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# Introduction to statistical causal inference

A brief overview of how causal inference and causal effects are formalized. Say we observe a negative relationship between number of apples eaten per day and heart disease. Does this relationship mean that apples are protective against disease? Maybe. It is well known that correlation does not imply causation. Perhaps in this case number of apples eaten per day correlates with general diet, or general fitness, which instead are the cause of lower heart disease. Such factors are a source of confounding. How do we distinguish between these possibilities? A statistician answers these causal inference questions in two ways: by considering counterfactuals and interventions. A counterfactual is simply a potential event that did not occur. A given patient either does or does not receive the treatment on a given trial. Whichever event does not occur is the counterfactual. Under a counterfactual account of causality to claim that a proposed treatment causes disease remission is to claim that had the patient not received the treatment then the disease outcome would be different (greater). Interventionist accounts are similar but focus on the notion of manipulability. Here to claim that one variable causes another is to claim that if through intervention one variable is forced to a given state then a change in the other variable will be observed. Here the notion of intervention is treated as a primitive and causal relationships are derived from that. Thus, had a given patient *not* had so many apples per day, would their health be worse? And, if a patient was *forced* to eat many apples per day, would their health be better? Here we focus on frameworks that attempt to answer these questions in the presence of confounding. The basic idea is that if we do observe all the factors we reasonably consider to be confounding the estimate then we can correct for this. Learning causal relationships ============================= Randomized controlled trials (RCTs) are the gold standard for causal inference. The idea simply being that if assignment to a treatment group is randomized then the distribution of covariates in the control and treatment groups will be identical, and therefore any difference in outcome between the control and treatment groups can then only be attributed to the fact that one group received a treatment while the other did not. However, sometimes RCTs are difficult, expensive, or unethical to perform. This motivates considering when causal relationships can be inferred from observational data alone. In the absence of randomization, receiving treatment may be correlated with many other factors which could also impact the outcome. What are conditions in which the effects of confounding can be mitigated? Counterfactual outcomes are not observed for individual patients – they either receive a treatment or do not. This is known as the fundamental problem of causal inference. As a result often we need to (or in fact want to) consider aggregate causal effects estimated over a population. This has two consequences for analysis. The first is that in considering causal relationships in this aggregate sense, the timing of pairs of events is often unspecified, vaguely defined, or implicit in how data is collected. By losing this timing information it is harder to analyze cases of mutual causation. Thus the assumption made here is that one variable is the cause of another, or vice versa, or not at all – there is a directedness to the relationship over the time window in which observations are made. It addition to excluding mutual causation from consideration, it is simplest to further exclude cycles, or causal chains (e.g. $A \to B \to C \to A$). The second consequence is that, by considering a population of subjects/events, it becomes more necessary to allow for probabilistic causal relationships, in which one variable’s occurrence affects another’s probability of occurring and the relationship need in no way be deterministic. These considerations motivate summarizing causal relationships between a set of variables using directed acyclic graphs (DAGs), and using a probabilistic framework. Counterfactuals: the causal effect as difference in potential outcomes ====================================================================== Measuring causal effects in terms of counterfactuals is a relatively old idea (as far as statistics goes), dating back to 1923 from work of Neyman. The Neyman-Rubin causal model provides a framework for reasoning about causal effects with counterfactuals. In a simple setting, the model considers two *potential outcomes*: an outcome when a subject does receive a treatment, $Y(1)$, and an outcome when a subject does not receive a treatment, $Y(0)$ (i.e. a control subject). For a given subject, $i$, the *causal effect* is the difference in potential outcomes: \begin{align}E_i = Y_i(1)-Y_i(0).\end{align} If we let $W_i$ be a treatment random variable then assuming consistency between potential and observed outcome, $Y_i$, we have: \begin{align}\label{eq:consistent} Y_i = W_iY_i(0) + (1-W_i)Y_i(1).\end{align} As an aside, note that the potential outcomes $Y(i)$ are treated as kinds of hypothetical random variables. In a sense neither is observed, and they are only related to observation through the assumption that holds. This is a somewhat subtle point that is perhaps not well reflected in the notation. Equations in causal models can have quite different interpretations to standard statistical models, despite having similar notation, which is important to be aware of. Per the *fundamental problem of causal inference*, only one of these potential outcomes is ever observed. To get around this, causal effects can be measured over a population of subjects, some of which receive the treatment and some of which do not. Over a population we can consider the *average causal effect*: \begin{align}\tau = \mathbb{E}(Y_I(1)-Y_I(0)).\end{align} If $W_i$ is assigned to each subject at random then $\tau$ can be computed directly from the treatment and control subpopulation means. In randomized cases, $W_i$ is independent from the potential outcomes. If $W_i$ were not independent from the potential outcomes then the measured causal effect (difference in means) could simply be a result of this correlation. Causal assumptions for identifiability -------------------------------------- Being able to measure a causal effect in an unbiased (unconfounded) way means the effect is *identifiable*. Within this counterfactual framework, this linking of potential outcomes to causal effects relies on four *causal assumptions*. Some of these have been alluded to above. They are: 1. (**SUTVA**) Stable Unit Treatment Value Assumption. This means: - There is no interference in treatments – one subject receiving treatment does not affect others’ treatment. - There is only one form of treatment. 2. (**Consistency**) This assumption links the hypothetical potential outcomes to observed data. If we assume consistency then we are assuming: \begin{align}Y_i = W_iY_i(0) + (1-W_i)Y_i(1),\end{align} as discussed above. 3. (**No unmeasured confounders/ignorability**) The treatment assignment is independent of the potential outcomes: \begin{align}Y(1),Y(0) {\mathrel{\text{\perp\mkern-10mu\perp}}}W.\end{align} In most cases of interest both the outcome and treatment variable are related to a set of observed covariates, $X$. Causal inference then requires: \begin{align}Y(1),Y(0) {\mathrel{\text{\perp\mkern-10mu\perp}}}W | X.\end{align} In RCTs this assumption may be reasonable. This says that the distribution of potential outcomes $(Y(1), Y(0))$ is the same across treatment levels $W$, conditioned on $X$. In observational settings often this is the primary assumption that is a road block to identifiability. Another way to understand this is as follows. We want to relate observed quantities to hypothetical potential outcomes. We can do this if we assume ignorability: \begin{aligned} \mathbb{E}(Y|W=1)-\mathbb{E}(Y|W=0) &= \mathbb{E}(WY(1)+(1-W)Y(0)|W=1)-\mathbb{E}(WY(1)+(1-W)Y(0)|W=0)\\ &= \mathbb{E}(Y(1)|W=1)-\mathbb{E}(Y(0)|W=0)\\ \text{(ignorability)}\quad &= \mathbb{E}(Y(1) - Y(0)))\\ &= \tau\end{aligned} 4. (**Positivity**) Additionally, causal inference requires a non-zero probability of assignment to a treatment group for all subjects: \begin{align}0 < \mathbb{P}(W_i = 1|X_i = x) < 1, \quad \forall x.\end{align} This is known as the *positivity*, or overlap, assumption. Simply, a causal effect cannot be measured if no subjects receive the treatment, or they all do. Directed acyclic graphs and probability distributions ===================================================== In a sense the conditional independence between treatment and potential outcome is the main assumption that requires analysis in the above set of assumptions. This analysis can be aided by encoding our assumptions about the relations between different variables in a graph. This section defines and describes the behavior of these graphs. The following section contains criteria that can be used to identify sets of variables that are sufficient to act as controls, that remove the effect of confounding and hence that satisfy ignorability. These models are types of graphical models, sometimes known as *Bayesian networks*, and were first developed by Pearl in the 1980s. Here we will consider a set of random variables $\mathcal{X}$ as nodes on a directed acyclic graph $\mathcal{G}$. Let this graph have edges $\mathcal{E}$ that represent relations between the variables. Ignorability requires conditional independence of the outcome from the treatment variable, so here we will let the directed edges encode conditional independence assumptions. (A *causal Bayesian network* has additional semantics that are discussed below. For the moment the directed edges only encode information about conditional independence.) First note that the DAG imposes an ordering on the variables $\mathcal{X}$, from which we can talk about a node’s parents, children, ancestors or descendants. Note also that any multivariate distribution can be decomposed into a product of conditional probabilities for any ordering of the variables: \begin{align}P(X) = \prod_{j=1}^N P(X_j|\{X_k\}_{k>j}).\end{align} Given this, if we assume that the variables are ordered in a way that respects the ordering of the DAG, then we will say $\mathcal{X}$ is a Bayesian network with respect to $\mathcal{G}$ if the joint distribution over variables $\mathcal{X}$ factors according to: \begin{align}P(X) = \prod_{j=1}^N P(X_j|\text{Pa}(X_j)),\end{align} where $\text{Pa}(X_j)$ is the parents of node $X_j$. That is, each node $X_j$ is conditionally independent of its non-descendants given its parents: \begin{align}P(X_j|\{X_k\}_{k>j}) = P(X_j|\text{Pa}(X_j)).\end{align} This is the *Markov condition*, or Markov assumption, for a Bayesian network. A node is conditionally independent of the entire network given its *Markov blanket* – its parents, its children, and its children’s other parents. Often also invoked is the *faithfulness condition*, which is the condition that the conditional independencies implied by the graph are the only conditional independencies in the distribution. E.g. assuming faithfulness in the graph $A \to B$ says that there is in fact a dependence between $A$ and $B$. Some types of graphs -------------------- Some properties of a Bayesian network can be inferred graphically. For instance three basic components of DAGs are: 1. Chain: $A \to B \to C$ 2. Fork: $A \leftarrow B \to C$ 3. Collider (inverted fork): $A \to B \leftarrow C$ These graphs behave differently when conditioning on parts of them. Compare the fork and the inverted fork. - For the fork, $A$ and $C$ are dependent. Yet when conditioned on $B$, $A$ and $C$ become independent. - The converse is true for the inverted fork. Without conditioning, $A$ and $B$ are independent. Yet when conditioned on $B$, $A$ and $C$ become dependent. This may seem a little counter-intuitive. An example of this phenomenon is if $B$ is determined through tossing two independent coins, $A$ and $C$. If $B$ is determined as \begin{align}B = \begin{cases} 1, \quad A=H, C = H;\\ 0, \quad \text{else} \end{cases}\end{align} By itself, knowing $A$ tells you nothing about $C$. But knowing $B$ and $A$ now tells you something about $C$. Note that the fork and the chain have the same behavior: - For the fork, $A$ and $C$ are dependent. Yet when conditioned on $B$, $A$ and $C$ become independent. - For the chain, $A$ and $C$ are dependent. Yet when conditioned on $B$, $A$ and $C$ become independent. d-separation ------------ For more complicated graphs, are a given set of variables sufficient controls to render two nodes conditionally independent? Here the notion of *d-separation* is useful. The d stands for dependence. Let $P$ be a path from node $u$ to $v$. A path is a loop-free, undirected (i.e. all edge directions are ignored) path between two nodes. Then $P$ is said to be d-separated by a set of nodes $Z$ if any of the following conditions holds: - $P$ contains a directed chain such that the middle node $m$ is in $Z$, or - $P$ contains a fork, $u \cdots \leftarrow m \to \cdots v$, such that the middle node m is in Z, or - $P$ contains an inverted fork (or collider), $u \cdots \to m \leftarrow \cdots v$, such that the middle node $m$ is *not* in $Z$ and no descendant of $m$ is in $Z$. Nodes $u$ and $v$ are said to be d-separated by $Z$ if all paths between them are d-separated. If $u$ and $v$ are not d-separated, they are called d-connected. We have the result that $X_u$ and $X_v$ being d-separated by $Z$ tells us that $X_u$ and $X_v$ are conditionally independent given $Z$. Markov equivalence classes -------------------------- Note that a DAG may prescribe a factorization of the probability distribution, but the converse is not true. That is, knowing a factorization of the joint distribution does not always imply a unique DAG. Instead it prescribes a *Markov equivalence class* of DAGs. This means that if we want to think of the directed edges as representing causal relationships then knowing a joint distribution factorization does not always provide a unique graph of causal relationships. This limits what we can learn about causal relationships from a joint (observational) distribution alone. Two graphs are Markov equivalent iff they share the same conditional independencies. Equally, they are Markov equivalent iff they have the same d-separations. That is, if $u$ and $v$ are d-separated by $C$ in $\mathcal{G}_1$ then they are d-separated by $C$ in $\mathcal{G}_2$, and vice versa. Some examples of DAGs that are Markov equivalent are shown in Figure $fig:dags$. ![Examples of DAGs in the same Markov equivalence class.](./../images/dags.svg) In fact a simple graphical rule tells us if two DAGs are in the same Markov equivalence class. The *skeleton* of a network is the undirected graph. Two DAGs are in the same equivalence class (observationally equivalent) if they have the same skeleton and the same set of ‘v-structures’ – the same set of two converging arrows whose tails are not connected by an arrow. Controlling for confounders =========================== Now we know some of the behavior of Bayesian networks we can return to the question of identifying variables that can be controlled for to remove confounding. This means we want to identify variables $X$ such that ignorability holds: \begin{align}Y(1),Y(0) {\mathrel{\text{\perp\mkern-10mu\perp}}}W | X.\end{align} Note that the observed outcome is of the form $Y = W Y(1) + (1-W)Y(0)$, which induces a conditional dependence between $W$ and $Y$ – the corresponding DAG will have a directed edge from $W$ to $Y$. Ignorability requires essentially that any *other* paths from $W$ to $Y$ are blocked (i.e. controlled for, conditioned on). Which choices of $X$ achieve this? Three such criteria are identified below, stated without proof. An example of each is shown in Figure $fig:criteria$. Backdoor criterion ------------------ If a set of variables $X$ satisfy the following conditions: 1. $X$ blocks every path from $W$ to $Y$ that has an arrow into $W$ (blocks the back door), and 2. No node in $X$ is a descendant of $W$. then $X$ satisfies the backdoor criterion with respect to nodes $W$ and $Y$. Disjunctive cause criterion --------------------------- Sometimes simpler than using the backdoor criterion, which can involve analyzing the entire DAG is the disjunctive cause criterion. It is simply: - Control for all parents of the treatment variable, the effect variable (that are not descendants of the treatment), or both. Sometimes this is an easier set to identify than other (potentially smaller) sets that satisfy the backdoor criterion. Frontdoor criterion ------------------- If a set of variables $Z$ satisfy the following conditions: 1. $Z$ blocks all directed paths from $X_i$ to $X_j$, and 2. there is no backdoor path from $X_i$ to $Z$, and 3. all backdoor paths from $Z$ to $X_j$ are blocked by $X_i$ then $Z$ satisfies the frontdoor criterion with respect to nodes $X_i$ and $X_j$. ![Three criteria through which conditioning on $Z$ will render the effect of $X$ on $Y$ identifiable.](../../images/dags_criteria.svg) Some common methods =================== Once a set of variables to control for has been identified, how do we actually use this knowledge to identify causal effects? In theory, if we observe controls $X$ then we can measure the causal effect from: \begin{align}\tau = \mathbb{E}(\mathbb{E}(Y|W=1,X)-\mathbb{E}(Y|W=0,X)).\end{align} In practice however this requires a lot of data to get reliable estimates of each conditional expectation. In biomedical/social science settings this is often an issue. Generally each conditional expectation has to be estimated parametrically to capture the dependence on $X$. This introduces bias through choice of model, etc. Thus methods that can estimate causal effects without this modeling are attractive. A way of doing this is to effectively match the confound distribution $X$ between the control and treatment groups. Thereby making treatment independent of the covariates, and the data more like what is produced in a randomized control trial. This balancing of distributions among control and treatment groups is achieved through sampling subjects in different ways. Matching -------- The basic idea of matching is as follows. For each condition $W= 1$ and $W=0$ there are only a finite number of samples: \begin{align}\{y_i^{w=0}, x_i^{w=0}\}_{i=1}^{I_0} \text{ and } \{y_i^{w=1}, x_i^{w=1}\}_{i=1}^{I_1}.\end{align} Matching simply pairs one sample in the treatment group with one sample in the control group whose control covariates are close: \begin{align}(y_i^{w=0}, x_i^{w=0})\leftrightarrow (y_j^{w=1}, x_j^{w=1}), \quad x_i^{w=0}\sim x_j^{w=1}.\end{align} Since between treatment groups $X$ have roughly the same distribution, this dependence does not need to be modeled. This allows the above causal effect expectation to be approximated. Choices must be made about the metric that is used to decide when two points are similar. And choices must be made about how to deal with different treatment and control population sizes. One possibility is to discard all samples for which no match is made. Another possibility is to match one sample in the treatment group to more than one sample in the control group. A common way is to match on the treatment group. This then estimates what is known as the *causal effect of treatment on the treated*, often a quantity of interest. If we let $C(i)$ represent the sample index in the control population that is matched to sample $i$ in the treatment population then the causal effect is estimated from: \begin{align}\tau \approx \frac{1}{I_1}\sum_{i=1}^{I_1} y_i^{w=1} - y_{C(i)}^{w=0}.\end{align} Matching can be performed on all covariates, or just covariates that are identified as confounders, according to the backdoor or other criterion. Note that matching does not remove the need for ignorability – unmeasured confounders can still affect the analysis, thus $X$ still must satisfy the backdoor criteria. Propensity score matching ------------------------- Matching directly on controls $X$ can be difficult if $X$ is high-dimensional. Instead, we can match on what is called the propensity score, which is the probability of being treated given a set of controls: \begin{align}\pi(X) = P(W = 1| X).\end{align} Matching on $\pi(X)$ has the same effect as matching on $X$ directly. This is because subjects at the same propensity level have, by definition, the same probability of being assigned to the treatment group. Thus, for these subjects, treatment assignment is randomized (independent of $X$). In this way the distribution of $X$ in treatment and control groups are made to be the same. The propensity score is known, by definition, in randomized control trials. It has to be estimated in observational studies. But since it only involves observed data $X$ and $W$ this is straightforward. For example, one can use logistic regression. Again, propensity score matching still requires the ignorability assumption with controls $X$. Without it, even if the distribution of $X$ is balanced between control and treatment groups, unobserved confounders can still be different amongst control and treatment. Inverse probability of treatment weighting ------------------------------------------ Instead of matching on propensity score, which may discard some samples, we can simply reweight each subject by the inverse of its probability of receiving treatment – known as the inverse probability of treatment weighting (IPTW). This matches one unit in a treatment group with a certain number of ‘pseudo-units’ in the control group at a rate proportional to the relative probability of receiving treatment at a given level in $X$. In this way balance is achieved across levels. This is a type of importance sampling. Causal Bayesian networks {#sec:cbn} ======================== This is the framework developed most significantly by Pearl. A causal model is a Bayesian network along with a mechanism to determine how the model will respond to intervention. Now, rather than using the notion of potential outcomes and counterfactuals, causal effects are measured as the result of intervention. In addition to parents/children, we also think of the directed edges in the DAG as representing causal relationships, meaning a node’s parents and children are also its causes and effects. The *causal Markov condition* is the condition that all nodes are independent of their non-effects, given their direct causes. In the event that the structure of a Bayesian network accurately depicts causality, this is equivalent to the Markov condition. However, a network may accurately embody the Markov condition without depicting causality, in which case it should not be assumed to embody the causal Markov condition. Interventions and causal effects -------------------------------- An intervention on a single variable is denoted ${\text{do}}(X_i = y)$. Intervening on a variable removes the edges to that variable from its parents and forces the variable to take on a specific value: $$P(x_i|{\text{Pa}}_{X_i}=\mathbf{x_i}) = \delta(x_i = y)$$. The interventional joint distribution, $P_{X_i=y}$, is then defined as: \begin{align}P_{X_i=y}(\mathbf{x}) = \prod_{j\ne i}^N P(x_j | {\text{Pa}}_{X_j} = \mathbf{x}_j)\delta(x_i = y),\end{align} also abbreviated to $P_{X_i}$. Expectations under interventions then take the form: \begin{align}\mathbb{E}(X_j|{\text{do}}(X_i = y)) = \int x_j P_{X_i=y}(x_j)\,dx_j = \mathbb{E}_{X_i=y}(X_j).\end{align} Now, given the ability to intervene, the average causal effect between an outcome variable $X_j$ and a binary variable $X_i$ can be defined as: \begin{align}\tau = \mathbb{E}(X_j|{\text{do}}(X_i = 1)) - \mathbb{E}(X_j|{\text{do}}(X_i = 0)).\end{align} In general, the ‘do’ conditional is different to standard probabilistic conditioning. However criteria exist under which the interventional conditional distribution coincides with the probabilistic conditional distribution. The causal effect from node $X_i$ to $X_j$ can be inferred for conditional distributions that satisfy these criteria. These are actually the same criteria identified above in the counterfactual framework when searching for controls that provide ignorability. The interventional and counterfactual frameworks thus are compatible with one another. For instance, if $S_{ij}$ satisfy the backdoor criteria with respect to $X_i\to X_j$ then we can relate the interventional and observational expectations as follows: \begin{aligned} \nonumber \mathbb{E}(X_j|{\text{do}}(X_i = y)) &= \int x_j P_{X_i = y}(x_j)\,dx_j\\ \nonumber &= \int\int x_j P_{X_i = y}(x_j|\mathbf{s}_{ij})P_{X_i = y}(\mathbf{s}_{ij})\,dx_j d\mathbf{s}_{ij}\\ \nonumber &= \int\int x_j P(x_j|\mathbf{s}_{ij}, X_i = y)P(\mathbf{s}_{ij})\,dx_j d\mathbf{s}_{ij} \\ \label{eq:doce}&= \mathbb{E}\left(\mathbb{E}(X_j|\mathbf{S}_{ij}, X_i = y)\right),\end{aligned} from which a causal effect can be measured. Structural equation models ========================== The above frameworks are non-parametric, dealing simply with factorizations of joint distributions. The parametric form of a causal Bayesian network is the structural equation model (SEM). Each node is described by: \begin{align}X_j = f_j(\text{Pa}(X_j), \epsilon_j; \theta_j),\end{align} for some independent noise variable $\epsilon_j$, and parameters $\theta_j$. Note that the equality here is of a different nature to an algebraic equality. It conveys assignment rather than comparison. (Similar to the difference between = and == in programming languages.) Some authors use $\leftarrow$ instead of = to communicate this difference. This means that structural equation models have an invariance property that standard statistical models do not: the SEM is robust to intervention. The model should describe the data equally well regardless of whether it comes from observation or interventional experiments. Some further reading ==================== An overview of the counterfactual framework can be found in the short Coursera course. The interventionist framework of Pearl is described in his influential 2000 book. A more modern treatment, based on structural equation models, can be found in Peters et al 2017. - “A Crash Course in Causality: Inferring Causal Effects from Observational Data” Coursera course. By Jason Roy. [www.coursera.org/learn/crash-course-in-causality/](www.coursera.org/learn/crash-course-in-causality/) - “Causality: Models, Reasoning and Inference” Judea Pearl, 2000. - “Elements of Causal Inference: Foundations and Learning Algorithms” Jonas Peters, Dominik Janzing and Bernhard Schölkopf, 2017.

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# zbMATH — the first resource for mathematics

Existence and nonexistence of positive solutions for a higher-order three-point boundary value problem. (English) Zbl 07022524
Summary: This paper is concerned with the existence and nonexistence of positive solutions for a nonlinear higher-order three-point boundary value problem. The existence results are obtained by applying a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O’Regan.

##### MSC:
 34-XX Ordinary differential equations
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##### References:
 [1] Agarwal, R. P.; O’Regan, D.; Lakshmikantham, V., Singular $$\left(p, n - p\right)$$ focal and $$\left(n, p\right)$$ higher order boundary value problems, Nonlinear Analysis: Theory, Methods & Applications, 42, 2, 215-228, (2000) · Zbl 0977.34017 [2] Baxley, J. V.; Houmand, C. R., Nonlinear higher order boundary value problems with multiple positive solutions, Journal of Mathematical Analysis and Applications, 286, 2, 682-691, (2003) · Zbl 1046.34042 [3] Yang, B., Positive solutions for the $$\left(n, p\right)$$ boundary value problem, Electronic Journal of Qualitative Theory of Differential Equations, 31, 1-13, (2009) [4] Eloe, P. W.; Ahmad, B., Positive solutions of a nonlinear $$n$$th order boundary value problem with nonlocal conditions, Applied Mathematics Letters, 18, 5, 521-527, (2005) · Zbl 1074.34022 [5] Graef, J. R.; Moussaoui, T., A class of $$n$$th-order BVPs with nonlocal conditions, Computers & Mathematics with Applications, 58, 8, 1662-1671, (2009) · Zbl 1189.34033 [6] Hao, X.; Liu, L.; Wu, Y., Positive solutions for nonlinear $$n$$th-order singular nonlocal boundary value problems, Boundary Value Problems, 2007, (2007) · Zbl 1148.34015 [7] Graef, J. R.; Henderson, J.; Yang, B., Positive solutions of a nonlinear higher order boundary-value problem, Electronic Journal of Differential Equations, 2007, 45, 1-10, (2007) · Zbl 1117.34023 [8] Graef, J. R.; Henderson, J.; Wong, P. J. Y.; Yang, B., Three solutions of an $$n$$th order three-point focal type boundary value problem, Nonlinear Analysis: Theory, Methods & Applications, 69, 10, 3386-3404, (2008) · Zbl 1183.34027 [9] Zhang, X.; Feng, M.; Ge, W., Existence and nonexistence of positive solutions for a class of $$n$$th-order three-point boundary value problems in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 70, 2, 584-597, (2009) · Zbl 1176.34030 [10] Du, Z.; Liu, W.; Lin, X., Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations, Journal of Mathematical Analysis and Applications, 335, 2, 1207-1218, (2007) · Zbl 1133.34011 [11] Gao, Y., Existence and uniqueness of solutions for $$n$$th-order nonlinear three-point boundary value problems, Dynamics of Continuous, Discrete & Impulsive Systems A, 15, 2, 243-250, (2008) · Zbl 1148.34309 [12] Graef, J. R.; Qian, C.; Yang, B., A three point boundary value problem for nonlinear fourth order differential equations, Journal of Mathematical Analysis and Applications, 287, 1, 217-233, (2003) · Zbl 1054.34038 [13] Graef, J. R.; Yang, B., Positive solutions to a multi-point higher order boundary value problem, Journal of Mathematical Analysis and Applications, 316, 2, 409-421, (2006) · Zbl 1101.34004 [14] Ji, Y.; Guo, Y., The existence of countably many positive solutions for nonlinear $$n$$th-order three-point boundary value problems, Boundary Value Problems, 2009, (2009) · Zbl 1228.34041 [15] Karaca, I. Y., Positive solutions of an $$n$$th order three-point boundary value problem, The Rocky Mountain Journal of Mathematics, 43, 1, 205-224, (2013) · Zbl 1271.34031 [16] Liu, X.-J.; Jiang, W.-H.; Guo, Y.-P., Multi-point boundary value problems for higher order differential equations, Applied Mathematics E-Notes, 4, 106-113, (2004) · Zbl 1074.34013 [17] Liu, Y.; Ge, W., Positive solutions for $$\left(n - 1,1\right)$$ three-point boundary value problems with coefficient that changes sign, Journal of Mathematical Analysis and Applications, 282, 2, 816-825, (2003) · Zbl 1033.34031 [18] Palamides, P. K., Multi point boundary-value problems at resonance for $$n$$-order differential equations: positive and monotone solutions, Electronic Journal of Differential Equations, 2004, 25, 1-14, (2004) · Zbl 1066.34013 [19] Zhang, X.; Liu, L.; Zou, H., Positive solutions of fourth-order singular three point eigenvalue problems, Applied Mathematics and Computation, 189, 2, 1359-1367, (2007) · Zbl 1126.34321 [20] Agarwal, R. P.; O’Regan, D.; Wong, P. J. Y., Positive Solutions of Differential, Difference and Integral Equations, (1999), Boston, Mass, USA: Kluwer Academic, Boston, Mass, USA · Zbl 1157.34301 [21] Avery, R.; Henderson, J.; O’Regan, D., Functional compression-expansion fixed point theorem, Electronic Journal of Differential Equations, 2008, 22, 1-12, (2008) · Zbl 1149.47042 [22] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones. Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5, (1988), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0661.47045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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# Diagonal AC of a parallelogram ABCD bisects $$\angle{A}$$ (see figure). Show that (i) It bisects $$\angle{C}$$ also, (ii) ABCD is a rhombus.

Given, diagonal AC of a parallelogram ABCD bisects $$\angle{A}$$.

$$\therefore$$ $$\angle{DAC}$$ = $$\angle{BAC}$$ = $$\frac{1}{2}$$ $$\angle{BAD}$$ ...(i)
Here, AB || CD and AC is the transversal.
$$\angle{DCA}$$ = $$\angle{CAB}$$ ...(ii)(alternate angles)
$$\angle{BCA}$$ = $$\angle{DAC}$$ ...(iii)(alternate angles)

From eq. (i), (ii) and (iii), we get,
$$\angle{DAC}$$ = $$\angle{BAC}$$ = $$\angle{DCA}$$ = $$\angle{BCA}$$
Now, $$\angle{BCD}$$ = $$\angle{BCA}$$ + $$\angle{DCA}$$
$$\Rightarrow$$ $$\angle{BCD}$$= $$\angle{DAC}$$ + $$\angle{CAB}$$
$$\therefore$$ $$\angle{BCD}$$ = $$\angle{BAD}$$
Thus, we can say that, diagonal AC bisects $$\angle{C}$$.

Now, in $$\triangle{OAD}$$ and $$\triangle{OCD}$$, we have,
OA = OC
(since, diagonals bisects each other)
DO = OD ...(Common side)
$$\angle{AOD}$$ = $$\angle{COD}$$ = $$90^\circ$$
$$\therefore$$ $$\triangle{OAD}$$ $$\displaystyle \cong$$ $$\triangle{OCD}$$
(By SAS rule)
$$\therefore$$ AD = CD ...(By CPCT)

Now, AB = CD and AD = BC
(sides of parallelogram)
$$\therefore$$ AB = CD = AD = BC

Hence it is proved that, ABCD is a rhombus.


text:
# X^2 relation to 2x, 2x+1

1. Dec 4, 2015

### jaja1990

Is there a relation between the fact that the derivative of x^2 is 2x and that the difference between 1,4,9,16, ... is 3, 5, 7, 9, ...?

And why is the difference always 2?

2. Dec 4, 2015

### Samy_A

$(x+1)²-x²=2x+1$

This shows why the difference is always 2.

3. Dec 4, 2015

### PeroK

If you spotted that you should look for a pattern in the sequence $x^3: 1, 8, 27, 64 \dots$


text:
# 3D Mesh Plots in Julia

How to make 3D Mesh Plots

### Basic 3D Mesh example

The mesh3d trace type draws a 3D set of triangles with vertices given by x, y and z. If only coordinates are given, an algorithm such as Delaunay triangulation is used to draw the triangles. Otherwise the triangles can be given using the i, j and k parameters (see examples below).

using PlotlyJS, CSV, DataFrames, HTTP

df = CSV.File(
HTTP.get("https://raw.githubusercontent.com/plotly/datasets/master/mesh_dataset.txt").body;
delim=" ",
) |> DataFrame

plot(df, kind="mesh3d", x=:Column1, y=:Column2, z=:Column3, opacity=0.50)

### 3D Mesh example with Alphahull

The alphahull parameter sets the shape of the mesh. If the value is -1 (default value) then Delaunay triangulation is used. If >0 then the alpha-shape algorithm is used. If 0, the convex hull is represented (resulting in a convex body).

using PlotlyJS, CSV, DataFrames, HTTP

df = CSV.File(
HTTP.get("https://raw.githubusercontent.com/plotly/datasets/master/mesh_dataset.txt").body;
delim=" ",
) |> DataFrame

plot(df, x=:Column1, y=:Column2, z=:Column3,
kind="mesh3d",
alphahull=5,
opacity=0.4)

### Mesh Tetrahedron

In this example we use the i, j and k parameters to specify manually the geometry of the triangles of the mesh.

using PlotlyJS

plot(mesh3d(
x=[0, 1, 2, 0],
y=[0, 0, 1, 2],
z=[0, 2, 0, 1],
colorbar_title="z",
colorscale=[[0, "gold"],
[0.5, "mediumturquoise"],
[1, "magenta"]],
# Intensity of each vertex, which will be interpolated and color-coded
intensity=[0, 0.33, 0.66, 1],
# i, j and k give the vertices of triangles
# here we represent the 4 triangles of the tetrahedron surface
i=[0, 0, 0, 1],
j=[1, 2, 3, 2],
k=[2, 3, 1, 3],
name="y",
showscale=true
))

### Mesh Cube

using PlotlyJS

plot(mesh3d(
# 8 vertices of a cube
x=[0, 0, 1, 1, 0, 0, 1, 1],
y=[0, 1, 1, 0, 0, 1, 1, 0],
z=[0, 0, 0, 0, 1, 1, 1, 1],
colorbar_title="z",
colorscale=[[0, "gold"],
[0.5, "mediumturquoise"],
[1, "magenta"]],
# Intensity of each vertex, which will be interpolated and color-coded
intensity = 0:8,
# i, j and k give the vertices of triangles
i = [7, 0, 0, 0, 4, 4, 6, 6, 4, 0, 3, 2],
j = [3, 4, 1, 2, 5, 6, 5, 2, 0, 1, 6, 3],
k = [0, 7, 2, 3, 6, 7, 1, 1, 5, 5, 7, 6],
name="y",
showscale=true
))

### Intensity values defined on vertices or cells

The intensitymode attribute of mesh3d can be set to vertex (default mode, in which case intensity values are interpolated between values defined on vertices), or to cell (value of the whole cell, no interpolation). Note that the intensity parameter should have the same length as the number of vertices or cells, depending on the intensitymode.

Whereas the previous example used the default intensitymode='vertex', we plot here the same mesh with intensitymode='cell'.

using PlotlyJS

plot(mesh3d(
# 8 vertices of a cube
x=[0, 0, 1, 1, 0, 0, 1, 1],
y=[0, 1, 1, 0, 0, 1, 1, 0],
z=[0, 0, 0, 0, 1, 1, 1, 1],
colorbar_title="z",
colorscale=[[0, "gold"],
[0.5, "mediumturquoise"],
[1, "magenta"]],
# Intensity of each vertex, which will be interpolated and color-coded
intensity = 0:12,
intensitymode="cell",
# i, j and k give the vertices of triangles
i = [7, 0, 0, 0, 4, 4, 6, 6, 4, 0, 3, 2],
j = [3, 4, 1, 2, 5, 6, 5, 2, 0, 1, 6, 3],
k = [0, 7, 2, 3, 6, 7, 1, 1, 5, 5, 7, 6],
name="y",
showscale=true
))


text:
# spectroscopy: \#{} without non-breaking space

When using \#{} in the experimental environment of chemmacros it gives the number followed by the nucleus, but they are separated by a regular space, instead of a non-breaking space.

I don't know if this is intentional, but it can lead to the nucleus being printed in the next line, whereas it should be treated as a unit and therefore stick to the number.

Is there a way to redefine this macro by myself or will I have to manually type 2~H for now?

My MWE:

\documentclass{scrbook}

\usepackage{chemmacros}
\chemsetup{modules={spectroscopy}}

\begin{document}
\begin{experimental}
\NMR(300)[CD3COOD]
\val{9.87} (d, \J{1.2}, \#{1}),
\val{6.54} (d, \J{3.4}, \#{2}).
\end{experimental}
\end{document}

giving me (notice the H in the second line).

• the question would be much easier to answer if there was a MWE – David Carlisle Oct 12 '17 at 15:43

\# is defined in the spectroscopy module (the file chemmacros.module.spectroscopy.code.tex) to be equal to \chemmacros_nmr_number:n inside the experimental environment. This function is defined as

\cs_new_protected:Npn \chemmacros_nmr_number:n #1
{
\__chemmacros_nmr_number:n {#1}
\skip_horizontal:N \l__chemmacros_nmr_space_skip
\chemmacros_chemformula:V \g__chemmacros_nmr_element_tl
}

but probably should be defined like this:

\cs_new_protected:Npn \chemmacros_nmr_number:n #1
{
\__chemmacros_nmr_number:n {#1}
\chemmacros_skip_nobreak:N \l__chemmacros_nmr_space_skip
\chemmacros_chemformula:V \g__chemmacros_nmr_element_tl
}

\ExplSyntaxOn
\cs_set_protected:Npn \chemmacros_nmr_number:n #1
{
\__chemmacros_nmr_number:n {#1}
\chemmacros_skip_nobreak:N \l__chemmacros_nmr_space_skip
\chemmacros_chemformula:V \g__chemmacros_nmr_element_tl
}
\ExplSyntaxOff

after loading the spectroscopy module.

I'll change this in the package. The fix will be included starting with v5.8c.

• Perfect! I was actually just looking at that same snippet of code on GitHub trying to find an answer before posting my question, but I couldn't make any sense of it. :) – basseur Oct 12 '17 at 16:14


text:
Project

The last few weeks of the course will be devoted to a final project. You can work on the project in small groups (34 members). See below for due dates (or the main CourSys page for the course).

The project topic is your choice: we will be looking for you to demonstrate a good understanding of the big data tools and how they are used to analyze data. Some possible places where difficulty may occur, which will take into account when marking:

• Data analysis and difficulty of computation.
• Decent presentation, summarization, visualization of your results.
• Technical difficulty: new technologies/techniques that you had to deal with to complete the project.

The data analysis is definitely required. The others might be part of your project or not (but at least some should be there). Having some decent presentation of your results is recommended.

See the collected DataSets links for some possible inspiration (and if you have any data sets to add, post in the discussion forum). See also ProjectIdeas for some potential ideas.

Proposal

Before Monday October 25 2021, submit a brief proposal (1 page). Please indicate the proposed topic/analysis/product, as well as the technologies you intend to use. We will provide some feedback on your proposed project.

Create your group in CourSys (same group for all project-related stuff obviously) and submit to the CourSys activity Project Proposal

On Scale

I will point out two things on the subject of how big the project should be: (1) it is worth 25% of the final grade and each assignment is worth 7%, so the project should be around three assignments worth of work per person; (2) it is the final project for a 6-credit graduate-level course.

We will provide some feedback on scale along with your proposal.

Implementation

Implementation should use big data tools. This may include MapReduce or Spark, but not limited to those. You are welcome to explore new technologies and this will be considered to add to the difficulty of your project. See the The Big-Data Ecosystem Table for many amusing big-data technologies (and some not so big and others not even so much about data).

There is no requirement that your project process big data: many interesting data sets aren't that big. But your implementation should be scalable: if the data set grew to be big, it should be able to process it. Your project will be more interesting (and a better portfolio piece) if you have a good-sized data set to work on.

At the same time, please don't plan to do computations that will monopolize our (shared) cluster's resources unreasonably.

Source Control

You must use a Git repository for your project. The department's GitLab server is a good way to get one (instructions at that link). Group members must commit their own contributions to the repo. Please give the instructor and TAs (ggbaker, kaiyeec, bha59, vpa12, hjs10) developer access to your repository.

You are encouraged to publicize and open-source your work on GitHub or similar.

Computing Power

See ProjectTech and ProjectCluster.

Demos

You will be demonstrating your project in a short video, due with the rest of the project.

• A 510 minute video presentation.
• Submit a URL to the video: on your YouTube channel if you like, or a file in Google Drive or similar and we will post it on the PMP channel.
• No student numbers or other personal info in the video. Include your names if you want, but it's not required.
• It is not required for you to have a webcam or show your face in the video.
• It is not required to have all group members appear in the video: if your group is happy with the distribution of work, then it's fine to have one or two group members take care of the presentation.
• Make sure your audio is understandable and your on-screen content is visible. In particular, make sure your font size (in your slides, web browser, editor, terminal, etc) is large enough to be readable on screen: typically about twice the size of usual editor text. When you are recording, make sure you speak clearly and generally a little louder than you usually would sitting in front of your computer.

Final Implementation

The final implementation is due Friday December 10 2021. You will submit a tag from your repository (git tag final; git push --tags) to the CourSys activity Project.

In your repository, please include a file RUNNING.txt (or RUNNING.md if you prefer) indicating how we can actually test your project: commands on the cluster, input files, etc.

In your repository's README.md, you may include other notes about things we should look for, or be aware of when marking. If you created some kind of web frontend, please include a URL in the README.md as well.

Report

You will submit a report of at most 5 pages giving an overview of your project. (A little more is okay for bigger groups.)

• Problem definition: What is the problem that you are trying to solve? What are the challenges of this problem?
• Methodology: What is the problem that you are trying to solve? Briefly explain which tool(s)/technique(s) were used for which task and why you chose to implement that way.
• Problems: What problems did you encounter while attacking the problem? How did you solve them?
• Results: What are the outcomes of the project? What did you learn from the data analysis? What did you learn from the implementation?
• ProjectSummary: A summary of what you did to guide our marking.

This is also due Friday December 10 2021 and submitted to Project as a PDF or a URL to HTML (only one of those is necessary).

[Thanks to Jean-Pierre Lozi and Arash Vahdat who taught this course before me, and whose work I have based this and many assignment questions on.]

Updated Wed Sept. 08 2021, 14:52 by ggbaker.


text:
# AP Board 7th Class Maths Solutions Chapter 13 Area and Perimeter InText Questions

AP State Syllabus AP Board 7th Class Maths Solutions Chapter 13 Area and Perimeter InText Questions and Answers.

## AP State Syllabus 7th Class Maths Solutions 13th Lesson Area and Perimeter InText Questions

Try This

Question 1.
ABCD is a parallelogram with sides 8 cm and 6 cm. In Figure 1, what is the base of the parallelogram? What is the height? What is the area of the parallelogram? In Figure 2, what is the base of the parallelogram? What is the height? What is the area of the parallelogram? Is the area of Figure 1 and Figure 2 the same ? (Page No. 248)

Solution:
In fig (1)
Base of the parallelogram ABCD is 8cm.
Side of the parallelogram ABCD is 6cm.
Height of the parallelogram ABCD is 3cm.
Area of the parallelogram ABCD = base x height = 8×3 = 24cm2
In fig (2)
Base of the parallelogram = 6cm
Side of the.parallelogram = 8cm
Height of the parallelogram = 4cm
Area of the parallelogram = base x height = 6×4 = 24cm2
Yes. Areas of fig (1) & fig (2) are equal.

Do This

Question 1.
In parallelogram ABCD, AB = 10 cm and DE = 4cm. Find (i) the area of ABCD ii) the length of BF, if AD = 6cm (Page No. 248)
Solution:
Given AB = 10 cm, DE = 4 cm
Area of parallelogram ABCD = Base x Height
= 10 x 4 = 40 cm2
Also area of parallelogram ABCD = Base x Height
40 = 6 x BF
BF = $$\frac{40}{6}=\frac{20}{3}=6 \frac{2}{3}$$= 6.6cm

Do This

Question 1.
Carefully study the following parallelogram. (Page No. 248)

i) Find the area of the each parallelogram by counting the squares enclosed in it. For counting incomplete squares check whether two incomplete squares make a complete square in each parallelogram. Complete the following table accordingly.
Solution:

(ii) Do all parallelograms with equal base and equal heights have the same area ?
Solution:
Yes

Try This

Question

i) Why is the formula for finding the area of a rectangle related to the formula for finding the area of a parallelogram ? (Page No. 249)
Solution:
Any parallelogram can be modelled into a rectangle and hence, area of a parallelogram is related to the area of a rectangle.

ii) Explain why a rectangle is a parallelogram but a parallelogram may not be a rectangle.
Solution:
In a rectangle both pairs of opposite sides are parallel and hence it is a parallelogram. But each angle of a parallelogram is not always a right angle and hence it may not be a rectangle.

Try This

Question 1.
In the figure all triangles are on the base AB = 24 cm. Is the height of each of the triangles drawn on base AB, the same ? Will all the triangles have equal area ? Give reasons to support your answer. Are the triangles congruent also ? (Page No. 252)

Solution:
Heights of all triangles are equal, as the triangles are drawn between same parallels.
Areas of all triangles are equal, as all the triangles can be modelled as half of same parallelo¬gram. But triangles are not congruent.


text:
Infoscience

Journal article

# Determination of the branching fractions of B-s(0) -> D-s(-/+) K--/+ and B-0 -> Ds-K+

Measurements are presented of the branching fractions of the decays B-s(0) -> D-s(-/+) K--/+ and B-0 -> Ds-K+ relative to the decays B-s(0) -> D-s(-)pi(+) and B-0 -> D-s(-)pi(+), respectively. The data used correspond to an integrated luminosity of 3.0 fb(-1) of proton-proton collisions. The ratios of branching fractions are B(B-s(0) -> D-s(-/+) K--/+)/B(B-s(0) -> D-s(-)pi(+)) = 0.0752 +/- 0.0015 +/- 0.0019 and B(B-0 -> Ds-K+)/B(B-0 -> D-pi(+)) = 0.0129 +/- 0.0005 +/- 0.0008, where the uncertainties are statistical and systematic, respectively.


text:
# cyclotron class 12 notes

Energy levels of Hydrogen Atom. It is the class which decide your future. Carbon -14 and carbon -12 ions are obtained from a sample of the material to be dated and are accelerated in the cyclotron. We assure you that our solutions provide necessary guidelines to score good marks in the class . CBSE Class 12 Physics Notes. Free PDF download of Class 12 Physics revision notes & short key-notes for Chapter 4 - Moving Charges and Magnetism to score high marks in exams, prepared by expert Physics teachers from latest edition of CBSE (NCERT) books. 1) The frequency of oscillator of a cyclotron is 12MHz, and the radius of Ds is 70cm. CBSE Class 12 Notes for Physics, Chemistry, Maths and Biology subjects are provided in this article. Physics project on cyclotron class 12th.

CBSE Class 12 Notes for Physics, Chemistry, Maths and Biology subjects are provided in this article. An electron moving horizontally with a velocity of 4 x 10^4 enters a region of uniform magnetic field of 10^-5T , acting vertically downwards as shown in figure. (a) It will also be doubled. Every CBSE Class 12 student must be aware of the Physics syllabus in order to prepare an effective study plan and score well in the board examination. physics notes in hindi 12th Class; Biology notes in hindi class 12th; 11 ; 11; computer notes in hindi; ; B.SC (b sc) chemistry notes; 12th maths (mathematics) c . 4.4.2 Cyclotron. The topic of this project is "Optical Fiber and its Applications".

A uniform electric field and a uniform magnetic field exist in a region in the same direction. Answer: c. Cyclotron: It is a machine to accelerate charged particles or ions to . (a) E = 0, B 0. With decreasing temperatures from 300 K to 122 K, the peak intensity increased at first and then decreased. Friends, Here's a great Physics Investigatory Project for Class 12 students. To estimate a local average tune, Qz, the student notes the radial locations of . Introduce an important class of circular particle accelerators: Cyclotrons and Synchrocyclotrons Identify the key characteristics and performance of each type of cyclotron and discuss their primary applications Discuss the current status of an advance in both the science and engineering of these accelerators, including operation at Let's also learn more about light with ray optics and wave optics. Suppose a positive charge q is in motion in a uniform magnetic field B with velocity v . Uncharged particles cannot be accelerated by a cyclotron. It is used in hospitals to produce radioactive substances which can be used in diagnosis and treatment. Show that cyclotron frequency does not depend upon the speed of the particle? Diffraction. 8. a) 28MHz. When a strong magnetic field created by two large electromagnets applies to these charged particles, they accelerate and reach a very high speed due to the electromagnetic field. The maximum kinetic energy of the positive ion in the cyclotron is. The Cyclotron N [Class XII : Physics] 64 Nm-1 J magnetic field The potential energy associated with Full syllabus notes, lecture & questions for Irodov Solutions: Motion of Charged Particles In Electric And Magnetic Fields- 2 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE - JEE | Plus excerises question with solution to help you revise complete syllabus for I. E. Irodov Solutions for Physics Class 11 & Class 12 | Best notes, free PDF download An electron is projected with velocity pointed in the same direction.

This results in the production of high energy beam which is then used for artificial disintegration, etc. Q.7. electric field by crossing it again and again by use of strong magnetic field. CBSE Class 12th Physics Notes: Semiconductor Electronics. A cyclotron is a type of particle accelerator that accelerates charged particles using a strong magnetic field. At this instant, (a) the electric forces on both the particles cause identical accelerations. Uses of Cyclotron (i) The high energy particles produced in a Cyclotron are used to bombard nuclei. CBSE Class 12 Accountancy Deleted Syllabus Portion For 2020-21. CBSE Class 12th physics syllabus comprises a total of 9 units and 14 chapters and these units are divided into two terms. Revision notes Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. Class XII Physics 1 Revision Notes Class-12 Physics Chapter 4-Moving Charges And Magnetism 1. (c) energy is constant but momentum changes. Notesale is a site for students to buy and sell study notes online.

A proton of energy 100 eV is moving perpendicular to a magnetic field 10-4 T. Cyclotron Frequency Under the action of the given magnetic field (B fixed), the given positive ion (e/m fixed) will cover the semi-circular path in a fixed time only, if t in equation is equal to $\frac {T} {2}$Where T is the time period of the electric field. Cyclotron is a device used to accelerate charged particles to high energies. Along with Notes, we are also providing Online Quizzes for each Chapters of class 12 Physics and other subjects. Write the expression for the cyclotron frequency; A metal of length 17.01m is heated until it's temperature rises by 70^oc if it's new length is 18.02m .calculate it's linear expansivity? Biot - Savart law: The magnitude of Magnetic Field is proportional to the steady current I due to an element at a point P and inversely proportional to the distance r from the current element is, Magnetic field due to long straight current carrying conductor: It was devised by Lawrence.

Derive an expression for the electric field at a point on the axial position of an . Tweet.

Cyclotron. CBSE divided the whole syllabus into two terms and examination of both the terms will be conducted separately. Ask class: from the electromagnetic elds alone, does the particle energy change? system, k 1 = F=qBsin . (b) momentum as well as energy are constant. If the charges move, the electric as well as magnetic fields are produced. Be humble be teachable and always keep learning. The electron will. Raman Scattering. The core of 235 Uranium undergoes spontaneous fission into lighter fragments emitting two or three fission neutrons in the process These fission neutrons then interact with 235 U to produce the highly unstable 236 U which carries on the fission even in a self-sustaining nuclear chain reaction Materials can be lowered into ports in the reactor to be irradiated by the neutrons. Radius of circular path, r = mv / Bq Cyclotron frequency v = Bq / 2m Lorentz force between two charges q1 and q2 moving with velocity v1, v2 separated by distance r is given by. Also find the kinetic energy of electron beam accelerated by the cyclotron. The fitting results of the 32QL sample gives a total sheet carrier density 4.8 10 12 cm 2, effective transport mass 0.135 . Energy bands in conductors, semiconductors and insulators (qualitative ideas only) Intrinsic and extrinsic semiconductors- p and n-type, p-n junction. And gain the energy each time = q V. Ask your doubts for NIOS Board and CBSE through discussion forum. the energy spectrum would be a single line at the cyclotron frequency . Read also

Motional EMF, and .

Cyclotron cannot accelerate. The angular frequency of a cyclotron is independent of. () (p 1)/2 (12) implying a spectral index s = (p 1)/2.

Cyclotron resonance absorption peaks were observed even at 300 K and 118 T using light of 23.4 m wavelength as displayed in fig. DESCRIPTION & DESIGN 7 The cyclotron consists of two flat semicircular metallic boxes & called the Dees on account of their shape like the letter D. The two Dees are separated by a narrow parallel gap.

A cyclotron is used for accelerating positive ions so that they acquire energy large enough to carry out nuclear reactions. MCQ on Moving Coil Galvanometer. Answer. 71 likes 54,593 views. Get PDF download of Class 12 Physics Chapter 4 - Moving Charges and Magnetism Class 12 Notes, Video lectures and more made by expert teachers of Vidyakul to score high marks in your exams. CBSE Class 12 Physics Syllabus of chapter-14 Semiconductor Electronics 2022-23 pdf :-. The energy to which particles can be accelerated in a cyclotron is limited due to the variation of mass with velocity. Author notes. The topics which are included in class-12 physics chapter-14 Semiconductor Electronics are as follows :-. These are the revision notes . It consists of two D shaped metal boxes (about 21 in diameter) with their straight sides facing each other. Download CBSE Class 12 Physics Formulae Magnetic Effect Of Current And Magnetism Notes in PDF format. n F qBvsin F=kqBvsin k constant = Where in S.I. WF was the orbit stability . Physics Project for Class 12.

Cyclotron: A charge q executes a circular orbit in a plane normal to B with frequency called the cyclotron frequency given by, This cyclotron frequency is independent of the particle's speed and radius. Elements of a communication system (block diagram only); bandwidth of signals (speech, TV and digital data); bandwidth of transmission medium. Chapter Wise CBSE Class 12 English Flamingo Vistas Notes Summary of All Chapters in Hindi Pdf Download was designed by expert teachers from the latest edition of NCERT Textbook. Transcript. (iii) Neutrons being electrically neutral, cannot be accelerated in a Cyclotron. This peak disappeared completely at temperatures below 85 K. Instead of this peak, new absorption peaks . Our teachers have designed these concept notes for the benefit of Grade 12 students. Force on a moving charge: The source of the magnetic field is a moving charge. We attribute the sharp CR to two different topological surface states from both surfaces of the films.

1. Class 12th has a total of 15 chapters of Physics and the handwritten notes of all the chapter-wise are provided here and are carefully created by qualified teachers in simple language and in clear handwriting. F = qvB = mv2 R F = q v B = m v 2 R. R = mv qB R = m v q B. Angular velocity () = v R = qB m ( ) = v R = q B m. Time period of revolution T = 2 = 2m qB T = 2 = 2 m q B. Chapter-15: Communication Systems. To ensure the good marks to students we always stands behind them with all practical and theoretical support on Physics investigatory project. Uncharged particles cannot be accelerated by a cyclotron. system, k = 1 F=qBsin and F=q(v B) 2.

IIT JEE, Class 12 notes. . Artificial Intelligence syllabus for Class 10 Students by School Connect Online. The CBSE board has released the CBSE Class 12 Physics syllabus for the academic year 2022-2023 on their official website.

3. Solution: Given, f c = 12MHz, r = 0.7m, m e = 9.110 -31 kg, q = 1.610 -19 C The total energy of an electron in n th orbit of a hydrogen atom is given by En = (13.6/ n 2) eV. Energy bands in conductors, semiconductors and insulators (qualitative ideas only) Intrinsic and extrinsic semiconductors- p and n-type, p-n junction. Force on a moving charge:- The source of magnetic field is a moving charge. CBSE Class 12 Business Studies Syllabus 2020-21 (Revised): Download Pdf. Investigory project for class 12 board exam on the topic CYCLOTRON. b) 280MHz. = [1 - (v2 / c2)] m 0 The energy to which particles can be accelerated in a cyclotron is limited due to the variation of mass with velocity. If the cyclotron has a magnetic field of magnitude 2.40 T, what is the difference in cyclotron frequencies for the two ions? E.O Lawrence and M.S Livingston devised it in 1934 to examine the nuclear structure. Cyclotron Principle, Construction & Worki. Key topics presented in Optical Fibre - Class 12 Physics Investigatory Project Report: Write the expression for the cyclotron frequency; A metal of length 17.01m is heated until it's temperature rises by 70^oc if it's new length is 18.02m .calculate it's linear expansivity? Only charged particles like protons, deuterons and alpha particles can be accelerated by the cyclotron and electron . d) 28GHz. Q2.

.

Science. Interference of Light Waves and Young's Experiment. Principle Cyclotron works on the principle that a charged particle moving normal to a magnetic field experiences magnetic lorentz force due to which the particle moves in a circular path. Then, these particles are utilized in experiments where . August 5, 2021; Trending now Motional EMF in rotating rod Class 12.

Answer. The limitations of the cyclotron are listed below. Cyclotron is a device used to accelerate positively charged particles such as proton, deuteron etc. Finally, let's look at what happened in the last hundred years of physics by getting introduced to quantum mechanics with dual nature of . Cross fields are named such because both fields are perpendicular to each other. Jan. 14, 2017. Need for modulation, amplitude modulation. 4 Author to whom any correspondence should be . Then, these particles are utilized in experiments where . It is also used to implant ions into solids and modify their properties or even synthesise new materials. You will find step-by-step solutions for numerical value questions, answers for all logical reasoning questions and informative answers to all theoretical questions.

The cyclotron frequency of an electron grating in a magnetic field of 1 T is approximately _____.

Explain the principal and working of a cyclotron with the help of schematic diagram. BEAM PHYSICS DEMONSTRATIONS WITH THE RUTGERS 12-INCH CYCLOTRON* Timothy W. Koeth #, IREAP, University of Maryland, College Park, MD, 20742, U.S.A. . (b) turn to its left. Q1. Such notes will not be available to you anywhere else and after reading these notes you will not even need to go anywhere else.

1. Class 12 PCMB Notes have been prepared as per the latest CBSE Class 12 Syllabus.Important topics in each chapter, mark distribution, NCERT Solutions, Practicals, popular books and additional reference sources are also provided so that the students can have a clear understanding of the subject . Propagation of electromagnetic waves in the atmosphere, sky and space wave propagation, satellite communication. CBSE Class 12 Physics Syllabus of chapter-14 Semiconductor Electronics 2022-23 pdf :-. Description. Notesale: Turn your study into money. Page 1 Chapter wise Theoretical Important Questions in Physics for Class-XII Page 1 Electrostatics- 1. Download Offline Apps based on NCERT Books following the .

Cyclotron. It is now time to dive deeper into the world of physics with topics from class 12 (NCERT) such as electrostatics, electricity, magnetism, electromagnetic induction, and electromagnetic waves. Force between two parallel . The limitations of the cyclotron are listed below. 9,000 per year Select and buy Access Class 12 Physics Chapter 4 - Moving Charges and Magnetism Notes 1. CBSE Class 8 AI Syllabus. It is a device developed by Lawrence and Livingstone and is used to accelerate charged particles like protons and deuterons. The topics included are Electric Charges and Fields, Electrostatic Potential and Capacitance, Current Electricity, Moving Charges and Magnetism . 4.6 Magnetic Field on the Axis Of A Circular Current Loop. Force on a current-carrying conductor in a uniform magnetic field. Class 12 Physics Notes for Chapter 10: Wave Optics. (b) the magnetic forces on both the particles cause equal accelerations.

A charged particle is moving in a cyclotron, what effect on the radius of path of this charged particle will occur when the frequency of the ratio frequency field is doubled? The document Class 12 | Physics | Chapter Wise Derivation List Notes - Class 12 is a part of Class 12 category. (c) It will be increased by four times. (c) both particles gain or loose energy at the same rate. Suppose a positive charge q is in motion in a uniform magnetic field B with velocity v . Welcome to our class 12 Physics page. Download CBSE class 12 Physics study material in PDF format. this video explains Cyclotron working that is a charge accelerator working principle and construction an animation. [3] [4] A cyclotron accelerates charged particles outwards from the center of a flat cylindrical vacuum chamber along a spiral path. When a strong magnetic field created by two large electromagnets applies to these charged particles, they accelerate and reach a very high speed due to the electromagnetic field. Force between two parallel current-carrying conductors-definition of ampere. NCERT Solutions for Wave Optics.

A high Frequency of oscillator, which provides an alternating current is connected between the two Dees.

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c) 2.8 GHz. Explain the principal and working of a cyclotron with the help of schematic diagram. Xam Idea Class 12 Solutions for Physics are detailed and explanatory in a unique way. 4.5 Magnetic Field Due To a Current Element, Biot-Savart Law. 12th chemistry Notes. we observe prominent cyclotron resonances (CRs). As both the fields are perpendicular to each other, they are called cross fields.

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Interstellar space 10-12 4.2.3 Magnetic force on a current-carrying conductor We can extend the analysis for force due to magnetic field on a single moving charge to a straight rod carrying current. CBSE Class 12th Physics Notes: Ray Optics & Optical Instruments . (d) the motion of the centre of mass (CM) is determined by B alone. NCERT Solutions for Class 12 Physics Chapter 7 Alternating Current exercises are given here to use it online or download in PDF form free for new academic session 2022-23. (c) keep moving in the same direction but its speed will increase. A cyclotron is sometimes used for carbon dating. A cyclotron is an accelerator that produces radioactive isotopes that have therapeutic and medicinal values. . Moving Charges and Magnetism Class 12 Physics MCQs Pdf. . Principle of Cyclotron A positively charged particle can be accelerated through a modera. No Sub Class ISBN 978-3-95450-128-1 369 authors. A cyclotron is a type of particle accelerator that accelerates charged particles using a strong magnetic field.

Livingston invented it in 1934 to investigate the nuclear structure. View Answer. In this page you can find Physics Notes, assignments, concept maps and lots of study material for easy learning and understanding You can share the link with your classmates as all study material is free of cost. Class 12 PCMB Notes have been prepared as per the latest CBSE Class 12 Syllabus.Important topics in each chapter, mark distribution, NCERT Solutions, Practicals, popular books and additional reference sources are also provided so that the students can have a clear understanding of the subject .

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# Research Publications for 1996

## Books

A4. Phipps MC and Quine MP: A Primer of Statistics, Second Edition. Prentice Hall, NSW Australia, 1996, ISBN 0 7248 0946 5, iv + 141 pages.

## Chapters in Books

B1. Fackerell ED: Remarks on the Yilmaz and Alley papers, Proceedings of the First Australasian Conference on General Relativity and Gravitation. Australasian Conference on General Relativity and Gravitation, Institute for Theoretical Physics, University of Adelaide, Adelaide, 1996, ISBN 0 86396 430 3, 117–131.

B1. Henyey FS and Macaskill C: Sound through the internal wave field, Stochastic Modelling in Physical Oceanography. Progress in Probability, 39, Birkhäuser, Boston, 1996, ISBN 0-8176-3798-2, 141–184. MR96k:76106

B1. Hu BL and Matacz A: Quantum Noise in Gravitation and Cosmology, Fluctuations and Order: The New Synthesis. Springer, New York, 1996, ISBN 0-387-94602-0, 429–454. MR98c:83046

B1. Shannon AG, Loh RP, Melham RS and Horadam AF: A search for solutions of a functional equation, Applications of Fibonacci numbers. 6, Kluwer Academic Publishers, Dordrecht, 1996, 431–441. MR1393476

## Journal Articles

C1. Bennett MR, Farnell L and Gibson WG: Quantal transmitter release at somatic motor-nerve terminals: stochastic analysis of the subunit hypothesis. Biophysical Journal, 70 (1996), 654–668.

C1. Bennett MR, Farnell L and Gibson WG: Quantal transmitter release onto different patterns of receptor distributions at somatic neuromuscular junctions. Journal of Theoretical Biology, 183 (1996), 231–236.

C1. Bennett MR, Robinson J, Phipps MC, Karunanithi S, Lin YQ and Cottee L: Quantal components of spontaneous potentials at visualised varicosities. Journal of Autonomic Nervous System, 56 (1996), 161–174.

C1. Bloom SL, Sabadini N and Walters RFC: Matrices, machines and behaviors. Applied Categorical Structures, 4 (1996), 343–360. MR98a:68049

C1. Bolsinov AV, Dullin HR and Wittek A: Topology of energy surfaces and existence of transversal Poincaré sections. J. Phys. A,, 29 (1996), 4977–4985.

C1. Bosma W and Stevenhagen P: On the computation of quadratic 2-class groups. Journal de Théorie des Nombres, 8 (1996), 283–313. MR98e:11129a

C1. Bosma W and Stevenhagen P: Density computations for real quadratic units. Mathematics of Computation, 65 (1996), 1327–1337. MR96j:11171

C1. Brown G, Koumandos S and Wang KY: Positivity of Cotes numbers at more Jacobi abscissas. Monatshefte für Mathematik, 122 (1996), 9–19. MR97k:33006

C1. Brown G, Koumandos S and Wang KY: Positivity of more Jacobi polynomial sums. Mathematical Proceedings of the Cambridge Philosophical Society, 119 (1996), 681–694. MR97d:33002

C1. Brown SF: Formation of coronal loops with temperatures between $$10^5$$ and $$10^6$$ K. Astronomy and Astrophysics, 305 (1996), 649–659.

C1. Buchen PW and Ben Hador R: Free-mode surface wave computations. Geophysical Journal International, 124 (1996), 869–887.

C1. Buchen PW and Kelly MF: The maximum entropy distribution of an asset inferred from option prices. Journal of Financial and Quantitative Analysis, 31 (1996), 143–159.

C1. Cannon JJ and Playoust CE: Magma: a new computer algebra system. Euromath Bulletin, 2 (1996), 113–144. MR1413180

C1. Cao D, Dancer EN, Noussair ES and Yan S: On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete and Continuous Dynamical Systems, 2 (1996), 221–236. MR96m:35095

C1. Carboni A and Janelidze G: Decidable (=separable) objects and morphisms in lextensive categories. Journal of Pure and Applied Algebra, 110 (1996), 219–240. MR97d:18001

C1. Carboni A, Janelidze G and Magid AR: A note on Galois correspondence for commutative rings. Journal of Algebra, 183 (1996), 266–272. MR97g:13009

C1. Dancer EN: Some remarks on classical problems and fine properties of Sobolev spaces. Differential and Integral Equations, 9 (1996), 437–446. MR97e:35057

C1. Dancer EN: A counterexample on competing species equations. Differential and Integral Equations, 9 (1996), 239–246. MR97d:35220

C1. Dancer EN: Domain variation for certain sets of solutions and applications. Topological Methods in Nonlinear Analysis, 7 (1996), 95–113. MR97m:35079

C1. Dancer EN and Du Y: The generalized Conley index and multiple solutions of semilinear elliptic problems. Abstract and Applied Analysis, 1 (1996), 103–135. MR97i:35048

C1. Dancer EN and Toland JF: Real transformations with polynomial invariants. Journal of Geometry and Physics, 19 (1996), 99–122. MR97c:58120

C1. Daners D: Domain perturbation for linear and nonlinear parabolic equations. Journal of Differential Equations, 129 (1996), 358–402. MR97h:35071

C1. Dullin HR, Richter PH and Wittek A: A two parameter study of the extent of chaos in a billiard system. Chaos,, 6 (1996), 43–58.

C1. Easdown D and Gould V: Commutative orders. Proceedings of the Royal Society of Edinburgh, 126A (1996), 1201–1216.. MR97m:20069

C1. Easdown D and Gould V: Orders in power semigroups. Glasgow Mathematical Journal, 38 (1996), 39–47.. MR97a:20112

C1. Forbes LK, Gray BF and Sidhu HS: Periodically forced hydrocarbon oxidation in a CSTR. Bifurcation and Chaos, 6 (1996), 12.

C1. Fraser WB, Clark JD, Ghosh TK and Zeng Q: The effect of a control ring on the stability of the ring-spinning balloon. Royal Society of London. Proceedings, Series A. Mathematical and Physical Sciences, 452 (1996), 47–62.

C1. Gopalsamy K and He XZ: Dynamics of an almost periodic logistic integrodifferential equation. Methods in Applied Analysis, 2 (1996), 38–66.

C1. Graham JJ and Lehrer GI: Cellular algebras. Inventiones Mathematicae, 123 (1996), 1–34. MR97h:20016

C1. Gray BF, Forbes LK and Sidhu HS: A detailed study of the effects of forcing a hydrocarbon oxidation reaction. Mathematical Computation and Modelling, 24 (1996), no.8, 65–88.

C1. Gray BF and Jones JC: Futher evidence of anomolies in the self heating temperature histories of solid substrates. Journal of Fire Science, 14 (1996), 91–94.

C1. Gray BF, Mercer GN, Weber RO and Watt SD: Combustion pseudo-waves in a system with reactant consumption and heat loss. Mathematical Computation and Modelling, 24 (1996), no.8, 29–38. (Special issue: Nonlinear Dynamics of Combustion Systems) MR1422953

C1. Halburd RG and Joshi N: The coalescence limit of the second Painlevé equation. Studies in Applied Mathematics, 97 (1996), 1–15. MR1393931

C1. Hammer J: Orthogonal designs and generalised Fourier-Walsh transforms. Utilitas Mathematica, 50 (1996), 113–125.

C1. He XZ: Stability and delays in a predator-prey system. Journal of Mathematical Analysis and Applications, 198 (1996), 355–370. MR97a:92009

C1. Hillman JA: Embedding homology equivalent 3-manifolds in 4-space. Mathematische Zeitschrift, 223 (1996), 473–481. MR97h:57044

C1. Holst L, Quine MP and Robinson J: A general stochastic model for nucleation and linear growth. Annals of Applied Probability, 6 (1996), 903–921. MR97i:60093

C1. Hu H and Tholen W: A note on free regular and exact completions, and their infinitary generalizations. Theory and Applications of Categories, 2 (1996), 113–132. MR98e:18001

C1. Iltyakov AV: On rational invariants of the group $$E_6$$. Proceedings of the American Mathematical Society, 124 (1996), 3637–3640. MR97b:13008

C1. Iltyakov AV and Shestakov IP: On invariants of $$F_4$$ and the center of the Albert algebra. Journal of Algebra, 179 (1996), 838–851. MR96m:17058

C1. Ivanoff BG and Weber NC: Some characterisations of partial exchangeability. Journal of the Australian Mathematical Society, (Series A), 61 (1996), 345–359. MR97k:60099

C1. James GD and Mathas A: Hecke algebras of type A with q=-1. Journal of Algebra, 184 (1996), 102–158. MR97h:20017

C1. Joarder AH and Hossain MA: On the characteristic function of the multivariate Pearson type II distribution. Statistica, LVI 3 (1996), 313–319. MR1476909

C1. Joarder AH and Hossain MA: Estimation of the eigenvalues of the scale matrix of a class of elliptical distributions. Statistica Neerlandica, 56 (1996), no.3, 313–319. MR1476909

C1. John RD and Robinson J: Rates of convergence to normality for samples from a finite set of random variables. Journal of the Australian Mathematical Society, (Series A), 60 (1996), 355–362. MR97d:60040

C1. Johnstone PT: Remarks on quintessential and persistent localizations. Theory and Applications of Categories, 2 (1996), 90–99. MR97j:18004

C1. Joshi N and Petersen JA: Complex blow-up in Burgers' equation: an iterative approach. Bulletin of the Australian Mathematical Society, 54 (1996), 353–362. MR1419597

C1. Ku S and Seneta E: Quenouille-type theorem on autocorrelations. Annals of the Institute of Statistical Mathematics, 48 (1996), 621–630. MR98f:62259

C1. Kuchel PW, Lennon AJ and Durrant CJ: Analytical solutions and simulations for spin-echo measurements of diffusion of spins in a sphere with surface and bulk relaxation. Journal of Magnetic Resonance, Series B, 112 (1996), 1–17.

C1. Lavers TG: The monoid of ordered partitions of a natural number. Semigroup Forum, 53 (1996), 44–56. MR97b:20093

C1. Lehrer GI: A vanishing theorem for hyperplane cohomology. Bulletin of the Australian Mathematical Society, 53 (1996), 361–368. MR97d:52024

C1. Lehrer GI: The space of invariant functions on a Lie algebra. Transactions of the American Mathematical Society, 348 (1996), 31–50. MR96f:20070

C1. Ma C: Multivariate survival functions characterized by constant product of mean remaining lives and hazard rates. Metrika, 44 (1996), 71–83. MR98b:62183

C1. Ma C: A class of multivariate survival distributions characterized by minimum and marginal distributions. Biometrical Journal, 38 (1996), 647–652. MR98f:62025

C1. Macaskill C and Cao P: A new treatment of rough surface scattering. Royal Society of London. Proceedings, Series A. Mathematical and Physical Sciences, 452 (1996), 2593–2612. MR1427927

C1. Macaskill C and Ewart TE: Numerical solution of the fourth moment equation and comparison with the MATE data. Journal of the Acoustical Society of America, 99 (1996), 1419–1429.

C1. Macintosh AC and Gray BF: Self-heating of combustible vapour in porous material when the fibres are entirely covered by fluid. Combustion Science and Technology, 113–114 (1996), 503–515.

C1. Malin SRC and Winch DE: Annual variation of the geomagnetic field. Geophysical Journal International, 124 (1996), 170–174.

C1. Mathas A: On the left cell representations of Iwahori-Hecke algebras of finite Coxeter groups. Journal of Algebra, 54 (1996), 475–488. MR97h:20008

C1. Myerscough MR, Darwen MJ and Hogarth WL: Stability, persistence and structural stability in a classical predator-prey model. Ecological Modelling, 89 (1996), 31–42.

C1. Papadima S and Paunescu L: Reduced weighted complete intersection and derivations. Journal of Algebra, 183 (1996), 595–604. MR97e:13013

C1. Peiris MS: Improving the precision of forecasting. Microelectronics and Reliability, 36 (1996), 1375–1378.

C1. Peiris MS and Singha N: Predictors for seasonal and nonseasonal fractionally integrated arima models. Biometrical Journal, 38 (1996), 741–752. MR99c:62260

C1. Phipps MC and Robinson J: Non-randomness in time-ordered neurobiological data. Theory of Stochastic Processes, 2 (1996), 234–240.

C1. Poznanski RR and Peiris MS: Subthreshold response to white-noise current input in a tapering cable model of a neuron. IMA Journal of Mathematics Applied In Medicine and Biology, 13 (1996), 207–222.

C1. Quine MP and Law JS: Exact results for a secretary problem. Journal of Applied Probability, 33 (1996), 630–639. MR97k:60127

C1. Ramagge J: How to make life hell for the judges of the B.H. Neumann Prize. The Gazette of the Australian Mathematical Society, 23 (1996), 186–187.

C1. Ramagge J and Robertson G: Triangle Buildings and Actions of Type $$III_{1/q^2}$$. Journal of Functional Analysis, 140 (1996), 472–504.

C1. Rannard R: Incompressible surfaces in Seifert fibered spaces. Topology and its Applications, 72 (1996), 19–30. MR97g:57023

C1. Richter PH, Dullin HR, Waalkens H and Wiersig J: Spherical pendulum, actions and spin. J. Phys. Chem.,, 100 (1996), 19124–19135.

C1. Sabadini N, Vigna S and Walters RFC: A note on recursive functions. Mathematical Structures in Computer Science, 6 (1996), 127–139. MR97e:03061

C1. Seneta E: Markov and the birth of chain dependence. International Statistical Review, 64 (1996), 255–263.

C1. Seneta E and Chen TH: Frechet optimality of upper bivariate Bonferroni-type bounds. Theory of Probability and Mathematical Statistics, 52 (1996), 147–152. MR98a:60020

C1. Shneerson LM and Easdown D: Growth and existence of identities in a class of finitely presented inverse semigroups with zero. International Journal of Algebra and Computation, 6 (1996), 105–121. MR96m:20093

C1. Singh S, Joarder AH and King ML: Regression analysis using scrambled responses. The Australian Journal of Statistics, 38 (1996), 201–211. MR99c:62191

C1. Solomon AI: Catalan monoids, monoids of local endomorphisms and their presentations. Semigroup Forum, 53 (1996), 351–368. MR98c:20104

C1. Stump DM and Fraser WB: A simplified model of fabric drape based on ring theory. Textile Research Journal, 66 (1996), no.8, 506–514.

C1. Stump DM and Fraser WB: Transient solutions of the ring-spinning balloon equations. Journal of Applied Mechanics, 63 (1996), 523–528.

C1. Thavaneswaran A and Peiris MS: Nonparametric estimation for some nonlinear models. Statistics and Probability Letters, 28 (1996), 227–233. MR97e:62113

C1. Thomas CM, Gibson WG and Robinson J: Stability and bifurcations in an associative memory model. Neural Networks, 9 (1996), 53–66.

C1. Thompson MJ, Toomre J, Burtonclay D and Wilson PR: Differential Rotation and Dynamics of the Solar Interior. Science, 272 (1996), 1300–1305.

C1. Thompson RS and Aldis GK: Effect of a cylindrical refracting interface on ultrasound intensity and the CW Doppler spectrum. IEEE Transactions on Biomedical Engineering, 43 (1996), 451–459.

C1. Thompson RS and Wilson LS: The effect of variations in transducer position and sound speed in intravascular ultrasound: A theoretical study. Ultrasound in Medicine and Biology, 22 (1996), 719–734.

C1. van de Ven RJ and Weber NC: Percentile related bounds and estimator for the mean of the negative binomial distribution. Statistics, 27 (1996), 345–352. MR97k:60099

C1. Warren D and Seneta E: Peaks and eulerian numbers in a random sequence. Journal of Applied Probability, 33 (1996), 101–114. MR97b:60060

C1. Wilson PR: On the inference of the solar rotation profile from frequency splitting data. Bulletin of the Astronomical Society of India, 24 (1996), 251–260.

C1. Wilson PR, Burtonclay D and Li Y: On the inference of the solar internal rotation profile from frequency-splitting data. Astrophysical Journal, 470 (1996), 621–628.

C1. Wilson PR, Burtonclay D and Li Y: Calculations of the solar internal angular velocity for 1986-1990. Astrophysical Journal, 457 (1996), 440–446.

C3. Monro GP: Digital power - cheap. Sounds Australian, 14 (1996), 18–20. (This contribution to "Sounds Australian" was invited by the guest editor for the issue.)

C4. Phipps MC: The existence of an efficient estimator. Mathematical Scientist, 21 (1996), 129–131. MR97g:62036

## Reviews

D. Fraser WB: Ring spinning. Textile Horizons, April/May (1996), 37–39. (Review article contributed by invitation from the editor)

D. Gibson WG: Review of Neural Networks: An Introduction (second edition), by Muller, Reinhardt and Strickland. Australian and New Zealand Physicist, 33 (1996), 83–84.

## Conference Proceedings

E1. Charlton BG, Gray BF, Le BT, Jones JC and Foulds GA (1996) The use of a jet stirred CSTR to study the gas phase partial oxidation of methane to methanol. Fourth Natural Gas Conversion Symposium, Proceedings, Fourth Natural Gas Conversion Symposium, Elsevier, 00–00.

E1. Fraser WB (1996) A mathematical model of textile yarn ring-spinning. Australian Engineering Mathematics Conference 1996, Proceedings of the Second Biennial Australian Engineering Mathematics Conference, W.Y.D. Yuen, P. Broadbridge, J.M. Striner (ed.) , Institution of Engineers, Australia, ISBN 0 85825 653 3, 371–377.

E1. Fraser WB and Fackerell ED (1996) Modelling optical fibre cable. Mathematics-in-industry Study Group 1996, Proceedings of MISG96, John Hewitt (ed.) , MISG, ISBN 0 646 28979 9, 92–102.

E1. Gibson WG, Bennett MR and Farnell L (1996) Monte Carlo simulation of quantal transmission at purinergic junctions. Fourth Annual Computatinal Neuroscience Meeting, Monterey, CA, Computational Neuroscience: Trends in Research 1995, JM Bower (ed.) , Academic Press, 15–20. (Already reported as an unpublished conference paper (1995))

E1. Gibson WG, Bennett MR and Farnell L (1996) Synaptic conductances for neural networks. The Seventh Australian Conference on Neural Networks, ACNN96: Proceedings of the Seventh Australian Conference on Neural Networks, P. Bartlett, A. Burkitt, R.C. Williamson (ed.) , Australian National University, ISBN 0 7315 2429 2, 62–66.

E1. Gibson WG, Bennett MR and Robinson J (1996) Modeling the spontaneous release of quanta at active zones of varicosities, boutons and endplates. Fourth Annual Computational Neuroscience Meeting, Computational Neuroscience. Trends in Research 1995, JM Bower (ed.) , Academic Press, 9–14. (Aleady reported as an unpublished conference paper (1995))

E1. Joshi N and Vassiliou PJ (1996) Lie symmetries and linearizations of analytic discrete dynamical systems. Symmetries and Integrability of Difference Equations, Symmetries and Integrability of Difference Equations, Decio Levi, Luc Vinet and Pavel Winternitz (ed.) , CRM Proceedings & Lecture Notes, AMS, ISBN 0-8218-0601-7, 163–171. MR1416836

E1. Luckock HC (1996) Quantum cosmology, supersymmetry, and the problem of time. Australasian Conference on General Relativity and Gravitation, Proceedings of the First Australasian Conference on General Relativity and Gravitation, David L. Wiltshire (ed.) , Institute for Theoretical Physics, University of Adelaide, ISBN 0 86396 430 3, 175–190.

E1. Macaskill C and Cao P (1996) Numerical solution of the second kind integral equation for random rough surfaces with extreme RMS height. 3rd International Conference on Industrial and Applied Mathematics ICIAM 95, ZAMM Z.angew.Math.Mech. volume 76, Supplement 1: ICIAM/GAMM 95 Numerical Analysis, Scientific Computing, Computer Science, G.Alefeld, O.Mahrenholtz, R.Mennicken (ed.) , Akademie Verlag, ISBN 3-05-501761-7, 461–462. (This should REPLACE the 1995 entry under ICIAM 95 for Macaskill and Cao)

E1. Seneta E and Weber NC (1996) Time-series diagnostics and explosive processes. First Ukranian-Scandinavian conference on stochastic dynamical systems: theory and applications, Proceedings of the First Ukranian-Scandinavian conference on stochastic dynamical systems: theory and applications, Vladimir Korolyuk, Gunnar Kulldorff, Oleg Klesov, Dmitrii Silvestrov (ed.) , TBiMC Scientific Publishers MR1473291

E1. Tholen W (1996) Objects with dense diagonals. Workshop on Categorical Topology, Proceedings of the Workshop on Categorical Topology, Kluwer Academic Publishers, 213–220. MR1412586

E1. Thompson RS, Aldis GK and Wilson LS (1996) The effect on ultrasound intensity of refraction at a curved surface. 22nd International Symposium on Acoustical Imaging, Acoustical Imaging, P Tortoli, L Masotti (ed.) , Proceedings of the Twenty-Second International Symposium, Plenum Press, ISBN 0-306-45364-9, 371–376.

E1. Wen L, Chen Y and He XZ (1996) Asymptotic behavior of a class of fourth order nonlinear functional differential equation with delays. Differential Equations and Control Theory, Differential Equations and Control Theory, Zongqi Deng, Zhaojun Liang, Guang Lu and Shigui Ruan (ed.) , Marcel Dekker, ISBN 0-8247-9658-6, 371–376. MR96k:34156

E2. Edelman D and Gillespie T (1996) Improved volatility prediction using group analysis (empirical Bayes) methods. Workshop on Quantitative Methods in Finance, Financial Mathematics Research Report, E. Platen (ed.) , FMRR 008-96, The Australian National University, Volume 2, 345–354.

E2. Galloway DJ (1996) General Review of Dynamo Theory. Solar and Stellar Dynamos, Proceedings, third AFCOP meeting Solar and Stellar Dynamos'', J.-F. Donati (ed.) , Observatoire Midi-Pyrénées, 15–25.

E2. Galloway DJ (1996) Implications of Stellar Observations on Dynamo Theories. Solar and Stellar Dynamos, Proceedings, third AFCOP meeting Solar and Stellar Dynamos'', J.-F. Donati (ed.) , Observatoire Midi-Pyrénées, 117–122.

E2. Kelly MF and Buchen PW (1996) The simulation from option prices of joint distributions for asset prices, interest rates and volatility. ANU Workshop on Quantitative Methods in Finance, Financial Research Report, ANU Workshop on Quantitative Methods in Finance, Financial Research Report, FMRR-008-96, 329–344.

E2. Langton B and Fackerell ED (1996) Similarity reductions of the field equations for stationary axisymmetric rigidly rotating perfect fluids. Seventh Marcel Grossmann Meeting, The Seventh Marcel Grossmann Meeting, Robert T Jantzen, G. Mac Keiser (ed.) , World Scientific, ISBN 9810220642, 420–421.

E2. Luckock HC and Oliwa C (1996) Quantisation of Bianchi class A models in supergravity and the probability density function of the universe. The Seventh Marcel Grossman Meeting, The Seventh Marcel Grossman Meeting, Robert T Jantzen, G Mac Keiser and Remo Ruffini (ed.) , World Scientific Publishing Co. Pty Ltd, ISBN 9810220642, 1020–1022.

E3. Bennett MR, Farnell L, Gibson WG and Lavidis NA (1996) Synaptic transmission at visualized boutons on rat pelvic ganglion cells. Australian Neuroscience Society. Sixteenth Annual Meeting, Proceedings of the Australian Neuroscience Society, P. Pilowsky (ed.) , Australian Neuroscience Society, 86.

E3. Gibson WG (1996) Modelling some aspects of synaptic function in neurons. The 32nd Australasian Applied Mathematics Conference, ANZIAM'96 The 32nd Australasian Applied Mathematics Conference, G. Wake (ed.) , ANZIAM Conference Porceedings, Australian Mathematical Society: ANZIAM, ISBN 0725908696, (no pagination).

E3. Henery R, Gibson WG and Bennett MR (1996) Quantal currents and potential in the three-dimensional anisotropic bidomain of the vas deferens. Australian Neuroscience Society. Sixteenth Annual Meeting, Proceedings of the Australian Neuroscience Society, P. Pilowsky (ed.) , Australian Neuroscience Society, 148.

E3. Myerscough MR (1996) Bifurcations, structural stability and persistent populations—new insights into classical models with harvesting and stocking. The Nicholson Centenary Conference on Population Biology, Frontiers of Population Biology, R.B Floyd, A.W. Sheppard, P.J. de Barro (ed.) , CSIRO Publishing, ISBN 0 643 05781 1, 617–630.


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1 answer

# Kwak Motors preferred stock pays an annual dividend of $3.50 per share. If the stock is... ###### Question: Kwak Motors preferred stock pays an annual dividend of$3.50 per share. If the stock is selling for $50 per share, what would be the annual rate of return earned of Kwak Motors preferred stock? A) 1.43% B) 4.35% C) 7.00% D) 14.29% you expect that three years from today, periwinkle corporation will begin paying its first ever dividends of$4.00 a share, an amount that will be held constant. if you require a rate of return of 10%. what is the most that you should be willing to pay for a share of stock?

A) $35.05 B)$33.06 C) $37.68 D)$40.00

## Answers

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# If y=log[x+sqrt(x^2+a^2)] show that (x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0 - Mathematics

If y=log[x+sqrt(x^2+a^2)]  show that (x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0

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#### Solution

It is given that:

y=log[x+sqrt(x^2+a^2)]

Differentiating equation (1) with respect to x, we get

dy/dx=(1+x/sqrt(x^2+a^2))/(x+sqrt(x^2+a^2))

dy/dx=1/sqrt(x^2+a^2)..........(2)

xdy/dx=x/sqrt(x^2+a^2)...........(3)

Again differentiating equation (2) with respect to x, we get

(d^2y)/(dx^2)=-x/(x^2+a^2)^(3/2)

(x^2+y^2)(d^2y)/(dx^2)=-x/sqrt(x^2+a^2)..............(4)

Adding equation (3) and (4), we get

(x^2+y^2)(d^2y)/(dx^2)+xdy/dx=-x/sqrt(x^2+a^2)+x/sqrt(x^2+a^2)=0

(x^2+y^2)(d^2y)/(dx^2)+xdy/dx=0

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