Public Articles

Note: Series Reversion of Stieltjes Coefficients

If we take the Laurent expansion of the Riemann zeta function about s = 1 $$ \zeta(s) = {s-1} + ^\infty {n!}\gamma_n (s-1)^n $$ which defines γn, the Stieltjes constants, where γ₀ is the Euler-Mascheroni constant. Next perform a series reversion on this to give a series $$ \chi(s) = 1+{s}+^\infty {s^n} $$ which has expansion \chi(s) = 1 + {s} + {s^2} + {s^3} + {2s^4} + \cdots The coefficients κ(n) seem to decrease quite steadily, even up n being a few hundred, where the γn get large. n (n) ---- --------------- -- 0 1.000000000 1 1.000000000 2 0.5772156649 3 0.4059937693 4 0.3135616752 5 0.2556464523 6 0.2159181431 7 0.1869526867 8 0.1648872027 9 0.1475121704 10 0.1334717457 11 0.1218874671 12 0.1121649723 13 0.1038876396 14 0.09675470803 15 0.09054358346 : The first 16 coefficients of the inverse function. Letting $$ R_n=^n i_k $$ and $$ P_n=^n ki_k $$ and {i}n = {i₁, i₂, ⋯|P = n − 1}, I have observed the expression for κ(n) from series reversion to be $$ \kappa(n)=} (-1)^n\left[^{R-1}j-n\right]\left[^n {i_k!}\left({k!}\right)^{i_{k+1}}\right]\gamma_0^{i_1} $$ where we define κ(0)=1. Two examples $$ \kappa(3) = \gamma_0^2 - \gamma_1 = - \left[^{i_1+i_2+i_3-1} (j-n)\right]{1!}\right)^{i_2}\left({2!}\right)^{i_3}}{i_1!i_2!}\gamma_0^{i_1} $$ $$ \kappa(4) = \gamma_0^3 - 3 \gamma_0\gamma_1 + {2} = \left[^{i_1+i_2+i_3+i_4-1} (j-n)\right]{1!}\right)^{i_2}\left({2!}\right)^{i_3}\left({3!}\right)^{i_4}}{i_1!i_2!i_3!}\gamma_0^{i_1} $$ We can conjecture that $$ \kappa(n+1) < \kappa(n), \;\;\; n\in^{>0}? $$ Is this perhaps a more well behaved way to look at the Stieltjes constants?

Exploring a Zeta Function Extension and its Relation to Primes

An adapted version of the zeta function is defined as *θ*_{m}(*s*), this is used to explore the sequence of prime numbers.

The multiple meanings of "additivity" in quantitative behavioral genetics

AbstractThe first aim of this article is to shed light on the multiple definitions of the assumption of additivity in quantitative behavior genetics and its associated methodologies, such as heritability estimation. In addition, this article aims to assess the validity of this assumption, based on the multiple ways in which it has been defined. Suggestions for future research in quantitative genetic studies of complex traits are also provided.IntroductionIn a recent paper, Evan Charney argues that "...heritability estimates are frequently misinterpreted as showing that genetic similarities cause trait similarities in the study population" (p. 2, emphasis in original). The argument here is simply that a high heritability of a trait does not necessarily indicate that the trait in question is mostly caused by genetic factors, nor is a trait with a low heritability value necessarily under correspondingly low levels of genetic influence. This common, longstanding fundamental criticism stems from perceived limitations of the statistical technique of analysis of variance (or ANOVA for short) that has long been used to calculate heritability. It also concerns the frequent misinterpretation, noted by Charney in the quote above, that "heritability" is not the same as "genetically inheritable". This is partly because the technical definition of "heritability" (% of variation in a trait in a trait associated with genetic variation in a given population) is quite different from the genetics-based folk definition that most people assign to the same word.\cite{Stoltenberg1997} This frequent and understandable confusion has been noted many times, including very recently by Moore & Shenk (2017).In sum, critics of heritability calculations in general argue that, at least when conducted on complex human traits, they tell us essentially nothing--certainly nothing about how genetically influenced a trait is. Some behavioral genetics (BG) researchers seem to have no problem agreeing with this point. In fact, some of them wrote in 2011 that "...genetics is riddled with complexity of many degrees and kinds, and heritability is a poor indicator of either degree or kind of underlying genetic complexity."\cite{Johnson_2011} Otherwise, they tend to ignore criticisms of the underlying concept of heritability estimates, instead focusing on more superficial issues like whether or not twin studies generate valid estimates of heritability. However, there is at least one obvious exception to this tendency, and that is the work of Neven Sesardic. He has published several papers and a book (the latter called Making Sense of Heritability) attempting to defend the heritability statistic and its genetic-causative interpretation against its critics.But who is right? Those who argue that BG's dependence on the assumption of additivity renders the entire discipline and its conclusions untenable, or those who argue that this is not nearly so questionable or inaccurate an assumption, nor one whose violation has such serious consequences, as critics claim? I will try to address this question in the remainder of this paper.Defining the assumption of additivityFirst of all, it should be explained exactly what the assumption of "additivity" means here, because it can, in fact, have more than one meaning in BG. I will thus attempt to describe and distinguish between the two definitions that appear to have been used for the assumption of additivity in the context of genetics and heritability analyses. My goal of doing so is influenced by Moore, who recently described the different, often-confused meanings of the widely used phrase "gene-environment interaction".\cite{Moore_2018}Moore has defined the additivity assumption as the assumption "that genetic and environmental influences on phenotypic variation are additive".\cite{Moore_2006} This is what I will call "definition A" of the assumption of additivity: that phenotypic variance (V) can be accurately expressed as the sum of two separate components, one for genes and one for the environment. This may be written as the following equation (e.g., \cite{Tabery_2008}):V = VG + VEHere, V (also sometimes written VP, with the P standing for "phenotype") = total phenotypic variation in a given trait in a given population, VG = variation in that trait that is due to genetic factors, and VE = variation due to environmental factors. This definition of the assumption of additivity is only valid if equation 1 is accurate. It would not be accurate if, for instance, there is gene-environment interaction (G x E), in which case you would not be able to get the total variance just by adding together the genetic and environmental sources thereof.There is little question that it is really important for quantitative BG researchers that definition A of the assumption is true: it is essential to being able to interpret heritability estimates causally, as Lewontin noted in 1974. If this assumption is invalid, no functional conclusions can be drawn regarding the trait being analyzed.\cite{LEWONTIN_2006} As Oftedal explains, critics argue that this is because of the inherently local nature of heritability estimates if the additivity assumption is not met: "In situations of additivity, heritability estimates are no longer just local. The result from one environment can be extrapolated to other environments."\cite{Oftedal_2005} (p. 702) Similarly, Lynch (2016) has recently noted that the heritability statistic's validity "...relies on the assumption that VG and VE act additively, so that there is no interaction or correlation between the two terms."\cite{Lynch_2016} But there is also a second definition of the additivity assumption in regards to BG: namely, the assumption that the effects of genetic loci on variation in a complex trait are independent of each other and of the environment, so that you can add their individual effects together to get the total genetic effect. As Wahlsten (1994) wrote in describing the method of heritability analysis, "the effect of all polymorphic loci affecting a behaviour are combined by adding them to yield the total Gi for an individual, which assumes genotype at one locus does not influence the action of genes at other loci." \cite{Wahlsten_1994} (p. 245) Thus, I will refer to the assumption that most genetic variance in complex traits, behavioral or otherwise, is additive in nature as "definition B" of the additivity assumption.Defining heritabilityNext, it should be noted that there are two types of heritability: narrow-sense heritability (hB2) and broad-sense heritability (h2). The difference between the two is that hB2 is only based on additive genetic variance, whereas h2 is based on both additive and non-additive (i.e. total) genetic variance.\cite{edition} hB2, rather than h2, is the value that agricultural breeders care about, because it is used to predict what the fastest way will be to maximize a desired trait in the organism of interest through a specific selective breeding strategy.\cite{Feldman1975} It also needs to be explained just what heritability estimation actually is: as Oftedal has noted, it is "a statistical method based on a linear analysis of variance".\cite{Oftedal_2005} (p. 700)So first we will consider the first definition ("definition A") of the additivity assumption: that variation in a phenotype = genetically caused + environmentally caused + maybe a small interaction term. Lynch (2016) has recently highlighted the two ways that this assumption can be violated: gene-environment interaction (G x E) and gene-environment correlation (the latter also called gene-environment covariance, abbreviated G-E covariance).\cite{Lynch_2016}In addition, Wahlsten noted in a 1990 paper that "Additivity is often tested by examining the interaction effect in a two-way analysis of variance (ANOVA) or its equivalent multiple regression model. If this effect is not statistically significant at the α = 0.05 level, it is common practice in certain fields (e.g., human behavior genetics) to conclude that the two factors really are additive and then to use linear models, which assume additivity." But he reported in the same paper that "...ANOVA often fails to detect nonadditivity because it has much less power in tests of interaction than in tests of main effects. Likewise, the sample sizes needed to detect real interactions are substantially greater than those needed to detect main effects."\cite{Wahlsten_1990} In a subsequent paper, Wahlsten noted that the issue of non-additivity is often brushed aside by human BG researchers: "...it is asserted that the measured score (or phenotype) of an individual on a psychological test (Yi) is the sum of only two components, Gi determined by the genes and Ei specified by the person's environment; that is, G and E must not interact. Many sources introduce the model with a G x E interaction term attached to the end of the model, but they quickly drop this and proceed with the simple additive model as the basis for further analysis."\cite{Wahlsten_1994} (p. 245)He argues that there are fundamental biological reasons to believe that the assumption of additivity will almost always be false: "The additive model is not biologically realistic. There are so many instances where the response of an organism to a change in environment depends on its genotype or where the consequences of a genetic defect depend strongly upon the environment, that genuine additivity of the two factors is very likely the rare exception."\cite{Wahlsten_1994} (p. 249)And elsewhere, he contends that human BG faces unique obstacles in controlling for non-additivity that animal researchers (such as himself) do not have as much of a problem with: "To test interaction between genotype and environment, there must be many individuals with the same genotype who are reared in different environments. This is easily achieved with standard laboratory strains but not with humans. For our species, there is no valid test of gene x environment interaction, no matter what the sample size, unless distinct alleles of a specific gene in question can be identified...Because the additivity assumption cannot be tested empirically, the whole edifice of path models must be accepted on faith, if it is to be accepted at all."\cite{Wahlsten_2000}(p. 50)Many critics of BG have argued that definition A of the additivity assumption is untenable, and that the way in which genes and environments actually interact to produce phenotypes is just that--interactive, not additive. Thus this criticism alleges that heritability calculations are uninterpretable (at least in terms of the relative roles of genes vs. environment in causing phenotypic variation), because definition A is simply false. This criticism is well summed up in a paper by Vreeke: "The core of the critique of behavior genetics, as far as it relies on the analysis of variance, is thus that it conceptualises the relation between genes and the environment as (mainly) additive, whereas in fact development is interactive."\cite{Vreeke_2000} (p. 37) The same paper notes, "Experimental animal research shows that interaction between genotype and the environment occurs often. And if genes and the environment interact, it is not possible to separately weigh the effect of one of those factors: they depend on each other. There is no reason to expect that humans are different in this respect. An analysis of variance ignores those effects, so cannot provide a true account of the causes of behavior."\cite{Vreeke_2000} (p. 37)Locality and causalityAs noted above, critics of heritability analysis argue that the additivity assumption is false, and that heritability estimates are really just local. But which assumption is it that the critics claim is false? To some extent it is both, but definition A seems to be a more common target of such criticisms. Lewontin makes it clear that he considers the locality of heritability estimates to prevent them from allowing causal conclusions to be drawn: "There is one circumstance in which the analysis of variance can, in fact, estimate functional relationships...It is not surprising that the assumption of additivity is so often made, since this assumption is necessary to make the analysis of variance anything more than a local description."\cite{LEWONTIN_2006} This criticism is referred to by Oftedal as the "locality objection".\cite{Oftedal_2005} (p. 702)So what we have here are two BG responses to argument 5: 1) Actually, most (if not all) genetic variance is additive, so this assumption is going to be at least mostly correct, and 2) to the extent that the assumption of additivity is false, there are plenty of ways that we can successfully account for it already, thank you very much! I will now focus a bit more on the first of these responses: that most variation in the traits behavior geneticists are studying is actually additive, meaning that the assumption Charney criticizes so harshly is actually pretty accurate. One frequently cited study by those making this claim is that of Hill et al. (2008), entitled "Data and Theory Point to Mainly Additive Genetic Variance for Complex Traits". I noted above that Neiderhiser et al. (2017) cited this paper to justify their claim that genetic variation in complex traits is mostly additive. But is this conclusion really justified by the evidence presented by Hill et al. (2008)? Zuk et al. (2012) don't seem to think so: they argue that "...mistakenly assuming that a trait is additive can seriously distort inferences about missing heritability. From a biological standpoint, there is no a priori reason to expect that traits should be additive. Biology is filled with nonlinearity: The saturation of enzymes with substrate concentration and receptors with ligand concentration yields sigmoid response curves; cooperative binding of proteins gives rise to sharp transitions; the outputs of pathways are constrained by rate-limiting inputs; and genetic networks exhibit bistable states."\cite{Zuk_2012} In their supplementary information (p. 45), Zuk et al. go into more detail about why they consider Hill et al.'s claims not to stand up to scrutiny. First, Zuk et al. explain two key arguments made by Hill et al.: "(a) most variants in a large population will have extremely low minor allele frequency and (b) traits caused by low-frequency alleles will not have substantial variance due to interactions." But Zuk et al. don't find these arguments the least bit convincing: "Their claim is wrong, because the LP [linear pathway] models (a) can have substantial variance due to interactions (indeed, the majority) and yet (b) can involve any class of allele frequencies. (Specifically, LP models are defined as the minimum value of a set of traits, each of which is additive and normally distributed. There is no constraint on the allele frequencies of the variants that sum to yield these additive and normally distributed traits.)" And on the next page:"In effect, Hill et al.’s theory thus actually describes what happens for rare traits caused by a few rare variants. Not surprisingly, interactions account for a small proportion of the variance for such traits. Hill et al.’s model, however, is not pertinent to common traits. The interesting complex traits are those that have significant genetic variance in the population: these traits necessarily have higher allele frequencies (assuming they depend only on a few, e.g. two loci) and thus, under Hill et al.’s analysis, can involve larger interaction variance and a higher ratio VAA/VG." (Note: VAA = interaction variance and VG = total genetic variance.)Behavior geneticists respondTo the extent that BG heritability-estimation researchers have defended their practice against the charge that it inaccurately assumes additivity, they have made such arguments as this one, made by Michael Rutter in 2003: "Critics of behavior genetics are fond of attacking it on the grounds of the unwarranted presumption of additivity. However, behavior geneticists are well aware of this issue, and it is commonplace nowadays to make explicit tests for dominance or epistatic effects. Moreover, it is perfectly straightforward to include these in any overall model. There is a need to consider such effects, but their likely existence for some traits is not a justifiable reason for doubting behavior genetics."\cite{michael2003}So how exactly do behavior geneticists take the (non)existence of additivity/presence of non-additive effects into account? Rutter makes it sound really easy, but just how do they do it, and are their procedures for doing so adequate? It is important to keep in mind that many critics of BG argue that the techniques researchers in the field use to try to test and account for genetic non-additivity are woefully inadequate. In fact, such arguments have been made since at least 1973(!), when Willis Overton wrote, "...it does not change the situation any to maintain that this position does consider interactions by introducing an interaction term into the analysis of variance...As discussed by Overton and Reese [1972], such interaction effects, ‘are themselves linear, since they are defined as population cell means minus the sum of main effects (plus the population base rate)’ (p. 84). In fact, the very use of the term ‘interaction’ within this paradigm indicates that definitions of terms are not model independent".\cite{Overton_1973} Just fix the model!More recently, Partridge has argued that "Although these advances in GxE transactional models represent a substantial step forward for quantitative behavioral genetics models, there are inherent structural limitations to their analytic foundations...the nature of GxE transactions go much deeper than statistical interactionist models can accommodate. If structural sequences in the genome were isomorphic to genetic function and, more important, to protein function, then the inferred genetic variability assumed by behavioral genetic models might be more instrumental. However, genes, rather than being static structural entities, are dynamic processes."\cite{Partridge_2011}

Title

Resumen

En la siguiente practica se da solución a problemas en los que están involucradas las fuerzas y la cuales se representan mediante vectores.

Problema 1

muestra una fuerza que forma un angulo con la horizontal.

Esta fuerza tendrá componentes horizontales y verticales.

Fuerza que forma un angulo en la horizontal

¿Cuál de las siguientes opciones describe mejor la dirección de los componentes horizontal y vertical de esta fuerza?

Posibles respuestas Solución.

La respuesta es la opción d debido a que la fuerza se encuentra en el 3er cuadrante del plano cartesiano lo que nos indica que se encuentra a el lado negativo de las ”x”y a su vez lado negativo de las ”y”.

Problema 2

Cada velero experimenta la misma cantidad de fuerza, pero tiene Diferentes orientaciones a vela.

Posibles respuestas

¿En qué caso (A, B o C) es más probable que el velero se vuelque de lado?

Explique Solución La respuesta es Caso A ya que la fuerza aplicada se encuentra más inclinada hacia la componente y, lo que provocara que el velero gire y se vuelque de lado. Problema 3 Considere el camión de remolque a continuación. Si la fuerza de tensión en el cable es 1000 N y si el cable hace un ´angulo de 60 grados con la horizontal, entonces, ¿cu´al es el componente vertical de la fuerza que levanta el automóvil? ¿fuera de la Tierra? Figura 4: Remolque jalando un coche.

Solución

Paso 1: Hacer el diagrama de cuerpo libre.

Plantear ecuaciones de equilibrio. ΣF x = 0 2 T x = 0 ΣF y = 0 T y = 0 Haremos uso de la función trigonométrica sin θ para obtener la componente de la tensión T y. sin 60 = T y T (1) Se hace el despeje de T y de la siguiente manera. T y = T sin 60 (2)

Resolver ecuaciones y obtener el resultado. Se sustituye el valor de T y de θ en la ecuación 2. T y = (1000N) sin 60 = 866N (3) Se concluye que la componente vertical T y es igual a 866N.

Problema 4

Después de su entrega más reciente, la infame cigüeña anuncia la buena noticia. Si el cartel ¨ tiene una Masa de 10 kg, entonces ¿cuál es la fuerza de tensión en cada cable? Usa funciones trigonométricas y un croquis para ayudar en la solución.

Solución

Paso 1: Hacer el diagrama de cuerpo libre.

Paso 2: Plantear ecuaciones de equilibrio. ΣF x T DE = (10kg) 9,81 m s 2 T ACx − T ABx = 0 T AC cos θ − T AB cos θ = 0 ΣF y T ACy + T ABy = 0 T AC sin θ + T AB sin θ = 0

Paso 3: Resolver ecuaciones y obtener el resultado. Se igualan las ecuaciones de la componente x y se factorizan. T AC cos θ = T AB cos θ (4) T AC = T AB (5) Se sustituye el valor de T ED en las ecuaciones de la componente y y se minimiza la ecuación. T AC sin θ + T AB sin θ = 98.1N (6) 2T AC sin θ = 98.1N (7) Se despeja la T AC y se sustituyen los valores. T AC = 98.1N 2 sin 60 = 56.63N (8) La tensi´on de la cuerda es de 56.63N.

Problema 5

Si un bloque de 5kg se encuentra suspendido de la polea B y la cuerda a un distancia de 0.15m. Determine la fuerza en la cuerda ABC, desprecie el tamaño de la polea.

Bloque suspendido de una polea.

Solución

Paso 1: Hacer el diagrama de cuerpo libre.

Paso 2: Plantear ecuaciones de equilibrio. ΣF x = 0 T BCx − T BA = 0 T BC cos θ − T BA cos θ = 0 ΣF y = 0 T BD = (5kg) 9,81 m s 2 T BCy + T BAy = T BD T BC sin θ + T BA sin θ = 49,05N

Paso 3: Resolver ecuaciones y obtener el resultado. Para dar solución al problema es necesario obtener el ángulo de T BC para ellos utilizaremos la función tan θ. Se despeja θ y sustituimos los valores de x y y. El ángulo es de 36.86° tan θ = y x (9) θ = tan−1 y x (10) tan−1 0.15 0.2 = 36.86 (11) Se igualan las ecuaciones de la componente x. T BC cos θ − T BA cos θ (12) T BC = T BA (13) Obtenemos la tensión de T BD y lo sustituimos en la formula asi como el valor de θ, por ´ultimo despejamos T BC. T BC sin 36.86 + T BA sin 36.86 = 49.05N (14) 2T BC sin 36. = 49.05N (15) T BC = 49.05N 2 sin (36.86) (16) T BC = 40.88N (17) La tensión de la cuerda ABC es de 40.88N.

Problema 6

Si la masa del cilindro C es de 40kg, determine la masa del cilindro A para que el sistema se encuentre en una situación estática.

Cilindro C con una masa de 40kg.

Solución

Paso 1: Hacer el diagrama de cuerpo libre.

Paso 2: Plantear ecuaciones de equilibrio. ΣF x = 0 T EB − T ED = 0 ΣF y = 0 T EBy − T BA = 0 Utilizamos las funciones trigonométricas para calcular T EBx y T EBy. T EBx = T EB cos θ (18) T EBy = T EB sin θ (19) 6 Sustituimos los valores ya encontrados en las ecuaciones planteadas. T EB cos 30 − T ED = 0 (20) T EB sin 30 − W A = 0 (21) Dado que las cuerdas correspondientes a los segmentos EB y BC soportan la misma tensión y a la vez están en equilibrio con el cilindro C podemos concluir que: T EB = W C (22)

Paso 3: Resolver ecuaciones y obtener el resultado. Sustituimos los valores (40kg) 9.81 m s 2 cos 30 = T ED (23) T ED = 339.82 (24) W=ma (g),por ellos despejamos ma. (40kg) 9.81 m s 2 sin 30 = W A (25) ma = (40kg) 9.81 m s 2 sin 30 9.81 m s 2 (2

Calculus

# NoteUtil Parameters # Comments are '#' # Separator is '~' # Heading character is '|' # The 2 headings are 'Units' and 'Sub-units' # Extensions are: # Additional Information && && # Example %% %% # Mathbot LaTeX \[ \] # Image links ` `# Categories are: # ! Important definition

Multi-strain disease dynamics on a metapopulation network

and 1 collaborator

Many of the most impactful diseases that affect humans, livestock, and wildlife have clusters in their population-genetic variability that we classify as strains. Importantly, host immunity to one of these strains is neither independent from nor equivalent to immunity to related strains. This partial cross-protective immunity affects disease dynamics across the population as a whole and can dramatically influence intervention strategies. While the study of multi-strain diseases goes back decades, this work has not yet been generalized to a loosely connected collection of subpopulations, i.e. a metapopulation. Starting from the strain theory of host-pathogen systems proposed by \citet{Gupta_1998}, we simulate multi-strain disease dynamics on a network of interconnected populations, characterizing the effects of parameterization and network structures on these dynamics. We find that dynamics propagate through the metapopulation network, even if parameters vary between populations. Moreover, in chains of connected populations experiencing cyclical dynamics, the movement of (partially) immune individuals dampens the dynamics of populations further along the chain. This work serves as an important first step in extending prior results on multi-strain diseases to a generalized population structure. This extension is particularly apt in the case of livestock production, where a system of mostly isolated populations (farms) is connected through the forced movement of individuals.

Sensory signs in lumbar disc herniation related radicular pain: a scoping review protocol

and 2 collaborators

This document is a draft for a scoping review protocol concerning sensory signs in disc herniation related lumbar radicular pain. The protocol is open for all pre-submission feedback. Please feel free to use the text icon to the right to comment, suggest, or ask questions (see how here). Any comment are welcome. A PDF version is also available at the Open Science Framework website. The finalized protocol will be registered in the International prospective register of systematic reviews (PROSPERO).

Authorea - Wish List

This would be a great platform for professors or instructors that teach classes and create their own lecture notes. I have seen them having to manually create their own websites and post files for people to download. By using this platform, they have a dynamic platform on which to edit their class notes and bring in other collaborators (graduate students or teaching assistants) to collaborate.

Technologies such as LaTeXML and PreTeXt are OK -- but they require a lot of "havy lifting" to install. Plus, none of them are a platform for collaboration, such as what you have built.

- Why isn't the LaTeX section text output in a font that resembles Latin Modern Roman?

- Spacing:
- UPDATE: Can you change the spacing between multiline equations?

- Why doesn't \vspace{1\baselineskip} work? How do you add more space within a LaTeX block?

- Can you reference equations in another file? Right now, it looks like if you insert a reference, it only gives you the choice to reference an equation in the same document.

- Does it support highlighting certain formulas?

- Support for inserting PDF images - PENDING

- If I want to insert a figure in the middle of a LaTeX block, there is no way right now to "pinch off" that block and separate it into two blocks and insert a figure where I need it.
- Similarly, there is no way to combine two LaTeX blocks into one?

- Can you make the LaTeX editor block wider?

- Delete blocks - WORKS

- Can you group files / documents into certain folders or have some kind of structure?
- Can you use one header file for multiple documents?
- Can I build a page that acts as a link to all my other documents?

Arithmetic Sequences and Series

The goal of this section is a gentle introduction to arithmetic sequences and series. This will lead up to the more general discussion of arithmetic-geometric series.

Counting the Cost: A Report on APC-supported Open Access Publishing in a Research Library

and 2 collaborators

At one-hundred twenty-two articles published, the open access journal _Tremor and other Hyperkinetic Movements_ (tremorjournal.org, ISSN: 2160-8288), is growing its readership and expanding its influence among patients, clinicians, researchers, and the general public interested in issues of non-Parkinsonian tremor disorders. Among the characteristics that set the journal apart from similar publications, _Tremor_ is published in partnership with the library-based publications program at Columbia University’s Center for Digital Research and Scholarship (CDRS). The production of _Tremor_ in conjunction with its editor, a researching faculty member, clinician, and epidemiologist at the Columbia University Medical Center, has pioneered several new workflows at CDRS: article-charge processing, coordination of vendor services, integration into PubMed Central, administration of publication scholarships granted through a patient-advocacy organization, and open source platform development among them. Open access publishing ventures in libraries often strive for lean operations by attempting to capitalize on the scholarly impact available through the use of templated and turnkey publication systems. For CDRS, production on _Tremor_ has provided opportunity to build operational capacity for more involved publication needs. The following report introduces a framework and account of the costs of producing such a publication as a guide to library and other non-traditional publishing operations interested in gauging the necessary investments. Following a review of the literature published to date on the costs of open access publishing and of the practice of journal publishing in academic libraries, the authors present a brief history of the _Tremor_ and a tabulation of the costs and expenditure of effort by library staff in production. Although producing _Tremor_ has been more expensive than other partner publications in the center's portfolio, the experiences have improved the library's capacity for addressing more challenging projects, and developments for _Tremor_ have already begun to be applied to other journals.

Exercises in High-Dimensional Probability by Vershynin

and 2 collaborators

\({n \choose m}=\frac{n(n-1)\cdots (n-m+1)}{m!}\), so that \({n \choose m}=\left(\frac{n}{m}\right)^m\cdot \frac{m^m}{m!}\cdot\frac{n(n-1)\cdots(n-m+1)}{n^m}\). The ratio of the quantities is therefore the product \(\prod_{j=0}^{m-1}\frac{m\cdot (n-k)}{(m-k)n}\). Each factor in the products individually has value greater than or equal to one because \(n\geq m\). Therefore, the inequality is as claimed.

Second inequality is obvious because \({n \choose m}\) is one of the summands in the sum on the RHS (and all summands are nonnegative).

Third inequality (upper bound on the sum of binomial coefficients): following the hint, multiply both sides by \((m/n)^m\). Since \(m/n\leq1\),

\(\sum_{k=0}^m{n \choose k}\left(\frac{m}{n} \right)^m\leq\sum_{k=0}^m{n \choose k}\left(\frac{m}{n} \right)^k\leq\sum_{k=0}^n{n \choose k}\left(\frac{m}{n} \right)^k\). Apply the binomial theorem to rewrite this quantity as \(\left( 1+\frac{m}{n}\right)^n=\left[\left( 1+\frac{m}{n}\right)^{(n/m)}\right]^m<e^m\), from the familiar limit representation familiar limit representation of e (and the fact that the limiting expressions are bounded above by e).

\(X_+ := \max(0,X)\), \(X_-=-\min(0,X)\), so that both \(X_+, X_-\) are non-negative and \(X=X_+ - X_-\). By linearity of expectation, \(\mathbf{E}X=\mathbf{E}X_++\mathbf{E}[-X_-]\), and **Lemma 1.2.1** applies to \(X_+\), so we obtain the first term. The variable \(-X_-\) is a non-positive random variable, so for the second term we need to prove a version of **Lemma 1.2.1** which applies to non-positive random variables. We do this following the steps of **Lemma 1.2.1**, first by noting that for any non-positive real number \(x\), \(x=-\int_{-\infty}^0 \mathbf{1}_{t>x}\mathrm{d}t\). Next assume that \(X\) is a nonpositive (rather than non-negative) function proceed as in the proof of **Lemma 1.2.1** as follows: \(\mathbf{E}X=\mathbf{E}[-\int_{-\infty}^0\mathbf{1}_{\{t>X\}}\mathrm{d}t=-\int_{-\infty}^0\mathbf{E}\mathbf{1}_{\{t>X\}}\mathrm{d}t=-\int_{\infty}^0\mathbf{P}\left\{t>X\right\}\).

\(X_i=\left\{+1\ \text{incorrect with probability}\ \frac{1}{2}-\delta;\ -1\ \text{correct with probability }\ \frac{1}{2}+\delta\ \right\}\)

Because \(\mathbf{E}\ X_i=-2\delta\), \(\sum_{_{i=1}}^N\mathbf{E}X_i=-2\delta N\). Therefore, \(\mathbf{P\ \left\{\sum_{_{i=1}}^NX_i>0\right\}\ =P}\left\{\sum_{_{i=1}}^NX_i-\mathbf{E}X_i>2\delta N\right\}\). **Theorem 2.2.6**, Hoeffding's Inequality, implies that \(\mathbf{P}\left\{\sum_{_{i=1}}^NX_i>0\right\}\le\exp\left(\frac{-2\cdot4\delta^2N^2}{\sum_{_{i=1}}^N2^2}\right)=\exp\left(-2\delta^2N\right).\) In order to have \(\exp\left(-2\delta^2N\right)<\epsilon\) it is necessary and sufficient to have \(N>\frac{1}{2}\delta^2\ln\epsilon^{-1}\).

Implicit exercise: **Lemma 3.2.4**. The proof of the second part doesn't seem to use the hypothesis that \(X\) and \(Y\)are independent, but clearly the conclusion does not hold unless they are independent. Where is the hypothesis of independence actually used?

\(\mathbf{E}\left\langle X,Y\right\rangle^2=\mathbf{E\mathbf{_YE_{X\left|Y\right|}}}\left(\left\langle X,Y\right\rangle^2\left|Y\right|\right)\) is what the law of total expectation states in general. We can replace the inner conditional expectation with an unconditional expectation \(E_X\) only because of the independence.

Modification of ICU environment is associated with reduced incidence of delirium - Results from the VITALITY study

and 11 collaborators

This abstract was presented at the ESICM LIVES 2018 meeting and was published in Intensive Care Medicine Experimental 2018, 6(Suppl 2):1305.