text:
# Power control¶

This example is adapted from Boyd, Kim, Vandenberghe, and Hassibi,A Tutorial on Geometric Programming.”

The problem data is adapted from the corresponding example in CVX’s example library (Almir Mutapcic).

This example formulates and solves a power control problem for communication systems, in which the goal is to minimize the total transmitter power across n trasmitters, each trasmitting positive power levels $$P_1$$, $$P_2$$, $$\ldots$$, $$P_n$$ to $$n$$ receivers, labeled $$1, \ldots, n$$, with receiver $$i$$ receiving signal from transmitter $$i$$.

The power received from transmitter $$j$$ at receiver $$i$$ is $$G_{ij} P_{j}$$, where $$G_{ij} > 0$$ represents the path gain from transmitter $$j$$ to receiver $$i$$. The signal power at receiver $$i$$ is $$G_{ii} P_i$$, and the interference power at receiver $$i$$ is $$\sum_{k \neq i} G_{ik}P_k$$. The noise power at receiver $$i$$ is $$\sigma_i$$, and the signal to noise ratio (SINR) of the $$i$$th receiver-transmitter pair is

$S_i = \frac{G_{ii}P_i}{\sigma_i + \sum_{k \neq i} G_{ik}P_k}.$

The transmitters and receivers are constrained to have a minimum SINR $$S^{\text{min}}$$, and the $$P_i$$ are bounded between $$P_i^{\text{min}}$$ and $$P_i^{\text{max}}$$. This gives the problem

$\begin{split}\begin{array}{ll} \mbox{minimize} & P_1 + \cdots + P_n \\ \mbox{subject to} & P_i^{\text{min}} \leq P_i \leq P_i^{\text{max}}, \\ & 1/S^{\text{min}} \geq \frac{\sigma_i + \sum_{k \neq i} G_{ik}P_k}{G_{ii}P_i}. \end{array}\end{split}$
import cvxpy as cp
import numpy as np

# Problem data
n = 5                     # number of transmitters and receivers
sigma = 0.5 * np.ones(n)  # noise power at the receiver i
p_min = 0.1 * np.ones(n)  # minimum power at the transmitter i
p_max = 5 * np.ones(n)    # maximum power at the transmitter i
sinr_min = 0.2            # threshold SINR for each receiver

# Path gain matrix
G = np.array(
[[1.0, 0.1, 0.2, 0.1, 0.05],
[0.1, 1.0, 0.1, 0.1, 0.05],
[0.2, 0.1, 1.0, 0.2, 0.2],
[0.1, 0.1, 0.2, 1.0, 0.1],
[0.05, 0.05, 0.2, 0.1, 1.0]])
p = cp.Variable(shape=(n,), pos=True)
objective = cp.Minimize(cp.sum(p))

S_p = []
for i in range(n):
S_p.append(cp.sum(cp.hstack(G[i, k]*p[k] for k in range(n) if i != k)))
S = sigma + cp.hstack(S_p)
signal_power = cp.multiply(cp.diag(G), p)
inverse_sinr = S/signal_power
constraints = [
p >= p_min,
p <= p_max,
inverse_sinr <= (1/sinr_min),
]

problem = cp.Problem(objective, constraints)

problem.is_dgp()

True

problem.solve(gp=True)
problem.value

0.9615384629119621

p.value

array([0.18653846, 0.16730769, 0.23461538, 0.19615385, 0.17692308])

inverse_sinr.value

array([5., 5., 5., 5., 5.])

(1/sinr_min)

5.0


text:
## A finite group is nilpotent if and only if all pairs of elements of relatively prime order commute

Prove that a finite group $G$ is nilpotent if and only if whenever $a, b \in G$ with $\mathsf{gcd}(|a|,|b|) = 1$, then $ab = ba$. [Hint: Use Theorem 3.]

$(\Rightarrow)$ Suppose $G$ is a finite nilpotent group. Let $a,b \in G$ with $\mathsf{gcd}(|a|,|b|) = 1$. By Theorem 6.3, $G$ is the internal direct product $G = P_1P_2 \cdots P_k$ of its Sylow subgroups; thus we may write $a = a_1a_1\cdots a_k$ and $b = b_1b_2 \cdots b_k$, where $a_i,b_i \in P_i$. Moreover, $|a| = \mathsf{lcm}(|a_i|)$ and $|b| = \mathsf{lcm}(|b_i|)$; if some $a_i \neq 1$, then $p_i$ divides $|a|$. so $p_i$ does not divide $|b|$, and $b_i = 1$. Similarly, if $b_i \neq 1$ then $a_i = 1$.

By relabeling the Sylow subgroups of $G$ and collecting factors, we have the internal direct product $G = Q_1 \times Q_2$, with $a \in Q_1$ and $b \in Q_2$. Thus $ab = ba$.

$(\Leftarrow)$ Suppose $G$ is a finite group in which the hypothesis holds. Let $P \leq G$ be a Sylow $p$-subgroup. If $Q \leq G$ is a Sylow subgroup for some other prime $q$, then $Q \leq C_G(P) \leq N_G(P)$. In particular, $N_G(P)$ contains a subgroup of maximal $q$-power order for each $q$ dividing $|G|$; thus $N_G(P) = G$, so that $P \leq G$ is normal. Since $P$ is arbitrary, all Sylow subgroups of $G$ are normal. By Theorem 3, $G$ is nilpotent.

format: plain


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# Ruin-time in Gambler's Ruins

Could someone give me some tips?

Let $e_1,e_2,\dots$ be iid normal mean 0 variance 1. Let $X_t := e_1+\cdots+e_t$, for $t=1,2,\dots$ and $X_0 := 0$. (So we have a discrete-time random walk whose steps are iid $N(0,1)$)

Define first-hitting time $T_0 := \inf\{t>0 : X_t <0\}$.

What is the CDF of $T_0$, i.e. what is $\mathbb{P}(T_0 \leq t)$?

And second (probably more difficult) question: what is $\mathbb{E}( X_t \mid T_0 > t )$?

Thank you.

You are looking at a Brownian motion, evaluated at integer times $t$. Then defining $$T_0=\inf(t > 0, t\in {\mathbb Z}: B_t < 0)$$ and integer $t > 0$ we have $$P(T_0 > t)=P(B_1 > 0,B_2 > 0,\dots, B_t > 0)={2t\choose t}/4^t.$$
The key insight is that you should not to try to solve the probability exactly, but rather set up a recursion that the probabilities solve. Setting $p(t)=P(T_0>t)$ for integer $t\geq 0$, you can show that for any $t$ we have $\sum_{j=0}^t p(t-j)p(j)=1$.
The probabilistic justification for the recursion formula is that every path of a diffuse, symmetric, discrete-time random walk can be broken into two positive paths at the time $j$ where it achieves its minimum value.


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# Discontinuous plot when it should not be

I am trying to fix the plot in the manipulate to be continuous. I used Exclusion->None and adjusted the PlotPoints too. Nothing has worked so far. Could anybody help me to have a continuous plot?

R = 8.3145;
S = 0.2;
Solve[{(S - 2 vo)^2 (-3 + S - vo) +
Exp[(-i + (T + 273) j)/(R (T + 273))]*Sqrt[k] vo == 0}, {vo},
Reals][[3]][[1]][[2]]

Manipulate[
Plot[%444, {T, 200, 900},
AxesLabel -> {Temperature[\[Degree]C], \[Delta]},
Exclusions -> None], {{k, 0.21, "pO2"}, {0.21, 0.01, 0.0015,
0.00003},
Setter}, {{i, -500000,
"\[CapitalDelta]H\[Degree]"}, -500000, -50000}, {{j, -500,
"\[CapitalDelta]S\[Degree]"}, -500, -100}]

• Edit your post and include the code formatted code for the Manipulate. – Edmund May 7 '20 at 23:49
• Sorry about that, I just edited my question with Manipulate code. – user72564 May 8 '20 at 2:59

The problem isn't with Plot, it's with the function you're plotting. Note that %444 /. {k -> 0.21, i -> -500000, j -> -500, T -> 690} returns undefined. This is due to the condition that follows the Root function. All of the conditions return False so the equation is left unevaluated. The solution to your equation consists of 6 possible solutions. My guess is that one of the other solutions fills in the missing part.

It looks like the fifth solution fills in the missing part.

R = 8.3145;
S = 0.2;
nsol = vo /.
Solve[{(S - 2 vo)^2 (-3 + S - vo) +
Exp[(-i + (T + 273) j)/(R (T + 273))]*Sqrt[k] vo == 0}, {vo},
Reals]
Manipulate[
Plot[{
With[{k = k, i = i, j = j}, nsol[[3]]],
With[{k = k, i = i, j = j}, nsol[[5]]]
},
{T, 200, 900},
PlotStyle -> ColorData[97][1]
],
{{k, 0.21, "p02"}, {0.21, 0.01, 0.0015, 0.00003}},
{{i, -500000, "\[CapitalDelta]H"}, -500000, -50000},
{{j, -500, "dS"}, -500, -100}
]

I added in PlotStyle -> ColorData[97][1] so that both parts would plot in the same colour, but if you prefer that they plot in different colours, you can simply remove that line.

Also, as Edmund mentioned, please include all of the code whenever possible rather than a screenshot. If you forget, you can always edit the question later to include it. This makes it much easier for people to help out because we don't have to retype all of the code in the screenshot, we can just copy and paste.

• Thank you so much for helping me to solve this. Sorry about the coding I forgot to include. I learned a new thing today! – user72564 May 8 '20 at 2:56


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# 1D topological insulator

This question is inspired by another one about the simplest model of topological insulator, where 4tnemele showed a nice two band model in the answer.

I read that and am wondering if we and push that to one dimension.

For example, by analogy to the graphene case, if we have a Hamiltonian in 1D (say x) as $H(k_x)=(k_x-k_0)+m$ for $k_x>0$. When $k_x=k_0$, one has $m>0$. $H(k_x)=(k_x+k_0)+m$ for $k_x<0$. When $k_x=-k_0$, one has $m<0$. A smooth connection in between, we will have a conductive edge (two ends in the 1D structure), right?

If I want to make a intuitive picture like below, is it correct?

Any suggestion for real materials show this behavior?

-
I can't say anything too insightful as an answer to your actual question, but I think it's interesting to note that novel "edge modes" on the free ends of 1D systems are actually quite generic, two beautiful examples being emergent spin-1/2 excitations at the tips of S=1 Heisenberg magnets (see also the AKLT chain) or Majorana fermion modes on the ends of the Kitaev chain. – wsc Feb 26 '11 at 4:01

## 1 Answer

Topological insulators are gapped states of free fermions with particle number conservation and time-reversal symmetry. According to the K-theory classification, there is no Topological insulator in 1D.

However, 1D interacting fermions with time-reversal symmetry do have non-trivial symmetry protected topological phases if the particle number is conserved only mod n. The result can be obtained from group cohomology theory arXiv:1106.4772 of Chen, Gu, Liu, and Wen.

-
Dear Xiao-Gang Wen, if you cite yourself, it would be good if you could say so explicitly in your answers, not just in the links, cf. Physics.SE policy and SE policy. – Qmechanic May 30 '12 at 17:07
Hey Prof. Wen, how to understand the particle number conservation in topological insulators. Normally people only say it is time-reversal invariant. Thank you! – Timothy Oct 16 '12 at 13:02
@Qmechanic I thought "Chen, Gu, Liu, and Wen" explicitly contains my name Wen. Jeremy: Yes, normally people only say it is time-reversal invariant. But "insulator" implies particle number conservation. – Xiao-Gang Wen Oct 18 '12 at 5:34
Dear @Xiao-Gang Wen: What is meant is, that it would be best, if you could refer to yourself using a 1st person pronoun. – Qmechanic Oct 18 '12 at 5:44
@Qmechanic OK， Thanks – Xiao-Gang Wen Oct 18 '12 at 6:09


text:
## mckayla_04 Group Title Create your own radical equation. Describe in complete sentences and demonstrate the process in finding its solution(s). 2 years ago 2 years ago

The two legs of a right triangle are 3 and4 inches. Calculate the hypotenuse. Let x = hypotenuse then $x=\sqrt{3^{2}+4^{2}}$
$x=\sqrt{9 +16}=\sqrt{25}=5$


text:
# Home

Score : $100$ points

### Problem Statement

Given is an integer $x$ that is greater than or equal to $0$, and less than or equal to $1$. Output $1$ if $x$ is equal to $0$, or $0$ if $x$ is equal to $1$.

### Constraints

• $0 \leq x \leq 1$
• $x$ is an integer

### Input

Input is given from Standard Input in the following format:

$x$

### Output

Print $1$ if $x$ is equal to $0$, or $0$ if $x$ is equal to $1$.

### Sample Input 1

1

### Sample Output 1

0

### Sample Input 2

0

### Sample Output 2

1


text:
# National Health Survey: Persons accessing Pharmaceutical Benefits Scheme subsidised prescriptions, 2014-15

Exploratory analysis comparing medications self-reported in the National Health Survey, 2014-15 and accessed through Pharmaceutical Benefits Scheme

Released
9/10/2020

## Purpose, background, driver for change and scope

### Purpose

This research paper showcases the integration of the 2014-15 National Health Survey (NHS) with the Multi-Agency Data Integration Project (MADIP) asset. The MADIP asset includes data from the Pharmaceutical Benefits Scheme (PBS) which funds access to the supply of medications via subsidised prescriptions through the PBS and the Repatriation Pharmaceutical Benefits Scheme (RPBS) by the Australian Government. The analysis presented assesses the feasibility of using data from the PBS represented in the MADIP asset to replace the medications questionnaire module in future NHS cycles, and explores the use of PBS subsidised prescriptions by participants in the linked 2014-15 NHS dataset.

### Background

The NHS is an Australia-wide health survey conducted by the Australian Bureau of Statistics (ABS) which collects information about the health of people, including:

• prevalence of long-term health conditions
• health risk factors such as smoking, overweight and obesity, alcohol consumption and physical activity
• demographic and socioeconomic characteristics
• health related actions such as medication use.

MADIP is a partnership among Australian Government agencies to develop a secure and enduring approach for combining information on healthcare, education, government payments, personal income tax, and population demographics (including the Census) to create a comprehensive picture of Australia over time.

The 2014-15 NHS is the first ABS household survey to be linked to the MADIP asset and provides the opportunity to explore, investigate and better understand relationships between health status as measured in the NHS and use of government services. For more information regarding the integration of the 2014-15 NHS with the MADIP asset, see: 4321.0 - Research Paper: Integration of the National Health Survey with the Multi-Agency Data Integration Project (MADIP), 2014-15.

### Driver for change

The ABS is making more use of administrative data to help inform policy decisions and research questions. Using administrative data has many advantages compared with collecting data through surveys including: reducing respondent burden; minimising respondent recall bias; and reducing costs. The ABS uses supermarket scanner data to improve the processing of the Consumer Price Index, tax data to support business collections, and more recently has explored opportunities to replace some household survey questions with existing administrative data from the MADIP.

One such example is covered in this research paper, where the ABS proposes to replace the direct collection of medications information with the PBS subsidised prescriptions data in the MADIP asset.

### Scope

The 2014-15 NHS was conducted in all states and territories and across urban, rural and remote areas of Australia from July 2014 to June 2015. The survey included 19,257 people in 14,723 private dwellings.

Results in this research paper are based on the 18,287 2014-15 NHS records that were linked to the MADIP asset (a linkage rate of 95%). The linked records have been re-weighted to represent the population estimates of the NHS representing approximately 23 million people in Australia in 2014-15. For more information on weighting, see the Explanatory Notes.

As the scope of the 2014-15 NHS excluded very remote areas and discrete Aboriginal and Torres Strait Islander communities, there is no coverage of these populations in the linked dataset.

For the linked records, the analysis presented in this paper is based on PBS subsidised prescriptions for the 12 months prior to the NHS interview. PBS subsidised prescriptions are included based on the date of supply that a subsidised prescription medicine was filled at a Pharmacy. As a result, PBS records used have supply dates ranging from approximately July 2013 to June 2015. The 12 month data period allows for the analysis of patterns of medication use.

## Interpretation of results

There are a number of factors that should be considered when interpreting information presented in this research paper.

### 2014-15 National Health Survey (NHS)

Self-reported medication use was collected in the medications questionnaire module of the 2014-15 NHS and it is important to note the following:

• All medication use is captured in a face-to-face interview and represent medications used in the two weeks prior to interview.
• Medication use was self-reported by respondents, which may have implications for the extent to which certain types of medication were reported.
• Whilst respondents were encouraged to collect their medications to assist in recall, this did not always occur, which may have led to some medications not being reported, or being reported incorrectly.
• Some respondents may have provided responses that they felt were expected, rather than those that accurately reflected their own situation. Every effort has been made to minimise such bias through the development and use of appropriate survey methodology.
• Interviewers recorded the Australian Register of Therapeutic Goods Administration (TGA) identification number of each medication taken by the respondent. These were either:
• AUST R medicines - which include all prescription medications and many over-the-counter products such as those used for pain relief, coughs and colds and antiseptic creams; or
• AUST L medicines - which are generally lower risk self-medication products which include vitamins, minerals, and herbal and homoeopathic products.
• A small number of medications presented by respondents were not able to be coded to the World Health Organisation Anatomical Therapeutic Chemical (ATC) classification system[1], and were not included in output data. For example, for medications containing multiple active ingredients, or medications containing ingredients used for more than one therapeutic application. This is expected to have negligible effect on the accuracy of results.
• Information on what the medication was being taken for or prescribed for is not captured.
• For further information on the 2014-15 NHS, refer to the Explanatory Notes and User Guide.

### Pharmaceutical Benefits Scheme (PBS)

For PBS subsidised prescriptions represented in the MADIP asset, it is important to note the following:

• The Scheme is available to all Australian residents who hold a current Medicare card. For certain countries, the Australian Government’s Reciprocal Health Care Agreement (RHCA) allows international visitors to access PBS subsidised prescriptions.
• The subsidy of some prescriptions is restricted to particular population or clinical subgroups which may impact on the representation of the data available.
• There are programs operating under Section 100 of the National Health Act 1953 particularly in remote and very remote areas, such as the Aboriginal Medical Services, which receive access to free and subsidised medicines, at times on a bulk supply basis, that are not captured through the PBS data when distributed to patients[2].
• PBS subsidised prescriptions can be identified based on timeframes using the prescription supply date.
• PBS subsidised prescriptions do not include over-the-counter, private prescriptions, dietary supplements or medications supplied to public hospital in-patients.
• Some prescription medications are not captured through PBS data, these are referred to as non-PBS prescriptions. Examples include prescription medicines that are not listed on the PBS and medications that are being used outside of its PBS manner of administration (such as PBS eye ointment for topical non-ophthalmic use)[3].
• PBS subsidised prescriptions tell us when a prescription is filled but not whether the patients took the medication.
• The ABS update the MADIP asset on an annual basis with data on the PBS received from Services Australia. The MADIP asset (at the time of this research) contained PBS subsidised prescriptions from 2011 to 2016.

Self-reported medication use in the 2014-15 NHS and PBS subsidised prescriptions have the following in common:

• A person’s use of medication(s) does not imply a diagnosis of a condition.
• It is not known whether people actually used the medications they self-reported in the 2014-15 NHS or the medications for which subsidised prescriptions were filled in the PBS.

### Types of medications in Australia

There are three types of medications available in Australia[4]. These are:

• Complementary: non-prescription medicines available predominantly over the counter from health food shops, supermarkets and pharmacies.
• Over-the-counter: non-prescription medicines available over the counter or accessible from a pharmacist after a consultation. These medicines are available in pharmacies with selected products also available in supermarkets, health food stores and other retailers. Such medicines are typically used for mild health problems.
• Prescription: medicines that can only be accessed via written instruction of an authorised health professional, for example a General Practitioner. These medicines are only available for purchase from a pharmacy.

All medicines in Australia must be either registered or listed with the Australian Register of Therapeutic Goods (ARTG). All prescription medicines in Australia are registered as they carry higher risks when consumed, thus requiring a valid prescription from a health professional to gain access. The Australian Government subsidises the access to prescription medication through the PBS and RPBS, and it is such medicines that are represented in the MADIP asset.

Over-the-counter and Complementary medications are either register or listed, but can be accessed off the shelf (some requiring a discussion with a Pharmacist) as they carry lower risk upon consumption. When purchased off the shelf without a prescription these medicines are not subsidised through the PBS or RPBS and are not represented in the MADIP asset.

### Endnotes

1. World Health Organisation (WHO) Collaborating Centre for Drug Statistics Methodology www.whocc.no/atc_ddd_index/ last accessed 12/03/2020.
2. The Department of Health. Aboriginal Health Services and the Pharmaceutical Benefits Scheme (PBS) https://www1.health.gov.au/internet/main/publishing.nsf/Content/health-pbs-indigenous last accessed 19/05/2020.
3. Services Australia. Education guide - Writing PBS and RPBS prescriptions https://www.servicesaustralia.gov.au/organisations/health-professionals/services/medicare/highly-specialised-drugs/resources/education-guide-writing-pbs-and-rpbs-prescriptions#a6 last accessed 19/05/2020.
4. Therapeutic Goods Administration, The Department of Health. How we regulate medicine https://www.tga.gov.au/how-we-regulate-medicines last accessed 15/05/2020.

## Assessing the feasibility of using data from the PBS

This section compares the self-reported medication use data from the 2014-15 NHS to the PBS subsidised prescriptions data represented in the MADIP asset to determine the suitability of replacing the medications questions module in future NHS cycles with the rich PBS administration data source.

To compare the two data sources, comparability of reference periods was explored. The self-reported medications use data from the 2014-15 NHS represents medications used in the two weeks prior to the NHS interview. Using the supply date which represents when a PBS subsidised prescription was filled at a pharmacy, PBS subsidised prescriptions were categorised into two time periods representing:

• PBS subsidised prescriptions filled in the month of or month prior to interview (referred to as month prior to 2014-15 NHS interview)
• PBS subsidised prescriptions filled up to 12 months prior to interview (referred to as 12 months prior to 2014-15 NHS interview).

These categories were chosen because the PBS supply date available in MADIP is represented only by month and year, meaning data comparisons could not be made specifically for the two weeks prior to the 2014-15 NHS interview. The 12 month prior period was of interest to consider for medicine use for long-term health conditions.

Of the linked 2014-15 NHS population, there was generally more people self-reporting medication use compared with people who had at least one script filled for a PBS subsidised prescription in the month prior to the NHS interview (46.7% compared with 41.6%). Greater similarity can be observed between the 2014-15 NHS and PBS subsidised prescriptions typically for ATC Classification System groups where medications are only available via prescription for example, Cardiovascular system (18.0% compared with 17.8% respectively). However, this pattern is not evident across all of the ATC Classification Systems groups and is particularly inconsistent where over-the-counter medications are typically used, for example, Blood and blood forming organs (7.5% compared with 3.9% respectively) see Table 1 below.

Table 1: Persons medication use self-reported in the 2014-15 NHS and PBS subsidised prescription(s) use
People who self-reported medication use(a) in the 2 weeks prior to 2014-15 NHS interviewPeople who had at least one script filled for a PBS(b) subsidised medicine in the month prior 2014-15 NHS interviewPBS-NHS difference for medication use in the month prior to 2014-15 NHS interviewPeople who had at least one script filled for a PBS(b) subsidised medicine in the 12 months prior to 2014-15 NHS interviewPBS-NHS difference for medication use in the 12 months prior to 2014-15 NHS interview
Anatomical Therapeutic Chemical (ATC) Classification System(c)

Proportion(%)
Alimentary tract and metabolism11.411.90.519.98.5
Blood and blood forming organs7.53.9-3.67.3-0.2
Cardiovascular system18.017.8-0.221.03.0
Dermatologicals1.12.51.410.79.6
Genito-urinary system and sex hormones4.94.1-0.89.64.7
Systemic hormonal preparations, excluding sex hormones and insulins4.13.1-1.010.76.6
Antiinfectives for systemic use2.912.69.745.542.6
Antineoplastic and immunomodulating agents1.31.2-0.11.80.5
Musculo-skeletal system8.45.0-3.413.04.6
Nervous system21.214.9-6.326.25.0
Antiparasitic products, insecticides and repellents0.10.0-0.10.20.1
Respiratory system8.04.4-3.610.72.7
Sensory organs1.23.42.210.49.2
Various0.10.20.10.60.5
Total(d)46.741.6-5.170.523.8

a.   The National Health Survey (NHS) collects information on medications taken by respondents in the 2 weeks prior to the survey interview. The medications were coded, based on their active ingredient(s) and their therapeutic application, to the World Health Organisation Anatomical Therapeutic Chemical (ATC) classification system.
b.   The Pharmaceutical Benefits Scheme (PBS) is part of the Australian Government’s National Medicines Policy which aims to meet medication and related service needs so that optimal health outcomes and economic objectives are achieved.
c.   The ATC classification system has fourteen main anatomical or pharmacological groups (first level). These first level anatomical main groups are presented in this publication.
d.   Sum of component items do not equal totals as multiple responses are possible.

As expected, when the 2014-15 NHS two week self-reported medication use was compared to the PBS subsidised prescriptions filled up to 12 months prior to the 2014-15 NHS interview, a greater difference between the two data sources was observed for most ATC groups. However, where over-the-counter medications are readily used (for example, for Blood and blood forming organs, Respiratory system and Nervous System groups where aspirin, Ventolin, pain relief and cold medications are respectively used) the same pattern does not hold.

For information on over-the-counter and prescription medications in Australia, please refer to the Interpretation of results section.

### Agreement between 2014-15 NHS and the PBS subsidised prescriptions

To assess the impact of replacing the NHS medications questionnaire module with PBS subsidised prescriptions available via the MADIP asset, a proportion agreement between the two data sources was calculated as follows. The proportion agreement uses the frequency of agreements and disagreements observed for the linked records which have the presence (or non-presence) of a self-reported or PBS accessed medication in the month prior to interview, in a particular ATC group (i.e. a binary categorical variable).

$$\large\ Proportion\space Agreement = {\frac{Weighted\space number\space of\space records\space agreed\space between\space NHS\space and\space PBS}{Weighted\space number\space of\space records}}$$

Although test statistics such as Cohen’s ĸ could be used to statistically test the level of agreement between the two datasets, such statistics have several limitations, including difficulty of interpretation[5] therefore the decision was made to report proportion agreement, which is directly interpretable.

The proportion agreement was calculated for each ATC Classification System group (see Table 2). High levels of agreement were achieved for most ATC Classification System groups when comparing self-reported medication use from the 2014-15 NHS to the PBS subsidised prescriptions represented in the MADIP asset.

Like the raw proportion comparisons in Table 1, levels of agreement were not as strong for ATC Classification System groups where over-the-counter medications are readily used, such as the Nervous system group.

Caution should be used when interpreting the level of agreement for the various ATC Classification System groups, as groups with a very small proportion of usage will inherently have a high proportion agreement. The proportion agreement should be viewed in conjunction with Table 1 to assess the overall comparability of the two data sources.

Table 2: Agreement Statistic for each ATC Classification System group
Anatomical Therapeutic Chemical (ATC) Classification System(a)(b)Level of agreement Proportion (%)
Alimentary tract and metabolism94.0
Blood and blood forming organs94.7
Cardiovascular system96.4
Dermatologicals97.1
Genito-urinary system and sex hormones94.5
Systemic hormonal preparations, excluding sex hormones and insulins96.1
Antiinfectives for systemic use88.6
Antineoplastic and immunomodulating agents99.3
Musculo-skeletal system92.1
Nervous system86.3
Antiparasitic products, insecticides and repellents99.8
Respiratory system92.3
Sensory organs96.8
Various99.8

a.   The ATC classification system has fourteen main anatomical or pharmacological groups (first level). These first level anatomical main groups are presented in this publication.
b.   Persons who accessed a PBS subsidised medicine in the month prior to interview.

Generally, results showed a higher level of agreement between self‐report and prescription based data for medications used on a regular and ongoing basis (for example, for treatment of chronic conditions such as heart disease). In contrast, the difference between the two data sources for some medications may be explained, to a greater or lesser extent, by their availability over-the-counter and also, those medications that are used intermittently, infrequently or on a short-term basis (such as the use of antibiotics for acute infections). There have been similar results found in other research studies comparing self-reported and prescription data sources[6,7].

Overall, the linkage between the 2014-15 NHS and the PBS subsidised prescriptions as represented in the MADIP asset appears feasible to replace the survey data. For ATC Classification Groups where over-the-counter medications are commonly used, data could possibly be supplemented through specifically asking about the use of key over-the-counter medications in the survey.

Replacing survey content with the PBS subsidised prescriptions would help improve survey efficiency in future NHS collections. Whilst the PBS was not designed for research or planning purposes, the data are an objective and accurate record of medicines that were supplied and when they were supplied. Both data sources have pros and cons, but when taking into account the availability of existing data sources and respondent burden it appears feasible to use administrative data where possible as a survey data replacement. The ABS will endeavour to investigate whether additional administrative data sources exist that would help address over-the-counter medications, private prescriptions, dietary supplements and other data gaps.

After comparing time windows, it would be preferable to replace the NHS medications questionnaire module with PBS subsidised prescriptions as close to the survey reference period as possible. However, when analysing PBS subsidised prescriptions use, a longer time period would allow for more analysis of the patterns of medication use.

### Endnotes

1. McHugh M. L. (2012). Interrater reliability: the kappa statistic. Biochemia medica, 22(3), 276–282 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3900052/ accessed 21/04/2020.
2. Oksuzyan, A., Sauer, T., Gampe, H., Höhn, A., Wod, M., Christensen, K., and Wastesson, J. W. (2018). Is who you ask important? Concordance between survey and registry data on medication use among self- and proxy-respondents in the longitudinal study of aging Danish twins and the Danish 1905-cohort study. The Journals of Gerontology: Series A, Volume 74, Issue 5, May 2019, Pages 742–747 https://doi.org/10.1093/gerona/gly104 accessed 06/03/2020.
3. Hafferty, J. D., Campbell, A, I., Navrady, L. B., Adams, M. J., MacIntyre, D., Lawrie, S. M., Nicodemus, K., Porteous, D. J., and McIntosh, A. M. (2018). Self-reported medication use validated through record linkage to national prescribing data. Journal of Clinical Epidemiology 94: 132-142 https://www.sciencedirect.com/science/article/pii/S0895435617303062?via%3Dihub accessed 24/02/2020.

## Use of PBS subsidised prescriptions

This section explores the linked dataset from the integration between the 2014-15 NHS and the MADIP asset. It presents data on the use of PBS subsidised prescriptions by certain characteristics of people in the 2014-15 NHS.

Data is presented across a range of medications according to the World Health Organization Anatomical Therapeutic Chemical (ATC) Classification System[8]. It is a classification system that classifies the active ingredients of medications according to the organ or system on which they act and their therapeutic, pharmacological and chemical properties. The ATC Classification System categorises medicines into one of fourteen anatomical groups, which are presented in this research paper.

For the 12 months prior to the 2014-15 NHS interview:

• Seven in ten (70.5%) people had at least one script filled for a PBS subsidised prescription.
• Antiinfectives for systemic use, which include antibiotics, were the most commonly accessed type of medication, with almost half (45.5%) of all people having at least one script filled.
• Nervous system medications, which include mental health and psychoactive related medications, were the second most commonly prescribed medications. Just over one quarter (26.2%) of all people accessing at least one PBS subsidised prescription for this medication type.
• On average, the highest number of scripts filled per person was for Cardiovascular system medications with an average of 18.5 scripts per person.
Table 3: Persons who accessed (a) PBS subsidised prescription(s) up to 12 months prior to 2014-15 NHS interview
Estimate ('000)Proportion (%)Number of scripts ('000)Average number of scripts no. per person
Anatomical Therapeutic Chemical (ATC) Classification System(b)
Alimentary tract and metabolism4,579.719.939,492.58.6
Blood and blood forming organs1,667.97.310,901.36.5
Cardiovascular system4,825.021.089,476.218.5
Dermatologicals2,455.310.74,647.41.9
Genito-urinary system and sex hormones2,207.79.67,804.63.5
Systemic hormonal preparations, excluding sex hormones and insulins2,451.710.76,085.82.5
Antiinfectives for systemic use10,439.345.529,871.82.9
Antineoplastic and immunomodulating agents407.81.83,628.88.9
Musculo-skeletal system2,993.013.011,874.24.0
Nervous system6,010.726.257,594.59.6
Antiparasitic products, insecticides and repellents55.30.266.31.2
Respiratory system2,465.010.713,296.95.4
Sensory organs2,384.210.410,369.04.3
Various127.50.6479.63.8
Total people who used at least one PBS subsidised script up to 12 months prior to NHS interview16,192.570.5285,588.717.6
All people in 2014-15 NHS22,966.6100.0....

a.   Persons represented in the linked 2014-15 NHS dataset.
b.   The ATC classification system has fourteen main anatomical or pharmacological groups (first level). These first level anatomical main groups are presented in this publication.

### Age and sex

In terms of age, older people were more likely to access PBS subsidised prescriptions, which could be attributed to factors such as increasing life expectancy, prevalence of conditions and polypharmacy.

Of all people in the NHS who accessed at least one PBS subsidised prescription up to 12 months prior to their 2014-15 NHS interview:

• Females (75.9%) were more likely than males (65.0%), to access at least one PBS prescription medication.
• Generally, access to PBS subsidised prescriptions increased with age. The majority of people aged 75 years and over (97.8%) accessed at least one PBS subsidised prescription.
• The largest difference between females and males by age group was at 18-24 years (81.4% compared with 49.5% respectively). This difference generally reflects females’ gynaecological and reproductive health needs.

Across most age groups for males and females, Antiinfectives for systemic use medications were the most commonly accessed subsidised prescription in the linked dataset. However, the relative pattern of use for males and females differed across age, particularly in adults aged 18-44 years. The rate did not necessarily increase with age, as typically observed in other ATC medication groups.

Antiinfectives for systemic use medications include antibiotics which help stop infections that are caused by bacteria. Australia has a relatively higher rate of antibiotic prescribing compared with the Organisation for Economic Co-operation and Development (OECD) average (23.4 per 1,000 people, compared with an OECD average of 20.6 per 1,000 people)[9].

Nervous system medications were the second most commonly accessed subsidised prescription in the linked dataset and include antipsychotics, anxiolytics/hypnotics and sedatives, antidepressants and psychostimulants medications. Whilst young males (0-17 years) were slightly more likely than females to have accessed a Nervous system prescription, for all other age groups females were more likely than males. The largest difference between males and females was at ages 65-74 years (41.9% compared with 54.3% respectively). Generally, use of Nervous system medications increased with age and people aged 75 years and over had the highest rate of any age group, with six in ten people (61.7%) having accessed a Nervous system prescription via the PBS.

Cardiovascular system medications were the third most commonly accessed subsidised prescription in the linked dataset and are used in treating cardiovascular disease, such as heart, stroke and vascular disease. They are also used to prevent risk factors such as high blood pressure and high cholesterol. Rates of use generally increased with age for both males and females with a similar proportion of males and females in each age group. There was a slight difference at ages 55-64 years with 48.4% of males compared with 40.4% of females. Generally, use of Cardiovascular system medications increased with age and people aged 75 years and over had the highest rate of any age group at 85.1%.

### Self-reported long-term health conditions

Long-term health conditions are typically managed by prescribed medications. However, there is no detailed information regarding what conditions prescription medications are being prescribed for in the PBS and limited information from the NHS. To better understand the relationship between long-term health conditions and medication use, the integrated data can be used to cross-classify and highlight medication prescribing patterns in the PBS subsidised prescriptions to better understand comorbidity and polypharmacy.

This section looks at those people who self-reported a long-term health condition in the 2014-15 NHS and the types of PBS subsidised prescriptions they accessed in the 12 months prior to 2014-15 NHS interview. Long-term health conditions self-reported by respondents in the 2014-15 NHS are presented using a classification originally developed for the 2001 NHS by the Family Medicine Research Centre, University of Sydney, in conjunction with the ABS. The classification is based on the 10th revision of the International Classification of Diseases (ICD).

Of those people who self-reported a long-term health condition in the 2014-15 NHS, the proportion who accessed at least one PBS subsidised prescription were as follows:

• Hypertension and Diabetes mellitus – both at 97.3%
• Osteoporosis – 96.0%
• Heart, stroke and vascular disease – 95.6%
• Chronic obstructive pulmonary disease (COPD) – 92.7%
• Cancer (malignant neoplasms) – 91.5%
• Arthritis – 91.2%
• Kidney disease – 91.0%
• Asthma – 87.5%
• Mental and behavioural conditions – 84.6%
• Back problems (dorsopathies) – 80.0%
• Hayfever and allergic rhinitis – 72.2%

For further information on prevalence rates of self-reported long-term health conditions in the 2014-15 NHS, see 4364.0.55.001 - National Health Survey: First Results, 2014-15.

Of those who self-reported a long-term health condition, the following PBS subsidised prescriptions were accessed:

• Of those who self-reported having Hypertension, 93.5% had at least one PBS subsidised prescription for Cardiovascular system medications.
• More than four in five (85.2%) of those who self-reported having diabetes had at least one Alimentary tract and metabolism script, which comprises medications targeted for diabetes. This was closely followed by Cardiovascular system (83.0%) which could be explained by people with diabetes often taking blood pressure lowering medications in combination. People who have diabetes are at risk of heart disease. Heart attacks and strokes are up to four times more likely in people with diabetes[10].
• Seven in ten (70.3%) who self-reported having COPD had a PBS subsidised prescription for Antiinfectives for systemic use. This medication group includes antibiotics which are commonly used for the management of COPD.
• Four in five people (83.2%) who self-reported having Heart, stroke and vascular disease had a Cardiovascular system script. Around half of these people (50.4%) with this condition also had a PBS subsidised prescription for Blood and blood forming organs medication.
• Musculo-skeletal medications were most commonly accessed by people who self-reported having Osteoporosis at 51.4%.
• More than half of the people who self-reported having COPD (53.9%) and Asthma (53.0%) had a PBS subsidised prescription filled for a Respiratory system medication.

### Endnotes

1. World Health Organisation (WHO) Collaborating Centre for Drug Statistics Methodology www.whocc.no/atc_ddd_index/ last accessed 12/03/2020.
2. OECD (Organisation for Economic Co-operation and Development) 2017, Health at a glance 2017: OECD indicators. Paris: OECD Publishing https://www.oecd.org/australia/Health-at-a-Glance-2017-Key-Findings-AUSTRALIA.pdf accessed 31/03/2020.
3. Diabetes Australia, Living with Diabetes, Preventing Complications, Heart disease - https://www.diabetesaustralia.com.au/heart-disease accessed 17/04/2020.

## Future opportunities

The aim of this exploratory analysis was to showcase the use of PBS subsidised prescriptions with NHS data and to assess the feasibility of using PBS subsidised prescriptions to replace the medications questionnaire module in future cycles of the NHS. Based on the analysis, the administrative data provides similar or higher quality information for most ATC Classification System medications groups compared to survey data.

After comparing two time windows, the analysis illustrates the PBS subsidised prescriptions should align as close to the NHS reference period as possible in order to achieve conceptually comparable results. However, for analysis purposes utilising PBS subsidised prescriptions over a longer period would allow for a larger breadth of research to be undertaken. An advantage of the PBS subsidised prescriptions data is the availability of a range of time periods which can be chosen depending on the research question, whereas the NHS data is constrained to a two week window for the purposes of improving respondent recall and to reduce cognitive load.

In the NHS, respondents would still be asked about medications used for specific long-term health conditions of cardiovascular disease, diabetes and high sugar levels, and asthma medication use. This is due to the high priority data need to ascertain the treatment and management of these long-term health conditions to help improve outcomes.

As illustrated in the analysis, over-the-counter medications are a data gap in the PBS subsidised prescriptions. It would be worth investigating how to expand or complement the medications data represented in the MADIP asset to include over-the-counter medications, private prescriptions, dietary supplements and other data gaps. Investigations into questions related to over-the-counter medications, such as aspirin and Ventolin have already begun for use in future NHS cycles.

By using PBS subsidised prescriptions, there is an opportunity to expand the information available to be published such as: the number of prescriptions dispensed; the frequency of scripts; amount of medicine supplied; time between prescriptions; and costs associated, for example, benefit amount or patient contribution amount.

Future linkage between NHS and MADIP could also support additional analysis into relationship outcomes such as; mortality, barriers to health care, hospitalisations and visits to emergency departments with access to prescription medications. Such analysis would be valuable for better understanding and improving health outcomes in Australia.


text:
# Math Help - Induction

1. ## Induction

I don't need help with induction but just part of the question.

Prove for any real x and for all n > 1

that $x^n-1=(x-1)(x^{n-1}+x^{n-1}+\cdots +x^{n-r}+\cdots +x+1)$

It isn't stated but is n supposed to be Naturals or Integers? Or is n in Reals?

2. Since this is a problem on induction, I think n is a natural number.


text:
# American Institute of Mathematical Sciences

August & September  2019, 12(4&5): 929-940. doi: 10.3934/dcdss.2019062

## A nonlinear empirical analysis on influence factor of circulation efficiency

 1 College of Business Administration, Ningbo Dahongying University, No.899 Xueyuan Road, Haishu District, Ningbo 315175, Zhejiang, China 2 Department of Economic and Trade, Sejong University, 209, Neungdong-ro, Gwangjin-gu, Seoul 05006, Republic of Korea 3 School of Business Administration, Jiangxi University of Finance and Economics, No.168 Shuang Gang East Street, Nanchang 330013, Jiangxi, China

* Corresponding author: 18511513@qq.com

Received  June 2017 Revised  December 2017 Published  November 2018

A series of phenomena including lower circulation efficiency of Chinese fresh agricultural products, postharvest decay, damage and waste of agricultural products, regional and structural contradiction of supply and demand, drastic fluctuation in price and difficulty in buying and selling, etc. are serious, which has restricted the sound development of Chinese fresh agricultural product industry. To analyze and discuss main factors affecting circulation efficiency of Gannan navel orange, the methods, such as AHP (analytic hierarchy process) and Delphic method, etc., have been used for empirical analysis on Gannan navel orange, and it is found that fruit factors (including single structure and centralized mature period of navel orange, etc), infrastructure factors (including the lack standardization for construction of trading place, outdated warehousing facility and technology, insufficient input of infrastructure of cold chain, etc) and policy environment factors (including food safety, absence of relevant laws and regulations of market supervision, etc) are the existing main factors restricting high-efficient circulation of Gannan navel orange. Based on the conclusion of empirical research and beginning with main circulation links of production, storage and transportation as well as marketing, etc and supporting measures of brand building, product safety and policy service system, etc, the countermeasures and suggestions are proposed to improve circulation efficiency of Gannan navel orange.

Citation: Wu Chanti, Qiu Youzhen. A nonlinear empirical analysis on influence factor of circulation efficiency. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 929-940. doi: 10.3934/dcdss.2019062
##### References:

show all references

##### References:
Hierarch Structure Model for Evaluation on Level of Influence Factor of Circulation Efficiency of Gannan Navel Orange
Index System for Influence Factor of Circulation Efficiency of Gannan Navel Orange
 Class-1 index Class-2 index Evaluation on Level of Influence Factor of Circulation Efficiency of Gannan Navel Orange Fruit factor C$_{1}$ Fruit seasonality c$_{\mathbf{11}}$ Fruit storability c$_{12}$ Demand for initial processing of fresh fruit c$_{13}$ Consumption preference c$_{14}$ Quality level of participant C$_{2}$ Scale of participant c$_{21}$ Capability of getting information c$_{22}$ Logistics professionalization quality level c$_{23}$ Mutual cooperation level c$_{24}$ Infrastructure construction of logistics C$_{3}$ Traffic transportation facility construction c$_{31}$ Trading place construction c$_{32}$ Warehousing facility construction c$_{33}$ Information spreading network construction c$_{34}$ Infrastructure construction of cold chain c$_{\mathbf{35}}$ Market environment C$_{4}$ Degree of market opening c$_{41}$ Degree of fair transaction c$_{42}$ Degree of market activity c$_{\mathbf{43}}$ Policy environment C$_{5}$ Agricultural industry investment policy c$_{51}$ Charge policy for agricultural products circulation c$_{52}$ Food safety policy c$_{\mathbf{53}}$
 Class-1 index Class-2 index Evaluation on Level of Influence Factor of Circulation Efficiency of Gannan Navel Orange Fruit factor C$_{1}$ Fruit seasonality c$_{\mathbf{11}}$ Fruit storability c$_{12}$ Demand for initial processing of fresh fruit c$_{13}$ Consumption preference c$_{14}$ Quality level of participant C$_{2}$ Scale of participant c$_{21}$ Capability of getting information c$_{22}$ Logistics professionalization quality level c$_{23}$ Mutual cooperation level c$_{24}$ Infrastructure construction of logistics C$_{3}$ Traffic transportation facility construction c$_{31}$ Trading place construction c$_{32}$ Warehousing facility construction c$_{33}$ Information spreading network construction c$_{34}$ Infrastructure construction of cold chain c$_{\mathbf{35}}$ Market environment C$_{4}$ Degree of market opening c$_{41}$ Degree of fair transaction c$_{42}$ Degree of market activity c$_{\mathbf{43}}$ Policy environment C$_{5}$ Agricultural industry investment policy c$_{51}$ Charge policy for agricultural products circulation c$_{52}$ Food safety policy c$_{\mathbf{53}}$
1-9 Scale Method and Definition
 Scale $x_{i}, x_{j}$ pairwise comparison standard 1 X$_{i}$ and X$_{j}$ is equal in importance 3 X$_{i }$is weakly important than X$_{j}$ 5 X$_{i }$is obviously important than X$_{j}$ 7 X$_{i}$ is strongly important than X$_{j}$ 9 X$_{i }$is extremely important than X$_{j}$ 2, 4, 6, 8 Mid-value of the above two adjacent judgments $a_{ji} =1/a_{ij}$ Reciprocal $a_{ij}$ is the comparison judgment between factor i and j, and the comparison judgment between factor i and j that $a_{ji} =1/a_{ij}$.
 Scale $x_{i}, x_{j}$ pairwise comparison standard 1 X$_{i}$ and X$_{j}$ is equal in importance 3 X$_{i }$is weakly important than X$_{j}$ 5 X$_{i }$is obviously important than X$_{j}$ 7 X$_{i}$ is strongly important than X$_{j}$ 9 X$_{i }$is extremely important than X$_{j}$ 2, 4, 6, 8 Mid-value of the above two adjacent judgments $a_{ji} =1/a_{ij}$ Reciprocal $a_{ij}$ is the comparison judgment between factor i and j, and the comparison judgment between factor i and j that $a_{ji} =1/a_{ij}$.
Value of Average Random Consistency Index (0$^{th}$ and 10$^{th}$ order)
 n 1 2 3 4 5 6 7 8 9 10 $RI$ 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49
 n 1 2 3 4 5 6 7 8 9 10 $RI$ 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49
Results of Random Consistency Check
 A A$_{\mathbf{1}}$ A$_{\mathbf{2}}$ A$_{\mathbf{3}}$ A$_{\mathbf{4}}$ A$_{\mathbf{5}}$ $CI$ value 0.0013 0.0546 0.0103 0.0083 0.0429 0.0368 $RI$ value 1.12 0.90 0.90 1.12 0.58 0.58 CR value 0.0012 0.0607 0.0115 0.0074 0.0739 0.0634
 A A$_{\mathbf{1}}$ A$_{\mathbf{2}}$ A$_{\mathbf{3}}$ A$_{\mathbf{4}}$ A$_{\mathbf{5}}$ $CI$ value 0.0013 0.0546 0.0103 0.0083 0.0429 0.0368 $RI$ value 1.12 0.90 0.90 1.12 0.58 0.58 CR value 0.0012 0.0607 0.0115 0.0074 0.0739 0.0634
Summary of Data of Index System for Influence Factor of Circulation Efficiency of Gannan Navel Orange
 The first hierarchy The second hierarchy Class distribution Index Weight No. Index Weight No. Very good Good Common Poor Very poor Fruit factor A$_{1}$ Fruit seasonality a$_{11}$ 0.2599 0.00 0.00 0.17 0.47 0.37 Fruit storability a$_{12}$ 0.4502 0.03 0.07 0.40 0.33 0.17 Demand for initial processing of fresh fruit a$_{13}$ 0.1838 0.00 0.17 0.73 0.10 0.00 Consumption preference a$_{14}$ 0.1061 0.10 0.33 0.53 0.03 0.00 Quality level of participant A$_{2}$ Scale of participant a$_{21}$ 0.2776 0.00 0.27 0.50 0.17 0.07 Capability of getting information a$_{22}$ 0.1603 0.07 0.17 0.27 0.33 0.17 Logistics professionalization quality level a$_{23}$ 0.4668 0.03 0.23 0.23 0.33 0.17 Mutual cooperation level a$_{24}$ 0.0953 0.07 0.17 0.43 0.23 0.10 Infrastr ucture construction of logistics A$_{3}$ Traffic transportation facility construction a$_{31}$ 0.2154 0.10 0.33 0.27 0.20 0.10 Trading place construction a$_{32}$ 0.0735 0.00 0.13 0.30 0.43 0.13 Warehousing facility construction a$_{33}$ 0.1208 0.03 0.10 0.27 0.40 0.20 Information spreading network construction a$_{34}$ 0.3749 0.17 0.23 0.40 0.13 0.07 Infrastructure construction of cold chain a$_{35}$ 0.2154 0.00 0.00 0.23 0.50 0.27 Market environ ment A$_{4}$ Degree of market opening a$_{41}$ 0.2255 0.27 0.47 0.20 0.07 0.00 Degree of fair transaction a$_{42}$ 0.1007 0.23 0.53 0.20 0.03 0.00 Degree of market activity a$_{43}$ 0.6738 0.03 0.13 0.43 0.30 0.10 Policy enviro nment A$_{5}$ Agricultural industry investment policy a$_{51}$ 0.6144 0.13 0.40 0.33 0.10 0.03 Charge policy for agricultural products circulation a$_{52}$ 0.1172 0.10 0.23 0.40 0.20 0.07 Food safety policy a$_{53}$ 0.2684 0.00 0.10 0.30 0.40 0.20 Data source: calculation based on data of survey
 The first hierarchy The second hierarchy Class distribution Index Weight No. Index Weight No. Very good Good Common Poor Very poor Fruit factor A$_{1}$ Fruit seasonality a$_{11}$ 0.2599 0.00 0.00 0.17 0.47 0.37 Fruit storability a$_{12}$ 0.4502 0.03 0.07 0.40 0.33 0.17 Demand for initial processing of fresh fruit a$_{13}$ 0.1838 0.00 0.17 0.73 0.10 0.00 Consumption preference a$_{14}$ 0.1061 0.10 0.33 0.53 0.03 0.00 Quality level of participant A$_{2}$ Scale of participant a$_{21}$ 0.2776 0.00 0.27 0.50 0.17 0.07 Capability of getting information a$_{22}$ 0.1603 0.07 0.17 0.27 0.33 0.17 Logistics professionalization quality level a$_{23}$ 0.4668 0.03 0.23 0.23 0.33 0.17 Mutual cooperation level a$_{24}$ 0.0953 0.07 0.17 0.43 0.23 0.10 Infrastr ucture construction of logistics A$_{3}$ Traffic transportation facility construction a$_{31}$ 0.2154 0.10 0.33 0.27 0.20 0.10 Trading place construction a$_{32}$ 0.0735 0.00 0.13 0.30 0.43 0.13 Warehousing facility construction a$_{33}$ 0.1208 0.03 0.10 0.27 0.40 0.20 Information spreading network construction a$_{34}$ 0.3749 0.17 0.23 0.40 0.13 0.07 Infrastructure construction of cold chain a$_{35}$ 0.2154 0.00 0.00 0.23 0.50 0.27 Market environ ment A$_{4}$ Degree of market opening a$_{41}$ 0.2255 0.27 0.47 0.20 0.07 0.00 Degree of fair transaction a$_{42}$ 0.1007 0.23 0.53 0.20 0.03 0.00 Degree of market activity a$_{43}$ 0.6738 0.03 0.13 0.43 0.30 0.10 Policy enviro nment A$_{5}$ Agricultural industry investment policy a$_{51}$ 0.6144 0.13 0.40 0.33 0.10 0.03 Charge policy for agricultural products circulation a$_{52}$ 0.1172 0.10 0.23 0.40 0.20 0.07 Food safety policy a$_{53}$ 0.2684 0.00 0.10 0.30 0.40 0.20 Data source: calculation based on data of survey
Results for Evaluation on Level of Efficiency Influence Factor of Circulation Pattern of Gannan Navel Orange
 Factor set Very good Good Common Poor Very poor Good and over Common and over A 0.0727 0.1971 0.3439 0.4631 0.1270 0.2698 0.6137 A$_{1}$ 0.0256 0.0960 0.4148 0.2933 0.1703 0.1216 0.5364 A$_{2}$ 0.0326 0.2255 0.3318 0.2775 0.1326 0.2581 0.5899 A$_{3}$ 0.0881 0.1812 0.3119 0.2809 0.1379 0.2692 0.5811 A$_{4}$ 0.1061 0.2488 0.3572 0.2205 0.0674 0.3549 0.7121 A$_{5}$ 0.0936 0.2999 0.3322 0.1922 0.0820 0.3936 0.7258
 Factor set Very good Good Common Poor Very poor Good and over Common and over A 0.0727 0.1971 0.3439 0.4631 0.1270 0.2698 0.6137 A$_{1}$ 0.0256 0.0960 0.4148 0.2933 0.1703 0.1216 0.5364 A$_{2}$ 0.0326 0.2255 0.3318 0.2775 0.1326 0.2581 0.5899 A$_{3}$ 0.0881 0.1812 0.3119 0.2809 0.1379 0.2692 0.5811 A$_{4}$ 0.1061 0.2488 0.3572 0.2205 0.0674 0.3549 0.7121 A$_{5}$ 0.0936 0.2999 0.3322 0.1922 0.0820 0.3936 0.7258
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# Math Help - Differential equations

1. ## Differential equations

I would really appreciate it if someone could help me with how to solve these two differential equations.

1. ((dK)/(dt))=λ(P-K), given K(0)=0,2P

2. (1/K)((dK)/(dt))=λ(P-K), given K(0)=0,2P

2. ## Re: Differential equations

Originally Posted by GardsVision
I would really appreciate it if someone could help me with how to solve these two differential equations.

1. ((dK)/(dt))=λ(P-K), given K(0)=0,2P
$\frac{dK}{P- K}= \lambda dt$. Integrate.

2. (1/K)((dK)/(dt))=λ(P-K), given K(0)=0,2P
$\frac{dK}{K(P- K)}= \lambda dt$. Integrate. Use "partial fractions" on the left.

I don't know what you mean by the ",2P".


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Capacitor diode circuit

I have this circuit

My task is to match these four statements.

1. "$$\v_1=0\$$ for a very long time. The time constant for the charging time for the capacitor immediately after $$\v_1\$$ jumps to $$\v_1=10V\$$ is ...

2. "$$\v_1=10V\$$ for a very long time. The time constant for the discharge time for the capacitor immediately after $$\v_1\$$ jumps to $$\v_1=0\$$ is ...

3. $$\R_2\cdot C\$$

4. $$\(R_1+R_2)\cdot C\$$

I can't really find any arguments as to why these statements belong together.

Has it something to do with the diodes not being forward biased? I hope someone can clarify this for me.

Edit

The main problem I have with this, is that I don't exactly understand how the capacitor is charging and discharging through the resistors, and when and why the diodes are conducting and non-conducting.

I just saw the answers to this problem. For number 1 they have this drawing.

For number 2 they have this drawing.

• You should match "question" 1 with "answer" 3 or "answer" 4 and "question" 2 with the remaining "answer" (answer 3 or answer 4) – Huisman Dec 11 '19 at 14:02
• Has it something to do with the diodes not being forward biased? So all the diodes are in reverse mode? Are you sure? Consider the cases where C has no charge and V1 is 1) a negative voltage 2) V1 = 0 V and 3) When V1 = a positive voltage. – Bimpelrekkie Dec 11 '19 at 14:03
• Huisman, can you please explain why this is? – Carl Dec 11 '19 at 14:04
• There are other relations as well: All four statements relate to each other because they all address the same circuit. Statement 3 and 4 relate because the only consist of component value variables. But assuming this is a homework assignment, I think my first comment makes most sense. – Huisman Dec 11 '19 at 14:56
• Do you understand that V1 either changes from 0V to 10V, or from 10V to 0V, and you have to align these two conditions with the time constants given in 3. and 4.? – Chu Dec 11 '19 at 14:57


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1. ## piecewise function

$f(x) = \left\{ \begin{array}{lll}
x^2 + 2 x + 1 & \mbox{if} & x\le -2 \cr
ax + b & \mbox{if} & x>-2
\end{array} \right .$

This is differentiable at -2 .

(a) It is known that if a function is differentiable at a point c, then it is continuous at c. Using now the continuity of f at −2, we can establish a relationship between a and b. Find this relationship and express it in the form b=Aa+B, where A and B are constants.

b) Assuming x>-2

$\displaystyle \frac{f(x)-f(-2)}{x-(-2)}$

simplify into the form Ca+D, where C and D are constants. Find these constants.

c) assuming x<-2

$\displaystyle \frac{f(x)-f(-2)}{x-(-2)}$

simplify into the form Ex+F, where E and F are constants. Find these constants.

(d) Using the results of parts (a), (b) and (c), find the values of a and b.

This is the part I don't understand how to do. The answer is allegedly -2 & -3.

Can someone please explain how to do this part "D"?

Thank you!

2. Continuity gives $-2a+b=1$.

Differentiability gives $a=-2$

3. Got it, Thanks.


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## In reinforcement learning, all objective functions are not equal

Feb 12, 2018 (modified: Jun 04, 2018) ICLR 2018 Workshop Submission readers: everyone Show Bibtex
• Abstract: We study the learnability of value functions. We get the reward back propagation out of the way by fitting directly a deep neural network on the analytically computed optimal value function, given a chosen objective function. We show that some objective functions are easier to train than others by several magnitude orders. We observe in particular the influence of the $\gamma$ parameter and the decomposition of the task into subtasks.
• Keywords: reinforcement learning, deep learning
• TL;DR: In reinforcement learning, all objective functions are not equal
0 Replies


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• Create Account

### #Actualkd7tck

Posted 29 December 2012 - 01:28 PM

Abstract away.  Personaly now that I think about it, as long as you don't blit more than the monitor refresh rate you don't need to worry about this. This should only be implimented to slow down a fast program, v-sync can be used in place of it if you want. In the cases of slow programs just let er rip and use delta time to calculate movements based on time, rather than frame number.

class TimeDif(){
long range = 16;
long lastTime = GetTime();

bool getDif(){
long dif = GetTime() - lastTime;

if(dif >= range){
lastTime = GetTime();
return true;
}

return false;
}

}

### #1kd7tck

Posted 29 December 2012 - 01:26 PM

Abstract away.  Personaly now that I think about it, as long as you don't blit more than the monitor refresh rate you don't need to worry about this. This should only be implimented to slow down a fast program, v-sync can be used in place of it if you want. In the cases of slow programs just let er rip and use delta time to calculate movements based on time, rather than frame number.

class TimeDif(){
long range = 16;
long lastTime = GetTime();

long getDif(){
long dif = GetTime() - lastTime;

if(dif >= range)
lastTime = GetTime();

return dif;
}

}

PARTNERS


text:
China Jin-Ping underground Laboratory

Located in the middle of a 18-km tunnel under 2400 meters of rock overburden, in the Sichuan province of south-west China. As one of the deepest underground labs in the world, the CJPL has an extremely low flux of muon rate of less than $20/m^{2}/100$ day, which is about two orders of magnitude lower than the flux at the Gran Sasso underground laboratory (LNGS) in Italy. The low muon rate and the resulting background makes the lab ideal for a sensitive dark matter detection.


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# 2.7.1: Using Multiple Integrating Factors (Exercises)

## Q2.6.1

1.

1. Verify that $$\mu(x,y)=y$$ is an integrating factor for $y dx+\left(2x+\frac{1}{y} \right) dy=0 \tag{A}$ on any open rectangle that does not intersect the $$x$$ axis or, equivalently, that $y^{2} dx +(2xy+1) dy=0 \tag{B}$ is exact on any such rectangle.
2. Verify that $$y\equiv0$$ is a solution of (B), but not of (A).
3. Show that $y(xy+1)=c \tag{C}$ is an implicit solution of (B), and explain why every differentiable function $$y=y(x)$$ other than $$y\equiv0$$ that satisfies (C) is also a solution of (A).

2.

1. Verify that $$\mu(x,y)=1/(x-y)^2$$ is an integrating factor for $-y^{2}dx+x^{2}dy=0 \tag{A}$ on any open rectangle that does not intersect the line $$y=x$$ or, equivalently, that $-\frac{y^{2}}{(x-y)^{2}}dx + \frac{x^{2}}{(x-y)^{2}}dy=0 \tag{B}$ is exact on any such rectangle.
2. Use Theorem 2.2.1 to show that $\frac{xy}{(x-y)}=c\tag{C}$ is an implicit solution of (B), and explain why it is also an implicit solution of (A)
3. Verify that $$y=x$$ is a solution of (A), even though it can’t be obtained from (C).

## Q2.6.2

In Exercises 2.6.3-2.6.16 find an integrating factor; that is a function of only one variable, and solve the given equation.

3. $$ydx-xdy=0$$

4. $$3x^{2}ydx +2x^{3}dy=0$$

5. $$2y^{3}dx+3y^{2}dy=0$$

6. $$(5xy+2y+5)dx+2xdy=0$$

7. $$(xy+x+2y+1)\,dx+(x+1)\,dy=0$$

8. $$(27xy^2+8y^3)\,dx+(18x^2y+12xy^2)\,dy=0$$

9. $$(6xy^2+2y)\,dx+(12x^2y+6x+3)\,dy=0$$

10. $$y^2\,dx+\left(xy^2+3xy+{1\over y}\right)\,dy=0$$

11. $$(12x^3y+24x^2y^2)\,dx+(9x^4+32x^3y+4y)\,dy=0$$

12. $$(x^2y+4xy+2y)\,dx+(x^2+x)\,dy=0$$

13. $$-y\,dx+(x^4-x)\,dy=0$$

14. $$\cos x\cos y\,dx +(\sin x\cos y-\sin x\sin y+y)\,dy=0$$

15. $$(2xy+y^2)\,dx+(2xy+x^2-2x^2y^2-2xy^3)\,dy=0$$

16. $$y\sin y\,dx+x(\sin y-y\cos y)\,dy=0$$

## Q2.6.3

In Exercises 2.6.17-2.6.23 find an integrating factor of the form $$\mu (x,y)=P(x)Q(y)$$ and solve the given equation.

17. $$y(1+5\ln|x|)\,dx+4x\ln|x|\,dy=0$$

18. $$(\alpha y+ \gamma xy)\,dx+(\beta x+ \delta xy)\,dy=0$$

19. $$(3x^2y^3-y^2+y)\,dx+(-xy+2x)\,dy=0$$

20. $$2y\,dx+ 3(x^2+x^2y^3)\,dy=0$$

21. $$(a\cos xy-y\sin xy)\,dx+(b\cos xy-x\sin xy)\, dy=0$$

22. $$x^4y^4\,dx+x^5y^3\,dy=0$$

23. $$y(x\cos x+2\sin x)\,dx+x(y+1)\sin x\,dy=0$$

## Q2.6.4

In Exercises 2.6.24-2.6.27 find an integrating factor and solve the equation. Plot a direction field and some integral curves for the equation in the indicated rectangular region.

24. $$(x^4y^3+y)\,dx+(x^5y^2-x)\,dy=0; \quad \{-1\le x\le1,-1\le y\le1\}$$

25. $$(3xy+2y^2+y)\,dx+(x^2+2xy+x+2y)\,dy=0; \quad \{-2\le x\le2,-2\le y\le2\}$$

26. $$(12 xy+6y^3)\,dx+(9x^2+10xy^2)\,dy=0; \quad \{-2\le x\le2,-2\le y\le2\}$$

27. $$(3x^2y^2+2y)\,dx+ 2x\,dy=0; \quad \{-4\le x\le4,-4\le y\le4\}$$

## Q2.6.5

28. Suppose $$a$$, $$b$$, $$c$$, and $$d$$ are constants such that $$ad-bc\ne0$$, and let $$m$$ and $$n$$ be arbitrary real numbers. Show that

$(ax^my+by^{n+1})\,dx+(cx^{m+1}+dxy^n)\,dy=0$

has an integrating factor $$\mu(x,y)=x^\alpha y^\beta$$.

29. Suppose $$M$$, $$N$$, $$M_x$$, and $$N_y$$ are continuous for all $$(x,y)$$, and $$\mu=\mu(x,y)$$ is an integrating factor for $M(x,y)dx+N(x,y)dy=0.\tag{A}$

Assume that $$\mu_x$$ and $$\mu_y$$ are continuous for all $$(x,y)$$, and suppose $$y=y(x)$$ is a differentiable function such that $$\mu(x,y(x))=0$$ and $$\mu_x(x,y(x))\ne0$$ for all $$x$$ in some interval $$I$$. Show that $$y$$ is a solution of (A) on $$I$$.

30. According to Theorem 2.1.2, the general solution of the linear nonhomogeneous equation $y'+p(x)y=f(x)\tag{A}$

is $y=y_{1}x\left( c+\int f(x)/y_{1}(x) dx \right),\tag{B}$

where $$y_1$$ is any nontrivial solution of the complementary equation $$y'+p(x)y=0$$. In this exercise we obtain this conclusion in a different way. You may find it instructive to apply the method suggested here to solve some of the exercises in Section 2.1.

1. Rewrite (A) as $[p(x)y-f(x)]dx +dy =0,\tag{C}$ and show that $$\mu=\pm e^{\int p(x)\,dx}$$ is an integrating factor for (C).
2. Multiply (A) through by $$\mu=\pm e^{\int p(x)\,dx}$$ and verify that the resulting equation can be rewritten as $(\mu(x)y)'=\mu(x)f(x).$ Then integrate both sides of this equation and solve for $$y$$ to show that the general solution of (A) is $y={1\over\mu(x)}\left(c+\int f(x)\mu(x)\,dx\right).$ Why is this form of the general solution equivalent to (B)?

This page titled 2.7.1: Using Multiple Integrating Factors (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.


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# eigenvectors of orthogonal matrix

More... class Eigen::RealQZ< _MatrixType > Performs a real QZ decomposition of a pair of square matrices. But often, we can âchooseâ a set of eigenvectors to meet some specific conditions. Constructing an Orthogonal Matrix from Eigenvalues - Duration: 10:09. That's what I mean by "orthogonal eigenvectors" when those eigenvectors are complex. Yeah, that's called the spectral theorem. . Taking eigenvectors as columns gives a matrix P such that $$\displaystyle P^-1AP$$ is the diagonal matrix with the eigenvalues 1 and .6. Orthogonal matrices are the most beautiful of all matrices. The above matrix is skew-symmetric. So if I have a symmetric matrix--S transpose S. I know what that means. The normal modes can be handled independently and an orthogonal expansion of the system is possible. Proof that the eigenvectors span the eigenspace for normal operators. The extent of the stretching of the line (or contracting) is the eigenvalue. The matrix $$P$$ whose columns consist of these orthonormal basis vectors has a name. A symmetric matrix (in which a i j = a j i a_{ij}=a_{ji} a i j = a j i ) does necessarily have orthogonal eigenvectors. Perfect. eigenvectors of A are orthogonal to each other means that the columns of the matrix P are orthogonal to each other. That's just perfect. James Rantschler 9,509 views. Recall some basic de nitions. Suppose S is complex. So, citing the mathematical foundations of orthogonal axes doesn't really explain why we use this approach for PCA. Differential Equations and Linear Algebra, 6.5: Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors - Video - MATLAB & Simulink Symmetric Matrix , Eigenvectors are not orthogonal to the same eigenvalue. Every symmetric matrix is an orthogonal matrix times a diagonal matrix times the transpose of the orthogonal matrix. Eigenvalues and Eigenvectors The eigenvalues and eigenvectors of a matrix play an important part in multivariate analysis. Prove the eigenvectors of a reflection transformation are orthogonal. evp = NullSpace[(M - 3 IdentityMatrix[6])] evm = NullSpace[(M + 3 IdentityMatrix[6])] evp[[1]].evm[[1]] Orthogonalization of the degenerate subspaces proceeds without â¦ 2. Let be an complex Hermitian matrix which means where denotes the conjugate transpose â¦ Eigenvectors are not unique. Eigenvectors of The Lorentz Matrix We know that the eigenvectors associated with eigenvalues have to be linearly indepen-dent and orthogonal, which implies its determinant has to be not equal to zero, so nding the eigenvectors matrix and exam its linear independency will check the validity of the derived eigenvalues (Eq.(8)). }\) Furthermore, if we normalize each vector, then we'll have an orthonormal basis. In a Hermitian Matrix, the Eigenvectors of Different Eigenvalues are Orthogonal. An interesting property of an orthogonal matrix P is that det P = ± 1. Orthonormal eigenvectors. 0. More... class Eigen::HessenbergDecomposition< _MatrixType > Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation. But suppose S is complex. Left eigenvectors, returned as a square matrix whose columns are the left eigenvectors of A or generalized left eigenvectors of the pair, (A,B). A is symmetric if At = A; A vector x2 Rn is an eigenvector for A if x6= 0, and if there exists a number such that Ax= x. Orthogonal matrices are very important in factor analysis. MATH 340: EIGENVECTORS, SYMMETRIC MATRICES, AND ORTHOGONALIZATION Let A be an n n real matrix. Since a normal matrix has eigenvectors spanning all of R^n, I don't know why this wouldn't be the case. This factorization property and âS has n orthogonal eigenvectorsâ are two important properties for a symmetric matrix. It is easy to see that <1, 1> and <1, -1> are orthogonal. Eigenvectors of a matrix are also orthogonal to each other. Now without calculations (though for a 2x2 matrix these are simple indeed), this A matrix is . 0. The determinant of the orthogonal matrix has a value of ±1. The eigenvectors in W are normalized so that the 2-norm â¦ The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. Thus, if matrix A is orthogonal, then is A T is also an orthogonal matrix. Orthogonal Eigenvectors Suppose P1, P2 â¬ R2 are linearly independent right eigenvectors of A E R2x2 with eigenvalues 11, 12 E R such that 11 # 12. However eigenvectors w (j) and w (k) corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). A matrix P is orthogonal if P T P = I, or the inverse of P is its transpose. Orthogonality is a concept of two eigenvectors of a matrix being perpendicular to each other. saad0105050 Elementary, Expository, Mathematics, Matrix Analysis, Spectral Graph Theory September 21, 2016 November 18, 2020 1 Minute. 4. Prove that Composition of Positive Operators is Positive . . The fact that the eigenvectors and eigenvalues of a real symmetric matrix can be found by diagonalizing it suggests that a route to the solution of eigenvalue problems might be to search for (and hopefully find) a diagonalizing orthogonal transformation. 1. stuck in proof: eigenvalues of a self-adjoint compact operator on hilbertspace are postive. The proof assumes that the software for [V,D]=eig(A) will always return a non-singular matrix V when A is a normal matrix. Find the characteristic function, eigenvalues, and eigenvectors of the rotation matrix. We call the eigenvalue corresponding to x; We say a set of vectors v1;:::;vk in Rn is orthogonal if vi vj = 0 whenever i 6= j. Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices ... doomed because some eigenvectors of the initial matrix (corresponding to very close eigenvalues perhaps even equal to working accuracy) may be poorly determined by the initial representation L0D0Lt 0. Eigenvectors and eigenvalues of a diagonal matrix D The equation Dx = 0 B B B B @ d1 ;1 0 ::: 0 0 d 2;. And itâs very easy to see that a consequence of this is that the product PTP is a diagonal matrix. . Suppose that pÅ¿ p2 = 0, Ipil = 1, |p2| = 2 (a) (PTS: 0-2) Write an expression for a 2 x 2 matrix whose rows are the left-eigenvectors of A (b) (PTS: 0-2) Write an expression for a similarity transform that transforms A into a diagonal matrix. This is an elementary (yet important) fact in matrix analysis. Example The eigenvalues of the matrix:!= 3 â18 2 â9 are â.=â /=â3. If we futher choose an orthogonal basis of eigenvectors for each eigenspace (which is possible via the Gram-Schmidt procedure), then we can construct an orthogonal basis of eigenvectors for $$\R^n\text{. Thus, the inverse of an orthogonal matrix is simply the transpose of that matrix. In the same way, the inverse of the orthogonal matrix, which is A-1 is also an orthogonal matrix. Definition 4.2.3. We can say that when two eigenvectors make a right angle between each other, these are said to be orthogonal eigenvectors. Since !has two linearly independent eigenvectors, the matrix 6is full rank, and hence, the matrix !is diagonalizable. To explain this more easily, consider the following: That is really what eigenvalues and eigenvectors are about. 10:09 . I must remember to take the complex conjugate. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.Although we consider only real matrices here, the definition can be used for matrices with entries from any field.However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. . Consider the 2 by 2 rotation matrix given by cosine and sine functions. Alternatively, a matrix is orthogonal if and only if its columns are orthonormal, meaning they are orthogonal and of unit length. For this matrix A, is an eigenvector. Since you want P and \(\displaystyle P^{-1}$$ to be orthogonal, the columns must be "orthonormal". And then the transpose, so the eigenvectors are now rows in Q transpose. If a matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by â = â â If is a symmetric matrix, since is formed from the eigenvectors of it is guaranteed to be an orthogonal matrix, therefore â =.Furthermore, because Î is a diagonal matrix, its inverse is easy to calculate: The form and normalization of W depends on the combination of input arguments: [V,D,W] = eig(A) returns matrix W, whose columns are the left eigenvectors of A such that W'*A = D*W'. When I use [U E] = eig(A), to find the eigenvectors of the matrix. Then for a complex matrix, I would look at S bar transpose equal S. Overview. And I also do it for matrices. These eigenvectors must be orthogonal, i.e., U*U' matix must be Identity matrixâ¦ Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem. The most general three-dimensional improper rotation, denoted by R(nË,Î¸), consists of a product of a proper rotation matrix, R(nË,Î¸), and a mirror reï¬ection through a plane The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. Because the eigenvectors of the covariance matrix are orthogonal to each other, they can be used to reorient the data from the x and y axes to the axes represented by the principal components. You re-base the coordinate system for the dataset in a new space defined by its lines of greatest variance. matrices) they can be made orthogonal (decoupled from one another). The decoupling is also apparent in the ability of the eigenvectors to diagonalize the original matrix, A, with the eigenvalues lying on the diagonal of the new matrix, . Substitute. It's conventional for eigenvectors to be normalized to unit length, because a set of orthogonal unit vectors make a good basis for a vector space, but normalization is not strictly required. The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset. I think I've found a way to prove that the qr decomposition of the eigenvector matrix [Q,R]=qr(V) will always give orthogonal eigenvectors Q of a normal matrix A. The eigenvectors in one set are orthogonal to those in the other set, as they must be. 0 0 ::: 0 d n;n 1 C C C C A 0 B B B @ x1 x2 x n 1 C C C â¦ All the discussion about eigenvectors and matrix algebra is a little bit beside the point in my opinion (and also, I'm not that mathematically inclined)--orthogonal axes are just an inherent part of this type of matrix algebra. 1. Matrices of eigenvectors (discussed below) are orthogonal matrices. . Let us call that matrix A. Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues. Orthogonal eigenvectors in symmetrical matrices with repeated eigenvalues and diagonalization 2 Symmetric Matrix , Eigenvectors are not orthogonal to the same eigenvalue. The eigenvalues and eigenvectors of improper rotation matrices in three dimensions An improper rotation matrix is an orthogonal matrix, R, such that det R = â1. Statement. Eigenvectors and eigenspaces for a 3x3 matrix | Linear Algebra | Khan Academy - â¦ Make a right angle between each other a new space defined by its of. Of different eigenvalues are orthogonal the extent of the matrix of orthogonal axes does really., to find the characteristic function, eigenvalues, and eigenvectors of the stretching of the matrix diagonal. Important properties for a symmetric matrix is ) fact in matrix analysis Hessenberg form by orthogonal. 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## 170- Quiz 4

Quiz 4 is here. Please remember that the first midterm is this Wednesday.

Solutions follow.

Problem 1 asks for the derivative of the function $6/\root 3\of {x^5}$.

To solve this, remember the following laws of exponents:

• $\root 3\of t=t^{1/3}$,
• $(t^a)^b=t^{ab}$, and
• $1/t^a=t^{-a}$.

Using these laws we quickly obtain that $6/\root 3\of {x^5}=6/x^{5/3}=6x^{-5/3}$. Now, to take the derivative of this function, we apply two of the rules covered on lecture:

• $(cf)'=cf'$ when $c$ is a constant, and
• $(x^k)'=kx^{k-1}$ when $k$ is a constant.

We have $(6x^{-5/3})'=6\cdot(-5/3)\cdot x^{-8/3}=-10x^{-8/3}$.

Problem 2 asks to use the product rule to find the derivative of $(x^2+5x-3)(x^5 -6x^3+3x^2-7x+1)$.

The product rule says that

$(fg)'=f'g+fg'$,

and in this case it gives us that the derivative is

$(2x+5)(x^5-6x^3+3x^2-7x+1)$ $+(x^2+5x-3)(5x^4-18x^2+6x-7)$.


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Contents

model category

for ∞-groupoids

# Contents

## Definition

A Reedy category with fibrant constants is a Reedy category $R$ such that for every model category $M$ and every fibrant object $X \in M$ the constant $R$-diagram at $X$ is Reedy fibrant.

## Properties

###### Theorem

For a Reedy category $R$ the following are equivalent.

1. $R$ has fibrant constants.
2. For any model category $M$ the colimit functor $\colim_R \colon M^R \to M$ is a left Quillen functor.
3. All matching categories of $R$ are connected or empty.

This theorem is proven in Hirschhorn,Proposition 15.10.2 and Theorem 15.10.8.

###### Proposition
1. The product of two Reedy categories with fibrant constants is a Reedy category with fibrant constants.
2. If $R$ is an elegant Reedy category and $X$ is a presheaf on $R$, then the category of elements of $X$ is a Reedy category with fibrant constants.
###### Proof
1. This follows by combining point 3. of the theorem above with the fact that the matching categories satisfy the “Leibniz rule”, see Riehl, Verity, Observation 4.2.
2. This is proven in Hirschhorn, Proposition 15.10.4 in the case of $R = \Delta$, but the argument given there applies to any elegant Reedy category.

## Examples

• The simplex category $\Delta$ is a Reedy category with fibrant constants.

• More generally, Joyal’s disk categories $\Theta_n$ are Reedy categories with fibrant constants.

• Every direct category is a Reedy category with fibrant constants.

## References

Last revised on October 7, 2013 at 09:05:26. See the history of this page for a list of all contributions to it.


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# A spell stacking question

I've a couple of questions regarding stacking multiple area effect spells:

1. Does Celestial Brilliance spell (BoED p94) stack with itself? That is, if two instances of this spell are cast and an evil outsider is within both areas of effect, does it take 4d6 points of damage or 2d6? Why? (there are plenty of spells like that, e.g. wall of fire, it's just Celestial Brilliance which is easy to stack)

2. What if it was an emanation, instead of an unspecified area effect?

3. What if it was a spread?

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As a DM, I would break out the most resounding "NO" I had in my bag of tricks, answers, but at first glance I can't find a rule that allows or disallows it either way. – dpatchery Jul 5 '11 at 14:35
I think the part that says "different strength spells only use the best version" covers it. – Cthos Jul 5 '11 at 14:38
@Cthos yep, definitely. Good find. – dpatchery Jul 5 '11 at 14:40

They would not stack

It's covered by this sentence:

In most cases, modifiers to a given check or roll stack (combine for a cumulative effect) if they come from different sources and have different types (or no type at all), but do not stack if they have the same type or come from the same source (such as the same spell cast twice in succession). If the modifiers to a particular roll do not stack, only the best bonus and worst penalty applies. Dodge bonuses and circumstance bonuses however, do stack with one another unless otherwise specified.

Emphasis mine.

Edit: Note, that this is referring to Bonuses, but I believe the same applies to Damage as well, unless the spell says it stacks, it wouldn't stack.

Similar Reference in the Magic Section:

In cases when two or more identical spells are operating in the same area or on the same target, but at different strengths, only the best one applies.

Which I think covers the damage stacking aspect.

Source: d20srd.org

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@AceCalhoon - I edited with a more concrete reference. – Cthos Jul 5 '11 at 14:41
+1 The second quotation is what seals it. Note that the rule applies for all scenarios 1, 2 and 3 in the question. The type of area of effect has no influence on the stacking rules. – dpatchery Jul 5 '11 at 14:43
Thanks, your second quote is what I've overlooked. (being theoretical now) Still, since the damage is rolled anew each round, is it rolled twice and the larger applied? Or we should use Same Effect with Differing Results section rule here and apply only one instance of the spell? Or both spells have similar effects with similar (2d6) strength and, once again, only one of them applies? – Jeor Mattan Jul 5 '11 at 15:50
@Jeor I would say you would only roll it once. I've always read it as since they have similar effects only the stronger (pre calculation) applies. So you'd apply the "similar effect" rule before rolling. And since they're both 2d6, only 2d6 would ever get rolled. – Cthos Jul 5 '11 at 16:30