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Latest release

Household and Family Projections, Australia methodology

Reference period
2016 - 2041
Released
14/03/2019
Next release Unknown
First release

Explanatory notes

Introduction

1 This publication contains 2016-based projections of households, families and living arrangements for the period 2016 to 2041. The projections are available for Australia, states and territories, capital cities and rest of states. These projections supersede the 2011-based series published in Household and Family Projections, Australia, 2011 to 2036 (cat. no. 3236.0) in March 2015.

Geography definition

2 Capital cities and rest of states are the GCCSA level of the Australian Statistical Geography Standard (ASGS) 2016 (cat. no. 1270.0.55.001). The ASGS defines the whole ACT as a capital city area, thus projections were not produced below the Territory level for the ACT. The projections for Australia include Christmas Island, Cocos (Keeling) Islands, Jervis Bay Territory and Norfolk Island, collectively referred to as Other Territories.

Household definition

3 A household is defined as one or more persons, at least one of whom is at least 15 years of age, usually resident in the same private dwelling. Although in reality some living arrangements are fluid, the Census assigns each person to a single household. The projection method therefore treats households as discrete units.

Method

4 These projections use a propensity method based on that developed by McDonald and Kippen and used in Household Trends and Projections: Victoria, 1986-2011. The method identifies propensities from the Census of Population and Housing for persons to be in different living arrangements. Trends observed in the propensities over the last four Censuses are assumed to continue into the future, and applied to a projected population. Numbers of families are then derived from the projected living arrangements of the population, and numbers of households are derived from the projected numbers of families. For details see Technical Note - 'Method'.

Background

5 The projection results are not predictions or forecasts. They are illustrations of the change in the numbers of households and families that would occur if assumptions about future trends in living arrangements were to occur.

6 The assumptions are based on past trends in living arrangements. No assessment has been made of changes in social and economic conditions which may influence future living arrangements. There is no certainty that any of the assumptions will be realised.

Sources of data

7 Sources of data for the projections are:

• The 2001, 2006, 2011 and 2016 Censuses of Population and Housing
• Estimated resident population at 30 June 2016 and 30 June 2017
• Estimated households at 30 June 2016
• ABS population projections at 30 June 2018 to 2041

Census of population and housing

8 Three Census items are used to obtain living arrangement propensities. These are household composition, family composition, and relationship in household of the usual resident population.

9 The Census only collects information on the relationship of each household member to person 1 and/or person 2 on the Census form. Relationships between other members of the household (e.g. between person 3 and person 4) are not captured. This may lead to an underestimation of the number of families, as well as relationships within the household and family type.

10 Person-level ‘relationship in household’ data within the Census is based on place of enumeration. This means persons temporarily absent from the household were excluded from the propensities due to limited relationship information. Visitors to households on Census night were also excluded from the propensities because they are not included in family coding in the Census.

Estimated Resident Population

11 Estimated Resident Population (ERP) published in Australian Demographic Statistics (cat. no. 3101.0) is used as the 2016 and 2017 population base (projected population is used for later years).

12 ERP is based on Census counts of usual residents, with adjustments made for Census undercount (people present in Australia on Census night but missed by the Census count) and usual residents of Australia who were temporarily overseas at the time of the Census.

13 ERP does not distinguish between people who live in private versus non-private dwellings (NPDs). Household and family projections are concerned exclusively with the population usually resident in private dwellings. Therefore, for the purposes of these projections, the proportion of the Census count in NPDs was identified and excluded from the calculation of numbers of households and families. People living in NPDs are included in the analysis of living arrangements.

Estimated households

14 Estimated households are used to constrain the base year (2016) living arrangement propensities calculated from the 2016 Census. For details see Technical Note - 'Method'.

15 The 2016 estimated households in Australia in this publication (9,204,635) is greater than the 2016 Census count of households (8,286,077) published in various 2016 Census publications. The difference is a combination of dwelling undercount and dwelling misclassification in the Census and persons that were temporarily overseas on Census night. For the differences between Census counts and ERP see Explanatory Notes 11 to 13.

Populations projections

16 Projected number of persons reflect the Series B population projections published in Population Projections, Australia, 2017 (base) - 2066 (cat. no. 3222.0)

17 Series B assumes ‘medium’ levels of future births, deaths and migration, based recent historical trends:

• the total fertility rate for Australia declining to 1.8 babies per woman in 2027, and remaining constant thereafter;
• life expectancy at birth will reach 82.1 years for males and 85.5 years for females by 2041;
• net overseas migration will reach 225,000 people per year by 2027 and remain constant thereafter;
• net interstate migration gains for some states and territories, and losses for others.

18 Only one population projection series is used. This means that the differences between the three series of household, family and living arrangement projections only reflect the different living arrangement assumptions, rather than differences in the underlying population.

19 Chapter 9 – ‘What If . . .’ discusses the results of using two other projections of the population, the Series A and Series C projections from Population Projections, Australia, 2017 (base) - 2066 (cat. no. 3222.0), as the assumptions about the future population of Australia and the states and territories.

Rounding

21 Usually, in this publication, figures of less than one million are rounded in the text to the nearest hundred while figures of more than one million are rounded to the nearest hundred thousand.

Percentage growth

22 Changes in population over time are commonly discussed in terms of average annual growth rates. In this publication, however, changes are for the most part presented as percentage increases or decreases over the entire 25-year period (from 2016 to 2041), in order to make differences between projected numbers (of households, families and people in different living arrangements) more discernible.

Comparison with state government household projections

23 Some state and territory governments produce household projections, which may be a useful complement to the projections in this publication. These are typically based on different methods and are produced with specific attention to the observed and expected trends of that state, without the requirement of being constrained to Australia level data.

Acknowledgment

24 ABS publications draw extensively on information provided freely by individuals, businesses, governments and other organisations. Their continued cooperation is very much appreciated; without it, the wide range of statistics published by the ABS would not be available. Information received by the ABS is treated in strict confidence as required by the Census and Statistics Act 1905.

25 As well as the statistics included in this and related publications, the ABS may have other relevant data available on request. Inquiries should be made to the National Information and Referral Service on 1300 135 070.

Technical note - method

Introduction

1 These projections use a propensity method based on that developed by McDonald and Kippen and used in Household Trends and Projections: Victoria, 1986-2011. The method identifies propensities from the Census of Population and Housing for persons to be in different living arrangements. Trends observed in the propensities over the last four Censuses are assumed to continue into the future, and applied to a projected population. Numbers of families are then derived from the projected living arrangements of the population, and numbers of households are derived from the projected numbers of families.

Method

2 The method for obtaining household and family projections is outlined below. The method consists of five main steps:

• Step 1: Calculate living arrangement propensities from Census
• Step 2: Calculate assumed future living arrangement propensities based on past trends
• Step 3: Derive projected living arrangements from assumed future propensities
• Step 4: Derive projected numbers of families from the projected living arrangements
• Step 5: Derive projected numbers of households

Step 1: Calculate living arrangement propensities from census

3 2016 living arrangement propensities were calculated for each five-year age group (0–4, 5–9, …, 85 and over). The propensity is the Census counts of people (for each age group) in that living arrangement type, divided by the total Census count for each age group. There are 15 living arrangement types, as shown in summary tables in the page 'Types of Living Arrangements'.

4 Living arrangement propensities were calculated using a large proportion (91% in 2016) of the Census count of Australian usual residents. Categories of people included in the 2016 Census counts but excluded from the calculation of living arrangement propensities were:

• imputed records.
• people coded to 'other non-classifiable' relationship in household.
• visitors in the private dwelling in which they were enumerated, who were usually resident elsewhere in Australia.
• people enumerated in non-private dwellings who were usually resident elsewhere in Australia.

5 People included in the projected population but not included in the living arrangement calculation were proportionally distributed across the living arrangement types. This assumes that these people have the same living arrangement distribution as those included in the calculation of the propensities.

6 Distributing by this method is likely to lead to an underestimation of lone-person households. It is less likely that a lone-person household will be enumerated at home on Census night because there are fewer people in the household. This bias is addressed in part by replacing Census counts of lone-person households with lone-person household estimates in the propensity calculations.

7 Census-based living arrangement propensities in the base year (2016) are reconciled to align with the estimate of households published in Australian Demographic Statistics (cat. no. 3101.0) for both total households and lone-person households. These estimates are based on Census counts but have been adjusted to account for person and dwelling undercount, dwelling misclassification and residents temporarily overseas on Census night.

8 The difference between the total numbers of households and lone-person households was proportionally distributed to group and family households. The number of families and then persons by living arrangement were derived from the scaled household estimates, using the method outlined in step 4 (paragraphs 17 to 18) in reverse. For example, to obtain the number of persons in a group household, the ‘reconciled’ number of group households was multiplied by the average size of group households in the 2016 Census.

9 The 2016 reconciled counts by living arrangement were proportionally distributed across the five-year age groups according to the 2016 living arrangement propensities (as described in paragraph 3). The distribution was constrained to sum to 30 June 2016 ERP (by age) and the reconciled totals for each living arrangement.

10 Reconciled base year living arrangement propensities are used in step 2 as the basis for the projected propensities.

Step 2: Calculating assumed future living arrangement propensities

11 A line of best fit was calculated for each living arrangement (for each age group), using the observed 2001, 2006, 2011 and 2016 propensities. If the fitted straight line produced an r-squared value of less than 0.2, it was assumed there had been no change in propensity between 2001 and 2016, and therefore the 2016 propensity for that living arrangement for that age group was held constant throughout the projection period in all three assumption series. The annual rate of change for each line of best fit was calculated.

12 Assumed rates of change were applied to the 2016 reconciled propensities to give projected propensities for the period 2017 to 2041 for each living arrangement (for each age group). This produced three series of projected propensities. Series I holds 2016 propensities constant over the whole period, series II assumed the observed trend would continue at a decreasing rate, and series III continues at the full rate of change over the whole projection period. For more information see the page 'Assumptions'.

13 Projected propensities were adjusted to ensure they sum to 100% in each five-year age group using iterative proportional fitting.

Step 3: Derive projected persons by living arrangement from assumed future propensities

14 The projected propensities were then applied to the projected population of each age group to produce the projected number of people in each living arrangement, for each year from 2017 to 2041.

15 Propensities were produced independently for Australia, the states, territories, capital cities and each balance of state. They were then applied to the relevant projected population.

16 State-level data was constrained to sum to the Australian living arrangement totals and five-year age group totals. This was done using two-dimensional iterative proportional fitting. The process was then repeated to constrain capital city and rest of state data to state data. For a more detailed description of iterative proportional fitting, see Purcell, N. J. and Kish, L., (1979), Estimations for small domains, Biometrics, 35, pp. 365-384.

Step 4: Derive projected number of families from living arrangement projections

17 The number of couple families (with or without children) was calculated as half the number of partners in couple families (with or without children). The number of one-parent families was calculated as the number of male lone parents plus the number of female lone parents. The number of 'other families' was calculated by dividing the number of persons living in 'other families' by the average size of 'other families' at the 2016 Census. See paragraph 22 for the average size of ‘other families’ by part of state.

18 The 2016 estimate of the number of families in Australia in this publication (6,659,038) is greater than the 2016 Census count of families (6,070,316) published in various 2016 Census publications. The difference is a combination of net undercount in the Census and families that were temporarily overseas on Census night. For the differences between Census counts and ERP see Explanatory Notes 11 to 13.

Family households

19 Family households can contain more than one family. The number of family households was calculated as the number of families multiplied by the ratio in the 2016 Census of family households to families – see paragraph 22 for ratios by part of state.

Non family households

20 The number of lone-person households was calculated as the number of male lone persons plus the number of female lone persons. The number of group households was calculated by dividing the number of persons in group households by the average size of group households at the 2016 Census. See paragraph 22 for average group household size by part of state.

Total households

21 Total projected households is the sum of all household types.

Census ratios

22 The ratios mentioned in paragraphs 17, 19 and 20 above are presented in the following table:

2016 Census ratios
Average size of 'other families' (a) (people)Family households to families ratio (b) (people)Average size of group households (c) (people)
Sydney2.150.962.45
Rest of NSW2.080.982.17
Melbourne2.120.972.41
Rest of Vic.2.070.992.21
Brisbane2.110.972.35
Rest of Qld2.100.982.15
Rest of SA2.120.992.10
Perth2.120.982.29
Rest of WA2.180.982.11
Greater Hobart2.090.992.28
Rest of Tas.2.070.992.10
Darwin2.170.972.27
Rest of NT2.610.842.25
ACT2.090.982.37
1. The number of people living in 'other families', divided by the number of 'other families'
2. The number of family households, divided by the number of families
3. The number of people living in group households, divided by the number of group households.

Geographical areas

23 The propensities were derived using the area boundaries according to the relevant ABS Geography classification of each capital city and rest of state area as they existed at the relevant reference points (2001, 2006, 2011 and 2016). Previous analysis suggests that the use of such propensities (rather than based on revised areas) would not have had a significant effect on the comparability of the propensities over time.

Glossary

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Average annual growth rate

The average annual growth rate, r, is calculated as a percentage using the formula:

$$\large{r=\left[\left(\frac{P_{n}}{P_{0}}\right)^{\frac{1}{n}}-1\right] \times 100}$$

where $${P_0}$$ is the population at the start of the period, $${P_n}$$ is the population at the end of the period and n is the length of the period between $${P_n}$$ and $${P_0}$$ in years.

Average household size

Average household size refers to the number of persons per household in private dwellings.

Capital city

Refers to the Greater Capital City Statistical Areas of states and territories as defined in the Australian Statistical Geography Standard.

Child

A child is a person of any age who is a natural, adopted, step or foster son or daughter of a couple or lone parent, usually resident in the same household. A child is also any individual under 15 years, usually resident in the household, who forms a parent-child relationship with another member of the household. This includes otherwise related children under 15 years and unrelated children under 15 years.

In order to be classified as a child, the persons can have no partner or child of his/her own usually resident in the household. A separate family in the household is formed in this instance. If a person is aged under 15 years and has a partner/spouse these relationships are not recorded.

Couple family with children

A family based on two persons who are in a registered or de facto marriage, who are usually resident in the same household. The family must include one or more children usually resident in the same household. The family may include any number of other related individuals usually resident in the household.

Couple family without children

A family based on two persons who are in a registered or de facto marriage, who are usually resident in the same household and have no children usually resident in the same household. The family may include any number of other related individuals usually resident in the household.

Dwelling

A dwelling is a building or structure in which people live. Some examples of dwellings are a house, a block of flats, a caravan or tent, prison, humpy or houseboat. For the purposes of Census of Population and Housing, dwellings are classified into private and non-private dwellings. Each of these dwelling types is further divided into occupied and unoccupied dwelling categories.

Estimated households

Estimated households is a measure of the number of households, which corresponds to the usual resident population living in private dwellings. It is based on the Census count of households and adjusted to account for dwelling undercount, dwelling misclassification, residents counted away from home on Census night and residents temporarily overseas on Census night. It excludes groups of visitors (from overseas or elsewhere in Australia) counted by Census in the same dwelling, as they do not form a household.

Estimated resident population

The official measure of the population of Australia is based on the concept of usual residence. It refers to all people, regardless of nationality or citizenship, who usually live in Australia, with the exception of foreign diplomatic personnel and their families. It includes usual residents who are overseas for less than 12 months over a 16-month period. It excludes overseas visitors who are in Australia for less than 12 months over a 16-month period.

Estimates of the Australian resident population are generated on a quarterly basis by adding natural increase (births minus deaths) and net overseas migration (NOM) occurring during the period to the population at the beginning of each period. This is known as the cohort component method, and can be represented by the following equation:

P$$_{t+1}$$ = P$$_t$$ + B - D + NOM, where:

P$$_t$$ = the estimated resident population at time point $$t$$

P$$_{t+1}$$ = the estimated resident population at time point $$_{t+1}$$

B = the number of births occurring between $$t$$ and $$_{t+1}$$

D = the number of deaths occurring between $$t$$ and $$_{t+1}$$

NOM = net overseas migration occurring between $$t$$ and $$_{t+1}$$.

For state and territory population estimates, an additional term is added to the equation representing net interstate migration (NIM) occurring between $$t$$ and $$t+1$$, represented by the following equation:

P$$_{t+1}$$ = P$$_t$$ + B - D + NOM + NIM.

Family

A family is defined by the ABS as two or more persons, one of whom is at least 15 years of age, who are related by blood, marriage (registered or de facto), adoption, step or fostering, and who are usually resident in the same household. The basis of a family is formed by identifying the presence of either a couple relationship, lone parent-child relationship or other blood relationship. Some households will contain more than one family.

Family composition

A classification of families based on the presence, or absence, of a couple relationship, parent-child relationship, or other blood relationship.

Family growth

Growth in the number of families.

Group household

A group household is a household consisting of two or more unrelated people where all persons are aged 15 years or over. There are no reported couple relationships, parent-child relationships or other blood relationships in these households.

Household

A household is defined as one or more persons, at least one of whom is at least 15 years of age, usually resident in the same private dwelling.

Persons usually resident in non-private dwellings, such as hotels, motels, boarding houses, gaols and hospitals, are not included in household estimates.

This definition of a household is consistent with that used in the Census.

Household composition

Household composition is used to describe and categorise households on the basis of the number of families present, and whether or not unrelated household members are present (if it is a family household), or the number of household members (if it is a non-family household).

Household count

The count of households is the number of households counted in the Census. It is not adjusted for undercount, dwellings containing overseas visitors, households of Australian residents where all members were temporarily overseas at the time of the Census, households of Australian residents where all members were not home on Census night and spent Census night in a non-private dwelling in Australia, and households of Australian residents where some members were not at home on Census night and were counted as a separate household elsewhere. Characteristics of households are available according to place of enumeration.

Household growth

Growth in the number of households.

Household population

The household population is the estimated resident population (ERP) that usually lives in private dwellings. It is the ERP less the population that usually lives in non-private dwellings.

Life expectancy

Life expectancy refers to the average number of additional years a person of a given age and sex might expect to live if the age-specific death rates of the given period continued throughout his or her lifetime.

Living arrangement

Living arrangement combines the three concepts 'relationship in household', 'family type' and 'household type'. It is used to describe the familial and non-familial relationship type of each person, within each family type, and within each household type.

Lone parent

A person who has no spouse or partner present in the household but who forms a parent-child relationship with at least one child usually resident in the household.

Lone person

A person at least 15 years of age who lives in a dwelling on their own.

Non-private dwelling (NPD)

Non-private dwellings (NPDs) are residential dwellings with accommodation which are not included in the Census of Population and Housing list of private dwelling categories. NPDs are classified according to their function. They include hotels, motels, guest houses, prisons, religious and charitable institutions, military establishments, hospitals and other communal dwellings. Where this type of accommodation includes self-contained units (as provided by hotels, motels, homes for the elderly and guest houses), the units are enumerated as part of the NPD. Complexes such as retirement villages, which have a combination of self-contained units, hostel and/or nursing home accommodation, are enumerated as NPDs.

Occupied private dwelling

An occupied private dwelling is defined as the premises occupied by a household on Census night.

One-parent family

A family consisting of a lone parent with at least one child (regardless of age) who is also usually resident in the household. The family may also include other related individuals.

Other family

A family of related individuals residing in the same household. These individuals do not form a couple or parent-child relationship with any other person in the household and are not attached to a couple or one-parent family in the household. For example, a household consisting of a brother and sister only.

Other related individual

An individual who is related to at least one other member of the household, but who does not form a couple or parent-child relationship.

Partner

A person in a couple relationship with another person usually resident in the same household. The couple relationship may be in either a registered or de facto marriage, and includes same-sex couples.

Private dwelling

A private dwelling (PD) in the Census is defined as a house, flat, part of a house, or even a room; but can also be a house attached to, or rooms above shops or offices; an occupied caravan in a caravan park or boat in a marina, a houseboat, or its own block of land. A caravan situated on a residential allotment is also classed as a PD.

Relationship in household

Describes the familial and non-familial relationship of each person within each family in a given household. The familial relationship within each family is measured with reference to a family reference person chosen for that particular family.

Residents temporarily overseas

Residents temporarily overseas are Australian residents who are overseas for a period less than 12 months.

Rest of state or territory

Within each state and territory, except for the ACT, the area not defined as being part of the Greater Capital City Statistical Area is call the Rest of State region.

Total fertility rate

The sum of age-specific fertility rates (live births at each age of mother per female population at that age). It represents the number of children a female would bear during her lifetime if she experienced current age-specific fertility rates at each age of her reproductive life.

Unoccupied private dwellings

These are structures built specifically for living purposes which are habitable, but unoccupied at the time of the Census of Population and Housing. Vacant houses, holiday homes, huts, cabins (other than seasonal workers' quarters) and houseboats are counted as unoccupied dwellings. Also included are newly completed dwellings not yet occupied, dwellings which are vacant because they are due for demolition or repair, dwellings to let and dwellings where all members of the household were absent on Census night.

Usual residence

Usual residence within Australia refers to that address at which the person has lived or intends to live for a total of six months or more in a given reference year.

Quality declaration - summary

Institutional environment

The Australian Bureau of Statistics (ABS) household and family projections are produced from a combination of ABS data: the 5-yearly Census of Population and Housing, estimated resident population, household estimates and population projections.

For information on the institutional environment of the ABS, including the legislative obligations of the ABS, financing and governance arrangements, and mechanisms for scrutiny of ABS operations, please see ABS Institutional Environment.

Relevance

Household and family projections inform on future changes in numbers and types of households and families, and are used in a variety of planning decisions. Projections are published for Australia, the states, territories, capital cities and rest of state regions.

Timeliness

ABS household and family projections are compiled and published once every five years; typically, three years following the most recent ABS Census of Population and Housing.

Accuracy

ABS household and family projections are based on assumptions about future living arrangements of Australia's population. They are not intended as predictions or forecasts, but are illustrations of growth and change in numbers and types of households and families that would occur if assumptions about future living arrangements of Australia's population were to prevail over the projection period.

While the assumptions are formulated on the basis of an assessment of demographic trends, there is no certainty that any of the assumptions will be realised. In addition, no assessment has been made of possible future changes in non-demographic conditions.

It is important to also recognise that the future population of Australia underlying the household and family projections is itself based on a range of assumptions (about future fertility rates, mortality rates and levels of migration). For the purpose of this publication, the Series B projection from Population Projections, Australia, 2017 (base) - 2066 (cat. no. 3222.0) has been used as Australia's future population.

Coherence

As the assumptions used in each successive set of household and family projections incorporate recent trends, comparison of data across issues of this publication is not possible. However, projected numbers of households are compatible and comparable with ABS household estimates for 2016 and earlier years.

Estimates of the number of families in 2016 in this publication differ from counts of families from the 2016 Census of Population and Housing. This is due to estimates of families in this publication being based on 2016 estimated resident population as well as 2016 Census living arrangement propensities. The 2016 estimated resident population of Australia is greater than the 2016 Census count of people in Australia as it includes Census net undercount and residents temporarily overseas, and as a result, the number of families based on 2016 estimated resident population is greater than the number of families from the 2016 Census.

Interpretability

ABS household and family projections are not intended as predictions or forecasts, and should not be considered as such. Rather, they are illustrations of growth and change in numbers of households and families in Australia that would occur if the assumptions were to prevail over the projection period. As future trends in living arrangements are unpredictable, three assumptions have been made regarding these, to illustrate a range of possible future outcomes.

This publication contains notes on the assumptions and methods used to produce the household and family projections. It also contains Explanatory Notes, Appendices and a Glossary that provide information on the data sources, terminology, classifications and other technical aspects associated with these statistics.

Accessibility

ABS household and family projections are available in a variety of formats on the ABS website under the 3236.0 product family. The formats available are:

• Main Features, which contains key figures commentary;
• three data cubes (in Microsoft Excel format) containing numbers of households, families, and persons by living arrangement for Australia, the states, territories, capital cities and rest of state regions, for 2016 to 2041;
• one data cube (in Microsoft Excel format) containing living arrangement propensities for Australia for the 2001 to 2016 Censuses; and
• three ABS.Stat datasets containing numbers of households, families, and persons by living arrangement for Australia, the states, territories, capital cities and rest of state regions, for 2016 to 2041.

If the information you require is not available as a standard product, then ABS Consultancy Services can help you with customised services to suit your needs. For inquiries contact the National Information and Referral Service on 1300 135 070 or email client.services@abs.gov.au.

The ABS observes strict confidentiality protocols as required by the Census and Statistics Act, 1905. This may limit access to data at a detailed level.

Abbreviations

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 ABS Australian Bureau of Statistics ACT Australian Capital Territory ASGS Australian Statistical Geography Standard Aust. Australia Bal Balance ERP Estimated Resident Population GCCSA Greater Capital City Statistical Area no. number NPD Non-Private dwelling NSW New South Wales NT Northern Territory Qld Queensland SA South Australia Tas. Tasmania Vic. Victoria WA Western Australia


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Corpus ID: 204904238

# An Adaptive and Momental Bound Method for Stochastic Learning

@article{Ding2019AnAA,
title={An Adaptive and Momental Bound Method for Stochastic Learning},
author={Jianbang Ding and Xuancheng Ren and Ruixuan Luo and X. Sun},
journal={ArXiv},
year={2019},
volume={abs/1910.12249}
}
Training deep neural networks requires intricate initialization and careful selection of learning rates. The emergence of stochastic gradient optimization methods that use adaptive learning rates based on squared past gradients, e.g., AdaGrad, AdaDelta, and Adam, eases the job slightly. However, such methods have also been proven problematic in recent studies with their own pitfalls including non-convergence issues and so on. Alternative variants have been proposed for enhancement, such as… Expand
8 Citations

#### References

SHOWING 1-10 OF 43 REFERENCES

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# Result of constant current through DC motor under different loads

I've done some research on brushless DC motor and have a few questions what would happen when a constant current is sent to a DC motor.

From my understanding, the current is directly related to the torque that the motor supplies, while the voltage would relate to the RPM. My question is, what would happen when you try to keep the current supplied constant, regardless of the load/rpm.

For example, let's say you have a weight attached to the shaft of the motor, like in this picture:

You set current to be sent to the motor so that the torque applied is 10 in-lbs without any load. In this scenario, this current doesn't change. You attach a weight to the end of the pulley. What happens when the torque from the weight/load is 5, 10, and 15 in-lbs.

I assume, in the 5 in-lb scenario, the motor pull the weight up, at 10 in-lbs the motor would stall, and at 15 in-lbs, the weight would drop and it would actually be acting as a generator.

Now, I'm obviously ignoring the voltage and back-emf in this scenario. How would they play a role? Without any voltage, there won't be any current running through it, so there's no torque applied. But beyond that, would the voltage only affect the acceleration?

Sorry if this question has already been answered, but most of what I found related to a constant NET torque, not a constant applied torque from the generator.

• A few points worth looking up and then clarifying in the question: (1) Brushless DC motors aren't driven by constant current drives, whereas brushed DC motors could be. So which is it? (2) The effect of voltage and of current aren't quite as perceived: The relevant parameter is power, i.e. voltage x current, and power relates to the generated torque. (3) This torque needs to overcome stiction, inertia (during speed transition anyway), bearing losses, vibration losses, and the actual load torque. At steady state, the generated torque must match the sum of the last few factors. – Anindo Ghosh Jan 8 '14 at 17:45
• Useful reading on BLDC driving: BLDC Motor Control Algorithms, specially the first two lines: "Brushless motors are not self-commutating, and hence are more complicated to control. BLDC motor control requires knowledge of the rotor position and mechanism to commutate the motor." – Anindo Ghosh Jan 8 '14 at 17:53
• The motor doesn't have to be brushless. I'll take a look at that article, thanks. It certainly makes sense that the voltage would play a role in the torque produced, but nothing clickedfor me as to how. Would a constant torque (ie. Overcoming all the resistance mentioned in number 3 EXCEPT the load torque) be possible? It just would require a more complicated control than simply a constant current ? – user35266 Jan 8 '14 at 19:25
• Yes, a constant residual torque (after eliminating the various other forces opposing it) is certainly possible. Also, BLDC control algorithms are actually not very complex, as that same link above would have indicated. Constant torque would, simplistically, be easier to achieve than constant rotation speed, in some ways. The key part of the control for a BLDC is the commutation i.e. switching which coils get power in what sequence - one can't just be pushing a constant current through a single set of coils, else the motor would reach a static equilibrium and stop rotating. – Anindo Ghosh Jan 8 '14 at 19:31
• Also, constant torque, constant rotation speed, constant acceleration to target set-point, and several other such control algorithms are available out of the box from a number of different BLDC controller ICs - it's far simpler doing what you need using one of those, compared to rolling your own. You might want to look up Texas Instruments InstaSpin-BLDC. It's just the family I am most familiar with, several other manufacturers have easy to use, sophisticated motor controllers as well. Also, similar integrated controllers exist for brushed DC too. – Anindo Ghosh Jan 8 '14 at 19:35

A constant current means, for an ideal motor, a constant torque. This is approximately true for real motors. It doesn't matter what you attach to the motor, or how fast it's turning.

What you seem to be missing is Newton's second law of motion. It states that force is the product of mass and acceleration:

$$F = ma$$

The constant current you supply to the motor is one force. The weight opposes that force. The difference is the net force, $F$ in this equation, and $m$ is the mass of the weight, plus the mass of the rotor and the string and everything else the motor must move.

You set current to be sent to the motor so that the torque applied is 10 in-lbs without any load.

Not possible. There is nothing for the motor to "torque against". This is the mechanical equivalent of trying to develop 10 volts across a dead short. The motor will rapidly spin at its maximum speed, and the back-EMF will rise to the driving voltage such that your driving electronics are unable to supply enough voltage above the back-EMF to make enough current to have that much torque.

Let's just say you determine how much current is required for 10 in-lbs of torque, and you drive your motor with a constant-current supply set to that.

What happens when the torque from the weight/load is 5 in-lbs?

Assuming that the rotor and the string are massless and frictionless, the weight will be accelerated upwards by the net 5 in-lbs of torque (motor's 10 in-lbs, less 5 in-lbs from the weight). The rate of the acceleration is determined by the mass of the weight and Newton's law above.

As the speed of the motor changes (the weight is accelerating), the back-EMF also changes. Your constant-current supply to the motor will have to apply an increasing voltage to maintain the same current. Electrical power thus goes up, as does mechanical power.

What happens when the torque from the weight/load is 10 in-lbs?

Motor torque balances weight torque. However fast the weight is moving (if at all), it keeps doing that. Newton's first law applies.

What happens when the torque from the weight/load is 15 in-lbs?

The weight will accelerate downward, overpowering the motor. However, it won't be a free-fall. The motor cancels some of the force of the weight, resulting in a slower acceleration downwards.

If the weight overpowers the motor, then eventually it can get the motor to run backwards, relative to the way it would run if there were no load. When this happens, the back-EMF now adds (instead of subtracts) from the voltage you apply to the motor. At some point, your controller, which is attempting to maintain a constant current, must apply a negative voltage to maintain that current. In other words, the back-EMF is sufficient to create the necessary torque on its own: your controller must oppose it.

This is perfectly symmetrical with the first case, where the motor was overpowering the weight. In that case, electrical and mechanical power went up (without bound, if you let them). In this case, electrical and mechanical power go down (negative, if you let them). Energy is conserved because you are changing the gravitational potential of the weight.

The need to resist the back-EMF usually means storing electrical energy in a capacitor or battery, or using it to heat a resistor. If you can't do this fast enough, then the motor will create more torque than your desired 10 in-lbs, and you have hit the limits of your "constant current" driver.

• Thanks for the detailed response and clearing up many misconceptions I had. It seems like I wasn't too far off. Great explanation on the third scenario as well. I will read those sources. Thanks again! – user35266 Jan 9 '14 at 1:11

Assuming the commutation electronics didn't have problems with the voltages involved, a brushless motor fed with a constant current would behave much as would a brush motor, and supply constant torque. If the combination of torque due to weight and friction were equal and opposite the torque produced by the motor, the motor would turn at constant speed. There would be a certain voltage drop due to DC resistance (equal to resistance times current), plus an additional voltage drop proportional to speed. If the motor's torque exceeded the torque due to weight and friction, the motor would continuously accelerate until that ceased to be the case; the motor's voltage drop would increase as this occurred. Speed would likely be limited by the supply's (or electronics') inability to supply current at higher voltages, by limited travel of the mechanics, or by mechanical breakdown.

In the scenario where the motor torque plus friction are insufficient to prevent the weight from falling, the weight would accelerate downward (or reduce its upward speed, if any). If the weight started by traveling upward (because of some external force, or because current had initially been higher) the motor would add energy to the weight as long as it continued moving upward. Between the time the weight started downward and the time the voltage produced by its speed was equal to the voltage loss due to resistance, the motor would be "plugging"--a condition in which the motor absorbs all mechanical and electrical power that goes into it (as the motor speeds up, the amount of mechanical power would increase and electrical power would decrease). Once the speed reaches the point that the voltage at the motor terminals would be zero, the motor will start acting as a generator, trying to push current in the direction it's already flowing [note that a the polarity would be the opposite of what typical constant-current supply would be equipped to handle, so such a supply would likely burn off all the generated power as heat].

• Thanks for the response. Seems to be roughly what I expected. I am interestedabout the third case. Let's say the load is oscillating. Plugging would occur when the motor changes direction from positive to negative; would the same thing occur when the load decreases enough for it start spinning positively again? And it would it possible to set up a circuit that collects the power generated from the load reversing the direction of the shaft, correct? – user35266 Jan 8 '14 at 23:17


text:
# Scale¶

class astropy.modeling.functional_models.Scale(factor=1, **kwargs)[source]

Multiply a model by a dimensionless factor.

Parameters
factorfloat

Factor by which to scale a coordinate.

Notes

If factor is a Quantity then the units will be stripped before the scaling operation.

Attributes Summary

 factor fittable input_units This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None if any units are accepted). linear param_names

Methods Summary

 evaluate(x, factor) One dimensional Scale model function fit_deriv(x, \*params) One dimensional Scale model derivative with respect to parameter

Attributes Documentation

factor = Parameter('factor', value=1.0)
fittable = True
input_units

This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None if any units are accepted).

Model sub-classes can also use function annotations in evaluate to indicate valid input units, in which case this property should not be overridden since it will return the input units based on the annotations.

linear = True
param_names = ('factor',)

Methods Documentation

static evaluate(x, factor)[source]

One dimensional Scale model function

static fit_deriv(x, *params)[source]

One dimensional Scale model derivative with respect to parameter


text:
# Ways of spending money combinatorial problem

Suppose person X has $12$ dollars.In each of the first 5 days he buys one of the following items.

1.Item A for $1 2.Item B for$2

3.Item C for $3. In how many ways can he spend the money during the first five days? This is what I did. Each of the five days he can buy Item A without running out of money. Also each of the five days he can buy Item B without running out of money.Therefore based on these 2 choices he has 2 selections each day.Thus$2^5$selections. Now if he decides to buy item C, If he buys Item C for 1 day then, still all other four days have 2 choices each day.Ways of selecting the day to buy item C=${5 \choose 1}$.And each 4 days having 2 choices make the total number of selections=$2^4*5=80$. If he buys Item C for 2 days then ways of selecting these two days =${5 \choose 1}$.All other three days have 2 choices each day.Total selections in this case=$2^3*10=80$. If he buys Item C for 3 days then ways of selecting these two days =${5 \choose 2}$There are 3 ways of selecting items for the next two days.Thus total number of selections for this case=$3*10=30$Therefore total number of ways of spending 12 dollars is$2^5+80+80+30=222$. Is this correct. • Does order matter? Or does only the final shopping bag matter? – Alexandre Halm Oct 14 '14 at 11:19 • It is correct if order matters. – Coolwater Oct 14 '14 at 11:58 ## 3 Answers So basically if order matters your universe is$\{1,2,3\}^5$(cardinal$3^5=243$). I would simply count how many of these quintuplets$(a_1,...,a_5) \in \{1,2,3\}^5$do not satisfy$a_1+...+a_5 \le 12$by distinguishing how many "$3$" values they have : • 5 times "3" : only one tuple$(3,3,3,3,3)$• 4 times "3" : fives tuples where "1" is the fifth value (from$(1,3,3,3,3)$to$(3,3,3,3,1)$) and five ones where "2" is the last value • 3 times "3" : only those where the two other values are "2", I count$\binom{5}{2}$ways to place the two "2"s Answer :$243 - 1 - 5 - 5 - 10 = 222$. If we understand the probleme as this: What is the number of ways to spend 12 dollars in 5 days if we have to buy only one item a day, which costs 1 or 2 or 3 dollars? It is the number of ways to partition 12 in 5 parts, with each of these 5 parts are 1 or 2 or 3:$1 + 2 + 3 + 3 + 3 =12$all the permutations:$\frac{5!}{3!} =202 + 2 + 2 + 3 + 3 =12$all the permutations:$\frac{5!}{3!2!} =10$So,$20+10 =30$ways Cheers • you don't need to spend the whole 12 dollars in the problem – Alexandre Halm Oct 14 '14 at 12:57 This is trickier, are number of sums to a limited quantity. You can calculate it using a generating function or by hand knowing the limitations. Using a generating function will be the sum of the coefficients from the power 5 to 12 of this polynomial $$f(x)=(x+x^2+x^3)^5=x^{15}+5x^{14}+15x^{13}+30x^{12}+45x^{11}+51x^{10}+45 x^9+30 x^8+15x^7+5 x^6+x^5\\ \to f^*(x)=30x^{12}+45x^{11}+51x^{10}+45 x^9+30 x^8+15x^7+5 x^6+x^5\\ f^*(1)=222$$ Im not consumist so I prefer the version where you can choose DONT buy if you want, in this version the generating function will be $$g(x)=(1+x+x^2+x^3)^5=x^{15}+5 x^{14}+15 x^{13}+35x^{12}+65 x^{11}+101x^{10}+135 x^9+155 x^8+155x^7+135 x^6+101 x^5+65x^4+35 x^3+15 x^2+5 x+1\\ \to g^*(x)=35x^{12}+65 x^{11}+101x^{10}+135 x^9+155 x^8+155x^7+135 x^6+101 x^5+65x^4+35 x^3+15 x^2+5 x+1\\ g^*(1)=1003$$ ;) • Does g represent the ordered selection and f the unordered – clarkson Oct 14 '14 at 12:51 • @clarkson, all are ordered/unordered. Order doesnt matter for this problem.$g(x)$is a version of the original question but considering that you can NOT buy too, i.e., spent 0$ if you want any day. – Masacroso Oct 14 '14 at 12:54
• Everyday during these 5 days he buys something – clarkson Oct 14 '14 at 12:58
• Guy, you read what I write? – Masacroso Oct 14 '14 at 12:58


text:
Consider the following polynomials in P_3:q(x)=-x^3+3x^2-x+5, r(x)=-4x^3+7x^2-x+10, u(x)=-5x^3+8x^2+10

Consider the following polynomials in ${P}_{3}$:
$q\left(x\right)=-{x}^{3}+3{x}^{2}-x+5$
$r\left(x\right)=-4{x}^{3}+7{x}^{2}-x+10$
$u\left(x\right)=-5{x}^{3}+8{x}^{2}+10$
For each of the following polynomials, determine whether it is in {q,r,u}. If so, ecpress it as a lineare combination of the polynomials above. Use e.g. q rather than q(x) to represebt the polynomials in your linear combination.
${p}_{1}\left(x\right)=-3{x}^{2}-3x+10$

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Given the polynomials $q\left(x\right)=-{x}^{3}+3{x}^{2}-x+5,r\left(x\right)=-4{x}^{3}+7{x}^{2}-x+10$ and $u\left(x\right)=-5{x}^{3}+8{x}^{2}$.
Now, a polynomial p(x) is in span of {q,r,u} if there exist nonzero constants a,b and c such that $p\left(x\right)=aq+br+cu$
The polynomial is ${p}_{1}\left(x\right)=-3{x}^{2}-3x+10$
Then,
${p}_{1}=aq+br+cu$
$-3{x}^{2}-3x+10=a\left(-{x}^{3}+3{x}^{2}-x+5\right)+b\left(-4{x}^{3}+7{x}^{2}-x+10\right)+c\left(-5{x}^{3}+8{x}^{2}+10\right)-3{x}^{2}-3x+10$
$=\left(a-4b-5c\right){x}^{3}+\left(3a+7b+8c\right){x}^{2}+\left(-a-b\right)x+5a+10b+10c$
Equate the coefficients of both sides and find the values of a, b and c.
$a-4b-5c=0$ (1)
$3a+7b+8c=-3$ (2)
$-a-b=-3$ (3)
$5a+10b+10c=10$ (4)
From equation (3),
$a=-3+b$
$a=3-b$
Substitute $b=3-a$ in equation (1) and equation (2) and (3)
$-\left(3-b\right)-4b-5c=0$ (4)
$3\left(3-b\right)+7b+8c=-3$ (5)
$5\left(3-b\right)+10b+10c=10$ (6)
From equation (4),
$-3+b-4b-5c=0$
$-3-3b-5c=0$
$-3b=5c+3$
$b=-\frac{5c+3}{3}$
Substitute $b=-\frac{5c+3}{3}$ and in equation (5) and (6)
$-9+4\left(-\frac{5c+3}{3}\right)+8c=-3$ (7)
$15+5\left(-\frac{5c+3}{3}\right)+10c=10$


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0

Astonishingly, the solution to the problem is similar to that of calculating the angular sizes. After focusing on the problem I realized that: Center = Real Center / Depth. The real center is the center of the object at distance = 1. The same way: Angular Size = Real Size / Depth The real size is the size of the object at distance = 1. Center2 = ...

1

This of course has been solved many years ago in computer graphics. Look up something called perpsective projection. The simplest and most common way to do this is a point to plane projection. All parts of the scene are projected to a single point, usually called the view point or the eye point. You conceptually suspend the image plane at a fixed ...

-2

Observer, if you assume as someone watching the experiment or activity, is plainly wrong. Anything that can be detected and measured and thus, in principle, from infinitely hard calculation can tell us about the past or previous states, can be said to be information, and thus, entropy increased while the process was being carried out. Take for example, a box ...

1

Where's the flaw? Here's one: Let γ be the squareroot of 1−v²/c² but, in fact, $$\gamma_v = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$ But, there is a flaw in your reasoning too so correcting the error in the formula for $\gamma$ will not get you to the correct answer. Let's work it out with coordinates to see explicitly what's going on. Let the ...

0

I don't have the reputation to comment, so I'll comment about David Hammen's (accepted) answer here. His conclusion is correct, but mentions "Those formulae do imply a singularity for the clock that is closest to you" in reference to the forumale he thought you were referring to. He also mentions "In between, you'll get a nice continuous change from faster ...

0

I am assuming we are talking about the one dimensional case in which we can "move through" clocks. Assuming you synchronise the clocks in your own frame, those further away from you will show an older time than those closer because it takes longer for light to reach you from them. On top of that, all clocks will be equally time dilated since they are all ...

42

I assume you used the formulae $f_o = fs\sqrt{\frac{1+v/c}{1-v/c}}$ for the clocks ahead of you and $f_o = fs\sqrt{\frac{1-v/c}{1+v/c}}$ for the clocks behind you. Those formulae do imply a singularity for the clock that is closest to you. Which equation to use? The answer is neither. Those expressions assume the travel is along the line of sight to the ...

2

If you consider a straight-on trajectory, there really will be a discontinuity. That is the same as with the audible Doppler effect. There is a smooth drop in frequency of a fire truck's siren passing you on the street. The reason is, that there is a certain distance between you and the truck at the closest point. If that were not the case, i.e. the siren ...

3

The signal from the clock moving towards you is the Doppler shifted version of the value you "know" it to be - that is, first slow it down by gamma (clock moving relative to your frame of reference), then speed it up by Doppler shift. Ditto, with sign reversed, for clocks behind you. Now the clock moving at right angles shows what you expect and there is no ...

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G = Q8×C3×C18order 432 = 24·33

Direct product of C3×C18 and Q8

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C3×C18
 Chief series C1 — C3 — C6 — C3×C6 — C3×C18 — C3×C36 — Q8×C3×C9 — Q8×C3×C18
 Lower central C1 — C2 — Q8×C3×C18
 Upper central C1 — C6×C18 — Q8×C3×C18

Generators and relations for Q8×C3×C18
G = < a,b,c,d | a3=b18=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 190, all normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, Q8, C9, C32, C12, C2×C6, C2×C6, C2×Q8, C18, C3×C6, C3×C6, C2×C12, C3×Q8, C3×C9, C36, C2×C18, C3×C12, C62, C6×Q8, C6×Q8, C3×C18, C3×C18, C2×C36, Q8×C9, C6×C12, Q8×C32, C3×C36, C6×C18, Q8×C18, Q8×C3×C6, C6×C36, Q8×C3×C9, Q8×C3×C18
Quotients: C1, C2, C3, C22, C6, Q8, C23, C9, C32, C2×C6, C2×Q8, C18, C3×C6, C3×Q8, C22×C6, C3×C9, C2×C18, C62, C6×Q8, C3×C18, Q8×C9, C22×C18, Q8×C32, C2×C62, C6×C18, Q8×C18, Q8×C3×C6, Q8×C3×C9, C2×C6×C18, Q8×C3×C18

Smallest permutation representation of Q8×C3×C18
Regular action on 432 points
Generators in S432
(1 413 24)(2 414 25)(3 397 26)(4 398 27)(5 399 28)(6 400 29)(7 401 30)(8 402 31)(9 403 32)(10 404 33)(11 405 34)(12 406 35)(13 407 36)(14 408 19)(15 409 20)(16 410 21)(17 411 22)(18 412 23)(37 257 276)(38 258 277)(39 259 278)(40 260 279)(41 261 280)(42 262 281)(43 263 282)(44 264 283)(45 265 284)(46 266 285)(47 267 286)(48 268 287)(49 269 288)(50 270 271)(51 253 272)(52 254 273)(53 255 274)(54 256 275)(55 340 428)(56 341 429)(57 342 430)(58 325 431)(59 326 432)(60 327 415)(61 328 416)(62 329 417)(63 330 418)(64 331 419)(65 332 420)(66 333 421)(67 334 422)(68 335 423)(69 336 424)(70 337 425)(71 338 426)(72 339 427)(73 234 172)(74 217 173)(75 218 174)(76 219 175)(77 220 176)(78 221 177)(79 222 178)(80 223 179)(81 224 180)(82 225 163)(83 226 164)(84 227 165)(85 228 166)(86 229 167)(87 230 168)(88 231 169)(89 232 170)(90 233 171)(91 123 345)(92 124 346)(93 125 347)(94 126 348)(95 109 349)(96 110 350)(97 111 351)(98 112 352)(99 113 353)(100 114 354)(101 115 355)(102 116 356)(103 117 357)(104 118 358)(105 119 359)(106 120 360)(107 121 343)(108 122 344)(127 196 371)(128 197 372)(129 198 373)(130 181 374)(131 182 375)(132 183 376)(133 184 377)(134 185 378)(135 186 361)(136 187 362)(137 188 363)(138 189 364)(139 190 365)(140 191 366)(141 192 367)(142 193 368)(143 194 369)(144 195 370)(145 214 320)(146 215 321)(147 216 322)(148 199 323)(149 200 324)(150 201 307)(151 202 308)(152 203 309)(153 204 310)(154 205 311)(155 206 312)(156 207 313)(157 208 314)(158 209 315)(159 210 316)(160 211 317)(161 212 318)(162 213 319)(235 380 300)(236 381 301)(237 382 302)(238 383 303)(239 384 304)(240 385 305)(241 386 306)(242 387 289)(243 388 290)(244 389 291)(245 390 292)(246 391 293)(247 392 294)(248 393 295)(249 394 296)(250 395 297)(251 396 298)(252 379 299)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)(163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306)(307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324)(325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342)(343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378)(379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396)(397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414)(415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)
(1 248 123 175)(2 249 124 176)(3 250 125 177)(4 251 126 178)(5 252 109 179)(6 235 110 180)(7 236 111 163)(8 237 112 164)(9 238 113 165)(10 239 114 166)(11 240 115 167)(12 241 116 168)(13 242 117 169)(14 243 118 170)(15 244 119 171)(16 245 120 172)(17 246 121 173)(18 247 122 174)(19 290 104 232)(20 291 105 233)(21 292 106 234)(22 293 107 217)(23 294 108 218)(24 295 91 219)(25 296 92 220)(26 297 93 221)(27 298 94 222)(28 299 95 223)(29 300 96 224)(30 301 97 225)(31 302 98 226)(32 303 99 227)(33 304 100 228)(34 305 101 229)(35 306 102 230)(36 289 103 231)(37 137 147 423)(38 138 148 424)(39 139 149 425)(40 140 150 426)(41 141 151 427)(42 142 152 428)(43 143 153 429)(44 144 154 430)(45 127 155 431)(46 128 156 432)(47 129 157 415)(48 130 158 416)(49 131 159 417)(50 132 160 418)(51 133 161 419)(52 134 162 420)(53 135 145 421)(54 136 146 422)(55 262 193 203)(56 263 194 204)(57 264 195 205)(58 265 196 206)(59 266 197 207)(60 267 198 208)(61 268 181 209)(62 269 182 210)(63 270 183 211)(64 253 184 212)(65 254 185 213)(66 255 186 214)(67 256 187 215)(68 257 188 216)(69 258 189 199)(70 259 190 200)(71 260 191 201)(72 261 192 202)(73 410 390 360)(74 411 391 343)(75 412 392 344)(76 413 393 345)(77 414 394 346)(78 397 395 347)(79 398 396 348)(80 399 379 349)(81 400 380 350)(82 401 381 351)(83 402 382 352)(84 403 383 353)(85 404 384 354)(86 405 385 355)(87 406 386 356)(88 407 387 357)(89 408 388 358)(90 409 389 359)(271 376 317 330)(272 377 318 331)(273 378 319 332)(274 361 320 333)(275 362 321 334)(276 363 322 335)(277 364 323 336)(278 365 324 337)(279 366 307 338)(280 367 308 339)(281 368 309 340)(282 369 310 341)(283 370 311 342)(284 371 312 325)(285 372 313 326)(286 373 314 327)(287 374 315 328)(288 375 316 329)
(1 315 123 287)(2 316 124 288)(3 317 125 271)(4 318 126 272)(5 319 109 273)(6 320 110 274)(7 321 111 275)(8 322 112 276)(9 323 113 277)(10 324 114 278)(11 307 115 279)(12 308 116 280)(13 309 117 281)(14 310 118 282)(15 311 119 283)(16 312 120 284)(17 313 121 285)(18 314 122 286)(19 204 104 263)(20 205 105 264)(21 206 106 265)(22 207 107 266)(23 208 108 267)(24 209 91 268)(25 210 92 269)(26 211 93 270)(27 212 94 253)(28 213 95 254)(29 214 96 255)(30 215 97 256)(31 216 98 257)(32 199 99 258)(33 200 100 259)(34 201 101 260)(35 202 102 261)(36 203 103 262)(37 402 147 352)(38 403 148 353)(39 404 149 354)(40 405 150 355)(41 406 151 356)(42 407 152 357)(43 408 153 358)(44 409 154 359)(45 410 155 360)(46 411 156 343)(47 412 157 344)(48 413 158 345)(49 414 159 346)(50 397 160 347)(51 398 161 348)(52 399 162 349)(53 400 145 350)(54 401 146 351)(55 289 193 231)(56 290 194 232)(57 291 195 233)(58 292 196 234)(59 293 197 217)(60 294 198 218)(61 295 181 219)(62 296 182 220)(63 297 183 221)(64 298 184 222)(65 299 185 223)(66 300 186 224)(67 301 187 225)(68 302 188 226)(69 303 189 227)(70 304 190 228)(71 305 191 229)(72 306 192 230)(73 431 390 127)(74 432 391 128)(75 415 392 129)(76 416 393 130)(77 417 394 131)(78 418 395 132)(79 419 396 133)(80 420 379 134)(81 421 380 135)(82 422 381 136)(83 423 382 137)(84 424 383 138)(85 425 384 139)(86 426 385 140)(87 427 386 141)(88 428 387 142)(89 429 388 143)(90 430 389 144)(163 334 236 362)(164 335 237 363)(165 336 238 364)(166 337 239 365)(167 338 240 366)(168 339 241 367)(169 340 242 368)(170 341 243 369)(171 342 244 370)(172 325 245 371)(173 326 246 372)(174 327 247 373)(175 328 248 374)(176 329 249 375)(177 330 250 376)(178 331 251 377)(179 332 252 378)(180 333 235 361)

G:=sub<Sym(432)| (1,413,24)(2,414,25)(3,397,26)(4,398,27)(5,399,28)(6,400,29)(7,401,30)(8,402,31)(9,403,32)(10,404,33)(11,405,34)(12,406,35)(13,407,36)(14,408,19)(15,409,20)(16,410,21)(17,411,22)(18,412,23)(37,257,276)(38,258,277)(39,259,278)(40,260,279)(41,261,280)(42,262,281)(43,263,282)(44,264,283)(45,265,284)(46,266,285)(47,267,286)(48,268,287)(49,269,288)(50,270,271)(51,253,272)(52,254,273)(53,255,274)(54,256,275)(55,340,428)(56,341,429)(57,342,430)(58,325,431)(59,326,432)(60,327,415)(61,328,416)(62,329,417)(63,330,418)(64,331,419)(65,332,420)(66,333,421)(67,334,422)(68,335,423)(69,336,424)(70,337,425)(71,338,426)(72,339,427)(73,234,172)(74,217,173)(75,218,174)(76,219,175)(77,220,176)(78,221,177)(79,222,178)(80,223,179)(81,224,180)(82,225,163)(83,226,164)(84,227,165)(85,228,166)(86,229,167)(87,230,168)(88,231,169)(89,232,170)(90,233,171)(91,123,345)(92,124,346)(93,125,347)(94,126,348)(95,109,349)(96,110,350)(97,111,351)(98,112,352)(99,113,353)(100,114,354)(101,115,355)(102,116,356)(103,117,357)(104,118,358)(105,119,359)(106,120,360)(107,121,343)(108,122,344)(127,196,371)(128,197,372)(129,198,373)(130,181,374)(131,182,375)(132,183,376)(133,184,377)(134,185,378)(135,186,361)(136,187,362)(137,188,363)(138,189,364)(139,190,365)(140,191,366)(141,192,367)(142,193,368)(143,194,369)(144,195,370)(145,214,320)(146,215,321)(147,216,322)(148,199,323)(149,200,324)(150,201,307)(151,202,308)(152,203,309)(153,204,310)(154,205,311)(155,206,312)(156,207,313)(157,208,314)(158,209,315)(159,210,316)(160,211,317)(161,212,318)(162,213,319)(235,380,300)(236,381,301)(237,382,302)(238,383,303)(239,384,304)(240,385,305)(241,386,306)(242,387,289)(243,388,290)(244,389,291)(245,390,292)(246,391,293)(247,392,294)(248,393,295)(249,394,296)(250,395,297)(251,396,298)(252,379,299), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306)(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414)(415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,248,123,175)(2,249,124,176)(3,250,125,177)(4,251,126,178)(5,252,109,179)(6,235,110,180)(7,236,111,163)(8,237,112,164)(9,238,113,165)(10,239,114,166)(11,240,115,167)(12,241,116,168)(13,242,117,169)(14,243,118,170)(15,244,119,171)(16,245,120,172)(17,246,121,173)(18,247,122,174)(19,290,104,232)(20,291,105,233)(21,292,106,234)(22,293,107,217)(23,294,108,218)(24,295,91,219)(25,296,92,220)(26,297,93,221)(27,298,94,222)(28,299,95,223)(29,300,96,224)(30,301,97,225)(31,302,98,226)(32,303,99,227)(33,304,100,228)(34,305,101,229)(35,306,102,230)(36,289,103,231)(37,137,147,423)(38,138,148,424)(39,139,149,425)(40,140,150,426)(41,141,151,427)(42,142,152,428)(43,143,153,429)(44,144,154,430)(45,127,155,431)(46,128,156,432)(47,129,157,415)(48,130,158,416)(49,131,159,417)(50,132,160,418)(51,133,161,419)(52,134,162,420)(53,135,145,421)(54,136,146,422)(55,262,193,203)(56,263,194,204)(57,264,195,205)(58,265,196,206)(59,266,197,207)(60,267,198,208)(61,268,181,209)(62,269,182,210)(63,270,183,211)(64,253,184,212)(65,254,185,213)(66,255,186,214)(67,256,187,215)(68,257,188,216)(69,258,189,199)(70,259,190,200)(71,260,191,201)(72,261,192,202)(73,410,390,360)(74,411,391,343)(75,412,392,344)(76,413,393,345)(77,414,394,346)(78,397,395,347)(79,398,396,348)(80,399,379,349)(81,400,380,350)(82,401,381,351)(83,402,382,352)(84,403,383,353)(85,404,384,354)(86,405,385,355)(87,406,386,356)(88,407,387,357)(89,408,388,358)(90,409,389,359)(271,376,317,330)(272,377,318,331)(273,378,319,332)(274,361,320,333)(275,362,321,334)(276,363,322,335)(277,364,323,336)(278,365,324,337)(279,366,307,338)(280,367,308,339)(281,368,309,340)(282,369,310,341)(283,370,311,342)(284,371,312,325)(285,372,313,326)(286,373,314,327)(287,374,315,328)(288,375,316,329), (1,315,123,287)(2,316,124,288)(3,317,125,271)(4,318,126,272)(5,319,109,273)(6,320,110,274)(7,321,111,275)(8,322,112,276)(9,323,113,277)(10,324,114,278)(11,307,115,279)(12,308,116,280)(13,309,117,281)(14,310,118,282)(15,311,119,283)(16,312,120,284)(17,313,121,285)(18,314,122,286)(19,204,104,263)(20,205,105,264)(21,206,106,265)(22,207,107,266)(23,208,108,267)(24,209,91,268)(25,210,92,269)(26,211,93,270)(27,212,94,253)(28,213,95,254)(29,214,96,255)(30,215,97,256)(31,216,98,257)(32,199,99,258)(33,200,100,259)(34,201,101,260)(35,202,102,261)(36,203,103,262)(37,402,147,352)(38,403,148,353)(39,404,149,354)(40,405,150,355)(41,406,151,356)(42,407,152,357)(43,408,153,358)(44,409,154,359)(45,410,155,360)(46,411,156,343)(47,412,157,344)(48,413,158,345)(49,414,159,346)(50,397,160,347)(51,398,161,348)(52,399,162,349)(53,400,145,350)(54,401,146,351)(55,289,193,231)(56,290,194,232)(57,291,195,233)(58,292,196,234)(59,293,197,217)(60,294,198,218)(61,295,181,219)(62,296,182,220)(63,297,183,221)(64,298,184,222)(65,299,185,223)(66,300,186,224)(67,301,187,225)(68,302,188,226)(69,303,189,227)(70,304,190,228)(71,305,191,229)(72,306,192,230)(73,431,390,127)(74,432,391,128)(75,415,392,129)(76,416,393,130)(77,417,394,131)(78,418,395,132)(79,419,396,133)(80,420,379,134)(81,421,380,135)(82,422,381,136)(83,423,382,137)(84,424,383,138)(85,425,384,139)(86,426,385,140)(87,427,386,141)(88,428,387,142)(89,429,388,143)(90,430,389,144)(163,334,236,362)(164,335,237,363)(165,336,238,364)(166,337,239,365)(167,338,240,366)(168,339,241,367)(169,340,242,368)(170,341,243,369)(171,342,244,370)(172,325,245,371)(173,326,246,372)(174,327,247,373)(175,328,248,374)(176,329,249,375)(177,330,250,376)(178,331,251,377)(179,332,252,378)(180,333,235,361)>;

G:=Group( (1,413,24)(2,414,25)(3,397,26)(4,398,27)(5,399,28)(6,400,29)(7,401,30)(8,402,31)(9,403,32)(10,404,33)(11,405,34)(12,406,35)(13,407,36)(14,408,19)(15,409,20)(16,410,21)(17,411,22)(18,412,23)(37,257,276)(38,258,277)(39,259,278)(40,260,279)(41,261,280)(42,262,281)(43,263,282)(44,264,283)(45,265,284)(46,266,285)(47,267,286)(48,268,287)(49,269,288)(50,270,271)(51,253,272)(52,254,273)(53,255,274)(54,256,275)(55,340,428)(56,341,429)(57,342,430)(58,325,431)(59,326,432)(60,327,415)(61,328,416)(62,329,417)(63,330,418)(64,331,419)(65,332,420)(66,333,421)(67,334,422)(68,335,423)(69,336,424)(70,337,425)(71,338,426)(72,339,427)(73,234,172)(74,217,173)(75,218,174)(76,219,175)(77,220,176)(78,221,177)(79,222,178)(80,223,179)(81,224,180)(82,225,163)(83,226,164)(84,227,165)(85,228,166)(86,229,167)(87,230,168)(88,231,169)(89,232,170)(90,233,171)(91,123,345)(92,124,346)(93,125,347)(94,126,348)(95,109,349)(96,110,350)(97,111,351)(98,112,352)(99,113,353)(100,114,354)(101,115,355)(102,116,356)(103,117,357)(104,118,358)(105,119,359)(106,120,360)(107,121,343)(108,122,344)(127,196,371)(128,197,372)(129,198,373)(130,181,374)(131,182,375)(132,183,376)(133,184,377)(134,185,378)(135,186,361)(136,187,362)(137,188,363)(138,189,364)(139,190,365)(140,191,366)(141,192,367)(142,193,368)(143,194,369)(144,195,370)(145,214,320)(146,215,321)(147,216,322)(148,199,323)(149,200,324)(150,201,307)(151,202,308)(152,203,309)(153,204,310)(154,205,311)(155,206,312)(156,207,313)(157,208,314)(158,209,315)(159,210,316)(160,211,317)(161,212,318)(162,213,319)(235,380,300)(236,381,301)(237,382,302)(238,383,303)(239,384,304)(240,385,305)(241,386,306)(242,387,289)(243,388,290)(244,389,291)(245,390,292)(246,391,293)(247,392,294)(248,393,295)(249,394,296)(250,395,297)(251,396,298)(252,379,299), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306)(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414)(415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,248,123,175)(2,249,124,176)(3,250,125,177)(4,251,126,178)(5,252,109,179)(6,235,110,180)(7,236,111,163)(8,237,112,164)(9,238,113,165)(10,239,114,166)(11,240,115,167)(12,241,116,168)(13,242,117,169)(14,243,118,170)(15,244,119,171)(16,245,120,172)(17,246,121,173)(18,247,122,174)(19,290,104,232)(20,291,105,233)(21,292,106,234)(22,293,107,217)(23,294,108,218)(24,295,91,219)(25,296,92,220)(26,297,93,221)(27,298,94,222)(28,299,95,223)(29,300,96,224)(30,301,97,225)(31,302,98,226)(32,303,99,227)(33,304,100,228)(34,305,101,229)(35,306,102,230)(36,289,103,231)(37,137,147,423)(38,138,148,424)(39,139,149,425)(40,140,150,426)(41,141,151,427)(42,142,152,428)(43,143,153,429)(44,144,154,430)(45,127,155,431)(46,128,156,432)(47,129,157,415)(48,130,158,416)(49,131,159,417)(50,132,160,418)(51,133,161,419)(52,134,162,420)(53,135,145,421)(54,136,146,422)(55,262,193,203)(56,263,194,204)(57,264,195,205)(58,265,196,206)(59,266,197,207)(60,267,198,208)(61,268,181,209)(62,269,182,210)(63,270,183,211)(64,253,184,212)(65,254,185,213)(66,255,186,214)(67,256,187,215)(68,257,188,216)(69,258,189,199)(70,259,190,200)(71,260,191,201)(72,261,192,202)(73,410,390,360)(74,411,391,343)(75,412,392,344)(76,413,393,345)(77,414,394,346)(78,397,395,347)(79,398,396,348)(80,399,379,349)(81,400,380,350)(82,401,381,351)(83,402,382,352)(84,403,383,353)(85,404,384,354)(86,405,385,355)(87,406,386,356)(88,407,387,357)(89,408,388,358)(90,409,389,359)(271,376,317,330)(272,377,318,331)(273,378,319,332)(274,361,320,333)(275,362,321,334)(276,363,322,335)(277,364,323,336)(278,365,324,337)(279,366,307,338)(280,367,308,339)(281,368,309,340)(282,369,310,341)(283,370,311,342)(284,371,312,325)(285,372,313,326)(286,373,314,327)(287,374,315,328)(288,375,316,329), (1,315,123,287)(2,316,124,288)(3,317,125,271)(4,318,126,272)(5,319,109,273)(6,320,110,274)(7,321,111,275)(8,322,112,276)(9,323,113,277)(10,324,114,278)(11,307,115,279)(12,308,116,280)(13,309,117,281)(14,310,118,282)(15,311,119,283)(16,312,120,284)(17,313,121,285)(18,314,122,286)(19,204,104,263)(20,205,105,264)(21,206,106,265)(22,207,107,266)(23,208,108,267)(24,209,91,268)(25,210,92,269)(26,211,93,270)(27,212,94,253)(28,213,95,254)(29,214,96,255)(30,215,97,256)(31,216,98,257)(32,199,99,258)(33,200,100,259)(34,201,101,260)(35,202,102,261)(36,203,103,262)(37,402,147,352)(38,403,148,353)(39,404,149,354)(40,405,150,355)(41,406,151,356)(42,407,152,357)(43,408,153,358)(44,409,154,359)(45,410,155,360)(46,411,156,343)(47,412,157,344)(48,413,158,345)(49,414,159,346)(50,397,160,347)(51,398,161,348)(52,399,162,349)(53,400,145,350)(54,401,146,351)(55,289,193,231)(56,290,194,232)(57,291,195,233)(58,292,196,234)(59,293,197,217)(60,294,198,218)(61,295,181,219)(62,296,182,220)(63,297,183,221)(64,298,184,222)(65,299,185,223)(66,300,186,224)(67,301,187,225)(68,302,188,226)(69,303,189,227)(70,304,190,228)(71,305,191,229)(72,306,192,230)(73,431,390,127)(74,432,391,128)(75,415,392,129)(76,416,393,130)(77,417,394,131)(78,418,395,132)(79,419,396,133)(80,420,379,134)(81,421,380,135)(82,422,381,136)(83,423,382,137)(84,424,383,138)(85,425,384,139)(86,426,385,140)(87,427,386,141)(88,428,387,142)(89,429,388,143)(90,430,389,144)(163,334,236,362)(164,335,237,363)(165,336,238,364)(166,337,239,365)(167,338,240,366)(168,339,241,367)(169,340,242,368)(170,341,243,369)(171,342,244,370)(172,325,245,371)(173,326,246,372)(174,327,247,373)(175,328,248,374)(176,329,249,375)(177,330,250,376)(178,331,251,377)(179,332,252,378)(180,333,235,361) );

G=PermutationGroup([[(1,413,24),(2,414,25),(3,397,26),(4,398,27),(5,399,28),(6,400,29),(7,401,30),(8,402,31),(9,403,32),(10,404,33),(11,405,34),(12,406,35),(13,407,36),(14,408,19),(15,409,20),(16,410,21),(17,411,22),(18,412,23),(37,257,276),(38,258,277),(39,259,278),(40,260,279),(41,261,280),(42,262,281),(43,263,282),(44,264,283),(45,265,284),(46,266,285),(47,267,286),(48,268,287),(49,269,288),(50,270,271),(51,253,272),(52,254,273),(53,255,274),(54,256,275),(55,340,428),(56,341,429),(57,342,430),(58,325,431),(59,326,432),(60,327,415),(61,328,416),(62,329,417),(63,330,418),(64,331,419),(65,332,420),(66,333,421),(67,334,422),(68,335,423),(69,336,424),(70,337,425),(71,338,426),(72,339,427),(73,234,172),(74,217,173),(75,218,174),(76,219,175),(77,220,176),(78,221,177),(79,222,178),(80,223,179),(81,224,180),(82,225,163),(83,226,164),(84,227,165),(85,228,166),(86,229,167),(87,230,168),(88,231,169),(89,232,170),(90,233,171),(91,123,345),(92,124,346),(93,125,347),(94,126,348),(95,109,349),(96,110,350),(97,111,351),(98,112,352),(99,113,353),(100,114,354),(101,115,355),(102,116,356),(103,117,357),(104,118,358),(105,119,359),(106,120,360),(107,121,343),(108,122,344),(127,196,371),(128,197,372),(129,198,373),(130,181,374),(131,182,375),(132,183,376),(133,184,377),(134,185,378),(135,186,361),(136,187,362),(137,188,363),(138,189,364),(139,190,365),(140,191,366),(141,192,367),(142,193,368),(143,194,369),(144,195,370),(145,214,320),(146,215,321),(147,216,322),(148,199,323),(149,200,324),(150,201,307),(151,202,308),(152,203,309),(153,204,310),(154,205,311),(155,206,312),(156,207,313),(157,208,314),(158,209,315),(159,210,316),(160,211,317),(161,212,318),(162,213,319),(235,380,300),(236,381,301),(237,382,302),(238,383,303),(239,384,304),(240,385,305),(241,386,306),(242,387,289),(243,388,290),(244,389,291),(245,390,292),(246,391,293),(247,392,294),(248,393,295),(249,394,296),(250,395,297),(251,396,298),(252,379,299)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162),(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288),(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306),(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324),(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342),(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378),(379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396),(397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414),(415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)], [(1,248,123,175),(2,249,124,176),(3,250,125,177),(4,251,126,178),(5,252,109,179),(6,235,110,180),(7,236,111,163),(8,237,112,164),(9,238,113,165),(10,239,114,166),(11,240,115,167),(12,241,116,168),(13,242,117,169),(14,243,118,170),(15,244,119,171),(16,245,120,172),(17,246,121,173),(18,247,122,174),(19,290,104,232),(20,291,105,233),(21,292,106,234),(22,293,107,217),(23,294,108,218),(24,295,91,219),(25,296,92,220),(26,297,93,221),(27,298,94,222),(28,299,95,223),(29,300,96,224),(30,301,97,225),(31,302,98,226),(32,303,99,227),(33,304,100,228),(34,305,101,229),(35,306,102,230),(36,289,103,231),(37,137,147,423),(38,138,148,424),(39,139,149,425),(40,140,150,426),(41,141,151,427),(42,142,152,428),(43,143,153,429),(44,144,154,430),(45,127,155,431),(46,128,156,432),(47,129,157,415),(48,130,158,416),(49,131,159,417),(50,132,160,418),(51,133,161,419),(52,134,162,420),(53,135,145,421),(54,136,146,422),(55,262,193,203),(56,263,194,204),(57,264,195,205),(58,265,196,206),(59,266,197,207),(60,267,198,208),(61,268,181,209),(62,269,182,210),(63,270,183,211),(64,253,184,212),(65,254,185,213),(66,255,186,214),(67,256,187,215),(68,257,188,216),(69,258,189,199),(70,259,190,200),(71,260,191,201),(72,261,192,202),(73,410,390,360),(74,411,391,343),(75,412,392,344),(76,413,393,345),(77,414,394,346),(78,397,395,347),(79,398,396,348),(80,399,379,349),(81,400,380,350),(82,401,381,351),(83,402,382,352),(84,403,383,353),(85,404,384,354),(86,405,385,355),(87,406,386,356),(88,407,387,357),(89,408,388,358),(90,409,389,359),(271,376,317,330),(272,377,318,331),(273,378,319,332),(274,361,320,333),(275,362,321,334),(276,363,322,335),(277,364,323,336),(278,365,324,337),(279,366,307,338),(280,367,308,339),(281,368,309,340),(282,369,310,341),(283,370,311,342),(284,371,312,325),(285,372,313,326),(286,373,314,327),(287,374,315,328),(288,375,316,329)], [(1,315,123,287),(2,316,124,288),(3,317,125,271),(4,318,126,272),(5,319,109,273),(6,320,110,274),(7,321,111,275),(8,322,112,276),(9,323,113,277),(10,324,114,278),(11,307,115,279),(12,308,116,280),(13,309,117,281),(14,310,118,282),(15,311,119,283),(16,312,120,284),(17,313,121,285),(18,314,122,286),(19,204,104,263),(20,205,105,264),(21,206,106,265),(22,207,107,266),(23,208,108,267),(24,209,91,268),(25,210,92,269),(26,211,93,270),(27,212,94,253),(28,213,95,254),(29,214,96,255),(30,215,97,256),(31,216,98,257),(32,199,99,258),(33,200,100,259),(34,201,101,260),(35,202,102,261),(36,203,103,262),(37,402,147,352),(38,403,148,353),(39,404,149,354),(40,405,150,355),(41,406,151,356),(42,407,152,357),(43,408,153,358),(44,409,154,359),(45,410,155,360),(46,411,156,343),(47,412,157,344),(48,413,158,345),(49,414,159,346),(50,397,160,347),(51,398,161,348),(52,399,162,349),(53,400,145,350),(54,401,146,351),(55,289,193,231),(56,290,194,232),(57,291,195,233),(58,292,196,234),(59,293,197,217),(60,294,198,218),(61,295,181,219),(62,296,182,220),(63,297,183,221),(64,298,184,222),(65,299,185,223),(66,300,186,224),(67,301,187,225),(68,302,188,226),(69,303,189,227),(70,304,190,228),(71,305,191,229),(72,306,192,230),(73,431,390,127),(74,432,391,128),(75,415,392,129),(76,416,393,130),(77,417,394,131),(78,418,395,132),(79,419,396,133),(80,420,379,134),(81,421,380,135),(82,422,381,136),(83,423,382,137),(84,424,383,138),(85,425,384,139),(86,426,385,140),(87,427,386,141),(88,428,387,142),(89,429,388,143),(90,430,389,144),(163,334,236,362),(164,335,237,363),(165,336,238,364),(166,337,239,365),(167,338,240,366),(168,339,241,367),(169,340,242,368),(170,341,243,369),(171,342,244,370),(172,325,245,371),(173,326,246,372),(174,327,247,373),(175,328,248,374),(176,329,249,375),(177,330,250,376),(178,331,251,377),(179,332,252,378),(180,333,235,361)]])

270 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A ··· 4F 6A ··· 6X 9A ··· 9R 12A ··· 12AV 18A ··· 18BB 36A ··· 36DD order 1 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

270 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C2 C3 C3 C6 C6 C6 C6 C9 C18 C18 Q8 C3×Q8 C3×Q8 Q8×C9 kernel Q8×C3×C18 C6×C36 Q8×C3×C9 Q8×C18 Q8×C3×C6 C2×C36 Q8×C9 C6×C12 Q8×C32 C6×Q8 C2×C12 C3×Q8 C3×C18 C18 C3×C6 C6 # reps 1 3 4 6 2 18 24 6 8 18 54 72 2 12 4 36

Matrix representation of Q8×C3×C18 in GL4(𝔽37) generated by

 1 0 0 0 0 26 0 0 0 0 10 0 0 0 0 10
,
 36 0 0 0 0 10 0 0 0 0 21 0 0 0 0 21
,
 36 0 0 0 0 36 0 0 0 0 0 1 0 0 36 0
,
 1 0 0 0 0 1 0 0 0 0 3 29 0 0 29 34
G:=sub<GL(4,GF(37))| [1,0,0,0,0,26,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,10,0,0,0,0,21,0,0,0,0,21],[36,0,0,0,0,36,0,0,0,0,0,36,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,3,29,0,0,29,34] >;

Q8×C3×C18 in GAP, Magma, Sage, TeX

Q_8\times C_3\times C_{18}
% in TeX

G:=Group("Q8xC3xC18");
// GroupNames label

G:=SmallGroup(432,406);
// by ID

G=gap.SmallGroup(432,406);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,1037,512,528]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^18=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽


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A block of mass m is placed at...

# A block of mass m is placed at rest on a smooth wedge of mass M placed at rest on a smooth horizontal surface. As the system is released

Updated On: 27-06-2022

Solution : Out of two blocks, one block of mass m is moving in vertical direction also (downwards). Therefore COM is moving vertically downwards and momentum of the system is not conserved in vertical direction.


text:
### 最新内容

We present an $\tilde O(m+n^{1.5})$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on $m$-edge, $n$-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. $m = \Omega(n^{1.5})$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $O(m\sqrt{n})$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $\tilde O(n^\omega)$-time algorithms [Ibarra-Moran 1981; Mucha-Sankowski 2004] (where currently $\omega\approx 2.373$). On sparser graphs, i.e. when $m = n^{9/8 + \delta}$ for any constant $\delta>0$, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an $\tilde O(m^{4/3+o(1)})$ runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand et al.] and our new IPMs. Combining this general machinery yields a simpler $\tilde O(n \sqrt{m})$ time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art $\tilde O(m+n^{1.5})$ time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).

### 最新论文

We present an $\tilde O(m+n^{1.5})$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on $m$-edge, $n$-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. $m = \Omega(n^{1.5})$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $O(m\sqrt{n})$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $\tilde O(n^\omega)$-time algorithms [Ibarra-Moran 1981; Mucha-Sankowski 2004] (where currently $\omega\approx 2.373$). On sparser graphs, i.e. when $m = n^{9/8 + \delta}$ for any constant $\delta>0$, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an $\tilde O(m^{4/3+o(1)})$ runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand et al.] and our new IPMs. Combining this general machinery yields a simpler $\tilde O(n \sqrt{m})$ time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art $\tilde O(m+n^{1.5})$ time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).

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# Convergence of a sequence

1. Jan 27, 2014

### zelmac

1. The problem statement, all variables and given/known data
Check whether the sequence $a_{1}=\alpha ,\alpha > 0, a_{n+1}=6*\frac{a_{n}+1}{a_{n}+7}$ converges and find its limit if it does, depending on α.

2. Relevant equations

3. The attempt at a solution
I showed boundedness([0,6]) and found that in the case of convergence the limit is 2, but I dont know how to check its convergence... Any help is appreciated :)

2. Jan 27, 2014

### dirk_mec1

Show it's monotone.


text:
## Relay Expansion Python Module

The Onion PWM Expansion Python module, relayExp is based on the C Relay Expansion Library. Using this module, you will be able to control the Relay Expansion from within your Python program.

### Programming Flow

After each power-cycle, the chip that controls the Relay Expansion must be programmed with an initialization sequence. After the initialization, the relays can be turned on and off at will.

The Relay Expansion is the only expansion that has a configurable I2C device address. This was done so that up to eight Relay Expansions can be stacked on a single Omega, giving the user the ability to control 16 relay modules independently.

The base device address is 0x20, the dip switches control the offset added to the base address:

• The ‘Off’ position for each switch is when the toggle is close to the numbers on the switch, or away from the relay modules.
• The ‘On’ position is when the toggle is away from the numbers on the switch, or closer to the relay modules.

The table below defines the relationship:

Switch 1 Switch 2 Switch 3 I2C Device Address addr Argument
Off Off Off 0x27 7
Off Off On 0x26 6
Off On Off 0x25 5
Off On On 0x24 4
On Off Off 0x23 3
On Off On 0x22 2
On On Off 0x21 1
On On On 0x20 0

The addr argument is used for library functions to specify which Relay Expansion to act on.

#### Relay Module Select

Each relay expansion has two channel which can be called using binary values.

### The Python Module

The relayExp Python Module in the OmegaExpansion package provides a wrapper around the C library functions. The functions are largely the same as their C counterparts, including the arguments and return values. Any differences from the C library will be explicitly mentioned below.

#### Source Code

The source code can be found in the Onion i2c-exp-driver GitHub Repo.

#### Installing the Module

To install the Python module, run the following commands:

opkg update
opkg install python-light pyRelayExp

This will install the module to /usr/lib/python2.7/OmegaExpansion/

This only needs to be done once.

To add the Onion Relay Expansion Module to your Python program, include the following in your code:

from OmegaExpansion import relayExp

The functions are largely the same as their C counterparts, including the arguments and return values. Any differences from the C library will be explicitly mentioned below.

#### Example Code

Example code that uses the relayExp module can be found here, in the i2c-exp-driver Onion GitHub Repo.

#### The Channels

The Relay Expansion has two channels, the image below describes how they are enumerated:

#### Return Values

All functions follow the same pattern with return values:

If the function operation is successful, the return value will be 0.

If the function operation is not successful, the function will return 1.

### Functions

Function Prototype
Initialization Function driverInit(addr)
Check for Initialization checkInit(addr)
Set Relay State setChannel(addr, channel, state)
Set State for both Relays setAllChannels(addr, state)
Read Relay State readChannel(addr, channel)

### Initialization Function - driverInit()

To perform the initialization sequence on the Relay Expansion:

relayExp.driverInit(addr)

After this step is completed, the functions to set the relay states can be used with success.

#### Arguments

The addr argument is described above in the I2C Device Address section.

#### Examples

Initialize a Relay Expansion with all switches set to 0, meaning the I2C device address will be 0x27:

status  = relayExp.driverInit(7)

Initialize with switches set to on-on-off (device address: 0x24):

status  = relayExp.driverInit(4)

### Check for Initialization - checkInit()

Performs several reads to determine if the Relay Expansion requires the initialization sequence to be programmed before the relay states can be changed:

relayExp.checkInit(addr)

The return value of the function indicates the Initialization Status:

Return Value Initialization Status
0 Not Initialized
1 Initialized

#### Arguments

The addr argument is described above in the I2C Device Address section.

#### Example

Check if a Relay Expansion (with switches set to Off-On-On) is initialized:

bInit   = relayExp.checkInit(1)

if (bInit == 0):
print 'The Relay Expansion needs to be initialized'
else:
print 'The Relay Expansion is good to go!'
}

### Set Relay State - setChannel()

Use this function to change the state of the relay:

relayExp.setChannel(addr, channel, state)

#### Arguments

The addr argument is described above in the I2C Device Address section.

The channel argument selects the relay in question. See the diagram above for info on which channel corresponds to which relay.

The state argument allows the user to select if the relay will be turned on or off: * 0 turn the relay OFF * 1 turn the relay ON

#### Example

Set channel 0 to On (all switches are On):

status  = relayExp.setChannel(0, 0, 1)

### Set State for both Relays - setAllChannels()

In the event that both relays need to be turned on or off at the same time:

relayExp.setAllChannels(addr, state)

This is performed with a single register write so both relays should react at the same time.

#### Arguments

The addr argument is described above in the I2C Device Address section.

The state argument allows the user to select if the relays will be turned on or off: * 0 turn the relays OFF * 1 turn the relays ON

#### Example

Turn both channels off (all switches are On):

status  = relayExp.setAllChannels(0, 0)

Turn both channels on (all switches are On):

status  = relayExp.setAllChannels(0, 1)

Use this function to read the state of a specific relay:

state = relayExp.readChannel(addr, channel)

#### Arguments

The addr argument is described above in the I2C Device Address section.

The channel argument selects the relay in question. See the diagram above for info on which channel corresponds to which relay.

#### Return Value

The return value indicates the state of the relay channel in question: * 0 indicating the relay is OFF * 1 indicating the relay is ON

#### Example

Read the state of channel 0 (all switches are Off):

value   = relayExp.readChannel(0, 0)
if value == 1:
print 'Relay 0 is ON!'
else:
print 'Relay 0 is OFF'


text:
»  Home »  Resources & support »  FAQs »  Using a value label multiple times

## Can I apply the label values command to more than one variable at the same time? I want to attach the same value labels to several variables.

 Title Using a value label multiple times Author Paul Lin, StataCorp

Yes. Say you have a survey, and the codings for the variables q1, q2, ..., q10 are all the same. You create a value label,

        . label define yesno 0 "no" 1 "yes"

and you can use it ten times:

        . label values q1 yesno
. label values q2 yesno
...
. label values q10 yesno

Actually, you can save time by typing

        . foreach var of varlist q1-q10 {
label values var' yesno
}
`

See help foreach in Stata, or see [P] foreach.