text:
## Intro

When adding one number A with another number B the operation will produce a Sum S and a Carry C.The Operation of an adder is Shown below.

If there are two binary number A and B . The operation of adding two numbers can be shown below

A + B = S C
0 + 0 = Sum 0 Carry 0
0 + 1 = Sum 1 Carry 0
1 + 0 = Sum 1 Carry 0
1 + 1 = Sum 0 Carry 1

The operation of Half Adder can be summarized in the Truth Table below

${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle S}$
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0

From above, In term of logic gate XOR will produce a sum of two input . logic gate AND produce carry

${\displaystyle S=A\oplus B}$
${\displaystyle C=A\cdot B}$

Half Adder can constructed from AND gate and XOR gate as shown below

              ___________
A ------|           |
|   Half    |----- ${\displaystyle S=A\oplus B}$
|           |----- ${\displaystyle C=A\cdot B}$
B ------|___________|

A full adder is a logical circuit that performs an addition operation on three one-bit binary numbers. The full adder produces a sum of the two inputs and carry value. It can be combined with other full adders (see below) or work on its own.

Input Output
${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C_{i}}$ ${\displaystyle C_{o}}$ ${\displaystyle S}$
0 0 0 0 0
0 1 0 0 1
1 0 0 0 1
1 1 0 1 0
0 0 1 0 1
0 1 1 1 0
1 0 1 1 0
1 1 1 1 1
${\displaystyle S=(A\oplus B)\oplus C_{in}}$
${\displaystyle C_{out}=(A\cdot B)+(C_{in}\cdot (A\oplus B))}$

Note that the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. This is because the only difference between OR and XOR gates occurs when both inputs are 1; for the adder shown here, this is never possible. Using only two types of gates is convenient if one desires to implement the adder directly using common IC chips.

A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting Ci to the other input and OR the two carry outputs. Equivalently, S could be made the three-bit XOR of A, B, and Ci, and Co could be made the three-bit majority function of A, B, and Ci.

It is possible to create a logical circuit using multiple full adders to add N-bit numbers. Each full adder inputs a Cin, which is the Cout of the previous adder. This kind of adder is a ripple carry adder, since each carry bit "ripples" to the next full adder. Note that the first (and only the first) full adder may be replaced by a half adder.

The layout of ripple carry adder is simple, which allows for fast design time; however, the ripple carry adder is relatively slow, since each full adder must wait for the carry bit to be calculated from the previous full adder. The gate delay can easily be calculated by inspection of the full adder circuit. Each full adder requires three levels of logic. In a 32-bit [ripple carry] adder, there 32 full adders,so the critical path (worst case) delay is ${\displaystyle 32*3=96}$ gate delays.

To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry lookahead adders. They work by creating two signals (P and G) for each bit position, based on whether a carry is propagated through from a less significant bit position (at least one input is a '1'), a carry is generated in that bit position (both inputs are '1'), or if a carry is killed in that bit position (both inputs are '0'). In most cases, P is simply the sum output of a half-adder and G is the carry output of the same adder. After P and G are generated the carries for every bit position are created. Some advanced carry look ahead architectures are the Manchester carry chain, Brent-Kung adder, and the Kogge-Stone adder.

Some other multi-bit adder architectures break the adder into blocks. It is possible to vary the length of these blocks based on the propagation delay of the circuits to optimize computation time. These block based adders include the carry bypass adder which will determine P and G values for each block rather than each bit, and the carry select adder which pre-generates sum and carry values for either possible carry input to the block.

By combining multiple carry look-ahead adders even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses four 16-bit CLAs with two levels of LCUs.


text:
anonymous one year ago I need help with feeling in the missing boxes. For example mass= weight, volume(liquids), volume(solids) and weight= pound(lb) & fluid ounce(floz) . I need help finding 3 words for kilograms.

• This Question is Open
1. anonymous

2. anonymous

For weight, we know from Newton's laws that F=ma. What does this actually mean without numbers? Substituting a for the gravitational acceleration constant, we then have F=mg. Looking at the units, we have $mass*acceleration=kg*\frac{ m }{ s^2 }$ This combined unit of measurement is also called a Newton (N), a measurement of force. All forces in the metric system are denoted by N, whereas in the U.S. system it is denoted as pounds (lb) (when deriving pounds, we use the same process F=ma but the units for the mass and acceleration are in the U.S. system) The unit for a liquid volume is termed liter, and is broken down in SI base units as 0.001m^3. In other words, one liter is the volume of a liquid cube with side lengths of 10cm. The unit for a solid volume is simply the length cubed. ex: 10m^3 or 20mm^3. Sorry if this was more complicated than what you were looking for. I just think sometimes it's helpful to understand some logical thinking in case you forget what the units are.

3. anonymous

Also, the imperial (U.S.) system for mass is the slug. In all my 12 years of public schooling, I've never had to deal with units of slugs.

format: plain


text: Cleaning up some old stuff on my HD and just found this nice Beamer template: My preferred one is the “fi” variant. #latex


text:
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## Swipe om te navigeren naar een ander artikel

18-03-2019 | Uitgave 6/2019 Open Access

# Development and validation of the caregiver roles and responsibilities scale in cancer caregivers

Tijdschrift:
Quality of Life Research > Uitgave 6/2019
Auteurs:
Valerie Shilling, Rachel Starkings, Valerie Jenkins, David Cella, Lesley Fallowfield
Belangrijke opmerkingen

## Electronic supplementary material

The online version of this article (https://doi.org/10.1007/s11136-019-02154-4) contains supplementary material, which is available to authorized users.

## Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Background

Informal (i.e. unpaid) caregivers of patients, such as spouses, family members or friends, frequently provide significant amounts of care and support [ 1, 2] often alongside their many other responsibilities [ 36]. Informal caregiving contributes to patients’ overall experiences of treatment, therefore maintaining their wellbeing and satisfaction is essential [ 1, 2].
Caregivers must manage several factors relating directly to a patient’s illness together with other ‘secondary stressors’ [ 4] such as changing roles, disruption to lifestyle and family functioning, changes to relationships and sense of self, concerns regarding their own and the patient’s employment with the associated financial implications [ 4]. Additionally informal caregivers may also be caring for children and/or other family members, which can complicate and compound the more negative aspects and effects of being a primary caregiver to someone with cancer [ 3, 7, 8]. Informal caregivers play a crucial role, yet formal assessment of the effects on their lives and wellbeing is not routine.
Our systematic review [ 9] revealed that few of the measures used for caregiver quality of life and caregiver burden were developed and validated specifically in the cancer caregiver population. In addition, most were developed a long time ago, when the advances in medicine that now allow so many patients to live with cancer as a chronic condition, and the associated adaptations required to their own and their family’s lives, had not been made. We found notable gaps in item coverage in a concept mapping exercise to examine the content of these measures. Results suggested that currently available scales do not capture adequately changes in occupational, financial, household and family roles and responsibilities as a result of caregiving, or the broader impacts on the family unit. The influence on career aspiration and progression were not addressed at all; for example, the opportunity cost of a missed promotion at work [ 9]. Caregivers experience a sense that their own life is suspended, ‘on hold’ in some way; that the uncertainty around the length and quality of the patient's survival affects wide-ranging areas of life [ 1013]. It is vital that this lack of control and altered sense of agency is better captured in caregiver outcome measures.
Our systematic reviews of Patient Reported Outcome Measures in use in the cancer patient [ 14] and cancer caregiver [ 9] populations highlighted the need for new rigorously developed, well-validated quality of life measures to assess these broader impacts of disease and treatment.

## Overview

The PROACT (Patient Reported Outcomes, impact of Age and Carer role demands associated with Treatment) programme of work is a multi-phase project, informed by the Quality of Survival (QoS) concept framework [ 15]. The measures were developed to be used alongside the Functional Assessment of Chronic Illness Therapy (FACIT) measurement system and are intended to capture the broader impact of cancer and its treatment during the evaluation of new treatments in clinical trials. Results from questionnaire completion might also enable discussions between clinicians, patients and caregivers about potential treatments and supportive interventions.
Previously we described the development and initial validation of the Patient Roles and Responsibilities Scale (PRRS) [ 16]. This paper reports the four studies that comprised the development and evaluation of the Caregiver Roles and Responsibilities Scale (CRRS). The initial development studies were conducted with caregivers to people with stage III-IV cancer. This was to reflect the fact that due to scientific advances, there has been a large increase in the number of patients living with their cancer, rather than being cured of it, while trying to maintain ‘normal’ life including family, financial and employment responsibilities. These aspects of life are significantly impacted for caregivers as well as patients and we wanted to ensure the measure captured these. The measure was, however, developed with the intention to be appropriate for caregivers to patients with any stage or type of cancer and the later validation study includes caregivers to patients with stage I-II cancer.

## Ethics statement

Studies 1–3 received ethics approval from London Queen Square Research Ethics Committee (ref: 15/LO/1323; 14th September 2015 and ref: 16/LO/1125; 6th September 2016). Study 4 received ethics approval from London Central Research Ethics Committee (ref: 17/LO/1773; 12th October 2017). Signed informed consent was obtained from all participants.

### Eligibility criteria

Inclusion/exclusion criteria were broadly the same across all four studies. Eligible patients had advanced (stage III/IV) lung cancer, gynaecological cancer or melanoma. For studies 1 and 2, we required that patients nominate an informal caregiver also willing to take part. Both cohorts were required to be over 18 years of age, have capacity to give informed consent and be able to read and speak in English (essential for the purposes of scale development and evaluation). Individuals who were inpatients or acutely distressed for any reason were excluded. In studies 2 and 3 we expanded the patient population to include women with breast cancer. In study 3 we relaxed the requirement that both patient and caregiver consent to the study; either party could participate alone. In study 4 we expanded the population further to include patients with renal, head and neck and colorectal cancer and those whose cancer was stage I/II. This is because the questionnaire has been developed to be suitable for use by people supporting patients with cancer of all types and stages.

## Scale development: studies 1 and 2

### Study 1: item generation

We conducted in depth qualitative interviews with 24 patients with advanced cancer and their nominated informal caregivers ( N = 23) about the influence of extended cancer survival on broader aspects of life and wellbeing (see supplementary file S1 for participant demographics). Interviews were recorded and transcribed verbatim.
A thematic framework was developed from an initial process of open coding and tested iteratively as new data were collected. Thematic analysis identified 20 themes and 33 sub-themes from which a list of 181 potential items was generated for the caregiver scale. These were reviewed for relevance and redundancy by the authors before 51 potential items were viewed by our panel of advisors. 64 items were retained for evaluation in Study 2. The Caregivers Roles and Responsibilities Scale will sit within the FACIT measurement system. As such, when concepts were identified that were important to caregivers, we checked if existing items within the FACIT library were appropriate rather than generating new items; those items being already validated measures of the concept. It would be unnecessary and even inappropriate to generate new items on the same content when fully validated (and often translated into multiple languages) items exist within the measurement system that the CRRS will be part of. With permission, we included 3 items from the FACIT Comprehensive Score for Financial Toxicity (COST) measure [ 17], 3 items from the Functional Assessment of Cancer Therapy-General (FACT-G) [ 18] and 1 item from FACIT-Spirituality (FACIT-Sp) [ 19]. For further detail on the procedure and analysis of these interviews, see the related open access paper [ 16].

### Study 2: item reduction and scale construction

Cognitive interviews were conducted with a new cohort of 20 patients and their informal caregivers using a mixture of the ‘think aloud’ technique and specific probes around comprehension, retrieval, judgement and response options to assess each of the potential scale items. Items were also discussed in terms of acceptability, relevance, redundancy and importance. Participants were also given the opportunity to identify missing topics. See supplementary file S1 for participant demographics.
Scale items were revised, added and removed in an iterative fashion through the course of the study. Ninety-four changes were made to the caregiver scale including 80 wording revisions, 9 deletions and 5 additions resulting in a 60 item scale. In order to present the questionnaire in manageable ‘chunks’ for the participant we loosely grouped items under the headings: Family and Support ( N = 12); Relationships and Communication ( N = 7); Lifestyle and Outlook ( N = 12); Health and Wellbeing ( N = 14); Financial Wellbeing ( N = 6) and Jobs and Careers ( N = 9). We did not theorise that these groupings necessarily reflect the underlying factor structure.

### Evaluation and validation: studies 3 and 4

Studies 3 and 4 were conducted consecutively. In study 3, participants completed the full 60 item CRRS as described below. CRRS individual item data were then analysed for the purpose of item reduction. Participants in study 4 completed the reduced CRRS. Data from the reduced CRRS from both studies were analysed together for the Exploratory Factor Analysis, test retest reliability, criterion and convergent validity and sensitivity to change.

## Methods

### Population and procedure

Study 3 participants were recruited from 11 sites in England. Participants were the informal caregivers to patients with stage III/IV breast, gynaecological or lung cancer or malignant melanoma. Study 4 participants were recruited from 13 sites in England. Participants were the informal caregivers to patients with stage I-IV breast, colorectal, gynaecological, head and neck, lung or renal cancer or malignant melanoma.
Participants completed questionnaires at home either on paper or online, whichever their preference. Participants completing online were not forced to respond to all items, although an onscreen message notified them of omissions. Where participants missed multiple items, whether completing on paper or online, they were followed up by the study team to determine whether the omissions were intentional or if they had skipped them accidentally (with the option to provide a response at this point). Demographics and the full validation pack were completed at baseline, the CRRS was completed alone after 7 days (for test–retest) and the full pack completed again after 2 months (sensitivity to change).

### Measures

Participants completed the CRRS alongside a validation battery including the WHOQOL-BREF [ 20] and the Caregiver Quality of Life—Cancer [ 21]. They provided basic demographic information such as age, employment status, level of education, relationship status, caregiving responsibilities and information regarding the diagnosis and treatment of the person they support.

#### The CRRS

The CRRS as completed in Study 3 comprised 60 items: 51 Core Items formatted for the FACIT measurement system (responses as item applies to the past 7 days, 5 response options ranging from Not at All—Very Much) and 1 binary response item on whether the participant has stopped work due to their caregiving responsibilities. A standalone scale, Jobs and Careers comprising 2 binary response items and 7 Likert-scale items is completed only by participants currently in paid employment. Negatively worded items were reverse scored so that a higher score corresponds to better Quality of Life. Participants in Study 4 completed the reduced CRRS with 41 Core Items and the Jobs and Careers subscale.

#### The WHOQOL-BREF [20]

The WHOQOL-BREF is a 26 item scale that produces a quality of life profile with four domain scores (physical health, psychological health, social relationships and environment) and two individually scored items about an individual’s overall perception of quality of life and health. Each item is rated on a 5 point scale (scored 1–5). The four domain scores are scaled in a positive direction with higher scores indicating a higher quality of life.

#### The caregiver quality of life cancer scale (CQOLC) [21]

The CQOLC is a well-validated and extensively used quality of life measure developed specifically for caregivers of people with cancer. It has 35 items rated on a five point scale (not at all—very much). Scores range from 0 to 140 with higher scores indicating higher quality of life. It comprises domains relating to burden, disruptiveness, finances and positive adaptation. Our recent systematic review [ 22] found the CQOLC, alongside the Caregiver Reaction Assessment [ 23] had the strongest support for its psychometric performance.

### Data analysis

Analyses were conducted using the Statistical Package for Social Sciences (IBM SPSS; version25). Missing data from CRRS, although rare, were managed by prorating the total for selected analyses. Missing data from other questionnaires were managed as per the specific instrument guidance. We used guidelines from the International Society for Pharmacoeconomics and Outcomes Research (ISPOR) [ 24, 25] and COSMIN (Consensus-based Standards for the Selection of Health Measurement Instruments) [ 26] in the development and evaluation of this measure.
On the completion of Study 3, the CRRS was assessed in terms of acceptability (missing data, time to complete) and precision (floor/ceiling effects, skewness, inter-item correlations) and a number of items removed.
Combined data from Study 3 and 4 were then analysed using Exploratory Factor Analysis (PCA with oblique rotation); internal consistency (Cronbach’s alpha); criterion and convergent validity (correlation); test–retest reliability (Intraclass Correlation, two-way random, absolute agreement; weighted Kappa for individual items) and responsiveness to change (caregivers were categorised as improved, worsened or unchanged based on meaningful change in WHOQOL-BREF. Paired t-tests were used to determine if CRRS change within a group was significantly different from zero. Spearman correlation coefficients were calculated to explore the relationship between change on PRRS and change on WHOQOL-BREF and CQOLC). Data analysis methods, including thresholds, are explained in greater detail in the related open access paper [ 16].

## Results study 3

### Participants

Of the 119 caregivers who consented to take part, 110 completed baseline assessment (29 caregivers to patients with breast cancer, 31 gynaecological, 31 lung and 19 melanoma), 103 completed T2 and 96 completed T3. At baseline, age ranged from 18 to 88 (median 60 years); 57% were male. Relationship to patient was: partner/spouse ( N = 83, 75%); child of patient ( N = 14, 13%); friend ( N = 5, 4%); sibling ( N = 4, 4%) parent of patient ( N = 2, 2%) and ‘other’ ( N = 2, 2%).

### Acceptability and precision of CRRS

Missing data rates for the CRRS were extremely low; 0.3% at baseline, 0.1% at Time 2 and 0.1% at Time 3. Missing data were distributed across 11 of the 58 Likert-scale items at baseline, 8 items at Time 2 and 5 at Time 3 with no single item having more than 2 missing responses at any time point. We had set a threshold that items with > 15% missing data would be investigated for acceptability. The highest rate of missing data for a single item at any time point was 4% to an employment question with 45 responses of a possible 47; for core questions with the larger sample, the highest rate of missing data for a single item at any time point was 2%.
At baseline, 62 participants completed questionnaires online, 48 completed on paper, but only 43 recorded the time taken to complete the CRRS. Time to complete online ranged from 2.9 to 20.6 min with 1 extreme outlier at 34.4 min (mean excluding outlier = 9.3 min, SD = 4.0 min). On paper, time to complete ranged from 8 to 25 min, (excluding outliers above the 90th percentile), mean = 13.2 min, SD = 4.5 min.

### CRRS core items: item reduction

None of the individual items had missing data > 15%. Scores of minimum ‘0’ ranged from 0.9 to 90.9%; on 42/51 items less than 20% of responses were the minimum ‘0’. 1 item had floor effects exceeding 70% (90.9%—‘I take part in support groups and/or internet forums’).
Four items showed above threshold ceiling effects (72.7% ‘My caregiving responsibilities are a burden’ [reverse scored]; 77.8% ‘I feel appreciated by X’; 79.1% ‘I have difficulty meeting the additional costs of supporting X’ [reverse scored]; 83.6% ‘I feel close to X’). Scores of maximum ‘4’ ranged from 0.9 to 83.6%; 15/51 items had 20% or less responses at maximum ‘4.
Eleven items had an unacceptably low level of correlation with other items and low (< 0.3) corrected item-total correlations in reliability analysis and concerns around face validity. In addition five of these did not meet the measure of sampling adequacy (set at 0.7). Ten of the eleven were removed from further analysis including the item with above threshold floor effects and 2 of the 4 items with above threshold ceiling effects. The item Sp9 was retained because it was felt important to keep an item pertaining to spiritual wellbeing.
Of the 41 retained Likert-scale core items, 14 had z-score of skewness > ǀ4ǀ however their performance in terms of relationship to other items, corrected item-total correlation and/or conceptual significance to the measure ensured their retention at this stage. Total scores on the modified CRRS Core-41 ( N = 110) ranged from 33 to 148 (possible range 0-164), mean = 105.56, SD = 23.48 with skewness = − 0.73(SE = 0.23).

## Results study 3 and 4

### Participants

Of the 269 caregivers who consented to take part, 245 completed baseline assessment (59 caregivers to patients with lung cancer, 57 gynaecological, 43 breast, 38 melanoma, 19 colorectal, 17 renal and 12 head and neck), 227 completed T2 and 211 completed T3. At baseline, age ranged from 18 to 89 (median 60 years); 52% were male. Relationship to patient was: partner/spouse ( N = 192, 78%); child of patient ( N = 22, 9%); friend ( N = 12, 5%); sibling ( N = 8, 3%) parent of patient ( N = 8, 3%) and ‘other’ ( N = 3, 1%). See Fig.  1 for full details of caregivers approached, consented and completed questionnaires at each time point, along with reasons for decline/drop out and Table  1 for key participant characteristics.
Table 1
Key caregiver characteristics and CRRS-41 scores
aEthnicity is not reported as 94% of participants identified as White British
bThe question relating to education was posed differently in study 3 (highest level of education, one choice only including professional qualifications) and study 4 (education and professional/ vocational qualifications reported separately)
Missing data rates for the CRRS were very low: 0.6% at baseline; 0.4% at Time 2 and 0.4% at Time 3. Missing data were distributed across 27 of the 48 Likert-scale items at baseline; in the majority of cases with only one or two missing responses from 245 participants. Three items in particular had higher missing data rates of 10 (4%), 8 (3%) and 5 (2%). Missing data at T2 were distributed across 22/48 items. Two of the same items again had higher rates of missing data with 8 (4%) and 6 (3%) missing responses. At T3 missing data were distributed across 22/48 items.
We had set a threshold that items with > 15% missing data would be investigated for acceptability. While none of the items had missing data close to this level, the three items identified also showed a very low pattern of correlation with other items ( r < 0.3) and low corrected item-total correlations in reliability analysis. As such, they were not included in the exploratory factor analysis and do not sit within the subscales identified therein. They do however cover important aspects of wellbeing not measured by other items in the scale and are retained as standalone items (see Table  2 for details).
Table 2
Standalone items
Item
Inter-item correlation range
Median/mode
$$\left| {{\text{z-score}}\;{\text{skew}}} \right|$$
% Floor (score 0)
% Ceiling (score 4)
Weighted κ
I am less able to fulfil my other caregiving responsibilities (e.g. looking after children, grandchildren, another adult, pets)
0.01–0.29
4/4
10.16
2
63
0.43
I worry about the impact of X’s illness on my children and/or other family members
0.008–0.31
1/1
1.86
24
9
0.56
I find comfort in my faith or spiritual beliefs
0.002–0.24
0/0
10.27
63
7
0.75

### Exploratory factor analysis

Thirty-eight items were entered into the EFA. Following listwise deletion based on all variables, N = 228 (COSMIN guidelines [ 26] consider an item to participant ratio of 1:5 to be ‘good’; this ratio is 1:6). Bartlett’s test of sphericity was significant at < 0.0001 and the Kaiser–Meyer–Olkin (KMO) value was 0.87, confirming that the data were suitable for factor analysis. All items had above threshold (> 0.7) MSA suggesting adequate communality with other variables.
Nine eigenvalues were greater than 1.0, explaining a total of 64.3% of variance. Consideration of the scree plot suggested that 5 factors might be retained explaining 26.9, 8.8, 6.1, 5.2 and 4.8% of variance respectively (51.8% total variance). Items and their factor loadings are shown in Table  3. Most (33/38) items loaded clearly on one factor; the remaining had loadings < 0.4 on all factors (range 0.33–0.39).
Table 3

Item
Factor 1
Emotional health and wellbeing
Factor 2
Support and impact
Factor 3
Lifestyle
Factor 4
Financial wellbeing
Factor 5
Self-care
CH8
I feel depressed by the situation
0.77

GE1
0.77

CH6
I feel angry about the situation
0.73

CO8
I find it hard to think about my own future
0.73

CO10
I feel angry that our plans for the future have changed because of X’s illness
0.71

CO6
It is difficult to think about anything other than X’s illness
0.69

CH7
I feel overwhelmed by the situation
0.66

CH11
I feel resentful that my life has changed because of X’s illness
0.50

CR3
I find some of X’s decisions about treatment difficult to accept
0.38

CS56
Friends and family provide emotional support

0.86

CS22
Friends and family provide practical support

0.81

CS24
People are interested in how I am coping

0.74

CS55
My family and I support each other

0.71

CS21
I feel the cancer team recognise the impact on me

0.54

CS53
I feel that support is available from the health system

0.53

CO1
I keep to my normal routines and activities

0.76

CO2
I socialise less because of my caregiving responsibilities

0.74

CS2
I have taken on some of X’s responsibilities at home (e.g. cooking, cleaning, gardening, DIY)

0.72

GF6
I am enjoying the things I usually do for fun

0.62

CO4
I make time to do things for myself

0.58

CH4
My physical health has suffered as a result of the situation

0.53

CO7
My caregiving responsibilities are a burden

0.52

CO9
I feel as if my life is on hold

0.50

GF7
I am content with the quality of my life right now

0.42

CO5
I feel guilty taking time for myself

0.34

CH5
My mental health has suffered as a result of the situation

0.33

CF4
I have difficulty meeting the additional costs of supporting X

0.66

FT11
I feel in control of my financial situation

0.61

FT3
I worry about the financial problems I will have in the future as a result of X’s illness or treatment

0.61

CF3
The additional costs of supporting X are more than I thought they would be (e.g. travel and parking, heating, healthy eating, supplements, non-prescription medication, paying for help at home)

0.55

CF2
I have to help X financially as a result of his/her illness

0.50

CF1
I know where to access financial advice and support

0.41

CH9
I feel confident in my ability to support X

0.63
CR4
X and I speak openly with each other about his/her illness

0.60
CH3
I look after myself emotionally

0.51
CR6
I protect myself by not talking about X’s illness

0.42
CH1
I have a positive attitude to life

0.39
CH2
I look after myself physically a

0.37

0.37
Factor loadings < 0.4 are suppressed, unless an item does not load at > 0.4 on any factor. The highest factor loading is then shown in italics
aThe item CH2 loads equally on Factors 5 and 3. This item has been retained in Factor 5 (and the associated subscale self-care) as this is where it has most face validity
The five factors were labelled and evaluated as potential subscales: Emotional Health and Wellbeing (Cronbach’s α = 0.87, corrected item-total correlation range 0.33 to 0.72); Lifestyle (Cronbach’s α = 0.87, corrected item-total correlation range 0.49 to 0.70); Support and Impact (Cronbach’s α = 0.82, corrected item-total correlation range 0.52 to 0.69); Self-care (Cronbach’s α = 0.75, corrected item-total correlation range 0.36 to 0.68) and Financial Wellbeing (Cronbach’s α = 0.78, corrected item-total correlation range 0.43 to 0.68) Inter-item and inter-subscale correlations are shown in Table  4.
Table 4
Inter-item correlations and correlation between subscale scores
Subscale
Mean (SD)
Range
Inter-item correlation range
Correlations between subscale scores
Emotional health and wellbeing
Lifestyle
Support and impact
Self-care
Financial wellbeing
Emotional Health and Wellbeing (9 items, possible score 0–36)
25.14 (7.65)
3–36
N = 241
0.14 to 0.65
29/36 r > 0.3
N = 241
1.00

Lifestyle (11 items, possible score 0–44)
29.03 (8.49)
5–44
N = 237
0.22 to 0.60
45/55 r > 0.3
N = 237
0.62
N = 235
1.00

Support and Impact (6 items, possible score 0–24)
13.65 (5.47)
2–24
N = 237
0.22 to 0.65
12/15 r > 0.3
N = 237
0.17
N = 235
0.19
N = 230
1.00

Self-care (6 items, possible score 0–24)
18.02 (3.98)
4–24
N = 243
0.14 to 0.68
6/15 r > 0.3
N = 243
0.52
N = 240
0.48
N = 237
0.35
N = 235
1.00

Financial Wellbeing (6 items, possible score 0–24)
17.76 (5.02)
4–24
N = 242
0.12 to 0.59
10/15 r > 0.3
N = 242
0.50
N = 240
0.52
N = 235
0.22
N = 236
0.36
N = 240
1.00

### Internal consistency

Cronbach’s α for the CRRS full scale at baseline was 0.92 (corrected item-total correlation range 0.09–0.65); at T2 α = 0.94 (CITC range 0.10–0.71) and at T3 α = 0.93 (CITC range 0.12–0.71). If the three standalone items are excluded, α at baseline was 0.92 (corrected item-total correlation range 0.24–0.64); at T2 α = 0.94 (CITC range 0.32–0.71) and at T3 α = 0.93 (CITC range 0.25–0.71).

### Criterion and convergent validity

Total scores for participants with full data sets on the CRRS Core-41 ( N = 219) ranged from 33 to 152 (possible range 0–164), mean = 108.74, SD = 23.07 with skewness = − 0.58 (SE = 0.16). As predicted, total scores on the CRRS correlated strongly with scores on the gold standard measure, the Caregiver Quality of Life—Cancer ( r = 0.89, N = 217) and the other validation measure the WHOQOL-BREF ( r = 0.75, N = 214).
Removing the three standalone items from the total score results in a range from 32 to 148 (possible range 0–152), N = 228, mean = 103.04, SD = 22.39 with skewness = − 0.60 (SE = 0.16) correlating with CQOLC ( r = 0.88, N = 226) and WHOQOL-BREF ( r = 0.74, N = 223).

### Test–retest reliability

The median number of days between baseline and T2 was 9 (range 4–26). Two participants were excluded from analysis as T2 was completed too far from baseline. Total CRRS scores showed good test–retest reliability (two-way random, absolute agreement) (ICC = 0.91, 95% CI 0.88–0.93) as did the five potential subscales: Emotional health and wellbeing (ICC = 0.85, 95% CI 0.81–0.89); Lifestyle (ICC = 0.86, 95% CI 0.82–0.89); Support and impact (ICC = 0.77, 95% CI 0.71–0.82); Self-care (ICC = 0.79, 95% CI 0.73–0.84) and Financial Wellbeing (ICC = 0.88, 95% CI 0.85–0.91).
Weighted kappa for individual items were all in the acceptable range (κ ≥ 0.4) excepting one at κ = 0.39 (I find some of X’s decisions about treatment difficult to accept). Kappa ranged from 0.39 to 0.75 with 31 items in the ‘acceptable’ range (0.4 ≤ κ ≤ 0.59) and 9 in the ‘good’ range (κ ≥ 0.6).

### Responsiveness to change

We have calculated responsiveness to change using two methods. First, we have prorated total CRSS scores for those participants who missed one item or more at baseline or T3. This maximises the number of participants included in the analysis and prorating, if anything, will dull responsiveness to change. As some experts may query prorating in a validation study, we have also analysed the data excluding those with any missing item at either time point. The median number of days between baseline and T3 was 66 (range 56–103).
Using prorated total CRRS scores, N = 208. Participants were categorised as improved, worsened or unchanged on WHOQOL-BREF using an effect size of 0.2 as a minimum threshold for significant change. Twenty-six percent were categorised as ‘unchanged’, 25% as ‘improved’ and 49% as ‘worsened’. Participants who showed meaningful decline on WHOQOL-BREF also showed significant decline on CRRS scores ( p < 0.001). Those who improved showed significant improvement on CRRS ( p = 0.001). CRRS scores did not change significantly for those categorised as unchanged on WHOQOL-BREF ( p = 0.196). Using a more conservative effect size of 0.5 as a threshold for change, 65.5% were categorised as ‘unchanged’, 26.5% as ‘worsened’ and 8% as ‘improved’. Participants categorised as worsened showed significant decline on CRRS scores ( p < 0.001). Those categorised as improved showed significant improvement on CRRS ( p = 0.001). CRRS scores did not change significantly for participants categorised as unchanged ( p = 0.288).
Including only participants who had answered all items on the scale, N = 171. The pattern of results was very similar using the 0.2 effect size for change on WHOQOL-BREF: 25% were categorised as unchanged; 26% improved and 49% worsened with those showing meaningful decline also showing significant decline on CRRS scores ( p < 0.001), those who improved showed significant improvement on CRRS ( p = 0.003) and no significant change in CRRS scores for those categorised as unchanged on WHOQOL-BREF ( p = 0.371). Using the effect size of 0.5 as a threshold for change, 64% were categorised as ‘unchanged’, 27% as ‘worsened’ and 9% as ‘improved’. As before those who showed meaningful decline or improvement also showed significant decline or improvement on the CRRS ( p < 0.001 and p = 0.003 respectively). CRRS scores did not change significantly for those categorised as unchanged ( p = 0.496). Table  5 shows mean CRRS change by group for both analyses.
Table 5
CRRS change by change on WHOQOL-BREF
WHOQOL-BREF
N
Baseline CRRS mean (SD)
Mean CRRS change (SD)
p
Effect size a
Using prorated CRRS scores if missing data present N = 208
0.2 effect size
Improved
52
108.60 (25.00)
− 5.62 (11.16)
0.001
0.22
Unchanged
55
110.78 (20.84)
2.32 (13.13)
0.196
0.11
Worsened
101
109.62 (22.11)
7.75 (12.13)
< 0.001
0.35
0.5 effect size
Improved
17
98.75 (31.20)
− 12.04 (12.79)
0.001
0.39
Unchanged
136
110.14 (21.33)
1.05 (11.52)
0.288
0.05
Worsened
55
111.89 (21.53)
12.35 (11.25)
< 0.001
0.57
Using only participants with complete CRRS data N = 171
0.2 effect size
Improved
44
109.57 (26.14)
− 5.52 (11.64)
0.003
0.21
Unchanged
43
112.09 (21.33)
1.58 (11.48)
0.371
0.07
Worsened
84
107.31 (22.18)
8.18 (12.90)
< 0.001
0.37
0.5 effect size
Improved
15
101.40 (32.38)
− 12.60 (13.53)
0.003
0.39
Unchanged
110
109.93 (22.27)
0.71 (10.89)
0.496
0.03
Worsened
46
109.61 (21.21)
13.54 (11.43)
< 0.001
0.64
a $${{\left| {{\text{Mean}}\;{\text{change}}\;{\text{score}}} \right|} \mathord{\left/ {\vphantom {{\left| {{\text{Mean}}\;{\text{change}}\;{\text{score}}} \right|} {{\text{SD}}\,{\text{mean}}\;{\text{baseline}}\;{\text{score}}}}} \right. \kern-0pt} {{\text{SD}}\,{\text{mean}}\;{\text{baseline}}\;{\text{score}}}}$$
As predicted, CRRS change scores correlated significantly with change on CQOLC and WHOQOL-BREF whether using prorated CRRS scores ( N = 209, r = 0.62; N = 208 r = 0.53 respectively) or only including participants with complete CRRS data ( N = 173, r = 0.65; N = 171 r = 0.56 respectively).

### CRRS jobs and career scale

The Jobs and Career Scale was completed by 128, 113 and 109 caregivers at the different time points respectively. Table  6 shows baseline summary statistics for the items that make up the scale. Most of the items were skewed with respondents rating excellent Quality of Life; 2 had ceiling effects greater than 70%. Internal consistency did not reach an acceptable level at baseline (α = 0.62, corrected item-total correlation range 0.07–0.51), T2 (α = 0.58 CITC range 0.01–0.49) or T3 (α = 0.61 CITC range 0.17–0.49). Test–retest reliability was robust however: weighted Kappa ranged from 0.45 to 0.73; ICC for subscale total was 0.79 (95% CI 0.71–0.85).
Table 6
Baseline descriptive statistics and weighted kappa for items on the Jobs and Career Subscale ( N = 128)

Item
Missing data (%)
Mean ± SD
Median/mode
$$\left| {{\text{z-score}}\;{\text{skew}}} \right|$$
% Floor (score 0)
% Ceiling (score 4)
Inter-item correlations
CITC
Weighted κ
CE3
I have reduced my working hours because of X’s illness
0
3.31 ± 1.15
4.0/4
7.87
4.7
64.8
− 0.05 to 0.45
0.07
0.62
CE4
I have increased my working hours to compensate for loss of X’s income [reverse scored] a
0
3.89 ± 0.49
4.0/4
27.94
0.8
93.0
− 0.05 to 0.33
0.21
0.73
CE5
My working hours are flexible to accommodate my caregiving responsibilities (e.g. attending appointments)
0.8
2.39 ± 1.51
3.0/4
1.67
16.5
35.4
− 0.06 to 0.53
0.20
0.56
CE6
I feel I am able to do my job as well as I would like
0
2.59 ± 1.30
3.0/4
2.69
9.4
32.0
0.05–0.41
0.50
0.45
FT9
I am concerned about keeping my job and income [reverse scored]
0
3.30 ± 1.18
4.0/4
8.18
7.0
65.6
0.12–0.47
0.50
0.59
CE7
I feel that supporting X has limited my career opportunities [reverse scored]
0
3.57 ± 0.97
4.0/4
11.52
3.1
78.1
0.03–0.47
0.42
0.65
CE8
I feel supported by my employer
1.6
2.83 ± 1.30
3.0/4
4.05
8.7
42.9
− 0.15 to 0.53
0.51
0.71
aWhere items are reverse scored, the response ‘not at all’ is given the maximum score ‘4’

## Discussion

The CRRS was developed as a self-report instrument for informal caregivers. Informed by concept mapping of current PRO content, it was developed through extensive qualitative work to address the identified neglected areas. It is a comprehensive measure providing a profile of five subscales: Emotional Health and Wellbeing, Lifestyle, Support and Impact, Self-Care and Financial Wellbeing. The standalone subscale, Jobs and Careers is completed only by participants currently in paid employment. The format and structure are compatible with the FACIT Measurement System. Developed initially with the informal caregivers of people with advanced cancer, it is nevertheless sufficiently generic for use in early stage cancer and with other chronic conditions.
Our method of recruiting caregivers for all of the PROACT studies was to have patients nominate their primary informal caregiver. Defined as ‘the person who is your main source of support’, caregivers could be providing emotional and/or practical support, such as attending appointments, not necessarily more personal care tasks. Thus some participants might have been providing emotional support every few days by telephone, while others would be engaged in personal care such as help with dressing or washing. While the cancer stage is important, so is the level of caregiving that is provided, and this varied within both early and advanced groups. For example, a patient with early stage cancer may need considerable support in many aspects of life immediately after surgery and during treatment, while someone who had surgery for grade III melanoma some years previously and is currently only having regular scans may need relatively little support. Because the questionnaires have been developed and validated in such a range of caregivers, we feel confident that the items included are relevant and important at any level of caregiving and can provide a useful measure over time if the trajectory changes. Furthermore, the very low rate of missing data in study 4 leads us to believe that the items are acceptable, appropriate and possible to answer for caregivers to patients with early stage cancer.
The CRRS appears to be acceptable to participants both in terms of time to complete and the very low missing data rate in total and per item. Missing data were notably higher for three items, which have been excluded from the EFA and any subscale scores. No participants recorded minimum or maximum scores on the final scale.
The inclusion of a total score as well as a profile of subscale scores is a topic of discussion. We believe there is utility in a total score, particularly when the scale is being used as part of a clinical discussion for straightforward monitoring of global change over time and as a way to alert both the caregiver and healthcare professional that they may be experiencing difficulties and may benefit from further discussion and/or investigation. That said, some caution may be warranted in the use of a total score as the correlation between ‘emotional health and wellbeing’ and ‘lifestyle’ subscale scores is quite high ( r = 0.62) and there is a risk these factors might dominate the score.
The 41 core items yield a total score and 5 subscale scores. All demonstrated good internal consistency and test–retest reliability. The total score can be calculated using all 41 items or excluding the three standalone items, either score showed the predicted strength of correlation with measures for criterion and convergent validity. The scale demonstrated responsiveness to change. Whether a threshold effect size of 0.2 or a more conservative 0.5 was used, caregivers showing improvement on the anchor variable showed significant improvement in CRRS scores and those declining on the anchor variable showed significant decline in CRRS scores.
We did not make any a priori hypotheses on group differences on either the total or separate subscale scores. We found that age was a significant factor with significant post-hoc contrasts between those over 65 years and those aged 51–65, and those over 65 years and those 50 years or younger. Retired caregivers had significantly higher scores than those working and those not in paid employment. Those working had significantly higher scores than those not in paid employment. Caregivers who had given up work because of their caregiving responsibilities had significantly lower scores than those who had not (either because they were not working or had continued to work) and those caring for someone currently on treatment, significantly lower than those caring for someone not receiving treatment. There was no difference in scores for those who had additional caregiving responsibilities, such as for children or grandchildren, and those who did not, although previous research suggests that the impact for caregivers is likely to be related to the number of other social roles they fulfil [ 3]. Further exploration of these potential cumulative effects and the role of age and employment will be performed at a later date.
The instrument had good validity and reliability overall but there were features of the data that warrant caution. Although not displaying ceiling effects, the total score is negatively skewed. The Financial Wellbeing subscale is particularly skewed with 10% (still below our cut-off threshold) of participants having maximum score. This may reflect the fact that UK patients do not pay for medical treatment and that a large proportion of our sample were already retired so the financial implications due to loss of employment for example are less pronounced.
While internal consistency and test–retest reliability for the subscales is good, in some cases the corrected item-total and inter-item correlations are lower than 0.5 and 0.3 respectively raising the question that some items may not be contributing significantly to the subscales and the data might benefit from further item level analysis. Similarly while the test–retest reliability for the Jobs and Careers subscale and individual items was good; internal consistency for this scale was disappointing and warrants further investigation.

## Limitations and future research

We developed the measure in caregivers to people with stage III/IV cancer and in a limited number of tumour groups which although chosen to ensure a spread of key patient characteristics such as age, inevitably resulted in some sampling bias. Additionally while a majority (80.5%) of UK population are White British, 94% of our participants described themselves as this. Cultural validity needs to be established as informal caregiving and the expectations on family members are perceived differently across cultures.
We invited patients to nominate the person who they felt was their main source of support. This resulted in a limited range of relationship ‘types’ between patient and caregiver; 78% of caregivers were the partner or spouse of the patient. This means we are unable to make meaningful comparisons between different types of relationship, but reflects the pattern of support in this group of patients.
In this study we measured change in PRO scores over time but have not explicitly linked these to marked changes in trajectory such as transition to palliative care or from diagnosis through treatment. This is an avenue for future research as caregiving responsibilities are likely to vary in line with different lines of treatment [ 9]. Furthermore, Girgis et al. [ 27] report that caregiver unmet needs shift from cancer care-related needs towards their own needs with increasing time since diagnosis. In the current study there were no differences in CRRS total score by time since diagnosis (less than a year, 1–2 years or more than 2 years), however a more detailed analysis of responses to items which are more cancer-care centred versus those which are more caregiver centred may be valuable to explore any difference over time.
The CRRS is comprehensive. Future work will examine which items are most descriptive of the underlying dimensions. Using IRT methods we will investigate the possibility of reducing the number of items or producing useful short forms.

## Conclusion

The importance of considering the needs of caregivers as well as patients is recognised for the provision of good care [ 28, 29].The use of PROs for caregivers alongside those for patients might facilitate discussions about potential treatments and supportive interventions needed for both. There has been an increase in the development of services for caregivers, [ 30, 31] now there is need for robust measures to quantify the effectiveness of any such intervention. The CRRS and its subscale scores may provide useful indications as to how different domains are impacted for different individuals, allowing for a more tailored approach to supportive interventions [ 32]. This initial validation suggests that the CRRS shows promise. Further work, including item level analysis is being conducted.

## Acknowledgements

We would like to thank the patients, caregivers and clinic teams that participated in all of the PROACT research studies and our panel of patient and caregiver advisors. The PROACT studies were funded by Bristol-Myers Squibb.

## Compliance with ethical standards

### Conflict of interest

VS has received an honorarium from Bristol-Myers Squibb. RS declares no conflict of interest. VJ has received research grants from Novartis and Sanofi Aventis. DC has received research grants from Ipsen, Clovis, Novartis and has consulted for Pfizer, Novartis, Regeneration, Astellas, Abbvie. He is a member of the National Quality Forum Scientific Methods Panel. LF has received research grants from Boehringer Ingelheim, Novartis, Bristol-Myers Squibb and Amgen, consulted for AstraZeneca, Puma Biotechnology and Roche and received honoraria from Amgen, Teva, Boehringer Ingelheim, Voluntis Oncolia, Novartis, Astellas Pharma, Eisai and Exerpta Medica.

### Ethical approval

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

### Informed consent

Informed consent was obtained from all individual participants included in the study.

## Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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text:
# $(BP_*, BP_*BP)$

In this page, I hope to collect information regarding the Hopf algeboid structure on $(BP_*, BP_*BP)$. The main point of this page is the maxima code at the end that gives explicit formulas. Recall that the structure of a Hopf algebroid involves maps

• $\eta_R: BP_* \to BP_*BP$
• $\eta_L: BP_* \to BP_*BP$
• $\Delta: BP_*BP \to BP_*BP \otimes_{BP_*} BP_*BP$
• $\varepsilon: BP_*BP \to BP_*$
• $c: BP_*BP \to BP_*BP$.

All of these maps are ring maps. Moreover, $BP_*BP$ is a $BP_*$-bimodule via $\eta_L$ and $\eta_R$, and $\Delta$ and $\varepsilon$ are $BP_*$-bimodule maps. Recall also that

• $BP_* = \mathbb{Z}_{(p)}[v_1, v_2, \ldots]$ with $|v_n| = 2(p^n - 1)$.
• $BP_*BP = BP_*[t_1, t_2, \ldots]$ with $|t_n| = 2(p^n - 1)$.

# Choice of generators

The elements $v_n$ are not "well-defined". There are two standard choicse. Our formulas below will be expressed in terms of auxiliary variables $\lambda_n$, and the two choices correspond to how $\lambda_n$ is expressed in terms of $v_n$.

The Hazewinkel generators are defined by

$p \lambda_n = \sum_{0 \leq i < n} \lambda_i v_{n-i}^{p^i}.$

The Araki generators are defined by

$p \lambda_n = \sum_{0 \leq i \leq n} \lambda_i v_{n-i}^{p^i}.$

# Formulas

Explicit formulas for the Hopf algebroid structure are as follows:

\begin{aligned}\varepsilon(t_i) &= 0, \varepsilon(v_i) = v_i \\ \eta_L(v_n) &= v_n. \\ \eta_R (\lambda_n) &= \sum_{0 \leq i \leq n} \lambda_i t_{n-i}^{p^i}. \\ \sum_{i, j \geq 0} \lambda_i \Delta(t_j)^{p^i} &= \sum_{i, j, k \geq 0} \lambda_i t_j^{p^i} \otimes t_k^{p^{i + j}}. \\ \sum_{i, j, k \geq 0} \lambda_i t_j^{p^i} c(t_k)^{p^{i + j}} &= \sum_{i \geq 0} \lambda_i .\end{aligned}

For the purposes of applying the last two formulas, it is useful to note that $\lambda_i \Delta(t_j)^{p^i}$ has degree $2(p^{i + j} - 1)$; and $\lambda_i t_j^{p^i} \otimes t_k^{p^{i + j}}$ has degree $2(p^{i + j + k} - 1)$, etc.

# Maxima code

The following contains code for computing $\eta_R$, $\Delta$ and $c$ in maxima:

p : 3;

/* Define some edge cases */
t[0] : 1;
s[0] : 1;
c[0] : 1;
v[0] : p;

/* Hazewinkel */
l[n] := if is(equal(n, 0)) then 1 else 1/p * sum(l[i] * (v[n-i])^(p^i) ,i, 0, n-1);
m(n) := n-1;

/* Araki */
/* l[n] := if is(equal(n, 0)) then 1 else 1/(p - p^(p^n)) * sum(l[i] * (v[n-i])^(p^i) ,i, 0, n-1);
m(n) := n; */

/* Actual code */
etaRl[n] := sum(l[i] * t[n-i]^(p^i),i, 0, n);
etaR[n] := if is(equal(n, 0)) then p else expand(p * etaRl[n] - sum(etaRl[i] * (etaR[n-i]^(p^i)), i, 1, m(n)));

d[n] := expand(sum(sum(l[i] * t[j]^(p^i) * s[n-i-j]^(p^(i+j)), j, 0, n-i),i, 0, n) - sum(l[i] * d[n-i]^(p^i), i, 1, n));

c[n] := expand(l[n] - sum(sum(l[i] * t[n-k-i]^(p^i) * c[k]^(p^(n-k)), i, 0, n-k), k, 0, n-1));

To use this in maxima, Then import/paste in the above code. You can then type etaR[n] to compute $\eta_R(v_n)$, and d[n] and c[n] to compute $\Delta(t_n)$ and $c(t_n)$ respectively. Note that the computation time needed grows very quickly with n.

By default, this does calculations at the prime 3, which can be customized in the first line.

Also, by default, this uses the Hazewinkel generators. If you want to use the Araki generators, comment out the two lines following /* Hazewinkel */ and uncomment those following /* Araki */.


text:
EN FR
• Legal notice
• Accessibility - non conforme
##### POLSYS - 2016

New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography

## Section: Research Program

### Low level implementation and Dedicated Algebraic Computation and Linear Algebra.

Participants : Jean-Charles Faugère, Christian Eder, Elias Tsigaridas.

Here, the primary objective is to focus on dedicated algorithms and software for the linear algebra steps in Gröbner bases computations and for problems arising in Number Theory. As explained above, linear algebra is a key step in the process of computing efficiently Gröbner bases. It is then natural to develop specific linear algebra algorithms and implementations to further strengthen the existing software. Conversely, Gröbner bases computation is often a key ingredient in higher level algorithms from Algebraic Number Theory. In these cases, the algebraic problems are very particular and specific. Hence dedicated Gröbner bases algorithms and implementations would provide a better efficiency.

Dedicated linear algebra tools.FGb is an efficient library for Gröbner bases computations which can be used, for instance, via Maple . However, the library is sequential. A goal of the project is to extend its efficiency to new trend parallel architectures such as clusters of multi-processor systems in order to tackle a broader class of problems for several applications. Consequently, our first aim is to provide a durable, long term software solution, which will be the successor of the existing FGb library. To achieve this goal, we will first develop a high performance linear algebra package (under the LGPL license). This could be organized in the form of a collaborative project between the members of the team. The objective is not to develop a general library similar to the Linbox project but to propose a dedicated linear algebra package taking into account the specific properties of the matrices generated by the Gröbner bases algorithms. Indeed these matrices are sparse (the actual sparsity depends strongly on the application), almost block triangular and not necessarily of full rank. Moreover, most of the pivots are known at the beginning of the computation. In practice, such matrices are huge (more than ${10}^{6}$ columns) but taking into account their shape may allow us to speed up the computations by one or several orders of magnitude. A variant of a Gaussian elimination algorithm together with a corresponding C implementation has been presented. The main peculiarity is the order in which the operations are performed. This will be the kernel of the new linear algebra library that will be developed.

Fast linear algebra packages would also benefit to the transformation of a Gröbner basis of a zero–dimensional ideal with respect to a given monomial ordering into a Gröbner basis with respect to another ordering. In the generic case at least, the change of ordering is equivalent to the computation of the minimal polynomial of a so-called multiplication matrix. By taking into account the sparsity of this matrix, the computation of the Gröbner basis can be done more efficiently using a variant of the Wiedemann algorithm. Hence, our goal is also to obtain a dedicated high performance library for transforming (i.e. change ordering) Gröbner bases.

Dedicated algebraic tools for Algebraic Number Theory. Recent results in Algebraic Number Theory tend to show that the computation of Gröbner basis is a key step toward the resolution of difficult problems in this domain ( P. Gaudry, Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem, Journal of Symbolic Computation 44,12 (2009) pp. 1690-1702). Using existing resolution methods is simply not enough to solve relevant problems. The main algorithmic bottleneck to overcome is to adapt the Gröbner basis computation step to the specific problems. Typically, problems coming from Algebraic Number Theory usually have a lot of symmetries or the input systems are very structured. This is the case in particular for problems coming from the algorithmic theory of Abelian varieties over finite fields ( e.g. point counting, discrete logarithm, isogeny.) where the objects are represented by polynomial system and are endowed with intrinsic group actions. The main goal here is to provide dedicated algebraic resolution algorithms and implementations for solving such problems. We do not restrict our focus on problems in positive characteristic. For instance, tower of algebraic fields can be viewed as triangular sets; more generally, related problems (e.g. effective Galois theory) which can be represented by polynomial systems will receive our attention. This is motivated by the fact that, for example, computing small integer solutions of Diophantine polynomial systems in connection with Coppersmith's method would also gain in efficiency by using a dedicated Gröbner bases computations step.


text:
# Getting name of current structure level

In a document I'm currently working on, we often change the hierarchy of chapters/sections etc. So within the text I would like to reference the current level of hierarchy in a sentence like

In this chapter we will talk about ...

As the structure changes often, I would like to add the name of the current hierarchy level ("chapter" in the example above) to some command, so that it adapts automatically, when we change the structure again.

I could not find, however, a command for this. Most commands are focused towards getting either the current section's title or the reference to some other section including the section number.

So given this example, what would I need to write as \someCommand to get the current chapter/section/subsection/etc. name?

\documentclass[a4paper]{memoir}
\begin{document}

\chapter{Example-Chapter}

Current level: \someCommand - should be "chapter" and not "Example-Chapter"

\section{Example-Section}

Current level: \someCommand - should be "section" and not "Example-Section"

\end{document}

You can use the fact that \chapter resets the linked counters and the same happens for \section.

\documentclass[a4paper]{memoir}

\newcommand{\someCommand}{%
\ifnum\value{section}=0
chapter%
\else
\ifnum\value{subsection}=0
section%
\else
subsection%
\fi
\fi
}
\setsecnumdepth{subsection}

\begin{document}

\chapter{Example chapter}

Current level: \someCommand, should be chapter''

\section{Example section}

Current level: \someCommand, should be section''

\subsection{Example subsection}

Current level: \someCommand, should be subsection''

\section{Example section}

Current level: \someCommand, should be section''

\end{document}

This version also works for unnumbered structure levels. The structure macro definitions \chapter, \section etc. are extended at the begin with the definition of the structure level name and number.

\documentclass{report}
\setcounter{secnumdepth}{-1}

\usepackage{etoolbox}
\newcommand*{\tmp}[3]{%
\pretocmd#1{%
\gdef\StructureLevelNumber{#2}%
\gdef\StructureLevelName{#3}%
}{}{%
\errmessage{Cannot patch \string#1}%
}%
}
\tmp\part{-1}{part}
\tmp\chapter{0}{chapter}
\tmp\section{1}{section}
\tmp\subsection{2}{subsection}
\tmp\subsubsection{3}{subsubsection}
\tmp\paragraph{4}{paragraph}
\tmp\subparagraph{5}{subparagraph}

\newcommand*{\test}{%
We are in structure level \StructureLevelName'' %
(\StructureLevelNumber).%
}

\begin{document}
\part{Part title}
\test
\chapter{Chapter title}
\test
\section{Section title}
\test
\subsection{Subsection title}
\test
\subsubsection{Subsubsection title}
\test
\paragraph{Paragraph title.}
\test
\subparagraph{Subparagraph title.}
\test
\end{document}

Third page:

# References with structure levels

Package zref supports the definition of new properties that can be stored in labels (\zlabel, \zref@label...) and referenced (\zref, \zref@extract...).

The following example only used integer numbers to identify the structure levels and generates the name for the structure level via macro \StructureLevelName.

\documentclass{report}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{color}
\setcounter{secnumdepth}{-1}

\usepackage{etoolbox}
\newcommand*{\tmp}[2]{%
\pretocmd#1{%
\gdef\StructureLevel{#2}%
}{}{%
\errmessage{Cannot patch \string#1}%
}%
}
\tmp\part{-1}
\tmp\chapter{0}
\tmp\section{1}
\tmp\subsection{2}
\tmp\subsubsection{3}
\tmp\paragraph{4}
\tmp\subparagraph{5}
\newcommand*{\StructureLevel}{-2}

\newcommand*{\StructureLevelName}[1]{%
\ifcase\numexpr(#1)\relax
chapter%
\or
section%
\or
subsection%
\or
subsubsection%
\or
paragraph%
\or
subparagraph%
\else
\ifnum\numexpr(#1)\relax=-1 %
part%
\else
unknown%
\fi
\fi
}

\usepackage{zref-base, zref-titleref, zref-user}
\makeatletter
\zref@newprop{StructureLevel}[-2]{\StructureLevel}
\newcommand*{\zrefStructureLevelName}[1]{%
\zref@refused{#1}%
\StructureLevelName{%
\zref@extractdefault{#1}{StructureLevel}{-2}%
}%
}
\makeatother

\newcommand*{\test}{%
We are in structure level
\textcolor{red}{\StructureLevelName{\StructureLevel}}'' %
(\textcolor{blue}{\StructureLevel}).%
}
\newcommand*{\testref}[1]{%
\texttt{\textbackslash zlabel\{#1\}}:
Structure element is \textcolor{red}{\zrefStructureLevelName{#1}}'',
title: \textcolor{blue}{\ztitleref{#1}}\par
}

\begin{document}
\part{Part title}
\zlabel{a}
\test
\chapter{Chapter title}
\zlabel{b}
\test
\section{Section title}
\zlabel{c}
\test
\subsection{Subsection title}
\zlabel{d}
\test
$$\zlabel{eq} E=mc^2$$
\subsubsection{Subsubsection title}
\zlabel{e}
\test
\paragraph{Paragraph title.}
\zlabel{f}
\test
\subparagraph{Subparagraph title.}
\zlabel{g}
\test
\newpage
\testref{a}
\testref{b}
\testref{c}
\testref{d}
\testref{e}
\testref{f}
\testref{g}
\texttt{\textbackslash zlabel\{eq\}}:
Equation \zref{eq} is in structure element
\textcolor{red}{\zrefStructureLevelName{eq}}''.
\end{document}

Last page with the references:

While not being a unique new answer, I wanted to add this for reference:

After experimenting a little with the answer of @egreg, I also removed the "static" sectioning labels and replaced them by the ones hyperref uses, so it stays consistent with the use of \autoref in the document.

\documentclass[a4paper]{memoir}
\usepackage{hyperref}

\newcommand{\structuringLevel}{%
\ifnum\value{section}=0
\chapterautorefname%
\else
\ifnum\value{subsection}=0
\sectionautorefname%
\else
\subsectionautorefname%
\fi
\fi
}
\setsecnumdepth{subsection}

\begin{document}

\chapter{Example chapter}

Current level: \someCommand, should be chapter''

\section{Example section}

Current level: \someCommand, should be section''

\subsection{Example subsection}

Current level: \someCommand, should be subsection''

\section{Example section}

Current level: \someCommand, should be section''

\end{document}


text:
# Reading 10

Takagi, H., Asakawa, C., Fukuda, K., & Maeda, J. (2004). Accessibility designer: Visualizing Usability for the Blind. Paper presented at the Proceedings of the 6th international ACM SIGACCESS conference on Computers and accessibility. Retrieved from: http://doi.acm.org/10.1145/1028630.1028662
This purpose of this study is to develop software for improving web usability and blind user’s productivity called Accessibility Designer. The authors describe the factors that become problems beyond the software development which are accessibility checker, requirement of guidelines compliance, and syntactic checking of web page. For overcoming the problems, the author develop a new software that has three fundamental features: background color pattern for presenting reaching time to each part a web page, color filling analysis for indicating accessibility particular area of a web page, and visual layout analysis using text information that generated by standard screen reader. This new software is expected to improve usability of a web page.

In the use, the author found that the background-color-based analysis has the power to reveal several problems such as availability and appropriateness skip-links navigator, availability and appropriateness of  headings usage, and content order. The result of the analysis will contribute to development of web design that may decrease the time and cost maintenance, determine web usability even for high level accessibility, achieve real accessible and usable website by focusing on the user experience and actual productivity.

Even tough this article is technical oriented and focus on usability for user with visual impaired, but it has valuable knowledge for improving accessibility and usability of information displayed on a web page.


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# If 8^r/4^s =2^t, then what is r in terms of s and t ?

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Math Expert
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Posts: 50570
If 8^r/4^s =2^t, then what is r in terms of s and t ?  [#permalink]

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28 Oct 2018, 22:34
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Question Stats:

81% (00:51) correct 19% (01:18) wrong based on 31 sessions

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If $$\frac{8^r}{4^s}=2^t$$, then what is r in terms of s and t ?

A. $$s + t + 1$$

B. $$s + t + 5$$

C. $$\frac{2s+t}{3}$$

D. $$\frac{2st}{3}$$

E. $$\frac{s}{2}+\frac{t}{4}$$

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Re: If 8^r/4^s =2^t, then what is r in terms of s and t ?  [#permalink]

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28 Oct 2018, 22:40
IMO C

2^(3r-2s) = 2^t
=> r= (2s+t)/3

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Re: If 8^r/4^s =2^t, then what is r in terms of s and t ?  [#permalink]

### Show Tags

28 Oct 2018, 22:44
2^3r / 2^2s = 2^t
2 ^ (3r-2s) = 2^t
3r - 2s = t
3r = 2s+t
r = (2s+t)/3 ....... CORRECT ANSWER = C
Re: If 8^r/4^s =2^t, then what is r in terms of s and t ? &nbs [#permalink] 28 Oct 2018, 22:44
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text:
2017 East Central NA Regional Contest

#### Start

2017-10-28 14:00 UTC

## 2017 East Central NA Regional Contest

#### End

2017-10-28 19:00 UTC
The end is near!
Contest is over.
Not yet started.
Contest is starting in -511 days 1:06:42

5:00:00

0:00:00

# Problem ITwenty Four, Again

Yes, we know $\ldots$ we’ve used Challenge $24$ before for contest problems. In case you’ve never heard of Challenge $24$ (or have a very short memory) the object of the game is to take $4$ given numbers (the base values) and determine if there is a way to produce the value $24$ from them using the four basic arithmetic operations (and parentheses if needed). For example, given the four base values 3 5 5 2, you can produce $24$ in many ways. Two of them are: 5*5-3+2 and (3+5)*(5-2). Recall that multiplication and division have precedence over addition and subtraction, and that equal-precedence operators are evaluated left-to-right.

This is all very familiar to most of you, but what you probably don’t know is that you can grade the quality of the expressions used to produce $24$. In fact, we’re sure you don’t know this since we’ve just made it up. Here’s how it works: A perfect grade for an expression is $0$. Each use of parentheses adds one point to the grade. Furthermore, each inversion (that is, a swap of two adjacent values) of the original ordering of the four base values adds two points. The first expression above has a grade of $4$, since two inversions are used to move the $3$ to the third position. The second expression has a better grade of $2$ since it uses no inversions but two sets of parentheses. As a further example, the initial set of four base values 3 6 2 3 could produce an expression of grade $3$ — namely (3+6+3)*2 — but it also has a perfect grade $0$ expression — namely 3*6+2*3. Needless to say, the lower the grade the “better” the expression.

Two additional rules we’ll use: $1$) you cannot use unary minus in any expression, so you can’t take the base values 3 5 5 2 and produce the expression -3+5*5+2, and $2$) division can only be used if the result is an integer, so you can’t take the base values 2 3 4 9 and produce the expression 2/3*4*9.

Given a sequence of base values, determine the lowest graded expression resulting in the value $24$. And by the way, the initial set of base values 3 5 5 2 has a grade $1$ expression — can you find it?

## Input

Input consists of a single line containing $4$ base values. All base values are between $1$ and $100$, inclusive.

## Output

Display the lowest grade possible using the sequence of base values. If it is not possible to produce $24$, display impossible.

Sample Input 1 Sample Output 1
3 5 5 2
1
Sample Input 2 Sample Output 2
1 1 1 1
impossible


text:
Thread: Check my arithmetic View Single Post
2006-06-05, 23:49   #4
R.D. Silverman

Nov 2003

11101001001002 Posts

Quote:
 Originally Posted by R. Gerbicz There was also an error in N: N=2,1526L/2,218L=2^654+2^600+2^545-2^436-2^382-2^327-2^273-2^218+2^109+2^55+1 I think the correct polynom ( I've found this by hand ) is the following: (checking this by Pari-Gp ): (23:15) gp > f(x)=x^6+2*x^5-10*x^4-20*x^3+16*x^2+32*x+8 (23:15) gp > g(z)=substpol(f(x),x,2*z+1/z) %4 = (64*z^12 + 64*z^11 + 32*z^10 - 16*z^8 - 16*z^7 - 8*z^6 - 8*z^5 - 4*z^4 + 2*z^2 + 2*z + 1)/z^6 (23:15) gp > Further checking to see that g(z) has a root of 2^54 modulo N: (23:28) gp > lift(g(Mod(2^54,N))) %13 = 0 (23:28) gp >
I got the coefficient of x wrong, and that led to an incorrect constant as
well

Thanks.


text:
Conference paper Open Access

# Baryonic Angular Momentum in Simulated Disks: The CGM

DeFelippis, Daniel; Genel, Shy; Bryan, Greg

Physical processes that occur the circumgalactic medium (CGM) are largely responsible for setting the baryonic angular momentum of simulated galaxies. However, the angular momentum of the CGM has yet to be characterized in a large cosmological simulation, which are now capable of reproducing observed angular momentum properties for large populations of galaxies. We present a first step towards understanding the angular momentum properties of the CGM statistically for galaxies in the IllustrisTNG simulation. We focus on Milky Way-mass halos at $z=0$, but plan to extend this analysis to other masses and redshifts in future work.

Files (644.0 kB)
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defelippis.pdf
644.0 kB
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text:
# Tag Info

## New answers tagged packages

5

This is a known issue. It is not about package loading (or Nearest) but about Catch/Throw not working correctly when used during kernel initialization. For example, just putting the following in init.m Catch[Throw[1]] will result in similar behavior. The workaround I've been using is to run any init code depending on Catch/Throw as a scheduled task, e.g. ...

2

Try clearing the variables first, for example: Clear[a, b, x, y]; The result in the question can be reproduced exactly by evaluating a = (x + y)^20 // Expand; Solve[x^2 + a x + b == 0, x] and the package example notebook ToMatlabExamples.nb does contain such an assignment to a -- most likely it has been evaluated in the same kernel session before ...

5

AA and BB in the definition of f are local symbols of the Test1context. If you want to use them in the Global context you have to export them from the package or you have to precede them with their context: f[Test1PrivateAA[3]] 1 Or BeginPackage["Test1"]; ClearAll[f]; f::usage = ""; AA::usage = ""; BB::usage = ""; Begin["Private"]; f[x_AA] := ...

11

I am not really familiar with LISP macros but from what little I think I know these do not appear to be strongly related. Rather they appear to be utility functions for function authors; it seems that a number of recently added internal top-level definitions use them. Some of them clearly do use meta-programming type expansions so perhaps they are closer ...

1

This question has been solved with the help from comments above. In fact, I have to say,I don't really think how to use this function when I begin to edit.In the process of discussing with others, I think there are two ways to solve this problem. If I really want a symbolic output, I must define or declare X and Y at the beginning of the Private part.But I ...

15

In addition to the error messages quoted in the question the line returns: GeneralUtilitiesBenchmarkingPackagePrivate`plot[ IndexBy[{{{16, 9.37132*10^-6}, . . . IndexBy was removed from 10.1.0: Note that IndexBy will be removed in a future version of Mathematica. It was something that was considered for 10.0.0 but didn't make the cut. – Stefan R ...

2

It is a matter of having the right $ContextPath at the time n is referenced. This is slightly complicated by the fact that contexts are interpreted at parse time, so typical tricks such as Block do not work. The behavior of Begin and BeginPackage are also different with respect to$ContextPath, but this is all discussed in the documentation, so I will let ...

Top 50 recent answers are included


text:
Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# A point source S is placed midway between two converging

Question:

A point source $S$ is placed midway between two converging mirror having equal focal length $f$ as shown in fig. Find the values od d for which only one image is formed.

Solution:

Both mirror produces one image in following two cases:

(A) When Source is at distance $=2 f$ or at center of curvature then image is produce at $v=2 f+2 f=4 f$

(B) When source is at distance $f$ then rays become parallel after reflecting through one mirror, then it is reflected through other mirror and only one image is formed

$V=f+f=2 f$


text:
# Dice loss gives binary output whereas binary crossentropy produces probability output map

On recommendation of Kanak on stackoverflow I am posting this question here:

Currently I am experimenting with various loss functions and optimizers for my binary image segmentation problem. The loss functions that I use in my Unet however give different output segmentation maps.

I have a highly imbalanced dataset, thus I am trying dice loss for which the customized function is given below.

    def dice_coef(y_true, y_pred, smooth=1):
"""
Dice = (2*|X & Y|)/ (|X|+ |Y|)
=  2*sum(|A*B|)/(sum(A^2)+sum(B^2))
ref: https://arxiv.org/pdf/1606.04797v1.pdf
"""
intersection = K.sum(K.abs(y_true * y_pred), axis=-1)
return (2. * intersection + smooth) / (K.sum(K.square(y_true), -1) + K.sum(K.square(y_pred), -1) + smooth)

def dice_coef_loss(y_true, y_pred):
return 1 - dice_coef(y_true, y_pred)

Binary cross entropy results in a probability output map, where each pixel has a color intensity that represents the chance of that pixel being the positive or negative class. However, when I use the dice loss function, the output is not a probability map but the pixels are classed as either 0 or 1.

My questions are:

1.How is it possible that these different loss functions have these vastly different results?

1. Is there a way to customize the dice loss function so that the output segmentation map is a probability map similar to the one of binary crossentropy loss.


text:
# Beat frequency and length question. (grade 11)

1. May 23, 2015

### Ethan_Tab

1. The problem statement, all variables and given/known data
Two pipes, Identical in length and closed at one end are producing notes. You are very, very annoyed because the notes are creating an audible beat frequency of 20Hz. If one pipe is producing a note at 10Hz and the temperature is 15C (which correlates to a speed of sound of 340.39m/s), what is the length of the two pipes?

2. Relevant equations

Beat frequency= abs(ƒ_2-ƒ_1)
L(n)=[(2n-1)/4]λ --- (length as a function of the harmonic)
v=λƒ
v(t)=331.4+0.606T --- (velocity as a function of temperature)
T=1/f --- where T= period

3. The attempt at a solution

Since we can't have negative beat frequencies we know that the second pipe is creating a 30Hz sound.
λ_2=340.39/30=11.35meters (second pipe)
and
λ_1=340.39/10=34.04 meters (first pipe)

We also know that for a closed pipe, the closed end must have a node while the open end must have an antinode. But since we don't know the harmonic, theres not much we can do with that information.

Any ideas?

2. May 23, 2015

### BvU

You missed the bit of info saying the pipes are equal in length ?

3. May 23, 2015

### Ethan_Tab

Im not too sure I understand what you mean.

4. May 23, 2015

### BvU

Well, you have the wavelengths in both pipes and you have the fact that the lengths of the pipes themselves are equal.

5. May 23, 2015

### Ethan_Tab

So if I'm understanding you correctly, you're saying to set the two equation L(n)=[(2n-1)/4]λ_1 and L(n)=[(2n-1)/4]λ_2 equal
[(2n-1)/4]34.04=[(2n-1)/4]11.35
We can solve for the n and in this case
n=0.5

Is that what you mean?

6. May 23, 2015

### BvU

Very close, but no: first of all you need to find an integer number n. In fact you need to find two integer numbers, one for each pipe.
And it becomes easier if for the 34.04 you write $\lambda_1$ and for the other $\lambda_2 = \lambda_1/3$ !

7. May 23, 2015

### BvU

Yet another eureka moment eh ? !

 premature !

Last edited: May 23, 2015
8. May 23, 2015

### Ethan_Tab

Okay so doing that, another system of equation:

λ(2n-1)/4=L
and
λ(2n-1)/12=L

solving for n in equation 1:
n=4L/2λ+1
Putting that into equation 2:
[λ(2(4L/2λ+1)-1]/12
which simplifies into
4L/12=L

Now what?

9. May 23, 2015

### BvU

You need two n. $n_1$ for pipe 1 and $n_2$ for pipe 2. That way
becomes$$\lambda_1 {2n_1-1\over 4} = \lambda_1 {2n_2-1\over 12}$$

10. May 23, 2015

### Ethan_Tab

Im confused as to what I need to solve for.
Also, how would I solve for either n_1 or n_2 given that there is only one equation?

11. May 23, 2015

### BvU

There is one equation. Same as with the other guy, $\lambda_1$ divides out. You are left with one equation and two unknowns. Disaster ? No ! You also know that both are natural numbers.

12. May 23, 2015

### Ethan_Tab

I understand why the two n's must be natural. I don't understand however, how I can solve for either one algebraically given that there is only one equation.

13. May 23, 2015

### BvU

Work out the equation (eliminate the $\lambda_1$ and the factor 4) to its simplest form and see what you get -- in particular, consider which combinations $n_1 , n_2$ satisfy the equation. Then consider what pipe length goes with such solutions.

14. May 23, 2015

### Ethan_Tab

Using guess and check I can see that n_1= 1 when n_2=2

Theres the eureka moment!!!

If we use the λ2n-1/4 with the sets [n_1 and λ_1 or n_2 and λ_2] We get the same (presumably correct) answer of 8.51 meters!

Thanks for the help and patience BvU!

15. May 23, 2015

### BvU

Not done yet: what about higher numbers than 1 and 2 ?

16. May 23, 2015

### Ethan_Tab

Well, n_1=3 and n_2=8 also works

so using these numbers to find the new length I get: 42.55 meters.

17. May 23, 2015

### Ethan_Tab

It appears that there is more then one solution? as there are quite a lot (if not an infinite amount) of possible "n" combinations.

18. May 23, 2015

### BvU

Yes, and in between there is $n_1 =2,\ n_2 = 5$.

My impression is that 8.5 m is already pretty big for a pipe, so you're safe assuming the lowest possible combination is what they want for an answer. But if I were teacher, I'd sure reward the students who point out that there are bigger pipes that also qualify !

In addition, it's already pretty difficult to get such a big pipe to resonate at its second resonance frequency, let alone at its fifth. But that's something the experts should sort out, and I'm just an ordinary physicist...

Well done.

@idc: it's not polite to hijack a thread and interfere this way. Constructive contributions are always welcome, though, if they help the original poster.
--

19. May 23, 2015

### Ethan_Tab

Haha. Again, I appreciate your help immensely.

20. May 24, 2015