text:
STM32F0 Tutorial 4: Timer and Counter

In this post, we will explore the Timer and Counter of STM32F0 using CubeMX. In this STM32F0 timer tutorial, I will try to cover as many functions of the STM32F0’s Timer as possible because this peripheral may have the greatest features and functions among the other peripherals. If you have worked with Arduino before, you might realize the limitations of the arduino’s timers. The STM32’s timers give you way more functions that will be very useful for many of your applications that you may not think of before. For STM32F051, it has a total of 9 main timers including seven 16-bit timers, one 24-bit systick timer, and one 32-bit timer. Some of them have 6 channels advanced-control PWM output, deadtime generation, etc. It also has an independent timer and a watchdog timer for dealing with program hanging. This tutorial will cover the following function of the timer:

• Timebase interrupt function
• Counter with external input
• Input capture
• PWM generation

Time Base Interrupt

STM32F051 has several timers for you to play with including TIM1, TIM2, TIM3, TIM6, TIM14, TIM15, TIM16, and TIM17. Basically, the timer and counter are just different from the input clock signal. For the timer, the clock source is an internal clock that is generated from the external crystal internal RC circuit of the STM32F0 Discovery. The following video will show you how to config a timer in CubeMX in order to generate a time base interrupt for blinking the green Led (PC9).

Noticed the configurations that I have entered in the timer setting window of TIM3. There are some definitions that need to be mentioned here:

• Prescaler: can be understood as a frequency divider for the counter, with a range from 0 to 65535. In the video, I set it to 47999 (corrected from 48000) so it means that the frequency supply for the counter will be equal to 48MHz/(47999+1) = 1000Hz = 1ms.
• Counter mode: can be count-up or count-down. For count-up: the counter will count from 0 to the value stored in the Auto-reload register (Counter period) and then generate an Overflow interrupt and start all over again from 0. For count-down: the counter will count from the value stored in the Auto-reload register (Counter period) down to 0 and then generate Underflow interrupt and start all over again from the stored value.

Counter period: the value that the counter will count to or count from (depend on count-up or count-down). In the video I set it to 499, which means that the counter will count from 0 to 499 with a 1ms cycle, so totally the interrupt time will be 500*1ms =500ms.

Speaking of the interrupt handler, the HAL library helps us to handle all the checking and clearing status flag bits so we don’t have to worry about them again, just use the following function as an interrupt handler routine. Check the code in the video again:

void HAL_TIM_PeriodElapsedCallback(TIM_HandleTypeDef *htim)
{
if (htim->Instance==TIM3) //check if the interrupt comes from TIM3
{
HAL_GPIO_TogglePin(GPIOC, GPIO_PIN_9);
}
}

This Callback function is shared among all timers interrupt. If you are using more than one Timebase interrupt, you need to check the source of the interrupt before executing any command. For example, if you use TIM3 and TIM6 time base interrupt, the Callback function should be like this:

void HAL_TIM_PeriodElapsedCallback(TIM_HandleTypeDef *htim)
{
if (htim->Instance==TIM3)
{
//do something here
}
if (htim->Instance==TIM6)
{
//do something else here
}
}

External Input Counter

In the last tutorial on “External interrupt“, I have shown the example of using an external interrupt to count the input signal from the button. In this post, I will introduce another method to count the external input signal using Counter. So what is the difference between the two methods? While the external interrupt needs to jump into the interrupt routine to do the increment or decrement of a variable, the counter can handle the job nicely without jumping anywhere. Therefore, it will be obviously useful when your program has many types of interrupts running. Take a look at the F0’s reference manual, timer section:

We can easily see that the timer has many sources for the input clock. The first source is the “Internal clock”, which we discovered in the first part of this tutorial. Now, we’re gonna look at the second source, which is the “external input pin (TIx)”. With this source of the clock, the timer will apparently become a counter. The other two sources are out of the scope of this tutorial that you can discover by yourself later. So which pin is compatible with the counter mode? You can notice these pins with functions like “TIMx_CH1” and “TIMx_CH2” in which x represents the timer number (1,2,3….).

The picture above is an example of pin PA0, which has the “TIM2_CH1” function. It means that you can use this pin as an input signal for Counter TIM2. Watch the video below and follow yourself. Basically, I will set up Timer 2 as a counter to count the number of times that I press the blue button on the Discovery kit with CubeMX. I also used STMStudio to monitor the counter value.

Please make sure that you need to start the timer before doing anything else. The code in the video:

/* USER CODE BEGIN 2 */
HAL_TIM_Base_Start_IT(&htim3);
HAL_TIM_Base_Start(&htim2);    //Start TIM2 without interrupt
/* USER CODE END 2 */

The macro “HAL_TIM_GetCounter(timer)” is used to retrieve the timer counter.

/* USER CODE BEGIN WHILE */
while (1)
{
count = __HAL_TIM_GetCounter(&htim2);    //read TIM2 counter value
/* USER CODE END WHILE */

Input Capture

Another function of the Timer is to identify the width of input signals by using Input capture. It will record a timestamp in memory when an input signal is received. It will also set a flag indicating that an input has been captured so that you can read out the capture value easily through interrupt or event polling.

So what can it be used for? See the picture below, what if you want to determine the cycle period or the pulse width of the input pulse? Input capture will help you with that.

Similar to the external input counter function, the input capture function also uses these pins with the name TIMx_CH1 to TIMx_CH4. Each time the input capture is triggered, it will latch the counter value into the Capture/Compare register of the respective channel as seen in the picture below:

The video tutorial shows you how to configure a timer to be used with the input capture function. Here, I used TIM2_CH1 as the input capture channel because it is connected to the User button and I want to capture the time between two pushes (1 cycle).

The code in the video: There is a small modification in the code (thanks to Nathan to spot it out) as

input_capture= __HAL_TIM_GetCompare(&htim2, TIM_CHANNEL_1);

is placed into HAL_TIM_IC_CaptureCallback instead of being in the main loop.

-Timer Input capture callback:

void HAL_TIM_IC_CaptureCallback(TIM_HandleTypeDef *htim)
{
if (htim->Instance==TIM2)
{
input_capture= __HAL_TIM_GetCompare(&htim2, TIM_CHANNEL_1);    //read TIM2 channel 1 capture value
__HAL_TIM_SetCounter(&htim2, 0);    //reset counter after input capture interrupt occurs
}
}

-Start timer 2 in input capture interrupt mode, channel 1:

HAL_TIM_IC_Start_IT(&htim2, TIM_CHANNEL_1);

-Main loop

/* USER CODE BEGIN WHILE */
while (1)
{
count = __HAL_TIM_GetCounter(&htim2);    //read TIM2 counter value
/* USER CODE END WHILE */

/* USER CODE BEGIN 3 */
}
/* USER CODE END 3 */

PWM Generation

As you may hear about this function somewhere before, PWM (Pulse-Width Modulation) is mostly used for controlling the speed of a DC motor or changing the brightness of an LED, or even mixing colors for RGB LED. In this part, I’m going to show you how to initialize the PWM function to control the brightness of the Blue LED on the STM32 Discovery kit.

Digital control is used to create a square wave, a signal switched between on and off. This on-off pattern can simulate voltages between full-on (3.3 Volts) and off (0 Volts) by changing the portion of the time the signal spends on versus the time that the signal spends off. The duration of “on time” is called the pulse width. So basically, when talking about PWM, we need to know immediately 2 elements: Pulse width and Period. From these 2 elements, we can determine the Duty Cycle (%) of the PWM signal. A low-duty cycle will result in low brightness (LED) or low speed (DC motor) and vice versa.

Similar to the Input Capture function, PWM generation uses the same Capture/Compare register. The only difference is that one captures the pulse width and stores it into the register while the other compares the register with the counter to trigger the output pin. PWM also uses pins that have TIMx_CH1 to CH4 function. It means that one Timer can generate a maximum of 4 PWM outputs and the STM32F051 we are using here can generate a maximum of 16 PWM signals.

The following video shows how to setup in CubeMX:

How to calculate the PWM pulse width and period?

As you may see in the video, I set the Timer Prescaler at 24 and the Counter period (Autoreload register) at 200. What do these numbers mean? The STM32F051 chip currently runs at 48MHz then the clock frequency supplies for Timer 3 is: 48MHz/(24+1) = 1.92MHz ~ 0.5us. From that, Timer3 will take (0.5us * (200+1)) = 100us to finish one cycle counting ~ 10kHz. As a result, PWM Period relies on both Prescaler and Counter Period (Autoreload register). Besides, Pulse (Capture/Compare register) will determine the Pulse width.

To make it simple, just use the following steps to calculate Prescaler and counter period based on PWM resolution and frequency:

• Determine the desired PWM resolution (for example 100 steps, 200 steps, 1000…)
• Determine the desired PWM frequency (for example 1kHz, 10kHz, 40kHz, …)
• Set the Counter Period at the same value as the resolution
• Set the Prescaler = 48000000/(PWM frequency*PWM resolution) – 1 (thanks to Ruben and Avinash to spot the mistake)
• Set the Pulse equal to the desired Pulse width where the value varies from 0 to Counter Period.

The code in the video:

– Start PWM function:

/* USER CODE BEGIN 2 */
HAL_TIM_PWM_Start(&htim3, TIM_CHANNEL_3);
/* USER CODE END 2 */

/* USER CODE BEGIN WHILE */
while (1)
{
for (pwm=0;pwm<=200;pwm++)  //darkest to brightest: 0-100% duty cycle
{
__HAL_TIM_SetCompare(&htim3, TIM_CHANNEL_3, pwm); //update pwm value
HAL_Delay(10);
}
HAL_Delay(400);  //hold for 400ms
for (pwm=200;pwm>=0;pwm--)  //brightest to darkest: 100-0% duty cycle
{
__HAL_TIM_SetCompare(&htim3, TIM_CHANNEL_3, pwm);
HAL_Delay(10);
}
HAL_Delay(400);   //hold for 400ms

/* USER CODE END WHILE */
/* USER CODE BEGIN 3 */
}

Now, you can try making the Green LED fade in and out similarly to the Blue LED. Hope you enjoy the tutorial.

The next tutorial will be about UART.


text:
## ErinWeeks 2 years ago Find the surface area of the figure to the nearest whole number. PLEASE HELPPPP

1. ErinWeeks

A. 310 ft2 B. 158 ft2 C. 135 ft2 D. 74 ft2 I thought it was C but it was wrong -_-

2. ankolix

the back side in the image is very confusing :/

3. ErinWeeks

its 2.5 and 3.3

4. JayDS

hint: split it into two objects. find the surface area of the top triangle pyramid and add it to the surface area of the square prism

5. ErinWeeks

im still confused

6. JayDS

but you understand what I said?

7. ErinWeeks

i dont really -_-

8. JayDS

k wait, let me work through the whole thing first then I'll explain it to you.

9. ErinWeeks

okay thank you

10. JayDS

|dw:1357770316535:dw| see I split the two? this is what I meant and excuse my drawing. Note: we do not need to find the bottom base of the pyramid and top of the square prism, as it is not accounted towards the total surface area. (It's the part where they are glued together if you may).

11. ankolix

well i can to see 2 solids... the piramid and the cube.. so the piramid area is $\frac{ 1 }{ 2 }* perimetre*face length$. in your piramid $\frac{ 1 }{ 2 } \times 18\times 3.3$ and the cube surface

12. ErinWeeks

for the second formula i got 29.7 ... i dont know how to do the 1st one

13. JayDS

yes 29.7 is right for the top(pyramid).

14. ErinWeeks

okay now can you explain how to do the bottom

15. JayDS

for the bottom(square prism) it is even easier. 2(4.5 x 6 + 4.5 x 6) + (4.5 x 4.5)

16. JayDS

this is because the top of the square prism(where it needs to be used to be stuck against the bottom part of the pyramid) is not accounted for.

17. ErinWeeks

i got 74.25 now what ??

18. JayDS

19. JayDS

20. ErinWeeks

i got 103.95 but thats not a choice

21. JayDS

your 74.25 you got is wrong, try again.

22. JayDS

it should be 128.25

23. ankolix

no, it is $4(4.5*6)+ 2(4.5*4.5)$

24. ErinWeeks

with @ankolix formula i got 142.5

25. JayDS

@ankolix I don't think you understand the concept that you only need 4.5 x 4.5 once not twice.

26. ErinWeeks

27. ankolix

yeah.. ivemistake

28. JayDS

29. ErinWeeks

A. 310 ft2 B. 158 ft2 C. 135 ft2 D. 74 ft2

30. JayDS

and you made a calculation error before, so please be careful when entering it in the calculator.

31. ankolix

is 4(4.5*6)+(4.5*4.5)+29.7=157.95.. option B

32. JayDS

yep, option B.

33. JayDS

using my calculation you will get 157.95. and why is it not exact? because probably when they calculated they used more decimal places which resulted them to get a more accurate answer but we have been only given that, so we just those figures in the picture.

34. ankolix

medal for gratitude ;)

35. ErinWeeks

okay thank you so muchh..

36. JayDS

@ankolix, I would like to say you deserve the medal for trying but you have copied off me but good job.

format: plain


text:
# math homework 5 problems

One application of ‘Exponential Functions’ is the concept of compound interest.

In this case, we will use the following formula to determine the future value of an investment with monthly payments:

$A$

=

P

[

(

1

+

r

)

n

âˆ’

1

r

]

.

A=future value P=payment amount r=interest rate per period n=number of payment periods

For example: If \$100 were deposited monthly at annual interest rate of 12% for 20 years, then P = \$100, r = .12/12 = .01 and n = 12*20 = 240. The formula would look like A = 100[(1+.01)^240-1]/0.1 = \$98,925.54.

Write this formula down to use when answering the questions.

Question 1: Rachel deposits \$200 into her retirement plan each month and her company also deposits half of Rachelâ€™s monthly deposits into her retirement plan as well. Find the monthly total being deposited into Rachel’s retirement plan. The retirement plan earns 6% per year, compounded monthly, so the interest rate per period is 0.06/12 = 0.005. Rachel wants to retire in 30 years, so the number of monthly payments is 12 times 30 = 360. Using the formula stated in the instructions, how much money will Rachel have in her account when she wants to retire in 30 years? Round the answer to the nearest cent.

Question 2:How much money will Rachel have in her account if she decided to retire in 40 years in stead of 30 years? Round the answer to the nearest cent.

Question 3:How much more money would Rachel have by retiring in 40 years instead of 30 years? Round the answer to the nearest cent.

Question 4:How much money did Rachel and her company deposit into her account during the additional 10 years? Write the answer to the nearest whole dollar. Do not include any cents in the answer.

Question 5:How much extra money was in her account after 40 years as opposed to 30 years as a result of the compounded interest? Write the answer to the nearest cent.

0 replies


text:
In this post we would like to show the impossibility of having one certain lower bound for the first eigenvalue $\nu_1(\Omega)$ of the following problem:

Here $\Omega \subset \mathbb{R}^N$ is a bounded domain whose boundary $\partial \Omega$ consists of two connected components: the outer one - $\Gamma_0$, and the inner one - $\partial \Omega \setminus \Gamma_0$.

Let $\Omega^\#$ be the spherical shell (i.e., difference of two balls centred at the same point) of the same measure as $\Omega$ and such that the outer boundary of $\Omega^\#$ has the same $(N-1)$-measure as $\Gamma_0$.

It was proved by Payne & Weinberger [1] that in the planar case $N=2$, $\nu_1(\Omega)$ satisfies the following reversed Faber-Krahn inequality:

This result was generalized to the $N$-dimensional case, as well as to the $p$-Laplacian version of the problem, by Anoop & Kumar [2], under the assumption that $\Gamma_0$ is a sphere.

In Section 4 of the same paper [2], it is asked whether there exists a universal constant $C>0$ such that the following lower bound for $\nu_1(\Omega)$ takes place:

Let us show that such $C$ does not exists, in general. To this end, we will construct a sequence of equimeasurable domains $\Omega_n$ with equimeasurable outer boundary, such that $\nu_1(\Omega_n) \to 0$.

We construct $\Omega_n$ as follows. Let the outer boundary $\Gamma_0$ of $\Omega_n$ be a sphere $\partial B$. Now we consider a smaller concentric spherical shell $\Theta_n$ with a cylindrical hole $H_n$ of radius $1/n$ in it, as depicted in the following figure:

Clearly, the boundary of $\Theta_n \setminus H_n$ is connected. Moreover, we assume that the inner boundary of $\Theta_n$ is of a fixed radius. We set $\Omega_n = B \setminus (\Theta_n \setminus H_n)$, and we assume the width of $\Theta_n$ to be such that $\Omega_n$ is of a constant measure with respect to $n$.

Note that $\nu_1(\Omega)$ has the following variational characterization:

Let us take the following admissible function for this minimization problem. We set $u_n = 0$ in the “outer” part of $\Omega_n$, and $u_n=C=const$ in the “inner” part of $\Omega_n$. In the cylinder $H_n$ we assume $u_n$ to be, e.g., a linear function like $u_n(x_1,\dots,x_N) = ax_1+b$, where $a$ and $b$ are adjusted in such a way that $u_n$ is continuous. See the figure below.

Thus, we get

Clearly, the numerator goes to zero, while the denominator is bounded away from zero. Thus, the desired result follows.


text:
Article | Open | Published:

# Control of capillary behavior through target-responsive hydrogel permeability alteration for sensitive visual quantitative detection

## Abstract

DNA hydrogels have received considerable attention in analytical science, however, some limitations still exist in the applications of intelligent hydrogels. In this paper, we describe a way to prepare gel film in a capillary tube based on the thermal reversible principle of DNA hydrogel and the principle of capillary action. Because of the slight change in the internal structure of gel, its permeability can be increased by the addition of some specific targets. The capillary behavior is thus changed due to the different permeability of the hydrogel film. The duration time of the target solution flowing through the capillary tube with a specified length is used to quantify this change. With this proposed method, ultra-trace DNA hydrogel (0.01 μL) is sufficient to realize the sensitive detection of cocaine without the aid of other instruments, which has a low detection limit (1.17 nM) and good selectivity.

## Introduction

DNA hydrogels are based on the interlocking of assembled DNA secondary structures that undergo swelling in water1, where DNA not only serves as a reversible cross-linker modulating the mechanical and rheological properties of hydrogels but also selectively responds to a variety of different molecules1,2,3,4,5,6,7. Moreover, DNA hydrogels are convenient to store and can immobilize reactive functional groups, chemicals, or cells without losing their physicochemical property in changeable surrounding environments8,9. Over the last decade, aptamer-functionalized DNA hydrogel has elegantly been applied in analysis science due to its excellent portability and molecular recognition properties10,11,12. Recently, DNA hydrogels have been used for the in situ identification of circulating tumor cells by two-color confocal microscopy combined with an aptamer-trigger clamped hybridization chain reaction method10. Some DNA hydrogel-based sensors have also been proposed, where the analyte-induced chemical or physical changes of DNA hydrogel are transduced into an electrical or optical signal13,14,15,16,17,18,19. In addition, in order to reduce the dependence on other analytical instruments, Yang’s research group20,21,22,23,24,25 designed a series of bar-chart chips for diagnostics based on DNA hydrogel. In their way, they translated DNA hydrogel degradation events into distance-based signals just by monitoring with the naked eye.

However, all those published sensors were designed based on the idea of analyte-dependent formation or complete degradation of DNA hydrogel12,26,27. which can usually achieve qualitative detection or need other auxiliary means to help characterize the gel degradation to achieve precise detection. In addition, those studies only focused on macro-morphological changes, without considering micro-changes and processes. It is well known that any change in macroscopical performance comes from the accumulation of microscopic change. The complete degradation of gel into sol is the result of the accumulation of small changes of gel to a certain extent. A small number of target molecules can only cause small changes in DNA hydrogel, but will not cause the degradation of the whole gel. Therefore, the transformation from gel to sol requires more stimulations, resulting in high limit of detection (LOD) in previously published work based on gel degradation. In addition, in order to facilitate the observation and use of sol–gel transformation, gels are usually prepared in large containers and a large volume of DNA hydrogels were used, which would increase the cost of the measurement. These practical shortcomings limit the further application of DNA hydrogels in visual quantitative sensors, especially in rural clinics and patients’ homes9. How to transform such small changes in DNA hydrogel into readable, visual signals without relying on other equipment is a great challenge to further promote the application of DNA hydrogel.

Herein, we report a simple, cheap, sensitive capillary self-driven regulator sensor (CSDR-Sensor), which can transform the analyte-induced small changes inside the DNA hydrogel into visual signals. As shown in Fig. 1, DNA hydrogel film can be subtly prepared in capillary tube to obtain the CSDR-Sensor based on the thermal reversible principle of DNA hydrogel and the principle of capillary action. In the analysis process, the permeability of the DNA hydrogel film will increase because of the small structural changes in the gel induced by the interaction between target molecules and the aptamer linkers, thereby changing the flow velocity of the sample solution in the capillary tube. The duration time of the target solution flowing through the capillary tube with a specified length is used to characterize the concentration of different solutions. Compared with the previously reported DNA hydrogel sensors, this research has broken the dependence of DNA hydrogel qualitative transformation between gel and sol. By flexibly utilizing capillary tube and capillary action (self-driven motion), this sensor directly converts the target response into an easily measurable parameter by the slight microscopic changes of gel (i.e., permeability alteration) rather than macroscopic shape changes, which greatly reduces the detection limit of analysis. This sensor has also realized the visual quantitative detection without relying on other equipment. Furthermore, it can greatly reduce the usage of gel by using the hydrogel film in capillary tube. The principles and methods used in the preparation and use of CSDR-Sensor have opened a route for the design of visual quantitative sensors and greatly improved the practicability of DNA hydrogel.

## Results

### Design and fabrication of the CSDR-Sensor

In this work, a cocaine aptamer, which was proposed by Landry’s group and had been used to design various biosensors, is used in the design of DNA hydrogel for the detection of model target cocaine in our visual sensing devices28. Traditional hydrogel films, as reported in the literature, were normally prepared with a reaction mixture containing monomers and a crosslinking agent, which was either spin-coated onto a planar substrate or confined between two planar substrates13,14,15,29. In this work, the gel film was subtly prepared in capillary tube, which was based on the thermal reversible principle of DNA hydrogel and the principle of capillary action. As shown in Fig. 2, when the pre-prepared DNA hydrogel was in contact with the hot capillary tube, it turned into a liquid solution, which was then injected into the capillary tube by virtue of the capillary action. Subsequently, the DNA hydrogel film was formed in the capillary tube at room temperature to obtain the CSDR-Sensor. The gel film of 0.15 ± 0.01 mm was obtained by adjusting the time (3 s) of hot capillaries in the gel. A statistical analysis of the thickness of the gel film was made, and the yield of qualified product was about 90% as shown in Supplementary Figure 2. In order to ensure that the liquid velocity is not affected in our experiments, the error of the thickness of gel film is controlled within ±0.01 mm.

### Working principle of the CSDR-Sensor

In a typical test using CSDR-Sensor, the device was firstly placed horizontally. The sample solution was added onto a carrier plate, and then moved the carrier plate to touch the hydrogel film in capillary tube. A timer with a resolution of tens of milliseconds was used to record the flow time required for the sample through the capillary tube with certain length. Hydrogel films are very stable and sensitive and have attracted the attention of increasingly more researchers regarding their potential versatility for the efficient detection or sensing of target molecules13,14,15,16. However, these DNA hydrogel film sensors are not widely used by the public because they usually rely on expensive large-scale equipment and trained personnel for accurate preparation. Permeability is one of the characteristics of hydrogels that immediately change during gel degradation by external stimuli. The permeability of a gel film is difficult to measure and has not been used for DNA hydrogel sensors because traditional hydrogel films are very soft and easily break30,31. In this work, every CSDR-Sensor is composed of a DNA hydrogel film within a capillary tube and a substrate with scale, as shown in Fig. 3. Responsive DNA gels are used to block the capillary tube and regulate the capillary action via membrane permeability. The capillary tube protects the gel from destruction, and the capillary action allows the permeability of the gel to be visually measured. The cocaine aptamer is designed as a DNA hydrogel linker, which is a DNA molecule that can bind cocaine. DNA sequences used for this work are shown in Supplementary Table 1. In the presence of cocaine, the cocaine aptamer is able to bind the analyte to form the cocaine-stabilized conjugates. Subsequently, the permeability will be increased as a result of reducing the crosslinking density. In contrast, a blank sample (without cocaine) will not produce permeability changes in the DNA hydrogel film. Therefore, the different permeability of the DNA hydrogel results in different flow velocities of the sample solution in the capillary. Thus, the target can be visually and quantitatively detected without an external electronic device and power source. This ingenious method provides a portable, sensitive, and visual quantitative sensor that resolves the faced issues of DNA hydrogel applications.

### Optimization of the aptamer linker

In this sensing scheme, the pore size of DNA hydrogel network plays a vital role in its sensitivity. By regulating the concentration of DNA aptamer in the hydrogel, the pore size of the hydrogel can be changed, which can optimize the diffusion rate of target molecules in the hydrogel so that the hydrogel can be used for sensitive and rapid detection of targets in a range of concentrations. The pore size of the gel is determined by the concentration of the DNA linker. In order to get the proper gel, optimization experiments were carried out. The relevant data are shown in Fig. 4a–d. Three well-formed gels prepared from different concentrations of aptamer (50, 70, and 90 µM linker DNA in the hydrogel) were first selected for further optimization to detect 0 and 100 µM cocaine solutions. And the flow-through time over an 8 cm distance for 0 and 100 µM cocaine solutions are recorded as T(0 µM) and T(100 µM) (Fig. 4a, left y-axis), respectively. The signal-to-noise ratio (SNR) is defined as the ratio of {T(0 µM)–T(100 µM)} to T(0 µM), which distinguish the signal from background levels. As shown in the right y-axis in Fig. 4a, the SNRs of DNA crosslinking densities are 0.23 (50 µM linker DNA), 0.93 (70 µM linker DNA), and 0.76 (90 µM linker DNA hydrogel), respectively. Therefore, the DNA hydrogel with 70 µM linker concentration is chosen for the following experiments.

### Characterization of DNA hydrogels

The properties of the as-prepared hydrogel were then characterized using rheology. In order to obtain the linear viscoelastic region, amplitude scanning (shear strain range 0.01–1000%) was performed for 70 µM DNA hydrogels. Figure 4b shows that the storage modulus and loss modulus are constants when the shear strain range is 0.1–10%, and the characteristic of hydrogels can be obtained within this range. As the DNA hydrogel is based on the interlocking of assembled DNA secondary structures, which are only stable under their melting point, such DNA hydrogel should be temperature dependent. As shown in Fig. 4c, the storage modulus G′ value of hydrogel decreases with the increase of temperature and the gelling-transition point is about 40 °C at 70 µM aptamer linker concentration. Frequency sweep tests were performed between 0.1 and 50 rad s−1 with a fixed strain of 1% at 25 °C (Fig. 4d). In the frequency sweep, the storage modulus G′ value increased from 10 to 900 Pa with the increase of DNA linker concentration, which can be attributed to the densification of the hydrogel network32,33. Considering the result of Fig. 4a, the capillary flow velocity of the sample gradually decreases with the increase of aptamer concentration, which is consistent with the storage modulus analysis results. The larger the storage modulus is, the smaller the hydrogel mesh is, and the corresponding permeability becomes lower. The relationship among aptamer concentration, gel permeability and the flow time of samples confirmed the feasibility of controlling capillary flow rate by regulating the permeability of gel membrane, which is the theoretical basis of our experiments.

Subsequently, as shown in Supplementary Figure 1a, the volume of gel (thickness is 0.15 mm) is calculated to be 0.0106 μL based on the inner diameter (0.3 mm) of the capillary. Additionally, the DNA hydrogel shows well-defined interconnected 3D porous networks. From the highly magnified scanning electron microscopy (SEM) images in Supplementary Figure 1b, the network exhibits varying micrometer-sized pores, which suggests that the hydrogel can provide space for the reaction.

### Visual quantitative detection of cocaine

To further investigate the accuracy of this method based on the aptamer-functionalized DNA hydrogel film, different concentrations of cocaine are detected. Different concentrations of cocaine are spiked in a buffer and detected by the CSDR-Sensor based on the readout time. The cocaine solutions are placed in contact with the gel film in the capillary, and the flow-through time of the sample solution is recorded. The analysis results of the CSDR-Sensor for the detection of cocaine are shown in Fig. 5a–c. Cocaine concentrations ranging from 0 to 100 µM are tested with three repeated measurements, each in parallel. As shown in Fig. 5a, when the 100 µM cocaine sample is assayed, it takes 2′45″ for the sample solution to flow through a 6 cm length. The short flow time indirectly reflects an increasing permeability as a result of the reduction in the crosslinking density induced by high concentrations of the cocaine. On the other hand, for the 10 nM cocaine sample, it takes 15′30″ to allow the solution to flow through a 2.6 cm length, and the solution still does not reach the 6 cm position after 23′30″ (Fig. 5a). As shown in Fig. 5b and in the Movies (Supplementary Movie 1 (0 nM), Supplementary Movie 2 (1 µM). and Supplementary Movie 3 (100 µM) cocaine detection, respectively), the flow rate of the sample in the capillary gradually increases as the cocaine concentration increases. A high concentration of cocaine results in more cocaine molecules binding with the linker in the DNA hydrogel, which leads to an increasing permeability of the DNA hydrogel. Figure 5c shows that the flow-through time is directly proportional to the cocaine concentration. Therefore, the CSDR-Sensor has the excellent capability of quantitative cocaine detection in a linear range from 10 nM to 100 μM. The linear regression equation is obtained as y = −9.9692x + 51.0845 with a correlation coefficient of 0.9879. The LOD is calculated to be 1.17 nM based on the 3σ/slope rule (N = 3), where σ is the standard deviation of the blank samples. These results clearly demonstrate that the CSDR-Sensor based on the DNA hydrogel film in a capillary tube can achieve quantitative detection with the naked eye with excellent accuracy.

### Evaluation of selectivity and practicality

The potential false-positive signals will affect the practical performance of the sensor, which should be excluded. Benzoylecgonine (BE) and ecgonine methyl ester (EME) are the most common interfering molecules in the detection of cocaine. In order to test the anti-interference performance of the device in a complex environment, two sample solutions (containing 1 mM BE and 1 mM EME) and a blank sample were tested with our CSDR-Sensors, respectively. The results obtained were compared with those of samples containing 100 μM cocaine. As shown in Fig. 6a, the analytical results of the solution containing BE and EME are basically consistent with those of blank samples. DNA adapter molecules in hydrogels can only specifically recognize cocaine, but cannot recognize BE and EME, even at high BE and EME concentrations. Therefore, the sensor based on DNA hydrogel membrane has good ability to recognize target molecule cocaine in real detection scenarios, and the interfering molecules cannot cause the changes in the permeability of hydrogels.

To further assess the applicability and reliability of the method proposed, herein, we utilize the CSDR-Sensor for the detection of cocaine in urine. Ten diluted urine samples are assayed according to general procedures, and the determined concentrations are estimated from the corresponding signals of the flow-through time and the regression equations. As shown in Fig. 6b, the flow-through time is linearly correlated with the concentration of cocaine in urine. Moreover, an adjusted R-square value of 0.9829 and a slope of 0.9775 are obtained. The results prove that the CSDR-Sensor has high sensitivity for the detection of cocaine in real samples and would be well suited for highly sensitive detection in clinical and POCT settings. On the other hand, DNA-hydrogel can be easily stored, which lays the foundation of the applicability of the sensor for POCTs assays. As shown in Supplementary Figure 3, after storing the sensors for a time duration (2 weeks) under a certain temperature (4 °C) in an enclosed space with atmosphere of 2 mL buffer (77 mM NaHPO4, 23 mM NaH2PO4, 50 mM NaCl, 5 mM MgCl2), the sensors shall still be able to function. With the same concentration of cocaine solution (1 μM), similar results were obtained with RSD less than 3.4%, suggesting the fabrication reproducibility and working stability of the CSDR-Sensor.

## Discussion

In comparison with conventional quantitative cocaine assays based on DNA hydrogel using colorimetric26,34, glucose meter35, UV–vis spectra27, or naked eye12,21,22 measurements, the CSDR-Sensor based on the above principles and methods shows good performance and effectiveness (Supplementary Table 2). The analysis results of our work are quantitative, and the LOD of our method is extremely low compared with the reported methods previously. This result is comparable to the most sensitive methods based on hydrogel reported to date, and a small amount of DNA hydrogel is needed for the sensitive analysis without the aid of other instruments. Only 0.01 μL of DNA hydrogel (just a small, thin film) is required for the detection of 10 nM to 100 µM cocaine. Therefore, it is very cheap than previously reported methods. Moreover, this method is very simple for real POCT applications by just integrating the capillary action and DNA hydrogel film into the capillary tube.

In summary, a facile and sensitive visual quantitative sensor is developed by integrating the permeability change in a DNA hydrogel film with self-driven motion. The time of the sample solution flowing through a certain length of capillary tube is influenced by the different permeability of hydrogel film. The permeability has a linear relationship with the target concentration. This result is comparable to the most sensitive methods based on hydrogel reported to date, and it can be achieved by ultra-trace DNA hydrogel without the aid of other instruments. Moreover, the method developed here can be used as a powerful tool for the detection of other targets simply by replacing the linker sequences in the hydrogel. A variety of aptamers against a broad range of targets are either available or can be obtained via the systematic evolution of ligands by exponential enrichment (SELEX)36,37,38. With the advantages of high sensitivity, selectivity, and versatility, as well as the features of low cost, equipment-free, and portability, this method has great significance for developing smart hydrogel public applications.

## Methods

### Materials and reagents

Cocaine was obtained from National Institutes for Food and Drug Control (Beijing, China). Ammonium persulfate (APS), tetramethylethylenediamine (TEMED), and acrylamide were purchased from Sigma-Aldrich (St. Louis, MO, USA). 2-Cyanoethyl diisopropyl chlorophosphoramidite was purchased from Chem Genes (Wilmington, MA, USA). FC-40 (a mixture of perfluoro-tri-n-butylamine and perfluoro-di-n-butylmethylamine) was purchased from Minnesota Mining and Manufacturing Company (St. Paul, MN, USA). DNA synthesis reagents were purchased from Glen Research (Sterling, VA, USA). Other reagents were purchased from Sinopharm Chemical Reagent (Shanghai, China). The cocaine buffer contained 77 mM Na2HPO4, 23 mM NaH2PO4, 50 mM NaCl, 5 mM MgCl2 (pH 7.3). All solutions were prepared with ultra-pure Milli-Q water (resistance > 18 MΩ cm−1). Glass capillaries from Suzhou City Crystal Glass Co., Ltd. were used. The inner radius of these capillaries is constant and equal to 300 ± 5 μm.

Synthesis of Acrylic-DMT Phosphoramidite and DNA synthesis were prepared according to the reported methods. All oligonucleotides used in the work were synthesized on a PolyGen 12-Column DNA Synthesizer in-house according to the standard DNA synthesis protocol. Strand A and strand B were modified at the 5′-end using the acrylic phosphoramidite. The product was cleaved from the solid support, deprotection with ammonia treatment, and purified by LC3000 semi-preparative HPLC system on a reverse-phase C18 column (Chuang Xin Tong Heng, Beijing, China). A solution of 0.1 M triethylamine acetate (pH 6.5) was used as HPLC buffer A, and HPLC-grade acetonitrile (Fisher) was used as HPLC buffer B. After detritylation, the DNA were desalted with a NAP-5 column (GE Healthcare), quantified by UV–vis spectrometry, and stored at −20 °C for future use.

Stock solutions of strand A and B with the desired concentrations were prepared separately in centrifuge tubes containing 4% acrylamide. They were placed in a vacuum desiccator for 10 min at 25 °C to remove air. Then, 1.4% (v/v) of freshly prepared initiator (0.05 g of APS dissolved in 0.5 mL ultrapure water) and accelerator (25 μL TEMED dissolved in 0.5 mL ultrapure water) solution were added to both stock solutions immediately. The centrifuge tubes were placed in the vacuum desiccator again to polymerize under vacuum at 25 °C. Linear-chain PS-A and PS-B were produced after 15 min. To generate the hydrogel, 110 μM PS-A and 110 μM PS-B were mixed with a certain amount of aptamer linker in buffer (77 mM NaHPO4, 23 mM NaH2PO4, 50 mM NaCl, 5 mM MgCl2). The mixture was shaken vigorously to guarantee the homogeneity of the solution before incubating in a dry bath at 65 °C for 5 min, and then allowed to slowly cool to room temperature (repeat three times) to produce cocaine-aptamer cross-linked hydrogel.

### CSDR-Sensor design and fabrication

In order to get more silicon hydroxyl, glass capillary tube was cleaned by piranha solution (mixture of 7:3 (v/v) 98% H2SO4 and 30% H2O2) at 100 °C for 1 h and followed by thorough ultrasonication for 10 min in acetone and ethanol. Then the glass capillary tube was rinsed three times in Milli-Q water and blown dry with nitrogen gas. In order for the capillary to obtain the temperature which could disintegrate the DNA hydrogel of step 2, the capillary tube was heated at 90 °C for 10 min. Next, the hot capillary tubes were inserted into the DNA hydrogel by lifting machine (depth 1 mm) and taken out about 3 s later. The hydrogel film in capillary with thickness was quickly selected through a microscope. Subsequently, the capillary tube with DNA hydrogel film is fixed on a graduated plate to simply obtain CSDR-Sensor, and refrigerated storage.

### CSDR-Sensor assays

The use of CSDR-Sensor device is as simple as using a thermometer as shown in Supplementary Movie 4. In a typical test on CSDR-Sensor, the device is placed horizontally. And the sample solution (10 nM to 100 μM cocaine solution in buffer 77 mM NaHPO4, 23 mM NaH2PO4, 50 mM NaCl, 5 mM MgCl2) is added on a carrier plate, then move the carrier to touches hydrogel film in capillary tube. Then a timer with a resolution of tens of milliseconds was used to record the time required for the sample reagent to flow through the 8 cm-length in capillary tube.

### Rheological measurements

Rheological measurements were carried out using a rheometer model MCR-302 (Anton Paar, Graz, Austria) equipped with a Peltier device for temperature control. During all rheological measurements, silicone oil trap was used to minimize evaporation. For characterization of DNA hydrogels, a plate–plate geometry with a 12 mm diameter upper plate was used. The upper plate was made of stainless steel and the lower plate was a Teflon Peltier surface. The upper plate was set at the desired separation distance that was equivalent to a normal force (FN) of 5 N (within a linear range; the moduli are independent of the magnitude of imposed force) corrected with a “true gap” function of the instrument. The hydrogels were characterized under frequency sweep (ω = 0.1–50 rad s−1, 25 °C). The mesh size ξ of the hydrogels was calculated from the storage modulus G′ value as follows32:

$$\xi = \left( {\frac{{{RT}}}{{G^{\prime}{ N_{\mathrm{A}}}}}} \right)^{\frac{1}{3}}$$
(1)

where R is the gas constant, T is the measurement temperature, and NA is Avogadro’s constant.

## Data availability

Data within the manuscript and its Supplementary Information are available from the corresponding author upon reasonable request. A separate source data file is provided in a persistent repository (URL: https://pan.baidu.com/s/1Wb12QMsmxVjPcZLNaWHopg; Fetch Code: dij4). The source data underlying Figs. 4a–d, 5b, c, and 6a, b and Supplementary Figures 2 and 3 are provided as a Source Data file.

## References

1. 1.

Cheng, E. J. et al. A pH-triggered, fast-responding DNA hydrogel. Angew. Chem. Int. Ed. 48, 7660–7663 (2009).

2. 2.

Shao, Y., Jia, H., Cao, T. & Liu, D. Supramolecular hydrogels based on DNA self-assembly. Acc. Chem. Res. 50, 659–668 (2017).

3. 3.

Lilienthal, S., Shpilt, Z., Wang, F., Orbach, R. & Willner, I. Programmed DNAzyme-triggered dissolution of DNA-based hydrogels: means for controlled release of biocatalysts and for the activation of enzyme cascades. ACS Appl. Mater. Interfaces 7, 8923–8931 (2015).

4. 4.

Xiang, B. et al. Self-assembled DNA hydrogel based on enzymatically polymerized DNA for protein encapsulation and enzyme/DNAzyme hybrid cascade reaction. ACS Appl. Mater. Interfaces 8, 22801–22807 (2016).

5. 5.

Deshpande, S. R. et al. DNA-responsive polyisocyanopeptide hydrogels with stress-stiffening capacity. Adv. Funct. Mater. 26, 9075–9082 (2016).

6. 6.

Liu, D. S., Cheng, E. J. & Yang, Z. Q. DNA-based switchable devices and materials. NPG Asia Mater. 3, 109–114 (2011).

7. 7.

Luo, D. The road from biology to materials. Mater. Today 6, 38–43 (2003).

8. 8.

Miyata, T., Asami, N. & Uragami, T. A reversibly antigen-responsive hydrogel. Nature 399, 766 (1999).

9. 9.

Liu, J. Oligonucleotide-functionalized hydrogels as stimuli responsive materials and biosensors. Soft Matter 7, 6757–6767 (2011).

10. 10.

Song, P. et al. DNA hydrogel with aptamer-toehold-based recognition, cloaking, and decloaking of circulating tumor cells for live cell analysis. Nano Lett. 17, 5193–5198 (2017).

11. 11.

Zhang, L., Lei, J., Liu, L., Li, C. & Ju, H. Self-assembled DNA hydrogel as switchable material for aptamer-based fluorescent detection of protein. Anal. Chem. 85, 11077–11082 (2013).

12. 12.

Wei, X. et al. Target-responsive DNA hydrogel mediated “stop-flow” microfluidic paper-based analytic device for rapid, portable and visual detection of multiple targets. Anal. Chem. 87, 4275–4282 (2015).

13. 13.

Tellis, J. C., Strulson, C. A., Myers, M. M. & Kneas, K. A. Relative humidity sensors based on an environment-sensitive fluorophore in hydrogel films. Anal. Chem. 83, 928–932 (2011).

14. 14.

Biela, A. et al. Disposable MMP-9 sensor based on the degradation of peptide cross-linked hydrogel films using electrochemical impedance spectroscopy. Biosens. Bioelectron. 68, 660–667 (2015).

15. 15.

Men, D. et al. Optical sensor based on hydrogel films with 2D colloidal arrays attached on both the surfaces: anti-curling performance and enhanced optical diffraction intensity. J. Mater. Chem. C 3, 3659–3665 (2015).

16. 16.

Men, D. et al. Functional hydrogel film attached with 2D Au nanosphere array and its ultrahigh optical diffraction intensity as a visualized sensor. J. Mater. Chem. C 4, 2117–2122 (2016).

17. 17.

Sadman, K. et al. Quantitative rheometry of thin soft materials using the quartz crystal microbalance with dissipation. Anal. Chem. 90, 4079–4088 (2018).

18. 18.

Tokarev, I. & Minko, S. Stimuli-responsive hydrogel thin films. Soft Matter 5, 511–524 (2009).

19. 19.

Holtz, J. H. & Asher, S. A. Polymerized colloidal crystal hydrogel films as intelligent chemical sensing materials. Nature 389, 829 (1997).

20. 20.

Huang, Y. et al. Target-responsive DNAzyme cross-linked hydrogel for visual quantitative detection of lead. Anal. Chem. 86, 11434–11439 (2014).

21. 21.

Zhu, Z. et al. Au@ Pt nanoparticle encapsulated target‐responsive hydrogel with volumetric bar‐chart chip readout for quantitative point‐of‐care testing. Angew. Chem. Int. Ed. 53, 12503–12507 (2014).

22. 22.

Tian, T. et al. Integration of target responsive hydrogel with cascaded enzymatic reactions and microfluidic paper-based analytic devices (µPADs) for point-of-care testing (POCT). Biosens. Bioelectron. 77, 537–542 (2016).

23. 23.

Xie, Y. et al. A Shake&Read distance-based microfluidic chip as a portable quantitative readout device for highly sensitive point-of-care testing. Chem. Commun. 52, 13377–13380 (2016).

24. 24.

Zhang, H., Zhou, L., Zhu, Z. & Yang, C. Recent progress in aptamer-based functional probes for bioanalysis and biomedicine. Chem. Eur. J. 22, 9886–9900 (2016).

25. 25.

Tian, T. et al. Integrated distance-based Origami paper analytical device for one-step visualized analysis. Colloid Surf. B 159, 277 (2017).

26. 26.

Zhu, Z. et al. An aptamer cross-linked hydrogel as a colorimetric platform for visual detection. Angew. Chem. Int. Ed. 122, 1070–1074 (2010).

27. 27.

Mao, Y. et al. A portable visual detection method based on a target-responsive DNA hydrogel and color change of gold nanorods. Chem. Commun. 53, 6375–6378 (2017).

28. 28.

Zhu, Z. et al. An aptamer cross-linked hydrogel as a colorimetric platform for visual detection. Angew. Chem. Int. Ed. 49, 1052–1056 (2010).

29. 29.

Harmon, M. E., Jakob, T. A. M., Knoll, W. & Frank, C. W. A surface plasmon resonance study of volume phase transitions in N-isopropylacrylamide gel films. Macromolecules 35, 5999–6004 (2002).

30. 30.

Li, S. et al. Hydrogels with precisely controlled integrin activation dictate vascular patterning and permeability. Nat. Mater. 16, 953 (2017).

31. 31.

Tokarev, I., Orlov, M. & Minko, S. Responsive polyelectrolyte gel membranes. Adv. Mater. 18, 2458–2460 (2006).

32. 32.

Wang, J. & Ugaz, V. M. Using in situ rheology to characterize the microstructure in photopolymerized polyacrylamide gels for DNA electrophoresis. Electrophoresis 27, 3349–3358 (2006).

33. 33.

Colby, R. H., Fetters, L. J. & Graessley, W. W. The melt viscosity–molecular weight relationship for linear polymers. Macromolecules 20, 2226–2237 (1987).

34. 34.

Yin, B.-C., Ye, B.-C., Wang, H., Zhu, Z. & Tan, W. Colorimetric logic gates based on aptamer-crosslinked hydrogels. Chem. Commun. 48, 1248–1250 (2012).

35. 35.

Yan, L. et al. Target-responsive “sweet” hydrogel with glucometer readout for portable and quantitative detection of non-glucose targets. J. Am. Chem. Soc. 135, 3748–3751 (2013).

36. 36.

Stoltenburg, R., Reinemann, C. & Strehlitz, B. SELEX—a (r) evolutionary method to generate high-affinity nucleic acid ligands. Biomol. Eng. 24, 381–403 (2007).

37. 37.

Daniels, D. A., Chen, H., Hicke, B. J., Swiderek, K. M. & Gold, L. A tenascin-C aptamer identified by tumor cell SELEX: systematic evolution of ligands by exponential enrichment. Proc. Natl Acad. Sci. USA 100, 15416–15421 (2003).

38. 38.

Sefah, K., Shangguan, D., Xiong, X., O’donoghue, M. B. & Tan, W. Development of DNA aptamers using Cell-SELEX. Nat. Protoc. 5, 1169 (2010).

## Acknowledgements

This work was supported by Beijing Natural Science Foundation (2172039), National Key Research and Development Program of China (2018YFD0401302), National S&T Major Project of China (2018ZX10301201), the National Science Foundation of China (21735004, 21890742, 21727815, 21521004, 21435004), and the Fundamental Research Funds for the Central Universities and USTB.

## Author information

Yansheng Li and Yongqiang Wen designed the experiments project. Yansheng Li carried out the measurements. Yanli Ma, Tingyu Li, Zhehao Lv, and Chaoyong James Yang prepared the DNA materials. Yongqiang Wen, Yansheng Li, and Xiangyu Jiao discussed and analyzed the data. Xueji Zhang assisted in evaluating the results. Yansheng Li and Yongqiang Wen wrote and revised the paper.

Correspondence to Chaoyong James Yang or Xueji Zhang or Yongqiang Wen.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Journal peer review information: Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


text:
It takes 23 hours 56 minutes and 4 seconds for the earth to make one revolution (mean sidereal day). What is the angular speed of the earth?
7.29×10-5 rad/s (i got this part...now the below i cant get the right answer for)
1)Assume the earth is spherical. Relative to someone on the rotation axis, what is the linear speed of an object on the surface if the radius vector from the center of the earth to the object makes an angle of 59.0° with the axis of rotation. The radius of the earth is 6.37×103 km. (Linear speed for motion in a circle is the radius times the angular velocity. Note, the radius of the earth is given in kilometers)

2)What is the acceleration of the object on the surface of the earth in the previous problem?


text:
## Note on Determination of Latent Heat of Steam by the Method of Mixture

• Note
• Things to remember

#### Determination of Latent Heat of Steam by the Method of Mixture

Following are the steps to determine the latent heat of vaporization of water:

1. A calorimeter with stirrer is taken and it is weighted.
2. Some water is poured into a calorimeter then mass and temperature of a stirrer, calorimeter and water is taken.
3. Steam is generated in a steam generator and the temperature of steam is noted.
4. For a certain time, the steam is passed into the calorimeter containing water.
5. The final mixture is stir then the temperature and mass of final mixture are taken.

m1= mass of calorimeter and stirrer

m2 = mass of calorimeter, stirrer and water

mw = m1 - m2 = mass of water only

m3 = mass of calorimeter, stirrer, water and steam

m= m3 - m2 = mass of steam only

= temperature of water, stirrer and calorimeter

= temperature of steam

= temperature of mixture

sc = specific heat capacity of a calorimeter

sw = specific heat capacity of water

lv = Latent heat of vaporization = ?

Heat loss by water and calorimeter$$= m l_v +ms_c(\theta_2 -\theta)$$

Heat gain by ice when 0ice and 0water and 0water to water$$= m_ws_w(\theta_1 - \theta) + m_1s_c(\theta_1 - \theta)$$

From principle of calorimetry

\begin{align*} \text {Heat loss} &= \text{Heat gain} \\(m l_v +ms_w(\theta_2 -\theta) &= m_ws_w(\theta_1 - \theta) + m_1s_c (\theta_1 - \theta) \\m l_v &= (m_ws_w + m_1s_c)(\theta - \theta_1)- ms_w(\theta_2 -\theta) \\\therefore l_v &= \frac {(m_ws_w + m_1s_c)(\theta_1 - \theta)- ms_w (\theta_2 -\theta) }{m} \\ \end{align*}

#### Melting and Freezing

When a solid changes into liquid or liquid changes into solid, the change of state take place. The change of state from solid to liquid is called melting while the change of state from liquid to solid is called freezing. During melting and freezing, the latent heat of fusion is lost to the surroundings while during the melting same amount of heat is absorbed.

#### Super Cooling

When a liquid is cooled gently, it is found that the liquid may cool even several degrees below its freezing point without solidification. Such a liquid at a temperature below its freezing point is called super cold liquid. Water can be cooled up to the temperature of -20 oCwithout forming ice. But if we drop a small piece of ice into super-cooled water, the water at once will start solidification and immediately the temperature of thewater rises to 0, so this super-cooled state is unstable.

Pressure and Melting Point

At normal temperature, the melting point of a solid or freezing point of a liquid is fixed. But due to increase in pressure, the melting point or freezing point of the substance can change. The melting point can increase or decrease the melting point. In the case of a substance which expands on solidification, the increase of pressure lowers their melting point and in the case of a substance which contracts on solidification the increase of pressure increase their melting point.

#### Regelation

The phenomenon of melting of ice under the application of pressure and its solidification after the removal of pressure is called regelation. A string can pass through the block of ice without cutting it into two pieces due to regeletion.

Freezing solutions

When a solution is made on dissolving some common salt in water, it lowers the freezing point of water. The higher the quantity of common salt per unit volume of water, the greater is the fall in freezing point. When ice and common salt are mixed, the mixture cools below 0but remains in a liquid. The mixture of salt and ice provides a simple means of reaching a temperature below 0i.e. freezing mixture. This phenomena is important in chemistry especially in metallurgy.

Evaporation

The slow and silent conversion of a liquid into a gaseous state at all temperature is called evaporation. The rate of evaporation depends on following factors:

1. Area of the liquid surface
2. Temperature and pressure of the liquid
3. Nature of the liquid
4. Effect of wind
5. Humidity of air

Boiling or Ebullition: It is the rapid and noisy process in which conversion of a liquid into the vapour state takes place. During boiling, the temperature of the liquid remains constant.

Difference between evaporation and boiling

 S.N. Evaporation S.N. Boiling 1. It is the slow and silent process of converting liquid into the vapour state. 1. It is the rapid and noisy process of converting liquid into water vapour. 2. It occurs at the surface of the liquid. 2. It occurs throughout the volume of the liquid. 3. It takes place at all temperature. 3. It takes place at a fixed temperature. 4. Cooling effect is observed. 4. No cooling is observed.

When a solid changes into liquid or liquid changes into solid, the change of state take place. The change of state from solid to liquid is called melting.

During melting and freezing, the latent heat of fusion is lost to the surroundings while during the melting same amount of heat is absorbed.

The phenomenon of melting of ice under the application of pressure and its solidification after the removal of pressure is called regelation.

When a solution is made on dissolving some common salt in water, it lowers the freezing point of water.

The slow and silent conversion of a liquid into a gaseous state at all temperature is called evaporation.

Ebullition is the rapid and noisy process in which conversion of a liquid into the vapour state takes place.

.

0%


text:
# How do we know that radioactive decay is memoryless?

Let $\tau$ be the random variable that describes the lifetime of a given particle. It seems to conform to common-sense that $\mathbf{P}(\tau>t+s|\tau>s)=\mathbf{P}(\tau>t)$, as it would be weird if the particle could keep track" of how long it has been around.

My question is, do we have any good reason to believe that $\tau$ is memoryless in this way, besides the common-sense reason I have provided? I would appreciate references.

This is because this result, together with the assumptions that the particles are not "communicating" with one another and that they are identical confirms that the probability density of the time to decay for each particle is exponential. If we assume no "communication" i.e. no interaction so that a particle's decay does not influence another's, then the times to decay are statistically independent variables, so that the rate of decay is proportional to $p(\tau)$, the pdf of the time to decay.


text:
ISSN 1671-3710
CN 11-4766/R

Advances in Psychological Science ›› 2021, Vol. 29 ›› Issue (12): 2161-2171.

• Regular Articles •

### The relation between non-symbolic magnitude representation and symbolic fraction representation

MAO Huomin1, LIU Qin2, LV Jianxiang2, MOU Yi2

1. 1Zhu Hai Sun Yat-sen University Primary School, Zhuhai 519031, China;
2Department of Psychology, Sun Yat-sen University, Guangzhou 51006, China
• Received:2021-02-08 Online:2021-12-15 Published:2021-10-26

Abstract: A fundamental research question in numerical development concerns the relation between early emerging non-symbolic magnitude representation and symbolic numbers and mathematics learning. Especially, whether non-symbolic magnitude representation is a cognitive foundation for symbolic numbers and mathematic learning. Since early infants, one can represent the magnitude of a set of items (single magnitude) and the proportion between two magnitudes non-verbally (i.e., non-symbolic magnitude representation). Many of existing studies have examined the relation between non-symbolic representations for single magnitudes and symbolic number or mathematics learning. In contrast, few studies have investigated whether and how non-symbolic representations for the proportion between magnitudes relate with symbolic numbers or mathematics. Given that fraction is a critical concept of symbolic proportion, and it is an important concept taught in elementary mathematics, the present article reviewed and summarized studies on the relations between the representations for non-symbolic proportions and symbolic fractions. Previous studies showed that the precision of individuals’ non-symbolic magnitude representations for proportions were correlated with the precision of representations for symbolic fractions. In addition, both non-symbolic proportions and symbolic fractions activate some common brain regions, suggesting their common neural foundations. However, these correlational findings may not necessarily mean that non-symbolic magnitude representations for proportions provide a cognitive foundation for learning of symbolic fractions. First, when examining the relation between representations for non-symbolic proportions and symbolic fractions, most of existing studies did not rigorously control for general cognitive abilities. Therefore, it is not clear if non-symbolic proportion representations are uniquely correlated with symbolic fraction representations. Second, while most studies found the concurrent correlations between non-symbolic proportion and symbolic fraction representations, little has been done to examine if the two are related longitudinally. Third, besides demonstrating the correlations, more studies are needed to reveal how symbolic fraction representations are built on non-symbolic proportion representations. These discussions are also informative for mathematics education.

CLC Number:


text:
# Given the set of information, find a linear equation satisfying the conditions, if possible. (If not...

###### Question:

Given the set of information, find a linear equation satisfying the conditions, if possible. (If not possible, enter IMPOSSIBLE.) passes through (x, y) = (1, 6) and (x, y) = (7, 18) Additional Materials eBook Linear Functions Example Video 17. (-/1 Points] DETAILS OSCAT1 5.1.028. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use the vertex (h, k) and a point on the graph (x, y) to find the general form of the equation of the quadratic function. (h, k) = (0,7), (x, y) = (2, 11) f(x)

#### Similar Solved Questions

##### Suppose that at the Reedley Jack-In-The Box; sample of 81 drive thru customers revealed that their wait times (food order until food pickup) had mean of 87 minutes with standard deviation 0.52 minutes The answer for part dependent entirely upon the result from Port and NO additional calculations need be done_ 20 points-17 points for Part and points for part bPlease construct 95% confidence interval estimate (2 dp) of the meon customer drive - thru wait time for the Reedley Jack-In-The: Box_ and
Suppose that at the Reedley Jack-In-The Box; sample of 81 drive thru customers revealed that their wait times (food order until food pickup) had mean of 87 minutes with standard deviation 0.52 minutes The answer for part dependent entirely upon the result from Port and NO additional calculations nee...
##### PS, For part a, please include: who to report, who to filter, who receive the information,...
PS, For part a, please include: who to report, who to filter, who receive the information, when, where, how, the nature of the incident, what kind of immediate management intervention For part b, please briefly explain the Kurt Lewin's force field model of change and how to apply it into this ca...
##### Ath 126 3. a. Solve the equation V5 - 2x - 2 = 0, algebraically. Give...
ath 126 3. a. Solve the equation V5 - 2x - 2 = 0, algebraically. Give exact answers. b. Which feature of the calculator will you be using to solve the equation graphically? (Circle one) INTERSECT or ZERO c. Define the equation(s) you are using for your graphical solution(s). Y1 = (if applicable) Y2 ...
##### 10. The half-life of francium is about 1600 years. If a rock initially contains a 2...
10. The half-life of francium is about 1600 years. If a rock initially contains a 2 g of francium, the amount left after 16000 years will be about...
##### Given the functons:f6)-r+6xg6)-V2xh6)-Sr-7Evaluate the functicn (h %f) (x) for x = Wrlte vour answyer exact simplified form. Sclect "Undclined" applicable:2)(s) Is 0O/0 Undelneu
Given the functons: f6)-r+6x g6)-V2x h6)-Sr-7 Evaluate the functicn (h %f) (x) for x = Wrlte vour answyer exact simplified form. Sclect "Undclined" applicable: 2)(s) Is 0 O/0 Undelneu...
##### What are the industry trends that are associated with your area of interest? Research the past...
What are the industry trends that are associated with your area of interest? Research the past five years, and the projections for next decade, and list them along with the employment requirements necessary to work in that field. Include any licensure and professional certifications that are necessa...
##### 2 1 51 A=1-14 313 L-10-1 13 | Give your answer in the form A = XDX-1 , where X is s and D = | 0 λ2 01 where the λ, the eigenbas are ordered increasingly. 0 0A3 1-1 TT1 0 0 0 0.5 0.5 0.5 0.5 -0.5...
2 1 51 A=1-14 313 L-10-1 13 | Give your answer in the form A = XDX-1 , where X is s and D = | 0 λ2 01 where the λ, the eigenbas are ordered increasingly. 0 0A3 1-1 TT1 0 0 0 0.5 0.5 0.5 0.5 -0.5 2 1 51 A=1-14 313 L-10-1 13 | Give your answer in the form A = XDX-1 , where X is s and D...
##### Show that F is conservative vector field and use this fact t0 evaluate fFadr along the given curve. F(x,Y)-(4x'y?-2xy' )I +(xty-3x?yl+4y')j, t: ( +sinTt) â‚¬ + (2tt cstt)5 02+41
Show that F is conservative vector field and use this fact t0 evaluate fFadr along the given curve. F(x,Y)-(4x'y?-2xy' )I +(xty-3x?yl+4y')j, t: ( +sinTt) â‚¬ + (2tt cstt)5 02+41...
##### Let tan Awith A in QH and find the exact value forsin(2A)PreviewTipEntcr = vour answer aS a number (like 5-32.2172) or as a
Let tan A with A in QH and find the exact value for sin(2A) Preview Tip Entcr = vour answer aS a number (like 5-32.2172) or as a...
##### Question 35confidence interval for the and sample data to construct a decimal places: Use the given degree of confidence interval. Round to three Write down the confidence population proportion p. of an issue before the legislature in one state reveals that 408 favor approval. A survey of 865 voters all voters in the state who favor the true proportion of the 95% confdence interval for Construct approval: digits, e g.1.21 Round to 3 between the decimal point = and (Type your answers without an
Question 35 confidence interval for the and sample data to construct a decimal places: Use the given degree of confidence interval. Round to three Write down the confidence population proportion p. of an issue before the legislature in one state reveals that 408 favor approval. A survey of 865 vote...
##### Given the vectors v = i + 2j and w = 61 + 3j, find the angle between the vectors and determine ifthe vectors are parallel, orthogonal, or neither:90 degrees; orthogona70 dcgrecs; patallel0,orthogonalPjrallcl:nonc of tnosc
Given the vectors v = i + 2j and w = 61 + 3j, find the angle between the vectors and determine ifthe vectors are parallel, orthogonal, or neither: 90 degrees; orthogona 70 dcgrecs; patallel 0,orthogonal Pjrallcl: nonc of tnosc...
##### (1 point) Here we investigate whether the register balance at a local retail store is better...
(1 point) Here we investigate whether the register balance at a local retail store is better on days with a manager than days without a manager. This evidence might be used to determine whether or not you should always schedule a manager. The table gives the register balance for a sample of 10 days ...
##### Determine if each is a function: 20. y13 = x + 5- 21. x4 = y...
Determine if each is a function: 20. y13 = x + 5- 21. x4 = y +45 22. x19.+ 4y18 = 16 23. xº + y2 = 10...
##### 3. Solve the following differential equation: xy' +y = sin(x) +et
3. Solve the following differential equation: xy' +y = sin(x) +et...
##### 04in Inanetnaou} fEie 0f Gunt oye(r_ 1x' - 3x + 1)wah resped to X at * =2
04in Inanetnaou} fEie 0f Gunt oye(r_ 1x' - 3x + 1)wah resped to X at * =2...
##### Machine Learning 4·Let (X.J) be a instance-label set with χ e Rn and let χ1ram be...
Machine Learning 4·Let (X.J) be a instance-label set with χ e Rn and let χ1ram be a sample set (of training data). I (is linearly separable, then there is some logistic regression model Hsig that will produce a loss of 0 on xtrain (a) Explain why the last sentence above is true. (b...
##### Recordxour data in the table below: Calculate the averages and convert to mm/s Temperature No_ Lines Avg: Lines Speed in Crossedlmin CrossedImin mmls10 033,31504os200250
Recordxour data in the table below: Calculate the averages and convert to mm/s Temperature No_ Lines Avg: Lines Speed in Crossedlmin CrossedImin mmls 10 0 33,3 150 4os 200 250...
##### Draw out the two-step mechanism and reaction energy diagram for the following reaction."OCHgNHzNHz"OCHa
Draw out the two-step mechanism and reaction energy diagram for the following reaction. "OCHg NHz NHz "OCHa...
##### 4. [201 Use Prolog (e.g., SWI-Prolog) to create a knowledge base for the family tree of...
4. [201 Use Prolog (e.g., SWI-Prolog) to create a knowledge base for the family tree of Figure 1 and then ask queries about the family tree. Assume the intended interpretation of all predicates of the form p(x.y) is that "x is the p of y" George eMum r-Kydd Elizabeth Philip Margaret Diana-Ch...
##### [2] Steve has created a "grab the marble" game. If you grab a green marble you...
[2] Steve has created a "grab the marble" game. If you grab a green marble you get a dollar, if you grab a yellow marble you get nothing. Out of a total of 100 marbles, there are 31 green marbles, and 69 yellow ones. You decide to reach into the bag and grab a marble 39 times, replacing the ...
##### 0 <x<2 A. Find the fourier cosine and sine series representations of f (x) = 2, 2<*<3 where f has period Determine the values to which each series converges to atx-0,2,-2, Sketch the even and odd 2L periodic extensions of f for three periods.
0 <x<2 A. Find the fourier cosine and sine series representations of f (x) = 2, 2<*<3 where f has period Determine the values to which each series converges to atx-0,2,-2, Sketch the even and odd 2L periodic extensions of f for three periods....
##### DETEenije z6c : 2 Suck tmT e=-4
DETEenije z6c : 2 Suck tmT e=-4...
##### A radioactive isotope ""_Z^XA decay and emits one alpha particle and two beta particles consecutively. How to write the formula for the nuclides produced?
A radioactive isotope ""_Z^XA decay and emits one alpha particle and two beta particles consecutively. How to write the formula for the nuclides produced?...
##### 1 Sketch the root locus for the unity feedback systems that have an open-loop transfer function...
1 Sketch the root locus for the unity feedback systems that have an open-loop transfer function of: 6. G(S) = k(s? +1) (s - 1)(8 + 2)(8+3)...
##### Silver ion is being considered as a reagent for separating 1037 from CO3 in a solution...
Silver ion is being considered as a reagent for separating 1037 from CO3 in a solution that is 0.060 M in K2CO3 and 0.070 M in Nalo. Which anion will precipitate first when silver ion is added to the solution and what will its concentration be when the second anion begins to precipitate? Can the two...
##### Evaluate the integral. $\int \frac{x e^{2 x}}{(1+2 x)^{2}} d x$
Evaluate the integral. $\int \frac{x e^{2 x}}{(1+2 x)^{2}} d x$...
##### (b) Let y In(r3 tan-1 (e")) to find y Put a box around
(b) Let y In(r3 tan-1 (e")) to find y Put a box around...
##### Please read case above and answer the question , thank you . what organizational processes does c...
Please read case above and answer the question , thank you . what organizational processes does compensation affect and what is costco's rationale for having an exceptional compensation plan ? Case 11-1 Discounting Everything but Compensation at Costco Costco has gone head to head with the likes...
##### Solve the linear system by elimination:T + 42 + 213 = ~ 212 +343 = 301 Tx +43 = 102x1 +212 + 213 ~2n + S42 + 213 8x + 42+4,=-12b + 3c =3a + 6b _ 3c = -2 6a + 6b + 3c =T - V+2 _ W =- 2r + > - 2- _ Zw = -2 =+2-4+ W = 31" 3w =-
Solve the linear system by elimination: T + 42 + 213 = ~ 212 +343 = 301 Tx +43 = 10 2x1 +212 + 213 ~2n + S42 + 213 8x + 42+4,=-1 2b + 3c = 3a + 6b _ 3c = -2 6a + 6b + 3c = T - V+2 _ W =- 2r + > - 2- _ Zw = -2 =+2-4+ W = 31" 3w =-...
##### Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation. y' + (x+6)y =0
Find the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation. y' + (x+6)y =0...
##### @ 0 Then land Given] M 2 U Itwolsets
@ 0 Then land Given] M 2 U Itwolsets...
##### ProbLed_ indicate how they can be distinguished by the carbonyl stretching For cach pair of structurcs, frequcncy;COOHHCOOHGive the structure of the molecule that is consistent with cach set of = spectral data gven below. CHO,2980,2860, 1712 cm ' CHO, 3070,2810,2704,1705, 1615 cm ' CHOCL, 2975,2864, 1785 cm ' & CHOz, 3070,2983,2846, 1761, 1640 cm ' Place the following types of non-conjugated carbonyls into ascending order of frequency: aldehyde; acid anhydride; ester; ca
ProbLed_ indicate how they can be distinguished by the carbonyl stretching For cach pair of structurcs, frequcncy; COOH HCOOH Give the structure of the molecule that is consistent with cach set of = spectral data gven below. CHO,2980,2860, 1712 cm ' CHO, 3070,2810,2704,1705, 1615 cm ' CHOC...
QUESTION 2 1 points How much is over- or underapplied overhead using the following information: Estimated Actual Total overhead cost $100,000$110,250 Machine hours $20,000 21.000$10.250 overapplied 5 $10.250 underapplied$5,250 overapplied d. \$5,250 underapplied None of the above QUESTION 3 1 poi...


text:
# Revision history [back]

### Finding roots of a polynomial over an extension field

Apologies if I am not articulating this very well. I am still in an early learning phase.

I am trying to find the roots of a polynomial s^37 + 7 over a finite field of size 241 and modulus x^32 + 44.

My code is as follows:

K.<x> = GF(241)[]

T.<a> = K.quotient(x^32 + 44)

J.<s> = T[]

G = s^32 + 7

G.roots()

and here I get : 'root finding with multiplicities for this polynomial not implemented (try the multiplicities=False option)'

I think I expressed this correctly as some sanity-checking with known roots is correctly evaluated as follows:

rootList = [20a^29, 51a^13, 221*a^13]

for r in rootList: assert(G(r) == 0)

It does not appear root finding is implemented for G.

Is there some alternative way to find the roots of G?

Thank you.


text:
7.4 Power  (Page 2/8)

 Page 2 / 8

Check Your Understanding Estimate the power expended by a weightlifter raising a 150-kg barbell 2 m in 3 s.

980 W

The power involved in moving a body can also be expressed in terms of the forces acting on it. If a force $\stackrel{\to }{F}$ acts on a body that is displaced $d\stackrel{\to }{r}$ in a time dt , the power expended by the force is

$P=\frac{dW}{dt}=\frac{\stackrel{\to }{F}·d\stackrel{\to }{r}}{dt}=\stackrel{\to }{F}·\left(\frac{d\stackrel{\to }{r}}{dt}\right)=\stackrel{\to }{F}·\stackrel{\to }{v},$

where $\stackrel{\to }{v}$ is the velocity of the body. The fact that the limits implied by the derivatives exist, for the motion of a real body, justifies the rearrangement of the infinitesimals.

Automotive power driving uphill

How much power must an automobile engine expend to move a 1200-kg car up a 15% grade at 90 km/h ( [link] )? Assume that 25% of this power is dissipated overcoming air resistance and friction.

Strategy

At constant velocity, there is no change in kinetic energy, so the net work done to move the car is zero. Therefore the power supplied by the engine to move the car equals the power expended against gravity and air resistance. By assumption, 75% of the power is supplied against gravity, which equals $m\stackrel{\to }{g}·\stackrel{\to }{v}=mgv\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ where $\theta$ is the angle of the incline. A 15% grade means $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =0.15.$ This reasoning allows us to solve for the power required.

Solution

Carrying out the suggested steps, we find

$0.75\phantom{\rule{0.2em}{0ex}}P=mgv\phantom{\rule{0.2em}{0ex}}\text{sin}\left({\text{tan}}^{-1}\phantom{\rule{0.2em}{0ex}}0.15\right),$

or

$P=\frac{\left(1200\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}9.8\phantom{\rule{0.2em}{0ex}}\text{N}\right)\left(90\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}3.6\phantom{\rule{0.2em}{0ex}}\text{s}\right)\text{sin}\left(8.53\text{°}\right)}{0.75}=58\phantom{\rule{0.2em}{0ex}}\text{kW,}$

or about 78 hp. (You should supply the steps used to convert units.)

Significance

This is a reasonable amount of power for the engine of a small to mid-size car to supply $\left(1\phantom{\rule{0.2em}{0ex}}\text{hp}=0.746\phantom{\rule{0.2em}{0ex}}\text{kW}\text{).}$ Note that this is only the power expended to move the car. Much of the engine’s power goes elsewhere, for example, into waste heat. That’s why cars need radiators. Any remaining power could be used for acceleration, or to operate the car’s accessories.

Summary

• Power is the rate of doing work; that is, the derivative of work with respect to time.
• Alternatively, the work done, during a time interval, is the integral of the power supplied over the time interval.
• The power delivered by a force, acting on a moving particle, is the dot product of the force and the particle’s velocity.

Key equations

 Work done by a force over an infinitesimal displacement $dW=\stackrel{\to }{F}·d\stackrel{\to }{r}=|\stackrel{\to }{F}||d\stackrel{\to }{r}|\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$ Work done by a force acting along a path from A to B ${W}_{AB}=\underset{\text{path}AB}{\int }\stackrel{\to }{F}·d\stackrel{\to }{r}$ Work done by a constant force of kinetic friction ${W}_{\text{fr}}=\text{−}{f}_{k}|{l}_{AB}|$ Work done going from A to B by Earth’s gravity, near its surface ${W}_{\text{grav,}AB}=\text{−}mg\left({y}_{B}-{y}_{A}\right)$ Work done going from A to B by one-dimensional spring force ${W}_{\text{spring,}AB}=\text{−}\left(\frac{1}{2}k\right)\left({x}_{B}^{2}-{x}_{A}^{2}\right)$ Kinetic energy of a non-relativistic particle $K=\frac{1}{2}m{v}^{2}=\frac{{p}^{2}}{2m}$ Work-energy theorem ${W}_{\text{net}}={K}_{B}-{K}_{A}$ Power as rate of doing work $P=\frac{dW}{dt}$ Power as the dot product of force and velocity $P=\stackrel{\to }{F}·\stackrel{\to }{v}$

Conceptual questions

Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, it is a zero-watt device.) Explain in terms of the definition of power.

Appliances are rated in terms of the energy consumed in a relatively small time interval. It does not matter how long the appliance is on, only the rate of change of energy per unit time.

lists 5 drawing instruments and their uses
that is a question you can find on Google, anyway of top of my head, compass, ruler, protractor, try square, triangles.
Rongfang
A force F is needed to break a copper wire having radius R. The force needed to break a copper wire of radius 2R will be
2F
Jacob
The difference between vector and scaler quantity
vector has both magnitude & direction but scalar has only magnitude
Manash
my marunong ba dto mag prove ng geometry
ron
how do I find resultant of four forces at a point
Inusah
use the socatoa rule
kingsley
draw force diagram, then work out the direction of force.
Rongfang
In a closed system of forces... Summation of forces in any direction or plane is zero... Resolve if there is a need to then add forces in a particular plane or direction.. Say the x direction... Equate it tk zero
Jacob
define moment of inertia
what is Euler s theorem
what is thermocouple?
joining of two wire of different material forming two junctions. If one is hot and another is cold the it will produce emf...
joining of two metal of different materials to form a junction in one is hot & another is cold
Manash
define dimensional analysis
mathematical derivation?
Hira
explain what Newtonian mechanics is.
a system of mechanics based of Newton laws motion this is easy difenation of mean...
Arzoodan
what is the meaning of single term,mechanics?
jyotirmayee
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
Lalita
SO ASK What is Newtonian mechanics in physics? Newtonian physics, also calledNewtonian or classical mechanics, is the description of mechanical events—those that involve forces acting on matter—using the laws of motion and gravitation formulated in the late seventeenth century by English physicist
Suleiman
can any one send me the best reference book for physics?
Prema
concept of physics by HC verma, Fundamentals of Physics, university of physics
tq u.
Prema
these are the best physics books one can fond both theory and applications.
can any one suggest best book for maths with lot of Tricks?
Vivek
what is the water height in barometer?
SUNEELL
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
LOVE
water is 13.5 times dense than the Mercury
LOVE
plz tell me frnds the best reference book for physics along with the names of authors.
Prema
i recomended the reference book for physics from library University of Dublin or library Trinity college
Arzoodan
A little help here... . 1. Newton's laws of Motion, are they applicable to motions of all speeds? 2.state the speeds which are applicable to Newtons laws of Motion
Derek
mechanics which follows Newtons law
Manash
The definition of axial and polar vector .
Arpita
polar vector which have a starting point or pt. of applications is,force,displacement
jyotirmayee
axial vector represent rotational effect and act along the axis of rotation b
jyotirmayee
explain the rule of free body diagram
The polar coordinates of a point are 4π/3 and 5.50m. What are its Cartesian coordinates?
application of elasticity
good
Anwar
a boy move with a velocity of 5m/s in 4s. What is the distance covered by the boy?
What is the time required for the sun to reach the earth?
anthony
24th hr's, your question is amazing joke 😂
Arzoodan
velocity 20 m, s
Ahmed
the sun shines always and the earth rotates so the question should specify a place on earth and that will be 24hrs
Opoku
20m
Gabriel
good nice work
Anwar
20m
Evelyn
why 20?.
Arzoodan
v =distance/time so make distance the subject from the equation
Evelyn
20m
Olaide
exatly
Arzoodan
what is differemce between principles and laws
plz
Anwar
how can a 50W light bulb use more energy than a 1000W oven?
That depends on how much time we use them
Phrangsngi
Define vector law of addition
It states that, " If two vectors are represented in magnitude and direction by the two sides of a triangle, then their resultant is represented in magnitude and direction by the third side of the triangl " .
Nabin
thanks yaar
Pawan
And it's formula
Pawan
Manash


text:
# [Xy-pic] Fw: Xy-pic and Qcircuit

Thu Mar 10 01:40:34 CET 2011

```Hi all,

I am Daniel Müllner, the developer of Xy-pic's recent PDF driver. At
this stage, it is not clear to me which of these errors are caused by
the PDF driver and which by the the other extensions or the core Xy-pic.

To facilitate my part, Bryan and Steve, could you maybe isolate the
errors that are caused by the PDF driver. Those are:

1) Errors that occur when you compile the document with pdfLaTeX but
don't occur with LaTeX.

or

2) Errors that vanish when you write \xypdfoff in the document preamble.

As Kris mentioned, a minimal TeX file (as short as possible but with a
preamble so that I can run it through LaTeX and reproduce the errors)


text:
# This isn't a circle

Algebra Level pending

Find the area under the graph of $$\left\lfloor { x }^{ 2 } \right\rfloor +\left\lceil { y }^{ 2 } \right\rceil =1$$ from -1 to 1. over the x axis

×


text:
Difference between true & pseudo random numbers

I recently heard that in Linux, /dev/random and /dev/urandom are pseudo random number generators but not true random number generator.

So I am curious to know the difference between pseudo & true random number? In other words What is the difference between pseudo-randomness & true-randomness?

• This could largely be answered by a Google search. Do you have a more specific question about this related to computational science? – horchler Mar 25 '16 at 15:07
• @horchler If this question is more suitable on other (SE) site (example-Unix & Linux) then tell/suggest me – Pandya Mar 29 '16 at 5:09
• Wikipedia article Pseudorandom number generator – Steve May 26 '16 at 10:51


text:
# semimartingale topology

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}}),\mathbb{P})$ be a filtered probability space and $(X^{n}_{t})$, $(X_{t})$ be cadlag adapted processes. Then, $X^{n}$ is said to converge to $X$ in the semimartingale topology if $X^{n}_{0}\rightarrow X_{0}$ in probability and

 $\int_{0}^{t}\xi^{n}\,dX^{n}-\int_{0}^{t}\xi^{n}\,dX\rightarrow 0$

in probability as $n\rightarrow\infty$, for every $t>0$ and sequence of simple predictable processes $|\xi^{n}|\leq 1$.

This topology occurs with stochastic calculus where, according to the dominated convergence theorem (http://planetmath.org/DominatedConvergenceForStochasticIntegration), stochastic integrals converge in the semimartingale topology. Furthermore, stochastic integration with respect to any locally bounded (http://planetmath.org/LocalPropertiesOfProcesses) predictable process $\xi$ is continuous under the semimartingale topology. That is, if $X^{n}$ are semimartingales converging to $X$ then $\int\xi\,dX^{n}$ converges to $\int\xi\,dX$, a fact which does not hold under weaker topologies such as ucp convergence.

Also, for cadlag martingales, $L^{1}$ convergence implies semimartingale convergence.

It can be shown that semimartingale convergence implies ucp convergence. Consequently, $X^{n}$ converges to $X$ in the semimartingale topology if and only if

 $X^{n}_{0}-X_{0}+\int\xi^{n}\,dX^{n}-\int\xi^{n}\,dX\xrightarrow{\rm ucp}0$

for all sequences of simple predictable processes $|\xi^{n}|\leq 1$.

The topology is described by a metric as follows. First, let $D^{\rm ucp}(X-Y)$ be a metric defining the ucp topology. For example,

 $D^{\rm ucp}(X)=\sum_{n=1}^{\infty}2^{-n}\mathbb{E}\left[\min\left(1,\sup_{t

Then, a metric $D^{\rm s}(X-Y)$ for semimartingale convergence is given by

 $D^{\rm s}(X)=\sup\left\{D^{\rm ucp}(X_{0}+\xi\cdot X):|\xi|\leq 1\textrm{ is % simple previsible}\right\}$

($\xi\cdot X$ denotes the integral $\int\xi\,dX$). This is a proper metric under identification of processes with almost surely equivalent sample paths, otherwise it is a pseudometric.

If $\lambda_{n}\not=0$ is a sequence of real numbers converging to zero and $X$ is a cadlag adapted process then $\lambda_{n}X\rightarrow 0$ in the semimartingale topology if and only if

 $\lambda_{n}\int_{0}^{t}\xi^{n}\,dX\rightarrow 0$

in probability, for every $t>0$ and simple predictable processes $|\xi^{n}|\leq 1$. By the sequential characterization of boundedness (http://planetmath.org/SequentialCharacterizationOfBoundedness), this is equivalent to the statement that

 $\left\{\int_{0}^{t}\xi\,dX:|\xi|\leq 1\textrm{ is simple predictable}\right\}$

is bounded in probability for every $t>0$. So, $\lambda_{n}X\rightarrow 0$ in the semimartingale topology if and only if $X$ is a semimartingale. It follows that semimartingale convergence only becomes a vector topology (http://planetmath.org/TopologicalVectorSpace) when restricted to the space of semimartingales. Then, it can be shown that the set of semimartingales is a complete topological vector space (http://planetmath.org/CompletenessOfSemimartingaleConvergence).

Title semimartingale topology SemimartingaleTopology 2013-03-22 18:40:41 2013-03-22 18:40:41 gel (22282) gel (22282) 6 gel (22282) Definition msc 60G48 msc 60G07 msc 60H05 semimartingale convergence UcpConvergence UcpConvergenceOfProcesses


text:
# Decision trees for survival analysis

November 2014

Survival analysis is an interesting problem in machine learning, but it doesn’t get nearly as much attention as the usual classification and regression tasks, so there aren’t as many tools for it. Here I describe a nifty reduction that allows us to bring more traditional machine-learning tools to bear on the problem. Combined with the view of decision trees as greedy piecewise-constant loss-minimizing classifiers, it enables a number of powerful and flexible algorithms for large-scale discrete survival analysis.

## Setup: discrete survival analysis

Discrete survival analysis is motivated by problems like disease prognosis. Suppose we’re trying to predict survival times from a rare diseas. We enroll a bunch of subjects who were just diagnosed, and we do check up on them monthly until they die. Then we try to predict the time until recurrence, $$Y$$, from data about the patient, $$X$$.

We could use standard regression algorithms, but there are a couple wrinkles:

• We only know the recurrence times up to the month, because that’s how often we check up on them, so $$Y$$ isn’t actually continuous. This is where the “discrete” part of the name comes from. (We could probably get the exact death date if we tried, but that’s too complex for us right now. Plus, it’ll make the analysis harder later, as we’ll see.)

• $$Y$$ is known to be greater than zero. Most regression algorithms don’t have a natural way to take this into account.

• Most importantly, we won’t be able to observe a recurrence for everyone. Some people will be lost to follow-up. Some people will be cured. Even some of those who die might outlast the study period. This leads to a problem called right-censored data, where for certain patients we know that, say, $$Y > 60$$ (months), but we don’t know the exact value of $$Y$$.

We could get arount the last one by discarding all data points that don’t experience a recurrence, but this would introduce severe selection bias. Plus, it would throw away a lot of info on healthy people. So instead we’ll do something more clever.

## The hazard function

If we were doing ordinary regression, where we had observed the $$Y$$ for each $$X$$, we would try to learn a function $$f(X) = E(Y|X)$$ (or sometimes some other property of the conditional distribution of $$Y$$, like the median). But we don’t have enough info about $$Y$$ for that to work.

So instead of thinking in terms of $$E(Y|X)$$, we’ll instead think about something more granular called the hazard function. The hazard function is $$h(X, t) = P(Y=t | Y \ge t, X),$$ –the probability that, if someone has survived up until month $$t$$, they will die in that month.

In fact, learning $$h$$ will tell us more about the disease than learning $$f$$, because it will give us not only the expected survival time but the entire distribution of survival times! You can verify for yourself (it’s not hard) that $$P(Y = t|X) = h(X, t)\prod_{i=1}^{t-1} (1 - h(X, i)),$$ so we can recover the entire probability mass function.

## A loss function

So we want to learn some approximation $$\hat h(X, t) \approx h(X, t)$$. But how do we decide what a good approximation is? The obvious thing to try is maximizing the expected log-likelihood.

The other wrinkle is that we don’t actually see $$Y$$. Instead we see some related variables–how long each subject was in the study ($$T$$), and whether they exited by dying or being censored ($$D$$). ($$D$$ is a Boolean variable, so $$D=1$$ if $$T=Y$$, and $$D=0$$ if $$T < Y$$.)

We could try to evaluate the log-likelihood of the exact data we see–that is, $$P(T, D|X)$$. But that would require us to build a model not only of the time of death but also the time of censorship (because we’d need the likelihood of getting censored at time $$T$$). Usually the censorship distribution is a property of the data-collection mechanism, not the underlying reality–for instance, in clinical studies it’s a property of the follow-up time. So a prediction of it probably won’t generalize well to other settings.

For that reason, instead of evaluating the likelihood of the entire data point $$(T, D)$$, we pretend that censorship just exogenously removes some information about $$Y$$. So we’ll split our likelihood into two pieces: $$P(Y=T)$$ for uncensored data, and $$P(Y > T)$$ for censored data. Given this, we can write down the log-likelihood of seeing a particular datum if $$\hat h$$ were correct:

$$L(T, D|X, h = \hat h) = \log \left[ \prod_{i=1}^{T-1}(1 - \hat h(X, i)) \right]\left[\hat h(X, T)^D (1 - \hat h(X, T))^{1-D}\right]$$

where the first factor in the log is the probability that the patient survives until month $$T$$, and the second is $$\hat h(X, T) = P(Y=T|X)$$ if $$D=1$$ (the probability under $$\hat h$$ that the patient died, if we observed them to die at $$T$$) and $$(1 - \hat h(X, T))$$ if $$D=0$$ (the probability under $$\hat h$$ that the patient survived, if we observed them to survive at $$T$$, i.e., they exited by censoring).

Now that we know what the log-likelihood of a particular datum looks like, we can use our dataset to calculate the empirical expected log-likelihood of $$\hat h$$ by taking the average log-likelihood over each data point. With some simplification, this becomes:

$$L(\hat h) = \frac1N \sum_{i=1}^N \left[ D_i \log \hat h(X_i, T_i) + (1 - D_i) \log (1 - \hat h(X_i, T_i)) + \sum_{t=0}^{T_i-1} \log (1 - \hat h(X_i, T_i)) \right]$$

## The trick

Of course, finding the $$\hat h$$ that maximizes this log-likelihood loss function is still a challenge. But we can make one key observation, which is that if you treat $$T$$ as another covariate, $$\hat h$$ and $$L(\hat h)$$ have the exact same structure as the classifier function and loss, $$\hat f$$ and $$L(\hat f)$$, of a maximum-likelihood estimator for a classifier. In other words, this problem is just classification-in-disguise! Each datapoint in the hidden classification problem is the combination of an $$X_i$$ in our original dataset plus some month $$t$$, and the classification problem is “did point $$X_i$$ die in month $$t$$”. We’ll call this new variable $$D_{it}$$, and rewrite our loss function in terms of it:

$$L(\hat h) = \frac1N \sum_{i=1}^N \sum_{t=1}^{T_i} D_{it} \log \hat h(X_i, t) + (1 - D_{it}) \log (1 - \hat h(X_i, t))$$

It’s now obvious that this is proportional to the classifier likelihood function I wrote down earlier. This means that we can transform our original data set into a new one, with one row for each month that each $$X_i$$ is in the sample; train a standard maximum-likelihood classifier on this new dataset with $$D_{it}$$ as the target; and extract a perfectly good survival model for our original dataset!

## The punchline

This reduction will work with any classifier-training algorithm that tries to maximize the likelihood of the data. Combined with my earlier reframing of decision trees as greedy piecewise-constant maximum-likelihood approximations, this means that decision trees will work with it–and by extension, any algorithm that uses decision trees as a building block, like like random forests or gradient machines. But you could also use the same transformation with logistic regression or any other maximum-likelihood method.

Some caveats need to be applied, though. With bagging-based methods like random forests or subsampled gradient trees, the default behaviour will be to bag the transformed dataset, but you probably want to make sure that each bag either contains every month’s point for each $$X_i$$, or no points at all for that $$X_i$$. With logistic regression, you should make sure to include interaction terms between $$t$$ and the other covariates; otherwise the hazard function will be forced to have the same general shape for every datapoint.

Once those caveats are taken care of, though, this reduction is extremely helpful for developing survival models quickly. Survival analysis doesn’t get as much attention as standard classification or regression tasks, so it’s convenient that so many more frequently-used models can be essentially reused wholesale.

## Appendix: pseudocode for the reduction

def train(X, T, D)
// X, T, D are the original dataset
X' = []
D' = []

// the transformation
for each index i in X:
for t=1 to T[i]:
new_D = (0 if t < T[i], else D[i])
append new_D to D'
new_X = (X[i], t)
append new_X to X'

return a decision tree trained on (X', D')

def pmf(h, X)
// X is a single datapoint
// returns an array A where A[i] = P(Y = i | X)
A = []
p_so_far = 1 // this is p(T >= t | X)
for t = 1 to (the last month where h has any data):
// h knows p(T = t | T >= t, X), we call this p_cur
p_cur = h's prediction for (X, t)
append (p_so_far * p_cur) to A
p_so_far *= (1 - p_cur)

Enjoyed this post? Get notified of new ones via email or RSS. Or comment:

email me replies