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How do you simplify i^21 + i^30?
1 Answer
Nov 13, 2015
$i - 1$
Explanation:
Observe that the powers of $i$ are cyclic:
${i}^{0} = 1$
${i}^{1} = i$
${i}^{2} = - 1$
${i}^{3} = - i$
${i}^{4} = 1$
$\ldots$
So, to find high powers of $i$, we can calculate the modulus $4$ Since $21 = 5 \cdot 4 + 1$, and $30 = 7 \cdot 4 + 2$, we have that ${i}^{21} = {i}^{1} = i$, and ${i}^{30} = {i}^{2} = - 1$. So,
${i}^{21} + {i}^{30} = i - 1$. | {} |
# Does either Quantum Field Theory or the Standard Model of Particle Physics predict the maximum number of particles or fields that can exist?
Perhaps since the Standard Model relies heavily on the symmetries of chosen groups do either or both theories demand a maximum number of particles or fields that are allowed to exist? My guess is that someone has asked this but I cannot find it in the "Similar Questions" section or the "Questions that may already have your answer" section.
• Nope. In fact a huge deal of modern theoretical physics is to randomly assume another bunch of particles and see what that would predict. – image Sep 24 '17 at 10:59
• I fear you have fallen into the trap of considering the SM as a perfect theory that came out of a received QFT text and "predicts" everything. It is a minimal, remarkable!, pragmatic fit of experimental facts, essentially a vehicle for systematic abuse of a terminology expressly invented for that very purpose. There is a standard theory involving the gauge groups (not a "model", but a solid theory, like the "theory of relativity"); however the representations and extensions of it are open-ended. So, e.g., new pieces like sterile neutrinos are neither predicted nor excluded by it. – Cosmas Zachos Sep 25 '17 at 15:55
• If experiment confirms them, (sterile neutrinos, singlet Higgses, BSM entities) so be it, but leave QFT predictions out of it: trying to model-build out of hyperbolic totemic QFT properties gives both QFT and model-building a bad name. – Cosmas Zachos Sep 25 '17 at 15:58
• A framework is incomplete, by definition. Nobody is trying to break anything--one looks for anything new that extends knowledge, like anywhere else in physics. If you actually read the books, instead of irresponsible science reporting you might appreciate the actual beauty and the gaps of the framework. – Cosmas Zachos Sep 26 '17 at 13:44
• Short answer, No. – Rexcirus Sep 26 '17 at 19:20
There is a limit on the number of flavors in Quantum Chromodynamics, behind that limit Color Confinement can no longer exist. The Beta-function that describes the interaction strength at different scales (at one loop) is:
$$\beta(g) = \frac{g^3}{16 \pi^2} \left( - \frac{11}{3} N_c + \frac{2}{3} N_f \right)$$
This is negative for $N_f = 6$ quark flavors and $N_c = 3$ colors which leads to the confinement phenomena. A positive value would mean that quarks must interact more at very small distances, which contradicts confinement.
Another interesting aspect of this is the unitarity of the CKM matrix. If there are more quark families than what is presently known, the measurements should eventually show violations of unitarity of this 3x3 matrix.
• The constraint from the $\beta$ function boils down to $N_f<11N_c/2$. Since $N_c$ has no apriori upper bound, neither does $N_f$. – user154997 Sep 23 '17 at 22:00
• Furthermore, I don't get your argument about unitary CKM: with a 4th family we would have a 4x4 matrix and we can require it to be unitary. – user154997 Sep 23 '17 at 22:07
• I might be wrong, but isn't $N_c$ fixed by our observations of hadron production? If $N_c$ would be bigger - wouldn't we see larger scattering rates in existing processes? – Darkseid Sep 23 '17 at 22:19
• @LucJ.Bourhis : the number of colors $N_{c}$ is fixed experimentally because of comparison the experimental neutral pion decay width with the SM prediction based on the chiral anomaly. – Name YYY Sep 23 '17 at 22:49
• I know $N_c\ne 3$ is falsified experimentally but we have a direct observational constraint on $N_f$ through $Z$ pole physics: by measuring the widths and computing $\Gamma(\text{invisible})=\Gamma_Z-3\Gamma(\text{leptons})-\Gamma(\text{hadron})=499.0\pm 1.5 \text{MeV}$, the number $N_\nu$ of neutrinos much lighter than the mass of $Z$ can then be fitted and the result is $N_\nu=2.992\pm0.007$. I.e. $N_f=3$. So as @ACuriousMind wrote, if we allow observations for one side of the coin, the OP game is over. – user154997 Sep 24 '17 at 5:02
It depends on what kinds of "Quantum Field Theory or the Standard Model of Particle Physics" you are focusing on.
Other than the familiar quarks+leptons+ gauge bosons+ Higgs particle sectors in the Standard Model, there could be other sectors that are topological, that could have extended objects like strings (Cosmic strings) or different kinds of topological defects, or anyon particles, or even anyonic strings (that are neither bosonic nor fermionic), for examples see Ref. 1 and Ref. 2. You can describe some of these topological objects by fields of higher form gauge fields, etc.
In general, you can imagine there are other Topological sectors that somehow couple to the underlying standard model in some way. And there is no limit but many many number of anyon particles that you can construct, and you can see for example Ref. 3.
It just if the mother Nature uses these topological sectors as fundamental as the known Standard Model, then the Nature must weave her puzzle in a non-contrived but elegant way. | {} |
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# Extremely ($90$%) biased coins. What information can we derive/assume based on results of only $10$ coin flips?
Let's suppose there are $2$ heavily biased coins such that coin A has a bias of coming up $90$% heads and coin B has a bias of coming up $90$% tails. Both coins are placed in a bag and one is randomly chosen in a way that either coin is equally likely to be chosen and cannot be identified as either A or B. The coin is tossed fairly $10$ times and it is observed that $10$ heads came up.
Then if someone were to ask that if that same coin is tossed $10$ more times, what is the number of heads expected, can we assume at the point that it is coin A or can we not assume that and just say we would expect $5$ heads on average? That is, since the $10$ "given" heads is not a $100$% definitive indication of what coin we have, must we say that it could be either coin A or coin B or can we "bias" our answer towards coin A and say that we expect something like $9$ out of $10$ heads instead of just $5$? However, if we do that and we "guess" wrongly, (that is it was actually coin B), then our estimate will likely be WAY off! Should we assume that it is more likely that coin A was chosen than coin B and thus our answer will be affected? If so, then how do we compute the number of expected heads?
The "problem" is that in the shortrun, even a heavy bias may not "pan out" to the expected outcome. For example, maybe this experiment was tried millions of times and this short (relative to millions) observed outcome just happened to be $10$ heads in a row but it was actually coin B that did this.
• You are asking if you should input a hypothesis test into computations for the expected value. This is not really it. You can compute $P(10H | A) = 0.9^{10}$ and $P(10H | B) = 0.1^{10}$ but you want $P(A | 10H)$ or $P(B | 10H) = 1 - P(A | 10H)$. – AlexR Oct 16 '14 at 22:14
• I am asking how do we accurately compute the number of expected heads on the 2nd group of $10$ coin tosses given that the first group of $10$ were all heads but the coin selected in unknown. – David Oct 16 '14 at 22:20
• That's the same as $$E(X | 10H) = \underbrace{E(X | 10H \cap B)}_{=E(X|B)=1} \cdot P(B | 10H) + \underbrace{E(X | 10H \cap A)}_{=E(X|A)=9} \cdot P(A | 10H)\\=1+8 P(A|10H)$$ or am I missing something? – AlexR Oct 16 '14 at 22:22
• Oh my. That means that one SHOULD assume that based on the results that it IS coin A and thus report a high number of expected heads on the next $10$ coin flips. So generalizing, if there is ANY bias in the $2$ coins then a similar thing should be done, even if the coins are say $51$% and $49$% biased towards heads. – David Oct 16 '14 at 22:40
• Not quite. "should" is always a question of taste. If you want to incorporate the event $10H$ in your calculations, you'll be more or less forced to or just forget that it all happened and assume $A$ and $B$ equally likely. What changes is that $P(A|10H) \approx 0.5$ the slighter the bias is. And thus $E(X|10H) \approx 1 + 8\cdot 0.5 = 5$ as expected. – AlexR Oct 16 '14 at 22:43
\begin{align} \text{From the given:} &\quad C\, (\text{coin identity}), E\, (\text{evidence}) \\ \mathsf P(E=10H\mid C=A) & = 0.9^{10} \\ & = 0.3486784401 \\[1ex] \mathsf P(E=10H\mid C=B) & = 0.1^{10} \\ & = 0.0000000001 \\[2ex] \text{We can find:} \\[1ex] \mathsf P(C=A\mid E=10H) & = \frac{\mathsf P(E=10H\mid C=A)\mathsf P(C=A)}{\mathsf P(E=10H\mid A)\mathsf P(C=A)+\mathsf P(E=10H\mid B)\mathsf P(C=B)} \\[1ex] & = \frac{0.9^{10}0.5}{0.9^{10}0.5+0.1^{10}0.5} \\[1ex] & \approx 0.9{\small 9999999971320280100300850204388...} \end{align}
So we can say that, given the evidence, it's quite highly probable that the coin is A. We cannot assume that it is certainly so, but we can be fairly confident about it.
However, we then use the Law of Iterated Expectation to find the expected number of heads on subsequent tosses.
\begin{align}\mathsf E(X\mid E=10H) & =\mathsf E_C[\mathsf E[X\mid E=10H, C]] \\[1ex] & = \mathsf E[X\mid C=A] \mathsf P(C=A\mid E=10H) + \mathsf E[X\mid C=B]\mathsf P(C=B\mid E=10H) \\[2ex] & = \frac{0.9^{10}\mathsf E[X\mid C=A]+ 0.1^{10}\mathsf E[X\mid C=B]}{0.9^{10}+0.1^{10}} \\[2ex] & = \frac{0.9^{10}\cdot 10\cdot 0.9+ 0.1^{10}\cdot 10\cdot 0.1}{0.9^{10}+0.1^{10}} \\[2ex] & = \frac{0.9^{11}\cdot 10+ 0.1^{11}\cdot 10}{0.9^{10}+0.1^{10}} \\ & \approx 8.9{\small 999999977056224080240680163511...} \end{align}
To use the information $10H$ (abbr. $H$) we compute $$E(X|H) = 1+8P(A|H) = 1 + 8 \underbrace{P(H|A)}_{=0.9^{10}} \frac{P(A)}{P(H)} = 1 + 8 \cdot 0.9^{10} \cdot 0.5 \cdot \frac1{0.9^{10} + 0.1^{10}}\\ =4.999999998852811204012034008175536171278306641914362905882931\ldots$$
You can see it's extremely close to $5$, I don't know why the expected value comes out $<5$, but maybe I got a term wrong there.
I have used Bayes' theorem there and $P(H) = P(H|A) + P(H|B) = 0.9^{10} + 0.1^{10}$
$$P(\text{10 heads initially}|A)=0.9^{10}$$ $$P(\text{10 heads initially}|B)=0.1^{10}$$ $$P(A|\text{10 heads initially})=\frac{\frac12\times 0.9^{10}}{\frac12\times 0.9^{10}+\frac12\times 0.1^{10}}=\frac{9^{10}}{9^{10}+1} \approx 0.99999999971$$ $$P(B|\text{10 heads initially})=\frac{1}{9^{10}+1} \approx 0.00000000029$$ $$E[\text{number of heads in second 10}|A]=10\times 0.9 = 9$$ $$E[\text{number of heads in second 10}|B]=10\times 0.1 = 1$$ $$E[\text{number of heads in second 10}|\text{10 heads initially}] = 9 \times \frac{9^{10}}{9^{10}+1} + 1 \times \frac{1}{9^{10}+1}$$ $$= \frac{9^{11}+1}{9^{10}+1} \approx 8.9999999977$$ so given the $10$ heads initially, it is much more likely that the coin chosen was $A$, and the expected number of heads in the second $10$ is almost, but not quite, the same as the expectation with coin $A$. | {} |
# balancing redox reactions half reaction method
In the above equation, there are $$14 \: \ce{H}$$, $$6 \: \ce{Fe}$$, $$2 \: \ce{Cr}$$, and $$7 \: \ce{O}$$ on both sides. not have to do anything. The OH- ions, Then, on that side of the equation which contains both (OH. This is called the half-reaction method of balancing redox reactions, or the ion-electron method. the number of electrons lost equals the number of electrons gained we do Balancing it directly in basic seems fairly easy: Fe + 3OH¯ ---> Fe(OH) 3 + 3e¯ And yet another comment: there is an old-school method of balancing in basic solution, one that the ChemTeam learned in high school, lo these many years ago. 2. reducing to the smallest whole number by cancelling species which on both Balancing Redox Reactions. KBr (aq) • First, is this even a REDOX reaction? dichromate ethanol C2H4O Sixth, equalize the number of electrons lost with the number of electrons Is there a species that is being reduced and a species that is being oxidized? You then use some arrows to show your half-reactions. Balancing Redox Equations: Half-Reaction Method. Half reactions are often used as a method of balancing redox reactions. Oxidation number method is based on the difference in oxidation number of oxidizing agentand the reducing agent. In an acidic medium, add hydrogen ions to balance. Balancing Redox Equations via the Half-Equation Method is an integral part of understanding redox reactions. Equation balancing & stoichiometry lectures » half reaction method » Equation balancing and stoichiometry calculator. and the ethanol/acetaldehyde as the oxidation half-reaction. For example, this half-reaction: Fe ---> Fe(OH) 3 might show up. Each half-reaction is then balanced individually, and then the half-reactions are added back together to form a new, balanced redox equation. Balancing redox reactions is slightly more complex than balancing standard reactions, but still follows a relatively simple set of rules. To indicate the fact that the reaction takes place in a basic solution, Separate the equation into half-reactions (i.e., oxidation half and reduction half). Redox reactions are commonly run in acidic solution, in which case the reaction equations often include H 2 O(l) and H + (aq). And that is wrong because there is an electron in the final answer. For the reduction half-reaction above, seven H 2 O molecules will be added to the product side, The electrons must cancel. If necessary, cancel out $$\ce{H_2O}$$ or $$\ce{H^+}$$ that appear on both sides. NO → NO 3-6. Simplify the equation by subtracting out water molecules, to obtain the use hydrogen ions (H. The fifth step involves the balancing charges. Cr 2O 7 2 - → Cr3+ 5. To do this one must One major difference is the necessity to know the half-reactions of the involved reactants; a half-reaction table is very useful for this. SO 4 2- → SO 2 7. By adding one electron to the product side of the oxidation half-reaction, there is a $$2+$$ total charge on both sides. Divide the complete equation into two half reactions, one representing oxidation and the other reduction. Balance the O by adding water as needed. Each electron has a charge equal to (-1). by reduction with the number of electrons produced by oxidation. This method of balancing redox reactions is called the half reaction method. For the reduction half-reaction, the electrons will be added to the reactant side. Chemists have developed an alternative method (in addition to the oxidation number method) that is called the ion-electron (half-reaction) method. Step 4: Balance oxygen atoms by adding water molecules to the appropriate side of the equation. To determine the number The half-reaction method for balancing redox equations provides a systematic approach. H 2O 2 + Cr 2O 7 2- → O 2 + Cr 3+ 9. (H, The fourth step involves balancing the hydrogen atoms. records this change. Balance the atoms in each half reaction separately according to the following steps: 1. Balancing Redox Reactions: The Half-Reaction Method Balanced chemical equations accurately describe the quantities of reactants and products in chemical reactions. Balance the unbalanced redox reaction without any complications by using this online balancing redox reactions calculator. final, balanced equation. Now the hydrogen atoms need to be balanced. Legal. Here are the steps for balancing in an acid solution (adding H+ and H2O). Balancing Redox Reactions via the Half-Reaction Method Redox reactions that take place in aqueous media often involve water, hydronium ions (or protons), and hydroxide ions as reactants or products. For more information contact us at [email protected] or check out our status page at https://status.libretexts.org. For reactions occurring in acidic medium, add H2O to balance O atoms and H+ to balance H atoms. To do this, add water This ion is a powerful oxidizing agent which oxidizes many substances this guideline, the oxidation half reaction must be multiplied by "3" to The sixth step involves multiplying each half-reaction by the smallest In general, the half-reactions are first balanced by atoms separately. Redox equations are often so complex that fiddling with coefficients to balance chemical equations doesn’t always work well. The half-reaction method works better than the oxidation-number method when the substances in the reaction are in aqueous solution. Organic compounds, called alcohols, are readily oxidized by acidic solutions of dichromate ions. of dichromate ions. The following reaction, written in net ionic form, BALANCING REDOX REACTIONS. Balancing Redox Equations for Reactions in Acidic Conditions Using the Half-reaction Method. $6 \ce{Fe^{2+}} \left( aq \right) \rightarrow 6 \ce{Fe^{3+}} \left( aq \right) + 6 \ce{e^-}$. Worksheet # 5 Balancing Redox Reactions in Acid and Basic Solution Balance each half reaction in basic solution. Equation balancing & stoichiometry lectures » half reaction method » Equation balancing and stoichiometry calculator. We need the balanced equation to compare mole ratio in scenarios such as this redox reaction worked example.. Each of these half-reactions is balanced separately and then combined to give the balanced redox equation. To balance the equation, use Recall that a half-reaction is either the oxidation or reduction that occurs, treated separately. Another method for balancing redox reactions uses half-reactions. Let’s take a look at a simple reaction WITHOUT HYDROGEN OR OXYGEN to balance: K (s) + Br 2 (l) ! Zn(s) -----> Zn(OH)42- (aq) NO31- -----> NH3. MnO 2 → Mn 2O 3 Balance each redox reaction in acid solution using the half reaction method. The dichromate ions are reduced to $$\ce{Cr^{3-}}$$ ions. MnO 2 → Mn 2O 3 Balance each redox reaction in acid solution using the half reaction method. For the reduction half-reaction above, seven $$\ce{H_2O}$$ molecules will be added to the product side. CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon. Balancing Redox Equations for Reactions in Acidic Conditions Using the Half-reaction Method. This method of balancing redox reactions is called the half reaction method. Here are the steps for balancing redox reactions using the oxidation state method (also known as the half-equation method): Identify the pair of elements undergoing oxidation and reduction by checking oxidation states; Write two ionic half-equations (one of the oxidation, one for the reduction) \begin{align} &\text{Oxidation:} \: \ce{Fe^{2+}} \left( aq \right) \rightarrow \ce{Fe^{3+}} \left( aq \right) \\ &\text{Reduction:} \: \overset{+6}{\ce{Cr_2}} \ce{O_7^{2-}} \left( aq \right) \rightarrow \ce{Cr^{3+}} \left( aq \right) \end{align}. and hydrogen atoms. $\ce{Fe^{2+}} \left( aq \right) \rightarrow \ce{Fe^{3+}} \left( aq \right) + \ce{e^-}$. Balancing Redox Reactions via the Half-Reaction Method Redox reactions that take place in aqueous media often involve water, hydronium ions (or protons), and hydroxide ions as reactants or products. Draw an arrow connecting the reactant a… 4. listed in order to identify the species that are oxidized and reduced, Each half-reaction is balanced separately and then the equations are added together to give a balanced overall reaction. The equation is balanced. Since the following steps: The electrons must always be added to that side which has the greater Half-reaction method depends on the division of the redox reactions into oxidation half and reduction half. and the other a reduction half- reaction, by grouping appropriate species. The reduction half-reaction needs to be balanced with the chromium atoms, Step 4: Balance oxygen atoms by adding water molecules to the appropriate side of the equation. This is done by adding electrons First, separate the equation into two half-reactions: the oxidation portion, and the reduction portion. Basic functions of life such as photosynthesis and respiration are dependent upon the redox reaction. By following this guideline in the example Let's dissect an equation! (e-). (There are other ways of balancing redox reactions, but this is the only one that will be used in this text. The method that is used is called the ion-electron or "half-reaction" method. There are two ways of balancing redox reaction. H 2O 2 + Cr 2O 7 2- → O 2 + Cr 3+ 9. Step 6: Add the two half-reactions together. \begin{align} 6 \ce{Fe^{2+}} \left( aq \right) &\rightarrow 6 \ce{Fe^{3+}} \left( aq \right) + \cancel{ 6 \ce{e^-}} \\ \cancel{6 \ce{e^-}} + 14 \ce{H^+} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) &\rightarrow 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right) \\ \hline 14 \ce{H^+} \left( aq \right) + 6 \ce{Fe^{2+}} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) &\rightarrow 6 \ce{Fe^{3+}} \left( aq \right) + 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right) \end{align}. Example 1 -- Balancing Redox Reactions Which Occur in Acidic Solution. The seventh and last step involves adding the two half reactions and Another method for balancing redox reactions uses half-reactions. Using (You can in a half-reaction, but remember half-reactions do not occur alone, they occur in reduction-oxidation pairs.) In the ion-electron method, the unbalanced redox equation is converted to the ionic equation and then broken […] Another method for balancing redox reactions uses half-reactions. Have questions or comments? Here, you do all the electron balancing on one line. (You can in a half-reaction, but remember half-reactions do not occur alone, they occur in reduction-oxidation pairs.) $6 \ce{e^-} + 14 \ce{H^+} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) \rightarrow 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right)$. The Half-Reaction Method . is Determine the oxidation numbers first, if necessary. 22.10: Balancing Redox Reactions- Half-Reaction Method, [ "article:topic", "showtoc:no", "license:ccbync", "program:ck12" ], https://chem.libretexts.org/@app/auth/2/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FIntroductory_Chemistry%2FBook%253A_Introductory_Chemistry_(CK-12)%2F22%253A_Oxidation-Reduction_Reactions%2F22.10%253A_Balancing_Redox_Reactions-_Half-Reaction_Method, 22.9: Balancing Redox Reactions- Oxidation Number Change Method, 22.11: Half-Reaction Method in Basic Solution, Balancing Redox Equations: Half-Reaction Method, information contact us at [email protected], status page at https://status.libretexts.org. This method involves the following steps : 1. Add electrons to one side of the half reaction to balance … gained by multiplying by an appropriate small whole number. Step 5: Balance the charges by adding electrons to each half-reaction. Separate the reaction into two half-reactions, one for Zn and one for N in this case. Chemists have developed an alternative method (in addition to the oxidation number method) that is called the ion-electron (half-reaction) method. The reason for this will be seen in Chapter 14 “Oxidation and Reduction” , Section 14.3 “Applications of Redox Reactions… In the ion-electron method, the unbalanced redox equation is converted to the ionic equation and then broken […] Just enter the unbalanced chemical equation in this online Balancing Redox Reactions Calculator to balance the reaction using half reaction method. The oxidation states of each atom in each compound Hello, Slight road-bump while doing my … Pigments of these colors are often made with a dichromate salt (usually sodium or potassium dichromate). Fourth, balance any hydrogen atoms by using an (H+) for each hydrogen atom. 2) Here are the correct half-reactions: 4e¯ + 4H + … First, divide the equation into two halves; one will be an oxidation half-reaction An examination of the oxidation states, indicates that carbon is being The reduction First, divide the equation into two halves by grouping appropriate species. $\ce{Cr_2O_7^{2-}} \left( aq \right) \rightarrow 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right)$. BALANCING REDOX REACTIONS. The half-reaction method for balancing redox equations provides a systematic approach. Balancing redox reactions is slightly more complex than balancing standard reactions, but still follows a relatively simple set of rules. In this method, the overall reaction is broken down into its half-reactions. sides of the arrow. In this example, the oxidation half-reaction will be multiplied by six. This will be resolved by the balancing method. The reduction half-reaction needs to be balanced with the chromium atoms, Step 4: Balance oxygen atoms by adding water molecules to the appropriate side of the equation. The half-reaction method works better than the oxidation-number method when the substances in the reaction are in aqueous solution. In this method, the overall reaction is broken down into its half-reactions. Finally, the two half-reactions are added back together. half-reaction requires 6 e-, while the oxidation half-reaction produces note: the net charge on each side of the equation does not have to An unbalanced redox reaction can be balanced using this calculator. Balance any remaining substances by inspection. The following reaction, written in … To balance the charge, six electrons need to be added to the reactant side. Although these species are not oxidized or reduced, they do participate in chemical change in other ways (e.g., by providing the elements required to form oxyanions). When balancing redox reactions we have always - apart from all the rules pertaining to balancing chemical equations - additional information about electrons moving. Organic compounds, called alcohols, are readily oxidized by acidic solutions The Half-Reaction Method . Cr3+ + Electrons are included in the half-reactions. (I-) ions as shown below in net ionic form. above, only the, The third step involves balancing oxygen atoms. ion. chromium(III) acetaldehyde. respectively. Use this online half reaction method calculator to balance the redox reaction. It can be done via the following systematic steps. The half-reaction method of balancing redox equations is described. whole number that is required to equalize the number of electrons gained both equations by inspection. Recall that a half-reaction is either the oxidation or reduction that occurs, treated separately. Let's dissect an equation! For example, this half-reaction: Fe ---> Fe(OH) 3 might show up. The half-reaction method works better than the oxidation-number method when the substances in the reaction are in aqueous solution. The aqueous solution is typically either acidic or basic, so hydrogen ions or hydroxide ions are present. Step 3: Balance the atoms in the half-reactions other than hydrogen and oxygen. equation. Now equalize the electrons by multiplying everything in one or both equations by a coefficient. Worksheet # 5 Balancing Redox Reactions in Acid and Basic Solution Balance each half reaction in basic solution. Assign oxidation numbers 2. Example #4: Sometimes, the "fake acid" method can be skipped. They serve as the basis of stoichiometry by showing how atoms and mass are conserved during reactions. The picture below shows one of the two Thunder Dolphin amusement ride trains. This page will show you how to write balanced equations for such reactions even when you do not know whether the H 2 O(l) and H + (aq) are reactants or products. 2. 2) Here are the correct half-reactions: 4e¯ + 4H + … Balancing Redox Reactions: Redox equations are often so complex that fiddling with coefficients to balance chemical equations. and non-oxygen atoms only. For the reduction half-reaction above, seven H 2 O molecules will be added to the product side, Example 1 -- Balancing Redox Reactions Which Occur in Acidic Solution. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 8. For oxidation-reduction reactions in acidic conditions, after balancing the atoms and oxidation numbers, one will need to add H + ions to balance the hydrogen ions in the half reaction. under basic conditions. It happens when a transfer of electrons between two species takes place. Balance the atoms other than H and O in each half reaction individually. One major difference is the necessity to know the half-reactions of the involved reactants; a half-reaction … When presented with a REDOX reaction in this class, we will use the “half-reactions” method to balance the reaction. Check to make sure the main atoms, Zn and N are balanced. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 2 e-. 1. The net charge is $$24+$$ on both sides. of electrons required, find the net charge of each side the equation. Each half-reaction is then balanced individually, and then the half-reactions are added back together to form a new, balanced redox equation. If you have properly learned how to assign oxidation numbers (previous section), then you can balance redox equations using the oxidation number method. In other words, balance the non-hydrogen In the ion-electron method (also called the half-reaction method), the redox equation is separated into two half-equations - one for oxidation and one for reduction. First, separate the equation into two half-reactions: the oxidation portion, and the reduction portion. $\ce{Fe^{2+}} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) \rightarrow \ce{Fe^{3+}} \left( aq \right) + \ce{Cr^{3+}} \left( aq \right)$. And that is wrong because there is an electron in the final answer. Redox reactions are commonly run in acidic solution, in which case the reaction equations often include H 2 O(l) and H + (aq). Step 7: Check the balancing. oxidized, and chromium, is being reduced. How to Balance Redox Reactions by Half Reaction Method - Tutorial with Definition, Equations, Example Definition Redox Reaction is a chemical reaction in which oxidation and reduction occurs simultaneously and the substance which gains electrons is termed as oxidizing agent. positive charge as shown below. You establish your two half reactions by looking for changes in oxidation numbers. The example is the oxidation of $$\ce{Fe^{2+}}$$ ions to $$\ce{Fe^{3+}}$$ ions by dichromate $$\left( \ce{Cr_2O_7^{2-}} \right)$$ in acidic solution. Second, if necessary, balance all elements except oxygen and hydrogen in Balancing Redox Equations: Half-Reaction Method. Oxidation half reaction: l (aq) → l 2(s) +7 +4. Balancing Redox Reactions: The Half-Reaction Method Balanced chemical equations accurately describe the quantities of reactants and products in chemical reactions. $14 \ce{H^+} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) \rightarrow 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right)$. You cannot have electrons appear in the final answer of a redox reaction. Recall that a half-reaction is either the oxidation or reduction that occurs, treated separately. A reaction in which a reducing agent loses electrons while it is oxidized and the oxidizing agent gains electrons while it is reduced is called as redox (oxidation – reduction) reaction. This train has an orange stripe while its companion has a yellow stripe. When balancing redox reactions, the overall electronic charge must be balanced in addition to the usual molar ratios of the component reactants and products. one must now add one (OH-) unit for every (H+) present in the equation. Step 1: Write the unbalanced ionic equation. Redox equations are often so complex that fiddling with coefficients to balance chemical equations doesn’t always work well. Third, balance the oxygen atoms using water molecules . They serve as the basis of stoichiometry by showing how atoms and mass are conserved during reactions. The product side has a total charge of $$6+$$ due to the two chromium ions $$\left( 2 \times 3 \right)$$. give the 6 electrons required by the reduction half-reaction. Balancing Redox Reaction. NO → NO 3-6. Watch the recordings here on Youtube! $\ce{Cr_2O_7^{2-}} \left( aq \right) \rightarrow 2 \ce{Cr^{3+}} \left( aq \right)$. It depends on the individual which method to choose and use. The reduction half-reaction needs to be balanced with the chromium atoms. Another method for balancing redox reactions uses half-reactions. Calculator of Balancing Redox Reactions 8. by the ion-electron method. Balancing Redox Reactions with Half-Reaction Method? Balancing Redox Reactions. SO 4 2- → SO 2 7. Balancing a redox reaction requires identifying the oxidation numbers in the net ionic equation, breaking the equation into half reactions, adding the electrons, balancing the charges with the addition of hydrogen or hydroxide ions, and then completing the equation. In this example, fourteen $$\ce{H^+}$$ ions will be added to the reactant side. In the ion-electron method (also called the half-reaction method), the redox equation is separated into two half-equations - one for oxidation and one for reduction. These brightly colored compounds serve as strong oxidizing agents in chemical reactions. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Redox Reactions: It is the combination oxidation and reduction reactions. Oxidation and reduction reactions need to be bal- Example #4: Sometimes, the "fake acid" method can be skipped. These are then balanced so that the number of electrons lost is equal to the number of electrons gained. Steps for balancing redox reactions. There is a total charge of $$12+$$ on the reactant side of the reduction half-reaction $$\left( 14 - 2 \right)$$. They already are in this case. 4. A typical reaction is its behavior with iodide Cr 2O 7 2 - → Cr3+ 5. (There are other ways of balancing redox reactions, but this is the only one that will be used in this text. Below is the modified procedure for balancing redox reactions using the oxidation number method. To achieve balanced redox reaction, simply add balanced oxidation and reduction half reactions in order to cancel unwanted electrons: $$\ce{2MnO4- + 8H+ + 6I- -> 2MnO2 + 3I2 + 4H2O }$$ This redox reation is forward reaction because it has a net positive potential (refer reduction potentials of two half reactions). Missed the LibreFest? by the ion-electron method. Recall that a half-reaction is either the oxidation or reduction that occurs, treated separately. One method is by using the change in oxidation number of oxidizing agent and the reducing agent and the other method is based on dividing the redox reaction into two half reactions-one of reduction and other oxidation. The chromium reaction can now be identified as the reduction half-reaction The two half reactions involved in the given reaction are: -1 0. Second, if needed, balance both equations, by inspection ignoring any oxygen Balancing it directly in basic seems fairly easy: Fe + 3OH¯ ---> Fe(OH) 3 + 3e¯ And yet another comment: there is an old-school method of balancing in basic solution, one that the ChemTeam learned in high school, lo these many years ago. When balancing redox reactions we have always - apart from all the rules pertaining to balancing chemical equations - additional information about electrons moving. The reason for this will be seen in Chapter 14 “Oxidation and Reduction” , Section 14.3 “Applications of Redox Reactions: Voltaic Cells” .) This page will show you how to write balanced equations for such reactions even when you do not know whether the H 2 O(l) and H + (aq) are reactants or products. Cr2O72- + C2H6O This is called the half-reaction method of balancing redox reactions, or the ion-electron method. Each half-reaction is balanced separately and then the equations are added together to give a balanced overall reaction. Note; each electron (e-) represents a charge of (-1). The nature of each will become evident in subsequent steps. First of all balance the atoms other than H and O. In the oxidation half-reaction above, the iron atoms are already balanced. Oxidation and reduction reactions need to be bal- This example problem illustrates how to use the half-reaction method to balance a redox reaction in a solution. 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# What is the value of –[(s + t)0] if t + s ≠ 0?
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What is the value of –[(s + t)0] if t + s ≠ 0? [#permalink]
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29 Sep 2018, 06:12
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What is the value of -[$$(s+t)^0$$] if $$t+s\neq{0}$$?
(A) -1
(B) 0
(C) 1
(D) s+t
(E) -s+t
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Re: What is the value of –[(s + t)0] if t + s ≠ 0? [#permalink]
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29 Sep 2018, 09:45
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Important number properties to remember is that any number to the power of zero will equal 1 except for 0. Because we cannot divide by 0.
Since we are given that s+t does not equal zero
Then we know that s+t will be some number
We are given - [(s+t)^0] so - (1) = -1
Some theory on this to justify why it becomes 1
say we have x = 3 which is 3^1
If we divide 3^1/3^1 = we get 3^(1-1) = 3^0 = 1
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Re: What is the value of –[(s + t)0] if t + s ≠ 0? [#permalink] 29 Sep 2018, 09:45
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### A probability question
Two analysts make prediction of the movement of a stock separately. Analyst A have an accuracy of $p_1$, which means his prediction is correct at the probability of $p_1$, while analyst B have an accuracy of $p_2$. Now given that both analysts predict the stock price will rise tomorrow, what is the probability for the stock price to rise tomorrow? What if A predicts rising and B predicts falling? (Assuming the stock either rise or fall but won't stay the same tomorrow.)
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### A probability question
Insufficient information. For example, it could be that analyst 1 is from Goldman and analyst 2 is from UBS. The UBS analyst has stolen the Goldman excel model which makes these predictions, so the predictions are the same except for the 20% of the time that the UBS analyst gets the sign wrong on the model's output. In this case the probability is p_1 because you gain no additional information from the second prediction.On the other hand, if we assume that the accuracy of the analysts' predictions are independent then you get a trivial probability question - the probability is simply the conditional probability of two independent events given that they either both happen or both do not, i.e. p_1 * p_2 / (p_1 * p_2 + (1 - p_1) * (1 - p_2)) | {} |
## Abstract
This study empirically investigates the relationship between retirement duration and cognition among older Irish women using microdata collected in the third wave of The Irish Longitudinal Study on Ageing. Ordinary least squares (OLS) regression estimates indicate that the longer an individual has been retired, the lower the cognitive functioning, with other factors thought to affect cognition held constant (e.g., age, education, and early-life socioeconomic conditions). However, retirement is potentially endogenous with respect to cognition because cognition may affect decisions relating to retiring. If so, the OLS estimates will be biased. To test for this possibility, instrumental variable (IV) estimation is used. This method requires an IV that is highly correlated with retirement duration but not correlated with cognition. The instrument used in this study is based on the so-called marriage bar, the legal requirement that women leave paid employment upon getting married, which took effect in Ireland in the 1930s and was abolished only in the 1970s. The IV regression estimates, along with formal statistical tests, provide no evidence in support of the view that cognition affects retirement decisions. The finding of a small negative effect of retirement duration on cognition is robust to alternative empirical specifications. These findings are discussed in the wider context of the effects of work-like and work-related activities on cognition.
## Introduction
Many cognitive abilities decline in old age, and age-related declines in cognitive abilities are correlated with declines in the ability to perform everyday tasks (Tucker-Drob 2011). Researchers, however, are showing greater recognition of individual differences in the extent to which cognitive abilities are lost or maintained in old age (Tucker-Drob and Salthouse 2011). Two questions are of particular importance. First, Who are the people who are able to maintain cognitive abilities (e.g., memory, reasoning, and processing speed) in old age? Second, How do these people differ from those who cannot maintain cognitive abilities in old age?
Two main (and somewhat competing) hypotheses have been used to explain observed heterogeneity across individuals with respect to the decline in or preservation of cognitive abilities in old age. The cognitive reserve hypothesis suggests that the advantages afforded by early-life socioeconomic opportunities serve to slow the rate of age-related cognitive decline (Stern 2002, 2003). Individuals who experience more enriched socioeconomic environments during childhood and early adulthood have more resilient cognitive and/or neurobiological architectures in adulthood and, in turn, experience less cognitive decline as they age. Research suggests that one of the best indicators of socioeconomic advantages is educational attainment (Tucker-Drob and Salthouse 2011).
The mental exercise hypothesis—also known as cognitive enrichment or cognitive use hypothesis (Hertzog et al. 2008; Hultsch et al. 1999; Salthouse 2006)—suggests that individuals who engage in mental exercise and maintain an engaged lifestyle experience relatively less cognitive decline. More specifically, it is argued that high levels of neuronal activation brought about by mental stimulation can buffer against neuro-degeneration and cognitive decline in old age (Churchill et al. 2002; van Praag et al. 2000). An early proponent of this position (Sorenson 1938) suggested that to prevent cognitive decline, people should order their lives such that they constantly find themselves in new situations and confronted with novel problems. The view that keeping mentally active will maintain one’s level of cognitive functioning—and possibly prevent cognitive decline—is so pervasive in contemporary culture that it is frequently expressed in curt terms: “use it or lose it” (Salthouse 2006:70).
If engaging in mental exercise can indeed help maintain cognitive functioning and possibly prevent cognitive decline (as suggested by the mental exercise hypothesis), the next logical question is, Which types of activities are most beneficial? Mentally stimulating activities hypothesized as protective against age-related declines in cognition include recreational activities, such as doing crossword puzzles and playing chess, or learning a new skill or how to speak a foreign language (Tucker-Drob and Salthouse 2011). A small albeit growing body of research has suggested that another way to preserve cognition is to delay retirement and continue to work into the later years. The hypothesis is that workers engage in more mental exercise than retirees because work environments provide more cognitively challenging and stimulating environments than do nonwork environments. Thus, perhaps a negative relationship exists between retirement and cognitive functioning (cognition).
Our study adds to the small but growing body of research that empirically tests the validity of this specific instantiation of the “use it or lose it” hypothesis, using data from Ireland. The relationship between retirement and cognitive functioning is investigated using data for older Irish women collected in the third wave of The Irish Longitudinal Study on Ageing (TILDA). Ordinary least square (OLS) regressions are used in first instance. Because retirement is potentially endogenous with respect to cognition, instrumental variable (IV) estimation is also used. The identifying instrument in the IV estimation is the abolition of the so-called marriage bar, which was the legal requirement that women leave paid employment on getting married.
Our analysis suggests a negative effect of retirement duration on cognitive functioning. That is, the longer an individual has been retired, the lower the cognitive functioning, holding other factors constant with multiple regression. However, this effect is small. We see no evidence that retirement is endogenous, in the sense that there is no evidence that cognitive functioning has an effect on retirement duration. The findings are found to be robust to alternative empirical specifications.
## Contribution to the Literature
Several recent studies in the fields of both health and economics have investigated the effect of retirement on cognition. However, the focus is somewhat different between the two sets of studies. Health-based studies have mainly used longitudinal data and methods to investigate the effect of retirement on cognitive decline, concerned primarily with intraindividual changes in cognitive functioning over time. Most of these studies explored whether the nature of employment in the preretirement occupation affects the rate of cognitive decline after retirement. On the other hand, economics-based studies have argued that the main challenge to identifying the effect of retirement on cognition is that the decision to retire might itself be affected by cognition (Rohwedder and Willis 2010). In other words, the direction of causation between retirement duration and cognition may be two-way. If this were the case, the key empirical challenge is to determine which causal direction dominates. Most of the economics-based studies have used a statistical technique known as the instrumental variable estimation to address this issue. We describe the technique in detail later.
The first study to investigate the effect of retirement on cognitive decline in an epidemiological sample was Roberts et al. (2011). Using data spanning a five-year period from the UK Whitehall II Study, they found that individuals who retired in the study period showed a trend toward smaller cognitive test score increases than those who were still working at follow-up. Using data spanning a six-year period from the Swedish National Study on Aging and Care, Rennemark and Berglund (2014) found that participants who retired prior to age 60 experienced cognitive decline in the study period. Cognitive decline was not found for those who worked in the study period.
Finkel et al. (2009), Fisher et al. (2014), and Andel et al. (2015) employed latent growth models to investigate whether job characteristics during one’s time of employment moderate the association between retirement and cognitive decline. Using data from a subset of twins from the population-based Swedish Twin Registry, Finkel et al. (2009) found larger negative effects of retirement on cognitive decline for individuals whose preretirement jobs were characterized by high levels of “complexity” for some (but not all) measures of cognition included in their data set. Fisher et al. (2014), using longitudinal data spanning 18 years from the U.S. Health and Retirement Study (HRS), found that individuals with preretirement jobs that were characterized by higher “mental demands” had less steep cognitive declines after retirement. Likewise, Andel et al. (2015), using multiple waves of the HRS, found that individuals whose preretirement jobs were characterized by “less control” and “greater strain” had steeper cognitive declines after retirement.
To our knowledge, only five economics-based studies have investigated the effect of retirement on cognition. Four of these studies used IV estimation to explore the endogeneity of retirement. The IV approach requires a variable (instrument) that is correlated with the retirement decision but not correlated with cognition. It also needs to be exogenous in the sense that it is not a direct outcome of individual decision-making.
Using data collected in the HRS, the English Longitudinal Study on Ageing (ELSA) and the multicountry Survey of Health, Retirement and Ageing in Europe (SHARE), Rohwedder and Willis (2010) and Mazzonna and Peracchi (2012) employed cross-country and temporal changes in policies affecting the age at which individuals are entitled to receive a state-supplied pension and other age-related benefits. The expectation is that this variability would have a sizable effect on retirement decisions but have no direct effect on cognition. Before and after controlling for endogeneity, both studies found sizable negative effects of retirement on cognition. Bonsang et al. (2012), using data from the HRS, reached a similar conclusion following a similar approach. de Grip et al. (2015), using Dutch data from the Maastricht Aging Study, found large negative effects of retirement on cognitive decline for some (but not all) measures of cognition included in their data set. Finally, Coe et al. (2012), also using HRS data, used early retirement offers (which are legally required to be nondiscriminatory) as a source of exogenous variation, and found no support that retirement affects cognition.
Our study differs from the previous studies in three main ways. First, our analysis focuses on women. The employment histories for men and women are generally different. In most high-income countries, men typically work uninterruptedly from when they complete schooling until retirement, with ill health and unemployment being the main factors causing deviation from this pattern. The pattern for women is typically different because childbearing and child-rearing frequently result in mothers leaving the labor force, often for considerable periods of time. With the exception of Mazzonna and Peracchi (2012), the existing studies focused only on men or did not disaggregate the analysis by sex. Grouping men and women may mask important differences. For all these reasons, we believe it important to analyze women separately—and even more important, not to exclude them.
Second, the differences in the findings of the economics-based studies may be a product of differences in the exogenous variation used in the statistical models. Basically, this variation is caused by policy changes that should affect retirement decisions. However, it assumes that individuals are rational and fully understand these changes. Considerable evidence shows that this is not the case (see, e.g., Hancock et al. 2004). Therefore, we exploit an alternative source of exogenous variation unique to the Irish context caused by the abolition of the so-called marriage bar. The marriage bar was the legal requirement that women leave paid employment—in a sense, retire from paid work—upon marrying. It was established in the 1930s and abolished in the 1970s. The TILDA data used here surveyed women who were required to leave paid employment—retire—because of the marriage bar. Many of these women spent a significant proportion of their lives after getting married in retirement.
Third, the TILDA data include measures of cognition that are novel in the context of other large-scale, nationally representative studies on aging. One unique feature is that they are administered and scored by nurses trained specifically for this purpose. Therefore, they should be subject to less measurement error compared with self-assessed or interviewer-administered measures. The four measures of cognition employed in the analysis of our study capture processing speed and mental switching, which are central to effective cognitive functioning. Crucially, both processing speed and mental switching require effortful processing at the time of assessment and do not require production of previously acquired knowledge (Tucker-Drob and Salthouse 2011).
## Methodology
### Data
The data we use are from the third wave of TILDA, which is a nationally representative sample of community-dwelling individuals aged 50 or older in Ireland. The survey collects detailed information on the economic, health, and social aspects of the respondents’ lives. It is modeled closely on HRS, ELSA, and SHARE. At the Wave 3 interview (2014/2015), 6,566 respondents completed a computer-assisted personal interview (CAPI) in their homes and were invited to travel to a dedicated health center based in Trinity College Dublin for a comprehensive health assessment. If unable or unwilling to travel to the health center, respondents were offered a modified assessment in their home. All assessments were carried out by qualified and trained research nurses. A total of 5,395 respondents underwent a health assessment: 80 % in the Trinity College Dublin health center and 20 % in their home. Although the main analysis of this article is based on data from the third wave of TILDA, data on labor market circumstances from the first (2009/2011) and second (2012/2013) waves were also employed to construct the relevant labor market variables or for robustness checks. For more detail about TILDA, see Cronin et al. (2013), Kearney et al. (2011), and Whelan and Savva (2013).
### Statistical Model
In our statistical model, we assume that cognition (Cog) is a function of retirement duration (RetDur), a vector of other controls (Xj) (such as j = age and education), and an error term (u). In regression form,
$Cogi=β0+β1RetDuri+∑jβjXij+ui,$
1
where the subscript i denotes the individual, i = 1, 2, . . . , N. If RetDur is correlated with u, then OLS estimates of β1 will be biased and inconsistent. IV estimation can be used to purge the relationship between RetDur and Cog of this bias. Key to IV estimation is the availability of at least one variable, Z (instrument), which has the following three key properties: (1) variation in Z is associated with variation in RetDur; (2) variation in Z is not associated with variation in Cog (apart from the indirect route via RetDur); and (3) variation in Z is not associated with variation in unmeasured variables that affect RetDur and Cog. If one has available a variable that satisfies these properties, then one can estimate the following regression:
$RetDuri=π0+π1Zi+∑jπjXij+wi,$
2
where RetDur is as a function of Z, Xj, and an error term w. By estimating this first-stage regression, one can then form predictions for RetDur:
$RetDur^i=π0^+π1^Zi+∑jπj^Xij.$
3
One can use OLS to estimate the second-stage regression:
$Cogi=b0+b1RetDur^i+∑jbjXij+ei,$
4
where predicted values of RetDur from Eq. (3) are used. Assuming that all assumptions are met, the error term in this regression, e, is random and not correlated with RetDur. If this is the case, Eq. (4) will provide an unbiased estimate, b1, of the relationship between retirement duration and cognition. On the other hand, if b1 = β1 (which is a testable hypothesis), retirement duration is exogenous, and OLS provides such an estimate.
A note of caution is needed when using IV estimation. For all analyses using IV estimation, generalizability is a concern because the IV estimation recovers what in the literature is referred to as the local average treatment effect (LATE) (Angrist and Imbens 1994). The LATE is the average effect of the treatment among only the group affected by the instrument. In our analysis, IV estimates the average effect of retirement duration on cognition for the group of women who were affected by the instrument Marriage Bar because the law was in place but would not have not been affected had the law not been in place.
### Variables
#### Cognition
The four cognition variables are tests of processing speed and mental switching that have been widely used and validated in clinical studies. The Colour Trail Task 1 test (CTT1) captures mainly visual scanning and mental processing speed. The Colour Trail Task 2 test (CTT2) captures additional executive functions, such as task switching (D’Elia et al. 1996). The Choice Reaction Time (CRT) and Choice Reaction Time Variability (CRT_VAR) tests capture processing speed and concentration. Importantly, these tests require effortful processing at the time of assessment and do not require production of previously acquired knowledge (Tucker-Drob and Salthouse 2011).
In TILDA, cognitive tests are administered and scored by trained and qualified nurses during the health assessment. Focusing on the four tests employed in this study, respondents are first passed a sheet of paper containing numbers in yellow or pink circles. For the CTT1, respondents are instructed to rapidly draw a line with a pencil, connecting the circles numbered 1–25 in consecutive order. In the CTT2, respondents are asked to connect numbered circles alternating between pink and yellow circles (e.g., pink 1, yellow 2, pink 3, and so on). The performance indicator for both CTT1 and CTT2 is the time taken (in seconds) to successfully complete the test, with shorter completion times indicative of better performance.
Respondents are then required to perform a computer-based task. They are asked to depress a central button until a stimulus appears on-screen: either the word YES or the word NO. Each time a stimulus appears, respondents are required to press the corresponding button. A return to the central button is necessary after each response for the next word to appear on-screen. There are approximately 100 repetitions. The task variables of interest are the mean intraindividual CRT and the standard deviation of individual CRT, the latter providing a measure of variability (CRT_VAR). CRT and CRT_VAR are measured in milliseconds.
In Fig. 1, panels a–d plot the relationship between age and the four cognition measures. For each measure, respondents were ranked from slowest to fastest based on the time taken to complete the task. Then the mean ranking position by year of age was computed for each of the four cognitive measures. Figure 1 shows a clear negative relationship between age and cognition. For completeness, the relationship between age and the four cognitive measures expressed in the original metric (i.e., time taken to complete the task) is illustrated in Fig. S1 in Online Resource 1. The relationship between age and the standardized values (z scores) of the cognition variables is also shown in the same figures.
#### Retirement Duration
In the CAPI interview, respondents are asked to report the status that best describes their current labor market situation: (1) retired, (2) employed, (3) self-employed, (4) unemployed, (5) permanently sick or disabled, (6) looking after home or family, (7) in education or training, and (8) other. Respondents can select only one choice because the options are designed to be mutually exclusive. At the Wave 3 interview, 34.5 % of women in the sample are employed or self-employed, and another 40.5 % are retired. Nearly one-fifth (19.4 %) are looking after home or family; 3.1 % are permanently sick or disabled, and 2 % are unemployed. We classify an individual as working if she reports to be currently in employment, or retired otherwise. Working individuals are, therefore, those who chose categories (2) and (3), and retired individuals are those who chose categories (1) and (4)–(8). Robustness checks concerned with the reliability of our definition of retirement are reported later herein.
Respondents not working at the time of the interview are then asked whether they have done any paid work in the week prior to the interview. Individuals who reported to have done some paid work in that week (n = 56) are excluded. A total of 160 respondents reported to have never done any paid work. Some of these respondents may have engaged in unpaid work at some point over their lifetime—for example, on the family farm or in the family business. Unfortunately, additional information on the employment history of respondents who report never having done any paid work is not collected in TILDA. For this reason, these respondents are excluded from the analysis. Only respondents who report having done paid work at some point in their life are kept in the sample.
Respondents in categories (1) and (4)–(8) are asked to report the month and year when they stopped working. For example, respondents who report being retired (i.e., in category (1)) are asked the following question: “In what month/year did you stop working?” Similarly, respondents who report being unemployed (i.e., in category (4)) are asked the following question: “In what month/year did you become unemployed?” We define retirement duration as the time elapsed between the date the respondent stopped working and the date of the health assessment for that respondent. Retirement duration in full months is calculated and converted to years of retirement for ease of interpretation. For those at work, retirement duration is set to 0.
Because information on labor market status is also collected at Waves 1 and 2 with the same questions, this information is used to construct a more robust measure of retirement duration. If inconsistent answers are provided across the three waves, we consider as most reliable the measure of retirement duration constructed based on Wave 1 reports, followed by Wave 2 reports and Wave 3 reports. This should minimize recall bias: the time elapsed between the date of retirement and the date of interview is shorter because Wave 1 occurs before Waves 2 and 3. Retirement duration cannot be calculated for 117 women because of missing information, and these individuals are excluded from the sample.
Panels a–d of Fig. 2 plot the relationship between retirement duration and the four cognitive measures, showing that respondents who have retired for longer are, on average, slower at completing the cognition tasks. The relationship between retirement duration and the four cognitive measures expressed in the original time metric and between age and the standardized values (z scores) of the cognition variables is shown in Fig. S2 in Online Resource 1.
#### Controls
Additional variables thought to affect cognition are included. These variables include the key factors of age and education, as well as a set of variables aimed at capturing childhood characteristics. The main aim is to restrict the list of control variables to those that are clearly exogenous and not subject to same endogeneity considerations as retirement duration. We achieve this aim by selecting variables measured when the respondent was young.
The relationship between education and cognition has been studied. A number of studies have found evidence that education positively affects cognition in later life (e.g., Banks and Mazzonna 2012; Schneeweis et al. 2014). Because most schooling among older Irish women is completed when they are young and before they enter the labor market, it is exogenous. Education (School) is measured as the number of years of schooling completed.
Several childhood characteristics have been shown to be associated with cognition in later life (Borenstein et al. 2006; Brown 2010; Everson-Rose et al. 2003). We employ a set of dummy variables based on respondent’s self-reporting of childhood conditions before age 14: NoBook = 1 if there were no or very few books in the home where respondent grew up (0 = otherwise); PoorHealth = 1 if respondent was in fair/poor health (0 = otherwise); PoorFam = 1 if respondent grew up in a poor family (0 = otherwise); MotherNotWork = 1 if respondent’s mother never worked outside the home (0 = otherwise); and FatherNotWork = 1 if respondent’s father never worked outside the home (0 = otherwise). For 37 women, information is missing on one or more of these variables, and these individuals are excluded from the sample.
The final samples are 2,519 women for the model based on CTT1; 2,481 women for the model based on CTT2; and 2,383 women for the models based on CRT and CRT_VAR. Table 1 displays descriptive statistics for all independent variables based on the sample including 2,519 women. The average age is 65.8 years, and the average retirement duration is 12 years.
#### Instrumental Variable: The Marriage Bar
We believe that the abolition of the so-called marriage bar in Ireland caused exogenous variation in retirement decisions. The marriage bar was the legal requirement that women leave their paid employment after getting married. It was established for primary school teachers in 1933 and for civil servants in 1956. Although not legally obliged to do so, many semi-state and private organizations—including banks, utility companies, and large manufacturers—also dismissed women when they married. Private sector employers dismissed women working in primarily clerical and skilled jobs, but in some cases, they dismissed unskilled workers (Kiely and Leane 2012:91).
The marriage bar for primary school teachers was lifted in 1958, and lifted for civil servants in 1973. Discrimination in employment on the grounds of sex or marital status was made illegal in 1977. Unsurprisingly, the labor force participation rate of married women aged 15 and older increased from 7.5 % in 1971 to 14.5 % in 1975 (Pyle 1990). For more on the Irish marriage bar, see Connolly (2003), Cullen Owens (2005), Kiely and Leane (2012), and O’Connor (1998).
Crucially, no evidence exists that the marriage bar forced women to choose between paid employment or getting married. For example, Fig. 3 shows female activity rates for married and single women in 1970 in Ireland and other countries. Clearly, although activity rates of single women in Ireland were closely aligned to activity rates of single women in other countries, married women in Ireland were significantly less likely to be active than those in other countries. This suggests that an exogenous factor preventing married women from working in Ireland was present, which we believe is the marriage bar.
Additional evidence consistent with this view is shown in Figs. 4 and 5. Figure 4 shows the proportions of never-married and married women calculated from the TILDA and SHARE surveys by birth cohort. In Ireland, like in many other countries, the proportion of never-married women is very small, suggesting that marriage was the norm for women born in the first half of the twentieth century. Figure 5 shows the historical crude marriage rate and the general marriage rate for Ireland (1926–1996). One would expect that if women were forced to choose between marriage and paid employment, the marriage rate would increase after the abolition of the marriage bar. Figure 5 shows that, if anything, the marriage rate stabilized and then decreased after the abolition of the marriage bar: that is, it moved in the opposite direction.
Ireland is not the only country where women were dismissed from employment at marriage. For example, marriage bars survived up to the 1950s in the United States (Goldin 1990), England (Smith 1986), the Netherlands (Boeri and van Ours 2013), and Germany (Kolinsky 1989). Ireland is, however, unique in the duration of the enforcement of the marriage bar. Many Irish women who were affected are still alive and are in the TILDA sample. Comparatively, most of the women affected by the marriage bar in the other countries are likely to have died or to be very old.
TILDA is the first large-scale longitudinal study on aging to include specific questions on the marriage bar. In TILDA Wave 3, women are asked the following question: “Did you ever have to leave a job because of the Marriage Bar?” The instrument used is a dummy variable, MarBar, coded 1 if a woman reported having to leave employment on getting married and 0 otherwise. It is also coded 0 for the few women in the sample who reported never marrying. Of the 2,519 women in the final sample, 318 reported that they had to leave a job because of the marriage bar. Some of these women subsequently returned to work. For these women, the instrument is coded 1, and RetDur is defined as the time elapsed between the date the respondent stopped working in her final job and the date of the health assessment for that respondent.
## Results
### Main Empirical Findings
Columns 1 and 2 in Tables 2 and 3 show the OLS regression estimates for CTT1 and CTT2, and CRT and CRT_VAR, respectively. We transform the four outcome variables by taking the natural logarithm in order to ensure normality of the residuals. We then multiply the transformed scores by –1. Therefore, a higher value of these transformed variables suggests a higher level of cognitive functioning and vice versa, which makes interpretation of the estimates more intuitive.
Since the cognition measures are transformed into natural logarithms, the regression coefficients can be easily transformed into percentage effects. For example, %RetDur = [exp(β1) – 1].
The coefficient of RetDur is negative for the four cognition measures, which is consistent with the hypothesis that a longer retirement duration is associated with lower cognition. Even though these associations are statistically significant at the 5 % level or lower, the magnitude is small. An additional year of retirement corresponds to a 0.2 % reduction in CTT1, a 0.1 % reduction in CTT2, a 0.1 % reduction in CRT, and a 0.3 % reduction in CRT_VAR. As expected, the coefficient of Age is negative for all four cognition measures and is statistically significant at the 1 % level. An additional year of age is associated with a reduction of 2.1 % in CTT1, 1.7 % in CTT2, 0.8 % in CRT, and 2.1 % in CRT_VAR.
The coefficient of School is positive and statistically significant for all cognition measures. An additional year of schooling is associated with a 1.1 % increase in CTT1, a 1.3 % increase in CTT2, a 0.5 % increase in CRT, and a 1.6 % increase in CRT_VAR. As a group, the remaining variables should proxy well the socioeconomic conditions in the home where the respondent grew up. Strong support for the hypothesis that early-life conditions’ effects on later-life cognition is found for the variable growing up in a household with no or few books. The coefficient of NoBooks is negative and statistically significant at the 1 % level for all four cognition variables. The magnitude of this association is sizable: cognition is approximately 5.7 % lower for CTT1, 8.5 % lower for CTT2, 4.7 % lower for CRT, and 9.2 % lower for CRT_VAR growing up in a household with no or few books. It is not clear, however, whether this is a socioeconomic effect or an early reading effect. Self-reported health is also important. However, the reasons behind poor childhood health can be caused not only by socioeconomic conditions but also by factors largely independent of socioeconomic conditions (such as contagious disease).
The association of RetDur with CTT1, CTT2, CRT, and CRT_VAR before and after the control variables are added is visually depicted in Fig. 6. Larger symbols are used to depict the RetDur coefficient before the control variables are added. Smaller symbols are used to depict the RetDur coefficient after the control variables are added. The 95 % confidence interval of each coefficient is also shown. Figure 6 shows that after the control variables are added, the size of the RetDur coefficient is approximately 20 % to 25 % of the size of the initial coefficient.
The estimates of columns 1 and 2 in Tables 2 and 3 and Fig. 6 are based on the assumption that retirement duration is exogenous. The IV estimates that test for the potential endogeneity are shown in columns 3–8 in Tables 2 and 3. These columns show the first-stage IV estimates, the reduced-form estimates, and the second-stage IV estimates. As discussed in the previous section, the instrument employed is whether the woman reported having to leave a job because of the marriage bar. Columns 3 and 4 in Table 2 show the first-stage estimates for CTT1 and CTT2. There are only slight differences between the two columns because of the small differences in sample sizes. Columns 3 and 4 in Table 3 show the first-stage estimates for CRT and CRT_VAR. The two columns are identical because the sample size is the same in the two regressions.
Clearly, MarBar is an important predictor of RetDur. The coefficient of MarBar in all equations is positive, large in magnitude, and statistically significant at well below the 1 % level. The statistics from the first-stage equations reported at the bottom of Tables 2 and 3 confirm that the instrument is not weak (see Bound et al. 1995; Hernan and Robins 2006; Murray 2006; Staiger and Stock 1997; Stock and Yogo 2005). For example, the F statistics range between 33.4 and 35.1. According to Staiger and Stock’s (1997) rule of thumb, the F statistics should be at least 10 for the instrument not to be weak. Similarity, the Stock-Yogo tests of weak identification reject the null hypothesis that the instrument is weak given that the F statistics exceed the selected critical values. In short, women who had to leave work because of the marriage bar have a longer retirement duration—or more correctly, a longer current period of not working—even after we control for age and education. The requirement that the instrument is a strong predictor of the potentially endogenous variable is satisfied.
Unfortunately, we cannot directly test the requirement that there is no relationship between MarBar and Cog, apart from the indirect route via RetDur. However, we can obtain some information by considering the reduced-form regressions. In these regressions, CTT1, CTT2, CRT, and CRT_VAR are expressed as a function of the MarBar and of the other variables. These estimates are shown in columns 5 and 6 of Tables 2 and 3. MarBar is not statistically significant in any regression. In fact, the t statistics range between 0.2 and 0.7. This lack of statistical significance is encouraging and suggests that a relationship between the IV and the outcome of interest is unlikely to exist (Angrist and Krueger 2001; French and Popovici 2011).
Finally, columns 7 and 8 in Tables 2 and 3 show the estimates of the second-stage regression results. For all cognition measures, the coefficient of RetDur is statistically insignificant. We compare differences between the estimators of the OLS and IV by employing the Hausman test. If OLS and IV estimators are found to have a different probability limit, then there is evidence that endogeneity is present, and OLS estimators will be inconsistent. If OLS and IV estimators are found to have the same probability limit, then there is no evidence that endogeneity is present. Both estimators will be consistent, and OLS estimation is preferred. The results of the Hausman test are given at the bottom of Tables 2 and 3. For all four cognition measures, the χ2 values are not statistically significant, implying that the null hypothesis that retirement duration is exogenous cannot be rejected at any level of statistical significance. This leads us to conclude that the OLS estimates are preferred. More generally, there is no statistical evidence that retirement duration is endogenous. Therefore, if retirement duration and cognition are causally related, then retirement affects cognition and not the other way round.
### Robustness Checks and Model Extensions
To consider the robustness of the estimates, five sets of additional regressions are estimated (results available in Online Resource 1). The main conclusion is that the magnitude of the relationship between retirement duration and cognition remains small and statistically significant for all cognition measures.
The first set of regressions employ three alternative IVs. As explained earlier, the marriage bar was not enforced universally. It was enforced by law in the public sector and mimicked by many, but not all, private sector employers. One cannot exclude that women with certain characteristics that are not measured in the TILDA data set selected into jobs that were affected, or not affected, by the marriage bar. For example, perhaps women with an innate desire to be active in the labor force opted for jobs that would allow them to work after marriage, primarily in the private sector. If this unmeasured variable innate desire to be active in the labor force is also correlated with employment/retirement duration and cognition, then the IV used in the analysis is not valid.
Other unobservable characteristics that are potentially correlated with occupational choice at labor market entry and retirement duration are risk aversion and family preferences. For example, perhaps women who were more risk-averse and more family oriented opted for jobs in the public sector given that retiring at marriage was enforced by law. Similarly, perhaps women who were less risk-averse and less family-oriented opted for jobs in the private sector given that not all private sector employers enforced the marriage bar. In other terms, career prospects might have been better in the private sector. Although it is difficult to argue that traits such as risk aversion and family preferences are also correlated with cognition, one cannot exclude this might be the case.
Three IVs that are clearly independent of the occupation the woman had are constructed. The first two instruments are proxies for the number of years a woman was exposed to the marriage bar. The first instrument, MarBarBirth, is the time elapsed between a woman’s year of birth and 1977, which was the year when discrimination in employment on the grounds of sex or marital status was made illegal in Ireland. The second instrument, MarBar18, is the time elapsed between the year in which a woman turned 18 years of age and 1977. The third instrument, PropMarBar, is equal to the proportion of women in the TILDA sample who reported having been affected by the marriage bar by birth cohort.
The second set of regressions focus on whether the coefficient of RetDur is significantly different in magnitude under alternative specifications compared with what is found in the OLS baseline regressions of Tables 2 and 3. Five tests are employed. First, older women are excluded from the sample because employment rates among “older” women are very low. Second, women who performed the health assessment in their homes are excluded because they might differ from those who travelled to Trinity College Dublin to undertake the health assessment. Third, the unemployed and the sick and disabled are excluded to examine how robust the estimate of RetDur is to different definitions of retirement. Fourth, only those who have a retirement duration of at least one year are considered as retired. Fifth, quadratic and cubic terms in age are added to the list of explanatory factors.
The third set of regressions investigate the role of “nonwork substitution activities.” It is reasonable to hypothesize that women who retired around the time of marriage or in early adulthood substituted work activities with nonwork activities. If such activities are mentally stimulating, one would expect to find a smaller and potentially insignificant effect of retirement duration on later-life cognition for this group of women. Three tests are employed. The first test is an investigation of whether the time spent out of the labor force—associated with having children—affects later-life cognition. Perhaps the positive effect that child-rearing has on cognition outweighs the negative effect of time not working. The second test is an investigation of whether there is an association between current nonwork activities—such as volunteering—and cognition. The (untestable) assumption is that women who engage more into nonwork activities at present are more likely to have engaged in such activities in the past. The third test employs additional information on employment histories collected for women who had to leave a job because of the marriage bar.
The fourth set of regressions investigate whether the relationship between retirement and cognition can be explained by the nature of employment during one’s working life. Two tests are employed. The first test is to add an interaction term between RetDur and a dummy variable capturing the occupational sector of the preretirement job to the list of explanatory factors. If the cognitive stimulating nature of work is what improves cognitive function, then one can expect that the largest effects of retirement are for women in more cognitively stimulating jobs. The second test is to add an interaction term between RetDur and a dummy variable capturing whether employment is performed on a part-time or full-time basis. If there is a dose-response relationship between hours worked in a typical week and cognitive stimulation, then one can expect that the largest negative effects of retirement are for women in full-time jobs. However, another possibility is that women working part-time engage in equally cognitively stimulating activities when they are not in work—particularly for women who choose to retire gradually from work.
The fifth set of regressions investigates the role of cohort differences in cognitive functioning because one cannot exclude that lower duration of retirement is simply a marker for being born in a more recent birth cohort. If, ceteris paribus, individuals born in later generations begin adulthood with higher overall levels of performance than those born in earlier generations, then these younger participants will outperform older participants at any given time point—not because of aging-related changes but because of historical differences in, for example, nutrition or education (Tucker-Drob and Salthouse 2011). To test this hypothesis, we add an interaction term between RetDur and age at retirement to the list of explanatory factors.
## Conclusion
In this study, we empirically investigated the relationship between retirement duration and cognitive functioning using data for older Irish women collected in the third wave of The Irish Longitudinal Study on Ageing. Because retirement is potentially endogenous with respect to cognition, we used IV estimation. The identifying instrument was the abolition of the so-called marriage bar, which was the legal requirement that women leave paid employment upon getting married. We found a robust negative effect of retirement duration on cognition but found no support for the alternative causal direction. The finding of a negative effect of retirement duration on cognition supports the mental exercise—the “use it or lose it”—hypothesis. However, the effect of retirement duration on cognition was small in magnitude. At least three possible explanations account for our finding of a small effect.
The first explanation is that our measure of retirement duration is possibly prone to measurement error, which in turn could reduce the predictive power of the effect of retirement duration on cognition. Respondents were asked to report the date they ceased working. These self-reported responses may be subject to recall bias. In addition, there might be substantial heterogeneity in what women perceive as being work. Finally, questions on timing of labor market exit were asked slightly differently to respondents according to whether they reported to be retired, unemployed, or disabled, or looking after family. This may have created some distortion in respondents’ self-reports as to when they stopped working. TILDA data might not be of the sufficient quality needed to support the rigorous statistical analysis of the relationship between retirement and cognition.
The second explanation is that the calculation of retirement duration as “time elapsed since last stopped working” likely masks important aspects of employment histories. For example, it is reasonable to hypothesize that the estimated cognitive disadvantage associated with longer retirement duration is a lower bound of the true effect if women who retire gradually (i.e., who reduce hours of work before retirement) engage in equally stimulating cognitive activities in the newly available time before and after retirement. Similarly, it is also reasonable to hypothesize that women who have been retired for longer substituted work activities with equally cognitively stimulating nonwork activities. Information collected in TILDA on part-time versus full-time employment, current nonwork activities, and childbearing and child-rearing was used to test these hypotheses. We did not find strong evidence in favor of the substitution hypothesis. However, to investigate this with rigor would require the collection of detailed employment and life histories, which are not currently a feature of TILDA.
The third explanation is that the cognition variables employed in the analysis are based on cognitive tests that capture processing speed and mental switching, which are central to effective cognitive functioning. These tests have two important advantages. First, they are administered and scored by nurses trained specifically for this purpose. Second, they require effortful processing at the time of assessment and do not require production of previously acquired knowledge (Tucker-Drob and Salthouse 2011). However, these tests have a clear limitation. Previous investigations of aging trajectories for the processing speed factor have reported strong genetic influences on rates of cognitive decline, with little contribution from environmental factors (Finkel et al. 2005; Reynolds et al. 2005). If the validity of this finding is confirmed by future research, then it will not be surprising that the effects of retirement duration on cognition—measured by tests capturing processing speed—are small.
Another finding of our study was that the effects of education and other favorable early-life indicators on later-life cognition were positive and large in magnitude. This finding is encouraging because it suggests that educational attainment and early-life conditions may have important real-world implications for cognitive functioning in adulthood and old age (Tucker-Drob and Salthouse 2011). Whether these factors also protect from age-related cognitive decline is still the subject of debate in the literature and is beyond the scope of this study.
Our analysis was based on older Irish women. It is reasonable to hypothesize that the effects of retirement on cognition might be greater among older Irish men perhaps as a result of men being more oriented toward paid work than women or perhaps as a result of women experiencing very heterogeneous life trajectories. As a consequence, some of the analysis for women was repeated for men using TILDA data. However, we could not investigate the potential endogeneity of retirement among men because the abolition of the marriage bar is only a sensible IV for women. These estimates are not reported here but are available on request. The estimates confirm a similar relationship for men. The magnitude of the relationship is larger for men but is still small. Because it was not possible to explore the endogeneity issue for men, these estimates, albeit encouraging, are only indicative and far from conclusive.
In closing, we believe that our findings are generalizable to other high-income countries. Our analysis confirmed findings of research from other countries regarding the effect of age, education, and early-life socioeconomic conditions on later-life cognition. In this respect, Irish women appear to be no different. For the same reason, we do not believe that the key finding of a small, negative relationship between retirement duration and later-life cognition is not generalizable. However, further research based on additional data—and possibly on alternative sources of exogenous variation—is needed to further clarify the relationship between retirement and later-life cognition. Distinguishing the relative importance of the work environment and the alternative uses of time during retirement for maintaining levels of cognition in later life should be a priority.
## Acknowledgments
The authors would like to thank the funders of TILDA, the Irish Department of Health, the Atlantic Philanthropies, and Irish Life plc for supporting this research. Researchers interested in using TILDA data may access the data at no charge from the following sites: Irish Social Science Data Archive (ISSDA) at University College Dublin (http://www.ucd.ie/issda/data/tilda/), and Interuniversity Consortium for Political and Social Research (ICPSR) at the University of Michigan (http://www.icpsr.umich.edu/icpsrweb/ICPSR/studies/34315).
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# Generate the shortest De Bruijn
A De Bruijn sequence is interesting: It is the shortest, cyclic sequence that contains all possible sequences of a given alphabet of a given length. For example, if we were considering the alphabet A,B,C and a length of 3, a possible output is:
AAABBBCCCABCACCBBAACBCBABAC
You will notice that every possible 3-character sequence using the letters A, B, and C are in there.
Your challenge is to generate a De Bruijn sequence in as few characters as possible. Your function should take two parameters, an integer representing the length of the sequences, and a list containing the alphabet. Your output should be the sequence in list form.
You may assume that every item in the alphabet is distinct.
An example generator can be found here
Standard loopholes apply
• Can the integer representing the length of the sequences be larger than the number of unique letters? Dec 23 '14 at 21:17
• Yes. A binary sequence of length 4 would be 0000111101100101 Dec 23 '14 at 21:18
• "Your challenge is to generate a De Bruijn sequence in as few characters as possible" - Does this mean "code golf" or "get the shortest possible De Bruijn sequence length"? Dec 23 '14 at 21:46
• Both. To qualify, your program must output the shortest sequence possible, but to win, your program must be the shortest. Dec 23 '14 at 21:59
• @xem: usually De Bruijn sequences include wraparound, which is where those missing sequences appear. Dec 25 '14 at 4:14
# Pyth, 31 bytes
This is the direct conversion of the algorithm used in my CJam answer. Tips for golfing welcome!
Mu?G}H+GG+G>Hefq<HT>G-lGTUH^GHk
This code defines a function g which takes two arguments, the string of list of characters and the number.
Example usage:
Mu?G}H+GG+G>Hefq<HT>G-lGTUH^GHkg"ABC"3
Output:
AAABAACABBABCACBACCBBBCBCCC
Code expansion:
M # def g(G,H):
u # return reduce(lambda G, H:
?G # (G if
}H # (H in
>H # slice_end(H,
e # last_element(
f # Pfilter(lambda T:
q # equal(
<HT # slice_start(H,T),
>G # slice_end(G,
-lGT # minus(Plen(G),T))),
UH # urange(H)))))),
^GH # cartesian_product(G,H),
k # "")
Try it here
# CJam, 52 49 48 bytes
This is surprisingly long. This can be golfed a lot, taking in tips from the Pyth translation.
q~a*{m*:s}*{:H\:G_+\#)GGHH,,{_H<G,@-G>=},W=>+?}*
The input goes like
3 "ABC"
i.e. - String of list of characters and the length.
and output is the De Bruijn string
AAABAACABBABCACBACCBBBCBCCC
Try it online here
• Gosh CJam should be banned, it is not just made for one golfing task but it seems for every possible golfing task... Dec 23 '14 at 21:42
• @flawr you should wait for a Pyth answer then :P Dec 23 '14 at 21:44
# CJam, 52 49 bytes
Here is a different approach in CJam:
l~:N;:L,(:Ma{_N*N<0{;)_!}g(+_0a=!}g]{,N\%!},:~Lf=
Takes input like this:
"ABC" 3
and produces a Lyndon work like
CCCBCCACBBCBACABCAABBBABAAA
Try it here.
This makes use of the relation with Lyndon words. It generates all Lyndon words of length n in lexicographic order (as outlined in that Wikipedia article), then drops those whose length doesn't divide n. This already yields the De Bruijn sequence, but since I'm generating the Lyndon words as strings of digits, I also need to replace those with the corresponding letters at the end.
For golfing reasons, I consider the later letters in the alphabet to have lower lexicographic order.
# Jelly, 15 bytes
ṁL*¥Œ!;wⱮẠɗƇṗḢ
Try it online!
Pretty slow, uses a brute force approach with around an $$\O(n \times m \times (n \times m)!)\$$ complexity, where $$\n\$$ is the integer input and $$\m\$$ is the length of the string. Times out if both $$\n\$$ and $$\m\$$ are greater than 3 on TIO
## How it works
The length of the De Bruijn sequence will always be $$\m^n\$$ and each symbol in the provided alphabet will occur the same number of times, $$\m^{n-1}\$$. Therefore, we generate the string with that many symbols, then filter its permutations to find valid De Bruijn sequences.
ṁL*¥Œ!ẋ2wⱮẠʋƇṗḢ - Main link. Takes an alphabet A on the left and n on the right
L - Length of A
* - Raised to the power n
ṁ - Mold A to that length
Œ! - All permutations
ṗ - Powerset; Get all length n combinations of A. Call that C
ʋƇ - Filter the permutations P on the following dyad g(P, C):
ẋ2 - Repeat P twice
Ɱ - For each element E in C:
w - Is it a sublist of P?
Ạ - Is this true for all elements of C?
Ḣ - Take the first one
# JavaScript (ES6) 143
Using Lyndon words, like Martin's aswer, just 3 times long...
F=(a,n)=>{
for(w=[-a[l='length']],r='';w[0];)
{
n%w[l]||w.map(x=>r+=a[~x]);
for(;w.push(...w)<=n;);
for(w[l]=n;!~(z=w.pop()););
w.push(z+1)
}
return r
}
Test In FireFox/FireBug console
console.log(F("ABC",3),F("10",4))
Output
CCCBCCACBBCBACABCAABBBABAAA 0000100110101111
# Python 2, 114 bytes
I'm not really sure how to golf it more, due to my approach.
def f(a,n):
s=a[-1]*n
while 1:
for c in a:
if((s+c)[len(s+c)-n:]in s)<1:s+=c;break
else:break
print s[:1-n]
Try it online
Ungolfed:
This code is a trivial modification from my solution to more recent challenge.
def f(a,n):
s=a[-1]*n
while 1:
for c in a:
p=s+c
if p[len(p)-n:]in s:
continue
else:
s=p
break
else:
break
print s[:1-n]
The only reason [:1-n] is needed is because the sequence includes the wrap-around.
# Powershell, 164 96 bytes
-68 bytes with -match O($n*2^n) instead recursive generator O(n*log(n)) param($s,$n)for(;$z=$s|% t*y|?{"$($s[-1])"*($n-1)+$x-notmatch-join"$x$_"[-$n..-1]}){$x+=$z[0]}$x Ungolfed & test script: $f = {
param($s,$n) # $s is a alphabet,$n is a subsequence length
for(; # repeat until...
$z=$s|% t*y|?{ # at least a character from the alphabet returns $true for expression: "$($s[-1])"*($n-1)+$x-notmatch # the old sequence that follows two characters (the last letter from the alphabet) not contains -join"$x$_"[-$n..-1] # n last characters from the new sequence
}){
$x+=$z[0] # replace old sequence with new sequence
}
$x # return the sequence } @( ,("ABC", 2, "AABACBBCC") ,("ABC", 3, "AAABAACABBABCACBACCBBBCBCCC") ,("ABC", 4, "AAAABAAACAABBAABCAACBAACCABABACABBBABBCABCBABCCACACBBACBCACCBACCCBBBBCBBCCBCBCCCC") ,("ABC", 5, "AAAAABAAAACAAABBAAABCAAACBAAACCAABABAABACAABBBAABBCAABCBAABCCAACABAACACAACBBAACBCAACCBAACCCABABBABABCABACBABACCABBACABBBBABBBCABBCBABBCCABCACABCBBABCBCABCCBABCCCACACBACACCACBBBACBBCACBCBACBCCACCBBACCBCACCCBACCCCBBBBBCBBBCCBBCBCBBCCCBCBCCBCCCCC") ,("ABC", 6, "AAAAAABAAAAACAAAABBAAAABCAAAACBAAAACCAAABABAAABACAAABBBAAABBCAAABCBAAABCCAAACABAAACACAAACBBAAACBCAAACCBAAACCCAABAABAACAABABBAABABCAABACBAABACCAABBABAABBACAABBBBAABBBCAABBCBAABBCCAABCABAABCACAABCBBAABCBCAABCCBAABCCCAACAACABBAACABCAACACBAACACCAACBABAACBACAACBBBAACBBCAACBCBAACBCCAACCABAACCACAACCBBAACCBCAACCCBAACCCCABABABACABABBBABABBCABABCBABABCCABACACABACBBABACBCABACCBABACCCABBABBABCABBACBABBACCABBBACABBBBBABBBBCABBBCBABBBCCABBCACABBCBBABBCBCABBCCBABBCCCABCABCACBABCACCABCBACABCBBBABCBBCABCBCBABCBCCABCCACABCCBBABCCBCABCCCBABCCCCACACACBBACACBCACACCBACACCCACBACBACCACBBBBACBBBCACBBCBACBBCCACBCBBACBCBCACBCCBACBCCCACCACCBBBACCBBCACCBCBACCBCCACCCBBACCCBCACCCCBACCCCCBBBBBBCBBBBCCBBBCBCBBBCCCBBCBBCBCCBBCCBCBBCCCCBCBCBCCCBCCBCCCCCC") ,("01", 3, "00010111") ,("01", 4, "0000100110101111") ,("abcd", 2, "aabacadbbcbdccdd") ,("0123456789", 3, "0001002003004005006007008009011012013014015016017018019021022023024025026027028029031032033034035036037038039041042043044045046047048049051052053054055056057058059061062063064065066067068069071072073074075076077078079081082083084085086087088089091092093094095096097098099111211311411511611711811912212312412512612712812913213313413513613713813914214314414514614714814915215315415515615715815916216316416516616716816917217317417517617717817918218318418518618718818919219319419519619719819922232242252262272282292332342352362372382392432442452462472482492532542552562572582592632642652662672682692732742752762772782792832842852862872882892932942952962972982993334335336337338339344345346347348349354355356357358359364365366367368369374375376377378379384385386387388389394395396397398399444544644744844945545645745845946546646746846947547647747847948548648748848949549649749849955565575585595665675685695765775785795865875885895965975985996667668669677678679687688689697698699777877978878979879988898999") ,("9876543210", 3, "9998997996995994993992991990988987986985984983982981980978977976975974973972971970968967966965964963962961960958957956955954953952951950948947946945944943942941940938937936935934933932931930928927926925924923922921920918917916915914913912911910908907906905904903902901900888788688588488388288188087787687587487387287187086786686586486386286186085785685585485385285185084784684584484384284184083783683583483383283183082782682582482382282182081781681581481381281181080780680580480380280180077767757747737727717707667657647637627617607567557547537527517507467457447437427417407367357347337327317307267257247237227217207167157147137127117107067057047037027017006665664663662661660655654653652651650645644643642641640635634633632631630625624623622621620615614613612611610605604603602601600555455355255155054454354254154053453353253153052452352252152051451351251151050450350250150044434424414404334324314304234224214204134124114104034024014003332331330322321320312311310302301300222122021121020120011101000") ) |% {$s,$n,$e = $_$r = &$f$s $n "$($r-eq$e): \$r"
}
Output:
True: AABACBBCC
True: AAABAACABBABCACBACCBBBCBCCC
True: AAAABAAACAABBAABCAACBAACCABABACABBBABBCABCBABCCACACBBACBCACCBACCCBBBBCBBCCBCBCCCC
True: AAAAABAAAACAAABBAAABCAAACBAAACCAABABAABACAABBBAABBCAABCBAABCCAACABAACACAACBBAACBCAACCBAACCCABABBABABCABACBABACCABBACABBBBABBBCABBCBABBCCABCACABCBBABCBCABCCBABCCCACACBACACCACBBBACBBCACBCBACBCCACCBBACCBCACCCBACCCCBBBBBCBBBCCBBCBCBBCCCBCBCCBCCCCC
True: AAAAAABAAAAACAAAABBAAAABCAAAACBAAAACCAAABABAAABACAAABBBAAABBCAAABCBAAABCCAAACABAAACACAAACBBAAACBCAAACCBAAACCCAABAABAACAABABBAABABCAABACBAABACCAABBABAABBACAABBBBAABBBCAABBCBAABBCCAABCABAABCACAABCBBAABCBCAABCCBAABCCCAACAACABBAACABCAACACBAACACCAACBABAACBACAACBBBAACBBCAACBCBAACBCCAACCABAACCACAACCBBAACCBCAACCCBAACCCCABABABACABABBBABABBCABABCBABABCCABACACABACBBABACBCABACCBABACCCABBABBABCABBACBABBACCABBBACABBBBBABBBBCABBBCBABBBCCABBCACABBCBBABBCBCABBCCBABBCCCABCABCACBABCACCABCBACABCBBBABCBBCABCBCBABCBCCABCCACABCCBBABCCBCABCCCBABCCCCACACACBBACACBCACACCBACACCCACBACBACCACBBBBACBBBCACBBCBACBBCCACBCBBACBCBCACBCCBACBCCCACCACCBBBACCBBCACCBCBACCBCCACCCBBACCCBCACCCCBACCCCCBBBBBBCBBBBCCBBBCBCBBBCCCBBCBBCBCCBBCCBCBBCCCCBCBCBCCCBCCBCCCCCC
True: 00010111
True: 0000100110101111
f=lambda a,n:(a[:n]in a[1:])*a[n:]or max((f(c+a,n)for c in{*a}),key=len)
` | {} |
#### Volume 12, issue 2 (2012)
Recent Issues
The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Free group automorphisms with parabolic boundary orbits
### Arnaud Hilion
Algebraic & Geometric Topology 12 (2012) 933–950
##### Abstract
For $N\ge 4$, we show that there exist automorphisms of the free group ${F}_{N}$ which have a parabolic orbit in $\partial {F}_{N}$. In fact, we exhibit a technology for producing infinitely many such examples.
##### Keywords
automorphism of free group, fixed point, symbolic dynamics
##### Mathematical Subject Classification 2010
Primary: 20E05, 20E36, 37B25, 37E15
Secondary: 20F65, 37B05
##### Publication
Received: 22 September 2011
Revised: 28 January 2012
Accepted: 28 January 2012
Published: 24 April 2012
##### Authors
Arnaud Hilion Mathematiques LATP - UMR 7353 Aix-Marseille Université Avenue de l’escadrille Normandie-Niémen 13397 Marseille Cedex 20 France http://junon.u-3mrs.fr/hilion/ | {} |
# A failure of formats
I have a very long diatribe on the state of document editors that I am working on, and almost anyone that knows me has heard parts of it, as I frequently rail against all of them: MS Word, Google Docs, Dropbox Paper, etc. In the process of writing up my screed I started thinking what a proper document format would look like and realized the awfulness of all the current document formats is a much more pressing problem than the lack of good editors. In fact, it may be that failing to have a standard, robust, and extensible file format for documents might be the single biggest impediment to having good document editing tools. Seriously, even if you designed the world’s best document editor but all it can output is PNGs then how useful if your world’s best document editor?
Let’s start with HTML, as that is probably the most successful document format in the history of the world outside of plain old paper. I can write an HTML page, host it on the web, and it can be read on any computer in the world. That’s amazing reach and power, but if you look at HTML as a universal document format you quickly realize it is stunted and inadequate. Let’s look at a series of comparisons between HTML and paper, starting with simple text, a note to myself:
Now this is something that HTML excels at:
<p>Get Milk</p>
And here HTML is better than paper because that HTML document is easily machine readable. I throw in the word “easily” because I know some ML/AI practitioner will come along and claim they can also “read” the image, but we know there are many orders of magnitude difference in processing power and complexity of those two approaches, so let’s ignore them.
So far HTML is looking good. Let’s make our example a little more complex; a shopping list.
This again is something that HTML is great at:
<h1>Shopping list</h1>
<ul>
<li>Milk</li>
<li>Eggs</li>
</ul>
Not only can HTML represent the text that’s been written, but can also capture the intended structure by encoding it as a list using the <ul> and <li> elements. So now we are expressing not only the text, but also the meaning, and again this representation is machine readable.
At this point we should take a small detour to talk about the duality we are seeing here with HTML, between the markup and the visual representation of that markup. That is, the following HTML:
<h1>Shopping list</h1>
<ul>
<li>Milk</li>
<li>Eggs</li>
</ul>
Is rendered in the browser as:
# Shopping list
• Milk
• Eggs
The markup carries not only the text, but also the semantics. I hesitate to use the term ‘semantics’ because that’s an overloaded term with a long history, particularly in web technology, but that is what we’re talking about. The web browser is able to convert from the markup semantics, <ul> and <li>, into the visual representation of a list, i.e. vertically laying out the items and putting bullets next to them. That duality between meaningful markup in text, distinct from the final representation, is important as it’s the distinction that made search engines possible. And we aren’t restricted to just visual representations, screen readers can also use the markup to guide their work of turning the markup into audio.
But as we make our example a little more complex we start to run into the limits of HTML, for example when we draw a block diagram:
When the web was first invented your only way to add such a thing to web page would have been by drawing it as an image and then including that image in the page:
<img src="server.png" title="Server diagram with two disk drives.">
The image is not very machine readable, even with the added title attribute. HTML didn’t initially offer a native way to create that visualization in a semantically more meaningful way. About a decade after the web came into being SVG was standardized and became available, so you can now write this as:
<svg width="580" height="400" xmlns="http://www.w3.org/2000/svg">
<title>Server diagram with two disk drives.</title>
<g>
<rect height="60" width="107" y="45" x="215" stroke-width="1.5" stroke="#000" fill="#fff"/>
<text font-size="24" y="77" x="235" stroke-width="0" stroke="#000" fill="#000000" id="svg_3">
Server
</text>
<line y2="204" x2="173" y1="105" x1="267" stroke-width="1.5" stroke="#000" fill="none" id="svg_4"/>
<rect height="59" width="125" y="205" x="98" stroke-width="1.5" stroke="#000" fill="#fff" id="svg_5"/>
<rect height="62" width="122" y="199" x="342" stroke-width="1.5" stroke="#000" fill="#fff" id="svg_6"/>
<line y2="197" x2="403" y1="103" x1="268" stroke-width="1.5" stroke="#000" fill="none" id="svg_7"/>
<text font-size="24" y="240" x="119" fill-opacity="null" stroke-opacity="null" stroke-width="0" stroke="#000" fill="#000000" id="svg_8">
Disk 1
</text>
<text stroke="#000" font-size="24" y="236" x="361" fill-opacity="null" stroke-opacity="null" stroke-width="0" fill="#000000" id="svg_9">
Disk 2
</text>
</g>
</svg>
This is a slight improvement over the image. For example, we can extract the title and the text found in the diagram from such a representation, but the markup isn’t what I would call human readable. To get a truly human readable markup of such a diagram we’d need to leave HTML and write it in Graphviz dot notation:
graph {
Server -- "Disk 1";
Server -- "Disk 2";
}
So we’ve already left the capabilities of HTML behind and we’ve only just begun, what about math formulas?
Again, about a decade after the web started MathML was standardized as a way to add math to HTML pages. It’s been 20 years since the MathML specification was released and you still can’t use MathML in your web pages because browser support is so bad.
But even if MathML had been fully adopted and incorporated in to all web browsers, would we be done? Surely not, what about musical notations?
If we want to include notes in a semantically meaningful way on a web page do we have to wait another 10 years for standardization and then hope that browsers actually implement the spec?
You see, the root of the issue is that humans don’t just communicate by text, we communicate by notation; we are continually creating new new notations, and we will never stop creating them. No matter how many FooML markup languages you standardize and stuff into a web browser implementation you will only ever scratch the surface, you will always be leaving out more notations than you include. This is the great failing of HTML, that you cannot define some squiggly set of lines as a symbol and then use that symbol in your markup.
## Is such a thing even possible?
The only markup language that comes even close to achieving this universality of expression is TeX. In TeX it is possible to create you own notations and define how they are rendered and then to use that notation in your document. For example, there’s a TeX package that enables Feynman diagrams:
\feynmandiagram [horizontal=a to b] {
i1 -- [fermion] a -- [fermion] i2,
a -- [photon] b,
f1 -- [fermion] b -- [fermion] f2,
};
Note that both TeX and the \feynmandiagram notation are both human readable, which is an important distinction, as without it you could point at Postscript or PDF as a possible solution. While PDF may be able to render just about anything, the underlying markup in PDF files is not human readable.
I’m also not suggesting we abandon HTML in favor of TeX. What I am pointing out is that there is a serious gap in the capabilities of HTML: the creation and re-use of notation, and if we want HTML to be a universal format for human communication then we need to fill this gap. | {} |
# Chapter 16 - Questions and Problems - Page 771: 16.29
$[H_3O^+] = 2.51 \times 10^{- 5}M$
#### Work Step by Step
1. Calculate the pH. pOH + pH = 14 9.4 + pH = 14 pH = 4.6 2. Find $[H_3O^+]$ $[H_3O^+] = 10^{- 4.6}$ $[H_3O^+] = 2.51 \times 10^{- 5}$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {} |
# Why does bromine water make the viscosity of olive oil increas? [duplicate]
1. Olive oil consists of almost 100% of fat. As you can see in the table of contents below, it includes both unsaturated, monounsaturated and polyunsaturated fats in the oil.
Nutrition declaration per 100 ml (= 92 g) Energy 3390kJ / 823 Kcal Fat 92 g Carbohydrate 0 g Of which saturated fat is 13 g Of which sugars are 0 g Monounsaturated fat 68 g Protein 0 g Polyunsaturated fat 7.2 g Salt 0 g
If you add bromine water, Br2 (aq), to the olive oil, its consistency changes significantly and it becomes more viscous. What is it that happens chemically with the olive oil when adding bromine water and why does it become more viscous?
## marked as duplicate by Karsten Theis, Ivan Neretin, Mithoron, Todd Minehardt, Mathew MahindaratneMay 1 at 1:13
It is a similar process to hydrogenation of plant oil on Raney nickel, producing margarines.
The saturated fats have zig-zag structure of $$\ce{C-C}$$ chain, what makes them good in "molecule spooning", leading to high melting point.
OTOH the cus double bonds of oils are obstacles for "spooning" and the menting point is low.
The bromine is, AFAIK, used to determine the amount of unsaturated bonds in edible oils.
\begin{align} \small \ce{ \\ \small -CH2-CH=CH-CH_2 - + H2 &->[Ni] \small -CH2-CH2-CH2-CH_2 -\\ \small -CH2-CH=CH-CH_2 - + Br2 &-> \small -CH2-CHBr-CHBr-CH_2 -\\ }\end{align}
Fatty acids are components of many types of lipids. Fatty acids are carboxylic acids with very long hydrocarbon chains, usually consists of 12-18 carbon atoms long hydrophobic chain. Olive oil is a liquid obtained from olives, which contains mixture of fatty acids. According to Wikipedia:
The composition of olive oil varies with the cultivar, altitude, time of harvest and extraction process. It consists mainly of oleic acid (up to 83%), with smaller amounts of other fatty acids including linoleic acid (up to 21%) and palmitic acid (up to 20%).
The main component of olive oil, oleic acid, is a monounsaturated fatty acid (FA; See Table) and generally average about 70% in olive oil. Another monounsaturated FA, palmitoleic acid averages in the range of 0.3-3.5%. Olive oil also contains polyunsaturated FAs such as linoleic acid (~15%) and $$\alpha$$-linolenic acid (~0.5%). Only about 20% saturated FAs (see Table)) are present in olive oil and most of them are palmitic acid (~13.0%) and stearic acid (~1.5%).
Structures of Common Fatty Acids (saturated), of which, palmitic (~13.0%) and stearic acid (~1.5%) are present in olive oil: $$\begin{array}{ccc} \text{Name of FA} & \text{# of carbons} & \text{Structure} & \text{Melting Point} \\ \hline \text{Lauric acid} & 12 & \ce{CH3(CH2)10CO2H} & \pu{44 ^{\circ}C} \\ \text{Myristic acid} & 14 & \ce{CH3(CH2)12CO2H} & \pu{58 ^{\circ}C} \\ \text{Palmitic acid } & 16 & \ce{CH3(CH2)14CO2H} & \pu{63 ^{\circ}C} \\ \text{Stearic acid} & 18 & \ce{CH3(CH2)16CO2H} & \pu{70 ^{\circ}C} \\ \hline \end{array}$$
Structures of Common Fatty Acids (unsaturated), three of which, oleic acid (~70%, linoleic acid (~15%), and $$\alpha$$-linolenic acid (~0.5%) are present in olive oil:
$$\begin{array}{ccc} \text{Name of FA} & \text{# of carbons} & \text{Structure} & \text{Melting Point} \\ \hline \text{ Palmitoleic acid} & 16 & \ce{CH3(CH2)5CH=CH(CH2)7CO2H} & \pu{-1 ^{\circ}C} \\ \text{Oleic acid} & 18 & \ce{CH3(CH2)7CH=CH(CH2)7CO2H} & \pu{4 ^{\circ}C} \\ \text{Linoleic acid} & 18 & \ce{CH3(CH2)4CH=CHCH2CH=CH(CH2)7CO2H} & \pu{-5 ^{\circ}C} \\ \text{ Linolenic acid} & 18 & \ce{CH3CH2(CH=CHCH2)2CH=CH(CH2)7CO2H} & \pu{-11 ^{\circ}C} \\ \hline \end{array}$$
According to the melting points of FAs in olive oil, one can conclude that it contains a lot of double bonds (mostly mono-unsaturated). These unsaturated fatty acids contain only cis-double bonds (e.g., oleic and linoleic acid). The presence of cis-double bonds has an important lowering effect on the melting point of the fatty acid (see Table), because cis-double bonds form rigid kinks in the fatty acid chains (see the picture of oleic acid):
Keep in mind that there is no rotation around a double bond, and as a result, the unsaturated fatty acids cannot line up very well to give a regularly arranged crystal structure. On the other hand, saturated fatty acids would line up in a very regular manner, better than that of unsaturated, leading to better van der Waals-forces, thus get closer together (thicker). This is one of the reasons why they are solids in ambient conditions with high melting points (see Table).
Now, what happens when you add bromine water to olive oil? Bromine added to double bond (same reaction as clarification test for unsaturation), making the fatty acid saturated. For example, when $$\ce{Br2}$$ added to $$\ce{C18}$$ oleic acid (m.p.: $$\pu{4 ^{\circ}C}$$), it'd become a resemblance of $$\ce{C18}$$ stearic acid (m.p.: $$\pu{70 ^{\circ}C}$$) with two extra $$\ce{Br}$$ atoms replacing 2 $$\ce{H}$$ atoms. Now you see why olive oil getting thicker! | {} |
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# 11: Chi-Square and ANOVA Tests
This chapter presents material on three more hypothesis tests. One is used to determine significant relationship between two qualitative variables, the second is used to determine if the sample data has a particular distribution, and the last is used to determine significant relationships between means of 3 or more samples.
• 11.1: Chi-Square Test for Independence
• 11.2: Chi-Square Goodness of Fit
• 11.3: Analysis of Variance (ANOVA)
There are times where you want to compare three or more population means. One idea is to just test different combinations of two means. The problem with that is that your chance for a type I error increases. Instead you need a process for analyzing all of them at the same time. This process is known as analysis of variance (ANOVA). The test statistic for the ANOVA is fairly complicated, you will want to use technology to find the test statistic and p-value. | {} |
+0
# precalc
0
44
2
+4
Can someone help me with this very complex question? I don't even know where to start.
Jan 27, 2023
#1
0
Using row properties of matrices:
$$\mathbf{A} \mathbf{u} = (2, 6, -2)$$
$$\mathbf{A} \mathbf{v} = (-2, 8, 12)$$
Jan 27, 2023
#2
0
Hi!
In these type of problems, you should start by writing the givens.
That is, we are given u and v which are "three dimensional vectors", and so we can write:
Let
$$u=\begin{pmatrix} u_1\\ u_2 \\ u_3 \end{pmatrix}$$ , and $$v=\begin{pmatrix} v_1\\ v_2 \\ v_3 \end{pmatrix}$$
Now, we are given that the length of u and v is 2 and 4 respectively. Moreover, the angle between u and v is 120.
These givens should be familiar right? Yes: $$u*v = \left | u \right |\left | v\right |cos(\theta)$$; this is the geometric definition of the dot product right?
Moreover, $$u*v=u_1v_1+u_2v_2+u_3v_3$$ so we can calculate the dot product between u and v using the givens as follows:
$$u*v=u_1v_1+u_2v_2+u_3v_3=2(4)cos(120)=-4$$ (Calculated from the geometric definition).
Now, how to continue this? Well, we are given a matrix's A rows. So why not construct this matrix?
So:
$$A:=\begin{bmatrix} u_1 && u_2 && u_3 \\ v_1 && v_2 && v_3 \\ 3u_1+2v_1 && 3u_2+2v_2 && 3u_3+2v_3 \end{bmatrix}$$
Now, what does the question wants? Au and Av right? Let's start with Au, and writing them yields:
$$A*u:=\begin{bmatrix} u_1 && u_2 && u_3 \\ v_1 && v_2 && v_3 \\ 3u_1+2v_1 && 3u_2+2v_2 && 3u_3+2v_3 \end{bmatrix} *\begin{pmatrix} u_1\\ u_2 \\ u_3 \end{pmatrix}$$
Well, this is a matrix multiplication (3*3 matrix with 3*1 matrix) so the resultant matrix will be (3*1) (A vector).
You may be used to multiply matrices that has numbers right? However, this matrices do not have numbers, rather variables.
But, it is the same procedure u_1 * u_1, u_2*u_2 etc.. , similarly for the second row, third row, and so we get:
$$\implies Au=\begin{pmatrix} u_1^2+u_2^2+u_3^2\\ u_1v_1+u_2v_2+u_3v_3 \\ 3(u_1^2+u_2^2+u_3^2)+2(u_1v_1+u_2v_2+u_3v_3) \end{pmatrix}$$
For the last row, expand everything and group like terms to get what we got above.
(Multiply u by (3u+2v) etc.. and add all of these then factor the common factors).
But notice, the first row in Au is just the length of vector u squared!
And, the second row is the dot product of u and v (which we found above to be -4)
And the third row is just a linear combination of both.
Hence,
$$Au=\begin{pmatrix} 2^2\\ -4\\ 3(2^2)+2(-4)\\ \end{pmatrix} = \begin{pmatrix} 4\\ -4 \\ 4 \end{pmatrix}$$
Now try to do the same steps for Av.
I hope this helps, and if you need further help do not hesitate to ask!
Jan 27, 2023
edited by Guest Jan 27, 2023
edited by Guest Jan 27, 2023 | {} |
# Analysis of Discrete Data #1 – Overview
Posted on September 11, 2014 by
This lesson is an overview of the course content as well as a review of some advanced statistical concepts involving discrete random variables and distributions, relevant for STAT 504 — Analysis of Discrete Data. This Lesson assumes that you have glanced through the Review Materials included in the Start Here! block.
#### Key concepts:
• Discrete data types
• Discrete distributions: Binomial, Poission, Multinomial
• Likelihood & Loglikelhood
• Observed & Expected Information
• Likelihood based Confidence Intervals & Tests: Wald Test, Likelihood-ratio test, Score test
#### Objectives:
• Learn what discrete data are and their taxonomy
• Learn the properties of Binomial, Poission and Multinomial distributions
• Understand the basic principles of likelihood-based inference and how to apply it to tests and intervals regarding population proportions
• Introduce the basic SAS and R code
• Ch.1 Agresti (2007)
• If you are using other textbooks or editions: Ch.1 Agresti (2013, 2002, 1996)
____________________________________________________________________________________________________________________________
The outline below can be viewed as a general template of how to approach data analysis regardless of the type of statistical problems you are dealing with. For example, you can model a continuous response variable such as income, or a discrete response such as a true proportion of U.S. individuals who support new health reform. This approach has five main steps. Each step typically requires an understanding of a number of elementary statistical concepts, e.g., a difference between a parameter to be estimated and the corresponding statistic (estimator).
## 1.1 – Focus of this Course
The focus of this class is multivariate analysis of discrete data. The modern statistical inference has many approaches/models for discrete data. We will learn the basic principles of statistical methods and discuss issues relevant for the analysis of Poisson counts of some discrete distribution, cross-classified table of counts, (i.e., contingency tables), binary responses such as success/failure records, questionnaire items, judge’s ratings, etc. Our goal is to build a sound foundation that will then allow you to more easily explore and learn many other relevant methods that are being used to analyze real life data. This will be done roughly at the introductory level of the required textbook by A. Agresti (2007).
Basic data are discretely measured responses such as counts, proportions, nominal variables, ordinal variables, continuous variables grouped into a small number of categories, etc. Data examples will be used to help illustrate concepts. The “canned” statistical routines and packages in R and SAS will be introduced for analysis of data sets, but the emphasis will be on understanding the underlying concepts of those procedures. For more detailed theoretical underpinnings you can read A. Agresti (2012).
We will focus on two kinds of problems.
1) The first broad problem deals with describing and understanding the structure of a (discrete) multivariate distribution, which is the joint and marginal distributions of multivariate categorical variables. Such tasks may focus on displaying and describing associations between categorical variables by using contingency tables, chi-squared tests of independence, and other similar methods. Or, we may explore finding underlying structures, possibly via latent variable models.
2) The second problem is a sort of “generalization” of regression with a distinction between response and explanatory variables where the response is discrete. Predictors can be all discrete, in which case we may use log- linear models to describe the relationships. Predictors can also be a mixture of discrete and continuous variables, and we may use something like logistic regression to model the relationship between the response and the predictors. We will explore certain types of Generalized Linear Models, such as logistic and Poisson regressions.
The analysis grid below highlights the focus of this class with respect to the models that you should already be familiar with.
## 1.2 – Discrete Data Types and Examples
### Categorical/Discrete/Qualitative data
Measures on categorical or discrete variables consist of assigning observations to one of a number of categories in terms of counts or proportions. The categories can be unordered or ordered (see below).
#### Counts and Proportions
Counts are variables representing frequency of occurrence of an event:
• Number of students taking this class.
• Number of people who vote for a particular candidate in an election.
Proportions or “bounded counts” are ratios of counts:
• Number of students taking this class divided by the total number of graduate students.
• Number of people who vote for a particular candidate divided by the total number of people who voted.
Discretely measured responses can be:
• Nominal (unordered) variables, e.g., gender, ethnic background, religious or political affiliation
• Discrete interval variables with only a few values, e.g., number of times married
• Continuous variables grouped into small number of categories, e.g., income grouped into subsets, blood pressure levels (normal, high-normal etc)
We we learn and evaluate mostly parametric models for these responses.
#### Measurement Scale and Context
Interval variables have a numerical distance between two values (e.g. income)
Measurement hierarchy:
• nominal < ordinal < interval
• Methods applicable for one type of variable can be used for the variables at higher levels too (but not at lower levels). For example, methods specifically designed for ordinal data should NOT be used for nominal variables, but methods designed for nominal can be used for ordinal. However, it is good to keep in mind that such analysis method will be less than optimum as it will not be using the fullest amount of information available in the data.
• Nominal: pass/fail
• Ordinal: A,B,C,D,F
• Interval: 4,3,2.5,2,1
Note that many variables can be considered as either nominal or ordinal depending on the purpose of the analysis. Consider majors in English, Psychology and Computer Science. This classification may be considered nominal or ordinal depending whether there is an intrinsic belief that it is ‘better’ to have a major in Computer Science than in Psychology or in English. Generally speaking, for a binary variable like pass/fail ordinal or nominal consideration does not matter.
Context is important! The context of the study and the relevant questions of interest are important in specifying what kind of variable we will analyze. For example,
• Did you get a flu? (Yes or No) — is a binary nominal categorical variable
• What was the severity of your flu? ( Low, Medium, or High) — is an ordinal categorical variable
Based on the context we also decide whether a variable is a response (dependent) variable or an explanatory (independent) variable.
Discuss the following question on the ANGEL Discussion Board:
Why do you think the measurement hierarchy matters and how does it influence analysis? That is, why we recommend that statistical methods/models designed for the variables at the higher level not be used for the analysis of the variables at the lower levels of hierarchy?
#### Contingency Tables
• A statistical tool for summarizing and displaying results for categorical variables
• Must have at least two categorical variables, each with at least two levels (2 x 2 table)May have several categorical variables, each at several levels (I1 × I2 × I3 × … × Ik tables) Place counts of each combination of the variables in the appropriate cells of the table.
Here are a few simple examples of contingency tables.
A university offers only two degree programs: English and Computer Science. Admission is competitive and there is a suspicion of discrimination against women in the admission process. Here is a two-way table of all applicants by sex and admission status. These data show an association between the sex of the applicants and their success in obtaining admission.
Male Female Total Admit 35 20 55 Deny 45 40 85 Total 80 60 140
#### Example: Number of Delinquent Children by the County and the Head of Household Education Level
This is another example of a two-way table but in this case 4×4 table. The variable County could be treated as nominal, where as the Education Level of Head of Household can be treated as ordinal variable. Questions to ask, for example: (1) What is the distribution of a number of delinquent children per county given the education level of the head of the household? (2) Is there a trend of where the delinquent children reside given the education levels?
County Low Medium High Very High Total Alpha 15 1 3 1 20 Beta 20 10 10 15 55 Gamma 3 10 10 2 25 Delta 12 14 7 2 35 Total 50 35 30 20 135
• Ordinal and nominal variables
• Fixed total
#### Example: Census Data
Source: American Fact Finder website (U.S. Census Bureau: Block level data)
This is an example of a 2×2×4 three-way table that cross-classifies a population from a PA census block by Sex, Age and Race where all three variables are nominal.
#### Example: Clinical Trial of Effectiveness of an Analgesic Drug
Source: Koch et al. (1982)
• This is a four-way table (2×2×2×3 table) because it cross-classifies observations by four categorical variables: Center, Status, Treatment and Response
• Fixed number of patients in two Treatment groups
• Small counts
We will see throughout this course that there are many different methods to analyze data that can be represented in coningency tables.
### Example of proportions in the news
You should be already familiar with a simple analysis of estimating a population proportion of interest and computing a 95% confidence interval, and the meaning of the margin or error (MOE).
Notation:
• Population proportion = p = sometimes we use π
• Population size = N
• Sample proportion = $\hat{p}=X/n$=# with a trait / total #
• Sample size = n
• X is the number of units with a particular trait, or number of success.
The Rule for Sample Proportions
• If numerous samples of size n are taken, the frequency curve of the sample proportions $(p^\prime s)$ from the various samples will be approximately normal with the mean p and standard deviation $\sqrt{p(1-p)/n}$
• $\hat{p}\sim N(p,p(1-p)/n)$
## 1.3 – Discrete Distributions
Statistical inference requires assumptions about the probability distribution (i.e., random mechanism, sampling model) that generated the data. For example for a t-test, we assume that a random variable follows a normal distribution. For discrete data key distributions are: Bernoulli, Binomial, Poisson and Multinomial. A more or less thorough treatment is given here. The mathematics is for those who are interested. But the results and their applications are important.
Recall, a random variable is the outcome of an experiment (i.e. a random process) expressed as a number. We use capital letters near the end of the alphabet (X, Y, Z, etc.) to denote random variables. Random variables are of two types: discrete and continuous. Here we are interested in distributions of discrete random variables.
A discrete random variable X is described by a probability mass functions (PMF), which we will also call “distributions,” f(x)=P(X =x). The set of x-values for which f (x) > 0 is called the support. Support can be finite, e.g., X can take the values in {0,1,2,…,n} or countably infinite if X takes values in {0,1,…}. Note, if the distribution depends on unknown parameter(s) θ we can write it as f (x; θ) (preferred by frequentists) or f(x| θ) (preferred by Bayesians).
Here are some distributions that you may encounter when analyzing discrete data.
#### Bernoulli distribution
The most basic of all discrete random variables is the Bernoulli. X is said to have a Bernoulli distribution if X = 1 occurs with probability π and X = 0 occurs with probability 1 − π ,
$f(x)=\left\{\begin{array} {cl} \pi & x=1 \\ 1-\pi & x=0 \\ 0 & \text{otherwise} \end{array} \right.$
Another common way to write it is: $f(x)=\pi^x (1-\pi)^{1-x}\text{ for }x=0,1$
Suppose an experiment has only two possible outcomes, “success” and “failure,” and let π be the probability of a success. If we let X denote the number of successes (either zero or one), then X will be Bernoulli. The mean of a Bernoulli is
$E(X)=1(\pi)+0(1-\pi)=\pi$
and the variance of a Bernoulli is
$V(X)=E(X^2)-[E(X)]^2=1^2\pi+0^2(1-\pi)-\pi^2=\pi(1-\pi)$
#### Binomial distribution
Suppose that $X_1, X_2,\dots,X_n$ are independent and identically distributed (iid) Bernoulli random variables, each having the distribution
$f(x_i|\pi)=\pi^{x_i}(1-\pi)^{1-x_i}\text{ for }x_i=0,1\; \text{and }\; 0\leq\pi\leq 1$
Let $X=X_1+X_2+\dots+X_n$. Then X is said to have a binomial distribution with parameters $n$ and $p$,
$X\sim \text{Bin}(n,\pi)$
Suppose that an experiment consists of n repeated Bernoulli-type trials, each trial resulting in a “success” with probability π and a “failure” with probability 1 − π . For example, toss a coin 100 times, n=100. Count the number of times you observe heads, e.g. X=# of heads. If all the trials are independent—that is, if the probability of success on any trial is unaffected by the outcome of any other trial—then the total number of successes in the experiment will have a binomial distribution, e.g, two coin tosses do not affect each other. The binomial distribution can be written as
$f(x)=\dfrac{n!}{x!(n-x)!} \pi^x (1-\pi)^{n-x} \text{ for }x_i=0,1,2,\ldots,n,\; \text{and }\; 0\leq\pi\leq 1.$
The Bernoulli distribution is a special case of the binomial with n = 1. That is, XBin(1,π) means that X has a Bernoulli distribution with success probability π.
One can show algebraically that if X∼Bin(1,π) then E(X)=nπ and V(X)=nπ(1−π). An easier way to arrive at these results is to note that $X=X_1+X_2+\dots+X_n$ where $X_1,X_2,\dots,X_n$ are (iid) Bernoulli random variables. Then, by the additive properties of mean and variance,
$E(X)=E(X_1)+E(X_2)+\cdots+E(X_n)=n\pi$
and
$V(X)=V(X_1)+V(X_2)+\cdots+V(X_n)=n\pi(1-\pi)$
Note that X will not have a binomial distribution if the probability of success π is not constant from trial to trial, or if the trials are not entirely independent (i.e. a success or failure on one trial alters the probability of success on another trial).
$\text{if }X_1\sim \text{Bin}(n_1,\pi) \text{ and }X_2\sim \text{Bin}(n_2,\pi),\text{ then }X_1+X_2 \sim \text{Bin}(n_1+n_2,\pi)$
As n increases, for fixed π, the binomial distribution approaches normal distribution N(nπ,nπ(1−π)).
For example, if we sample without replacement from a finite population, then the hypergeometric distribution is appropriate.
#### Hypergeometric distribution
Suppose there are n objects. n1 of them are of type 1 and n2 = n − n1 of them are of type 2. Suppose we draw m (less than n) objects at random and without replacement from this population. A classic example, is having a box with n balls, n1 are red and n2 are blue. What is the probability of having t red balls in the draw of m balls? Then the PMF of N1=t is
$p(t) = Pr(N_1 = t) =\frac{\binom{n_1}{ t}\binom{n_2}{m-t}}{\binom{n}{m}},\;\;\;\; t \in [\max(0, m-n_2); \min(n_1, m)]$
The expectation and variance of $N_1$ are given by: $E(N_1) =\frac{n_1 m}{n}$ and $V(N_1)=\frac{n_1n_2m(n-m)}{n^2(n-1)}$
#### Poisson distribution
Let XPoisson(λ) (this notation means “X has a Poisson distribution with parameter λ”), then the probability distribution is
$f(x|\lambda)= Pr(X=x)= \frac{\lambda^x e^{-\lambda}}{x!}, x=0,1,2,\ldots, \mbox{and}, \lambda>0.$
Note that E(X)=V(X)=λ, and the parameter λ must always be positive; negative values are not allowed.
The Poisson distribution is an important probability model. It is often used to model discrete events occurring in time or in space.
The Poisson is also limiting case of the binomial. Suppose that XBin(n,π) and let n and π0 in such a way that nπλ where λ is a constant. Then, in the limit, XPoisson(λ). Because the Poisson is limit of the Bin(n,π), it is useful as an approximation to the binomial when n is large and π is small. That is, if n is large and π is small, then
$\dfrac{n!}{x!(n-x)!}\pi^x(1-\pi)^{n-x} \approx \dfrac{\lambda^x e^{-\lambda}}{x!}$
where λ=nπ. The right-hand side of (1) is typically less tedious and easier to calculate than the left-hand side.
For example, let X be the number of emails arriving at a server in one hour. Suppose that in the long run, the average number of emails arriving per hour is λ. Then it may be reasonable to assume XP(λ). For the Poisson model to hold, however, the average arrival rate λ must be fairly constant over time; i.e., there should be no systematic or predictable changes in the arrival rate. Moreover, the arrivals should be independent of one another; i.e., the arrival of one email should not make the arrival of another email more or less likely.
When some of these assumptions are violated, in particular if there is a presence of overdispersion (e.g., observed variance is greater than what the model assume), the Negative Binomial distribution can be used instead of Poisson.
Overdispersion
Count data often exhibit variability exceeding that predicted by the binomial or Poisson. This phenomenon is known as overdispersion.
Consider, for example the number of fatalities from auto accidents that occur next week in the Center county, PA. The Poisson distribution assumes that each person has the
same probability of dying in an accident. However, it is more realistic to assume that these probabilities vary due to
• whether the person was wearing a seat belt
• time spent driving
• where they drive (urban or rural driving)
Person-to-person variability in causal covariates such as these cause more variability than predicted by the Poisson distribution.
Let X be a random variable with conditional variance V(X|λ). Suppose λ is also a random variable with θ=E(λ). Then E(X)=E[E(X|λ)] and V(X)=E[V(X|λ)]+V[E(X|λ)]
For example, when X|λ has a Poisson distribution, then E(X)=E[λ]=θ (so mean stays the same) but the V(X)=E[λ]+V(λ)=θ+V(λ)>θ (the variance is no longer θ but larger).
When X|π is a binomial random variable and πBeta(α,β). Then $E(\pi)=\frac{\alpha}{\alpha+\beta}=\lambda$ and $V(\pi)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$.
Thus, E(X)=nλ (as expected the same) but the variance is larger V(X)=nλ(1λ)+n(n1)V(π)>nλ(1λ).
#### Negative-Binomial distribution
When the data display overdispersion, the analyst is more likely to use the negative-binomial distribution instead of Poission to model the data.
Suppose a random variable X|λPoisson(λ) and λGamma(α,β). Then the joint distribution of X and λ is:
$p(X=k,\lambda)=\frac{\beta^\alpha}{\Gamma(\alpha)k!}\lambda^{k+\alpha-1}\exp^{-(\beta+1)\lambda}$
Thus the marginal distribution of X is negative-binomial (i.e., Poisson-Gamma mixture):
$\begin{eqnarray} p(X=k)&=&\frac{\beta^\alpha}{\Gamma(\alpha)k!}\int^{\infty}_0\lambda^{k+\alpha-1}\exp^{-(\beta+1)\lambda} d\lambda\\ & = & \frac{\beta^\alpha}{\Gamma(\alpha)k!} \frac{\Gamma(k+\alpha)}{(\beta+1)^{(k+\alpha)}} \\ & = & \frac{\Gamma(k+\alpha)}{\Gamma(\alpha)\Gamma(k+1)}(\frac{\beta}{\beta+1})^\alpha(\frac{1}{\beta+1})^k \end{eqnarray}$
with $E(X)=E[E(X|\lambda)]=E[\lambda]=\frac{\alpha}{\beta}$
$V(X)=E[var(X|\lambda)]+var[E(X|\lambda)]=E[\lambda]+var[\lambda]=\frac{\alpha}{\beta}+\frac{\alpha}{\beta^2}=\frac{\alpha}{\beta^2}(\beta+1).$
#### Beta-Binomial distribution
A family of discrete probability distributions on a finite support arising when the probability of a success in each of a fixed or known number of Bernoulli trials is either unknown or random. For example, the researcher believes that the unknown probability of having flu π is not fixed and not the same for the entire population, but it’s yet another random variable with its own distribution. For example, in Bayesian analysis it will describe a prior belief or knowledge about the probability of having flu based on prior studies. Below X is what we observe such as the number of flu cases.
Suppose X|πBin(n,π) and πBeta(α,β). Then the marginal distribution of X is that of beta-binomial random variable
$\begin{eqnarray} P(X=k)& = & \binom{n}{k}\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\int^{1}_0\pi^{k+\alpha-1}(1-\pi)^{n+\beta-k-1} d\pi\\ & = & \frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)}\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\frac{\Gamma(\alpha+k)\Gamma(n+\beta-k)}{\Gamma(n+\alpha+\beta)} \end{eqnarray}$
with
$E(X)=E(n\pi)=n\frac{\alpha}{\alpha+\beta}$
$Var(X)=E[n\pi(1-\pi)]+var[n\pi]=n\frac{\alpha\beta(\alpha+\beta+n)}{(\alpha+\beta)^(\alpha+\beta+1)}$ | {} |
# Is ESG a Factor?
July 2020
Read Time: 20 min
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Key Points
• Increasingly, investors are asking if ESG is a factor. We answer this question using the criteria set forth by our Research Affiliates colleagues in their 2016 Graham and Dodd Scroll–winning article, “Will Your Factor Deliver? An Examination of Factor Robustness and Implementation Costs.” We conclude that ESG is not a factor.
• We do believe, however, that ESG could be a powerful theme as new owners of capital—in particular, women and millennials—prioritize ESG in their portfolios over the next two decades. Progress in aligning definitions of “good” and “bad” ESG companies will also enhance the ability of the ESG theme to deliver positive investor outcomes.
• We conclude that ESG does not need to be a factor for investors to achieve their ESG and performance goals.
Abstract
As we hit the halfway point of this remarkable year, the health of our planet, the well-being of our communities, and the necessity for meaningful societal change are all top of mind and assuming a greater sense of urgency. Accordingly, many investors desire to take personal action by incorporating environmental, social, and governance (ESG) considerations into their investment portfolios. Unfortunately, they are confronted by a confusing ESG landscape with conflicting claims—similar to the multitude of competing health care studies. This confusion may be slowing down their good intentions. As a factor index provider with ESG offerings, we attempt to answer the question “Is ESG a factor?” by synthesizing what we, and our colleagues, have discovered over the years.
”Can drinking red wine daily stave off heart disease? A newly released study answers that very question. We’ll cover it right after this commercial break.” Does this teaser sound familiar from your favorite morning television or radio show? Does this attention-grabber peak your interest to wait a few minutes, bear the commercials, and hear the story? The media has a natural inclination to use science to engage audiences. Consequently, we’re bombarded with new studies, especially as they relate to our health, a topic of interest to everyone and, of course, top of mind today.
The findings may entertain, but do they inform? Nagler (2014) finds that contradictory scientific claims on red wine, coffee, fish, and vitamins, all touted by the media, led to substantial confusion on the part of consumers. Indeed, the claims led to such confusion that many consumers grew skeptical of even vetted health advice such as exercising and eating fruits and vegetables. Ironically, learning more about nutrition via competing claims led to more confusion, lack of trust, and less likely adoption of better eating and exercising habits.
As we hit the halfway point of a remarkable 2020 and become more acclimated to our new circumstances, we’re concerned about the health of our planet, the well-being of our communities, and the necessity for meaningful societal change. Accordingly, many desire to put these concerns into their investment portfolios using environmental, social, and governance (ESG) considerations and tools. Investors, however, find a confusing ESG landscape with conflicting claims—similar to the multitude of competing health care studies—that may be slowing down good intentions.
As an example, it was John’s turn to represent Research Affiliates at the annual Inside ETFs event in Florida earlier this year.1 The overwhelming points of emphasis from both ETF and index providers throughout the presentations were factor investing and ESG. Several sessions covered one or the other, often both. Regardless of whether the headliner was ESG or factor investing, inevitably a question popped up at the end of the session from either the moderator or the audience: “Is ESG a factor?” If John heard the question six times, he’d venture to guess he heard more than 12 answers! Accordingly, as a factor index provider with ESG offerings, we attempt to answer the question, synthesizing what we, and our colleagues, have discovered over the years.
## What Is a Factor, and Can We Count on It in the Future?
Factors are stock characteristics associated with a long-term risk-adjusted return premium. An example is the value premium, which rewards investors who buy stocks that have a low price relative to their fundamentals. Two theories are advanced to explain the value effect: one is risk based and the other is behavior based.
The risk-based explanation posits that value companies are cheap for a reason, such as lower profitability and/or greater leverage, and thus investors require that they earn a premium to compensate for the risk of investing in them (a risk premium). The behavioral-based explanation posits that investor biases, such as being overly pessimistic about value companies and overly optimistic about growth companies, create stock mispricings, and that value stocks outperform once investors’ expectations are not met and mean reversion occurs.
Popular factors, such as value, low beta, quality, and momentum, have been well documented and vetted by both academics and practitioners. Research by Beck et al. (2016) provides a useful framework for determining if a factor is robust. For ESG to be a factor, it should satisfy these three critical requirements:
1. A factor should be grounded in a long and deep academic literature.
2. A factor should be robust across definitions.
3. A factor should be robust across geographies.
A factor should be grounded in a long and deep academic literature. Traditional factors, such as value, low beta, and momentum, have been thoroughly researched and have a track record spanning several decades; very little debate currently exists regarding their robustness. Beyond the size factor,2 all of the factors in the following table have a positive CAPM alpha and are statistically significant at the 95% t-stat level (1.96).
In examining the vast body of research on ESG, we find little agreement regarding its robustness in earning a return premium for investors. Research by Clark, Feiner, and Viehs (2015), Friede, Busch, and Bassen (2015), and Khan, Serafeim, and Yoon (2016) finds that ESG is additive to returns, while research by Brammer, Brooks, and Pavelin (2006), Fabozzi, Ma, and Oliphant (2008), and Hong and Kacperczyk (2009) demonstrates that ESG detracts from returns. Neither is there evidence to suggest a risk-based or behavioral-based explanation for the ESG factor.
Arguments are put forth that certain situations could lead to positive ESG-related stock price movements, such as increased popularity of strong ESG companies as more investors adopt ESG (more on this topic later). These price movements, however, would be one-time adjustments and cannot be expected to deliver a reliable and robust premium over time.
ESG is not an equity return factor in the traditional, academic sense.
Factors should be robust across definitions. Slight variations in the definition of a factor should still produce similar performance results. Using the value factor as an example, the three valuation metrics of price-to-book ratio, price-to-earnings ratio, and price-to-cash flow ratio all yield similar performance results in assessing the factor’s long-horizon performance.
ESG has no common standard definition and is a broad term that encapsulates a range of themes and subthemes.3 ESG ratings providers examine hundreds of metrics when determining a company’s ESG score. Conducting a quick web search yields several ESG strategies whose underlying themes are quite distinct and different. These index strategies align more closely with investor preferences than with a particular factor.
To illustrate this, we construct a simple test on four variants of ESG definitions. We build long–short portfolios by selecting the top 30% and bottom 30% of US companies by market capitalization each year, after ranking by overall ESG rating. We also build three similarly constructed long–short portfolios, ranking companies on each individual ESG characteristic of environmental, social, and governance.4 None of these strategies displays a materially positive CAPM alpha except for the environmental long–short strategy, and no strategy is statistically significant at the 95% t-stat level (1.96).
Unfortunately, none of the simulated strategies we tested has a long track record because the ESG data history is quite short. This lack of history is a significant impediment to conducting research in ESG investing, limiting our study period to 11 years from July 2009 to June 2020. Because multiple decades of data are needed to conduct a proper test, the lack of significance in the t-values is not surprising. Only after several decades of quality ESG data will it be possible to accurately test the claim that ESG is a robust factor.
In addition to the problem of a short data history, the lack of consistency among ESG ratings providers also hinders our ability to determine if ESG is a robust factor. Research Affiliates published findings earlier this year that showed the correlation of company ratings between ESG ratings providers is low (Li and Polychronopoulos, 2020). We illustrated this by comparing two companies, Wells Fargo and Facebook, and showed that one ESG ratings provider rates Wells Fargo positively and Facebook negatively, while a second ratings provider ranked them the opposite way. In addition, we demonstrated that a portfolio construction process using the same methodology, but different ESG ratings providers, can yield different results. While beyond the scope of this article, had we used a different ESG ratings provider for the analysis in the preceding table, we likely would have gotten different results!
Factors should be robust across geographies. We conduct the same study using European companies. The results are largely consistent with the US results. None of the strategies tested has a materially positive CAPM alpha except for the environmental strategy, and no strategy tested exhibits statistically significant CAPM alpha at the 95% t-stat level.
We should note that for the US and European analysis we conduct a simple single-factor linear regression against the market return. In the appendix we present the results of a stricter test using a multi-factor regression that incorporates the value, size, profitability, investment, momentum, and low beta factors. The multi-factor regression results indicate low or negative alpha for the majority of the strategies.
Having put ESG investing strategies through a framework to assess factor robustness, we find that ESG fails all three tests outlined by Beck et al. (2016): 1) evidence of an ESG return premium is not supported by a long and deep academic literature, 2) ESG performance results are not robust across definitions, and 3) ESG performance results are not robust across regions.
## ESG Is Not a Factor, but Could Be a Powerful Theme
Even though we are unable to apply the factor framework to ESG, these strategies, however heterogeneous, may still produce superior returns. Non-robust, and even robustly negative, strategies will invariably cycle through periods—think three-to-five year stretches—of outperformance. And over the very long term, possibly decades, stocks that rank well on ESG criteria may also outperform.
We witness two principle arguments in favor of superior risk-adjusted returns for companies that rate well on ESG metrics. First, as some claim, there may be latent risks in companies that rate poorly on ESG metrics (Orsagh et al., 2018). In other words, ESG risk needs to be incorporated into security selection.
Let’s consider carbon. Historical fundamental analysis developed during a predominantly stable climate backdrop may miss the investment risk associated with carbon and thereby deliver poor results if the risk materializes. Coal has been a declining source of energy production in the United States for years, accounting for 52% of the nation’s total electricity generation in 1990, but just 23% at the end of 2019.5 The percentage will continue to decline as energy providers move toward cleaner and more-energy-efficient alternatives to combat climate change, leaving coal companies with assets of decreasing value. Investment managers who do not consider and integrate the ESG risk of, in this case, climate change may be blindsided.
“The theme is the massive coming adoption of ESG investing on the part of new owners of capital.”
Not recognizing a specific type of risk implies a mispricing effect. This mispricing seems to be highly idiosyncratic in nature and probably best exploited via the forward-looking framework of active management. Such “ESG alpha” has the potential to be sizeable, especially if very few managers are incorporating ESG criteria into their investment processes—but that’s not the case. According to Cerulli, 83% of investment managers are embedding ESG criteria into their fundamental processes.6
At the time of this publication, over 2,200 investment managers have signed on to the United Nations (UN) PRI | Principles for Responsible Investing, which encourages signatories to “incorporate ESG issues into investment analysis and decision-making processes.” Indeed, investment manager signatories managed approximately US$80 trillion as of March 31, 2020.7 Such widespread use of ESG criteria in the investment management process means that identifying ESG skill will likely be as difficult as identifying other types of investor skill.8 Neither does it speak to the ability of investors to harvest the alpha, if found. Will investors have the patience to wait out manager ESG risk assessments, especially given the very long horizon for some of these risks? A large shift in investor preference toward ESG is occurring as two distinct groups—women and millennials—take greater control of household assets. Accordingly, Bank of America (2019) recently noted a “tsunami of assets is poised to invest in ‘good’ stocks” and concluded that “three critical investor cohorts care deeply about ESG: women, millennials, and high net worth individuals. Based on demographics, we conservatively estimate over$20tn of asset growth in ESG funds over the next two decades—equivalent to the S&P 500 today.”9 Similarly, an Accenture study concluded that US30 trillion in assets will change hands, a staggering amount which, at its peak between 2031 and 2045, will witness 10% of total US wealth transferred every five years.10 Not only are investor preferences shifting in favor of ESG strategies, regulatory efforts in Europe aim to bring greater standardization and transparency to ESG products, which is likely to increase demand. As of 2019, UK government pension funds are required to integrate ESG considerations into their investment management approach (McNamee, 2019). Starting in March 2021, the European Union will require investment managers to provide ESG disclosures related to their investment products. The effort “aims to enhance transparency regarding integration of environmental, social, and governance matters into investment decisions and recommendations” (Maleva-Otto and Wright, 2020). In 2018, the European Commission set up a Technical Expert Group tasked with several ESG initiatives including creating index methodology requirements for low carbon benchmarks, increasing transparency in the green bond market, and creating an EU taxonomy to help companies transition to a low carbon economy. Outside of Europe there has been less movement on the regulatory front, but good progress made on standard setting. In the United States, public pension funds have taken the lead on ESG integration and in 2018 held 54% of all ESG-related investments in the United States (Bradford, 2019). The UN has created the Sustainable Development Goals, a blueprint for improving the planet, both environmentally and socially, by 2030. The 17 goals—including reducing poverty, improving education, creating affordable and clean energy, and creating sustainable cities and communities—have been adopted by all UN member states. “ESG does not need to be a factor for investors to achieve their ESG and performance goals.” The numbers are large, and the implication that the new owners of wealth will favor "good" ESG stocks will in turn likely lead to a very different supply–demand dynamic than in the past. More demand for good ESG companies may result in an upward, one-time positive shock to relative valuations of these companies and the funds that invest in them. We previously discussed that factors and smart beta strategies can experience such a revaluation alpha (Arnott et al., 2016).11 This is classic thematic investing, following in the footsteps of cloud, artificial intelligence, and robotics themes, but it’s not factor investing. The theme in this case is the massive coming adoption of ESG investing on the part of new owners of capital. Getting ahead of that demand could be substantially profitable on two conditions. First, the perceived demand is not already reflected in stock prices. Second, the market’s perception of good ESG companies is fairly consistent so that these inflows more or less benefit the same companies. As we have explained, we currently see incredibly inconsistent definitions of good and bad ESG companies. Yes, a rising tide lifts all boats, but they all have to be in the water and in the same harbor! It may very well be that the best options for thematic investing in ESG are for narrower—and therefore homogenous—groups of securities. Low carbon, sustainable forestry, or gender equality may be easier to exploit in a thematic manner than the entire ESG company universe. ## Incorporate ESG into a Variety of Equity Index Strategies At Research Affiliates, we believe ESG is an important investing consideration despite dismissing it as a factor or lacking confidence in its ability to currently deliver as a theme. One of our core investment beliefs is that investor preferences are broader than risk and return. As value investors, we believe that prices vary around fair value and that investing in unpopular companies and not following the herd is a strategy that will be rewarded as prices mean revert over a market cycle (Brightman, Masturzo, and Treussard, 2014). Of course, investor preferences extend beyond value investing, and as we have shown, many investors have a preference for ESG strategies for many reasons, such as the desire to bring about societal change, mindfulness of the environment, promotion of good corporate governance, or all of the above. Investors can satisfy their ESG preferences while still maintaining the characteristics of their preferred investment strategy. We illustrate this by comparing the characteristics of three strategies: RAFI Fundamental Developed Index, RAFI ESG Developed Index, and RAFI Diversity & Governance Developed Index. All three strategies utilize the Fundamental Index approach, which selects and weights companies by fundamental measures of company size rather than market capitalization. The RAFI Fundamental Developed Index does not incorporate any ESG considerations. The RAFI ESG Developed Index is a broad-based ESG index that tilts toward companies with strong overall ESG scores. The RAFI Diversity & Governance Developed Index reflects a preference for companies that score well across several metrics of gender diversity and strong corporate governance. All three strategies share similar characteristics. The Fundamental Index methodology is a contrarian approach that uses fundamental weights to act as rebalancing anchors against market price movements. Fundamental Index strategies typically trade at a discount to cap-weight. All three strategies maintain similar valuation discounts and dividend yields, with the only noticeable differences being index concentration. Given that the ESG and Diversity & Governance indices exclude many securities that perform poorly across multiple ESG considerations, they have a much higher active share. In addition, all three strategies maintain similar factor exposures, mainly positive loadings on value and negative loadings on momentum. The Diversity & Governance index, which incorporates a tilt toward lower-volatility companies, also has a high exposure to the low beta factor. The bottom line is that investors who would like to incorporate ESG into their investment decisions can do so and retain their desired investment characteristics. Accordingly, they likely maintain a similar expected return outcome (although with some short-term deviations in performance) whether their preferred approach is traditional passive, smart beta, or active. ESG does not need to be a factor for investors to achieve their ESG and performance goals. ## Conclusion Let’s hope the events of 2020—Australian wildfires, a global pandemic, a searing recession, and social protests denouncing racial inequality—lead to positive societal changes and perhaps more refinement to and greater consistency in ESG ratings. Indeed, once the dust settles, we expect these forces to accelerate an already simmering ESG investment movement—but action will require clarity around exactly what ESG is and what it is not. Currently, various stakeholders are sending a whole host of mixed messages. Investors, particularly fiduciaries, need education and alignment. If ESG remains a heterogeneous basket of claims, we will likely never see it fulfill its vast promise. We have debunked one of these messages: ESG is not an equity return factor in the traditional, academic sense. We have shown that, unlike vetted factors such as value, low beta, quality, or momentum, ESG strategies lack sufficient historical data, impeding our ability to make a similar conclusion of robustness. Nevertheless, ESG can be a very powerful theme in the portfolio management process in the years ahead. Furthermore, we believe a variety of equity styles can very effectively capture ESG criteria. We believe our conclusions will add clarity around the question “Is ESG a factor?” and therefore quicken the pace of ESG integration in equity portfolios. ## Appendix We examine the results of a multi-factor regression compared to long–short ESG portfolios in the United States and Europe. This approach results in low or negative alpha from the majority of the strategies. The environmental strategy in Europe is the only strategy with annual alpha greater than 1.0%, however, the results are not statistically significant at the 95% t-stat level (1.96). Most of the strategies exhibit positive loadings on the low beta, profitability, and investment factors, meaning that ESG portfolios tend to exhibit low-volatility and high-quality characteristics, bringing merit to the argument of ESG as a risk mitigation strategy. FEATURED TAGS Learn More About the Author ## Endnotes 1. The notion of hundreds of attendees gathering in a ballroom and congregating around coffee and snack tables seems, quoting George Lucas, like a long time ago in a galaxy far, far away. 2. Although size is a commonly accepted factor by many investors, Research Affiliates has expressed concern that the size factor may lack robustness (Kalesnik and Beck, 2014). 3. The attention given to specific ESG considerations has varied over time. For example, climate change has been a leading ESG issue for several years, while gender equality and even more recently racial equality, are issues now starting to gain momentum. Discussing whether gender or racial inequality was a priced factor decades in the past is irrelevant if we wish to support investors who desire to have an impact today. 4. We use the Russell 1000 Index as the starting universe for selection within the US, and we use the FTSE All World Developed Europe Index as the starting universe for selection within Europe. We exclude companies without an ESG rating, and strategies rebalance once a year on June 30. We use ESG ratings data from Vigeo Eiris. 5. Source is US Energy Information Administration. 6. Source is “Environmental, Social, and Governance (ESG) Investing in the United States,” Cerulli Associates (2019). 7. Source is UN PRI accessed on July 9, 2020. 8. In 2018, responsible investment strategies used in actively managed equity assets included US7.4 trillion for integration; US$4.4 trillion for screening and integration; US$1.8 trillion for screening, thematic, and integration; and US\$0.4 trillion for thematic and integration. Source is UN PRI.
9. Source is Bank of America Merrill Lynch (September 23, 2019).
10. Source is “The ‘Greater’ Wealth Transfer: Capitalizing on the Intergenerational Shift in Wealth,” Accenture (2015).
11. Arnott et al. (2016) note that revaluation alpha can cut both ways in that a strategy trading at a substantial premium to the market might perform poorly if valuations mean revert toward market multiples.
## References
Arnott, Robert, Noah Beck, Vitali Kalesnik, and John West. 2016. “How Can ‘Smart Beta’ Go Horribly Wrong?” Research Affiliates Publications (February).
Arnott, Robert, Campbell Harvey, Vitali Kalesnik, and Juhani Linnainmaa. 2019. “Alice’s Adventures in Factorland.” Research Affiliates Publications (February). Available at SSRN.
Bank of America. 2019. “10 Reasons You Should Care about ESG.” ESG Matters–US (September 23).
Beck, Noah, Jason Hsu, Vitali Kalesnik, and Helge Kostka. 2016. “Will Your Factor Deliver? An Examination of Factor Robustness and Implementation Costs.” Financial Analysts Journal, vol. 72, no. 5 (September/October):58–82.
Bradford, Hazel. 2019. “Public Funds Taking the Lead in Spectacular Boom of ESG.” Pensions and Investments (April 19).
Brammer, Stephen, Chris Brooks, and Stephen Pavelin. 2006. “Corporate Social Performance and Stock Returns: UK Evidence from Disaggregate Measures.” Financial Management, vol. 35, no. 3 (September):97–116.
Brightman, Chris, James Masturzo, and Jonathan Treussard. 2014. “Our Investment Beliefs.” Research Affiliates Fundamentals (October).
Clark, Gordon, Andreas Feiner, and Michael Viehs. 2015. “From the Stockholder to the Stakeholder: How Sustainability Can Drive Financial Outperformance.” Available on SSRN.
Fabozzi, Frank, K.C. Ma, and Becky Oliphant. 2008. “Sin Stock Returns.” Journal of Portfolio Management, vol. 35, no. 1 (Fall):82–94.
Friede, Gunnar, Timo Busch, and Alexander Bassen. 2015. “ESG and Financial Performance: Aggregated Evidence from More Than 2000 Empirical Studies.” Journal of Sustainable Finance and Investment, vol. 5, no. 4:210–233.
Hong, Harrison, and Marcin Kacperczyk. 2009. “The Price of Sin: The Effects of Social Norms on Markets.” Journal of Financial Economics, vol. 93, no. 1 (July):15–36.
Kalesnik, Vitali, and Noah Beck. 2014. “Busting the Myth about Size.” Research Affiliates Simply Stated (November).
Khan, Mozaffar, George Serafeim, and Aaron Yoon. 2016. “Corporate Sustainability: First Evidence on Materiality.” Accounting Review, vol. 91, no. 6 (November):1697–1724.
Li, Feifei, and Ari Polychronopoulos. 2020. “What a Difference an ESG Ratings Provider Makes!” Research Affiliates Publications (January).
Maleva-Otto, Anna, and Joshua Wright. 2020. “New ESG Disclosure Obligations.” Harvard Law School Forum on Corporate Governance (March 24).
McNamee, Emmet. 2019. “UK’s New ESG Pension Rules: Four Measures to Ensure Their Success.” PRI Blog (October 1).
Nagler, Rebekah. (2014). “Adverse Outcomes Associated with Media Exposure to Contradictory Nutrition Messages.” Journal of Health Communication, vol. 19, no. 1:24–40.
Orsagh, Matt, James Allen, Justin Sloggett, Anna Georgieva, Sofia Bartholdy, and Kris Duoma. 2018. Guidance and Case Studies for ESG Integration: Equities and Fixed Income. CFA Institute: Charlottesville, VA. | {} |
# Prove sequent using natural deduction
I need to prove the following predicate logic sequent using natural deduction:
$\exists y \forall x (P(x) \rightarrow x = y) \vdash \forall x \forall y (P(x) \land P(y) \rightarrow x = y)$
This is my half-finished proof. I hope I'm on the right track but there is something about box packing/unpacking I don't understand yet:
1. $\exists y \forall x (P(x) \rightarrow x = y) \quad\mathrm{Premise}$
2. $y_0: \forall x P(x) \rightarrow x = y_0) \quad\mathrm{Assumption}$
3. $x_0: P(x_0) \rightarrow x_0 = y_0 \quad \forall x e2$
4. $P(x_0) \land P(y_0) \quad \mathrm{Assumption}$
5. $P(x_0) \quad \land e_1 4$
6. $x_0 = y_0 \quad \rightarrow e 3, 5$
7. $P(x_0) \land P(y_0) \rightarrow x_0 = y_0 \quad \rightarrow i 4-6$
8. $\forall x (P(x) \land P(y_0) \rightarrow x = y_0 \quad \forall x i 3-7$
Then I can't go further because I can't use universal reintroduction on the $y$ variable.
Edit: I managed to finish it with the help from the answer!
1. $\exists y \forall x (P(x) \rightarrow x = y) \quad\mathrm{Premise}$
2. $z: \forall x P(x) \rightarrow x = z) \quad\mathrm{Assumption}$
3. $a: P(a) \rightarrow a = z \quad \forall x e2$
4. $b: P(b) \rightarrow b = z \quad \forall x e2$
5. $P(a) \land P(b) \quad \mathrm{Assumption}$
6. $P(a)\quad \land e_1 5$
7. $P(b)\quad \land e_2 5$
8. $a = z \quad \rightarrow e3,6$
9. $b = z \quad \rightarrow e4,7$
10. $b = b \quad =i$
11. $z = b \quad =e 9,10$
12. $a = b \quad =e 8,11$
13. $P(a) \land P(b) \rightarrow a = b \quad \rightarrow i 5-12$
14. $\forall y (P(a) \land P(y) \rightarrow a = y) \quad \forall y i 4-13$
15. $\forall x \forall y (P(x) \land P(y) \rightarrow x = y) \quad \forall x i 3-14$
16. $\forall x \forall y (P(x) \land P(y) \rightarrow x = y) \quad \exists z e 1,2-15$
• In short: Since you wish to use Universal (Re)Introduction twice, this is a clue to first use Universal Elimination twice. – Graham Kemp Dec 28 '17 at 4:54
• Short answer is, you want step 4 to be $P(x_0) \land P(z_0)$ – DanielV Dec 28 '17 at 7:20
$$\begin{array}{r|l:l}1&\exists y\forall x~(P(x)\to x=y)\\ 2&\quad\forall x~(P(x)\to x=c) & 1,\exists \text{Elimination }[y\backslash c]\\[0ex] 3 & \qquad P(a)\to a=c & 2,\forall\text{Elimination }[x\backslash a]\\[0ex] 4 & \qquad\quad P(b)\to b=c & 2,\forall\text{Elimination }[x\backslash b] \\[0ex] 5 & \qquad\qquad P(a)\wedge P(b) & \text{Assumption} \\[-1ex] 6 & \qquad\qquad\vdots & \\[-1ex] 7 & \qquad\qquad\vdots & \\[-1ex] 8 &\qquad\qquad\vdots & \\[-1ex] 9 & \qquad\qquad\vdots & \\[0ex] 10 & \qquad\qquad a=b & ~,~,=\text{Elimination}\\[0ex] 11 & \qquad\quad (P(a)\wedge P(b))\to(a=b) & 5,10,\to\text{Introduction}\\[-1ex] \vdots\end{array}$$ | {} |
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proof-pile-2のopen-web-mathからランダムに所得したデータセット
https://huggingface.co/datasets/EleutherAI/proof-pile-2
https://huggingface.co/datasets/open-web-math/open-web-math
ライセンスは上記に準じます
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