text:
## Internet of Things (IoT) System and Field Sensors for Exercise Intensity Measurements

### Abstract

Although exercise training according to individual peak aerobic capacity ( o2peak) has been recommended at all ages, sensors available in the field are limited. The most popular sensors in the field are pedometers, but they cannot be used to monitor exercise intensity. Instead, although heart rate (HR) monitors are broadly available in the field to estimate exercise intensity, HR responses to exercise vary by individual according to physical fitness and environmental conditions, which hinders the precise measurement of energy expenditure. These issues make it difficult for exercise physiologists to collaborate with geneticists, nutritionists, and clinicians using the internet of things (IoT). To conquer these problems, we have developed a device that is equipped with a triaxial accelerometer and a barometer to measure energy expenditure during interval walking training (IWT) in the field with inclines. IWT is a training regimen to repeat fast and slow walking for 3 min each, equivalent to greater than 70% and approximately 40% of individual o2peak, respectively. Additionally, we developed an IoT system that enables users to receive instructions from trainers according to their walking records even if they live far away. Since the system is available at low cost with minimum personnel, we can investigate any factors affecting the adherence to and effects of IWT in a large population for a long period. This system was also used to verify any effects of nutritional supplements during IWT and to examine the value of applying IWT to clinical medicine. © 2020 American Physiological Society. Compr Physiol 10:1207‐1240, 2020.

 Figure 1. Our experimental strategy. We first conducted the walking program by instructing middle‐aged and older individuals to walk 10,000 steps per day. However, physical fitness did not increase after training, and adherence to the program was very low. On the other hand, standard exercise training tailored to an individual's fitness level is generally performed at a gymnasium using machines but is costly and of limited availability. To extend individualized training beyond the gymnasium for middle‐aged and older individuals, we developed a graded walking test to determine peak aerobic capacity ($V˙$o2peak) for walking using an accelerometer 125. Based on each subject's results, we then instructed them to perform interval walking training, which was remotely supervised via the internet. Our hypothesis is that an exercise training system composed of interval walking training and the IoT can increase physical fitness in older people. Reused, with permission, from Masuki S, et al., 2017 97. Figure 2. Comparisons between peak aerobic capacity ($V˙$o2peak) values determined by accelerometry (JD Mate) during the graded walking test (W$V˙$o2peak) and by respiratory gas analysis during the graded cycling test (C$V˙$o2peak) 125. Figure 3. Interval walking training and the e‐health link named “e‐Health Promotion System.” Participants visit a health care institution, a drug store, or a local community office every 2 weeks to transfer their walking records from the tracking device, JD Mate, to a central server computer via the internet. The server computer then provides them with a trend graph of their records along with advice automatically generated by the server. Based on this report, the staff, nurses, dietitians, or trainers give them individualized training advice. By anonymizing and combining the DNA data stored in a separate offline computer and the clinical data stored in the central server computer, we have begun to explore the genomic variations explaining inter‐individual variations in training response. The data from the research may be used to revise the e‐Health Promotion System to develop an algorithm to predict the effects of interval walking training on physical fitness and the indices of lifestyle‐related diseases in individuals with different physical and genetic characteristics. An e‐Key is used to limit a person's access to the database (DB). Squares on the internet circle indicate the firewall function. Solid arrows indicate online communications between users and the server via the internet, and dotted arrows indicate offline communications. Reused, with permission, from Nose H, et al., 2009 130. Figure 4. Percentage changes in isometric knee extension (FEXT) and flexion forces (FFLX) (A) and peak aerobic capacity by the graded walking test (W$V˙$o2peak) (B) after training in three groups: no‐walking training (no‐WT; males = 9, females = 37, total = 46); moderate‐intensity continuous walking (WCNT; males = 8, females = 43, total = 51); and high‐intensity interval walking (WINT; males = 11, females = 31, total = 42). *P < 0.05, **P < 0.01, ***P < 0.001: significant differences from the pretraining values. ††P < 0.01, †††P < 0.001: significant differences from the corresponding values in no‐WT. #P < 0.05. ##P < 0.01, ###P < 0.001: significant differences from the corresponding values in WCNT. Reused, with permission, from Nemoto K, et al., 2007 125. Figure 5. Changes in systolic (SBP) and diastolic blood pressures (DBP) at rest after training. †Significant differences from the corresponding values in no‐WT at the level of P < 0.05. The number of subjects and other symbols are the same as in Figure 4. Reused, with permission, from Nemoto K, et al., 2007 125. Figure 6. Total lifestyle‐related disease (LSD) score and peak aerobic capacity for walking ($V˙$o2peak) before and after interval walking training in males (A) and females (B). When the subjects were divided equally into three groups according to $V˙$o2peak, the score was lower in higher $V˙$o2peak groups. After interval walking training for 4 months, the score decreased as $V˙$o2peak increased in every group. *Versus before training at P < 0.05, #versus Middle group at P < 0.05, †versus High group at P < 0.05. Reused, with permission, from Morikawa M, et al., 2011 114. Figure 7. Relationship between total lifestyle‐related disease (LSD) score and peak aerobic capacity ($V˙$o2peak) in the low (circle), middle (square), and high $V˙$o2peak groups (triangle). The mean and SE bars for 66 males and for 156 females in each group are presented. The open and closed symbols indicate the values before and after interval walking (IWT), respectively, and data pairs are shown with arrows. Total LSD score (y) was significantly correlated with $V˙$o2peak (x) in males (R2 = 0.71, P < 0.001) and females (R2 = 0.94, P < 0.001) according to the mean values for 66 males and 156 females in each group before and after IWT. *Versus before training at P < 0.05, #versus middle group at P < 0.05, †versus high group at P < 0.05, before and after training, respectively. Reused, with permission, from Morikawa M, et al., 2011 114. Figure 8. Fast, slow, and total walking time over 5 months versus changes in peak aerobic capacity for walking (Δ$V˙$o2peak, A) and in lifestyle‐related disease score (ΔLSD score, B) after training in 679 middle‐aged and older subjects. Values are means ± SE. The number of subjects that were included in each bin of training achievements was also presented in panel (A). *Significant differences from pretraining values, P < 0.05. Reused, with permission, from Masuki S, et al., 2019 98. Figure 9. One representative gene, NFKB2, influenced by the IWT. (A) The genome‐wide methylation microarray assay was performed with microarray slides containing individual oligonucleotide probes (1–17). The ratio of methylation levels post‐/pre‐IWT is shown at sites corresponding to each probe. Open and closed bars indicate control and exercise groups, respectively. (B) The location of 17 probes used in a genome‐wide microarray assay to evaluate the methylation level of the NFKB2 promoter. (C‐E) Regression analysis of methylation status of the NFKB2 gene promoter region versus total energy consumption divided by body weight. The vertical axis shows the ratio value of the post‐IWT methylation/pre‐IWT methylation in the promoter region for probes 9 (C), 10 (D), and 12 (E). The correlation between methylation and energy consumption was analyzed using the Spearman nonparametric test. TSS, transcription start site; ○, nonexercise group; Δ, exercise group. Reused, with permission, from Zhang Y, et al., 2015 195. Figure 10. Subjects with type 2 diabetes were randomized to a control group (CON, white bars), continuous walking training group (CWT, striated bars), or interval walking training group (IWT, black bars). Intervention‐induced changes in glycemic control were assessed by examining post–preintervention changes in the following variables: mean 48‐h continuous glucose monitoring (CGM) glucose (A), minimum 48‐h CGM glucose (B), and maximum 48‐h CGM glucose (C). Data are presented as mean Δ values (post–preintervention values) ± SE. Significant differences from the pre‐intervention values, *P < 0.05 and **P < 0.01; significant differences between groups (a connecting line between bars), **P < 0.01. Reused, with permission, from Karstoft K, et al., 2013 71. Figure 11. Adherence to prescribed walking days (APWD, A) and fast walking time (APFWT, B) over the 22‐month training period versus change in lifestyle‐related disease score from baseline to 22 months (ΔLSD score). First, we calculated the LSD scores based on the Japanese and US healthcare guidelines 96. We assigned one point when a value met any of the following four criteria: (i) BMI ≥25 kg/m2; (ii) systolic blood pressure ≥130 mmHg or diastolic blood pressure ≥85 mmHg; (iii) triglycerides ≥150 mg/dL, high‐density lipoprotein cholesterol less than 40 mg/dL or low‐density lipoprotein cholesterol ≥130 mg/dL; and (iv) blood glucose ≥110 mg/dL. The maximum total score was 4 points. Subjects were then pooled according to their ΔLSD score from baseline to 22 months as follows: ≥1 (men = 30; women = 74), 0 (men = 93; women = 270), −1 (men = 50; women = 128), −2 (men = 19; women = 25), and ≤ −3 (men = 4; women = 3). Significant differences in adherence from those with the worst ΔLSD score (≥1), *P < 0.05 and **P < 0.01. APWD was calculated as the number of walking days completed divided by the total number of walking days prescribed for each month (4 days/week), and then these monthly ratios were averaged over 22 months. APFWT was calculated as the fast walking time completed divided by the total fast walking time prescribed for each month (60 min/week), but time in excess of 60 min/week was regarded as 100%, and then these monthly ratios were averaged over 22 months. Reused, with permission, from Masuki S, et al., 2015 96. Figure 12. Adherence to prescribed walking days (APWD, A) and fast walking time (APFWT, B) over the 22‐month training period versus change in peak aerobic capacity for walking from baseline to 22 months (Δ$V˙$o2peak). Subjects were pooled according to their Δ$V˙$o2peak: ≤−2.5 (men = 8; women = 39), −2.4 to 0 (men = 46; women = 116), 0.1 to 2.5 (men = 52; women = 111), 2.6 to 5.0 (men = 52; women = 135), and greater than 5.0 mL/kg/min (men = 38; women = 99). Significant differences in adherence from those with the lowest Δ$V˙$o2peak (≤−2.5 mL/kg/min), **P < 0.01 and ***P < 0.001. APWD and APFWT determinations were the same as those described in Figure 11. Reused, with permission, from Masuki S, et al., 2015 96. Figure 13. Monthly adherence rate over the 22‐month training period in men according to the presence of the 334 allele of RS3 and C/T alleles of rs1042615 after adjustment for possible covariates. RS3 and rs1042615 denote a microsatellite and single‐nucleotide polymorphism in AVPR1A, respectively. Group 1, zero 334 alleles and CC or CT genotype (n = 89); group 2, one or two 334 alleles and CC/CT (n = 47); group 3, zero 334 alleles and TT (n = 31); and group 4, one or two 334 alleles and TT (n = 29). Adherence rate was calculated as the number of walking days completed divided by the total number of walking days prescribed for the month (4 days/week) and then adjusted for baseline BMI and smoking status using multiple regression analysis. §Group 1 to 3 versus 4. Reused, with permission, from Masuki S, et al., 2015 96. Figure 14. Percentage changes after training in hamstring muscle tissue area and isometric knee flexion force. Data are presented as the mean ± SE. CNT, interval walking training group; NUT, interval walking training plus macronutrient supplement group. *P < 0.05. Reused, with permission, from Okazaki K, et al., 2013 136. Figure 15. Changes in plasma volume (ΔPV′, A), plasma albumin content (ΔAlbcont′, B), and hemoglobin A1c (ΔHbA1c′, C) from baselines before starting additional 5‐month IWT. These values are corrected by ANCOVA with baseline plasma glucose concentration as a covariate. The means and SE bars are presented for 12 and 14 subjects in the CHO and the carbohydrate + whey protein (Pro‐CHO) groups, respectively. ##Versus baseline at P < 0.01. *Versus the CHO group at P < 0.05. Reused, with permission, from Uchida K, et al., 2018 186. Figure 16. Percentage changes after training in muscle strength (A) and methylation of the NFKB1 and NFKB2 promoter regions as assessed by pyrosequencing (B). The data were adjusted for pretraining values by ANCOVA. The mean and SE bars are presented for 12, 12, and 13 subjects in the IWT control (CNT), IWT + low‐dose (LD) and IWT + high‐dose milk product intake (HD) groups, respectively. **Significant differences from pretraining values, P < 0.01. Significant differences from the CNT group, †P < 0.05 and †††P < 0.001. Significant differences from the LD group, ‡P < 0.05 and ‡‡P < 0.01. (A) Average percent changes in isometric knee extension (ΔFEXT) and flexion (ΔFFLX) forces are presented. (B) Average percentage changes in CpG sites 1 to 7 for NFKB1 (upper) and average percentage changes in CpG sites 1 to 6 for NFKB2 (lower) are presented. Reused, with permission, from Masuki S, et al., 2017 100. Figure 17. Oxygen consumption rate ($V˙$o2), carbon dioxide production rate ($V˙$co2), ventilation volume ($V˙$E) (A), and plasma lactate concentration ([Lac−]p) (B) responses during graded cycling exercise under placebo (PLC; left) and ALA + SFC supplement intake conditions (right). The average value per minute is presented from rest to the highest workload of 75 W, at which all subjects could maintain the rhythm. Open symbols, before supplement intake; solid symbols, after supplement intake. Values are means ± SE of 10 subjects. *P < 0.05, **P < 0.01, and ***P < 0.001 versus before supplement intake. Reused, with permission, from Masuki S, et al., 2016 99. Figure 18. Training days (A), training impulse (B), and training time (C) at total, fast, and slow walking during the supplement intake period. Values are means ± SE of nine subjects. *P < 0.05 and ***P < 0.001 between the PLC and ALA + SFC trials. Reused, with permission, from Masuki S, et al., 2016 99. Figure 19. Montgomery‐Åsberg Depression Rating Scale (MADRS) sum score in the PLC trial (left) and the ALA + SFC trial (right) before and after a supplement intake period. Values are the means ± SE for nine subjects. **P < 0.01 versus before the supplement intake period. Reused, with permission, from Suzuki H, et al., 2018 175. Figure 20. Relationship between the oxygen consumption rate ($V˙$o2, mL/kg/min) and the vector magnitude of triaxial accelerations (VM) during graded walking tests on land and in a pool (water) before IWT. The values of 31 subjects from the WG and the LG in the land trial and the values of 16 subjects in the water trial from the WG for the last 30 s of the rest, slow, moderate, and fastest walking periods (3 min each) were pooled. Regression lines (solid lines) and area for 95% confidence limits (dotted lines) are shown. Reused, with permission, from Handa S, et al., 2016 51. Figure 21. Fast‐walking exercise intensity measured on the last walking day of every week during training. The means and SE bars include the results of 16 subjects in the WG and 15 subjects in the LG. Significant differences in $V˙$o2 between the WG and the LG, †P < 0.05 and ††P < 0.01; ##significant interactive effect of [training weeks × groups] on $V˙$o2 for the fast walking at P < 0.01. Reused, with permission, from Handa S, et al., 2016 51. Figure 22. Isometric knee extension (FEXT) (A) and flexion (FFLX) (B) forces before and after training, presented as the means and SE bars including the results from 16 and 15 subjects in the WG and the LG, respectively. Significant differences versus the pretraining values for each group, ***P < 0.001; significant interactive effects of [before and after training × groups] on FEXT and FFLX, #P < 0.05 and ##P < 0.01, respectively. Reused, with permission, from Handa S, et al., 2016 51. Figure 23. The relationship between the hourly frequency of starting IWT versus atmospheric temperature (Ta) above 25 °C during summer vacation. The hourly frequency was negatively correlated with Ta (r = −0.81, P = 0.008). Each symbol indicates the mean value for 20 subjects. CD, critical difference in Ta is shown on the x‐axis, and hourly frequency of starting IWT shown is on the y‐axis at the level of P < 0.05 using Fisher's LSD test. Reused, with permission, from Tanabe A, et al., 2018 179. Figure 24. An example of the contents of a smartphone application to motivate participants to continue IWT. (1) Recognition of progress: Based on individual walking and health records returned to each individual by the server computer, participants can recognize their achievements and the effects of training, which encourages them to continue IWT with the confidence that their efforts are being rewarded. (2) Comparison to others: If participants recognize an individual's training achievement in the up‐to‐date histograms made from all participants' data, then this knowledge stimulates their competitive spirit. (3) Community promotion: If participants see an up‐to‐date ranking of the training achievements in their community, then they may encourage colleagues in their own community to elevate the ranking.


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# Math in MMC 4

#### mariano.iannuzzi

27 Oct, 2017 02:11 AM

Hello,

I installed MMC today on macOS High Sierra. Even after I enabled the MathJax CDN, mathematical expression do not render in the preview as you can see in the attached screen capture.

Is this a known issue or am I doing something wrong?

Thank you very much

Mariano

1. Support Staff Posted by Fletcher on 27 Oct, 2017 03:22 PM

It works fine for me on 10.12 and 10.13.

Have you tried refreshing the preview to reload MathJax?

2. Posted by mariano.iannuzz... on 28 Oct, 2017 02:26 AM

Indeed. Also, closing the app and reopening it. I also played with the different Mathjax notations without success.

Marked 2 displays the same text just fine.

Thanks for the response.

Cheers

Sent from my iPhone

3. Support Staff Posted by Fletcher on 28 Oct, 2017 02:12 PM

Try rebooting to refresh the OS' web cache to ensure that MathJax is being loaded.

4. Support Staff Posted by Fletcher on 05 Nov, 2017 07:29 PM

F-

5. Posted by mariano.iannuzz... on 06 Nov, 2017 01:31 AM

Hello,

I apologize, I forgot to respond. It is working now. I believe the problem was not related to MathJax. I gave the equation line an extra space and then it rendered properly (see the two screen captures before and after). Would you know why would this be?

Thank you very much for following up.

Cheers

Mariano

6. Support Staff Posted by Fletcher on 06 Nov, 2017 02:32 AM

The second image is not math -- only has a single slash before each bracket. ( \[ instead of \\[).

7. Fletcher closed this discussion on 06 Nov, 2017 02:32 AM.

8. mariano.iannuzzi re-opened this discussion on 06 Nov, 2017 03:10 AM

9. Posted by mariano.iannuzz... on 06 Nov, 2017 03:10 AM

sorry, wrong one.

10. Support Staff Posted by Fletcher on 06 Nov, 2017 03:22 AM

Works for me (unless I typed it wrong). If you send me source text, I can take a look.

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This is the review of the paper “Knowledge and Common Knowledge in a Distributed Environment” by Joseph Halpern and Yoram Moses, who won the 2009 Dijkstra Prize in Distributed Computing for their research of knowledge in the multi-agent setting.

Designing distributed systems is hard. Hard, because the majority of specifications imposes global constraints on the system, but which independent agents must co-operate to satisfy using only local information. Hard, because communication is the modus operandi of distributed systems: often all that the agents can do is to send or receive messages.

The inability to gauge the global state of the system, synchronization problems and faulty communication channels are just a few of the challenges that any developer meets in practice. Dealing with them requires the ability to track the diffusion of knowledge among the nodes and differentiate between states which appear similar on the first sight.

Bounding this uncertainty requires understanding how system controls affect its behavior, and quantifying the feedback from groups of agents emerging in the process of solving a global problem. Halpern and Moses formalize the change in the knowledge state and progression up the knowledge hierarchy, building a solid ground for the design and analysis of distributed protocols.

The authors provide two environments where knowledge management proves to be crucial for understanding the solution or its impossibility: the muddy children puzzle and the coordinated attack problem.

The muddy children puzzle

Distributed protocols often involve agents with some information about other nodes in the system. The following vignette captures this common layout:

A group of $$n$$ children are playing together. During play, $$k$$ of them, where $$k > 1$$, get mud on their foreheads, but all stay silent as one of the parents approaches and says: "At least one of you has mud on your face". The parent then continues to ask the following question again and again: "Can any of you prove that you have mud on your forehead?". Assuming that all children understand the situation, have strong deductive abilities and tell the truth simultaneously, what happens next?

You might want to think about the problem on your own. The authors give the solution: all the kids say no $$k - 1$$ times, but on the $$k$$th time all the dirty children say yes.

Before the kids start answering, each one of them already knows the truth of what the parent is going to say. Nevertheless, repeating the same question multiple times still manages to provide useful information. Halpern and Moses proceed to lay out a logical framework and define a knowledge hierarchy which enables rigorous reasoning about knowledge in distributed systems.

Building the knowledge hierarchy

Shedding light on the muddy children puzzle requires us to consider the collective knowledge of independent agents. Here we use two properties of knowledge as formalized in the framework:

• what an agent knows depends only on its local history: the initial information and any events observed since the beginning
• an agent knows only the truth

Then we are ready to classify the types of group knowledge:

• Unknown facts: none of the group members know the fact
• Distributed knowledge: someone learning everything that each member of the group knows would know the fact
• Someone knows: some members of the group know the fact
• Everyone knows: every member of the group knows the fact
• Everyone knows $$k$$-recursively: every member knows (that every member knows)$$^{k-1}$$ the fact
• Common knowledge: everyone knows $$k$$-recursively for all $$k \geq 1$$

Despite the clear hierarchical nature of the classification, these notions might coincide with each other, as in the case of a distributed system consisting of parallel processes with the common memory. On the other hand, if the processors are instead connected via some communication channel and each one of them has its own dedicated memory, then the distinction between classes is strict in the sense that there exists a task which requires an action such that one level of the hierarchy would suffice, but no other lower level would. In the case of the muddy children puzzle, everyone knows $$k$$-recursively is enough to obtain confessions from all of the kids with dirty faces, but everyone knows $$(k-1)$$-recursively is not. Indeed, the parent's phrase guarantees that the children have common knowledge of the fact and that each child knows it.

This is a very common situation in distributed systems: communication helps the agents climb up the knowledge hierarchy. Two concepts in this view are of utmost importance: fact discovery, which promotes distributed knowledge to the explicit state (so that at least someone knows the fact) and fact publication, which makes the fact common knowledge. Halpern and Moses focus on the latter.

Common knowledge and fact publication are fundamental to reaching an agreement, starting a convention and deciding on a coordinated action. Halpern and Moses discuss two major ways of obtaining the common knowledge of a fact. The first one, group membership, is evident on the roads: law-abiding drivers must obtain a license before embarking on a trip. The second one is co-presence with the occurrence of the fact. If instead the parent told the same information in private, without other children taking a notice, then no common knowledge is available to the kids for deduction.

In the context of distributed systems, the fact that each agent is a node in the distributed protocol implies community membership, with common knowledge preset before the protocol starts running. Ensuring co-presence through messages, however, is not a trivial task.

The coordinated attack problem is a nice example of all the subtleties that arise from using communication channels for simulating co-presence:

Two generals, each leading a separate army, are preparing to attack the city in the valley below the hilltops where the armies stay. If one army attacks without the other, they lose, but they have not agreed on the initial plan on where to launch the assault. Neither general is willing to attack until receiving the confirmation from the other army, sent by a messenger through the valley filled with the enemy. There are no other means of communication. The messenger might get lost or captured by the defenders of the city, but on this particular night, the communication is smooth. How long will the coordination take?

The fact that the coordination takes forever is a well-known result: the generals never reach an agreement and never attack. Proving such impossibility results is often a daunting task without formal tools that capture all the possibilities. The logical framework for knowledge that the authors develop provides all the necessary components.

The notion of a trace (called run by the authors) is the building block of the model we are going to consider. The trace is a description of how the system evolves, from start to the end of execution. The paper focuses on deterministic protocols, and thus the internal state of each processor is a function of the history that we can recover from the trace.

Halpern and Moses derive two key results, giving a logical interpretation of the model, which not only gives us the power of propositional logic, but also enables us to reason about the knowledge in a system in terms of modal operators. First of all, achieving common knowledge requires simultaneity: whenever a group learns a new fact, the change in local histories of its members must be simultaneous. For the coordinated attack problem, any protocol must account for the fact that the decision to attack, if ever reached, is common knowledge, and thus the generals must act simultaneously. Finally, the authors prove that nothing can become common knowledge unless the fact is common knowledge in absence of communication. Thus, no army ever launches an assault in the coordinated attack problem, given a correct protocol. Even when the channel of communication is reliable, but the message delivery time is unbounded, agents still cannot achieve common knowledge.

Bringing common knowledge to practice

Relaxing the requirement of simultaneity to account for temporal imprecision, the authors show that, in the strict sense, practical distributed systems still cannot attain common knowledge in the absence of a global clock. When considering the muddy children puzzle, for example, we can say that some children could overhear what the parent says in the beginning, so they never attain common knowledge to start with. We seem to get stuck in the paradox: agreement and consensus are key to many real well-performing distributed systems on the one hand, and on the other — each of us is likely to have the gut feeling that the intuitive common knowledge is not a rare phenomenon.

The issue is that the notion of coordination that we have used so far is too strong. Many applications do not require simultaneous coordination. The authors offer three ways of introducing weaker states of common knowledge.

Suppose that all nodes receive a message within some epsilon time interval after it is sent. This $$\epsilon$$-common knowledge is attainable, but may require additional overhead for deciding the optimal value of epsilon.

Now, consider the case when the communication is asynchronous: the delivery times have no bounds, but channels guarantee that messages eventually reach the destinations. Unfortunately, this eventual common knowledge does not resolve the issue of the coordinated attack: any protocol that guarantees that whenever either army attacks, the other also eventually attacks, is such that neither party necessarily attacks. If you want to perform an action at all sites within a preset time period, asynchronous communication is of no use for coordination.

Finally, the authors suggest using relativistic time for attaining timestamped common knowledge in many practical cases by associating a timestamp with all knowledge. For example, distributed protocols often work in phases, which provide a natural interpretation as internal clocks. Paxos and atomic broadcast are just two algorithms that leverage timestamped common knowledge for consensus.

What's next?

The proposed framework suffers from the curse of generality: in the attempt to capture the complexity of the largest possible scope, the model makes several simplifying assumptions and discards specifics which may prove to be useful in designing a more optimal solution. For example, the paper looks only at deterministic protocols, which do a poor job at reflecting most real systems, where crashes and faults are common.

Despite these limitations, the paper has proven to be seminal not only in the distributed systems research, but also in AI security and game theory, showing that knowledge management on a group level is a viable approach for revealing the implementation subtleties and analyzing the performance of multi-agent systems.


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Article Contents
Article Contents

# Optimal control of system governed by the Gao beam equation

• In this contribution several optimal control problems are mathematically formulated and analyzed for a nonlinear beam which was introduced in 1996 by David Y. Gao. The beam model is given by a static nonlinear fourth-order differential equation with some boundary conditions. The beam is here subjected to a vertical load and possibly to an axial tension load as well. A cost functional is constructed in such a way that the lower its value is, the better model we obtain. Both existence and uniqueness are studied for the solution to the proposed control problems along with optimality conditions. Due to the fact that analytical solution is not available for the nonlinear Gao beam, a finite element approximation is provided for the proposed problems. Numerical results are compared with Euler-Bernoulli beam as well as the authors' previous considerations.
Mathematics Subject Classification: Primary: 49J15, 65L60; Secondary: 49N10.

 Citation:

•  [1] A. Borzi, V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations, SIAM, Philadelphia, 2012. [2] I. Ekeland, R. Témam, Convex Analysis and Variational Problems, SIAM, Philadelphia, 1999. [3] D.Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches. Mechanics Research Communications, 23, (1), pp. 11-17, 1996. [4] D.Y. Gao, Finite deformation beam models and triality theory in dynamical post-buckling analysis. Int. J. of Non-Linear Mechanics, 35, pp. 103-121, 2000. [5] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. [6] J.-L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems, SIAM, Philadelphia, 1972. [7] Reddy, J.N.:, An Introduction to the Finite Element Method. Third edition. McGraw-Hill Book Co., New York, 2006. [8] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, AMS, Providence, Rhode Island, 2010.
Open Access Under a Creative Commons license


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# to_start_stop function¶

(Shortest import: from brian2.groups.neurongroup import to_start_stop)

brian2.groups.neurongroup.to_start_stop(item, N)[source]

Helper function to transform a single number, a slice or an array of contiguous indices to a start and stop value. This is used to allow for some flexibility in the syntax of specifying subgroups in NeuronGroup and SpatialNeuron.

Parameters: item : slice, int or sequence The slice, index, or sequence of indices to use. Note that a sequence of indices has to be a sorted ascending sequence of subsequent integers. N : int The total number of elements in the group. start : int The start value of the slice. stop : int The stop value of the slice.

Examples

>>> from brian2.groups.neurongroup import to_start_stop
>>> to_start_stop(slice(3, 6), 10)
(3, 6)
>>> to_start_stop(slice(3, None), 10)
(3, 10)
>>> to_start_stop(5, 10)
(5, 6)
>>> to_start_stop([3, 4, 5], 10)
(3, 6)
>>> to_start_stop([3, 5, 7], 10)
Traceback (most recent call last):
...
IndexError: Subgroups can only be constructed using contiguous indices.


text:
# Math Help - Help to find solutions

1. ## Help to find solutions

1. suppose that f(x) and g(x) are twise differentiable at x=a,
show that
(fg)"(a)= f(a) g"(a) + 2f'(a) g'(a) + f"(a) g(a)
__________________________________________________ _____________

2. Use the definition of derivative to show that for the function

f(x)=3(x)^(1/3)

(f of x is equal to 3 x to the power 1/3)

we have f'(a)=a^(2/3), for every "a" belongs to "R" (Set of real numbers)

[Use the identity (a^3)-(b^3)= (a-b)((a^2)+ab+(b^2))]
__________________________________________________ _____________

3. Differentiate the following and prove the necessary expressions as given

a) If y=xtanx

show that

x sin²x (dy/dx)=x²tan²x +y sin²x
_____
b) y= (x+√ x²-1 )ⁿ
prove that

(x²-1) (dy/dx)²=n²y²

Plz help me to solve these.

2. nobody

3. Originally Posted by chanaka89
2. Use the definition of derivative to show that for the function

f(x)=3(x)^(1/3)

(f of x is equal to 3 x to the power 1/3)

we have f'(a)=a^(2/3), for every "a" belongs to "R" (Set of real numbers)

[Use the identity (a^3)-(b^3)= (a-b)((a^2)+ab+(b^2))]
_________________________________________________
$f(x) = 3x^{\frac{1}{3}}$.

$f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

$= \lim_{h \to 0}\frac{3(x + h)^{\frac{1}{3}} - 3x^{\frac{1}{3}}}{h}$

$= \lim_{h \to 0}\frac{3[(x + h)^{\frac{1}{3}} - x^{\frac{1}{3}}]}{h}$

$= \lim_{h \to 0}\frac{3\left[\frac{(x + h) - x}{(x + h)^{\frac{2}{3}} + + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\right]}{h}$

$= \lim_{h \to 0}\frac{\frac{3h}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}}{h}$

$= \lim_{h \to 0}\frac{3}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}$

$= \frac{3}{x^{\frac{2}{3}} + x^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}$

$= \frac{3}{3x^{\frac{2}{3}}}$

$= \frac{1}{x^{\frac{2}{3}}}$

$= x^{-\frac{2}{3}}$

4. Originally Posted by Prove It
$f(x) = 3x^{\frac{1}{3}}$.

$f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

$= \lim_{h \to 0}\frac{3(x + h)^{\frac{1}{3}} - 3x^{\frac{1}{3}}}{h}$

$= \lim_{h \to 0}\frac{3[(x + h)^{\frac{1}{3}} - x^{\frac{1}{3}}]}{h}$

$= \lim_{h \to 0}\frac{3\left[\frac{(x + h) - x}{(x + h)^{\frac{2}{3}} + + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\right]}{h}$

$= \lim_{h \to 0}\frac{\frac{3h}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}}{h}$

$= \lim_{h \to 0}\frac{3}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}$

$= \frac{3}{x^{\frac{2}{3}} + x^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}$

$= \frac{3}{3x^{\frac{2}{3}}}$

$= \frac{1}{x^{\frac{2}{3}}}$

$= x^{-\frac{2}{3}}$
Thanks

5. Originally Posted by chanaka89
1. suppose that f(x) and g(x) are twise differentiable at x=a,
show that
(fg)"(a)= f(a) g"(a) + 2f'(a) g'(a) + f"(a) g(a)
Use the product rule- twice.

__________________________________________________ _____________

2. Use the definition of derivative to show that for the function

f(x)=3(x)^(1/3)

(f of x is equal to 3 x to the power 1/3)

we have f'(a)=a^(2/3), for every "a" belongs to "R" (Set of real numbers)

[Use the identity (a^3)-(b^3)= (a-b)((a^2)+ab+(b^2))]
__________________________________________________ _____________

3. Differentiate the following and prove the necessary expressions as given

a) If y=xtanx

show that

x sin²x (dy/dx)=x²tan²x +y sin²x
Okay, what did you get for dy/dx?
_____
b) y= (x+√ x²-1 )ⁿ
prove that

(x²-1) (dy/dx)²=n²y²
Again, what did you get for dy/dx?

Plz help me to solve these.


text:
# Average acceleration of a rocket given average mass and thrust

If we assume that a small scale rocket is launched perpendicularly to the ground and ignore the drag, how can I calculate the average acceleration given that I know the burn time, the average thrust, and the average mass?

update: There are some calculations in this model rocketry tutorial but I wonder how well this applies to real launch vehicles.

• Rockets are usually between 85% to 95% propellant by mass, so "average mass" as a concept is somewhat indirect. Acceleration can change from 0.2g to 5 or 10g over several minutes, so average acceleration is not the best way to look at it. Nonetheless once a propellant mass fraction is given or assumed, the Tsiolkovsky rocket equation can be used to generate a velocity vs time curve and can be differentiated to produce acceleration (or it can be numerically integrated directly) and then averaged, but fun is in the shape of the curve!
– uhoh
Dec 30 '20 at 22:59
• "but I wonder how well this applies to real launch vehicles." - the equations listed, (limited by the assumptions), are the same basic equations that go into real launch planning. (though simplified). The biggest difference here is they will only be applied to a short phase of flight from lift off to a low altitude burn out. The primary difference with real flight planning is in the details... such as atmospheric conditions, use of actual vehicle data (such as an actual Cd) and how each, thrust, attitude, etc.. vary with time, altitude, etc.. They are actual basic equations. Jan 2 at 10:45

I would like to give a simple example. Lets assume a rocket with 1 kN thrust and 1000 kg weight. I calculate the acceleration for some weights.

  1000 kg    1.000 m/s^2
875 kg    1.142 m/s^2
750 kg    1.333 m/s^2
625 kg    1.600 m/s^2
500 kg    2.000 m/s^2
375 kg    2.667 m/s^2
250 kg    4.000 m/s^2
125 kg    8.000 m/s^2

Do you really think an average acceleration is useful for such a non linear relation?

• Well, I have seen some resources on the internet using it for rough apogee estimates. This pdf nakka-rocketry.net/articles/altcalc.pdf kind of answered my question. Dec 30 '20 at 22:39
• @funeselmemorioso oh that's a great link, I'm going to add it to your question because it represents prior research (which is good to include when asking questions) and it makes answering easier. The rocket in the tutorial has a huge dry weight ("dead weight") and tiny propellant mass, and as mentioned here real launch vehicles are totally the opposite!
– uhoh
Dec 30 '20 at 23:10
• Oh, thank you for the edit! Dec 31 '20 at 7:36

Too long for a comment:

I wonder how meaningful an average would be considering a rocket goes from 0 to 6 gs during launch.

It doesn't help that acceleration depends on the wildly variable mass of the rocket. Fuel accounts for the bulk of the weight, and most of it is gone two minutes into flight.

Nor does it help that the rocket gradually pitches down from vertical to horizontal. Close to vertical you lose close to 1g just counteracting gravity, and close to horizontal you lose close to nothing.

So even when you simplify by ignoring drag and other details of launch, the basic physics you're left with still make it very challenging to estimate things like acceleration---and therefore averages of those things, to the extent that they're meaningful :)

• I meant this as a comment, sorry. I'll delete it in a bit so please don't add more comments below. And yes, calculating averages is straightforward---if you have data to calculate them from. But acceleration during launch is nonlinear and very dependent on variables that are themselves nontrivial to estimate. But even if you get your average, remember you're not dealing with some statistical distribution. How meaninful is it to say that your average is 3gs when the instantaneous values vary rapidly from 0 to 6gs? But you do you.
– user36480
Jan 2 at 2:45
• I think your comment is quite reasonable, no reason for "sorry" or deletion! In Stack Exchange we can discuss stuff openly and others may disagree with me completely! From time to time answer posts are used for longish, helpful comments. Let's try this edit and see how it goes?
– uhoh
Jan 2 at 2:53


text:
REPORTS > DETAIL:

### Paper:

TR04-012 | 19th December 2003 00:00

#### The Resolution Complexity of Random Graph $k$-Colorability

TR04-012
Authors: Paul Beame, Joseph Culberson, David Mitchell, Cristopher Moore
Publication: 4th February 2004 16:51
We consider the resolution proof complexity of propositional formulas which encode random instances of graph $k$-colorability. We obtain a tradeoff between the graph density and the resolution proof complexity.
For random graphs with linearly many edges we obtain linear-exponential lower bounds on the length of resolution refutations. For any $\epsilon>0$, we obtain sub-exponential lower bounds of the form $2^{n^\delta}$ for some $\delta>0$ for non-$k$-colorability proofs of graphs with $n$ vertices and $O(n^{\frac{3}{2}-\frac{1}{k}-\epsilon}$ edges. We obtain sharper lower bounds for Davis-Putnam-DPLL proofs and for proofs in a system considered by McDiarmid.


text:
AJack Partnership manufactures jackhammers. AJack Partnership is looking for guidance in the... 1 answer below »

AJack Partnership manufactures jackhammers. AJack Partnership is looking for guidance in the variances of its standard cost system. It would like you to assist in understanding material price, material quantity, rate, efficiency, and overhead variances.

The standard cost card information for unit of product is below.

 Standard cost card per unit of product Direct materials: 4 pounds at $9.00 per pound of steel$36.00 Direct labor: 2.0 direct labor hours at $20.00 per hour 40.00 Variable overhead: 100% of a direct labor hour at$10.00 per hour 20.00 Fixed overhead: 100% of a direct labor hour at $20.00 per hour 40.00 Standard cost per unit$136.00

The following information is available in the year just finished:

• AJack Partnership manufactured 10,000 jackhammers during the year.
• The total purchases of steel in the year at a cost of $8.75 per pound were 45,000 pounds. • All of the material was used to manufacture the 10,000 jackhammers. • There was no beginning or ending inventory. • The material was purchased on January 15, 20XX • AJack Partnership incurred 21,000 direct labor hours at$19.50 per hour.
• During the year, one production order was issued on February 15 20XX, number 789, for 10,000 jackhammers.
• Actual variable overhead was $210,000. • Actual fixed overhead was$405,000.

Assignment Guidelines:

Material

1. Compute the material price variance for Jan 15, 20XX.
2. Provide the accounting entry for the price variance.

Labor

1. Compute the labor rate variance.
2. Compute the labor efficiency variance.
3. Provide the accounting entry for the labor rate and efficiency variances.

1. Compute the variable overhead rate variance.
2. Compute the variable overhead efficiency variance.
3. Provide the accounting entry for the overhead rate and efficiency variances.


text:
# Expanding Measurable Sets

Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that $$\|f(x)-f(y)\| \;\geq\; \|x-y\|$$ for all $x,y \in S$. Does it follow that $\mu(S) \leq \mu(T)$?

-

It follows from two observations:

• For Hausdorff meeasure your statement follows from the definition.
• Hausdorff measure = Lebesgue measure (up to constant).
-
Oh, great. Thanks! –  Jim Belk Jun 9 '11 at 22:33
Note that the assumption that the bijection f is measurable is not needed for this argument. –  Alex Simpson Jun 10 '11 at 7:20
@Alex, you need $T$ to be measurable; otherwise you can not write the inequality. –  Anton Petrunin Jun 10 '11 at 13:49
I wasn't questioning the measurability of S and T. But the measurability of the bijection f is an additional assumption in the question as formulated and is unnecessary. –  Alex Simpson Jun 12 '11 at 19:11


text:
# 235_sample_mid_2_soln.pdf - Math 235 Sample Midterm 2 1...

• Test Prep
• 6
• 100% (1) 1 out of 1 people found this document helpful

This preview shows page 1 - 3 out of 6 pages.

Math 235Sample Midterm 21.Short Answer Problemsa) SupposeL:VWis a linear mapping, whereVis a vector space with a basisB={~v1, . . . ,~vn}andWis a vector space with a basisC={~w1, . . . , ~wm}. DefineC[L]B, thematrix ofLwith respect to basesBandC.
b) LetB={~v1, . . . ,~vn}be orthonormal in an inner product spaceVand let~vVsuch that~v=a1~v1+· · ·+an~vn. Prove thatai=h~v,~vii.
6
c) ConsiderP2with inner producthp(x), q(x)i=p(-1)q(-1) +p(0)q(0) +p(1)q(1). Findthe value ofh1-x-x2,1 +x2i.
d) Write a basis for the four fundamental subspaces ofA=
1
1
2.Suppose the matrix of a linear mappingL:R3P2(R) with respect to the standardbases isA=1111010-11.Find the inverse ofAand use it to define the inverse linear mappingL-1:P2(R)R3.-12-11-101-11. This is the matrix ofL-1with respect to the standard basis. From here, we get


text:
Reference documentation for deal.II version 9.3.3
Householder< number > Class Template Reference

#include <deal.II/lac/householder.h>

Inheritance diagram for Householder< number >:
[legend]

## Public Types

using size_type = types::global_dof_index

## Public Member Functions

Householder ()=default

template<typename number2 >
Householder (const FullMatrix< number2 > &A)

template<typename number2 >
void initialize (const FullMatrix< number2 > &A)

template<typename number2 >
double least_squares (Vector< number2 > &dst, const Vector< number2 > &src) const

template<typename number2 >
double least_squares (BlockVector< number2 > &dst, const BlockVector< number2 > &src) const

template<class VectorType >
void vmult (VectorType &dst, const VectorType &src) const

template<class VectorType >
void Tvmult (VectorType &dst, const VectorType &src) const

## Private Attributes

std::vector< number > diagonal

FullMatrix< double > storage

## Detailed Description

template<typename number>
class Householder< number >

QR-decomposition of a full matrix.

This class computes the QR-decomposition of given matrix by the Householder algorithm. Then, the function least_squares() can be used to compute the vector $$x$$ minimizing $$\|Ax-b\|$$ for a given vector $$b$$. The QR decomposition of $$A$$ is useful for this purpose because the minimizer is given by the equation $$x=(A^TA)^{-1}A^Tb=(R^TQ^TQR)^{-1}R^TQ^Tb$$ which is easy to compute because $$Q$$ is an orthogonal matrix, and consequently $$Q^TQ=I$$. Thus, $$x=(R^TR)^{-1}R^TQ^Tb=R^{-1}R^{-T}R^TQ^Tb=R^{-1}Q^Tb$$. Furthermore, $$R$$ is triangular, so applying $$R^{-1}$$ to a vector only involves a backward or forward solve.

### Implementation details

The class does not in fact store the $$Q$$ and $$R$$ factors explicitly as matrices. It does store $$R$$, but the $$Q$$ factor is stored as the product of Householder reflections of the form $$Q_i = I-v_i v_i^T$$ where the vectors $$v_i$$ are so that they can be stored in the lower-triangular part of an underlying matrix object, whereas $$R$$ is stored in the upper triangular part.

The $$v_i$$ vectors and the $$R$$ matrix now are in conflict because they both want to use the diagonal entry of the matrix, but we can only store one in these positions, of course. Consequently, the entries $$(v_i)_i$$ are stored separately in the diagonal member variable.

Note
Instantiations for this template are provided for <float> and <double>; others can be generated in application programs (see the section on Template instantiations in the manual).

Definition at line 79 of file householder.h.

The documentation for this class was generated from the following file:


text:
# Particle moving around the inside of a semicone - how to model its position up incline?

An inverted hollow cone (cut in half) is set up with its vertex at the origin $$O$$ and an angle $$\alpha$$ between the horizontal ($$x$$ and $$y$$-axes) and the cone (so $$\alpha$$ close to $$0$$ would be a flat circle while close to $$\pi/2$$ would be a cylinder).

A particle is then launched and moves around the inside of the semicone (slipping), we could assume friction is negligible. Of course, there is a specific initial speed $$v$$, angle $$\alpha$$, and starting location of particle up along the incline of the cone $$s$$ which would produce circular motion if specific conditions are met. However, at differing initial conditions the motion will oscillate around some equilibrium (never reaching such). I have looked into the Lagrangian equations one could form for the particles based on a cylindrical coordinate system, and while this is all good I still cannot understand how to clearly model some aspects of the particle's motion

I'd like to be able to quantitatively model the displacement of the particle up the incline $$s$$ as a function of its angular displacement around the semicone $$\phi$$ ($$0\leq\phi\leq\pi$$), so that I could say, for example, at $$\phi=\pi/4$$, the ball will be $$s$$ cm up the incline (depending on $$\alpha, v,$$ and $$s_{\textrm{initial}}$$).

I know that this would have to do with formulating the Lagrangian of the particle, but then I have gotten stuck on having a bunch of differential equations with derivatives whose values are unknown to me.

How can this be represented in a calculable way, preferably not involving simulations? (I'm very new to DEs and Lagrangians, sorry...)

• You should really provide your working on the problem so we have a better handle on where you are going wrong. What's specifically about the differential equations was difficult to solve? May 13 at 21:57
• Would you be interested in building some sort of computational model? There doesn't seem to be an explicit closed form solution in terms of compositions of simple functions, so somewhere one probably needs to draw the line and move on to numerical implementations. I assume, for small oscillations around a circular orbit, one could carry out some reasonable series approximations and calculate a more explicit pair of equations linking the variables $s, \,\phi$ and time $t$. But are you interested in that? May 17 at 20:51
• Well, I very much underestimated how hard modeling motion like this would be. But, as this is not pure math, approximating a solution or a function with respect to a variable is my next goal - and hoping that this can be applied to real life! I'm not sure where exactly to start. May 19 at 0:27

In polar coordinates, the constraint fixing the particle on the cylinder reads : $$z = r\tan(\alpha)$$

Therefore, the kynetic energy is (setting $$m = 1$$ for simplicity) : $$T = \frac{1}{2}(\dot r^2 + \dot z^2+r^2\dot \theta^2)=\frac{1}{2}((1+\tan(\alpha)^2)\dot r^2 + r^2\dot\theta^2)$$ The potential energy is : $$V = gz = gr\tan(\alpha)$$

Therefore, the Lagrangian is : $$L = \frac{1}{2}((1+\tan(\alpha)^2)\dot r^2 + r^2\dot\theta^2)-mgr\tan(\alpha)$$ Using this, we can derive the Euler-Lagrange equations : \begin{align} (1+\tan^2(\alpha))\ddot r &= g\tan(\alpha)-r\dot \theta^2 \\ \frac{d}{dt}(r^2\dot\theta)&=0 \end{align}

The second equation is the conservation of angular momentum. It implies that there is a cosntant $$C$$ such that $$r^2 \dot \theta = C$$. Then, plugging this into the other equation, we get a differential equation for $$r$$ : $$(1+\tan^2(\alpha))\ddot r -g\tan(\alpha) + \frac{C}{r^3} = 0$$

This is equivalent to a particle with mass $$1$$ and moving in an effective potential : $$V_{\text{eff}}(r) = \frac{1}{1+\tan^2(\alpha)}\left(g\tan(\alpha)r + \frac{C}{2r^2}\right)$$

I don't think that there is an analytical solution to this equation. However, we can derive some qualitative properties of this potential. For example, for a given value of $$C$$, the radial coordinate will oscillate around the equilibrium $$r_* = \sqrt[R]{\frac{C}{g\tan(\alpha)}}$$, which is the radius of the corresponding circular orbit.

• Sorry, just wanted to notify you that you have a typo for the effective potential. i think, the last term should be something like $+ \frac{C}{2 r^2}$.. May 14 at 0:31
• Thanks ! I edited the mistake (that's what I get from doing physics past midnight :) May 14 at 7:50
• Thank you so much for this! To be honest, I'm not sure how the Euler-Lagrange equation is actually derived, but that is not as relevant. But more importantly, is there a non-analytical approach to solving the differential equation? I don't want the work done for me, but is there a direction I can go in - perhaps using Newtonian mechanics instead - such that I can obtain a computable relationship between, say, $r$ or $s$ with respect to $t$ or the angular displacement $\phi$? May 14 at 16:25
• For a non-analytical approach use Runge-Kutta numerical integration. May 14 at 21:59


text:
# Parametric curve on plane fitting with PyTorch

This post is part of a series of posts on the fitting of mathematical objects (functions, curves and surfaces) through a MLP (Multi-Layer Perceptron) neural network; for an introduction on the subject please see the post Fitting with highly configurable multi layer perceptrons.

The topic of this post is the fitting of a parametric curve on a Cartesian reference system $Oxy$ with parameter $t$ within a closed interval of reals is defined with a pair of real functions $x(t) \colon [a,b] \to \rm I\!R$ and $y(t) \colon [a,b] \to \rm I\!R$ that return respectively the values of $x$ and $y$ coordinates as the parameter $t$ changes; equivalently a parametric curve on plane is also defined with a vector function $f(t) \colon [a,b] \to {\rm I\!R x \rm I\!R}$ in this way: $$f(t) = \begin{bmatrix} x(t) \\ y(t) \\ \end{bmatrix}$$ where the two components of the vectors of the function image are respectively the $x$ and $y$ coordinates of the curve on Cartesian reference system.

Note that the two definitions of parametric curve, albeit mathematically equivalent, suggest two different architectures of neural networks: the first definition leads to a pair of independent MLPs, one to fit $x(t)$ and the other one to fit $y(t)$ to then combine the results from the two networks and obtain the $(x,y)$ pairs that fit the curve on plane; the second definition instead suggests a single MLP where the input layer contains only one neuron because the dimension of domain is 1 instead the output layer contains 2 neurons because the dimension of codomain is 2.

As part of the fitting is to allow the user to test different combinations of MLP architectures, their own activation functions, training algorithm and loss function without writing code but working only on the command line of the four Python scripts (plus two variants) which separately implement the following features:

• Dataset generation
• MLP architecture definition + Training
• Prediction
• Visualization of the result
The related code requires the version 3 of Python and it uses PyTorch technology; it requires also NumPy and MatPlotLib libraries.
To get the code please see the paragraph Download of the complete code at the end of this post.

The exact same mechanism was created using TensorFlow 2 (with Keras) technology; see the post Parametric curve on plane fitting with TensorFlow always published on this website.

## Dataset generation

Goal of the pmc2t_gen.py Python program is to generate datasets (both training and test ones) to be used in later phases; it takes in command line the two component functions to be approximated (in lambda body syntax), the interval of independent parameter $t$ (begin, end and discretization step) and it generates the dataset in an output csv file applying the two function to the passed interval of the parameter $t$.
In fact the output csv file has three columns (without header): first column contains the sorted values of independent parameter $t$ within the passed interval discretized by discretization step; second and third columns contain respectively the values of the components $x$ and $y$, ie the values of functions $x(t)$ and $y(t)$ correspondent to values of $t$ of first column.

To get the program usage you can run this following command:

$python pmc2t_gen.py --help and the output got is: usage: pmc2t_gen.py [-h] -h, --help show this help message and exit --dsout DS_OUTPUT_FILENAME dataset output file (csv format) --xt FUNCX_T_BODY x=x(t) body (lamba format) --yt FUNCY_T_BODY y=y(t) body (lamba format) --rbegin RANGE_BEGIN begin range (default:-5.0) --rend RANGE_END end range (default:+5.0) --rstep RANGE_STEP step range (default: 0.01) Please read the file README.md for a complete detail of the semantics of the parameters supported on the command line. ### An example of using the program pmc2t_gen.py Suppose you want to approximate in the range$[0,4\pi]$the Lissajous curve that is described by the following function: $$f(t) = \begin{bmatrix} x(t)=2 \sin (\frac{t}{2} + 1) \\ y(t)=3 \sin t \\ \end{bmatrix}$$ Keeping in mind that np is the alias of NumPy library, the translation of this function in lambda body Python syntax is: 2 * np.sin(0.5 * t + 1) 3 * np.sin(t) To generate the training dataset, run the following command: $ python pmc2t_gen.py \
--dsout mytrain.csv \
--xt "2 * np.sin(0.5 * t + 1)" \
--yt "3 * np.sin(t)" \
--rbegin 0.0 \
--rend 12.56 \
--rstep 0.01
instead to generate the test dataset, run the following command:

$python pmc2t_gen.py \ --dsout mytest.csv \ --xt "2 * np.sin(0.5 * t + 1)" \ --yt "3 * np.sin(t)" \ --rbegin 0.0 \ --rend 12.56 \ --rstep 0.0475 Note that the discretization step of the test dataset is larger than that of training and it is a normal fact because the training, to be accurate, it must be run on more data. Also note that it is appropriate for the discretization step of the test dataset is not a multiple of the training one in order to ensure that the test dataset contains most of the data that is not present in training dataset, and this makes prediction more interesting. ## MLP architecture definition + Training This feature has two different implementations according to the two mathematical definitions of parametric curve on plane: the official implementation is the one that fits the vector function using a single MLP instead the Twin variant implementation fits the component functions separately using two twin MLPs. The official version implemented by Python program pmc2t_fit.py creates dynamically an MLP and performs its training according to the passed parameters through the command line in order to fit the vector function that defines the curve on plane. The Twin variant implemented Python program pmc2t_fit_twin.py creates dynamically an pair of twin MLPs and performs their training according to the passed parameters through the command line in order to fit the separately the component functions that defines the curve on plane. To get the official program usage you can run this following command: $ python pmc2t_fit.py --help
and the output got is:

usage: pmc2t_fit.py [-h]
--trainds TRAIN_DATASET_FILENAME
--modelout MODEL_PATH
[--epochs EPOCHS]
[--batch_size BATCH_SIZE]
[--hlayers HIDDEN_LAYERS_LAYOUT [HIDDEN_LAYERS_LAYOUT ...]]
[--hactivations ACTIVATION_FUNCTIONS [ACTIVATION_FUNCTIONS ...]]
[--optimizer OPTIMIZER]
[--loss LOSS]
[--device DEVICE]
To get the Twin variant usage you can run this following command:

$python pmc2t_fit_twin.py --help and the output got is (excluding the program name) equal to the official version usage. Please read the file README.md for a complete detail of the semantics of the parameters supported on the command line of both programs. ### An example of using the program pmc2t_fit.py Suppose you have a training dataset available (for example generated through pmc2t_gen.py program as shown in the previous paragraph) and you want the MLP to have two hidden layers with 150 neurons each and you want to use the ReLU activation function exiting the two layers; moreover you want to perform 500 training epochs with a 200 items batch size using the Adamax with learning rate equal to 0.02 and loss function equal to MSELoss. To put all this into action, run the following command: $ python pmc2t_fit.py \
--trainds mytrain.csv \
--modelout mymodel.pth \
--hlayers 150 150 \
--hactivation 'ReLU()' 'ReLU()' \
--epochs 500 \
--batch_size 200 \
--loss 'MSELoss()'
at the end of which the file mymodel.pth will contain the MLP model trained on mytrain.csv dataset according to the parameters passed on the command line.

## Prediction

As with training, this feature also has two different implementations according to the two mathematical definitions of parametric curve on plane: the official implementation that makes the prediction using a single pre-trained MLP that fits the curve defined by a vector function and the Twin variant that makes the prediction using a pair of pre-trained twin MLPs that fit separately the two component functions.

Goal of the official version implemented by pmc2t_predict.py Python program is to apply the MLP model generated through pmc2t_fit.py to an input dataset (for example generated through pmc2t_gen.py program as shown in a previous paragraph); the execution of the program produces in output a csv file with three columns (without header): the first column contains the values of indepedent parameter $t$ taken from input dataset and the second and third columns contain the predicted values of the two components, ie the values of the prediction that fits the parametric curve $f(t)$ correspondent to values of $t$ of first column.

Goal of the Twin variant implemented by pmc2t_predict_twin.py Python program is to apply the pair of MLPs generated through pmc2t_fit_twin.py to an input dataset (for example generated through pmc2t_gen.py program as shown in a previous paragraph); the execution of the program produces in output a file structurally equal to the output one of official version; the way to compute the values of second and thrid columns is different because these are the values of predictions coming out from the two MLPs that fit separately the component functions $x(t)$ and $y(t)$.

To get the official program usage you can run this following command:

$python pmc2t_predict.py --help and the output got is: usage: pmc2t_predict.py [-h] --model MODEL_PATH --ds DATASET_FILENAME --predictionout PREDICTION_DATA_FILENAME [--device DEVICE] To get the Twin variant usage you can run this following command: $ python pmc2t_predict_twin.py --help
and the output got is (excluding the program name) equal to the official version usage.

Please read the file README.md for a complete detail of the semantics of the parameters supported on the command line.

### An example of using the program pmc2t_predict.py

Suppose you have the test dataset mytest.csv available (for example generated through pmc2t_gen.py program as shown in a previous paragraph) and the trained model of MLP in the file mymodel.pth (generated through pmc2t_fit.py program as shown in the example of previous paragraph); run the following command:

$python pmc2t_predict.py \ --model mymodel.pth \ --ds mytest.csv \ --predictionout myprediction.csv  at the end of which the file myprediction.csv will contain the fitting of the initial function. ## Visualization of the result Goal of the pmc2t_plot.py Python program is to visualize the prediction curve superimposed on initial dataset curve (test or training, as you prefer) and it allows the visual comparison of the two curves. To get the program usage you can run this following command: $ python pmc2t_plot.py --help
and the output got is:

usage: pmc2t_plot.py [-h]
--ds DATASET_FILENAME
--prediction PREDICTION_DATA_FILENAME
[--savefig SAVE_FIGURE_FILENAME]
Please read the file README.md for a complete detail of the semantics of the parameters supported on the command line.

### An example of using the program pmc2t_plot.py

Having the test dataset mytest.csv available (for example generated through pmc2t_gen.py program as shown in a previous paragraph) and the prediction csv file (generated through pmc2t_predict.py program as shown in the previous paragraph), to generate the two xy-scatter charts, execute the following command:

$python pmc2t_plot.py \ --ds mytest.csv \ --prediction myprediction.csv which shows the two xy-scatter charts superimposed: in blue the input dataset, in red the prediction one. Note: Given the stochastic nature of the training phase, your specific results may vary. Consider running the example a few times. Chart generated by the program pmc2t_plot.py that shows the fitting done by the MLP of the Lissajous curve$f(t) = \begin{bmatrix} x(t)=2 \sin (\frac{t}{2} + 1) \\ y(t)=3 \sin t \\ \end{bmatrix}$## Examples of cascade use of the four programs In the folder parametric-curve-on-plane-fitting/examples there are five shell scripts that show the use in cascade of the four programs, with training and prediction in the official version, in various combinations of parameters (MLP architecture, activation functions, optimization algorithm, loss function, training procedure parameters). Also there are five other shell scripts that reproduce the same examples but with with training and prediction in the Twin variant. To run the five examples for official version, run the following commands: $ cd parametric-curve-on-plane-fitting/examples
$sh example1.sh$ sh example2.sh
$sh example3.sh$ sh example4.sh
$sh example5.sh  Instead to run the five examples for the Twin variant, run the following commands: $ cd parametric-curve-on-plane-fitting/examples
$sh example1_twin.sh$ sh example2_twin.sh
$sh example3_twin.sh$ sh example4_twin.sh
\$ sh example5_twin.sh

Note: Given the stochastic nature of these examples (relative to the training phase), your specific results may vary. Consider running the example a few times.


text:
# Leetcode 526 - Beautiful Arrangement

Note:

• Use backtracking to find all permutations.
• While doing dfs, pre-check if it meets the requirement of beautiful arrangements.

Question:

Suppose you have n integers labeled 1 through n. A permutation of those n integers perm (1-indexed) is considered a beautiful arrangement if for every i (1 <= i <= n), either of the following is true:

• perm[i] is divisible by i.
• i is divisible by perm[i].

Given an integer n, return the number of the beautiful arrangements that you can construct.

Example:

Code:


text:
# Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:

1. R is a local principal ideal domain, and not a field.
2. R is a valuation ring with a value group isomorphic to the integers under addition.
3. R is a local Dedekind domain and not a field.
4. R is a noetherian local ring with Krull dimension one, and the maximal ideal of R is principal.
5. R is an integrally closed noetherian local ring with Krull dimension one.
6. R is a principal ideal domain with a unique non-zero prime ideal.
7. R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
8. R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
9. R is not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as finite intersection of fractional ideals properly containing it.
10. There is some discrete valuation ν on the field of fractions K of R, such that R={x : x in K, ν(x) ≥ 0}.

## Examples

Let Z(2)={ p/q : p, q in Z, q odd }. Then the field of fractions of Z(2) is Q. Now, for any nonzero element r of Q, we can apply unique factorization to the numerator and denominator of r to write r as 2kp/q, where p, q, and k are integers with p and q odd. In this case, we define ν(r)=k. Then Z(2) is the discrete valuation ring corresponding to ν. The maximal ideal of Z(2) is the principal ideal generated by 2, and the "unique" irreducible element (up to units) is 2.

Note that Z(2) is the localization of the Dedekind domain Z at the prime ideal generated by 2. Any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings Z(p) for any prime p in complete analogy.

For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.

Another important example of a DVR is the ring of formal power series R = K[[T]] in one variable T over some field K. The "unique" irreducible element is T, the maximal ideal of R is the principal ideal generated by T, and the valuation ν assigns to each power series the index (i.e. degree) of the first non-zero coefficient.

If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that converge in a neighborhood of 0 (with the neighborhood depending on the power series). This is also a discrete valuation ring.

Finally, the ring Zp of p-adic integers is a DVR, for any prime p. Here p is an irreducible element; the valuation assigns to each p-adic integer x the largest integer k such that pk divides x.

## Uniformizing parameter

Given a DVR R, then any irreducible element of R is a generator for the unique maximal ideal of R and vice versa. Such an element is also called a uniformizing parameter of R (or a uniformizing element, a uniformizer, or a prime element).

If we fix a uniformizing parameter t, then M=(t) is the unique maximal ideal of R, and every other non-zero ideal is a power of M, i.e. has the form (t k) for some k≥0. All the powers of t are distinct, and so are the powers of M. Every non-zero element x of R can be written in the form αt k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = k. So to understand the ring completely, one needs to know the group of units of R and how the units interact additively with the powers of t.

The function v also makes any discrete valuation ring into a Euclidean domain.[citation needed]

## Topology

Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. The distance between two elements x and y can be measured as follows:

$|x-y| = 2^{-\nu(x-y)}$

(or with any other fixed real number > 1 in place of 2). Intuitively: an element z is "small" and "close to 0" iff its valuation ν(z) is large. The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring.

A DVR is compact if and only if it is complete and its residue field R/M is a finite field.

Examples of complete DVRs include the ring of p-adic integers and the ring of formal power series over any field. For a given DVR, one often passes to its completion, a complete DVR containing the given ring that is often easier to study. This completion procedure can be thought of in a geometrical way as passing from rational functions to power series, or from rational numbers to the reals.

Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of Z(p) (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Zp.


text:
## 9.37 d

$\Delta S = \frac{q_{rev}}{T}$

Christina Cen 2J
Posts: 53
Joined: Sat Jul 22, 2017 3:01 am

### 9.37 d

For 37 part d) 4KClO3(s) => 3KClO4(s) + KCl(s) why are the products more ordered than the reactants, resulting in a negative entropy?

Mitch Mologne 1A
Posts: 74
Joined: Fri Sep 29, 2017 7:04 am

### Re: 9.37 d

I believe it is because KCl is a very ordered crystal, giving it a lower entropy than KClO4 and KClO3 has less variability in where the Cl can be than KClO4, thus also resulting in lower entropy.


text:
# Converting polar to cartesian?

So far I got

\begin{align} r & = 7 / (4 - 2 \cos\theta) \\ r (4 & - 2\cos\theta) = 7 \\ r (4 & - 2( x / r ) ) = 7 \end{align}

I apologize in advance for the bad formatting.

-

## 1 Answer

You are on the right track. So we get $$4r-2x=7.$$ This implies that $$4r=7+2x.$$ Taking square on both sides, we get $$16r^2=(7+2x)^2.$$ Now note that $x^2+y^2=r^2$. Put this into the last equation, we have $$16(x^2+y^2)=(7+2x)^2.$$

-
Thanks Paul. Is it appropriate to ask questions like this on here? I am new and here to learn. Thanks –  user1530249 Feb 5 '13 at 6:53
Yes, here you can ask any question related to math. And it's good to let people know where you are stuck and what you don't understand, as you have already done. But you may want to learn how to type the equation using TeX. See here: meta.math.stackexchange.com/questions/107/… –  Paul Feb 5 '13 at 6:57


text:
# How can I show ithat a language is regular?

I have a very quick question about regular languages, I think $\{a^{2n}| n\geq 1\}$ is regular. I do know that pumping lemma can be used to show something that is not regular. I am wondering what should I use in order to show that $\{a^{2n}| n\geq 1\}$ is regular.

You can show that a language, $L$, is regular with one of those:
• Make an NFA that recognizes $L$
• Make a DFA that recognizes $L$
• Determine a primitive recursive function $f:\Sigma^*\to\{0,1\}$with the property $$f(x)=\begin{cases}1\text{, if } x\in L \\0\text{, else}\end{cases}$$ where $\Sigma$ is the alphabet of the language.


text:
# Find Missing Vector

## Homework Statement

When adding Vectors A & B; Vector A = 26a 35 deg Resultant = 50a 12 deg

## Homework Equations

None, no idea how to start

## The Attempt at a Solution

Identify Vector B

Yes, first class on Physics and I am stumbling big time

#### Attachments

• Vectors.doc
68 KB · Views: 370
Last edited:

PeterO
Homework Helper

## Homework Statement

When adding Vectors A & B; Vector A = 26a 35 deg Resultant = 50a 12 deg

## Homework Equations

None, no idea how to start

## The Attempt at a Solution

Identify Vector B

Yes, first class on Physics and I am stumbling big time

Do you understand what "Vector A = 26a 35 deg" means?

Well, Vector A is a measure of 26a, with A being a variable which could be pounds, millisecond, etc and it is at 35 degrees.

Am I on track?

It has been 40 years since I finished college, and now taking a crash course in Physics since my son has his first class. He has given up out of frustration and gone to bed. I want to give him a hand, but it is a lot of work trying to remember Trig along with understanding Physics.

PeterO
Homework Helper
Well, Vector A is a measure of 26a, with A being a variable which could be pounds, millisecond, etc and it is at 35 degrees.

Am I on track?

Getting there.

at 35o to what? ie. making an angle of 35o with what?

Well, since I am adding 2 vectors, I am guessing they are going in the same direction, but at different degrees.

PeterO
Homework Helper
Well, Vector A is a measure of 26a, with A being a variable which could be pounds, millisecond, etc and it is at 35 degrees.

Am I on track?

It has been 40 years since I finished college, and now taking a crash course in Physics since my son has his first class. He has given up out of frustration and gone to bed. I want to give him a hand, but it is a lot of work trying to remember Trig along with understanding Physics.

Best way here is to draw the two vectors with a common start point - it would look like the hands on a clock.

The 26a 35 deg could be an hour hand pointing at approx 2'oclock
The 35a 12 deg could be a minute hand pointing at about 13mins past

what vector is needed to go from the end of the first to the end of the second?
You could use the cosine rule, or do x- and y- components of them to work it out.
The angle will be negative, that is for sure.

PeterO, no question you are trying to help. Problem is I am just now re-learning the concepts and terminology.

I read the "why aren't I getting anwers" post, and thankfully you were willing to provide asssistance. Clearly, I am not ready to be posting on this forum since I am trying to solve a problem I do not understand.

My hopes were for a quick fix that someone would show me the solution steps, then I would discuss with my son.

Clearly this is not the intent of this forum and for this reason, I appreciate your willingness to provide assistance.

ehild
Homework Helper
Anester,

Look at the books and notes of your son. They had to learn "vector addition". What method did they use? Is it geometry, drawing a vector triangle or parallelogram or using x and y components of the vectors?

ehild

PeterO
Homework Helper
PeterO, no question you are trying to help. Problem is I am just now re-learning the concepts and terminology.

I read the "why aren't I getting anwers" post, and thankfully you were willing to provide asssistance. Clearly, I am not ready to be posting on this forum since I am trying to solve a problem I do not understand.

My hopes were for a quick fix that someone would show me the solution steps, then I would discuss with my son.

Clearly this is not the intent of this forum and for this reason, I appreciate your willingness to provide assistance.

One system of defining vector - especially in 2 dimensions - is like polar co-ordinates.
You give the amplitude, and the angle above the positive x-axis of the Cartesian system.

I wonder whether 26a 35 deg was really meaning 26 (amplitude) 35 deg
As I said the hour hand on a clock at 2:00 pm does a pretty good impression of [26, 35o] , especially of it is 26 cm long. [the angle is slightly wrong] note Cartesian co-ordinates (x,y) are written in round brackets. Polar Co-ordinates eg [25,34o] are written in square brackets.

In some courses you learn how to work with polar co-ordinates as polar co-ordinates. Other wise it is common to convert them to a Cartesian equivalent, add them then convert back.

ehild,

This is an introductory Physics class at his Community College. First class was last week and he has not picked up the book yet. He "lost" his notes. Hit me at 8pm tonight with the problem, now it is 1am. Class at 9:30am so I wanted to get him some confidence before he turned in his first assignment.

Fortunately, I figured out the Resultant steps; have a few hours left to get the Equilibrant. Will have to give up on the missing Vector question which are 2 of the 9 questions. Not terrible for his first assignment. Obviously, we are going to talk about getting an earlier start on these assignments.

Last edited:
PeterO,

I won't begin to attempt to use words to explain my ignorance, so this is the problem directly from the worksheet:

PROCEDURE
1. Use appropriate math processes to provide missing information in the table and requested information in the problems following.

DATA -- Be sure to include appropriate units of measure.

VECTOR A VECTOR B CALCULATED CALCULATED
RESULTANT EQUILIBRANT
6 26 a 35*________ 50 a 12* ________

rl.bhat
Homework Helper
You should have given the full text of the problem.
Is it given in the problem that the angle between vector A and vector B is 35 degrees and the angle between the vector R and vector B is 12 degrees?

rl.bhat,

Actually figured out how to attach the assignment - Vectors.doc.

ehild
Homework Helper
Equilibrant is the opposite of the original vector. The magnitude is the same and it makes an angle of 180° with the original. So its angle is 180°+ the angle of the original vector.
As for vector addition, the simplest way is using the coordinates. They are the projections of the vector to the x and y axis (ax and ay ). See figure attached.
If you have vector with length a and angle $\alpha$ with respect to the positive x axis, ax=acos($\alpha$)
and ay=asin($\alpha$)

You get the components of the sum of two vectors a and b by adding the components :c=a+b ---> cx=ax+bx, cy=ay+by.

If you have the components of a vector, the length is √(ax2+ay2)
and tan($\alpha$)=ay/ax.

Can I give a piece of advice: no reason to help if your son does not listen to the lectures. He must make notes. Do not try to help without seeing what they learn and how. You might explain on a different way than his teacher does, and you will confuse the boy.

ehild

#### Attachments

• vector.JPG
5.2 KB · Views: 520
ehild,

Your advice is well taken - guessing you may be a Professor with first hand knowledge.

I won't be able to be a long term solution, just wanted to kick start his first weeks. You are correct, I cannot replace his Prof.

I appreciate your detail and will be able to work with it. One question - do you know of a website that has an Equilibrant calculator, then I can validate my steps are correct?

ehild
Homework Helper
You can add two numbers without calculator, do you not? Add or subtract 180° from the angle of the resultant vector (so as the result is less than 360°) to get the angle of the equilibrant. The magnitudes are the same.

ehild

#### Attachments

• equilibrant.JPG
6 KB · Views: 638
Got it, and my highest gratitude for the Physics Forums team assistance.

ehild, loved the jpg; will actually get a few hours extra sleep with the assistance from the forum.

Thanks to all


text:
### Book Store

Currently only available for.
CBSE

# CBSE Physics 2012 Exam Questions

1.

An electric dipole is held in a uniform electric field.

(i) Show that the net force acting on it is zero.

(ii) The dipole is aligned parallel to the field. Find the work done in rotating it through the angle of 180°.

i)  Dipole moment of dipole is given by,

i)

Force on –q at A = -qE

Force on +q at B = + qE

So,

Net force = qE – qE = 0

ii)

Work done on dipole is given by,

4459 Views

2.

A charge ‘q’ is placed at the centre of a cube of side l. What is the electric flux passing through each face of the cube?

Using Gauss theorem,

1034 Views

3.

A bar magnet is moved in the direction indicated by the arrow between two coils PQ and CD.

Predict the directions of induced current in each coil.

In the figure, N pole is receding away coil (1), so in coil (1), the nearer faces will act as S-pole and in coil (2) the nearer face will also act as S-pole to oppose the approach of magnet towards coil (2), so current in coils will flow clockwise as seen from the side of magnet. The direction of current will be from P to Q in coil (1) and from C to D in coil (2).

2509 Views

4.

A light bulb is rated 100 W for 220 V ac supply of 50 Hz. Calculate

(a) the resistance of the bulb;

(b) the rms current through the bulb

OR

An alternating voltage given by V = 140 sin 314 t is connected across a pure resistor of 50 W. Find

(i) the frequency of the source.

(ii) the rms current through the resistor.

OR

1321 Views

5.

Two wires of equal length, one of copper and the other of manganin have the same resistance. Which wire is thicker?

Resistance of a material is given by,

Resistance for copper wire and manganin wire is given by,

That is, manganin wire is thicker.

3421 Views

6.

State the underlying principle of a transformer. How is the large scale transmission of electric energy over long distances done with the use of transformers?

The principle of transformer is based upon the principle of mutual induction which states that due to continuous change in the current in the primary coil an emf gets induced across the secondary coil. At the power generating station, the step up transformers step up the output voltage which reduces the current through the cables and hence reduce resistive power loss. Then, at the consumer end, a step down transformer steps down the voltage. Therefore, the large scale transmission of electric energy over long distances is done with the uses of transformers.

1716 Views

7.

What are the directions of electric and magnetic field vectors relative to each other and relative to the direction of propagation of electromagnetic waves?

Both electric and magnetic field vectors are perpendicular to each other and perpendicular to the direction of propagation.

618 Views

8.

A test charge ‘q’ is moved without acceleration from A to C along the path from A to B and then from B to C in electric field E as shown in the figure. (i) Calculate the potential difference between A and C. (ii) At which point (of the two) is the electric potential more and why?

i) Potential difference between A and C

ii) Direction of electric field is in higher to lower potential. So, Vc > VA

2803 Views

9.

A capacitor of capacitance of ‘C’ is being charged by connecting it across a dc source along with an ammeter. Will the ammeter show a momentary deflection during the process of charging? If so, how would you explain this momentary deflection and the resulting continuity of current in the circuit? Write the expression for the current inside the capacitor.

When capacitor is charged by a dc source, ammeter will show a momentary deflection because of the presence of displacement current.

Conduction current is equal to the displacement current.

i.e.,

Resulting continuity of current becomes zero because,  is maximum when, capacitor is fully charged.

1310 Views

10.

Mention the two characteristic properties of the material suitable for making core of a transformer.

Two characteristic properties are:

i) Low hysteresis loss

ii)  Low coercivity.

1377 Views

Do a good deed today
Refer a friend to Zigya


text:
Making a real matrix positive definite by replacing nonzero and nondiagonal entries with arbitrary nonzero reals

Let $A$ be real square matrix.

Let $\mathcal{F}(A)$ be the set of real matrices $A'$ of the same size such that $A'_{ii}=A_{ii}$ for all $i$, and for all $i,j$, $A_{ij}=0\Rightarrow A'_{ij}=0 \land A_{i,j} \ne 0 \implies A'_{i,j} \ne 0$: in other words, obtained from $A$ by modifying nonzero nondiagonal entries.

Q1 What are sufficient conditions for $A$ s.t. exists a positive definite $A'$ in $\mathcal{F}(A)$?

Since there are several definitions of positive definite, we don't require neither $A$ or $A'$ to be symmetric.

In practice this is possible for some $A$.

Partial answers are welcome. The definition of positive definite from mathworld

$M$ is positive definite if for all nonzero real vectors $x$, $x^T M x >0$.

• Please indicate what you mean by positive definite (I understand it means $X^tAX>0$ for every nonzero column vector $X$?)
– YCor
Dec 31, 2015 at 11:02
• @YCor Exactly, will edit soon, citing mathworld (wikipedia requires symmetric).
– joro
Dec 31, 2015 at 11:20

The answer is quite boring... Necessary and sufficient conditions are $A_{ii}>0$ for each $i$. In this case, let $B_\varepsilon$ the matrix with entries $$(B_\varepsilon)_{ij} = \begin{cases} A_{ij} & i=j,\\ \varepsilon & i\neq j, A_{ij}\neq 0,\\ 0 & i\neq j, A_{ij}=0. \end{cases}$$ The matrix $B_0$ is diagonal and posdef, so for any sufficiently small value of $\varepsilon$ the matrix $B_\varepsilon$ (which belongs to $\mathcal{F}(A)$) is posdef, too, since positive definiteness is an open condition.

On the other hand, if $A_{ii}\leq 0$ for some $i$ there is no hope, since for each $A'\in\mathcal{F}(A)$ we have $x^\top A'x \leq 0$ when $x$ is the vector $e_i$ of the canonical basis.

• Thanks. Ycor misquoted my definition. I edited: $A_{i,j} \ne 0 \implies A'_{i,j} \ne 0$ (I had this in wording). Does your answer still applies?
– joro
Dec 31, 2015 at 11:22
• @Joro Yes. Just put epsilons instead of zeros in the entries where you need nonzeros. Any sufficiently small modification of $D$ is posdef, since posdefness is an open condition. Dec 31, 2015 at 11:26
• Thank you. I suggest to edit, so I accept your answer (The title doesn't accept zeros and I had explicitly nonzero, but Ycor's edit missed that).
– joro
Dec 31, 2015 at 11:37
• I discarded the condition $A_{ij}\neq 0\Rightarrow A'_{ij}\neq 0$ because it does not affect the existence of a positive-definite matrix in $F(A)$, by the openness condition Federico mentioned.
– YCor
Dec 31, 2015 at 11:41
• Happy New year celebration!
– joro
Dec 31, 2015 at 11:48


text:
# cutting an equilateral triangle to $n$ equal pieces

We have an equilateral triangle and we want to cut it into $n$ equal pieces.

For which $n$ is it possible?

My Attempt: I found these possible numbers $2,3,4,6$ and also I proved every $n$ of the form $4^n$

note: I don't mean equal areas I mean equal triangles

• By equal do you mean congruent? You can always cut it into pieces that are equal in area. Given any $n$ for which you are successful, you can do all numbers of the form $n\cdot 4^k$ by cutting the triangle into $4^k$ equilateral triangles, then cutting each into $n$ – Ross Millikan Jun 15 '16 at 16:03
• All $n$ of the form $k^2$ would do. So would $2k^2$, and probably many more. Your findings leave 5 as the smallest unknown number; turns out that it kinda can be done, too! (The text is in Russian, but you don't really need it; the picture pretty much says it all. Yes, the pieces are not connected, but still congruent.) – Ivan Neretin Jun 15 '16 at 16:13
• – Rahul Jun 15 '16 at 16:36
• Do you mean congruent pieces or do you also require that the pieces be triangles? – almagest Jun 15 '16 at 17:57
• no it should be triangle – Taha Akbari Jun 15 '16 at 17:58


text:
# zbMATH — the first resource for mathematics

On the search for the definition of the space complexity for the calculation of polynomials using Valiant’s method. (À la recherche de la définition de la complexité d’espace pour le calcul des polynômes à la manière de Valiant.) (French) Zbl 1160.03020
Boolean and arithmetic circuits have been used extensively in current literature to analyze the complexity of map calculations. Valiant circuits are arithmetic circuits (inner nodes correspond to either addition or multiplication of two arguments provided by previous nodes) enlarged with nodes able to apply summation operators indexed by Boolean words. The input of the circuits are either of integer or Boolean type.
In this paper, several classes of maps are defined: VPdl, VPdb,VPmd and VPt, consisting, respectively, of maps calculated by sequences of arithmetic circuits of polynomial sizes, of polynomial sizes and polynomial circuit semidegrees, of polynomial sizes and polynomial circuit degrees, and of logarithmic heights. If $$(u,v,x)\mapsto f(u,v,x)$$ is a map in a class VP, where $$u$$ and $$v$$ are finite lists of Boolean variables, then let $$\sigma(f):(u,v,x)\mapsto \sum_vf(u,v,x)$$. For any class VP, the class VNP consists of maps $$\sigma(f)$$ with $$f$$ in VP; and the class VSP is defined similarly as VP but considering Valiant circuits. The author claims that one main goal in considering such complexity classes lies in the construction of calculation models, in order to analyze the plausibility of those calculations, rather than in the construction of realistic models.
Several relations among these classes are examined (e.g. VNPt = VNPmd entails the #P-completeness of the calculation of the permanent) and the implausibility of some other are discussed (e.g. any of the relations $$\text{VNPmd}=\text{VPmd}$$, $$\text{VNPdl}=\text{VPdl}$$ or $$\text{VNPmd}\subset\text{VPdl}$$ would imply $$\text{P}=\#$$P).
Let $$P_n$$ be the map that, given two Boolean strings, produces the binomial coefficient of the integers whose base-2 representation are the input strings, and let $$F_n$$ be the map defined similarly considering the factorial integer map. It is proved that $$P_n$$ is in VSPdl, hence $$F_n$$ is also in this class. Then, one may expect that VSPdl will differ from VPdl, because an efficient calculation of the factorial may solve efficiently the Integer Factorization Problem. It is also shown that if VNPmd = VPmd and P = PSPACE then $$F_n$$ will fall into VPdl.
The above defined circuit classes are relativized to the arithmetic modulo a prime number, and some relations on the resulting classes are given. It is seen that P coincides with VPdl (modulo 2), while PSPACE coincides with VSPdl (modulo 2).
Then the author lists some sufficient conditions in order to have VSPdl = VPdl. However, the conditions indicate that it will not be possible to suppress summation operators in Valiant circuits in order to build equivalent circuits using just arithmetical nodes.
There are many unconventional notations the reader should be aware of. For instance, exponentiation, $$+^+$$, is denoted via $$\land$$, $$+\land+$$. This is confusing in a first reading of the paper. Some intended exponentiations, $$2^j$$, are displayed as products, $$2j$$. And several intended subindexes appear as seemingly product factors .

##### MSC:
 03D15 Complexity of computation (including implicit computational complexity) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
##### Keywords:
circuit complexity; classes of functions; Valiant circuits
Full Text:
##### References:
 [1] Accessible telephone directories 59 pp 92– (1994) [2] On defining integers in the counting hierarchy and proving lower bounds in algebraic complexity 4393 (2007) [3] Completeness and reduction in algebraic complexity theory (2000) · Zbl 0948.68082 [4] DOI: 10.1007/BF01200057 · Zbl 0774.68040 · doi:10.1007/BF01200057 [5] DOI: 10.1137/0212043 · Zbl 0524.68028 · doi:10.1137/0212043 [6] L’Enseignement Mathématique 30 pp 365– (1982) [7] Soviet Physics – Doklady 7 pp 595– (1963) [8] Les petits cailloux (1995) [9] Characterizing Valiant’s algebraic complexity classes 4162 pp 267– (2006) [10] DOI: 10.1145/321958.321973 · Zbl 0335.68022 · doi:10.1145/321958.321973 [11] VPSPACE and a transfer theorem over the Reals 4393 (2007) · Zbl 1146.68379 [12] Computational Complexity 13 pp 131– (2004) · Zbl 1087.68566 [13] Proceedings of the 11th ACM STOC pp 249– (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# What it means "TypeError: 'list' object is not callable"?

This post is a wiki. Anyone with karma >750 is welcome to improve it.

1) For example, how can I extract the divisors < 10 from the list divisors(n) ? I try to create a def with S=sum(d(i)) , d(i) verifying d(i)<10, but I get that TypeError

2) More generally, how can I do sum(d(i) with condition d(i)<10) ?

3) Where can I find a reference document for SAGE and an elementary tutorial?

Thanks

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I do not see the link between the title of the post and the three questions above...

( 2014-11-23 06:37:57 +0200 )edit

You are right, I forgot to make clear that I had that error text because I was looking for a conditioned sum. Anyway the given answer guessed what I was meaning. Sorry

( 2014-11-24 15:39:40 +0200 )edit

Then, it would be helpful for other people looking for help if you change the title.

( 2014-11-25 21:52:49 +0200 )edit

Sort by » oldest newest most voted

To extract the divisors less than 10, you can do this:

sage: k = 720
sage: sum(d for d in k.divisors() if d < 10)

See this tutorial.

The error you got is probably because of the following:

If you defined d as

sage: d = divisors(k)

Then d is a list. You can extract element number i of d as d[i], but you cannot do d(i) which would be "calling the function d with argument i". Sage tells you d is not "callable".

If you want to sum the first 10 divisors, do:

sage: sum(d[i] for i in range(10))

or

sage: sum(d[:10])

more

Thank you so much!

( 2014-11-22 22:32:57 +0200 )edit

If this answered your question, don't forget to accept it so that someone else coming to this forum sees quickly that it has an accepted, correct answer. And have fun learning more Python and Sage - it's fun to see how elegantly many such problems can be solved!

( 2014-11-23 03:34:22 +0200 )edit