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Brilliant mathematician John Nash worked on many theory such as nonlinear parabolicpartial differential equations theory

and singularity theory. The most important work of John Nash is Nash equilibrium which is related to the non-cooperative

games. Nash won the Nobel Prize with the following study: .

# Nash Equilibrum

A Nash equilibrium of a game G in strategic form is defined as any outcome $\left({a}_{1}^{*},..,{a}_{n}^{*}\right)$ such that ${u}_{i}\left({a}_{i}^{*},{a}_{-i}^{*}\right)\ge {u}_{i}\left({a}_{i},{a}_{-i}^{1}\right)$ holds for each player i: The set of all Nash equilibria of G is denoted N(G): In a two player game, for example, an action profile $\left({a}_{1}^{*},{a}_{2}^{*}\right)$ is a Nash equilibrium if the following two conditions hold

$\begin{array}{l}{a}_{1}^{*}\in \text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{arg}\text{\hspace{0.17em}}\underset{{a}_{1}\in {A}_{1}}{\underbrace{\mathrm{max}}}{u}_{1}\left({a}_{1},{a}_{2}^{*}\right)\\ {a}_{2}^{*}\in \text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{arg}\text{\hspace{0.17em}}\underset{{a}_{2}\in {A}_{2}}{\underbrace{\mathrm{max}}}{u}_{2}\left({a}_{1}^{*},{a}_{2}\right)\end{array}$

Therefore, we may say that, in a Nash equilibrium, each players choice of action is a best response to the actions actually taken by his opponents. This suggests, and sometimes more useful, definition of Nash equilibrium, based on the notion of the best response correspondance. We define the best response correspondance of player i in a a strategic form game as the correspondence

$\begin{array}{l}{B}_{i}:\text{\hspace{0.17em}}{A}_{-i}\to {A}_{i\text{\hspace{0.17em}}}\text{\hspace{0.17em}}given\text{\hspace{0.17em}}by\\ {B}_{i}\left({a}_{-i}\right)=\left\{{a}_{i}\in {A}_{i}:{u}_{i}\left({a}_{i},{a}_{-i}\right)\ge {u}_{i}\left({b}_{i},{a}_{-i}\right)\text{\hspace{0.17em}}for\text{\hspace{0.17em}}all\text{\hspace{0.17em}}{b}_{i}\in {A}_{i}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{arg}\underset{{a}_{i}\in {A}_{i}}{\underbrace{\mathrm{max}}}{u}_{i}\left({a}_{i},{a}_{-i}\right)\\ \end{array}$

(Notice that, for each ${a}_{-i}\in {A}_{-i},{B}_{i}\left({a}_{-i}\right)\text{\hspace{0.17em}}is\text{\hspace{0.17em}}a\text{\hspace{0.17em}}set\text{\hspace{0.17em}}which\text{\hspace{0.17em}}may\text{\hspace{0.17em}}or\text{\hspace{0.17em}}may\text{\hspace{0.17em}}not\text{\hspace{0.17em}}be\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\mathrm{sin}\text{}gleton\right).$ So, for example, in a 2-person game, if player 2 plays ${a}_{2}$ player 1s best choice is to play some action in ${B}_{1}\left({a}_{2}\right)$ $\begin{array}{l}{B}_{1}\left({a}_{2}\right)=\left\{{a}_{1}\in {A}_{1}:{u}_{1}\left({a}_{1},{a}_{2}\right)\ge {u}_{2}\left({b}_{1},{a}_{2}\right)\text{\hspace{0.17em}}for\text{\hspace{0.17em}}all\text{\hspace{0.17em}}{b}_{1}\in {A}_{1}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

For instance, in the game :

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{ccc}\text{\hspace{0.17em}}L& M& R\end{array}\\ U\text{\hspace{0.17em}}\text{}\overline{)1,0}\overline{)1,2}\overline{)0,2}\\ D\text{\hspace{0.17em}}\overline{)0,3}\text{}\text{\hspace{0.17em}}\overline{)1,1}\overline{)2,0}\\ We\text{\hspace{0.17em}}have\text{\hspace{0.17em}}{B}_{1}\left(L\right)=\left\{U\right\},\text{\hspace{0.17em}}{B}_{1}\left(M\right)=\left\{U,D\right\}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{B}_{1}\left(R\right)=\left\{D\right\},while\text{\hspace{0.17em}}{B}_{2}\left(U\right)=\left\{M,R\right\}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{B}_{2}\left(D\right)=\left\{L\right\}\end{array}$


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# fa.functional analysis – A locally convex \$C^*\$ algebraic structure on the disk algebra Answer

Hello dear visitor to our network We will offer you a solution to this question fa.functional analysis – A locally convex \$C^*\$ algebraic structure on the disk algebra ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

## fa.functional analysis – A locally convex \$C^*\$ algebraic structure on the disk algebra

A locally convex $$C^*$$ algebra is a locally convex topological vector space $$A$$ whose topology is generated by a family of complete $$C^*$$ semi normal and $$A$$ is a $$*$$ algebra. Moreover all algebra operations and involution are continuous operators(A complete $$C^*$$ semi norm is a semi norm which satisfies $$|xx^*|=|x|^2$$ and is a complete semi-norm).

Questions: Is there a locally convex $$C^*$$ algebraic structure on the disk algebra $$A(\mathbb{D})$$?

The disk algebra is the Banach algebra of all holomorphic functions on the interior of the unit disk with continuous extention the boundary of the disk.

A motivation for this question is the following post:

A locally convex \$C^*\$ algebra without zero divisor

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## Abstract

With a combined carrier frequency of 1:200, heteroplasmic mitochondrial DNA (mtDNA) mutations cause human disease in ∼1:5000 of the population. Rapid shifts in the level of heteroplasmy seen within a single generation contribute to the wide range in the severity of clinical phenotypes seen in families transmitting mtDNA disease, consistent with a genetic bottleneck during transmission. Although preliminary evidence from human pedigrees points towards a random drift process underlying the shifting heteroplasmy, some reports describe differences in segregation pattern between different mtDNA mutations. However, based on limited observations and with no direct comparisons, it is not clear whether these observations simply reflect pedigree ascertainment and publication bias. To address this issue, we studied 577 mother–child pairs transmitting the m.11778G>A, m.3460G>A, m.8344A>G, m.8993T>G/C and m.3243A>G mtDNA mutations. Our analysis controlled for inter-assay differences, inter-laboratory variation and ascertainment bias. We found no evidence of selection during transmission but show that different mtDNA mutations segregate at different rates in human pedigrees. m.8993T>G/C segregated significantly faster than m.11778G>A, m.8344A>G and m.3243A>G, consistent with a tighter mtDNA genetic bottleneck in m.8993T>G/C pedigrees. Our observations support the existence of different genetic bottlenecks primarily determined by the underlying mtDNA mutation, explaining the different inheritance patterns observed in human pedigrees transmitting pathogenic mtDNA mutations.

## Introduction

First described in 1988, point mutations of mitochondrial DNA (mtDNA) have emerged as a major cause of maternally inherited human disease (1,2). Pathogenic mtDNA mutations causing a severe multisystem phenotype are usually heteroplasmic, with a mixture of mutated and wild-type mtDNA present in the same individual. The percentage level of mutated mtDNA largely determines whether a biochemical defect is expressed on a cellular level, and the inherited level of heteroplasmy correlates more or less with the severity of the clinical phenotype (3).

Rapid intergenerational shifts in the level of mtDNA heteroplasmy levels were first observed in Holstein cows (4,5) and subsequently documented in human pedigrees transmitting pathogenic mtDNA mutations. A restriction in the number of mitochondrial genomes repopulating the female germ line (the mtDNA bottleneck) is thought to explain this phenomenon, supported by observations in mice (6,7). Although the initial analysis of human pedigrees implied differences in the mtDNA bottleneck between different families (8), the analysis of aggregate data did not support earlier reports (9). Limited data analysis pointed towards a random genetic drift mechanism acting on all heteroplasmic mtDNA mutations (9), but in some instances, there appeared to be evidence of selection in favour of mutated genomes (10), even after minimising the effects of ascertainment bias. These conflicting data lead to two fundamental unanswered questions: do all mtDNA mutations behave the same during inheritance, and is there selection for or against different levels of heteroplasmy? Clarifying the underlying trends is key to providing reliable recurrence risks for patients with mtDNA diseases, and understanding the underlying mechanisms involved may open new avenues for preventative treatment in the future (11).

To address this issue, we have performed the largest analysis of inherited mtDNA heteroplasmic mutations in humans to date. Our findings show that differences in the behaviour of the mtDNA bottleneck between specific pathogenic mtDNA mutations, explaining the variability in clinical inheritance pattern observed in human pedigrees transmitting different mtDNA mutations.

## Results

### Determining the potential impact of ascertainment bias

Given previous concerns about ascertainment bias when studying the inheritance of heteroplasmy in human pedigrees (9), first we performed a simulation experiment to determine the possible consequences of identifying pedigrees through a clinically affected child. We then determined whether the standard approach of omitting the affected proband minimizes any bias to an acceptable level.

The simulations were based on an established model for the mtDNA genetic bottleneck using measurements of heteroplasmy made in human oocytes for neutral alleles (i.e. with no selection) (12,13). We studied the difference in heteroplasmy level between a mother and child (ΔM-O) in simulated pedigrees in silico, with three possible strengths of the mtDNA genetic bottleneck (where bottleneck parameter, b= 0.9, is a weak bottleneck; b= 0.7 is an intermediate strength bottleneck; and b= 0.2 is a strong bottleneck). We then modelled the effects of ascertaining the whole pedigrees (Fig. 1) using three forms of ascertainment: (1) through a mother (AAM), (2) through an affected child (AAC) or (3) through the other unaffected child, but disregarding data contributed by the affected child (AOC)—the latter mimicking the conventional approach used to minimize ascertainment bias by omitting to analyse data from an affected proband. For a full description, see the Materials and Methods and Figure 1.

Figure 1.

Overview of the inheritance model. Simulations used to determine the potential effects of ascertainment bias through a mother, and affected child, or an unaffected child. Mothers= red; offspring= blue; unaffected offspring= green. Darker distributions represent ascertained pedigrees; lighter distributions represent non-ascertained pedigrees. The probability of a family being recruited depends on an individual developing the disease, which in turn is correlated with the heteroplasmy level. The probability that an individual is recruited, pr, based on a distribution that is zero below a level of heteroplasmy d1, and one above heteroplasmy d2, with a linear increase between these points. The probability of being recruited with a heteroplasmy of (d1 + d2)/2 is 0.5, reflecting the increased likelihood of developing symptoms and thus presenting clinically. AAM= families ascertained through an affected mother; AAC families ascertained through an affected child; AOC families ascertained through an affected child, only but including the transmission from the mother to the other child, thus mimicking the effect of deleting probands from the ascertainment of real pedigrees.

Figure 1.

Overview of the inheritance model. Simulations used to determine the potential effects of ascertainment bias through a mother, and affected child, or an unaffected child. Mothers= red; offspring= blue; unaffected offspring= green. Darker distributions represent ascertained pedigrees; lighter distributions represent non-ascertained pedigrees. The probability of a family being recruited depends on an individual developing the disease, which in turn is correlated with the heteroplasmy level. The probability that an individual is recruited, pr, based on a distribution that is zero below a level of heteroplasmy d1, and one above heteroplasmy d2, with a linear increase between these points. The probability of being recruited with a heteroplasmy of (d1 + d2)/2 is 0.5, reflecting the increased likelihood of developing symptoms and thus presenting clinically. AAM= families ascertained through an affected mother; AAC families ascertained through an affected child; AOC families ascertained through an affected child, only but including the transmission from the mother to the other child, thus mimicking the effect of deleting probands from the ascertainment of real pedigrees.

The simulations confirmed previous assumptions (9) that ascertainment through an affected child (AAC) leads to a skewed distribution of ΔM-O, creating the false impression that there is selection bias in favour of the mtDNA mutation (Fig. 2, blue line compared with the grey histogram). This was most prominent when the bottleneck was narrow (i.e. b was low, a strong bottleneck). However, when families were ascertained through the mother (AAM), or the data from the affected child (AOC) were not included, the ΔM-O distribution resembled the entire data set before sampling (Fig. 2, red and green lines compared with the grey histogram), showing that omission of the proband (AAC) minimized the ascertainment bias.

Figure 2.

Modelling the effects of ascertainment bias on simulated pedigrees transmitting mtDNA heteroplasmy. Frequency distribution histograms for the difference between maternal and offspring heteroplasmy levels (ΔM-O) for three different values of the bottleneck strength (bottleneck parameter, b= 0.9 a weak bottleneck; b= 0.7 an intermediate bottleneck; and b= 0.2 a strong bottleneck), and for three different methods of ascertainment: sampling the families where there was an affected mother—(ascertainment − affected mother, AAM, red); sampling the families where there was an affected child—(ascertainment − affected child, AAC, blue); and sampling the families through an affected child, only but including the transmission from the mother to the other child—(ascertainment − other child, AOC, green). The grey histogram gives the simulated distribution before any ascertainment, thus corresponding to data without any ascertainment bias. Each row reflects different thresholds heteroplasmy values required to cause disease. The parameter d models the threshold for disease resulting from the pathogenic mutation. The range given for d is the range of heteroplasmies where the probability of disease increases linearly from 0 (lower level) to 1 (upper level) (see legend to Fig. 1 for details of the simulation model).

Figure 2.

Modelling the effects of ascertainment bias on simulated pedigrees transmitting mtDNA heteroplasmy. Frequency distribution histograms for the difference between maternal and offspring heteroplasmy levels (ΔM-O) for three different values of the bottleneck strength (bottleneck parameter, b= 0.9 a weak bottleneck; b= 0.7 an intermediate bottleneck; and b= 0.2 a strong bottleneck), and for three different methods of ascertainment: sampling the families where there was an affected mother—(ascertainment − affected mother, AAM, red); sampling the families where there was an affected child—(ascertainment − affected child, AAC, blue); and sampling the families through an affected child, only but including the transmission from the mother to the other child—(ascertainment − other child, AOC, green). The grey histogram gives the simulated distribution before any ascertainment, thus corresponding to data without any ascertainment bias. Each row reflects different thresholds heteroplasmy values required to cause disease. The parameter d models the threshold for disease resulting from the pathogenic mutation. The range given for d is the range of heteroplasmies where the probability of disease increases linearly from 0 (lower level) to 1 (upper level) (see legend to Fig. 1 for details of the simulation model).

### Analysis of human pedigree data

To generate the largest data set possible, we studied pedigrees derived from a meta-analysis of published data [n= 532 mother–child pairs transmitting five common heteroplasmic mtDNA mutations: m.11778G>A (n= 117), m.3460G>A (n= 74), m.8344A>G (n= 96), m.8993T>G/C (n= 117) and m.3423A>G (n= 128), citations shown in the Material and Methods]; and 45 new unpublished mother–child pairs [m.8344A>G (n= 9), m.8993T>G/C (n= 2) and m.3243A>G (n= 34)] measured in two centres. Given that there was no obvious difference between the two data sets (Fig. 3), we merged all of the data and minimized ascertainment bias by analysing separately data with and without the clinically affected probands. The final data set included 467 mother–child pairs for the uncorrected data. m.3243A>G was analysed before and after correcting for the known decrease in leucocyte heteroplasmy levels for this specific mutation using the published approach (14). For this correction, only individuals with a heteroplasmy level of <95% were included to avoid pairs where the mother or offspring values corrected to >100%. This reduced the sample size of m.3243A>G from 137 to 99 pairs. The entire data set was used to determine the likely size of the bottleneck parameter, b, using the model described previously (12,13), incorporating the laboratory assay and laboratory site as covariates. Bayesian statistical analyses were performed using JAGS (15).

Figure 3.

Relationship between the level of mtDNA heteroplasmy in mothers and offspring for five pathogenic mtDNA mutations. Maternal and offspring heteroplasmy levels are displayed as a proportion. Red symbols= affected proband, green symbols= mother of the proband, blue= other relatives. Circles= meta-analysis data, triangles= new pedigree data measured at Centre 1, squares= new pedigree data measured at Centre 2. m.3243A>G corrected refers to heteroplasmy values after a correction for the known decline in heteroplasmy levels with age (14).

Figure 3.

Relationship between the level of mtDNA heteroplasmy in mothers and offspring for five pathogenic mtDNA mutations. Maternal and offspring heteroplasmy levels are displayed as a proportion. Red symbols= affected proband, green symbols= mother of the proband, blue= other relatives. Circles= meta-analysis data, triangles= new pedigree data measured at Centre 1, squares= new pedigree data measured at Centre 2. m.3243A>G corrected refers to heteroplasmy values after a correction for the known decline in heteroplasmy levels with age (14).

For each mutation, the average change in heteroplasmy was not significantly different from zero, consistent with no selection for or against the mtDNA mutations during transmission. The posterior differences in bottleneck strength, b, are shown in Figure 4. As predicted from the simulations, the bottleneck strength was overestimated (i.e. b is low, a narrower bottleneck) when we included pairs ascertained through a clinically affected proband. However, after the exclusion of affected probands, we observed a difference in the strength of the bottleneck parameter, b, estimated for different mtDNA mutations. m.8993T>G/C showed the largest difference, closely followed by m.3460A>G. A two tailed Bayesian hypothesis test was used to determine whether the bottleneck parameter, b, was significantly stronger for m.8993T>G/C than for m.11778G>A, m.8344A>G and m.3460G>A. This test gave estimated posterior probabilities of 0.031, 0.044 and 0.54. The bottleneck parameter distribution for the uncorrected m.3243A>G data was distorted, likely due to the known confounding effect of changing heteroplasmy levels with age (Fig. 4, far right hand panel) (14). We therefore used age-corrected data in the Bayesian hypothesis test comparing the bottleneck parameter, b, for m.8993T>G/C to m.3243A>G, which revealed an estimated posterior probability of 0.001. No other tests reached statistical significance at the 5% level.

Figure 4.

Differences in the size of the mitochondrial DNA genetic bottleneck of five pathogenic mtDNA mutations. Violin plots show the probability density for a given bottleneck strength parameter, b, calculated from actual heteroplasmy measurements in human pedigrees shown in Figure 3. All= all of the mother–child pairs. No probands= mother–child pairs remaining after the exclusion of an affected proband, thus minimising ascertainment bias. ‘m.3243A>G corrected’ refers to heteroplasmy values after a correction for the known decline in heteroplasmy levels with age (14).

Figure 4.

Differences in the size of the mitochondrial DNA genetic bottleneck of five pathogenic mtDNA mutations. Violin plots show the probability density for a given bottleneck strength parameter, b, calculated from actual heteroplasmy measurements in human pedigrees shown in Figure 3. All= all of the mother–child pairs. No probands= mother–child pairs remaining after the exclusion of an affected proband, thus minimising ascertainment bias. ‘m.3243A>G corrected’ refers to heteroplasmy values after a correction for the known decline in heteroplasmy levels with age (14).

## Discussion

Our analysis accounted for several sources of potential variability, including different analytical techniques to measure heteroplasmy levels, different laboratories, differences between individual pedigrees themselves and differences in maternal age (16). Having minimised the likely effects of ascertainment bias, we observed different rates of heteroplasmy segregation during maternal transmission between mtDNA mutations. Patients harbouring the m.8993T>G/C mutation showed more rapid segregation of heteroplasmy levels than any of the other mutations. Although this trend has been suspected for some time (17), the analysis we present here provides the first direct comparison and a formal demonstration that m.8993T>G/C behaves differently to other pathogenic mtDNA mutations. Given that the size of the mitochondrial genetic bottleneck is the primary factor determining the rate of segregation during transmission, our findings indicate differences in the behaviour of the mtDNA bottleneck based on the underlying mtDNA genotype.

The more rapid segregation observed in m.8993T>G/C pedigrees is consistent with earlier reports of major shifts in heteroplasmy observed in pedigrees transmitting this mutation (17,18). This explains why severely affected children are often the only affected individuals in families transmitting this mutation, and why their mothers are usually asymptomatic. Conversely the less dramatic rates of segregation seen for other mtDNA mutations explain why a wide range of heteroplasmy values are seen in different family members, with oligo-symptomatic individuals transmitting the mutation in larger pedigrees, with multiple moderate and severely affected family members (19,20). Our findings also provide a potential explanation for the relative frequency of different pathogenic mutations in epidemiological studies (for example, m.3243A>G being much more common than m.8993T>G/C), despite the same background frequency of healthy carriers in the population with low heteroplasmy levels (21,22). It should be noted, however, that this discussion relates to the observed statistical trends, and although uncommon, rapidly segregating families with m.3243A>G (23) and slowly segregating families with m.8993T>G/C have been described (24,25) and are also consistent with our findings.

Studies in mice have shown a dramatic reduction in the amount of mtDNA within single cells at an early stage in mammalian germ cell development (6,7). This reduction is sufficient to explain the segregation of mtDNA heteroplasmy in mice (6). Mathematical models predict that subtle differences in the size of the mtDNA genetic bottleneck at this critical period will have a dramatic impact on the rate of segregation of mtDNA heteroplasmy (26). Given emerging evidence that the amount of mtDNA within cells can be influenced by the mtDNA sequence (27,28), it is plausible that the different mtDNA mutations we have studied cause differences in the amount of mtDNA within the developing germ line (29), either through a replication advantage or as a compensation for the lower rates of oxidative phosphorylation and higher levels of reactive oxygen species production (28,30). The differences in mtDNA level would lead to differences in the rate of segregation through the genetic bottleneck. Alternatively, selection either for or against a particular mutation would lead to differences in the rate of segregation (31). First described by population genetic theory (13,32), the effective bottleneck size, Ne, can be defined like the effective population size during genetic drift. Studies in isolated reduced populations have shown that selective pressures can influence Ne, without directly influencing the true population size. The apparent difference in bottleneck size between the different mtDNA mutations could thus reflect the selection pressure and need not necessarily be caused by a difference in the actual amount of mtDNA during germ cell development. Although it will be fascinating to determine the underlying molecular mechanisms, this will not alter our conclusions, or the relevance of our findings for women transmitting the mtDNA mutations we have studied.

These findings have important implications for our understanding for the recurrence risks of heteroplasmic mtDNA diseases, for the prevention of mtDNA disease using conventional techniques such as prenatal and pre-implantation diagnosis, and also suggest that therapies aimed at manipulating a germ cell mtDNA content could influence the underlying rate of segregation of pathogenic mtDNA mutations in humans. On the one hand, increasing the mtDNA content in germ cells could slow down segregation and thereby prevent the expression of severe disease in subsequent generations. On the other hand, factors known to reduce mtDNA content could lead to more rapid segregation of heteroplasmic alleles increasing the probability of generating germ cells with very low heteroplasmy levels (as well as very high heteroplasmy levels, which may not be viable at an early stage in pregnancy). Finally, environmental factors leading to selection pressure could also alter the behaviour of the genetic bottleneck, as described earlier. Given the recent observation of widespread low-level mtDNA heteroplasmy (33), these reductions in mtDNA content could lead to the emergence of pre-existing pathogenic variants, previously present at a very low level in the female germline. With recent evidence that common genetic polymorphisms of mtDNA can influence mtDNA levels (28), it is conceivable that geographic, ethnic and even inter-familial differences in segregation occur, influenced by non-synonymous single base-pair substitutions on the background mtDNA haplotype. This could explain differences in the rates of segregation of the same pathogenic mtDNA mutation, which have been described in different families, and also influence the segregation of low-level heteroplasmy, providing a potential mechanism for the association of mtDNA haplogroups with common late-onset human diseases (34).

## Materials and Methods

### Heteroplasmy levels in mothers and offspring

We studied the transmission of mtDNA heteroplasmy in both published and new unpublished pedigrees.

#### Ascertainment of published data

Published pedigrees transmitting m.11778G>A, m.3460G>A, m.8344A>G, m.8993T>G/C and m.3423A>G were identified through a systematic review of the literature. We included all papers where the age, clinical status and laboratory methods were clearly recorded, and there was at least one heteroplasmic individual. These parameters, along with the laboratory location, were incorporated in subsequent analyses. We identified 77 publications, containing blood DNA heteroplasmy levels from 532 mother–child pairs transmitting 5 common heteroplasmic mitochondrial DNA mutations: m.11778G>A (n= 117 mother–child pairs) (35–45), m.3460G>A (n= 74) (35,37,43,46–51), m.8344A>G (n= 96) (52–62), m.8993T>G/C (n= 117) (24,62–84) and m.3423A>G (n= 128) (20,54,85,86,87,88,89,90,91,92,93,94,95–105).

#### Unpublished data

Unpublished pedigrees transmitting the m.8344A>G (n= 9 mother–child pairs), m.8993T>G/C (n= 2) and m.3423A>G (n= 34) mutations were identified from the following accredited diagnostic laboratories: Newcastle, UK; Maastricht, NL; Milan and Bologna, Italy; Bergen, Norway and Munich, Germany. Genomics DNA was analysed in two laboratories: Newcastle, UK; and Maastricht, NL.

#### Quantification of heteroplasmy: Newcastle

Heteroplasmy was measured by quantitative pyrosequencing. Pyromark Assay Design Software v.2.0 (Qiagen) was used to design primers for template generation, one of which contains a biotinylated tag (*BIO), and the pyrosequencing reaction (Seq). For m.3243A>G: F-*BIO-TAAGGCCTACTTCACAAAGCG, R-GCGATTAGAATGGGTACAATGAG, Seq-ATGCGATTACCGGGC; for m.8344A>G: F-*BIO-CATGCCCATCGTCCTAGAAT, R-TTTTTATGGGCTTTGGTGAGG, Seq-TAAGTTAAAGATTAAGAGA; and for m.8993T>G F-AGGCACACCTACACCCCTTA, R-*TGTGAAAACGTAGGCTTGGAT, Seq-CATTCAACCAATAGCCC. Product templates were generated with a GoTaq® DNA Polymerase (Promega) reaction according to manufacturer's protocol. Pyrosequencing was performed using the Pyromark Q24 platform according to the manufacturer's protocol, using the designed pyrosequencing primers for each mutation. Pyromark Q24 software was used to quantify the heteroplasmy levels of each mutation through comparison of the relevant peak heights of both wild-type and mutant mtDNA. The accuracy of the pyrosequencing assay was determined by generating wild-type and mutant clones, which when mixed at the correct proportions mimicked a range of heteroplasmy levels between 0 and 100% for each mutation. Each mixed sample was assessed for mutation load using pyrosequencing and used to generate a standard curve (Supplementary Material, Fig. S1).

#### Quantification of heteroplasmy: Maastricht

Heteroplasmy was measured by semiquantitative restriction fragment length polymorphism (RFLP) analysis using mutation-specific restriction enzymes. Primers used for m.3243A>G were as follows: CAACTTAGTATTATACCCACACCAACTTAGTATTATACCCACAC (forward) and TTTCGTTCGGTAAGCATTAG (reverse); for m.8344A>G: TCGTCCTAGAATTAATTCCC (forward) and GTAGTATTTAGTTGGGGCATTTCACTGTAAAGCCGTGTTG (reverse); and for m.8993T>C/G: CACACCTACACCCCTTATCCC (forward) and TCATTATGTGTTGTCGTGCAG (reverse). For each forward primer, an unlabelled and FAM-labelled primer was available, polymerase chain reaction (PCR) was performed on the GeneAmp PCR System 9700 (Perkin–Elmer Applied Biosystems) in a total volume of 50 µl, containing 1× PCR buffer (Invitrogen) and 1 U of Taq DNA polymerase (Invitrogen), 2 mm MgCl2 (Invitrogen) and 0.1 mm dNTP (Pharmacia). Unlabelled forward (15 pmol) and reverse primer (3pmal) sets were used in the first round. First round PCR started with 5-min denaturation at 94°C followed by 32 cycles of 1-min denaturation at 92°C, 45-s annealing at 53°C and 45-s elongation at 72°C, followed by a final elongation step of 7 min at 72°C. 15 ml of first round amplification product was adjusted to 50 ml with second round PCR mix, containing 1× PCR buffer (Invitrogen) 1 U of DNA polymerase, 2 mm MgCl2 (Invitrogen) and a fluorescently labelled forward primer (15 pmol). One final amplification cycle was performed of 5-min denaturation at 94°C, 1-min annealing at 53°C and 7-min elongation at 72°C. The labelled second round PCR product (15 ml) was digested in the appropriate digestion buffer in a total volume of 50 ml containing 10 U HaeIII (10 U/µl; Roche) for the m.3243A>G mutation, 10 U BglI (10 U/µl) for the m.8344A>G mutation and 10 U HpaII (10 U/ µl; Roche) for the m.8993T>G mutation. The m.3243A>G-amplicon contains an additional HaeIII restriction site as an internal control for restriction enzyme digestion completion. For each mutation, a sample with known mutation load was included as reference. After digestion, samples were purified using a QIAquick PCR purification kit (Qiagen). Samples were analysed by capillary electrophoresis on an ABI Prism 3730 Genetic Analyser followed by GeneScan analysis. To calculate the mutation load, the area of the mutation peak was divided by the sum of the peak area of the wild-type and mutation peak. Each forward primer contains a biotinylated tag. PCR was performed using Taq DNA Polymerase (Invitrogen) according to manufacturer's protocol. The PCR fragments were digested by mutation-specific restriction enzymes: Hae III (10 U/µl) Roche for m.3243A>G; Bgl I (10 U/µl) Biolabs for m.8344A>G and Hpa II (10 U/µl) Roche for m.8993T>C/G. Digestion products were purified (Qiagen Qiaquick PCR Purification). Fragments were separated and analysed by capillary electrophoresis (automatic DNA Sequencer 3730, Applied Biosystems) to determine mutation load.

### Simulation of mtDNA heteroplasmy inheritance

An overview of the simulations is shown in Figure 1. First, we generated a population of simulated mothers with different heteroplasmy levels. We then modelled the inheritance of heteroplasmy from each of these mothers to two offspring (see below). Next we determined the likelihood of the individuals being clinically affected. We then sampled the simulated families based on the following ascertainment criteria, in the following order: This sampling algorithm mimicked the different ways that mother–child pairs could be identified in a study of real human pedigrees, and the resulting difference in distributions for the three different pairs (Supplementary Material, Fig. S2) resembles those from the real data from the human pedigrees shown in Figure 3.

1. Sampling the families where there was an affected mother—(AAM). Here, we included the transmission from the mother to both offspring.

2. Sampling the families where there was an affected child—(AAC), but the mother was not affected. Here, we also included the transmission from the mother to both offspring.

3. Sampling the families through an affected child, only but including the transmission from the mother to the other child—(AOC).

We then compared the sampled mother–child pairs with the original simulated pedigrees to determine whether the ascertainment method influenced the observed distribution of heteroplasmy transmissions.

#### Simulating the transmission of mtDNA heteroplasmy (the mtDNA genetic bottleneck)

We modelled the mtDNA genetic bottleneck using established model derived from measurements of heteroplasmy made in human oocytes (12,13). Here, the bottleneck is modelled as an infinite population of mtDNA molecules that was instantaneously reduced to the minimum bottleneck size N for g generations, before expanding back to a large size. We simulated the transmission of mtDNA heteroplasmy of a neutral allele from the mother to two offspring through the bottleneck by approximating with a beta distribution with parameters α= pMb/(1−b) and β= (1−pM)b/(1−b), where b is the drift parameter b= exp(−g/N). A large b indicates a weak (or wide) genetic bottleneck. When g is large compared with N, b approaches 0, corresponding to a very strong (or narrow) bottleneck. Using this model, we generated a series of simulated heteroplasmy transmissions from mothers starting with a range of heteroplasmy values.

### Statistical methods

Changes in heteroplasmy between mother and child were modelled using the same beta distribution as for the simulation, using a hierarchical model with a different bottleneck parameter b for each mtDNA mutation. Measurements of heteroplasmy were assumed to be normally distributed about the true values, with censored at 0 and 1, and with an error rate σi for publication i, to allow for different measurement technologies. Writing mi for the observed maternal heteroplasmy for mother i, and oij for the heteroplasmy for child j of mother i, where j= 1, , ni, and ni is the number of children for Mother I,

$mi∼N(μi,σri2),oij∼N(ηij,σri2),andηij∼Betaμibmk1−bmk,(1−μi)bmk1−bmk,$

where $bk=exp(−(g/Nk)),$mi gives the mutation for Mother i and ri gives the publication for mother i. To test for systematic increases or decreases in the level of heteroplasmy (i.e. selection), an additional model was built. This allowed for changes in the expected heteroplasmy in the offspring, by multiplying the expected child heteroplasmy, $μi,$ by a random factor, $sk,$ which models selection by increasing or decreasing the average heteroplasmy value. The JAGS statistical package (15) was used to make inferences about the strength of the bottleneck using the statistical model above. A Bayesian framework was used with uniform priors between 0 and 1 for the bk, and broad priors for the error parameters $σk2.$ The model and all JAGS and R scripts are available through the following URL: github.com/ijwilson/mtBottleneck.

### Bayesian comparison of bottleneck sizes

Maternal heteroplasmy values were corrected for age as described (14). Bayesian hypothesis tests of bj versus bk were calculated using the empirical joint posterior distributions of bj and bk from JAGS (15) and using $2max(P(bj>bk),P(bk>bj)).$ The JAGS script is available through the following URL: github.com/ijwilson/mtBottleneck.

## Funding

Funding to pay the Open Access publication charges for this article was provided by the Wellcome Trust.

## Acknowledgements

P.F.C. is a Wellcome Trust Senior Fellow in Clinical Science (101876/Z/13/Z), and a UK NIHR Senior Investigator, who receives support from the Medical Research Council Mitochondrial Biology Unit (MC_UP_1501/2), the Wellcome Trust Centre for Mitochondrial Research (096919Z/11/Z), the Medical Research Council (UK) Centre for Translational Muscle Disease research (G0601943), EU FP7 TIRCON, and the National Institute for Health Research (NIHR) Biomedical Research Centre based at Cambridge University Hospitals NHS Foundation Trust and the University of Cambridge. The views expressed are those of the author(s) and not necessarily those of the NHS, the NIHR or the Department of Health. R.H. is supported by the European Research Council (309548). P.F.C. and R.H. are supported by the Mitochondrial European Educational Training, ITN Marie Curie People (317433). R.W.T. also receives support from The Lily Foundation and the UK NHS Specialist Commissioners, which funds the ‘Rare Mitochondrial Disorders of Adults and Children’ Diagnostic Service in Newcastle upon Tyne. C.L.A. is the recipient of an NIHR doctoral fellowship (NIHR-HCS-D12-03-04). We acknowledge the ‘Cell lines and DNA Bank of Paediatric Movement Disorders and Neurodegenerative Diseases’ of the Telethon Network of Genetic Biobanks (grant GTB12001J) and the EurobiobanK Network. This work was supported by the Medical Research Council, the Pierfranco and Luisa Mariani Foundation, Telethon Grant GGP11011 and ERC Advanced Grant FP7-322424. The publication is the work of the authors, and Patrick Chinnery will serve as guarantor for the contents of this paper.

Conflict of Interest statement. None declared.

## References

1
Vafai
S.B.
,
Mootha
V.K.
(
2013
)
Medicine. A common pathway for a rare disease?
Science
,
342
,
1453
1454
.
2
Gorman
G.S.
,
Schaefer
A.M.
,
Ng
Y.
et al
. (
2015
)
Prevalence of nuclear and mtDNA mutations related to adult mitochondrial disease
.
Ann. Neurol.
,
77
,
753
759
.
3
DiMauro
S.
,
Schon
E.A.
,
Carelli
V.
,
Hirano
M.
(
2013
)
The clinical maze of mitochondrial neurology
.
Nat. Rev. Neurol.
,
9
,
429
444
.
4
Upholt
W.B.
,
Dawid
I.B.
(
1977
)
Mapping of mitochondrial DNA of individual sheep and goats: rapid evolution in the D loop region
.
Cell
,
11
,
571
583
.
5
Olivo
P.D.
,
Van de Walle
M.J.
,
Laipis
P.J.
,
Hauswirth
W.W.
(
1983
)
Nucleotide sequence evidence for rapid genotypic shifts in the bovine mitochondrial DNA D-loop
.
Nature
,
306
,
400
402
.
6
Cree
L.M.
,
Samuels
D.C.
,
de Sousa Lopes
S.C.
et al
. (
2008
)
A reduction of mitochondrial DNA molecules during embryogenesis explains the rapid segregation of genotypes
.
Nat. Genet.
,
40
,
249
254
.
7
Wai
T.
,
Teoli
D.
,
Shoubridge
E.A.
(
2008
)
The mitochondrial DNA genetic bottleneck results from replication of a subpopulation of genomes
.
Nat. Genet.
,
40
,
1484
1488
.
8
Howell
N.
,
Halvorson
S.
,
Kubacka
I.
,
McCullough
D.A.
,
Bindoff
L.A.
,
Turnbull
D.M.
(
1992
)
Mitochondrial gene segregation in mammals: is the bottleneck always narrow?
Hum. Genet.
,
90
,
117
120
.
9
Chinnery
P.F.
,
Thorburn
D.R.
,
Samuels
D.C.
et al
. (
2000
)
The inheritance of mtDNA heteroplasmy: random drift, selection or both?
Trends Genet.
,
16
,
500
505
.
10
Uusimaa
J.
,
Moilanen
J.S.
,
Vainionpaa
L.
et al
. (
2007
)
Prevalence, segregation, and phenotype of the mitochondrial DNA 3243A>G mutation in children
.
Ann. Neurol.
,
62
,
278
287
.
11
Hellebrekers
D.M.
,
Wolfe
R.
,
Hendrickx
A.T.
et al
. (
2012
)
PGD and heteroplasmic mitochondrial DNA point mutations: a systematic review estimating the chance of healthy offspring
.
Hum. Reprod.
,
update
18
,
341
349
.
12
Brown
D.T.
,
Samuels
D.C.
,
Michael
E.M.
,
Turnbull
D.M.
,
Chinnery
P.F.
(
2001
)
Random genetic drift determines the level of mutant mitochondrial DNA in human primary oocytes
.
Am. J. Hum. Genet.
,
68
,
553
536
.
13
Wright
S.
(
1969
)
Evolution and the Genetics of Populations
.
University of Chicago Press
,
Chicago
.
14
Rajasimha
H.K.
,
Chinnery
P.F.
,
Samuels
D.C.
(
2008
)
Selection against pathogenic mtDNA mutations in a stem cell population leads to the loss of the 3243A-->G mutation in blood
.
Am. J. Hum. Genet.
,
82
,
333
343
.
15
Plummer
M
. (
2003
)
JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling
. In
Proceedings of the 3rd International Workshop on Distributed Statistical Computing
.
Technische Universit at Wien, Vol. 124, p. 125
.
16
Rebolledo-Jaramillo
B.
,
Su
M.S.
,
Stoler
N.
et al
. (
2014
)
Maternal age effect and severe germ-line bottleneck in the inheritance of human mitochondrial DNA
.
,
111
,
15474
15479
.
17
Blok
R.B.
,
Gook
D.A.
,
Thorburn
D.R.
,
Dahl
H.H.
(
1997
)
Skewed segregation of the mtDNA nt 8993 (T to G) mutation in human oocytes
.
Am. J. Hum. Genet.
,
60
,
1495
1501
.
18
White
S.L.
,
Collins
V.A.
,
Woolfe
R.
et al
. (
1999
)
Genetic counseling and prenatal diagnosis for the mitochondrial DNA mutations at nucleotide 8993
.
Am. J. Hum. Genet.
,
65
,
474
482
.
19
N.G.
,
Tulinius
M.H.
,
Holme
E.
et al
. (
1992
)
Segregation and manifestations of the mtDNA tRNA(Lys) A-->G(8344) mutation of myoclonus epilepsy and ragged-red fibers (MERRF) syndrome
.
Am. J. Hum. Genet.
,
51
,
1201
1212
.
20
Ciafaloni
E.
,
Ricci
E.
,
Shanske
S.
et al
. (
1992
)
MELAS: clinical features, biochemistry, and molecular genetics
.
Ann. Neurol.
,
31
,
391
398
.
21
Elliott
H.R.
,
Samuels
D.C.
,
Eden
J.A.
,
Relton
C.L.
,
Chinnery
P.F.
(
2008
)
Pathogenic mitochondrial DNA mutations are common in the general population
.
Am. J. Hum. Genet.
,
83
,
254
460
.
22
Chinnery
P.F.
,
Johnson
M.A.
,
Wardell
T.M.
et al
. (
2000
)
Epidemiology of pathogenic mitochondrial DNA mutations
.
Ann. Neurol.
,
48
,
188
193
.
23
Hammans
S.R.
,
Sweeney
M.G.
,
Brockington
M.
et al
. (
1993
)
The mitochondrial DNA transfer RNA(Lys)A-->G(8344) mutation and the syndrome of myoclonic epilepsy with ragged red fibres (MERRF). Relationship of clinical phenotype to proportion of mutant mitochondrial DNA
.
Brain
,
116
,
617
632
.
24
Uziel
G.
,
Moroni
I.
,
Lamantea
E.
et al
. (
1997
)
Mitochondrial disease associated with the T8993G mutation of the mitochondrial ATPase 6 gene: a clinical, biochemical, and molecular study in six families
.
J. Neurol. Neurosurg. Psychiatry
,
63
,
16
22
.
25
Pfeffer
G.
,
Blakely
E.L.
,
Alston
C.L.
et al
. (
2012
)
Adult-onset spinocerebellar ataxia syndromes due to MTATP6 mutations
.
J. Neurol. Neurosurg. Psychiatry
,
83
,
883
886
.
26
Wonnapinij
P.
,
Chinnery
P.F.
,
Samuels
D.C.
(
2010
)
Previous estimates of mitochondrial DNA mutation level variance did not account for sampling error: comparing the mtDNA genetic bottleneck in mice and humans
.
Am. J. Hum. Genet.
,
86
,
540
550
.
27
Moreno-Loshuertos
R.
,
Acin-Perez
R.
,
Fernandez-Silva
P.
et al
. (
2006
)
Differences in reactive oxygen species production explain the phenotypes associated with common mouse mitochondrial DNA variants
.
Nat. Genet.
,
38
,
1261
1268
.
28
Gomez-Duran
A.
,
Pacheu-Grau
D.
,
Lopez-Gallardo
E.
et al
. (
2010
)
Unmasking the causes of multifactorial disorders: OXPHOS differences between mitochondrial haplogroups
.
Hum. Mol. Genet.
,
19
,
3343
3353
.
29
Monnot
S.
,
Samuels
D.C.
,
Hesters
L.
et al
. (
2013
)
Mutation dependance of the mitochondrial DNA copy number in the first stages of human embryogenesis
.
Hum. Mol. Genet.
,
22
,
1867
1872
.
30
Pello
R.
,
Martin
M.A.
,
Carelli
V.
et al
. (
2008
)
Mitochondrial DNA background modulates the assembly kinetics of OXPHOS complexes in a cellular model of mitochondrial disease
.
Hum. Mol. Genet.
,
17
,
4001
4011
.
31
Stewart
J.B.
,
N.G.
(
2014
)
Keeping mtDNA in shape between generations
.
PLoS Genet.
,
10
,
e1004670
.
32
Solignac
M.
,
Monnerot
M.
,
Mounolou
J.C.
(
1983
)
Mitochondrial DNA heteroplasmy in Drosophila mauritiana
.
,
80
,
6942
6946
.
33
Payne
B.A.
,
Wilson
I.J.
,
Yu-Wai-Man
P.
et al
. (
2013
)
Universal heteroplasmy of human mitochondrial DNA
.
Hum. Mol. Genet.
,
22
,
384
390
.
34
Hudson
G.
,
Gomez-Duran
A.
,
Wilson
I.J.
,
Chinnery
P.F.
(
2014
)
Recent mitochondrial DNA mutations increase the risk of developing common late-onset human diseases
.
PLoS Genet.
,
10
,
e1004369
.
35
Carelli
V.
,
Ghelli
A.
,
Ratta
M.
et al
. (
1997
)
Leber's hereditary optic neuropathy: biochemical effect of 11778/ND4 and 3460/ND1 mutations and correlation with the mitochondrial genotype
.
Neurology
,
48
,
1623
1632
.
36
Chuenkongkaew
W.L.
,
Suphavilai
R.
,
Vaeusorn
L.
,
Phasukkijwatana
N.
,
Lertrit
P.
,
Suktitipat
B.
(
2005
)
Proportion of 11778 mutant mitochondrial DNA and clinical expression in a thai population with leber hereditary optic neuropathy
.
J. Neuro-Ophthalmol.
,
25
,
173
175
.
37
Harding
A.E.
,
Sweeney
M.G.
,
Govan
G.G.
,
Riordan-Eva
P.
(
1995
)
Pedigree analysis in Leber hereditary optic neuropathy families with a pathogenic mtDNA mutation
.
Am. J. Hum. Genet.
,
57
,
77
86
.
38
Holt
I.J.
,
Miller
D.H.
,
Harding
A.E.
(
1989
)
Genetic heterogeneity and mitochondrial DNA heteroplasmy in Leber's hereditary optic neuropathy
.
J. Med. Genet.
,
26
,
739
743
.
39
Howell
N.
,
Xu
M.
,
Halvorson
S.
,
Bodis-Wollner
I.
,
Sherman
J.
(
1994
)
A heteroplasmic LHON family: tissue distribution and transmission of the 11778 mutation
.
Am. J. Hum. Genet.
,
55
,
203
206
.
40
Juvonen
V.
,
Nikoskelainen
E.
,
Lamminen
T.
,
Penttinen
M.
,
Aula
P.
,
Savontaus
M.L.
(
1997
)
Tissue distribution of the ND4/11778 mutation in heteroplasmic lineages with Leber hereditary optic neuropathy
.
Hum. Mutat.
,
9
,
412
417
.
41
Phasukkijwatana
N.
,
Chuenkongkaew
W.L.
,
Suphavilai
R.
,
Luangtrakool
K.
,
Kunhapan
B.
,
Lertrit
P.
(
2006
)
Transmission of heteroplasmic G11778A in extensive pedigrees of Thai Leber hereditary optic neuropathy
.
J. Hum. Genet.
,
51
,
1110
1117
.
42
Simon
D.K.
,
Pulst
S.M.
,
Sutton
J.P.
,
Browne
S.E.
,
Beal
M.F.
,
Johns
D.R.
(
1999
)
Familial multisystem degeneration with parkinsonism associated with the 11778 mitochondrial DNA mutation
.
Neurology
,
53
,
1787
1793
.
43
Sweeney
M.G.
,
Davis
M.B.
,
Lashwood
A.
,
Brockington
M.
,
Toscano
A.
,
Harding
A.E.
(
1992
)
Evidence against an X-linked locus close to DXS7 determining visual loss susceptibility in British and Italian families with Leber hereditary optic neuropathy
.
Am. J. Hum. Genet.
,
51
,
741
748
.
44
Tanaka
A.
,
Kiyosawa
M.
,
Mashima
Y.
,
Tokoro
T.
(
1998
)
A family with Leber's hereditary optic neuropathy with mitochondrial DNA heteroplasmy related to disease expression
.
J Neuro-Ophthalmol.
,
18
,
81
83
.
45
Zhu
D.P.
,
Economou
E.P.
,
Antonarakis
S.E.
,
Maumenee
I.H.
(
1992
)
Mitochondrial DNA mutation and heteroplasmy in type I Leber hereditary optic neuropathy
.
Am. J. Med. Genet.
,
42
,
17317
9
.
46
Black
G.C.
,
Morten
K.
,
Laborde
A.
,
Poulton
J.
(
1996
)
Leber's hereditary optic neuropathy: heteroplasmy is likely to be significant in the expression of LHON in families with the 3460 ND1 mutation
.
Brit. J. Ophthalmol.
,
80
,
915
917
.
47
Ghosh
S.S.
,
Fahy
E.
,
Bodis-Wollner
I.
,
Sherman
J.
,
Howell
N.
(
1996
)
Longitudinal study of a heteroplasmic 3460 Leber hereditary optic neuropathy family by multiplexed primer-extension analysis and nucleotide sequencing
.
Am. J. Hum. Genet.
,
58
,
325
334
.
48
Howell
N.
,
Bindoff
L.A.
,
McCullough
D.A.
et al
. (
1991
)
Leber hereditary optic neuropathy: identification of the same mitochondrial ND1 mutation in six pedigrees
.
Am. J. Hum. Genet.
,
49
,
939
950
.
49
Howell
N.
,
McCullough
D.
,
Bodis-Wollner
I.
(
1992
)
Molecular genetic analysis of a sporadic case of Leber hereditary optic neuropathy
.
Am. J. Hum. Genet.
,
50
,
443
446
.
50
Kaplanova
V.
,
Zeman
J.
,
Hansikova
H.
et al
. (
2004
)
Segregation pattern and biochemical effect of the G3460A mtDNA mutation in 27 members of LHON family
.
J. Neurol. Sci.
,
223
,
149
155
.
51
Volod'ko
N.V.
,
L'Vova
M.A.
,
Starikovskaia
E.B.
et al
. (
2006
)
Spectrum of pathogenic mtDNA mutations in Leber hereditary optic neuropathy families from Siberia
.
Genetika
,
42
,
89
97
.
52
Campos
Y.
,
Esteban
J.
,
Cabello
A.
,
Arenas
J.
(
1994
)
Genetic analysis of one family with myoclonic epilepsy and ragged-red fibers (MERRF)
.
Muscle Nerve
,
17
,
1229
1231
.
53
Gamez
J.
,
Playan
A.
,
Andreu
A.L.
et al
. (
1998
)
Familial multiple symmetric lipomatosis associated with the A8344G mutation of mitochondrial DNA
.
Neurology
,
51
,
258
260
.
54
Hammans
S.R.
,
Sweeney
M.G.
,
Hanna
M.G.
,
Brockington
M.
,
Morgan-Hughes
J.A.
,
Harding
A.E.
(
1995
)
The mitochondrial DNA transfer RNALeu(UUR) A-->G(3243) mutation. A clinical and genetic study
.
Brain
,
118
(Pt 3)
,
721
734
.
55
Howell
N.
,
Kubacka
I.
,
Smith
R.
,
Frerman
F.
,
Parks
J.K.
,
Parker
W.D.
Jr
(
1996
)
Association of the mitochondrial 8344 MERRF mutation with maternally inherited spinocerebellar degeneration and Leigh disease
.
Neurology
,
46
,
219
222
.
56
Munoz-Malaga
A.
,
Bautista
J.
,
Salazar
J.A.
et al
. (
2000
)
Lipomatosis, proximal myopathy, and the mitochondrial 8344 mutation. A lipid storage myopathy?
Muscle Nerve
,
23
,
538
542
.
57
Piccolo
G.
,
Focher
F.
,
Verri
A.
et al
. (
1993
)
Myoclonus epilepsy and ragged-red fibers: blood mitochondrial DNA heteroplasmy in affected and asymptomatic members of a family
.
Acta. Neurol. Scand.
,
88
,
406
409
.
58
Seibel
P.
,
Flierl
A.
,
Kottlors
M.
,
Reichmann
H.
(
1994
)
A rapid and sensitive PCR screening method for point mutations associated with mitochondrial encephalomyopathies
.
Biochem. Biophys. Res. Commun.
,
200
,
938
942
.
59
Silvestri
G.
,
Ciafaloni
E.
,
Santorelli
F.M.
et al
. (
1993
)
Clinical features associated with the A-->G transition at nucleotide 8344 of mtDNA (‘MERRF mutation’)
.
Neurology
,
43
,
1200
1206
.
60
Traff
J.
,
Holme
E.
,
Ekbom
K.
,
Nilsson
B.Y.
(
1995
)
Ekbom's syndrome of photomyoclonus, cerebellar ataxia and cervical lipoma is associated with the tRNA(Lys) A8344G mutation in mitochondrial DNA
.
Acta Neurol. Scand.
,
92
,
394
397
.
61
van de Glind
G.
,
de Vries
M.
,
Rodenburg
R.
,
Hol
F.
,
Smeitink
J.
,
Morava
E.
(
2007
)
Resting muscle pain as the first clinical symptom in children carrying the MTTK A8344G mutation
.
Eur. J. Paed. Neurol.
,
11
,
243
246
.
62
Wong
L.J.
,
Wong
H.
,
Liu
A.
(
2002
)
Intergenerational transmission of pathogenic heteroplasmic mitochondrial DNA
.
Genet. Med.
,
4
,
78
83
.
63
Bartley
J.
,
D.
,
Park
P.
,
Brar
H.
,
D
A.
,
Wong
L.
(
1996
)
Prenatal diagnosis of T8993G mitochondrial DNA point mutation in amniocytes by heteroplasmy detection
.
Am. J. Hum. Genet.
,
59
.
64
Ciafaloni
E.
,
Santorelli
F.M.
,
Shanske
S.
et al
. (
1993
)
Maternally inherited Leigh syndrome
.
J. Pediatr.
,
122
,
419
422
.
65
Degoul
F.
,
Diry
M.
,
Rodriguez
D.
et al
. (
1995
)
Clinical, biochemical, and molecular analysis of a maternally inherited case of Leigh syndrome (MILS) associated with the mtDNA T8993G point mutation
.
J. Inher. Met. Dis.
,
18
,
682
688
.
66
Enns
G.M.
,
Bai
R.K.
,
Beck
A.E.
,
Wong
L.J.
(
2006
)
Molecular-clinical correlations in a family with variable tissue mitochondrial DNA T8993G mutant load
.
Mol. Genet. Metab.
,
88
,
364
371
.
67
Ferlin
T.
,
Landrieu
P.
,
Rambaud
C.
et al
. (
1997
)
Segregation of the G8993 mutant mitochondrial DNA through generations and embryonic tissues in a family at risk of Leigh syndrome
.
J. Pediatr.
,
131
,
447
449
.
68
Fryer
A.
,
Appleton
R.
,
Sweeney
M.G.
,
Rosenbloom
L.
,
Harding
A.E.
(
1994
)
Mitochondrial DNA 8993 (NARP) mutation presenting with a heterogeneous phenotype including ‘cerebral palsy
’.
Arch. Dis. Childhood
,
71
,
419
422
.
69
Holt
I.J.
,
Harding
A.E.
,
Petty
R.K.
,
Morgan-Hughes
J.A.
(
1990
)
A new mitochondrial disease associated with mitochondrial DNA heteroplasmy
.
Am. J. Hum. Genet.
,
46
,
428
433
.
70
Houstek
J.
,
Klement
P.
,
Hermanska
J.
et al
. (
1995
)
Altered properties of mitochondrial ATP-synthase in patients with a T-->G mutation in the ATPase 6 (subunit a) gene at position 8993 of mtDNA
.
Biochim. Biophys. Acta
,
1271
,
349
357
.
71
Lodi
R.
,
Montagna
P.
,
Iotti
S.
et al
. (
1994
)
Brain and muscle energy metabolism studied in vivo by 31P-magnetic resonance spectroscopy in NARP syndrome
.
J. Neurol. Neurosurg. Psychiatry
,
57
,
1492
1496
.
72
Mak
S.C.
,
Chi
C.S.
,
Liu
C.Y.
,
Pang
C.Y.
,
Wei
Y.H.
(
1996
)
Leigh syndrome associated with mitochondrial DNA 8993 T-->G mutation and ragged-red fibers
.
Pediatr. Neurol.
,
15
,
72
75
.
73
Makela-Bengs
P.
,
Suomalainen
A.
,
Majander
A.
et al
. (
1995
)
Correlation between the clinical symptoms and the proportion of mitochondrial DNA carrying the 8993 point mutation in the NARP syndrome
.
Pediatric Res.
,
37
,
634
639
.
74
Mkaouar-Rebai
E.
,
Chaari
W.
,
Younes
S.
,
Bousoffara
R.
,
Sfar
M.T.
,
Fakhfakh
F.
(
2009
)
Maternally inherited Leigh syndrome: T8993G mutation in a Tunisian family
.
Pediatr. Neurol.
,
40
,
437
442
.
75
Pastores
G.M.
,
Santorelli
F.M.
,
Shanske
S.
et al
. (
1994
)
Leigh syndrome and hypertrophic cardiomyopathy in an infant with a mitochondrial DNA point mutation (T8993G)
.
Am. J. Med. Genet.
,
50
,
265
271
.
76
Porto
F.B.
,
Mack
G.
,
Sterboul
M.J.
et al
. (
2001
)
Isolated late-onset cone-rod dystrophy revealing a familial neurogenic muscle weakness, ataxia, and retinitis pigmentosa syndrome with the T8993G mitochondrial mutation
.
Am. J. Ophthalmol.
,
132
,
935
937
.
77
Puddu
P.
,
Barboni
P.
,
Mantovani
V.
et al
. (
1993
)
Retinitis pigmentosa, ataxia, and mental retardation associated with mitochondrial DNA mutation in an Italian family
.
Brit. J. Ophthalmol.
,
77
,
84
88
.
78
Sakuta
R.
,
Goto
Y.
,
Horai
S.
et al
. (
1992
)
Mitochondrial DNA mutation and Leigh's syndrome
.
Ann. Neurol.
,
32
,
597
598
.
79
Santorelli
F.M.
,
Shanske
S.
,
Macaya
A.
,
DeVivo
D.C.
,
DiMauro
S.
(
1993
)
The mutation at nt 8993 of mitochondrial DNA is a common cause of Leigh's syndrome
.
Ann. Neurol.
,
34
,
827
834
.
80
Shoffner
J.M.
,
Fernhoff
P.M.
,
Krawiecki
N.S.
et al
. (
1992
)
Subacute necrotizing encephalopathy: oxidative phosphorylation defects and the ATPase 6 point mutation
.
Neurology
,
42
,
2168
2174
.
81
Steffann
J.
,
Gigarel
N.
,
Corcos
J.
et al
. (
2007
)
Stability of the m.8993T->G mtDNA mutation load during human embryofetal development has implications for the feasibility of prenatal diagnosis in NARP syndrome
.
J. Med. Genet.
,
44
,
664
669
.
82
Tatuch
Y.
,
Christodoulou
J.
,
Feigenbaum
A.
et al
. (
1992
)
Heteroplasmic mtDNA mutation (T----G) at 8993 can cause Leigh disease when the percentage of abnormal mtDNA is high
.
Am. J. Hum. Genet.
,
50
,
852
858
.
83
Tulinius
M.H.
,
Houshmand
M.
,
N.G.
et al
. (
1995
)
De novo mutation in the mitochondrial ATP synthase subunit 6 gene (T8993G) with rapid segregation resulting in Leigh syndrome in the offspring
.
Hum. Genet.
,
96
,
290
294
.
84
White
S.L.
,
Shanske
S.
,
McGill
J.J.
et al
. (
1999
)
Mitochondrial DNA mutations at nucleotide 8993 show a lack of tissue- or age-related variation
.
J. Inher. Metab. Dis.
,
22
,
899
914
.
85
Cervin
C.
,
Liljestrom
B.
,
Tuomi
T.
et al
. (
2004
)
Cosegregation of MIDD and MODY in a pedigree: functional and clinical consequences
.
Diabetes
,
53
,
1894
1899
.
86
Chinnery
P.F.
,
Zwijnenburg
P.J.
,
Walker
M.
et al
. (
1999
)
Nonrandom tissue distribution of mutant mtDNA
.
Am. J. Med. Genet.
,
85
,
498
501
.
87
Chou
Y.J.
,
Ou
C.Y.
,
Hsu
T.Y.
et al
. (
2004
)
Prenatal diagnosis of a fetus harboring an intermediate load of the A3243G mtDNA mutation in a maternal carrier diagnosed with MELAS syndrome
.
Prenatal Diagn.
,
24
,
367
370
.
88
Dubeau
F.
,
De Stefano
N.
,
Zifkin
B.G.
,
Arnold
D.L.
,
Shoubridge
E.A.
(
2000
)
Oxidative phosphorylation defect in the brains of carriers of the tRNAleu(UUR) A3243G mutation in a MELAS pedigree
.
Ann. Neurol.
,
47
,
179
185
.
89
Fabrizi
G.M.
,
Cardaioli
E.
,
Grieco
G.S.
et al
. (
1996
)
The A to G transition at nt 3243 of the mitochondrial tRNALeu(UUR) may cause an MERRF syndrome
.
J. Neurol. Neurosurg. Psychiatry
,
61
,
47
51
.
90
Hotta
O.
,
Inoue
C.N.
,
Miyabayashi
S.
,
Furuta
T.
,
Takeuchi
A.
,
Taguma
Y.
(
2001
)
Clinical and pathologic features of focal segmental glomerulosclerosis with mitochondrial tRNALeu(UUR) gene mutation
.
Kidney Int.
,
59
,
1236
1243
.
91
Huang
C.C.
,
Chen
R.S.
,
Chen
C.M.
et al
. (
1994
)
MELAS syndrome with mitochondrial tRNA(Leu(UUR)) gene mutation in a Chinese family
.
J. Neurol. Neurosurg. Psychiatry
,
57
,
586
589
.
92
Huang
C.C.
,
Chen
R.S.
,
Chu
N.S.
,
Pang
C.Y.
,
Wei
Y.H.
(
1996
)
Random mitotic segregation of mitochondrial DNA in MELAS syndrome
.
Acta Neurol. Scand.
,
93
,
198
202
.
93
Iwanishi
M.
,
Obata
T.
,
S.
et al
. (
1995
)
Clinical and laboratory characteristics in the families with diabetes and a mitochondrial tRNA(LEU(UUR)) gene mutation
.
Diabetes Res. Clin. Pract.
,
29
,
75
82
.
94
Jansen
J.J.
,
Maassen
J.A.
,
van der Woude
F.J.
et al
. (
1997
)
Mutation in mitochondrial tRNA(Leu(UUR)) gene associated with progressive kidney disease
.
J. Am. Soc. Nephrol. JASN
,
8
,
1118
1124
.
95
Ko
C.H.
,
Lam
C.W.
,
Tse
P.W.
,
Kong
C.K.
,
Chan
A.K.
,
Wong
L.J.
(
2001
)
De novo mutation in the mitochondrial tRNALeu(UUR) gene (A3243G) with rapid segregation resulting in MELAS in the offspring
.
J. Paedr. Child Health
,
37
,
87
90
.
96
Lien
L.M.
,
Lee
H.C.
,
Wang
K.L.
,
Chiu
J.C.
,
Chiu
H.C.
,
Wei
Y.H.
(
2001
)
Involvement of nervous system in maternally inherited diabetes and deafness (MIDD) with the A3243G mutation of mitochondrial DNA
.
Acta Neurol. Scand.
,
103
,
159
165
.
97
Liou
C.W.
,
Huang
C.C.
,
Chee
E.C.
et al
. (
1994
)
MELAS syndrome: correlation between clinical features and molecular genetic analysis
.
Acta Neurol. Scand.
,
90
,
354
359
.
98
Lu
C.Y.
,
Tso
D.J.
,
Yang
T.
,
Jong
Y.J.
,
Wei
Y.H.
(
2002
)
Detection of DNA mutations associated with mitochondrial diseases by Agilent 2100 bioanalyzer
.
Clin. Chim. Acta
,
318
,
97
105
.
99
Martinuzzi
A.
,
Bartolomei
L.
,
Carrozzo
R.
et al
. (
1992
)
Correlation between clinical and molecular features in two MELAS families
.
J. Neurol. Sci.
,
113
,
222
229
.
100
Morovvati
S.
,
Nakagawa
M.
,
Sato
Y.
,
K.
,
Higuchi
I.
,
Osame
M.
(
2002
)
Phenotypes and mitochondrial DNA substitutions in families with A3243G mutation
.
Acta Neurol. Scand.
,
106
,
104
108
.
101
Olsson
C.
,
Zethelius
B.
,
Lagerstrom-Fermer
M.
,
Asplund
J.
,
Berne
C.
,
Landegren
U.
(
1998
)
Level of heteroplasmy for the mitochondrial mutation A3243G correlates with age at onset of diabetes and deafness
.
Hum. Mutat.
,
12
,
52
58
.
102
Onishi
H.
,
Hanihara
T.
,
Sugiyama
N.
et al
. (
1998
)
Pancreatic exocrine dysfunction associated with mitochondrial tRNA(Leu)(UUR) mutation
.
J Med Genet
,
35
,
255
257
.
103
Rusanen
H.
,
Majamaa
K.
,
Tolonen
U.
,
Remes
A.M.
,
Myllyla
R.
,
Hassinen
I.E.
(
1995
)
Demyelinating polyneuropathy in a patient with the tRNA(Leu)(UUR) mutation at base pair 3243 of the mitochondrial DNA
.
Neurology
,
45
,
1188
1192
.
104
Vilarinho
L.
,
Santorelli
F.M.
,
Rosas
M.J.
,
Tavares
C.
,
Melo-Pires
M.
,
DiMauro
S.
(
1997
)
The mitochondrial A3243G mutation presenting as severe cardiomyopathy
.
J. Med. Genet.
,
34
,
607
609
.
105
Wilichowski
E.
,
Korenke
G.C.
,
Ruitenbeek
W.
et al
. (
1998
)
Pyruvate dehydrogenase complex deficiency and altered respiratory chain function in a patient with Kearns-Sayre/MELAS overlap syndrome and A3243G mtDNA mutation
.
J. Neurol. Sci.
,
157
,
206
213
.

## Author notes

I.J.W. and P.J.C. contributed equally to the study.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.


text:
# integrating the discord api into Angular

So I recently (yesterday) found this npm package for discord: https://discord.js.org/#/, I run npm install on it and it installs correctly. However it does not seem to recognise any of the classes made by the library. I have the library listed in my package.json but I cannot find the way to get it to work…

import { Component, OnInit } from '@angular/core';
import * as Discord from 'discord.js'

@Component({
selector: 'app-discord',
templateUrl: './discord.component.html',
styleUrls: ['./discord.component.scss']
})
export class DiscordComponent implements OnInit {

constructor() { }

ngOnInit() {
console.log(new Discord.Client())
}

}

The errors this is throwing all seem to refer to missing dependencies in my project, but some of them seem to be NodeJS which is confusing me. I will post the code on hatebin for brevity, as it would clutter my post up quite a bit. https://hatebin.com/spiirlcopq

My question is, can I use this api in the front-end? or do I need a different api? There is a webpack version but I have yet to find how I can add a webpack to my code and have a tangible intellisense.

Edit: my VSCode does register the package because my intelissense is understanding what I am trying to do and typing the Discord classname gives me access to those classes, but for some reason they won’t load.

Source: New feed
Source Url integrating the discord api into Angular


text:
# nLab ordinal subdivision

## Idea

The most usual method of subdivision for a simplicial complex as used in elementary algebraic and geometric topology is the barycentric subdivision. There is however another very well structured subdivision construction encountered which can be useful. The basic geometric construction involves chopping up a geometric $n$-simplex by $n$-planes parallel to a face and halfway between that face and the opposite vertex.

## Categorical descriptions

We first define the ordinal subdivision on the simplices, $\Delta[n]$. (Here $\oplus$ denotes the ordinal sum

The ordinal subdivision of $\Delta[n]$, the standard $n$-simplex in simplicial sets, is denoted by $Sd(\Delta[n])$, and is defined as follows:-

$Sd(\Delta[n]) := \int^{p,q} \Delta([p]\oplus[q],[n]) \cdot (\Delta[p] \times \Delta[q]).$

For a general simplicial set we can then define:

the ordinal subdivision of a simplicial set, $X$, is denoted by $Sd(X)$ and is defined by

$Sd(X) := \int^{n} X_n \cdot Sd(\Delta [n]).$

This expands to

$Sd(X) := \int^{n} X_n \cdot \left(\int^{p,q} \Delta([p]\oplus[q],[n]) \cdot (\Delta[p] \times \Delta[q]) \right).$

This is related to the total décalage of $X$ by

$Sd(X) = \cong \int^{p,q} DEC(X)_{p,q} \cdot (\Delta[p] \times \Delta[q]).$

## References

• P. J. Ehlers and Tim Porter, Ordinal subdivision and special pasting in quasicategories, Advances in Math. 217 (2007), No 2. pp 489-518.Delta

Last revised on July 28, 2014 at 23:17:28. See the history of this page for a list of all contributions to it.


text:
## November 27, 2011

### Harking Back

After my recent spate of “you can do mathematics in your blog” posts (I am no mathematician, but I do know computers) I remembered that some time ago I did a post about Euler's identity, and at the time the best I could do was a graphic with the equation in it. I reproduce the graphic in large format below, to demonstrate how clearly unsatisfactory the graphic truly was at higher resolution (you may notice that at this size the pixellation of the font has become clearly visible, though some browsers make valiant efforts to disguise it):

Of course I could have taken more care, and produced a higher-resolution graphic, but the fact remains that ultimately this representation cannot be copied and pasted as part of another expression, because the bitmap representation is completely unrelated to the structure of the equation. And eventually, no matter how high the resolution at which you generate, someone will come up with an application that demands more resolution. More of everything cheaper is kind of a clarion call in the computing world, and by and large the engineers have done their best to deliver.

Now, though, with the magic of TeX at my blogging fingertips, I can display the same equation (or at least the same identity: most people seem to prefer the zero identity as the canonical form, but I am persisting in my original bloody-minded mistake for backwards compatibility reasons) with ease using the TeX formula

$e^{\pi i} = -1$

which renders (using standard font sizes) as:
$e^{\pi i} = -1$
Thanks to MathJax, when I blow the TeX formula up it renders without the unpleasant pixellation effects of my previous attempt:
$e^{\pi i} = -1$

To me this is clearly an improvement over my original method (though I must confess that I cannot remember at this distance how I produced the graphic for the original blog entry).

If you hover the cursor over the MathJax equation you will get a context menu from it (Windows and Ubuntu users use your right mouse button, recent Mac users use a two-finger click) and ask to see the equation source code using the Show Source option:
The first time I used this feature it was in TeX mode, and what I saw was the TeX code I used to create the expression.  On my Mac that window looked like this:
So now your mathematics can be copied by others, for example to paste into their own blogs or other papers. People can now copy and paste mathematics freely. I think that's exciting news! Of course you can also repay the favor by copying TeX source from other blogs (as well as the many excellent TeX web papers and articles) into your own blogs.

MathML Too

If, like me, you are an inveterate tinkerer then you might wonder whether it's worth learning MathML. If you already know TeX then it would appear that the answer is “no,” because MathJax can translate between TeX and MathML format for you. If you want to see the mathML equivalent you first have to change MathJax's display format. This is again just a matter of using the MathJax context menu:

If you repeat your request to see the equation source you will now see the MathML equivalent to the TeX source I wrote in the first place. This neatly gives you the answer to your question “is mathML representation much more long-winded than TeX markup,” to which the answer is “yes, and then some” but you don't need to worry about that. Just be happy that MathJax can take care of them both:

I know it is equally possible to author for MathJax in MathML, so I would assume (but have not verified) that translation to TeX would similarly be possible. You'll have to switch back to TeX mode to start seeing TeX sources again, by the way.

Experiment

Someone wanted to know if superscripts can have superscripts. The answer, naturally, is yes. You can use the context menu to examine the source.

$e^{{\pi i}^{\pi i}}$

How did that work for you? Welcome to the world of copy and paste mathematics!


text:
43oh

[Energia Library] Petit FatFS SD Card Library

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You can use SD cards, but you must write on-the-fly. Since (as stated) you cannot cache a sector (which are 512 bytes for almost all cards), since the msp430g2553 has only 256 bytes of RAM.

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Hello,   This is a port of ChaN Petit FatFS SD library targeted to low RAM microcontrollers and offers a minimal FAT implementation, with limited write support. For more details see http://elm-chan.

A small update: - all config settings in one file - pffconf.h - a soft SPI driver ( shuftIn/shiftOut ) that can be enabled in pffconf.h. It defaults on HW SPI pins, but these can be changed using Fa

Hi, i just wanted to report that the library won't compile with Energia release 0101E0011 and the F5529 Launchpad because "DIR" is already defined in \energia-0101E0011\hardware\tools\msp430\msp430\in

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Sorry, I still don't get it. Is it possible to write anything on SD card with msp430g2553? In this case I will store every log string in a separate file. Or maybe I need to find SD card with <512 bytes sectors? I'm currently using very old 16mb Toshiba card. Mb it is the right one...

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Sorry, I still don't get it. Is it possible to write anything on SD card with msp430g2553? In this case I will store every log string in a separate file. Or maybe I need to find SD card with <512 bytes sectors? I'm currently using very old 16mb Toshiba card. Mb it is the right one...

SD cards with sectors under 512 bytes do not exist, some with 1024 bytes or 2048 bytes sectors exist, albeit somewhat out of spec (for the 2048 bytes version).

The only real way to write data to the SD card is to prepare the card in a PC to have a hard sequential order of sectors within one file. Then write to that sector in a sequential way. You will keep the writing cycle open this way so you can push more data to it before finishing the write cycle.

This is quite a crude and inelegant way, but it may very well work.

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roadrunner84, thank you! Now the idea is clear. Could you please give me an idea how to implement write to sd card in this way? Is it possible to do it in Energia or something more sophisticated is required?

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roadrunner84, thank you! Now the idea is clear. Could you please give me an idea how to implement write to sd card in this way? Is it possible to do it in Energia or something more sophisticated is required?

Everything is possible in Energia, it's just C++ with some framework.

I cannot help you to fix your issue, you'll have to rewrite large portions of the SD library to get thing working this way.

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Everything is possible in Energia, it's just C++ with some framework.

I cannot help you to fix your issue, you'll have to rewrite large portions of the SD library to get thing working this way.

Considering the fact that I'm just a newbie, it's impossible for me to rewrite large portions of the SD library. So it looks more reasonable to use a bluetooth module instead of SD card and just send temp. sensor data right to the server.

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The only file on sdcard is "SD card test.txt",

PFatFS uses the 8.3 naming convention.That means, a longer name will be "truncated" to 8 characters. In the case of a file with a name as you used ("SD card test.txt"), PFatFS shows me it as "SDCARD~1.TXT".

Maybe an ideea will be to start keeping just the directory listing part from the test sketch, to see if you have communication and just the name is wrong, or you don't have communication at all.

/*-----------------------------------------------------------------------*/
/* Program Main                                                          */
/*-----------------------------------------------------------------------*/

void loop()
{
Serial.println();
Serial.println("Open root directory.");
delay(100);
rc = FatFs.opendir(&dir, "");
if (rc) die(rc);

Serial.println();
Serial.println("Directory listing...");
delay(100);
for (; {
if (rc || !fno.fname[0]) break;    /* Error or end of dir */
if (fno.fattrib & AM_DIR) {Serial.print("<dir>\t"); Serial.println(fno.fname);delay(100);}
else {Serial.print(fno.fsize);Serial.print("\t"); Serial.println(fno.fname);delay(100);}
}
if (rc) die(rc);

Serial.println();
Serial.print("Test completed.");
//for (; ;
}

PS I'm using a tivac board.

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PFatFS uses the 8.3 naming convention.That means, a longer name will be "truncated" to 8 characters. In the case of a file with a name as you used ("SD card test.txt"), PFatFS shows me it as "SDCARD~1.TXT".

Maybe an ideea will be to start keeping just the directory listing part from the test sketch, to see if you have communication and just the name is wrong, or you don't have communication at all.

/*-----------------------------------------------------------------------*/
/* Program Main                                                          */
/*-----------------------------------------------------------------------*/

void loop()
{
Serial.println();
Serial.println("Open root directory.");
delay(100);
rc = FatFs.opendir(&dir, "");
if (rc) die(rc);

Serial.println();
Serial.println("Directory listing...");
delay(100);
for (; {
if (rc || !fno.fname[0]) break;    /* Error or end of dir */
if (fno.fattrib & AM_DIR) {Serial.print("<dir>\t"); Serial.println(fno.fname);delay(100);}
else {Serial.print(fno.fsize);Serial.print("\t"); Serial.println(fno.fname);delay(100);}
}
if (rc) die(rc);

Serial.println();
Serial.print("Test completed.");
//for (; ;
}

PS I'm using a tivac board.

omg! Thank you so much!

Open root directory.

Directory listing...
4096 ~1.TRA
<dir> TRASHE~1
<dir> FSEVEN~1
37 SDCARD~1.TXT
4096 _SDCAR~1.TXT

Test completed.

Now I'm gonna try to write on SD card

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It is possible to write into my file ("SDCARD~1.TXT"), but by some reason I can't exceed initial size (37 bytes). Could someone please explain me the reason why? Is it related with msp430g2553 ram size?

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There are some limitation in using the write function. Calin has listed them in the post 4.

Impossibility to change the size of the file is one of them.

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There are some limitation in using the write function. Calin has listed them in the post 4.

Impossibility to change the size of the file is one of them.

Hopefully, it is possible to live with. Could you evaluate my plan?

1) Create two 2mb "empty" files on SD card (fileA and fileB).

2) Write log data to fileA and remember how much space already have taken.

3) When fileA is almost full with log data, start writing to fileB

4) When fileB is almost full with log data, start writing to fileA and so on.

Considering the fact that 1 line of log less then 32 bytes (YYYY.MM.DD HH.MM Temp) and mcu check temperature once in a minute, 2mb file can store data for one month and a half. File size could be increased, 2mb files are just for test purposes.

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Hello,

Unfortunattely I can't help you any furter. All I've done with this library was to connect a card to my Launchpad and make some communication tests.

But in my oppinion your plan has big chances to work.

Success,

Liviu

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• 1 month later...

To select another SPI port on the LM4F120 or TM4C123, have a look at the Sd2Card::init function.

uint8_t Sd2Card::init(uint8_t chipSelectPin, uint8_t sckRateID, int8_t SPI_Port, int8_t cardDetectionPin, int8_t level)

#if defined(__LM4F120H5QR__) || defined(__TM4C1230C3PM__) || defined(__TM4C123GH6PM__) || defined(__TM4C129XNCZAD__) || defined(__TM4C1294NCPDT__)
// LM4F and TM4C specific
if (SPI_Port >= 0) {
SPI_for_SD.setModule(SPI_Port);
}
#endif

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Hi all,

I have a question that's directed a little more towards the hardware side of interfacing my TM4C123G launchpad with my SD card module using this library. If anyone has any thoughts or ideas, please let me know here: http://forum.stellarisiti.com/topic/2205-help-with-sd-card-module-on-tm4c123gxl/

Thanks so much.

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• 2 months later...

Hello, I'm also getting the messgage Failed with rc=6, please someone help us out!

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text:
# multivariable critical points calculator

Exponents Rational Exponents Factoring Polynomials Axis of Symmetry and Vertex of a Parabola 3. Solving Exponential Equations Evaluatefxx, fyy, and fxy at the critical points. The main purpose for determining critical points is to locate relative maxima and minima, as in single-variable calculus. endstream endobj startxref Critical/Saddle point calculator for f(x,y) No related posts. Factoring General Polynomials Rationalizing the Denominator Polar Representation of Complex Numbers Simplifying Fractions 1 0 While this may seem like a silly point, after all in each case $$t = 0$$ is identified as a critical point, it is sometimes important to know why a point is a critical point. A critical point of a multivariable function is a point where the partial derivatives of first order of this function are equal to zero. 0 â® ... Use the gradient function to calculate the derivative. Dividing Rational Expressions Find more Mathematics widgets in Wolfram|Alpha. 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Decimals and Fractions By using this website, you agree to our Cookie Policy. Positive Integral Divisors Observe that the constant term, c, â¦ It only says that in some region around the point (a,b)(a,b) the function will always be larger than f(a,b)f(a,b). Adding Rational Expressions with the Same Denominator Graphing Linear Inequalities Percents and Fractions We all wished we had it 4 months ago. hÞbbdbf! Mathematical Terms hÞbf2ÌÞ ÈÀO A critical value is the image under f of a critical point. Radical Notation Standard Form of a Line %%EOF Home A critical point of a function of a single real variable, f(x), is a value x 0 in the domain of f where it is not differentiable or its derivative is 0 (f â²(x 0) = 0). Adding Fractions Multiplying Monomials Each component in the gradient is among the function's partial first derivatives. Follow 73 views (last 30 days) PJ on 18 Jan 2018. Adding and Subtracting Fractions Dividing Polynomials by Monomials and Binomials The internet calculator will figure out the partial derivative of a function with the actions shown. Vote. Quadratic Equations with Imaginary Solutions Multiplying and Dividing Fractions ¢ H°>GË($¸ZA,o !Ö R$N÷20120¸Ô$þ3æ| 0 é ! Subtracting Rational Expressions with the Same Denominator Critical/Saddle point calculator for f(x,y) Added Aug 4, 2018 by Sharonhahahah in Mathematics Rationalizing the Denominator From Multivariable Equation Solver to scientific notation, we have got all kinds of things covered. Solving Quadratic Equations by Factoring Such points are called critical points. saddle point calculator multivariable. Factoring Trinomials Most other programs just give you the answer, which did not help me when it come to test time, Algebrator helped me through each problem step by step. Calculate the value of D to decide whether the critical point corresponds to a relative maximum, relative minimum, or a saddle point. Subtracting Mixed Numbers with Renaming But now, we see that the minimum is actually Thanks for contributing an answer to â¦ Free partial derivative calculator - partial differentiation solver step-by-step This website uses cookies to ensure you get the best experience. Powers of Complex Numbers Fractions Outside of that region it is completely possible for the function to be smaller. Critical Points and Extrema Calculator The calculator will find the critical points, local and absolute (global) maxima and minima of the single variable function. Please use this form if you would like to have this math solver on your website, free of charge. Like Radical Terms By using this website, you agree to our Cookie Policy. Multiplication by 572 Multiplying Monomials Multiplying Polynomials Imaginary Solutions to Equations Order and Inequalities Again, outside of tâ¦ example. Simplifying Radicals Solving Rational Equations It is a number 'a' in the domain of a given function 'f'. The point $$(a,b)$$ is a critical point for the multivariable function $$f(x,y)\text{,}$$ if both partial derivatives are 0 at the same time. Wiki says: March 9, 2017 at 11:14 am Here there can not be a mistake? Solving Quadratic Equations by Using the Quadratic Formula A function z=f(x,y) has critical points where the gradient del f=0 or partialf/partialx or the partial derivative partialf/partialy is not defined. Simplifying Products and Quotients Involving Square Roots is a twice-differentiable function of two variables and In this article, we â¦ FOIL Multiplying Polynomials Midpoint of a Line Segment Computes and visualizes the critical points of single and multivariable functions. Factors and Prime Numbers The above calculator is an online tool which shows output for the given input. Equations of a Line - Point-Slope Form Differentiate any single or multivariable function; Find the critical points and saddle points of a function; Calculate the gradient of a function; Identify the local extrema of a function; Find the single, double, or triple integral of a function; Determine the dot or cross product of two vectors; Calculate the divergence or curl of a vector field Variables and Expressions Solve these equations to get the x and y values of the critical point. Fields Medal Prize Winners (1998). Rational Expressions TUTORIALS: Find the critical points by setting the partial derivatives equal to zero. 0.1 Reminder For a function of one variable, f(x), we ï¬nd the local maxima/minima by diï¬erenti- ation. Example 2 Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by f(x , y) = 2x 2 - 4xy + y 4 + 2 . Rationalizing the Denominator seventh Edition, Mcdougal Littell Algebra 2 math book problems, interpreting slope for quadratic equation, proving trigonometric identities (example problems), adding and subtracting integers worksheets, negative exponents practice "free worksheets". how to solve linear system of equations in programming by gauss elimination method in c language? Anthony Washington, MO. Powers Quadratic Expressions Completing Squares I have showed the program to a couple of classmates during a study group and they loved it. Calculus: Integral with adjustable bounds. Rational Exponents Calculus: Integral with adjustable bounds. Fractional Exponents Ratios and Proportions Solution to Example 1: We first find the first order partial derivatives. Simplifying Square Roots Solving Quadratic Inequalities Multiplying and Dividing Monomials Solving Quadratic Equations by Completing the Square Finding the Least Common Denominator Rationalizing the Denominator Solving Linear Systems of Equations by Elimination %PDF-1.5 %âãÏÓ Similarly, with functions of two variables we can only find a minimum or maximum for a function if both partial derivatives are 0 at the same time. Then I got Algebrator, and it helped me not only with quadratic but also with pretty much any equation or expression I could think of! 1. It is 'x' value given to the function and it is set for all real numbers. Simple Partial Fractions Khan Academy is a 501(c)(3) nonprofit organization. The Function Analysis Calculator computes critical points, roots and other properties with the push of a button. Rationalizing the Denominator Dana White, IL, I think it is great! A First Course in Abstract Algebra. fx(x,y) = 2x = 0 fy(x,y) = 2y = 0 The solution to the above system of equations is the ordered pair (0,0). Complex Numbers Reply. 100% Free. Critical Number: It is also called as a critical point or stationary point. example. Calculus: Fundamental Theorem of Calculus Mathematical Terms Graphing Solutions of Inequalities Critical points of multivariable functions calculator Critical points of multivariable functions We recall that a critical point of a function of several variables is a point at which the gradient of the function is either the zero vector 0 or is undefined. Critical Points Critical points: A standard question in calculus, with applications to many ï¬elds, is to ï¬nd the points where a function reaches its relative maxima and minima. How to find and classify the critical points of multivariable functions. Simplifying Square Roots That Contain Whole Numbers Ò¥îÚîzâ!o)¤ºòïwÆN ÛìéíñÌØùâØKB ã!aJm® ÷¡ö)J@ÍÔJBí5Ô¨Óø&)ZHÂ©ç>ãÐPKMÑp  Äç#~ÀQÂ¯Àé/Þ0K³. Multiplying Radicals The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6). : we first find the first partial derivatives solve these equations to get the x and f y to function. Each component in the domain of a differentiable function of a differentiable function a. GË ($ ¸ZA, o! Ö R $N÷20120¸Ô$ 0. Min read found Algebrator maximum, relative minimum, or a saddle point is... An online tool which shows output for the given input multivariable functions the domain of critical... One variable, f ( x ), we ï¬nd the local maxima/minima by diï¬erenti-.. Also called as a critical point Bravo, your idea simply excellent fact, in a couple of during... Push of a critical point of a differentiable function of one variable, f ( x, y 1. 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text:
Is the finite dimension required in this proof?

Let $V$ and $W$ be vector spaces over a field $K$. If a linear map $L:V \rightarrow W$ is surjective then its dual is injective. If $V$ and $W$ are finite dimensional then the converse holds, i.e. $L^*:W^* \rightarrow V^*$ injective implies $L$ surjective.

I have proved both statements but I don't see where I used the finite dimensional requirement for the second. Here is my proof:

Assume $L$ is not surjective, say the element $e_i$ of the basis of $W$ is not in the image of $L$. Take its corresponding dual $\alpha_i \in W^*$, then $L^*(\alpha_i)=\alpha_i \circ L =0$ so the kernel of $L^*$ is not 0 and therefore $L^*$ is not injective.

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Your argument seems correct, but I might be missing something. Where did you see this statement? Do you know that every vector space has a basis (i.e. are you allowed to use the Axiom of Choice)? – Michael Albanese Aug 31 '13 at 20:35
You claimed that $e_i\not\in L(V)$ implies $\alpha_i\circ L=0$. I don't think this is true, even for finite-dimensional spaces. – Julian Rosen Aug 31 '13 at 20:40
I think you need to make sure to first choose a basis of the image of $L$, and then extend it to a basis of $W$. If you don't do that, there is no guarantee that your dual basis element acts properly on the image of $L$. – Carl Aug 31 '13 at 21:31
It is problem 10.5 of Tu's book "An introduction to Manifolds". Yes the axiom of choice is assumed so every vector space has a basis. @Pink Elephants the only elements which are not mapped to $0$ by $\alpha_i \circ L$ are those whose image by $L$ is a multiple of $e_i$. – inquisitor Sep 1 '13 at 0:36
@inquisitor Let $e_1,\ldots,e_n\in W$ be a basis, $\alpha_1,\ldots,\alpha_n\in W^*$ the dual basis. It is not true that the only elements of $W$ not mapped to 0 by $\alpha_i$ are scalar multiples of $e_i$. For example, $\alpha_1(e_1+e_2)=1$. So if $e_1$ is not in the image of $L$ but $e_1+e_2$ is, then $\alpha_1$ is not in the kernel of $L^*$. – Julian Rosen Sep 1 '13 at 0:55

My first argument has to be changed as the comments of @Julian Rosen show. Finite dimension is not needed:

Take a basis $B'$ of the image of $L$ in $W$ and complete it to a basis $B$ of $W$ (by assumption $B \setminus B'$ is not empty). Define the linear functional $\alpha$ by $\alpha(e)=1$ where $e\in B\setminus B'$ and $\alpha(v)=0$ for $v \in B'$ (and extend linearly). Then $L^*(\alpha)=\alpha \circ L = 0$, and $\alpha$ is not the $0$ linear functional, hence $L^*$ is not injective.

For the proof of $L^*$ surjective iff $L$ injective see the last answer of: Are injectivity and surjectivity dual?

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I think Pete L. Clark's answer is intuitive and I don't mean to obfuscate the problem, but could I add some abstract nonsense (which obviously one could ignore)?

In the category of vector spaces, we can easily show every mono (injective) and epi (surjective) splits. The dual construction is faithfully functorial (another straightforward exercise), hence it both preserves and reflects split monos and split epis, i.e. monos iff the dual is epi and epi iff dual is mono.

Thank you for the indulgence.

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text:
# Displacement Power

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

Displacement power refers to what the quality of work has to do with a force causing a displacement.

## formula

$$P = \frac {F}{\delta} t$$

Where:

$$P$$ or $$P_d$$ = power

$$F$$ = force

$$d$$ or $$\delta$$ (Greek symbol delta) = displacement

$$t$$ = time

Solve for:

$$F = \frac {Pt}{\delta}$$

$$\delta = \frac {Pt}{f}$$

$$t = \frac {F \delta}{P}$$


text:
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I will assume that you do not have a control group (otherwise, you could proceed as similarly but add a group variable to the fit). This makes this kind of tricky, but here's an idea: Denote with $h_{p,t}$ the heart rate of patient $p \in \mathcal{P}$ at time $t \in \{1, ..., 10\}$, and with $a_{t}$ an indicator variable that takes value one after treatment (...