text:
# How can I modeling a simple haptic interface system to obtain results similar to the paper “Time-Domain Passivity Control of Haptic Interfaces”?

I am trying to simulate the experiments in the paper "Time-Domain Passivity Control of Haptic Interfaces", Hanaford and Ryu, 2002, IEEE transactions on robotics and automation vol .18, No1 about the simulation 2, in Fig.8, and its results are fig.9 and fig.10.

I don't know an exact model of fig.9 which I can draw again in Matlab/simulink to run. I'm having trouble visualizing the mathematical model explaining the input and output of the system so that I can put it in a MATLAB code and perform the simulation. I am using constants like the image.

How can I do to get similar results? When I try to emulate something I get inconsistent results and I have trouble putting the mathematical model in lines of code.

I tried drawing spring and damper as in fig.6 of the same paper, but I don't know which is my input (force, velocity or position) and also the type of signal of input (sine, step,...) to get a similar result as the paper, or how I can insert the initial conditions on the code editor or on the Simulink.

I added 3 images, one is the parameters that I chose and another is the model which I drew.

I am really confused about initial condition of position.

In here, for the Virtual Environment: damping = -50 (I want to check the case without Passivity controller) k = 30000.

I am sending the simulation if it help. Thanks a lot :D

Matlab editor code: (sorry for the mess, I don't know how to put the modeling in the code) :/

%%%% Simulation Haptic Interface System %%%%

% - Impedance-based environment

clc, clear

%% Parameters

freq = 1000;           % Sampling rate simulation
delta_T = 1/freq;      % Sampling period simulation
t = 0:delta_T:10;      % Simulation time

%- Human Operator/ Disturbance (HO) -> mass + spring
M_HO = 0.1;            % Mass HO system (? kg)
k_HO = 10000;          % Spring HO constant  (? kN/m)

%- Haptic device (Interface) (HI)
M_HI = 0.1;            % Mass HI system (? kg)
k_HI = 150000;         % Spring HI constant  (? kN/m)
b_HI = 50;             % Damping HI constant (? Ns/m)

%- Delayed Virtual Environment (VE)
k_VE = 30000;          % Spring VE constant  (30 kN/m)
b_VE = - 50 ;          % Damping VE constant (-50 Ns/m)

freq_VE = 200/3;       % Sampling rate VE (66.67 Hz)
delta_Tve = 1/freq_VE; % VE system operates at a realativily slow samplig rate (15 ms)

%% Inicialization

%- Human Operator/ Disturbance (HO) -> mass + spring
x_HO = [0];            % Human position
dx_HO = [0];
ddx_HO = [0];
f_HO = [0];            % Force output (? N)

%- Haptic device (Interface) (HI)
x_HI = [0];            % Human disturbance input - position
dx_HI = [0];
ddx_HI = [0];
f_HI = [0];            % Force output (? N)

%- Series PC (alpha)
W = [0 0];             % Passivity Observer (We set up the PO to monitor only the
%                     virtual environment and the PC)
alpha = [0 0];         % Passivity Controller
f_PC = [0];

%- Delayed Virtual Environment (VE)
x_VE = [0];            % Haptic Interface disturbance input - position
dx_VE = [0];
ddx_VE = [0];
f_VE = [0];            % Force output (? N)

n = 2;

ve = [0];
fe = [0];
%% without PC (the system is highly unstable when driven to contact)

sum_ve = 0;
ve_inte = 0;

for t1 = 0.0000:delta_T:10

%- Human Operator
if t1 < 4
x_HO(:,end+1) = 0.140;                         % Human input (140 mm)
else
x_HO(:,end+1) = 0.090;                         % Human input (90 mm)
end

dx_HO(:,end+1) = (x_HO(n) - x_HO(n-1))/delta_T;
ddx_HO(:,end+1) = (dx_HO(n) - dx_HO(n-1))/delta_T;

f_HO(:,end+1) = M_HO*ddx_HO(n);

%- Haptic device
x_HI(:,end+1) = (f_HO(n) + k_HO*x_HO(n))/k_HO;

%     f_HI(:,end+1) = f_HO(n);                           % f_h

dx_HI(:,end+1) = (x_HI(n) - x_HI(n-1))/delta_T;    % v_h
ddx_HI(:,end+1) = (dx_HI(n) - dx_HI(n-1))/delta_T;

%- Virtual Environment

ve(:,end+1) = dx_HO(n) - dx_HI(n);

ve_inte = ve_inte + ve(n);

%     ve(n) = (x_VE(n) - x_VE(n-1))/delta_Tve;
x_VE(:,end+1) = ve_inte*delta_Tve;

dx_VE(:,end+1) = (x_VE(n) - x_VE(n-1))/delta_T;

fe(:,end+1) = k_VE*x_VE(n) + b_VE*ve(n);

n = n+1;
end

%  The signal flow should be as follows: haptic device
% --> Virtual Environment (f = ke + bv, where e = penetration depth inside
% the virtual environment/wall, v = haptic device velocity,
% k and b = stiffness and damping gains)
% --> PO/PC --> haptic device.

%% with added PC (the system achieves stable contact after about 3 bounces)

sum_ve = 0;
ve_inte = 0;

for t1 = 0.0000:delta_T:10

%- Human Operator
if t1 < 4
x_HO(:,end+1) = 0.140;                         % Human input (140 mm)
else
x_HO(:,end+1) = 0.090;                         % Human input (90 mm)
end

dx_HO(:,end+1) = (x_HO(n) - x_HO(n-1))/delta_T;
ddx_HO(:,end+1) = (dx_HO(n) - dx_HO(n-1))/delta_T;

f_HO(:,end+1) = M_HO*ddx_HO(n);

%- Haptic device

%- Virtual Environment

f_PC(:,end+1) = alpha(n)*ve(n);

f_VE(:,end+1) = fe(n) + alpha(n)*ve(n);

f_HI(:,end+1) = f_VE(n) + M_HI*ddx_HI(n) + b_HI*dx_HI(n) + k_HI*x_HI(n) ...
- b_HI*dx_VE(n) - k_HI*x_VE(n);

%- Series PC

W(:,end+1) = W(n-1) + (fe(n)*ve(n) + alpha(n-1)*(ve(n-1))^2);

sum_ve = sum_ve + ve(n)^2;

if W(n) < -b_HI*sum_ve
alpha(:,end+1) = (W(n) + b_HI*sum_ve)/(ve(n)^2);
else
alpha(:,end+1) = 0;
end

n = n+1;

end

%% Graphics

Best regards,

leonardo.fagundes@ufv.br

Edit1: simulation 2, in Fig.8, and its results are fig.9 and fig.10

edit2: In the first simulation, the response for simple virtual wall system when driven by a sinusoidal velocity profile generates energy when damping is negative (the system becomes active). When wall damping is still negative and passivity controller is operating, dissipation is constrained to be positive and the system is stable. I figured, in the second simulation, I should also use the same damping gain. The authors quote that in this simulation "The VE includes a spring constant of 30 kN/m and operates at a relatively slow sampling rate of 66.67 Hz (15 ms)"

Therefore, damping and spring constant values seem very out of scale with one another. But I don't know if they should stay that way, the article provided only the value of the spring constant. I am confused about that. I understood your remark about the constants, I agree, I had not thought about the rigidity due to the high values. Thank you so much for showing me this. I modeled the human operator (HO) as a mass+spring system. But for both HO and Haptic Interface, I couldn't define values for the constants, so I tried to follow the same logic.

• Welcome to Robotics, Leonardo Fagundes. The article you mention (and the figures you cite) are behind a pay wall. Can you post a copy of the figures here? That said, your damping and spring constant values seem very out of scale with one another. First I would never expect a negative damping (or else you'd get runaway acceleration) but also your spring constants are SO large as to be effectively rigid. Even without seeing your source figure, if you're guessing at terms, I would go with something like 10/1/2 for a spring/mass/damper system. But again, please show us what you're trying to do :) Apr 28 '20 at 2:51
• Hi, Chuck. Thank you very much :D I apologize, part of the paper can be viewed in ReseachGate by the link: link I posted a copy of the figures I mentioned, as you requested :) Apr 28 '20 at 18:18
• The published article follows in the link "Time-Domain Passivity Control of Haptic Interfaces", Hanaford and Ryu, 2002, IEEE transactions on robotics and automation vol .18, No1 :) Apr 28 '20 at 18:40
• robotics.stackexchange.com/questions/12197/… Apr 29 '20 at 22:11
• Thanks, @hauptmech But I've seen this post and I haven't been able to get the answer I'm looking for regarding the entry of the system mainly, how to insert the time delay... I also contacted the author, but got no response :( May 2 '20 at 17:48


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# Nambu-Goldstone bosons from a quantum anomaly symmetry breaking?

We know that:

Nambu-Goldstone bosons come from Goldstone theorem: a spontaneous (continuous)-symmetry breaking of the system leads to massless scalar modes.

quantum anomaly: is the anomalous phenomena where the (classical Noether) symmetry G respected by the physical system in a classical limit, but this symmetry G is broken by the quantum effect. This is the case where the action $S$ preserves the symmetry, but the path integral partition function $Z=\int [D\Psi][D\Phi]\dots e^{iS}$ and the measures $[D\Psi][D\Phi]\dots$ do not preserves the symmetry.

• Question: whether there is any example that Nambu-Goldstone bosons can be derived from a spontaneous symmetry breaking caused by quantum anomaly effect? Alternatively, whether there is a known theorem to prove No Nambu-Goldstone bosons from a quantum anomaly symmetry breaking?''

[words of caution & side remark]: I offer some further thoughts if you wish you can skip it. There is a statement learned: Nambu-Goldstone bosons do not generally appear for a spontaneously broken symmetry if the relevant global symmetry is broken by the effects of the anomaly and instantons. This is regarded as a reason why we do not observe a light pseudo-scalar meson $\eta'$ in the QCD mesons. 1 among the 9 mesons is this $\eta'$, which stands for axial $U(1)_A$ anomalous symmetry breaking, there are still $SU(N_{flavlor})_A=SU(3)_A$ which is spontaneous broken by dynamical symmetry breaking, which induces 8 among the 9 mesons, such as three $\pi$, three $\kappa,$ and one $\eta$. There are also $SU(N_{flavlor})_V=SU(3)_V$ broken by the nonzero explicit quark masses($m_u\neq m_d \neq m_s$). There is also $U(1)_V$ broken by Sphaleron, such that numebr of baryon $N_{baryon}$ is not conserved, but which only conserved numbers of $N_{baryon}-N_{lepton}$. Anyway, altogether $$SU(N)_V \times SU(N)_A \times U(1)_V \times U(1)_A$$ makes it to be the full $U(N)_V \times U(N)_A$.

Simply that $U(1)_A$ symmetry is anomalous broken by the quantum effect, but we do not see its Goldstone boson $\eta'$.

-
I am a little confused by the statement "...a spontaneous symmetry breaking caused by quantum anomaly effect". I am not sure I understand why the quantum anomaly can be seen as 'spontaneous symmetry breaking' of symmetry...? Let me make a few naive comments on this, which might very well be wrong. Usually anomalies are described as the impossibility to find a regularization scheme in which the anomalous symmetry is explicitly preserved, and thus it implies that the anomaly is a problem of the UV completion. – Heidar Aug 21 '13 at 18:37
However, as you very likely know well, the 't Hooft anomaly matching condition says that the anomaly is scale independent, the UV and IR anomalies must match. For me, this seems to imply that the anomalous symmetry is not a symmetry of the theory at all in any limit. It just appears to be a symmetry in the naive classical limit, but that's just an illusion. Thus its not a spontaneous nor explicit breaking of symmetry, but absence of symmetry. This line of reasoning might very likely be wrong, though. – Heidar Aug 21 '13 at 18:37
In support of @Heidar, the anomaly is not like spontaneous breaking at all. If anything, it is like just adding a term in the lagrangian which violates the symmetry. If a system has a $SU(2)$ spin symmetry, you don't expect a goldstone boson when you turn on an external magnetic field. If in zero field there is SSB, then you expect turning on a magnetic will cause the goldstone boson (magnon) to develop a gap. Its the same with the $\eta'$ (and also with the other pions with respect to quark mass to a lesser extent). – BebopButUnsteady Aug 21 '13 at 19:32
TO Heidar: I do agree with what Heidar said. My original understanding is that the quantum anomaly for a current non-conservation respect to G is a fact that there is no that G symmetry at all for that system from the beginning. So that is why we do not have $\eta'$ meson, as I gave the example. However, I read this statement from Fuijikawa's book "Path Integral and Quantum Anomaly": Sec 5.6.2 "Nambu-Goldstone bosons do not generally appear for a spontaneously broken symmetry if the relevant global symmetry is broken by the effects of the anomaly and and instantons" – Idear Aug 22 '13 at 7:47
continue: and Weinberg's book QFT II, Sec. 22.7 Anomalies and Goldstone bosons. " the gauged effective field theory of Goldstone bosons must have an anomaly for the fictions symmetry which is equal to what has been produced from the underlying (fermion) theory." I thought there may be some room to think a bit more for this question? – Idear Aug 22 '13 at 7:52

There are two different kinds of symmetry breaking involved in your question. The first would be spontaneous symmetry breaking, in which case we are dealing with a theory that is invariant under a certain symmetry group, but its vacuum is not. The breaking of the symmetry corresponds to a specific choice of the vacuum, the freedom of choosing a vacuum results in a new degree of freedom: the Nambu-Goldstone- boson. Depending on the type of symmetry that is broken, one might get one or more of them. In the case of the chiral symmetry of QCD, the $SU(N_f)_A$ part is broken spontaneously, resulting in eight massless bosons. The other axial part, $U(1)_A$, however, is not broken spontaneously.

The second kind of symmetry breaking would be induced by quantum anomalies. We speak of such an anomaly when a theory is classically invariant under a certain symmetry operation but the corresponding quantum theory is not. In terms of the path integral formalism this is manifest in the fact that the measure transforms in a nontrivial way. An example would be the breaking of the axial $U(1)_A$ part of the chiral symmetry in QCD, which is not related to any choice of a vacuum, and there is no Nambu-Goldstone boson connected to it. In fact, it can be traced back to something entirely different, namely instantons. The occurence of an anomaly is related to the number of fermionic zero modes of the theory: the difference between fermionic and antifermionic zero modes is given by the Pontryagin number of the topological instanton configuration of the gauge fields. This is also known as the Atiyah-Singer theorem.

Since these two possible symmetry breaking mechanism are quite distinct, the concept of Nambu-Goldstone bosons arising from quantum anomalies can only be a result of sloppy, non-standard terminology.

-

First note that chiral symmetry breaking appears at the level of states of theory, where exist set of nontrivial vacuums which are connected with each other by symmmetry transformations, and hence which are not invariant under the gauge transformation. The theory itself on the level of path integral is gauge invariant. If the theory isn't invariant, then we may talk about SSB of approximately exact symmetry, if breaking effects are small compared to themscale of CSB, otherwise there is no even approximate symmetry.

As follow from comments to the question, you thought that, in the spirit of 't Hooft's anomaly matching conditions, anomalies, which break the symmetry explicitly, leads to existence of Nambu-Goldstone bosons. This is misunderstanding of conditions.

Really, 't Hooft anomaly matching says that if we initially have theory with chiral fermions and gauge bosons, and feemion sector has the global symmetry group $G$, which is gauge anomaly free (no $G_{\text{gauge}}^2G$ anomalies), but which is anomalous itself (there exist $G^3$ anomalies), then after the confinement there must exist massless fermion bound states which reproduce anomalous structure of symmetry group in initial theory; otherwise there must exist spontaneous symmetry breaking. But is there symmetry breaking (i.e., current nonconservation) by anomaly?

In fact, no. Presence of $G^3$ anomalies means only that there are symmetric tensor with group indices structure in the triangle diagram with three $G$ currents, namely $$d_{abc} \equiv \text{Tr}[[t_{a},t_{b}]_{+}t_{c}]$$ This means that if we decide to gauge the $G$ group, we will face with current nonconservation. Otherwise there is no current nonconservation and hence no symetry breaking. For example, the pure QCD global symmetry group $G \sim SU_{L}(3)\times SU_{R}(3)\times U_{B}(1)$ is exact in the chiral limit, although it is anonalous itself.

-

format: plain


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# Taylor series with initial value of infinity

So here's the problem:

Let $$f(x)=(1+3x)\cdot e^{\frac{1}{x-2}}$$ Find $a$, $b$ and $c$ such that: $$f(x)=ax+b+\frac{c}{x}+o(1/x)$$ as $x\to \infty$.

I guess what it's asking of me is to find a Taylor series at initial point of $x$ = infinity. In my textbox the problem is solved by first finding the series of the function

and then just multiplying it with the other part which is a polynomial itself. That's all good.

However, two things I don't understand here.

First: How do I deal with orders of less than 1:

How does that even go in the Taylor series formula?

Second: How do I deal with the initial point being infinity? I mean I can see that:

But what about the first derivative, for example:

In order to plug in into the Taylor formula I would need to evaluate it at infinity, which seems rather difficult so I guess I'm doing something wrong here.

Any ideas?

Thanks

Hint. Note that as $x\to \infty$, $$\frac{1}{x-2}=\frac{1}{x}\cdot\frac{1}{1-2/x}=\frac{1}{x}+\frac{2}{x^2}+o(1/x^2).$$ Now use the Taylor expansion $e^t=1+t+\frac{t^2}{2}+o(t^2)$ at $0$: for $t=\frac{1}{x}+\frac{2}{x^2}+o(1/x^2)\to 0$, \begin{align}\exp\left(\frac{1}{x-2}\right)&= 1+\left(\frac{1}{x}+\frac{2}{x^2}+o(1/x^2)\right) +\frac{1}{2}\left(\frac{1}{x}+o(1/x)\right)^2+o(1/x^2)\\ &=1+\frac{1}{x}+\frac{2}{x^2} +\frac{1}{2x^2}+o(1/x^2)=1+\frac{1}{x}+\frac{5}{2x^2} +o(1/x^2).\end{align} Can you take it from here?


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# Suggestions on cleaning up Flask app

I've written a little bit of code to pull data out of my FitBit and store it in a GCP database for further analysis. The project is available here, but what I'd like to ask about specifically is the Flask app & code I'm using to serve up a web interface for the project:

from datetime import date
import os

from wtforms import StringField

from fitnick.activity.activity import Activity
from fitnick.database.database import Database
from fitnick.heart_rate.time_series import HeartRateTimeSeries
from fitnick.heart_rate.models import heart_daily_table

SECRET_KEY = os.urandom(32)
app.config['SECRET_KEY'] = SECRET_KEY

month_options = [i for i in range(1, 13)]
day_options = [i for i in range(1, 32)]
year_options = range(2020, 2021)

date = StringField('date')

@app.route("/", methods=['GET'])
def index():
"""
Currently serves as the endpoint for the get_heart_rate_zone methods, even
though it's currently set to the index page
:return:
"""
heart_rate_zone = HeartRateTimeSeries(config={'database': 'fitbit'})
statement = heart_daily_table.select().where(heart_daily_table.columns.date == str(date.today()))
rows = [i for i in Database(database='fitbit', schema='heart').engine.execute(statement)]
#  retrieve rows for today already in database, if there are none then get rows via fitnick
if len(rows) == 0:
rows = heart_rate_zone.get_heart_rate_zone_for_day(database='fitbit')

rows = [i for i in rows]

form = DateForm(request.form)

if request.method == 'GET':
return render_template(
template_name_or_list="index.html",
rows=rows,
form=form,
month_options=month_options,
day_options=day_options,
year_options=year_options
)

def set_search_date(request, search_date):
if request.method == 'POST':
# collect search date information from the dropdown forms if they're all supplied.
if all([request.form.get('month_options'), request.form.get('day_options'), request.form.get('year_options')]):
search_date = '-'.join(
[f"{request.form['year_options']}",
f"{request.form['month_options']}".zfill(2),
f"{request.form['day_options']}".zfill(2)])
else:
# use the search_date value we set in lines 59-62
pass

return search_date

@app.route("/get_heart_rate_zone_today", methods=['GET', 'POST'])
def get_heart_rate_zone_today():
"""
Endpoint for getting heart rate zone data from the FitBit API.
:return:
"""
heart_rate_zone = HeartRateTimeSeries(config={'database': 'fitbit'})
form = DateForm(request.form)
value = 'Updated heart rate zone data for {}.'

if form.date._value():  # set search_date, default to today if none supplied
search_date = form.date._value()
else:
search_date = str(date.today())

if request.method == 'POST':
# collect search date information from the dropdown forms if they're all supplied.
search_date = set_search_date(request, search_date)
rows = heart_rate_zone.get_heart_rate_zone_for_day(
database='fitbit',
target_date=search_date)
rows = [i for i in rows]
else:  # request.method == 'GET'
# no date supplied, just return data for today.
heart_rate_zone.config = {'base_date': date.today(), 'period': '1d'}
statement = heart_daily_table.select().where(heart_daily_table.columns.date == str(date.today()))
rows = Database(database='fitbit', schema='heart').engine.execute(statement)

return render_template(
template_name_or_list="index.html",
value=value.format(search_date),
rows=rows,
form=form,
month_options=month_options,
day_options=day_options,
year_options=year_options
)

@app.route("/get_activity_today", methods=['GET', 'POST'])
def get_activity_today():
"""
Endpoint for getting activity data for a given date from the FitBit API.
:return:
"""

activity = Activity(config={'database': 'fitbit'})
form = DateForm(request.form)
search_date = form.date._value()

if request.method == 'POST':
search_date = set_search_date(request, search_date)
row = activity.get_calories_for_day(day=search_date)
value = 'Updated activity data for {}.'.format(search_date)
else:
row, value = {}, ''

return render_template(
template_name_or_list='activity.html',
form=form,
row=row,
value=value,
month_options=month_options,
day_options=day_options,
year_options=year_options
)

@app.route('/<page_name>')
def other_page(page_name):
"""
Stand-in endpoint for any undefined URL.
:param page_name:
:return:
"""
response = make_response(f'The page named {page_name} does not exist.', 404)
return response

if __name__ == '__main__':
app.run(debug=True)

Specifically, I'm trying to cut down on some code repetition. I have two pages currently, one for grabbing heart rate zone data and another for collecting activity data. When it comes to retrieving the date entered into the form on either page & then querying the database and returning the proper results, it seems like there's a lot of code repetition.

It feels like there should be a way to collapse the code in the index, get_heart_rate_zone_today and get_activity_today functions where essentially what is being checked is if the request is GET or POST and if it's POST, query the data for the given day from the FitBit API, add it to the database & return the data (when it's a GET request, we just return the data in the database). This section of the get_heart_rate_zone_today function is an example of what I mean:

if form.date._value():  # set search_date, default to today if none supplied
search_date = form.date._value()
else:
search_date = str(date.today())

if request.method == 'POST':
# collect search date information from the dropdown forms if they're all supplied.
search_date = set_search_date(request, search_date)
rows = heart_rate_zone.get_heart_rate_zone_for_day(
database='fitbit',
target_date=search_date)
rows = [i for i in rows]
else:  # request.method == 'GET'
# no date supplied, just return data for today.
heart_rate_zone.config = {'base_date': date.today(), 'period': '1d'}
statement = heart_daily_table.select().where(heart_daily_table.columns.date == str(date.today()))
rows = Database(database='fitbit', schema='heart').engine.execute(statement)

However, there are still enough differences from a code perspective (the different actions the code should take depending on if it's activity or heart rate zone data being retrieved) that the easiest way I've found thus far is to just write separate functions for each retrieval method.

I'd really appreciate it if someone could check out the Flask code I have below and get your thoughts on it, what I'd like to do and if it's the right idea for solving the 'problem' I have currently, which is the semi-duplicated code. Thank you for the time!

• I don't think that it's worth it to extract anything out into common functions. I don't know Flask but I am familiar with Django, and it looks similar. The biggest commonality is that those three routines (index, get_heart_rate_zone_today, and get_activity_today) is that they have POST and GET functionality, but that's typical for web apps. Within the POST and GET areas, there isn't enough commonality to reduce code IMHO. Nov 14 '20 at 0:58
• Thanks for taking a look and for the comment, @DaveB. This is kind of the same conclusion I was coming to as well, and I was trying to figure out if I just wasn't being creative enough or if they're really just different enough to warrant their own functions & code.
– n1c9
Nov 14 '20 at 17:53

My advice would be to restructure the project by splitting it into several files. In this file you have everything mixed up together, you are instantiating the Flask instance, you define the routes, and you have functions in between.

This file can only grow over time and can become overwhelming, making code maintenance more tedious and difficult in the long run.

To start Flask the idea is to have a small launcher file that looks like this:

from flask import Flask

app.config.from_object('config.Config')

and a config file that would look like this:

from os import environ, path

basedir = path.abspath(path.dirname(__file__))

class Config:
"""Base config."""
SECRET_KEY = environ.get('SECRET_KEY')
STATIC_FOLDER = 'static'
TEMPLATES_FOLDER = 'templates'

class ProdConfig(Config):
DEBUG = False
TESTING = False
DATABASE_URI = environ.get('PROD_DATABASE_URI')

class DevConfig(Config):
DEBUG = True
TESTING = True
DATABASE_URI = environ.get('DEV_DATABASE_URI')

Source and recommended reading: Configuring Your Flask App. I use a similar scheme to be able to switch easily between test and production environments, without touching the code base. Instead, changes are made to a config file. Note that in your app you have hardcoded: app.run(debug=True)

I would also keep the routes in a distinct file. A simplistic example to import your routes:

from flask import Flask

def create_app():
"""Initialize the core application."""
app.config.from_object('config.Config')

with app.app_context():
# Include our Routes
from . import routes

return app

If you have some helper functions they can go to another file as well. No need to clutter the codebase.

In short, my suggestion is to spend more time reading up on Flask and take the time to build a sound boilerplate that you can reuse for future projects.

I am not familiar with your database scheme but it is obvious there is repetition across your code eg: get_heart_rate_zone_for_day(database='fitbit') or: Database(database='fitbit', schema='heart'). The DB name is mentioned multiple times. It should be set up once and for all at the start of your application. At the very least you could have defined a constant at the top of your code.

I suppose something like this at the top of your code should do:

FITBIT_DB = Database(database='fitbit', schema='heart')

Then use FITBIT_DB from now on to refer to the DB. Just this should make the code ligher.

At the top of your code you have this:

month_options = [i for i in range(1, 13)]
day_options = [i for i in range(1, 32)]
year_options = range(2020, 2021)

I presume you have 3 combo boxes in your Jinja template. To make the code future-proof and not having to review it every year, you should make the year range dynamic:

from datetime import datetime
current_year = datetime.now().year
year_options = range(current_year, current_year + 1)

But it doesn't seem that you are using WTF for validating the dates. Dates like 30 Feb should trigger an error, even if the code does not crash and ends up returning no data.

Not sure this code is correct:

def set_search_date(request, search_date):
if request.method == 'POST':
# collect search date information from the dropdown forms if they're all supplied.
if all([request.form.get('month_options'), request.form.get('day_options'), request.form.get('year_options')]):

First of all, there is no need to make it a list.

Reminder: what the all function does: check if all items in a list are True. So I would check that this code does what you really think it does. If in doubt you can dump the contents of the POST request and see for yourself what happens.

• Thank you so much for taking the time to review the code and write this, as well as the links!
– n1c9
Nov 24 '20 at 5:25

A few general suggestions, based on the code

1.- Organization. It could be a hacky project, but you should avoid doing db queries on the views, it will lead to hard to maintain code, as you are noticing now. Try to have a separate file to handle all queries, and then use if you want a function to communicate between them.stants defined above, which may suit bett

2.- Keep as little logic in the views as possible. Create functions you can reuse, for example to retrieve data from db, or to populate the search data

3.- Constants. If the months, years, etc.. are not going to change consider using tuples for them. Also the range you are creating only has a year on it

With this, it should lead to places where you can see reusable code, specially db queries

Good luck!

## List comprehensions

This pattern is seen throughout your code:

[i for i in range(1, 13)]

In the case above, there's no need to make a list comprehension; simply hold onto the range directly. In a case like this:

rows = [i for i in rows]

you're better off calling the list constructor, i.e.

rows = list(rows)

or better, if you're able, the tuple constructor for immutable variables:

rows = tuple(rows)

## Redundant method check

This:

if request.method == 'GET':

is written on an index method that only accepts GET, so the if is redundant and can be dropped.

## set_search_date

This is a curious method. First, it doesn't set anything - it returns a date. Unless I'm missing something, rename this to be get_search_date. You can also combine your predicates:

if (
request.method == 'POST'
and all(
f'{k}_options' in request.form
for k in (
'month', 'day', 'year',
)
)
):
return '-'.join # ...

return search_date


text:
Matter Waves - INFN

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Barrier tunneling levels of the hydrogen atom. De Broglie wavelength and Heisenberg´s uncertainty relations, π. 2/ h. m/s) where λ is the Broglie wavelength of the particle in meters and h is Schrödinger's equation.

60. 3.2 Since ψ must be single-valued, the number of de Broglie wavelengths. Kuvahaun tulos haulle Schrödinger equation explanation Internet, Astronomi mechanics physics with wave particle duality solved | De Broglie Wavelength. particular, when the Fermi-wavelength of the conduction electrons is comparable The wave nature of the electron was predicted by L. de Broglie in 1924[32]: a In the BdG equations, the superconducting pair-potential is given by a complex. av H Fashandi · 2016 · Citerat av 1 — proportional to the square magnitude of the structure factor, (see equation (3.3)).

De Broglie Wavelength Formula is used to calculate the wavelength and momentum in any  Compton's formula established that an electromagnetic wave can behave like a Calculate the de Broglie wavelength of: (a) a 0.65-kg basketball thrown at a  Since the circumference of a cirle is 2pi times the radius, the general formula relating deBroglie wavelength to orbit radius is just the following: lambda(n) = 2pi x r(  The wavelength λ = h/p is called the de Broglie wavelength, and the relations λ = h/p and f = E/h are called the de Broglie relations. These are the same relations  The wavelength associated with an electron is related to the momentum of the electron by the de Broglie relation, λ = h/p. For electrons to probe the inner structure  De Broglie Wavelength Formula · The unit of the de Broglie wavelength is meters (m), though it is often very small, and so expressed in nanometers (1 nm = 10(-9)   The deBroglie wavelength is given by our equation above.

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(This is why the limiting resolution of an electron microscope is much higher than that of an optical microscope.) De Broglie Wavelength Formula. The following equation is used to calculate a de broglie wavelength.

Matter Waves - INFN

Spektrum Fysik Finalen Facit Kap 10 Referenser. Spektrum Fysik Facit Finalen Kapitel 10 Or ゴールデンカムイ Rar. Finally, because momentum  p  is equal to mass  m  times velocity  v:  \lambda = \frac {h} {p} λ = ph This is known as the de Broglie equation. As with any wavelength, standard unit of measure for the de Broglie wavelength is meters (m). de Broglie equation states that a matter can act as waves much like light and radiation, which also behave as waves and particles. The equation further explains that a beam of electrons can also be diffracted just like a beam of light. In essence, the de Broglie equation helps us understand the idea of matter having a wavelength. De Broglie Wavelength Formula De-Broglie waves explain about the nature of the wave related to the particle.

For particles with mass like electrons, protons, etc., but not the photons, there is another form of the de Broglie wavelength formula. At non-relativistic speeds, the momentum of a particle will be equal to its rest mass m, multiplied by its velocity v. The unit of the de Broglie wavelength is in meters. Thermal de Broglie Wavelength The thermal de Broglie wavelength (λ th) is approximately the average de Broglie wavelength of the gas particles in an ideal gas at the specified temperature. The thermal de Broglie wavelength is given by the expression: λ D = h / √ 2 π m k B T In this video, I'll explain how to properly use de Broglie's equation to find the wavelength and frequency. I'll also go over a really important unit convers De Broglie Wavelength Formula. The following equation is used to calculate a de broglie wavelength.
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The wavelength of these 'material waves' - also known as the de Broglie wavelength - can be calculated from Planks constant $$h$$ divided by the momentum $$p$$ of the particle. For particles with mass like electrons, protons, etc., but not the photons, there is another form of the de Broglie wavelength formula.
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2018-02-01 Solution: According to the de Broglie equation. Illustration: Calculate the de Broglie wavelength of a ball of mass 0.1kg moving at a speed of . According to Bohr’s model, the electron revolves around the nucleus in circular orbits. According to the de Broglie concept, the electron is not only a … de Broglie's wavelength of matter waves: de Broglie equated the energy equations of Planck(wave) and Einstein(particle).

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Svenska Fysikersamfundet - NanoPDF

These are the same relations  The wavelength associated with an electron is related to the momentum of the electron by the de Broglie relation, λ = h/p. For electrons to probe the inner structure  De Broglie Wavelength Formula · The unit of the de Broglie wavelength is meters (m), though it is often very small, and so expressed in nanometers (1 nm = 10(-9)   The deBroglie wavelength is given by our equation above. Solution: = = = 6.03 x 10-38 m. The diameter of an atomic nucleus  Instead, their motion is governed by a wave equation. Thus, the wave-particle duality first observed with photons is a fundamental behavior, intrinsic to all quantum  Apply the de Broglie wave equation λ=hmv λ = h m v to solve for the wavelength of the moving electron. Step 2: Calculate. λ=h  Both light and matter behave like a wave on a large scale and like a particle on a small scale.

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(v.) = moverse, agitarse. de broglie wave length density wave density wave quantum wave equation radio wave sawtooth  Which of the following equations represents Biot-savart law? If the kinetic energy of the moving particle is E, then the de Broglie wavelength is (a) λ = h / (b) λ  little optical absorption at wavelengths where stimulated emission may occur;. they are therefore very calculation/evaluation of SPR-dip movement have also been developed. This Broglie wavelength (~50 nm in silicon).

You just clipped your first slide! Clipping is a handy way to collect important slides you want to go back to later. Now customize the name of a clipboard to store your clips. De Broglie’s equation may be put in words as: The momentum of a particle in motion is inversely proportional to wavelength, Planck’s constant (h) being the constant of proportionality. Dec 28, 2020 French physicist Louis de Broglie won the Nobel Prize in 1929 for groundbreaking work in quantum mechanics.


text:
# types of binary relations

To understand the contemporary debate about relations we will need tohave some logical and philosophical distinctions in place. Some important types of binary relations R over sets X and Y are listed below. Theorem.If $R$ and $S$ are relations on $X$, then $(R\cup S)^{-1}=R^{-1}\cup S^{-1}$. Properties are “one-place” or“m… The inverse of $R$ is the relation $$R^{-1}=\{(b,a)\in X\times X : (a,b)\in R\}.$$. Theorem. The degree of a relationship is the number of entity types that participate(associate) in a relationship. If $R$ and $S$ are relations on $X$, then $(R\circ S)^{-1}=S^{-1}\circ R^{-1}$. Theorem. Proof. JavaTpoint offers too many high quality services. The basis step is obvious: $(R^{1})^{-1}=(R^{-1})^1$. Bases case, $i=1$ is obvious. I first define the composition of two relations and then prove several basic results. A Binary relation R on a single set A is defined as a subset of AxA. If (a, b) ∈ R and R ⊆ P x Q then a is related to b by R i.e., aRb. In fact, $(R^2)^{-1}=(R\circ R)^{-1}=R^{-1}\circ R^{-1}=(R^{-1})^2$. Let $R$ and $S$ be relations on $X$. Symmetric Relation 1. Let $R$ and $S$ be relations on $X$. Type 1: Divide and conquer recurrence relations – Following are some of the examples of recurrence relations based on divide and conquer. De nition: A binary relation from a set A to a set Bis a subset R A B: If (a;b) 2Rwe say ais related to bby R. Ais the domain of R, and Bis the codomain of R. If A= B, Ris called a binary … Composition of functions and invertible functions 5. The first of our 3 types of relations, we’ll start with is one-to-many. If sets P and Q are equal, then we say R ⊆ P x P is a relation on P e.g. JavaTpoint offers college campus training on Core Java, Advance Java, .Net, Android, Hadoop, PHP, Web Technology and Python. Let us discuss the concept of relation and function in detail Very useful concept in describing binary relationship types. If a is an element of a set A, then we write a A∈ and say a belongs to A or a is in A or a is a member of A.If a does not belongs to A, we write Copyright © 2021 Dave4Math, LLC. 1 Sets, Relations and Binary Operations Set Set is a collection of well defined objects which are distinct from each other. If $R$, $S$ and $T$ are relations on $X$, then $R\circ (S\cup T)=(R\circ S)\cup (R\circ T)$. If $R\subseteq S$, then $R^{-1}\subseteq S^{-1}$. \begin{align*} & (x,y)\in (R\cup S)^{-1} \Longleftrightarrow (y,x)\in R\cup S \Longleftrightarrow (y,x)\in R \lor (y,x)\in S \\ & \qquad  \Longleftrightarrow (x,y)\in R^{-1} \lor (x,y)\in S^{-1}  \Longleftrightarrow (x,y)\in R^{-1}\cup S^{-1} \end{align*}. Relation or Binary relation R from set A to B is a subset of AxB which can be defined as aRb ↔ (a,b) € R ↔ R (a,b). Theorem. It is also possible to have some element that is not related to any element in $X$ at all. Then $R^{-1}(A\cap B)\subseteq R^{-1}(A)\cap R^{-1}(B)$. Theorem. Introduction 2. Theorem. A binary relation R is defined to be a subset of P x Q from a set P to Q. So, go ahead and check the Important Notes for Class 11 Maths Sets, Relations and Binary Operations from this article. Proof. Theorem. \begin{align*} (x,y)\in & R\circ (S\circ T) \\ & \Longleftrightarrow \exists z\in X, (x,z)\in S\circ T \land (z,y)\in R\\ & \Longleftrightarrow \exists z\in X, [ \exists w\in X, (x,w)\in T \land (w,z)\in S ] \land  (z,y)\in R \\ & \Longleftrightarrow \exists w, z\in X, (x,w)\in T \land (w,z)\in S \land (z,y)\in R\\ & \Longleftrightarrow \exists w\in X, [\exists z\in X, (w,z)\in S \land (z,y)\in R] \land (x,w)\in T\\ & \Longleftrightarrow \exists w\in X, (x,w)\in T \land (w,y)\in R\circ S \\ & \Longleftrightarrow (x,y)\in (R\circ S) \circ T  \end{align*}. David is the founder and CEO of Dave4Math. Transitive Relation 1. We can say that the degree of relationship i… If $R$, $S$ and $T$ are relations on $X$, then $R\subseteq S \implies R\circ T \subseteq S\circ T$. Proof. Theorem. Let $R$ and $R_i$ be relations on $X$ for $i\in I$ where $I$ is an indexed set. Examples: < can be a binary relation over ℕ, ℤ, ℝ, etc. \begin{align*} & (x,y)\in (R\cap S)^{-1}  \Longleftrightarrow (y,x)\in R\cap S  \Longleftrightarrow (y,x)\in R \land (y,x)\in S \\ &  \qquad  \Longleftrightarrow (x,y)\in R^{-1} \land (x,y)\in S^{-1}  \Longleftrightarrow (x,y)\in R^{-1}\cap S^{-1} \end{align*}. Let $X$ be a set and let $X\times X=\{(a,b): a,b \in X\}.$ A (binary) relation $R$ is a subset of $X\times X$. If sets P and Q are equal, then we say R ⊆ P x P is a relation on P e.g. Theorem. For example, If we have two entity type ‘Customer’ and ‘Account’ and they are linked using the primary key and foreign key. Proof. Theorem. So, there are 24= 16 relations from A to A. i.e. In this article, I discuss binary relations. Proof. If $R$ and $S$ are relations on $X$ and $R(x)=S(x)$ for all $x\in X$, then $R=S$. Candidates who are pursuing in CBSE Class 11 Maths are advised to revise the notes from this post. The image of $A\subseteq X$ under $R$ is the set $$R(A)=\{y\in X : \exists \, x\in A, (x,y)\in R\}.$$. Theorem. Let P and Q be two non- empty sets. \begin{align*} (x,y)\in \left(\bigcup_{i\in I} R_i\right)\circ R & \Longleftrightarrow \exists z\in X, (x,z)\in R \land (z,y)\in \bigcup_{i\in I} R_i \\ & \Longleftrightarrow \exists z\in X, \exists i\in I, (x,z)\in R \land (z,y)\in R_i \\ & \Longleftrightarrow (x,y)\in \bigcup_{i\in I}(R_i\circ R) \end{align*}. The relationship from Driver_License to Person is optional as not al… Theorem. If $R$ and $S$ are relations on $X$ and $A, B\subseteq X$, then $A\subseteq B \implies R(A)\subseteq R(B)$. With the help of Notes, candidates can plan their Strategy for particular weaker section of the subject and study hard. For binary relationships, the cardinality ratio must be one of the following types: 1) One To One An employee can work in at most one department, and a department can have at most one employee. If $R$ and $S$ are relations on $X$, then $(R^c)^{-1}=(R^{-1})^c$. Examples: Some examples of binary relations are provided in an appendix. There are 8 main types of relations which include: 1. ↔ can be a binary relation over V for any undirected graph G = (V, E). Range of Relation: The range of relation R is the set of elements in Q which are related to some element in P, or it is the set of all second entries of the ordered pairs in R. It is denoted by RAN (R). Proof. So, there are 2n2 relations from A to A. Example2: If A has m elements and B has n elements. In this type the primary key of one entity must be available as foreign key in other entity. One-to-many relation. Person has the information about an individual and Driver_License has information about the Driving License for an individual. Universal Relation. Assume $R(x)=S(x)$ for all $x\in X$, then $$(x,y)\in R \Longleftrightarrow y\in R(x) \Longleftrightarrow y\in S(x) \Longleftrightarrow (x,y)\in S$$ completes the proof. Proof. Proof. Then $R^{-1}(A)\setminus R^{-1}(B)\subseteq R^{-1}(A\setminus B)$. If $R$, $S$ and $T$ are relations on $X$, then $(S\cup T)\circ R=(S\circ R)\cup (T\circ R)$. Definition (binary relation): A binary relation from a set A to a set B is a set of ordered pairs where a is an element of A and b is an element of B. \begin{align*} & x\in R^{-1}(A\cup B)  \Longleftrightarrow \exists y \in A\cup B, (x,y)\in R  \\ & \qquad \Longleftrightarrow \exists y\in A, (x,y)\in R \lor \exists y\in B, (x,y)\in R \\ & \qquad  \Longleftrightarrow x\in R^{-1}(A)\lor R^{-1}(B)  \Longleftrightarrow x\in R^{-1}(A)\cup R^{-1}(B) \end{align*}. The relations we are interested in here are binary relations on a set. Here one role group of one entity is mapped to one role group of another entity. Let $R$ and $S$ be relations on $X$. How many relations are there from A to B and vice versa? Theorem. Compositions of binary relations can be visualized here. If $R$ and $S$ are relations on $X$, then $(R\cap S)^{-1}=R^{-1}\cap S^{-1}$. Then $(R^n)^{-1}=(R^{-1})^n$ for all $n\geq 1$. \begin{align*} \qquad  y\in R(A) \Longleftrightarrow \exists x\in A, (x,y)\in R \implies \exists x\in B, (x,y)\in R \Longleftrightarrow y\in R(B) \end{align*}. \begin{align*} \qquad \quad  & (x,y) \in R\circ (S\cap T)  \\& \qquad  \Longleftrightarrow \exists z\in X, (x,z)\in S\cap T \land (z,y)\in R  \\& \qquad  \Longleftrightarrow \exists z\in X, [(x,z)\in S \land (x,z)\in T] \land (z,y)\in R  \\& \qquad  \Longleftrightarrow \exists z\in X, [(x,z)\in S \land (z,y)\in R] \land (x,z)\in T \\& \qquad  \Longleftrightarrow \exists z\in X, [(x,z)\in S \land (z,y)\in R] \land [(x,z)\in T \land (z,y)\in R] \\& \qquad  \Longrightarrow [\exists z\in X, [(x,z)\in S \land (z,y)\in R]   \land [ \exists w\in X, (x,w)\in T \land (w,y)\in R] \\& \qquad  \Longleftrightarrow (x,y)\in R\circ S \land (x,y)\in R\circ T  \\& \qquad  \Longleftrightarrow (x,y)\in (R\circ S) \cap (R\circ T) \end{align*}. Let’s start with a real-life problem. Solution: There are m x n elements; hence there are 2m x n relations from A to A. Example3: If a set A = {1, 2}. Example1: If a set has n elements, how many relations are there from A to A. Dave4Math » Introduction to Proofs » Binary Relations (Types and Properties). Sets of ordered pairs are called binary relations.Let A and B be sets then the binary relation from A to B is a subset of A x B. Theorem. 2) One To Many Theorem. In other words, a binary … We assume the claim is true for $j$. \begin{align*} \qquad & y\in R(A\cap B) \Longleftrightarrow \exists x\in X, x\in A\cap B \land (x,y)\in R \\ & \qquad \Longleftrightarrow  \exists x\in X, (x\in A \land x\in B) \land (x,y)\in R \\ & \qquad \Longrightarrow  \exists x\in A, (x,y)\in R \land \exists x\in B, (x,y)\in R \Longleftrightarrow  y\in R(A) \cap R(B) \end{align*}. Theorem. Let $R$ and $S$ be relations on $X$. Types of Functions 4. The induction step is (R^n)^{-1}=(R^{-1})^n\implies (R^{n+1})^{-1}=(R^{-1})^{n+1}. In simple terms one instance of one entity is mapped with only one instance of another entity. Proof. Let $R$ be a relation on $X$. If $R$ and $S$ are relations on $X$, then $(R\setminus S)^{-1}=R^{-1}\setminus S^{-1}$. so with the of help Binary operations we can solve such problems, ... HOW TO UNDERSTAND BINARY OPERATIONS IN RELATIONS … Theorem. Proof. Solution: If a set A has n elements, A x A has n2 elements. All rights reserved. A relation r from set a to B is said to be universal if: R = A * B. We’ll explain each of these relations types separately and comment on what is their actual purpose. The proof follows from the following statements. © Copyright 2011-2018 www.javatpoint.com. Thesedistinctions aren’t to be taken for granted. Another Example of Binary Relations In our phone number example, we defined a binary relation, L, from a set M to a set N. We can also define binary relations from a … Consider a relation R from a set A to set B. Relations and Their Properties 1.1. There are many types of relation which is exist between the sets, 1. Proof. Also, Parallel is symmetric, since if a line a is ∥ to b then b is also ∥ to a. Antisymmetric Relation: A relation R on a set A is antisymmetric iff (a, b) ∈ R and (b, a) ∈ R then a = b. Here we are going to learn some of those properties binary relations may have. Please mail your requirement at hr@javatpoint.com. \begin{align*} & (x,y)\in (R\circ S)^{-1}  \Longleftrightarrow (y,x)\in R\circ S \\ & \qquad  \Longleftrightarrow \exists z\in X, (y,z)\in S \land (z,x)\in R \\ &  \qquad  \Longleftrightarrow \exists z\in X, (z,y)\in S^{-1} \land (x,z)\in R^{-1} \\ & \qquad  \Longleftrightarrow \exists z\in X, (x,z)\in R^{-1} \land (z,y)\in S^{-1} \\ &  \qquad  \Longleftrightarrow (x,y)\in S^{-1} \circ R^{-1} \end{align*}. Then the complement, image, and preimage of binary relations are also covered. A woman who can be someone’s mother 2. Certain important types of binary relation can be characterized by properties they have. \begin{align*} & x\in R^{-1}(A)\setminus R^{-1}(B)\Longleftrightarrow  x\in R^{-1}(A) \land \neg(x\in R^{-1}(B))\\ & \qquad \Longleftrightarrow  x\in R^{-1}(A)\land [\forall y\in B, (x,y)\not\in R] \\ & \qquad \Longleftrightarrow  \exists y\in A, (x,y)\in R \land [\forall y\in B, (x,y)\not\in R]\\ & \qquad \Longrightarrow \exists y\in A\setminus B, (x,y)\in R \Longleftrightarrow x\in R^{-1}(A\setminus B)\end{align*}. By seeing an E-R diagram, we can simply tell the degree of a relationship i.e the number of an entity type that is connected to a relationship is the degree of that relationship. Part of thedevelopment of the debate has consisted in the refinement of preciselythese distinctions. \end{align*}. Proof. All rights reserved. Binary Relations Intuitively speaking: a binary relation over a set A is some relation R where, for every x, y ∈ A, the statement xRy is either true or false. The reason for that is that it’s the most commonly used and the remaining two are “subtypes” of this one. Then $R\circ \left(\bigcup_{i\in I} R_i\right)=\bigcup_{i\in I}(R\circ R_i)$. Reflexive Relation 1. The most important types of binary relations are equivalences, order relations (total and partial), and functional relations. A binary relation between members of X and members of Y is a subset of X ×Y — i.e., is a set of ordered pairs (x,y) ∈ X ×Y. Then $R^{-1}(A\cup B)=R^{-1}(A)\cup R^{-1}(B)$. A person that is a someone’s child 3. \begin{align*} y\in R(A)\setminus R(B)  & \Longleftrightarrow y\in R(A)\land y\not\in R(B) \\ & \Longleftrightarrow \exists x\in A, (x,y)\in R \land \forall z\in B, (z,y)\not\in R \\ & \Longleftrightarrow \exists x\in A\setminus B, (x,y)\in R \Longleftrightarrow y\in R(A\setminus B) \end{align*}. and M.S. A binary relation R is defined to be a subset of P x Q from a set P to Q. Duration: 1 week to 2 week. CS340-Discrete Structures Section 4.1 Page 1 Section 4.1: Properties of Binary Relations A “binary relation” R over some set A is a subset of A×A. The topics and subtopics covered in relations and Functions for class 12 are: 1. If $R$ and $S$ are relations on $X$, then $(R^{-1})^{-1}=R$. By induction. Domain of Relation: The Domain of relation R is the set of elements in P which are related to some elements in Q, or it is the set of all first entries of the ordered pairs in R. It is denoted by DOM (R). \begin{align*} (x,y)\in R\circ \left(\bigcup_{i\in I} R_i\right)  & \Longleftrightarrow  \exists z\in X, (x,z)\in \bigcup_{i\in I} R_i \land (z,y)\in R \\ & \Longleftrightarrow \exists z\in X, \exists i\in I, (x,z)\in R_i \land (z,y)\in R \\ & \Longleftrightarrow \exists i\in I, (x,y)\in R\circ R_i  \\ & \Longleftrightarrow (x,y) \in \bigcup_{i\in I}(R\circ R_i) \end{align*}. T (n) = 2T (n/2) + cn T (n) = 2T (n/2) + √n These types of recurrence relations can be easily solved using Master Method. If $(a,b)\in R$, then we say $a$ is related to $b$ by $R$. Binary Relation. Proof. Theorem. Proof. The proof follows from the following statements. Proof. If (a, b) ∈ R and R ⊆ P x Q then a is related to b by R i.e., aRb. Proof. Universal Relation 1. Then $R^n \cup S^n\subseteq (R\cup S)^n$ for all $n\geq 1$. After that, I define the inverse of two relations. \begin{align*} & (x,y)\in (R\setminus S)^{-1}  \Longleftrightarrow (y,x)\in R\setminus S  \Longleftrightarrow (y,x)\in R \land (y,x)\notin S \\ & \qquad \Longleftrightarrow (x,y)\in R^{-1} \land (y,x)\notin S \Longleftrightarrow (x,y)\in R^{-1} \land (x,y)\notin S^{-1} \\ & \qquad  \Longleftrightarrow (x,y)\in R^{-1}\setminus S^{-1} \end{align*}, Definition. \begin{align*} & x\in R^{-1}(A\cap B)  \Longleftrightarrow \exists y\in A \cap B, (x,y)\in R \\ & \qquad \Longleftrightarrow \exists y\in X, y\in A \land y\in B \land (x,y)\in R \\ & \qquad \Longrightarrow  x\in R^{-1}(A) \land x\in R^{-1}(B) \Longleftrightarrow x\in R^{-1}(A) \cap x\in R^{-1}(B)\end{align*}. X and Y are listed below two sets and powers only one instance of entity! And Y are listed below 8 main types of relation R from set a to B said. P and Q be two sets, with it ’ S mother 2 B be two non- empty sets a! 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# Simplify the expression (4x+8)+(-6x). Explain how the associative and commutative properties were used to solve the expression?

Jul 14, 2018

$- 2 \left(x - 4\right)$

#### Explanation:

1. use the distribution property to change $+ \times \left(- 6 x\right)$ to -6x  ( + xx - = -)
2. remove the parenthesis giving 4x +8
( there are several ways to proceed from here my choice is)
3. use the commutative property to move the +8 and - 6x
$+ 4 x + 8 - 6 x = 4 x - 6 x + 8$ commutative property

4. Use the associative property to group + 4x -6x
$+ 4 x - 6 x + 8 = \left(+ 4 x - 6 x\right) + 8$ associative property

5. Use algebraic addition to solve for $\left(+ 4 x - 6 x\right)$
$\left(+ 4 x - 6 x\right) + 8 = - 2 x + 8$

6. Use the reverse distribution principal to simply and remove common terms

$- 2 \times \left\{\frac{- 2 x}{- 2} + \frac{8}{- 2}\right\} = - 2 \left(x - 4\right)$


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# Lesson 9

Constant Speed

### Lesson Narrative

In the previous lesson, students used the context of shopping to explore how equivalent ratios and ratios involving one can be used to find unknown amounts. In this lesson, they revisit these ideas in a new context—constant speed—and through concrete experiences. Students measure the time it takes them to travel a predetermined distance—first by moving slowly, then quickly—and use it to calculate and compare the speed they traveled in meters per second.

Here, double number lines are used to represent the association between distance and time, and to convey the idea of constant speed as a set of equivalent ratios (e.g., 10 meters traveled in 20 seconds at a constant speed means that 0.5 meters is traveled in 1 second, and 5 meters is traveled in 10 seconds). Students come to understand that, like price, speed can be described using the terms per and at this rate.

The idea of a constant speed relating the quantities of distance and time is foundational for the later, more abstract idea of a constant rate, and is important in the development of students’ ability to reason abstractly about quantities (MP2).

### Learning Goals

Teacher Facing

• Calculate the distance an object travels in 1 unit of time and express it using a phrase like “meters per second” (orally and in writing).
• For an object moving at a constant speed, use a double number line diagram to represent equivalent ratios between the distance traveled and elapsed time.
• Justify (orally and in writing) which of two objects is moving faster, by identifying that it travels more distance in the same amount of time or that it travels the same distance in less time.

### Student Facing

Let’s use ratios to work with how fast things move.

### Required Preparation

Before class, set up 4 paths with a 1-meter warm-up zone and a 10-meter measuring zone.

### Student Facing

• I can choose and create diagrams to help me reason about constant speed.
• If I know an object is moving at a constant speed, and I know two of these things: the distance it travels, the amount of time it takes, and its speed, I can find the other thing.

Building On

### Glossary Entries

• double number line diagram

A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but the other numbers are usually different.

• meters per second

Meters per second is a unit for measuring speed. It tells how many meters an object goes in one second.

For example, a person walking 3 meters per second is going faster than another person walking 2 meters per second.

• per

The word per means “for each.” For example, if the price is $5 per ticket, that means you will pay$5 for each ticket. Buying 4 tickets would cost $20, because $$4 \boldcdot 5 = 20$$. • unit price The unit price is the cost for one item or for one unit of measure. For example, if 10 feet of chain link fencing cost$150, then the unit price is $$150 \div 10$$, or \$15 per foot.

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