096 & 306,252 & 112,391 \\\hline Exptmod & 512 & 30,497,732 & 7,222,872 \\Exptmod & 768 & 98,943,020 & 16,474,567 \\Exptmod & 1024 & 221,123,749 & 30,070,883 \\Exptmod & 2048 & 1,694,796,907 & 154,697,320 \\Exptmod & 2560 & 3,262,360,107 & 318,998,183 \\Exptmod & 3072 & 5,647,243,373 & 494,313,122 \\Exptmod & 4096 & 13,345,194,048 & 1,036,254,558 \end{tabular}\end{center}\end{small}\subsection{Digit Size}The first major constribution to the time savings is the fact that 28 bits are stored per digit instead of the MPI defualt of 16. This means in many of the algorithms the savings can be considerable. Consider a baseline multiplier with a 1024-bit input. With MPI the input would be 64 16-bit digits whereas in LibTomMath it would be 37 28-bit digits.A savings of $64^2 - 37^2 = 2727$ single precision multiplications. \subsection{Multiplication Algorithms}For most inputs a typical baseline $O(n^2)$ multiplier is used which is similar to that of MPI. There are two variants of the baseline multiplier. The normal and the fast variants. The normal baseline multiplier is the exact same as thealgorithm from MPI. The fast baseline multiplier is optimized for cases where the number of input digits $N$ is lessthan or equal to $2^{w}/\beta^2$. Where $w$ is the number of bits in a \textbf{mp\_word}. By default a mp\_word is64-bits which means $N \le 256$ is allowed which represents numbers upto $7168$ bits.The fast baseline multiplier is optimized by removing the carry operations from the inner loop. This is often referredto as the comba'' method since it computes the products a columns first then figures out the carries. This has theeffect of making a very simple and paralizable inner loop.For large inputs, typically 80 digits\footnote{By default that is 2240-bits or more.} or more the Karatsuba method is used. This method has significant overhead but an asymptotic running time of $O(n^{1.584})$ which means for fairly largeinputs this method is faster. The Karatsuba implementation is recursive which means for extremely large inputs theywill benefit from the algorithm.MPI only implements the slower baseline multiplier where carries are dealt with in the inner loop. As a result even atsmaller numbers (below the Karatsuba cutoff) the LibTomCrypt multipliers are faster.\subsection{Squaring Algorithms}Similar to the multiplication algorithms there are two baseline squaring algorithms. Both have an asymptotic runningtime of $O((t^2 + t)/2)$. The normal baseline squaring is the same from MPI and the fast is a comba'' squaringalgorithm. The comba method is used if the number of digits $N$ is less than $2^{w-1}/\beta^2$ which by default covers numbers upto $3584$ bits. There is also a Karatsuba squaring method which achieves a running time of $O(n^{1.584})$ after considerably largeinputs.MPI only implements the slower baseline squaring algorithm. As a result LibTomMath is considerably faster at squaringthan MPI is.\subsection{Exponentiation Algorithms}LibTomMath implements a sliding window $k$-ary left to right exponentiation algorithm. For a given exponent size $L$ anappropriate window size $k$ is chosen. There are always at most $L$ modular squarings and $\lfloor L/k \rfloor$ modularmultiplications. The $k$-ary method works by precomputing values $g(x) = b^x$ for $0 \le x < 2^k$ and a given base $b$. Then the multiplications are grouped in windows of $k$ bits. The sliding window technique has the benefit that it can skip multiplications if there are zero bits following or preceding a window. Consider the exponent $e = 11110001_2$ if $k = 2$ then there will be a two squarings, a multiplication of $g(3)$, two squarings, a multiplicationof $g(3)$, four squarings and and a multiplication by $g(1)$. In total there are 8 squarings and 3 multiplications. MPI uses a binary square-multiply method. For the same exponent $e$ it would have had 8 squarings and 5 multiplications. There is a precomputation phase for the method LibTomCrypt uses but it generally cuts down considerably on the numberof multiplications. Consider a 512-bit exponent. The worst case for the LibTomMath method results in 512 squarings and 124 multiplications. The MPI method would have 512 squarings and 512 multiplications. Randomly every $2k$ bits another multiplication is saved via the sliding-window technique on top of the savings the $k$-ary method provides.Both LibTomMath and MPI use Barrett reduction instead of division to reduce the numbers modulo the modulus given. However, LibTomMath can take advantage of the fact that the multiplications required within the Barrett reductiondo not to give full precision. As a result the reduction step is much faster and just as accurate. The LibTomMath codewill automatically determine at run-time (e.g. when its called) whether the faster multiplier can be used. Thefaster multipliers have also been optimized into the two variants (baseline and fast baseline).As a result of all these changes exponentiation in LibTomMath is much faster than compared to MPI. \end{document}
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# How could you improve the yield of diels alder reactions?

Mar 1, 2016

I can think of four possible methods.

#### Explanation:

1. Increase the temperature

Although heat is not often required in Diels-Alder reactions, increasing the temperature usually improves the yield.

2. Increase the amount of catalyst

You usually use a catalyst to increase the rate of a reaction.

Using a molar ratio of 10% catalyst often gives a large increase in both the selectivity and the yield of the reaction.

3. Do the reaction in LPDE

Many Diels-Alder reactions using ${\text{5.0 mol/L LiClO}}_{4}$ in diethyl ether (LPDE) can be carried out at ambient temperature and pressure, go faster, give better endo/exo selectivity, and give improved yields.

4. Do the reaction in a microwave

Many common Diels-Alder reactions give better yields and are effectively complete after 5 to 10 min in the microwave at 125 °C to 150 °C.


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# Why does the naive choice of homogeneous coordinate ring of a product of projective schemes not work?

Let $S$ be a graded ring with $A := S_0$. Set $X := \textrm{Proj} (S)$. Then the projective coordinate ring of $X \times_A X$ is the graded ring $\bigoplus_{n \geq 0} S_n \otimes_A S_n$, cf Hartshorne, Exercise II.5.11. (This matches with geometric intuition because this ring corresponds [at least when $S$ is nice enough, eg if $X$ is projective over $A$ and $S$ is the homogeneous coordinate ring of an embedding into $\mathbf P^m_A$] to the section ring of a very ample sheaf on $X \times_A X$ coming from the exterior product of very ample sheaves on $X$.)

However, there is another natural graded ring that we can construct here, namely $S \otimes_A S$ with the obvious grading coming from total degree. This ring is clearly not the same as the graded ring in the first paragraph above. Naively, though, by analogy to the affine case one might still expect that $\textrm{Proj}(S \otimes_A S) \cong X \times_A X$. Is this true? Probably this is not true, although I don't know how to show it; so a followup question is: what scheme does this produce?

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For one thing, the dimension is wrong. And, have you tried computing an example? –  Angelo Apr 6 '13 at 18:17
It's helpful to remember that a graded ring S gives not only a projective scheme but also an embedding (via S(1)), and lots of rings can give the same projective scheme. If S is the ring you cite from Hartshorne, then S(1) gives the Segre embedding of your projective variety. I'm also interested in knowing if there are other easy rings to write down whose Proj gives you X x X. In the case when S is the polynomial ring k[x,y], then the total tensor product just gives you k[x,y,z,w], so you just get P^3 which isn't what we want. –  David Dynerman Apr 6 '13 at 18:24
Ah yes, that makes sense. So I guess this boils down to the obvious fact that a product of affine cones over a projective scheme will never be an affine cone over the product (eg by dimension considerations), which immediately shows that the naive hypothesis is false. –  Chuck Hague Apr 6 '13 at 18:36
There's a construction called "multiproj" for rings with several $\mathbb N$ gradings. If you think of $S\otimes S$ as doubly graded and take multiproj, then you'll get the product. You can pass from a multiproj to a project by restricting to a generic ray (if you're lucky), which is what you did. –  Ben Webster Apr 6 '13 at 18:51
It is maybe worth mentioning that for projective varieties, the full tensor product corresponds to the JOIN of X with another copy of itself after appropriate embeddings. Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" has a helpful discussion on page 304: books.google.ie/… –  Oliver Nash Apr 10 '13 at 18:16

A ($\mathbb{Z}$-)grading on a ring $S$ is equivalent to an action of the group $\mathbb{G}_m$ on $U=\operatorname{Spec} S$. This is an exercise in algebra; the statement is that the symmetric monoidal category of comodules over the Hopf algebra $A[t,t^{-1}]$ is equivalent to the category of graded $A$-modules. The construction $\operatorname{Proj} S$ then corresponds to taking the quotient of $U\setminus U_0$ by the $\mathbb{G}_m$ action, where $U_0$ is the subscheme corresponding to the "irrelevant ideal". This is really just the familiar fact that $\mathbb{P}^n$ is the quotient of $\mathbb{A}^{n+1}\setminus 0$ by rescaling.
What happens when we take a product of two such schemes $U$ and $V$ with corresponding graded rings $S$ and $T$? I'm going to ignore $U_0$ and $V_0$ in this discussion to simplify notation (i.e., I'm going to pretend that $\operatorname{Proj} S$ is just $U/\mathbb{G_m}$). Well, $U\times V$ will naturally have an action of $\mathbb{G}_m\times \mathbb{G}_m$; this corresponds to the natural bigrading on $S\otimes T$. By taking the diagonal $\mathbb{G}_m\to\mathbb{G}_m\times\mathbb{G}_m$, we get an action of $\mathbb{G}_m$ on the product, and this corresponds to the "total" grading on $S\otimes T$. But if we mod out $U\times V$ by the diagonal action of $\mathbb{G}_m$, we don't get $U/\mathbb{G}_m\times V/\mathbb{G}_m$; to get that, we would need to mod out the entire action of $\mathbb{G}_m\times \mathbb{G}_m$. Alternatively, instead of modding out the whole action of $\mathbb{G}_m\times\mathbb{G}_m$ at once, we could first mod out the action of the antidiagonal $\alpha:\mathbb{G}_m\to\mathbb{G}_m\times\mathbb{G}_m$ given by $\alpha(t)=(t,t^{-1})$ and then mod out the action of the diagonal; this works since the antidiagonal and the diagonal together generate the whole group. What do we get when we mod out the antidiagonal? Well, it turns out that in this case the quotient will still be affine, and its coordinate ring can be found by taking invariants of the action on $S\otimes T$. If $x\in S_n$ and $y\in T_m$, then the the antidiagonal action of $t\in\mathbb{G}_m$ sends $x\otimes y$ to $t^{n-m}x\otimes y$, so the invariants are exactly $\bigoplus S_n\otimes T_n$. Thus the product $U/\mathbb{G}_m\times V/\mathbb{G}_m$ can be obtained as $\operatorname{Proj}(\bigoplus S_n\otimes T_n)$.


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## Wednesday, 16 January 2013

### The myth of the harmlessness of 'taking profits'

Thee percentage of winning trades - in the broadest sense of the word -  is something every trader ought to be familiar with in his own trading.  Let's call this percentage (on a 0-1 scale, so that it also doubles up as a probability of winning) $p_w$.  0.5 is the score a coin tossing machine gets.  Anything consistently lower than that, then you're really on to something - just start doing the opposite of what you are doing and soon untold wealth will be yours.  0.6 I think is a decent batting rate.  0.7 and 0.8 very good.  I doubt many human beings have ever consistently achieved 0.8.

So, given all that, in the following analysis, let's consider an extraordinary trader who scores 0.9.  Unbelievable, I know, but 9 out of 10 of all his trades win, in the technical sense of return > 0.  One out of ten returns <= 0.

Next, let's take a principle you see mentioned in a lot of books on trading: don't risk more than 1% of your total capital on any one trade.  Let's assume mister 9-out-of-10 sticks to that rule religiously, and indeed is happy to stay in the position on the downside as long as he isn't stopped out at a level which triggers a stop, closing the trader for a loss of 1%.  I will call the monetary value of a failed trade as $0.01 P$, or 1% of AUM.

So the set up is in place - our trader wins nine times out of 10, and has a 1% of AUM stop loss.  Sound plausible?  Happy?  Even more so, let us force him to only put on is best trade at any given time.  That is, a maximum of only 1 trade on at any moment.  He can be holding no positions if he likes, for as long as he likes.  Nothing forces him to open a position unless he feels compelled to.

The point of all this set up, which sounds rather wonderful, is to show that it all comes to nothing if you set your trades to get you out of the trade on a +10 basis points profit.  That is, if you close your winners, all 90% of them, for a 10 basis point gain, you've effectively imposed a rule on yourself which prevents you from making any money at all, since $(p_w \times 0.001 \times P) + (1-p_w \times - 0.01 \times P)$.  I assume your 1 win out of 10 comes on average at any point in a series of trials, and isn't clustered and otherwise exhibits no informational characteristics within the series.  On average this set up will bleed 1 basis point of your AUM every time.  You will go bust.  You can't abandon the 1% stop loss, due to the more violent downside of many markets (equities, bonds) on occasion, so get rid of the safe-sounding 'sell out on a 10 basis point profit' rule.

With a more realistic 6 or 7 out of 10 batting average, you need to make them all count by at least 20 basis points for a break even strategy.  That's a lot.  And if you're thinking 1% AUM can be tightened to a smaller loss, then remember that this transformation of a fraction of AUM into a stop loss on a specific market leaves you open to the volatility of that market.  If your trading account is under-capitalised, then that stop loss is way too close to the market, meaning there's not much chance at all that you'll hit your winning out-target price before you hit your loss making out target price.  Strictly speaking, it is a function of the minimal trade size and the deposit requirement of that market.  Additionally, the outcome is partly a function of the volatility of tick-by-tick returns.  A handy rule of thumb is: if your ideal place to put a stop means you could lose 1% or more of your capital, then you're under-capitalised.