Partial Contribution By One Parent

Partial Contribution By One Parent

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For flowering plants, what might cause the genome size of offspring from a cross between 2 diploid parent of differing genome size to reflect only a partial contribution by the larger genome size parent?

Many hybridizers of Rhododendron attempt wide crosses between two sections of the genus. In some cases the resulting hybrids show physical characteristics of both parents. However, flow cytometry indicates that instead of the offspring being half the sum of the 2 parents genome sizes the genome size is actually twice the genome size of the smaller genome size parent.

See Evergreen azalea X deciduous azalea

Can a successful cross result in a reduction in normal contribution of one of the diploid parents after fertilization?

Partial Contribution By One Parent - Biology

How are ABO alleles inherited by our children?

Problem set Each biological parent donates one of their two ABO alleles to their child. A mother who is blood type O can only pass an O allele to her son or daughter. A father who is blood type AB could pass either an A or a B allele to his son or daughter. This couple could have children of either blood type A (O from mother and A from father) or blood type B (O from mother and B from father).

Since there are 4 different maternal blood types and 4 different paternal blood types possible, there are 16 different combinations to consider when predicting the blood type of children.

The following Blood Type Calculator lets you determine the *possible* blood type of a child, given the blood types of the two biological parents or the *possible* blood types of one biological parent, given the blood types of the child and the other biological parent. We emphasize "possible" because, in most cases, blood typing is not conclusive when attempting to determine, include or exclude an individual as the parent of an offspring.

This calculator is based solely on theoretical principles. It would be a mistake to use this information to make any conclusions about your own family tree. Anyone wishing personal information about their own blood type inheritance is encouraged to contact their health care provider.

Scientists have noted the blending of traits back into ancient times, although until Mendel, no one used the words "incomplete dominance." In fact, Genetics was not a scientific discipline until the 1800s when Viennese scientist and friar Gregor Mendel (1822–1884) began his studies.

Like many others, Mendel focused on plants and, in particular, the pea plant. He helped define genetic dominance when he noticed that the plants had either purple or white flowers. No peas had lavender colors as one might suspect.

Up to that time, scientists believed that physical traits in a child would always be a blend of the traits of the parents. Mendel proved that in some cases, the offspring can inherit different traits separately. In his pea plants, traits were visible only if an allele was dominant or if both alleles were recessive.

Mendel described a genotype ratio of 1:2:1 and a phenotype ratio of 3:1. Both would be consequential for further research.

While Mendel's work laid the foundation, it was German botanist Carl Correns (1864–1933) who is credited with the actual discovery of incomplete dominance. In the early 1900s, Correns conducted similar research on four o'clock plants.

In his work, Correns observed a blend of colors in flower petals. This led him to the conclusion that the 1:2:1 genotype ratio prevailed and that each genotype had its own phenotype. In turn, this allowed the heterozygotes to display both alleles rather than a dominant one, as Mendel had found.

Does Marriage Matter?

Some of the current debate presumes that being born to unmarried parents is more harmful than experiencing parents' divorce and that children of divorced parents do better if their mother remarries. Our evidence suggests otherwise.

Children born to unmarried parents are slightly more likely to drop out of school and become teen mothers than children born to married parents who divorce. But the difference is small compared to the difference between these two groups of children and children who grow up with both parents. What matters for children is not whether their parents are married when they are born, but whether their parents live together while the children are growing up.

Children who grow up with widowed mothers, in contrast, fare better than children in other types of single-parent families, especially on measures of educational achievement. Higher income (due in part to more generous social polices toward widows), lower parental conflict, and other differences might explain this apparent anomaly.

Remarriage is another instance where the conventional wisdom is wrong. Children of stepfamilies don't do better than children of mothers who never remarry. Despite significantly higher family income and the presence of two parents, the average child in a stepfamily has about the same chance of dropping out of high school as the average child in a one-parent family.

Some people believe that single fathers are better able to cope with family responsibilities because they have considerably more income, on average, than the mothers. However, our evidence shows that children in single-father homes do just as poorly as children living with a single mother.

Galton's derivations of the ancestral law

Galton formulated the ancestral law in 1885, and he derived the law by a plausible, though faulty, mathematical argument in an appendix to that paper, which was repeated in almost identical terms in Natural Inheritance (Galton, 1889). He returned to the subject in 1897 with two new arguments, which are even less convincing than the first one (Galton, 1897). These arguments will now be considered in turn.

Derivation of the law in 1885

As a first step in framing the ancestral law, Galton tried to determine what could be inferred about the deviates of more remote ancestors given the deviate of the mid-parent. To do this, he used his bivariate frequency distribution of the heights of offspring and mid-parents to plot the average mid-parental height against the height of the offspring the idea was that this would give the same regression as that of mid-grandparent on mid-parent. He found a straight line with a slope of 1/3, so that ‘the most probable mid-parentage of a man is one that deviates only one-third as much as the man does’. In modern terminology:

Galton used the properties of the bivariate normal distribution to understand the relation between the regressions in eqns (1) and (8). Today the following argument should apply. Under random mating, which held approximately for Galton's data, it is expected that Var(D1)=Var(D0)/2, because D1 is the mean of two randomly chosen heights and this was empirically true. It is a standard result from regression theory that the slope of the regression of D0 on D1 is Cov(D0, D1)/Var(D1), whereas that of D1 on D0 is Cov(D0, D1)/Var(D0), so that the first slope should be twice the second, as observed.

Galton derived the ancestral law in an appendix to his paper, which is reprinted in the Appendix to this paper and which can be rephrased as follows. The parental, grandparental and more distant ancestral deviations may all affect the offspring deviation because of reversion caused by latent elements this can be expressed in the multiple regression formula (eqn 4). The regression coefficient β1 reflects the direct effect of the mid-parent on the offspring, β2 reflects the direct effect of the mid-grandparent, and so on.

The regression of offspring on mid-parent is E(D0|D1)=β*D1, say. It is expected that β*>β1 because β* will be influenc ed not only by the direct effect of the mid-parent but also by the indirect effects of more remote ancestors above-average parents are themselves likely to have above-average parents (grandparents of the offspring), and so on. From eqn (4) can be written:

Galton had found empirically for human stature that the regression of mid-parent on offspring is 1/3 (eqn 8). This must be the same as that of mid-grandparent on mid-parent, so that:

and he assumed by analogy that:

To find a relationship between the total regression coefficient β* and the partial regression coefficients βi, Galton considered two limiting hypotheses. Under the constant hypothesis, βi=β for all i, so that:

because he had found empirically that β*=2/3 for human stature. Under the geometric decrease hypothesis, βii , so that:

Galton now remarks that the two estimates of β are nearly the same, and that their average is nearly 1/2, and he concludes that β1=1/2, β2=1/4, β3=1/8, and so on. This leads to his final result for the multiple regression, the law of ancestral inheritance:

Unfortunately, there are several problems in the derivation of this law. First, the coefficients in eqn (11) should be 1/6, 1/12, and so on, rather than 1/9, 1/27, and so on. Secondly, the two results β=4/9 and β=6/11 are obtained under different models, so that there is little logic in averaging them to obtain a value of 1/2 furthermore, Galton abandons the constant hypothesis in favour of the geometric decrease hypothesis as soon as he has obtained the average value of 1/2. Thirdly, neither the constant hypothesis nor the geometric decrease hypothesis is generally true under Galton's model of inheritance the appropriate hypothesis is βi=cβ i (eqn 7). If Galton's argument is reworked under the latter hypothesis, with the coefficients in eqn (11) corrected to 1/6, 1/12, and so on, and with the assumption that the coefficients in the multiple regression formula sum to unity so that c=(1−β)/β, it leads to the result obtained from eqn (7) with p=2/3:

It should also be noted that these coefficients do not, as Galton assumed, reflect the contributions of the different ancestors, which are 2/3 for the two parents, 2/9 for the four grandparents, 2/27 for the eight great-grandparents, and so on. Hence the assumption that they sum to unity needs separate justification.

Galton was a pioneer with a very powerful intuition, but he lacked the mathematical skill to develop the technique of multiple regression to its logical conclusion. In view of his mathematical limitations, it is remarkable how close he came to the correct answer under the model he had adopted, which was quite plausible until it was displaced by Mendelism.

Derivation of the law in 1897

Galton returned to the subject with two new arguments in 1897 (Galton, 1897). He still did not distinguish between the use of the law as a prediction formula and as a representation of ancestral contributions. He presented data verifying the validity of the law as a prediction formula (see Table 1), but his main argument for the law regarded it as representing ancestral contributions:

‘A wide though limited range of observations assures us that the occupier of each ancestral place may contribute something of his own peculiarity, apart from all others, to the heritage of the offspring. Further, it is reasonable to believe that the contributions of parents to children are in the same proportion as those of the grandparents to the parents, of the great-grandparents to the grandparents, and so on in short, that their total amount is to be expressed by the sum of the terms in an infinite geometric series diminishing to zero. Lastly, it is an essential condition that their total amount should be equal to 1, in order to account for the whole of the heritage. All these conditions are fulfilled by the series of 1/2+(1/2) 2 +(1/2) 3 +&c., and by no other.’

In other words, he argues that it is plausible to assume the geometric relationship βii , and that the terms must sum to unity, Σβi=1. Hence β=1/2, giving the ancestral law in eqn (16). This is a different justification of the law from that given in 1885. Galton has abandoned the use of the empirically determined regression of offspring on mid-parent in its derivation and has, instead, adopted a completely a priori approach. In fact, the contribution of the ith ancestral generation under Galton's model is p(1−p) i−1 , a modified geometric series with a free parameter to be empirically estimated.

To this argument Galton added another of even more dubious logic:

‘It should be noted that nothing in this statistical law contradicts the generally accepted view that the chief, if not the sole, line of descent runs from germ to germ and not from person to person. The person may be accepted on the whole as a fair representative of the germ, and, being so, the statistical laws which apply to the persons would apply to the germs also, though with less precision in individual cases. Now this law is strictly consonant with the observed binary subdivisions of the germ cells, and the concomitant extrusion and loss of one-half of the several contributions from each of the two parents to the germ-cell of the offspring. The apparent artificiality of the law ceases on these grounds to afford cause for doubt its close agreement with physiological phenomena ought to give a prejudice in favour of its truth rather than the contrary.’

He is appealing to recent discoveries about the reduction division of the germ cells. He seems to be arguing as follows: (i) parents transmit half of their germ-plasm to their offspring, grandparents one-quarter to their grandchildren, and so on (ii) therefore an individual receives one-half of his germ-plasm from his parents, one-quarter from his grandparents, and so on (iii) therefore the same law applies to the inheritance of personal characteristics because the same statistical laws apply to phenotypic and genotypic values. If this is his argument, it is a bad one. The first statement is true, but the second does not follow from it, and the premise of the third statement is false. It is not clear how seriously he intended this argument to be taken.

Thus Galton had come to believe in 1897 that the ancestral law was a logical necessity which could be derived by a priori arguments, although it required empirical verification. In the introduction to this paper he wrote: ‘I stated [the law] briefly and with hesitation in my book ‘Natural Inheritance’, because it was then unsupported by sufficient evidence. Its existence was originally suggested by general considerations, and it might, as will be shown, have been inferred from them with considerable assurance’ (Galton, 1897). After presenting the above two arguments, he concluded: ‘These and the foregoing considerations were referred to when saying that the law might be inferred with considerable assurance a priori’.

Timing Your Roth IRA Contributions

Although you can own separate traditional IRAs and Roth IRAs, the dollar limit on annual contributions applies collectively to all of them. If an individual under 50 deposits $2,500 in one IRA for the tax year 2020, then that individual can only contribute $3,500 to another IRA in that tax year.  

Contributions to a Roth IRA can be made up until tax filing day of the following year. So contributions to a Roth IRA for 2021 can be made through the deadline on April 15, 2022, for filing income tax returns. Obtaining an extension of time to file a tax return does not give you more time to make an annual contribution.  

  • In extending the deadline to file Form 1040 series returns to May 17, the IRS is automatically postponing to the same date the time for individuals to make 2020 contributions to their individual retirement arrangements (IRAs and Roth IRAs), health savings accounts (HSAs), Archer Medical Savings Accounts (Archer MSAs), and Coverdell education savings accounts (Coverdell ESAs). This postponement also automatically postpones to May 17, 2021, the time for reporting and payment of the 10% additional tax on amounts includible in gross income from 2020 distributions from IRAs or workplace-based retirement plans.
  • On February 22, 2021, the Internal Revenue Service (IRS) announced that victims of the 2021 winter storms in Texas will have until June 15, 2021, to file various individual and business tax returns and make tax payments. Among other things, this also means that affected taxpayers will have until June 15 to make 2020 IRA contributions.

If you're a real early-bird filer, and you received a tax refund, you can apply some or all of it to your contribution. You will have to instruct your Roth IRA trustee or custodian that you want the refund used in this way.

Conversion to a Roth IRA from a taxable retirement account, such as a 401(k) plan or a traditional IRA, has no impact on the contribution limit. However, making a conversion adds to MAGI, and may trigger or increase a phase-out of your Roth IRA contribution amount. Also, rollovers from one Roth IRA to another are not taken into account for purposes of making annual contributions.

What Was Rudolf Virchow's Contribution to Cell Theory?

The German doctor Rudolf Virchow proposed that all cells result from the division of previously existing cells, and this idea became a key piece of modern cell theory. Virchow also founded the discipline of cellular pathology based on the idea that diseases do not affect an entire organism but are instead localized to certain groups of cells. This made it easier to diagnose and treat diseases.

Virchow was appointed as the chair of pathological anatomy at the University of Wurzburg in 1849 and carried out a great deal of research. In 1855, he first published his idea that all cells arise from other cells. Rather than being formed by the action of a life force or spontaneously crystallizing from other matter, Virchow argued that cells only formed from the division of other cells. This idea is one of the key principles of cell theory, along with the idea that the cell is the basic unit of organization for living organisms.

During this period, he also proposed the basic ideas of cellular pathology. Rather than being the result of changes in an organism as a whole, Virchow believed that diseases result from changes in specific groups of cells. By examining cells for certain changes or alterations, doctors can more precisely identify and diagnose a disease.

What You Need

You should use your family's most recent year’s income and asset information (converted to U.S. dollars if applicable). We will only request tax forms and other financial documents to verify this information when you complete a financial aid application.

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  1. Children’s Savings Accounts (CSAs): Long-term, restricted savings accounts established for first-grade students enrolled in participating LAUSD schools to support student’s post-secondary education.
  2. County Student (“County Student”): CSA Program-eligible first graders enrolled in LAUSD schools that reside in the unincorporated areas of the County and cities outside the City of Los Angeles.
  3. City Student (“City Student”): CSA Program-eligible first graders enrolled in LAUSD schools that reside within the boundaries of the City of Los Angeles.
  4. Custodial Account (“Custodial Account”): Financial account that incorporates CSAs, established for the benefit of the Student. Administered and managed by the CSA Program in accordance with the program Memorandum Of Agreement (MOA).
  5. Incentive (“Incentive”): Any additional CSA funding, not including family or friend’s contributions, beyond the CSA Seed Funds that may be contributed to CSAs based upon specific criteria established and mutually agreed upon by all the Parties, (as defined in the MOA). Students may earn Incentives only while enrolled in an eligible LAUSD school. Use of all Incentives are subject to the requirements as set forth in Exhibit A, which is attached and incorporated by reference.
  6. Legal Guardian (“Legal Guardian”): Someone who is not the child’s parent, however, has legal custody of the child and can provide health, education, and financial decisions on behalf of the child.
  7. Non-Program Contributions or Non-Program Funds (“Non-Program Contribution or Non-Program Funds”): Deposits of funds contributed to a CSA by a person or organization not at the direction or associated with the CSA Program for the benefit of the student.
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  9. Program Contributions or Program Funds (“Program Contribution or Program Funds”): Initial Seed Deposit, Incentives, and any growth amount accrued by the Custodial Account for the benefit of the Students.
  10. School Selection Plan (“School Selection Plan”): Yearly plan used to identify LAUSD schools to participate in the CSA Program as mutually agreed upon by all the Parties and in accordance with Section VII – School Selection Plan below.
  11. Seed Funding (“Seed Funding”): Initial funding to establish CSAs for the benefit of Students as agreed upon by all the Parties.
  12. Student or Participant (“Student or Participant”): A City or County Student that is the intended beneficiary of the CSA and a) currently attends an LAUSD school or b) graduated from an LAUSD high school.
  13. Year (“Year”): July 1st to June 30th.
  14. Escheat: A legal process that transfers ownership of abandoned property in this case CSA program funds, to the state in accordance with state law. Before the state can take full ownership, it must attempt to find the owners (Student, Student parent or legal guardian) and provide an opportunity for the Student to claim their funds.

About the Program

Opportunity LA is the first children’s savings account (CSA) program in the greater Los Angeles area. Beginning in the spring, accounts will be automatically opened with a $50 deposit for first-grade students enrolled in select schools.

Program Overview

The Los Angeles Housing, Community + Investment Department (HCIDLA) has established Opportunity LA (OLA) to reduce financial barriers for Los Angeles Unified School District (LAUSD) Students planning to attend a higher education institution. HCIDLA, in its capacity as the Custodian of the Opportunity LA Program, is the administrator of funds contributed to and managed by the OLA Program which is referred to herein as the “Program.”
The Program is a partnership among the City of Los Angeles, County of Los Angeles, and LAUSD. The Program period is from July 1 through June 30 (school year). The Program is offered on a year-to-year basis and will continue beyond the current school year at the discretion of the Program.

Participation and Eligibility

In order to be eligible for participation in the Program, a Student must be currently enrolled at a school that is participating in the Program. Exceptions to this eligibility requirement may be allowed at the Program’s sole discretion. An account card with a pre-preprinted account number is included in this welcome packet.

Account Registration

A parent or legal guardian must register the Student’s account using the pre-printed account number provided in the welcome packet, at, to gain online access to the Student’s account record.

Program Contributions and Incentives

A Contribution is a bank deposit of funds by a Student, family, friend, or another person for the benefit of the Student. Contributions will be credited to the Student’s OLA account, and Contributions will be distributable to the Student or to the Student’s parent/guardian if the student is under the age of 18.

Incentives are funds that the Student can earn as part of the Program if the Student completes the Program requirements. There are several forms of Incentives in the Program. One-time incentive or Performance incentive: An amount which can be earned by a Student on a one-time basis pursuant to an offer of the Program. One-time incentives are conditional and determined by the Program. Match incentive: An amount equal to or based upon multiple Contributions in a Program Account, which can be earned by a Student pursuant to an offer of the Program. Matches are conditional and are paid by the Program. Incentives also include the first deposit of $50 dollars by the Program and any growth amount accrued.

In no event will any Contributions and Incentives be credited to the OLA Account or a Student’s Incentive Ledger for the benefit of the Student following the Student’s graduation from high school.

Incentives are not guaranteed and are subject to change based on the funding availability of the program.


Program disbursements are done upon the completion of the Program (which is high school graduation determined to be the last day of June any given year or the next business day if the last day of June falls on a weekend).

Contributions Disbursement: The Program will disburse funds directly to the Student (or in the case of a Student under the age of 18, to the parent or guardian for the benefit of the Student). The Student’s record in the OLA Program Account will be closed upon completion of the disbursement of all Contributions. Any Student who is in a high school graduation year may request their contribution disbursement between March 1st and June 30th of that year. Any Student withdrawing from the Program will automatically receive all Contributions (or the parent or legal guardian for the benefit of the Student if the Student is under the age of 18).

Contributions shall escheat to the State in accordance with State law upon the expiration of three (3) years in the event that the Program cannot reach the Student or parent or guardian for the disbursement of the funds or if the check for
Contributions disbursement has not been cashed.

Incentives Disbursement: Program Incentives are conditional and are distributed by the Program according to these Program Rules. Students are eligible for the Incentives in the form of a College Scholarship. Incentives shall be disbursed directly to the qualifying educational institution unless other approved Incentive arrangements have been made. For purposes of disbursement of Incentives, a qualifying educational institution is defined as a two-year or four-year accredited college or university that offers credit toward an undergraduate degree, vocational school, and some private or trade schools. The Program will disburse Incentives directly to the qualifying educational institution.

Failure to Satisfy Program Requirements

If Incentive documentation is not completed by a Student prior to the second anniversary of a Student’s projected high school graduation date, the Student will have no right to receive any Incentives, and the Student’s record in the Program Account will be closed and the Incentive Ledger will be terminated. Please note that Contributions will be distributed by the Program to the Student (or any parent/guardian for the benefit of the Student if the Student is under the age of 18).
Any extension of the Termination Date may be granted at the Program’s sole discretion. Extensions to the Termination Date may be granted prior to the Termination Date if a Student applies for an extension of time to complete the Program’s requirements as a result of military service, medical emergency, AmeriCorps Service commitment or other circumstances to be determined by the Program on a case-by-case basis.

Early Withdrawal from Program

At any time following the first anniversary of the Student’s enrollment in the Program, a Student may elect to end participation in the Program. A Student will be deemed to have elected to end participation in the Program if the Student leaves his/her school prior to graduating from that school. The death of a Student will be treated as an early withdrawal from the Program.

Leaving the School District

If the Student leaves an LAUSD school, the Student account record will be closed. Access to the online Student account will be terminated as of the date of account closure. If any contributions were made to the account on behalf of the Student, a check payable to the Student’s name will be mailed to the address on record within twenty (20) business days of account closure. All Incentives and contributions made by the Program are returned to the Program.
If the Student returns to an LAUSD school participating in the Program, a new account will be created for the Student and the initial $50 deposit will be credited to the new account.
Consequences of an early withdrawal include the following: all Contributions will be distributed by the to the Student (or the parent or legal guardian for the benefit of the Student if the Student is under the age of 18) the Student will forfeit all Incentives the Student’s Program account will be closed, and the Incentive Ledger will terminate.

Account Information and Inquiries

Please contact the Program by mail:
HCIDLA-Opportunity LA Program
1200 W. 7th St Suite #900
Los Angeles, CA 90017

Please contact the Program by email at:
[email protected]

Tax Matters

The information contained in the Program materials or on the Program’s site is not intended to constitute tax advice. The tax consequences of participation in the Program will depend on a Student’s particular tax circumstances. Students and their parent(s)/legal guardian(s) are responsible for obtaining their own tax advice with respect to participation in the Program. The Program shall not have any liability for any information contained in or omitted from the Program materials or the Program site with respect to tax matters.

Investment Advice

Nothing contained on the Program site or in the Program materials is intended to constitute investment advice, nor does the Program give advice or offer any opinion or recommendation on the suitability of any investment strategy. Any investment decision Students make will be based solely on their own evaluation of the merits of the particular investment decision in light of their financial circumstances and investment objectives.

Program Modifications and Termination

The Program reserves the right to add to, delete and change the terms of the Student Agreement and these Program Rules from time to time at its sole discretion. If this were to occur, the Program will notify Students of any change that affects their participation. A Student’s failure to terminate participation in the Program after notice of any modification of the Student Agreement or these Program Rules will constitute affirmative acceptance by such Student of such modification and such Student’s consent to abide by those terms as modified. It is also possible that the Program may decide to terminate. If this were to occur, the Program will provide notice to Students of the decision and Students would be eligible to receive a distribution of an amount equal to all Contributions in the Program Account as of the termination date.

Limitations on Liability


Appendix F


In this section we show that the rapid convergence of an individual’s reproductive value proved in Appendix D for a panmictic population is not greatly slowed by two simple forms of structure: partial selfing and subdivision.

Partial selfing:

Consider a population in which a fraction α of offspring in the population is produced by self-fertilization and the remaining 1 − α by random mating.

Lemma F.1. In the notation of Appendix D, under partial selfing (F1) and (F2) In particular, Proof. Still using the notation of Appendix D, (2m1, … , 2mN) will once again be an exchangeable random vector. It is tedious but not difficult to check that Substituting in our previous proof yields the result.▪

Apart from the initial behavior, Nordborg and Donnelly (1997) and Möhle (1996) show that for a population with partial selfing, under the Wright–Fisher model, the Kingman coalescent remains a valid model for the genealogy of a “small” sample, but the rate of coalescence is increased by a factor 2/(2 − α). For the exact coalescent for the Wright–Fisher model too, the effect of selfing will be to increase the rate of coalescence and so, using the notation of the proof of Lemma E.1, for with high probability and we can approximate the coalescent after that time by the (time-changed) Kingman coalescent. The proof of Lemma E.1 will then carry over to this setting.

The island model:

In this subsection we consider an island model in which the population is subdivided into D demes, each with N0 occupants. Mathematically, it is convenient to separate the steps of reproduction and migration. Thus in a reproductive step, each deme (separately) undergoes the diploid Wright–Fisher reproduction that we have seen above. Between reproductive steps a number of migration steps take place in which two demes are chosen at random and an individual from deme i is exchanged with one in deme j.

Again we trace the matrix Mst whose (i, j)th entry records the probability that a gene in individual i at time s is derived from one in individual j at time t in the past. It is convenient to label individuals so that labels 1, … , N0 lie in the first deme, N0 + 1, … , 2N0 lie in the second, and so on. In place of the matrix Mt we now have two sorts of matrix. The first, corresponding to reproduction, is block diagonal, with each block a copy of the Mt corresponding to the diploid Wright–Fisher model for a population of size N0. Premultiplication by the second type of matrix corresponds to exchanging two randomly chosen rows of Mst.

We examine the rate of decay of the variance of the entries in the first column to obtain the analog of Equation D4. We denote the entries of the first column of Mst by mij, where 1 ≤ iD refers to the number of the deme and 1 ≤ jN0 to the number of the individual within that deme. Write We now write the variance of the entries in the first column of Est as Now note that we can rewrite the second term as The variance of the entries in the first column of our matrix then becomes Let us write var1(s) and var2(s) for these two terms in the variance of the first column in Mst. In a reproduction event, by (D4), the term var1 is reduced by a factor The term var2 on the other hand can increase. Let us write Then by Equation D3, (independently for each i) and E[εi] = 0. Thus E[var2(s − 1) − var2(s)] becomes In a migration step involving the interchange of just two columns, the overall variance cannot change (we are merely shuffling the entries in the column, not changing them), but the expected value of the change in the second term is easily checked to be Combining these, if a proportion m of offspring migrates immediately after each reproduction step, the change in variance over a whole cycle of reproduction and migration is From this we see that The first part of the variance is reduced by a factor of 2 in each cycle and, once this has been repeated often enough that var2(s) > var1(s), mass from var2(s) is transfered to var1(s − 1) so that it, in turn, can be reduced.


However, a couple of issues might arise for taxpayers claiming the $300 above-the-line deduction. First, low-income taxpayers whose AGI does not exceed the standard deduction will largely fail to realize the deduction's intended benefit. Even if these individuals do have any taxable income before credits, nonrefundable credits (e.g., the child tax credit, child and dependent care credit, etc.) may reduce their taxable income — and, in turn, their tax liability — to zero.

The second issue might arise when a married individual filing a separate return whose spouse itemizes deductions is not eligible for the standard deduction (or has a zero standard deduction), raising the question of whether such an individual may claim the above-the-line charitable deduction(Sec. 63(c)(6)(A)). Assuming such an individual does not also itemize deductions, an above-the-line charitable deduction would seem to be available, since ineligibility for a full standard deduction is not, per se, an election to itemize (Sec. 63(e)(1)). But some IRS guidance on this point would be welcome.

Practitioners with clients who do not regularly itemize post-TCJA should consider alerting these taxpayers to the above-the-line charitable contribution deduction. The amount may be relatively small, but in the throes of a health care crisis, every bit can make a difference — in this case, for both the donor and the donee.