text: When you reach an indeterminant form you need to try someting else. They are listed for standard, two-sided limits, but they work for all forms of limits. It is a relation of the type Y ¼ kXa, where Yand X are variables of interest, a is the PL exponent, and k is typically an unremarkable constant.1 For example, At first, they could increase their strength by as much as 10% per session; now it takes months to improve by even 1%. lim x→axn = an, n = 0,±1,±2,… and a ≠ 0, if n ≤ 0. $x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are defined in terms of $x$. Legal officials at various timesand in various places have objectives and they need to find the bestway of achieving them. The limits of functions $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$ are $f{(a)}$ and $g{(a)}$ respectively. Limits of Polynomials and Rational Functions. lim x → a f ( x) g ( x) = lim x → a f ( x) lim x → a g ( x) It is a property of power rule, used to find the limit of an exponential function whose base and exponent are in a function form. Thanks to limit laws, for instance, you can find the limit of combined functions (addition, subtraction, multiplication, and division of functions, as well as raising them to powers). Learn cosine of angle difference identity, Learn constant property of a circle with examples, Concept of Set-Builder notation with examples and problems, Completing the square method with problems, Evaluate $\cos(100^\circ)\cos(40^\circ)$ $+$ $\sin(100^\circ)\sin(40^\circ)$, Evaluate $\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{bmatrix}$ $\times$ $\begin{bmatrix} 9 & 8 & 7\\ 6 & 5 & 4\\ 3 & 2 & 1\\ \end{bmatrix}$, Evaluate ${\begin{bmatrix} -2 & 3 \\ -1 & 4 \\ \end{bmatrix}}$ $\times$ ${\begin{bmatrix} 6 & 4 \\ 3 & -1 \\ \end{bmatrix}}$, Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sin^3{x}}{\sin{x}-\tan{x}}}$, Solve $\sqrt{5x^2-6x+8}$ $-$ $\sqrt{5x^2-6x-7}$ $=$ $1$. Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising. Use the limit laws to evaluate the limit of a polynomial or rational function. Because power laws usually describe systems where the larger events are more rare than smaller events (i.e. Evaluate the limit of a function by using the squeeze theorem. Some might seek to end casual street violence,so impose stiff legal penalties on anyone caught engaging in suchconduct. Critics fear that executive orders sometimes claim too much power for presidents to interpret law, essentially making new laws in the process, which is constitutionally the role of Congress. This ensures that the the power law is a monotonically decreasing function. We'll use the Constant Multiple Rule on this limit. Check it out: a wild limit appears. All you have to be able to do is find the limit of each individual function separately. Power of attorney is a legal document giving a person broad or limited legal authority to make decisions about the principal's property, finances, or medical care. It is a property of power rule, used to find the limit of an exponential function whose base and exponent are in a function form. Discovering the Limits of a Durable Power of Attorney March 26 . Estimated wait time: 50 minutes. The limit rule is completely in exponential notation. Discovering the Limits of a Durable Power of Attorney. The idea is that we can "pull a constant multiple out" of any limit and still be able to find the solution. It's just the limit of f of x as x approaches c, minus the limit of g of x as x approaches c. So it's just going to be L minus M. And we also often call it the difference rule, or the difference property, of limits. Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them. A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of fractals, and scaling laws in biological systems. Now we can use the Composition Limit Law to bring the limit "inside" the exponential function: but only if we can find a limit for the inside function. Law has limits. The limit of a sum equals the sum of the limits. A better method is to use the following properties of limits called Limit Laws . The ubiquity of power-law relations in physics is partly due to dimensional constraints, while in complex systems, power laws are often thought to be signatures of hierarchy or of specific stochastic processes. If f (x) = xn, then. $=L+(-1)M$ $=L-M$ House Speaker Nancy Pelosi, D-Calif., seeks to write a law to limit presidential pardon power after President Donald Trump commuted the sentence of Roger Stone. Now that we've found our constant multiplier, we can evaluate the limit and multiply it by our constant: All you have to be able to do is find the limit of each individual function separately. Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. $\,\,\, \therefore \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{g{(x)}}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a} \, {\normalsize g{(x)}}}}$. Consider the following limit:. However, note that if a limit is infinite, then the limit does notexist. Math131 Calculus I The Limit Laws Notes 2.3 I. Sum of Functions. The limits of $f{(x)}$ and $g{(x)}$ as $x$ closer to $a$ are written mathematically in calculus as follows. I’m on hold with the Social Security Administration right now. Some might seek to end demonstrable harms caused by alcohol ordrugs through prohibiting their sale and consumption. Consider a person who begins weightlifting for the first time. CONCEAL YOUR INTENTIONS. Perhaps they resort to taking performance-enhancing drug… First, we will use property 2 to break up the limit into three separate limits. Limits to Federal Judicial Power Over the States, States have given away through the amendment process their sovereignty, Decisions by the Supreme Court are not the supreme law of … For a while, they make huge improvements. In particular, lim x→a p√f (x) = p√lim x→af (x). 3. $(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $\,=\,$ $f{(a)}$, $(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$ $\,=\,$ $g{(a)}$. ... (even though in many states they are required by law to do so.) The law is derived from the principles of conservation of mass and momentum of the air stream flowing through an idealized "actuator disk" that extracts energy from the wind stream. We will now proceed to specifically look at the limit constant multiple and power laws (law 5 and law 6 from the Limit of a Sequence page) and prove their validity. If you know the limits of two functions, you know the limits of them added, subtracted, multiplied, divided, or raised to a power. We will then use property 1 to bring the constants out of the first two limits. In other words, figure out the limit … That is obvious. Therefore, it can be written that $f{(a)}$ and $g{(a)}$ as the limits of functions $f{(x)}$ and $g{(x)}$ respectively. It was published in 1919 by the German physicist Albert Betz. Limit Constant Multiple/Power Laws for Convergent Sequences. The first 6 Limit Laws allow us to find limits of any polynomial function, though Limit Law 7 makes it a little more efficient. you can use the limit operations in the following ways. laws which, when used in conjunction, can find most limits. Find the limit of the exponential function by substituting $x$ by $a$. The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above. Thanks to limit laws, for instance, you can find the limit of combined functions (addition, subtraction, multiplication, and division of functions, as well as raising them to powers). Formula. Never put too much trust in friends; learn how to use enemies. $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{g{(x)}}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a} \, {\normalsize g{(x)}}}}$. Root Law. The Limit Laws Assumptions: c is a constant and f x lim ( ) →x a and g x lim ( ) →x a exist Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f, then = f x lim ( ) x a Power Rule. more Power … Betz concluded that this value is 59.3%, meaning that at most only 59.3% of the kinetic energy from wind can be used to spin the turbine and generate electricity.In reality, turbines cannot reach the Betz limit, and common efficiencies are in the 35-45% range. Constant Multiple Law for Convergent Sequences Constant Rule for Limits If a , b {\displaystyle a,b} are constants then lim x → a b = b {\displaystyle \lim _{x\to a}b=b} . In the previous section, we evaluated limits by looking at … So, it is called as the power rule of limit in calculus. What are the constitutional limits of the state’s power to restrict fundamental rights during a pandemic? The Betz limit is the theoretical maximum efficiency for a wind turbine, conjectured by German physicist Albert Betz in 1919. Power Law. The following example makes use of the subtraction, division, and power laws: Finding the limit using limit laws really is that easy! As written in Article II, Section 2 of the U.S. Constitution, the President's power to pardon seems nearly limitless: " [The President] shall have Power to Grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment." Section 7-1 : Proof of Various Limit Properties. For polynomials and rational functions, $\lim_{x→a}f(x)=f(a).$ You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. Use the exponential function to change the form of your limit. Schumpeter (1949, p. 155), discussing the Pareto law 1. Now, find the limit of this exponential function as $x$ approaches $a$. $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{g{(x)}}}$. This first time through we will use only the properties above to compute the limit. Eventually, however, their progress slows down. The limit of a positive integer power of a function is the power of the limit of the function: Example: Evaluate . • Hiring friends will limit your power. Now, we can rewrite the limit as follows: Now that we have the formal definition of a limit, we can set about proving some of the properties we stated earlier in this chapter about limits. Be wary of friends – they will betray … magnitude 8 earthquakes happen much less often than magnitude 2) α is positive. Actually, the value of $f{(a)}$ is raised to the power of $g{(a)}$ is determined as the limit of the $f{(x)}$ is raised to the power of $g{(x)}$ as $x$ closer to $a$. $=\lim\limits_{x\to c} [f(x)]+\lim\limits_{x\to c} [(-1)g(x)]$ We can write the expression above as the sum of two limits, because of the Sum Law proven above. • An enemy at your heels keeps you sharp, alert, and focused. Limits: Limit Laws Graphs and tables can be used to guess the values of limits but these are just estimates and these methods have inherent problems. If you know the limit laws in calculus, you’ll be able to find limits of all the crazy functions that your pre-calculus teacher can throw your way. During their initial sessions, they can lift only a small amount of weight. Using the regular limit laws, we cannot find this limit. $=\lim\limits_{x\to c} f(x)+(-1)\lim\limits_{x\to c} g(x)$ Then we rewrite the second term using the Scalar Multiple Law, proven above. lim x→a[f (x)]p = [lim x→af (x)]p, where the power p can be any real number. Never put too much trust in friends, learn how to use enemies. by Karen Purze. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics, computer science, linguistics, geophysics, neuroscience, systematics, sociology, economics and more. $\implies {f{(a)}}^{g{(a)}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a} \, {\normalsize g{(x)}}}}$. (7) Power Law: lim x → a(f(x))n = (lim x → af(x))n provided lim x → af(x) ≠ 0 if n < 0 When taking limits with exponents, you can take the limit … INTRODUCTION A power law (PL) is the form taken by a remarkable number of regularities, or laws, in economics and finance. But as they invest more time, they find that for each training session, their strength increases a surprising amount. Limit Rules example lim x!3 x2 9 x 3 =? The limit of a positive integer root of a function is the root of the limit of the function: It is assumed that if is even. If the. Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. • Transgression of the Law. But hire a former enemy and he will … In this section we are going to prove some of the basic properties and facts about limits that we saw in the Limits chapter. Writing all the letters down is the key to understanding the Laws So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it. • You destroy an enemy when you make a friend of him. This is a special case of the previous property. rst try \limit of ratio = ratio of limits rule", lim x!3 x2 9 x 3 = lim x!3 x 2 9 lim x!3 x 3 = 0 0 0 0 is called an indeterminant form. Doing this gives us, She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. Therefore, the limit property is proved that the limit of $f{(x)}$ is raised to the power of $g{(x)}$ as $x$ approaches $a$ equals to the limit of $f{(x)}$ as $x$ approaches $a$ is raised to the power of the limit of $g{(x)}$ as $x$ closer to $a$. 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# General Chemistry/The Quantum Atom

 ← The Quantum Model · General Chemistry · Shells and Orbitals → Book Cover · Introduction ·  v • d • e

## The Quantum Numbers

These four numbers are used to describe the location of an electron in an atom.

Number Symbol Possible Values
Principal Quantum Number ${\displaystyle n\,}$  ${\displaystyle \displaystyle 1,2,3,4,\ldots }$
Angular Momentum Quantum Number ${\displaystyle \ell \,}$  ${\displaystyle \displaystyle 0,1,2,3,\ldots ,(n-1)}$
Magnetic Quantum Number ${\displaystyle m_{\text{l}}\,}$  ${\displaystyle \displaystyle -\ell ,\ldots ,-1,0,1,\ldots ,\ell \,}$
Spin Quantum Number ${\displaystyle m_{\text{s}}\,}$  ${\displaystyle \displaystyle +1/2,-1/2}$

### Principal Quantum Number (n)

Determines the shell the electron is in. The shell is the main component that determines the energy of the electron (higher n corresponds to higher energy), as well as size of the orbital, corresponding to maximum nuclear distance (higher n means further possible distance from the nucleus). The row that an element is placed on the periodic table tells how many shells there will be. Helium (n = 1), neon (n = 2), argon (n = 3), etc. Note that the shells will have different numbers, as described by the table above; for example, argon will contain the ${\displaystyle n=1}$ , ${\displaystyle n=2}$ , and ${\displaystyle n=3}$  subshells, for that total of 3.

### Angular Momentum Quantum Number (l)

Also known as azimuthal quantum number. Determines the subshell the electron is in. Each subshell has a unique shape and a letter name. The s orbital is shaped like a sphere and occurs when l = 0. The p orbitals (there are three) are shaped like teardrops and occur when l = 1. The d orbitals (there are five) occur when l = 2. The f orbitals (there are seven) occur when l = 3. (By the way, when l = 4, the orbitals are "g orbitals", but they (and the l = 5 "h orbitals") can safely be ignored in general chemistry.). The numbers of the subshells in each shell can be calculated using the principal quantum number like so. ${\displaystyle l=0,1,...(n-2),(n-1).}$  For example, in the ${\displaystyle n=2}$  shell, the subshells are an ${\displaystyle l=0}$  subshell, and 3 ${\displaystyle l=1}$  subshells. You will learn how to determine the number of orbitals for each subshells in the next section.

This number also gives information as to what the angular node of an orbital is. A node is defined as a point on a standing wave where the wave has minimal amplitude. When applied to chemistry this is the point of zero-displacement and thus where no electrons are found. In turn angular node means the planar or conical surface in which no electrons are found or where there is no electron density. The models shown on this page show the most simple representations of these orbitals and their nodes. More accurate, but more complex depictions are not necessary for the scope of this book.

Here are pictures of the orbitals. Keep in mind that they do not show the actual path of the electrons, due to the Heisenberg Uncertainty Principle. Instead, they show the volume where the electron is most likely to occur, i.e. the probability amplitude is largest. The two colors represent two signs (phases) of the wave function (the choice is arbitrary). Each of the depicted orbitals is a superposition of two opposite m quantum numbers (see below).

ml 0 -1 and 1 -2 and 2 -3 and 3
S orbital →
P orbitals →
D orbitals →
F orbitals →

### Magnetic Quantum Number (ml)

ml -3 -2 -1 0 1 2 3
S orbital →
P orbitals →
D orbitals →
F orbitals →

Magnetic quantum number determines the orbital in which the electron lies. The number of orbitals in each subshell can be calculated like so: ${\displaystyle m}$ ${\displaystyle l}$ ${\displaystyle =-l,-(l-1),...,-1,0,1,...(l-1),l}$ . ml determines how rapidly the complex phase increases around the z-axis. Without magnetic field, these orbitals all have the same energy, they are degenerate and can be combined into different shapes and spatial orientations. The orbitals in a subshell with degeneracy are called degenerate orbitals. This simply means that the orbitals in each p subshell all have the same energy level. The difference in shapes as well as orientation of higher ${\displaystyle l}$  subshells is not important during general chemistry, and the orbitals in the same higher ${\displaystyle l}$  subshells are still degenerate regardless of shape differences.

### Spin Quantum Number (ms)

Does not determines the spin on the electron. +½ corresponds to the up arrow in an electron configuration box. If there is only one electron in an orbital (one arrow in one box), then it is always considered +½. The second arrow, or down arrow, is considered -½. Every orbital can contain one "spin up" electron, and one "spin down" electron.

### Some Examples

Let's examine the quantum numbers of electrons from a magnesium atom, 12Mg. Remember that each list of numbers corresponds to (n, l, ml, ms).

 Two s electrons: (1, 0, 0, +½) (1, 0, 0, -½) Two s electrons: (2, 0, 0, +½) (2, 0, 0, -½) Six p electrons: (2, 1, -1, +½) (2, 1, -1, -½) (2, 1, 0, +½) (2, 1, 0, -½) (2, 1, 1, +½) (2, 1, 1, -½) Two s electrons: (3, 0, 0, +½) (3, 0, 0, -½)

### The Periodic Table

Notice a pattern on the periodic table. Different areas, or blocks, have different types of electrons. The two columns on the left make the s-block. The six columns on the right make the p-block. The large area in the middle (transition metals) makes the d-block. The bottom portion makes the f-block (Lanthanides and Actinides). Each row introduces a new shell (aka energy level). Basically, the row tells you how many shells of electrons there will be, and the column tells you which subshells will occur (and which shells they occur in). The value of ml can be determined by some of the rules we will learn in the next chapter. The value of ms doesn't really matter as long as there are no repeating values in the same orbital.

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시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 256 MB 12 10 5 100.000%

문제

A sequence of digits usually represents a number, but we may define an alternative interpretation. In this problem we define a new interpretation with the order relation ≺ among the digit sequences of the same length defined below.

Let s be a sequence of n digits, d1d2···dn, where each di (1 ≤ i ≤ n) is one of 0, 1, . . . , and 9. Let sum(s), prod(s), and int(s) be as follows:

• sum(s) = d1 + d2 + · · · + dn
• prod(s) = (d1 + 1) × (d2 + 1) × · · · × (dn + 1)
• int(s) = d1 × 10n−1 + d2 × 10n−2 + · · · + dn × 100

int(s) is the integer the digit sequence s represents with normal decimal interpretation.

Let s1 and s2 be sequences of the same number of digits. Then s1 ≺ s2 (s1 is less than s2) is satisfied if and only if one of the following conditions is satisfied.

1. sum(s1) < sum(s2)
2. sum(s1) = sum(s2) and prod(s1) < prod(s2)
3. sum(s1) = sum(s2), prod(s1) = prod(s2), and int(s1) < int(s2)

For 2-digit sequences, for instance, the following relations are satisfied.

00 ≺ 01 ≺ 10 ≺ 02 ≺ 20 ≺ 11 ≺ 03 ≺ 30 ≺ 12 ≺ 21 ≺ · · · ≺ 89 ≺ 98 ≺ 99

Your task is, given an n-digit sequence s, to count up the number of n-digit sequences that are less than s in the order ≺ defined above.

입력

The input consists of a single test case in a line.

d1d2···dn

n is a positive integer at most 14. Each of d1, d2, . . . , and dn is a digit.

출력

Print the number of the n-digit sequences less than d1d2···dn in the order defined above.

예제 입력

20

예제 출력

4

예제 입력 2

020

예제 출력 2

5

예제 입력 3

118

예제 출력 3

245

예제 입력 4

11111111111111

예제 출력 4

40073759

예제 입력 5

99777222222211

예제 출력 5

23733362467675


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# Microdata: Pregnancy and Employment Transitions, Australia

Enables detailed analysis of employment for mothers and partners during pregnancy and after birth

## Introduction

This product provides information about the release of microdata from the November 2011 Pregnancy and Employment Transitions Survey including details about the survey methodology and how to use the Microdata. Also provided are the data item list, a glossary and information on the conditions of use and the quality of the microdata.

Microdata are the most detailed information available from a survey and are generally the responses to individual questions on the questionnaire or data derived from two or more questions and are released with the approval of the Australian Statistician.

### Available products

The following microdata product is available from this survey:

• Detailed Microdata via DataLab.

Further information about these services, and other information to assist users in understanding and accessing Microdata in general, is available from the Microdata Entry Page on the ABS web site.

Before you apply for access, users should read and familiarise themselves with the information contained in this product and the Responsible Use of ABS Microdata, User Guide.

### Further information

Further information about the survey and the microdata products can be found in this product:

• A detailed list of data items for the Microdata are available in the Data downloads section;
• The Abbreviations and Glossary relating to these products can be found in the left navigation menu.

### Data available on request

Data obtained in the survey but not contained in the Microdata may be available from the ABS, on request, as statistics in tabulated form.

Subject to confidentiality and sampling variability constraints, special tabulations can be produced incorporating data items, populations and geographic areas selected to meet individual requirements. These are available on request, on a fee for service basis. For more information, contact the ABS by visiting www.abs.gov.au/about/contact-us.

## Survey methodology

### Scope and coverage

The scope of the Labour Force Survey (LFS) is restricted to persons aged 15 years and over and excludes the following people:

• members of the permanent defence forces;
• certain diplomatic personnel of overseas governments, customarily excluded from the Census and estimated populations; and
• overseas residents in Australia; and members of non-Australian defence forces (and their dependants).

Students at boarding schools, patients in hospitals, residents of homes (e.g. retirement homes, homes for people with disabilities), and inmates of prisons are excluded from all supplementary surveys.

This supplementary survey was conducted in both urban and rural areas in all states and territories, but excluded persons living in Aboriginal and Torres Strait Islander communities in very remote parts of Australia.

Only women aged 15 years and over who had given birth to a child less than two years ago and were living with that child were included in the survey. The survey excluded:

• women who were members of the Australian permanent defence forces;
• women living in non-private dwellings;
• visitors to private dwellings; and
• all males.

In addition, for those women whose partner at the time of interview was not in scope of the survey (e.g. the partner was in the permanent defence forces), the details of the partner's job at November 2011 was not collected. For these partners, job details at November 2011 were classified as not applicable.

In the LFS, coverage rules are applied which aim to ensure that each person is associated with only one dwelling and hence has only one chance of selection in the survey. See Labour Force, Australia (cat. no. 6202.0) for more details.

### Survey design

Supplementary surveys are not conducted on the full LFS sample. Since 1994 the sample for supplementary surveys has been restricted to no more than seven-eighths of the LFS sample. The sample for PaETS is a subsample of 36,604 private dwelling households and special dwelling units included in the ABS Monthly LFS in November 2011. The final sample on which estimates are based is composed of 1,351 birth mothers aged 15 years and over, usually resident in these private and special dwellings in Australia, with at least one child less than two years of age living with them at the time of interview. In cases where women had more than one child under the age of two, data was collected in respect of the employment transitions of women around the birth of their most recent child (i.e. the youngest child).

### Data collection methodology

Data were collected by trained interviewers, who conducted computer-assisted personal and telephone interviews at selected dwellings throughout Australia. These interviews were primarily conducted over two weeks in November 2011, with any follow up activity necessary undertaken during the following two weeks.

The publication, Labour Force, Australia (cat. no. 6202.0), contains information about survey design, sample redesign, scope, coverage and population benchmarks relevant to the monthly LFS, which also applies to the supplementary surveys. It also contains definitions of demographic and labour force characteristics, and information about interviewing which are relevant to both the monthly LFS and supplementary surveys.

## Weighting, benchmarking and estimation

#### Weighting

Weighting is the process of adjusting results from a sample survey to infer results for the total population. To do this, a 'weight' is allocated to each sample unit. The weight is a value which indicates how many population units are represented by the sample unit.

The first step in calculating weights for each person is to assign an initial weight, which is equal to the inverse of the probability of being selected in the survey. For example, if the probability of a person being selected in the survey was 1 in 300, then the person would have an initial weight of 300 (that is, they represent 300 people).

Separate weights were calculated for LFS and Pregnancy and Employment Transitions Survey samples (as some units were in scope for LFS but not for the Pregnancy and Employment Transitions Survey). The LFS weighting method ensures that LFS estimates conform to the benchmark distribution of the population by age, sex and geographic area, and also LFS region by sex (two sets of benchmarks). Weights are allocated to each sample respondent according to their state/territory of selection, state/territory of usual residence, part of state of usual residence, age group and sex.

#### Benchmarking

The weights were calibrated to align with independent estimates of the population, referred to as benchmarks, in designated categories of sex by age by area of usual residence. Weights calibrated against population benchmarks ensure that the survey estimates conform to the independently estimated distribution of the population rather than to the distribution within the sample itself. Calibration to population benchmarks helps to compensate for over or under-enumeration of particular categories of persons which may occur due to either the random nature of sampling or non-response.

The Pregnancy and Employment Transitions Survey is benchmarked to LFS estimates for the following variables: state or territory of usual residence, part of state or territory of usual residence, sex, age group, social marital status.

Benchmarking to LFS estimates accounts for the one-eighth of the sample where the Pregnancy and Employment Transitions Survey is not conducted and for non-respondents to the Pregnancy and Employment Transitions Survey. The Pregnancy and Employment Transitions Survey weighting excludes all residents in institutions, boarding schools, and very remote areas because the sample scope excludes these people.

Due to differences in scope and sample size between this supplementary survey and that of the LFS, the estimation procedure may lead to some small variations between labour force estimates from this survey and those from the LFS.

#### Estimation

Survey estimates of counts of persons are obtained by summing the weights of persons with the characteristic of interest. Estimates of non-person counts (e.g. days away from work) are obtained by multiplying the characteristic of interest with the weight of the reporting person and aggregating.

### Comparison with 2005 Microdata

The ABS conducted the first Pregnancy and Employment Transitions Survey in November 2005. November 2011 was the second time this survey has been conducted. Results from most recent survey, Pregnancy and Employment Transitions, Australia (cat. no. 4913.0) were released in November 2012.

Care should be taken when comparing results from the November 2011 Pregnancy and Employment Transitions Survey to the November 2005 Pregnancy and Employment Transitions Survey.

The November 2011 Pregnancy and Employment Transitions Survey was redeveloped to better capture information on:

• job details and types of leave taken while women were pregnant;
• job details about women's first job started or returned to since the birth of the child;
• women with a child under 2 years, job at November 2011;
• partner's job details and types of leave taken while women were pregnant;
• job details about partner's first job started or returned to since the birth of the child; and
• partner's job details at November 2011.

Information on key changes made to the Pregnancy and Employment Transitions Survey can be found in the Explanatory Notes of the publication, Pregnancy and Employment Transitions, Australia (cat. no. 4913.0).

As a result of the redevelopment, the data items on the 2011 Microdata will not align with the data items on the 2005 Microdata.

### Reliability of estimates

All sample surveys are subject to error which can be broadly categorised as either sampling error or non-sampling error.

• Sampling error occurs because only a small proportion of the total population is used to produce estimates that represent the whole population. Sampling error can be reliably measured as it is calculated based on the scientific methods used to design surveys.
• Non-sampling error can occur at any stage throughout the survey process. For example, persons selected for the survey may not respond (non-response); survey questions may not be clearly understood by the respondent; responses may be incorrectly recorded by interviewers; or there may be errors when coding or processing the survey data.

#### Sampling error

One measure of the likely difference between an estimate derived from a sample of persons and the value that would have been produced if all persons in scope of the survey had been included, is given by the Standard Error (SE) which indicates the extent to which an estimate might have varied by chance because only a sample of persons was included. There are about two chances in three (67%) that the sample estimate will differ by less than one SE from the number that would have been obtained if all persons had been surveyed and about 19 chances in 20 (95%) that the difference will be less than two SEs.

Another measure of the likely difference is the Relative Standard Error (RSE), which is obtained by expressing the SE as a percentage of the estimate.

Generally, only estimates (numbers, percentages, means and medians) with RSEs less than 25% are considered sufficiently reliable for most purposes. In ABS publications, estimates with an RSE of 25% to 50% are preceded by an asterisk (e.g. *15.7) to indicate that the estimate should be used with caution. Estimates with RSEs over 50% are indicated by a double asterisk (e.g.**2.8) and should be considered unreliable for most purposes.

In addition to the main weight (as outlined earlier), each record on the Microdata also contains a set of 30 'replicate weights' applicable to the Pregnancy and Employment Transitions Survey. The purpose of these replicate weights is to enable the calculation of the standard error on each estimate produced. This method is known as the 30 group Jackknife variance estimator.

The basic concept behind this replication approach is to select different sub-samples repeatedly (30 times) from the whole sample. For each of these sub-samples the statistic of interest is calculated. The variance of the full sample statistics is then estimated using the variability among the replicate statistics calculated from these sub-samples. As well as enabling variances of estimates to be calculated relatively simply, replicate weights also enable unit record analyses such as chi-square and logistic regression to be conducted which take into account the sample design.

Further information about RSEs and how they are calculated can be referenced in the section on Standard Errors under File Structure in this product and the Technical Note of Pregnancy and Employment Transitions, Australia (cat. no. 4913.0).

#### Non-sampling error

Non-sampling error may occur in any collection, whether it is based on a sample or a full count such as a census. One of the main sources of non-sampling error is non-response by persons selected in the survey. Non-response occurs when persons cannot or will not cooperate, or cannot be contacted. Non-response can affect the reliability of results and can introduce a bias. The magnitude of any bias depends upon the rate of non-response and the extent of the difference between the characteristics of those persons who responded to the survey and those that did not.

Every effort was made to reduce non-response and other non-sampling errors by careful design and testing of the questionnaire, training and supervision of interviewers, and undertaking extensive editing and quality control procedures at all stages of data processing.

### Seasonal factors

Estimates are based on information collected in the survey month, and, due to seasonal factors, they may not be representative of other months of the year.

## File structure

### Weights and estimation

As the survey was conducted on a sample of households in Australia, it is important to take account of the method of sample selection when deriving estimates from the Microdata. This is particularly important as a person's chance of selection in the survey varied depending on the state or territory in which they lived.

Each person record contains a weight which for the Pregnancy and Employment Transitions Survey is called FINPRSWT. The weight indicates the number of people in the civilian population represented by that person.

The Pregnancy and Employment Transitions Survey weight, FINPRSWT, appears on all 1,351 records. The estimates in the Pregnancy and Employment Transitions Survey publication are based on these records, that is women with a child under the age of two years during November 2011. Therefore when using FINPRSWT, to match the published Pregnancy and Employment Transitions Survey total estimate, the filter POP1=1 must be used.

Where estimates are derived from the Microdata, it is essential that they are calculated by adding the weights of persons in each category and not just by counting the number in each category. If each person's 'weight' is ignored, then no account would be taken of a person's chance of selection or of different response rates across population groups, and the resulting estimates could be significantly biased and would only represent distributions within the actual selected sample and not the population of interest. The application of weights will ensure that the subsequent estimates conform to an independently estimated distribution of the population by age and sex, rather than to the age and sex distribution within the sample itself.

For further information see the Explanatory Notes in the publication Pregnancy and Employment Transitions, Australia (cat. no. 4913.0).

### Standard errors

Standard errors for each estimate produced from this Microdata can be calculated using the replicate weights provided on the file.

Each person record contains a set of 30 replicate weights. Replicate weights applicable to the Pregnancy and Employment Transitions Survey data items, contain the prefix 'WPX02'. By using these weights, it is possible to calculate standard errors for weighted estimates produced from the microdata. This method is known as the 30 group Jackknife variance estimator. For data items that are only applicable to the Pregnancy and Employment Transitions Survey, refer to Data Item List.

Under the Jackknife method of replicate weighting, weights were derived as follows:

• 30 replicate groups were formed with each group formed to mirror the overall sample (where units from a collection district all belong to the same replicate group and a unit can belong to only one replicate group)
• one replicate group was dropped from the file and then the remaining records were weighted in the same manner as for the full sample
• records in that group that were dropped received a weight of zero.

This process was repeated for each replicate group (i.e. a total of 30 times). Ultimately each record had 30 replicate weights attached to it with one of these being the zero weight.

Replicate weights enable variances of estimates to be calculated relatively simply. They also enable unit records analyses such as chi-square and logistic regression to be conducted which take into account the sample design. Replicate weights for any variable of interest can be calculated from the 30 replicate groups, giving 30 replicate estimates. The distribution of this set of replicate estimates, in conjunction with the full sample estimate (based on the general weight) is then used to approximate the variance of the full sample.

To obtain the standard error of a weighted estimate y, the same estimate is calculated using each of the 30 replicate weights. The variability between these replicate estimates (denoting y(g) for group number g) is used to measure the standard error of the original weighted estimate y using the formula:

$$SE(y) = \sqrt {(29/30) \sum \limits^{30} _{g=1} (y _{(g)} - y)^2 }$$

Where:

$$g$$ = the replicate groups

$$y_{(g)}$$ = the weighted estimate, having applied the weights for replicate group 'g'

$$y$$ = the weighted estimate from the full sample.

The 30 group Jackknife method can be applied not just to estimates of population total, but also where the estimate y is a function of estimates of population total, such as a proportion, difference or ratio. For more information on the 30 group Jackknife method of SE estimation, see Research Paper: Weighting and Standard Error Estimation for ABS Household Surveys (Methodology Advisory Committee), July 1999 (cat. no. 1352.0.55.029).

Use of the 30 group Jackknife method for complex estimates, such as regression parameters from a statistical model, is not straightforward and may not be appropriate. The method as described does not apply to investigations where survey weights are not used, such as in unweighted statistical modelling.

The following table has been provided to enable Microdata users to check some of the relative standard errors they have produced.

Women with child under 2 years - by selected characteristics
WomenRelative Standard Error
'000%
Age of women (years)
15 - 2459.410.9
25 - 29130.35.3
30 - 34175.84.7
35 - 39112.95.8
40 and over44.910.3
Social marital status
Married459.63.1
Not married63.79.2
Country of birth and period of arrival
Born in Australia385.63.5
Born overseas137.75.0
Arrived before 200154.910.2
Arrived 2001 to the date of interview82.85.2
Total523.33.2

### Not applicable categories

Many data items included in the Microdata include a 'Not applicable' category. The classification value of the 'Not applicable' category, where relevant, is shown in the relevant data item lists available in the Data downloads section.

## Using the Microdata

The data included in the Pregnancy and Employment Transition Survey, November 2011 Detailed Microdata are released under the provisions of the Census and Statistics Act 1905. This legislation allows the Australian Statistician to release unit record data, or microdata, provided this is done “in a manner that is not likely to enable the identification of a particular person or organisation to which it relates.”

The ABS ensures the confidentiality of the data by:

• removing name, address and any other information that might uniquely identify any individual
• changing a small number of values - particularly unusual values - and removing very unusual records
• controlling the detail available for all records on the Detailed Microdata
• excluding some data items that were collected
• placing restrictions on how the data are used, supported by both information in the Responsible Use of ABS Microdata, User Guide,  the undertaking signed by the head of each organisation and the terms and conditions signed by each user.

As a result, data on the Detailed Microdata will not exactly match other previously published estimates. Any changes to the distribution of values are not significant and the statistical validity of aggregate data is not affected.

### Identifiers

Each person has a unique random identifier - ABSPID.

### Detailed Microdata file contents

The Pregnancy and Employment Transitions Survey November 2011 Detailed Microdata can be accessed in DataLab and is available in SAS, SPSS and STATA formats. Detailed Microdata comprises the following files:

### Data files

• SAS FILE: PAET11E.SAS7BDAT contains the Microdata in SAS format
• SPSS FILE: PAET11E.SAV contains the Microdata in SPSS format
• STATA FILE: PAET11E.DTA contains the Microdata in STATA format

### Information files

• DATA ITEMS LIST: contains all the data items, including details of categories and code values
• FORMATS FILE: FORMATS.sas7bcat is a SAS library containing formats
• FREQUENCY FILES: The following frequency files contain documentation of the person level data. Data item code values and category labels are provided with frequencies of each value. These files are in plain text format
• FREQUENCIES_UNWTD_PAET11E.txt
• FREQUENCIES_SUPWTD_PAET11E.txt

## Data item list

The Pregnancy and Employment Transitions Survey comprised a questionnaire containing 4 modules and approximately 170 questions.

All data items are available for the Pregnancy and Employment Transitions Survey sample. Users intending to purchase the Microdata should ensure that the data they require, and the level of detail they need, are available in this product.

The Detailed Microdata file contains 166 data items. The data items list for the Detailed Microdata, including relevant population and classification details, can be found in the Excel spreadsheet available from the Data downloads section. The data items are sorted by like items in 10 worksheets contained in the spreadsheet.

These worksheets are named as follows:

• Populations;
• Record Identifiers & Weights;
• Demographic;
• Income;
• Womens job while pregnant;
• Partners job during pregnancy;
• Womens first job after birth;
• Partners first job after birth;
• Womens current job; and
• Partners current job.

The Pregnancy and Employment Transitions Survey Detailed Microdata contain 1,351 confidentialised respondent records from the survey. Subject to the limitations of the sample size and the data classifications used, it is possible to interrogate the Microdata, produce tabulations and undertake statistical analyses to individual specifications.

Data files

## Previous releases

Pregnancy and Employment Transitions, 2005  Detailed microdata

## Glossary

### Show all

##### Away from job/business throughout pregnancy

Women who had a job while pregnant, but who were on leave/away from their job or business for the full period of the pregnancy.

##### Baby Bonus

Baby Bonus is paid by the Australian Government to eligible families following the birth of a child or for adopted children who enter the family's care before they turn 16 years old. A family can not receive both Paid Parental Leave and Baby Bonus for the one birth or adoption.

##### Birth mother

The natural mother of a child, i.e. the woman who gave birth to that child. Birth mothers residing with at least one child aged under two years were in scope for this survey.

##### Born in Australia

Includes persons born in Australia, Norfolk Island and Australian External Territories.

##### Contributing family worker

A person who works without pay, in an economic enterprise operated by a relative.

##### Current main job

The main job which the person is currently employed at.

##### Employed

People aged 15 years and over who, during the reference week:

• worked for one hour or more for pay, profit, commission or payment in kind, in a job or business or on a farm (comprising employees, employers and own account workers); or
• worked for one hour or more without pay in a family business or on a farm (i.e. contributing family workers); or
• were employees who had a job but were not at work and were:
• away from work for less than four weeks up to the end of the reference week; or
• away from work for more than four weeks up to the end of the reference week and received pay for some or all of the four week period to the end of the reference week; or
• away from work as a standard work or shift arrangement; or
• on strike or locked out; or
• on workers' compensation and expected to return to their job; or
• were employers or own account workers who had a job, business of farm, but were not at work.
##### Employees

People who:

• worked for a public or private employer; and
• received remuneration in wages or salary; or were paid a retainer fee by their employer and worked on a commission basis, or for tips, piece-rates or payment in kind; or
• operated their own incorporated enterprise with or without hiring employees.
##### Employers

People who operate their own unincorporated economic enterprise or engage independently in a profession or trade, and hire one or more employees.

##### Employment type

Classifies employed people according to the following categories on the basis of their main job (that is, the job in which they usually worked the most hours):

• Employees (excluding owner managers of incorporated enterprises);
• with paid leave entitlements;
• without paid leave entitlements;
• Owner managers of incorporated enterprises;
• Owner managers of unincorporated enterprises;
• Contributing family workers.
##### Equivalised household income

Total household income that has been adjusted using an equivalence scale. Equivalence scales are used to adjust the actual incomes of households in a way that enables the analysis of the relative well-being of people living in households of different size and composition. For a household comprising more than one person, it is an indicator of the total household income that would need to be received by a lone person household to enjoy the same level of economic well-being as the household in question. If one or more persons in the household has refused to provide income, the household has been excluded from this category.

The equivalence scale is built up by allocating points to each person in a household. Taking the first adult in the household as having a weight of 1 point, each additional person who is 15 years or older is allocated 0.5 points, and each child under the age of 15 is allocated 0.3 points. Equivalised household income is derived by dividing total household income by a factor equal to the sum of the equivalence points allocated to the household members. The equivalised income of a lone person household is the same as its unequivalised income.

##### Family

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 a couple relationship, lone parent-child relationship or other blood relationship. Some households will, therefore, contain more than one family

##### Family day care

A type of formal care provided in caregivers' homes.

##### First job started or returned to

The first job in which the person had started or returned to since the birth of child.

##### Formal care

Regulated care away from the child's home. The main types of formal care are long day care, family day care and occasional care.

##### Full-time workers

Employed persons who usually worked 35 hours or more a week.

##### Full-time workers in main job

Employed people who usually worked 35 hours or more a week (in their main job) and others who, although usually worked less than 35 hours a week, worked 35 hours or more during the reference week.

##### Had a job while pregnant

Women who had a job for some or all of the period during which they were pregnant. This includes women who were away from their job or business throughout their pregnancy.

##### Household

A group of two or more related or unrelated people who usually reside in the same dwelling.

##### Incorporated enterprise

A enterprise which is registered as a separate legal entity to its members or owners. Also known as a limited liability company.

##### Industry

An industry is a group of businesses or organisations that undertake similar economic activities to produce goods and/or services. In this publication, industry refers to ANZSIC Division as classified according to the Australian and New Zealand Standard Industrial Classification (ANZSIC), 2006 (cat. no.1292.0).

##### Informal care

Unregulated care either in the child's home or elsewhere. It includes care by (step) brothers or sisters, care by grandparents, care by other relatives (including the other parent) and care by other (unrelated) people such as friends, neighbours, nannies or babysitters. It may be paid or unpaid.

##### Job

In this survey a 'job' is defined as a set of tasks designed to be performed by one person either:

• for an employer in return for pay, commission or payment in kind; or
• as an operator of an enterprise (either incorporated or unincorporated); or
• when engaging independently in a profession or trade; or
• as a contributing family worker working without pay in an enterprise operated by a relative.
##### Job held during pregnancy

The main job in which the person was working in before the birth of child.

##### Job sharing

Job sharing is an arrangement in which two or more people share the one full-time job, each working part time. Job sharing is available in a wide range of industries, and is in place to help employees achieve a balance between work and other aspects of their life.

##### Leave

Paid or unpaid time away from work taken by employed persons. An individual's specific work arrangements will determine the particular lengths and types of leave to which they have access.

##### Length of leave

Total amount of paid or unpaid leave or time away from a job for the child's birth and subsequent care until the mother returns or joins the workforce after the birth of the child or until the date of interview. It refers to the total amount of leave taken by the women or partner up until the date of interview.

##### Long day care

Regulated care that is provided to children in a dedicated centre.

##### Long service leave

A period of paid leave granted to an employee in recognition of a long period of service to an employer.

##### Main job

The job in which the most hours were usually worked.

##### Married

Marital status relates to a 'social marital status' where married is classified as a person who is living with another person in a couple relationship. This relationship is either a registered marriage, or a de facto marriage.

##### Not in the labour force

People who were not in the categories 'employed' or 'unemployed' as defined.

##### Occasional care

A type of formal care provided mainly for families who require short term care for their children.

##### Occupation

An occupation is a collection of jobs that are sufficiently similar in their title and tasks, skill level and skill specialisation which are grouped together for the purposes of classification. In this publication occupation refers to Major Group as defined by ANZSCO - Australian and New Zealand Standard Classification of Occupations, First Edition, Revision 1, 2009 (cat. no. 1220.0).

##### One parent family

For the purpose of this survey, a family consisting of a lone female parent with at least one natural child aged under two years of age who is also usually resident in the family. The family may also include any number of other dependants, non-dependants and other related individuals.

##### Other paid leave (women)

For the purpose of this survey, all types of paid leave other than paid maternity leave, paid holiday leave or long service leave.

##### Other person care

Informal care by people who are not related to the child including friends, babysitters and nannies.

##### Other relative care

Informal care by relatives of the child excluding parents, not otherwise categorised.

##### Other unpaid leave

This includes all other types of unpaid leave that has not been stated which the women or their partner took for the birth of child.

##### Own account workers

People who operated their own unincorporated economic enterprise or engaged independently in a profession or trade, and hired no employees.

##### Owner managers

People who work in their own business, with or without employees, whether or not the business is an incorporated enterprise. Comprises owner managers of incorporated enterprises and owner managers of unincorporated enterprises.

##### Owner managers of incorporated enterprises (OMIEs)

People who work in their own incorporated enterprise, that is, a business entity which is registered as a separate legal entity to its members or owners (may also be known as a limited liability company). These people are classified as employees under 'status in employment'.

##### Owner managers of unincorporated enterprises (OMUEs)

People who operated their own unincorporated enterprise, that is, a business entity in which the owner and the business are legally inseparable, so that the owner is liable for any business debts that are incurred. Includes those engaged independently in a trade or profession. These people are classified as employers under 'status in employment' if their business has employees, or own account workers if it does not.

##### Paid maternity leave

Paid leave that a woman receives for the birth of her child. It is generally for a period before the due date and just after the birth of the child. Depending on the woman's workplace, she may be entitled to paid maternity leave for a number of months.

##### Paid Parental Leave

For the purpose of this survey mothers of children born on or after 1 January 2011, were asked about their entitlement to Paid Parental Leave, subject to government eligibility conditions. The Paid Parental Leave scheme provides financial support to eligible working parents of newborn or recently adopted children. If eligible, persons may receive up to 18 weeks of Parental Leave Pay at the rate of the National Minimum Wage.

##### Partner

For the purpose of this survey, a person who was:

• in a couple relationship with the selected respondent (either a registered or de facto marriage; including same-sex couples); and
• a usual resident of the same household as the respondent; and
• was the same partner during the women's pregnancy.
##### Part-time workers

Employed persons who usually worked less than 35 hours a week.

##### Part-time workers in main job

Employed people who usually worked less than 35 hours a week (in their main job) and who did so during the reference week.

##### Paternity/parenting leave

Paternity/parenting leave is leave provided to employees to care for their newborn child and during the first year of the child's life. Some workplaces offer this paid leave anywhere from 1-14 weeks. This leave does not break continuity of service.

##### Permanently left job

Ceasing employment in a particular job with no intention of returning.

##### Public/private sector

The public/private classification is used to identify whether an enterprise is a public or private unit. The public sector includes all government units, such as government departments, non-market non-profit institutions that are controlled and mainly financed by government, and corporations and quasi-corporations that are controlled by government. All other enterprises are classified to the private sector.

##### Reference week

The week preceding the week in which the interview was conducted.

##### Status in employment

Employed people classified by whether they were employees, employers, own account workers, or contributing family workers.

##### Time away from work (partner)

Time away from work taken by owner managers following the birth of a child. Time away from work will generally be unpaid.

##### Unemployed

People aged 15 years and over who were not employed during the reference week, and:

• had actively looked for full-time or part-time work at any time in the four weeks up to the end of the reference week and were available for work in the reference week; or
• were waiting to start a new job within four weeks from the end of the reference week, and could have started in the reference week if the job had been available then.
##### Unincorporated enterprise

A business entity in which the owner and the business are legally inseparable, so that the owner is liable for any business debt that are incurred.

##### Unpaid maternity leave

Unpaid maternity leave is leave without pay specifically designed so that women can take time off work to care for their child in their first year, without having to permanently leave their job. This period of time away from work does not count as a break in service, as it would for some other types of unpaid leave such as 'leave without pay'.

##### Usual weekly hours of work

Usual weekly hours of work refers to a typical period rather than to a specified reference period. The concept of usual hours applies both to people at work and to people temporarily absent from work, and is defined as the hours worked during a typical week. The time includes all regular paid and unpaid overtime.

##### Usual hours worked

The number of hours usually worked.

##### With paid leave entitlements

The entitlement of employees (excluding OMIEs) to either paid holiday leave or paid sick leave (or both) in their current job. People employed in their own business or who were contributing family workers were not asked questions about paid leave entitlements.

##### Without paid leave entitlements

Employees (excluding OMIEs) who were not entitled to paid holiday leave and paid sick leave, or did not know whether they were entitled to paid sick leave and paid holiday leave in their current job.

##### Women with children less than two years old

The birth mother of a child living in the same household who was under the age of two years at the date of interview. If the birth mother has more than one child under the age of two years, data relates to the pregnancy and work arrangements for the most recent birth.

##### Worked in a job while pregnant

Women who worked in a job or business for some or all of their pregnancy. This includes women who took paid or unpaid leave and women who had no leave entitlements for the birth depending on their individual employment status.

## Abbreviations

### Show all

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# Math Help - Directly proportional to the square

1. ## Directly proportional to the square

What does this mean (question in the title). I need this to work out a question

2. Originally Posted by Mukilab
What does this mean (question in the title). I need this to work out a question
if y is directly propotional to the square of x then

y=kx^2 where k is a constant

3. Originally Posted by jiboom
if y is directly propotional to the square of x then

y=kx^2 where k is a constant
What do you mean by constant.

4. A constant is a fixed number. Like 2, or -56, or $\pi$. In this case, the expression $y=kx^2$ where $k$ is constant, it is asking you to consider only one value for $k$ for all values of $x$.

5. Ok, I got that before... sorry I should have updated the thread.


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# Force Required to move a 63.5 kg mass. (1 Viewer)

### Users Who Are Viewing This Thread (Users: 0, Guests: 1)

1. The problem statement, all variables and given/known data
What is the smallest possible mass needed to move a 63.5 kg mass? To find this, I need to know the force need to move the known mass, time the unknown mass stays in contact with the known mass, and velocity at which the unknown mass is traveling. I believe I can find out the rest of this information if I know the average force required to move a 63.5 kg mass.

2. Relevant equations
F = ma, I = Fav$$\Delta$$t = $$\Delta$$p, p = mv

3. The attempt at a solution
What I have done so far....nothing really. I need this information to move forward, and I don't know of a way to calculate it with the information available.

#### mgb_phys

Homework Helper
Whats the rest of the question?
From what you have you can't really get anywhere

I'm not sure if there is a rest of the question. We were just assigned to perform an experiment. We decided on seeing how little a mass we need to move a person who is 63.5 kg and I've been trying to use physics to develop a supported hypothesis before we started. That is all the information I could think of.

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Which of the following is corr...

# Which of the following is correct order of packing efficiency?

### Ab Padhai karo bina ads ke

Khareedo DN Pro and dekho sari videos bina kisi ad ki rukaavat ke!

Updated On: 27-06-2022

Text Solution

HCP=F C C gt B C C gt SCSC gt B C C gt HCP=F CC B C Cgt SC gt HCP lt F C CF C C=HCP gt SC gt B C C

Solution : packing HCP 74%
 efficiency=FCC 74%
 Sc=52%
 BCC=68%

Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

Transcript

Vivan the question which of the following is correct order of packing efficiency and we have been given for option for comparison between http and https basically it close close packing basically is packing efficiency that is denoted by 74% similarly Na movie on to the next one that is fcc which is centred cubic structure of

packing efficiency of 74% and next one is BCC that this board centre body centred cubic structure packing efficiency of and now next one simple cubic structure of simple cubic structure has a packing efficiency http and https have same packing efficiency at T = 2 STC which is greater than that is BCC this both are greater than and greater than that

is simple to so this is the correct order of parking at least for now let's have a look at the option that are option number one is correct match that is equal or greater and greater than the correct answer is option number 1st can't be greatest this is from DC also can't be created this is also the next is fcc hcp and can't be greater than vcc to this is also wrong answer is option number 1


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# Algebra of matrices

## Algebra of Matrices

Algebra of matrices is a basic things to work with matrices. Algebra of matrices contains addition, subtraction, scalar multiplication and multiplication of matrices. Let us understand algebra of matrices in details with examples.

## Equal Matrices.

Two matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ are called equal matrices if

• Number of rows of $A=$ Number of rows of $B$.

• Number of columns of $A=$ Number of columns of $B$.

• Corresponding entries are equal. i.e. $a_{ij}=b_{ij}$ for all $i,j$.

### Examples of Equal Matrices and not Equal Matrices.

Let $A=\begin{bmatrix} 2&3&0\\ 5&1&2\end{bmatrix}$, $B=\begin{bmatrix}1&0 \\-1&\dfrac{1}{\sqrt{2}}\end{bmatrix}$
and $C=\begin{bmatrix}\sin \pi/2 &\log 1 \\\cos\pi&\sin\pi/4\end{bmatrix}$. Then

• Matrices $A$ and $B$ are not equal matrices because number of columns of $A\ne$ number of columns of $B$.

• Matrices $A$ and $C$ are not equal matrices because number of columns of $A\ne$ number of columns of $C$.

• Matrices $B$ and $C$ are equal matrices because number of rows and columns of $B$ are equal to number of rows and columns of $C$ respectively, and

 \begin{aligned} c_{11}& =\sin\pi/2=1=b_{11},\\ c_{12}&=\log 1=0=b_{12},\\ c_{21}&=\cos\pi=-1=b_{21},\\ c_{22}&=\sin\pi/4=\dfrac{1}{\sqrt{2}}=b_{22} \end{aligned}

Two matrices can be added if and only if they have the same number of rows and same number of columns. That is the order of both the matrices are same.

In order to find addition of two matrices, you just add the corresponding entries of the matrices.
The resultant matrix of addition of two matrices is also of the same order.

Let $A=[a_{ij}]_{m\times n}$ and $B=[b_{ij}]_{m\times n}$ be two matrices then the addition of $A$ and $B$ is defined as

 \begin{alignat*}{2} A+B &=\left[\begin{matrix} a_{11}&a_{12}&\ldots&a_{1j}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{1j}&\ldots&a_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{i1}&a_{i2}&\ldots&a_{ij}&\ldots&a_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{m1}&a_{m2}&\ldots&a_{mj}&\ldots&a_{mn}\\ \end{matrix}\right]_{m\times n}+\;\;\;\;\; \left[ \begin{matrix} b_{11}&b_{12}&\ldots&b_{1j}&\ldots&b_{1n}\\ b_{21}&b_{22}&\ldots&b_{1j}&\ldots&b_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ b_{i1}&b_{i2}&\ldots&b_{ij}&\ldots&b_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ b_{m1}&b_{m2}&\ldots&b_{mj}&\ldots&b_{mn}\\ \end{matrix}\right]_{m\times n}\\ &\\ &=\left[\begin{matrix} a_{11}+b_{11}&a_{12}+b_{12}&\ldots&a_{1j}+b_{1j}&\ldots&a_{1n}+b_{1n}\\ a_{21}+b_{21}&a_{22}+b_{22}&\ldots&a_{2j}+b_{2j}&\ldots&a_{2n}+b_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{i1}+b_{i1}&a_{i2}+b_{i2}&\ldots&a_{ij}+b_{ij}&\ldots&a_{in}+b_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{m1}+b_{m1}&a_{m2}+b_{m2}&\ldots&a_{mj}+b_{mj}&\ldots&a_{mn}+b_{mn}\\ \end{matrix}\right]_{m\times n} \end{alignat*}
That is

 \begin{aligned} A+B&=[a_{ij}]_{m\times n}+[b_{ij}]_{m\times n}\\ &=[a_{ij}+b_{ij}]_{m\times n}. \end{aligned}

### Examples of Addition of two Matrices.

Let us understand addition of two matrices with following examples.

1. Let $A=\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}$ and $B=\begin{bmatrix}5&1&-6\\2&3&-2\end{bmatrix}$. Then

 \begin{alignat*}{2} A+B&=\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}+\begin{bmatrix}5&1&-6\\2&3&-2\end{bmatrix}\\ &\\ &=\begin{bmatrix}2+5&3+1&6+(-6)\\0+2&(-1)+3&2+(-2)\end{bmatrix}\\ &\\ &=\begin{bmatrix}7 & 4& 0\\2& 2&0\end{bmatrix} \end{alignat*}

1. Let $A=\begin{bmatrix}2&1\\-5&3\\1&7\end{bmatrix}$ and $B=\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}$. Then

 \begin{aligned} A+B&=\begin{bmatrix}2&1\\-5&3\\1&7\end{bmatrix}+\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}\\ &\\ &=\begin{bmatrix}2+3&1+5\\-5+(-2)&3+6\\1+4&7+1\end{bmatrix}\\ &\\ &=\begin{bmatrix}5 & 6\\-7&9\\5&8\end{bmatrix} \end{aligned}

## Subtraction of two Matrices.

Two matrices can be subtract if and only if they have the same number of rows and same number of columns. That is the order of both the matrices are same.

In order to find subtraction of two matrices, you just subtract the corresponding entries of the matrices.
The resultant matrix of subtraction of two matrices is also of the same order.

Let $A=[a_{ij}]_{m\times n}$ and $B=[b_{ij}]_{m\times n}$ be two matrices then the subtraction of $A$ and $B$ is defined as

 \begin{aligned} A-B&=\left[\begin{matrix} a_{11}&a_{12}&\ldots&a_{1j}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{1j}&\ldots&a_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{i1}&a_{i2}&\ldots&a_{ij}&\ldots&a_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{m1}&a_{m2}&\ldots&a_{mj}&\ldots&a_{mn}\\ \end{matrix}\right]_{m\times n}-\;\;\;\;\; \left[\begin{matrix} b_{11}&b_{12}&\ldots&b_{1j}&\ldots&b_{1n}\\ b_{21}&b_{22}&\ldots&b_{1j}&\ldots&b_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ b_{i1}&b_{i2}&\ldots&b_{ij}&\ldots&b_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ b_{m1}&b_{m2}&\ldots&b_{mj}&\ldots&b_{mn}\\ \end{matrix}\right]_{m\times n}\\ &\\ &=\left[\begin{matrix} a_{11}-b_{11}&a_{12}-b_{12}&\ldots&a_{1j}-b_{1j}&\ldots&a_{1n}-b_{1n}\\ a_{21}-b_{21}&a_{22}-b_{22}&\ldots&a_{2j}-b_{2j}&\ldots&a_{2n}-b_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{i1}-b_{i1}&a_{i2}-b_{i2}&\ldots&a_{ij}-b_{ij}&\ldots&a_{in}-b_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{m1}-b_{m1}&a_{m2}-b_{m2}&\ldots&a_{mj}-b_{mj}&\ldots&a_{mn}-b_{mn}\\ \end{matrix}\right]_{m\times n} \end{aligned}

That is

 \begin{aligned} A-B&=[a_{ij}]_{m\times n}-[b_{ij}]_{m\times n}\\ &=[a_{ij}-b_{ij}]_{m\times n} \end{aligned}

### Examples of Subtraction of two Matrices.

Let us understand subtraction of two matrices with following examples.

1. Let $A=\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}$ and $B=\begin{bmatrix}5&1&-6\\2&3&-2\end{bmatrix}$. Then

 \begin{aligned} A-B&=\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}-\begin{bmatrix}5&1&-6\\2&3&-2\end{bmatrix}\\ &\\ &=\begin{bmatrix}2-5&3-1&6-(-6)\\0-2&(-1)-3&2-(-2)\end{bmatrix}\\ &\\ &=\begin{bmatrix}-3 & 2& 12\\-2& -4&4\end{bmatrix} \end{aligned}

1. Let $A=\begin{bmatrix}2&1\\-5&3\\1&7\end{bmatrix}$ and $B=\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}$. Then

 \begin{aligned} A-B&=\begin{bmatrix}2&1\\-5&3\\1&7\end{bmatrix}-\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}\\ &\\ &=\begin{bmatrix}2-3&1-5\\-5-(-2)&3-6\\1-4&7-1\end{bmatrix}\\ &\\ &=\begin{bmatrix}-1 & -4\\-3&-3\\-3&6\end{bmatrix} \end{aligned}

## Multiplication of a Matrix by a Scalar

Let $A=[a_{ij}]_{m\times n}$ be a matrix and $k\in\mathbb{R} (or\; \mathbb{C})$ be a scalar. Then to find the multiplication of $A$ by a scalar $k$, multiply every entry of the matrix by $k$.

 \begin{aligned} kA&=k\;\left[\begin{matrix} a_{11}&a_{12}&\ldots&a_{1j}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{1j}&\ldots&a_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{i1}&a_{i2}&\ldots&a_{ij}&\ldots&a_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{m1}&a_{m2}&\ldots&a_{mj}&\ldots&a_{mn}\\ \end{matrix}\right]_{m\times n}\\ &=\left[\begin{matrix} k\cdot a_{11}&k\cdot a_{12}&\ldots&k\cdot a_{1j}&\ldots&k\cdot a_{1n}\\ k\cdot a_{21}&k\cdot a_{22}&\ldots&k\cdot a_{1j}&\ldots&k\cdot a_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ k\cdot a_{i1}&k\cdot a_{i2}&\ldots&k\cdot a_{ij}&\ldots&k\cdot a_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ k\cdot a_{m1}&k\cdot a_{m2}&\ldots&k\cdot a_{mj}&\ldots&k\cdot a_{mn}\\ \end{matrix}\right]_{m\times n} \end{aligned}

That is

 \begin{aligned} kA&=k[a_{ij}]\\ &=[k\cdot a_{ij}] \end{aligned}

### Examples of Multiplication of a Matrix by a Scalar

Let us understand multiplication of a matrix by a scalar with following examples.

1. Let $A=\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}$. Then

 \begin{aligned} 3A&=3\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}\\ &\\ &=\begin{bmatrix}3\cdot 2&3\cdot3&3\cdot 6\\3\cdot 0&3\cdot (-1)&3\cdot 2\end{bmatrix}\\ &\\ &=\begin{bmatrix}6 & 9& 18\\0& -3&6\end{bmatrix} \end{aligned}

1. Let $B=\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}$. Then

 \begin{aligned} (-5)B&=(-5)\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}\\ &\\ &=\begin{bmatrix}(-5)\cdot3&(-5)\cdot5\\(-5)\cdot(-2)&(-5)\cdot 6\\(-5)\cdot 4&(-5)\cdot 1\end{bmatrix}\\ &\\ &=\begin{bmatrix}-15 & -25\\10& -30\\-20&-5\end{bmatrix} \end{aligned}

By multiplication of a matrix by a scalar does not change the order of the matrix.

## Multiplication of two Matrices

Multiplication of two matrices can be possible if and only if the number of columns of first matrix is equal to the number of rows of the second matrix. In other words, if $A=[a_{ij}]_{m\times n}$ and $B=[b_{ij}]_{p\times q}$ be two matrices, then the multiplication $AB$ of matrices $A$ and $B$ is defined iff $n=p$.

Let $A=[a_{ij}]_{m\times n}$ and $B=[b_{ij}]_{n\times q}$ be two matrices then the multiplication $AB$ of $A$ and $B$ is defined as

$$AB=[c_{ij}]_{m\times q}$$

where $c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{in}b_{nj}=\sum\limits_{k=1}^n a_{ik}b_{kj}$ for all $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$.

### Examples of Multiplication of Matrices

Let us understand multiplication of matrices with following examples.

1. If $A=\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}$ and $B=\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}$, then

 \begin{aligned} AB&=\begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}_{2\times 3} \cdot \begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}_{3\times 2}\\ &\\ &=\begin{bmatrix} 2\cdot 3+3\cdot(-2)+6\cdot 4 & 2\cdot 5+3\cdot 6+6 \cdot 1\\ 0\cdot 3+(-1)\cdot(-2)+2\cdot 4 & 0\cdot 5+(-1)\cdot 6+2 \cdot 1 \end{bmatrix}\\ &\\ &=\begin{bmatrix} 24 & 24\\ 9& -4 \end{bmatrix}_{2\times 2} \end{aligned}

 \begin{aligned} BA&=\begin{bmatrix}3&5\\-2&6\\4&1\end{bmatrix}_{3\times 2}\cdot \begin{bmatrix}2&3&6\\0&-1&2\end{bmatrix}_{2\times 3}\\ &\\ &=\begin{bmatrix} 3\cdot 2+5\cdot 0 &3\cdot 3+5\cdot (-1)&3\cdot 6+5\cdot 2\\ -2\cdot 2+6\cdot 0&-2\cdot 3+6\cdot (-1)&-2\cdot 6+6\cdot 2\\ 4\cdot 2+1\cdot 0 &4\cdot 3+1\cdot (-1)&4\cdot 6+1\cdot 2\\ \end{bmatrix}\\ &\\ &=\begin{bmatrix} 6 & 4 & 28\\ -4 & -12&0\\ 8&11&26 \end{bmatrix}_{3\times 3} \end{aligned}

Order of matrix $A$ is $2\times 3$ and order of $B$ is $3\times 2$.

The number of columns of the matrix $A$=The number of rows of the matrix $B=3$

The order of the resultant matrix $AB$ is $2\times 2$.

The order of the resultant matrix $AB$ is (The number of rows of the first matrix) $\times$ (The number of columns of the second matrix).

$BA$ is also defined and it is of order $3\times 3$.

Here $AB\ne BA$. That is matrix multiplication is not commutative.

1. If $X=\begin{bmatrix}1&3&7\\0&1&2\\2&4&1\end{bmatrix}$ and $Y=\begin{bmatrix}2&5\\1&6\\4&-3\end{bmatrix}$, then

 \begin{aligned} XY&=\begin{bmatrix}1&3&7\\0&1&2\\2&4&1\end{bmatrix}_{3\times 3} \cdot \begin{bmatrix}2&5\\1&6\\4&-3\end{bmatrix}_{3\times 2}\\ &\\ &=\begin{bmatrix} 1\cdot 2+3\cdot 1+7\cdot 4 & 1\cdot 5+3\cdot 6+7 \cdot (-3)\\ 0\cdot 2+1\cdot 1+2\cdot 4 & 0\cdot 5+1\cdot 6+2 \cdot (-3)\\ 2\cdot 2+4\cdot 1+1\cdot 4 & 2\cdot 5+4\cdot 6+1 \cdot (-3)\\ \end{bmatrix}\\ &\\ &=\begin{bmatrix} 33 & 2\\ 9& 0\\ 12&31 \end{bmatrix}_{3\times 2} \end{aligned}

Order of matrix $X$ is $3\times 3$ and order of $Y$ is $3\times 2$.

The number of columns of the matrix $X$=The number of rows of the matrix $Y=3$

The order of the resultant matrix $XY$ is $3\times 2$.

The order of the resultant matrix $XY$ is (The number of rows of the first matrix) $\times$ (The number of columns of the second matrix).

$YX$ is not defined because number of columns of matrix $Y$ is not equal to number of rows of matrix $X$.

Milan holds Ph.D. Degree in Mathematics, his area of research is Banach Algebra and Harmonic Analysis, a branch of pure mathematics. He has cleared NET and GPSC (Gujarat Public Service Commission) with first rank. Milan enjoys teaching and problem solving in Mathematics. He has good hand on Real Analysis, Complex Analysis, Linear Algebra, Abstract Algebra and Differential Equation. Apart from teaching Milan spends his time to guide the students for various competitive examinations in Mathematics like JAM, CSIR NET-JRF, GSET, GATE, NBHM, TIFR etc. More than 29 students had cleared such type of examinations under his guidance. He has teaching experience of more than 13 years in the field of Mathematics at graduate level in Engineering and Science College.