MAIN If we have ^\infty {n} try introducing a parameter r → 1, this gives ^\infty {n} then try an Abel transform with respect to r, knowing that if r = 1 then r² = 1, K = 2\int_y^\infty {} \; dr = }{4}{n(n^2 x^2)^{1/4}} Then sum over this ^\infty K = {8}-{8})}{(x^2)^{1/4}} x y(_{3/2}(e^{-2ixy^2})+i _{3/2}(e^{2ixy^2})) by the inverse transform we should then have ^\infty {n} = -{2}+{2})}{} \int_r^\infty _{1/2}(e^{-2ixy^2})-i_{1/2}(e^{2ixy^2}))}{(x^2)^{1/4}} \; dy ^\infty {n} = -{2}+{2})}{} \int_1^\infty _{1/2}(e^{-2ixy^2})-i_{1/2}(e^{2ixy^2}))}{(x^2)^{1/4}} \; dy

This case commentary concerns a 56-year-old patient with stage III malignant melanoma which was removed but within 7 months the patient developed secondary tumours in his vertebrae. A multi-epitope vaccination was given after which the patient developed vitiligo and posterior uveitis. The topics addressed include the different types of antigens that can be expressed by malignant melanoma, how immune surveillance fails, the immunological mechanisms involved in the side effects of vaccination, and a comparison of peptide vaccination with dendritic cell immunotherapies. This commentary was created in 2015 as part of the masters programme in Molecular and Cellular Biochemistry at the University of Oxford.

This case commentary concerns a 3-month old female infant (of parents who were first cousins) who presented with persistent diarrhoea, failure to thrive, and a low lymphocyte count. The topics addressed include the typical histories for severe combined immune deficiencies (SCID), how SCID can be confirmed, and the relevance of immunophenotyping circulating lymphocytes in the diagnosis of different types of SCID. This commentary was created in 2015 as part of the masters programme in Molecular and Cellular Biochemistry at the University of Oxford.

Mendelian randomisation is a technique which, fuelled by the results of GWA studies, can be used to determine causal relationships between intermediate phenotypes such as metabolite levels and outcomes such as cardiovascular disease . Much faster and cheaper than randomised controlled trials, and relatively free from the biases of observational studies, it has the potential to identify new drug targets and reduce attrition rates in the pharmaceutical development pipeline.

Dr. John SextonRelationship  of Government and Religion11  March 2017Assignment 8Supreme Court of the United States of AmericaWisconsin v.  YoderConcurring Opinion by Gurgen Tadevosyan, undergraduate student at New York University.Respondents are members of the Amish religious group. They prevented  their children from getting a state-recognized education after they completed  the eighth grade at the ages of 14 and 15. Yoder 207. Wisconsin  statute requires parents to “cause their children to attend schools until  reaching the age of 16.” Ibid. The issue raised by the respondents is  that the state statute puts a burden on their free exercise rights, secured by  the Free Exercise Clause of the First Amendment.  The trial court concluded  that the reasonable governmental exercise of power overweighs the burden placed  on the Amish religion; thus, held in favor of the state. Although the Appellate  court affirmed, the Supreme Court of Wisconsin held that the state interest  does not overweighs the Amish’s freedom of exercise of religion; thus,  reversed.  The question presented is  whether a universally applicable law of the State of Wisconsin that requires  parents to send their children to governmentally approved schools up until the  age of 16 violates the Free Exercise Clause when applied to Amish parents for  refusing to obey it due to religious beliefs. The majority holds that, as  applied, the statute violates the free exercise of the Respondents’ religion,  as the burden placed on it overweighs the state’s interest in compulsory  education. I am writing to implement the Sherbert  test and to prove that, in this case, the rights of the Amish people to freely  exercise their religion should be respected. As applied, the Wisconsin Statute  of the Compulsory Education violates the Free Exercise Clause of the First  Amendment as it puts an unjustified burden on the bona fide religious  actions of the members of the Amish community.The case will be discussed under the Sherbert Test. It[N3]  analyzes the  constitutionality of the statute applications under two main questions. First -  does the challenged statute put a burden on the honest and good bona fide  religious beliefs of the Amish people. And second - is the compelling state  interest outweighed by the resultant burden. Cite.  The order of the prongs should be kept.  The application of a statute is impermissible  if the answers to both of the questions is ‘yes.’The Amish refusal of compelling [N5] with the  Wisconsin statute is a religious action and forbidding it is a burden on their  free exercise rights.     The evidence shows that the religious belief of the Amish people  that forces them to violate the statute is an honest and a good[N6]  part of their  identity. Desires to “return to the early, simple, Christian life deemphasizing  material success, rejecting the competitive spirit, and seeking to insulate  themselves from the modern world” are the reasons why this religion  was established in the 16th century. Yoder, 210. In our times, the Amish  community is guided by the literal reading of the Bible. Their conduct is  regulated by the Ordnung – “rules of the church community” and in the  adulthood they take a voluntary heavy obligation of following the rules of the Church.  Id. The religious values of these people are the determinants of their  actions; these rules dictate them to be a cooperative and an interdependent  community. Overall, one comes to two conclusions. Firstly, the Amish conduct  and lifestyle is deeply tied to their beliefs and secondly, these beliefs are  based on an honest, good and bona fide religion.   Unfortunately, the religious values of this group go in contrast  with the ones that the American educational system promotes.  Therefore, the compulsory education law  places a burden on the free exercise rights of the Amish respondents.  The latter emphasizes “intellectual and  scientific accomplishments, self-distinction, competitiveness, worldly success,  and social life with other students.” Id. at 211. On the contrary, the  Amish society promotes “learning through doing." Ibid. Their  community identity, which is inseparable from their religion, depends on values  such as goodness, intellect, community welfare and isolation from the  contemporary ‘worldly society.’ Ibid.  Due to these differences, the Amish students,  attending government-recognized schools, are subject to exposition and  education of values that are considered ‘spoilt’ for the sect  they belong to. Making the attendance compulsory results in a state-promoted  and enforced corruption of the religious values and beliefs that this community  is based on. If imposed, the statute and the state will suppress Amish  religious doctrine of lifestyle, which will prevent their younger generations  from learning their traditional ways of life that are inseparable from their  religion. The result will be the loss of the Amish religion and thus – the  identity. Overall, the Wisconsin statute could cause the destruction of an  entire sect because of the suppression of their freedom to exercise their own religion and  dissent from participating in activities that endanger its existence. In order to survive this intolerance, the Amish people were  forced to disobey the Wisconsin statute. The reason for their actions, backed  by religious demands, are ones aiming at freely exercising their religion,  which, in itself is their constitutional right. By requiring the parents to send their children to school until age  16, the state burdens the right of Amish people to freely exercise their religion.               The Wisconsin statute was enacted to satisfy a state interest of  the ‘highest order;’ however, as the Amish community serves those interests in  an alternative way, the burden placed on their freedom of exercise is not of a  ‘sufficient magnitude .’  The state’s asserted  interest in education is one of the cornerstones of the operation of our  democratic system. Thomas Jefferson has pointed out that some degree of  education is necessary to prepare citizens to effectively and intelligently  participate in our open political system. Yoder, 221. Educated people  have higher chances of making right[N13]  political decisions, and, most importantly, participating in our  system not only rationally, but also strategically – making choices that will both  reflect and maximize the individual’s impact on the outcome of the process.  Furthermore, it is also necessary for the  development of self-reliant and self-sufficient individuals. Ibid. Employment is  one of the ways of achieving such aims and education is key to employment and to  the personal success; it ensures the base on which people can build specialized  skills. These are the pillars on which the idea of labor division stands  and without them our society will not be able to provide the vital goods of the  cooperation; thus, of the public welfare. This chain events that education  causes of depicts how the lack of it can bring to the destruction of the state.However, the reality shows that even without attending a high  school, the Amish community manages to achieve the highest-order targets of the  state. By isolating themselves from the public welfare in any of its  usual forms, Amish have proven to be productive, self-sufficient and  law-abiding; even the Congress itself has authorized them to receive an  exemption from the obligation of paying social security taxes. Id. at  222. The experts also claim that in small communities they achieve  such interrelations that ensure their self-sufficiency and self-reliance. Yoder 223. This is a result of their interdependent agrarian lifestyle,  which is based on their religious values. By encouraging each member of the  society to forgo their individual benefit and work for the greater good, these  values ensure cooperation and maximal benefit of the collective action.  Thus, the state’s interest in using education  to promote good citizenship can be satisfied without the additional two years  of schooling.   In order to make this prosperous lifestyle sustainable, the passage  of these traditions from generation to generation should be ensured. The youth  of the Amish community is its future and educating them to these values is of  crucial importance for their existence. The decision of parents not to send their  children to school after the age of 14 and help them acquire these values,  leading to cooperative and bible-based life, is a is important in achieving  this aim. This itself represents an alternative way of educating new  generations the values that Jefferson was referring to as vital bases of our  system; thus, the objectives of the state in demanding a compulsory education  are being met by the Amish. Consequently, the Amish community “otherwise  serves” the required “state interests of the highest order” and proves that the  burden placed on their free exercise rights is not justified by a state  interest of sufficient magnitude.     The case in front of us today is one  that arose under the Free Exercise Clause of the First Amendment. As applied,  the Wisconsin Statute prohibits the Amish to prevent their children from attending  high schools; however, the members of this religious group claim that it  suppresses their freedom to exercise their religion. After analyzing the issue  under the Sherbert test, four essential answers were determined. (1) The  beliefs that the Amish people hold are religious; (2) their actions that are  challenged are strictly and tightly interconnected with their religion; (3) the  interest of the state that the Wisconsin statute aims to satisfy are of the  highest order and (4) the Amish community serves these interests in an  alternative way. Overall, the answer to the test is that the burden placed on  the free exercise of these people is not overweighed by the state interest.  Based on this, a conclusion can be drawn. The scale of justice leans towards  the Free Exercise and the Amish people should be granted it. As applied, the  Wisconsin statute cannot prohibit them from refusing to send their children to  any schools. The holding of the Appellate Court was affirmed and I agree.

MAIN Take the elliptic integral E(ϕ, k) and expand about ϕ = 0. Then E(\phi,k) = \phi - {3!} + {5!} - \cdots we can express this as E(\phi,k) = ^\infty {(2i-1)!}\phi^{2i-1} where the Qi(k) are polynomials in k which begin Q_0(k)=& 1 \\ Q_1(k)=& -k \\ Q_2(k)=& 4k-3k^2 \\ Q_3(k)=&-16k+60k^2-45k^3 these polynomials can be generated with the following expression Q_n(m) = ^n ^{k-1} {j}(-1)^{n-j+k+1}2^{1+2n-3k}(j-k)^{2n}}{k!} m^k, \;\;\; n>0 which could also be written as Q_n(m) = 2(-4)^n^n {k!}\left({8}\right)^k ^{k-1} {j}(-1)^{1-j}(j-k)^{2n}, \;\;\; n>0 the reversion of E(ϕ, k) looks very similar to the function itself \phi(E,k) = E + {3!} + {5!} + {7!} + \cdots so we can write \phi(E,k) = ^\infty {(2i-1)!}E^{2i-1} where the polynomials begin R_0(k) =& 1 \\ R_1(k) =& k \\ R_2(k) =& -4k+13k^2 \\ R_3(k) =& 16k-284k^2+493k^3 in principle we should be able to write the series reversion of the first expansion using the formula (Morse and Fesbach) we know the nth coefficient {3!} = {3}(-1)^{k_1+k_2} ^{k_1+k_2-1}3+k_0}{k_1!k_2!}\left(\epsilon\right)^{k_1}\left({3!}\right)^{k_2} {5!} = {5}(-1)^{k_1+k_2+k_3+k_4} ^{k_1+k_2+k_3+k_4-1}5+k_0}{k_1!k_2!k_3!k_4!}\left(\epsilon\right)^{k_1}\left({3!}\right)^{k_2}\left(\epsilon\right)^{k_3}\left(\epsilon{5!}\right)^{k_4} where ϵ vanishes to zero, meaning any combination with k₁, k₃, k₅, ⋯ non-zero, will give no contribution (even powers in the expansion). We can essentially relabel the terms then to give a more general formula R_n(k) = {2n+1} (-1)^{k_1+k_2+k_3+\cdots} ^{k_1+k_2+k_3+\cdots-1}2n+1+k_0}{k_1!k_2!k_3!\cdots}\left({3!}\right)^{k_1}\left({5!}\right)^{k_2}\left({7!}\right)^{k_3}\cdots which can be simplified R_n(k) = (2n)! (-1)^{k_1+k_2+k_3+\cdots} ^{k_1+k_2+k_3+\cdots}2n+k_0}{k_1!k_2!k_3!\cdots}\left({3!}\right)^{k_1}\left({5!}\right)^{k_2}\left({7!}\right)^{k_3}\cdots so let σn = k₁ + k₂ + k₃ + ⋯ + kn, and τn = k₁ + 2k₂ + 3k + 3 + ⋯ + nkn R_n(k) = (2n)! (-1)^{\sigma_n} ^{\sigma_n}2n+i}{^n k_j!}^n \left({(2l+1)!}\right)^{k_l}

The power of the atomAt the beginning of the 20th century, major advancements in our understanding of fundamental physics led scientists to the discovery of nuclear energy. An unprecedented amount of power could in principle be released by combining (nuclear fusion) or breaking (nuclear fission) certain atomic species under special conditions. Nuclear fusion in particular was understood to be the process powering the immense luminosity of stars, including our Sun. Nuclear fusion is the light-bulb illuminating the vast living room of our Universe.Why so much energy?Burning fossil fuels releases chemical energy. This chemical energy is stored in the mild electromagnetic interactions between atoms in a compound. Nuclear energy, on the other hand, comes from the very central regions of the atom.  As the name suggests, it is stored in the nuclei, which are kept together by the strong force. The strong force is much stronger than all the other forces, including the electromagnetic one. As a result, nuclear fuel has an energy density about ten million times larger than chemical fuel. If your car was running on nuclear fuel, its gas mileage would be something like hundreds of millions of MPG. From light to darknessThe physics revolution that characterized the first three decades of the 20th century and led to the development of quantum mechanics and nuclear physics, was followed by the second World War. In 1942, the United States started a very ambitious project to build a nuclear weapon. The Manhattan Project, led by Robert Oppenheimer and gathering some of the best physicists on the planet, culminated with the successful Trinity experiment in 1945 (Fig.\ref{982837}). The first detonation of a nuclear weapon was the most shocking demonstration of the great power of science and the scientific method. Only less than a month later, two nuclear bombs were dropped over the Japanese cities of Hiroshima and Nagasaki, resulting in the end of WWII and the death of hundreds of thousands of people. The sheer destruction inflicted by the atomic bomb left an indelible mark on humankind's consciousness, formally starting a new era in the history of man. An era of greater responsibility. While no nuclear weapons have been purposely used in war ever since, more than 2000 nuclear tests have been performed after the Trinity, Hiroshima and Nagasaki explosions.  

Exposure and response prevention (ERP) is a first-line behavior therapy for obsessive-compulsive disorder, and has also been tested in Tourette syndrome (TS). However, ERP for tic disorders requires intentional tic suppression, which for some patients is difficult even for brief periods. Additionally, practical access to behavior therapy is difficult for many patients, especially those in rural areas. The authors present a simple, working web platform (TicTrainer) that implements a strategy called reward-enhanced exposure and response prevention (RE–ERP). This strategy sacrifices most expert therapist components of ERP, focusing only on increasing the duration of time for which the user can suppress tics through automated differential reinforcement of tic-free periods (DRO). RE–ERP requires an external tic monitor, such as a parent, during training sessions. The user sees increasing digital rewards for longer and longer periods of successful tic suppression, similar to a video game score. TicTrainer is designed for security, storing no personally identifiable health information, and has features to facilitate research, including optional masked comparison of tics during DRO _vs._ noncontingent reward conditions.

Woods and Himle developed a standardized tic suppression paradigm (TSP) for the experimental setting to quantify the effects of intentional tic suppression in Tourette syndrome. The present article describes a Java program that automates record keeping and reward dispensing during the several experimental conditions of the TSP. The software optionally can be connected to a commercial reward token dispenser to further automate reward delivery to the participant. The timing of all tics, 10-second reward-free intervals, and dispensed rewards is recorded in plain text files for later analysis. Expected applications include research on Tourette syndrome and related disorders.

The arXiv is the most popular preprint repository in the world. Since its inception in 1991, the arXiv has allowed researchers to freely share publication-ready articles prior to formal peer review. The growth and the popularity of the arXiv emerged as a result of  new technologies that made document creation and dissemination easy, and cultural practices where collaboration and data sharing were dominant. The arXiv represents a unique place in the history of research communication and the Web itself, however it has arguably changed very little since it's creation.  Here we look at the strengths and weaknesses of arXiv in an effort to identify what possible improvements can be made based on new technologies not previously available. Based on this, we argue that a modern arXiv might in fact not look at all like the arXiv of today.IntroductionThe arXiv, pronounced "archive", is the most popular preprint repository in the world.  Started in 1991 by physicist Paul Ginsparg, the arXiv allows researchers to freely share post publication-ready articles prior to formal peer review and publication. Today, the arXiv publishes over 10,000 articles each month from high-energy physics, computer science, quantitative biology, statistics, quantitative finance, and others (see Fig \ref{104668}). The early success of arXiv stems from the introduction of new technological advances paired to a well-developed culture of collaboration and sharing. Indeed, before the arXiv even existed, physicists were already physically sharing recently finished manuscripts via mail, first, and by email, later.  To understand the success of the arXiv it is important to understand the history of the arXiv. Below we highlight a brief history of technology, services, and cultural norms that predate the arXiv and were integral to its early and continued success.  The history of the arXivPrior to the arXiv, preprinting was performed by institutional repositories, such as the SPIRES-HEP database (Stanford Physics Information REtrieval System- High Energy Physics) at the Stanford Linear Accelerator Center (SLAC) and the Document Server at CERN. Developed in the early 70's, SPIRES created a bibliographic standard and centralized resource that allowed researchers across universities in high energy physics to email the database and request a list of preprints be sent to them.  Since papers themselves could not be emailed at the time, the system relied on traditional mail. This resource was immediately successful with requests numbering in the thousands within the first few years \cite{Elizalde_2017}.  While SPIRES greatly improved the flow of information, it still took weeks for articles (preprints) to be sent and received. A new typesetting system would soon emerge and change this.TeX, pronounced "tech", was developed by Donald Knuth in the late 70's as a way for researchers to write and typeset articles programmatically. Soon after the introduction of TeX, Leslie Lamport set a standard for TeX formatting, called LaTeX, which made it very easy for all researchers to professionally typeset their documents on their own.  This system made sharing papers easier and cheaper than ever before. Indeed, many, if not most, researchers at the time relied upon secretaries or typists to write their work, which then had to be photocopied in order to be sent via mail to a handful of other researchers. Tex allowed researchers to write their documents in a specified manner (binary) that could be emailed and then downloaded and compiled without the need for physical mail. Soon, physicists were emailing and downloading .tex files at great rates hastening the process of research communication like never before.Such a system immediately created a new problem for researchers: information overload. Researchers were exchanging emails containing preprints at great rates, and given the size of computer hard drives at the time, email servers were running out of space \cite{Ginsparg_2011}.  To address this problem, an automated email server, called arXiv, was set up in the early 90's. The arXiv would allow researchers to automatically request preprints via email as needed. It would soon become one of the world's first web servers and today still serves as one of the most open and efficient forms of research communication in the world.  The arXiv was a leader in introducing and utilizing new technology when it was launched, however it has arguably changed very little since its inception, despite a wealth of new technologies now available. Here we look at the strengths and weaknesses of the arXiv in an effort to identify what possible improvements can be made based on new technologies and tools and propose that a modern arXiv might in fact not look at all like the arXiv of today --- a development that will likely occur with or without arXiv.

Aside from Bell’s inequality, QM and local real theory have other specifications that can be observed in experiments. To explore these specifications, we re-examine EPR paradox to show that non-locality arises from the absence of location variable. Our analysis are then applied to several reported experiments. 1) In a known short range Bell experiment with high detection efficiency, portion of the presented data agrees more with local real model than with QM. 2) The so called non maximally entangled state in several experiments are essentially partially entangled photons, with a large local real part helping the violation of Bell’s inequality, and the reported event counts deviate from expected entanglement model. 3) In long range EPR experiments for closing locality loophole, interactions with local real apparatus prior to measurements put the entanglement in question.

ABSTRACT I introduce k-crossing paths and partitions and count the number of paths for each number of desired crossings k for systems with 11 points or less. I give some conjectures into the number of possible paths for certain numbers of crossings as a function of the number of points. INTRODUCTION A order n meandric partition is a set of the integers 1⋯n, such that a path from the south-west can weave through n points labeled 1⋯n without intersecting itself and finally heads east (examples are shown in Fig. 1). Counting the number of possible paths for n points is a tricky problem, and no recursion relation, generating function or explicit formula for the number of order n meandric partitions appears to have been found. This work is concerned with the number of paths that must intersect themselves exactly k times, where when k is 0, we have the meandric paths. It is possible to draw a line that deliberately crosses itself as many times as required, because of this we only consider a path to be k-crossing if k is the smallest number of crossings possible, that is a path that must cross itself k times (an example of a 3-crossing path over 9 points is given in Fig. 2). RESULTS Define ak(n) to be the number of configurations of n points where the path through them is forced to cross itself k times. For 0-crossings on n points we have the open meandric numbers, given in the OEIS as A005316 a_0(n) = 1, 1, 1, 2, 3, 8, 14, 42, 81, 262, 538, 1828, 3926, \cdots, \;\; n=0,1,\cdots this work has counted this for k > 0 by calculating all n! permutations of the n integers and checking to see the minimal number of crossings for each, we then have n =&0&1&2&3&4&5&6&7&8&9&10&11\cdots\\ a_0(n) =&1,& 1,& 1,& 2,& 3,& 8,& 14,& 42,& 81,& 262,& 538,& 1828,\cdots\\ a_1(n) =&0,&0,& 1,& 4,& 10,& 36,& 85,& 312,& 737,& 2760,& 6604, &25176,\cdots\\ a_2(n) =&0,&0,& 0,& 0,& 8,& 42,& 168,& 760,& 2418,& 10490,& 30842, &131676,\cdots\\ a_3(n) =&0,&0,& 0,& 0,& 2,& 16,& 164,& 944,& 4386,& 22240,& 83066, &398132,\cdots\\ a_4(n) =&0,&0,& 0,& 0,& 1,& 18,& 146,& 1076,& 6255,& 37250,& 168645, &908898,\cdots\\ a_5(n) =&0,&0,& 0,& 0,& 0,& 0,& 96,& 960,& 7388,& 51968,& 282122, &1711824, \cdots\\ a_6(n) =&0,&0,& 0,& 0,& 0,& 0,& 30,& 440,& 6472,& 55140,& 384065, &2642444,\cdots\\ a_7(n) =&0,&0,& 0,& 0,& 0,& 0,& 14,& 368,& 5176,& 53920,& 455944, &3575040,\cdots\\ a_8(n) =&0,&0,& 0,& 0,& 0,& 0,& 2,& 66,& 3542,& 45960,& 484058, &4336734,\cdots\\ a_9(n) =&0,&0,& 0,& 0,& 0,& 0,& 1,& 72,& 2011,& 32280,& 452504, &4661756,\cdots\\ a_{10}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1172,& 25066,& 396493, &4709856,\cdots\\ a_{11}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 420,& 11840,& 309696, &4291440,\cdots\\ a_{12}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 201,& 8930,& 225754, &3661348,\cdots\\ a_{13}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 40,& 2240,& 151849, &2947392,\cdots\\ a_{14}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 18,& 2040,& 91147, &2103648,\cdots\\ a_{15}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 2,& 224,& 55030, &1575744,\cdots\\ a_{16}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1,& 270,& 26762, &915924,\cdots\\ a_{17}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 14627, &665088,\cdots\\ a_{18}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 5405, &295956,\cdots\\ a_{19}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2642, &218508,\cdots\\ a_{20}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 641, &63522,\cdots\\ a_{21}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 293, &54672,\cdots\\ a_{22}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 48, &8964,\cdots\\ a_{23}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 22, &9552,\cdots\\ a_{24}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2, &706,\cdots\\ a_{25}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 1, &972,\cdots where the vertical sum over columns of terms gives n!. CONJECTURES The above information has lead to a few conjectures. a_{n^2}(2n) = 1 this can be converted to words as, there is exactly one path through 2n points that crosses n² times. The partitions associated with these paths are (2,1)\\ (3,1,4,2)\\ (4,1,5,2,6,3)\\ (5,1,6,2,7,3,8,4)\\ (6,1,7,2,8,3,9,4,10,5) and a clear interlaced pattern can be seen (an example is given in Fig. 3). a_{n^2-1}(2n) = 2, \; n>1 a_{n^2-2}(2n) = 4n+2, \; n>2 a_{n^2-3}(2n) = 8n+8, \; n>3 a_{n^2}(2n+1) = 2(n+1)3^{n-1}, \; n>1

ABSTRACT Consider the Lambert-W function and some analogues MAIN The Lambert-W function W(x) satisfies W(x) e^{W(x)} = x we can write the coefficients of W(x) in an expansion ^\infty }{n!}x^n we could also consider the function that satisfies f_1^2(x) e^{f_1(x)} = x^2 we find that it seems to have an expansion f_1(x) = -x -^\infty \left(^n k {k} n^{n-1-k} \right) }{2^n n!} which can also be written as f_1(x) = -x -^\infty {n})^n n^n}{n+1}}{n! 2^n}

This is a story of _semantics-by-convention_ gone wrong that hit us at Authorea last week.

RECORDING FROM SINGLE NEURONS IN THE BRAIN FOR LONG PERIODS OF TIME HAS BEEN A CENTRAL GOAL OF BOTH BASIC NEUROSCIENCE AND TRANSLATIONAL NEUROLOGY IN ORDER TO UNDERSTAND BRAIN PROCESSES SUCH AS LEARNING CITEPLACEHOLDER1ID. RECENT ADVANCES IN MATERIALS ENGINEERING, DIGITAL SIGNAL ACQUISITION, AND ANALYSIS ALGORITHMS HAVE BROUGHT US CLOSER TO ACHIEVING THIS GOAL, AND THE POSSIBILITY HAS GATHERED MUCH PUBLIC ATTENTION CITEPLACEHOLDER2ID. HOWEVER, IT REMAINS A CHALLENGE TO RECORD FROM THE SAME UNITS FOR WEEKS TO MONTHS. HERE, WE RECORD MANY HIGH-QUALITY JUXTACELLULAR TETRODE NEURONAL SIGNALS RELIABLY OVER LONG PERIODS OF TIME IN BOTH DEEP AND SUPERFICIAL AREAS OF THE BRAIN. WE ACHIEVE THIS BY COMBINING ELECTROCHEMICAL ROUGHENING AND CARBON NANOTUBE COATING OF A FLEXIBLE PLATINUM/IRIDIUM SUBSTRATE. THE MATERIAL PROPERTIES OF THIS COATING WERE CHARACTERIZED, AND PACKAGING AND INSERTION TECHNIQUES WERE OPTIMIZED TO MINIMIZE TIP MOVEMENT WITH BRAIN PULSATION. THIS "MAGDEBURGER" PROBE ENABLES RECORDINGS WITH LONG-TERM SIGNAL STABILITY AND HIGH SIGNAL-TO-NOISE RATIO AT A REASONABLE COST IN BOTH RODENT AND PRIMATE BRAINS. ROBUST TRACKING OF IDENTIFIED NEURONS OVER LONGER TIME PERIODS, IN MULTIPLE INDEPENDENTLY TARGETED AREAS OF THE BRAIN, WILL ALLOW FUNDAMENTAL ADVANCES IN THE STUDY OF COGNITIVE LEARNING, AGING, AND PATHOGENESIS, AND OPENS NEW POSSIBILITIES FOR BRAIN INTERFACES IN HUMANS.

ABSTRACT I will write about a methodology for creating integer sequences by ’power replacement rules’ (PRR). These arise by using a special character in a series expansion of a certain function. Then powers of this character are mapped to constants (or later more complex objects i.e functions etc.). A target function is stated and then the constants are assigned such that the series expansion of the initial function match the target function. MAIN An example is follows, define a character χ, we can use an example function f(x,\chi) = \log\left({1+2x+x^2+x^3}\right) we have a target function g(x) = {1-x} = x+x^2+x^3+x^4+x^5+\cdots the expansion of f(x, χ) with respect to x about x = 0 is f(x,\chi) = (\chi-2)x +{2}(4-\chi^2)x^2 + {3}(\chi^3-3\chi-5)x^3+{4}(8+4\chi^2-\chi^4)x^4+\cdots we now pick the values of powers of χ such that the expansion of f equals the expansion of g term for term. The appropriate values are \chi&=3\\ \chi^2&=2\\ \chi^3&=17\\ \chi^4&=12\\ \chi^5&=92\\ \chi^6&=79 and we see a new integer sequence a(n)=χn, defined for n = 1, 2, 3, ⋯. It seems there doesn’t always exist a χ mapping to transform every function to each other. If we can find a function χn = a(n), then we can state f(x,\chi\|\chi^n=a(n))=g(x) RESULTS To map $f(x,\chi)=\exp(\chi x) \to g(x)={1-x}$, we need χn = n! To map $f(x,\chi)=\log(\chi x+1) \to g(x)={1-x}$, we need χn = ( − 1)n + 1n \log(1+\chi x^2\|\chi^n=n)= {1+x^2} For the example \left({(1-\chi x)^2}\bigg\|\chi^n=n\right)={(x-1)^3} an actual solution does exist to equate the two if χ is \chi \to \chi(x) = \cdots

ABSTRACT I note down a log expansion formula that seems to hold MAIN It seems that for odd n > 0 we have -\log\left(^{n-1}x^k\right)=^\infty \left(2^{(n-1)/2} \cos\left({n}\right)\right) {k} and for even n > 0 we have -\log\left(^{n-1}x^k\right)=^\infty \left( e^{ik\pi}+ 2^{(n-1)/2} \cos\left({n}\right)\right) {k} we have ^\infty x^k = {1-x} for convergent x, and also that -\log\left({1-x}\right)= -^\infty {k} which would then imply 2^{(n-1)/2} \cos\left({n}\right) = -1, \; \forall k \in ^+ EXPONENTIATE We can turn this into an expression for all n, with an iverson bracket [ ⋅ ] -\log\left(^{n-1}x^k\right)=^\infty \left([n\bmod2 \equiv 0]e^{i \pi k}+ 2^{(n-1)/2} \cos\left({n}\right)\right) {k} and convert into ^{n-1}x^k=\exp\left(-^\infty \left( [n\bmod2 \equiv 0]e^{i \pi k}+ 2^{(n-1)/2} \cos\left({n}\right)\right) {k}\right) then ^{n-1}x^k=^\infty \exp\left(- [n\bmod2 \equiv 0]e^{i \pi k} -2^{(n-1)/2} \cos\left({n}\right) {k}\right) ^{n-1}x^k=^\infty \exp\left(- [n\bmod2 \equiv 0]e^{i \pi k}\right)\exp\left( -2^{(n-1)/2} \cos\left({n}\right) {k}\right) ^{n-1}x^k=^\infty \exp\left(- [n\bmod2 \equiv 0]e^{i \pi k}\right) ^{{2}}\exp\left(-2\cos\left({n}\right) {k}\right) This formula doesn’t seem to work for n = 1 and n = 2, however beyond that we get relationships such as 1+x+x^2 = \exp\left(x+{2}-{3}+{4}+{5}-\cdots\right) This isn’t too surprising, This is purely a statement of 1+x+x^2 = \exp(\log(1+x+x^2)) however this is a nice way to think about polynomial division, when we take the ratio of polynomials we can now equate this to their exponential series expansions {1+2x+x^2+x^3} = {2}-{3}+{4}+{5}-\cdots\right)}{\exp\left(2x-x^2+{3}-{2}+{5}-\cdots\right)} but due to the nature of exponentials we can write this as an infinite product {1+2x+x^2+x^3} = e^{-2x^3/3}e^{x^4/4}\cdots}{e^{2x}e^{-x^2}e^{5x^3/3}e^{-5x^4/2}\cdots}

Many (though not all) of my patients who have tried marijuana have felt that their tics improved after using it. Such self-treatment is not rare (poster P94 here), and other doctors report similar results (see for example poster P6 here). Pharmacological benefits from cannabis products are plausible, since cannabinoid receptors in the brain's basal ganglia are well positioned to affect movement . Of course, in addition to any real benefit from marijuana, there could be expectation effects, or one could simply care less about tics when high. Random allocation clinical trials with blind rating of benefit (RCTs) are essential to demonstrating whether marijuana has any true benefit for tics. Müller-Vahl and colleagues carried out two RCTs about 15 years ago in Tourette syndrome (TS) using THC (tetrahydrocannabinol), the main intoxicating ingredient in cannabis . Both trials showed benefit, but the trials were relatively small. Two to 3 years ago, the Tourette Association of America funded two pilot studies in this field, but results have not yet been reported. One trial, at Yale, was to study the FAAH (fatty acid amide hydrolase) inhibitor PF-04457845 in TS , but the trial was placed on clinical hold pending results from a different trial. Investigators at Toronto Western Hospital were funded for a trial in TS of medical cannabis products with varying concentrations of THC and cannabidiol . Cannabidiol is being studied in several brain disorders, including epilepsy, with hopes that it may provide benefit without the psychological side effects of THC. Not surprisingly, the paucity of data has led to different viewpoints. Müller-Vahl has argued that THC may be appropriate in some TS patients , whereas an American Academy of Neurology review and a Cochrane-style review in JAMA concluded that the evidence was insufficient to recommend THC for tic disorders . The clinical utility of cannabinoids in TS was one of two clinical controversies debated at the 2015 First World Congress on Tourette Syndrome and Tic Disorders .

 Kenneth D. Royal and Melanie Lybarger The topic of automation replacing human jobs has been receiving a great deal of media attention in recent months. In January, the McKinsey Global Institute (Manyika et al., 2017) published a report stating 51% of job tasks (not jobs) could be automated with current technologies. The topic of ‘big data’ and algorithms was also briefly discussed on the Rasch listserv last year and offered a great deal of food-for-thought regarding the future of psychometrics in particular. Several individuals noted a number of automated scoring procedures are being developed and fine-tuned, and each offer a great deal of promise. Multiple commenters noted the potential benefits of machine scoring using sophisticated algorithms, such as power, precision, and reliability. Some comments even predicted humans will become mostly obsolete in the future of psychometrics. Certainly, there is much to get excited about when thinking about the possibilities. However, there remain some issues that should encourage us to proceed with extreme caution. The Good For many years now algorithms have played a significant role in our everyday lives. For example, if you visit an online retailer’s website and click to view a product, you will likely be presented a number of recommendations for related products based on your presumed interests. In fact, years ago Amazon employed a number of individuals whose job was to critique books and provide recommendations to customers. Upon developing an algorithm that analyzed data about what customers had purchased, sales increased dramatically. Although some humans were (unfortunately) replaced with computers, the ‘good’ was that sales skyrocketed for both the immediate and foreseeable long-term future and the company was able to employ many more people. Similarly, many dating websites now use information about their subscribers to predict matches that are likely to be compatible. In some respects, this alleviates the need for friends and acquaintances to make what are often times awkward introductions between two parties, and feel guilty if the recommendation turns out to be a bad one. The ‘good’, in this case, is the ability to relieve people that have to maintain relationships with each party of the uncomfortable responsibility of playing matchmaker. While the aforementioned algorithms are generally innocuous, there are a number of examples that futurists predict will change most everything about our lives. For example, in recent years Google’s self-driving cars have gained considerable attention. Futurists imagine a world in which computerized cars will completely replace the need for humans to know how to drive. These cars will be better drivers than humans - they will have better reflexes, enjoy greater awareness of other vehicles, and will operate distraction-free (Marcus, 2012). Further, these cars will be able to drive closer together, at faster speeds, and will even be able to drop you off at work while they park themselves. Certainly, there is much to look forward to when things go as planned, but there is much to fear when things do not. The Bad Some examples of algorithmic failures are easy to measure in terms of costs. In 2010, the ‘flash crash’ occurred when an algorithmic failure from a firm in Kansas who ordered a single mass sell and triggered a series of events that led the Dow Jones Industrial Average into a tailspin. Within minutes, nearly $9 trillion in shareholder value was lost (Baumann, 2013). Although the stocks later rebounded that day, it was not without enormous anxiety, fear and confusion. Another example involving economics also incorporates psychosocial elements. Several years ago, individuals (from numerous countries) won lawsuits against Google when the autocomplete feature linked libelous and unflattering information to them when their names were entered into the Google search engine. Lawyers representing Google stated "We believe that Google should not be held liable for terms that appear in autocomplete as these are predicted by computer algorithms based on searches from previous users, not by Google itself." (Solomon, 2011). Courts, however, sided with the plaintiffs and required Google to manually change the search suggestions. Another example involves measures that are more abstract, and often undetectable for long periods of time. Consider ‘aggregator’ websites that collect content from other sources and reproduces it for further proliferation. News media sites are some of the most common examples of aggregators. The problem is media organizations have long been criticized with allegations of bias. Cass Sunstein, Director of the Harvard Law School's program on Behavioral Economics and Public Policy, has long discussed the problems of ‘echo chambers’, a phenomenon that occurs when people consume only the information that reinforces their views (2009). This typically results in extreme views, and when like-minded people get together, they tend to exhibit extreme behaviors. The present political landscapes in the United States (e.g., democrats vs. republicans) and Great Britain (e.g., “Brexit” - Britain leaving the European Union) highlight some of the consequences that result from echo chambers. Although algorithms may not be directly responsible for divisive political views throughout the U.S. (and beyond), their mass proliferation of biased information and perspectives certainly contributes to group polarization that may ultimately leave members of a society at odds with one another. Some might argue these costs are among the most significant of all. The Scary Gary Marcus, a professor of cognitive science at NYU, has published a number of pieces in The New Yorker discussing what the future may potentially hold if (and when) computers and robots reign supreme. In a 2012 article he presents the following scenario: Your car is speeding along a bridge at fifty miles per hour when an errant school bus carrying forty innocent children crosses its path. Should your car swerve, possibly risking the life of its owner (you), in order to save the children, or keep going, putting all forty kids at risk? If the decision must be made in milliseconds, the computer will have to make the call. Marcus’ example underscores a very serious problem regarding algorithms and computer judgments. That is, when we outsource our control we are also outsourcing our moral and ethical judgment. Let us consider another example. The Impermium corporation, which was acquired by Google in 2014, was essentially an anti-spam company whose software purported to automatically “identify not only spam and malicious links, but all kinds of harmful content—such as violence, racism, flagrant profanity, and hate speech—and allows site owners to act on it in real-time, before it reaches readers.” As Marcus (2015) points out, how does one “translate the concept of harm into the language of zeroes and ones?” Even if a technical operation was possible to do this, there remains the problem that morality and ethics is hardly a universally agreed upon set of ideals. Morality and ethics are, at best, a work-in-progress for humans, as cultural differences and a host of contextual circumstances presents an incredibly complex array of confounding variables. These types of programming decisions could have an enormous impact on the world. For example, algorithms that censor free speech in democratic countries could spark civil unrest among people already suspicious of their government; individuals flagged to be in violation of an offense could have his/her reputation irreparably damaged, be terminated by an employer, and/or charged with a crime(s). When we defer to computers and algorithms to make our decisions for us, we are entrusting that they have all the ‘right’ answers. This is a very scary proposition given the answers fed to machines come from data, which are often messy, out-of-date, subjective, and lacking in context. An additional concern involves the potential to program evil into code. While it is certainly possible that someone could program evil as part of an intentional, malicious act (e.g., terrorism), we are referring to evil in the sense of thoughtless actions that affect others. Melissa Orlie (1997), expanding on the idea of “ethical trespassing” as originally introduced by political theorist Hannah Arendt, discusses the notion of ‘ordinary evil’. Orlie argues that despite our best intentions, humans inevitably trespass on others by failing to predict every possible way in which our decisions might impact others. Thoughtless actions and unintended consequences must, therefore, be measured, included, and accounted for in our calculations and predictions. That said, the ability to do this perfectly in most contexts can never be achieved, so it would seem each day would present a new potential to open Pandora’s Box. Extensions to Psychometrics Some believe the ‘big data’ movement and advances in techniques designed to handle big data will, for the most part, make psychometricians obsolete. No one knows for sure what the future holds, but at present that seems to be a somewhat unlikely proposition. First, members of the psychometric community are notorious for being incredibly tedious with respect to not only the accuracy of information, but also the inferences made and the way in which results are used. Further, it is apparent that the greatest lessons learned from previous algorithmic failures pertains to the unintended consequences, albeit economically, socially, culturally, politically, and legally that may result (e.g., glitches that result in stock market plunges, legal liability for mistakes, increased divisions in political attitudes, etc.). Competing validity conceptualizations aside, earnest efforts to minimize unintended consequences is something most psychometricians take very seriously and already do. If anything, it seems a future in which algorithms are used exclusively could only be complemented by psychometricians who perform algorithmic audits (Morozov, 2013) and think meticulously about identifying various ‘ordinary evils’. Perhaps instead of debating whether robots are becoming more human or if humans are becoming more robotic, we would be better off simply appreciating and leveraging the strengths of both? References Baumann, N. (2013). Too fast to fail: How high-speed trading fuels Wall Street disasters. Mother Jones. Available at: http://www.motherjones.com/politics/2013/02/high-frequency-trading-danger-risk-wall-street Manyika, J., Chui, M., Miremadi, M., Bughin, J., George, K., Willmott, P., & Dewhurst, M. (2017). A future that works: Automation, employment, and productivity. The McKinsey Global Institute. Available at: http://www.mckinsey.com/global-themes/digital-disruption/harnessing-automation-for-a-future-that-works Marcus, G. (2012). Moral machines. The New Yorker. Available at: http://www.newyorker.com/news/news-desk/moral-machines Marcus, G. (2015). Teaching robots to be moral. The New Yorker. Available at: http://www.newyorker.com/tech/elements/teaching-robots-to-be-moral Morozov, E. To Save Everything, Click Here: The Folly of Technological Solutionism (2013). PublicAffairs Publishing, New York, NY. Orlie, M. (1997). Living ethically, acting politically. Cornell University Press, Ithaca, NY. Solomon, K. (2011). Google loses autocomplete lawsuit. Techradar. Available at: http://www.techradar.com/news/internet/google-loses-autocomplete-lawsuit-941498 Sunstein, C. R. (2009). Republic.com 2.0. Princeton University Press, Princeton, NJ.    

ABSTRACT I show that odd (and other) zeta constants such as Apery’s constant can be obtained from integrating a continued fraction function. MAIN We start with the identity for the Mellin transform of csch(x), \int_0^\infty x^{s-1}(x)\;dx = 2^{1-s}(2^s-1)\Gamma(s)\zeta(s),\;\;(s)>1 allowing use to get a formula for ζ(n) with n > 1 an integer, for example n = 3 will give Apery’s constant. \zeta(3) = {7} \int_0^\infty x^2 (x) \; dx the next step is to convert the csch(x) function into a continued fraction, from the reference below we may write (x) = {x} - {T_0(x)}\\ T_n(x) = S_n(ix) \\ S_n(x) = (2n+2)(2n+3) - x^2 + {S_{n+1}(x)} where i is the imaginary unit. Then we may write \zeta(3) = {7} \int_0^\infty x - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx where the sequence of numbers 6, 20, 42, 72, 110, 156, 210, ⋯ is A068377, and have the general form (2n − 2)(2n − 1) here starting from n = 2. ALL POWERS We can of course repeat this for all of the odd (and even) values of ζ(n) if we wish. Does this show that these kind of constants are all irrational as they have an infinite continued fraction? Because it is being integrated over the answer is probably no. \zeta(5) = {93} \int_0^\infty x^3 - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx \zeta(7) = {5715} \int_0^\infty x^5 - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx with the general case \zeta(n) = {2^{1-n}(2^n-1)\Gamma(n)}\int_0^\infty x^{n-2} - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx N.B. If you decide to truncate the infinite continued fraction, a small divergence about x = d appears which shrinks are more terms are used. If one wanted to numerically integrate to within some accuracy, the integrals from 0 to d − ε and from d + ε to ∞ can be computed. If we do take the infinite case, we can see that ζ(n), is purely given by the integral of the closing difference between two functions, we can truncate the integral \zeta(n) = {2^{1-n}(2^n-1)\Gamma(n)}\int_0^y x^{n-2} - {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx and separate \zeta(n) = {2^{1-n}(2^n-1)\Gamma(n)}\left( }{n-1} -\int_0^y {6+x^2-{20+x^2-{42+x^2-{\cdots}}}}\;dx \right) REFERENCES Continued Fraction Expansion of csch(x) (in Russian) РАЗЛОЖЕНИЕ В ЦЕПНУЮ ДРОБЬ ФУНКЦИЙ cos(x), sec(x), ch(x), sch(x), sin(x), cosec(x), sh(x), csch(x), sec2(x), cosec2(x), sch2(x), csch2 (x) С.Н. Гладковский https://ia601406.us.archive.org/10/items/ContinuedFractionExpansionForFunctionsCosxSecxChxSchx/CFMZ_sin_x_PDFreview02.pdf Mellin Transform Identity From Table of Series and Integrals ()

ABSTRACT I highlight a set of interwoven subsequences in some continued fraction representations of well known constants. MAIN It is well known that the continued fraction expansion of e is e-1=[1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,\cdots] we can see there are three interwoven subsequences S_1(n)=1\\ S_2(n)=1\\ S_3(n)=2n and then the continued fraction takes the form e-1=[S_1(1),S_2(1),S_3(1),S_1(2),S_2(2),S_3(2),\cdots] so we can say the ’period’ of this constant is 3, as it takes three sequences to describe. We also have the constant e^2-1 = [6,2,1,1,3,18,5,1,1,6,30,8,1,1,9,\cdots] is period 5 with S_1(n) = 12(n-1)+6\\ S_2(n) = 3(n-1)+2\\ S_3(n) = 1 \\ S_4(n) = 1 \\ S_5(n) = 3n\\ Together this makes a sequence with generating function e^2-1 \to {(x^5-1)^2}=6+2x+x^2+x^3+3x^4+18x^5+\cdots so we can link constants and generating functions.

ABSTRACT I briefly investigate a series expansion in x which seems to always have 2 as the coefficient of prime powers of x. MAIN I had discovered that the expansion of \Xi(x) = -\log ^\infty \left( 1 - {k} \right) = x + x^2 + {3}x^3 + {8}x^4 + {5}x^5 + \cdots always appeared to have a coefficient of 2/p for prime powers p of x. We can manipulate \Xi(x) = -^\infty \log \left(1 - {k}\right) then -\log\left( 1 - {m} \right) = ^\infty }{km^k} giving \Xi(x) = ^\infty ^\infty }{km^k} but for a given power of x, say n, if we want to fully evaluate the coefficient we can truncate the function \Xi_n(x) = ^n ^{\lfloor n/m \rfloor} }{km^k} for only the coefficient we change it to (x) = ^n ^{\lfloor n/m \rfloor} }{km^k} which can be re-written with an Iverson Bracket [ ⋅ ] (x) = ^n ^{\lfloor n/m \rfloor} [mk=n]}{km^k} we then see it is obvious that a similar function \Xi^*_{[n]}(x) = ^n ^{\lfloor n/m \rfloor} [mk=n]x^{mk} using mk = n, m = n/k \Xi^*_{[n]}(x) = x^{n} so all of the primes have only 2 divisors. We could then write a measure of a number n \lambda(n)={(n-1)!}{dx^n}\Xi_n(x) where if λ(n)=2, we have that n ∈ ℙ.

ABSTRACT I investigate how functions may be defined as an integral over an Iverson Bracket, where the integral varies some parameter in the bracket. MAIN The Iverson Bracket is shown by a logical statement in square parenthesis, which is 1 if the logical statement is true and 0 otherwise. If the statement is an inequality we might have a bracket [f(x)≤g(x)], for example if f(x)=x⁴ − x², and g(x)=x², then the bracket will be 1 for $-\le x \le $. We could then say ^\infty [x^4-x^2\le x^2] \; dx = 2 However, if we insert another parameter in the bracket, say a, then the value of the integral will change as a changes. For example, if change the integral we get q(a)=^\infty [x^4-x^2\le ax^2] \; dx = 2 we have some fun ways to generate constants \int_0^{2\pi} [\sin x < a\cos x] \; dx = \pi which seems to hold for all a. \int_0^\infty [x^2 < ax] \; dx = a and it seems \int_0^\infty [x^{b+1} < ax^b] \; dx = a and more generally \int_0^\infty [x^{c} < ax^b] \; dx = a^{1/(c-b)} for c > b. OTHER DEFINITIONS There are some functions that can be defined \int_1^\infty {k^2} \; dk = {x} + [x\le 1]

ABSTRACT I note down some interesting conditional sums that produce interesting prime sums. PRIME CONDITIONAL First a sum }^\infty x^{i+j} = } (p-1)x^p and similar }^\infty x^{i+j} = } {2}x^p then \in }^\infty x^{i+j} = \in} 2x^{p_{t>}} where pt> are the greater of twin primes. Of course, this always requires that i or j is 2. }^\infty x^j = ^\infty (\pi(2n)-\pi(n))x^n this is strongly linked to Bertrands Postulate which would state every power of x has a positive non-zero coefficient in this expansion. It is easy to rephrase this as ^\infty } x^{(p-k)^+}=^\infty (\pi(2n)-\pi(n))x^n where (..)+ indicates the term is only summed if the contents is greater than 0. Likewise }^\infty x^j = ^\infty (2n-\pi(2n)-n+\pi(n))x^n we also have } x^j = } \pi(p)x^p = \pi(n)(\pi(n)-\pi(n-1))x^n \;j\in} x^j = |n\pi(n-1)-(n-1)\pi(n)|x^n = } (p-\pi(p))x^p + } (\pi(q))x^q } x^j = \pi(n)x^n } x^k = \left[^n\pi(2k)-\pi(k)\right]x^n It is not entirely clear what } x^k = ? is. we have } x^k = \left[^n \pi(k) \right]x^n } x^k = \left[^n p_k \right]x^n where pk is the kth prime. } x^j = (\pi(2n-2)-\pi(n))x^n which is even more closely linked to Bertrand’s Postulate. INTEGER CONDITIONAL } x^j = \tau(n)x^n where τ(n) is the number of divisors of n. } x^{i+j} = (\tau(n)-1)x^n \&i+j\in} x^{i+j} = }(n)x^n where χℙ(n) is the characteristic function of primes. \&i\&j\in} x^{i+j} = } x^{2p} \&i\in} x^j = \omega(n)x^n where ω(n) is the number of distinct primes dividing n. } x^{j} = a(n)x^n where here specifically a(n) is the number of divisors of 2n excluding 1. ADDITIONAL EXPERIEMENTAL We can try some more exotic forms, conjecture {(i+j)}\in} x^j = {1-x} \\ {(i+j)}\in} x^j= {1-x^2} \\ {(i+j)}\in}x^j= {1-x^3} \\ {(i+j)}\in} x^j= {1+x+x^2-x^4-x^5-x^6} \\ {2}+{3}\right)\in} x^j = {(1-x)^2(1+x+x^2+x^3+x^4+x^5)} It seems that for n > 1 we have \varphi(n) = ^n 1 where φ(n) is the Euler totient function. This is the same as the number of matrices that satisfy for a given n n & a \\ b & c =1, \; 1\le a,b,c \le n along these lines we also seem to have ^{n-1}\tau(k) = ^n 1 ^{n}(k,n) = ^n 1 \varphi(n)-1= ^n 1 A053570(n+1)=^n \varphi(b)