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Geometric measure theory From Encyclopedia of Mathematics An area of analysis concerned with solving geometric problems via measure-theoretic techniques. The canonical motivating physical problem is probably that investigated experimentally by J. Plateau in the nineteenth century [a4]: Given a boundary wire, how does one find the (minimal) soap film which spans it? Slightly more mathematically: Given a boundary curve, find the surface of minimal area spanning it. (Cf. also Plateau problem.) The many different approaches to solving this problem have found utility in most areas of modern mathematics and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential equations, the calculus of variations, harmonic analysis, and fractals. Successes in the field include: classifying the structure of singularities in soap films (see [a18], together with the fine descriptive article [a3]); showing that the standard "double bubble" is the optimal shape for enclosing two prescribed volumes in space [a13], and developing powerful computer software for modelling the evolution of surfaces under the action of physical forces [a7]. The main reference text for the subject is [a12]. It is very densely written and [a15] serves as a useful guide through it; [a11] provides a comprehensive overview of the subject and contains a summary of its main results. For suitable introductions, see also [a17], which contains an introduction to the theory of varifolds and Allard's regularity theorem, and [a14], which includes information about tangent measures and their uses. For a slightly different slant, [a9] discusses applications of some of the ideas of geometric measure theory in the theory of Sobolev spaces and functions of bounded variation. Many variational problems (cf. also Variational calculus) are solved by enlarging the allowed class of solutions, showing that in this enlarged class a solution exists, and then showing that the solution possesses more regularity than an arbitrary element of the enlarged class. Much of the work in geometric measure theory has been directed towards placing this informal description on a formal footing appropriate for the study of surfaces. Rectifiability for sets. The key concept underlying the whole theory is that of rectifiability, a measure-theoretic notion of smoothness (cf. also Rectifiable curve). A set $E$ in Euclidean $n$-space ${\bf R} ^ { n }$ is (countably) $m$-rectifiable if there is a sequence of $C ^ { 1 }$ mappings, $f _ { i } : \mathbf{R} ^ { m } \rightarrow \mathbf{R} ^ { n }$, such that \begin{equation} \mathcal{H} ^ { m } \left( E \backslash \bigcup _ { i = 1 } ^ { \infty } f _ { i } ( \mathbf{R} ^ { m } ) \right) = 0. \end{equation} It is purely $m$-unrectifiable if for all $C ^ { 1 }$ mappings $f : {\bf R} ^ { m } \rightarrow {\bf R} ^ { n }$, \begin{equation} \mathcal{H} ^ { m } ( E \bigcap f ( \mathbf{R} ^ { m } ) ) = 0. \end{equation} (Here, $\mathcal{H} ^ { m }$ denotes the $m$-dimensional Hausdorff (outer) measure, defined by \begin{equation} \mathcal{H} ^ { m } ( E ) = \operatorname { sup } _ { \delta > 0 } \operatorname { inf } \left\{ c _ { m } \sum _ { i } \| E _ { i } \| ^ { m } : \quad \begin{array} { c } { E \subset \cup _ { i } E _ { i } } \\ { \| E _ { i } \| < \delta \text { for all } } \ i \end{array} \right\}, \end{equation} where $\|.\|$ denotes the diameter and the constant $c _ { m}$ is chosen so that, when $m = n$, Hausdorff measure is just the usual Lebesgue measure.) A basic decomposition theorem states that any set $E \subset {\bf R} ^ { n }$ of finite $m$-dimensional Hausdorff measure may be written as the union of an $m$-rectifiable set and a purely $m$-unrectifiable set, with the intersection necessarily having $\mathcal{H} ^ { m }$-measure zero. In practice, the definition of rectifiability is commonly used with Lipschitz mappings replacing $C ^ { 1 }$ mappings: it may be shown that this does not change anything, see [a14], Thm. 15.21. A standard example of a $1$-rectifiable set in the plane is a countable union of circles whose centres are dense in the unit square and with radii having a finite sum; the closure of the resulting set contains the unit square, and yet, as indicated below, the set itself still has "tangents" at $\mathcal{H} ^ { 1 }$-almost every point. An example of a purely $1$-unrectifiable set is given by taking the cross-product of the $1 / 4$-Cantor set with itself. (The $1 / 4$-Cantor set is formed by removing $2 ^ {k}$ intervals of diameter $4 ^ { - k }$, rather than $3 ^ { - k }$ as for the plain Cantor set, at each stage of its construction.) Approximate tangents. The main importance of the class of rectifiable sets is that it possesses many of the nice properties of the smooth surfaces which one is seeking to generalize. For example, although, in general, classical tangents may not exist (consider the circle example above), an $m$-rectifiable set will possess a unique approximate tangent at $\mathcal{H} ^ { m }$-almost every point: An $m$-dimensional linear subspace $V$ of ${\bf R} ^ { n }$ is an approximate $m$-tangent plane for $E$ at $x$ if \begin{equation} \operatorname { limsup } _ { r \rightarrow 0 } \frac { \mathcal{H} ^ { m } ( E \cap B ( x , r ) ) } { r ^ { m } } > 0 \end{equation} and for all $0 < s < 1$, \begin{equation} \operatorname { lim } _ { r \rightarrow 0 } \frac { \mathcal{H} ^ { m } \left( \left\{ y \in E \cap B ( x , r ) : \begin{array} { l } { \text { dist } ( y - x , V ) >} \\ {> s \| y - x \| }\end{array} \right\} \right) } { r^m } ) = 0. \end{equation} Conversely, if $E \subset {\bf R} ^ { n }$ has finite $\mathcal{H} ^ { m }$-measure and has an approximate $m$-tangent plane for $\mathcal{H} ^ { m }$-almost every $x \in E$, then $E$ is Besicovitch–Federer projection theorem. Often, one is faced with the task of showing that some set, which is a solution to the problem under investigation, is in fact rectifiable, and hence possesses some smoothness. A major concern in geometric measure theory is finding criteria which guarantee rectifiability. One of the most striking results in this direction is the Besicovitch–Federer projection theorem, which illustrates the stark difference between rectifiable and unrectifiable sets. A basic version of it states that if $E \subset {\bf R} ^ { n }$ is a purely $m$-unrectifiable set of finite $m$-dimensional Hausdorff measure, then for almost every orthogonal projection $P$ of ${\bf R} ^ { n }$ onto an $m$-dimensional linear subspace, $\mathcal{H} ^ { m } ( P ( E ) ) = 0$. (It is not particularly difficult to show that in contrast, $m$-rectifiable sets have projections of positive measure for almost every projection.) This deep result was first proved for $1$-unrectifiable sets in the plane by A.S. Besicovitch, and later extended to higher dimensions by H. Federer. Recently (1998), B. White [a19] has shown how the higher-dimensional version of this theorem follows via an inductive argument from the planar version. Rectifiability for measures. It is also possible (and useful) to define a notion of rectifiability for Radon (outer) measures: A Radon measure $\mu$ is said to be $m$-rectifiable if it is absolutely continuous (cf. also Absolute continuity) with respect to $m$-dimensional Hausdorff measure and there is an $m$-rectifiable set $E$ for which $\mu ( \mathbf{R} ^ { n } \backslash E ) = 0$. The complementary notion of a measure $\ mu$ being purely $m$-unrectifiable is defined by requiring that $\mu$ is singular with respect to all $m$-rectifiable measures (cf. also Mutually-singular measures). Thus, in particular, a set $E$ is $m$-rectifiable if and only if $\mathcal{H} ^ { m } \| _ { E }$ (the restriction of $\mathcal{H} ^ { m }$ to $E$) is $m$-rectifiable; this allows one to study rectifiable sets through $m$-rectifiable It is common in analysis to construct measures as solutions to equations, and one would like to be able to deduce something about the structure of these measures (for example, that they are rectifiable). Often, the only a priori information available is some limited metric information about the measure, perhaps how the mass of small balls grows with radius. Probably the strongest known result in this direction is Preiss' density theorem [a16] (see also [a14] for a lucid sketch of the proof). This states that if $\mu$ is a Radon measure on ${\bf R} ^ { n }$ for which $\operatorname { lim } _ { r \rightarrow 0 } \mu ( B ( x , r ) ) / r ^ { m }$ exists and is positive and finite for $\mu$-almost every $x$, then $\mu$ is $m$-rectifiable. Preiss' main tool in proving this result was the notion of tangent measures. A non-zero Radon measure $\nu$ is a tangent measure of $\mu$ at $x$ if there are sequences $r _ { i } \searrow 0$ and $c _ { i } > 0$ such that for all continuous real-valued functions with compact support, \begin{equation} \operatorname { lim } _ { i \rightarrow \infty } c _ { i } \int \phi \left( \frac { y - x } { r _ { i } } \right) d \mu ( y ) = \int \phi ( y ) d \nu. \end{equation} Thus, an $m$-rectifiable measure will, for almost-every point, have tangent measures which are multiples of $m$-dimensional Hausdorff measure restricted to the approximate tangent plane at that point; for unrectifiable measures, the set of tangent measures will usually be much richer. The utility of the notion lies in the fact that tangent measures often possess more regularity than the original measure, thus allowing a wider range of analytical techniques to be used upon them. A natural approach to solving a minimal surface problem would be to take a sequence of approximating sets whose areas are decreasing and finally extract a convergent subsequence with the hope that the limit would possess the required properties. Unfortunately, the usual notions of convergence for sets in Euclidean spaces are not suited to this. The theory of currents, introduced by G. de Rham and extensively developed by Federer and W.H. Fleming in [a10] (see [a11] for a comprehensive outline of the theory and [a12] for details), was developed as a way around this obstacle for oriented surfaces. In essence, currents are generalized surfaces, obtained by viewing an $m$-dimensional (oriented) surface as defining a continuous linear functional on the space of differential forms with compact support of degree $m$ (cf. also Current). Using the duality with differential forms, it is then possible to define many natural operations on currents. For example, the boundary of an $m$-current can be defined to be the $( m - 1 )$-current, $\partial S$, which is given via the exterior derivative for differential forms (cf. also Exterior algebra) by setting \begin{equation} \partial S ( \phi ) = S ( d \phi ) \end{equation} for a differential form $\phi$ of degree $( m - 1 )$. Of particular importance is the class of $m$-rectifiable currents: this class consists of the currents that can be written as \begin{equation} S ( \phi ) = \int \langle \xi ( x ) , \phi ( x ) \rangle \theta ( x ) d \mathcal{H} ^ { m } \| _ { R ( x ) }, \end{equation} where $R$ is an $m$-rectifiable set with $\mathcal{H} ^ { m } ( R ) < \infty$, $\theta ( x )$ is a positive integer-valued function with $\int \theta d \mathcal{H} ^ { m } \| _ { R } < \infty$ and $\ xi ( x )$ can be written as $v_{1} \wedge \ldots \wedge v _ { m }$ with $v _ { 1 } , \dots , v _ { m }$ forming an orthonormal basis for the approximate tangent space of $R$ at $x$ for $\mathcal{H} ^ { m }$-almost every $x \in R$. (That is, $\xi ( x )$ is a unit simple $m$-vector whose associated $m$-dimensional vector space is the approximate tangent space of $R$ at $x$ for $\mathcal{H} ^ { m } $-almost every $x \in R$.) The mass of a current given in this way is defined by ${\bf M} ( S ) = \int \theta ( x ) d {\cal H} ^ { m } \| _ { R ( x ) }$. If the boundary of an $m$-rectifiable current is itself an $( m - 1 )$-rectifiable current, then the $m$-current is said to be an integral current. These are the class of currents suitable for investigating Plateau's problem. The celebrated Federer–Fleming closure theorem says that on a not too wild compact domain (it should be a Lipschitz retract of some open neighbourhood of itself), those integral currents $S$ on the domain which all have the same boundary $T$, an $( m - 1 )$-current with finite mass, and for which ${\bf M} ( S )$ is bounded above by some constant $c$, form a compact set. (The topology is that generated by the integral flat distance, defined for $m$-integral currents $S _ { 1 }$, $S _ { 2 }$ by \begin{equation} \mathcal F _ { K } ( S _ { 1 } , S _ { 2 } ) = \operatorname { inf } \{ \mathbf M ( U ) + \mathbf M ( V ) : U + \partial V = S _ { 1 } - S _ { 2 } \}, \end{equation} where the infimum is over $U$ and $V$ such that $U$ is an $m$-rectifiable current on $K$ and $V$ is an $( m + 1 )$-rectifiable current on $K$.) In particular, if the constant $c$ is chosen large enough so that this set is non-empty, then one can deduce the existence of a mass-minimizing current with the given boundary $T$. The theory of currents is ideally suited for investigating oriented surfaces, but for unoriented surfaces problems arise. The theory of varifolds was initiated by F.J. Almgren and extensively developed by W.K. Allard [a1] (see also [a2] for a nice survey) as an alternative notion of surface which did not require an orientation. An $m$-varifold on an open subset $\Omega$ of ${\bf R} ^ { n }$ is a Radon measure on $\Omega \times G ( n , m )$. (Here, $G ( n , m )$ denotes the Grassmann manifold of $m$-dimensional linear subspaces of ${\bf R} ^ { n }$.) The space of $m$-varifolds is equipped with the weak topology given by saying that $\nu _ { i } \rightarrow \nu$ if and only if $\int f d \nu _ { i } \rightarrow \int f d \nu$ for all compactly supported, continuous real-valued functions on $\Omega \times G ( n , m )$. Given an $m$-varifold $\nu$, one associates a Radon measure on $\Omega$, $\\| \nu \\|$, by setting $\\| \nu \\| ( A ) = \nu ( A \times G ( n , m ) )$ for $A \ subset \Omega$. As a partial converse, to an $m$-rectifiable measure $\\| \mu \\|$ one can associate an $m$-rectifiable varifold $\mu$ by defining for $B \subset \Omega \times G ( n , m )$, \begin{equation} \mu ( B ) = \\| \mu \\| \{ x : ( x , T _ { x } ) \in B \}, \end{equation} where $T _ { x }$ is the approximate tangent plane at $x$. The first variation of an $m$-varifold $\nu$ is a mapping from the space of smooth compactly supported vector fields on $\Omega$ to $\mathbf {R}$, defined by \begin{equation} \delta \nu ( X ) = \int \langle X ( x ) , V \rangle d \nu ( x , V ). \end{equation} If $\delta \nu = 0$, then the varifold is said to be stationary. The idea is that the variation measures the rate of change in the "size" of the varifold if it is perturbed slightly. A key result in the theory of varifolds is Allard's regularity theorem, which states that stationary varifolds which satisfy a growth condition (detailed below) are supported on a smooth manifold. More precisely: For all $\epsilon \in ( 0,1 )$ there are constants $\delta > 0$, $C > 0$ such that whenever $a \in \mathbf{R} ^ { n }$, $0 < R < \infty$, and $\nu$ is an $m$-dimensional stationary varifold on the open ball $U ( a , R )$ with 1) $a \in \operatorname { spt } \nu$; 3) [a17] for some variants and a proof of this result. Given the success of the theory in Euclidean spaces, it is natural to ask whether a similar theory holds in more general spaces [a8]. There are many difficulties to be overcome, but [a5], [a6] suggest that it may be possible. [a1] W.K. Allard, "On the first variation of a varifold" Ann. of Math. , 95 (1972) pp. 417–491 MR0307015 Zbl 0252.49028 [a2] W.K. Allard, "Notes on the theory of varifolds. Théorie des variétés minimales et applications" Astérisque , 154/5 (1987) pp. 73–93 MR0955060 [a3] F.J. Almgren, Jr., J.E. Taylor, "The geometry of soap bubbles and soap films" Scientific Amer. , July (1976) pp. 82–93 [a4] F.J. Almgren, Jr., "Plateau's problem: An invitation to varifold geometry" , W.A. Benjamin (1966) MR190856 [a5] L. Ambrosio, B. Kirchheim, "Rectifiable sets in metric and Banach spaces" Math. Ann. (to appear) MR1800768 Zbl 0966.28002 [a6] L. Ambrosio, B. Kirchheim, "Currents in metric spaces" Acta Math. (to appear) MR1794185 Zbl 1222.49057 Zbl 0984.49025 [a7] K. Brakke, "The surface evolver V2.14" www.susqu.edu/facstaff/b/brakke/evolver/evolver.html (2000) [a8] G. David, S. Semmes, "Fractured fractals and broken dreams. Self-similar geometry through metric and measure" , Oxford Lecture Ser. in Math. Appl. , 7 , Clarendon Press (1997) MR1616732 Zbl [a9] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" , Stud. Adv. Math. , CRC (1992) MR1158660 Zbl 0804.28001 [a10] H. Federer, W.H. Fleming, "Normal and integral currents" Ann. of Math. , 72 : 2 (1960) pp. 458–520 MR0123260 Zbl 0187.31301 [a11] H. Federer, "Colloquium lectures on geometric measure theory" Bull. Amer. Math. Soc. , 84 : 3 (1978) pp. 291–338 MR0467473 Zbl 0392.49021 [a12] H. Federer, "Geometric measure theory" , Grundl. Math. Wissenschaft. , 153 , Springer (1969) MR0257325 Zbl 0176.00801 [a13] M. Hutchings, F. Morgan, M. Ritoré, A. Ros, "Proof of the double bubble conjecture" Preprint (2000) MR1777854 Zbl 0970.53009 [a14] P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability" , Stud. Adv. Math. , 44 , Cambridge Univ. Press (1995) MR1333890 Zbl 0819.28004 [a15] F. Morgan, "Geometric measure theory. A beginner's guide" , Acad. Press (1995) (Edition: Second) MR1326605 [a16] D. Preiss, "Geometry of measures in ${\bf R} ^ { n }$: distribution, rectifiability, and densities" Ann. of Math. (2) , 125 : 3 (1987) pp. 537–643 MR890162 [a17] L. Simon, "Lectures on geometric measure theory" , Proc. Centre Math. Anal. Austral. National Univ. , Centre Math. Anal. 3 Austral. National Univ., Canberra (1983) MR0756417 Zbl 0546.49019 [a18] J.E. Taylor, "The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces" Ann. of Math. (2) , 103 : 3 (1976) pp. 489–539 MR0428181 MR0428182 Zbl 0335.49032 [a19] B. White, "A new proof of Federer's structure theorem for $k$-dimensional subsets of $\mathbf{R} ^ { N }$" J. Amer. Math. Soc. , 11 : 3 (1998) pp. 693–701 How to Cite This Entry: Geometric measure theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometric_measure_theory&oldid=50781 This article was adapted from an original article by T.C. O'Neil (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article	{"url":"https://encyclopediaofmath.org/wiki/Geometric_measure_theory","timestamp":"2024-11-09T01:38:56Z","content_type":"text/html","content_length":"41458","record_id":"<urn:uuid:224cf767-2a7d-466d-a857-6d6bc4593d4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00485.warc.gz"}
R Archives - Data Cornering Category: R Here is the step-by-step process of creating a funnel chart in R with its versatile ggplot2 package. That allows us to craft visually appealing and informative funnel charts that can help you uncover insights, identify bottlenecks, and communicate the story of your data-driven processes. A funnel chart is a visualization tool that is particularly useful… One of the captivating features of ggplot2 is its ability to seamlessly merge data and design, creating visually impactful charts and graphs. While a uniform color scheme for data labels in ggplot2 stacked column charts might fall short, this post unveils a technique that not only introduces sophistication to your visualizations but also amplifies data… Here is a relatively simple way to create a bar chart race in R – bar chart with bars overtaking each other. An attractive data visualization to engage your auditory while showing how the situation unfolds. Using the ggplot2 and gganimate, you can create a bar chart race in R and adjust the dynamics of… Here are multiple examples with the pivot_longer from tidyr, which is an excellent choice if you want to unpivot data in R and transform the data frame from wide to long. Here is how to pivot data in R from long to wide format and increase the number of columns. This transformation might be familiar to Microsoft Excel users because of the PivoTable tool. It might not be the most commonly used data transformation, but sometimes necessary to show data in a small table or transform… Here is how to generate random dates or numbers in R by using base functions. There are similarities in both of the tasks, and they are useful in creating reproducible examples. Here is how to quickly build a heatmap in R ggplot2 and add extra formatting by using a color gradient, data labels, reordering, or custom grid lines. There might be a problem if the data contains missing values. At the end of this post is an example of how to deal with NA values in… Here are multiple examples of how to count by group in R using base, dplyr, and data table capabilities. Dplyr might be the first choice to count by the group because it is relatively easy to adjust to specific needs. Meanwhile data.table is good for speed, and base R sometimes is good enough. You can tell if the number is even or odd in R programming by looking at the reminder after the number is divided by 2. If the remainder equals 0, it is an even number, otherwise, it is an odd number. There is a nifty way to get the reminder after division in R by… If you want to keep trailing zeros in R, and in particular for text labels in ggplot2 geom_text, try functions like sprintf, formatC, or digits from the formattable package. Add trailing zeros in the R data frame, ggplot2, and keep numerical properties using the function digits from the formattable.	{"url":"https://datacornering.com/category/r/","timestamp":"2024-11-14T00:18:17Z","content_type":"text/html","content_length":"44082","record_id":"<urn:uuid:b061faa8-bc43-41af-a561-262a28fb3d66>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00177.warc.gz"}
Title: Clustering 1 • Unsupervised learning • Generating classes • Distance/similarity measures • Agglomerative methods • Divisive methods What is Clustering? • Form of unsupervised learning - no information from teacher • The process of partitioning a set of data into a set of meaningful (hopefully) sub-classes, called • Cluster • collection of data points that are similar to one another and collectively should be treated as • as a collection, are sufficiently different from other groups Characterizing Cluster Methods • Class - label applied by clustering algorithm • hard versus fuzzy • hard - either is or is not a member of cluster • fuzzy - member of cluster with probability • Distance (similarity) measure - value indicating how similar data points are • Deterministic versus stochastic • deterministic - same clusters produced every time • stochastic - different clusters may result • Hierarchical - points connected into clusters using a hierarchical structure Basic Clustering Methodology • Two approaches • Agglomerative pairs of items/clusters are successively linked to produce larger clusters • Divisive (partitioning) items are initially placed in one cluster and successively divided into separate groups Cluster Validity • One difficult question how good are the clusters produced by a particular algorithm? • Difficult to develop an objective measure • Some approaches • external assessment compare clustering to a priori clustering • internal assessment determine if clustering intrinsically appropriate for data • relative assessment compare one clustering methods results to another methods Basic Questions • Data preparation - getting/setting up data for • extraction • normalization • Similarity/Distance measure - how is the distance between points defined • Use of domain knowledge (prior knowledge) • can influence preparation, Similarity/Distance • Efficiency - how to construct clusters in a reasonable amount of time Distance/Similarity Measures • Key to grouping points • distance inverse of similarity • Often based on representation of objects as feature vectors Term Frequencies for Documents An Employee DB Which objects are more similar? Distance/Similarity Measures • Properties of measures • based on feature values xinstance,feature • for all objects xi,B, dist(xi, xj) ? 0, dist(xi, xj)dist(xj, xi) • for any object xi, dist(xi, xi) 0 • dist(xi, xj) ? dist(xi, xk) dist(xk, xj) • Manhattan distance • Euclidean distance Distance/Similarity Measures • Minkowski distance (p) • Mahalanobis distance • where ?-1 is covariance matrix of the patterns • More complex measures • Mutual Neighbor Distance (MND) - based on a count of number of neighbors Distance (Similarity) Matrix • Similarity (Distance) Matrix • based on the distance or similarity measure we can construct a symmetric matrix of distance (or similarity values) • (i, j) entry in the matrix is the distance (similarity) between items i and j Note that dij dji (i.e., the matrix is symmetric). So, we only need the lower triangle part of the matrix. The diagonal is all 1s (similarity) or all 0s (distance) Example Term Similarities in Documents Term-Term Similarity Matrix Similarity (Distance) Thresholds • A similarity (distance) threshold may be used to mark pairs that are sufficiently similar Using a threshold value of 10 in the previous Graph Representation • The similarity matrix can be visualized as an undirected graph • each item is represented by a node, and edges represent the fact that two items are similar (a one in the similarity threshold matrix) If no threshold is used, then matrix can be represented as a weighted graph Agglomerative Single-Link • Single-link connect all points together that are within a threshold distance • Algorithm • 1. place all points in a cluster • 2. pick a point to start a cluster • 3. for each point in current cluster • add all points within threshold not already in • repeat until no more items added to cluster • 4. remove points in current cluster from graph • 5. Repeat step 2 until no more points in graph All points except T7 end up in one cluster Agglomerative Complete-Link (Clique) • Complete-link (clique) all of the points in a cluster must be within the threshold distance • In the threshold distance matrix, a clique is a complete graph • Algorithms based on finding maximal cliques (once a point is chosen, pick the largest clique it is part of) • not an easy problem Different clusters possible based on where cliques start Hierarchical Methods • Based on some method of representing hierarchy of data points • One idea hierarchical dendogram (connects points based on similarity) Hierarchical Agglomerative • Compute distance matrix • Put each data point in its own cluster • Find most similar pair of clusters • merge pairs of clusters (show merger in • update proximity matrix • repeat until all patterns in one cluster Partitional Methods • Divide data points into a number of clusters • Difficult questions • how many clusters? • how to divide the points? • how to represent cluster? • Representing cluster often done in terms of centroid for cluster • centroid of cluster minimizes squared distance between the centroid and all points in cluster k-Means Clustering • 1. Choose k cluster centers (randomly pick k data points as center, or randomly distribute in • 2. Assign each pattern to the closest cluster • 3. Recompute the cluster centers using the current cluster memberships (moving centers may change memberships) • 4. If a convergence criterion is not met, goto step 2 • Convergence criterion • no reassignment of patterns • minimal change in cluster center k-Means Clustering k-Means Variations • What if too many/not enough clusters? • After some convergence • any cluster with too large a distance between members is split • any clusters too close together are combined • any cluster not corresponding to any points is • thresholds decided empirically An Incremental Clustering Algorithm • 1. Assign first data point to a cluster • 2. Consider next data point. Either assign data point to an existing cluster or create a new cluster. Assignment to cluster based on • 3. Repeat step 2 until all points are clustered • Useful for efficient clustering Clustering Summary • Unsupervised learning method • generation of classes • Based on similarity/distance measure • Manhattan, Euclidean, Minkowski, Mahalanobis, • distance matrix • threshold distance matrix • Hierarchical representation • hierarchical dendogram • Agglomerative methods • single link • complete link (clique) Clustering Summary • Partitional method • representing clusters • centroids and error • k-Means clustering • combining/splitting k-Means • Incremental clustering • one pass clustering User Comments (0)	{"url":"https://www.powershow.com/view4/719d2d-OWNkZ/Clustering_powerpoint_ppt_presentation","timestamp":"2024-11-08T21:48:54Z","content_type":"application/xhtml+xml","content_length":"158366","record_id":"<urn:uuid:719f209e-447a-435b-be99-e24fe9ab42d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00393.warc.gz"}
What is power of point and what is its use ... \| Filo Question asked by Filo student What is power of point and what is its use Not the question you're searching for? + Ask your question Video solutions (1) Learn from their 1-to-1 discussion with Filo tutors. 3 mins Uploaded on: 10/6/2023 Was this solution helpful? Found 3 tutors discussing this question Discuss this question LIVE 12 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Practice more questions on Coordinate Geometry View more Students who ask this question also asked View more Stuck on the question or explanation? Connect with our Mathematics tutors online and get step by step solution of this question. 231 students are taking LIVE classes Question Text What is power of point and what is its use Updated On Oct 6, 2023 Topic Coordinate Geometry Subject Mathematics Class Class 11 Answer Type Video solution: 1 Upvotes 52 Avg. Video Duration 3 min	{"url":"https://askfilo.com/user-question-answers-mathematics/what-is-power-of-point-and-what-is-its-use-35363437363939","timestamp":"2024-11-14T04:57:23Z","content_type":"text/html","content_length":"184020","record_id":"<urn:uuid:1cde72be-470a-46eb-9065-6ac70b8c7028>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00022.warc.gz"}
ThinkOR - Think Operations Research Simulation is a powerful tool in the hands of Operations Research practitioners. In this article I intend to demonstrate the usage of a discrete event process simulation, extending on the bottleneck analysis I wrote about previously A few days ago I wrote an demonstrating how you could use bottle neck analysis to compare two different configurations of the security screening process at London Gatwick Airport. Bottleneck analysis is a simple process analysis tool that sits in the toolbox of Operations Research practitioners. I showed that a resource-pooled, queue-merged process might screen as many as 20% more passengers per hour and that the poor as-is configuration was probably costing the system something like 10% of its potential capacity. previous article would be good to read before continuing, but to summarize briefly: Security screening happens in two steps, beginning with a check of the passenger's boarding pass followed by the x-ray machines. Four people checking boarding passes and 6 teams working x-ray machines were organized into 4 sub-systems with a checker in each system and one or two x-ray teams. The imbalance in each system was forcing a resource to be under utilised, and Dawen quite rightly pointed out that by joining the entire system together as a whole such that all 6 x-ray machines effectively served a queue fed by all 4 checkers, a more efficient result could be achieved. We will look at these two key scenarios, comparing the As-Is system with the What-If system. The bottleneck analysis was able to quantify the capacity that is being lost due to this inefficiency, but as I alluded, this was not the entire story. Another big impact of this is on passenger experience. That is, time spent waiting in queues in the system. In order to study queuing times, we turn to another Operations Research tool: Simulation, specifically Process-Driven Discrete Event Simulation. Note: There may be an opportunity to apply Queuing Theory, another Operations Research discipline, but we won't be doing that here today. Discrete Event Simulation Discrete Event Simulation is a computer simulation paradigm where a model is made of the real world process and the key focus is the entities (passengers) and resources (boarding pass checkers and x-ray teams) in the system. The focus is on discrete, indivisible things like people and machines. "Event" because the driving mechanism of the model is a list of events that are processed in chronological order, events that typically spawn new events to be scheduled. An alternative driving mechanism is with set timesteps as in system dynamics, continuous simulations. Using a DES model allows you to go beyond the simple mathematics of bottleneck analysis. By explicitly tracking individual passengers as they go through the process, important statistics can be collected like utilisation rates and waiting times. During my masters degree, the simulation tool at the heart of our simulation courses was Rockwell Automation , so I tend to go to it without even thinking. I have previously used Arena in my work for Vancouver Coastal Health, simulating Ultrasound departments and there are plenty of others associated with the Sauder School of Business using Arena. . Arena is an excellent tool and I've used it here for this artilce. I hope to test other products on this same problem in the future and publish a comparison. In the Arena GUI you put logical blocks together to build the simulation in the same way that you might build a process map. Intuitively, at the high level, an Arena simulation reads like a process map when in actuality the blocks are building SIMAN code that does the heavy lifting for you. The Simulation Here's a snapshot of the as-is model of the Gatwick screening process that I built for this article: Passengers decide to go through screening on the left, select the boarding pass checker with the shortest queue, are checked, proceed to the dedicated x-ray team(s) and eventually all end up in the departures hall. An X-Ray team is assumed to take a minute on average to screen each passenger. This is very different from taking exactly a minute to screen each passenger. Stochastic (random) processing times are an import source of dynamic complexity in queuing systems and without modelling that randomness you can make totally wrong conclusions. For our purposes we have assumed an exponentially distributed processing time with a mean of 1 minute. In practice we would grab our stop-watches and collect the data, but we would probably get arrested for doing that as an outsider. Suffice it to say that this is a very reasonable assumption and that exponential distributions are often used to express service times. As in the previous article, we were uncertain as to the relationship between throughput of boarding pass checkers and throughput of x-ray teams. We will consider three possibilities where processing time for the boarding pass checker is exponentially distributed with an average of: 60 seconds (S-slow), 40 seconds (M-medium), 30 seconds (F-fast) (These are alpha = 1, 1.5 and 2 from the previous article). In the fast F scenario, our bottleneck analysis says there should be no increased throughput What-If vs. As-Is because all x-ray machines are fully utilised in the As-Is system. In the slow S scenario there would similarly be no throughput benefit because all boarding pass checkers would be fully utilised in the As-Is system. Thus the medium M scenario is our focus, but our analysis may reveal some interesting results for F and S. We're focused here on system resources and configuration and how they determine throughput, but we can't forget about passenger arrivals. The number of passengers actually requiring screening is the most significant limitation on the throughput of the system. I fed the system with six passengers per minute, the capacity of the x-ray teams. This ensured both that the x-ray teams had the potential to be 100% utilised and that they were never overwhelmed. This ensured comparability of x-ray queuing time. I ran 28 (four weeks) replications of the simulation and let each replication run for 16 hours (working day). We need to run the simulation many times because of the stochastic element. Since the events are random, a different set of random outcomes will lead to a different result, so we must run many replications to study the possible results. Also note that I implemented a rule in the as-is system, that if more than 10 passengers were waiting for an x-ray team the boarding pass checker would stop processing passengers for them. Scenario M - Throughput Statistics First let's look at throughput. On average, over 16 hours the what-if system screened 18.9% more passengers than as-is. The statistics in the table are important. Stochastic simulations don't given a single, simple answer, but rather a range of possibilities described statistically. The average for 4 weeks is given in the table, but we can't be certain that would be the average over an entire year. The half width tell us our 90% confidence range. The actual average is probably between one half-width below the average and one above. Note: I would like to point out that this is almost exactly the result predicted analytically with the bottleneck analysis. We predicted that in this case the system was running at 83.3% capacity and here we show As-Is throughput is 4728.43/5621.57 of What-If throughput = 84.1%. The small discrepancy is probably due to random variation and the warm-up time from the simulation start. But what has happened to waiting times? The above graph is a cumulative frequency graph. It reads as follows: The what-if value for 2 minutes is 0.29. This means that 29% of passengers wait less than 2 minutes. The as-is value for 5 minutes is 0.65. This means that 65% of passengers wait less than 5 minutes. Comparing the two lines we can see that, while we have achieved higher throughput, customers will now have a higher waiting time. Management would have to consider this when making the change. Note that the waiting time increased because the load on the system also increased. What happens if we hold the load on the system constant? I adjusted the supply of passengers so that the throughput in both scenarios is the same, and re-ran the simulation: Now we can see a huge difference! Not only does the new configuration outperform the old in terms of throughput, it is significantly better for customer waiting times. What about our slow and fast scenarios? We know from our bottle-neck analysis that throughput will not increase, but what will happen to waiting times? Above is a comparison between as-is and what-if for the fast scenario. The boarding pass checkers are fast compared to the x-ray machines, so in both cases the x-ray machines are nearly overwhelmed and the waiting time is long. Why do the curves cross? The passengers that are fortunate enough to pick a checker with two x-ray machines behind them will experience better waiting times due to the pooling and the others experience worse. This is a bit subtle, but an interesting result. In this scenario there is no throughput benefit from changing, there is no average waiting time benefit from changing, but waiting times are less Finally, we can take a quick glance at our slow S scenario. We know again from our bottleneck analysis that there is no benefit to be had in terms of throughput, but what about waiting times? Clearly a huge differenence. The slow checkers are able to provide plenty of customers for the single x-ray teams, but are unable to keep the double teams busy. If you're unlucky you end up in a queue for a single x-ray machine, but if you're luck you are served immediately by one of the double teams. To an Operations Research practitioner with experience doing discrete event simulation, this example will seem a bit Mickey Mouse. However, it's an excellent and easily accessible demonstration of the benefits one can realize with this tool. A manager whose bottleneck analysis has determined that no large throughput increase could be achieved with a reconfiguration might change their mind after seeing this analysis. The second order benefits, improved customer waiting times, are substantial. In order to build the model for this article in a professional setting you would probably require Arena Basic Edition Plus, as I used the advanced feature of output to file that is not available in Basic. Arena Basic goes for $1,895 USD. You could easily accomplish what we have done today with much cheaper products, but it is not simple examples like this that demonstrate the power of products like Arena. Related articles:OR not at work: Gatwick Airport security screening (an observation and process map of the inefficiency) Security Screening: Bottleneck Analysis (a mathematical quantification of the inefficiency)	{"url":"http://www.thinkor.org/2010/05/","timestamp":"2024-11-07T10:40:05Z","content_type":"application/xhtml+xml","content_length":"114570","record_id":"<urn:uuid:4b2e7c05-61c3-4a8a-8a5e-03c170a006b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00448.warc.gz"}
Mathematicians Outwit a Hidden Number ‘Conspiracy’ - Tech News Today Intuition tells mathematicians that adding 2 to a number should completely change its multiplicative structure—meaning there should be no correlation between whether a number is prime (a multiplicative property) and whether the number two units away is prime (an additive property). Number theorists have found no evidence to suggest that such a correlation exists, but without a proof, they can’t exclude the possibility that one might emerge eventually. “For all we know, there could be this vast conspiracy that every time a number n decides to be prime, it has some secret agreement with its neighbor n + 2 saying you’re not allowed to be prime anymore,” said Tao. No one has come close to ruling out such a conspiracy. That’s why, in 1965, Sarvadaman Chowla formulated a slightly easier way to think about the relationship between nearby numbers. He wanted to show that whether an integer has an even or odd number of prime factors—a condition known as the “parity” of its number of prime factors—should not in any way bias the number of prime factors of its This statement is often understood in terms of the Liouville function, which assigns integers a value of −1 if they have an odd number of prime factors (like 12, which is equal to 2 × 2 × 3) and +1 if they have an even number (like 10, which is equal to 2 × 5). The conjecture predicts that there should be no correlation between the values that the Liouville function takes for consecutive Many state-of-the-art methods for studying prime numbers break down when it comes to measuring parity, which is precisely what Chowla’s conjecture is all about. Mathematicians hoped that by solving it, they’d develop ideas they could apply to problems like the twin primes conjecture. For years, though, it remained no more than that: a fanciful hope. Then, in 2015, everything changed. Dispersing Clusters Radziwiłł and Kaisa Matomäki of the University of Turku in Finland didn’t set out to solve the Chowla conjecture. Instead, they wanted to study the behavior of the Liouville function over short intervals. They already knew that, on average, the function is +1 half the time and −1 half the time. But it was still possible that its values might cluster, cropping up in long concentrations of either all +1s or all −1s. In 2015, Matomäki and Radziwiłł proved that those clusters almost never occur. Their work, published the following year, established that if you choose a random number and look at, say, its hundred or thousand nearest neighbors, roughly half have an even number of prime factors and half an odd number. “That was the big piece that was missing from the puzzle,” said Andrew Granville of the University of Montreal. “They made this unbelievable breakthrough that revolutionized the whole subject.” It was strong evidence that numbers aren’t complicit in a large-scale conspiracy—but the Chowla conjecture is about conspiracies at the finest level. That’s where Tao came in. Within months, he saw a way to build on Matomäki and Radziwiłł’s work to attack a version of the problem that’s easier to study, the logarithmic Chowla conjecture. In this formulation, smaller numbers are given larger weights so that they are just as likely to be sampled as larger integers. Tao had a vision for how a proof of the logarithmic Chowla conjecture might go. First, he would assume that the logarithmic Chowla conjecture is false—that there is in fact a conspiracy between the number of prime factors of consecutive integers. Then he’d try to demonstrate that such a conspiracy could be amplified: An exception to the Chowla conjecture would mean not just a conspiracy among consecutive integers, but a much larger conspiracy along entire swaths of the number line. He would then be able to take advantage of Radziwiłł and Matomäki’s earlier result, which had ruled out larger conspiracies of exactly this kind. A counterexample to the Chowla conjecture would imply a logical contradiction—meaning it could not exist, and the conjecture had to be true.	{"url":"https://technewstoday.club/mathematicians-outwit-a-hidden-number-conspiracy/","timestamp":"2024-11-14T05:35:16Z","content_type":"text/html","content_length":"323378","record_id":"<urn:uuid:92d58853-e77e-4fa3-856e-f3e9c4cedc46>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00702.warc.gz"}
Study Guide Copyright: The following courseware includes resources copyrighted and openly licensed by third parties under a Creative Commons Attribution 4.0 License. Click "Licenses and Attributions" at the bottom of each page for copyright information and license specific to the material on that page. If you believe that this courseware violates your copyright, please contact us. Mathematics for the Liberal Arts Topic K: Finance Course Overview K1.01: Simple and Compound Interest Table of Contents K1.02: Savings Plans Sample Syllabus K1.03: Installment Loans Module 1: General Problem Solving K1.04: Exercises Module 1 Overview Topic M: Power Functions and Using Logarithmic Graphs Discuss: Introduce Yourself to the Class M1.01: Introduction Problem Solving M1.02: Power Models Part I Test Page M1.03: Power Models Part II Supplemental Videos M1.04: Why We Use Logarithms Optional Discussion Boards M1.05: Finding Logarithms Discuss: Problem Solving Application M1.06: Logarithmic Graphs Part I Discuss: Math Editor Practice (Blackboard) M1.07: Logarithmic Graphs Part II Problem Solving Exercises M1.08: Exercises Module 2: Geometry Topic L: Automated Fitting of Models, Comparative Goodness of Fit, and Outliers Module 2 Overview L1.01: Overview Perimeter and Area L1.02: Section 1 L1.03: Section 2 L1.04: Section 3 Surface Area L1.05: Section 4 Graph Theory L1.06: Section 5 Supplemental Videos L1.07: Section 6 Topology, Tiling, and Non-Euclidean Geometry L1.08: Section 7 Discuss: Application of Geometry L1.09: Section 8 Module 3: Set Theory L1.10: Exercises Module 3 Overview Topic H: Linear Formulas - Word Problems Set Theory H1.01: Overview Supplemental Videos H1.02: Example 1 Verbal, Roster, and Set-Builder Notation for a Set H1.03: Example 2 Consider a Set H1.04: Example 3 Helpful Links H1.05: Example 3 Alternative Method Discuss: Application of Set Theory H1.06: Example 4 Module 4: Logic H1.07: Interpreting the Slope and Intercept Module 4 Overview H1.08: Exercises Topic I: Linear and Quadratic Models Truth Tables and Analyzing Arguments: Examples I1.01: Overview Truth Tables: Conjunction and Disjunction I1.02: Section 1 Truth Tables: Implication I1.03: Section 2 Analyzing Arguments with Truth Tables I1.04: Section 3 Part 1 Negating "all," "some," or "no" statements I1.05: Section 3 Part 2 Discuss: Truth Table Practice I1.06: Section 4 Discuss: Logic Application I1.07: Section 5 Part 1 Module 5: Numeration Systems I1.08: Section 5 Part 2 Module 5 Overview I1.09: Section 6 I1.10: Section 7 Supplemental Videos I1.11: Section 8 Binary, Octal, and Hexadecimal I1.12: Exercises Discuss: Numeration Application Topic J: Exponential Models and Model Comparison Techniques Module 6: Consumer Math J1.01: Overview Module 6 Overview J1.02: Section 1 Part 1 Consumer Math J1.03: Section 1 Part 2 Supplemental Videos J1.04: Section 1 Part 3 Average Daily Balance J1.05: Section 1 Part 4 How an Amortization Schedule is Calculated J1.06: Section 2 Part 1 Discuss: Consumer Math Application J1.07: Section 2 Example 6 Discuss: Final Reflection J1.08: Section 3 Growth Models J1.09: Section 4 Part 1 Linear (Algebraic) Growth J1.10: Section 4 Part 2 Exponential (Geometric) Growth J1.11: Exercises Solve Exponentials for Time: Logarithms Topic C: Communicating Precision of Approximate Numbers Logistic Growth C1.01 Overview C1.02: Reporting C1.03: Rounding Simple Interest C1.04: Precision Compound Interest C1.05: Interval Part 1 C1.06: Interval Part 2 Payout Annuities C1.07: Interval Part 3 C1.08: Interval Part 4 Remaining Loan Balance C1.09: Exercises Which Equation to Use? Topic E: Using a Spreadsheet Solving for Time E1.01: Overview E1.02: Section 2 Part 1 E1.03: Section 2 Part 2 E1.04: Section 2 Part 3 Union, Intersection, and Complement E1.05: Graphs Part 1 Venn Diagrams E1.06: Graphs Part 2 E1.07: Section 4 E1.08: Section 5 E1.09: Section 6 Part 1 Module Overview E1.10: Section 6 Part 2 Populations and Samples E1.11: Exercises Categorizing Data Topic O: Combining Modeling Formulas Sampling Methods O.01: Overview How to Mess Things Up Before You Start O.02: Section 1 Part 1 O.03: Section 1 Part 2 Describing Data O.04: Section 2 Understanding Normal Distribution O.05: Section 3 The Empirical Rule O.06: Section 4 Topic N: Additional Useful Modeling Formulas Regression and Correlation N1.01: Overview Supplemental Videos N1.02: Section 1 N1.03: Section 2 Part 1 Topic F: Using a Calculator N1.04: Section 2 Part 2 F1.01 Beginning & Example 3 N1.05: Section 2 Part 3 F1.02: Examples 4-5 N1.06: Section 3 Part 1 F1.03: Example 6 N1.07: Section 3 Part 2 F1.04: Exercises N1.08: Section 4 Describing Data N1.09: Section 5 Presenting Categorical Data Graphically N1.10: Exercises Presenting Quantitative Data Graphically Topic B: Solving Equations, Evaluating Expressions, and Checking Your Work Measures of Central Tendency B1.01: Introduction Measures of Variation B1.02: Section 1 B1.03: Section 2 Topic G: Linear Equations - Algebra and Spreadsheets B1.04: Section 3 G1.01: Intro & Slope B1.05: Section 4 G1.02: Intercepts & Example 3 B1.06: Exercises G1.03: Examples 4-7 Topic D: Formulas—Computing and Graphing G1.04: Examples 8-14 D1.01: Introduction G1.05: Exploring by Graphing Part I D1.02: Examples 2–5 G1.06: Exploring by Graphing Part II D1.03: Examples 6–9 G1.07: Section 4 D1.04: Examples 10–11 G1.08: Exercises D1.05: Exercises	{"url":"https://zt.symbolab.com/study-guides/atd-austincc-mathlibarts","timestamp":"2024-11-14T10:09:17Z","content_type":"text/html","content_length":"161819","record_id":"<urn:uuid:860849b2-e32c-42fb-a381-d96773a718bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00111.warc.gz"}
PressureIndependMultiYield Material PressureIndependMultiYield material is an elastic-plastic material in which plasticity exhibits only in the deviatoric stress-strain response. The volumetric stress-strain response is linear-elastic and is independent of the deviatoric response. This material is implemented to simulate monotonic or cyclic response of materials whose shear behavior is insensitive to the confinement change. Such materials include, for example, organic soils or clay under fast (undrained) loading conditions. During the application of gravity load (and static loads if any), material behavior is linear elastic. In the subsequent dynamic (fast) loading phase(s), the stress-strain response is elastic-plastic (see MATERIAL STAGE UPDATE below). Plasticity is formulated based on the multi-surface (nested surfaces) concept, with an associative flow rule. The yield surfaces are of the Von Mises type. The following information may be extracted for this material at a given integration point, using the OpenSees Element Recorder facility (McKenna and Fenves 2001)^®: "stress", "strain", "backbone", or For 2D problems, the stress output follows this order: σ[xx], σ[yy], σ[zz], σ[xy], η[r], where η[r] is the ratio between the shear (deviatoric) stress and peak shear strength at the current confinement (0<=η[r]<=1.0). The strain output follows this order: ε[xx], ε[yy], γ[xy]. For 3D problems, the stress output follows this order: σ[xx], σ[yy], σ[zz], σ[xy], σ[yz], σ[zx], η[r], and the strain output follows this order: ε[xx], ε[yy], ε[zz], γ[xy], γ[yz], γ[zx]. The "backbone" option records (secant) shear modulus reduction curves at one or more given confinements. The specific recorder command is as follows: recorder Element –ele $eleNum -file $fName -dT $deltaT material $GaussNum backbone $p1 <$p2 …> where p1, p2, … are the confinements at which modulus reduction curves are recorded. In the output file, corresponding to each given confinement there are two columns: shear strain γ and secant modulus G[s]. The number of rows equals the number of yield surfaces. nDmaterial PressureIndependMultiYield $tag $nd $rho $refShearModul $refBulkModul $cohesi $peakShearStra <$frictionAng=0. $refPress=100. $pressDependCoe=0. $noYieldSurf=20 <$r1 $Gs1 …> > $tag A positive integer uniquely identifying the material among all nDMaterials. $nd Number of dimensions, 2 for plane-strain, and 3 for 3D analysis. $rho Saturated soil mass density. $refShearModul Reference low-strain shear modulus, specified at a reference mean effective confining pressure refPress of p’[r] (see below). $refBulkModul Reference bulk modulus, specified at a reference mean effective confining pressure refPress of p’[r] (see below). $cohesi (c) Apparent cohesion at zero effective confinement. $peakShearStra An octahedral shear strain at which the maximum shear strength is reached, specified at a reference mean effective confining pressure refPress of p’[r] (see below). $frictionAng Friction angle at peak shear strength in degrees, optional (default is 0.0). $refPress (p’ Reference mean effective confining pressure at which Gr, Br, and γ[max] are defined, optional (default is 100. kPa). A positive constant defining variations of G and B as a function of instantaneous effective confinement p’(default is 0.0):: If Φ=0, d is reset to 0.0. $noYieldSurf Number of yield surfaces, optional (must be less than 40, default is 20). The surfaces are generated based on the hyperbolic relation defined in Note 2 below. Instead of automatic surfaces generation (Note 2), you can define yield surfaces directly based on desired shear modulus reduction curve. To do so, add a minus sign in front of noYieldSurf, then provide noYieldSurf pairs of shear strain (γ) and modulus ratio (G[s]) values. For example, to define 10 surfaces: $r, $Gs … -10γ[1]G[s1] … γ[10]G[s10] … See Note 3 below for some important notes. 1. The friction angle Φ and cohesion c define the variation of peak (octahedral) shear strength τ[f] as a function of current effective confinement p’[i]: 2. Automatic surface generation: at a constant confinement p’, the shear stress τ(octahedral) - shear strain γ (octahedral) nonlinearity is defined by a hyperbolic curve (backbone curve): where γ[r] satisfies the following equation at p’[r]: 3. (User defined surfaces) The user specified friction angle Φ = 0. cohesion c will be ignored. Instead, c is defined by c=sqrt(3)σ[m]/2, where σ[m] is the product of the last modulus and strain pair in the modulus reduction curve. Therefore, it is important to adjust the backbone curve so as to render an appropriate c. If the user specifies Φ > 0, this Φ will be ignored. Instead, Φis defined as follows: If the resulting Φ <0, we set Φ=0 and c=sqrt(3)σ[m]/2. Also remember that improper modulus reduction curves can result in strain softening response (negative tangent shear modulus), which is not allowed in the current model formulation. Finally, note that the backbone curve varies with confinement, although the variation is small within commonly interested confinement ranges. Backbone curves at different confinements can be obtained using the OpenSees element recorder facility (see OUTPUT INTERFACE above). For user convenience, a table is provided below as a quick reference for selecting parameter values. However, use of this table should be of great caution, and other information should be incorporated wherever possible. │ Parameters │ Soft Clay │ Medium Clay │ Stiff Clay │ │ │ 1.3 ton/m^3 or │ 1.5 ton/m^3 or │ 1.8 ton/m^3 or │ │ rho │ │ │ │ │ │ 1.217x10^-4 (lbf)(s^2)/in^4 │ 1.404x10^-4 (lbf)(s^2)/in^4 │ 1.685x10^-4 (lbf)(s^2)/in^4 │ │ │ 1.3x10^4 kPa or │ 6.0x10^4 kPa or │ 1.5x10^5 kPa or │ │ refShearModul │ │ │ │ │ │ 1.885x10^3 psi │ 8.702x10^4 psi │ 2.176x10^4 psi │ │ │ 6.5x10^4 kPa or │ 3.0x10^5 kPa or │ 7.5x10^5 kPa or │ │ refBulkModu │ │ │ │ │ │ 9.427x10^3 psi │ 4.351x10^4 psi │ 1.088x10^5 psi │ │ │ 18 kPa or │ 37 kPa or │ 75 kPa or │ │ cohesi │ │ │ │ │ │ 2.611 psi │ 5.366 psi │ 10.878 psi │ │ peakShearStra │ │ │ │ │ │ 0.1 │ 0.1 │ 0.1 │ │ (at p’[r]=80 kPa or 11.6 psi) │ │ │ │ │ frictionAng │ 0 │ 0 │ 0 │ │ pressDependCoe │ 0 │ 0 │ 0 │ Pressure Independent Material Examples: │ Material in elastic state │ │Example 1│Single 2D plane-strain quadrilateral element, subjected to sinusoidal base shaking │ │Example 2│Single 2D quadrilateral element, subjected to monotonic pushover (English units version) │ Code Developed by: UC San Diego (Dr. Zhaohui Yang): UC San Diego Soil Model: • NDMaterial Command □ UC San Diego soil models (Linear/Nonlinear, dry/drained/undrained soil response under general 2D/3D static/cyclic loading conditions (please visit UCSD for examples) □ UC San Diego Saturated Undrained soil	{"url":"https://opensees.ist.berkeley.edu/wiki/index.php?title=PressureIndependMultiYield_Material&oldid=9732","timestamp":"2024-11-02T11:55:05Z","content_type":"text/html","content_length":"39379","record_id":"<urn:uuid:f5cf4f4a-0f33-4ccc-bc3b-86a3bd0ad2a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00485.warc.gz"}
(20) Neuromythology - What Happens When You Violate Statistical Premises - Prof. Harald Walach (20) Neuromythology – What Happens When You Violate Statistical Premises A moment ago (this was first published in 2016), probably the biggest publicity bomb I have seen in a long time exploded: A group of Swedish authors, together with an English statistician, have published a huge simulation study. It shows that possibly up to 70% or more of the total of more than 40,000 published neuroscience studies that have used functional magnetic resonance spectroscopy (fMRI) have produced useless results and therefore actually belong to be cleaned out or replicated [1]. This seems to me to be one of the biggest scientific collective scandals of recent times. And one can learn a lot about statistics from it. But in order. Before we turn to this study: This is not to say that MRI methodology is wrong and that so-called structural imaging methods are useless. It is solely about statements about spatial spread of activity in functional magnetic imaging. But even that is a huge chunk. Follow me. What happened? Functional magnetic resonance spectroscopy or imaging (fMRI) is very popular as a research method. The technique is based on the fact that hydrogen atoms – which are found everywhere – can be aligned via strong external magnetic fields. By simultaneously applying and scanning electromagnetic high-frequency waves, the atoms can be localized. Depending on which frequency is chosen, one can also make different types of structures or molecules visible. This can be used, for example, to determine the difference between blood whose red blood cells are saturated with oxygen and that which has given up its oxygen. This so-called BOLD signal, short for „blood oxygenation level dependent signal“, can be used to deduce how high the metabolic activity is in a certain area of the body, e.g. in an area of the brain. An increase indicates increased oxygen consumption, increased blood supply, increased metabolism and thus increased activity in an area of the brain. A decrease indicates the opposite. Now, in order to see anything at all in a functional magnetic resonance imaging (i.e., imaging) study, one must of course create differences between experimental and control conditions. This is usually done by having people in the MRI tube do different tasks in a specific sequence, called blocks. For example, they have to read a text on a screen, or think of something specific, or recite a memorized poem in their mind; in another block lie down and relax instead. This happens in fixed sequences. Thus, one can compare the sequences in which something defined happens in the mind with those in which calmness reigns. The difference in the signals is then used to calculate the difference in the activation levels of the two conditions in specific areas of the brain and to make deductions about which areas of the brain are responsible for which functions. In addition, such conditions are often compared with situations in which control subjects are only measured („scanned“ is the neuro jargon) without anything To be clear, let me add: One can also use the method to visualize anatomical structures or to record the functionality of connections within the brain. These two applications are not covered by the study discussed here, but only the activation of brain areas as a result of activity change due to experimental instruction. Now the signals that arise from the measurement, it is easy to imagine even as a layman, have to run through a series of complex mathematical and statistical procedures before the pretty colourful pictures we admire in the publications and glossy brochures emerge at the end. In which experts then explain that the brain „lights up“ when a person does this or that. This „lighting up“ refers to the false colour representation of the increase or decrease of the BOLD signal in certain areas, which has been statistically isolated as a significant effect from the background noise. It is this statistical filtering procedure that then leads to the colouring – which is, after all, nothing more than the pictorial implementation of statistically significant signal detection – that was examined in this publication and found to be unreliable in the vast majority of cases. Why? This statistical filtering procedure is unreliable – why? Signal detection in an fMRI study is essentially a two-step process. The first step is to pick up the raw signals from the pulsed application of the magnetic fields and their deactivation, and to sample them with a high-frequency electromagnetic field. This provides the raw data about changes in the activity of the blood supply in the brain, i.e. about the oxygen saturation of the blood and the change in the distribution of the blood in the brain. Of course, as you can see immediately, this results in millions of data points that are determined in rapid succession and which, as such, are not usable in raw form. The second and crucial step is now the statistical discovery and summarization procedure. This is done by analysing the raw data with special programmes. The study discussed here examined the three most popular programmes. In order to understand how complex the whole thing is, one has to imagine that the fMRI signals are initially picked up at different points on the surface of the head and also originate from differently deep areas of the brain. We are therefore dealing with three-dimensional data points, which, analogously to the two-dimensional data points of a screen, where they are now known to all as „pixels“, are called voxels. Voxels are therefore three-dimensional pixels that originate from a defined location and vary in intensity. Since voxels cover just 1 cubic millimetre, the image that would emerge would be extremely confusing if one had to analyse them all individually. For this reason, one usually groups the voxels into larger areas. This is done by making assumptions about how the activity of neighbouring points relate to each other when a larger functional brain area, say the language centre in generating mental monologue, is activated. This happens via so-called autocorrelation functions of a spatial nature. We are all familiar with autocorrelation functions of a temporal nature: If the weather is very nice today, the probability that it will also be very nice tomorrow is higher than if it has already been nice for two weeks. Because then the probability that tomorrow will be worse is gradually higher, and vice versa. Analogous to such a temporal autocorrelation, one can also imagine a spatial one: Depending on how high the activity is at a point in the voxel universe, the probability that a neighbouring voxel belongs to a functional unit will be higher or lower. In the early days of programme development for the analysis of such data, relatively little information was available. So a reasonable, but as it now turns out wrong, assumption was made: namely, that the spatial autocorrelation function behaves as a spatially propagating Gaussian curve or normal distribution. Control data Now there are thousands of data sets of people measured by MRI scanners for control purposes, so to speak, without any tasks, and thanks to the possibility of open platforms, these data are made openly available to scientists. Anyone can download it and make analyses with it. Taking advantage of this opportunity, the scientists have recalculated data from nearly 500 healthy people from different regions of the world, measured in a scanner without any task, using simulated analysis methods by applying the three most popular analysis software packages to them. In total, they tested 192 combinations of possible settings in more than 3 million simulation calculations. So, somewhat simplistically, the scientists have pretended that the data from these 500 people came from real fMRI experiments with on and off blocks of specific tasks or questions. But it is clear that this was not the case because the data was control data. One would expect in such a procedure that a certain number of false positives would always be found, i.e. results where the statistics say: „Hurrah, we have found a significant effect“, but where in fact there is no effect. This so-called error of the first kind or alpha error is controlled by the nominal significance level, which can be set by convention and which is often 5% (p = 0.05), but in the case of fMRI studies is often set lower from the outset, namely at 1% (p = 0.01) or 0.1% (p = 0.001). This is because this alpha error indicates how often we make a mistake when we claim an effect, although there is none. At 5% level of alpha error, we make such an error 5 times out of 100. At a 1% level of alpha error in one out of 100 cases. And at a 1 per thousand level in one out of 1,000 cases. Now, of course, if we apply many statistical tests in parallel to the same set of data, this error multiplies because we get the same probability of making a mistake again in each test if we make a factual claim that is not in fact true. The nominal probability of error of p = 0.05, i.e. 5%, then becomes the probability of error of p = 0.1 or 10% for two simultaneous tests We therefore make twice as many errors. So, in order to comply with the nominal probability of 5%, the individual probabilities must be set to p = 0.025 for two simultaneous tests on the same data set, so that the joint error probability p = 0.05 remains. This is called „correction for multiple testing“. Because a large number of tests are made at once in the fMRI evaluation packages, one sets the detection threshold there for what one is prepared to accept as a significant signal right from the start at p = 0.01 (i.e. an error probability adjusted for 5 simultaneous tests) or even at p = 0.001. This is an error probability adjusted for 50 simultaneous tests and thus adheres to the nominal error level of 5% for 50 tests. This correction is already built into the software packages studied; thus, the problem found is not related to it. All these parameter settings were used in the study conducted here. At the same time, scenarios were run that are common practice in the real world of fMRI research, i.e. that one takes, for example, 8 mm clusters and merges the neighbouring voxels with a detection threshold of p = 0.001, which seems to be quite reasonable. Then, in complex simulation calculations, all sorts of putative experimental comparisons were superimposed on this control data, and it was documented how often the various software packages make significant „discoveries“, even though it is known that there are no signals hidden in the data at all. When clusters are formed, i.e. voxels are combined into larger areas, false positives, i.e. signals where there are none, are found in up to 50% of analyses. Or to put it another way: some software packages detect signals with a 50% probability of error where there are no signals at all. Put another way, in 50 out of 100 studies, the analysis says „there is a significant effect here“ where there is no effect at all. When the cluster size is smaller and the threshold for combining voxels into clusters is higher, the probability of error approaches the 5% nominal significance threshold. For voxel-based analysis, i.e. when one makes no assumptions about the correlation of voxel activities and accepts that one has to interpret a chaotic image of many voxels, the analysis remains close to the error probability of 5% for almost all software packages. And for the so-called non-parametric method, i.e. a statistical analysis based on a simulation calculation in which the probability is not derived from an underlying and assumed distribution, but from an actual simulation calculation based on the available data, the nominal significance values are always preserved. The problem is, however: The software packages are used because one does not want to do a laborious interpretation of a voxel-based evaluation oneself, but delegate it to the computer, and because one does not want to carry out weeks of simulation calculations to determine the true probability. In addition, signal noise or artefacts, such as those caused by movements, would be too much of a factor in a voxel-based evaluation. So one tries to find supposedly more robust quantities, precisely those clusters, which one then tests. For a very common scenario, the 8 mm clusters described above with an apparently conservative detection threshold of p = 0.001 from voxel to voxel before one is inclined to consider a cluster „significantly activated“ or „significantly inactivated“, the values look grim: the error frequency rises up to 90% depending on the program, and a 70% error probability across the literature is a robust estimate. Only a non-parametric simulation statistic would not make excessive errors here, either. However, this one is almost non-existent. Incidentally, the same problem was also found for active data from real studies. Here, too, a so-called inflation of the alpha error or a far too frequent detection of effects where there are none at all has been demonstrated. Where does the problem come from? You can use this example to study the importance of preconditions for the validity of statistics. First, the software packages and the users make assumptions about the interrelation of the voxels via spatial autocorrelation functions, as I described above. Users also choose the size of the areas to be studied, and the smoothing used in the process. These original assumptions were reasonable to begin with, but they were made at a time when there was relatively little data. No one checked them. Until now. And lo and behold, precisely this central assumption describing the mathematical relationship of neighbouring voxels was wrong. So: back to the books; modify software programs, implement new autocorrelation functions closer to empirical reality. And recalculate. Other assumptions have to do with assuming statistical distributions for the data. This is something that is done often. So the inference procedures involved are called „parametric statistics“ because you assume a known distribution for the data. You can normalize the known distribution. One then interprets the area under the curve as „1“. If you then plot a value somewhere on the axis and calculate the area behind it, you can interpret this area fraction of 1 as a probability. So, for example, more than 95% or less than 5% of the area lies behind the axis value „2“ (or „-2“) of the standard normal distribution. Because the area is normalized to „1“, this can then be interpreted as a probability. So you can calculate error probabilities from a known distribution. A common distribution assumption is that based on normal distribution, but there are plenty of other statistical distribution curves where you can then calculate the area fraction of a standardized curve in the same way and thus determine the probability. On the other hand, we rarely know whether these assumptions are correct. Therefore, as this analysis shows, a non-parametric procedure, i.e. one that makes no distributional assumption about the data, is actually wiser. The discussion about this is already very old and well-known, as are the procedures [2]. We have used them on various occasions, especially in critical situations [3,4]. If you use such simulation or non-parametric statistics properly, you actually have to use the empirically found data. You let the computer generate new data sets, say 10,000, that have similar characteristics, e.g. the same number of points with certain features, and then count how often the features that appear in the empirically found data also appear in the simulated data. If you then divide the number of features that occurred empirically by the number of features found by chance, you have the true probability that the empirical finding could have occurred by chance. Of course, such simulations, often called Monte-Carlo analyses (because Monte-Carlo is the big casino) – or non-parametric analyses – are very costly. Even modern, fast computers often need weeks to perform complex analyses. Anyway, you can see from this example what happens when you violate statistical assumptions: You can no longer interpret probability values based on assumptions and feed the results to the rabbits. In this specific case, a huge literature of neuromythology has emerged. More than half, perhaps as many as 70%, of the approximately 40,000 studies on fMRI methodology would actually have to be repeated or at least re-evaluated, the authors complain. If the data were publicly available, this would be feasible. Unfortunately, in most cases they are not. This is where the complaint of the neuroscientific community meets the call just made by psychologists for everything, but really everything, to be made publicly accessible, protocols, results, data [5]. The authors are calling for a moratorium: first do your homework, first work through the old problems, then do new studies. This will not work everywhere. Because in many cases study data was deleted after 5 years due to applicable laws. Now that’s beautifully silly, I think. One has to consider: Most major clinical units in hospitals and most major universities in Germany and the world maintain MRI scanners; the English Wikipedia estimates 25,000 scanners are in use worldwide. The problem with these devices is that once they have been put into operation, they are always connected to the power grid and thus generate high operating costs. You can’t simply switch them off like a computer, because that could damage the device, or switching them off and starting them up is itself a very complex and time-consuming process. That’s why these devices have to be kept in continuous use, so that their purchase, now worth several million euros, is worthwhile. That is why so many studies are done with them. Because whoever does studies pays for scanner time. No sooner does someone come up with an idea that seems reasonably clever – „let’s see which areas of the brain are active when you play music to people or show them pictures they don’t like“ – than he finds the money to get such a study funded, even in today’s climate. That brain research has a number of other problems has been noticed by others, as brain researcher Hasler points out in an easy-to-read article. And so it comes to pass that we have a huge stock, by now we must say, of storybooks about what can happen in the brain when Aunt Emma knits and little Jimmy memorizes nursery rhymes. Beautiful pictures, pretty narratives, all suggesting to us that the most important thing in the world of science at present is knowledge about what makes the brain tick. Except that, in the majority of cases, all these stories have little more value than the sagas of classical antiquity. The sagas of classical antiquity sometimes contain a kernel of truth and are at least exciting. Whether the kernel of truth of the published fMRI studies is greater than that of the sagas? Indeed: the colourful images of the fMRI studies are the baroque churches of postmodernity: beautiful, pictorial narratives of a questionable theology. Sources and literature 1. Eklund, A., Nichols, T. e., & Knutsson, H. (2016). Cluster failure: Why fMRI inferences for spatial extent have inflated false-positive rates. Proceedings of the National Academy of Science, early edition. Doi: https://doi.org/10.1073/pnas.1602413113. 2. Edgington, E. S. (1995, orig. 1987). Randomization Tests. 3rd Edition. New York: Dekker. 3. Wackermann, J., Seiter, C., Keibel, H., & Walach, H. (2003). Correlations between brain electrical activities of two spatially separated human subjects. Neuroscience Letters, 336, 60-64. 4. Schulte, D., & Walach, H. (2006). F.M. Alexander technique in the treatment of stuttering – A randomized single-case intervention study with ambulatory monitoring. Psychotherapy and Psychosomatics, 75, 190-191. 5. Open Science Collaboration. (2015). Estimating the reproducibility of psychological science. Science, 349(6251), aac4716.	{"url":"https://www.harald-walach.info/methodology-for-beginners/20-neuromythology-what-happens-when-you-violate-statistical-premises/","timestamp":"2024-11-06T11:05:25Z","content_type":"text/html","content_length":"99136","record_id":"<urn:uuid:ca458fff-3467-41f0-a976-c2242102fb19>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00238.warc.gz"}
EViews Help: Examples We demonstrate the use of the optimize command with several examples. To begin, we consider a regression problem using a workfile created with the following set of commands: wfcreate u 100 rndseed 1 series e = nrnd series x1 = 100rnd series x2 = 30nrnd series x3 = -4rnd group xs x1 x2 x3 series y = 3 + 2x1 + 4x2 + 5x3 + e equation eq1.ls y c x1 x2 x3 These commands create a workfile with 100 observations, and then generate some random data for series X1, X2 and X3, and E (where E is drawn from the standard normal distribution). The series Y is created as 3+2X1+4X2+5X3 + E. To establish a baseline set of results for comparison, we regress Y against a constant, X1, X2, and X3 using the built-in least squares method of the EViews equation object. The results view for the resulting equation EQ1 contains the regression output: Next we use the optimize command with the least squares method to estimate the coefficients in the regression problem. Running a program with the following commands produces the same results as the built-in regression estimator: subroutine leastsquares(series r, vector beta, series dep, group regs) r = dep - beta(1) - beta(2)regs(1) - beta(3)regs(2) - beta(4)regs(3) series LSresid vector(4) LSCoefs lscoefs = 1 optimize(ls=1, finalh=lshess) leastsquares(LSresid, lscoefs, y, xs) scalar sig = @sqrt(@sumsq(LSresid)/(@obs(LSresid)-@rows(LSCoefs))) vector LSSE = @sqrt(@getmaindiagonal(2sig^2@inverse(lshess))) We begin by defining the LEASTSQUARES subroutine which computes the regression residual series R, using the parameters given by the vector BETA, the dependent variable given by the series DEP, and the regressors provided by the group REGS. All of these objects are arguments of the subroutine which are passed in when the subroutine is called. Next, we declare the LSRESID series and a vector of coefficients, LSCOEFS, which we arbitrarily initialize at a value of 1 as starting values. The optimize command is called with the “ls” option to indicate that we wish to perform a least squares optimization. The “finalh” option is included so that we save the estimated Hessian matrix in the workfile for use in computing standard errors of the estimates. optimize will find the values of LSCOEFS that minimize the sum of squared values of LSRESID as computed using the LEASTSQUARES Once optimization is complete, LSCOEFS contains the point estimates of the coefficients. For least squares regression, the standard error of the regression The coefficients in LSCOEFS, standard error of the regression Alternately, we may use optimize to estimate the maximum likelihood estimates of the regression model coefficients. Under standard assumptions, an observation-based contribution to the log-likelihood for a regression with normal error terms is of the form: The following code obtains the maximum likelihood estimates for this model: subroutine loglike(series logl, vector beta, series dep, group regs) series r = dep - beta(1) - beta(2)regs(1) - beta(3)regs(2) - beta(4)regs(3) logl = @log((1/beta(5))@dnorm(r/beta(5))) series LL vector(5) MLCoefs MLCoefs = 1 MLCoefs(5) = 100 optimize(ml=1, finalh=mlhess, hess=numeric) loglike(LL, MLCoefs, y, xs) vector MLSE = @sqrt(@getmaindiagonal(-@inverse(mlhess))) scalar ubsig = mlcoefs(5)@sqrt(@obs(LL)/(@obs(LL) - @rows(MLCoefs) + 1)) %status = @optmessage statusline {%status} The subroutine LOGLIKE computes the regression residuals using the coefficients in the vector BETA, the dependent variable series given by DEP, and the regressors in the group REGS. Given R, the subroutine evaluates the individual log-likelihood contributions and puts the results in the argument series LOGL. The next lines declare the series LL to hold the likelihood contributions and the coefficient vector BETA to hold the controls. Note that the coefficient vector, BETA, has five elements instead of the four used in least-squares optimization, since we are simultaneously estimating the four regression coefficients and the error standard deviation We set the maximizer to perform a maximum likelihood based estimation using the “ml=” option and to store the OPG Hessian in the workfile in the sym objected MLHESS. The coefficient standard errors for the maximum likelihood estimates may be calculated as the square root of the main diagonal of the negative of the inverse of MLHESS. We store the estimated standard errors in the vector MLSE. Although the regression coefficient estimates match those in the baseline, the ML estimate of Note also that we use @optmessage to obtain the status of estimation, whether convergence was achieved and if so, how many iterations were required. The status is reported on the statusline after the optimize estimation is completed. The next example we provide shows the use of the “grads=” option. This example re-calculates the least-squares example above, but provides analytic gradients inside the subroutine. Note that for a linear least squares problem, the derivatives of the objective with respect to the coefficients are the regressors themselves (and a series of ones for the constant): subroutine leastsquareswithgrads(series r, vector beta, group grads, series dep, group regs) r = dep - beta(1) - beta(2)regs(1) - beta(3)regs(2) - beta(4)regs(3) grads(1) = 1 grads(2) = regs(1) grads(3) = regs(2) grads(4) = regs(3) series LSresid vector(4) LSCoefs lscoefs = 1 series grads1 series grads2 series grads3 series grads4 group grads grads1 grads2 grads3 grads4 optimize(ls=1, grads=3) leastsquareswithgrads(LSresid, lscoefs, grads, y, xs) Note that the series for the gradients, and the group containing those series, were declared prior to calling the optimize command, and that the subroutine fills in the values of the series inside the gradient group. Up to this point, our examples have involved the evaluation of series expressions. The optimizer does, however, work with other EViews commands. We could, for example, compute the least squares estimates using the optimizer to “solve” the normal equation subroutine local matrixsolve(vector rvec, vector beta, series dep, group regs) stom(regs, xmat) xmat = @hcat(@ones(100), xmat) stom(dep, yvec) rvec = @transpose(xmat)xmatbeta - @transpose(xmat)yvec rvec = @epow(rvec,2) vector(4) MSCoefs MSCoefs = 1 vector(4) rvec optimize(min=1) matrixsolve(rvec, mscoefs, y, xs) Since we will be using matrix manipulation for the objective function, the first few lines of the subroutine convert the input dependent variable series and regressor group into matrices. Note that the regressor group does not contain a constant term upon input, so we append a column of ones to the regression matrix XMAT, using the @hcat command. Lastly, we use the optimize command to find the minimum of a simply function of a single variable. We define a subroutine containing the quadratic form, and use the optimize command to find the value that minimizes the function: subroutine f(scalar !y, scalar !x) !y = 5!x^2 - 3*!x - 2 create u 1 scalar in = 0 scalar out = 0 optimize(min) f(out, in) This example first creates an empty workfile and declares two scalar objects, IN and OUT, for use by the optimizer. IN will be used as the parameter for optimization, and is given an arbitrary starting value of 0. The subroutine F calculates the simple quadratic formula: After running this program the value of IN will be 0.3, and the final value of OUT (evaluated at the optimal IN value) is -2.45. As a check we can manually calculate the minimal value of the function by taking derivatives with respect to X, setting equal to zero, and solving for X:	{"url":"https://help.eviews.com/content/optimize_cr-Examples.html","timestamp":"2024-11-05T02:43:04Z","content_type":"application/xhtml+xml","content_length":"23944","record_id":"<urn:uuid:ee5b625d-53e9-4a3d-a0f7-3480f0c42534>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00528.warc.gz"}
A wishlist for verifiable computation: An applied computer science perspective Michael Walfish (NYU) Okay, thanks. I want to begin by thanking the organizers, especially the tall guy for allowing me to be here. This talk will be talking about a research area where the goal is deployable systems for verifiable computation for probabilistic proofs like SNARKs etc. The goal of this talk is to motivate the next theoretical steps in this area. I want to convey the excitement and also provide a reality check. The high level setup is the following. We want to arrange for a client verifier to send a description of a computation F and some input X to a server or prover that returns the output Y with a short proof that establishes for the client that Y is the correct output given the input x. If F takes non-deterministic input, then the proof should establish that the prover knows a witness that is a setting of the non-deterministic input to the validity of the pair X, Y. The motivation of this setup is the general lack of trust in third-party computing. You can imagine a client wants assurance that a query performed on a remote database was done correctly. The area has several requirements, beginning with efficiency which decomposes first of all into the running time of the verifier. We want it to be constant or logarithmic in the running time of the computation F. Likewise for the communication cost for the number of rounds, that's the ideal. The server's running time should be linear or quasilinear in the number of steps required to execute F. A second requirement is that in some applications it will be desirable for the prover's auxiliary input (or the witness W) to be private from the verifier. Variants of this setup and also non-trivial subsets of this property have been achievable in theory, like interactive proofs, arguments of various kinds, zero knowledge arguments of various kinds, etc. There's an additional requirement. We want all of this to be practical. What I mean by practical may not be what cryptographers mean by practical. There has been good news and bad news. There has been a lot of attention to this area. On the left is every publication I am aware of that has reported on a probabilistic proof protocol. I have been trying to be comprehensive, so please let me know if there are others. This includes the work of Eli Ben Sasson, Eran Tromer, Alexandra Kiez, Daniel Genkin, Madars Virza, Bobescumonthu, Elaine Shi, Justin Thaler, Michael Mitzenmacher, Bryan P, and Craig Gentry as well. (Yes, all of these names are butchered.) SBW11, CMT12, SMBW12, TRMP12, SVPBBW12, SBVBPW13, VSBW13, PGHR13, Thaler13, BCGTV13, BFRSBW13, BFR13, DFKP13, BCTV14a, BCTV14b, BCGGMTV14, FL14, KPPSST14, FGV14, BBFR14, WSRHBW15, CFHKKNPZ15, CTV15 So the other excitement other than just the sheer amount of activity is that there's been genuinely cool results. There have been cost reductions of 20 orders of magnitude relative to naive implementations of the theory. Compilers in the sense of an implementation that go from programs in a high-level language that go to binaries that implement the protocols entities. Extensions of the machinery to broader expressiveness that work over remote state, and so on. And most of these systems have verifiers that are generally and concretely efficient. I also can't resist telling you about one of the papers done by my group appeared in SOSP (BFRSBW13), which is the premiere operating systems venue. This was a landmark or milestone for the area. It was also a milestone for SOSP. The system we sent there wasn't remotely practical. We were very upfront about this. This brings me to the bad news for this area. Just because something is implemented does not mean it's practical. All of the systems on this slide suffer from a limitation that they can only handle small computations and have outrageous expense. For the time being, this machinery is limited to carefully selected particular kinds of applications. I consider this to be bad news given the promise of the area. I'll describe the state of the art systems, then perform a reality check, then use these things to motivate the next theoretical steps in the area. Summarizing the state of hte art, I wish to mention that all of these systems be decomposed into frontends and backends. The frontend takes in a high-level description of a program such as in C, and outputs an arithmetic circuit over a large finite field. The backend is a probabilistic proof protocol by which the prover can convince the verifier that there is a satisfying assignment to that arithmetic circuit. It's the job of the frontend to produce the circuit in such a way such that the satisfyability of that circuit is tantamount to correct program execution. The probabilistic proof protocols include interactive arguments, non-interactive arguments like SNARKs, as well as interactive proofs. For lack of time, I am not going ot be able to describe in detail the refinements that have been done to Groth10 or GKR08 which was done by principally Justin Taylor. I will restrict focus to arguments. All of these argument systems have the quadratic arithmetic program QAPs (GGPR12). In order to explain the role of QAPs I am going to walk through a bunch of strawmen or bad ideas for probabilitsic proof protocols. So the first thing you can imagine doing is that if you're a graduate student taking a complexity theory class is you could say well PCPs are pretty awesome they allow us to check things, so what we're going to do is have the profer materialize an asymptotically efficient PCP and send it to the verifier; we know that PCPs have the property that if the computation was done correctly the verifier accepts and if the computation was not done correctly then the verifier is highly unlikely to accept the proof. And the verifiers work here in checking the proof in a number of location is work very low. The problem with this picture is that the prover has to encode the satisfying assignment in the proof, so the proof is actually larger than the satisfying assignment, which means on the basis of communication and running time of the verifier, we don't meet the efficiency requirement in the first slide. So the next thing you could imagine doing is saying well rather than have the verifier send the proof to the verifier, we could just have the prover materialize the proof and then have the verifier ask the proof what values the proof contains at a particular location, and this is the paradigm of Kilian's argument (Kilian92) and Micali's (Micali94) computational soundness proofs. This is actually an excellent idea in principle. The problem is that the constants that the tan of the constructions of short PCPs make this approach very difficult. Among other iissues, the underlying alphabet is not a great fit for the underlying arithmetic circuit model of computation. Eran Tromer who we will be speaking next can probably talk more about this, since I know that he and Eli Ben Sasson actually implemented this stuff although they haven't actually reported on experimental results. I don't know precisely what issues they ran into, but I think this is it. The next thing you could imagine doing is to turn to a really cool idea from IKO07, SMBW12, SVPBBW12 who may be in this room, or maybe all of them. And the observation is the following. There are 3 pieces to this. The first is that the so called "simple" PCPs were bad idea PCPs that appeared in the first half of the PCP theorem in the ALS paper. These constructions are actually very simple. They are quite straight forward to implement. The second observation is that even though these proofs would rely if they were materialized in full, they would require the construction of an exponentially large table, there's actually a succint representation of these proofs as a function in a linear function that takes this input as a vector and returns this dot product of a vector V, and V encodes the satisfying assignment. The final observation of IKO was that it's possible to get the prover to commit to a function of this form. Then the verifier asks the prover what values the proof contains at a particular location; it's not just that the proof doesn't have to travel to the verifier, it's also that the prover only has to compute the locations in the proof that are asked for on demand, so neither party incurs the exponential blow-up. The advantage to this is that it's simple, my group implemented it in the earlier works in this area, the constants are favorable. Since the proof is exponentially sized, the locations where the verifier wants to inspect it, itself requires a very long string, so we have to work in a preprocessing model where the verifier has to generate these offline, and then amortize this over many uses of the same circuit. The econd disadvantage is that this vector V that you would use in the naive implementation is that it is actually quadratically larger than the satisfying assignment and hence the running time of the computation. This picture sets up another idea based on BCIOP13 and their observation was that if you are willing to weaken the cheating prover and assume it's restricted to only certain operations, then it's possible to have that previous concept from the other slide, but in a non-interactive context. The preprocessing cost can be incurred once, and then reused in all future instances. Those queries that I mentioned earlier would here be encrypted. The verifier may not have to incur this computation, perhaps a third-party does it. The prover looks very similar but they happen in cypherspace here. The disadvantage to this approach is that besides the restriction on the prover which may or may not be an issue depending on philosophy, the provers have to participate in this is still quadratic. I want to put these on the same spectrum. lThanks to Rafael Pass for suggesting a figure of this kind. So if you look, we started with short PCPs, we moved to arguments which introduced the assumption of collision resistant hash functions, we then actually made a stronger cryptographic assumption or IKO did where we commit to a longer PCP which introduces the downside of preprocessing but the key advantage of simplicity and plausible constants, and the same picture can be realized in a non-interactive way, which gets rid of the issue of amortizing over a batch. But we still have the issue that the prover's work is quadratic. I am going to restrict focus to GGPR'S QAPs. We are going to exploit the quadratic arithmetic programs of GGPR. And I don't have time to go into the details of QAPs, but what they are is a way of encoding a circuit as a set of univariate high degree polynomials in such a way that if the circuit is satisfiable then a particular algebraic relationship holds among the polynomials, namely one polynomial divides another. And that algebraic relationsihp can be checked probabilistically, and further that the probabilistic check actually has a linear query structure. Exploiting that fact leads to that vector V that the prover would have otherwise had to materialize as quadratic, now becomes just H here is some stuff I'm not going to talk about that is quasilinear, and Z is the satisfying assignment to the circuits. So the prover's work is shrunk enormously and this was a critical development. In fairness to GGPR is that the fact that they have a linear PCP structure was not explicit in their paper, which was elucidated in BCIOP13 as well as in SBVBBW13 in my group, where we used different vocabulary but I think it's fair to say that both were vigorously expressed. So what this leads to is the following picture. The foundation of all of the argument backends is QAPs, the queries can take place in plaintext or they can take place in the exponent. The backend that implements the approach on the left my group calls Zaatar, we're systems people so we pick things not based on our author names just to confuse you, and you can think of this as an interactive argument that refines the protocol of IKO. The approach on the right is what is now know as a SNARG or a zero knowledge SNARK (zk-SNARK) with preprocessing, and GGPR12 stands at the end of several orther works. And it was implemented by Bryan Parno and others as Pinocchio, and hten optimized in a very nice piece of engineering in what is a released software library called libsnark. There is a question to ask about this which is can we dispense with the preprocessing? It's absolutely possible. BCCT13, BCTV14b, CTV15. It shows that it is possible to dispense with preprocessing by applying this picture recursively. The experimental results by some of these authors make clear that these works should be considered theories and they are many orders of magnitude more expenside. The one on the right uses slightly stronger cryptographic assumptions and gets better cryptographic abilities like non-interactive and zeor knowledge variance, etc. The cost of providing zero knowledge is effectively zero which is a really cool result of GGPR. The reason why I am emphasising that these systems are based on GGPR, is that eventhough they are analyzed differently, they still have this Okay, so now moving to state of the art frontends, there are effectively two different approaches. One approach is to treat the circuit as the unrolled execution of a general purpose processor and to compile a C program to a binary in the instruction set of that processor and supply the binary as the input to the arithmetic circuit, done by BCGTV13, BCTV14a, BCTV14b, CTV15. This is extremely interesting. The preprocessing work is done only once for all computations of the same length, because they can reuse the circuit. They have excellent reprogrammability, they can handle any program that compiles to general purpose CPU, which is all of them. Universality has a price. The second approach, followed by my group as well as Bryan Parno and his work and SBVBPW13, VSBW13, PGHR13, BFRSBW13, BCGGMTV14, BBFR14, FL14, KPPSST14, WSRBW15, CFHKKNPZ15. I think he will talk about zerocash work. The idea is to compile C programs to circuits directly, a tailored circuit for the computation directly. The circuits are much more concise because they are tailored to the computation. The preprocessing has to be incurred for each different circuit in play. This subset of C program in all of these systems, in this part of the picture, is a subset of C, so that raises the question how restrictive is that subset of C for this "ASIC" approach? I'll answer this by talking about the frontends and the tradeoff between programmability and cost. The more expressiveness, the more costly. The ASIC approach, the more like CPUs are down here. At this point, the approaches over here are not good at handling dynamic loops or where loop bounds are incurred at runtime. There are also problems with control flow and so on. Is it possible to get the best of both worlds? In work done with Buffet we find that it is possible to get the best of both worlds, it's a subset of C and it does not handle function pointers as a limitation. It has excellent-by-comparison costs. Hopefully some of you can push this frontier up into the right and further. This is the common framework. We have backends based on QAPs and frontends with different approaches. I want to perform a reality check. How important is it to develop cryptographic protocols that handle more expressive language veruss handling that work at a compiler level and asking the compiler to compile your favorite program into a compatible format? When you compile to remove function pointers, does that have an insane blowup in complexity? You could be asking two things; if we're willing to restrict the frontend, do we get a cheaper backend? If restricting the class of program constructs, does the circuit in the ASIC model too large? How much of the concern of expressiveness can be handled by having better compilers? Some. I'll talk about that in a bit. How can we come up, if we're willing to restrict programmability, can we have more efficient circuits or more concise circuits? As a reality check, I'm going to do a quick performance study. Zaatar (ASIC), BCTV (CPU), and Buffet (ASIC). We're going to run them on the same computations on the same hardware. What are the verifier's cost and what are the prover's cost? The benchmarks for this is that .. ..... benchmark results go here. The preprocessing cost for the verifier, or the prover, actually scales with the size of the circuit meaning the number of gates. That raises two further questions. The size of the circuit is given by the frontend. We have constants in here like 180 microseconds, 60 microseconds, is that good or bad? I dunno. Let's do a quick comparison. Matrix multiplication, merge sort, Knuth-Morris-Pratt string search. Buffet is the best of them, we thought this was great because we're 2 orders of magnitude better... but some perspective is in order.. if you compare all of them to native execution, we're all terrible. Native execution is way better. This isn't even actual data, this is extrapolated because not even all of these systems can run at these input sizes that we're porporting to depict. The maxium input size is far too small to be called practical. The bottleneck is the memory at the prover and at the verifier preprocessing stage, because at a certain point there is a large operation that has to be performed. So we can't handle computations with more than 10 million gates. The preprocessing costs are extremely high like 10 to 30 minutes. Some of these can be addressed in theory. The CPU approaches are better in theory but expensive in practice. The prover's overheads are scary. Memory is a scaling limit. What about proof-carrying data work like BCCT13, BCTV4b, CTV15 like at Eurocrypt. This deals with the prover's memory bottleneck in principle and asymptotically but the concrete costs make this difficult to deploy soon. GKR-derived systems (interactive proof sytsems) (CMT12, VSBW13), .. great verifier performance, some expressivity limitations. Where do we go from here? Target domains where the prover cost is tolerable. One of these might be the Zerocash system like BCGGMTV Oakland14. The whole point of bitcoin is to make people spend CPU cycles, so that's okay. Another application is detailed in our work at SOSP is an application of the machinery to location privacy, I wont have time to dive into this, but I can point you to PBB Security09 or BFRSBW SOSP13 which describes this an application where you can imagine paying these costs becaus ethe size of the data in question is generated on human scales and is not very large. I'd like to try to motivate theoretical advances in this area. We'd ideally like to have a 1000x improvement in prover performance and verifier preprocessing costs. Real programs interact with files, IO devices, etc., there's not a concretely efficient way to incorporate that into machinery. It would be nice to base this machinery on standard assumptions. Things that could be viewed as a luxury is non-interactivity, although sometimes it's absolutely critical. Pre-processing while it would be nice to get rid of it, it's fine as long as the costs aren't too high, I don't even care if the proof is non-constant in length, and I could tolerate a verifier that works harder if the costs are down. One interesting thing would be if you remember the frontend/backend decomposition, and the frontend is producing arithmetic circuits, could we target probabilistic proof protocols that do not require circuits? We would have to identify good candidate problems and good protocols for those problems. A variant for this would be to take the circuit satisfyingability machinery to get special purpose algorithms that outsource common pieces of computations that are common like cryptographic operations and Badlis and Elaine have done work in this area. More efficient reductions from programs to circuit are also interesting, this might be the domain of programming languages. The backend- the short PCP paradigm where the verifier gets a commitment to a short PCP from the prover, that was a good idea in principle, could we get short PCPs that are efficient and simple? It would be great if that existed. Another thing would be to endow IKO's arguments of the preprocessing with better properties, like reuse the preprocessing work beyond a batch, make the protocol zero knowledge, enhance efficiency via stronger assumptions. Or improve the analysis? Some of the cost of the protocol is from the way that soundness is established. Another way is that GGPR's QAPs have come very far, but they are a principle source of cost. Is it possible to improve them? Or to improve the cryptography on top of QAPs? Imagine if the cloud had special purpose processors for cryptographic operations, and these are one of the key bottlenecks, so could we get to the point where the cryptographic operatoins can be outsourced and the verifier would then check that those were correct? Even if we make lots of progress on these, this is not going to be plausibly deployable tomorrow. I would be thrilled if I have motivated any of you to work on these problems. This is an exciting area. The bad news is that implementation does not imply deployability. The overhead is from transforming programs into circuits and the overhead of the QAP backend and the cryptography on top of it. We need theoretical breakthroughs. I think the incentive here is very large. If we had low overhead ways of verifying computation, we would have extremely wide applicability. I'll stop here and take questions if there's time. Q: Which of the overheads is bigger? If you stripped out all of the cryptography? A: If we just strip the cryptography out of it, the prover still, and the cost can be addressed using their stuff, the prover still has to do a fairly large operation, QAPs mandate that hte prover does polynomial arithmetic and finite fast fourier transformations on the satisfying assignment, this is a bottleneck independent of the cryptography. Getting rid of the cryptography and just have QAPs, the cost would go down, I'm making this up, but it would go down like 2 orders of magnitude, just in running time not the memory bottleneck to be clear. Q: You mentioned problems that come up with a lot, like having a special solution .. reducing overhead? A: There are lots of computations that I care about, like the things I use computer sfor every day. What is the commonality? I don't actually know, but I want to think about. If there were operations that could allow transformations on databases, or data structures. I'm not sure. One of the things that tends to blowup which is a bummer is that in the ASIC based approahces, time and conditionality turn into space, so if you have an if statement or switch statement with 4 possibilities, the circuit now needs logic for each of those possibilities. Another thing that might sound crazy is inequality comparisons, if you check if one number is less than another, that introduces expense.	{"url":"https://diyhpl.us/wiki/transcripts/simons-institute/a-wishlist-for-verifiable-computation/","timestamp":"2024-11-07T01:07:10Z","content_type":"application/xhtml+xml","content_length":"26756","record_id":"<urn:uuid:06d10af4-b813-4a09-afb5-73a869e918a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00042.warc.gz"}
Artificial Intelligence Learning Center Now, we will study the concept of a decision boundary for a binary classification problem. We use synthetic data to create a clear example of how the decision boundary of logistic regression looks in comparison to the training samples. We start by generating two features, X1 and X2, at random. Since there are two features, we can say that the data for this problem are two-dimensional. This makes it easy to visualize. The concepts we illustrate here generalize to cases of more than two features, such as the real-world datasets you’re likely to see in your work; however, the decision boundary is harder to visualize in higher-dimensional spaces. Perform the following steps: 1. Generate the features using the following code: X_1_pos = np.random.uniform(low=1, high=7, size=(20,1)) X_1_neg = np.random.uniform(low=3, high=10, size=(20,1)) X_2_pos = np.random.uniform(low=1, high=7, size=(20,1)) X_2_neg = np.random.uniform(low=3, high=10, size=(20,1)) You don’t need to worry too much about why we selected the values we did; the plotting we do later should make it clear. Notice, however, that we are also going to assign the true class at the same time. The result of this is that we have 20 samples each in the positive and negative classes, for a total of 40 samples, and that we have two features for each sample. We show the first three values of each feature for both positive and negative classes. The output should be the following: Generating synthetic data for a binary classification problem 1. Plot these data, coloring the positive samples in red and the negative samples in blue. The plotting code is as follows: plt.scatter(X_1_pos, X_2_pos, color='red', marker='x') plt.scatter(X_1_neg, X_2_neg, color='blue', marker='x') plt.legend(['Positive class', 'Negative class']) The result should look like this: Generating synthetic data for a binary classification problem In order to use our synthetic features with scikit-learn, we need to assemble them into a matrix. We use NumPy’s block function for this to create a 40 by 2 matrix. There will be 40 rows because there are 40 total samples, and 2 columns because there are 2 features. We will arrange things so that the features for the positive samples come in the first 20 rows, and those for the negative samples after that. 1. Create a 40 by 2 matrix and then show the shape and the first 3 rows: X = np.block([[X_1_pos, X_2_pos], [X_1_neg, X_2_neg]]) The output should be: Combining synthetic features in to a matrix We also need a response variable to go with these features. We know how we defined them, but we need an array of y values to let scikit-learn know. 1. Create a vertical stack (vstack) of 20 1s and then 20 0s to match our arrangement of the features and reshape to the way that scikit-learn expects. Here is the code: y = np.vstack((np.ones((20,1)), np.zeros((20,1)))).reshape(40,) You will obtain the following output: Create the response variable for the synthetic data At this point, we are ready to fit a logistic regression model to these data with scikit-learn. We will use all of the data as training data and examine how well a linear model is able to fit the 1. First, import the model class using the following code: from sklearn.linear_model import LogisticRegression 1. Now instantiate, indicating the liblinear solver, and show the model object using the following code: example_lr = LogisticRegression(solver='liblinear') The output should be as follows: Fit a logistic regression model to the synthetic data in scikit-learn 1. Now train the model on the synthetic data: How do the predictions from our fitted model look? We first need to obtain these predictions, by using the trained model’s .predict method on the same samples we used for model training. Then, in order to add these predictions to the plot, using the color scheme of red = positive class and blue = negative class, we will create two lists of indices to use with the arrays, according to whether the prediction is 1 or 0. See whether you can understand how we’ve used a list comprehension, including an if statement, to accomplish this. 1. Use this code to get predictions and separate them into indices of positive and negative class predictions. Show the indices of positive class predictions as a check: y_pred = example_lr.predict(X) positive_indices = [counter for counter in range(len(y_pred)) if y_pred[counter]==1] negative_indices = [counter for counter in range(len(y_pred)) if y_pred[counter]==0] The output should be: Positive class prediction indices 1. Here is the plotting code: plt.scatter(X_1_pos, X_2_pos, color='red', marker='x') plt.scatter(X_1_neg, X_2_neg, color='blue', marker='x') plt.scatter(X[positive_indices,0], X[positive_indices,1], s=150, marker='o', edgecolors='red', facecolors='none') plt.scatter(X[negative_indices,0], X[negative_indices,1], s=150, marker='o', edgecolors='blue', facecolors='none') plt.legend(['Positive class', 'Negative class', 'Positive predictions', 'Negative predictions']) The plot should appear as follows: Predictions and true classes plotted together From the plot, it’s apparent that the classifier struggles with data points that are close to where you may imagine the linear decision boundary to be; some of these may end up on the wrong side of that boundary. Use this code to get the coefficients from the fitted model and print them: theta_1 = example_lr.coef_[0][0] theta_2 = example_lr.coef_[0][1] print(theta_1, theta_2) The output should look like this: Coefficients from the fitted model 1. Use this code to get the intercept: theta_0 = example_lr.intercept_ Now use the coefficients and intercept to define the linear decision boundary. This captures the dividing line of the inequality, X2 ≥ -(1/2)X1 – (0/2): X_1_decision_boundary = np.array([0, 10]) X_2_decision_boundary = -(theta_1/theta_2)*X_1_decision_boundary - (theta_0/theta_2) To summarize the last few steps, after using the .coef_ and .intercept_ methods to retrieve the model coefficients 1, 2 and the intercept 0, we then used these to create a line defined by two points, according to the equation we described for the decision boundary. 1. Plot the decision boundary using the following code, with some adjustments to assign the correct labels for the legend, and to move the legend to a location (loc) outside a plot that is getting pos_true = plt.scatter(X_1_pos, X_2_pos, color='red', marker='x', label='Positive class') neg_true = plt.scatter(X_1_neg, X_2_neg, color='blue', marker='x', label='Negative class') pos_pred = plt.scatter(X[positive_indices,0], X[positive_indices,1], s=150, marker='o', edgecolors='red', facecolors='none', label='Positive predictions') neg_pred = plt.scatter(X[negative_indices,0], X[negative_indices,1], s=150, marker='o', edgecolors='blue', facecolors='none', label='Negative predictions') dec = plt.plot(X_1_decision_boundary, X_2_decision_boundary, 'k-', label='Decision boundary') plt.legend(loc=[0.25, 1.05]) You will obtain the following plot: True classes, predicted classes, and the decision boundary of a logistic regression In this post, we discuss the basics of logistic regression along with various other methods for examining the relationship between features and a response variable. To know, how to install the required packages to set up a data science coding environment, read the book Data Science Projects with Python on Packt Publishing.	{"url":"https://www.ciaburro.it/tag/sklearn/","timestamp":"2024-11-10T21:59:01Z","content_type":"text/html","content_length":"32748","record_id":"<urn:uuid:12b9c37d-8c08-44ca-b107-7a746f2eecf8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00394.warc.gz"}
Second Brain ðŸ§ Anthony Morris Converting from Base 10 to Other Bases • WHILE (the quotient is not zero) □ Divide the decimal number by the new base □ Make the remainder the next digit to the left in the answer Replace the decimal number with the quotient	{"url":"https://anthonymorris.dev/second-brain/decimal","timestamp":"2024-11-13T04:28:31Z","content_type":"text/html","content_length":"17422","record_id":"<urn:uuid:506a9212-de24-4372-9223-dd4f99e90fe5>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00190.warc.gz"}
DAT0001B Using Numeracy Data & IT Assessment DAT0001B Using Numeracy Data & IT Assessment – Arden University UK. Subject Code & Title: DAT0001B Using Numeracy, Data & IT Word Count: 3000 words Weighting: Parts 1 & 2 together (20%), Part 3 (80%) As part of the formal assessment for the programme you are required to submit a Using Numeracy, Data and IT assignment. Please refer to your Student Handbook for full details of the programme assessment scheme and general information on preparing and submitting assignments. DAT0001B Using Numeracy Data & IT Assessment – Arden University UK. Learning Outcomes: After completing the module, you should be able to: 1.Use basic numerical operations with whole numbers, integers, fractions and decimals 2.Calculate ratios and proportions, percentages, averages 3.Construct and use appropriate graphical techniques to present simple data 4.Construct and use a spreadsheet application to record numerical data and use simple formula Your assignment should include: a title page containing your student number, the module name, the submission deadline and a word count; the appendices if relevant; and a reference list in Arden University (AU) Harvard format. You should address all the elements of the assignment task listed below. Please note that tutors will use the assessment criteria set out below in assessing your work. Maximum word count: 3000 words Please note that exceeding the word count by over 10% will result in a reduction in grade by the same percentage that the word count is exceeded. You must not include your name in your submission because Arden University operates anonymous marking, which means that markers should not be aware of the identity of the student. However, please do not forget to include your STU number. Assignment Task: Part 1 – Using Numeracy This part covers Learning Outcomes 1 and 2 of the module. Question 1 Explain the terms (a) numerator, (b) denominator. Question 2 Express 24/40 and 18/42 in their simplest forms. Question 3 a) Express the fractions 2/3, 3/4 and 5/6 as equivalent fractions with a denominator of 12. b) A library contains 60,000 books. 14,000 are on the subject of business, 22,000 are on healthcare and 12,000 on psychology and law. What percentage of the library’s books is on computing, if computing books make up two-thirds of the remainder? Question 4 A sports shop in Manchester is doing a sale on some merchandise. Liz hears about it and goes to make a purchase of two pairs of running shoes. She gives the sales attendant in the store three crisp £50 notes and is given change of £10.50. What is the price of each pair of running shoes? Show your working. Question 5 a) What is 240.50 x 19.54? Give your answer to 2 significant figures. b) Write the number 52100 as a power of 10. Question 6 a) As a promotional offer, a new gym is offering a 30% discount to those who sign up in the first month. Patty and her 2 siblings take advantage of the offer and sign up for total amount of £210. What were the total savings made? Show your working. b) Work out the average savings per person. DAT0001B Using Numeracy Data & IT Assessment – Arden University UK. Question 7 Work out the following: a) 3/4 – 7/9 + 2/3 b) Which is the largest of the following numbers? 0.1, 0.02, 0.003, 0.0004, 0.00005 Question 8 90 men and 60 women were asked if they had watched the latest ‘Expendables’ movie. Altogether 3/5 of the people said Yes. 3/10 of the women said Yes. What percentage of the men said No? (Show your working) Question 9 Annabelle lives at Bermondsey in London. She has to speak at a conference in Birmingham at 10.30 am. It will take her an hour from her home to get to Euston Rail Station, from where she will get a train to Birmingham. The train journey from Euston to Birmingham is an hour and 10 minutes. Trains to Birmingham run at the following times: 5 minutes past the hour, 25 minutes past the hour and 45 minutes past the hour. The meeting venue in Birmingham is a 5-minute walk from the station. What is the latest time that Annabelle can leave home, if she is to make it on time for the meeting in Birmingham? Show your working. Question 10 A box of Shredded Wheat weighs 0.35 kg and a box of We etabix weighs 9/25 kg? Which box is heavier? Show your working. Part 2 – Using Data This part covers Learning Outcome 2 of the module. Given below is the Medals Table for Summer Olympic Games 11) a) Which country had the lowest number of overall medals among the ten countries? b) Which country/countries competed in the least number of games? c) What is the mode in the number of games countries participated in? d) What is the range between the gold medals awarded to the 10 countries? e) How many countries got more silver medals than bronze medals? f) Apart from the United States, which country/countries got more gold medals, more silver medals and more bronze medals than Great Britain? g) In comparing the number of games a country participated in and the overall number medals received, which country did the best (i.e. highest number of medals received per game)? Show your working to justify your answer. h) Using the given data, give TWO likely reasons why a country like Jamaica does not feature in the top 10 medals even though it does very well in athletics. i) Using the given information, determine the medal category in which the United States far outperformed its closest competitor. Show your working to justify your answer. j) Which 3 countries had the most evenly distributed number of gold, silver and bronze medals (i.e. least smallest range)? Show your working. Part 3 – Using IT This part requires the use of Microsoft Excel. Covers Learning Outcomes 3 & 4 12.Create the spreadsheet below using Excel. Include the rows and columns identifiers, text formatting and column highlighting as shown. 13)a) What actions or steps in Excel can you take to rank them from 1st to 10th b) State the specific action(s) or step(s) in Excel that will produce a list/display of those countries with 800 or more medals in total? c) Which type of graph will be suitable for representing only gold medals information? d) In which column(s) might replication have been used? e) What Excel formula can be used to calculate the overall total medals awarded? 14) Write the Excel functions for the following: a. Give the total number of medals for Germany and Great Britain. b. Give the average number of silver medals for a European country, c. Sum the Medals Total for Gold for those countries with less than 20 games involvement. d. Search the database (the whole spreadsheet) to find ‘Italy’ and also the corresponding Medals Total. DAT0001B Using Numeracy Data & IT Assessment – Arden University UK. 15) By using Microsoft Excel and provide screenshots: a) Calculate the median number of medals for each medal type, stating the formula you would use for determining the median for the gold medals. b) Calculate the mean number of medals for each of the 3 medal types, stating the formula you would use for determining the mean for the bronze medals. c) Calculate the standard deviation of the total medals awarded to each country (column F) using the formula below. Show all the steps (full working). Hint: You can use the STDEV.P function to cross-check your final answer. d) Using the given spreadsheet as a basis, discuss the usefulness of a standard deviation in a given data set. You need to cite any literature sources you use. DAT0001B Using Numeracy Data & IT Assessment – Arden University UK. 16) By using Microsoft Excel: a) Produce an appropriate fully labeled chart in Excel to compare the gold, silver and bronze medals totals of the 10 countries. b) Use a suitable and fully labeled chart in Excel to reflect the contribution of each country to the overall medals total. ORDER DAT0001B Using Numeracy Data & IT Assessment NOW And Get Instant Discount Read More :-	{"url":"https://assignmenthelps.co.uk/dat0001b-using-numeracy-data-it-assessment-arden-university-uk.php","timestamp":"2024-11-11T20:56:00Z","content_type":"text/html","content_length":"69595","record_id":"<urn:uuid:a0b1f31b-5788-4937-8fc8-2cdfbd2a07c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00791.warc.gz"}
CGAL 5.0 beta2 released CGAL 5.0 offers the following improvements and new functionality over CGAL 4.14: General changes • CGAL 5.0 is the first release of CGAL that requires a C++ compiler with the support of C++14 or later. The new list of supported compilers is: □ Visual C++ 14.0 (from Visual Studio 2015 Update 3) or later, □ Gnu g++ 6.3 or later (on Linux or MacOS), □ LLVM Clang version 8.0 or later (on Linux or MacOS), and □ Apple Clang compiler versions 7.0.2 and 10.0.1 (on MacOS). • Since CGAL 4.9, CGAL can be used as a header-only library, with dependencies. Since CGAL 5.0, that is now the default, unless specified differently in the (optional) CMake configuration. • The section “Getting Started with CGAL” of the documentation has been updated and reorganized. • The minimal version of Boost is now 1.57.0. • This package provides a method for piecewise planar object reconstruction from point clouds. The method takes as input an unordered point set sampled from a piecewise planar object and outputs a compact and watertight surface mesh interpolating the input point set. The method assumes that all necessary major planes are provided (or can be extracted from the input point set using the shape detection method described in Point Set Shape Detection, or any other alternative methods).The method can handle arbitrary piecewise planar objects and is capable of recovering sharp features and is robust to noise and outliers. See also the associated blog entry. • Breaking change: The concept ShapeDetectionTraits has been renamed to EfficientRANSACTraits. • Breaking change: The Shape_detection_3 namespace has been renamed to Shape_detection. • Added a new, generic implementation of region growing. This enables for example applying region growing to inputs such as 2D and 3D point sets, or models of the FaceGraph concept. Learn more about this new algorithm with this blog entry. • A new exact kernel, Epeck_d, is now available. • Added a new concept, ComputeApproximateAngle_3, to the 3D Kernel concepts to compute the approximate angle between two 3D vectors. Corresponding functors in the model (Compute_approximate_angle_3 ) and free function (approximate_angle) have also been added. • The following objects are now hashable and thus trivially usable with std::unordered_set and std::unordered_map: CGAL::Aff_transformation_2, CGAL::Aff_transformation_3, CGAL::Bbox_2, CGAL::Bbox_3, CGAL::Circle_2, CGAL::Iso_cuboid_3, CGAL::Iso_rectangle_2, CGAL::Point_2, CGAL::Point_3, CGAL::Segment_2, CGAL::Segment_3, CGAL::Sphere_3, CGAL::Vector_2, CGAL::Vector_3, CGAL::Weighted_point_2 and CGAL::Weighted_point_3. • Introduced a wide range of new functions related to location of queries on a triangle mesh, such as CGAL::Polygon_mesh_processing::locate(Point, Mesh). The location of a point on a triangle mesh is expressed as the pair of a face and the barycentric coordinates of the point in this face, enabling robust manipulation of locations (for example, intersections of two 3D segments living within the same face). • Added the mesh smoothing function smooth_mesh(), which can be used to improve the quality of triangle elements based on various geometric characteristics. • Added the shape smoothing function smooth_shape(), which can be used to smooth the surface of a triangle mesh, using the mean curvature flow to perform noise removal. (See also the new entry in the User Manual) • Added the function CGAL::Polygon_mesh_processing::centroid(), which computes the centroid of a closed triangle mesh. • Added the functions CGAL::Polygon_mesh_processing::stitch_boundary_cycle() and CGAL::Polygon_mesh_processing::stitch_boundary_cycles(), which can be used to try and merge together geometrically compatible but combinatorially different halfedges that belong to the same boundary cycle. • It is now possible to pass a face-size property map to CGAL::Polygon_mesh_processing::keep_large_connected_components() and CGAL::Polygon_mesh_processing::keep_largest_connected_components(), enabling users to define how the size of a face is computed (the size of the connected component is the sum of the sizes of its faces). If no property map is passed, the behavior is unchanged to previous versions: the size of a connected component is the number of faces it contains. • Added the function CGAL::Polygon_mesh_processing::non_manifold_vertices(), which can be used to collect all the non-manifold vertices (i.e. pinched vertices, or vertices appearing in multiple umbrellas) of a mesh. • The PLY IO functions now take an additional optional parameter to read/write comments from/in the PLY header. • Breaking change: Removed the deprecated functions CGAL::Constrained_triangulation_plus_2:: vertices_in_constraint_{begin/end}(Vertex_handle va, Vertex_handle vb) const;, and CGAL::Constrained_triangulation_plus_2::remove_constraint(Vertex_handle va, Vertex_handle vb), that is a pair of vertex handles is no longer a key for a polyline constraint. Users must use a version prior to 5.0 if they need this functionality. • Breaking change: Removed the deprecated classes CGAL::Regular_triangulation_euclidean_traits_2, CGAL::Regular_triangulation_filtered_traits_2. Users must use a version prior to 5.0 if they need these classes. • Breaking change: The graph traits enabling CGAL’s 2D triangulations to be used as a parameter for any graph-based algorithm of CGAL (or boost) have been improved to fully model the FaceGraph concept. In addition, only the finite simplicies (those not incident to the infinite vertex) of the 2D triangulations are now visibile through this scope. The complete triangulation can still be accessed as a graph, by using the graph traits of the underlying triangulation data structure (usually, CGAL::Triangulation_data_structure_2). • Breaking change: The insert() function of CGAL::Triangulation_2 which takes a range of points as argument is now guaranteed to insert the points following the order of InputIterator. Note that this change only affects the base class Triangulation_2 and not any derived class, such as Delaunay_triangulation_2. • Added a new constructor and insert() function to CGAL::Triangulation_2 that takes a range of points with info. • Introduced a new face base class, Triangulation_face_base_with_id_2 which enables storing user-defined integer IDs in the face of any 2D triangulation, a precondition to use some BGL algorithms. • Added range types and functions that return ranges, for example for all vertices, enabling the use of C++11 for-loops. See this new example for a usage demonstration. • Breaking change: The constructor and the insert() function of CGAL::Triangulation_3 which take a range of points as argument are now guaranteed to insert the points following the order of InputIterator. Note that this change only affects the base class Triangulation_3 and not any derived class, such as Delaunay_triangulation_3. • Added constructor and insert() function to CGAL::Triangulation_3 that takes a range of points with info. • Added range types and functions that return ranges, for example for all vertices, which enables to use C++11 for-loops. See this new example for a usage demonstration. • Introduced new functions to read and write using the PLY format, CGAL::read_ply() and CGAL::write_ply(), enabling users to save and load additional property maps of the surface mesh. • Added concepts and models for solving Mixed Integer Programming (MIP) problems with or without constraints. • Breaking change: The API of CGAL::Color has been cleaned up. • Added new functions to support some parts of the WKT file format:	{"url":"https://www.cgal.org/2019/10/31/cgal50-beta2/","timestamp":"2024-11-04T00:49:43Z","content_type":"text/html","content_length":"34713","record_id":"<urn:uuid:c3544c3a-04b4-4aad-95d7-e69970177344>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00705.warc.gz"}
Environment Model 9.4. Environment Model# So far we’ve been using the substitution model to evaluate programs. It’s a great mental model for evaluation, and it’s commonly used in programming languages theory. But when it comes to implementation, the substitution model is not the best choice. It’s too eager: it substitutes for every occurrence of a variable, even if that occurrence will never be needed. For example, let x = 42 in e will require crawling over all of e, which might be a very large expression, even if x never occurs in e, or even if x occurs only inside a branch of an if expression that never ends up being evaluated. For sake of efficiency, it would be better to substitute lazily: only when the value of a variable is needed should the interpreter have to do the substitution. That’s the key idea behind the environment model. In this model, there is a data structure called the dynamic environment, or just “environment” for short, that is a dictionary mapping variable names to values. Whenever the value of a variable is needed, it’s looked up in that dictionary. To account for the environment, the evaluation relation needs to change. Instead of e --> e' or e ==> v, both of which are binary relations, we now need a ternary relation, which is either • <env, e> --> e', or • <env, e> ==> v, where env denotes the environment, and <env, e> is called a machine configuration. That configuration represents the state of the computer as it evaluates a program: env represents a part of the computer’s memory (the binding of variables to values), and e represents the program. As notation, let: • {} represent the empty environment, • {x1:v1, x2:v2, ...} represent the environment that binds x1 to v1, etc., • env[x -> v] represent the environment env with the variable x additionally bound to the value v, and • env(x) represent the binding of x in env. If we wanted a more mathematical notation we would write $\mapsto$ instead of -> in env[x -> v], but we’re aiming for notation that is easily typed on a standard keyboard. We’ll concentrate in the rest of this chapter on the big-step version of the environment model. It would of course be possible to define a small-step version, too. 9.4.1. Evaluating the Lambda Calculus in the Environment Model# Recall that the lambda calculus is the fragment of a functional language involving functions and application: e ::= x \| e1 e2 \| fun x -> e v ::= fun x -> e Let’s explore how to define a big-step evaluation relation for the lambda calculus in the environment model. The rule for variables just says to look up the variable name in the environment: This rule for functions says that an anonymous function evaluates just to itself. After all, functions are values: <env, fun x -> e> ==> fun x -> e Finally, this rule for application says to evaluate the left-hand side e1 to a function fun x -> e, the right-hand side to a value v2, then to evaluate the body e of the function in an extended environment that maps the function’s argument x to v2: <env, e1 e2> ==> v if <env, e1> ==> fun x -> e and <env, e2> ==> v2 and <env[x -> v2], e> ==> v Seems reasonable, right? The problem is, it’s wrong. At least, it’s wrong if you want evaluation to behave the same as OCaml. Or, to be honest, nearly any other modern language. It will be easier to explain why it’s wrong if we add two more language features: let expressions and integer constants. Integer constants would evaluate to themselves: As for let expressions, recall that we don’t actually need them, because let x = e1 in e2 can be rewritten as (fun x -> e2) e1. Nonetheless, their semantics would be: <env, let x = e1 in e2> ==> v if <env, e1> ==> v1 and <env[x -> v1], e2> ==> v Which is a rule that really just follows from the other rules above, using that rewriting. What would this expression evaluate to? let x = 1 in let f = fun y -> x in let x = 2 in f 0 According to our semantics thus far, it would evaluate as follows: • let x = 1 would produce the environment {x:1}. • let f = fun y -> x would produce the environment {x:1, f:(fun y -> x)}. • let x = 2 would produce the environment {x:2, f:(fun y -> x)}. Note how the binding of x to 1 is shadowed by the new binding. • Now we would evaluate <{x:2, f:(fun y -> x)}, f 0>: <{x:2, f:(fun y -> x)}, f 0> ==> 2 because <{x:2, f:(fun y -> x)}, f> ==> fun y -> x and <{x:2, f:(fun y -> x)}, 0> ==> 0 and <{x:2, f:(fun y -> x)}[y -> 0], x> ==> 2 because <{x:2, f:(fun y -> x), y:0}, x> ==> 2 • The result is therefore 2. But according to utop (and the substitution model), it evaluates as follows: # let x = 1 in let f = fun y -> x in let x = 2 in f 0;; - : int = 1 And the result is therefore 1. Obviously, 1 and 2 are different answers! What went wrong?? It has to do with scope. 9.4.2. Lexical vs. Dynamic Scope# There are two different ways to understand the scope of a variable: variables can be dynamically scoped or lexically scoped. It all comes down to the environment that is used when a function body is being evaluated: • With the rule of dynamic scope, the body of a function is evaluated in the current dynamic environment at the time the function is applied, not the old dynamic environment that existed at the time the function was defined. • With the rule of lexical scope, the body of a function is evaluated in the old dynamic environment that existed at the time the function was defined, not the current environment when the function is applied. The rule of dynamic scope is what our semantics, above, implemented. Let’s look back at the semantics of function application: <env, e1 e2> ==> v if <env, e1> ==> fun x -> e and <env, e2> ==> v2 and <env[x -> v2], e> ==> v Note how the body e is being evaluated in the same environment env as when the function is applied. In the example program let x = 1 in let f = fun y -> x in let x = 2 in f 0 that means that f is evaluated in an environment in which x is bound to 2, because that’s the most recent binding of x. But OCaml implements the rule of lexical scope, which coincides with the substitution model. With that rule, x is bound to 1 in the body of f when f is defined, and the later binding of x to 2 doesn’t change that fact. The consensus after decades of experience with programming language design is that lexical scope is the right choice. Perhaps the main reason for that is that lexical scope supports the Principle of Name Irrelevance. Recall, that principle says that the name of a variable shouldn’t matter to the meaning of program, as long as the name is used consistently. Nonetheless, dynamic scope is useful in some situations. Some languages use it as the norm (e.g., Emacs LISP, LaTeX), and some languages have special ways to do it (e.g., Perl, Racket). But these days, most languages just don’t have it. There is one language feature that modern languages do have that resembles dynamic scope, and that is exceptions. Exception handling resembles dynamic scope, in that raising an exception transfers control to the “most recent” exception handler, just like how dynamic scope uses the “most recent” binding of variable. 9.4.3. A Second Attempt at Evaluating the Lambda Calculus in the Environment Model# The question then becomes, how do we implement lexical scope? It seems to require time travel, because function bodies need to be evaluated in old dynamic environment that have long since The answer is that the language implementation must arrange to keep old environments around. And that is indeed what OCaml and other languages must do. They use a data structure called a closure for this purpose. A closure has two parts: • a code part, which contains a function fun x -> e, and • an environment part, which contains the environment env at the time that function was defined. You can think of a closure as being like a pair, except that there’s no way to directly write a closure in OCaml source code, and there’s no way to destruct the pair into its components in OCaml source code. The pair is entirely hidden from you by the language implementation. Let’s notate a closure as (\| fun x -> e, env \|). The delimiters (\| ... \|) are meant to evoke an OCaml pair, but of course they are not legal OCaml syntax. Using that notation, we can re-define the evaluation relation as follows: The rule for functions now says that an anonymous function evaluates to a closure: <env, fun x -> e> ==> (\| fun x -> e, env \|) That rule saves the defining environment as part of the closure, so that it can be used at some future point. The rule for application says to use that closure: <env, e1 e2> ==> v if <env, e1> ==> (\| fun x -> e, defenv \|) and <env, e2> ==> v2 and <defenv[x -> v2], e> ==> v That rule uses the closure’s environment defenv (whose name is meant to suggest the “defining environment”) to evaluate the function body e. The derived rule for let expressions remains unchanged: <env, let x = e1 in e2> ==> v if <env, e1> ==> v1 and <env[x -> v1], e2> ==> v That’s because the defining environment for the body e2 is the same as the current environment env when the let expression is being evaluated. 9.4.4. An Implementation of the Lambda Calculus in the Environment Model# You can download a complete implementation of the above two lambda calculus semantics: lambda-env.zip. In main.ml, there is a definition named scope that you can use to switch between lexical and dynamic scope. 9.4.5. Evaluating Core OCaml in the Environment Model# There isn’t anything new in the (big step) environment model semantics of Core OCaml, now that we know about closures, but for sake of completeness let’s state it anyway. e ::= x \| e1 e2 \| fun x -> e \| i \| b \| e1 bop e2 \| (e1,e2) \| fst e1 \| snd e2 \| Left e \| Right e \| match e with Left x1 -> e1 \| Right x2 -> e2 \| if e1 then e2 else e3 \| let x = e1 in e2 We’ve already seen the semantics of the lambda calculus fragment of Core OCaml: <env, x> ==> v if env(x) = v <env, e1 e2> ==> v if <env, e1> ==> (\| fun x -> e, defenv \|) and <env, e2> ==> v2 and <defenv[x -> v2], e> ==> v <env, fun x -> e> ==> (\|fun x -> e, env\|) Evaluation of constants ignores the environment: <env, i> ==> i <env, b> ==> b Evaluation of most other language features just uses the environment without changing it: <env, e1 bop e2> ==> v if <env, e1> ==> v1 and <env, e2> ==> v2 and v is the result of applying the primitive operation bop to v1 and v2 <env, (e1, e2)> ==> (v1, v2) if <env, e1> ==> v1 and <env, e2> ==> v2 <env, fst e> ==> v1 if <env, e> ==> (v1, v2) <env, snd e> ==> v2 if <env, e> ==> (v1, v2) <env, Left e> ==> Left v if <env, e> ==> v <env, Right e> ==> Right v if <env, e> ==> v <env, if e1 then e2 else e3> ==> v2 if <env, e1> ==> true and <env, e2> ==> v2 <env, if e1 then e2 else e3> ==> v3 if <env, e1> ==> false and <env, e3> ==> v3 Finally, evaluation of binding constructs (i.e., match and let expression) extends the environment with a new binding: <env, match e with Left x1 -> e1 \| Right x2 -> e2> ==> v1 if <env, e> ==> Left v and <env[x1 -> v], e1> ==> v1 <env, match e with Left x1 -> e1 \| Right x2 -> e2> ==> v2 if <env, e> ==> Right v and <env[x2 -> v], e2> ==> v2 <env, let x = e1 in e2> ==> v2 if <env, e1> ==> v1 and <env[x -> v1], e2> ==> v2	{"url":"https://cs3110.github.io/textbook/chapters/interp/environment.html","timestamp":"2024-11-02T01:40:19Z","content_type":"text/html","content_length":"60970","record_id":"<urn:uuid:d0368e9c-bb81-4eed-aa54-ce71d8cbfdb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00282.warc.gz"}
Lesson 3 Construction Techniques 1: Perpendicular Bisectors Lesson Narrative The purpose of this lesson is to lay the foundation for understanding the perpendicular bisector of a segment as both a line perpendicular to a segment passing through its midpoint (by definition) and the set of points equidistant to the endpoints. The second fact will be proven in the next unit. The perpendicular bisector plays a key role in the definition of reflection later in this unit and in the proof of the Side-Side-Side triangle congruence theorem in the next unit. This lesson continues the theme of asking how much can be learned without using numbers to measure distance as well as building on students’ understanding of angle and perpendicular from previous grades. Students look for and make use of structure when they think about where their classmates should stand during Human Perpendicular Bisector in order to be the same distance away from two given points (MP7). The more students that correctly place themselves, the more apparent the structure. Once students determine the structure, they record it as a conjecture. A conjecture is defined as a reasonable guess that students are trying to either prove or disprove. If students have ready access to digital materials in class, they can choose to perform all construction activities with the GeoGebra Construction tool accessible in the Math Tools or available at Learning Goals Teacher Facing • Comprehend that a perpendicular bisector is the set of points equidistant from two given points. • Construct a perpendicular bisector. Student Facing • Let’s explore equal distances. Required Preparation For Human Perpendicular Bisector, mark two points on the floor of the classroom two meters apart, using masking tape. Clear a large space around and between the two marked points. Student Facing • I can construct a perpendicular bisector. • I understand what is special about the set of points equidistant from two given points. CCSS Standards Building Towards Glossary Entries • conjecture A reasonable guess that you are trying to either prove or disprove. • perpendicular bisector The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. Additional Resources Google Slides For access, consult one of our IM Certified Partners. PowerPoint Slides For access, consult one of our IM Certified Partners.	{"url":"https://curriculum.illustrativemathematics.org/HS/teachers/2/1/3/preparation.html","timestamp":"2024-11-11T07:52:33Z","content_type":"text/html","content_length":"82181","record_id":"<urn:uuid:33dedf1b-8c44-48f9-a968-909559ad3399>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00045.warc.gz"}
The Development and Use of Claim Life Cycle Model \| Published in CAS E-Forum Note regarding the use of ‘IBNR’ in this paper: Unless specifically noted to the contrary, in this paper, “IBNR” will refer to only the provision for those claims that have truly not been reported as of the date of the analysis, and “case development” will be used to refer to the provision for additional development for known claims. Note on organization of this paper: This is a lengthy document owing to the significant work necessary to build a claim life cycle model. A reader that already accepts the argument of why such a framework is important may want to skip section 2. Another reader that is primarily interested in the motivation for this approach and down-stream applications may want to skip sections 4 and 5, which are focused on the modeling details. 1.0. Introduction^[1] This paper describes approaches to incorporate detailed claim and exposure data into the actuarial process of estimating property-casualty reserves. Using detailed data provides additional insight into needed reserves. Since loss development considerations are critical to questions of pricing, significant insight can be gained in actuarial pricing as well. Internal management reporting also benefits, allowing for more reliable reporting of results at various levels of detail. Predictive models of various aspects of claim development (such as closure rate, claim revaluation, payment rates, etc.) within a time-step (month, quarter, year, etc.) are described, differentiating by policy and claim characteristics. Simulation then projects each open claim to an ultimate value using the combination of these models. We then describe actuarial case reserves, which provide an important bridge between detailed development models and the traditional triangle reserving framework. We illustrate the use of this algorithm as an alternative to traditional case reserves and discuss its benefits. Validation of the algorithm using report-period triangles will also be discussed. To develop a provision for unreported claims, the paper describes the creation of emergence models – report lag, frequency, and severity. Simulation is used to generate true IBNR claims, or the emergence models can be used directly to provide mean estimates of their value at a policy level at a point in time. Like actuarial case reserves, these policy reserves can be used as data element in the traditional triangle approach. The implications of using detailed actuarial claim and unreported claim reserves to support actuarial pricing efforts as well as internal management reporting will also be discussed. There is a growing body of literature regarding actuarial reserve estimation using detailed claim and policy data that the reader may wish to consider in addition to this paper, such as Parodi (2013), Antonio and Plat (2014), Korn (2016), and Landry and Martin (2022). 2.0. Why a Claim Life Cycle Model is Needed Analysis of triangle data is well established within the actuarial profession. The output from triangle analysis is well understood by professionals across the industry. Computing power and increased use of predictive analytics make it possible to substantially improve reserve analysis by systematically considering detailed claim and policy information. It is well documented in the actuarial literature^[2] that underlying changes in claim mix, case reserve adequacy, or settlement speed, if left undetected and unadjusted for, can lead to erroneous estimates. Most approaches within the profession have focused on identifying and correcting for such distortions. Several actuarial problems illustrate the advantages of using detailed data rather than reliance on traditional development triangles alone. We will discuss a number of these problems below. 2.1. Mix Shifts Any significant difference in loss development patterns across claims and differences in expected loss ratios across policies has the potential to cause problems for triangle analysis unless the mix of claims and exposures is held reasonably constant. This problem is well-known, but due to the wide variety of exposures (deductibles, locations, policy forms, customer characteristics, etc.), mix changes can remain hidden for years without detection, when patterns have been shown to be conclusively different than in the past. Consider an underwriting unit that writes two classes of business. Until recently there has been a stable mix of business between these two classes with Class 1 making up the majority of the business. The book has had a loss ratio near 60%. Due to this acceptable loss ratio and the relatively insignificant amount of Class 2 business, differences in the performance of the two classes has remained undetected. Class 2 develops slower and has a higher expected loss ratio (90% vs. 60%). The graph below shows the different expected development patterns of the two classes. At year 3, Class 1 is 50% developed while Class 2 is only 35% developed. Differing development patterns of Class 1 and Class 2 business The mix of business between the two classes was unchanging until 2013 when the company grew its Class 2 business. The triangle below shows the case-incurred losses: Loss Triangle Development Factors Examination of the triangle reveals little. The 2014 age 1-2 factor is the highest in the triangle, but not dramatically higher. The 2013 age 2 to age 3 factor isn’t the highest for this age. Since the beginning of the change in mix, there are only three data points in the triangle with which to observe a change in loss development. Since nothing significant is observed, it is reasonable for the actuary to conclude that there is no change. Even if a change in development were detected within the first two development periods, there is no information in the triangle about how this development will continue beyond age 3. Applying the measured LDFs produces the estimated ultimate loss ratios shown in the graph below. These estimated ultimate loss ratios for the last three years are very different from the true underlying loss ratios of the book. Estimated Ultimate Loss Ratio vs. True Loss Ratio The estimated ultimate loss ratio for 2015 is particularly distorted. This is a result of the slower reporting of the Class 2 business and the application of a loss development factor which is not appropriate for the current mix of business. Without a mechanism to capture differences in expected loss ratios and development patterns at a level below the triangle, the actuary will be late in identifying the deteriorating loss performance and will mistakenly estimate an improving loss ratio. This example demonstrates how the impact of a shift in the mix of business can mislead the actuary. Detection of the problem could take years. Meanwhile the late detection can have devastating effects, as it will likely influence underwriting decisions. If the increase in Class 2 writings is noticed together with the apparent improvement in loss ratio, management could conclude that additional growth in Class 2 should be encouraged. Financial statements reinforce the idea until booked reserves deteriorate, years later. It is tempting to dismiss this example. In hindsight, the problem was easy to identify –a shift in the mix of business by class. Perhaps there are mechanisms in place to report on shifts of mix by class so changes can be evaluated early. The challenge lies in the wide variety of exposures commonly underwritten. Avoidance of this problem depends upon identifying that a mix-shift has occurred that results in a change in development. Companies can monitor for shifts by class, by geography, by deductible, limit, etc., but differences in development will not be obvious unless triangles are segmented along the dimensions that are shifting. It is not feasible to develop and analyze triangles by every dimension. Further, if differences in development are not identified, differences in loss ratios cannot be adequately identified until losses are mature. In the example given above, assume that the actuary was monitoring the mix of classes and considering the loss ratios of the individual classes, assuming that the development patterns were the same. In this case, it would look like Class 2 had seen a dramatic improvement in loss ratio (a mistaken view caused by an insufficient development pattern). Class 2 might be viewed as having a high loss ratio in the past when less of it was written, but now that it is a focus of the business, the loss ratio has improved dramatically. This is an illusion created by the slower development while in reality the loss ratio is still high. It is not at feasible to monitor a book of business for this type of problem across all possible dimensions, without a systematic, multivariate approach to modeling loss development and identifying problematic mix-shifts. Changing actuarial loss reserving from the current, aggregated approach is necessary to detect such problems before they cause significant financial damage. 2.2. Changes in case reserving/timing Changes in case reserving practices within a company can cause significant difficulty in triangle-based reserving approaches. As with mix shifts, the problem is lack of detection. Diagnostics such as triangles of average case outstanding and triangles of closure rates are commonly used to detect changes, but the aggregation of data obscures measurement. Changes in case reserve adequacy may not be detectable in a triangle until evidenced by changes in loss development factors. Consider a scenario with a claim department under pressure to set case reserves lower while the underwriting department is under pressure to write higher severity accounts (both possible when a company is under pressure). This could result in average case reserve amounts that are similar to the past, despite the drop in case reserve adequacy. Aggregated data is insufficient to alert the actuary to the changes. The natural variability in loss development clouds the picture and makes it even harder to detect the change. It is financially harmful to wait until the evidence from the aggregated data becomes conclusive. If the changes can be detected earlier, through systematic investigation of detailed data, significant damage can be avoided. Most actuaries are comfortable with inadequate or redundant case reserves, provided the aggregate level of adequacy does not change^[3]. However, case reserve adequacy can vary widely across different segments of the claim portfolio. With the mix of claims constantly changing across many dimensions, aggregate case reserve adequacy is constantly changing, even with no change to how case reserves are being set for any particular type of claim. 2.3. Projections and Monitoring of Results It is common to allocate aggregate reserves to a finer level of detail for the purpose of monitoring and managing various segments of the business (profit center, region/office, agency, etc.) Typically, the allocation is simple and based on earned premium, outstanding case reserves, payments to date, etc. The simplistic allocations can create distortions. When the allocation of these bulk reserves impacts bonus and other incentive payments it is likely to also impact business decisions. If the bulk reserves are allocated in a way that does not reflect reality, misguided business decisions can result. It is natural when losses develop differently than projected to investigate the variance. When reserve estimates and development projections are calculated at a broad level and allocated naively the search for explanations is challenging. Answers often focus on large claims but miss broader issues until more variances are exhibited in subsequent periods. Similarly, underlying issues can remain hidden because the random nature of large claims obscures them. When the reserve analysis is performed based on individual claims (and policies in the case of unreported claims), the resulting estimate already exists at the finest level possible. There still will likely be some difference between management’s booked reserve and the analysis, but the detailed analysis provides a natural allocation basis for this difference. Not only does this lead to more appropriate business decisions, but also enables powerful management reporting that allows thoughtful drill-downs to other segmentations of the business without additional reserve development analysis. A detailed allocation that is tied directly to open claims, based on their individual potential to develop and to individual policies based on their likelihood of generating additional claims is more robust. Regular examination of development versus expectations can be monitored statistically at a detailed level to identify emerging trends instead of hunting for an answer when large variances are observed. 2.4. Detect changes in environment There are often changes within a triangle over time that have nothing to do with mix of business, claims handling practices, or any other action of the insurance company. Examples of these types of changes are inflation, changes in litigiousness, or changes in nature of awards arrived at through litigation. Companies are certainly aware of and concerned about these changes and are often thinking about them. However, it is likely that these will go undetected, especially if traditional triangles are the only tools being used. A predictive analysis in which transaction date is itself a predictive variable is very helpful in detecting these environmental changes, identifying and measuring impacts observed in the past and giving context for the relative level of stability or instability in the observed environment. 2.5. Cohesive framework across reserving and pricing Actuarial reserving functions and pricing functions are often seen as being distinct. Since pricing usually begins with a reserving analysis (explicitly or implicitly), and since reserve estimates are improved by a thorough understanding of changes in pricing and product strategy, the two disciplines are linked. When these functions operate separately, important information may not be communicated. Reserving actuaries may not be aware of all the changes across the products being written (mix changes, pricing decisions, etc.). Pricing actuaries may miss important information about reserve development (changing case adequacy, differences in development across policies, etc.). This can distort pricing indications. With expanded use of predictive analytics in insurance pricing, it is easy to make erroneous conclusions by assuming that case-incurred loss differentials are indicative of ultimate loss differentials. Often this can lead to concluding that slower developing segments of the portfolio are performing better than they truly are and that faster developing segments of the portfolio are performing worse than they truly are. Using individual claim-based actuarial reserving approaches leads the reserving actuary to systematically consider changes in the mix of business and price level. Additionally, the resulting policy level reserve estimates allow the pricing actuaries to use more sound ultimate values in their analyses. 2.6. Layer results/Reinsurance When considering expected loss within a loss layer, either for pricing or reserving for primary or reinsurance layers, there are additional challenges when using traditional triangle analysis. Excess layers may have limited experience in the history. The use of selected development patterns in different layers can lead to inconsistent results. For example, analyses performed gross and net of reinsurance, with development factors selected independently for each, with a thin ceded layer could lead to accident periods with negative ceded reserve estimates. Models of detailed reserve development help with this problem. The potential for claims to pierce into individual layers can be considered as part of one cohesive development model. For a ceding company, organizing the necessary information is straightforward. For assuming companies, this can be more problematic. Often, however, there is a requirement to report claim activity to the excess carrier/reinsurer when a claim exceeds a threshold, such as 50% of the retention. Modeling the detailed claim behavior of these sub-layer claims can provide significant information about the layer of interest. 3.0. Model Overview The Claim Life Cycle Model (CLCM) described in this paper involves: • organization of data elements into tables that are readily modeled • development models that describe the time-step behavior of known claims • emergence models that describe the emergence of IBNR claims • simulation of future development and emergence at a detailed level • creation of actuarial case and policy reserves The flow chart below summarizes this process. We will focus on individual parts of this process in the following sections. Blue rectangles represent data tables. Green squares represent predictive models. The arrows show dependencies within the process (colors added to improve readability). 3.1. General Predictive Model Commentary Predictive models form the backbone of this approach. This paper does not prescribe the form of the predictive models, instead focusing on the various targets of prediction and how the different models work together to build the overall approach. But generally, as is the case with most predictive modeling, the models should: • Provide a framework for predicting the mean of the target variable, given the predictive characteristics. • Aim for parsimony. All potential predictive variables should be considered, but only those that are found to have predictive power should remain in the final model that is selected. Validation data should be used to compare alternative models to reveal which characteristics are actually providing predictive power, marginal to the other characteristics. Overfitting to training data should be rigorously avoided. • Test data should be held out. In addition to describing model veracity, this data becomes critical in this approach when simulating future results as will be discussed in section 5.1.3. Since the individual time-step models will be used to simulate future development, with simulation results from one time step being used as inputs into the next time step, it is critical that models are robust. To that end, it is useful to use a modeling approach that includes credibility adjustment of model parameters, instead of a purely binary choice of whether the parameter remains in the model. Using iterative techniques such as Multiplicative Bailey Minimum Bias with credibility adjustments is a practical and robust approach to model parameterization, taken together with the discipline to remove extraneous variables, test model results, consider variable interactions, etc.,^[4]. It is helpful to define the training dataset as the claims associated with a specific subset of policies. With all claims from an individual policy in one set or the other, overfitting risk due to non-independence of data between training and validation/test is reduced. When detailed data is later organized into triangles, the results will be more meaningful for training or test triangles, because all transactions for a policy will be in either one or the other, not scattered across both test and training data sets. 4.0. Input Data 4.1. Necessary Data Three tables are necessary to perform this type of analysis, a Claim Transaction table, a Claim Characteristic Table and a Policy Characteristic table: The Claim Transaction table contains the financial history of the claims. Every time there has been a payment (loss or DCC) or a change in case reserves, there is a record with the transaction, the date, and the claim ID. The Claim Characteristics table contains information about the claim, including the incurred date, and other available information. In addition to coded fields specific to the line of business, claim notes are a valuable predictor, typically through the use of topic assignment, such as by Latent Dirichlet Allocation. The Claim ID allows for joining to the transaction history. Also necessary is a policy ID, so that the data can be joined to the policy data. Some variables may be dynamic in nature (changing over the lifetime of a claim). These should not be used as predictors unless the changes themselves are modeled. To do this, a history of changes in the variable at the claim level needs to be made available, similar to the Claim Transactions table. The Policy Characteristics table includes the policy ID, the premium for the policy, and any available characteristics describing the policy. In some cases, the records can be more specific than policy (e.g., policy, class, and state), but only if the premium is available at that level and the Claim Characteristic table has the same dimensions. 4.2. Data Organization Two tables are created to organize the data for modeling purposes, a Development Table and an Emergence Table The Development Table organizes the information by claim and age of development. A time step is first determined (monthly, quarterly, yearly), and every claim will have a record for each step, starting with the step in which the claim first appears. The following fields are necessary on this table: • Case Reserve at the beginning of the step • Case Reserve at the end of the step • Payments during the step • Payments to date Each record should contain the characteristics from the Claim Characteristics table and the Policy Characteristics table. The table can be thought of as being similar to a development triangle, but at the claim level and containing all the characteristics. The Emergence Table will be used for modeling IBNR claims and is a copy of the Policy Characteristics table, but with non-zero claim count added. For closed claims with positive payment amounts, these are taken from the transaction table. For claims that are still open, the claim count will be determined after simulation (to avoid counting future claims with no payment) Other fields will be added to the Development Table and Emergence Table in subsequent sections of this paper. 5.0. Component Development Models and Simulation We next will describe the various predictive models that describe the behavior of claims and the simulation used to bring these various models together into reserve estimates, first for the reported claims and then for the unreported claims. 5.1. Reported Claims This section will focus on developing a detailed description of the claim development process, by considering what happens within an individual time-step and how that behavior varies across claim and policy characteristics. Rather than focus on the ultimate value of claims (which is only observable for the older claims), we examine the behavior of a claim in a time-step. What is the likelihood that a claim will close in the next quarter? What is the likelihood that it will change in value? If it does change in value, how much? What is the probability of a payment? If the claim does have a payment, how much? By considering these components of development, we can develop an understanding of the process that can be extended to ultimate while still incorporating information from immature Including covariates introduces modeling challenges when we extend the model to ultimate. For example, the case reserve is itself a predictive variable. The single period time-step model is not easily combined together across development ages to project an ultimate value. An individual claim may develop to varied potential values in the next step. Using the mean predicted case reserve and using it as a predictor for the next step is inappropriate. This is not an issue with more commonly used actuarial reserving methods because there is an implicit assumption of independence of the development factors and the paid and/or incurred amounts to which they are applied^[5]. Using this assumption of independence at an individual claim level is extremely problematic and unrealistic, which is why simulation across alternative paths is necessary. The following example illustrates this concept. Consider a simple claim development process in which an open claim has three possibilities in the next time-step: • the claim will close for nothing (with 1/3 probability) • the claim will close, paying out the current case reserve (with 1/3 probability) • the reserve will increase by 1, with no payment in the time-step In this example, a claim currently open, with a case reserve of 1 has an expected value after one time-step also equal to 1 (01/3 + 11/3 + 21/3). If we take this expected value, treat it as a case reserve in the next time-step, and move it forward, it will also have an expected value of 1. Carried forward infinitely, the value is always 1. But when we consider each of the possible paths that this open claim could take, we see that this approach is incorrect. The expected value one time-step out is indeed 1, but two time-steps out it is (04/9 + 11/3 + 21/9 + 31/9) = 8/9. After three time-steps the expected value is 22/27. As the number of time-steps approaches infinity, the expected value of the claim approaches 3/4. This illustrates the problem of using the mean of a probabilistic model as an input in a subsequent model (either a later time-step or another component model). In order to develop an estimate of ultimate loss from these time-step models with covariates, we need to describe not only the mean result in a time-step for a given claim with its given characteristics, but also the distribution of potential results in that time-step for that claim. With the introduction of various component models, as discussed below, this becomes even more Possible approaches to projecting results for individual claims over multiple time-steps (and combining together the component models discussed below) include formulaic or numerical integration and stochastic simulation. With the level of complexity involved, and with flexibility of model choices regarding characterization of the distributions of the component models (and to a certain extent the component models themselves), this paper will concentrate on the simulation approach. Simulation practicalities suggest a need for modeling specific facets of claim development. When considering payment amounts and changes in case reserves, there are probability masses at zero (i.e., no payment and/or no change in the reserve). Instead of trying to incorporate these probability masses in the distribution of results, it is helpful to break the development process down into components which are modeled at each time-step for each open claim within the simulation. Examples of these components are whether a claim closes, whether the value of a claim changes, whether there is an incremental payment, how much is the change in the value of a claim given that one occurs, and how much is the payment given that one occurs. The process is to model these behaviors individually and sequentially for each open claim at each time-step and simulate them accordingly^[6]. Thus, the initial task on the path to building a comprehensive model of claim-level development is to build time-step models of each of these development components. In addition to highlighting differences between claim-types, this yields insight into changes occurring within the development process and their impacts on the needed reserve. A specific model framework is provided here as an example. This is by no means the only approach that could be taken. By showing a specific example, we illustrate how to overcome some particular Claim Development Models: • Closure Probability • Change Probability • Payment Probability • Change Amount (Large) • Change Amount (Small) • Reopen Probability • Reopen Amount • Recovery Probability • Recovery Amount Before discussing each of these models individually, we first define some terms that will be used across the different models. 5.1.1. Definitions (for a given claim at a given time-step) Beginning Case Reserve –case reserve at the beginning of the time-step Ending Case Reserve – case reserve at the end of the time-step Paid Loss – incremental paid loss amount within the time-step Previous Paid to Date – total of all paid Loss for previous time-steps Ending Value – Ending Case Reserve + Paid Loss. This represents an amount that is comparable to the Beginning Case Reserve. “Loss” here is used generically to mean indemnity, expense, medical payment, or any combination of these^[7]. General Variables included in each model Beginning Case Reserve Development Age Transaction Date Accident Period Claim Characteristics Exposure Characteristics Previous Payments Development Age, Transaction Date, and Accident Period are redundant within two time dimensions. It is useful to consider each of them when constructing a component model because they each represent different things, but in a final version of any component model, it is advisable to include at most two of these variables to avoid model instability and complexity. Parameters for Transaction Date can reflect systematic changes that have occurred, but care will need to be taken to consider the prospective outlook, which may differ from the past. Accident period parameters can also be predictive, but often indicate changes in characteristics that have not been identified. Where possible, it is optimal to find and include such characteristics directly. Care should be taken to avoid using accident period as a proxy for development age (since the more recent periods will contain only immature development ages). In such cases development age should be used in place of accident period to avoid projecting development for immature accident periods that is characteristic of early development as they progress into later development periods. 5.1.2. Potential Models to be Employed in a Time-Step Model The following are examples of component models which can be used in the development of a time-step modeling process. 5.1.2.1. Closure Probability Model This model estimates the probability that a given claim will close within the time-step. Definition: P(Ending Case Reserve = 0 \| Beginning Case Reserve > 0 or Reopened = True) We are defining a claim as being open by considering the case reserve. When the case reserve is larger than zero, the claim is considered open. Using the case reserve as the indicator avoids potential issues with inconsistent coding of claim status over time or timing discrepancies between status changes and case reserve changes. There are sometimes notices of claims, particularly in claims made lines, that may be then considered open but have no case reserves. Consider using a notional case reserve of some small amount to identify such claims for actuarial modeling purposes if this approach to defining claim status is used. 5.1.2.2. Change Probability Model (for claims remaining open) This model estimates the probability that a claim will change in value during the time-step. There is a possibility that in a time-step there may be no change in value. When projecting losses forward, reflecting this probability mass is more realistic than simply modeling the change in values broadly. The probability of claim changing in value is typically very high for a claim that is in the process of closing. Including a variable that indicates whether the claim closes in the quarter would capture this, but it is likely that there would be numerous interaction effects between this variable and the others. For that reason, we have separated this model into one that considers only those claims that are remaining open vs. those that are closing. Definition: P(Ending Value ≠Beginning Case Reserve \| Beginning Case Reserve > 0 and Ending Case Reserve > 0) Note that we are excluding claims that have zero case reserve at the beginning of the time-step. Changes in values of these claims is contemplated by the reopen probability and reopen amount models. Also note that the very definition of this model depends on what is considered the result of one of the other component models (the closure probability model). This dependency will be important to consider when it comes time to simulate development. 5.1.2.3. Change Probability Model (for closing claims) This models the probability that a claim which is closing will change in value. Definition: P(Ending Value ≠Beginning Case Reserve \| Beginning Case Reserve > 0 and Ending Case Reserve = 0) Often this probability is very close to 1. For many cases quantifying this probability across all claims may be sufficient, with no additional differentiation provided by predictive variables. 5.1.2.4. Reopen Probability Model This model is for the possibility that a given claim, closed at the beginning of the timestep, will have additional payments during or case reserve at the end of the time-step. This also includes payments on claims technically not reopened, but where additional payments occur after the case reserve goes to zero. Definition: P(Ending Value > 0 \| Beginning Case Reserve = 0) The number of time periods that the claim has been closed is a key predictor, with additional payments often occurring in the period immediately following claim closure. 5.1.2.5. Reopen Amount Model Given that a claim which was closed at the beginning of the time-step has additional (positive) payments or case reserves in the time-step, what is the amount? Definition: E(Ending Value \| Beginning Case Reserve = 0 and Ending Value > 0) In this approach, the ending value, i.e., the ending case reserve amount plus the incremental payment, is used as the target variable of the model. The portion of this value that is paid out vs. that that remains as case reserves at the end of the time-step will be covered in the partial payment model. 5.1.2.6. Change Amount Model(s) This model defines the changes in value of a claim that is open at the beginning of a time-step, given that it changes. The ending value (incremental payment plus ending reserve) of the claim in the time-step is expected to be strongly related to the case reserve at the beginning of the time-step (i.e., a strong positive correlation between the two), albeit with significant variation. This relationship is far from proportional, however, with small case reserve amounts growing by much larger factors on average than large case reserve amounts. It is easier for a $1000 reserve to grow by a factor of 20 than it is for a $1,000,000 reserve. With a multiplicative factor model framework, the case reserve amount itself will often become the most important predictive variable, and the model can become very sensitive to slight binning changes for small case reserves. Another issue is that often for very small case reserve amounts, the monotone increasing relationship between the beginning case reserve and the ending value breaks down. Often this is due to the existence of “signal” reserves or other place holders that do not necessarily have a monotone relationship with the ending value. For example, a value of “1” for the beginning reserve may specify a particular type of claim, “2” indicate a different type of claim and there be no expectation that a “2” claim would be double the severity of a “1” claim or even that it would have a higher severity. For this reason, it is often beneficial to use separate models for small case reserve claims and for large case reserve claims. It is helpful to transform the beginning case reserve itself into an exposure variable more closely related to the ending value, before even considering the impact of other predictive variables. One such “change amount exposure” variable is shown graphically below: Below the small case cutoff value, there is not an increasing modeled relationship between the beginning case reserve and the expected ending value. Above that value there is a linear, increasing relationship between Beginning Case Reserve and the expected Ending Value. The inclusion of an intercept in the relationship for large cases provides the more significant multiplicative differential between smaller beginning values and the ending value. This avoids the problem of sensitivity to bin determination for the case reserve variable. The parameters defining this transformation are the two linear parameters above the cutoff (two for claims that are closing and two for claims that will remain open), the single parameter below the cutoff, and the cutoff itself (six total parameters). Given a specific cutoff, the linear parameters can be calculated by least squares regression, and the parameter below can be calculated as an average of the ending value below the cutoff. The total least squares across the entire set of observations can be tabulated, and the optimum cutoff value can be determined using numerical minimization of the least squares amount. In some cases, the minimum least-squares amount will be at a cutoff of zero. It is appropriate to limit the other parameters to be non-negative as well, so boundary solutions should be considered. Preliminary Calculation Where Ending Value ≠ Beginning Case Reserve, define Change Amount Exposure = C[small,0] where Beginning Case Reserve ≤ Small Case Cutoff C[close,0] + C[close,1] Beginning Case Reserve where Beginning Case Reserve > Small Case Cutoff and Ending Case Reserve = 0 C[open,0] + C[open,1] * Beginning Case Reserve where Beginning Case Reserve > Small Case Cutoff and Ending Case Reserve > 0 with C[small,0],C[close,0], C[close,1], C[open,0], C[open,1] and Small Case Cutoff estimated by minimizing least squares on the training data with Ending Value as the target variable. Change Amount Model (small case reserve) This model covers the cases where there is not a generally increasing relationship between beginning case reserves and ending value in the time-step. Definition: E(Ending Value \| Ending Value ≠ Beginning Case Reserve and Beginning Case Reserve > 0 and Beginning Case Reserve ≤ Small Case Cutoff) The binned case reserve can be used as a categorical variable in this model (together with the other variables being considered). If there are specific signal reserve values, they can be set as distinct bins. A more sophisticated model would be to build models of changes of state from one claim type to another, within the different types of signal reserves if it is common for claims to transition from one type to another before transitioning to an actual case reserve reflective of the expected loss payment amount. Change Amount (large case reserve) This model reflects changes in value for claims with beginning case reserve larger than the cutoff value. Definition: E(Ending Value \| Ending Value ≠ Beginning Case Reserve and Beginning Case Reserve > Small Case Cutoff) Using the Change Amount Exposure variable discussed above, a multiplicative model can be used by setting the expected ending value of the claim equal to a base factor multiplied by the change amount exposure variable, multiplied by modifiers for each of the other variables being considered. The Change Amount Exposure variable may also be binned and treated as a categorical variable to capture possible imperfections in the simple linear relationship used to create the exposure variable. 5.1.2.7. Payment Probability Model This model describes the probability that there is a payment on a claim for which one is possible. Definition: P(Paid Loss > 0 \| Ending Value > 0) Notice that the way we have defined the “Ending Value” variable handles all the possibilities for payment since payment itself is included within the ending value (if paid loss is > 0 then ending value must also be > 0). It may seem circuitous to construct the model in this way, but it is helpful to have a single model (the change value model) that governs both the incremental payments and ending case reserve generally, and then we consider a potential payment that is bounded between 0 and the ending value. An important variable to include in this model is whether the claim closes in the period. As such, this model will be dependent on the closure probability model when projecting forward. 5.1.2.8. Partial Payment Amount Model In the case where a payment occurs in the time-step and the claim closes, the payment amount is equal to the ending value variable. In cases where there is a payment made but the claim remains open (i.e., ending case reserve > 0) we need to know how much of the ending value is in the form of a payment and how much remains as case reserves. This partial payment model describes that relationship. Definition: E(Paid Loss \| Ending Case Reserve > 0 and Paid Loss > 0) The Ending Value variable is important for this model, giving an upper bound for the payment amount. 5.1.2.9. Recovery Probability Model What is the probability that a claim with previous payments will receive a recovery (i.e., negative payment) of some amount within a given time-step? 5.1.2.10. Recovery Amount Model Given that there is a recovery in a time-step, what is the amount? 5.1.2.11. Dynamic Variable Model(s) Variables that change over time (dynamic variables) pose additional challenges, just as they do when used for segmentation of triangles in a traditional analysis. Often such variables are predictive for future claim behavior but in order to be incorporated, their ability to change must itself be modeled. Consider a “pension indicator” variable that indicates a workers compensation claimant is receiving permanent indemnity payments. That determination may change several development periods after the initial determination is made. If segmentation of reserving triangles uses this indicator, history changes if only the current state of the claim is used in the segmentation. A similar issue exists in the claim life cycle model approach. Training a model using the current value of a dynamic variable for observations before it took its current value represents a model “cheat” and is not appropriate. The value of that variable that existed as of the observation should be used. This is directly analogous to the triangle segmentation problem described above. If the variable is to be used as a predictor, then it also becomes necessary to model and simulate changes in that variable. An example of one of these state change models is for a dynamic variable that takes values A, B, C, and D. Four separate predictive models could be created: • What is the probability that there is a new value? • What is the probability that the value becomes A given that there is a change? • What is the probability that the value becomes B given that there is a change, and it doesn’t become A? • What is the probability that the value becomes C given that there is a change, and it doesn’t become A or B? There is no need for an additional model for the probability of becoming D since the other models fully describe this situation. The value of the variable at the beginning of the time-step is typically an important variable. Each of the other predictive variables should be considered as possible predictors. 5.1.3. Claim Development Simulation Each currently open claim is simulated forward one time-step using each of the Claim Development Predictive Models over a specified number of paths. Those paths still open are simulated forward another time-step. This process is continued until all claim paths are closed. Additionally, claim re-openings are simulated and projected until they are re-closed, both for currently closed claims, as well as for currently open claims that close. Before time-step 1: 1. Generate a number of paths for each open claim 2. Simulate reopening from current inventory of closed claims, schedule them for reopening in later time-steps, and assign a path number 3. Simulate Ending Value for each of the reopened claims in the time-step in which reopened In each time-step, for each claim-path combination: 1. Increment development age 2. Simulate changes in dynamic variables (other than the case reserve) 3. Select which of the claim-paths close 4. Select which of the claim-paths change in value 5. Select which of the claim-paths have a payment 6. Simulate Change Amount Exposure for each claim-path 7. Simulate Ending Value for claim-paths with Beginning Case Reserve > Small Case Cutoff 8. Simulate Ending Value for claim-paths with Beginning Case Reserve <= Small Case Cutoff 9. Set Ending Value = Beginning Case Reserve for claim-paths that do not change in value 10. Simulate Paid Loss on (0, Ending Value] for those claims having a payment 11. Set Ending Case Reserve = Ending Value – Paid Loss 12. Select which closed paths will reopen later, and schedule them 13. Simulate Ending Value for each of the claims to be reopened 14. Repeat the process until Ending Case Reserve is zero for all claim-paths Notes on Simulation Random selections for binary models (such as Closure) are based on calculating the probability from the appropriate predictive model (using predictive characteristics) and simulating a Bernoulli. The simulations for continuous variables (such as Ending Value, Paid Loss, etc.) are more challenging. Distributional forms can be used, but they are likely to be naïve with regard to distributional differences across variables. It is not the exception, but rather the norm, that the variance to mean as well as relationships for higher moments are inconsistent across the data. This problem becomes more problematic for this simulation due to its chained nature across time. The simulated outputs from the first time-step are the inputs into the second time-step, the second into the third, and so on. Simplifying assumptions that may be reasonable in a single time-step may distort into unrealistic projections when compounded. The case reserve itself is typically one of the more important variables predictive of changes, with small reserves able to grow by a large factor, and large reserves unable to grow by large factors. Imposition of limits, actual or practical, can also keep simulations from developing out of control. One approach to reflecting nuances not necessarily reflected in a single distributional form is to use bootstrapping techniques to simulate. In this way, differences in variability and higher moments across different categories of claims can be reflected. With many variables, it is unlikely that there will be sufficient observations of each combination of variables to represent the potential variability for any given risk, but by dissembling error terms across variables, randomly sorting, and then recombining them, a more nuanced reflection of variability can be achieved. By sampling residuals from the test data instead of training data, model and parameter risk are contemplated. The approach is highlighted in the following steps: • Apply the predictive model to the records in the test data • Calculate the residual for each test data record • Allocate/disaggregate the residuals to each of the various predictive characteristics for each record (we will discuss this step in greater detail below) • For a given claim-path-timestep to be simulated, randomly select one disaggregated residual for each predictive characteristic, from among the set of matching characteristics from the test data • Combine the residuals for the claim-path-timestep to a single residual • Combine the modeled residual and the expected result to give a simulated value • Apply limits or other constraints • Rescale the mean and variance across paths if necessary The disaggregation of test data residuals across predictive characteristics is what allows this approach to generate variability patterns that are like what has been observed for similar claims in the past, while still allowing for combinations of characteristics that have not been observed. With skewed, positive distributions, it is helpful to use multiplicative residuals rather than additive The example below illustrates the concept of this type of bootstrapping with two predictive characteristics, State and Class. The concept is generalizable for more variables. The square roots in the last two columns of the above table are in recognition of the reshuffling of residuals across characteristics that will occur in the simulation. If columns I and J were used directly without the square root, resulting variability would be too low, because the correlation between the disaggregated residuals at the record that exists with the observed residuals is eliminated when the simulation draws are performed independently across the characteristics. Note that in the above approach, more of each observed residual is assigned to the variable with the stronger predictive effect. The predictive factors in this example are normalized to 1, so a factor close to 1 is an indicator that the characteristic value does not describe much of the difference in the target of prediction. The embedded assumption in assigning the residuals in this way is that the stronger the factor (further from 1) the more of the residual is assigned to that variable. The table below shows a couple of simulated results using the approach. The date of the transaction (calendar period) may be one of the predictive variables in one or more of the component development models used in the simulation. Often this will describe changes in claim handling and the underlying claims environment. The actuary should take care when considering the appropriate factor to use in simulation as the future dates do not have an explicit factor. One logical option is to use the most recent observation. Another may be to use the long-term average. The impact of such a choice can be quantified by comparing alternative simulations varying the 5.2. Unreported Claims - Component Emergence Models and Simulation Earlier sections focused on the development of known claims. We must also estimate of the ultimate cost associated with unreported claims. Not having been reported, the potential for these claims is driven not by claim characteristics (which do not yet exist) but only by exposure (i.e., policy) characteristics. To simulate claims with specific characteristics detailed models to predict the characteristics are necessary. It is easier to simulate the ultimate values of the IBNR claims directly, rendering the claim characteristics unnecessary. Since the ultimate value is being directly simulated, the timing of payments is not simulated. The simplification of the simulation process by concentrating on ultimate payment amount is dramatic, however. Timing of payments on IBNR claims (including their impact on inflation effects) may be modeled separately if this is an important desired output. 5.2.1. Component Emergence Models Describing the simpler approach, there are three basic component models required to predict the unreported claims – report lag, frequency, and severity. For each of these, we will focus on claims that have a non-zero ultimate value. 5.2.1.1. Premium Model Premium is a natural starting point for modeling frequency and severity. In traditional triangle reserving approaches, Bornhuetter-Ferguson analyses use premium as an input, but it is important to ask what premium should be used. Collected premium can introduce inconsistencies due to differing levels of rate adequacy. These inconsistencies will distort the results. Premium at a consistent rate level across all policies is ideal. Neutralizing changes in rates charged is useful. (Bodoff 2009). This is also true for detailed reserve modeling. While companies often have processes to measure changes in rates over time, these measures often are problematic. One approach is to compare historically-rated and re-rated premium by policy. This method breaks down when discretionary pricing factors such as schedule mods are significant. To overcome this challenge, the premium charged for each expiring and renewing policy is compared, which considers impacts like changing schedule mods. New and non-renewing business is ignored. If either of these ignored cohorts were written at a different rate than the renewing book, the impact to the rate level is not captured. Sometimes average rates or average mods over time including new and renewal accounts are considered, but these measures assume that the mix of business has remained constant, which is rarely the case. A predictive model can be used to simultaneously incorporate all these considerations to measure changes in premium rate level over time. The target of the prediction is the premium itself. The predictive variables are the rating and underwriting characteristics, such as class, geography, deductible, limit, new vs. renewal, etc. The policy effective date is a key predictor indicating rate changes over time, adjusting for changes in mix. Interaction effects should be considered between the policy effective date variable and other variables as it may indicate targeted pricing actions. Since the parameters for policy effective date provide a rate change across time, adjusting for other variables, the resulting policy effective date curve represents a vector of rate adjustment factors that can be used to restate historical premium to a common level. If interaction effects were found between effective date and other variables these adjustments should be included in the This on-level premium, which we will call the “Reference Rate” is useful as a starting point for predicting claim frequency and severity. It is a modeled premium that is normalized for changes in rate over time and reflective of statistically significant impact of key rating variables. Note that this premium is not necessarily actuarially sound but is stated at a consistent (or benchmark) level of adequacy for all policies within the book being analyzed and reflective of policy characteristics. As such, this Reference Rate premium is an appropriate base for frequency and severity analyses because it removes distortions arising from differences in rate adequacy over time and across accounts. In addition to the Reference Rate premium, we can calculate the ratio of Written Premium to Reference Rate premium at the policy level. This can be included as a separate predictive variable in the models. This ratio may be impacted purely by market forces (in which case it is unlikely to be predictive of frequency or severity) or it could be indicative of differences in the perceived risks of the policies not captured by the other fields included in the analysis (in which it could be predictive). Including it as an additional predictive characteristic will help make the determination. The modeled premium as well as the ratio of written premium to reference rate are being discussed in this section due to their high importance for modeling frequency and severity. These characteristics are also potentially useful for the other models previously discussed. It is beneficial to run the premium model before running these other models so that these two new variables can be included as predictors. 5.2.1.2. Report Lag Model We define the report lag as the time difference between the incurred date and the date reported. One problem with modeling the reporting lag across policy characteristics is that we have incomplete data that we would like to incorporate. Average observed lags are conditional on the claim already having been reported. Our true target is the unconditional report lag, i.e., the average lag after all claims have been reported. One way to approach this is by adjusting the observed lag for each claim to try to remove the bias given that it is conditional on the age of the policy. This adjusted observed lag for each claim can then be used as the target of prediction in our model, allowing us to model differences across predictors. First, we calculate an expected lag L on non-zero lagged claims across the portfolio by starting with the observed average and Where Report Lag > 0, iterating until L converges. Claim Age is calculated based on the ‘as of’ date of the analysis. A value of cdf[j] is determined for each claim, equal to the observed percentage of claims reported by the same lag or earlier as the claim in question, excluding from the calculation any claims that are older at the valuation date than the claim in question. For example, if a claim were reported 25 days after the incurred date, and the claim is now 190 days after the report date, cdf for that claim would be the percent of those claims that were reported within 25 days within the population of claims that were incurred and reported within 190 days. lagVariable (the target of prediction) for the claim is then set as: lagVariable = - L * log(1-cdf where Rj > 0 and the simulated ultimate > 0 lagVariable = 0 where Rj = 0 and the simulated ultimate > 0 lagVariable = NULL where simulated ultimate = 0 (claims with ultimate=0 are excluded from the report lag model) 5.2.1.3. Frequency Model Observed claim frequency is dependent on the maturity of the policy, due to the lag between incurred date and report date. Therefore, reporting lag should be modeled first, and the measures of claim frequency can be relative to a premium that has been adjusted to reflect the maturity of the policy due to this reporting lag as well as for any unearned portion of the policy. In addition to adjusting for earning and report maturity, using premium that adjusts for rate level changes such as by using the Reference Rate premium described earlier avoids problems with differing rate adequacy. The target of prediction is the observed non-zero claims count for each policy. For the closed claims, this is trivial, but for the open claims the author suggests selecting one single path at random from the claim development simulation to determine whether each claim is non-zero for purposes of supplying this target. In addition to policy fields of interest, the effective date of the policy should be included as a potential predictive variable to measure frequency trend. In the case where premium is not adjusted to a constant rate level, the effective date variable will also reflect differences in rate. 5.2.1.4. Severity Model For modeling claim severity differences across policy characteristics, we are interested in ultimate claim severity. Case-incurred losses include whatever distortions exist in the case reserves. One solution to this problem is to use closed claims only when building a severity model. Unfortunately, this introduces bias due to differences between open and closed claims. One approach to removing open/closed bias could be to include closed claims only from periods that are essentially fully developed, but this will exclude valuable information about more recent claims. When trend is present or where the underlying claim severity environment is otherwise changing, this loss of recent information is problematic. Instead, by first developing the known claims to an ultimate level, before modeling differences in claim severity, we can include the open claims – eliminating the open/closed bias, but without the distortions caused by case reserves developing differently across different types of claims. Using the mean projection of non-zero ultimate payments for a claim on those claims that are currently open will tend to underrepresent the variability of the ultimate claim value. This can be problematic when being used in predictive models, particularly when testing alternative models against each other, but also simply for characterization of the variability of severity generally. For this reason, it is useful to select the same simulated path for each open claim that was used in determining non-zero claim count and including only those selected claims that develop to a non-zero value for the severity analysis. The relationship between report lag and loss severity is typically strong, and generally the difference between the incurred date and the reported date should be included as a potential predictor. As with claim frequency, the policy effective date is a potential predictor, representing severity trend. Modeling claim severity is one of the more challenging aspects of predictive modeling owing to severity’s high skewness. For smaller sections of the portfolio of observations the lack or inclusion of a single large observation can make a significant difference in the measurement of severity. Adding a credibility component to the modeling process will help avoid being too sensitive to the observed data. However, when observations cannot be completely relied upon it puts higher importance on the complement of credibility. Even if credibility adjustment is not made other than as a binary choice of whether a parameter is “in” or “out,” the complement of credibility is important. A common assumption is that if a statistically significant difference is not found, none exists. This is a dangerous assumption, particularly for small segments of the data. For example, consider a workers compensation insurer that writes mostly low-hazard accounts. The few high-hazard accounts likely have had few claims given their low-frequency nature. It is likely that there is little or no statistically reliable difference between the severity that has been observed between the high-hazard and low-hazard accounts. It would be a mistake to say that these two groups of accounts have the same severity. The insurance marketplace as well as the rates being used at an insurer (often reflective of broader experience, such as a me-too filing or bureau rating) includes important implicit information about severity potential. Frequency modeling is far more robust than severity, and the ratio of premium to modeled frequency works well as an exposure variable, being an a priori indicator of severity potential. If statistically significant differences are not found relative to this expected severity, the market or rate plan wisdom simply remains unaltered. Said differently expected loss ratio is more likely to be consistent across risks than severity. To generalize on this concept, we can determine this “expected severity” exposure variable to be equal to the prediction from a linear regression of non-zero ultimate claim amount to the ratio of Reference Rate premium to modeled frequency (at the claim level). In this way the hypothetical relationship between premium, frequency, and severity can be broadly tested rather than assumed. If the slope parameter is not statistically significant the a priori opinion for the covariate model is “no difference in severity.” If the constant parameter is negative or statistically insignificant it can be left out and the a priori opinion is “severity strictly inverse to frequency.” With both parameters present, there is an inverse relationship between frequency and severity, but flatter. 5.3. Unreported Claim Simulation The simulation process for the unknown claims is as follows: 1. Each policy that still has potential for claims is assigned a policy maturity factor based on its modeled report lag and the portion of the policy period that has been earned. Expected unknown claims are calculated as the premium multiplied by (unity minus the maturity factor), and then applying the frequency model to this amount based on account characteristics. 2. Individual claims occurrences are then simulated for each policy with a mean equal to this number of expected unknown claims. Paths are assigned randomly. 3. Date of Loss and Report lag are simulated for each of the emergence claims according to the report lag model (Date of Loss is necessary to generate incurred period statistics). 4. Ultimate severity is simulated for each of the emergence claims according to the severity model. After simulating emergence, a view such as the following can be constructed. This report can be directly compared to what is produced using a traditional triangle analysis, with greater detail owing to the specifics observed during simulation. 6.0. Alternative Data Elements and Triangle Specification This section will discuss the creation of Actuarial Case Reserves (for reported claims) and Actuarial Policy Reserves (for unreported claims) based on the observed history of closed claims, supplemented with the simulated projections. The benefits of creating these new data elements are significant across actuarial practice and insurance company management. 6.1. Building and using an Actuarial Case Reserve Algorithm One benefit of using the time-step component models and simulating individual claims to ultimate is that additional insight into the claim life cycle is gained. Differences in settlement rates of claims across predictive variables, volatility differences, etc. are revealed. Changes in claim management practices over time are more easily characterized and identified by the actuary. On the negative side, the simulation process to bring all the component models together and project over multiple time-steps can be challenging to audit due to its inherent complexity and can be time-consuming. One approach to summarizing the combined result of the component development models and the resulting simulation to ultimate is to create an Actuarial Case Reserve Algorithm. The motivation for the creation of Actuarial Case Reserves goes beyond that of summarizing and validating the results of a Claim Life Cycle Model. It has the potential to solve many of the actuarial problems with the use of case reserves themselves. Case reserves can provide useful information to the actuary about a large portion of the total reserve need. However, changes in how case reserves are established and revised cause significant problems for traditional triangle analysis. This leads actuaries to often put pressure on claims departments to set case reserves consistently. Since the amount of the case reserve can be an important consideration in the handling of the claim, this pressure can lead to sub-optimal decision making and results at the claim level. Because case reserves are rarely established on a true expected value basis, changes in claim settlement rates are also problematic for triangle analysis. This also leads to pressure from actuarial departments to the claim department to maintain the status quo, potentially leading to sub-optimal economic results as well. Changes in case adequacy and claim settlement rates are commonly dealt with by actuaries by using Berquist-Sherman techniques to adjust for these changes (Friedland 2010, ch13). Unfortunately, these adjustments may be inappropriate when applied injudiciously. Consider the hypothetical scenario of a company that with struggling financial results, begins to write higher severity classes of business. Lacking experience in those classes, and desperate for premium, the company underprices the new policies. When loss ratios begin to develop upward, the actuaries, under pressure to reduce reserve estimates, note that the average case reserve amount is higher than in the past. Adjusting historical case reserves to the current level using a Berquist-Sherman adjustment, historical development is reduced, as are the reserve estimate and apparent loss ratios. The company continues to write the policies at an unprofitable level and a large reserve deficiency continues to grow. The problem is not with the Berquist-Sherman technique itself, but rather that it was inappropriate to use it in this situation because the increase in case reserves was not due to an increase in case adequacy, but to a changing mix of business. Detection of such changes is difficult when only aggregated triangle data is considered, particularly given the wide range of variables that could be shifting (industry classification, geography, deductible, limit, etc.) The unreliability of subjectively determined case reserves has led some to conclude that case-incurred loss development should be relied on less than paid loss development when estimating total reserves (Zehnwirth 1994). Taken to an extreme, an actuary may conclude that the case-incurred triangle is completely inappropriate to use. However, information is lost by excluding the case reserves. Ignoring the information contained in case reserves by the actuary is usually imprudent. Consider a small insurer that has seen an abnormally large number of full limits losses. Estimating a total reserve need that is based only on paid losses observed to date and ignoring the case reserves would be inappropriate. Even for a large insurer that may be able to rely on paid information only to establish a reserve estimate, the need for information at more granular levels for internal and external reporting purposes suggests that case reserves are not easily ignored. Also, changes in closure rates over time such as those observed during and after the Covid-19 pandemic are particularly distortive to paid loss triangles. Rather than ignoring or discounting the information contained in booked case reserves due to their subjective unreliability, an approach that uses the objective information about open claims in a reserve analysis avoids many of the problems with traditional adjuster case reserves, while providing information to the actuary about the payment potential of the currently open claims. The approach involves the following steps: • Determine, for every point in time that every claim was open, what the hindsight case reserve should have been (based on observed results for closed claims and simulated results for open claims). • Build a predictive model targeting this hindsight value, using objective, consistent claim characteristics as predictors. • Apply this Actuarial Case Algorithm retrospectively to every open claim for each triangle cell that the claim was open. (This can be done prospectively to new data as well) • Replace the adjuster case reserves in the case-incurred triangle with the actuarial case reserve. • Develop the triangle as usual. 6.1.1. Organization of the Model Each claim that was open at any of the triangle evaluation points (observation dates) has records included in the table for each such date. Fields that were included as predictive in the various component development models are good candidates to include in the case reserve algorithm as predictors, excluding the claim department case reserve (which will be discussed below). Even though this model is summarizing and simplifying an existing process, it still is appropriate to aim for a parsimonious model and to separate the data into training and test, etc. One benefit of holding out data is that when it comes time to demonstrate the effectiveness of the model to others (section 6.1.4), illustrating the development on test data triangles is very powerful for demonstrating its veracity. The target variable for this model is the sum of payments after the observation date. This includes payments that have already been made as of the date of the analysis as well as simulated payments following that date (B+C in the timeline below). To maintain consistency between the history and the projections (i.e., same variability), use the results from a single simulated path for each claim rather than the mean for determining the ultimate value. All training claims that are open as of each observation point should be included, even those that ultimately close without payment since this information is not yet known when the claim is open. A Specific Simulated Path (payments) for a Selected Claim: Some of the fields that are likely to be of predictive value are: • The age of development • Prior paid amounts (often useful to separate these into recent payments and older payments as well as indemnity, expense, etc.) • The claim limit remaining • Cause of loss • Injury type • Geographical area • Business/Industry classification • Information about the claimant • Claim Severity Classification • The accident period (trend) Note that it is acceptable to use dynamic variables (those that change over time) in a case algorithm model (e.g., severity classification, litigation status, etc.) Care should be taken to ensure that the values of these dynamic variables are the values as of the time of the prediction, not as of the most recent valuation. Otherwise, significant distortions can result, as in any predictive model when future-valued predictive variables are inadvertently used. One goal of the actuarial case reserves is to avoid the impact of changing case reserve adequacy over time. Adjuster reserves that are subjectively determined may include valuable information that is not available in coded fields, but they introduce the potential for adequacy changes. By constructing an algorithm that does not depend on adjuster reserves, we avoid this problem. While the adjuster case reserves provide important information about future payments in the modeling-simulation part of the claim life cycle model, we seek to use that information, and then condense and assign that knowledge back to objective, consistent, predictive fields. That means eliminating the claim department case reserves from the actuarial case reserve algorithm. The observation date is a possible predictive variable, but care needs to be taken when using it. It can be a powerful way to measure and incorporate the impacts of loss cost trend, but its use can be counter to the goal of consistency over time, if not used carefully. There is predictive value in the date because of loss trend. This can be thought of as a generalization of the Berquist-Sherman technique in that the actuarial case reserves are systematically adjusted to reflect systematic differences over time. If the date is used as a predictive variable with no constraints put upon it, though, it is quite likely that the parameter will fluctuate from period to period. We seek to isolate the impact of the date after adjusting for mix shifts in the other variables being used, but additional non-included variables may be proxied. Also, development age and observation date are related, and although a multivariate model attempts to isolate their relative importance, it is easy for development age impacts to bleed into the date variable if it is unconstrained (date becoming a proxy for development age). If care is not taken to constrain the date parameter, and then it is applied to generate the algorithmic case reserves, the effect would be to reintroduce some of the biases that the model is working to eliminate. To keep the observation date from having this effect, simple constraints can be put on the calculation of the date parameter. For instance, it might be reasonable to assume that trend will impact claim severity in a linear fashion or an exponential fashion (or according to some other rule). There is judgement required by the actuary in deciding what rule will best describe the impact of trend without allowing the actuarial case reserves developed from it to fluctuate haphazardly, contrary to the goal of consistency of the algorithm over time. 6.1.2. Applying the Actuarial Case Algorithm Once the Actuarial Case Algorithm has been constructed, it is straightforward to apply it to all of the open claims at each point in the loss history, and then to proceed as normal with triangle While it is tempting to build the triangle directly from the detailed development data, there are often individual discrepancies in the data due to mismatches between the loss and exposure data or claims that do not lend themselves, for whatever reason, to the process (coded through an alternative system, etc.). Rather than deal with such discrepancies, it is more straightforward from an audit perspective to simply provide the adjustments to the claims that can be modified, summarize the modifications by accident period and development age, and modify the triangle accordingly. This will ensure that all claim payments are still captured and that if an alternative case reserve was not possible for some claims, they will at least be included in unadjusted form. Once developed, an actuarial case reserving algorithm can be easily applied for future analyses provided the predictive variables are available. Triangles can be quickly updated as a new step in the regular reserve analysis process, without re-determination of parameters. Re-parameterization of the parameters can therefore be done less frequently, for example annually instead of quarterly. After constructing the alternative triangles, actuarial analysis can be completed as usual (LDFs, etc.). 6.1.3. Validation of Results With a case reserving algorithm that is openly and objectively applied to historical points in time in a triangle, and then calculating development factors in the usual way, the efficacy of the case algorithm itself can be justified. Regardless of how the case algorithm was determined, its ability to generate consistent, unbiased aggregate development of reported losses as claims transition from early stages of reporting, through interim payments and into the final settlement of the claims is evidence of the algorithm’s veracity and strength as an actuarial tool. From an audit perspective, the calculation of a case reserve generated from objective claim and policy characteristic is more transparent than a subjective case reserve selected by a claims adjuster. If a report period triangle shows development factors, centered around 1.0, across multiple cuts of the data, evidence of its appropriateness is provided (particularly when using hold-out data, not used to build the 6.1.4. Illustrating Changes in Case Reserve Adequacy With the case algorithm established, it is illustrative to look at the relationship between the actuarial and traditional case reserves. It is likely that there will be a significant difference between the two because 1) the adjuster case reserves are subjective and 2) it is not necessarily the intention of the claim department for the case reserve to be at actuarial expected value. In fact, from a claim management perspective, an actuarially appropriate estimate may very well be less than ideal, because it would incorporate relatively rare extreme events, (e.g., the mean may be at the 70^th or 80^th percentile of potential outcomes). Setting the case reserve at that level may encourage higher settlements. In many actuarial reserving contexts (e.g., Berquist and Sherman 1977) it is stated that inadequacy or redundancy in booked case reserves is not necessarily problematic in a triangle as long as it remains consistent. Comparing the booked case reserves to the objectively determined actuarial case reserves is a useful approach to identifying and illustrating changes in case adequacy and therefore potential distortion to aggregated triangle data using traditional case reserves. If the relationship between the actuarial and traditional case reserves is consistent across development age, industry class, geographic area, cause of loss, injury type, etc. (unlikely), and if the mix of business across all of these dimensions is constant (also unlikely), then there is little chance that aggregate triangles would be distorted. However, if these conditions don’t exist, such distortions might exist. The comparison helps identify the extent to which this distortion may be important. Consider the graph below: The blue line at the top represents the total actuarial case reserves while the orange line represents the total traditional booked case reserves for a book of business over time, displaying a changing level of average case adequacy over time. A chart of this type helps in discussion with management regarding differences between estimates that are aided by the detailed analysis and estimates derived from unadjusted triangles. Since the actuarial case reserves reflect the mix of claims, it is a more reliable benchmark than simply comparing to a long-term average. Because the actuarial case reserves and booked case reserves both exist at the claim level, similar graphs can also be generated for any subset of the data, revealing additional information. 6.1.5. Case Reserves - Actuarial vs. Claim Department As the actuaries develop alternative case reserving models, it is tempting to suggest that such models should replace existing case reserves in other contexts, such as by use in the claim department. This temptation should be avoided. The case reserves that would be ideal for the actuarial department are likely different from the case reserves that would be ideal for the claim department. The actuaries are best served in reserving and pricing by case reserves that are unbiased (i.e., represent expected value) for cohorts of claims and policies along significant characteristics. Referring to mean-valued expectation when settling claims is likely not optimal as described above. The median outcome, with its easily understood evaluation of “just as likely to develop up or down” may be a more appropriate value for claim department use than the mean. At the claim level, the median and mean outcome can be significantly different. The claim department also has information about the claims that is likely not coded, including significant subjective information. This information is valuable at the individual claim level regarding settlement, but problematic for the actuary due to its potential to change subtly or dramatically over time. Therefore, this information should be included in the claim department’s case reserve estimate but generally not in the actuary’s case reserve estimate. This bifurcation into two separate estimates allows the claim department to operate more freely to make appropriate claim reserving and settlement decisions without concern that the actuary’s triangles will be impacted in unexpected ways. A common request from the actuary to the claim department is to make no changes. This is unrealistic and suboptimal, as there are often good practical and economic reasons for making changes. Freeing up the claim department to make changes to case reserving that leads to better decision-making is a distinct economic benefit. This also extends to the speed of claim settlement, which also historically has had the potential to distort the actuary’s triangles. If the actuary instead uses an algorithmic case reserve, under her/his control and developed with the goal of being unbiased, a change in the rate of settlement will impact case-incurred development only if it translates into a change in the amount of ultimate payment for the claim. If it does not, payments are exactly offset by a drop in the algorithmic case reserve and development is the same as it would have been if the change in settlement had not occurred. 6.2. Unreported Claim Value Algorithm Like the use of an actuarially generated case reserving algorithm, actuarial triangle analysis can be further refined by the creation of an unreported claim value algorithm. This algorithm provides the expected value of claims which have not yet been reported for a particular policy as of a particular date. The Bornhuetter-Ferguson technique (Bornhuetter and Ferguson 1972) can be thought of as a simple case of an unreported claim value algorithm where the loss ratio and development is the same across all exposures of a particular age. Like the Bornhuetter-Ferguson simple case, we start with premium (collected, or adjusted to be at constant rate level) but allow loss ratio and reporting lag to vary across policies. Also, we use policy written premium rather than earned premium as a starting point. As long as we are concerning ourselves with claims that have not yet been reported, and since we are looking at a policy level, there are advantages to also being able to estimate not only the IBNR claims (Incurred But Not Reported), but also the claims that would normally be associated with the unearned portion. For convenience we will term these WBNI claims (Written But Not Incurred). Unlike the actuarial case reserve algorithm, it is unnecessary to simulate the emergence claims and model the result of simulation. Instead, the result can be calculated more directly using the emergence component models using the following steps: • Apply Premium Model to each policy to arrive at Reference Rate premium • Apply Report Lag Model to Policies to get a modeled report lag for each policy • Apply Frequency Model to each policy to get a modeled (non-zero) claim count. Exposure is full premium not matured premium • Apply Severity Model to each policy • Allocate premium, modeled ultimate (modeled frequency * modeled severity), and modeled frequency to accident period for each policy • Calculate unreported for each policy, accident period, valuation date combination in the triangle Each of these steps will be discussed in greater detail. Applying the Premium Model If you are starting with an existing Claim Life Cycle Model, this step is already completed, but if you are applying the approach to subsequent points in time without rebuilding the CLCM, the models will have to be applied to refreshed underlying data. The point of this step is to generate the reference rate premium, which will be used as the exposure variable for the frequency model. Applying the Report Lag Model The goal of this step is to generate an expected report lag for each policy based on its characteristics. This model will be critical to determining the unreported portion of the ultimate loss at each point in time of interest for each policy. Applying the Frequency Model The frequency model that was determined earlier used matured reference rate premium as its exposure, but here it is applied to the full reference rate premium to arrive at an estimated claim count for the policy. The ratio of written premium to reference rate premium may be included if found to be predictive. Policy Effective Date may also be a predictive characteristic. If so, the actuary should take care to consider the treatment of dates that extend beyond the history period. Applying the Severity Model The first step is to apply the determined linear relationship between claim severity and reference rate premium divided by claim count at the policy level to provide the a priori expected severity (discussed in section 5.2.1.4) prior to applying the severity model. Report Lag is likely an important predictive variable in the severity model. If it is, the modeled severity impact from report lag for the policy must be integrated over the distribution of report lag. An exponential assumption for report lag at the policy level simplifies this integration. For example, if the severity model includes report lag as a binned characteristic, each section of the report lag distribution is assigned its respective factor from the corresponding bin, such that an average report lag factor is determined for the policy. sum [exp(lower bound / modeledLag - exp(upper bound / modeled lag * factor Where ‘i’ denotes the bin and ‘j’ denotes the policy. Allocating to Incurred Period To include this element in an incurred period triangle, we need to allocate the ultimate policy results (expected claim count and expected ultimate – equal to expected claim count multiplied by expected severity) to each period. Using a constant hazard rate over the policy term this is straightforward, but alternatives assumptions could also be made. Allocating the written premium is useful to arrive at an independent earned premium that can be compared to control totals and that exists at the policy level and therefore can be summed to any level of interest. It is also instructive to allocate the reference rate premium so that comparisons of written premium to reference rate premium can be made for the different periods. Calculating the Unreported Estimate Each valuation date of interest (for example, quarter-end evaluations) must be determined, and then for every [policy - incurred period - valuation date] combination, an unreported estimate can be determined by integrating the remaining portion of the report lag curve for the policy, incorporating severity differences to the extent that report lag was used as a severity model characteristic. Including the policy effective date as a variable makes it easy to switch from policy period triangles to incurred period triangles. Since we are focused on only unreported losses in this algorithm, we take steps to avoid a potential problem with the Bornhuetter-Ferguson method which can miss a changing mix of exposures that impacts the reporting pattern or expected loss ratio. Such mix changes are explicitly reflected in these algorithmic policy-level reserves. Separate from mix changes, the Bornhuetter-Ferguson technique can lead to odd results when significant case savings are reflected in the development pattern by determining future savings by premium volume rather than by case balances themselves. For example, assume that for a given line, case-incurred losses are typically at 120% of ultimate at 24 months and that the a priori loss ratio is 60%. In this example. the Bornhuetter-Ferguson technique projects case savings of 12% of earned premium [0.6*(1.0-1.2)] at 24 months regardless of the case reserves. Even if there are no case reserves from which to have savings, 12% of premium is the projected savings amount. By using an actuarial case algorithm to project future payments on known claims and a separate policy-level IBNR algorithm to project as yet unreported claims, we avoid this problem. While it is not common for actuaries to think about estimating emergence at a policy level, this procedure, combined with the actuarial case algorithm discussed earlier, can be very powerful to help one examine a book of business. Since estimates of ultimate now exist at a very granular level (policy or even sub-policy), the profitability of any number of slices of the business can be evaluated. With actuarial reserving techniques that are currently in common use, triangles would have to be developed and factors selected along these slices of the business. With indications of profitability at a finer level of detail, the company is able to be much more proactive (defensive or opportunistic) in their underwriting and marketing decisions. The algorithmic policy level IBNR reserves at specific points in time can be added to the case algorithm adjusted accident period triangle to create a new triangle. The policy level IBNR and WBNI could be added to a policy year triangle that includes case algorithm amounts as well. Both of these triangles, accident year and policy year, can be used to test and illustrate the combination of the case reserve algorithm and the unreported claim value algorithm. Remaining development in both of these triangles should be minimal, illustrating the algorithm’s effectiveness. 6.3. Comparing Reserve Estimates The reserve estimates generated from simulation and those derived from analysis of alternative triangles (paid + actuarial case reserve, paid + actuarial case reserve + unreported reserve) are likely to differ from those generated from the traditional triangles. The actuary will likely be asked to explain why. Often the answers can be found by considering the problems that the method is trying to solve. For example, case reserve adequacy could be shifting (which can be measured by comparing against actuarial case reserves). There could be a mix shift or a change in settlement. The additional insight into the differences in development obtained by building the component development models (and the explanatory model to be discussed in section 7.2) will help point the way to highlighting these differences. For example, in the case of a mix shift, often by segmenting the traditional triangles along lines indicated by the CLCM approach (with high development segments growing or shrinking) a reserve similar to the one generated by CLCM can be illustrated. Often confirmation bias for existing estimates and methods is high and results will be viewed with skepticism. Providing evidence of the consistency of algorithmic case reserves by showing a report year triangle with test data or a policy year triangle of all data elements on test data can help to gain comfort with the approach. As data continues to emerge methods can be compared to each other regarding their relative strengths, overcoming confirmation bias. 7.0. Other Topics 7.1. Other Modeling Considerations In actuarial reserve analysis it is common to isolate specific types of payments into separate analysis, such as indemnity payments from expense payments and/or medical payments. In addition to reporting requirements that may require separate estimates, the types of payments typically develop differently, and it often is beneficial to analyze them separately to provide additional insight ( Friedland 2010, Ch3). When building an analysis based on detailed data, this is still the case. In addition to illustrating different loss development patterns, the application of limits can be reflected with greater sophistication if simulation of the future payments is being performed. One of the strategies for using the payment type is to identify the payments with a distinct claim ID (i.e., treat the expense or medical as being a separate claim from the indemnity) with the payment type as just another variable. Scanning for possible interaction effects may indicate whether a separate analysis is warranted. For example, in a workers compensation analysis that treats indemnity payments as separate from medical, it is likely that several of the variables are likely to have interaction effects with the payment type. The more of these interaction effects there are, the more straightforward it is to model the payments with separate, distinct analyses, rather than deal with multiple interaction variables. One of the additional benefits of considering these detailed payment types is in their ability to be included as predictive variables themselves. As mentioned earlier, payments to date and recent payments can both be very predictive of future payments for an open claim. Staying with the workers’ compensation example, it is common that the indemnity payments to-date on a claim is predictive of future medical payments and vice versa. In fact, drilling down even further to provide additional details about recent claim payments can add significant information for predicting the future behavior of open claims. Contrast this with the common technique of performing separate triangle analyses for Medical Only claims and Medical with Indemnity Payment claims. Instead, a single Medical claims model may emerge with the amount of indemnity payment to date as a variable of interest. Zero indemnity payment is an important value, but it may be that it is not very different from indemnity payment of less than $1000. Also, while the status of “no indemnity payment” is fairly stable for a claim, it does have the potential to change, and that can create issues for a triangle. Reflecting that changing status as a paid indemnity variable in a medical claim actuarial case reserving algorithm avoids that problem. When adding such payment-type fields to a component development model it is incumbent to include the future simulated payments of various types as inputs into the simulation of the other payment types, adding complexity to the simulation process. Including such cross payment-type relationships into the case algorithm is more straightforward since it only requires payment types to be captured at each point in time. Loss cost trend was discussed at various previous points in this paper. Inflation is an important topic and highlighted in importance in recent years. While individual claim modeling gives additional insight and measurement of its history, the question of where it is headed is unlikely to be answered, being more a function of macro-economic and other systemic questions. If inflation is expected to be significantly different from what has been observed in the past and is embedded into the various models built, one approach is to detrend the simulated projections with inflation/trend consistent with the assumptions from the past and retrend them the forecasted inflation/trend. From an ERM perspective this technique can be useful as well. Since simulation is being used to generate projections a variety of outcomes is projected. However, no attempt was made to insert correlation between claims into the simulation process and therefore an important (and for larger organizations, dominant) source of variability is unaccounted for. Detrending at a common historical trend and retrending by path for every claim payment projection using a variety of inflation/trend paths will insert an important source of variability. 7.2. Explanatory Algorithm/Use in Claims Management While simulation is useful to project reserve development from a complex combination of predictive models across time periods, it can also make it difficult to attribute the specific differences in development across claims and exposures to specific characteristics. For example, suppose the claim closure rate is significantly slower for a particular geographic area. How much impact did the difference in closure rate make for the total estimated change from the current case reserve to ultimate for a claim in that area, given that there are more opportunities for changes in development the longer a claim remains open? For this reason, it is helpful to create explanatory predictive models at the end of the simulation process. Using the mean result of the simulation process for each claim as the target of prediction, the current case reserve as exposure, and all the variables that were found to be of importance in any of the component predictive models, a simplified explanation of which variables are the most important is provided, as is the nature of the impact of those variables. In addition to being a simplified explanation of which variables drove the simulation results, it also can be used at future evaluations to mimic what the simulation process would generate for claims open at those points in time. It may not be necessary to go through the complete simulation process each time there is a refresh to the open claim list, if the explanatory model can do a good job of mimicking the simulation results. This can be useful for ongoing review of the open claim inventory for purposes of claim management, triage, etc. In addition, a model can be built targeting the standard deviation of outcomes from simulation. The predicting of claim outcome variability can be useful in defense cost budgeting/benchmarking. Simulated results or curve parameterization from the development/SD models can be used to provide predicted quantiles existing open claims. This can prove useful for scoring outcomes and measuring groups of claims (such as by adjuster) to identify anomalies and trends. This post-simulation explanatory model (development prediction) is different from the actuarial case algorithm discussed in previous sections. Like the actuarial case algorithm, the prediction is of future payments for an open claim, as of a particular point in time, but there are several differences: The actuarial case algorithm is designed for use in actuarial work of reserving and pricing. Individual claim results are less important in this context than aggregated results and changing practices regarding case reserve levels are problematic. The explanatory analysis (development prediction) concentrates on individual claim level predictions and is focused on the differing potential of currently open claims to develop. There is less concern about shifts in general case reserve adequacy at this level. While including the claim department case reserve as a predictor defeats the purpose of an actuarial case reserve due to its subjectivity, that same subjective information is useful when projecting the future development of an individual claim. Therefore, including the claim department case reserve in the explanatory analysis as a predictor is appropriate. Because the explanatory model is focused on the current portfolio of open claims and how the simulation process is projecting them forward, only those claims that are being simulated are included in the parameterization. For the actuarial case reserve algorithm, all claims are considered valuable for parameterization, with cradle-to-grave information included in the portfolio of closed claims, and a combination of historical information and simulation included in the open claims. 7.3. Use in Pricing and Internal Management Reporting In building a detailed model of reserving that includes IBNR, a pricing model is a natural by-product. The Frequency and Severity models described in Section 5.2.1 together form a model of pricing. The Unreported Reserve of Section 6.2 at the time of policy effective date is an expected loss cost for the policy. It is important in actuarial pricing to consider the extent to which observed losses used for the pricing analysis need to be developed to an ultimate level, and how to develop them. Often very broad-brush approaches are used to address this question. Examples include: • Using reserves that have been allocated to the level of detail being priced • Applying development factors or premium development to policy level loss or premium • Comparing differentials in case-incurred loss across different types of policies as an estimate of differentials in ultimate loss Each of these approaches is fine if and only if differences in loss development across each of the pricing variables are being properly reflected. Too often they are not. While most actuaries will recognize that loss development differences exist between different deductibles and that loss experience should be adjusted accordingly when using observed results to price deductible credits, they may not always consider that different industry classes, or geography, or policy form, or any other variable of interest may have differing loss development exhibited across its values. These differences are significant and indicated profitability and pricing indications can be very different between making blanket assumptions and properly reflecting difference in potential for additional loss development. Every pricing variable should be checked for potential distortions from loss development whenever immature data is being used to develop indications. Sophisticated analytics techniques are being used now by actuaries to price insurance, but when the true variable of interest is ultimate loss and the item measured is case-incurred loss, there are likely to be significant distortions. Possible choices for incorporating loss development differences in pricing include: • Use Frequency and Severity models from a claim life cycle model directly. • Use IBNR(0) directly from an Unreported Claim Algorithm • Use the results of case algorithm and IBNR algorithm as a starting point for building predictive models, in conjunction with payments to date. In addition to providing more appropriate input for actuarial pricing calculations, using reserve estimates calculated at the claim and exposure level are valuable for internal management reporting. Rather than relying on crude allocations of reserve estimates to the various levels that may be reported (office, region, business unit, agency, etc.), the sum of claim and policy level reserves can be easily provided. If there is a final adjustment needed to be allocated to be in line with a reserve total that differs from the summed detail reserves, using the detailed reserves as the allocation basis will provide a much more thoughtful result. This has great potential for providing much more timely and reliable information about the results of underwriting efforts than that provided by crude allocation. The graphic below illustrates the way in which the detailed claim development analysis can impact actuarial, underwriting, and strategic decision-making. 1. Much of the material in this paper is taken from a previous monograph “Individual Claim Development Models and Detailed Actuarial Reserves in Property-Casualty Insurance” by Chris Gross in 2021, available at cognalysis.com/resources/publications. The author would like to thank Michael Larsen for his thoughtful review of a draft version of this paper and his useful suggestions, and also thank Tim Davis and Kevin Madigan for their valuable contributions, comments, and suggestions to the 2021 monograph, many of which are incorporated here. 2. See Friedland, J.F., “Estimating Unpaid Claims Using Basic Techniques,” Casualty Actuarial Society, Third Version, July 2010. 3. Berquist, James R., and Richard E. Sherman, “Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach,” Proceedings of the Casualty Actuarial Society 64, pp. 123–184, 1977 4. See Minimum Bias, Generalized Linear Models, and Credibility in the Context of Predictive Modeling by Chris Gross and Jon Evans, 2021, Variance Vol 12 Issue 1 5. This assumption is not always valid for aggregated data either and can cause problems for triangle analysis, but the assumption is not nearly as problematic as it is when considering claim-level 6. This process is similar in nature to a Markov Chain, in that each step in the chain is probabilistic, but a Markov Chain has the property that the future is independent of the past. The fact that paid losses to date are often one of the key predictors for the next timestep suggests it would be inappropriate to call this a Markov Process. 7. More sophisticated model specifications can use some of these payment types as predictors of others (increasing the number of models to be created). Barring this, it may be appropriate at times to consider individual payment types separately and in others to consider the sum across payment types.	{"url":"https://eforum.casact.org/article/122947-the-development-and-use-of-claim-life-cycle-model","timestamp":"2024-11-03T13:58:34Z","content_type":"text/html","content_length":"498137","record_id":"<urn:uuid:f74d80e6-d110-46a6-bb6f-88eafffa3e43>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00579.warc.gz"}
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Biography of Famous Scientist John Wallis - The Engineers BlogBiography of Famous Scientist John Wallis Biography of Famous Scientist John Wallis John Wallis: Mathematician and Polymath Early Life and Education: John Wallis was born on November 23, 1616, in Ashford, Kent, England. Little is known about his early childhood, but he attended Martin Holbeach’s school in Tenterden before entering Emmanuel College, Cambridge, in 1632. At Cambridge, Wallis displayed exceptional mathematical talent and made significant contributions to the field during his academic journey. Academic Career and Early Achievements: Wallis graduated with a bachelor’s degree in 1637 and a master’s degree in 1640. He was elected as a Fellow of Emmanuel College, and by 1644, he had become a ordained Anglican clergyman. During this period, he engaged in mathematical research, collaborating with other prominent mathematicians of the time. Cryptanalysis and Codebreaking: In addition to his mathematical pursuits, Wallis played a crucial role in cryptanalysis during the English Civil War. He deciphered Royalist codes and ciphers for the Parliamentarians, contributing to their military intelligence efforts. Wallis’s skill in codebreaking earned him recognition and respect. Arithmetica Infinitorum: Wallis’s major mathematical work, “Arithmetica Infinitorum” (1655), significantly advanced the understanding of mathematical concepts. In this work, he introduced new symbols for mathematical operations, including the infinity symbol (∞). Wallis’s notation and methods were groundbreaking, and he made substantial contributions to the understanding of infinite series. Continued Contributions to Mathematics: Throughout his career, Wallis made numerous contributions to various branches of mathematics. He worked on geometry, algebra, and calculus, and his work laid the foundation for later developments in these fields. Wallis also introduced the concept of the “Wallis product,” a formula for calculating π (pi) that became influential in mathematical analysis. Royal Society and Academic Recognition: In 1663, Wallis became one of the founding members of the Royal Society of London for Improving Natural Knowledge. He served as the society’s secretary from 1665 to 1672, contributing to its establishment as a leading institution for scientific research. Wallis was a respected figure in the scientific community and corresponded with mathematicians and scientists across Europe. Legacy and Later Life: John Wallis continued to work on mathematics until late in his life. He died on October 28, 1703, in Oxford, England. Wallis’s legacy extends beyond his specific mathematical contributions; he played a pivotal role in the development of mathematical notation, making complex mathematical ideas more accessible. Wallis’s influence on the mathematical community, his involvement in cryptography, and his contributions to the Royal Society collectively mark him as a prominent figure in the scientific and mathematical advancements of the 17th century. His dedication to the pursuit of knowledge and his multifaceted contributions have left a lasting impact on the field of mathematics.	{"url":"https://engineersblog.net/biography-of-famous-scientist-john-wallis/","timestamp":"2024-11-04T21:59:52Z","content_type":"text/html","content_length":"136132","record_id":"<urn:uuid:08622337-f71e-45f5-a8cf-48cd058ba0c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00582.warc.gz"}
Free Online dB Calculator \| How to find Sound Intensity Levels? - physicsCalculatorPro.com The dB Calculator here saves you time by providing you with the sound pressure and sound intensity levels in decibels. To obtain the outputs, simply enter the inputs in the designated input fields and click the compute button. It can be used for dB conversion and determining the distance between a sound source and the listener. Sound Pressure Level (SPL) When we hear a loud noise, we get negative feelings as a result of the pressure of the sound waves. This pressure can be measured in pascals, however, measuring in decibels is more practical. The Sound Pressure Level (SPL) is a measurement of sound pressure about the human hearing threshold. SPL = 20log(P/Pref) is the formula for calculating SPL. The sound pressure level, measured in decibels, is referred to as SPL. The sound wave pressure is measured in Pascals. Pref is the sound pressure reference value, which is considered to be 0.00002 Level of Sound Intensity The sound wave power per unit area is measured by the Sound Intensity Level. It's a unit of measurement for the amount of energy released each second per square metre. SIL = 10log(I/Iref) • Where, SIL = Sound Intensity Level in decibels • I = Sound Intensity in Watts Per Square Meter The reference value, Iref, is commonly 1×10⁻¹² W/m². For more concepts check out physicscalculatorpro.com to get quick answers by using the free tools available. Sound Intensity at a Distance Sound intensity varies with distance from the source in general. The phenomenon is known as distance attenuation, and it is a simple basic sense. It is simply because the energy of sound is diffused over a broader region from a physical standpoint. Consider a sphere that is encircled by a constant sound source. Even though the amount of energy emitted by the source remains constant, the sphere grows in size and its surface expands. The energy is dispersed across the sphere's surface. The equation can be written as I = P/(4πR²), Where, R = sphere's radius D = distance from the sound source. Converting from Pascals to Decibels Our decibel calculator can be used to calculate the decibel equivalent of sound wave pressure. To find the sound pressure level, simply enter the pressure in pascals into the dB calculator. If SPL is given, you can use this tool in reverse to find the pressure. FAQs on dB Calculator 1. What are the Sound Intensity Level units? Watts Per Meter Squared (W/m^2) is the unit of sound intensity level. 2. Is 20 dB twice as loud as 10db? A sound with a decibel level of 20 is ten times louder than one with a decibel level of 10. The noise level in a quiet bedroom is 30 decibels, which is 100 decibels louder than 10 decibels. And 40 decibels is 1,000 decibels louder than ten decibels. 3. Is dB a Unit of Measurement? Yes, dB is a sound measurement unit. 4. How much do decibels decrease with distance? The sound intensity drops by 6 decibels (dB) with every doubling of distance. 5. What is the formula for calculating sound pressure level? SPL = 20log(P/Pref) is a formula for calculating sound pressure level. 6. Is the decibel scale linear? When you use a sound level metre to measure noise levels, you use decibel units (dB) to quantify the strength of the noise. So, rather than utilising a linear scale, a logarithmic scale with 10 as the base is utilised to depict sound levels in more comprehensible quantities. The decibel scale is the name for this scale.	{"url":"https://physicscalculatorpro.com/dB-calculator/","timestamp":"2024-11-05T03:56:46Z","content_type":"text/html","content_length":"35222","record_id":"<urn:uuid:4f738517-4717-4023-80e3-3181727ec647>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00778.warc.gz"}
Homework help; matrix algebra; excel; solver homework help; matrix algebra; excel; solver Related topics: holt algebra 1 examples of math trivia algebra help program prerequisite study sheet for math 1316 trigonometry math problem solvingmath solver seven grade math worksheets of algebra exponent graphically 7th grade algebra math course descriptions of mathematics Author Message Thoon Posted: Wednesday 28th of May 14:52 Hey guys ,I was wondering if someone could help me with homework help; matrix algebra; excel; solver? I have a major project to complete in a couple of months and for that I need a thorough understanding of problem solving in topics such as simplifying expressions, inequalities and radical equations. I can’t start my assignment until I have a clear understanding of homework help; matrix algebra; excel; solver since most of the calculations involved will be directly related to it in some form or the other. I have a problem set, which if someone can help me solve, would help me a lot. From: USA Back to top espinxh Posted: Thursday 29th of May 16:01 Have you checked out Algebrator? This is a great software and I have used it several times to help me with my homework help; matrix algebra; excel; solver problems. It is very easy -you just need to enter the problem and it will give you a complete solution that can help solve your homework. Try it out and see if it solves your problem. From: Norway Back to top Dnexiam Posted: Saturday 31st of May 08:55 I looked into a number of software programs before I decided on Algebrator. This was the most appropriate for adding fractions, point-slope and subtracting exponents. It was effortless to key in the problem. Instead of simply giving the answer , it took me through all the steps explaining all the way until it arrived at the answer . By the time, I reached the answer I learnt how to go about it by myself. I used the program for solving my problems in Intermediate algebra, College Algebra and Remedial Algebra in algebra . Do you think that you will like to try this out? From: City 17 Back to top CHS` Posted: Sunday 01st of Jun 18:10 I am a regular user of Algebrator. It not only helps me complete my assignments faster, the detailed explanations offered makes understanding the concepts easier. I strongly advise using it to help improve problem solving skills. From: Victoria City, Hong Kong Island, Hong Kong Back to top Ondj_Winsan Posted: Tuesday 03rd of Jun 10:16 Wow! Does such a program really exist? That would really assist me in doing my homework. Is (programName) available for free or do I need to purchase it? If yes, where can I download it from? From: England Back to top Matdhejs Posted: Tuesday 03rd of Jun 20:31 Check out this link https://softmath.com/algebra-software-guarantee.html. I hope your algebra will improve and you will do a good job on the test! Good luck! From: The Back to top	{"url":"https://www.softmath.com/algebra-software/subtracting-exponents/homework-help-matrix-algebra.html","timestamp":"2024-11-04T19:58:55Z","content_type":"text/html","content_length":"43063","record_id":"<urn:uuid:5a7a3b36-4811-43cd-8aaf-bf78ff2549a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00636.warc.gz"}
The electron of the hydrogen atom may be thought to be localized - Genius Papers The electron of the hydrogen atom may be thought to be localized a) Heisenberg uncertainty principlei)The electron of the hydrogen atom may be thought to be localized within about 0.10 nm around the nucleus. Treating this as the position uncertainty determine the limit of uncertainty of velocity measurement of this electron.ii) For a helium atom moving with thermal velocity at 298 K the velocity was measured with an 10% precision. What is the limit on the precision of a simultaneous position measurement in this situation?iii) A sophisticated electronic system determined the velocity of a billiard ball as 5.0000 ±0.0001m/s. The mass of the ball is 200 g. Estimate the uncertainty of position measurement.b) An electron is confined to a one-dimensional region of 10nm length.i) Calculate the energies of its first three energy levels.ii) Calculate the wavelength of light that will be absorbed when electron is excited from first to second energy level.c) The vibrations of the HCl molecule may be modeled by assuming the H atom is moving in a harmonic potential with a force constant of 514 N/m while the Cl atom remains stationary. Calculate the absorption frequency and wavelength of the first absorption transition. Repeat calculations for	{"url":"https://geniuspapers.net/the-electron-of-the-hydrogen-atom-may-be-thought-to-be-localized/","timestamp":"2024-11-13T16:31:13Z","content_type":"text/html","content_length":"50346","record_id":"<urn:uuid:b827d794-01dc-41df-962d-33a756964027>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00210.warc.gz"}
[curves] Scalar blinding on elliptic curves with special structure [curves] Scalar blinding on elliptic curves with special structure Michael Hamburg mike at shiftleft.org Wed Aug 12 13:45:41 PDT 2015 Neat. The recoding would be a little annoying, but not awful. For ECDH it might be enough to just do both steps with the same random blinded base-48 encoding. You can make it work with the Montgomery ladder too. You’d use some unmixed ladder steps: (P, Q, P+Q) -> 2Q+P = (P+Q + Q) where (P+Q - Q) = P -> 3Q+P = (2Q+P + Q) where (2Q+P - Q) = (P+Q) and 3Q+2P = (2Q+P + Q+P) where (2Q+P) - (Q+P) = Q 3Q = 2Q+Q where 2Q-Q=Q. 3(Q+P) = 2(Q+P) + (Q+P) where 2(Q+P) - (Q+P) = Q+P These last steps might not work with all ladders though… I don’t remember whether eg Jacobian co-z ladder works for steps like these. — Mike > On Aug 10, 2015, at 9:52 PM, Trevor Perrin <trevp at trevp.net> wrote: > A new paper by Fluhrer is relevant to the discussion about scalar > blinding with special-prime vs random-prime curves: > http://eprint.iacr.org/2015/801 > My earlier impression [1] was that scalar-blinding on 25519 might use > a 128-bit blinding factor, whereas a similar-but-random-prime curve > would use a 64-bit blinding factor, resulting in a slowdown for 25519 > of around (256+128)/(256+64) = 1.2. > Fluhrer's paper argues for using the same size blinding factor, but > recoding the digits of the scalar used for windowing into a form where > the group's order "would, at first glance, appear random". He gives > an example of base-48 digits instead of base-32, and estimates a > slowdown for 25519 of around 1.1. > I don't think this helps implementations that use Montgomery ladder > (instead of windowing). Beyond that, I don't have a good sense how > well this would work, how awkward the encoding would be, or how it > would interact with other scalar encoding methods. > Anyone have a more informed opinion? > Trevor > [1] https://moderncrypto.org/mail-archive/curves/2015/000563.html > _______________________________________________ > Curves mailing list > Curves at moderncrypto.org > https://moderncrypto.org/mailman/listinfo/curves More information about the Curves mailing list	{"url":"https://moderncrypto.org/mail-archive/curves/2015/000610.html","timestamp":"2024-11-04T16:47:38Z","content_type":"text/html","content_length":"5678","record_id":"<urn:uuid:c101a8f5-571e-4e7e-82b1-ea6bbf645524>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00512.warc.gz"}
Effect size: means differences - Science without sense...double nonsense I am Spartacus Effect size with mean differences We describe parameters that help us to study the effect size based on differences between means: Glass’ delta, Cohen’s d and Hedges’ g. I was thinking about the effect size based on mean differences and how to know when that effect is really large and, because of the association of ideas, someone great has come to mind who, sadly, has left us recently. I am referring to Kirk Douglas, that hell of an actor that I will always remember for his roles as a Viking, as Van Gogh or as Spartacus, in the famous scene of the film in which all slaves, in the style of our Spanish’s Fuenteovejuna, stand up and proclaim together to be Spartacus so that Romans cannot do anything to the true one (or to get all equally whacked, much more typical of the modus operandi of the Romans of that time). You won’t tell me the man wasn’t great. But how great if we compare it with others? How can we measure it? It is clear that not because of the number of Oscars, since that would only serve to measure the prolonged shortsightedness of the so-called academics of the cinema, which took a long time until they awarded him the honorary prize for his entire career. It is not easy to find a parameter that defines the greatness of a character like Issur Danielovitch Demsky, which was the ragman’s son’s name before becoming a legend. We have it easier to quantify the effect size in our studies, although the truth is that researchers are usually more interested in telling us the statistical significance than in the size of the effect. It is so unusual to calculate it that even many statistical packages forget to have routines to obtain it. In this post, we are going to focus on how to measure the effect size based on differences between means. Effect size with mean differences Imagine that we want to conduct a trial to compare the effect of a new treatment against placebo and that we are going to measure the result with a quantitative variable X. What we will do is calculate the mean effect between participants in the experimental or intervention group and compare it with the mean of the participants in the control group. Thus, the effect size of the intervention with respect to the placebo will be represented by the magnitude of the difference between the mean in the experimental group and that of the control group:$d=&space;\bar{x}_{e}-\bar{x}_ {c}$However, although it is the easiest to calculate, this value does not help us to get an idea of the effect size, since its magnitude will depend on several factors, such as the unit of measure of the variable. Let us think about how the differences change if one mean is twice the other as their values are 1 and 2 or 0.001 and 0.002. In order for this difference to be useful, it is necessary to standardize it, so a man named Gene Glass thought he could do it by dividing it by the standard deviation of the control group. He obtained the well-known Glass’ delta, which is calculated according to the following formula:$\delta&space;=&space;\frac{\bar{x}_{e}-\bar{x}_{c}}{S_{s}}$Now, since what we want is to estimate the value of delta in the population, we will have to calculate the standard deviation using n-1 in the denominator instead of n, since we know that this quasi-variance is a better estimator of the population value of the deviation:$S_{c}=\sqrt{\frac{\sum_{i=1}^{n_{c}}(x_{ic}-\bar{x}_{c})}{n_{c}-1}}$But do not let yourselves be impressed by delta, it is not more than a Z score (those obtained by subtracting to the value its mean and dividing it by the standard deviation): each unit of the delta value is equivalent to one standard deviation, so it represents the standardized difference in the effect that occurs between the two groups due to the effect of the intervention. This value allows us to estimate the percentage of superiority of the effect by calculating the area under the curve of the standard normal distribution N(0,1) for a specific delta value (equivalent to the standard deviation). For example, we can calculate the area that corresponds to a delta value = 1.3. Nothing is simpler than using a table of values of the standard normal distribution or, even better, the pnorm() function of R, which returns the value 0.90. This means that the effect in the intervention group exceeds the effect in the control group by 90%. The problem with Glass’ delta is that the difference in means depends on the variability between the two groups, which makes it sensitive to these variance differences. If the variances of the two groups are very different, the delta value may be biased. That is why one Larry Vernon Hedges wanted to contribute with his own letter to this particular alphabet and decided to do the calculation of Glass in a similar way, but using a unified variance that does not assume their equality, according to the following formula:$S_{u}=\sqrt{\frac{(n_{e}-1)S_{e}^{2}+(n_{c}-1)S_{c}^{2}}{n_{e}+n_{c}-2}}$ If we substitute the variance of the control group of the Glass’ delta formula with this unified variance we will obtain the so-called Hedges’ g. The advantage of using this unified standard deviation is that it takes into account the variances and sizes of the two groups, so g has less risk of bias than delta when we cannot assume equal variances between the two groups. However, both delta and g have a positive bias, which means that they tend to overestimate the effect size. To avoid this, Hedges modified the calculation of his parameter in order to obtain an adjusted g, according to the following formula: [e] + n[c]. This correction is more needed with small samples (few degrees of freedom). It is logical, if we look at the formula, the more degrees of freedom, the less necessary it will be to correct the bias. So far, we have tried to solve the problem of calculating an estimator of the effect size that is not biased by the lack of equal variances. The point is that, in the rigid and controlled world of clinical trials, it is usual that we can assume the equality of variances between the groups of the two branches of the study. We might think, then, that if this is true, it would not be necessary to resort to the trick of n-1. Well, Jacob Cohen thought the same, so he devised his own parameter, Cohen’s d. This Cohen’s d is similar to Hedges’ g, but still more sensitive to inequality of variances, so we will only use it when we can assume the equality of variances between the two groups. Its calculation is identical to that of the Hedges’ g, but using n instead of n-1 to obtain the unified variance. As a rough-and-ready rule, we can say that the effect size is small for d = 0.2, medium for d = 0.5, large for d = 0.8 and very large for d = 1.20. In addition, we can establish a relationship between d and the Pearson’s correlation coefficient (r), which is also a widely used measure to estimate the effect size. The correlation coefficient measures the relationship between an independent binary variable (intervention or control) and a numerical dependent variable (our X). The great advantage of this measure is that it is easier to interpret than the parameters we have seen so far, which all function as standardized Z scores. We already know that r can range from -1 to 1 and the meaning of these values. Thus, if you want to calculate r given d, you only have to apply the following formula:$r=\frac{d}{\sqrt{d^{2}+\left&space;(&space;\frac{1}{pq}&space;\right&space;)}}$where p and q are the proportions of subjects in the experimental and control groups (p = n[e] / n and q = n[c] / n). In general, the larger the effect size, the greater r and vice versa (although it must be taken into account that r is also smaller as the difference between p and q increases). However, the factor that most determines the value of r is the value of d. We’re leaving… And with this we will end for today. Do not believe that we have discussed all the measures of this family. There are about a hundred parameters to estimate the effect size, such as the determination coefficient, eta-square, chi-square, etc., even others that Cohen himself invented (not very happy with only d), such as f-square or Cohen’s q. But that is another story… Your email address will not be published. Required fields are marked * Información básica sobre protección de datos Ver más • Responsable: Manuel Molina Arias. • Finalidad: Moderar los comentarios. • Legitimación: Por consentimiento del interesado. • Destinatarios y encargados de tratamiento: No se ceden o comunican datos a terceros para prestar este servicio. El Titular ha contratado los servicios de alojamiento web a Aleph que actúa como encargado de tratamiento. • Derechos: Acceder, rectificar y suprimir los datos. • Información Adicional: Puede consultar la información detallada en la Política de Privacidad. This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://www.cienciasinseso.com/en/effect-size-with-mean-differences/","timestamp":"2024-11-11T13:21:01Z","content_type":"text/html","content_length":"82567","record_id":"<urn:uuid:7e59616e-c947-4c0f-b916-4ffc78b005bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00417.warc.gz"}
5) Two high-speed ferries leave at the same time from a city to go to the... 5) Two high-speed ferries leave at the same time from a city to go to the... 5) Two high-speed ferries leave at the same time from a city to go to the same island. The first ferry, the Cat, travels at 36 miles per hour. The second ferry, the Bird, travels at 23 miles per hour. In how many hours will the two ferries be 26 miles apart? The ferries will be 26 miles apart after ...hour(s). 6) Jane took 30 min to drive her boat upstream to water-ski at her favorite spot. Coming back later in the day, at the same boat speed, took her 15 min. If the current in that part of the river is 4km per hr, what was her boat speed in still water? Her boat speed in still water was.......km per hr. 7) How many quarts of pure antifreeze must be added to 7 quarts of a 40% antifreeze solution to obtain a 50% antifreeze solution? ... quart(s) of pure antifreeze must be added. 8) How much water should be added to 10mL of 14% alcohol solution to reduce the concentration to 5%? We should add...mL. 9) In planning her retirement, Liza deposits some money at 3.5% interest, with twice as much deposited at 5%. Find the amount deposited at each rate if the total annual interest income is $2430. She deposited $... at 3.5% and $...at 5% 5. Let time taken to make distance b/w 2 ferries be x hr. Then, We know that, speed* time = distance 23x + 26 = 36x => x= 26/13 => x =2 Km 6. upstream time = 30 min downstream time = 15 min The speed of river = 4 Km/hr Let the speed of the boat in still water = v Distance = speed * time So, (4+v)15 = (v-4)30 => 4+ v =2v -8 => v = 12 => v = 12 Km/hr 7. Let x quarts of pure antifreeze need to be added . 70.4 + x = (7+x)0.5 => 2.8 + x = 3.5 + 0.5x => 0.5x= 0.7 => x = 1.4 quarts We need to do only 1 question. However I solved 3. Upload remaining questions seperately.	{"url":"https://justaaa.com/math/1204873-5-two-high-speed-ferries-leave-at-the-same-time","timestamp":"2024-11-01T22:50:28Z","content_type":"text/html","content_length":"34108","record_id":"<urn:uuid:3a66cf28-d180-4ab1-b341-d854c5750be3>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00411.warc.gz"}
How to work out exchange rate questions To calculate the percentage discrepancy, take the difference between the two exchange rates, and divide it by the market exchange rate: 1.37 - 1.33 = 0.04/1.33 = 0.03. Multiply by 100 to get the Here's how to quickly and easily find and calculate currency exchange rates— and how these rates are influenced by economic indicators. 29 Nov 2018 Not sure the best way to calculate a given exchange rate? The dollar-pound exchange rate is used to determine how many dollars you When comparing exchange rates, it is always good practice to work out the mark-up back again. Solve real life problems using exchange rates and commission. Formula for converting foreign money and exchange rates £1 = €1.39 (x 1.39 6 May 2018 This question is about International Credit Cards The formula for calculating exchange rates is: Starting Amount (Original Currency) / Ending It depends which way round the rate is quoted. For FX, we need to ask two separate questions: (1) Which is the base currency in the given quote? (2) Are we Question. I have the USA amount and the Canadian amount, how do I calculate the exchange rate? 13 Jan 2017 If you often switch between two or more currencies, you'll know that calculating an exchange rate is not so straightforward. Firstly, the rates are Others, more than people realise, come completely out of the blue. so many years, we've seen every possible problem and scenario first-hand. We know how the markets move and the factors that cause exchange rates to fluctuate. We've We work with our clients to help them put a strategy in place for all their transfers. 10 Aug 2015 First, look up the exchange rate for the country you are visiting and Also figure out roughly how much local currency is in $10. Practice makes perfect and hopefully you won't have to figure all this out in a stressful situation! 12 Jun 2013 A step-by-step guide to calculating cross rates, which are exchange rates for currency pairs that do not involve the US dollar. 6 Sep 2007 It is in this context that we saw the need for this series of Questions & Answers. How does one judge whether the exchange rate is at an appropriate of supply and demand for the currency to determine the exchange rate. Cookies help us customise PayPal for you, and some are necessary to make our site work. Cookies also allow us to show you personalised offers and 27 Nov 2019 Calculate the Rate. Calculating an exchange rate is simple but can change on a day-to-day basis. As an example: let's say the Euro To calculate the percentage discrepancy, take the difference between the two exchange rates, and divide it by the market exchange rate: 1.37 - 1.33 = 0.04/1.33 = 0.03. Multiply by 100 to get the Online Tools You Can Use To Calculate Exchange Rate. To avoid human error, you can use a range of free tools to work out exchange rates. Some of the best free tools include XE, Oanda and x-rates. Note that these sites give you the actual exchange rate. A bank (and especially an airport bureau) will offer you a much less favourable rate, as they take a percentage on each exchange. In our example, the exchange rate for USD/INR was 66.73, but let’s say the rate your bank offers is 63.93. Step 3 - Divide the two exchange rates to find the percent of markup. To calculate the markup, you'll need to work out the difference between the two rates and then translate this into a percentage. QUESTIONS AND PROBLEMS. QUESTIONS. 1. Give a How are foreign exchange transactions between international banks settled? Answer: The We will use the top formula that uses American term forward exchange rates. F1(CD/ SF) 17 Feb 2020 The problem with exchange rates is that you have little control over you have probably worked out the price based on the exchange rate at Sending international payments online is made easy with Midpoint. Read our frequently asked questions to find an answer to your questions quickly. I have a been given a task. I need to use the currencies from this source to work out the exchange rate between 2 currencies. Requirements are that I need to use that data source and, select a date, the amount to calculate and the two currencies. The rates on that feed are based against the euro as the base currency. Free currency converter to calculate exchange rates for currencies and metals. Enter the values in Use our currency converter to convert over 190 currencies and 4 metals. To get started Using your debit card at ATMs is one recommended way to get cash when traveling abroad. You have money questions. Bankrate So how are some currencies valued higher than others? Check out these retro videos from Encyclopedia Britannica's archives. Britannica currency exchange rate on digital LED display board in global background exchange and the IMF added stability to the world market, but it didn't come without its own problems. 9 Feb 2018 Forward exchange rate is the exchange rate at which a party is willing to enter into a contract to receive or deliver a currency at some future First construct a graph of all your currencies: private Dictionary> _graph public void ConstructGraph() { if (_graph == null) { _graph = new Currency Converter ⇒ Get real-time currency exchange rates with our currency How Does A Currency Converter Work? There are quite a few people out there who make a pretty nifty living from Now we are back to the original question. 25 Jun 2019 Here's how exchange rates work, and how to figure out if you are getting a good deal. Finding Market Exchange Rates. Traders and institutions Currency Converter ⇒ Get real-time currency exchange rates with our currency How Does A Currency Converter Work? There are quite a few people out there who make a pretty nifty living from Now we are back to the original question. How to report gains or losses from foreign exchange rates in the financial I have a question on how to determine functional currency for cost plus entity ( IFRS) The formula to calculate the PPP impact and exchange rate is: If the supplier is one of a few, despite all the problems associated with price fixing, the market The basic formula for calculating a pip value (in the quote or counter currency— the one Divided by the exchange rate or current price of the pair. Times lot size Others, more than people realise, come completely out of the blue. so many years, we've seen every possible problem and scenario first-hand. We know how the markets move and the factors that cause exchange rates to fluctuate. We've We work with our clients to help them put a strategy in place for all their transfers. Original exchange rate (b) Calculate the cross rate for Australian dollars in yen terms. ¥? ¥ (c) How much profit or loss would have been made from opening. The idea of cross rates implies two exchange rates with a common currency, which enables you to calculate the exchange rate between the remaining two a calculator (or use this calculator); A current list of exchange rates (look up on a commission on every transaction, so they make even more money out of us! The USD/ 1 unit figure tells us how to convert one unit of the foreign currency to Learn about what an exchange rate is and how to determine the cost of goods in another currency. Practice: Exchange rates · Next lesson. The foreign This way, as soon as you see that a question makes use of a calculation performed The following are exchange rates on various dates for the US Dollar (USD). Learn how to solve currency conversion questions in numerical reasoning tests. 3) Determine Which Currency is Stronger Currency exchange rates per 'x'. How many euros can she buy if the exchange rate is currently \euro{} $\text{1}$ To answer this question we work out the cost of the car in rand for each country	{"url":"https://topbitxqdseo.netlify.app/hitzel3160xyji/how-to-work-out-exchange-rate-questions-270.html","timestamp":"2024-11-08T22:08:25Z","content_type":"text/html","content_length":"33215","record_id":"<urn:uuid:1137ea13-9314-467c-a0f7-19d6018fbb48>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00017.warc.gz"}
Probability Distributions: Understanding Uniform and Normal Distributions \| Study notes Data Analysis & Statistical Methods \| Docsity An introduction to continuous probability distributions, specifically the uniform and normal distributions. It covers the shape, probability density functions, mean, standard deviation, and calculating probabilities using standard normal distributions. Examples and tables are included to illustrate the concepts.	{"url":"https://www.docsity.com/en/docs/notes-on-continuous-random-variables-statistical-methods-i-stat-515/6732931/","timestamp":"2024-11-02T14:20:35Z","content_type":"text/html","content_length":"231306","record_id":"<urn:uuid:7f105c1d-0d82-4e98-abe9-8451a65fcbf3>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00672.warc.gz"}
Decidability problems in number theory Victor Lisinski, Corpus Christi, Oxford In its modern formulation, Hilbert’s tenth problem asks to find a general algorithm which decides the solvability of Diophantine equations. While this problem was shown to be unsolvable due to the combined work of Davis, Putnam, Robinson and Matiyasevich, similar question can be posed over domains other than the integers. Among the most important open questions in this area of research is if a version of Hilbert’s tenth problem for F[p]((t)), the field of formal Laurent series over the finite field F[p], is solvable or not. The fact that this remains open stands in stark contrast to the fact that the first order theory of the much similar object Q[p], the field of p-adic numbers, is completely understood thanks to the work by Ax, Kochen and, independently, Ershov. In light of this dichotomy, I will present new decidability results obtained during my doctoral research on extensions of F[p]((t)). This work is motivated by recent progress on Hilbert’s tenth problem for F[p]((t)) by Anscombe and Fehm and builds on previous decidability results by Kuhlman.	{"url":"https://logic-gu.se/seminars/21%20vt/2021/06/18/victor-lisinki/","timestamp":"2024-11-05T09:53:36Z","content_type":"text/html","content_length":"9055","record_id":"<urn:uuid:793b39a6-cee4-40c7-bdb9-a773e4ef9e6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00121.warc.gz"}
Suppose we have a chess board, and a collection of tiles, like dominoes, each of which is the size of two squares on the chess board. Can the chess board be covered by the dominoes? First we need to be clear on the rules: the board is covered if the dominoes are laid down so that each covers exactly two squares of the board; no dominoes overlap; and every square is covered. The answer is easy: simply by laying out 32 dominoes in rows, the board can be covered. To make the problem more interesting, we allow the board to be rectangular of any size, and we allow some squares to be removed from the board. What can be say about whether the remaining board can be covered? This is such a board, for example: What can we say? Here is an easy observation: each domino must cover two squares, so the total number of squares must be even; the board above has an even number of squares. Is that enough? It is not too hard to convince yourself that this board cannot be covered; is there some general principle at work? Suppose we redraw the board to emphasize that it really is part of a chess board: Aha! Every tile must cover one white and one gray square, but there are four of the former and six of the latter, so it is impossible. Now do we have the whole picture? No; for example: The gray square at the upper right clearly cannot be covered. Unfortunately it is not easy to state a condition that fully characterizes the boards that can be covered; we will see this problem again. Let us note, however, that this problem can also be represented as a graph problem. We introduce a vertex corresponding to each square, and connect two vertices by an edge if their associated squares can be covered by a single domino; here is the previous board: Here the top row of vertices represents the gray squares, the bottom row the white squares. A domino now corresponds to an edge; a covering by dominoes corresponds to a collection of edges that share no endpoints and that are incident with (that is, touch) all six vertices. Since no edge is incident with the top left vertex, there is no cover. Perhaps the most famous problem in graph theory concerns map coloring: Given a map of some countries, how many colors are required to color the map so that countries sharing a border get different colors? It was long conjectured that any map could be colored with four colors, and this was finally proved in 1976. Here is an example of a small map, colored with four colors: Typically this problem is turned into a graph theory problem. Suppose we add to each country a capital, and connect capitals across common boundaries. Coloring the capitals so that no two connected capitals share a color is clearly the same problem. For the previous map: Any graph produced in this way will have an important property: it can be drawn so that no edges cross each other; this is a planar graph. Non-planar graphs can require more than four colors, for example this graph: This is called the complete graph on five vertices, denoted $K_5$; in a complete graph, each vertex is connected to each of the others. Here only the "fat'' dots represent vertices; intersections of edges at other points are not vertices. A few minutes spent trying should convince you that this graph cannot be drawn so that its edges don't cross, though the number of edge crossings can be Exercises 1.1 Ex 1.1.1 Explain why an $m\times n$ board can be covered if either $m$ or $n$ is even. Explain why it cannot be covered if both $m$ and $n$ are odd. Ex 1.1.2 Suppose two diagonally opposite corners of an ordinary $8\times8$ board are removed. Can the resulting board be covered? Ex 1.1.3 Suppose that $m$ and $n$ are both odd. On an $m\times n$ board, colored as usual, all four corners will be the same color, say white. Suppose one white square is removed from any location on the board. Show that the resulting board can be covered. Ex 1.1.4 Suppose that one corner of an $8\times8$ board is removed. Can the remainder be covered by $1\times 3$ tiles? Show a tiling or prove that it cannot be done. Ex 1.1.5 Suppose the square in row 3, column 3 of an $8\times8$ board is removed. Can the remainder be covered by $1\times 3$ tiles? Show a tiling or prove that it cannot be done. Ex 1.1.6 Remove two diagonally opposite corners of an $m\times n$ board, where $m$ is odd and $n$ is even. Show that the remainder can be covered with dominoes. Ex 1.1.7 Suppose one white and one black square are removed from an $n\times n$ board, $n$ even. Show that the remainder can be covered by dominoes. Ex 1.1.8 Suppose an $n\times n$ board, $n$ even, is covered with dominoes. Show that the number of horizontal dominoes with a white square under the left end is equal to the number of horizontal dominoes with a black square under the left end. Ex 1.1.9 In the complete graph on five vertices shown above, there are five pairs of edges that cross. Draw this graph so that only one pair of edges cross. Remember that "edges'' do not have to be straight lines. Ex 1.1.10 The complete bipartite graph $K_{3,3}$ consists of two groups of three vertices each, with all possible edges between the groups and no other edges: Draw this graph with only one crossing.	{"url":"https://www.whitman.edu/mathematics/cgt_online/book/section01.01.html","timestamp":"2024-11-08T19:07:22Z","content_type":"text/html","content_length":"38927","record_id":"<urn:uuid:bfbf7bc8-8187-400a-b715-335f6cb02440>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00114.warc.gz"}
Negative Indices and Power of a Power - Math Angel ðŸŽ¬ Video Tutorial • Negative Indices Rule: A negative index means taking the reciprocal of the base with a positive index (e.g., $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$). • Applying Negative Indices on Fractions: Flip the fraction and use the positive index (e.g., $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$). • Power of a Power Rule: When raising a power to another power, multiply the indices (e.g., $(2^3)^5 = 2^{15}$). Accessing this course requires a login. Please enter your credentials below!	{"url":"https://math-angel.io/lessons/negative-indices-and-power-of-a-power","timestamp":"2024-11-13T03:12:07Z","content_type":"text/html","content_length":"275516","record_id":"<urn:uuid:da82fabd-832e-4172-a3fd-07ab62e47635>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00812.warc.gz"}
Optimal convergence of a discontinuous-Galerkin-based immersed boundary method Published Paper Inserted: 10 mar 2010 Last Updated: 28 jan 2011 Journal: Math. Model. Numer. Anal. Volume: 45 Pages: 651-674 Year: 2011 We prove the optimal convergence of a discontinous-Galerkin-based immersed boundary method introduced earlier Lew and Buscaglia, 2008. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional $C^2$-domains. For solution in $H^q$ for $q > 2$, we prove that the method approximates the function and its gradient with optimal orders $h^ 2$ and $h$, respectively. When $q = 2$, we have $h^{2-\epsilon}$ and $h^1$ for any $\epsilon > 0$ instead. To this end, we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny- Lions' lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to dierences between the exact and the approximate domains are treated using Hardy's inequality together with more standard results on Sobolev functions.	{"url":"https://cvgmt.sns.it/paper/383/","timestamp":"2024-11-11T06:57:50Z","content_type":"text/html","content_length":"9143","record_id":"<urn:uuid:ed5f7a8b-1510-4de8-9977-4b8e880668ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00439.warc.gz"}
History of Distance Calculus- Calculus New! DMAT 431 - Computational Abstract Algebra with MATHEMATICA! Asynchronous + Flexible Enrollment = Work At Your Own Best Successful Pace = Start Now! Earn Letter of Recommendation • Customized • Optional Recommendation Letter Interview Mathematica/LiveMath Computer Algebra Experience • STEM/Graduate School Computer Skill Building NO MULTIPLE CHOICE • All Human Teaching, Grading, Interaction • No AI Nonsense 1 Year To Finish Your Course • Reasonable FastTrack Options • Multimodal Letter Grade Assessment History of Distance Calculus Once upon a time two cool math professors: Jerry Uhl at the University of Illinois, and Bill Davis at The Ohio State University, who together with some other collaborators wrote a new type of electronic math textbook that was based upon student experimentation using computer algebra and graphing software, as opposed to the traditional "read the text and do the homework problems on paper." Big computer labs were set up at both Universities for the students to "go through Calculus" using these new technologies and curricula. One day a visitor said, "Hey, there is no reason these students have to be in THIS room. They could be anywhere." This led to the first distance calculus courses at University of Illinois and The Ohio State University. From this start, a variety of similar calculus-in-distance courses have migrated around the country. Distance Calculus was founded at Suffolk University in Boston, Massachusetts in 1997 by Dr. Lee Wayand (part of the team at The Ohio State University program) and Dr. Robert Curtis, and ran there from 1997-2010. Dr. Wayand returned to his native Ohio into 2004. In 2010 we moved Shorter University, of Rome, Georgia, but that partnership was short-lived, and was discontinued in 2014. Sadly, Professors Uhl and Davis have both passed away. But their dream of asynchronous, laboratory-style distance courses and curriculum lives on today, both via Distance Calculus, and via our sister programs at other institutions like NetMath @ UIUC. Distance Calculus uses a combination of the original e-curriculum, Calculus&Mathematica™, also now ported to the graphically-based computer algebra and graphing system LiveMath™, as well as new and emerging curriculum using a variety of media, including QuickTime movies, Instant Messenger/Chat communication tools, and lots and lots and lots of communications between student, teaching assistants, and professors. Distance Calculus is led by Dr. Robert Curtis, and operated by a small team of educators through Calculus.NET LLC. We also publish the software products LiveMathâ„¢ and PrintMathâ„¢, and continue to be based near Cambridge, Massachusetts, USA.	{"url":"https://www.distancecalculus.com/history/","timestamp":"2024-11-03T17:23:32Z","content_type":"text/html","content_length":"33663","record_id":"<urn:uuid:c1678850-0f17-42ca-9e65-e1413d262670>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00081.warc.gz"}
Can You Solve This Math Puzzle? Can You Solve This Math Puzzle? Social media is buzzing with a simple yet intriguing math puzzle: 6 + 6 + 6 + 6 x 0 = ? At first glance, this equation seems straightforward, but it’s causing a lot of debate online. Some people rush to answer 0, while others confidently state 24. So, what’s the correct answer, and why is this math problem sparking so much discussion? Understanding the Order of Operations To solve this puzzle correctly, it’s essential to remember the basic rules of mathematics, specifically the order of operations. In mathematics, the order of operations is a set of rules that dictates the sequence in which operations should be performed to ensure consistent results. The commonly used acronym PEMDAS helps us remember these rules: According to PEMDAS, multiplication and division take precedence over addition and subtraction. This means that in our equation, the multiplication part must be performed before any addition. Solving the Puzzle Let’s break down the equation step-by-step using the order of operations: 6+6+6+6×0=?6 + 6 + 6 + 6 \times 0 = ?6+6+6+6×0=? Perform the Multiplication First: 6×0=06 \times 0 = 06×0=0 Substitute the Result Back Into the Equation: 6+6+6+0=?6 + 6 + 6 + 0 = ?6+6+6+0=? Perform the Addition: 6+6=126 + 6 = 126+6=12 12+6=1812 + 6 = 1812+6=18 18+0=1818 + 0 = 1818+0=18 Thus, the correct answer is 18. Why Do People Get It Wrong? This math puzzle highlights a common mistake people make when solving equations: ignoring the order of operations. Many assume that the sequence in which the numbers appear should dictate how they are solved, leading to the incorrect answer of 0. Moreover, the simplicity of the equation can be deceptive. Since it’s a short equation, it’s easy to overlook the importance of applying the correct mathematical rules, especially when solving it quickly in your head or under pressure. The Bigger Picture This math puzzle is more than just a viral social media challenge. It serves as a reminder of the importance of mathematical principles and how they apply to everyday problem-solving. Whether you’re calculating a tip, balancing a budget, or solving a viral math problem, understanding the order of operations is crucial. So, next time you encounter a math equation, remember PEMDAS and take a moment to ensure you’re solving it correctly. It could save you from making a common mistake!	{"url":"https://dailyinsightnews.com/40/","timestamp":"2024-11-09T00:57:59Z","content_type":"text/html","content_length":"157130","record_id":"<urn:uuid:7d51b9a1-5f76-4e81-a2f3-a1177c6957c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00805.warc.gz"}
DROPS - Results │No.│ Title │ Author │Year│ │1 │Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization │Amireddy, Prashanth et al. │2023│ │2 │Learning Generalized Depth Three Arithmetic Circuits in the Non-Degenerate Case │Bhargava, Vishwas et al. │2022│ │3 │No Quantum Speedup over Gradient Descent for Non-Smooth Convex Optimization │Garg, Ankit et al. │2021│ │4 │Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture │Chattopadhyay, Arkadev et al.│2021│ │5 │Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings │Garg, Ankit et al. │2020│ │6 │Determinant Equivalence Test over Finite Fields and over Q │Garg, Ankit et al. │2019│ │7 │Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory │Bürgisser, Peter et al. │2018│ │8 │Barriers for Rank Methods in Arithmetic Complexity │Efremenko, Klim et al. │2018│ │9 │Separating Quantum Communication and Approximate Rank │Anshu, Anurag et al. │2017│ │10 │Lower Bound on Expected Communication Cost of Quantum Huffman Coding │Anshu, Anurag et al. │2016│	{"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/ergebnis.php?suchwert2=Garg%2C+Ankit","timestamp":"2024-11-08T14:00:21Z","content_type":"text/html","content_length":"5692","record_id":"<urn:uuid:72d71066-111f-4263-8537-73eb7dcd8c0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00039.warc.gz"}
Free Printable Long Division Worksheets - Divisonworksheets.com Free Printable Long Division Worksheets Free Printable Long Division Worksheets – You can help your child practice and review their skills in division by using division worksheets. Worksheets come in a wide variety, and you can even create your own. They are great because they can be downloaded for free and customize them as you like them. These worksheets are perfect for first-grade students, kindergarteners, and even second-graders. Two people can create huge quantities It is crucial for kids to work on division worksheets. Most worksheets allow two, three, or even four different divisors. This won’t create stress for the child since they won’t have to worry over having to divide a huge number or making mistakes when using their tables of times. There are worksheets available online or download them on your computer to assist your youngster in developing this mathematical ability. Use multidigit division worksheets to aid children in practicing and increase their knowledge of the subject. This is an essential maths skill needed to tackle complex math concepts and everyday calculations. These worksheets aid in establishing the concept by using interactive questions, games, and exercises based upon the division of multidigit integers. Students often have trouble dividing huge numbers. These worksheets are often constructed using a similar algorithm that provides step-by-step instructions. They may not provide the understanding they seek from these worksheets. Long division can be taught using the base ten blocks. After students have learned the steps, long division should be a common thing for them. Students can practice division of large numbers by using various worksheets and practice questions. In addition, fractional data expressed in decimals can be found in the worksheets. Additionally, you can find worksheets for hundredsths, which are especially useful in learning to divide large amounts of money. Divide the numbers into smaller ones. It can be difficult for smaller groups to make use of numbers. Although it appears good on paper, many small group facilitators detest the process. It genuinely reflects the way that human bodies develop, and the procedure can aid in the Kingdom’s endless growth. It encourages people to reach out and help the less fortunate, as well as the new leadership team to take over the reins. It can be useful to brainstorm ideas. You can form groups of individuals with similar traits and experience. This allows you to generate innovative ideas. Reintroduce yourself to each person after you’ve formed your groups. This is an excellent exercise that stimulates creativity and new thinking. To divide large numbers into smaller chunks of information, the basic division operation is utilized. This is extremely useful when you need to create equal quantities of things for multiple groups. For example, a huge class can be split into five classes. The groups are then added to give the original thirty pupils. If you are dividing numbers, there are two kinds of numbers to be aware of either the divisor or the quotient. Dividing one number by another produces “ten/five,” while divising two by two produces the same result. Ten power should be used to calculate huge numbers. The splitting of huge numbers into powers can make it easier to compare them. Decimals are an essential part of shopping. You can see them on receipts, price tags, and food labels. They are utilized by petrol pumps to display the cost per gallon as well as the quantity of fuel being transported via the pipe. It is possible to split large numbers into their powers of ten using two different methods by shifting the decimal point to the left or multiply by 10-1. The other method uses the powers of ten’s associative feature. Once you’ve learned how to utilize the power of ten associative feature, you can divide huge numbers into smaller powers. Mental computation is utilized in the initial method. Divide 2.5 by 10 to find a pattern. As a power of ten is increased, the decimal point will shift to the left. Once you’ve mastered this concept, it’s feasible to apply it to solve any problem. The second method is mentally dividing very large numbers into powers of 10. It is then possible to quickly express large numbers using scientific notation. If you are using scientific notation, big numbers should be written using positive exponents. To illustrate, if we shift the decimal points five spaces to your left, you can convert 450,000 into 4.5. You can also divide an enormous number into smaller power than 10, or split it into smaller power of 10. Gallery of Free Printable Long Division Worksheets The Long Division Printable Division Worksheet For Kids Math Blaster Free Printable Long Division Worksheets 5Th Grade Free Printable Long Division 3 Digits By 1 Digit Without Remainders 20 Leave a Comment	{"url":"https://www.divisonworksheets.com/free-printable-long-division-worksheets/","timestamp":"2024-11-04T03:48:13Z","content_type":"text/html","content_length":"66108","record_id":"<urn:uuid:9462893f-f20c-4f45-8071-f0474a7663b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00020.warc.gz"}
HDM: High Dimensional Data Mining The 2^nd International Workshop on High Dimensional Data Mining (HDM’14) In conjunction with the IEEE International Conference on Data Mining (IEEE ICDM 2014) The workshop will take place on 14 December 2014, in room Madrid 5. Accepted Papers SC219 Adaptive Semi-Supervised Dimensionality Reduction Jia Wei SC207 High dimensional Matrix Relevance Learning Frank-Michael Schleif, Thomas Villmann, and Xibin Zhu SC203 Boosting for Vote Learning in High-dimensional kNN Classification Nenad Tomasev DM515 Random KNN Shengqiao Li, James Harner, Donald Adjeroh SC204 SUBSCALE: Fast and Scalable Subspace Clustering for High Dimensional Data Amardeep Kaur and Amitava Datta SC208 Who Wrote This? Textual Modeling with Authorship Attribution in Big Data Naruemon Pratanwanich and Pietro Lio SC211Renyi Divergence based Generalization for Learning of Classification Restricted Boltzmann Machines Qian Yu, Yuexian Hou, Xiaozhao Zhao, and Guochen Cheng SC216 Two approaches of using heavy tails in high dimensional EDA Momodou Sanyang, Hanno Muehlbrandt, and Ata Kaban SC201 Out-of-Sample Error Estimation: the Blessing of High Dimensionality Luca Oneto, Alessandro Ghio, Sandro Ridella, Jorge Luis Reyes Ortiz, and Davide Anguita DM283 Dimensionality reduction based similarity visualization for neural gas Kadim Tasdemir Invited Talk Bob Durrant: The Unreasonable Effectiveness of Random Projections in Computer Science [slides] Abstract: Random projection is fast becoming a workhorse approach for high-dimensional data mining, with applications in clustering, regression, classification and low-rank matrix approximation amongst others. In the first half of this talk I will briefly survey some of the historical motivations for random projection and the applications these have inspired. In the second half I will focus on my work with Ata Kaban and Jakramate Bootkrajang, which takes some different perspectives leading to some novel theory and simple, yet effective, algorithms for classification and unconstrained real-valued optimization specifically aimed at very high-dimensional domains. │ 9:00- 9:10│Welcome message │ │ │HDM’14 Chairs │ │ 9:10-10:00│The Unreasonable Effectiveness of Random Projections in Computer Science (Invited Talk) slides │ │ │Bob Durrant │ │10:00-10:15│Coffee break │ │ │Morning session: Reducing the curses of high dimensionality │ │10:15-10:40│Dimensionality Reduction Based Similarity Visualization for Neural Gas │ │ │Kadim Tasdemi │ │10:40-11:05│Adaptive Semi-Supervised Dimensionality Reduction │ │ │Jia Wei │ │11:05-11:30│SUBSCALE: Fast and Scalable Subspace Clustering for High Dimensional Data │ │ │Amardeep Kaur and Amitava Datta │ │11:30-11:55│Who Wrote This? Textual Modeling with Authorship Attribution in Big Data │ │ │Naruemon Pratanwanich and Pietro Lio │ │11:55-12:20│High dimensional Matrix Relevance Learning │ │ │Frank-Michael Schleif, Thomas Villmann, and Xibin Zhu │ │12:20-14:05│Lunch break │ │ │Afternoon session: In search of the blessings of high dimensionality │ │14:05-14:30│Vote Learning in High-dimensional kNN Classification │ │ │Nenad Tomasev │ │14:30-14:55│Random KNN │ │ │Shengqiao Li, James Harner, Donald Adjeroh │ │14:55-15:20│Out-of-Sample Error Estimation: the Blessing of High Dimensionality │ │ │Luca Oneto, Alessandro Ghio, Sandro Ridella, Jorge Luis Reyes Ortiz, and Davide Anguita │ │15:20-15:45│Renyi Divergence based Generalization for Learning of Classification Restricted Boltzmann Machines │ │ │Qian Yu, Yuexian Hou, Xiaozhao Zhao, and Guochen Cheng │ │15:45-16:00│Coffee break │ │16:00-16:25│Two Approaches of Using Heavy Tails in High dimensional EDA │ │ │Momodou Sanyang, Hanno Muehlbrandt, and Ata Kaban │ │16:25-16:50│Discussion & Closing │ Description of Workshop Stanford statistician David Donoho predicted that the 21st century will be the century of data. "We can say with complete confidence that in the coming century, high-dimensional data analysis will be a very significant activity, and completely new methods of high-dimensional data analysis will be developed; we just don't know what they are yet." -- D. Donoho, 2000. Beyond any doubt, unprecedented technological advances lead to increasingly high dimensional data sets in all areas of science, engineering and businesses. These include genomics and proteomics, biomedical imaging, signal processing, astrophysics, finance, web and market basket analysis, among many others. The number of features in such data is often of the order of thousands or millions - that is much larger than the available sample size. A number of issues make classical data analysis methods inadequate, questionable, or inefficient at best when faced with high dimensional data spaces: 1. High dimensional geometry defeats our intuition rooted in low dimensional experiences, and this makes data presentation and visualisation particularly challenging. 2. Phenomena that occur in high dimensional probability spaces, such as the concentration of measure, are counter-intuitive for the data mining practitioner. For instance, distance concentration is the phenomenon that the contrast between pair-wise distances may vanish as the dimensionality increases. This makes the notion of nearest neighbour meaningless, together with a number of methods that rely on a notion of distance. 3. Bogus correlations and misleading estimates may result when trying to fit complex models for which the effective dimensionality is too large compared to the number of data points available. 4. The accumulation of noise may confound our ability to find low dimensional intrinsic structure hidden in the high dimensional data. 5. The computation cost of processing high dimensional data or carrying out optimisation over a high dimensional parameter spaces is often prohibiting. This workshop aims to promote new advances and research directions to address the curses and uncover and exploit the blessings of high dimensionality in data mining. Topics of interest include (but are not limited to): - Systematic studies of how the curse of dimensionality affects data mining methods - New data mining techniques that exploit some properties of high dimensional data spaces - Theoretical underpinning of mining data whose dimensionality is larger than the sample size - Stability and reliability analyses for data mining in high dimensions - Adaptive and non-adaptive dimensionality reduction for noisy high dimensional data sets - Methods of random projections, compressed sensing, and random matrix theory applied to high dimensional data mining and high dimensional optimisation - Models of low intrinsic dimension, such as sparse representation, manifold models, latent structure models, and studies of their noise tolerance - Classification of high dimensional complex data sets - Functional data mining - Data presentation and visualisation methods for very high dimensional data sets - Data mining applications to real problems in science, engineering or businesses where the data is high dimensional Paper submission High quality original submissions are solicited for oral and poster presentation at the workshop. Papers should not exceed a maximum of 8 pages, and must follow the IEEE ICDM format requirements of the main conference. All submissions will be peer-reviewed, and all accepted workshop papers will be published in the proceedings by the IEEE Computer Society Press. Submit your paper here. Important dates Early cycle submission deadline: August 17, 2014; Late-cycle submission deadline: 26 September. Notifications to authors: October 10, 2014 Workshop date: December 14, 2014 Program committee Robert J. Durrant - University of Waikato, NZ Barbara Hammer - Clausthal University of Technology, Germany Ata Kaban - University of Birmingham, UK John A. Lee - Universite Catholique de Louvain, Belgium Milos Radovanovic - University of Novi Sad, Serbia Stephan Gunnemann - Carnegie Mellon University Yiming Ying - University of Exeter, UK Michael Biehl - University of Groningen Carlotta Domeniconi - George Mason University Mehmed Kantardzic - University of Louisville Udo Seiffert - University of Magdeburg Frank-Michael Schleif - University of Birmingham, UK Peter Tino - University of Birmingham, UK Guoxian Yu - Southwest University Thomas Villmann - University of Applied Science Mittweida Michel Verleysen - Universite Catholique de Louvain, Belgium Workshop organisers School of Computer Science, University of Birmingham, UK School of Computer Science, University of Birmingham, UK University of Applied Sciences Mittweida, (Saxonia) Germany	{"url":"https://www.cs.bham.ac.uk/~axk/HDM14.htm","timestamp":"2024-11-07T03:30:27Z","content_type":"text/html","content_length":"138670","record_id":"<urn:uuid:fba9f30d-6d8b-4ee1-a87b-77f4f2d2fc40>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00280.warc.gz"}
Sp(6, 2)’s Family, Plots, and Ramsey Numbers Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36, 10, 4, 2), but there are 32548 non-isomorphic graphs with parameters (36, 15, 6, 6). Peter Cameron, Random Strongly Regular Graphs? This a shorter version of this report which I just put on my homepage. But I added more links. I assume that one is familiar with strongly regular graphs (SRGs). One particular SRG, the collinearity graph of [latex]Sp(6, 2)[/latex], has parameters [latex](63, 30, 13, 15)[/latex]. A very simple technique, Godsil-McKay (GM) switching, can generate many non-isomorphic graphs with the same parameters. More specifically, there are probably billions such graphs and I generated 13 505 292 of them. This is the number of graphs which you obtain by applying a certain type of GM switching (i.e. using a bipartition of type 4, 59) at most 5 times to [latex]Sp(6,2)[/latex]. Plots of the number of cliques, cocliques, and the size of the autmorphism group are scattered throughout this You can download the list of all graphs from my Dropbox folder here. Be aware that it is about 2 GB in size (gzip, in graph6 format). Let us give a short description of the collinearity graph of $ Sp(2d, q)$: Vertices are 1-dimensional subspaces of $ \mathbb{F}_q^{2d}$. Two 1-dimensional subspaces are adjacent if they are perpendicular with respect to the bilinear form $ x_1y_2 – x_2y_1 + \ldots + x_{2d-1}y_{2d} – x_{2d}y_{2d-1}$. For $ (d,q) = (3, 2)$, this graph has 63 vertices, is 30-regular, two adjacent vertices have 13 common neighbors, and two non-adjacent vertices have 15 common neighbors. It is long known that GM switching can produce graphs non-isomorphic to the collinearity graph of $ Sp(2d, 2)$. A variation of it works for $ Sp(2d, q)$ and even in more general settings. Why do I summarize my computations? The collinearity graph of the polar space $ Sp(6, 2)$ is in many ways the smallest really interesting representative of the family of collinearity graphs coming from finite classical polar spaces. Thus it is a good toy model to investigate general behavior. Some basic facts about $ Sp(6, 2)$: It has an automorphism group of size $ 1451520$, clique number 7 and coclique number 7. SRGs with the same parameters as $ Sp(6, 2)$ have spectrum $ (30,3^{35},-5^ {27})$, clique number at most 7 and coclique number at most 9. The clique and coclique bounds follow from Hoffman’s ratio bound (due to the name of my blog, I have to mention this). In the following we discuss some questions associated with SRGs with the same parameters as the collinearity graph of $ Sp(6, 2)$ and, more generally, $ Sp(6, q)$. Anurag Bishnoi, Valentina Pepe and I are implicitly asking if graphs such as the collinearity graph of $ Sp(2d, q)$ can be modified such that they are clique-free. Anurag discusses this here a bit for a special case. A far more specific question is if there exists an SRG with parameters the same as $ Sp(6, q)$ which is $ K_4$-free. If one can show this for infinitely many $ q$, then this essentially determines the asymptotic behavior of the Ramsey number $ r(4, n)$. Again, Anurag discussed this connection in his blog. This is still too general, so we are stuck with the following: Question 1. Is there a strongly regular graph with parameters $ (63, 30, 13, 15)$ which does not contain a clique of size $ 4$? Probable answer: no. Even a targeted threshold based computer search could only find a $ K_6$-free graph. Most SRGs with these parameters seem to be $ I_8$-free. As Anurag Bishnoi pointed out to me, we currently only know that the Ramsey number $ r(4, 8)$ is at least 56. Therefore the following is variation of the question above, even so the answer is still probably no: Question 2. Is there a strongly regular graph with parameters $ (63, 30, 13, 15)$ which contains no clique of size 4 and no coclique of size 8? Update (10.04.2020): Alexander Gavrilyuk pointed out to me that Bondarenko, Prymak, and Radchenko showed in 2014 that any SRG with parameters $ (63, 30, 13, 15)$ has at least 2354 cliques of size 4. Therefore, the answer to Question 1 and 2 is now surely no. Update (12.04.2020): I found a second (embarrisingly easy) argument to show that such an SRG always contains many cliques of size 4. Now I wondering if one can show that it contains a clique of size I was wondering before if the number of SRGs with the same parameters as the collinearity graph of $ Sp(2d, q)$ growth hyperexponentially (in $ q$ or $ d$, your choice). Similarly, Bill Kantor is interested in questions such as the following, see for instance here: For any given group G, can we find an SRG $ \Gamma$ with the same parameters as the $ Sp(2d, q)$ for some (alternatively: infinitely many) $ (d, q)$ such that $ Aut(\Gamma) = G$? The following question goes into the same direction: Question 3. Do almost all SRGs with the same parameters as $ Sp(2d, q)$ have a trivial automorphism group? You can choose what “almost all” means here. In any interpretation, I suspect the answer to be yes. We used nauty-traces, cliquer, a tiny C program, and standard GNU tools for this small investigation. 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Optimal Rent Calculator Author: Neo Huang Review By: Nancy Deng LAST UPDATED: 2024-10-02 21:21:20 TOTAL USAGE: 2310 TAG: Economics Finance Real Estate Unit Converter ▲ Unit Converter ▼ Powered by @Calculator Ultra Find More Calculator☟ Historical Background The concept of optimal rent involves assessing the best possible rent that a property owner can charge, maximizing rental income while remaining attractive to potential tenants. The formula allows landlords to estimate a fair rent amount based on the property’s value. To calculate the optimal rent, use the following formula: \[ OR = \frac{0.20 \cdot HV}{12} \] • $OR$ is the optimal rent ($/month), • $HV$ is the home value ($). Example Calculation If the home value is $360,000, the optimal rent is calculated as: \[ OR = \frac{0.20 \cdot 360,000}{12} \approx 6000 \text{ } \text{$/month} \] Importance and Usage Scenarios Calculating optimal rent helps landlords set a reasonable rent that maximizes their rental revenue while considering vacancy rates and the local rental market. It provides a baseline to help adjust the rent according to competitive rates and rental demand. Common FAQs 1. Is the optimal rent the maximum rent I should charge? Not necessarily. The optimal rent provides a baseline estimate, but it's crucial to consider market conditions, vacancy rates, and the property’s specific location. 2. How frequently should I adjust the optimal rent? Review and adjust the rent annually or when market conditions change, as real estate values and rental markets fluctuate. 3. What other factors should I consider besides the formula? Market trends, neighborhood demand, the condition of the property, and amenities all play significant roles in setting competitive rent. Understanding optimal rent allows property owners to balance profit margins with affordability and market competitiveness, leading to better tenant retention and sustainable rental income.	{"url":"https://www.calculatorultra.com/en/tool/optimal-rent-calculator.html","timestamp":"2024-11-04T01:13:38Z","content_type":"text/html","content_length":"46037","record_id":"<urn:uuid:e9495941-cc1c-45f4-8180-204138b3de16>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00120.warc.gz"}
Mr. Nickels College: SUNY New Paltz Teaching : 10+ years Math Courses: A, B, 3C Computer Science: Visual Basic/ Java AP Graduate Degree : Instructional. - ppt download Presentation is loading. Please wait. To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy , including cookie policy. Ads by Google	{"url":"http://slideplayer.com/slide/6383296/","timestamp":"2024-11-03T07:37:13Z","content_type":"text/html","content_length":"161189","record_id":"<urn:uuid:5bbc04f2-fb34-45c6-8505-979df4131dda>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00353.warc.gz"}
Next: Resistor Model Up: Passive Element Lines Previous: Coupled (Mutual) Inductors Contents Index General Form: rname n1 n2 [value \| modname] [options] [r=expr \| poly c0 [c1 ...]] Options: [m=mult] [temp=temp] [tc1=tcoeff1] [tc2=tcoeff2] [l=length] [w=width] rload 2 10 10k rmod 3 7 rmodel l=10u w=1u The n1 and n2 are the two element nodes, and value is the resistance, for a constant value resistor. A resistor model modname can alternatively be specified and allows the calculation of the actual resistance value from strictly geometric information and the specifications of the process. If a resistance is specified after modname, it overrides the geometric information (if any) and defines the nominal-temperature resistance. If modname is specified, then the resistance may be calculated from the process information in the model modname and the given length and width. In any case, the resulting value will be adjusted for the operating temperature temp if that is specified, using correction factors given. If value is not specified, then modname and length must be specified. If width is not specified, then it will be taken from the default width given in the model. If the resistance can not be determined from the provided parameters, a fatal error results. This behavior is different from traditional Berkeley SPICE, which provides a default value of 1K. The paramaters that are understood are: This is the parallel multiplication factor, that represents the number of devices effectivly connected in parallel. The effect is to multiply the conductance by this factor, so that the given resistance is divided by this value. This overrides the `m' multiplier found in the resistor model, if any. The temp is the Celsius operating temperature of the resistor, for use by the temperature coefficient parameters. The first-order temperature coefficient. This will override the first-order coefficient found in a model, if given. The keyword ``tc'' is an alias for ``tc1''. The second-order temperature coefficient. This will override the second-order coefficient found in a model, if given. The length of the resistor. This applies only when a model is given, which will compute the resistance from geometry. The width of the resistor. This applies only when a model is given, which will compute the resistance from geometry. The mult is a real number which will multiply the linear conductance used in the noise equations. Probably the major use is to give noise=0.0 to temporarily remove a resistor from a circuit noise This can also be given as ``res=expr'' or ``resistance=expr'', where expr is an expression giving the nominal-temperature device voltage divided by device current (``large signal'' resistance) in ohms, possibly as a function of other variables. This form is applicable when the first token following the node list is not a resistance value or model name. It also applies when a model is given, it overrides the geometric resistance value. poly c0 [c1 ...] This form allows specification of a polynomial resistance, which will take the form Resistance = c0 + c1^ . v + c2^ . v^2... where v is the voltage difference between the positive and negative element nodes. There is no built-in limit to the number of terms. Next: Resistor Model Up: Passive Element Lines Previous: Coupled (Mutual) Inductors Contents Index Stephen R. Whiteley 2024-10-26	{"url":"http://wrcad.com/manual/wrsmanual/node86.html","timestamp":"2024-11-10T11:26:44Z","content_type":"text/html","content_length":"7976","record_id":"<urn:uuid:e44ca736-e6b8-4d1e-98be-baa18c48e9c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00385.warc.gz"}
\cot x = 12$ find the value of $\ Hint: For the given question we have to divide denominator and numerator with $\sin x$. It is known that $\dfrac{{\cos x}}{{\sin x}} = \cot x$. Then it is given that $16\cot x = 12$, we will substitute the value of $\cot x$ and after simplifying we the value of the given question. Complete step-by-step answer: Given $16\cot x = 12$simplifying, we get, $ \Rightarrow \cot x = \dfrac{3}{4}$ ……………. $\left( i \right)$ Now according to question $\left[ {\dfrac{{\sin x + \cos x}}{{\sin x - \cos x}}} \right]$ Dividing denominator and numerator with $\sin x$,we get, $ \Rightarrow \left[ {\dfrac{{1 + \dfrac{{\cos x}}{{\sin x}}}}{{1 - \dfrac{{\cos x}}{{\sin x}}}}} \right]$ Substituting $\dfrac{{\cos x}}{{\sin x}} = \cot x$, we get, $ \Rightarrow \left[ {\dfrac{{1 + \cot x}}{{1 - \cot x}}} \right]$ Now, substituting $\left( i \right)$and on simplifying we get, $ \Rightarrow \left[ {\dfrac{{1 + \dfrac{3}{4}}}{{1 - \dfrac{3}{4}}}} \right]$ $ = 7$ $\therefore \left[ {\dfrac{{\sin x + \cos x}}{{\sin x - \cos x}}} \right] = 7$ Answer is option (A) Note: Alternative method $\left[ {\dfrac{{\sin x + \cos x}}{{\sin x - \cos x}}} \right]$ Dividing denominator and numerator with$\cos x$we get $ \Rightarrow \left[ {\dfrac{{\tan x + 1}}{{\tan x - 1}}} \right]$ We know that $\cot x = \dfrac{3}{4}$and $\tan x = \dfrac{1}{{\cot x}}$, we get, $\therefore \tan x = \dfrac{4}{3}$………….. (ii) Now substituting (ii) and simplifying we get $ \Rightarrow \left[ {\dfrac{{\dfrac{4}{3} + 1}}{{\dfrac{4}{3} - 1}}} \right] = 7$ $\therefore \left[ {\dfrac{{\sin x + \cos x}}{{\sin x - \cos x}}} \right] = 7$	{"url":"https://www.vedantu.com/question-answer/if-16cot-x-12-find-the-value-of-left-dfracsin-x-class-11-maths-cbse-5f8a837977bf9c142d52554c","timestamp":"2024-11-06T20:11:58Z","content_type":"text/html","content_length":"161883","record_id":"<urn:uuid:481e1e8a-9aef-4b0a-afc9-608e100a19cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00551.warc.gz"}
Ascension math question Has anyone done the math on value per cover for the different star tiers as you ascend characters? Like how much cp, iso, hp, shards, per cover across different tiers comparing ascending vs just champ recycling. I have tried searching and I am struggling to find what I am looking for. • I pride myself as a mathy person (I was at the forefront of the Champions 1.0 Farming debate way back when), but I haven't bothered to run the numbers for myself yet because my desire to Ascend characters to 5★, Levels 450-550 is so much more powerful than knowing that I might be missing out on ±10% of one resource or another. That's also why I've been Ascending everyone as Max-Min combinations, because I want to minimize roster slots in use and minimize the time / covers to get characters to 550. I'm not even sure this game will still exist in the time it will take me to 550 anyone, but that's why I also don't want to slow it down any. @meadowsweet said: I pride myself as a mathy person (I was at the forefront of the Champions 1.0 Farming debate way back when), but I haven't bothered to run the numbers for myself yet because my desire to Ascend characters to 5★, Levels 450-550 is so much more powerful than knowing that I might be missing out on ±10% of one resource or another. That's also why I've been Ascending everyone as Max-Min combinations, because I want to minimize roster slots in use and minimize the time / covers to get characters to 550. I'm not even sure this game will still exist in the time it will take me to 550 anyone, but that's why I also don't want to slow it down any. Agreed 100%. Also, a ton of characters are getting +100 levels every week, so if you can get them to 450...well, when they're boosted, they're 550. • The math was done. The covers have been counted. The rewards per level have been looked at. Rewards are greater the higher you go for your characters. But if you want character specific rewards don't ascend your toons. If you ascended toons at max champ you lose covers and levels...ie higher level rewards. Jumping a 1 to a 5 or a 2 to a 5 will take thousands of covers over the life...but the rewards get better the higher they get.....ie pull a 1* and get 25 cp type of situation. The short game is painful the long game is bountiful. All of this math has been done and posted when ascending became a thing. It is on the forums. go find it. I have tried searching and I am struggling to find what I am looking for. It is on the forums. go find it. Not super helpful? • I wish I could find a post someone had done but everyone who has dug into it says you come out ahead even when you consider the fact that the number of rewards per cover drop. The main question would be what you want to get; what thing will make the biggest difference to your roster? and what tier are you focused on? You have basically asked if anyone has done the math on every base tier and every level they get ascended to and what the ROI per cover, or per 100 or 200 or 300 or 400 covers, is. One thing you get with ascension - once you get someone into the 3/4/5 tier - is some QOL return when you don't need to farm and free up roster slots for a while as you just add covers/levels. AKA initial roster setup can be a bit higher but then you have less to worry about for a longer period of time while collecting covers for that character vs farming. Also the lowest tiers can be downright annoying to build when RNG isn't being your friend and you can't shard them and you have a 5/5/2 with 13 saved covers. Ascension reduces the frequency of that circumstance. • I guess cp and iso would be the biggest mark on my roster. I'm in 5* land with my highest leveled being 467, and have only maxed one 4, coulson. I am working on his dupe and trying to fill out my 5 tier to be able to one day champ everyone. I'm not really looking to 550 anyone. @tiomono said: I guess cp and iso would be the biggest mark on my roster. I'm in 5* land with my highest leveled being 467, and have only maxed one 4, coulson. I am working on his dupe and trying to fill out my 5 tier to be able to one day champ everyone. I'm not really looking to 550 anyone. My math (hopefully right) says ascending 1-3 means you get 65000 iso from 200 1* covers. Vs 20,000 from selling them. You get no cp from 1s unless you ascend them. From a 1-3 you get 6 LTs and 41 cp from those 200 covers you use to get them to 266. Hope that helps a little. • So the big complication in the math on how to optimally ascend characters is that covers are converted into levels at a set rate but are never converted back when you ascend more than the minimum. For example, converting two 2★s to a 3★ if they're both 144 you get an extra 25 levels on the new 3★ -- all well and good for the first one, but then you level them up to 266 and you need to get a new 3★. You take two 144 2★ again and get that 191 3★ and combine it with the 266, instead of the 191 counting as 50 extra covers, it just counts as 25 (as if it was a "native" 3★,) so that 25 gets divided by 3 to make a 278 with one cover left over. That's what they mean by "tax." If you just made a 166 champ, Ascended the max and new 3★, then applied 50 covers, you'd end up with a 286 4★ with two covers left over so you're losing 12 1/2 2★ covers in this deal. The question is are the extra rewards worth losing those covers? I'd say probably yes unless you really want to level your 2★ up to 4★ level as fast as possible. You also definitely want to get at least the LT from level 167 -- that's a no-brainer. OTOH I wouldn't always combine max-champs either -- it gets worse from there since you'd be spending the 3★ levels at 2:1 and then getting a 3:1 return on it. So you'll basically be spending (at least) 200 covers on getting that new 3★ to max champ, if you do that with the second champ you'll then get 33.3 levels out of 200 covers, which would normally give you 66.6 levels. @tiomono said: Has anyone done the math on value per cover for the different star tiers as you ascend characters? Like how much cp, iso, hp, shards, per cover across different tiers comparing ascending vs just champ recycling. I have tried searching and I am struggling to find what I am looking for. Hello tiomono, I have exactly this. I built a sprawling model to study various approaches to ascension. It calculates the total of every resource from any level to any other level, and the avg of each per cover invested. But, at 12,000 cells, it's not easily share ... uh ... share ... uh ... bull. ... and there are multiple versions, for considering the various varieties of feeders and nonfeeders, particularly in the 4* tier. What I was looking for was an optimal, or near-enough-to-optimal solution that was an offramp short of max-max'ing everything, hopefully as close to min-max'ing as possible. My thought was, that, if you took an approach that had the highest percentage of the coverflow "landing" in the 5* tier, that is, the most covers possible were chasing the highest value rewards ... which sounds a lot like min-max'ing ... that would be the approach that resulted in the highest avg per cover of each resource, or near enough ... highest avg per cover of the resources that mattered most. And ... the answer WAS ... nope. Max-max everything, from lvl 1 all the way to 550, for the highest avg return per cover. To me that was a very boring result ... and counter-intuitive! After all, consider a 1, which takes the most covers to go all the way and so many of them get pumped into low tiers if max-max'ing ... only 300 of 1908 covers are spent at the 5 tier, only 15.7% of all the covers chasing those sweet 5* rewards. If min-max'ing, 400 of 1508 covers go toward 5s, 26.5% ... how does it not result in higher avgs per cover ... ? It's all the double-dipping, all those champ level counting toward the next tier and getting paid twice for them. If that wasn't part of the scheme, then only sadistic suboptimal fools would ever max-max anything. But, it is part of the system ... and it ensures that max-max'ing IS the optimal "min-max" (harhar) solution. _... UNLESS ... _ The above is only true if you take the totals and avgs of each resource separately and at face value. If those numbers were weighted (with weights reflecting your personal situation) and indexed, you might get a different answer. For instance, if I become ridiculously post-Iso (again), I would give greater weight to HP, CP, and/or LTs, and greatly discount iso. That could result in a mathematical endorsement of min-max • I posted the following on reddit: https://www.reddit.com/r/MarvelPuzzleQuest/comments/173562p/ascended_farming_analysis/ I wrote it just after ascension rolled out, so some of it is out of date now. Specifically, everything that refers to 2 power 1s. The ascended champ rewards also turned out to be slightly different than advertised. The basic conclusions are sound though and the overall rewards relationships are basically the same. @JoeHandle said: @tiomono said: Has anyone done the math on value per cover for the different star tiers as you ascend characters? Like how much cp, iso, hp, shards, per cover across different tiers comparing ascending vs just champ recycling. I have tried searching and I am struggling to find what I am looking for. Hello tiomono, I have exactly this. I built a sprawling model to study various approaches to ascension. It calculates the total of every resource from any level to any other level, and the avg of each per cover invested. But, at 12,000 cells, it's not easily share ... uh ... share ... uh ... bull. I would just like to say that I salute you for your efforts. I've done a little of that, mostly to refute arguments for non-max-max ascension strategies, and have a full appreciation for your efforts. I never went as far as you did, but would like to thank you for doing so. It makes me much more confident in the max-max strategy. ... and there are multiple versions, for considering the various varieties of feeders and nonfeeders, particularly in the 4* tier. What I was looking for was an optimal, or near-enough-to-optimal solution that was an offramp short of max-max'ing everything, hopefully as close to min-max'ing as possible. My thought was, that, if you took an approach that had the highest percentage of the coverflow "landing" in the 5* tier, that is, the most covers possible were chasing the highest value rewards ... which sounds a lot like min-max'ing ... that would be the approach that resulted in the highest avg per cover of each resource, or near enough ... highest avg per cover of the resources that mattered most. And ... the answer WAS ... nope. Max-max everything, from lvl 1 all the way to 550, for the highest avg return per cover. To me that was a very boring result ... and counter-intuitive! After all, consider a 1, which takes the most covers to go all the way and so many of them get pumped into low tiers if max-max'ing ... only 300 of 1908 covers are spent at the 5 tier, only 15.7% of all the covers chasing those sweet 5* rewards. If min-max'ing, 400 of 1508 covers go toward 5s, 26.5% ... how does it not result in higher avgs per cover ... ? It's all the double-dipping, all those champ level counting toward the next tier and getting paid twice for them. If that wasn't part of the scheme, then only sadistic suboptimal fools would ever max-max anything. But, it is part of the system ... and it ensures that max-max'ing IS the optimal "min-max" (harhar) solution. _... UNLESS ... _ The above is only true if you take the totals and avgs of each resource separately and at face value. If those numbers were weighted (with weights reflecting your personal situation) and indexed, you might get a different answer. For instance, if I become ridiculously post-Iso (again), I would give greater weight to HP, CP, and/or LTs, and greatly discount iso. That could result in a mathematical endorsement of min-max ascension. I'm pretty sure max-max will still be best even if you don't care about Iso. CP and LT awards all come from champ progression, so moving through as much champ progression as possible, i.e. max-max ascension strategy, will still be the optimal solution. That said I did do a bit of analysis recently that showed that, at the 3 level the HP rewards for maxing the duplicate are not worth the "cover tax" on a per cover basis. Basically to max the HP /cover value when ascending a 1* or 2* you should max-max at 2, max-min at 3 and max-max at 4. The downside is it does cost you on CP and LTs. • I appreciate the info and replies from everybody. I think I have a more clear decision in my head now. • Isn’t max/max the best option if you want the most rewards per cover but min max if you want to get to 550 fastest? @Daredevil217 said: Isn’t max/max the best option if you want the most rewards per cover but min max if you want to get to 550 fastest? Max+min will get you to 450 the fastest. Max+max returns the most rewards overall. If asking "per cover" tho that can get murky. Lots of devil's in lots of details! Generally yes max+max maxes returns but if focused on a particular reward type that may not be true and the answer can also vary based on starting tier. @Daredevil217 said: Isn’t max/max the best option if you want the most rewards per cover but min max if you want to get to 550 fastest? I don't think Min Max gets you to 550 any faster. It just gets you a 5 faster. Max-Max should give you a 475 (+100 covers /4) so you still need 300 covers to get you to 550. Either way it's 400 covers beyond your 2nd champ. KGB Posts: 3,223 Chairperson of the Boards @Scofie said: @Daredevil217 said: Isn’t max/max the best option if you want the most rewards per cover but min max if you want to get to 550 fastest? I don't think Min Max gets you to 550 any faster. It just gets you a 5* faster. Max-Max should give you a 475 (+100 covers /4) so you still need 300 covers to get you to 550. Either way it's 400 covers beyond your 2nd champ. For 4 stars that's true. They are talking about 1, 2 and 3 stars where you have to ascend multiple times and when doing that there is the cover tax starting on the 2nd ascension if you go the max/max route. • Always is not correct. For base level ascending yes. So ascending two Polaris max/max. Ascending two 3* Psylocke max/max. Two 2* Moonstones max/max. There's no cover tax in these situations. And actually any base 4* always max/max. But for instance once you have a 370 and a 270 Psylocke you should just ascend them min/max. In order to take that 270 Psylocke to 370 would take 300 covers and you'd get 112,500 iso 4,000 HP 250 CP 10 LL tokens From all the 4* rewards and you'd have a level 475. But if instead you min/maxed them you'd not get any of those 4* rewards but would have a 450 Psylocke and 300 covers now to apply at a 4 covers to 1 level ratio or 75 levels for a 5* and get a 525 Psylocke and... 112,000 iso 350 CP 4,500 HP 12 LL tokens From all the rewards from 476 to 525 This is a copy paste for technically a different question but still applies here. @JoeHandle said: Max+min will get you to 450 the fastest. @Scofie said: I don't think Min Max gets you to 550 any faster. It just gets you a 5* faster. Max-Max should give you a 475 (+100 covers /4) so you still need 300 covers to get you to 550. Either way it's 400 covers beyond your 2nd champ. @Pantera236 said: Always is not correct. For base level ascending yes. So ascending two Polaris max/max. Ascending two 3* Psylocke max/max. Two 2* Moonstones max/max. There's no cover tax in these situations. Number of covers required to get characters to Levels 450 & 550, by original Tier: (In the time it takes to get a Max-Max character Ascended to 5★ Level 450, the Max-Min equivalent 1★ & 2★ would already be at Level 550, the 3★ at Level 525, and the 4★ at Level 475.) • One of the fun things about a system like ascension, and the larger topic of roster management in general, is that there is no one right, optimal approach for everyone. Everyone's "optimal" depends on their situation, resources, goals, preferences, blah blah blah. But then everyone can pretend that everone else is asserting that their take is the best take for everyone, which leads to unnecessary misunderstanding and confrontation so the internet can happen. Usually, if you can stay disengaged and appreciate the potential perspectives of others, you can see where they are coming from and why. Why they arrived at their 'optimal' approach to whatever it is. And thereby see why they're wrong and what flavor of psychoses they are afflicted with, harhar. But sometimes ... not. For instance, I see this "max base tier, then max-min, then 5* " asserted as 'best' often, in many places, from many people. but ... uhhh ... it don't make no sense (to me?). What's the focus? What's the rationale? Most approaches are focused either on performance (increasing utility (read: level) of the character as fast as possible), or resource(s) (how can I max the return of resource(s) while ascending [characters] ? ). But this advice is at odds with itself if it's focused on performance or resources ... is it something else? Or a not so great attempt to split the difference? If trying to get the best resource return from a given # of covers, well, this is not the way. It seems to be a misfire reaction to the increasing # of covers required to advance in ascended tiers (1/ lvl, 2/lvl, 3/lvl, 4/lvl ... argh). Without understanding what a person wants to accomplish and why they think their chosen method is a good one for achieving that accomplishing, it's impossible to tell them why they are wrong and Is there a white paper on this "base-tier max, then max-min then 5* " school of thought I can read? @Pantera236 said: Always is not correct. For base level ascending yes. So ascending two Polaris max/max. Ascending two 3* Psylocke max/max. Two 2* Moonstones max/max. There's no cover tax in these situations. And actually any base 4* always max/max. But for instance once you have a 370 and a 270 Psylocke you should just ascend them min/max. In order to take that 270 Psylocke to 370 would take 300 covers and you'd get 112,500 iso 4,000 HP 250 CP 10 LL tokens From all the 4* rewards and you'd have a level 475. But if instead you min/maxed them you'd not get any of those 4* rewards but would have a 450 Psylocke and 300 covers now to apply at a 4 covers to 1 level ratio or 75 levels for a 5* and get a 525 Psylocke and... 112,000 iso 350 CP 4,500 HP 12 LL tokens From all the rewards from 476 to 525 This is a copy paste for technically a different question but still applies here. • 44.7K Marvel Puzzle Quest • 13.5K Magic: The Gathering - Puzzle Quest • 548 Other 505 Go Inc. Games • 381 Other Games • 7 505 Go Inc. Forum Rules	{"url":"https://forums.d3go.com/discussion/89786/ascension-math-question","timestamp":"2024-11-09T10:34:41Z","content_type":"text/html","content_length":"406729","record_id":"<urn:uuid:4c87f873-a05e-45c2-8f79-58d7f0279531>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00456.warc.gz"}
Relations and Functions In this discussion, you will be assigned two equations where you will complete a variety of math work related to mathematical functions. Read the following instructions in order and view the example (available for download in your online classroom) to complete this discussion. Please complete the following problems according to your assigned number. (Instructors will assign each student their number).Solve the following ProblemsFor Question 1 See Attachment, problem #2 For Question 2: See 2nd Attachment. Problem is for #37, the second one with y=There are many ways to go about solving math problems. To solve this problem, you will be required to do some work that will not be included in the discussion point.First, graph your functions so that you can clearly describe the graphs in your post. Your graph itself is not required in your post, although a discussion of the graph is required. Make sure you have at least five points for each equation to graph. Show all math work for finding the points.Mention any key points on the graphs, including intercepts, vertex, or start/end points. Points with decimal values need not be listed, as they might be found in a square root function. Stick to integer value points.Discuss the general shape and location of each of your graphs.State the domain and range for each of your equations. Write them in interval notation.State whether each of the equations is a function or not giving your reasons for the answer.Select one of your graphs and assume it has been shifted three units upward and four units to the left. Discuss how this transformation affects the equation by rewriting the equation to incorporate those numbers.Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in sentences describing the thought behind your math work.FunctionRelationVertical Line testTransformationYour initial post should be at least 250 words in length.	{"url":"https://coursescholars.com/relations-and-functions/","timestamp":"2024-11-05T12:23:36Z","content_type":"text/html","content_length":"81232","record_id":"<urn:uuid:8666e888-b9ec-4f74-a54a-24ead24a8496>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00232.warc.gz"}
The Life and Times of Cyrus Cousins Abridged Biography I am Cyrus Cousins, a postdoctoral scholar at the Duke and Carnegie Mellon Universities, with professors Walter Sinnott-Armstrong, Jana Schaich Borg, Vincent Conitzer, and Hoda Heidari, and formerly a visiting assistant professor at Brown University, where I also completed my doctoral studies under the tutelage of Eli Upfal. Before arriving at Brown University, I earned my undergraduate degree in computer science, mathematics, and biology from Tufts University. Known to some as The Count of Monte Carlo, I study all manner of problems involving sampling, randomization, and learning in data science and beyond, with a particular interest in uniform convergence theory and the rigorous treatment of fair machine learning. My work studies theoretical bounds on generalization error in exotic settings, and applies such bounds to tasks of real-world interest, most notably in data science, empirical game theory, and fair machine learning. My dissertation Bounds and Applications of Concentration of Measure in Fair Machine Learning and Data Science received the Joukowsky Outstanding Dissertation Prize, and I have been awarded the Dean's Faculty Fellowship visiting assistant professorship at Brown University and the CDS Postdoctoral Fellowship postdoctoral scholarship at the University of Massachusetts Amherst. My favorite theorem is the Dvoretzky-Kiefer-Wolfowitz Inequality, and my favorite algorithm is simulated annealing. Research Statements and Application Materials For posterity and the benefit of future applicants, my older research statements (graduate school, etc.) are also available. Research Overview In my research, I strive for a delicate balance between theory and practice. My theory work primarily lies in sample complexity analysis for machine learning, as well as time complexity analysis and probabilistic guarantees for efficient sampling-based approximation algorithms and data science methods [1] [2] [3] [4]. In addition to statistical analysis, much of my work deals with delicate computational questions, like how to optimally characterize and bound the sample-complexity of estimation tasks from data (with applications to oblivious algorithms, which achieve near-optimal performance while requiring limited a priori knowledge), as well as the development of fair-PAC learning, with the accompanying computational and statistical reductions between classes of learnable On the practical side, much of my early work was led by the observation that modern methods in statistical learning theory (e.g., Rademacher averages) often yield vacuous or unsatisfying guarantees, so I strove to understand why, and to show sharper bounds, with particular emphasis on constant factors and performance in the small sample setting. From there, I have worked to apply statistical methods developed for these approaches to myriad practical settings, including statistical data science tasks, the analysis of machine learning, and fairness sensitive machine learning algorithms. By blurring the line betwixt theory and practice, I have been able to adapt rigorous theoretical guarantees to novel settings. For example, my work on adversarial learning from weak supervision stemmed from a desire to apply statistical learning theory techniques in absentia of sufficient labeled data. Conversely, I have also motivated novel theoretical problems via practical considerations and interdisciplinary analysis; my work in fair machine learning led to the fair-PAC learning formalism, where power-means over per-group losses (rather than averages) are minimized. The motivation to optimize power-means derives purely from the economic theory of cardinal welfare, but the value of this learning concept only becomes apparent when one observes that many of the desirable (computational and statistical) properties of risk minimization translate to power-mean minimization.	{"url":"https://www.cyruscousins.online/","timestamp":"2024-11-05T00:50:02Z","content_type":"text/html","content_length":"50813","record_id":"<urn:uuid:701dabab-c8fc-4d24-bf10-c62196a5b1ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00171.warc.gz"}
Dodecahedron Facts Notice these interesting things: • It has 12 Faces • Each face has 5 edges (a pentagon) • It has 30 Edges • It has 20 Vertices (corner points) • and at each vertex 3 edges meet • It is one of the Platonic Solids Volume and Surface Area Volume = (15+7×√5)/4 × (Edge Length)^3 Surface Area = 3×√(25+10×√5) × (Edge Length)^2 It is called a dodecahedron because it is a polyhedron that has 12 faces (from Greek dodeca- meaning 12). When we have more than one dodecahedron they are called dodecahedra When we say "dodecahedron" we often mean "regular dodecahedron" (in other words all faces are the same size and shape), but it doesn't have to be - this is also a dodecahedron, even though all faces are not the same. 12-Sided Dice? Yes! A dodecahedron which has 12 equal faces has an equal chance of landing on any face. In fact, you can make fair dice out of all of the Platonic Solids. Make your own Dodecahedron, cut out the shape and glue it together. 1848, 1849, 1850, 1851	{"url":"http://wegotthenumbers.org/dodecahedron.html","timestamp":"2024-11-08T12:40:32Z","content_type":"text/html","content_length":"5316","record_id":"<urn:uuid:2e72f4a2-429c-43b3-9304-b72056ea84f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00342.warc.gz"}
No fault lines Build a rectangle with multiple 2x1 bricks such that the rectangle has no "fault lines". A fault line is a straight line through the rectangle that does not cut through any of the bricks. For example, the following rectangle fails because of the fault line indicated in red. What's the smallest rectangle you can make without fault lines? You may send solutions to skepticsplay at gmail dot com. Can you do the same with 3x1 bricks? This puzzle is taken from Polyominoes: Puzzles, Patterns, Problems, and Packings , by Solomon W. Golomb. See solutions 11 comments: a single 2x1 brick But you need to use multiple bricks. More than three are necessary. If you need to verify a solution you can send it to me. I wanna say 5x4... It's not enough to guess the best size, you have to construct a solution! 5x4 is not possible. is a 5x6 case. Smallest? Yes, 5x6 is the smallest possible solution! I believe there are other unique forms though. Oh hey, 5x6 is the smallest? I found a 5x6 solution. No, nevermind, now that I look closely it's just like Eduard Baumann's solution, only flipped. This comment has been removed by the author. A B C C D D A B E F F G H H E I J G K L L I J M K N O OM	{"url":"https://skepticsplay.blogspot.com/2012/05/no-fault-lines.html","timestamp":"2024-11-12T00:43:40Z","content_type":"application/xhtml+xml","content_length":"86904","record_id":"<urn:uuid:9b0bdfc1-6b20-464e-9031-f813acc8dda6>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00821.warc.gz"}
Runing a FoxH code in Mathematica Ah, thank you. I apologize for not seeing your post with the attached notebook. If I open that notebook and evaluate the large cell containing the definition of FoxH and then evaluate your cell trying to use FoxH I get the same result that you do. In[9]:= FoxH[{{{-2,1}}},{{{0,1}}},-0.5, Out[9]= FoxH[{{{-2,1}}},{{{0,1}}},-0.5,OptionsPattern[{FoxHFractionTolerance,FoxHDuplicationLimit,FoxHWorkingPrecision}]] When Mathematica returns the expression you just evaluated and does not change it that usually means that Mathematica does not understand how to do that. Many new users are confused when it does this and does not print a message explaining.I am very puzzled by this and spend time trying to find what mistake I might have made. I carefully check spelling and capitalization and {} and find nothing. I find and read this and I change the seven "right arrow" unicode characters in the code to the two characters -> which Mathematica will understand as a "Rule" construct. I read this and look at the examples I then try this, also using -> which Mathematica will understand as a "Rule" construct. and it gives a variety of errors, but I suspect those are "good" errors and reflect the very simple example data that you used. Please make these changes and see if you get the same results. Then try using data which will give a result that you can verify and see if this works. There are more things about the Options which I think may need further changes, but you should be able to get started using the code. I believe it is possible, if you wish to use the default values for options, to do this FoxH[{{{-2, 1}}}, {{{0, 1}}}, -0.5]	{"url":"https://community.wolfram.com/groups/-/m/t/221962?sortMsg=Replies","timestamp":"2024-11-13T18:31:01Z","content_type":"text/html","content_length":"302277","record_id":"<urn:uuid:40ba70f8-3968-4bcf-87c4-353f7975a2f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00164.warc.gz"}
High Tech High TouchÂ (contributor), October 2017, Intentional Futures and the Bill and Melinda Gates Foundation Personalized Learning at WGU, EDUCAUSE, April 2016 (with David Leasure) The World is My School: Welcome to the Era of Personalized Learning, Jan/Feb 2011 issue of The Futurist Open Faculty, EDUCAUSE, July/August 2010 Busynessgirl.comÂ contains more than a thousandÂ short articles about teaching ideas, resources for learning, tutorials for using technology, teaching in the digital age, game design, and general adventures.Â This might not be considered a â scholarlyâ publication, but it was a heck of a lot of work and gets viewed by thousands of people every week. Algebra Activities, a 1000-page Instructor resource binder of activities and teaching guides for algebra, published January 2010 by Cengage Learning.Â This is accompanied by approximately 30 individual Algebra Activities student workbooks for the entire Cengage Learning Algebra textbook line.Â (working on republication, send an email if youâ re interested) Knowledge, Attitudes, and Instructional Practices of Community College Mathematics Instructors: The Search for a KAP Gap in Collegiate Math, Dissertation for Western Michigan University, April 2011 Bringing Color to the Classroom, MAA FOCUS, August/September 2012 Phone Cameras Handle Information in a Snap, MAA FOCUS, June/July 2012 The Secret Technology Club, MAA FOCUS, April/May 2012 Can Math and Discussion Boards Compute?, MAA FOCUS, December 2011 / January 2012 Abandon the Red Pen, MAA FOCUS, October/November 2011 Become a Screencasting Star, MAA FOCUS, August/September 2011 Organize Your Online Course Shell, MAA FOCUS. June/July 2011 Online Course Shell: Worth the Effort, MAA FOCUS, April/May 2011 Move Your Lecture into the Digital Age, MAA FOCUS, Feb/March 2011 Take Another Shot at Your Document Camera, MAA FOCUS, Dec 2010/Jan 2011 A â Really Simple Solutionâ to Information Overload, MAA FOCUS, Oct/Nov 2010 Use Online Video to Create Context in Math Courses, MAA FOCUS, June/July 2010 (p.5) Tips for Effective Webinars, eLearn Magazine, January 21, 2010 eLearning Tools for STEM, eLearn Magazine, October 6, 2009 Jing and Math, MathAMATYC Educator, September 2009 Wolfram Alpha: What IS it and How Will It Affect You? AMATYC News, Volume 24, Number 4 (August 2009). Instructor Resource Binders and Workbooks for Tussy/Gustafson Algebra Series, Andersen, M. H., Cengage Learning, 2009. An extensive collection of teaching guides, assessments, and concept-oriented activities for the algebra classroom.Â This set of publications includes three workbooks and three Instructor Resource Sample Chapter or Teasers to Try (these may be used in your classrooms free of charge) Student Solutions Manual and Instructorsâ Solutions Manual for Mathematical Applications for the Management, Life, and Social Sciences, 8th edition, Harshbarger, Reynolds, & Andersen, Cengage Learning, 2009. Back to the Board, NISOD Innovation Abstracts, Vol. 28, No. 19, 2006. PreAlgebra Instructorâ s Solutions Manual with Testing, 4th Edition, Aufmann, Barker, Lockwood, Verity, & Andersen, Houghton Mifflin Company, 2005. Instructorâ s Resource Manual for Basic College Mathematics, 8th edition, Aufmann, Barker, Lockwood, & Andersen, Houghton Mifflin Company, 2005. Instructorâ s Resource Manual for Introductory Algebra, 7th edition, Aufmann, Barker, Lockwood, & Andersen, Houghton Mifflin Company, 2005. Student Solutions Manual and Instructorsâ Solutions Manual for Mathematical Applications for the Management, Life, and Social Sciences, 7th edition, Harshbarger, Reynolds, & Andersen, Houghton Mifflin Company, 2004. Instructorâ s Resource Manual for College Algebra and Trigonometry, 5th edition, Aufmann, Barker, Nation, & Andersen, Houghton Mifflin Company, 2004. Selecting Applicants and Employees for Math Competency, Jackson, J. M. & Andersen, M. H., World at Work Journal, Vol. 12, No. 1, 2003. Basic Algebra Review, Andersen, M. H., McGraw-Hill, 1998. NOTE:Â There are many other textbook supplements that Iâ ve authored, but at some point I lost track and this is simply a list of the ones I can remember that are relatively recent.	{"url":"https://edgeoflearning.com/about-me/publications/","timestamp":"2024-11-11T14:42:59Z","content_type":"text/html","content_length":"183459","record_id":"<urn:uuid:6b0988c9-f398-43bf-b88e-9e0f3339de90>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00391.warc.gz"}
Spin Group of Prof. Justin Dressel Weak measurements of a quantum system allow us to monitor a quantum state in real time with only a small disturbance. Finding the quantum state from a series of weak measurements typically involves a “quantum filter” derived from basic laws of quantum mechanics. This traditional weak measurement approach works well if the quantum state changes slowly compared with the detector response time. However, if the qubit changes rapidly, traditional methods that reconstruct the quantum state fail because the detector affects our best estimate of the quantum state. In our experiment, we use weak measurements to monitor fast dynamics of a superconducting qubit coupled to a readout resonator. Instead of using a traditional method to monitor the qubit state, we develop a new method with a long short-term memory neural network, which learns the quantum mechanics responsible for the state trajectories by itself. The long short-term memory neural network also learns an unexpected correction to the standard quantum filter, which is most clearly visible in the stochastic measurement disturbance of the fast qubit trajectories. Our newly developed theory shows that this correction can be well explained by the memory effect of the detector. With our ability to accurately track fast qubit dynamics, we expect to see new applications of weak measurements such as diagnosing qubit gates in quantum processors and continuous measurements for quantum error correction.	{"url":"https://justindressel.com/","timestamp":"2024-11-14T11:55:09Z","content_type":"text/html","content_length":"39875","record_id":"<urn:uuid:41a005cf-47f8-4d11-9c41-950a10c67fa2>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00623.warc.gz"}
criteria for proper working if ball mill WEBMar 1, 2014 · Analysis of ball mill grinding operation using mill power specific kinetic parameters. March 2014. Advanced Powder Technology 25 (2):625–634. DOI: / Authors: Gupta ... WhatsApp: +86 18838072829 WEBAug 4, 2023 · A SAG mill, or semiautogenous grinding mill, operates on the principle of autogenous or selfgrinding. It uses a combination of ore and grinding media to break down larger rocks into smaller pieces. Understanding the working principles of a SAG mill is crucial for anyone involved in the mining industry. In this section, we will delve deeper ... WhatsApp: +86 18838072829 WEBBond models, SGI, Morrell Mi. The Bond ball mill work index is one of the most commonly used grindability tests in mining, and is often referred to as the Bond work index . The test is a 'lockedcycle' test where ground product is removed from test cycles and replaced by fresh feed. The test much achieve a steadystate before completion. WhatsApp: +86 18838072829 WEBJan 1, 2017 · An increase of over 10% in mill throughput was achieved by removing the ball ss from a single stage SAG mill. These ss are non spherical ball fragments resulting from uneven wear of balls ... WhatsApp: +86 18838072829 WEBThe length of the mill is approximately equal to its diameter. Principle of Ball Mill : Ball Mill Diagram. • The balls occupy about 30 to 50 percent of the volume of the mill. The diameter of ball used is/lies in between 12 mm and 125 mm. The optimum diameter is approximately proportional to the square root of the size of the feed. WhatsApp: +86 18838072829 WEBJul 15, 2013 · The basis for ball mill circuit sizing is still B ond's methodology (Bond, 1962). The Bond ball and rod. mill tests are used to determine specific energy c onsumption (kWh/t) to grind from a ... WhatsApp: +86 18838072829 WEBTypically R = 8. Rod Mill Charge: Typically 45% of internal volume; 35% – 65% range. Bed porosity typically 40%. Height of bed measured in the same way as ball mills. Bulk density of rods = tons/m3. In wet grinding, the solids concentration 1s typically 60% – 75% by mass. A rod in situ and a cutaway of a rod mill interior. WhatsApp: +86 18838072829 WEBFigure 1: Drill out the hole using a MicroDrill System or Tool Grip and ball mill. Figure 2: The eyelet flange can be used to secure a new circuit in place. Figure 3: Set the eyelet using an Eyelet Press or using a Tool Grip and Setting Tools. Figure 4: The small cone tip end of the lower Setting Tool faces up. WhatsApp: +86 18838072829 WEBMay 25, 2023 · Ball mill trunnion diameter and length are critical considerations, particularly in relation to the size of the mill and load distribution. The diameter should be proportionate to the mill's size to ensure proper loadbearing capacity. The length of the trunnion should be sufficient to support the mill shell and provide stability during ... WhatsApp: +86 18838072829 WEBThe ball mill is a rotating cylindrical vessel with grinding media inside, which is responsible for breaking the ore particles. Grinding media play an important role in the comminution of mineral ores in these mills. This work reviews the appliion of balls in mineral processing as a function of the materials used to manufacture them and the ... WhatsApp: +86 18838072829 WEBJul 14, 2016 · A Cerro Verde expansion used a similar flowsheet as the 2006commissioned circuit to triple circuit capacity. The expansion circuit includes eight MP1250 cone crushers, eight HPGRs (also x units, with 5 MW each), and six ball mills (22 MW each), for installed comminution power of 180 MW. and a nameplate capacity of 240,000 tpd. WhatsApp: +86 18838072829 WEBIf a ball mill uses little or no water during grinding, it is a 'dry' mill. If a ball mill uses water during grinding, it is a 'wet' mill. A typical ball mill will have a drum length that is 1 or times the drum diameter. Ball mills with a drum length to diameter ratio greater than are referred to as tube mills. WhatsApp: +86 18838072829 WEBJan 30, 2024 · The most common diameters are found in increments starting from 1/64 inch ( mm) to 1 inch ( mm) for precision to general milling appliions. Metric sizes, ranging from 1 mm to 25 mm, are also widely used. Furthermore, end mills can be acquired with different lengths, such as stub, regular, long, and extralong, which offer varying ... WhatsApp: +86 18838072829 WEBNever operate the mill without the cover securely in place. This will help to prevent injuries and damage to the equipment. b. Never add material to the mill while it is running. This can cause damage to the equipment and can be dangerous. c. Always use the correct grinding media for the specific material being ground. WhatsApp: +86 18838072829 WEBJan 1, 2021 · The grinding process in ball mills and vertical roller mills is fundamentally different [15]. Following are advantages of VRM over Ball Mills with reference to these issues: • Strong drying ability Inlet hot air from Kiln can dry materials with 20% water content. (max moisture 20% vs. 3% in ball mill) [15], [16]. • WhatsApp: +86 18838072829 WEBApr 1, 2005 · Process description. The laboratory grinding circuit consists of an overflow type ball mill (30 cm × 30 cm), a sump fitted with a variable speed pump and a hydrocyclone classifier (30 mm). The schematic diagram of the circuit is shown in Fig. 1. There are two local controllers that form part of the process: Sump level is maintained . WhatsApp: +86 18838072829 WEBBall mill is a grinder for reducing hard materials to powder. A ball mill grinds material by rotating a cylinder with steel grinding balls/ceramic balls causing the balls to fall back into the cylinder and onto the material to be ground. The cylinder rotates at a relatively slow speed, allowing the balls to cascade through the mill base, thus ... WhatsApp: +86 18838072829 WEBWhat is a ball mill? A ball mill is a size reduction or milling equipment which uses two grinding mechanisms, namely, impact and shear. 1 Unlike other size reduction equipment, such as breakers and jaw crushers, ball mills are capable of producing powders and particulate materials of mean particle sizes within the 750 to 50 micron (µm) range.. Ball . WhatsApp: +86 18838072829 WEBApr 17, 2018 · Product Size: All passing 6 mesh with 80% passing 20 mesh. Net power from pilot plant: KWH/ST. Mill, gear and pinion friction multiplier: Mill Power required at pinionshaft = (240 x x ) ÷ = 5440 Hp. Speed reducer efficiency: 98%. 5440 Hp ÷ = 5550 HP (required minimum motor output power). WhatsApp: +86 18838072829 WEBOct 19, 2016 · On Mill Installation and Maintenance. Before starting the erection of the mill, adequate handling facilities should be provided or made available, bearing in mind the weights and proportions of the various parts and subassemblies. This information can be ascertained from the drawings and shipping papers. WhatsApp: +86 18838072829 WEBJul 1, 2009 · Ball mill grinding circuits are essentially multivariable systems characterized with couplings, timevarying parameters and time delays. The control schemes in previous literatures, including detuned multiloop PID control, model predictive control (MPC), robust control, adaptive control, and so on, demonstrate limited abilities in control ball mill . WhatsApp: +86 18838072829 WEBApr 22, 2017 · When a ball mill having a proper crushing load is rotated at the critical speed, the balls strike at a point on the periphery about 45° below horizontal, or S in Fig. 1. ... The ratio of moisture to solids is important in ball mill work. From actual operation it has been observed that fine grinding is best done when water constitutes 33 to 40 ... WhatsApp: +86 18838072829 WEBAug 5, 2021 · The Ball mill pulveriser is basically horizontal cylindrical tube rotating at low speed on its axis, whose length is slightly more to its diameter. The inside of the Cylinder shell is fitted with heavy cast liners and is filled with cast or forged balls for grinding, to approximately 1/3 of the diameter. Raw coal to be ground is fed from the ... WhatsApp: +86 18838072829 WEBHere's a stepbystep guide to help make the process easier: Prepare the Mill: Before removing the old liners, ensure that the mill is shut down and all power is disconnected. This will ensure that the process is safe and that there is no risk of injury. Remove the Old Liners: Using a pry bar or other appropriate tool, carefully remove the ... WhatsApp: +86 18838072829 WEBOct 29, 2022 · Principle: Fluid mills operate on the principle of impact and attrition. Working of fluid energy mill: Start the fluid energy mill and pass the feed through the inlets of the venturi. the air entering through the grinding nozzle transport the power in the elliptical or circular truck of the mill. the highvelocity air stream passed and the milling . WhatsApp: +86 18838072829 /radial runout. of the drive trains. power splitting. distances variable. load distribution. of the girth gear. gear is through hardened only, fatigue strength is limited. Dynamic behaviour. A lot of individual rotating masses risk of resonance vicinities. WhatsApp: +86 18838072829 WEBApr 22, 2015 · In the present research work, the mixture of boron carbide and graphite ceramic powders with a theoretical composition of 50 %each by. weight were mechanically alloyed in a laboratory ball mill ... 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Who provides solutions for problems related to hash functions and collision resolution in algorithms assignments for cloud computing? Computer Science Assignment and Homework Help By CS Experts Who provides solutions for problems related to hash functions and collision resolution in algorithms assignments for cloud computing? For large infrastructure, we provide solutions to the following issues. – Our solution should be easy to find, reliable and reliable. – We first consider the case when the instance is fixed. This leads us to the following two goals: 1. – Find the number of buckets of the instance with the most buckets are set. 2. – Choose the single input key for hashing and compute the hash value. – The general case is when the number of objects is infinite. An infinite array of buckets changes its size. The actual change is divided among buckets in just 1 step. When the instance is created though a certain number – For a cloud, we randomly assign the buckets whose size is the sub problem of the instance – in each step. We then compute the value of the bucket twice. If two of the buckets are available in our case, they will only appear if they are in the unique key of the instance. – The number of objects (buckets) will grow across sizes in proportion to the number of the instance sizes. For this reason, we can use this solution for a large instance. The implementation uses a histogram of buckets whose length will grow as the number of objects is increased. As a result, the value of the bucket is significantly increased and we can obtain less buckets. – By the way, if the instance is a “hashing pack”, then it does not depend so much on compute speed (more times the number of instances is fixedWho provides solutions for problems related to hash functions and collision resolution in algorithms assignments for cloud computing? Although they can be related to click for more other through their names, these notions are generally not mutually exclusive. Many solutions, though, can further benefit from sharing the same idea of sharing the concept of a hash function. In 2012, researchers first presented the problem of finding a solution to an OAC problem in a high-risk environment. Complete My Online Class For Me The problem is referred to as a critical dimension, according to which the search space needs to be filled up, and the space to contain most of the information needed to solve it. Many problems were solved using this kind of solution; it is difficult for experts to be sure that another solution is the solution, let alone be robust against algorithms not described in a solution. Even with the simplicity and security of the hash functions, that is surely the case. In order to avoid false conclusions about algorithms themselves based on a single tool, researchers have developed some very specialized algorithms that can be used to carry out actual problems. This enables researchers to evaluate solutions that are potentially difficult to solve and find an experiment that will help them later. It also opens a potential path for researchers to conduct experiment-like research before pursuing software development. There are many algorithms out there for finding an OAC algorithm. The so-called OAC algorithms are a kind of hash function and have been used extensively, some of which are proposed in many different papers. They can be analyzed, for example, to find the solution to a major problem. When a researcher is surprised by this kind of analysis, he or she should decide to work with someone who knows how to do the analysis in another area, and he or she will use the tool he or she had developed years ago and use the solutions he or she found while writing a paper on that problem. However, the most appropriate approach to deal with such papers is the one proposed by the like it researchers. Some known very frequently used hash function algorithms include: using data in a hash function analyzing an OWho provides solutions for problems related to hash functions and collision resolution in algorithms go right here for cloud computing? Read our paper titled “Elastic algebra as a solution for collision analysis”, which is a full text paper dedicated to this issue. Abstract Global-free algorithms solvers for hash functions are at the point where they are the next best choice since they provide only a static solution, which is more resilient to many mutations than heuristics, but gives him click to find out more performance over heuristics on a wide variety of instances. This is demonstrated by the use of an inbred algorithm, an approach that works on very large instances, without taking into account the mutations required to design the solution from scratch in order to adapt it to a different problem. Here, we obtain solutions using tools used in the above literature, with a particular focus on large look at more info where each mutation is present and is quite often relevant at a fixed number of instances. Introduction A hash function encodes an algorithm’s input values using an integer-coded algorithm. Two such algorithms: “uniform” (called RSP from now on just “exponential”) and “hyperbolic” (also called “semi-littrowd”) are mentioned, which are considered to be compatible with, and useful wherever, in some way, one is able to work with more than one parameter. In such cases, hyperbolic algorithms are a good fit to a common basic problem, though they mostly extend to the case of large data sets, such as real-time SUT applications. The approach generally relies on an algorithm, known as the universal hashing algorithm, which computes the algorithm simply algorithmically. But, an algorithm can actually be executed in many different ways in the world. Homework For You Sign Up The best known algorithms for this problem, RSP, are actually linear extensions of a recently proved linear system, called the linear sparse algorithm. We will refer to this algorithm as the universal GK algorithm; while we would like to	{"url":"https://csmonsters.com/who-provides-solutions-for-problems-related-to-hash-functions-and-collision-resolution-in-algorithms-assignments-for-cloud-computing","timestamp":"2024-11-10T19:19:47Z","content_type":"text/html","content_length":"86596","record_id":"<urn:uuid:cfd97835-6710-4852-a77c-2b303334d02e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00146.warc.gz"}
Statistics/Introduction/Why - Wikibooks, open books for an open world Imagine reading a book for the first few chapters and then becoming able to get a sense of what the ending will be like - this is one of the great reasons to learn statistics. With the appropriate tools and solid grounding in statistics, one can use a limited sample (e.g. read the first five chapters of Pride & Prejudice) to make intelligent and accurate statements about the population (e.g. predict the ending of Pride & Prejudice). This is what knowing statistics and statistical tools can do for you. In today's information-overloaded age, statistics is one of the most useful subjects anyone can learn. Newspapers are filled with statistical data, and anyone who is ignorant of statistics is at risk of being seriously misled about important real-life decisions such as what to eat, who is leading the polls, how dangerous smoking is, etc. Knowing a little about statistics will help one to make more informed decisions about these and other important questions. Furthermore, statistics are often used by politicians, advertisers, and others to twist the truth for their own gain. For example, a company selling the cat food brand "Cato" (a fictitious name here), may claim quite truthfully in their advertisements that eight out of ten cat owners said that their cats preferred Cato brand cat food to "the other leading brand" cat food. What they may not mention is that the cat owners questioned were those they found in a supermarket buying Cato. “The best thing about being a statistician is that you get to play in everyone else’s backyard.” John Tukey, Princeton University More seriously, those proceeding to higher education will learn that statistics is the most powerful tool available for assessing the significance of experimental data, and for drawing the right conclusions from the vast amounts of data faced by engineers, scientists, sociologists, and other professionals in most spheres of learning. There is no study with scientific, clinical, social, health, environmental or political goals that does not rely on statistical methodologies. The basic reason for that is that variation is ubiquitous in nature and probability and statistics are the fields that allow us to study, understand, model, embrace and interpret variation. See Also UCLA Brochure on Why Study Probability & Statistics	{"url":"https://en.m.wikibooks.org/wiki/Statistics/Introduction/Why","timestamp":"2024-11-02T08:49:48Z","content_type":"text/html","content_length":"41535","record_id":"<urn:uuid:13241d61-7b00-4d83-b10d-60c6cad1d906>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00573.warc.gz"}
Equation of a Straight Line: Parametric Form Lesson Video: Equation of a Straight Line: Parametric Form Mathematics • First Year of Secondary School In this video, we will learn how to find the equation of a straight line in parametric form using a point on the line and the vector direction of the line. Video Transcript In this video, we will learn how to find the equation of a straight line in parametric form using a point on the line and the direction vector of the line. Let’s begin by recalling the vector form of a straight line. The vector form of a straight line passing through the point 𝐴 and parallel to the direction vector 𝐝 is 𝐫 is equal to 𝐎𝐀 plus 𝑡 multiplied by 𝐝. This can be represented on the two-dimensional coordinate plane as shown. We recall that the position vector of a point is the vector starting from the origin and ending at that point. The vector form of the equation of a line describes each point on the line as its position vector 𝐫. Each value of the parameter 𝑡 gives the position vector of one point on the line. If we consider a line passing through the point 𝐴 with coordinates 𝑥 sub zero, 𝑦 sub zero and parallel to the direction vector 𝐝 with components 𝑎 and 𝑏, then the vector form of the equation of the line is given by 𝐫 is equal to 𝑥 sub zero, 𝑦 sub zero plus 𝑡 multiplied by 𝑎, 𝑏. Simplifying the right-hand side of our equation, we get the components 𝑥 sub zero plus 𝑎𝑡 and 𝑦 sub zero plus 𝑏𝑡. We can then write the position vector on the left-hand side in terms of its 𝑥- and 𝑦-components. This leads us to the parametric form of the equation of a straight line. The parametric form of the equation of a line passing through the point 𝐴 with coordinates 𝑥 sub zero, 𝑦 sub zero and parallel to the direction vector 𝐝 is 𝑥 is equal to 𝑥 sub zero plus 𝑎𝑡, 𝑦 is equal to 𝑦 sub zero plus 𝑏𝑡. In our first question, we will look at an example of this in practice. Straight line 𝐿 passes through the point 𝑁 with coordinates three, four and has a direction vector 𝐮 equal to two, negative five. Then the parametric equations of line 𝐿 are what. We begin by recalling that the parametric form of the equation of a line passing through the point 𝑥 sub zero, 𝑦 sub zero and parallel to the direction vector 𝑎, 𝑏 is 𝑥 is equal to 𝑥 sub zero plus 𝑎𝑡 and 𝑦 is equal to 𝑦 sub zero plus 𝑏𝑡. We are told in the question that the straight line 𝐿 passes through the point with coordinates three, four. This means that our value of 𝑥 sub zero is three and 𝑦 sub zero is equal to four. We are also given a direction vector 𝐮 such that 𝑎 is equal to two and 𝑏 is negative five. Substituting in our values of 𝑥 sub zero and 𝑎, we get 𝑥 is equal to three plus two 𝑡. And substituting the values of 𝑦 sub zero and 𝑏, we get 𝑦 is equal to four minus five 𝑡. It is important to note that we could replace the letter 𝑡 with any other letter as the parameter. For example, 𝑥 is equal to three plus two 𝑘 and 𝑦 is equal to four minus five 𝑘 is also a valid solution. We can therefore conclude that these are the parametric equations of line 𝐿. We will now look at another example looking at the process of converting the vector form to the parametric form. The vector equation of a straight line is given by 𝐫 of 𝑡 is equal to 𝑡 multiplied by five, two plus negative one, three. Which of the following pairs of parametric equations represents this straight line? Is it (A) 𝑥 is equal to five 𝑡 plus two, 𝑦 is equal to negative 𝑡 plus three? (B) 𝑥 is equal to five 𝑡 minus one, 𝑦 is equal to two 𝑡 plus three. (C) 𝑥 is equal to three 𝑡 plus two, 𝑦 is equal to negative 𝑡 plus five. (D) 𝑥 is equal to negative 𝑡 plus five, 𝑦 is equal to three 𝑡 plus two. Or (E) 𝑥 is equal to two 𝑡 plus three, 𝑦 is equal to five 𝑡 minus one. We begin this question by recalling that the vector form of the equation of a line is 𝐫 is equal to 𝑥 sub zero, 𝑦 sub zero plus 𝑡 multiplied by 𝑎, 𝑏, where 𝑥 sub zero, 𝑦 sub zero is the position vector of a point on the line and 𝑎, 𝑏 is a direction vector of the line. Comparing this to the equation given, we see that the direction vector for our line is five, two. The position vector when 𝑡 equals zero is negative one, three. This means that our line passes through the point one, three and is parallel to the direction vector five, two. Next, we recall that the parametric form of the equation of a line is 𝑥 equals 𝑥 sub zero plus 𝑎𝑡 and 𝑦 equals 𝑦 sub zero plus 𝑏𝑡. Substituting in the values from this question, we have 𝑥 is equal to negative one plus five 𝑡 and 𝑦 is equal to three plus two 𝑡. Noting the way the equations have been written in the five options, we have 𝑥 is equal to five 𝑡 minus one and 𝑦 is equal to two 𝑡 plus three. The correct answer is option (B). In our next example, we will apply the definition for the parametric form to obtain the direction vector. The direction vector of the straight line whose parametric equations are 𝑥 equals two and 𝑦 equals negative two 𝑘 plus four is what. We begin by recalling that the parametric form of the equation of a line passing through the point with coordinates 𝑥 sub zero, 𝑦 sub zero and parallel to the direction vector with components 𝑎, 𝑏 is 𝑥 is equal to 𝑥 sub zero plus 𝑎𝑡 and 𝑦 is equal to 𝑦 sub zero plus 𝑏𝑡. We are given the parametric equations 𝑥 equals two and 𝑦 equals negative two 𝑘 plus four. Noting that the parameter here is 𝑘, we can rewrite the general equations as shown. Comparing terms, we see that 𝑥 sub zero is equal to two. 𝑎 is equal to zero, as there is no 𝑘 term in our first parametric equation. We also have 𝑦 sub zero is equal to four and 𝑏 is equal to negative two. This means that our line passes through the point two, four. It is also parallel to the direction vector zero, negative two. We can therefore conclude that the direction vector of the straight line whose parametric equations are 𝑥 equals two and 𝑦 equals negative two 𝑘 plus four is zero, negative two. Whilst it is not required in this question, we could use this information to write the vector equation of the line. The position vector 𝐫 is equal to two, four plus 𝑘 multiplied by zero, negative two. As we have seen in this question, instead of giving the direction vector directly, a problem may provide this indirectly. In fact, the direction vector of a line may be given indirectly in three possible ways: firstly, by providing two points that lie on the line; secondly, by providing the angle between the line and the positive 𝑥-axis; and thirdly, by providing the slope or gradient of the line. We will now consider a couple of these scenarios. Find the parametric equations of the straight line that makes an angle of 135 degrees with the positive 𝑥-axis and passes through the point one, negative 15. Is it (A) 𝑥 is equal to one plus 𝑘, 𝑦 is equal to negative 15 minus 𝑘? (B) 𝑥 is equal to one plus 𝑘, 𝑦 is equal to one minus 15𝑘. Is it (C) 𝑥 is equal to negative 15 minus 𝑘, 𝑦 is equal to one plus 𝑘? Or (D) 𝑥 is equal to one, 𝑦 is equal to negative 15 minus 𝑘. Let’s begin by sketching the straight line, noting it makes an angle of 135 degrees with the positive 𝑥-axis. We are told in the question that the line passes through the point with coordinates one, negative 15. And we recall that the slope of the line that makes an angle 𝜃 with the positive 𝑥-axis is given by tan 𝜃. As already mentioned, this angle 𝜃 is 135 degrees. And tan of 135 degrees is equal to negative one. This means that the slope of our line is negative one. This slope is equal to the rise over the run, which leads to a rise of negative one and a run of one. As the run is the change in the 𝑥-values and the rise is the change in the 𝑦-values, this gives us a direction vector of one, negative one. The parametric form of the equation of a line passing through a point with coordinates 𝑥 sub zero, 𝑦 sub zero and parallel to any direction vector 𝑎, 𝑏 is 𝑥 is equal to 𝑥 sub zero plus 𝑎𝑘 and 𝑦 equals 𝑦 sub zero plus 𝑏𝑘. As we now have these two pieces of information, we can substitute in our values. Firstly, we have 𝑥 is equal to one plus 𝑘. And secondly, we have 𝑦 is equal to negative 15 minus 𝑘. The correct answer is therefore option (A). The parametric equations of the straight line that makes an angle of 135 degrees with the positive 𝑥-axis and passes through the point one, negative 15 is 𝑥 equals one plus 𝑘 and 𝑦 equals negative 15 minus 𝑘. In our final question, we are given the direction vector of the line by means of the slope. A straight line passes through the point one, six and has a slope of one-half. Which of the following pairs of parametric equations represents this straight line? Is it (A) 𝑥 is equal to 𝑡 plus one, 𝑦 is equal to two 𝑡 plus six? (B) 𝑥 is equal to 𝑡 plus two, 𝑦 is equal to six 𝑡 plus one. (C) 𝑥 is equal to four 𝑡 plus one, 𝑦 is equal to two 𝑡 plus six. (D) 𝑥 is equal to six 𝑡 plus one, 𝑦 is equal to 𝑡 plus two. Or (E) 𝑥 is equal to four 𝑡 plus one, 𝑦 is equal to 𝑡 plus six. We begin by recalling that the parametric form of the equation of a line passing through a point with coordinates 𝑥 sub zero, 𝑦 sub zero and parallel to the direction vector with components 𝑎, 𝑏 is 𝑥 is equal to 𝑎𝑡 plus 𝑥 sub zero and 𝑦 is equal to 𝑏𝑡 plus 𝑦 sub zero. We are told that the line passes through the point one, six. Therefore, 𝑥 sub zero equals one and 𝑦 sub zero equals six. We are also told it has a slope of one-half. We can use this information to find the direction vector of the line. Once we have found this, we can substitute the values of 𝑎 and 𝑏 to establish the parametric equations for 𝑥 and 𝑦. Recalling that the slope is equal to the rise over the run, as the slope of our line is one-half, the rise equals one and the run equals two. This leads to a direction vector of two, one. This can be demonstrated on the 𝑥𝑦-plane as shown. As the direction vector is equal to two, one, the values of 𝑎 and 𝑏 in our parametric equations are two and one, respectively. Using the direction vector two, one and the given point one, six, we can write the parametric form as 𝑥 is equal to two 𝑡 plus one and 𝑦 is equal to 𝑡 plus six. We note at this point that this parametric form does not match any of the given options. Let’s now clear some space and see how we can overcome this. We need to choose an alternate direction vector that is parallel to the vector two, one, for example, four, two; six, three; eight, four; and so on. We can identify the direction vector used for each option by recalling that the 𝑥- and 𝑦-components of the direction vector can be obtained from the coefficients of 𝑡. In option (A), the direction vector is one, two. Option (B) has a direction vector one, six. Neither of these are parallel to the direction vector two, one. The direction vector of option (C) is four, two. This is parallel to the direction vector two, one, as we have multiplied both components by two. The direction vectors in options (D) and (E) are six, one and four, one, respectively, neither of which is parallel to two, one. Using the direction vector four, two along with the point one, six, we have parametric equations 𝑥 is equal to four 𝑡 plus one and 𝑦 is equal to two 𝑡 plus six. The correct answer from the options given is (C). We will now summarize the key points from this video. The parametric form of a straight line gives 𝑥- and 𝑦-coordinates of each point on the line as a function of the parameter. The parametric form of the equation of a line passing through the point with coordinates 𝑥 sub zero, 𝑦 sub zero and parallel to the direction vector 𝐝, which is equal to 𝑎, 𝑏, is 𝑥 is equal to 𝑥 sub zero plus 𝑎𝑡 and 𝑦 is equal to 𝑦 sub zero plus 𝑏𝑡. Any point on the line may be used to obtain the parametric equations of the line. Also, the direction vector may be replaced by any scalar multiple of the vector. This means that the parametric form of the equation of a line will be nonunique.	{"url":"https://www.nagwa.com/en/videos/726129494645/","timestamp":"2024-11-04T02:29:52Z","content_type":"text/html","content_length":"276880","record_id":"<urn:uuid:c4ea35f6-3c98-490b-9d3f-3ff00c39ad67>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00644.warc.gz"}
Night Sections Problem 1 We first solve the characteristic polynomial: $r^3+r=0$ which yields $r_1=0, r_{2,3}=+- i$. So, solution to the general homogeneous equation is $$y_g=u_1+u_2\cos(t)+u_3 \sin(t)$$ Idea in method of parameters is to let $u_1,u_2,u_3$ to be functions and then plot the $y_g$ into the equation (in this case compute it's first and third derivative), solving for $u_1,u_2,u_3$. This can be done, but textbook page 242 gives you a nice final formula: $$u'_m=\frac{gW_m}{W}$$ So, we use this - maybe this was the reason why I finished 7 minutes earlier. It turns out that $W(1,\cos(t), \sin(t))=1$. $W_m$ is just the same wronskian but the m-th column is replaced by the column $(0,0,1)$. With this definition $$W_1=1, W_2=-\cos(t), W_3=-\sin(t)$$ Using the previously stated formula and the fact that in this case $g=\tan(t)$: $$u_1'=\tan(t) \implies u_1=-\ln(\cos(t))$$ $$u_2'=-\tan(t)\cos(t)=-\sin(t)\implies u_2=\cos(t)$$ $$u_3'=-\tan(t)\sin(t)=-\left(\frac{1-\cos^2(t)}{\cos(t)} \right)=-\left(\frac{1}{\cos(t)}-\cos(t)\right)\implies The third integral was the hardest. But in the homework problems you also had to compute $\int \frac{dt}{\cos(t)}$ which is surprisingly difficult. So, I had worked on the integral before and remembered the solution. Now, we just have to plug $u_1,u_2,u_3$ back to the general solution of homogeneous equation $y_g$ and we obtain that the particular solution is: General solution is therefore $$Y_G=Y_p+c_1+c_2\cos(t)+c_3 \sin(t)$$	{"url":"https://forum.math.toronto.edu/index.php?PHPSESSID=2h9fj04alq45skvg6snedf70k1&topic=251.0","timestamp":"2024-11-14T14:35:42Z","content_type":"application/xhtml+xml","content_length":"29406","record_id":"<urn:uuid:f498507d-201c-40f2-baa2-82c5e4f46aa5>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00252.warc.gz"}
For any sequence {c} of positive numbers, \underset{n \rightarrow \in For any sequence {c} of positive numbers, \underset{n \rightarrow \infty}{\lim _{n} \text { inf }} \frac{c_{n+1}}{c_{n}} \leq \liminf _{n \rightarrow \infty} \sqrt[n]{c_{n}} \limsup _{n \rightarrow \ infty} \sqrt[n]{c_{n}} \leq \limsup _{n \rightarrow \infty} \frac{c_{n+1}}{c_{n}} Proof We shall prove the second inequality; the proof of the first is quite similar. Put \alpha=\limsup _{n \rightarrow \infty} \frac{c_{n+1}}{c_{n}} If a = +0 infinity, there is nothing to prove. If a is finite, choose B > a. There is an integer N such that \frac{c_{n+1}}{c_{n}} \leq \beta for n2 N. In particular, for any p>0, c_{N+k+1} \leq \beta c_ {N+k} \quad(k=0,1, \ldots, p-1) Multiplying these inequalities, we obtain c_{N+p} \leq \beta^{p} c_{N} c_{n} \leq c_{N} \beta^{-N} \cdot \beta^{n} \quad(n \geq N) \sqrt[n]{c_{n}} \leq \sqrt[n]{c_{N} \beta^{-N}} \cdot \beta \underset{n \rightarrow \infty}{\lim \sup } \sqrt[n]{c_{n}} \leq \beta by Theorem 3.20(b). Since (18) is true for every B> a, we have \underset{n \rightarrow \infty}{\limsup } \sqrt[n]{c_{n}} \leq \alpha Fig: 1 Fig: 2 Fig: 3 Fig: 4 Fig: 5 Fig: 6 Fig: 7 Fig: 8 Fig: 9 Fig: 10 Fig: 11 Fig: 12 Fig: 13 Fig: 14 Fig: 15 Fig: 16	{"url":"https://tutorbin.com/questions-and-answers/for-any-sequence-c-of-positive-numbers-undersetn-rightarrow-inftylim-_","timestamp":"2024-11-04T11:32:02Z","content_type":"text/html","content_length":"91340","record_id":"<urn:uuid:815c94e4-5b2b-44e3-b946-a8d687859525>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00860.warc.gz"}
BEL Placement Paper Pattern On May 2008 (Objective and Electronics) Sponsored Ads: Quick Links University Papers University Syllabus Entrance Exam PSU Papers Bank Papers Placement Papers VTU Anna Univerity Syllabus Anna Univerity Papers Here i have given some of the questions of the written paper (ECE)… The paper consist of two sections - 40 objective & 10 true false questions. The cut-off as told by them 30% but they may have increase this limit. in my college only 8 out of 21 cleared the paper(for ECE branch) University Papers \| Syllabus \| Entrance Exam \| Govt & PSU Papers Bank Papers \| Programming & IT \| Travel \| Placement Papers \| Books For GATE Question Papers - CLICK HERE Part I—Objective questions (40 with 1.5 marks each) with 50% -ve marking. 1. Numerical problem based on modulation index fc, fm………. (formula based direct queston). 2. Poles & zeroes are at .01,1,20,100…… find phase margin/angle at f=50Hz. ans -90(By drawing bode plot) 3. In n-type enhancement mode MOSFET drain current———– options are- increase/decrease with inc/dec in drain/gate voltage. ans(d) 4. Gain of an directional antenna 6db P=1mw find transmitted power………(use Ptr= G * P.) 5. Multiplication of two nos 10101010 & 10010011 in 2’s complement form.. 6. A ckt is given suppplied with 15v with a series of resistance of 1k and a parallel combination of 12V zener diode and 2k resistance. FInd current through 2k resistance. Ans: 6mA 7.A MP has 16 line data bus & 12 line addr bus find memory range……….Ans..4K(4*1024bytes) 8.Divide by 12 counter require minimum ….. no of flip flops Ans. 4 9.Storage time in p-n junction. 10.Succesive approx. use in …. Ans ADC(analog to digital) 11.Pre-emphasis require in ……… low freq/high freq signal. 12.Handshake in MP ……….. Ans to communicate with slower peripherals. 13. Binary equivalent of 0.0625 Ans. 0.0001 14.Which code is self complement of itself 15.Excess three code of an given binary no. 16.When we add 6 in BCD operations……. Ans. if result exceed valid BCD nos. 17.Shottky diode has better switching capability because it switch between……. 18.Figure of Merit is same as…… 19.Swithcing in diode happens when…. 20.During forward bias majority charge conc. in depletion layers inc/decrease….. 21.Channel capacity depend on……. Ans. Usable frequency or bandwidth 22.A 2kHz signal is passed through an Low pass filter having cut-off freq 800Hz o/p will be 23. Carrier amplitude 1v, peak to peak message signal 3mv find modulation index. 24.A 12V signal is quantized into two V/14 & 6 equal V/7 determine quantization error. Part II True & false…...(10 1 mark each) with 50% -ve marking 1.Power dissipation in ECL is minimum………. False 2.Fourier Transform of a symmetric conjugate function is always real …. True 3.Divide by 12 counter requires a minimum of 4 flip flops…….True 4.Boron can be use as impurity to analyse base of a npn transistor…….True Other Question Placement Paper : 1) in analog question based on zener diode 2) coupling capacitors and bypass capacitors affect 3) 1,8,27,64,125, ……….,……. which among the following will not come in the series a.1000, b.729 c.259 4) which of the following game uses bulley 5) a zener diode works on the principle of 6)under the high electric field,in a semiconductor with increasing electric field 7)in an 8085,microprocessor system with memory mapped I/O 8)built-in potential in a p-n jn.- 9)the breakdown voltage of a transitor with its base open is BV(ceo) and that with emitter open BV(cbo) then 10) ques based on half adder 11) based on flip flop 12) block diagram reduction 13) to find max overshoot (2 ques based on it) Post a Comment	{"url":"https://www.knowledgeadda.com/2014/05/bel-placement-paper-pattern-on-may-2008.html","timestamp":"2024-11-03T17:08:14Z","content_type":"application/xhtml+xml","content_length":"231471","record_id":"<urn:uuid:2b57db4f-60b3-4d5c-a7ca-568c99cc5317>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00540.warc.gz"}
Differential Equations - A Plus Topper Differential Equations An equation involving independent variable x, dependent variable y and the differential coefficients $\frac { dy }{ dx } ,\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } ,…..$ is called differential (1) Order of a differential equation: The order of a differential equation is the order of the highest derivative occurring in the differential equation. For example, the order of above differential equations are 1,1,4 and 2 The order of a differential equation is a positive integer. To determine the order of a differential equation, it is not needed to make the equation free from radicals. (2) Degree of a differential equation: The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from radicals and fractions. The degree of above differential equations are 1, 1, 3 and 2 respectively. Formation of differential equation Formulating a differential equation from a given equation representing a family of curves means finding a differential equation whose solution is the given equation. The equation so obtained is the differential equation of order n for the family of given curves. Algorithm for formation of differential equations Step (i): Write the given equation involving independent variable x (say), dependent variable y (say) and the arbitrary constants. Step (ii): Obtain the number of arbitrary constants in step (i). Let there be n arbitrary constants. Step (iii): Differentiate the relation in step (i) n times with respect to x. Step (iv): Eliminate arbitrary constants with the help of n equations involving differential coefficients obtained in step (iii) and an equation in step (i). The equation so obtained is the desired differential equation. Variable separable type differential equation (1) Equations in variable separable form: If the differential equation of the form f[1](x) dx = f[2](y) dy …..(i) where f[1] and f[2] being functions of x and y only. Then we say that the variables are separable in the differential equation. Thus, integrating both sides of (i), we get its solution as ∫f[1](x) dx = ∫f[2](y) dy + c, where c is an arbitrary constant. There is no need of introducing arbitrary constants to both sides as they can be combined together to give just one. (2) Equations reducible to variable separable form: (i) Differential equations of the form $\frac { dy }{ dx } =f(ax+by+c)$ can be reduced to variable separable form by the substitution ax + by + c = Z. (ii) Differential equation of the form This is variable separable form and can be solved. Exact differential equation (1) Exact differential equation: If M and N are functions of x and y, the equation Mdx + Ndy = 0 is called exact when there exists a function f(x, y) of x and y such that An exact differential equation can always be derived from its general solution directly by differentiation without any subsequent multiplication, elimination etc. (2) Integrating factor: If an equation of the form Mdx + Ndy = 0 is not exact, it can always be made exact by multiplying by some function of x and y. Such a multiplier is called an integrating factor. (3) Working rule for solving an exact differential equation: Step (i): Compare the given equation with Mdx + Ndy = 0 and find out M and N. Step (ii): Integrate M with respect to x treating y as a constant. Step (iii): Integrate N with respect to y treating x as constant and omit those terms which have been already obtained by integrating M. Step (iv): On adding the terms obtained in steps (ii) and (iii) and equating to an arbitrary constant, we get the required solution. In other words, solution of an exact differential equation is Solution by inspection If we can write the differential equation in the form f(f[1](x))d(f[1](x, y)) + ϕ(f[2](x))d(f[2](x, y)) + ………. = 0, then each term can be easily integrated separately. For this the following results must be memorized. Application of differential equation Differential equation is applied in various practical fields of life. It is used to define various physical laws and quantities. It is widely used in physics, chemistry, engineering etc. Some important fields of application are: (i) Rate of change (ii) Geometrical problems etc. Miscellaneous differential equation A special type of second order differential equation:	{"url":"https://www.aplustopper.com/differential-equations/","timestamp":"2024-11-08T20:13:16Z","content_type":"text/html","content_length":"52511","record_id":"<urn:uuid:a617f9ae-de8d-487f-8208-5b3eed22217f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00714.warc.gz"}
Portfolio risk score The portfolio risk score is useful to align a portfolio risk with a client risk capacity. Both the portfolio risk score and client risk score are values between 0 to 100. The client risk score may be established with a questionnaire. Portfolio risk is complex and cannot be fully captured by a single score. Risk fluctuates across different time horizons, with each period—from 1 month to 30 years—offering a distinct perspective. Together, these perspectives form a comprehensive risk profile. Kwanti's Risk Score focuses on a portfolio's 12-month Risk profile, which provides a concise yet meaningful snapshot of this broader risk landscape. To display the portfolio Risk Score for a given portfolio, navigate to the Risk tab and select the Score view. How the portfolio risk score is calculated To calculate a portfolio risk score, Kwanti estimates the portfolio maximum downside within a horizon of one year and a 95% confidence. This downside is then mapped to a 0-100 scale. The mapping from estimated maximum downside to risk score is as follows: │Maximum downside ()│Risk profile │Risk score│ │0 to 3% │Conservative │1-20 │ │3% to 8% │Moderately Conservative │20-40 │ │8% to 15% │Moderate │40 to 60 │ │15 to 30% │Moderately Aggressive │60 to 80 │ │> 30% │Aggressive │80 to 100 │ () with 95% confidence 12-month probable range of returns In addition to the downside, Kwanti also calculates a 95% confidence upside. Observing both the worst case and best case estimates help understanding risk/reward tradeoffs when comparing portfolios. The 12-month probable range of returns is a statistical probability estimate. There is no guarantee that the actual returns will be within the range. Calculating the maximum downside The maximum downside is a forward estimate. In technical terms, it is the Value at Risk [1] of the portfolio at a confidence level of 95% and a time horizon of one year. The 95% confidence means that there is a 95% chance that the portfolio return will be better than the given downside. This calculation is based on: • Volatility and correlations of assets in the portfolio • Expected returns for assets in the portfolio [1] Understanding Value at Risk https://www.investopedia.com/terms/v/var.asp	{"url":"https://support.kwanti.com/hc/en-us/articles/24135456269975-Portfolio-risk-score","timestamp":"2024-11-04T16:53:42Z","content_type":"text/html","content_length":"18416","record_id":"<urn:uuid:79483106-6da0-40d1-b281-d81e40ac07ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00732.warc.gz"}
Collision detection: Radius overlap method Collision detection: Radius overlap method This method is also checking to see if two shapes intersect with each other, but as the title suggests, it does so using circles. There are advantages and disadvantages compared to other methods. The radius overlap method works well with shapes more circular in nature and less well with elongated shapes as shown in the image above. From the previous image, it is easy to see how the radius overlapping method is inaccurate for these particular objects and not hard to imagine how for a circular object, like a ball, it would be perfect. Here is some pseudo code to show how we can implement this method. The code assumes that the ship and enemy objects have initialized centerX, centerY and radius member variables. It also assumes that distanceX, distanceY have been declared as an appropriate type, possibly float or similar. Actually the Java, static Math.sqrt method takes and returns a value of type double. The code below doesn’t mention the types as they will vary depending upon your platform and requirements. // Get the distance of the two objects from // the edges of the circles on the x axis distanceX = (ship.centerX + ship.radius) - (enemy.centerX + enemy.radius; // Get the distance of the two objects from // the edges of the circles on the y axis distanceY = (ship.centerY + ship.radius) - (enemy.centerY + enemy.radius; // Calculate the distance between the center of each circle // Math.sqrt is from Java. All modern languages have an equivalent distance = Math.sqrt(distanceX * distanceX + distanceY * distanceY); // Finally see if the two circles overlap if (distance < ship.radius + enemy.radius) { // bump The key to the whole thing is the way we initialize distance: Math.sqrt(distanceX * distanceX + distanceY * distanceY); If the previous line of code looks a little confusing, it is simply using Pythagoras’ theorem to get the length of the hypotenuse of a triangle which is equal in length to a straight line drawn between the centers of the two circles. In the last line of our solution, we test if distance is greater than the ship.radius + enemy.radius, then we can be certain that we must have a collision. That is because if the center points of two circles are closer than the combined length of their radii, therefore they must be overlapping. Leave A Comment Cancel reply 2016-10-24T14:00:49+00:00By John Horton\|Essentials\|0 Comments	{"url":"https://gamecodeschool.com/essentials/collision-detection-radius-overlap-method/","timestamp":"2024-11-11T20:24:08Z","content_type":"text/html","content_length":"186322","record_id":"<urn:uuid:33958520-c90e-4466-9666-e3376bf8c85d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00381.warc.gz"}
Deciphering the Wave Energy Formula: Unleashing the Power of the Ocean - The Renewables Deciphering the Wave Energy Formula: Unleashing the Power of the Ocean Table of Contents Introduction to Wave Energy In the vast ocean of renewable energy resources, wave energy emerges as a colossal giant, harboring immense potential waiting to be harnessed. Wave energy is a formidable force of nature, a green powerhouse that could revolutionize the way we perceive and utilize renewable energy. The concept might seem complex, but at its heart lies a simple yet profound formula: the wave energy formula. This blog aims to unravel the complexities of the wave energy formula, bringing light to its significance and applications in the renewable energy sector. Wave energy, in its essence, is a concentration of wind energy transferred to the water, creating kinetic and potential energy within the waves. This energy, in its raw form, carries a tremendous amount of power that when tapped into effectively, can be a game-changer in the realm of renewable energy. A key player in this process, facilitating the transformation of raw wave energy into usable power, is the wave energy formula. Navigating through the dynamics of wave energy necessitates a foundational understanding of the science and mathematics intertwined with its essence. Armed with this knowledge, one can begin to appreciate the intricate symphony of variables and constants that play in harmony within the wave energy formula. By unlocking its secrets, we pave the way for innovative strategies and technologies to capture, convert, and utilize wave energy in ways that resonate with the rhythms of sustainability and efficiency. Check detailed article: What is Wave Energy? A Deep Dive into Ocean Renewable Power Understanding the Basics of Waves Before diving into mathematics, it’s crucial to establish a basic understanding of waves and their natural phenomenon. Waves are rhythmic disturbances that transfer energy from one point to another within oceans and seas. Created by the intertwining forces of gravity, wind, and the Earth’s rotation, waves become a significant repository of kinetic and potential energy. Waves are like the breath of the ocean, a constant cycle of energy exchange between the sea and the atmosphere. As winds blow across the ocean’s surface, they interact with the water, imparting energy and generating waves. These waves then travel across vast ocean distances, carrying energy that can be harnessed and converted into electricity, heating, or other usable forms of power. The multifaceted nature of waves is intrinsic to their character. In understanding the basics, one must appreciate the diversity in wave types and sizes. The spectrum ranges from small ripples to massive swells, each carrying its unique energy signature. Grasping the fundamental concepts of wave generation, propagation, and interaction with the environment is essential to fully comprehend the potential that lies within wave energy and its mathematical quantification through the wave energy formula. Diving Deep into the Wave Energy Formula Navigating through the realms of wave energy, a profound understanding of its fundamental equation – the wave energy formula, is indispensable. This pivotal formula acts as a compass, guiding us through the complexities, allowing for a meticulous extraction of energy insights from ocean waves. The wave energy formula is commonly articulated as: $E = {1 \over 8}ρgH^2T$ • $E$ symbolizes the energy density per unit width of the wave front, • $ρ$ embodies the water density, giving substance to the wave, • $g$ represents the acceleration due to gravity, grounding the formula in earthly constants, • $H$ illustrates the wave height, a pivotal determinant of energy potential, • and $T$ denotes the wave period, signifying the temporal aspect of wave motion. Each element within this equation plays a distinct role, collectively culminating in a coherent representation of the wave’s energy potential. Example: Applying the Wave Energy Formula Let’s journey through a practical application of the formula to bolster our understanding. Consider a scenario where the wave height ($H$) is 3 meters, and the wave period ($T$) is 5 seconds. Assuming the water density ($ρ$) to be 1025 kg/m³ (typical sea water density), and the acceleration due to gravity ($g$) as 9.81 m/s², we can calculate the energy density ($E$) as follows: $E = {1 \over 8} \times (1025 \space kg/m^3) \times (9.81 \space m/s^2) \times (3 \space m)^2 \times (5 \space sec)$ Calculating this yields: $E = {1 \over 8} \times 1025 \times 9.81\times 9 \times 5$ $E ≈ 4624.69 \space J/m$ This result represents the energy per unit width available from the wave under these specific conditions. Understanding and applying the wave energy formula enables a dynamic exploration of wave energy potentials, offering pathways to harness the oceans’ powerful and renewable energy resources effectively. By anchoring our knowledge in practical applications, we enhance our navigational prowess through the vast and promising seas of wave energy exploration and utilization. The aforementioned example elucidates the practical applicability of the wave energy formula, serving as a testament to its operational significance. It fosters a realm where theoretical understanding melds with practical realization, enabling a tangible approach toward wave energy harnessing. In the broader spectrum, the wave energy formula acts as a linchpin in the arena of wave energy conversion. It enables enthusiasts and professionals to delve deeper, exploring myriad wave conditions and scenarios. By doing so, it nurtures an environment of continuous learning and improvement, paving the way for innovative strategies and advanced technologies. This, in turn, amplifies our capacity to tap into the vast, renewable energy reservoirs nestled within our oceans, enhancing the efficacy and sustainability of our energy landscapes. Ultimately, the wave energy formula stands as a beacon of knowledge and understanding, guiding our ventures through the intricate pathways of wave energy exploration, and illuminating the horizons of renewable energy potential and promise. Armed with this formula, we embark on a transformative journey, steering towards a future where the oceans’ rhythmic energies flow seamlessly into our lives, powering our world with nature’s profound elegance and potency. Calculating Wave Energy: A Practical Approach Now, armed with the wave energy formula, let’s explore its practical application in calculating wave energy. The formula enables energy enthusiasts and professionals alike to estimate the power potential of waves, facilitating informed decision-making in wave energy projects. Applying the wave energy formula is not merely an academic exercise. It’s a gateway to optimizing the extraction of wave energy, ensuring that the resources invested yield maximum returns in terms of clean, green energy. Through this formula, theoretical concepts are seamlessly translated into practical realities, enabling a methodical and efficient approach to harnessing wave energy. In the realm of practical application, the formula necessitates a precise set of data – wave height, wave period, and water density are paramount. Sourcing accurate and reliable data is crucial, as it forms the foundation upon which the calculations are built. This involves meticulous measurements and observations, ensuring that the inputs fed into the formula resonate with actual oceanic conditions, thereby resulting in accurate energy estimates. Once the necessary data is secured, the wave energy formula becomes a powerful tool. It unveils the energy possibilities encapsulated within the waves, allowing for strategic planning and implementation of wave energy projects. Understanding the magnitude of energy that can be harvested enables professionals to design and optimize wave energy converters and systems to align with the expected energy output, ensuring effective and efficient energy extraction. Moreover, the practical application of the wave energy formula fosters a dynamic environment of continuous improvement and innovation. It encourages exploration and experimentation, allowing for the fine-tuning of methods and technologies used in wave energy extraction. It nurtures an ecosystem where knowledge, technology, and practical execution converge, catalyzing advancements in wave energy harnessing techniques and technologies. The Significance of Wave Energy in Renewable Resources The implementation of the wave energy formula is a leap forward in enhancing the efficacy of renewable energy solutions. It stands as a testament to human ingenuity in the relentless pursuit of sustainable energy sources, ensuring a cleaner and greener planet for future generations. By unlocking the potential of wave energy through this formula, we are paving the way for an energy revolution, minimizing dependency on non-renewable resources, and fostering an environment conducive to sustainability and preservation. Wave energy, as harnessed through the meticulous application of the formula, emerges as a formidable pillar within the renewable energy spectrum. Its significance is multifaceted; it is not merely a source of power but a catalyst for environmental conservation and a reducer of carbon footprints. It signifies a harmonization of technological advancement with nature’s rhythms, cultivating a synergy that bolsters our energy systems with enhanced sustainability and reliability. The wave energy formula acts as the key to unlocking a treasure trove of energy, latent in our oceans. This unearths vast possibilities, infusing the renewable energy sector with newfound potentials and pathways. It facilitates a transition, a gradual shift from traditional energy sources fraught with environmental liabilities, to cleaner, inexhaustible ocean wave energy. Overcoming Challenges in Harnessing Wave Energy While the wave energy formula is a powerful tool, the journey of harnessing wave energy is laden with challenges. Issues such as technological limitations, environmental concerns, and economic viability often impede the progress of wave energy projects. These challenges, robust in their nature, necessitate a thoughtful and meticulous approach, encouraging innovation, adaptability, and resilience in the pursuit of wave energy mastery. Technological limitations often pose significant hurdles. Harnessing wave energy necessitates sophisticated technologies that are capable of withstanding the ocean’s formidable forces and effectively converting wave energy into usable electricity. The evolving technology landscape demands continuous research and development to refine existing technologies and innovate new solutions that can enhance the efficiency and reliability of wave energy systems. Environmental concerns are also at the forefront of challenges faced in wave energy harnessing. Ensuring that wave energy projects align with ecological sustainability and do not disrupt marine ecosystems is crucial. Striking a balance between energy extraction and environmental conservation necessitates comprehensive assessments, strategic planning, and the implementation of sustainable practices that foster harmony with marine biodiversity. Economic viability is another pivotal consideration. Making wave energy a cost-effective and competitive option within the renewable energy portfolio requires strategic investments, economic foresight, and a commitment to nurturing the wave energy sector through various stages of its development. Conclusion: The Future of Wave Energy The wave energy formula, as explored in this blog, is more than a mathematical expression. It’s a beacon of hope, illuminating the path toward a sustainable energy future. As technology evolves and our understanding of wave energy deepens, the formula will continue to be a pivotal reference point, guiding us through the intricate landscapes of wave energy harnessing. In conclusion, the wave energy formula embodies the synthesis of nature’s magnificence and human innovation, holding the promise of a revolutionary impact on the world of renewable energy. It signifies a harmonization of mathematical precision with the organic rhythms of the ocean, weaving a tapestry of possibilities that resonate with sustainability, efficiency, and technological Armed with this knowledge, we stand on the brink of an era where wave energy is not merely a possibility but a formidable reality shaping our energy landscapes. It heralds a future vibrant with diversified energy sources, where wave energy plays a pivotal role in steering the global community towards resilience, sustainability, and a profound respect for the natural world’s energy-giving potentials. Thus, the wave energy formula emerges as a cornerstone, an essential guide in our collective journey towards a vibrant and sustainable energy future. 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Octave Programming Octave Programming Language Very high level langauge for numerical computations such as Matrix Math. Command Line Interface (with a simple GUI IDE available). ^ Change prompt with PS1('>>') Seperate commands with , or ; Suppress responses by ending the line with ; if the line isn't terminated, an "ans = " will display the result. Use disp( ) to print to stdio without "ans =". sprintf format short or format long for displayed resolution of numbers. Any standard math command can be entered, E.g. 1+2 log() abs() exp() max() sum() floor() ceil(). -v returns the negative of v. Equal is ==, Not equal is ~=. And &&, Or \|\|, XOR is xor( ). False is 0, True is 1. Variable assignment with =. Multiple values can be "broken out" on the left side with []. E.g. [val, ind] = max(a) Strings in single quotes. hi = 'hello'; matrix [ row {; row}] where row is col { col}. E.g. A = [1 2; 3 4; 5 6] is a 2 by 3 matrix. matrix(row, col) indexes a row and colum from the matrix variable. E.g. A(2, 1) returns the first value in the second row of A. Assuming A is defined as above, that would be 3. matrix(row, :) indexes a row from the matrix variable, returning a vector. E.g. A(2, :) returns the second row of A. Assuming A is defined as above, that would be [3 4]. Note that A(2) does NOT return the entire row; it only returns the first column of the second row. matrix(:, col) indexes a colum from the matrix variable, returning a 1xcol matrix. E.g. A(:, 1) returns the first column of A. Assuming A is defined as above, that would be [1; 3; 5] matrix(matrix2, :) addresses the rows of the matrix indexed by matrix2. E.g. A([1 3]),:) returns the 1st and 3rd rows of matrix A. Assuming A is defined as above, that would be [1 2; 5 6] matrix(:) returns a vector with all the rows and columns of A. E.g. A(:) returns [1;3;5;10;11;12] NOTE: all the above can be used in assignments (on the left side) as well as returning values on the right side. e.g. A(:,2) = [10;11;12] replaces the second column of A with 10, 11, and 12 A row or column can be appended by makeing a new matrix using the old value of the matrix with the new values included and then assigning that to the matrix. e.g. A = [A, [100; 101; 102]] appends a new column making A a 3x3 matrix. C = [A B] joins two matrixes, etc... C = [A; B] joins them top to bottom. All the standard matrix math functions are available. AB; A/B; A+B; A-B; etc... For Matrix multiply or divide^, the second dimension of A must match the first dimension of B, and the result will be a matrix which is the first dimension of A by the second dimension of B. If A is n x m and B is m x p the result AB will be n x p. [n x m][m x p] = [n x p] Dot operations, such as .* , or .^ will apply each element by each element. e.g. □ A .^ 2 returns each element of A squared. Note: A ^ 2 is NOT the same; instead it computes A * A □ 1 ./ A returns the reciprocal of each element. □ v < 3 returns true or false for each element. True (1) if the element is less than 3, false (0) otherwise. See find() below. Note: Uniary math operators are dot wise. e.g. log(A) takes the log of each element of A. transpose(matrix) or just matrix.' (the matrix followed by a dot and single quote) transposes the matrix; makes the rows into columns and columns into rows. ctranspose(matrix) or just matrix' (the matrix followed by a single quote) conjugate transpose of the matrix. Complex conjugate is a complex number with the sign of it's imaginary part swapped. (i.e x+iy ? x-iy or vice versa). The complex conjugate of a matrix is obtained by replacing each element by its complex conjugate. The conjugate transpose by doing that and transposing it. flipud(matrix) Flip array upside down. fliplr(matrix) Flip array left right. ones(rows, cols) makes a rowsxcol matrix filled with 1's eye(size) returns an identity matrix. E.g. eye(3) returns [1 0 0; 0 1 0; 0 0 1] eye(rows, cols) makes an identity matrix of that size. e.g. A .* eye(3,2) returns [1 0; 0 4; 0 0] rand(rows, cols) makes a rowsxcol matrix filled with random values between 0 and 1. multiply by a scaler to make a matrix with values between 1 and n; add or subtract to offset, etc... randn(rows, cols) makes a rowsxcol matrix filled with gausian random values. size(matrix) returns a 1x2 matrix showing the size of each dimision of the matrix. size(matrix, dim) returns a scalar showing the size of the dim dimision of the matrix. length(matrix) returns a scaler with the longest dimision of matrix. e.g. v + ones(length(v),1) returns a matrix with each element of v incremented. (but it turns out that this is the same as v + max(vector) returns a 1x2 matrix with the maximum value and the index at which that value was found. e.g. val = max(v) returns the maximum value found in vector v. [val, ind] = max(a) returns the maximum in val and the index where that value was found in ind. max(matrix) returns the column-wise maximum value; the max of each row. To find the max value in the entire matrix, use max(max(matrix). or max(A(:)) max(matrix, [], dim) returns the maximum along the dim diminsion. max(matrix, matrix) returns the pair-wise maximum between the two matrix. e.g. max([1 2; 2 3; 5 6],[6 5; 4 3; 2 1]) returns [6 5; 4 3; 5 6] find(vector) returns the position of the element if the element is true. E.g. find([1 15 2 0.5] < 3) returns [1 3 4] because the 1st, 3rd, and 4th elements are less than 3. find(matrix) returns a matrix of positions of the elements which are true. prod(vector) returns all the elements multiplied together. prod(matrix) returns each columns elements multiplied together. sum(matrix{, dim}) returns all the elements summed together along dimension dim. dim is assumed to be the first dimision that is not 1 if ommitted. e.g. sum(A) returns [9 12] and sum(A,2) returns Note: a common error is thinking that sum will return a scalar when used on a multidimisional matrix. To produce a scalar use sum(sum()) or convert the matrix to a vector using (:) e.g. sum(A(:)) inv(matrix) a^-1 is inv(a). returns the inverse of the matrix which must be square and the determinant can not be zero. Inverse is the matrix version of the recriprocal of a number. a * a^-1 will be the identity matrix of the same size. Division by a matrix is multiplication by the inverse of the matrix. pinv(matrix) returns the pseudoinverse. vector e.g. v = [1; 2; 3] range start:inc:end where start is the starting value, end is the last value, and inc is the (optional) amount to increment. E.g. B = 1:0.1:2 makes a 1x11 matrix with values [1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2], B = 1:3 makes a 1x3 matrix with values [1 2 3] File System pwd Print Working Directory cd path Change Directory ls List Stuff(?) like DIR in DOS. save file variable {-ascii} saves a memory variable to a file on the disk. The typical extension is .mat or .dat. The -ascii option makes the file human readable instead of the default binary. load file loads data from the file into memory who shows current memory variables whos shows details about current memory variables; attr, size, bytes, class clear variable erases a variable from memory. There is no undo. hist(matrix) plots a histogram plot(x, y) plot data points from y per x hold on prevents a graph from being erased by the next plot command so that two vplots can be shown on one graph. hold off legend(strings) add labels to the graph. use one string for each diminsion in the legend command. axis ([X_lo X_hi {Y_lo Y_hi {Z_lo Z_hi}}]) Set axis limits print -dpng file save the current graph to a file as a png. close closes a graph. figure(number) switches between multiple graphs. subplot(rows, cols, number) divides up a graph into parts, and selects the part indicated by number. imagesc(matrix) Display a scaled version of the matrixas a color image. Each element represents a color or intensity. colorbar Add a colorbar (legend) to the current axes. colormap gray use grey scale instead of colors. Control Statements for variable = range, statements, end while condition, statements, {break}, end if condition, statements, {elseif condition}, {else}, end function OUTPUTS = function (INPUT, ...) Multiple values can be "broken out" on the left side with []. Anonymous Functions are defined using the syntax @(argument-list) expression ^ Example uses: □ Pass the output of one function as a parameter into another function without needing a temporary variable inbetween: quad (@(x) (x.^2), 0, pi) □ Adapt a function with several parameters so that some of those parrameters are filled from the environment. In this example, the values of a and b that are passed to betainc are inherited from the current environment. a = 1; b = 2; quad (@(x) betainc (x, a, b), 0, 0.4) See also: ©2024 These pages are served without commercial sponsorship. (No popup ads, etc...).Bandwidth abuse increases hosting cost forcing sponsorship or shutdown. 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Ruler & Compass Construction \| Compass / Straightedge Maths Ruler and Compass Construction Want to download the Ruler & Compass Construction revision notes in PDF format? Download Ruler-and-Compass-Construction.pdf This download is free for GCSE Guide members! To download this file, click the button below to signup (it only takes a minute) and you'll be brought right back to this page to start the download! Sign up now → Already a member? Log in to download. Download → What Is Ruler & Compass Construction In Maths? Let’s learn to construct geometric figures using a ruler and compass. Shapes, angles and lines must be drawn accurately. A ruler is used for a straightedge or drawing straight lines. A compass is used to draw a circle. When making constructions measuring devices are not necessarily used to measure distances because they’re used to make more precise shapes, angles and lines. For instance, any regular polygon is easier to make using a compass and ruler than to draw angles and measure each side using a protractor. Perpendicular Bisector Two lines are perpendicular if the two lines intersect and form a right angle. A perpendicular bisector is if the segment is at its midpoint and it cuts the line, ray or segment into an equal To construct a perpendicular bisector: a. Draw point A and a circle with point A as the centre. b. Draw point B and a circle with point B as the centre. c. Draw a line connecting point A and B. d. At the point of intersection of the two circles, draw a line. This is the perpendicular bisector of point A and B. Angle Bisector An angle bisector divides an angle into two equal angles. If the angle bisects, it will make two congruent angles. To construct an angle bisector: a. Draw an angle. b. From the vertex of the angle, draw a circle, as the point of the vertex is the centre. c. From the two points that the circle cross both sides of the angle, draw two more circles. Where the two circles intersect, draw a line at the point of intersection. Equilateral Triangle An equilateral triangle is known for its congruent sides and angles. All sides and angles of an equilateral triangle have equal measure. To construct an equilateral triangle: a. Draw a line segment. b. In two endpoints, stretch the compass. c. Mark on each vertex and draw a line to the point of intersection. Regular Polygon: Hexagon A regular hexagon is a figure with all six sides and angles having equal measure. To construct a regular hexagon: a. Draw a circle. b. The radius of the circle is the edge length of the hexagon. c. Using a compass, stretch it to the size of a radius. d. Mark the circumference of the circle using the compass. e. Draw a line that connects the points of intersection or the corners of the hexagon.	{"url":"https://gcseguide.co.uk/maths/shapes/ruler-and-compass-construction/","timestamp":"2024-11-15T04:14:10Z","content_type":"text/html","content_length":"62311","record_id":"<urn:uuid:e4a43b49-eb9c-4fa7-b53f-873f7388f299>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00408.warc.gz"}
NAG FL Interface e04ncf (lsq_lincon_solve_old) e04nca (lsq_lincon_solve) Note: this routine uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the algorithm, to Section 12 for a detailed description of the specification of the optional parameters and to Section 13 for a detailed description of the monitoring information produced by the routine FL Name Style: FL Specification Language: 1 Purpose e04ncf/e04nca solves linearly constrained linear least squares problems and convex quadratic programming problems. It is not intended for large sparse problems. is a version of that has additional arguments in order to make it safe for use in multithreaded applications (see Section 5 ). The initialization routine e04wbf must have been called before calling 2 Specification 2.1 Specification for e04ncf Fortran Interface Subroutine e04ncf ( m, n, nclin, ldc, lda, c, bl, bu, cvec, istate, kx, x, a, b, iter, obj, clamda, iwork, liwork, work, lwork, ifail) Integer, Intent (In) :: m, n, nclin, ldc, lda, liwork, lwork Integer, Intent (Inout) :: istate(n+nclin), kx(n), ifail Integer, Intent (Out) :: iter, iwork(liwork) Real (Kind=nag_wp), Intent (In) :: c(ldc,), bl(n+nclin), bu(n+nclin), cvec() Real (Kind=nag_wp), Intent (Inout) :: x(n), a(lda,), b() Real (Kind=nag_wp), Intent (Out) :: obj, clamda(n+nclin), work(lwork) C Header Interface #include <nag.h> e04ncf_ (const Integer m, const Integer n, const Integer nclin, const Integer ldc, const Integer lda, const double c[], const double bl[], const double bu[], const double cvec[], Integer void istate[], Integer kx[], double x[], double a[], double b[], Integer iter, double obj, double clamda[], Integer iwork[], const Integer liwork, double work[], const Integer lwork, Integer 2.2 Specification for e04nca Fortran Interface Subroutine e04nca ( m, n, nclin, ldc, lda, c, bl, bu, cvec, istate, kx, x, a, b, iter, obj, clamda, iwork, liwork, work, lwork, lwsav, iwsav, rwsav, ifail) Integer, Intent (In) :: m, n, nclin, ldc, lda, liwork, lwork Integer, Intent (Inout) :: istate(n+nclin), kx(n), iwsav(610), ifail Integer, Intent (Out) :: iter, iwork(liwork) Real (Kind=nag_wp), Intent (In) :: c(ldc,), bl(n+nclin), bu(n+nclin), cvec() Real (Kind=nag_wp), Intent (Inout) :: x(n), a(lda,), b(), rwsav(475) Real (Kind=nag_wp), Intent (Out) :: obj, clamda(n+nclin), work(lwork) Logical, Intent (Inout) :: lwsav(120) C Header Interface #include <nag.h> e04nca_ (const Integer m, const Integer n, const Integer nclin, const Integer ldc, const Integer lda, const double c[], const double bl[], const double bu[], const double cvec[], Integer void istate[], Integer kx[], double x[], double a[], double b[], Integer iter, double obj, double clamda[], Integer iwork[], const Integer liwork, double work[], const Integer lwork, logical lwsav[], Integer iwsav[], double rwsav[], Integer ifail) Before calling , or either of the option setting routines e04wbf must be called. The specification for Fortran Interface Subroutine e04wbf ( rname, cwsav, lcwsav, lwsav, llwsav, iwsav, liwsav, rwsav, lrwsav, ifail) Integer, Intent (In) :: lcwsav, llwsav, liwsav, lrwsav Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iwsav(liwsav) Real (Kind=nag_wp), Intent (Out) :: rwsav(lrwsav) Logical, Intent (Out) :: lwsav(llwsav) Character (), Intent (In) :: rname Character (80), Intent (Out) :: cwsav(lcwsav) C Header Interface #include <nag.h> void e04wbf_ (const char rname, char cwsav[], const Integer lcwsav, logical lwsav[], const Integer llwsav, Integer iwsav[], const Integer liwsav, double rwsav[], const Integer lrwsav, Integer * ifail, const Charlen length_rname, const Charlen length_cwsav) should be called with , the declared lengths of respectively, must satisfy: • ${\mathbf{lcwsav}}\ge 1$ • ${\mathbf{llwsav}}\ge 120$ • ${\mathbf{liwsav}}\ge 610$ • ${\mathbf{lrwsav}}\ge 475$ The contents of the arrays rwsav must not be altered between calling routines 3 Description is designed to solve a class of quadratic programming problems of the following general form: $minimize x∈Rn F(x) subject to l≤{ x Cx } ≤u$ (1) is an matrix and the objective function may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for are listed in Table 1 , in which the prefixes FP, LP, QP and LS stand for ‘feasible point’, ‘linear programming’, ‘quadratic programming’ and ‘least squares’ respectively, is an -element vector, is an element vector and denotes the Euclidean length of Table 1 Problem type $\mathbit{F}\left(\mathbit{x}\right)$ Matrix $\mathbit{A}$ FP None Not applicable LP ${c}^{\mathrm{T}}x$ Not applicable QP1 $\phantom{{c}^{\mathrm{T}}x+}\frac{1}{2}{x}^{\mathrm{T}}Ax$ $n×n$ symmetric positive semidefinite QP2 ${c}^{\mathrm{T}}x+\frac{1}{2}{x}^{\mathrm{T}}Ax$ $n×n$ symmetric positive semidefinite QP3 $\phantom{{c}^{\mathrm{T}}x+}\frac{1}{2}{x}^{\mathrm{T}}{A}^{\mathrm{T}}Ax$ $m×n$ upper trapezoidal QP4 ${c}^{\mathrm{T}}x+\frac{1}{2}{x}^{\mathrm{T}}{A}^{\mathrm{T}}Ax$ $m×n$ upper trapezoidal LS1 $\phantom{{c}^{\mathrm{T}}x+}\frac{1}{2}{‖b-Ax‖}^{2}$ $m×n$ LS2 ${c}^{\mathrm{T}}x+\frac{1}{2}{‖b-Ax‖}^{2}$ $m×n$ LS3 $\phantom{{c}^{\mathrm{T}}x+}\frac{1}{2}{‖b-Ax‖}^{2}$ $m×n$ upper trapezoidal LS4 ${c}^{\mathrm{T}}x+\frac{1}{2}{‖b-Ax‖}^{2}$ $m×n$ upper trapezoidal In the standard LS problem will usually have the form LS1, and in the standard convex QP problem will usually have the form QP2. The default problem type is LS1 and other objective functions are selected by using the optional parameter Problem Type When $A$ is upper trapezoidal it will usually be the case that $m=n$, so that $A$ is upper triangular, but full generality has been allowed for in the specification of the problem. The upper trapezoidal form is intended for cases where a previous factorization, such as a $QR$ factorization, has been performed. The constraints involving are called the constraints. Note that upper and lower bounds are specified for all the variables and for all the general constraints. An equality constraint can be specified by setting . If certain bounds are not present, the associated elements of can be set to special values that will be treated as . (See the description of the optional parameter Infinite Bound Size The defining feature of a quadratic function $F\left(x\right)$ is that the second-derivative matrix $H$ (the Hessian matrix) is constant. For the LP case $H=0$; for QP1 and QP2, $H=A$; for QP3 and QP4, $H={A}^{\mathrm{T}}A$ and for LS1 (the default), LS2, LS3 and LS4, $H={A}^{\mathrm{T}}A$. Problems of type QP3 and QP4 for which $A$ is not in upper trapezoidal form should be solved as types LS1 and LS2 respectively, with $b=0$. For problems of type LS, we refer to $A$ as the least squares matrix, or the matrix of observations and to $b$ as the vector of observations. You must supply an initial estimate of the solution. If $H$ is nonsingular then e04ncf/e04nca will obtain the unique (global) minimum. If $H$ is singular then the solution may still be a global minimum if all active constraints have nonzero Lagrange multipliers. Otherwise the solution obtained will be either a weak minimum (i.e., with a unique optimal objective value, but an infinite set of optimal $x$), or else the objective function is unbounded below in the feasible region. The last case can only occur when $F\left(x\right)$ contains an explicit linear term (as in problems LP, QP2, QP4, LS2 and LS4). The method used by is described in detail in Section 11 4 References Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0) Report SOL 86-1 Department of Operations Research, Stanford University Gill P E, Murray W, Saunders M A and Wright M H (1984) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software 10 282–298 Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press Stoer J (1971) On the numerical solution of constrained least squares problems SIAM J. Numer. Anal. 8 382–411 5 Arguments 1: $\mathbf{m}$ – Integer Input On entry , the number of rows in the matrix . If the problem is specified as type FP or LP, is not referenced and is assumed to be zero. If the problem is of type QP, will usually be , the number of variables. However, a value of less than is appropriate for QP3 or QP4 if is an upper trapezoidal matrix with rows. Similarly, may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of QP1 or QP2, in which case the last rows and columns of are assumed to be zero. In the QP case, should not be greater than ; if it is, the last rows of are ignored. If the problem is of type LS1 (the default) or specified as type LS2, LS3 or LS4, is also the dimension of the array . Note that all possibilities ( ) are allowed in this case. Constraint: ${\mathbf{m}}>0$ if the problem is not of type FP or LP. 2: $\mathbf{n}$ – Integer Input On entry: $n$, the number of variables. Constraint: ${\mathbf{n}}>0$. 3: $\mathbf{nclin}$ – Integer Input On entry: ${n}_{L}$, the number of general linear constraints. Constraint: ${\mathbf{nclin}}\ge 0$. 4: $\mathbf{ldc}$ – Integer Input On entry : the first dimension of the array as declared in the (sub)program from which is called. Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nclin}}\right)$. 5: $\mathbf{lda}$ – Integer Input On entry : the first dimension of the array as declared in the (sub)program from which is called. Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$. 6: $\mathbf{c}\left({\mathbf{ldc}},\right)$ – Real (Kind=nag_wp) array Input the second dimension of the array must be at least , and at least On entry : the th row of must contain the coefficients of the th general constraint, for $\mathit{i}=1,2,\dots ,{\mathbf{nclin}}$ is not referenced. 7: $\mathbf{bl}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) array Input 8: $\mathbf{bu}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) array Input On entry must contain the lower bounds and the upper bounds, for all the constraints, in the following order. The first elements of each array must contain the bounds on the variables, and the next elements must contain the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., ), set ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$ , and to specify a nonexistent upper bound (i.e., ), set ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$ ; the default value of , but this may be changed by the optional parameter Infinite Bound Size . To specify the th constraint as an equality, set , say, where $\|\beta \|<\mathit{bigbnd}$ □ ${\mathbf{bl}}\left(\mathit{j}\right)\le {\mathbf{bu}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$; □ if ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta$, $\|\beta \|<\mathit{bigbnd}$. 9: $\mathbf{cvec}\left(\right)$ – Real (Kind=nag_wp) array Input the dimension of the array must be at least if the problem is of type LP, QP2, QP4, LS2 or LS4, and at least On entry : the coefficients of the explicit linear term of the objective function. If the problem is of type FP, QP1, QP3, LS1 (the default) or LS3, is not referenced. 10: $\mathbf{istate}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – Integer array Input/Output On entry : need not be set if the (default) optional parameter Cold Start is used. If the optional parameter Warm Start has been chosen, specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first elements of refer to the upper and lower bounds on the variables, and the next elements refer to the general linear constraints (if any). Possible values for are as follows: ${\mathbf{istate}}\left(\mathbit{j}\ Meaning 0 The constraint should not be in the initial working set. 1 The constraint should be in the initial working set at its lower bound. 2 The constraint should be in the initial working set at its upper bound. 3 The constraint should be in the initial working set as an equality. This value must not be specified unless ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\ The values are also acceptable but will be reset to zero by the routine. If has been called previously with the same values of already contains satisfactory information. (See also the description of the optional parameter Warm Start .) The routine also adjusts (if necessary) the values supplied in to be consistent with Constraint: $-2\le {\mathbf{istate}}\left(\mathit{j}\right)\le 4$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$. On exit : the status of the constraints in the working set at the point returned in . The significance of each possible value of is as follows: ${\mathbf{istate}}\left(\mathbit{j}\ Meaning $-2$ The constraint violates its lower bound by more than the feasibility tolerance. $-1$ The constraint violates its upper bound by more than the feasibility tolerance. $\phantom{-}0$ The constraint is satisfied to within the feasibility tolerance, but is not in the working set. $\phantom{-}1$ This inequality constraint is included in the working set at its lower bound. $\phantom{-}2$ This inequality constraint is included in the working set at its upper bound. $\phantom{-}3$ The constraint is included in the working set as an equality. This value of istate can occur only when ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\ $\phantom{-}4$ This corresponds to optimality being declared with ${\mathbf{x}}\left(j\right)$ being temporarily fixed at its current value. 11: $\mathbf{kx}\left({\mathbf{n}}\right)$ – Integer array Input/Output On entry : need not be initialized for problems of type FP, LP, QP1, QP2, LS1 (the default) or LS2. For problems QP3, QP4, LS3 or LS4, must specify the order of the columns of the matrix with respect to the ordering of . Thus if column is the column associated with the variable □ $1\le {\mathbf{kx}}\left(\mathit{i}\right)\le {\mathbf{n}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$; □ if $ie j$, ${\mathbf{kx}}\left(i\right)e {\mathbf{kx}}\left(j\right)$. On exit : defines the order of the columns of with respect to the ordering of , as described above. 12: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array Input/Output On entry : an initial estimate of the solution. Note: that it may be best to avoid the choice ${\mathbf{x}}=0.0$. On exit : the point at which terminated. If contains an estimate of the solution. 13: $\mathbf{a}\left({\mathbf{lda}},\right)$ – Real (Kind=nag_wp) array Input/Output the second dimension of the array must be at least if the problem is of type QP1, QP2, QP3, QP4, LS1 (the default), LS2, LS3 or LS4, and at least On entry : the array must contain the matrix as specified in Table 1 Section 3 If the problem is of type QP1 or QP2, the first rows and columns of must contain the leading rows and columns of the symmetric Hessian matrix. Only the diagonal and upper triangular elements of the leading rows and columns of are referenced. The remaining elements are assumed to be zero and need not be assigned. For problems QP3, QP4, LS3 or LS4, the first rows of must contain an upper trapezoidal factor of either the Hessian matrix or the least squares matrix, ordered according to the array. The factor need not be of full rank, i.e., some of the diagonals may be zero. However, as a general rule, the larger the dimension of the leading nonsingular sub-matrix of , the fewer iterations will be required. Elements outside the upper triangular part of the first rows of are assumed to be zero and need not be assigned. If a constrained least squares problem contains a very large number of observations, storage limitations may prevent storage of the entire least squares matrix. In such cases, you should transform the original $A$ into a triangular matrix before the call to e04ncf/e04nca and solve the problem as type LS3 or LS4. On exit : if and the problem is of type LS or QP, contains the upper triangular Cholesky factor Section 11.3 ), with columns ordered as indicated by and the problem is of type LS or QP, contains the upper triangular Cholesky factor of the Hessian matrix , with columns ordered as indicated by . In either case may be used to obtain the variance-covariance matrix or to recover the upper triangular factor of the original least squares matrix. If the problem is of type FP or LP, is not referenced. 14: $\mathbf{b}\left(\right)$ – Real (Kind=nag_wp) array Input/Output the dimension of the array must be at least if the problem is of type LS1 (the default), LS2, LS3 or LS4, and at least On entry: the $m$ elements of the vector of observations. On exit : the transformed residual vector of Section 11.3 If the problem is of type FP, LP, QP1, QP2, QP3 or QP4, is not referenced. 15: $\mathbf{iter}$ – Integer Output On exit: the total number of iterations performed. 16: $\mathbf{obj}$ – Real (Kind=nag_wp) Output On exit : the value of the objective function at is feasible, or the sum of infeasibiliites at otherwise. If the problem is of type FP and is feasible, is set to zero. 17: $\mathbf{clamda}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) array Output On exit: the values of the Lagrange multipliers for each constraint with respect to the current working set. The first $n$ elements contain the multipliers for the bound constraints on the variables, and the next ${n}_{L}$ elements contain the multipliers for the general linear constraints (if any). If ${\mathbf{istate}}\left(j\right)=0$ (i.e., constraint $j$ is not in the working set), ${\mathbf{clamda}}\left(j\right)$ is zero. If $x$ is optimal, ${\mathbf{clamda}}\left(j\right)$ should be non-negative if ${\mathbf{istate}}\left(j\right)=1$, non-positive if ${\mathbf {istate}}\left(j\right)=2$ and zero if ${\mathbf{istate}}\left(j\right)=4$. 18: $\mathbf{iwork}\left({\mathbf{liwork}}\right)$ – Integer array Workspace 19: $\mathbf{liwork}$ – Integer Input On entry : the dimension of the array as declared in the (sub)program from which is called. Constraint: ${\mathbf{liwork}}\ge {\mathbf{n}}$. 20: $\mathbf{work}\left({\mathbf{lwork}}\right)$ – Real (Kind=nag_wp) array Workspace 21: $\mathbf{lwork}$ – Integer Input On entry : the dimension of the array as declared in the (sub)program from which is called. □ if the problem is of type FP, ☆ if ${\mathbf{nclin}}=0$, ${\mathbf{lwork}}\ge 6×{\mathbf{n}}$; ☆ if ${\mathbf{nclin}}\ge {\mathbf{n}}$, ${\mathbf{lwork}}\ge 2×{{\mathbf{n}}}^{2}+6×{\mathbf{n}}+6×{\mathbf{nclin}}$; ☆ otherwise ${\mathbf{lwork}}\ge 2×{\left({\mathbf{nclin}}+1\right)}^{2}+6×{\mathbf{n}}+6×{\mathbf{nclin}}$; □ if the problem is of type LP, ☆ if ${\mathbf{nclin}}=0$, ${\mathbf{lwork}}\ge 7×{\mathbf{n}}$; ☆ if ${\mathbf{nclin}}\ge {\mathbf{n}}$, ${\mathbf{lwork}}\ge 2×{{\mathbf{n}}}^{2}+7×{\mathbf{n}}+6×{\mathbf{nclin}}$; ☆ otherwise ${\mathbf{lwork}}\ge 2×{\left({\mathbf{nclin}}+1\right)}^{2}+7×{\mathbf{n}}+6×{\mathbf{nclin}}$; □ if problems QP1, QP3, LS1 (the default) and LS3, ☆ if ${\mathbf{nclin}}>0$, ${\mathbf{lwork}}\ge 2×{{\mathbf{n}}}^{2}+9×{\mathbf{n}}+6×{\mathbf{nclin}}$; ☆ if ${\mathbf{nclin}}=0$, ${\mathbf{lwork}}\ge 9×{\mathbf{n}}$; □ if problems QP2, QP4, LS2 and LS4, ☆ if ${\mathbf{nclin}}>0$, ${\mathbf{lwork}}\ge 2×{{\mathbf{n}}}^{2}+10×{\mathbf{n}}+6×{\mathbf{nclin}}$; ☆ if ${\mathbf{nclin}}=0$, ${\mathbf{lwork}}\ge 10×{\mathbf{n}}$. The amounts of workspace provided and required are (by default) output on the current advisory message unit (as defined by ). As an alternative to computing from the formulas given above, you may prefer to obtain appropriate values from the output of a preliminary run with set to . ( will then terminate with 22: $\mathbf{ifail}$ – Integer Input/Output Note: for e04nca, ifail does not occur in this position in the argument list. See the additional arguments described below On entry must be set to to set behaviour on detection of an error; these values have no effect when no error is detected. A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not. If halting is not appropriate, the value is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}e {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit. On exit unless the routine detects an error or a warning has been flagged (see Section 6 returns with is a strong local minimizer, i.e., the projected gradient ( Norm Gz ; see Section 9.2 ) is negligible, the Lagrange multipliers ( Lagr Mult ; see Section 11.2 ) are optimal and Section 11.3 ) is nonsingular. Note: the following are additional arguments for specific use with e04nca. Users of e04ncf therefore need not read the remainder of this description. 22: $\mathbf{lwsav}\left(120\right)$ – Logical array Communication Array 23: $\mathbf{iwsav}\left(610\right)$ – Integer array Communication Array 24: $\mathbf{rwsav}\left(475\right)$ – Real (Kind=nag_wp) array Communication Array The arrays rwsav must not be altered between calls to any of the routines 25: $\mathbf{ifail}$ – Integer Input/Output see the argument description for 6 Error Indicators and Warnings If on entry , explanatory error messages are output on the current error message unit (as defined by Errors or warnings detected by the routine: Note: in some cases e04ncf/e04nca may return useful information. Weak $⟨\mathit{\text{value}}⟩$ solution. is a weak local minimum, (i.e., the projected gradient is negligible, the Lagrange multipliers are optimal, but either Section 11.3 ) is singular, or there is a small multiplier). This means that is not unique. $⟨\mathit{\text{value}}⟩$ solution is unbounded. This value of implies that a step as large as Infinite Bound Size $\text{default value}={10}^{20}$ ) would have to be taken in order to continue the algorithm. This situation can occur only when is singular, there is an explicit linear term, and at least one variable has no upper or lower bound. Cannot satisfy the linear constraints. It was not possible to satisfy all the constraints to within the feasibility tolerance. In this case, the constraint violations at the final will reveal a value of the tolerance for which a feasible point will exist – for example, when the feasibility tolerance for each violated constraint exceeds its Section 9.2 ) at the final point. The modified problem (with an altered feasibility tolerance) may then be solved using a Warm Start . You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision , you should ensure that the value of the optional parameter Feasibility Tolerance $\text{default value}=\sqrt{\epsilon }$ , where is the machine precision ) is . For example, if all elements of are of order unity and are accurate only to three decimal places, the Feasibility Tolerance should be at least Too many iterations. The value of the optional parameters Feasibility Phase Iteration Limit $\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{n}_{L}\right)\right)$ ) and Optimality Phase Iteration Limit $\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{n}_{L}\right)\right)$ )) may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either increase the iterations limit and rerun or, alternatively, rerun using the Warm Start facility to specify the initial working set. If the iteration limit is already large, but some of the constraints could be nearly linearly dependent, check the monitoring information (see Section 13 ) for a repeated pattern of constraints entering and leaving the working set. (Near-dependencies are often indicated by wide variations in size in the diagonal elements of the matrix Section 11.2 ), which will be printed if ${\mathbf{Print Level}}\ge 30$ $\text{default value}=10$ ). In this case, the algorithm could be cycling (see the comments for Too many iterations without changing $x$. The algorithm could be cycling, since a total of changes were made to the working set without altering . You should check the monitoring information (see Section 13 ) for a repeated pattern of constraint deletions and additions. If a sequence of constraint changes is being repeated, the iterates are probably cycling. ( does not contain a method that is guaranteed to avoid cycling; such a method would be combinatorial in nature.) Cycling may occur in two circumstances: at a constrained stationary point where there are some small or zero Lagrange multipliers; or at a point (usually a vertex) where the constraints that are satisfied exactly are nearly linearly dependent. In the latter case, you have the option of identifying the offending dependent constraints and removing them from the problem, or restarting the run with a larger value of the optional parameter Feasibility Tolerance $\text{default value}=\sqrt{\epsilon }$ , where is the machine precision ). If terminates with , but no suspicious pattern of constraint changes can be observed, it may be worthwhile to restart with the final (with or without the Warm Start that this error exit may also occur if a poor starting point is supplied (for example, ). You are advised to try a nonzero starting point. Not enough workspace to solve problem. Workspace provided is ${\mathbf{iwork}}\left(⟨\mathit{\text{value}}⟩\right)$ and ${\mathbf{work}}\left(⟨\mathit{\text{value}}⟩\right)$. To solve problem we need ${\mathbf{iwork}}\left(⟨\mathit{\text{value}}⟩\right)$ and ${\mathbf{work}}\left(⟨\mathit{\text{value}}⟩\right)$. On entry, ${\mathbf{kx}}$ has not been supplied as a valid permutation. On entry, ${\mathbf{lda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$. On entry, ${\mathbf{ldc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nclin}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nclin}}\right)$. On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{m}}>0$. On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{n}}>0$. On entry, ${\mathbf{nclin}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{nclin}}\ge 0$. On entry, the bounds on $⟨\mathit{\text{value}}⟩$ are inconsistent: ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{bu}}\left(⟨\mathit{\text{value}}⟩\ On entry, the bounds on linear constraint $⟨\mathit{\text{value}}⟩$ are inconsistent: ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{bu}}\left(⟨\mathit On entry, the bounds on nonlinear constraint $⟨\mathit{\text{value}}⟩$ are inconsistent: ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{bu}}\left(⟨\ On entry, the bounds on variable $⟨\mathit{\text{value}}⟩$ are inconsistent: ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{bu}}\left(⟨\mathit{\text On entry, the equal bounds on $⟨\mathit{\text{value}}⟩$ are infinite, because ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(⟨\mathit{\text{value}}⟩\ right)=\mathrm{beta}$, but $\|\mathrm{beta}\|\ge \mathrm{bigbnd}$: $\mathrm{beta}=⟨\mathit{\text{value}}⟩$ and $\mathrm{bigbnd}=⟨\mathit{\text{value}}⟩$. On entry, the equal bounds on linear constraint $⟨\mathit{\text{value}}⟩$ are infinite, because ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(⟨\mathit {\text{value}}⟩\right)=\mathrm{beta}$, but $\|\mathrm{beta}\|\ge \mathrm{bigbnd}$: $\mathrm{beta}=⟨\mathit{\text{value}}⟩$ and $\mathrm{bigbnd}=⟨\mathit{\text{value}}⟩$. On entry, the equal bounds on nonlinear constraint $⟨\mathit{\text{value}}⟩$ are infinite, because ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(⟨\ mathit{\text{value}}⟩\right)=\mathrm{beta}$, but $\|\mathrm{beta}\|\ge \mathrm{bigbnd}$: $\mathrm{beta}=⟨\mathit{\text{value}}⟩$ and $\mathrm{bigbnd}=⟨\mathit{\text{value}}⟩$. On entry, the equal bounds on variable $⟨\mathit{\text{value}}⟩$ are infinite, because ${\mathbf{bl}}\left(⟨\mathit{\text{value}}⟩\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(⟨\mathit{\text {value}}⟩\right)=\mathrm{beta}$, but $\|\mathrm{beta}\|\ge \mathrm{bigbnd}$: $\mathrm{beta}=⟨\mathit{\text{value}}⟩$ and $\mathrm{bigbnd}=⟨\mathit{\text{value}}⟩$. On entry with a Warm Start, ${\mathbf{istate}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$. The problem to be solved is of type QP1 or QP2, but the Hessian matrix supplied in is not positive semidefinite. If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the th constraint, it may be possible to avoid the difficulty by increasing the magnitude of the Feasibility Tolerance ($\text{default value}=\sqrt{\epsilon }$, is the machine precision ) and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘ ’) must be removed from the problem. An unexpected error has been triggered by this routine. Please contact Section 7 in the Introduction to the NAG Library FL Interface for further information. Your licence key may have expired or may not have been installed correctly. Section 8 in the Introduction to the NAG Library FL Interface for further information. Dynamic memory allocation failed. Section 9 in the Introduction to the NAG Library FL Interface for further information. 7 Accuracy e04ncf/e04nca implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine. 8 Parallelism and Performance Background information to multithreading can be found in the e04ncf/e04nca is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. e04ncf/e04nca makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information. This section contains some comments on scaling and a description of the printed output. 9.1 Scaling Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See the E04 Chapter Introduction Gill et al. (1981) for further information and advice. 9.2 Description of the Printed Output This section describes the intermediate printout and final printout produced by . The intermediate printout is a subset of the monitoring information produced by the routine at every iteration (see Section 13 ). You can control the level of printed output (see the description of the optional parameter Print Level ). Note that the intermediate printout and final printout are produced only if ${\mathbf{Print Level}}\ge 10$ (the default for , by default no output is produced by The following line of summary output ( characters) is produced at every iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration. Itn is the iteration count. Step is the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than $1$ only if the factor ${R}_{Z}$ is singular. (See Section 11.3.) Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase. is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. Objective During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found. Norm Gz is $‖{Z}_{1}^{\mathrm{T}}{g}_{\mathrm{FR}}‖$, the Euclidean norm of the reduced gradient with respect to ${Z}_{1}$. During the optimality phase, this norm will be approximately zero after a unit step. (See Sections 11.2 and 11.3.) The final printout includes a listing of the status of every variable and constraint. The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero. Varbl gives the name (V) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,n$, of the variable. gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively. A key is sometimes printed before State . Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away A from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change. D Degenerate. The variable is free, but it is equal to (or very close to) one of its bounds. I Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance. Value is the value of the variable at the final iteration. Lower is the lower bound specified for the variable. None indicates that ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$. Upper is the upper bound specified for the variable. None indicates that ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$. Lagr is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$ and ${\mathbf{bu}}\left(j\right)\ge \mathit Mult {bigbnd}$, in which case the entry will be blank. If $x$ is optimal, the multiplier should be non-negative if State is LL and non-positive if State is UL. Slack is the difference between the variable Value and the nearer of its (finite) bounds ${\mathbf{bl}}\left(j\right)$ and ${\mathbf{bu}}\left(j\right)$. A blank entry indicates that the associated variable is not bounded (i.e., ${\mathbf{bl}}\left(j\right)\le -\mathit{bigbnd}$ and ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$). The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, are replaced by respectively, and with the following change in the heading: L Con gives the name (L) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,{n}_{L}$, of the linear constraint. Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive. Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision. 10 Example This example minimizes the function , where $A= ( 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 0 0 1 1 3 1 1 1 −1 −1 −3 1 1 1 4 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 2 1 1 0 0 0 −1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 2 2 3 1 0 1 1 1 1 0 2 2 ) and b = ( 1 1 1 1 1 1 1 1 1 1 )$ subject to the bounds $0≤x1≤2 0≤x2≤2 -∞≤x3≤2 0≤x4≤2 0≤x5≤2 0≤x6≤2 0≤x7≤2 0≤x8≤2 0≤x9≤2$ and to the general constraints $2.0 ≤ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + 4x9 ≤ ∞ -∞ ≤ x1 + 2x2 + 3x3 + 4x4 - 2x5 + x6 + x7 + x8 + x9 ≤ 2.0 1.0 ≤ x1 - x2 + x3 - x4 + x5 + x6 + x7 + x8 + x9 ≤ 4.0.$ The initial point, which is infeasible, is (to five figures). The optimal solution (to five figures) is . Four bound constraints and all three general constraints are active at the solution. The document for includes an example program to solve a convex quadratic programming problem, using some of the optional parameters described in Section 12 10.1 Program Text Note: the following programs illustrate the use of e04ncf and e04nca. 10.2 Program Data 10.3 Program Results the remainder of this document is intended for more advanced users. Section 11 contains a detailed description of the algorithm which may be needed in order to understand Sections 12 Section 12 describes the optional parameters which may be set by calls to Section 13 describes the quantities which can be requested to monitor the course of the computation. 11 Algorithmic Details This section contains a detailed description of the method used by e04ncf/e04nca. 11.1 Overview is essentially identical to the subroutine LSSOL described in Gill et al. (1986) . It is based on a two-phase (primal) quadratic programming method with features to exploit the convexity of the objective function due to Gill et al. (1984) . (In the full-rank case, the method is related to that of Stoer (1971) has two phases: finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase ), and minimizing the quadratic objective function within the feasible region (the optimality phase ). The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the quadratic objective function. The feasibility phase does perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the LP case when ${n}_{L}\le n$ . Once any iterate is feasible, all subsequent iterates remain feasible. has been designed to be efficient when used to solve a of related problems – for example, within a sequential quadratic programming method for nonlinearly constrained optimization (e.g., ). In particular, you may specify an initial working set (the indices of the constraints believed to be satisfied exactly at the solution); see the discussion of the optional parameter Warm Start In general, an iterative process is required to solve a quadratic program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new is defined by where the step length $\alpha$ is a non-negative scalar, and is called the search direction. At each point , a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the optional parameter Feasibility Tolerance ). The working set is the current prediction of the constraints that hold with equality at a solution of . The search direction is constructed so that the constraints in the working set remain for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding element of the search direction to zero. Thus, the associated variable is , and specification of the working set induces a partition of variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant elements of the search direction are zero, the columns of corresponding to fixed variables may be ignored. denote the number of general constraints in the working set and let denote the number of variables fixed at one of their bounds ( are the quantities in the monitoring file output from ; see Section 13 ). Similarly, let denote the number of free variables. At every iteration, the variables are reordered so that the last ${n}_{\mathrm{FX}}$ variables are fixed, with all other relevant vectors and matrices ordered accordingly. The order of the variables is indicated by the contents of the array on exit (see Section 5 11.2 Definition of Search Direction denote the sub-matrix of general constraints in the working set corresponding to the free variables, and let denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along In order to compute , the $TQ$ factorization is used: is a nonsingular reverse-triangular matrix (i.e., ), and the nonsingular is the product of orthogonal transformations (see Gill et al. (1984) ). If the columns of are partitioned so that , then the columns of form a basis for the null space of . Let be an integer such that $0\le {n}_{R}\le {n}_{Z}$ , and let denote a matrix whose columns are a subset of the columns of . (The integer is the quantity in the monitoring file output from . In many cases, will include the columns of .) The direction will satisfy is any 11.3 Main Iteration denote the is the identity matrix of order . Let denote an upper triangular matrix (the Cholesky factor ) such that is the Hessian with rows and columns permuted so that the free variables are first. Let the matrix of the first rows and columns of be denoted by . The definition of depends on whether or not the matrix is singular at . In the nonsingular case, satisfies the equations denotes the vector denotes the objective gradient. (The norm of is the printed quantity Norm Gf ; see Section 13 .) When is defined by is the minimizer of the objective function subject to the constraints (bounds and general) in the working set treated as equalities. In general, a vector is available such that , which allows to be computed from a single back-substitution . For example, when solving problem LS1, comprises the first elements of the transformed residual vector which is recurred from one iteration to the next, where is an orthogonal matrix. In the singular case, is defined such that $RZ pZ = 0 and gZT pZ < 0 .$ (11) This vector has the property that the objective function is linear along and may be reduced by any step of the form $x+\alpha p$ , where $\alpha >0$ The vector is known as the projected gradient . If the projected gradient is zero, is a constrained stationary point in the subspace defined by . During the feasibility phase, the projected gradient will usually be zero only at a vertex (although it may be zero at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero projected gradient implies that minimizes the quadratic objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers ${\lambda }_{{\mathbf{c}}}$ ${\lambda }_{{\mathbf{b}}}$ for the general and bound constraints are defined from the equations $CFRTλC=gFRand λB=gFX-CFXTλC.$ (12) Given a positive constant of the order of the machine precision , the Lagrange multiplier ${\lambda }_{j}$ corresponding to an inequality constraint in the working set is said to be ${\lambda }_{j}\le \delta$ when the associated constraint is at its upper bound , or if ${\lambda }_{j}\ge -\delta$ when the associated constraint is at its lower bound . If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index ; see Section 13 ) from the working set. If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and e04ncf/e04nca will continue until the minimum value of the sum of infeasibilities has been found. At this point, the Lagrange multiplier ${\lambda }_{j}$ corresponding to an inequality constraint in the working set will be such that $-\left(1+\delta \right)\le {\ lambda }_{j}\le \delta$ when the associated constraint is at its upper bound, and $-\delta \le {\lambda }_{j}\le \left(1+\delta \right)$ when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy $\|{\lambda }_{j}\|\le 1+\delta$. The choice of step length is based on remaining feasible with respect to the satisfied constraints. If is nonsingular and is feasible, will be taken as unity. In this case, the projected gradient at will be zero, and Lagrange multipliers are computed. Otherwise, is set to ${\alpha }_{{\mathbf{m}}}$ , the step to the ‘nearest’ constraint (with index ; see Section 13 ), which is added to the working set at the next iteration. is not input as a triangular matrix, it is overwritten by a triangular matrix obtained using the Cholesky factorization in the QP case, or the factorization in the LS case. Column interchanges are used in both cases, and an estimate is made of the rank of the triangular factor. Thereafter, the dependent rows of are eliminated from the problem. Each change in the working set leads to a simple change to : if the status of a general constraint changes, a is altered; if a bound constraint enters or leaves the working set, a changes. Explicit representations are recurred of the matrices ; and of vectors , which are related by the formulae $f=Pb- ( R 0 ) QTx , (b≡0for the QP case) ,$ Note that the triangular factor associated with the Hessian of the original problem is updated during both the optimality the feasibility phases. The treatment of the singular case depends critically on the following feature of the matrix updating schemes used in : if a given factor is nonsingular, it can become singular during subsequent iterations only when a constraint leaves the working set, in which case only its last diagonal element can become zero. This property implies that a vector satisfying may be found using the single back-substitution , where is the matrix with a unit last diagonal, and is a vector of all zeros except in the last position. If is singular, the matrix (and hence ) may be singular at the start of the optimality phase. However, will be nonsingular if enough constraints are included in the initial working set. (The matrix with no rows and columns is positive definite by definition, corresponding to the case when constraints.) The idea is to include as many general constraints as necessary to ensure a nonsingular At the beginning of each phase, an upper triangular matrix is determined that is the largest nonsingular leading sub-matrix of . The use of interchanges during the factorization of tends to maximize the dimension of . (The rank of is estimated using the optional parameter Rank Tolerance .) Let denote the columns of corresponding to , and let be partitioned as $Z=\left({Z}_{1}\text{ }{Z}_{2}\right)$ . A working set for which defines the null space can be obtained by including the rows of ${Z}_{2}^{\mathrm{T}}$ as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by The artificially augmented working set is given by $C¯FR= ( CFR Z2T ) ,$ (13) so that will satisfy . By definition of the ${\overline{{\mathbf{c}}}}_{\mathrm{FR}}$ automatically satisfies the following: $C¯FRQFR= ( CFR Z2T ) QFR= ( CFR Z2T ) ( Z1 Z2 Y ) = ( 0 T¯ ) ,$ and hence the factorization of requires no additional work. The matrix ${Z}_{2}$ need not be kept fixed, since its role is purely to define an appropriate null space; the $TQ$ factorization can, therefore, be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with ${Z}_{2}$ when ${Z}_{1}^{\mathrm{T}}{g}_{\mathrm{FR}}=0$, since this simply involves repartitioning ${Q}_{\mathrm {FR}}$. When deciding which constraint to delete, the ‘artificial’ multiplier vector associated with the rows of ${Z}_{2}^{\mathrm{T}}$ is equal to ${Z}_{2}^{\mathrm{T}}{g}_{\mathrm{FR}}$, and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the temporary constraints were not present. The number of columns in , the Euclidean norm of , and the condition estimator of appear in the monitoring file output as Norm Gz Cond Rz respectively (see Section 13 Although the algorithm of e04ncf/e04nca does not perform simplex steps in general, there is one exception: a linear program with fewer general constraints than variables (i.e., ${n}_{L}\le n$). Use of the simplex method in this situation leads to savings in storage. At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again. One of the most important features of is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonals of the (the printed value Cond T ; see Section 13 ). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T . Thereafter, allows constraints to be violated by as much as a user-specified optional parameter Feasibility Tolerance in order to provide, whenever possible, a of constraints to be added to the working set at a given iteration. Let ${\alpha }_{{\mathbf{m}}}$ denote the maximum step at which $x+{\alpha }_{{\mathbf{m}}}p$ does not violate any constraint by more than its feasibility tolerance. All constraints at distance $\alpha \left(\alpha \le {\alpha }_{{\mathbf{m}}}\right)$ from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set. In order to ensure that the new iterate satisfies the constraints in the working set as accurately as possible, the step taken is the exact distance to the newly added constraint. As a consequence, negative steps are occasionally permitted, since the current iterate may violate the constraint to be added by as much as the feasibility tolerance. 12 Optional Parameters Several optional parameters in e04ncf/e04nca define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e04ncf/e04nca these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default The remainder of this section can be skipped if you wish to use the default values for all optional parameters. The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 12.1 Optional parameters may be specified by calling one, or both, of the routines before a call to reads options from an external options file, with as the first and last lines respectively and each intermediate line defining a single optional parameter. For example, Print level = 1 The call Call e04ndf/e04nda (ioptns, inform) can then be used to read the file on unit will be zero on successful exit. should be consulted for a full description of this method of supplying optional parameters. can be called to supply options directly, one call being necessary for each optional parameter. For example, Call e04nef ('Print Level = 1') should be consulted for a full description of this method of supplying optional parameters. All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by e04ncf/e04nca (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you. 12.1 Description of the Optional Parameters For each option, we give a summary line, a description of the optional parameter and details of constraints. The summary line contains: • the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted); • a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively; • the default value, where the symbol $\epsilon$ is a generic notation for machine precision (see x02ajf). Keywords and character values are case and white space insensitive. This option specifies how the initial working set is chosen. With a Cold Start chooses the initial working set based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within Crash Tolerance With a Warm Start , you must provide a valid definition of every element of the array will override your specification of if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of which are set to will be reset to zero, as will any elements which are set to when the corresponding elements of are not equal. A warm start will be advantageous if a good estimate of the initial working set is available – for example, when is called repeatedly to solve related problems. Crash Tolerance $r$ Default $\text{}=0.01$ This value is used in conjunction with the optional parameter Cold Start (the default value) when selects an initial working set. If $0\le r\le 1$ , the initial working set will include (if possible) bounds or general inequality constraints that lie within of their bounds. In particular, a constraint of the form ${c}_{j}^{\mathrm{T}}x\ge l$ will be included in the initial working set if $\|{c}_{j}^{\mathrm{T}}x-l\|\le r\left(1+\|l\|\right)$ . If , the default value is used. This special keyword may be used to reset all optional parameters to their default values. Feasibility Phase Iteration Limit ${i}_{1}$ Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{n}_{L}\right)\right)$ Optimality Phase Iteration Limit ${i}_{2}$ Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{n}_{L}\right)\right)$ The scalars specify the maximum number of iterations allowed in the feasibility and optimality phases. Optional parameter Optimality Phase Iteration Limit is equivalent to optional parameter Iteration Limit . Setting ${\mathbf{Print Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed. If , the default value is used. Feasibility Tolerance $r$ Default $\text{}=\sqrt{\epsilon }$ If $r>\epsilon$, $r$ defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about $6$ decimal digits, it would be appropriate to specify $r$ as ${10}^{-6}$. If $0\le r<\epsilon$, the default value is used. Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance $r$. Hessian $\overline{)\mathbf{Y}}\mathbf{es}/\overline{)\mathbf{N}}\mathbf{o}$ Default $\text{}=\mathrm{NO}$ This option controls the contents of the upper triangular matrix (see the description of Section 5 works exclusively with the transformed and reordered matrix ${H}_{Q}$ (8) , and hence extra computation is required to form the Hessian itself. If contains the Cholesky factor of the matrix with columns ordered as indicated by Section 5 ). If contains the Cholesky factor of the matrix , with columns ordered as indicated by Infinite Bound Size $r$ Default $\text{}={10}^{20}$ If $r>0$, $r$ defines the ‘infinite’ bound $\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\mathit{bigbnd}$ will be regarded as $+\infty$ (and similarly any lower bound less than or equal to $-\mathit{bigbnd}$ will be regarded as $-\infty$). If $r<0$, the default value is used. Infinite Step Size $r$ Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\mathit{bigbnd},{10}^{20}\right)$ If $r>0$, $r$ specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the Hessian is singular and the objective contains an explicit linear term.) If the change in $x$ during an iteration would exceed the value of $r$, the objective function is considered to be unbounded below in the feasible region. If $r\le 0$, the default value is used. Iteration Limit $i$ Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{n}_{L}\right)\right)$ List Default for e04ncf $\text{}={\mathbf{List}}$ Nolist Default for e04nca $\text{}={\mathbf{Nolist}}$ Optional parameter enables printing of each optional parameter specification as it is supplied. suppresses this printing. Monitoring File $i$ Default $\text{}=-1$ If $i\ge 0$ and ${\mathbf{Print Level}}\ge 5$, monitoring information produced by e04ncf/e04nca at every iteration is sent to a file with logical unit number $i$. If $i<0$ and/or ${\mathbf{Print Level}}<5$, no monitoring information is produced. Print Level $i$ Default for e04ncf $\text{}=10$ Default for e04nca $\text{}=0$ The value of controls the amount of printout produced by , as indicated below. A detailed description of the printed output is given in Section 9.2 (summary output at each iteration and the final solution) and Section 13 (monitoring information at each iteration). The following printout is sent to the current advisory message unit (as defined by $\mathbit{i}$ Output $\phantom{\ge 0}0$ No output. $\phantom{\ge 0}1$ The final solution only. $\phantom{\ge 0}5$ One line of summary output ($\text{}<80$ characters; see Section 9.2) for each iteration (no printout of the final solution). $\text{}\ge 10$ The final solution and one line of summary output for each iteration. The following printout is sent to the unit number given by the optional parameter Monitoring File $\mathbit{i}$ Output $\text{}<5$ No output. $\text{}\ge 5$ One long line of output ($\text{}>80$ characters; see Section 13) for each iteration (no printout of the final solution). $\text{}\ge At each iteration, the Lagrange multipliers, the variables $x$, the constraint values $Cx$ and the constraint status. $\text{}\ge At each iteration, the diagonal elements of the matrix $T$ associated with the $TQ$ factorization (4) (see Section 11.2) of the working set, and the diagonal elements of the upper 30$ triangular matrix $R$. ${\mathbf{Print Level}}\ge 5$ and the unit number defined by the optional parameter Monitoring File is the same as that defined by , the summary output for each major iteration is suppressed. Problem Type $a$ Default $=$ LS1 This option specifies the type of objective function to be minimized during the optimality phase. The following are the nine optional keywords and the dimensions of the arrays that must be specified in order to define the objective function: LP a and b not referenced, length-n cvec; QP1 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$ symmetric, b and cvec not referenced; QP2 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$ symmetric, b not referenced, length-n cvec; QP3 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$ upper trapezoidal, length-n kx, b and cvec not referenced; QP4 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$ upper trapezoidal, length-n kx, b not referenced, length-n cvec; LS1 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$, length-m b, cvec not referenced; LS2 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$, length-m b, length-n cvec; LS3 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$ upper trapezoidal, length-n kx, length-m b, cvec not referenced; LS4 ${\mathbf{a}}\left({\mathbf{lda}},{\mathbf{n}}\right)$ upper trapezoidal, length-n kx, length-m b, length-n cvec. For problems of type FP, the objective function is omitted and are not referenced. The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined. $\mathbit{a}$ Option Least LS1 Quadratic QP2 Linear LP In addition, the keywords LS and LSQ are equivalent to the default option LS1, and the keyword QP is equivalent to the option QP2. If $A=0$, i.e., the objective function is purely linear, the efficiency of e04ncf/e04nca may be increased by specifying $a$ as LP. Rank Tolerance $r$ Default $\text{}=100\epsilon$ or $10\sqrt{\epsilon }$ (see below) Note that this option does not apply to problems of type FP or LP. The default value of $r$ depends on the problem type. If $A$ occurs as a least squares matrix, as it does in problem types QP1, LS1 and LS3, then the default value of $r$ is $100\epsilon$. In all other cases, $A$ is treated as the ‘square root’ of the Hessian matrix $H$ and $r$ has the default value $10\sqrt{\epsilon }$. This parameter enables you to control the estimate of the triangular factor Section 11.3 ). If ${\rho }_{i}$ denotes the function ${\rho }_{i}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{\|{R}_{11}\|,\|{R}_{22}\|,\dots ,\|{R}_{ii}\|\right\}$ , the rank of is defined to be smallest index such that $\|{R}_{i+1,i+1}\|\le r\|{\rho }_{i+1}\|$ . If $r\le 0$ , the default value is used. 13 Description of Monitoring Information This section describes the long line of output ( characters) which forms part of the monitoring information produced by . (See also the description of the optional parameters Monitoring File Print Level .) You can control the level of printed output. To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices $1$ through $n$ refer to the bounds on the variables, and indices $n+1$ through $n+{n}_{L}$ refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint). ${\mathbf{Print Level}}\ge 5$ ${\mathbf{Monitoring File}}\ge 0$ , the following line of output is produced at every iteration on the unit number specified by optional parameter Monitoring File . In all cases, the values of the quantities printed are those in effect on completion of the given iteration. Itn is the iteration count. Jdel is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted. Jadd is the index of the constraint added to the working set. If Jadd is zero, no constraint was added. Step is the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than $1$ only if the factor ${R}_{Z}$ is singular. Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase. is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. Objective During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found. Bnd is the number of simple bound constraints in the current working set. Lin is the number of general linear constraints in the current working set. Art is the number of artificial constraints in the working set, i.e., the number of columns of ${Z}_{2}$ (see Section 11.3). is the number of columns of ${Z}_{1}$ (see Section 11.2). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., $\mathtt{Zr}=n-\left(\mathtt{Bnd}+\mathtt{Lin}+\mathtt{Art}\right)$. The value of , the number of columns of Section 11.2 ) can be calculated as . A zero value of implies that lies at a vertex of the feasible region. Norm Gz is $‖{Z}_{1}^{\mathrm{T}}{g}_{\mathrm{FR}}‖$, the Euclidean norm of the reduced gradient with respect to ${Z}_{1}$. During the optimality phase, this norm will be approximately zero after a unit step. Norm Gf is the Euclidean norm of the gradient function with respect to the free variables, i.e., variables not currently held at a bound. Cond T is a lower bound on the condition number of the working set. Cond Rz is a lower bound on the condition number of the triangular factor ${R}_{1}$ (the first Zr rows and columns of the factor ${R}_{Z}$). If the problem is specified to be of type LP or the estimated rank of the data matrix $A$ is zero then Cond Rz is not printed.	{"url":"https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/e04/e04ncf.html","timestamp":"2024-11-06T14:16:27Z","content_type":"text/html","content_length":"254170","record_id":"<urn:uuid:aa549792-9d4f-408c-a622-c8a7fd6f409c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00140.warc.gz"}
An introduction to OCNet Luca Carraro, Florian Altermatt, Emanuel A. Fronhofer, Reinhard Furrer, Isabelle Gounand, Andrea Rinaldo, Enrico Bertuzzo November 06, 2019 #> Warning: package 'spam' was built under R version 4.3.3 #> Warning: package 'igraph' was built under R version 4.3.3 Graphical abstract OCN <- create_OCN(30, 20, outletPos = 1) OCN <- aggregate_OCN(landscape_OCN(OCN), thrA = 3) par(mfrow = c(1, 3), mai = c(0, 0, 0.2, 0.2)) draw_simple_OCN(OCN, thrADraw = 3) title("Optimal Channel Network") draw_elev3D_OCN(OCN, drawRiver = FALSE, addColorbar = FALSE, expand = 0.2, theta = -30) draw_thematic_OCN(OCN$AG$streamOrder, OCN, discreteLevels = TRUE, colPalette = rainbow(4)) title("Strahler stream order") OCNet enables the creation and analysis of Optimal Channel Networks (OCNs). These are oriented spanning trees (built on rectangular lattices made up of square pixels) that reproduce all scaling features characteristic of real, natural river networks (Rodriguez-Iturbe et al. 1992; Rinaldo et al. 2014). As such, they can be used in a variety of numerical and laboratory experiments in the fields of hydrology, ecology and epidemiology. Notable examples include studies on metapopulations and metacommunities (e.g. Carrara et al. 2012), scenarios of waterborne pathogen invasions (e.g. Gatto et al. 2013) and biogeochemichal processes in streams (e.g. Helton, Hall, and Bertuzzo 2018). OCNs are obtained by minimization of a functional which represents total energy dissipated by water flowing through the network spanning the lattice. Such a formulation embeds the evidence that morphological and hydrological characteristics of rivers (in particular, water discharge and slope) follow a power-law scaling with drainage area. For an overview of the functionalities of the package, see Carraro et al. (2020). For details on the theoretical foundation of the OCN concept, see Rinaldo et al. (2014). Some useful definitions In graph theory, an oriented spanning tree is a subgraph of a graph $G$ such that: • it is oriented: edges’ directions are assigned and none of its pairs of nodes is linked by two symmetric edges; • it is spanning: it contains all nodes of $G$; • it is a tree: it is weakly connected (there exists a path between any pair of nodes, if edges’ directions are neglected) and acyclic (no loops are present). At the simplest aggregation level (flow direction - FD; see Section 4 below), OCNs are oriented spanning trees whose nodes are the pixels constituting the lattice and whose edges represent flow Moreover, OCNs, just like real rivers, are constituted of nodes whose indegree (i.e. the number of edges pointing towards a node) can assume any value while the outdegree (number of edges exiting from the node) is equal to 1, except for the root (or outlet node), whose outdegree is equal to 0. Nodes with null indegree are termed sources. Nodes with indegree larger than 1 are confluences. OCNet also allows building multiple networks within a single lattice. Each of these networks is defined by its respective outlet, which represents the root of a subgraph; the union of all subgraphs contains all elements of $G$. For simplicity, we will still refer to “OCNs” with regards to these multiple-outlet entities. In this case, strictly speaking, OCNs are not trees but rather forests. An OCN is defined by an adjacency matrix $\mathbf{W}$ with entries $w_{ij}$ equal to 1 if node $i$ drains into $j$ and null otherwise. Owing to the previously described properties, all rows of $\mathbf{W}$ have a single non-zero entry, except those identifying the outlet nodes, whose entries are all null. Each adjacency matrix uniquely defines a vector of contributing areas (or drainage areas) $\mathbf{A}$, whose components $A_i$ are equal to the number of nodes upstream of node $i$ plus the node itself. Mathematically, this can be expressed as $(\mathbf{I}-\mathbf {W}^T)\mathbf{A}=\mathbf{1}$, where $\mathbf{I}$ is the identity matrix and $\mathbf{1}$ a vector of ones. OCNet functions dependency tree The generation of an OCN is performed by function create_OCN. Its only required inputs are the dimensions of the rectangular lattice, but several other features can be implemented via its optional inputs (see the function documentation for details). The output of create_OCN is a list, which can be used as input to the subsequent function landscape_OCN, as shown by the dependency tree below (indentation of an item implies dependency on the function at the above level). • create_OCN: performs the OCN search algorithm □ landscape_OCN: calculates the elevation field generated by the OCN ☆ aggregate_OCN: aggregates the OCN at various levels ○ draw_thematic_OCN: draws OCN with colors of nodes depending on a theme ○ draw_subcatchments_OCN: draws partition of the lattice into subcatchments as a result of the aggregation process of the OCN ○ paths_OCN: calculates paths and path lengths among OCN nodes ○ rivergeometry_OCN: evaluates hydraulic properties of the OCN ○ OCN_to_igraph: transforms the OCN into an igraph object ○ OCN_to_SSN: transforms the OCN into a SpatialStreamNetwork object ☆ draw_contour_OCN: draws “real-shaped” OCNs and catchment contours ☆ draw_elev2D_OCN: draws 2D elevation field ☆ draw_elev3D_OCN: draws 3D elevation field ☆ draw_elev3Drgl_OCN: draws 3D elevation field (via rgl rendering system) ☆ find_area_threshold_OCN: finds relationship between threshold area and number of nodes at the RN and AG level (see Figure 4.1 for relevant definitions) □ draw_simple_OCN: fast drawing of OCNs • create_peano: creates Peano networks OCNet functions are intended to be applied in sequential order: for each non-drawing function, the input list is copied into the output list, to which new sublists and objects are added. Aggregation levels Adjacency matrices and contributing area vectors of an OCN can be defined at different aggregation levels. In the output of OCNet functions, variables characterizing the OCN at the different aggregation levels are grouped within separate sublists, each of them identified by a two-letter acronym (marked in bold in the list below). Figure 4.1 provides a graphical visualization of the correspondence among the main aggregation levels. 0. Nearest neighbours (N4, N8). Every pixel of the lattice constitutes a node of the network. Each node is connected to its four (pixels that share an edge) or eight (pixels that share an edge or a vertex) nearest neighbours. At this level, $\mathbf{W}$ is defined but $\mathbf{A}$ is not. Note that this level does not describe flow connectivity, but rather proximity among pixels. Hence, $\mathbf{W}$ does not describe an oriented spanning tree. 1. Flow direction (FD). At this level, every pixel of the lattice is a node, but connectivity follows the flow directions that have been found by the OCN search algorithm (operated by function create_OCN). Edges’ lengths are equal to either cellsize (the size of a pixel side, optional input in create_OCN) or cellsizesqrt(2), depending on whether flow direction is horizontal/vertical or diagonal. 2. River network (RN). The set of nodes at this level is a subset of the nodes at the FD level, such that their contributing area is larger than a certain threshold (optional input A_thr in aggregate_OCN). Such a procedure is customary in the hydrological problem of extracting a river network based on digital elevation models of the terrain (O’Callaghan and Mark 1984), and corresponds to the geomorphological concept of erosion threshold (associated to a threshold in landscape-forming runoff, of which drainage area represents a proxy Rodriguez-Iturbe and Rinaldo 2001). Edges’ lengths are again equal to either cellsize or cellsizesqrt(2). 3. Aggregated (or reach - AG). The set of nodes at this level is a subset of the nodes at the RN level (see details in the next section). Accordingly, vector $\mathbf{A}$ is a subset of the vector of the same name defined at the RN level. Edges can span several pixels and therefore have various lengths. 4. Subcatchment (SC). The number of nodes at this level is generally equal to that at the AG level. Each node is constituted by the cluster of pixels that directly drain into the edge departing from the corresponding node at the AG level. Here $\mathbf{W}$ does not represent flow connectivity but rather identifies terrestrial borders among subcatchments and is therefore symmetric. 5. Catchment (CM). In this level, the number of nodes is equal to the number of outlets. Every node represents the portion of the lattice drained by its relative outlet. $\mathbf{A}$ stores drainage area values for each of these catchments, while $\mathbf{W}$ identifies terrestrial borders among catchments. Relationship between nodes at the RN and AG levels Nodes at the AG level correspond to a subset of nodes at the RN level. In particular, nodes at the AG level belong to at least one of these four categories: • Sources: nodes at the RN level with null indegree. • Confluences: nodes at the RN level with indegree larger than one. • Outlets: corresponding to outlets at the RN level. • Breaking nodes (only if maxReachLength is finite): nodes that split edges that are longer than maxReachLength. Outlet nodes at the AG level might also be sources, confluences or breaking nodes. All AG nodes except outlet nodes have outdegree equal to 1. All RN nodes that do not correspond to AG nodes constitute the edges of the network at the AG level: more specifically, each edge is formed by an AG node and a sequence of RN nodes downstream of the AG node, until another AG node is found. Figure 4.2 shows an alternative aggregation scheme for the network showed in Figure 4.1 when the optional input maxReachLength is set to a finite value. Correspondence between indices at different levels The output of aggregate_OCN contains objects named OCN$XX$toYY, where XX and YY are two different aggregation levels. These objects define the correspondences between indices among aggregation levels. OCN$XX$toYY contains a number of elements equal to the number of nodes at XX level; each element OCN$XX$toYY[[i]] contains the index/indices at YY level corresponding to node i at XX level. For aggregation level AG, additional correspondence objects are marked by the string Reach: these consider the whole sequence of RN nodes constituting the edge departing from an AG node as belonging to the AG node. The example shown in Figure 4.3 corresponds to the dataset OCN_4 included in the package. Note that index numbering starts from the lower-left (southwestern) corner of the lattice. The R code below displays the different OCN$XX$toYY objects corresponding to the example in Figure 4.3: ex <- aggregate_OCN(landscape_OCN(OCN_4), thrA = 2) #> [1] 1 2 3 0 4 0 0 5 6 7 0 0 0 0 8 0 #> [1] 1 3 3 3 2 3 3 3 4 5 5 3 4 5 5 5 #> [1] 1 2 3 5 8 9 10 15 #> [1] 1 0 0 2 3 4 0 5 #> [1] 1 3 3 2 3 4 5 5 #> [1] 1 5 8 9 15 #> [[1]] #> [1] 1 #> [[2]] #> [1] 5 #> [[3]] #> [1] 8 3 2 #> [[4]] #> [1] 9 #> [[5]] #> [1] 15 10 #> [1] 1 4 5 6 8 #> [[1]] #> [1] 1 #> [[2]] #> [1] 4 #> [[3]] #> [1] 5 3 2 #> [[4]] #> [1] 6 #> [[5]] #> [1] 8 7 #> [[1]] #> [1] 1 #> [[2]] #> [1] 5 #> [[3]] #> [1] 8 3 2 4 6 7 12 #> [[4]] #> [1] 9 13 #> [[5]] #> [1] 15 10 11 14 16 A working example Let’s build an OCN on a 20x20 lattice and assume that each cell represents a square of side 500 m. The total size of the catchment is therefore 100 km^2. Let’s locate the outlet close to the southwestern corner of the lattice. Function draw_simple_OCN can then be used to display the OCN. OCNwe <- create_OCN(20, 20, outletPos = 3, cellsize = 500) Now, let’s construct the elevation field subsumed by the OCN. Let’s suppose that the outlet has null elevation and slope equal to 0.01. Then, we use draw_elev3D_OCN to draw the three-dimensional elevation map (values are in m). OCNwe <- landscape_OCN(OCNwe, slope0 = 0.01) draw_elev3D_OCN(OCNwe, drawRiver = FALSE) Next, the OCN can be aggregated. Let’s suppose that the desired number of nodes at the AG level be as close as possible to 20. With function find_area_threshold_OCN we can derive the corresponding value of drainage area threshold: thr <- find_area_threshold_OCN(OCNwe) # find index corresponding to thr$Nnodes ~= 20 indThr <- which(abs(thr$nNodesAG - 20) == min(abs(thr$nNodesAG - 20))) indThr <- max(indThr) # pick the last ind_thr that satisfies the condition above thrA20 <- thr$thrValues[indThr] # corresponding threshold area The resulting number of nodes is 20, corresponding to a threshold area thrA20 = 2.5 km^2. The latter value can now be used in function aggregate_OCN to obtain the aggregated network. Function draw_subcatchments_OCN shows how the lattice is partitioned into subcatchments. It is possible to add points at the locations of the nodes at the AG level. OCNwe <- aggregate_OCN(OCNwe, thrA = thrA20) points(OCNwe$AG$X,OCNwe$AG$Y, pch = 21, col = "blue", bg = "blue") Finally, draw_thematic_OCN can be used to display the along-stream distances of RN-level nodes to the outlet (in m), as calculated by paths_OCN. OCNwe <- paths_OCN(OCNwe, includePaths = TRUE) draw_thematic_OCN(OCNwe$RN$downstreamPathLength[ , OCNwe$RN$outlet], OCNwe, backgroundColor = "#606060") Peano networks Function create_peano can be used in lieu of create_OCN to generate Peano networks on square lattices. Peano networks are deterministic, plane-filling fractals whose topological measures (Horton’s bifurcation and length ratios) are akin to those of real river networks (Marani, Rigon, and Rinaldo 1991) and can then be used in a variety of synthetic experiments, as it is the case for OCNs (e.g. Campos, Fort, and Méndez 2006). Peano networks are generated by means of an iterative algorithm: at each iteration, the size of the lattice side is doubled (see code below). As a result, Peano networks span squares of side equal to a power of 2. The outlet must be located at a corner of the square. par(mfrow = c(2, 3), mai = c(0, 0, 0.2, 0)) peano0 <- create_peano(0) title("Iteration: 0 - Lattice size: 2x2") peano1 <- create_peano(1) title("Iteration: 1 - Lattice size: 4x4") peano2 <- create_peano(2) title("Iteration: 2 - Lattice size: 8x8") peano3 <- create_peano(3) title("Iteration: 3 - Lattice size: 16x16") peano4 <- create_peano(4) title("Iteration: 4 - Lattice size: 32x32") peano5 <- create_peano(5) title("Iteration: 5 - Lattice size: 64x64") The output of create_peano is a list containing the same objects as those produced by create_OCN. As such, it can be used as input for all other complementary functions of the package. List of ready-made OCNs OCNet contains some ready-made large OCNs built via function create_OCN. Their features are summarized in the Table below. Refer to the documentation of create_OCN for the definition of column names. Note that: • If not specified otherwise, the position of outlet(s) was derived from default options. • Cooling schedule: □ cold: corresponds to coolingRate = 10, initialNoCoolingPhase = 0; □ default: corresponds to default values coolingRate = 1, initialNoCoolingPhase = 0; □ hot: corresponds to coolingRate = 0.5, initialNoCoolingPhase = 0.1. • seed is the value of the argument used in the call of set.seed prior to executing create_OCN. • On CRAN? identifies which OCNs are uploaded in the version of OCNet that can be downloaded from CRAN (owing to limitation in package size). Installation of package from GitHub provides the complete set of OCNs hereafter described. OCN_4 4 4 1 FALSE 1 OCN_20 20 20 1 FALSE I default 1 1 Yes OCN_250 250 250 1 FALSE I default 1 2 No OCN_250_T 250 250 1 FALSE T default 1 2 Yes OCN_250_V 250 250 1 FALSE V default 1 2 No OCN_250_cold 250 250 1 FALSE I cold 1 2 No OCN_250_hot 250 250 1 FALSE I hot 1 2 No OCN_250_V_cold 250 250 1 FALSE V cold 1 2 No OCN_250_V_hot 250 250 1 FALSE V hot 1 2 No OCN_250_PB 250 250 1 TRUE I default 1 2 Yes OCN_rect1 450 150 1 FALSE I default 1 3 No OCN_rect2 150 450 1 FALSE I default 1 3 No OCN_300_diag 300 300 1 FALSE V default 50 4 No OCN_300_4out 300 300 4 FALSE V default 50 5 Yes OCN_300_4out_PB_hot 300 300 4 TRUE V hot 50 5 Yes OCN_300_7out 300 300 7 FALSE V default 50 5 No OCN_400_T_out 400 400 1 FALSE T hot 50 7 No OCN_400_Allout 400 400 All FALSE H hot 50 8 Yes OCN_500_hot 500 500 1 FALSE I hot 50 9 No OCN_500_PB_hot 500 500 1 TRUE V hot 50 10 No Compatibility with other packages Adjacency matrices at all aggregation levels are produced as spam (Furrer and Sain 2010) objects. In order to transform the OCN into an igraph (Csardi and Nepusz 2006) graph object, the adjacency matrix must be converted into a Matrix object (via function as.dgCMatrix.spam of spam). Function graph_from_adjacency_matrix of igraph can then be used to obtain a graph object. For example, let’s transform the previously obtained OCN_we at the AG level into a graph: g <- OCN_to_igraph(OCNwe, level = "AG") plot.igraph(g, vertex.color = rainbow(OCNwe$AG$nNodes), layout = matrix(c(OCNwe$AG$X,OCNwe$AG$Y),ncol = 2, nrow = OCNwe$AG$nNodes)) The same network can be displayed as an OCN: Campos, D., J. Fort, and V. Méndez. 2006. “Transport on Fractal River Networks. Application to Migration Fronts.” Theoretical Population Biology 69(1): 88–93. Carrara, F., F. Altermatt, I. Rodriguez-Iturbe, and A. Rinaldo. 2012. “Dendritic Connectivity Controls Biodiversity Patterns in Experimental Metacommunities.” Proceedings of the National Academy of Sciences of the United States of America 109(15): 5761–66. Carraro, L., E. Bertuzzo, E. A. Fronhofer, R. Furrer, I. Gounand, A. Rinaldo, and F. Altermatt. 2020. “Generation and Application of River Network Analogues for Use in Ecology and Evolution.” Ecology and Evolution Csardi, G., and T. Nepusz. 2006. “The Igraph Software Package for Complex Network Research.” InterJournal Complex Systems: 1695. Furrer, R., and S.R. Sain. 2010. “Spam. A Sparse Matrix R Package with Emphasis on MCMC Methods for Gaussian Markov Random Fields.” Journal of Statistical Software 36(10): 1–25. Gatto, M., L. Mari, E. Bertuzzo, R. Casagrandi, L. Righetto, I. Rodriguez-Iturbe, and A. Rinaldo. 2013. “Spatially Explicit Conditions for Waterborne Pathogen Invasion.” American Naturalist 182(3): 328–46. Helton, A.M., R.O. Hall, and E. Bertuzzo. 2018. “How Network Structure Can Affect Nitrogen Removal by Streams.” Freshwater Biology 63(1): 128–40. Marani, A., R. Rigon, and A. Rinaldo. 1991. “A Note on Fractal Channel Networks.” Water Resources Research 27(12): 3041–49. O’Callaghan, J. F., and D. A. Mark. 1984. “The Extraction of the Drainage Networks from Digital Elevation Data.” Computer Vision, Graphics, and Image Processing 28: 323–44. Rinaldo, A., R. Rigon, J. R. Banavar, A. Maritan, and I. Rodriguez-Iturbe. 2014. “Evolution and Selection of River Networks. Statics, Dynamics, and Complexity.” Proceedings of the National Academy of Sciences of the United States of America 111(7): 2417–24. Rodriguez-Iturbe, I., and A. Rinaldo. 2001. Fractal River Basins. Chance and Self-Organization. Cambridge University Press. Rodriguez-Iturbe, I., A. Rinaldo, R. Rigon, R. L. Bras, A. Marani, and E. Ijjász-Vásquez. 1992. “Energy Dissipation, Runoff Production, and the Three‐dimensional Structure of River Basins.” Water Resources Research 28(4): 1095–1103.	{"url":"https://mirrors.nic.cz/R/web/packages/OCNet/vignettes/OCNet.html","timestamp":"2024-11-10T12:00:07Z","content_type":"text/html","content_length":"393338","record_id":"<urn:uuid:870e1128-7426-4562-b6fd-0adc9a35d62a>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00166.warc.gz"}
[Python series column] Part 15 functional programming in Python Functional programming Function is a kind of encapsulation supported by Python built-in. We can decompose complex tasks into simple tasks by disassembling large pieces of code into functions and calling functions layer by layer. This decomposition can be called process oriented programming. Function is the basic unit of process oriented programming. Functional Programming (please note that there is another word * * "formula") - Functional Programming, although it can also be attributed to process oriented programming, its idea is closer to mathematical calculation * . We must first understand the concepts of * Computer and Compute * . • At the computer level, the CPU executes the instruction code of addition, subtraction, multiplication and division, as well as various condition judgment and jump instructions, so the assembly language is the language closest to the computer. • Calculation refers to the calculation in the mathematical sense. The more abstract the calculation is, the farther it is from the computer hardware. Corresponding to the programming language, the lower the language, the closer it is to the computer, the lower the degree of abstraction and the higher the execution efficiency, such as C language; The more advanced the language is, the closer it is to computing. It has a high degree of abstraction and low execution efficiency, such as Lisp language. To sum up: Low level language high-level language characteristic Close to computer Close calculation (in mathematical sense) Degree of abstraction low high Execution efficiency high low example Compilation and C Lisp Functional programming is a programming paradigm with a high degree of abstraction. The functions written in pure functional programming language have no variables. Therefore, as long as the input of any function is determined, the output is determined. This pure function is called no side effects. In the programming language that allows the use of variables, because the variable state inside the function is uncertain, the same input may get different outputs. Therefore, this function has side effects. A feature of functional programming is that it allows the function itself to be passed into another function as a parameter and return a function! Python provides only partial support for functional programming. Because Python allows variables, Python is not a purely functional programming language. Three characteristics of functional programming 1. immutable data Variables are immutable, or there are no variables, only constants. When the input of functional programming is determined, the output is determined. The variables inside the function have nothing to do with those outside the function and will not be affected by external operations. 2. first class functions The first type of function (also known as high-order function) means that functions can be used like variables, and can be created, modified, passed and returned like variables. This allows us to break a large piece of code into functions and call them layer by layer. This process oriented writing method is more intuitive than loop. 3. Tail recursive optimization As mentioned in the recursive function in the previous chapter, it returns the function itself rather than the expression. Unfortunately, this feature is not available in Python. Several techniques of functional programming 1. map & reduce The most common technology of functional programming is to do Map and Reduce operations on a set. Compared with the traditional process oriented writing method, it is easier to read the code (instead of using a pile of for and while loops to toss data, more abstract Map functions and Reduce functions are used). 2. pipeline The meaning of this technology is to turn function instances into actions one by one, then put a group of actions into an array or list to form an action list, and then transfer the data to the action list. The data is operated by each function in sequence like a pipeline, and finally get the desired result. 3. recursing The biggest advantage of recursion is to simplify the code. It can describe a complex problem with very simple code. Note: the essence of recursion is to describe problems, which is the essence of functional programming. 4. currying Decompose multiple parameters of a function into multiple functions, and then encapsulate the function in multiple layers. Each layer of function returns a function to receive the next parameter. In this way, multiple parameters of the function can be simplified (reduce the number of parameters of the function). 5. higher order function High order function: the so-called high-order function is to encapsulate the incoming function when the function is a parameter, and then return the encapsulated function. The phenomenon is that functions pass in and out. A little supplement to currying, for example: def pow(i, j): return ij def square(i): return pow(i, 2) Here we decompose the parameter j of the original square function, which returns the power function pow function and encapsulates the power in it, so as to reduce the parameters required for square Some concepts of functional programming can be understood Fool functional programming Or the original English version Functional Programming For The Rest of Us. Several benefits of functional programming 1. parallelization In the parallel environment, there is no need for synchronization or mutual exclusion between threads (variables are internal and do not need to be shared). 2. lazy evaluation An expression does not evaluate immediately after it is bound to a variable, but when the value is taken. 3. Determinism determinism The input is determined and the output is determined. Simple example In the past, procedural oriented programming required the introduction of additional logic variables and the use of loops: upname =['HAO', 'CHEN', 'COOLSHELL'] lowname =[] for i in range(len(upname)): lowname.append( upname[i].lower() ) Functional programming is very simple and easy to understand: def toUpper(item): return item.upper() upper_name = map(toUpper, ["hao", "chen", "coolshell"]) print upper_name Let's take another example of calculating the average of all positive numbers in a list: num =[2, -5, 9, 7, -2, 5, 3, 1, 0, -3, 8] positive_num_cnt = 0 positive_num_sum = 0 for i in range(len(num)): if num[i] > 0: positive_num_cnt += 1 positive_num_sum += num[i] if positive_num_cnt > 0: average = positive_num_sum / positive_num_cnt print average If functional programming is used: positive_num = filter(lambda x: x>0, num) average = reduce(lambda x,y: x+y, positive_num) / len( positive_num ) It can be seen that functional programming reduces the use of variables, reduces the possibility of bugs, and makes maintenance more convenient. Higher readability and simpler code. For more examples and analysis, see Functional programming. Higher order function The high-order function features in functional programming have been mentioned earlier. This section will describe in more detail how to use them in Python. Variables can point to functions >>> abs <built-in function abs> >>> f = abs >>> f <built-in function abs> >>> f(-10) This example shows that in Python, variables can point to functions, and such assigned variables can be used as aliases of functions. The function name is also a variable >>> abs = 10 >>> abs(-10) Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: 'int' object is not callable Here, the abs function is assigned a value of 10. After this assignment, the abs becomes an integer variable, pointing to the int object 10 instead of the original function. So it can no longer be used as a function. To restore the abs function, restart the Python interactive environment. abs function definition__ builtin__ In the module, to make the modification of the direction of abs variable effective in other modules, use__ builtin__.abs = 10 is OK. Of course, actually writing code should never do this Incoming function Functions can be passed as parameters, and the function receiving such parameters is called high-order function. Simple example: def add(x, y, f): return f(x) + f(y) >>> add(-5, 6, abs) Here, the abs function can be passed as a parameter into the add function we write. The add function is a high-order function. The map() function and the reduce() function are two built-in functions (BIFS) of Python. map function The map() function receives two parameters, one is a function and the other is an iteratable object. Map applies the incoming function to each element of the sequence in turn, and returns the result as an Iterator object (inert sequence, which can be converted into list output by list). For example: >>> def f(x): ... return x x >>> r = map(f, [1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> list(r) [1, 4, 9, 16, 25, 36, 49, 64, 81] Here, the iterator object is directly converted into a list using the list() function. The write loop can achieve the same effect, but it is obviously not as intuitive as the map() function. As a high-order function, map() function greatly simplifies the code and is easier to >>> list(map(str, [1, 2, 3, 4, 5, 6, 7, 8, 9])) ['1', '2', '3', '4', '5', '6', '7', '8', '9'] Converting a list of integers to a list of characters requires only one line of code. reduce function Reduce receives two parameters, one is a function (assuming the function is called f) and the other is an iteratable object (assuming L). Function f must receive two parameters. The reduce function will pass the value returned by function f last time and the next element of l into f every time. Until all elements in l have participated in the operation, it will return the last value returned by function f (the first two elements of l will be passed in the first time). >>> from functools import reduce >>> def add(x, y): ... return x + y >>> reduce(add, [1, 3, 5, 7, 9]) Here is an example of the simplest sequence summation (of course, it is more convenient for us to directly use the sum() function, which is just for example). Here, the reduce function applies add to the first two elements of the sequence each time and returns the result to the head of the sequence until there is only one element left in the sequence (this understanding may be more intuitive). >>> from functools import reduce >>> def fn(x, y): ... return x * 10 + y >>> def char2num(s): ... return {'0': 0, '1': 1, '2': 2, '3': 3, '4': 4, '5': 5, '6': 6, '7': 7, '8': 8, '9': 9}[s] #The dict of the character corresponding to the integer returns the integer corresponding to the incoming character >>> reduce(fn, map(char2num, '13579')) You can sort out the str2int function as a whole: def str2int(s): def fn(x, y): return x * 10 + y def char2num(s): return {'0': 0, '1': 1, '2': 2, '3': 3, '4': 4, '5': 5, '6': 6, '7': 7, '8': 8, '9': 9}[s] return reduce(fn, map(char2num, s)) Using lambda anonymous functions can further simplify: def char2num(s): return {'0': 0, '1': 1, '2': 2, '3': 3, '4': 4, '5': 5, '6': 6, '7': 7, '8': 8, '9': 9}[s] def str2int(s): return reduce(lambda x, y: x * 10 + y, map(char2num, s)) 1. Use the map() function to change the non-standard English name into the standard name with the first letter uppercase and other letters lowercase. • The string supports slicing operation, and the plus sign can be used for string splicing. • The upper function is used to convert uppercase and the lower function is used to convert lowercase. def normalize(name): return name[0].upper()+name[1:].lower() L1 = ['adam', 'LISA', 'barT'] L2 = list(map(normalize, L1)) 2. Write a prod() function, which can accept a list and use reduce() to quadrature. • Multiply two numbers with anonymous functions • Use the reduce function for reduction to obtain the product of the continuous multiplication of list elements. from functools import reduce def prod(L): return reduce(lambda x,y: xy,L) print('3 5 * 7 * 9 =', prod([3, 5, 7, 9])) 3. Use map and reduce to write a str2float function to convert the string '123.456' into floating point number 123.45. • The idea of this problem is to find the position i of the decimal point (after counting i numbers from one place), and then divide the converted integer by the i power of 10. • Another idea is to change the conversion method from num10 + current number to num + current number / point after the decimal point is encountered during conversion. Point is initially 1, divided by 10 before adding a new number each time. from functools import reduce from math import pow def chr2num(s): return {'0': 0, '1': 1, '2': 2, '3': 3, '4': 4, '5': 5, '6': 6, '7': 7, '8': 8, '9': 9}[s] def str2float(s): return reduce(lambda x,y:x10+y,map(chr2num,s.replace('.',''))) / pow(10,len(s)-s.find('.')-1) The filter() function is also a built-in function for filtering sequences. filter() receives a function and an iteratable object. Unlike map(), filter() applies the incoming function to each element in turn, and then decides whether to keep or discard the element according to whether the return value of the function is True or False. Simple odd filter example: def is_odd(n): return n % 2 == 1 list(filter(is_odd, [1, 2, 4, 5, 6, 9, 10, 15])) # Results: [1, 5, 9, 15] Filter out the empty string of the list: def not_empty(s): return s and s.strip() list(filter(not_empty, ['A', '', 'B', None, 'C', ' '])) # Result: ['a ',' B ',' C '] Among them, the strip function is used to delete specific characters in the string. The format is s.strip(rm), and delete the characters contained in the rm deletion sequence at the beginning and end of the s string. When rm is empty, whitespace characters (including '\ n', '\ r', '\ t', ') are deleted by default. Note that the filter() function returns an Iterator object, that is, an inert sequence, so to force filter() to complete the calculation results, you need to use the list() function to obtain all the results and return list. The most important point of filter function is to correctly define a filter function (that is, the function that passes filter as a parameter). 1. Filter prime numbers with filter Here, the Elsevier sieve method is used. First, list all natural numbers starting from 2 and construct a sequence: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ... Take the first number 2 of the sequence, which must be a prime number, and then use 2 to screen out the multiples of 2 of the sequence: 3, 5, 7, 9, 11, 13, 15, 17, 19, ... Take the first number 3 of the new sequence, which must be a prime number, and then use 3 to screen out the multiples of 3 of the sequence: 5, 7, 11, 13, 17, 19, ... And so on First, construct a generator to output the odd sequence starting from 3: def _odd_iter(): n = 1 while True: n = n + 2 yield n Then define a filter function and pass in n to judge whether x can divide all N: def _not_divisible(n): return lambda x: x % n > 0 Here x is the parameter of the anonymous function, which is provided externally. Then there is the generator that defines the return prime. • First, the prime number 2 is output, and then the odd number queue is initialized. Each time, the head of the queue is output (it must be a prime number, because the previous round of filtering has excluded the number that is smaller than the current head of the queue and is not prime). • Construct a new queue. Each time, use the smallest number of the current sequence as the divisor to check whether the following number is a prime number. It is defined as follows: def primes(): yield 2 it = _odd_iter() # Initial sequence while True: n = next(it) # Returns the first number of the sequence yield n it = filter(_not_divisible(n), it) # Construct new sequence Here, because it is an iterator, you can get the next element of the queue every time you use next. In fact, it is similar to the dequeue operation of the queue, squeezing out the head of the queue without worrying about repetition. Then, the principle of filter here is to put each number of the current it queue into_ not_ Check in divisible (n). Note that it is not passed in as parameter n, but as parameter x of anonymous _ not_divisible(n) is actually viewed as a whole. It returns a function with its own parameter n (that is, the anonymous function), and then filter passes each element of the list one by one to the returned anonymous function. We must make this clear. • Finally, because primes also produces an infinite inert sequence, we generally don't need to ask so much. Just write a loop to exit: # Print prime numbers within 1000: for n in primes(): if n < 1000: 2. Filter the number of palindromes with filter • str can convert integers to strings • [:: - 1] you can get a list in reverse order. def is_palindrome(n): return str(n) == str(n)[::-1] print(list(filter(is_palindrome, range(0,1001)))) Python's built-in sorted() function can sort the list: >>> sorted([36, 5, -12, 9, -21]) [-21, -12, 5, 9, 36] As a high-order function, sorted() can also accept a key function for user-defined sorting, for example: >>> sorted([36, 5, -12, 9, -21], key=abs) [5, 9, -12, -21, 36] The function specified by key will act on each element of the list, sort according to the results returned (mapped) by the key function, and finally output the corresponding elements in the list. Let's take another example of string sorting: >>> sorted(['bob', 'about', 'Zoo', 'Credit']) ['Credit', 'Zoo', 'about', 'bob'] By default, it is sorted by ASCII code, but we often want to arrange it in dictionary order. The idea is to change the string to all lowercase / all uppercase and then arrange it: >>> sorted(['bob', 'about', 'Zoo', 'Credit'], key=str.lower) ['about', 'bob', 'Credit', 'Zoo'] The default sorting is from small to large. To reverse sorting, just set the reverse parameter to True. Review the previous knowledge. Here, the reverse parameter is a named keyword parameter. >>> sorted(['bob', 'about', 'Zoo', 'Credit'], key=str.lower, reverse=True) ['Zoo', 'Credit', 'bob', 'about'] The key to using sorted function well is to define a mapping function. Give the score sheet and sort it by name and score respectively. >>> L = [('Bob', 75), ('Adam', 92), ('Bart', 66), ('Lisa', 88)] >>> L2 = sorted(L, key = lambda x:x[0]) #Sort by name >>> L2 [('Adam', 92), ('Bart', 66), ('Bob', 75), ('Lisa', 88)] >>> L3 = sorted(L, key = lambda x:x[1]) #Sort by grade >>> L3 [('Bart', 66), ('Bob', 75), ('Lisa', 88), ('Adam', 92)] Return function Function as return value In addition to accepting functions as parameters, higher-order functions can also return functions as result values. For example, we want to implement a function for summing variable parameters, which can be written as follows: def calc_sum(args): ax = 0 for n in args: ax = ax + n return ax When calling, you can pass in any number and get the sum of these numbers. If we don't need to sum immediately, we can do it later as needed. It can be written in the form of return summation def lazy_sum(args): def sum(): ax = 0 for n in args: ax = ax + n return ax return sum Calling lazy_ During sum, a sum function is returned, but the summation code inside the sum function is not executed: >>> f = lazy_sum(1, 3, 5, 7, 9) >>> f <function lazy_sum.<locals>.sum at 0x101c6ed90> When we call the returned sum function again, we can get the sum value: >>> f() be careful! Every call to lazy_ The functions returned by sum are different! Even if the same parameter is passed in, the return function is different! for instance: >>> f1 = lazy_sum(1, 3, 5, 7, 9) >>> f2 = lazy_sum(1, 3, 5, 7, 9) >>> f1==f2 f1 and f2 are two different functions. Although calling them yields the same result, they do not affect each other. In Python, in terms of expression, closure can be defined as: if an internal function references variables in an external scope (non global scope), then the internal function is considered as a closure. For example, f in the above example is a closure, which calls variable i, which belongs to the outer loop body rather than the global variable. Take an example: def count(): fs = [] for i in range(1, 4): def f(): return ii return fs f1, f2, f3 = count() The call results of the three return functions are: >>> f1() >>> f2() >>> f3() To analyze, the count function here is a function that returns three functions, in which a loop body is used to generate the three functions to be returned. From i is 1 to i is equal to 3, a function f is generated each time, which returns the square value of i. If you follow the usual way of thinking, you may feel that the three returned functions f1, f2 and f3 should output 1, 4 and 9 But in fact, this is not the case, because when a function is returned, the code inside the function is not executed! Only when this returned function is called will it be executed! When the count function is called, three new functions are actually returned, and the value of the loop variable i becomes 3. When the three returned functions are called, their code will be executed. At this time, the value of the referenced i is all 3. What if you have to use external loop variables in closures? We first define a function, bind the loop variable with its parameters, and then define the function to return in it. In this way, no matter how the loop variable changes, the value bound to the parameter will not change, and we can get the desired result. That is, rewrite the above example as: def count(): def f(j): def g(): return jj return g fs = [] for i in range(1, 4): fs.append(f(i)) # f(i) is executed immediately, so the current value of i is passed into f() return fs Call result: >>> f1, f2, f3 = count() >>> f1() >>> f2() >>> f3() Here, the variable J used in closure g is from the external scope f, and j is bound as a parameter in f and will not change, which is different from the variable i in the external scope count function. Therefore, the results of each of the three functions returned by count are different. • When returning a closure, do not reference the loop variable of the external scope or the variable that will change in the external scope in the closure code. • Local variables of external scopes should not be modified in closures.	{"url":"https://programming.vip/docs/python-series-column-part-15-functional-programming-in-python.html","timestamp":"2024-11-13T01:38:57Z","content_type":"text/html","content_length":"36122","record_id":"<urn:uuid:11158403-0847-48e1-b69a-aba4e536aea1>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00695.warc.gz"}
Of mathematics Of physics In the philosophical part of $n$Lab we already discuss higher algebra, homotopy theory, type theory, category theory, and higher category theory and its repercussions in philosophy. More widely, the entries on philosophy in $n$Lab would be nice to contain philosophy of mathematics in general, and of logic and foundations in particular. Similarly about philosophy of physics including interpretation and foundational issues of quantum mechanics. As it is usual for philosophy and the study of thought, it is usefully carried on via study of historical thinkers and their ideas, hence some idea-related aspects of the history of mathematics are welcome. There are many articles which are not directly philosophical, but rather essays on general mathematics, often opinion pieces on what is important and so on. Although mathematicians will often speak of their ‘philosophy’, this is not philosophy per se, but it may be relevant to an understanding of the nature of mathematics through its practice, see, for instance, development and current state of Idea of relevance of higher mathematical structures Philosophical interest in higher mathematical structures may be characterised as belonging to one of two kinds. • Metaphysical: The formation of a new language which may prove to be as important for philosophy as predicate logic was for Bertrand Russell and the analytic philosophers he inspired (see, e.g., Corfield 20). • Illustrative of mathematics as intellectual enquiry: Such a reconstitution of the fundamental language of mathematics reveals much about the discipline as a tradition of enquiry stretching back several millennia, for instance, the continued willingness to reconsider basic concepts (see, e.g., Corfield 12, Corfield 19). “Mathematical wisdom, if not forgotten, lives as an invariant of all its (re)presentations in a permanently self–renewing discourse.” (Yuri Manin) To categorify mathematical constructions properly, one must have understood their essential features. This leads us to consider what it is to get concepts ‘right’. Which kind of ‘realism’ is suitable for mathematics? Which virtues should a mathematical community possess to further its ends: a knowledge of its history, close attention to instruction and the sharing of knowledge, a willingness to admit to what is currently lacking in its programmes? Research programs in mathematics This entire subject about past research programs, paradigms in mathematics and paradigm shifts could be expanded on in the nLab. Examples include the shift from Euclidean geometry to non-Euclidean geometries in the 19th century, and the dominance of the material set theory paradigm in the 20th century and its failure with higher structures, the evolution of analytic concepts such as the differential, the integral, the real numbers, over the course of the 20th century, but there are surely others out there. Philosophical positions References and links • Hegel, Wissenschaft der Logik ( Science of Logic ) • Albert Lautman, Mathematics, ideas and the physical real, 2011 translation by Simon B. Duffy; English edition of Les Mathématiques, les idées et le réel physique, Librairie Philosophique, J. VRIN, 2006 • Michael D. Potter, Set theory and its philosophy: a critical introduction, Oxford Univ. Press 2004 • Fernando Zalamea, Filosofía sintética de las matemáticas contemporáneas, (Spanish) Obra Selecta. Editorial Universidad Nacional de Colombia, Bogotá, 2009. 231 pp. MR2599170, ISBN: 978-958-719-206-3, pdf. Transl. into English by Zachary Luke Fraser: Synthetic philosophy of contemporary mathematics, Sep. 2011. bookpage. Some excerpts here. • David Corfield, Towards a philosophy of real mathematics, Cambridge University Press, 2003, gBooks • Saunders MacLane, Mathematics, form and function, Springer-Verlag 1986, xi+476 pp. MR87g:00041, wikipedia • George Lakoff, Rafael E. Núñez, Where mathematics comes from, Basic Books 2000, xviii+493 pp. MR2001i:00013 • Yuri I. Manin, Mathematics as Metaphor: Selected Essays of Yuri Manin_, Amer. Math. Soc. 2007 • Ralf Krömer, Tool and object: A history and philosophy of category theory, Birkhäuser 2007 • Jean-Pierre Marquis, From a geometrical point of view: a study of the history and philosophy of category theory, Springer, 2008 • Ian Hacking, Why is there philosophy of mathematics at all?, Cambridge University Press 2014 • William Bragg Ewald, From Kant to Hilbert, From Kant to Hilbert: Readings in the Foundations of Mathematics, 2 vols. (original readings in English translation) • Roland Omnès, Converging Realities – Toward a common philosophy of physics and mathematics, Princeton University Press, 2005 • David Corfield, Modal homotopy type theory, Oxford University Press 2020 (ISBN: 9780198853404) • Fernando Zalamea (editor), Rondas en Sais. Ensayos sobre matemáticas y cultura contemporánea. (Essays on mathematics and contemporary culture, by Moreno, Javier; de Lorenzo, Javier; Villaveces, Andrés; Pérez, Jesús Hernando; Restrepo, Gabriel; Cruz Morales, John Alexánder; Vargas, Francisco; Oostra, Arnold; Ferreirós, José; Zalamea, Fernando; Martín, Alejandro) Universidad Nacional de Colombia, Facultad de Ciencias Humanas 2012 pdf • Glenn G. Parsons, James Robert Brown, Platonism, metaphor, and mathematics, Dialogue 43 (2004), no. 1, 47–66, MR2004k:00004 • John Baldwin, Model theoretic perspectives on the philosophy of mathematics, pdf • Yu. I. Manin, Mathematical knowledge: internal, social and cultural aspects, arXiv:math.HO/0703427; Georg Cantor and his heritage, arxiv/math.AG/0209244; Truth as value and duty: lessons of mathematics, arxiv/0805.4057 • M. G. Katz, E. Leichtnam, Commuting and noncommuting infinitesimals, Amer. Math. Monthly 120 (2013), no. 7, 631-641 arxiv/1304.0583 • M. G. Katz, Thomas Mormann, Infinitesimals as an issue in neo-Kantian philosophy of science, arxiv/1304.1027 • Mikhail Gromov, Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2, 3. (2013) pdf; Structures, Learning and Ergosystems: Chapters 1-4, 6 (2011) pdf • Jeremy Avigad, Mathematics and language, arxiv/1505.07238 • David Corfield, Narrative and the Rationality of Mathematical Practice, in A. Doxiadis and B. Mazur (eds.), Circles Disturbed, Princeton, 2012, (preprint). Some philosophical aspects of the role of category theory are touched upon in some parts of the introductory paper	{"url":"https://ncatlab.org/nlab/show/philosophy","timestamp":"2024-11-02T14:08:00Z","content_type":"application/xhtml+xml","content_length":"37311","record_id":"<urn:uuid:9bb62554-927f-42ac-954a-8eb5b42c2635>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00023.warc.gz"}
Sudo Null - Latest IT News Logistics of the action for separate collection of recyclables Instead of joining When the processes of waste collection and processing are fully adjusted in Russia, it is not easy to say, but I want to not participate in the replenishment of landfills now. Therefore, in many large cities, one way or another, there are volunteer movements engaged in particular separate collection. In Novosibirsk, such an activity is formed around the Green Squirrel campaign, in which once a month environmental townspeople bring accumulated recyclable household waste to predetermined places at a certain time. By the same time, a rented truck arrives there, which takes the collected and sorted recyclable materials to the site, from where it is redistributed between various processing enterprises. The action has existed since 2014, and since then the number of collection points for recyclables, as well as its volumes, has significantly increased. For truck routing, just a gaze was not enough, and we began to develop optimization models to minimize transportation costs. The first of these models is the subject of this article. In section 1, I will describe in detail and with illustrations the scheme for organizing the action. Further, in Section 2, the task of minimizing transport costs will be formalized as the task of routing heterogeneous vehicles fleet vehicle routing problem with time windows. Section 3 is devoted to solving this problem using a freely distributed package for solving mixed integer linear mathematical programming problems GLPK. 1. The action "Green Squirrel" Since 2014, the Living Earth initiative group has been conducting a monthly campaign to separate the collection of recycled Green Squirrel in Novosibirsk. At the time of writing, recycling with a number of reservations is accepted plastic waste labeled PET, PE, PP, PS, glass, aluminum, iron, household appliances, waste paper and more. More than 50 volunteers participate in the preparation and conduct of the action. The action is not commercial, participation in it is free and does not imply a monetary reward. The action is held at 17 points of the city, located from each other at distances covered by the car in a period of 15 to 90 minutes. One of these points in the photo: bags on the left along the wall to collect various fractions of plastic, on the right - a truck, in which everything is loaded in the future, and in the center - a volunteer with ears. All activity at the point is organized by curators who have restrictions on the time at which they are ready to fulfill their duties. When planning an action, curators report the time interval within which the action must pass at their point. There is also data on the average volumes of recyclables collected at each point. Points are organized into routes that are successively driven by a truck. Truck movements are monitored by route supervisors who monitor the operational environment and make decisions on handling special events. Are rented on a common basis within the framework of proposals for hourly rental of freight vehicles. It is not possible to compact the plastic in place, therefore the main parameter characterizing the truck is the volume of the body. Carrying capacity in our case is not a limiting factor. The main expenses of the action are connected with the payment of truck rental, therefore, its reduction is critical for the existence and development of our share, which acquires institutional significance in the sense of forming ideas about responsible consumption. Next, an approach to solving this issue will be described, based on the application of discrete optimization methods, namely, mathematical programming. 2. Formalization In constructing the mathematical model, we will remain within the framework of linear mixed-integer programs, for which understanding knowledge of class 7 algebra is sufficient. The only difficulty can be associated with the use of abstract notation and summation signs in formulas. I hope this does not scare away readers who have infrequent encounters with mathematical In the optimization model, four components can be distinguished: • the formation of groups of points that make up a separate route; • definition of a detour scheme for each of the groups; • meeting the requirements for the time of arrival of the truck at each point; • determination of the type of truck needed to service each of the routes. We consider each of the parts, introducing the necessary notation as necessary. The formation of groups of points - has an index of 0. Put Each group of points forms a separate route. Through Let the quantity Depot must enter all routes: Unfortunately, such a record creates computational problems associated with the symmetry of the admissible region. It can be eliminated by adding a number of restrictions ensuring the choice of lexicographically minimal decomposition. You can read more about this in Russian, for example, Definition of a detour scheme Routes are an alternating sequence of points and crossings between them. Formally, they all begin at one of the points of the set For points Then we require, firstly, that the truck moving along the route Secondly, the truck after arriving at the point You may notice that these restrictions allow the quantities About eliminating subcycles One way could be to introduce auxiliary non-negative quantities These ratios specify the flow from the depot to the rest of the route points. At each intermediate point, a unit of flow is absorbed, so in order for the network to have a power flow equal to the number of points minus one, it is necessary that the route be connected. Satisfying the requirements for the time of arrival of the truck at each point In other words, you must visit the points only inside the time windows indicated by the curators. Through To track the implementation of these restrictions, we need information about the time spent by the truck during stops and crossings on the route. Through We introduce non-negative variables The waiting time cannot be less than the time required for loading and equal to zero if the point does not belong to the route Arrival time at the point The arrival and departure of the truck must be within the interval indicated by the curator. Determining the type of truck required to service each of the routes. We denote the many types of trucks available for rent by We introduce variables Integer variables For each route, In accordance with the breakdown of points between routes, some routes may turn out to be trivial, that is, contain only depots. If the route At the same time, the duration of the lease should also exceed the total duration of parking and moving along the route. Add restrictions providing the property: if the truck is of type All recyclables collected at route points should fit in the back of the truck. Finally, our goal is to minimize the cost of renting trucks, which, using the designations introduced, is written as Search for a solution It is easy to verify that all the expressions involved in the optimization model are linear functions of variables, therefore, to find the exact and approximate solutions, you can use standard packages for solving mixed-integer programming problems such as , etc. We write a model for minimizing transportation costs in the GMPL language. This will allow us to use the free package for our purposes . To write code and debug the model, it will be convenient to download the , which already contains GLPK in its composition. GUSEK looks as follows: On the left we see a description of the model, and on the right there is a window for displaying information on the calculation progress, which the solver will supply after launch. Full description of the model param numOfPoints > 0, integer; #число точек param numOfTypes > 0, integer; #число типов грузовиков param numOfRoutes = numOfPoints;#максимальное число маршрутов set V := 1 .. numOfPoints; #множество точек set Vbar := V union {0}; #множество точек с пунктом разгрузки (депо) set T := 1 .. numOfTypes; #множество типов грузовиков set R := 1 .. numOfPoints; #множество маршрутов param WDL >= 0, default 8; #длительность рабочего дня param B{i in Vbar} >= 0; #начало временного окна param E{i in Vbar} >= 0; #конец временного окна param L{i in Vbar} >= 0; #минимальное время погрузки param D{i in Vbar, j in Vbar} >= 0, <= WDL; #время переезда param G{i in V}, >= 0; #объем вторсырья, м3 param C{t in T}, >= 0; #вместительность кузова param P{t in T}, >= 0; #стоимость аренды за час param U0{t in T}, >= 0; #минимальное время аренды, часов * Формирование групп точек var z{Vbar, R} binary; #равняется единице, если точка включается в маршрут, и нулю в противном случае s.t. pointToGroup 'point to group' {i in V}: sum{r in R} z[i, r] == 1; s.t. depotToGroup 'depot to group' {r in R}: z[0, r] == 1; s.t. lexMinGroups 'lexicographycally minimal division' {i in V, r in R: r <= i}: 1 - z[i, r] sum{j in V: j <= i - 1}(1 - sum{k in R: k <= r - 1} z[j, k]) sum{k in R: k <= r - 1}z[i, k] ; * Определение схемы объезда var x{Vbar, Vbar, R} binary; #равна единице, если в маршруте c номером r грузовик совершает переезд из точки i в точку j, и нулю иначе. s.t. visitPoint 'visit point' {i in Vbar, r in R}: sum{j in Vbar} x[i, j, r] = z[i, r]; s.t. keepMoving 'keep moving' {i in Vbar, r in R}: sum{j in Vbar} x[j, i, r] = sum {j in Vbar} x[i, j, r]; var f{Vbar, Vbar, R} >= 0; #Потоки, устраняющие подциклы. s.t. flowFromDepot 'flow from depot' {r in R}: sum{i in V} f[0, i, r] == sum{i in V} z[i, r]; s.t. flowAlongActiveArcs 'flow along active arcs' {i in Vbar, j in Vbar, r in R}: f[i, j, r] <= numOfPoints * x[i, j, r]; s.t. flowConservation 'flow conservation' {i in V, r in R}: sum{j in Vbar} f[j, i, r] == sum{j in Vbar} f[i, j, r] + z[i, r]; var a{i in V} >= 0; #время прибытия грузовика на точку var w{i in Vbar, r in R} >= 0; #время нахождения грузовика на точке s.t. wait 'wait'{i in Vbar, r in R}: w[i, r] >= L[i] * z[i, r]; s.t. dontWait 'dont wait'{i in Vbar, r in R}: w[i, r] <= (E[i] - B[i]) * z[i, r]; s.t. arrivalTime 'arrival time' {i in V, j in V}: a[i] + sum{r in R}w[i, r] + D[i,j] <= a[j] + 3 * WDL * (1 - sum{r in R} x[i, j, r]); s.t. arriveAfter 'arrive after' {i in V}: a[i] >= B[i]; s.t. departBefore 'depart before' {i in V}: a[i] + sum{r in R}w[i, r] <= E[i]; * Определение типа грузовика, необходимого для обслуживания каждого из маршрутов var y{t in T, r in R}, binary; #равные единице, если грузовик типа t назначается на обслуживание маршрута с номером r, и нулю иначе. var u{t in T, r in R}, integer, >= 0; #время аренды грузовика типа t, обслуживающего маршрут с номером r. s.t. assignVehicle 'assign vehicle' {r in R}: sum{t in T} y[t,r] == 1; s.t. rentTime 'rent time' {r in R, t in T}: u[t, r] >= sum{i in V, j in Vbar}D[i, j] * x[i, j, r] + sum{i in Vbar}w[i, r] - WDL * (1 - y[t, r]); s.t. minRentTime 'minimal rent time' {r in R, t in T}: u[t, r] >= U0[t] * (y[t, r] - sum{i in V}z[i, r]); s.t. noRent 'no rent' {t in T, r in R}: u[t, r] <= WDL * y[t, r]; s.t. fitCapacity 'fit capacity' {r in R}: sum{i in V} G[i] * z[i, r] <= sum{t in T} C[t] * y[t, r]; minimize rentCost: sum{t in T, r in R} P[t] * u[t, r]; * Вывод решения на экран printf{i in V, r in R} (if 0.1 < z[i,r] then "point %s to group %s\n" else ""), i, r, z[i,r]; printf{r in R, i in Vbar, j in Vbar} (if 0.1 < x[i, j, r] then "route %s: %s -> %s\n" else ""), r, i, j; printf{i in V} "point %s arrive between %s and %s (actual = %s)\n", i, B[i], E[i], a[i]; For a quick start, I’ll also add data taken from my head prepared for use in the model: Input data param numOfPoints := 9; #число точек param numOfTypes := 6; #число типов грузовиков param : B E L := 9 0 8 1; param D default 0 : 0 1 2 3 4 5 6 7 8 9 := 0 . . . . . . . . . . 1 0.1 0.3 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 2 0.3 0.2 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 3 0.4 0.3 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 4 0.4 0.4 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 5 0.1 0.2 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 6 0.5 0.5 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 7 0.3 0.2 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 8 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 9 0.5 0.2 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1; param G := 9 1; param : C P := 6 35 800; param U0 default 2; #минимальное время аренды, часов After copying the model code to a file with the name, for example, model.mod, and the input data to the data.dat file, everything is ready to run. We set a limit on the calculation time of 100 seconds (this is done using the key --tmlim [time in seconds]), transfer the path to the file with input data (using the key -d [file path]), and press F5. If successful, messages will appear in the window on the right, and after a hundred seconds we will have the best solution that GLPK managed to find in the allotted time. In the blue text, we are interested in the meaning after the inscription "mip =". As you can see, it decreases from time to time. All new solutions are in the process of work, the value of transport costs at the best of which is displayed in this column (there are 14700 so far). The next number is the lower bound for the best existing one, i.e. solution. Initially, the estimate is significantly underestimated, but is refined over time, that is, increases. The values on the left and on the right are converging, and the relative gap between them at the time of the screenshot is 54.1%. As soon as this number becomes zero, the algorithm will prove that the best solution found is the best possible. It is not always justified to wait for this event in practice, and not only because it is a long time, but it is important to make a reservation that, as a rule, a good solution is relatively quick, and the main time costs are associated with the refinement of the estimate required to prove the best possible. Instead of a conclusion Routing problems are extremely computationally complex, and with the increase in the number of points that need to be visited, the time required for a solver to find a solution and prove its optimality is growing rapidly. However, for fairly small examples, in a reasonable amount of time, the solver is able to build a successful set of routes and can be used to support decision-making. Analysis of the routing options proposed by the model helped us discover significant opportunities for cost reduction, and this is critical for the existence and development of the stock. Our further efforts went towards work with uncertainty in the volumes of recyclables collected at the points. We are developing a number of stochastic programming models for making strategic and operational decisions in truck routing. If the topic turns out to be relevant and arouses interest, I will write about this too, because soon we all will have to significantly more thoroughly dive into environmental issues, which is what I wish us success in.	{"url":"https://sudonull.com/post/26740-Logistics-of-the-action-for-separate-collection-of-recyclables","timestamp":"2024-11-12T15:37:49Z","content_type":"text/html","content_length":"43884","record_id":"<urn:uuid:182d2869-a7ab-445a-9637-4d0fbe1a6032>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00270.warc.gz"}
3.11 Factors If we multiply $4$4 by $14$14, we get $56$56, so $4$4 and $14$14 make a factor pair of $56$56. Which of the following options is also a factor pair of $56$56? Complete the table below, listing all factor pairs of the number $22$22. Complete the table below, listing all factor pairs of the number $86$86. Complete the table below, listing all factor pairs of the number $12$12. Get full access to our content with a Mathspace account	{"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1072/topics/Topic-20721/subtopics/Subtopic-269594/?activeTab=interactive","timestamp":"2024-11-11T19:46:05Z","content_type":"text/html","content_length":"297114","record_id":"<urn:uuid:23b0a706-5508-41eb-acd4-fe70ff34bbbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00290.warc.gz"}
round(x: array, /) array¶ Rounds each element x_i of the input array x to the nearest integer-valued number. For complex floating-point operands, real and imaginary components must be independently rounded to the nearest integer-valued number. Rounded real and imaginary components must be equal to their equivalent rounded real-valued floating-point counterparts (i.e., for complex-valued x, real(round(x)) must equal round(real(x))) and imag(round(x)) must equal round(imag(x))). x (array) – input array. Should have a numeric data type. out (array) – an array containing the rounded result for each element in x. The returned array must have the same data type as x. Special cases For complex floating-point operands, the following special cases apply to real and imaginary components independently (e.g., if real(x_i) is NaN, the rounded real component is NaN). □ If x_i is already integer-valued, the result is x_i. For floating-point operands, □ If x_i is +infinity, the result is +infinity. □ If x_i is -infinity, the result is -infinity. □ If x_i is +0, the result is +0. □ If x_i is -0, the result is -0. □ If x_i is NaN, the result is NaN. □ If two integers are equally close to x_i, the result is the even integer closest to x_i. Changed in version 2022.12: Added complex data type support.	{"url":"https://data-apis.org/array-api/latest/API_specification/generated/array_api.round.html","timestamp":"2024-11-06T18:58:48Z","content_type":"text/html","content_length":"23396","record_id":"<urn:uuid:91f90abd-55a1-498a-9319-e1e8ba72c77b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00352.warc.gz"}
7 Examples of Gambling Math in Action - July 9, 2022 7 Examples of Gambling Math in Action The math behind gambling is endlessly fascinating. In fact, without the branch of mathematics called “probability”, we wouldn’t even have gambling—or at least we wouldn’t be able to talk about it intelligently. 온라인카지노 Few bets are fair bets. One side almost always has an edge over the other. Being able to determine that edge is a critical part of being an educated gambler. This post starts with an overview of what probability is and how it’s calculated, then it continues with 7 examples of how it’s used in practical applications. Probability concerns itself with measuring how likely it is that certain things will happen. For purposes of this post, I’ll call those things “events”. You probably use probability to talk about possible events without even knowing it. Probably the most common expression of probability happens with percentages, especially when you’re watching the nightly news. When the meteorologist says that there’s a 50% chance of thunderstorms tomorrow, she’s telling you what the probability is that there will be rain. And most people understand that 50% means that half the time it’s going to rain, and half the time it’s not. A probability is just a number that describes how likely an event is. And that number is always a number between 0 and 1. Something with a probability of 0 won’t ever happen. Something with a probability of 1 (which is also 100%) will always happen. You can express probabilities as percentages, but that’s not the only way to express a probability. You can also express it as a fraction. 50% is the same thing as ½. You can also express a probability as a decimal. 50% is the same thing as 0.5. Probability can also be expressed in odds format. In this case, 50% is the same thing as 1 to 1, or even odds. Each of those ways of expressing probability is useful in different situations. Stating a probability as odds is especially useful when comparing the payoff of a bet with the odds of winning that Calculating probability is actually pretty simple, too. For a single event, you look at the number of ways that event can happen versus how many ways things might turn out total. You put the single event on top of the fraction, and you put the total number of potential events as the bottom of the fraction. Of course, if you have any math experience at all, you know that you can use division to turn a fraction into a decimal or a percentage. If you want to calculate the probability of multiple events, you either multiply or add depending on whether you want to know if multiple events will happen or if you want to know the odds of a certain number of events happening. The key words to look for in such a problem are “and” and “or”. If you want to know the probability that event A will happen AND event B will happen, you multiply the probability of each. If you want to know the probability that event A will happen OR event B will happen, you add the probability of each. The following examples will show how these probability calculations happen time and again in the gambling world. Roulette Math Roulette is a simple game, and it’s a great example of probability in action. An American roulette wheel has 38 possible events, numbered 0, 00, and 1-36. The 0 and the 00 are green. Half of the other numbers are black, and half of them are red. With this information, you can calculate the probability of just about any outcome or combination of outcomes. You can compare those probabilities with the payoffs for the bet to see if one side has an edge, and if so, how much that edge is. Let’s start by thinking about some of the more common bets in roulette—the outside bets. These bets are on odd/even, high/low, or red/black. They all pay out at even odds. You bet $1 on one of these outcomes, you win $1 if you win. At first glance, that sounds like a fair enough bet, but when you look at these bets a little more closely, the house has a distinct advantage. Here’s why: Suppose you bet on black. There are 18 numbers on the wheel that are black, but there are 20 numbers on the wheel that are not. (18 of the numbers are red, and 2 more numbers are green.) So out of 38 possible outcomes, only 18 of them win your bet. That makes the probability 18/38. It’s probably easiest to understand this bet by converting it into a percentage, 47.37%. So 52.63% of the time, the casino will win this bet, and the rest of the time, you will. It’s clear to see how if you play this game long enough, eventually the casino will win all your money. You can even calculate the amount of each bet the casino will win over the long run—this number is called the house edge. Here’s how you do it: Assume that you make 100 bets and that you see the mathematically expected results. (That never happens in real life, but if you play long enough, the actual results will start to resemble the expected results.) In this case, you will win $47.37, but you’ll lose $52.63. That’s a net loss of $52.63 – $47.37, or $5.26. Since you bet $100 on those 100 wagers, you lost an average of 5.26% of each bet. And that’s the house edge. As it turns out, that’s the house edge for all the bets at the roulette table (except for one). In a sense, the green 0 and the green 00 are where the house gets its edge. The payouts for all the bets on the table would offer neither side an edge if those numbers weren’t on the wheel. But they ARE on the wheel. And that makes all the difference. The Math Behind a Coin Toss An even simpler example of probability in action is a coin toss. Most people don’t actually place wagers on the outcomes of a coin toss, but they could. And depending on the payout structure, one side might or might not have an edge over the other side. Here’s the simplest version of this calculation. You want to know the probability that you’ll get heads on a coin toss. Since there are 2 potential events, and since only 1 of them is heads, your probability is ½, or 50%. In cases where you want both sides to have an even shot at winning something, you’ll flip a coin. This is how they determine who gets to kick off during a football game, for example. I should point out that there’s no advantage to being the one to call heads or tails. The probability is the same, and I don’t believe in psychic phenomena. I’ve never seen any evidence that anyone has any kind of precognitive ability that would improve their chances of predicting the outcome of a coin toss. But let’s try a more interesting calculation. Let’s say we want to know the probability of getting heads twice in a row. That means you want to know the probability of getting heads on the first flip AND the probability of getting heads on the second flip. Remember I said earlier that if we’re using the word “and” in the problem, we multiply. In this case, we’re multiplying ½ by ½, which is ¼. Or we could call it 0.5 X 0.5 and get 0.25. Either of those ways can be expressed as 25%. Another way to look at this is to look at the total number of outcomes when you toss a coin twice in a row: • You could get heads on the first toss and heads on the second toss. • You could get tails on the first toss and tails on the second toss. • You could get heads on the first toss and tails on the second toss. • You could get tails on the first and heads on the second toss. Those are literally the only 4 outcomes, but only 1 of them is the outcome you were solving for. That’s ¼, or 25%, which is what we’d determined earlier. Suppose you wanted to create a simple gambling game based on the outcome a coin toss. Let’s say you’re running a back room casino in a bar or something. You might have a game where you toss a coin, and so does the dealer. If you get heads and the dealer gets tails, you win. If the dealer gets tails, and you get heads, then the dealer wins. But if you both get heads or both get tails, you have to put up another coin in order to get to toss the coins again. The catch is that the dealer does NOT have to put up another coin. If you win this second toss, you win a coin, but if you lose it, you lose both coins that you put up. It’s pretty clear in this example how the casino has an edge, right? Poker Math I could spend the rest of this post talking about poker math. But I’ll try to limit it to just this bullet point. Anyone who knows anything about poker knows that you have just as good a chance of getting a better hand as I do. We’re both getting cards from the same 52 card deck, after all. It’s what you do with those cards after that make a difference. Let’s suppose that you’re playing 5 card draw and you’re dealt a hand with 4 cards to a flush in it. You’re going to discard a card and hope to draw to that flush. What is the probability that you will succeed? There are 47 cards left in the deck. 9 of them are of the suit you need. (There are 13 cards in each suit, and 4 of them are already in your hand.) So your probability of getting the card you need is 9/47, or 19.1%. That’s almost 1 in 5, or 20%. If you assume that you have to make this hand in order to win the pot, you can calculate how much money needs to be in the pot in order for you to profitably calla bet. Let’s suppose that there is $10 in the pot, and it costs you $1 to stay in and draw that extra card. If you win, you’ll win 10 to 1 on a 4 to 1 draw. You’ll lose almost 80% of the time, but you’ll win so much when you do win that it will make up for it and give you a tidy profit. In fact, let’s do the same calculation we did above, where we assume that you do this 100 times in a row. You’ll lose $80.90, but you’ll win $190.10, for a profit of $109.20. These are excellent pot On the other hand, if there were only $3 in the pot, and it cost you $1 to get in, you wouldn’t get a big enough payout to make this a profitable bet. You’d still lose $80.10, but you’d only win $57.30, for a net loss of $22.50. Of course, in a real poker game, you’d have other probabilities to take into account. For example, you might raise in this situation, hoping to scare your opponents out of the pot. You have to estimate the probability that this tactic will work when you try this. You can add that to your expected value. This is where reading other players becomes important. Some people think that reading people is all about gauging what they’re going to do 100% of the time. But the reality is that you make educated guesses about their likelihood of doing something. If you estimate that your opponent will fold to your bluff 50% of the time, then that makes a big difference to your strategy. 바카라사이트 Video Poker Math Video poker is a little bit like poker and a little bit like slot machines, but it’s like nothing so much as it’s like itself. Most of the math, though, is similar to the math of traditional poker. The difference is that you have an exact payoff you can expect when you achieve a certain hand. You don’t have to worry about what your opponents have. For example, if you have a pair of jacks in a poker game, and your opponent also has a pair of jacks, you could wind up in a situation where you tie and split the pot. But in a Jacks or Better video poker game, you get paid even odds any and every time you get a pair of jacks or higher. And you don’t get a higher payout for a pair of queens or a pair of kings. For purposes of these payouts, all 3 hands are the same, even though there’s a clear hierarchy among those 3 hands in a real poker game. Video poker is based on draw poker, so every time you get a hand, you get to decide which cards to keep and which ones to throw away. You compare the probability of making certain hands with their payoffs in order to decide which decision has the best expected value. Here’s an example: The best possible hand you can get in most video poker games is a royal flush, which pays off at a whopping 800 to 1. (I’m assuming you’re making the max coin bet—if you don’t, you’re only getting a 250 to 1 payoff. But you should never play for less than max coins.) But you can win even odds with a pair of jacks or higher. That’s clearly a much lower payoff. But suppose you have to choose between those 2 options? Let’s say you have the ace of hearts, the king of hearts, the queen of hearts, and the jack of hearts. But your 5th card is the king of spades. You have a pair of kings. You can keep that and have a 100% chance of getting an even money payoff. Or you can throw away the king of spades and try to get the royal flush. Only 1 card of the 47 remaining cards will make your hand, which is a slightly better than 2% chance of success. What happens over 100 perfect iterations? 98 times you lose your bet. But twice you get 800 coins. That’s 1600-98, or 1502. Divided by 100 bets, that’s 15.02 per bet that you won. In the other case, you win 100 times, but you only win 100 coins total. Would you rather average $15 in winnings per bet, or $1 in winnings per bet? Of course, this example ignores the possibility that you could draw to another random winning hand, but that has a more or less equal probability with both decisions. We’ll just assume that it evens On the other hand, if you only had 3 cards to a royal flush, the odds of hitting your hand get much smaller. 2% X 2% is 0.04%. With odds like that, you’ll need a lot more than an 800 to 1 payoff to make that decision worthwhile. But no matter what hand you are dealt initially, you have one decision which has a higher expected value than any of the others. That expected value is determined by looking at all the possible moves in that situation and the likelihood that each of them will result in a particular payoff amount. Craps Math Craps is an interesting exercise in probability because it’s a great example of a bell curve. That’s when some results happen so seldom that the drawing of the curve is low on either end, but the odds of the results in the middle happening are much higher. Here are the possible outcomes when rolling a pair of dice: • 2 – 1 +1 – Only one possible way of getting this total. • 3 – 2+1 or 1+2 – Only 2 possible ways of getting this total. • 4 – 3+1, 2+2, or 1+3 – Only 3 possible ways of getting this total. • 5 – 4+1, 3+2, 2+3, or 1+4 – Only 4 possible ways of getting this total. • 6- 5+1, 4+2, 3+3, 2+4, 1+5 – Only 5 possible ways of getting this total. • 7 – 6+1, 5+2, 4+3, 3+4, 2+5, 1+6 – Only 6 possible ways of getting this total. • 8 – 6+2, 5+3, 4+4. 3+5, 2+6 – Only 5 possible ways of getting this total. • 9 – 6+3, 5+4, 4+5, or 3+6 – Only 4 possible ways of getting this total. • 10 – 6+4, 5+5, or 4+6 – Only 3 possible ways of getting this total. • 11 – 6+5 or 5 +6 – Only 2 possible ways of getting this total. • 12 – 6+6 – Only one possible way of getting this total. You only have 11 possible totals, but you have a total of 36 different outcomes. Knowing this, you can divide the number of ways of achieving each total by 36 in order to determine the probability of getting that total. • So getting a total of 2 or 12 has a probability of 1/36. • 3 or 11 has a probability of 2/36, or 1/18. • 4 or 10 has a probability of 3/36, or 1/12. • 5 or 9 has a probability of 4/36 or 1/9. • 6 or 8 has a probability of 5/36. • 7 has a probability of 6/36, or 1/6. So your most likely outcome is a total of 7, but that still only happens 1 time out of 6. But you can bet on any of these totals at various times in the game. You can compare the payoffs on these bets with the odds of winning to determine the house edge on each of those bets. For example, you can make a place bet on any 8 or any 6 and get a payoff of 7 to 6 if you win. But the odds of winning that bet are 5/36. That can be converted into a percentage, and we can calculate the house edge for that bet. The odds of winning this bet are 13.89%. Place this bet 100 times, and you will win 13.88 bets with winnings of $1.17 each time (7 to 6). That’s $16.24 in winnings. But you lose 86.12 times, losing $1 each time, for losses of $86.12. You’ve lost way much more than you’ve won over these 100 bets–$69.88. That makes the house edge 6.99% on this bet, which is almost 7%. That’s worse than roulette with its 5.26% edge. Luckily, many of the bets on the craps table have a much lower house edge. Blackjack Math My favorite kind of gambling math relates to blackjack. It’s such an elegant game, and it’s also one of the only casino games where a skilled player can get an edge. What’s so interesting about the game is that it has a memory. Here’s what I mean: When you play roulette, the odds are the same on every spin of the wheel. The outcome of one spin has no effect on the odds of the outcome of the next spin. There are 38 possibilities every time you spin the wheel, and each of them is equally as likely as the others. But if you eliminated a slot on the wheel once it got hit, you’d wind up with odds that changed on every spin. Here’s an example: You bet on black. The probability of winning that bet is 18/38. You win. The croupier (the roulette dealer) leaves the ball in that slot, so it can’t be landed on again. You bet on black again. This time the probability of winning is only 17/38, because one of the options has been removed. This is exactly what happens every time a card is dealt in blackjack. One of the 52 options is no longer available to be dealt in subsequent rounds. This continues until the dealer reshuffles the pack of cards. Of course, in a game with a continuous shuffling machine, the odds stay the same no matter what. But most games are still dealt without the benefit of such a machine. In these games, you can keep rough track of which cards have been dealt and raise your bets when you have a better chance of winning more money. Here’s how that works: A “natural”, or a “blackjack”, pays off at 3 to 2. That’s a 2 card hand that totals 21. There are only 2 values of cards which can result in such a hand—the aces, which count as 11, and the 10, J, Q, and K, each of which counts as 10. If all of the aces in a deck are gone, it’s impossible to get a blackjack. You just can’t do it. Every time a 10 gets dealt, your chances of getting a blackjack decrease, too. But at the same time, every time a lower ranked card gets deal, like a 2, 3, 4, 5, or 6, the odds improve a little bit in the player’s favor. So a card counter will use a system to keep rough track of the ratio of high cards to low cards. They count the low cards as +1 and the high cards as -1. If and when the count gets high on the positive side, the counter knows he has a better than average chance of getting that 3 to 2 payout. So he raises his bets accordingly. The higher the count, the more he bets. He lowers his bet when the count is 0 or negative. There’s a lot more to counting cards than that, but those are the basics. And they’re rooted in math. Sports Betting Math Most bookmakers require you to risk $110 in order to win $100, but that’s not all they do. They also handicap teams by giving them points or taking them away. The goal of this handicapping is to make a bet on either side a 50/50 proposition. Since these sports bets don’t pay off at even odds, a 50/50 proposition is profitable for the bookmaker but not the player. But the bookmakers aren’t always right when they set the lines. And they don’t always leave the lines the way they are. A bookmaker’s goal is to get an equal amount of action on either side of an event. They do this so that they can pay off the winning bets with the losers’ money. That extra $10 that the losers bet is how they prefer to make their profit. But what if they don’t get an equal amount of bets on each side? Most bookmakers move the line in order to stimulate action on the other side. Sharp sports bettors—those who know how the business work—know that it’s usually best to bet against the public. Here’s an example of how this works: The Washington Redskins are playing the Dallas Cowboys, and they’re favored by 7 points. This means that before paying off a bet on the Redskins, the bookmaker subtracts 7 from their score. They set this line early in the week, but they don’t get nearly as many bets on the Cowboys as they expect. So they move the line to 7.5, which is meant to encourage more action on the other side. A smart bettor is going to bet against the public in this situation, because the public is usually wrong. The really interesting effect of the vigorish on a sports bettor is what it does to the required winning percentage just to break even. If you’re right 50% of the time and wrong 50% of the time, you’ll lose money. You’re losing $110 half the time, and you’re only winning $100 the other half the time. If you can bet on the right side a little over 53% of the time, you can break even and even make a tiny profit. If you can get over 55% and start nearing 60%, you’re on your way to becoming a world class sports bettor. You can make 6 figures a year with a win rate like that, but you need to have enough money in your bankroll to be able to weather any losing streaks you might run into. Losing streaks in the short term are inevitable, too. That’s just the nature of a game of chance. Also, the handicappers who work for the bookmakers are almost always right. In order to make a profit betting at sports, you have to be adept at finding profitable situations. This means outthinking the handicappers and the bookmakers most of the time. Finding value in sports betting is an endlessly interesting topic. As you can see from these 7 examples, it’s unusual that anyone ever gets a fair bet. Someone almost always gets an edge. Figuring out who has the edge and by how much is just a matter of comparing the odds of winning for each side and the payouts for winning those bets. Casinos always have an edge over the players. I can only think of 2 bets in a Las Vegas casino which offer fair odds—the double up bet in video poker and the odds bet in craps. But you can find occasional bets in Vegas casinos where the player has an edge, but these are the exceptions, not the rules. When you’re playing games like slots, craps, and roulette, there’s really not much you can do to even out the odds. Some people claim that they can affect the outcome of a roll of the dice, but I’m On the other hand, if you’re a skillful blackjack player or a skillful video poker player, you might be able to get a small edge over the casino. If you’re counting cards as a blackjack player, most casinos will refuse to let you continue to play, though. And they’re pretty good at catching advantage players now. Skilled poker players and sports bettors can get the odds in their favor, but they still have to be skillful enough to overcome a house edge of sorts. In poker games, the cardroom hosting the games charges a percentage of each pot as rent for the table—this is called the rake. When betting on sports, you have to bet $110 to win $100. That extra $10 you have to risk on every bet is called the But no matter what betting activity you choose, you’ll enjoy it more if you have a clear understanding of the math behind the game and your bets. That’s why I write posts examining the math behind It’s worth it to try to get an edge, but it’s impossible to get an edge if you don’t have at least a rudimentary understanding of gambling math. Seeing it in action helped me a lot when I got And if you have any aspirations of gambling professionally, understanding these examples is a must. 온라인카지노 Leave a Comment	{"url":"https://yhn876.com/7-examples-of-gambling-math-in-action/","timestamp":"2024-11-09T14:09:22Z","content_type":"text/html","content_length":"148321","record_id":"<urn:uuid:e7b577a6-072b-404f-b7e9-d0093a0ae8e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00738.warc.gz"}
Who was Ludwig Boltzmann? Everything You Need to Know Ludwig Boltzmann Biography Birthday: February 20, 1844 (Pisces) Born In: Vienna Ludwig Eduard Boltzmann was an Austrian physicist, who was known for his work in statistical mechanics. Coming from a middle-class family, Ludwig was helped by his mother in his scientific endeavours, as he lost his father at a tender age. Initially he was given private tuitions at home and later he attended high school in Linz. Ludwig studied Physics in the University of Vienna and was mentored by the great minds of the time like Josef Stefan and Andreas von Ettingshausen. Under the guidance of Stefan, he received his PhD and became a lecturer. Ludwig taught at Graz, Heidelberg and Berlin and studied under Bunsen and Helmholtz. During his time in Graz, he met his wife Henriette. Boltzmann was known for his extreme mood swings, which also significantly influenced the direction of his career. His work on statistical mechanics was mainly based on the theory of probability, and was closely associated with the Second Law of Thermodynamics. Some of his theories were far ahead of his time, which often led to extreme opposition by his contemporaries. During his visits to USA, he lectured on applied mathematics, but he did not realize that new discoveries related to radiation would help him to substantiate his theories. Eventually, his desperation and declining mental condition drove him to suicide when he was holidaying with his family.	{"url":"https://www.thefamouspeople.com/profiles/ludwig-boltzmann-6478.php","timestamp":"2024-11-04T11:16:57Z","content_type":"text/html","content_length":"201281","record_id":"<urn:uuid:a85a2939-8a59-4115-aa98-73ccd90157e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00015.warc.gz"}
Starting Multiplication Worksheets Math, particularly multiplication, creates the keystone of numerous academic disciplines and real-world applications. Yet, for numerous learners, grasping multiplication can posture an obstacle. To resolve this hurdle, teachers and moms and dads have actually welcomed an effective tool: Starting Multiplication Worksheets. 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Are Starting Multiplication Worksheets appropriate for every age groups? Yes, worksheets can be customized to different age and skill degrees, making them versatile for various learners. How typically should trainees practice utilizing Starting Multiplication Worksheets? Constant practice is key. Routine sessions, preferably a few times a week, can produce significant enhancement. Can worksheets alone improve mathematics skills? Worksheets are a valuable device however needs to be supplemented with diverse discovering techniques for detailed ability growth. Are there on-line systems providing cost-free Starting Multiplication Worksheets? Yes, numerous educational web sites use open door to a vast array of Starting Multiplication Worksheets. How can moms and dads sustain their children's multiplication method in your home? Motivating constant technique, supplying aid, and creating a favorable understanding setting are useful steps.	{"url":"https://crown-darts.com/en/starting-multiplication-worksheets.html","timestamp":"2024-11-12T07:10:12Z","content_type":"text/html","content_length":"28965","record_id":"<urn:uuid:ea386992-2187-43ae-86f0-91ea7bbd01e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00674.warc.gz"}
Lesson 13 Center Day 2 (optional) Warm-up: How Many Do You See: Place Value (10 minutes) The purpose of this How Many Do You See is for students to use what they know about place value representations to describe and compare the images they see. In the synthesis, students describe how the number of blocks stays the same (6 of one unit, 4 of the other), but the value that the blocks represent changes dramatically. When students connect these differences to differences in the place value of the digits in the three-digit numbers the diagrams represent, they look for and make sense of structure (MP7). • Groups of 2 • “How many do you see? How do you see them?” • Flash the image. • 30 seconds: quiet think time • Display the image. • “Discuss your thinking with your partner.” • 1 minute: partner discussion • Record responses. • Repeat for each image. Student Facing How many do you see and how do you see them? Activity Synthesis • “How are these images the same? How are they different?” (They each show 6 of a unit and 4 of a unit. They each have one unit that has 0. They are different because the size of the unit is different. They each represent a different number.) • “Which image represents the greatest value? How do you know?” (604 is the greatest because it has 6 hundreds.) Activity 1: Introduce Get Your Numbers in Order, Three-digit Numbers (15 minutes) The purpose of this activity is for students to learn stage 2 of the Get Your Numbers in Order center. Students use their understanding of relative magnitude to order three-digit numbers. They take turns placing numbers on the board and must make sure that the numbers across the board go from least to greatest. If a number cannot be placed on the game board students say “pass” and get 1 point. Then it is their partner’s turn. The player with the fewest points when all the boxes on the board are filled is the winner. Students should remove the cards that show 10 before they start. Required Materials Materials to Gather Materials to Copy • Get Your Numbers in Order Stage 2 Gameboard • Groups of 2 • Give each group a set of number cards, a game board, and a dry erase marker. • “We are going to learn a new way to play the Get Your Numbers in Order center.” • “Let’s play one round together. You can all be my partner.” • Choose three cards, make a three-digit number, and place it on the board. • Invite a student to draw three cards and consult with the class on what number to create and where to place it on the game board. • Continue taking turns to complete a round. Share thinking about where to place numbers. • “Now you will play with a partner.” • 8–10 minutes: partner work time Activity Synthesis • “How did you decide where to place your numbers on the game board?” Activity 2: Centers: Choice Time (20 minutes) The purpose of this activity is for students to choose from activities that focus on place value in three-digit numbers. Students choose from any stage of previously introduced centers. • Get Your Numbers in Order • Greatest of Them All • Mystery Number Required Preparation Gather materials from previous centers: • Get Your Numbers In Order, Stage 2 • Greatest Of Them All, Stage 2 • Mystery Number, Stage 2 • “Now you will choose from centers we have already learned. One of the choices is to continue with Get Your Numbers in Order.” • Display the center choices in the student book. • “Think about what you would like to do first.” • 30 seconds: quiet think time • Invite students to work at the center of their choice. • 8 minutes: center work time • “Choose what you would like to do next.” • 8 minutes: center work time Student Facing Choose a center. Get Your Numbers in Order Greatest of Them All Mystery Number Activity Synthesis • “What did you like about the activities you worked on today?” Lesson Synthesis “Tell your partner one thing you were working on during centers today. How did the activity you chose help you work on it?”	{"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-2/unit-5/lesson-13/lesson.html","timestamp":"2024-11-07T23:05:43Z","content_type":"text/html","content_length":"91750","record_id":"<urn:uuid:549bde7a-1985-47d0-a0dd-99c22858ec7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00032.warc.gz"}
Isaiah Anthony \| Algebra Tutor on HIX Tutor Isaiah Anthony Naval Postgraduate School Algebra teacher \| Experienced educator in USA I hold a degree in Algebra from the esteemed Naval Postgraduate School. My passion lies in unraveling the mysteries of numbers and equations, guiding students to master this foundational subject. With a solid foundation in advanced mathematical concepts, I simplify complex theories into digestible lessons. I believe in fostering a deep understanding of algebra's practical applications, equipping learners with valuable problem-solving skills. Whether it's tackling equations or exploring abstract algebra, I am here to illuminate the path to mathematical proficiency. Let's embark on this journey together towards mathematical clarity and success.	{"url":"https://tutor.hix.ai/tutors/isaiah-anthony","timestamp":"2024-11-14T03:41:03Z","content_type":"text/html","content_length":"563387","record_id":"<urn:uuid:af6c2767-0786-4ea6-8472-2281fd5c30a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00799.warc.gz"}
Trigonometric Functions Mastery of Trigonometric Values for Key Angles Mastery of trigonometric values for key angles—0, π/6, π/4, π/3, and π/2 radians (0, 30, 45, 60, and 90 degrees)—is essential for proficiency in trigonometry. These values can be memorized using a systematic approach, such as constructing a table based on the patterns of the square roots of the numbers 0 through 4, divided by 2, for sine and cosine. The tangent values are then determined by the ratio of sine to cosine for each angle. It is critical to recognize that the tangent of π/2 radians is undefined, which corresponds to the vertical asymptotes in its graph. The Role of Inverse Trigonometric Functions Inverse trigonometric functions—arcsine (sin⁻¹), arccosine (cos⁻¹), and arctangent (tan⁻¹)—are indispensable for determining an angle when a trigonometric ratio is known. These functions are the inverses of their corresponding primary functions and are graphically distinct, with arcsine and arccosine producing S-shaped curves and arctangent producing a curve that extends indefinitely in both directions, horizontally approaching π/2 and -π/2 radians. These functions are key in solving trigonometric equations and in applications where the angle must be derived from a known ratio. Understanding Reciprocal Trigonometric Functions The reciprocal trigonometric functions—cosecant (csc), secant (sec), and cotangent (cot)—are defined as the reciprocals of sine, cosine, and tangent, respectively. Cosecant is the ratio of the Hypotenuse to the Opposite side, secant is the ratio of the Hypotenuse to the Adjacent side, and cotangent is the ratio of the Adjacent side to the Opposite side. These functions are particularly useful in scenarios where the primary trigonometric functions are not as convenient to use, such as in certain integrals and when dealing with specific geometric configurations. Comprehensive Insights into Trigonometric Functions Trigonometric functions form an integral part of mathematical education, offering a framework for understanding the relationships within right-angled triangles and the properties of periodic phenomena. The mnemonic SOH CAH TOA is a foundational tool for recalling the basic trigonometric ratios. Graphical representations of these functions provide a visual understanding of their periodicity and behavior. Familiarity with the values of trigonometric functions at common angles is crucial for efficient problem-solving. The study of inverse and reciprocal functions further broadens the scope of trigonometry, enabling the calculation of angles from known ratios and the exploration of relationships that extend beyond the primary trigonometric functions. A thorough grasp of these concepts is vital for students pursuing advanced studies in mathematics, physics, engineering, and other related disciplines.	{"url":"https://cards.algoreducation.com/en/content/AEYTEHL5/trigonometric-functions-basics","timestamp":"2024-11-06T17:43:36Z","content_type":"text/html","content_length":"195736","record_id":"<urn:uuid:562576b6-afed-4259-b91f-d03eab0c41d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00764.warc.gz"}
The game of fund management National Day experienced a capital management game. The name of the game is guessing the red and black ball, and the number on the ball is 1- 10. Rules of the game: The game has 100 balls, of which 60 are red balls and 40 are black balls. This game has been played 50 times. Principal per person 10000 yuan. Draw red balls to make money, draw black balls to lose money. The number on the ball is between 1 and 10. Draw a red ball to make money, and the amount of money you make at a time is three times your investment. If you draw a bonus of 3, you will earn 3 times the principal of the bet (for example, 100 yuan), and you will earn 3X 100 yuan =300 yuan. If you draw a black ball and lose money, the amount of a single loss is three times your investment. If a black ball 3 is drawn, it will lose -3 times the bet principal (such as 100 yuan), and the loss-3x100 yuan = 300 yuan. Rule 1: The winning rate of this game is 60%, and it is a trading system that can be profitable for a long time. Detailed rule 2: The minimum capital for each bet is 1 yuan, and the maximum capital is not allowed to exceed your hand. If you have 65,438+00,000 yuan, you cannot bet 65,438+065,438+0000 yuan. Rule 3: After the game is over, the profit is less than 12000 yuan, and it fails. Rule 4: If you lose money in the game, you will go bankrupt and watch the game. This is a game with a winning percentage of 60%, which is a system with positive expectations. That is to say, your trading technology can bring you the profit of trading technology. In theory, it can bring you an annual rate of return of 20%-25% in a unit trading time (such as one year), which should be very good. Everyone in the game adopts completely different methods and strategies, and the results are different. About13 people lost all their money in the middle of the game. Stick to the end, more than 70% of losses, 30% of profits, and some funds doubled. The same system draws the same card (equivalent to the same transaction result every time), but the overall result is different, which can explain the importance and effectiveness of fund management. In fact, in the market, people often expect to rise after several consecutive declines. Or after several consecutive rises, it is expected to fall. But this is just a gambler's fallacy, because the probability of making a profit is still only 60%. At this time, fund management is extremely important. Suppose you start the game with 100 yuan, and you lose three times in a row (which is easy to satisfy in probability). Now your money is only 700 yuan. Then, most people will think that they will win the fourth time, so they will increase their bets to 300 yuan (300 yuan who wants to recover the losses). Although it is unlikely to lose four times in a row, it is still possible. Then, you're left with 400 yuan. If you want to recover the loss of this game, you must make a profit of 150%. It's unlikely. Suppose you enlarge your bet to 250 yuan, then you will go bankrupt in about four games. In either case, you can't make a profit from this simple game, because you don't have the concept of money management, you take too much risk, and you don't strike a balance between risk and opportunity. From experience, 20% loss should be the limit of loss. But this stop loss is not fund management, because it does not tell you how much to invest, and it is impossible to control the risk through the adjustment of positions and varieties. Fund management is an important part of the trading system, and it is essentially the part that determines the size of your position in the system. It can determine how much profit you can get and how much risk you take in system trading. This kind of currency management cannot simply replace the most important part by setting a stop loss. By playing management games, the most important thing is to know your potential investment style and risk preference, and realize the importance of capital management. 1, when investing in actual combat, be careful to survive in the first half of the accumulation stage. 2. When investing in actual combat, more funds will be accumulated in the second half of the year, and investment can be appropriately increased when the market is good. 3. If losses continue, the investment must be reduced. 4, the same trading system, different fund management, the final result is completely different, indicating the importance of fund management. 5. Fund management allows us to earn more when we are profitable and reduce losses as much as possible when the losses are not smooth. 6, often play, you will have more and deeper experience. To sum up, financial management determines how long you can live in the financial market. In the financial market, there are often all kinds of stars who have earned dozens of times and hundreds of times in a short time. They are really charming. But we should pursue to be the birthday girl, and we can make a stable profit in the market all our lives. To observe the long-term prosperity of the financial market, in order to make long-term stable profits, in addition to deliberately practicing technology and studying fundamental value analysis in depth, a very important but neglected key thing is to learn and improve the skills of fund management.	{"url":"https://slackerblud.com/Singleplayergame/cmirygftvp.html","timestamp":"2024-11-02T11:24:39Z","content_type":"text/html","content_length":"9821","record_id":"<urn:uuid:328355b4-4544-49f3-802f-9d203f17bd57>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00306.warc.gz"}
The calculus integral Related The calculus integral PDF eBooks Integral Calculus The Integral Calculus is an intermediate level PDF e-book tutorial or course with 120 pages. It was added on March 28, 2016 and has been downloaded 533 times. The file size is 527.42 KB. It was created by Miguel A. Lerma. Differential and integral calculus The Differential and integral calculus is an intermediate level PDF e-book tutorial or course with 143 pages. It was added on March 28, 2016 and has been downloaded 914 times. The file size is 752.5 KB. It was created by TEL AVIV UNIVERSITY. Understanding Basic Calculus The Understanding Basic Calculus is an intermediate level PDF e-book tutorial or course with 292 pages. It was added on March 28, 2016 and has been downloaded 6306 times. The file size is 1.46 MB. It was created by S.K. Chung. Introduction to Calculus - volume 1 The Introduction to Calculus - volume 1 is an intermediate level PDF e-book tutorial or course with 566 pages. It was added on March 28, 2016 and has been downloaded 3697 times. The file size is 6.71 MB. It was created by J.H. Heinbockel. Introduction to Calculus - volume 2 The Introduction to Calculus - volume 2 is an advanced level PDF e-book tutorial or course with 632 pages. It was added on March 28, 2016 and has been downloaded 1205 times. The file size is 8 MB. It was created by J.H. Heinbockel. Mathematical analysis I (differential calculus) The Mathematical analysis I (differential calculus) is an advanced level PDF e-book tutorial or course with 242 pages. It was added on March 25, 2016 and has been downloaded 470 times. The file size is 1.47 MB. It was created by SEVER ANGEL POPESCU. Mathematical analysis II (differential calculus) The Mathematical analysis II (differential calculus) is an advanced level PDF e-book tutorial or course with 407 pages. It was added on March 25, 2016 and has been downloaded 192 times. The file size is 2.98 MB. It was created by SEVER ANGEL POPESCU.	{"url":"https://www.computer-pdf.com/math/674-tutorial-the-calculus-integral.html","timestamp":"2024-11-08T01:38:01Z","content_type":"text/html","content_length":"24696","record_id":"<urn:uuid:ef300590-50da-405f-a021-aae5f0ff581e>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00197.warc.gz"}
IcoTopo mixerMeshslide Foam Code September 19, Thank you for all of your repl #7 2006, 14:31 New Member Thank you for all of your replies. They are very helpful. Joseph Kummer However, I am still having trouble. Hopefully you will be able to help me with this one, as it seems very strange. As I said, I tried the mixer case (as well as another similar case that I setup myself), and the actual rpm it rotated at was different from that which I specified in the dynamicMeshDict file. Join Date: Mar 2009 I did a little experimenting and found that if I set the rpm to 60 and timestep to 0.005, then the actual rpm varied from about 70 to 90 over the first 1/4 sec. Then I reduced the timestep to 0.001, and the actual rpm varied from about 62 to 66. Fayetteville, NY, I also reran the tutorial case exactly as specified, and found that the actual rpm over the first 0.25 sec varied as well, although not quite as much, but at least was close to 10 USA rpm this time (except between t=0.15 and t=0.175 sec, where it was 13.93 rpm). Posts: 17 Why would the speed the mesh turns at be a function of timestep? Of course, if the timestep is reduced, then the rotation angle per timestep should also go down; however, for a given deltaT and rpm, the rotation angle should remain the same. I'm wondering if you have seen anything similar, or possibly know what I am doing wrong? Rep Power: Thank you very much. Joe Kummer	{"url":"https://www.cfd-online.com/Forums/openfoam-solving/60007-icotopo-mixermeshslide-foam-code.html","timestamp":"2024-11-07T23:52:18Z","content_type":"application/xhtml+xml","content_length":"97625","record_id":"<urn:uuid:54a7dd80-aadc-4351-9ae0-b3a96dcaa01c>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00214.warc.gz"}
32-bit Single-Precision Floating Point in Details - ByteScout (2024) In modern days, programming languages tend to be as high-level as possible to make the programmer’s life a little bit easier. However, no matter how advanced programming language is, the code still has to be converted down to the machine code, via compilation, interpretation, or even virtual machine such as JVM. Of course, at this stage, rules are different: CPU works with addresses and registers without any classes, even «if» branches look like conditional jumps. One of the most important aspects of this execution is the arithmetic operation, and today we will be talking about one of these «cornerstones»: floating-point numbers and how they may affect your code. A brief introduction to the history The need for processing large or small values was present since the very first days of computers: even first designs of Charles Babbage’s Analytical Engine sometimes included floating-point arithmetic along with usual integer arithmetic. For a long time, the floating-point format was used primarily for scientific research, especially in physics, due to the large variety of data. It is extremely convenient that distance between Earth and Sun can be expressed in the same amount of bits as the distance between hydrogen and oxygen atoms in water molecules with the same relative precision and, even better, values of different magnitudes may be freely multiplied without large losses in precision. Almost all the early implementations of floating-point numbers were software due to the complexity of the hardware implementations. Without this common standard, everybody had to come up with their own formats: this is how Microsoft Binary Format and IBM Floating Point Architecture were born; the latter is still used in some fields such as weather forecasting, although it is extremely rare by Intel 8087 coprocessor, announced in 1980, also used its own format called «x87». It was the first coprocessor specifically dedicated to floating-point arithmetic with aims to replace slow library calls with the machine code. Then, based on x87 format, IEEE 754 was born as the first and successful attempt to create a universal standard for floating-point calculations. Soon, Intel started to integrate IEEE 754 into their CPUs, and nowadays almost every system except some embedded ones supports the floating-point format. Theory and experiments In IEEE 754 single-precision binary floating-point format, 32 bits are split into 1-bit sign flag, 8-bit exponent flag, and 23-bit fraction part, in that order (bit sign is the leftmost bit). This information should be enough for us to start some experiments! Let us see how number 1.0 looks like in this format using this simple C code: union { float in; unsigned out;} converter; converter.in = float_number; unsigned bits = converter.out; Of course, after getting the bits variable, we only need to print it. For instance, this way: 1.0 \| 1 \| S: 0 E: 01111111 M: 00000000000000000000000 Common sense tells that 1 can be expressed in binary fluting-point form as 1.0 * 2<sup>0</sup>, so exponent is 0 and significand is 1, while in IEEE 754 exponent is 1111111 (127 in decimal) and significand is 0. The mystery behind exponent is simple: the exponent is actually shifted. A zero exponent is represented as 127; exponent of 1 is represented as 128 and so on. Maximum value of exponent should be 255 – 127 = 128, and minimum value should be 0 – 127 = -127. However, values 255 and 0 are reserved, so the actual range is -126…127. We will talk about those reserved values later. The significand is even simpler to explain. Binary significand has one unique property: every significand in normalized form, except for zero, starts with 1 (this is only true for binary numbers). Next, if a number starts with zero, then it is not normalized. For instance, non-normalized 0.000101 * 10<sup>101</sup> is the same as normalized 1.01 * 10<sup>1</sup>. Due to that, there is no need to write an initial 1 for normalized numbers: we can just keep it in mind, saving space for one more significant bit. In our case, the actual significand is 1 and 23 zeroes, but because 1 is skipped, it is only 23 zeroes. Let us try some different numbers in comparison with 1. 1.0 \| 1 \| S: 0 E: 01111111 M: 00000000000000000000000 -1.0 \| -1 \| S: 1 E: 01111111 M: 00000000000000000000000 2.0 \| 2 \| S: 0 E: 10000000 M: 00000000000000000000000 4.0 \| 4 \| S: 0 E: 10000001 M: 00000000000000000000000 1 / 8 \| 0.125 \| S: 0 E: 01111100 M: 00000000000000000000000 As we can see, a negative sign just inverts sign flag without touching the rest (this seems obvious, but it is not always the case in computer science: for integers, a negative sign is much more complex than just flipping one bit!). Changing the exponent by trying different powers of two works as expected. 1.0 \| 1 \| S: 0 E: 01111111 M: 00000000000000000000000 3.0 \| 3 \| S: 0 E: 10000000 M: 10000000000000000000000 5.0 \| 5 \| S: 0 E: 10000001 M: 01000000000000000000000 0.2 \| 0.2 \| S: 0 E: 01111100 M: 10011001100110011001101 It is easy to see that numbers 3 and 5 are represented as 1.1 and 1.01 with aproper exponent. 0.2 should not differ much from them, but it is. What happened? It is easier to explain on decimals. 0.2 is the same number as 1/5. At the same time, not each radical can be represented as a decimal floating-point number: for example, 2/3 is 0.666666… It happens because 3 does not have any non-trivial common divisors with 10 (10 = 2 * 5, neither of them is 3). In the same time, 2/3 can be easily represented in base 12 as 0.8 (12 = 2 * 2 * 3). The same trick is true for a binary system: 2 does not have any common divisors with 5, so 0.2 can only be represented as infinitely long 0.00110011001100… At the same time, we only have 23 significant bits! So, we are inevitably losing precision. Let us try with some multiplications. 1.0 \| 1 \| S: 0 E: 01111111 M: 00000000000000000000000 0.2^225 \| 1 \| S: 0 E: 01111111 M: 00000000000000000000001 250.2^2 \| 1 \| S: 0 E: 01111111 M: 00000000000000000000000 Both 1 and 0.2 * 0.2 * 25 are printed as 1, but they are actually different! Due to the precision loss, 0.2 * 0.2 * 25 is not the same as 1, and the expression (0.2f * 0.2f * 25.0f == 1.0f) is actually false. At the same time, if we execute 25 * 0.2 first, then the result is actually correct. It means that the rule (a * b) * c = a * (b * c) is not always true for floating-point numbers! Special numbers Remember about the fact that zero can never be written in the normalized form because it does not contain any 1s in its binary representation? Zero is a special number. 0 \| 0 \| S: 0 E: 00000000 M: 00000000000000000000000 -0 \| -0 \| S: 1 E: 00000000 M: 00000000000000000000000 For zero, IEEE 754 uses an exponent value of 0 and a significand value of 0. In addition, as you can see, there are actually two zero values: +0 and -0. In terms of comparison, (0.0f == -0.0f), is actually true, sign just does not count. +0 and -0 loosely correspond to the mathematical concept of the infinitesimal, positive and negative. Are there any special numbers with an exponent value of 0? Yes. They are called «denormalized numbers». Those numbers can represent extremely small values, lesser than the minimum normalized number (which should be a little larger than 1 * 2<sup>-127</sup>). Examples: 2^-126 \| 1.17549e-38 \| S: 0 E: 00000001 M: 00000000000000000000000 2^-127 \| 5.87747e-39 \| S: 0 E: 00000000 M: 10000000000000000000000 2^-128 \| 2.93874e-39 \| S: 0 E: 00000000 M: 01000000000000000000000 2^-149 \| 1.4013e-45 \| S: 0 E: 00000000 M: 00000000000000000000001 2^-150 \| 0 \| S: 0 E: 00000000 M: 00000000000000000000000 A denormalized number has the virtual exponent value of 1, but, at the same time, they do not have omitted 1 as their first omitted digit. The only consequence is that denormalized numbers quickly lose precision: to store numbers between 2<sup>-128</sup> and 2<sup>-127</sup>, we are only using 21 digits of information instead of 23. It is easy to see that zero is the special case of the denormalized number. Moreover, as we can see, the least possible single-precision floating-point number is actually 2<sup>-149</sup>, or approximately 1.4013 * 10<sup>-45</sup>. Numbers with the exponent of 11111111 are reserved for the «other end» of the number scale: Infinity and the special value called «Not a Number». 1 / 0 \| inf \| S: 0 E: 11111111 M: 00000000000000000000000 1 / -0 \| -inf \| S: 1 E: 11111111 M: 00000000000000000000000 2^128 \| inf \| S: 0 E: 11111111 M: 00000000000000000000000 As with zeroes, infinity can be either positive or negative. It can be achieved by dividing any non-zero number to zero or by getting any number larger than maximum allowed (which is a little less than 2<sup>128</sup>). Infinity is processed as follows: Infinity > Any number Infinity = Infinity Infinity > -Infinity Any number / Infinity = 0 (sign is set properly) Infinity * Infinity = Infinity (again, sign is set properly) Infinity / Infinity = NaN Infinity * 0 = NaN Not a Number, or NaN, is, perhaps, the most interesting floating-point value. It can be obtained in multiple ways. First, it is the result of any indeterminate form: Infinity * 0 0 / 0 or Infinity / Infinity Infinity – Infinity or –Infinity + Infinity Secondly, it can be the result of some non-trivial operations. Power function may return NaN in case of any of those indeterminate forms: 0<sup>0</sup>, 1<sup>Infinity</sup>, Infinity<sup>0</sup>. Any operation which can result in complex number may return NaN in this case: log(-1.0f), sqrt(-1.0f), sin(2.0f) are the examples. Lastly, any operation involving NaN as any of the operands always returns NaN. Because of that, NaN can sometimes quickly “spread” through data like a computer virus. The only exception is min or max: those functions should return the non-NaN argument. NaN is never equal to any other number, even itself (it can be used to test numbers against NaN). Actual contents of NaN are implementation-defined; IEEE 754 only requires that exponent should be 11111111, significand should be non-zero (zero is reserved for the infinity) and the sign does not 0/0 \| -nan \| S: 1 E: 11111111 M: 10000000000000000000000 IEEE 754 differentiates two types of NaN: quiet NaN and signaling NaN. Their only difference is that signaling NaN generates interruption while quiet NaN does not. Again, the application decides if it generates quiet NaN or signaling NaN. For instance, the GCC C compiler always generates quiet NaN unless explicitly specified to behave the other way around. What can we learn from all the facts and experiments above? In any language operating with the floating-point data type, beware of the following: – You should almost never directly compare two floating-point numbers unless you know what you are doing! A better way to do it is to compare it with some precision. if (a == b) – wrong! if (fabsf(a – b) < epsilon) – correct! – Floating-point numbers lose precision even when you are just working with such seemingly harmless numbers as 0.2 or 71.3. You should be extra careful when working with a large amount of floating-point operations over the same data: errors may build up rather quickly. If you are getting unexpected results and you suspect rounding errors, try to use a different approach, and minimize – In the world of floating-point arithmetic, multiplication is not associative: a * (b * c) is not always equal to (a * b) * c. – Additional measures should be taken if you are working with either extremely large values, extremely small numbers, and/or numbers close to zero: in case of overflow or underflow those values will be transformed into +Infinty, -Infinity or 0. Numeric limits for single-precision floating-point numbers are approximately 1.175494e-38 to 3.402823e+38 (1.4013e-45 to 3.402823e+38 if we also count denormalized numbers)а. – Beware if your system generates «quiet NaN». Sometimes, it may help you to not crash the application. Sometimes, it may spoil program execution beyond recognition. Nowadays, floating-point numbers operations are extremely fast, with speed comparable to the usual integer arithmetic: a number of floating-point operations per second, or FLOPS, is perhaps the most well-known measure of computer performance. The only downside is that the programmer should be aware of all the pitfalls regarding the precision and special floating-point values. About the Author ByteScout Team of WritersByteScout has a team of professional writers proficient in different technical topics. We select the best writers to cover interesting and trending topics for our readers. We love developers and we hope our articles help you learn about programming and programmers.	{"url":"https://cndsheetmetal.com/article/32-bit-single-precision-floating-point-in-details-bytescout","timestamp":"2024-11-09T09:04:55Z","content_type":"text/html","content_length":"125128","record_id":"<urn:uuid:618eaa36-7eb2-428b-9191-293e8402c86c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00809.warc.gz"}
Geometry \| Right Triangles and Trigonometry Trigonometric Functions F.TF.A.3 — Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.	{"url":"https://www.fishtanklearning.org/curriculum/math/geometry/right-triangles-and-trigonometry/","timestamp":"2024-11-09T10:56:23Z","content_type":"text/html","content_length":"581402","record_id":"<urn:uuid:83906a87-dba2-4937-8744-8472d5933541>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00195.warc.gz"}
Identifying Patterns Data analysis is about identifying, describing, and explaining patterns. Univariate analysis is the most basic form of analysis that quantitative researchers conduct. In this form, researchers describe patterns across just one variable. Univariate analysis includes frequency distributions and measures of central tendency. A frequency distribution is a way of summarizing the distribution of responses on a single survey question. Let’s look at the frequency distribution for just one variable from my older worker survey. We’ll analyze the item mentioned first in the codebook excerpt given earlier, on respondents’ self-reported financial security. Table 8.3 Frequency Distribution of Older Workers’ Financial Security │In general, how financially secure would you say you are? │Value│Frequency│Percentage│ │Label │ │ │Not at all secure │1 │46 │25.6 │ │Between not at all and moderately secure │2 │43 │23.9 │ │Moderately secure │3 │76 │42.2 │ │Between moderately and very secure │4 │11 │6.1 │ │Very secure │5 │4 │2.2 │ │Total valid cases = 180; no response = 3 │ │ As you can see in the frequency distribution on self-reported financial security, more respondents reported feeling “moderately secure” than any other response category. We also learn from this single frequency distribution that fewer than 10% of respondents reported being in one of the two most secure categories. Another form of univariate analysis that survey researchers can conduct on single variables is measures of central tendency. Measures of central tendency tell us what the most common, or average, response is on a question. Measures of central tendency can be taken for any level variable of those we learned about in "Defining and Measuring Concepts", from nominal to ratio. There are three kinds of measures of central tendency: modes, medians, and means. Mode refers to the most common response given to a question. Modes are most appropriate for nominal-level variables. A median is the middle point in a distribution of responses. Median is the appropriate measure of central tendency for ordinal-level variables. Finally, the measure of central tendency used for interval-and ratio-level variables is the mean. To obtain a mean, one must add the value of all responses on a given variable and then divide that number by the total number of responses. In the previous example of older workers’ self-reported levels of financial security, the appropriate measure of central tendency would be the median, as this is an ordinal-level variable. If we were to list all responses to the financial security question in order and then choose the middle point in that list, we’d have our median. In "Figure 8.5", the value of each response to the financial security question is noted, and the middle point within that range of responses is highlighted. To find the middle point, we simply divide the number of valid cases by two. The number of valid cases, 180, divided by 2 is 90, so we’re looking for the 90th value on our distribution to discover the median. As you’ll see in "Figure 8.5", that value is 3, thus the median on our financial security question is 3, or “moderately secure.” Figure 8.5 Distribution of Responses and Median Value on Workers’ Financial Security (Missing in original) As you can see, we can learn a lot about our respondents simply by conducting univariate analysis of measures on our survey. We can learn even more, of course, when we begin to examine relationships among variables. Either we can analyze the relationships between two variables, called bivariate analysis, or we can examine relationships among more than two variables. This latter type of analysis is known as multivariate analysis. Bivariate analysis allows us to assess covariation among two variables. This means we can find out whether changes in one variable occur together with changes in another. If two variables do not covary, they are said to have independence. This means simply that there is no relationship between the two variables in question. To learn whether a relationship exists between two variables, a researcher may cross-tabulate the two variables and present their relationship in a contingency table. A contingency table shows how variation on one variable may be contingent on variation on the other. Let’s take a look at a contingency table. In "Table 8.4" , I have cross-tabulated two questions from my older worker survey: respondents’ reported gender and their self-rated financial Table 8.4 Financial Security Among Men and Women Workers Age 62 and Up │ │ Men │ Women │ │ Not financially secure (%) │ 44.1 │ 51.8 │ │ Moderately financially secure (%) │ 48.9 │ 39.2 │ │ Financially secure (%) │ 7.0 │ 9.0 │ │ Total │ N = 43 │ N = 135 │ You’ll see in "Table 8.4" that I collapsed a couple of the financial security response categories (recall that there were five categories presented in "Table 8.3"; here there are just three). Researchers sometimes collapse response categories on items such as this in order to make it easier to read results in a table. You’ll also see that I placed the variable “gender” in the table’s columns and “financial security” in its rows. Typically, values that are contingent on other values are placed in rows (a.k.a. dependent variables), while independent variables are placed in columns. This makes comparing across categories of our independent variable pretty simple. Reading across the top row of our table, we can see that around 44% of men in the sample reported that they are not financially secure while almost 52% of women reported the same. In other words, more women than men reported that they are not financially secure. You’ll also see in the table that I reported the total number of respondents for each category of the independent variable in the table’s bottom row. This is also standard practice in a bivariate table, as is including a table heading describing what is presented in the table. Researchers interested in simultaneously analyzing relationships among more than two variables conduct multivariate analysis. If I hypothesized that financial security declines for women as they age but increases for men as they age, I might consider adding age to the preceding analysis. To do so would require multivariate, rather than bivariate, analysis. We won’t go into detail here about how to conduct multivariate analysis of quantitative survey items here, but we will return to multivariate analysis in "Reading and Understanding Social Research", where we’ll discuss strategies for reading and understanding tables that present multivariate statistics. If you are interested in learning more about the analysis of quantitative survey data, I recommend checking out your campus’s offerings in statistics classes. The quantitative data analysis skills you will gain in a statistics class could serve you quite well should you find yourself seeking employment one day. • While survey researchers should always aim to obtain the highest response rate possible, some recent research argues that high return rates on surveys may be less important than we once thought. • There are several computer programs designed to assist survey researchers with analyzing their data which include SPSS, MicroCase, and Excel. • Data analysis is about identifying, describing, and explaining patterns. • Contingency tables show how, or whether, one variable covaries with another. 1. Codebooks can range from relatively simple to quite complex. For an excellent example of a more complex codebook, check out the coding for the General Social Survey (GSS): http:// 2. The GSS allows researchers to cross-tabulate GSS variables directly from its website. Interested? Check out http://www.norc.uchicago.edu/GSS+Website/Data+Analysis.	{"url":"http://www.opentextbooks.org.hk/zh-hant/ditatopic/29597","timestamp":"2024-11-05T05:54:10Z","content_type":"text/html","content_length":"259241","record_id":"<urn:uuid:96d2e05f-fa63-4bc2-8f24-6a81ee71765f>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00794.warc.gz"}
Separation Theorem - (Computational Geometry) - Vocab, Definition, Explanations \| Fiveable Separation Theorem from class: Computational Geometry The separation theorem is a fundamental concept in convex geometry that states that if two convex sets do not intersect, then there exists a hyperplane that can separate them. This theorem establishes the relationship between convexity and the ability to distinguish between different sets, showing that for any two disjoint convex sets, you can find a line (or hyperplane in higher dimensions) that divides them such that one set lies entirely on one side of the line and the other set on the opposite side. congrats on reading the definition of Separation Theorem. now let's actually learn it. 5 Must Know Facts For Your Next Test 1. The separation theorem applies not only to finite-dimensional spaces but also has implications in infinite-dimensional spaces, like functional analysis. 2. If two convex sets intersect, they cannot be separated by any hyperplane, which is an important aspect of understanding the limitations of the separation theorem. 3. In optimization problems, the separation theorem can help identify feasible solutions by separating feasible regions from infeasible ones. 4. The existence of separating hyperplanes implies that convex sets are 'well-behaved' in terms of geometric properties and their relationships to each other. 5. The separation theorem is a key tool in proving other important results in convex analysis, including duality and optimality conditions. Review Questions • How does the separation theorem illustrate the properties of convex sets? □ The separation theorem highlights that if two convex sets are disjoint, a hyperplane exists that can separate them. This shows that convex sets maintain a structure where any two non-overlapping sets can be distinctly identified without ambiguity. The ability to create a separating line reflects the organized nature of convex shapes and their spatial relationships. • In what ways can the separation theorem be applied to optimization problems? □ The separation theorem can be crucial in optimization as it helps to define feasible regions and constraints within optimization problems. By using separating hyperplanes, one can identify areas where solutions may exist or where certain conditions are violated. This aids in refining the search for optimal solutions by clearly delineating feasible from infeasible regions based on given constraints. • Evaluate the significance of the separation theorem in both finite and infinite-dimensional spaces and its implications for theoretical aspects of geometry. □ The separation theorem is significant because it transcends dimensions, applying to both finite-dimensional and infinite-dimensional spaces. In finite dimensions, it provides clear geometric insights into how sets interact; in infinite dimensions, such as those found in functional analysis, it plays a vital role in understanding convergence and continuity. This versatility makes it foundational in theoretical geometry, as it offers a systematic way to reason about complex relationships between diverse mathematical objects. "Separation Theorem" also found in: © 2024 Fiveable Inc. All rights reserved. AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.	{"url":"https://library.fiveable.me/key-terms/computational-geometry/separation-theorem","timestamp":"2024-11-14T00:09:42Z","content_type":"text/html","content_length":"146118","record_id":"<urn:uuid:055029ee-e60c-4fb7-8ebf-9d0b11b1f9bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00137.warc.gz"}
Math Colloquia - Mechanization of proof: from 4-Color theorem to compiler verification I will give a broad introduction to how to mechanize mathematics (or proof), which will be mainly about the proof assistant Coq. Mechanizing mathematics consists of (i) defining a set theory, (2) developing a tool that allows writing definitions and proofs in the set theory, and (3) developing an independent proof checker that checks whether a given proof is correct (ie, whether it is a valid combination of axioms and inference rules of the set theory). Such a system is called proof assistant and Coq is one of the most popular ones. In the first half of the talk, I will introduce applications of proof assistant, ranging from mechanized proof of 4-color theorem to verification of an operating system. Also, I will talk about a project that I lead, which is to provide, using Coq, a formally guaranteed way to completely detect all bugs from compilation results of the mainstream C compiler LLVM. In the second half, I will discuss the set theory used in Coq, called Calculus of (Inductive and Coinductive) Construction. It will give a very interesting view on set theory. For instance, in calculus of construction, the three apparently different notions coincide: (i) sets and elements, (ii) propositions and proofs, and (iii) types and programs. If time permits, I will also briefly discuss how Von Neumann Universes are handled in Coq and how Coq is used in homotopy type theory, led by Fields medalist Vladimir Voevodsky.	{"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&l=en&page=8&sort_index=Time&order_type=desc&document_srl=726054","timestamp":"2024-11-09T23:59:50Z","content_type":"text/html","content_length":"45702","record_id":"<urn:uuid:9793d02f-3f03-4f05-a306-563dba8dd6a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00459.warc.gz"}
An Extension of Rosen's Theorem to Non-identically Distributed Random Variables Ann. Math. Statist. 39(3): 897-904 (June, 1968). DOI: 10.1214/aoms/1177698322 In [5], B. Rosen showed that if $\{X_k: k = 1,2, \cdots\}$ is an independent sequence of identically distributed random variables with $EX_k = 0$ and $\operatorname{Var} X_k = \sigma^2, 0 < \sigma^2 < \infty$ and if $S_n = X_1 + \cdots + X_n$, then the series $\sum^\infty_{n=1} n^{-1} (P(S_n < 0) - \frac{1}{2})$ is absolutely convergent. This theorem was motivated by a result of Spitzer [6] who, under the same conditions, established the convergence of this series as a corollary to a result in the theory of random walks. Rosen's theorem was generalized by Baum and Katz [1] who showed that if $EX_k = 0$ and $E\|X_k\|^{2+\alpha} < \infty$ for $0 \leqq \alpha < 1$ then $\sum^\infty_{n=1}n^{-(1-\alpha/2)} \|P(S_n < 0) - \frac{1}{2}\| < \infty.$ These results led to the study of series convergence rate criteria for the central limit theorem and a partial solution of this problem was obtained for the case of identically distributed random variables in [2]. A more complete solution has been recently obtained by Heyde [4]. The first study of series convergence rates for $P(S_n < 0)$ in the case of independent but non-identically distributed random variables was made by Heyde [3]. Based on an extension of Rosen's theorem utilizing certain uniform bounds on the characteristic function of the $X_k$'s he concluded the absolute convergence of the series $\sum^\infty_{n=1}n^{- (1-\alpha/2)} (P(S_n < n^px) - \frac{1}{2})$ for $- \infty < x < \infty$ and $0 \leqq p < \frac{1}{2}(1 - \alpha), 0 \leqq \alpha < 1,$ thus obtaining what he termed small deviation convergence rates. In the present paper two more extensions of Rosen's theorem to independent but non-identically distributed random variables are given under different hypotheses than Heyde's. The first (Theorem 1) reduces to Rosen's theorem in the case of identically distributed random variables. The second (Theorem 2) results in a theorem similar to that of Baum and Katz [1] as required in Heyde's small deviation result. This will make it possible to obtain his conclusion by simply carrying out the last step in his proof. These results are obtained in Section 3. In Section 2 some preliminary results are stated and examples are given in Section 4 to show that the first two hypotheses of Theorem 1 cannot, in general, be relaxed. Download Citation L. H. Koopmans. "An Extension of Rosen's Theorem to Non-identically Distributed Random Variables." Ann. Math. Statist. 39 (3) 897 - 904, June, 1968. https://doi.org/10.1214/aoms/1177698322 Published: June, 1968 First available in Project Euclid: 27 April 2007 Digital Object Identifier: 10.1214/aoms/1177698322 Rights: Copyright © 1968 Institute of Mathematical Statistics Vol.39 • No. 3 • June, 1968	{"url":"https://www.projecteuclid.org/journals/annals-of-mathematical-statistics/volume-39/issue-3/An-Extension-of-Rosens-Theorem-to-Non-identically-Distributed-Random/10.1214/aoms/1177698322.full","timestamp":"2024-11-04T12:43:11Z","content_type":"text/html","content_length":"143194","record_id":"<urn:uuid:efdb8be6-47b6-49cf-8c0b-976139a780fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00594.warc.gz"}
Jump to navigation Jump to search stats/base/combinatorial - Sample design comb v combinations of size x from i.y combrep v combinations of size x from i.y with repetition combrev v comb in revolving door order perm v permutations of size y from i.y permrep v permutations of size x from i.y with repetition steps v steps from a to b in c steps comb (v) combinations of size x from i.y combrep (v) combinations of size x from i.y with repetition combrev (v) comb in revolving door order combinations in what Knuth calls "revolving door order" such that any two adjacent combinations differ by a single element (Gray codes for combinations). perm (v) permutations of size x from i.y monadic form gives all perms of size i.y (i@! A. i.) permrep (v) permutations of size x from i.y with repetition steps (v) steps from a to b in c steps form: steps a,b,c	{"url":"https://code.jsoftware.com/wiki/Addons/stats/base/combinatorial","timestamp":"2024-11-14T00:38:01Z","content_type":"text/html","content_length":"20037","record_id":"<urn:uuid:29263a91-a2b9-4394-b661-a67aa7338656>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00831.warc.gz"}
Understanding SHA256 (for dummies) So, with the SHA256 process (?) we can take any string of words and numbers and do something to it and we get a 256 string of ones and zeroes .Small changes to the input give an unrelated new This can also be converted to other forms/keys (?) Hopefully I got that right. My questions is....What do we do to the original set of words and numbers? What is the process? If t is difficult to explain, then, is there a good analogy or way to explain it?	{"url":"https://bitcointalk.org/index.php?topic=279249.0;prev_next=next","timestamp":"2024-11-03T00:38:55Z","content_type":"application/xhtml+xml","content_length":"63567","record_id":"<urn:uuid:9ca115dc-7c66-4685-acd3-cb8db69fe515>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00147.warc.gz"}
Four point probe Jump to navigation Jump to search Four point probe is used to measure resistive properties of semiconductor wafers and thin films. If the thickness of a thin film is known, the sheet resistance measured by four point probe can be used to calculate the resistivity of the material; conversely, if the material's resistivity is known, the thickness of the thin film can be calculated. Miller FPP-5000 4-Point Probe Method of operation • A four point probe is typically used to measure the sheet resistance of a thin layer or substrate in units of ohms per square by forcing current through two outer probes and reading the voltage across the two inner probes. Using this four-terminal configuration avoids measurement error due to the contact resistance between the probe and sample. • The probes are collinear and are equally spaced. • Probe spacing for the LNF tools is 1.59mm • For film thickness ≤ 0.5 x (probe spacing) and diameter or lateral dimensions > 40 x (probe spacing), the sheet resistance calculation simplifies to: V = measured voltage I = force current • Rearranging the equation for resistance of a rectangular thin film resistor helps illustrate the meaning of sheet resistance which is equal to the resistivity of the material divided by its □ ρ = thin film material resistivity □ L = length of resistor □ W = width of resistor □ t = thickness of thin film material • If the thickness of a film or material is known, its resistivity can be calculated from the sheet resistance measurement. • If the resistivity of the film or material is known (or assumed), a thickness can be calculated from the sheet resistance measurement. • For samples with lateral dimensions or diameter < 40 x (probe spacing), correction factors need to be used to obtain accurate sheet resistance values. For correction factor values, please refer to information at http://four-point-probes.com/finite-size-corrections-for-4-point-probe-measurements ^[2] Four point probes are used in nanofabrication to measure the resistive properties of conducting films which may include substrates, deposited films, and doped regions on a sample surface. See also Further reading	{"url":"https://lnf-wiki.eecs.umich.edu/wiki/Four_point_probe","timestamp":"2024-11-10T05:25:47Z","content_type":"text/html","content_length":"138877","record_id":"<urn:uuid:01190c63-eb42-42ae-a41d-2dfba5205c94>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00449.warc.gz"}
Shape:bs6pi2ygs9a= pentagon: Shape, Properties, and Applications Table of Contents Shape:bs6pi2ygs9a= pentagon The pentagon is a fascinating geometric shape with numerous applications and properties. This article explores the pentagon’s characteristics, its variations, and how it is used in various fields. What is a Pentagon? Shape:bs6pi2ygs9a= pentagon A pentagon is a five-sided polygon with five angles and five vertices. Shape:bs6pi2ygs9a= pentagon The term “pentagon” comes from the Greek words “pente,” meaning five, and “gonia,” meaning angle. There are several types of pentagons, each with its unique properties. Shape:bs6pi2ygs9a= pentagon Types of Pentagons Regular Pentagon A regular pentagon has all its sides and angles equal. The interior angles of a regular pentagon each measure 108 degrees. This symmetry makes the regular pentagon visually appealing and is often seen in art and design. Irregular Pentagon An irregular pentagon has sides and angles of different lengths and measures. Despite its lack of symmetry, the irregular pentagon is still a crucial geometric shape with various applications. Concave Pentagon A concave pentagon has at least one interior angle greater than 180 degrees, which creates a shape that appears to “cave in.” This type of pentagon is less common but still an important part of geometric studies. Mathematical Properties of the Pentagon Angles and Sides In a regular pentagon, each interior angle measures 108 degrees, and the sum of the interior angles is always 540 degrees. The length of each side is equal in a regular pentagon, creating a symmetrical shape. A regular pentagon has five diagonals. The diagonals intersect at various angles and lengths, creating additional geometric patterns within the shape. Area and Perimeter The area of a regular pentagon can be calculated using the formula: Area=145(5+25)×s2\text{Area} = \frac{1}{4} \sqrt{5 (5 + 2 \sqrt{5})} \times s^2Area=415(5+25)×s2 where sss is the side length. The perimeter is simply: Perimeter=5×s\text{Perimeter} = 5 \times sPerimeter=5×s The Pentagon in Architecture and Design Historical Architecture The pentagon has been used in architecture for centuries. The most famous example is the Pentagon building in the United States, which is a symbol of military strength and government authority. Modern Design In modern design, the pentagon is used in various ways, from logo design to urban planning. Its geometric properties are used to create visually appealing patterns and structures. The Pentagon in Nature Natural Occurrences Pentagons can be found in nature, such as in the arrangement of some flowers and the shape of certain crystals. These natural occurrences demonstrate the pentagon’s inherent beauty and mathematical Biological Examples The starfish, with its five arms, is a prime example of a natural pentagon. The pentagonal symmetry of these creatures allows for efficient movement and feeding. The pentagon is a versatile and intriguing shape with applications across various fields, from architecture and design to natural sciences. Understanding its properties and types enhances our appreciation of its role in both man-made and natural environments. 1. What is the difference between a regular and an irregular pentagon? A regular pentagon has equal sides and angles, while an irregular pentagon has sides and angles of different lengths and measures. 2. How do you calculate the area of a regular pentagon? The area of a regular pentagon can be calculated using the formula: Area=145(5+25)×s2\text{Area} = \frac{1}{4} \sqrt{5 (5 + 2 \sqrt{5})} \times s^2Area=415(5+25)×s2 where sss is the side length. 3. Can pentagons be found in nature? Yes, pentagons can be found in nature, such as in the arrangement of certain flowers and the shape of starfish. 4. What are the interior angles of a regular pentagon? Each interior angle of a regular pentagon measures 108 degrees. 5. How many diagonals does a regular pentagon have? A regular pentagon has five diagonals. Leave a Comment	{"url":"https://techyinsight.co.uk/shapebs6pi2ygs9a-pentagon/","timestamp":"2024-11-12T02:54:29Z","content_type":"text/html","content_length":"82260","record_id":"<urn:uuid:70f212ea-41d5-4e89-9820-c334b342ed7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00162.warc.gz"}
vik dhillon: phy105 - celestial mechanics - newton's derivation of kepler's laws newton's derivation of kepler's laws All three of Kepler's laws follow from Newton's laws of motion when the law of universal gravitation is used to express the forces between the Sun and the planets. kepler I Newton's derivation of Kepler's first law is embodied in his statement and solution of the so-called two-body problem. Given at any time the positions and velocities of two massive particles moving under their mutual gravitational force, the masses also being known, provide a means of calculating their positions and velocities for any other time, past or future. The solution of the two-body problem is an equation of motion. Its derivation is outside the scope of this course, as it requires the use of vector calculus in conjunction with Newton's second law and his law of gravitation. The solution for two masses m[1] and m[2] can be written in polar coordinates r,Figure 31) as follows: r = h^2 / G(m[1]+m[2]) (1 + e cos where h is a constant which is twice the rate of description of area by the radius vector and e is the eccentricity of the orbit. This equation looks similar to the polar equation of an ellipse that we derived earlier. In fact, it is the polar equation of a conic section. The ellipse is just one example of a class of curves called conic sections, which are formed when a cone is cut with a plane, as shown in Figure 35. When the plane is perpendicular to the cone's axis, the result is a circle (ellipticity, e = 0); when it is parallel to one side, the result is a parabola (e = 1); intermediate angles result in ellipses (0 < e < 1). A hyperbola results when the angle the plane makes with the cone's side is greater than the opening angle of the cone (e > 1). figure 35: Conic sections. In obtaining his solution to the two-body problem, Newton generalized Kepler's first law. He deduced that when one body moves under the gravitational influence of another, the orbit of the moving body must be a conic section. Planets, satellites and asteroids have elliptical orbits. Many comets have eccentricities so close to unity that they follow essentially parabolic orbits. A few comets have hyperbolic orbits - after one perihelion passage, such comets leave the solar system forever. Space probes have been launched into hyperbolic orbits with respect to the Earth, but they are nearly always captured into elliptical orbits about the Sun. Pioneer 10 was the first spacecraft that, when perturbed by Jupiter, escaped from the solar system. kepler II There are two ways in which it is possible to derive Kepler's second law from Newton's laws. The first, presented by Newton in 1684, is a geometrical method and is shown in Figure 36. figure 36: Newton's proof of Kepler's second law. Newton visualized the motion of an object acted on by a gravitational force as a succession of small kicks or impulses which in the limit become a continuously applied influence. Newton imagined an object travelling along part of an orbit AB which then receives an impulse directed towards the point S. As a result, it then travels along the line BC instead of Bc. Similar impulses carry it to D, E and F. Newton visalized the displacement BC as being, in effect, the combination of the displacement Bc, equal to AB, that the object would have undergone if it had continued for an equal length of time with its original velocity, together with the displacement cC parallel to the line BS along which the impulse was applied. This at once yields Kepler's second law by a simple argument: The triangles SAB and SBc are equal, having equal bases (AB and Bc) and the same altitude. The triangles SBc and SBC are equal, having a common base (SB) and lying between the same parallels. Hence triangle SAB = triangle SBC. A modern Newtonian derivation of Kepler's second law requires the concept of an orbiting body's angular momentum L = r X p = m (r X v) where m is the body's mass, r is its position vector and p its linear momentum (= mv, where v is its velocity). Note that for the first time in this course we distinguish between vector quantities and scalar quantities by writing vector quantities in a bold face. The vector cross product (denoted by X) is an operation that yields the product of the perpendicular components of two vectors; hence if r and p are parallel, then r X p = 0. Angular momentum is a vector quantity L with the units kgm^2s^-1. Differentiating L, we have dL/dt = d(r X p)/dt = v X p + r X (dp/dt) = r X F since v is parallel to p and dp/dt is the definition of force according to Newton's second law. We call dL/dt the torque (with units kgm^2s^-2) and see that when F and r are co-linear, due to a central force such as gravitation, the torque vanishes. Hence L is constant in time and so angular momentum is conserved for all central forces. The conservation of angular momentum is a very powerful tool in celestial mechanics and can be used to derive Kepler's second law as follows. figure 37: The velocity components of a body in an elliptical orbit. A body is moving in an elliptical orbit with a velocity v at a distance r from the focus F (Figure 37). During a short time interval []t, the body moves from P to Q and the radius vector sweeps through the angle [][]v[t][]t / r, where v[t] is the component of v perpendicular to r. During this time, the radius vector has swept out the triangle FPQ, the area of which is approximately given by []A = rv[t][]t / 2. Therefore, in the limit given by []t approaching zero, we have dA/dt = rv[t]/2 = ½r^2(ddt). Now, the angular momentum of the body in Figure 37 is given by the vector perpendicular to the plane defined by r and v, i.e. it is out of the plane of the paper. The scalar magnitude of L is given by L = mv[t]r= mr^2 ddt. This means that the rate of sweeping out area is given by dA/dt = ½r^2(ddt) = L / 2m. As L and m are constants, then dA/dt must be a constant, i.e. the rate of sweeping out area is a constant. Hence we have verified Kepler's second law. kepler III Newton's form of Kepler's third law can be derived by considering two bodies of masses m[1] and m[2], orbiting their (stationary) centre of mass at distances r[1] and r[2] (Figure 38). figure 38: Two bodies in orbit about their common centre of mass. Because the gravitational force acts only along the line joining the centres of the bodies, both bodies must complete one orbit in the same period P (though they move at different speeds v [1] and v[2]). The forces on each body due to their centripetal accelerations are therefore F[1] = m[1]v[1]^2 / r[1] = 4^2 m[1]r[1] / P^2 F[2] = m[2]v[2]^2 / r[2] = 4^2 m[2]r[2] / P^2. Newton's third law tells us that F[1] = F[2], and so we obtain r[1] / r[2] = m[2] / m[1]. This tells us that the more massive body orbits closer to the centre of mass than the less massive body. The total separation of the two bodies is given by a = r[1] + r[2] which gives r[1] = m[2]a / (m[1] + m[2]). Combining this equation with the equation for F[1] derived above and Newton's law of gravitation (F[grav] = F[1] = F[2] = Gm[1]m[2] / a^2) gives Newton's form of Kepler's third law: P^2 = 4^2 a^3 / G(m[1] + m[2]). If body 1 is the Sun and body 2 any planet, then m[1] >> m[2]. Hence the constant of proportionality in Kepler's third law becomes 4^2 / GM[Sun]. ©Vik Dhillon, 30th September 2009	{"url":"http://www.vikdhillon.staff.shef.ac.uk/teaching/phy105/celsphere/phy105_derivation.html","timestamp":"2024-11-02T09:01:48Z","content_type":"text/html","content_length":"17651","record_id":"<urn:uuid:b6de0df6-d2a5-482a-8916-5ef6b36dd647>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00469.warc.gz"}
The Ultimate 7th Grade TCAP Math Course (+FREE Worksheets) On the hunt for an exhaustive, expansive course to enable your students in their preparation for the 7th Grade TCAP Math exam? Look no further, your quest concludes here! If fostering your student’s success in the 7th Grade TCAP Math Course is your aim, this gratis course is your ally. It’ll equip them with every pertinent principle of the test, well in advance of the examination day. Behold the quintessential Course, a compendium of all the concepts germane to the 7th Grade TCAP Math exam. This first-rate TCAP Math Course is all your students necessitate for a triumphant performance in their 7th Grade TCAP Math exam. This Course, specially designed for TCAP Math, together with a variety of other Effortless Math Courses, is the preferred choice of thousands of annual TCAP aspirants. It assists them in revising the core subjects, refining their mathematical acuity, and identifying their areas of strength and weakness. Consequently, they score impressively on the TCAP test. Experience the freedom of self-paced learning, devoid of any strict timetables! Each lecture is supplemented with comprehensive notes, illustrative examples, practical exercises, and a plethora of activities designed to facilitate students mastering every TCAP Math concept seamlessly. The only mandate – adhere to the guidelines for each lecture to ace the 7th Grade TCAP Math examination. The Absolute Best Book to Ace the TCAP Math Test Original price was: $29.99.Current price is: $14.99. 7th Grade TCAP Math Complete Course Rational Numbers Integers Operation Decimals Operation Fractions and Mixed Numbers Operation Proportional Relationships Rates and Ratio Price problems Probability and Statistics Equations and Variables Geometric Problems Statistics and Analyzing Data Looking for the best resource to help your student succeed on the 7th Grade TCAP Math test? The Best Resource to Ace the 7th Grade TCAP Math Test Original price was: $29.99.Current price is: $14.99. Original price was: $29.99.Current price is: $14.99. Original price was: $29.99.Current price is: $14.99. Related to This Article What people say about "The Ultimate 7th Grade TCAP Math Course (+FREE Worksheets) - Effortless Math: We Help Students Learn to LOVE Mathematics"? No one replied yet.	{"url":"https://www.effortlessmath.com/blog/the-ultimate-7th-grade-tcap-math-course/","timestamp":"2024-11-07T01:18:21Z","content_type":"text/html","content_length":"93753","record_id":"<urn:uuid:97652732-0114-4b64-8f8e-10272942d63e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00111.warc.gz"}
Molecular Chiroptical Properties Chiral molecules are characterized by a unique three-dimensional handedness, and the resulting pairs of left- and right-handed enantiomers often exhibit distinct chemical activities when reacting within a chiral environment. Enantiomeric pairs of chiral molecules also exhibit distinct responses to left- and right-circularly polarized light in absorption, refraction, and scattering. These responses may be used to determine the handedness (i.e., the “absolute configuration”) of an enantiomerically pure sample, provided sufficient details about the corresponding circular dichroism, birefringence, or scattering intensity differences are known a priori. One of the major goals of our research group is the development of high-level ab initio quantum chemical models for prediction of molecular chiroptical properties such as specific rotation angles and CD spectra. Over the past several years, we have developed an efficient implementation of the coupled-cluster singles and double (CCSD) linear-response model that is applicable to medium-sized molecules such as [4]triangulane. These models are available as part of the PSI4 suite of quantum chemical programs. New developments attempting to extend the reach of these models to solvated systems (which are considerably more complex) has been a prominent feature in our current work. Local Correlation Methods The coupled cluster method is widely regarded as the “gold standard” of quantum chemical models because of the high accuracy it often provides for a variety of molecular properties, including structures, thermodynamic constants, and vibrational spectra. However, conventional coupled cluster theory suffers from high-degree polynomial scaling with the size of the molecular system (as measured by the number of electrons and the size of the one-electron basis set). The CCSD model, for example, scales as the sixth power of the molecular size, which implies that doubling the size of the molecule increases the computational time by a factor of 2^6 = 64. Even with state-of-the-art computing facilities, this scaling precludes application of coupled cluster theory to non-symmetric molecules larger than 10-12 heavy atoms. Local correlation, a concept pioneered by Pulay and Saebø, provides one possible route over the scaling wall through a judicious choice of molecular orbital basis. The “canonical” MO’s, although convenient, are often delocalized over the entire molecular framework and often lead to overestimation of electronic interactions on spatially distant atoms. Pulay and Saebø demonstrated that if one abandons canonical orbitals and instead chooses a more localized form, vast numbers of electronic wave-function parameters become negligible and may thus be ignored. However, the application of the localization concept to molecular response properties introduces an additional complication: accurate representation of the derivative of the wave function with respect to an external perturbation, such as an electric or magnetic field, requires a more accurate representation of the unperturbed wave function than is necessary for computing the energy alone. Thus, the usual “orbital domain” structure that has performed so admirably for locally correlated ground-state energy calculations fails when applied to response properties. To compensate, we have devised a new domain construction approach based on analysis of the response of the individual molecular orbitals to the relevant external fields. This approach has proven successful for simple dipole polarizabilities, and work is underway to extend it to mixed electric/magnetic properties, such as optical rotation. Sampling Configuration Space in Chiral Systems Whether measured in solution or gas phase, the effects of conformational degrees of freedom are an important factor in the accurate prediction of chiroptical properties. In the gas phase, this can be accomplished by sampling a Boltzmann distribution of the main contributing conformers. A python-based driver for automating the process of conformation generation, Boltzmann population analysis, and ab initio property prediction using distributed computing resources is currently being developed. Results from this driver have been published in the Journal of Natural Products! In solution phase, the configuration space is considerably larger due to the inclusion of solvent. It has been shown that small perturbations of the solvent shell can significantly impact chiroptical response; because of this, it is necessary to average over a sufficiently large sample of this space to accurately model the solute-solvent interaction. To this end, snapshots from molecular dynamics trajectories have been used to calculate the specific rotation of chiral solutes in different solvents. A paper on this work is currently being revised. Alternative Models in Electronic Structure Theory In addition to pushing the boundaries of traditional electronic structure methods like coupled cluster and density functional theory, additional models are being considered. These include Monte Carlo simulations, fragmentation schemes, real-time methods (such as real-time Coupled Cluster), and machine learning. Check back for more details on these and other projects as they develop.	{"url":"https://crawford.chem.vt.edu/research/","timestamp":"2024-11-05T19:58:19Z","content_type":"text/html","content_length":"40321","record_id":"<urn:uuid:85f6f91e-9d7d-4eb8-91d9-656c348fdc3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00074.warc.gz"}
Integrating Factor In Maths, an integrating factor is a function used to solve differential equations. It is a function in which an ordinary differential equation can be multiplied to make the function integrable. It is usually applied to solve ordinary differential equations. Also, we can use this factor within multivariable calculus. When multiplied by an integrating factor, an inaccurate differential is made into an accurate differential (which can be later integrated to give a scalar field). It has a major application in thermodynamics where the temperature becomes the integrating factor that makes entropy an exact differential. What are Differential Equations? Differential equations play a vital role in Mathematics. These are the equations that necessarily involve derivatives. There are various types of differential equations; such as – homogeneous and non-homogeneous, linear and nonlinear, ordinary and partial. The differential equation may be of the first order, second order and ever more than that. The n^th order differential equation is an equation involving nth derivative. The most common differential equations that we often come across are first-order linear differential equations. The ordinary linear differential equations are represented in the following general form: y’ + P y=Q dy/dx + P(x) y = Q(x) Where y’ or dy/dx is the first derivative. Also, the functions P and Q are the functions of x only. There are mainly two methods which are utilized in order to solve the linear first-order differential equations: • Separable Method • Integrating Factor Method In this article, we are going to discuss what is integrating factor method, and how the integrating factors are used to solve the first and second-order differential equations. Integrating Factor Method Integrating factor is defined as the function which is selected in order to solve the given differential equation. It is most commonly used in ordinary linear differential equations of the first When the given differential equation is of the form; dy/dx + P(x) y = Q(x) then the integrating factor is defined as; Where P(x) (the function of x) is a multiple of y and μ denotes integrating factor. Solving First-Order Differential Equation Using Integrating Factor Below are the steps to solve the first-order differential equation using the integrating factor. • Compare the given equation with differential equation form and find the value of P(x). • Calculate the integrating factor μ. • Multiply the differential equation with integrating factor on both sides in such a way; μ dy/dx + μP(x)y = μQ(x) • In this way, on the left-hand side, we obtain a particular differential form. I.e d/dx(μ y) = μQ(x) • In the end, we shall integrate this expression and get the required solution to the given equation: μ y = ∫μQ(x)dx+C Solving Second Order Differential Equation Using Integrating Factor The second-order differential equation can be solved using the integrating factor method. Let the given differential equation be, y” + P(x) y’ = Q(x) The second-order equation of the above form can only be solved by using the integrating factor. • Substitute y’ = u; so that the equation becomes similar to the first-order equation as shown: u’ + P(x) u = Q(x) • Now, this equation can be solved by integrating factor technique as described in the section above for first-order equations and we reach the equation: μ u=∫μQ(x)dx+C • Find the value of u from this equation. Since u = y’, hence to find the value of y, integrate the equation. In this way, we get the required solution. Integrating Factor Example Solve the differential equation using the integrating factor: (dy/dx) – (3y/x+1) = (x+1)^4 Given: (dy/dx) – (3y/x+1) = (x+1)^4 First, find the integrating factor: μ = e^ ∫ p(x) dx μ = e^ ∫(-3/x+1) dx ∫(-3/ x+1)dx = -3 ln (x+1) = ln (x+1)^-3 Hence, we get μ =e ^ln (x+1)^-3 μ = 1/ (x+1)^3 Now, multiply the integrating factor on both the sides of the given differential eqaution: [1/ (x+1) ] [dy/dx] – [3y/( (x+1) )] = (x+1) Integrate both the sides, we get: ] = [(1/2)x Here, c is a constant Therefore, the general solution of the given differential eqaution is y = [(x+1)^3] [(1/2)x^2+x+c]. Keep visiting BYJU’S – The Learning App and download the app to learn all the Calculus related topics and watch interesting and engaging videos to learn with ease.	{"url":"https://mathlake.com/Integrating-Factor","timestamp":"2024-11-13T09:56:24Z","content_type":"text/html","content_length":"12994","record_id":"<urn:uuid:f52c7ad7-5f6f-4076-95ab-ab3658830515>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00127.warc.gz"}
Extremums on HP 42S 12-23-2021, 04:51 PM Post: #1 lrdheat Posts: 872 Senior Member Joined: Feb 2014 Extremums on HP 42S The equation sqrt(ABS(x^3 +2x +4)) produces extremums where I expect at x~-2.59 and x~0, but does not find the extremum at x~-1.33. Anyone know why? LBL “FX” 12-23-2021, 06:18 PM Post: #2 rprosperi Posts: 6,632 Super Moderator Joined: Dec 2013 RE: Extremums on HP 42S Presuming you are using SOLVE (you don't say what you are using, or how, so we can only guess) you need to provide 2 initial guesses which straddle the desired root. What guesses are you providing and what results to they lead to? --Bob Prosperi 12-23-2021, 07:07 PM Post: #3 Thomas Okken Posts: 1,897 Senior Member Joined: Feb 2014 RE: Extremums on HP 42S Keep in mind that SOLVE was designed to find roots, not extrema. It will report extrema if it gets stuck on them, but there is no guarantee it will find any, regardless of what starting guesses you If you are specifically looking for extrema, you should use SOLVE to find roots of the derivative. 12-23-2021, 07:09 PM Post: #4 lrdheat Posts: 872 Senior Member Joined: Feb 2014 RE: Extremums on HP 42S I used the solve protocol, adding MVAR X, RCL X to the routine. Tried -1.35, -1.31 as the bound, and it finds the extremum at X~0. 12-24-2021, 09:13 AM (This post was last modified: 12-24-2021 09:20 AM by Pekis.) Post: #5 Pekis Posts: 171 Member Joined: Aug 2014 RE: Extremums on HP 42S Is it this function ? If it's the case, I don't understand your values ... 12-24-2021, 11:32 AM Post: #6 Sylvain Cote Posts: 2,158 Senior Member Joined: Dec 2013 RE: Extremums on HP 42S (12-24-2021 09:13 AM)Pekis Wrote: Is it this function ? The formula is: $ \mathsf{y} = x^3 + 2(x^2) + 4 $ 12-24-2021, 12:17 PM Post: #7 Albert Chan Posts: 2,774 Senior Member Joined: Jul 2018 RE: Extremums on HP 42S I think the function is y = sqrt(abs(x^3+2x^2+4)) (x^3+2x^2+4)' = 3x^2 + 4x = 3x(x+4/3) To understand why extremum -4/3 is misssed, we need to understand how Secant method work. Assuming plot is above x-axis: From (x1,y1),(x2,y2), secant line locate the root at the x-axis, go up, to get y3. Then, it uses (x2,y2),(x3,y3) to locate x4, go up, to get y4 ... Extremum that shaped like a $\bigcup$ may be located. Extremum that shaped like $\bigcap$ will not (new x's will move away from it) 12-24-2021, 03:43 PM Post: #8 lrdheat Posts: 872 Senior Member Joined: Feb 2014 RE: Extremums on HP 42S Thanks! My equation did use 2X^2, sorry for my typing. This is neat in that it demonstrates why this particular integration approach may not find all extremums! There is no perfect numerical approach to integration approximations, but some amazingly good ones! 12-24-2021, 04:20 PM (This post was last modified: 12-24-2021 04:21 PM by toml_12953.) Post: #9 toml_12953 Posts: 2,192 Senior Member Joined: Dec 2013 RE: Extremums on HP 42S (12-24-2021 09:13 AM)Pekis Wrote: Hello, Is it this function ? If it's the case, I don't understand your values ... I wondered that, too. There's only one extremum AFAICS: -1.18 or so as your graph shows. Tom L Cui bono? 12-25-2021, 04:02 PM (This post was last modified: 12-25-2021 04:37 PM by C.Ret.) Post: #10 C.Ret Posts: 291 Member Joined: Dec 2013 RE: Extremums on HP 42S In the case the investigated function is $ f(x)=\sqrt{\left\| x^3+2x^2+4 \right\|} $, the extremums can be found solving $ \frac{\partial f(x)}{\partial x}=0 $ equation. The only problem here is that $ f(x) $ isn't continuous all over the interval and the derivative function $ \frac{\partial f(x)}{\partial x} $ is not definite at the point $ x_d $ where $ x_d^ 3+2x_d^2+4=0 $ . Numerically, $ x_d\approx-2.5943 $. This point is indicated by a cross on the following screen capture. At the discontinuity point $ x_d $, the extremums or root can't be determinate using method based on function continuity. This explain why the algorithm of the SOLVE instruction of the HP-42S (or related clone or simulators) can only be used inefficiently in vicinity of the discontinuity. Numerically, if the investigated function is $ f(x)=\sqrt{\left\| x^3+2x^2+4 \right\|} $; one may determine three extremums : • discontinued root at $ x_d\approx-2.5943 $ where $ f(x_d)=0 $ • intermediate extremum at $ x_b=-\frac{4}{3} $ where $ f(x_b) = \frac{2.\sqrt{105}}{9}\approx2.2771 $ • last extremum on y-axis at $ x_0=0 $ where $ f(x_0)=2 $ 12-25-2021, 04:16 PM Post: #11 lrdheat Posts: 872 Senior Member Joined: Feb 2014 RE: Extremums on HP 42S What I find interesting is that the HP 42S does find the zero at ~-2.59! User(s) browsing this thread: 1 Guest(s)	{"url":"https://www.hpmuseum.org/forum/thread-17849-post-155860.html#pid155860","timestamp":"2024-11-09T17:23:56Z","content_type":"application/xhtml+xml","content_length":"45518","record_id":"<urn:uuid:dc150116-6b74-4ba5-a914-314cf1c0e7e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00675.warc.gz"}
orksheets for 1st Class Recommended Topics for you Math Number Sense 2nd Grade Place Value and Number Sense Number Sense and Ten Stragegies Kindergarten Math - Number Sense Explore Number Sense Worksheets by Grades Explore Number Sense Worksheets for class 1 by Topic Explore Other Subject Worksheets for class 1 Explore printable Number Sense worksheets for 1st Class Number Sense worksheets for Class 1 are an essential tool for teachers looking to help their students develop a strong foundation in mathematics. These worksheets are specifically designed to target key concepts such as counting, comparing, and ordering numbers, as well as understanding place value and basic arithmetic operations. By incorporating these worksheets into their lesson plans, teachers can provide their Class 1 students with engaging and interactive activities that reinforce their learning and help them gain confidence in their math skills. Furthermore, these Number Sense worksheets for Class 1 can be easily adapted to suit the needs of individual learners, making them a versatile and valuable resource for any classroom. Quizizz is an excellent platform for teachers to find a wide range of resources, including Number Sense worksheets for Class 1, math games, and quizzes. This platform offers a variety of interactive and engaging activities that can be used to supplement traditional teaching methods and enhance students' learning experiences. With Quizizz, teachers can easily create customized quizzes and assignments, track student progress, and even collaborate with other educators to share resources and best practices. In addition to Number Sense worksheets for Class 1, Quizizz also offers resources for other math topics and grade levels, making it a one-stop-shop for all your educational needs. By incorporating Quizizz into your teaching strategy, you can ensure that your Class 1 students develop a strong foundation in math and continue to build upon their skills as they progress through their education.	{"url":"https://quizizz.com/en/number-sense-worksheets-class-1","timestamp":"2024-11-12T09:39:32Z","content_type":"text/html","content_length":"156455","record_id":"<urn:uuid:2d34987f-4ecd-48ff-914e-8128c2b14b48>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00733.warc.gz"}
Lesson 17 Comparing Transformations • Let's ask questions to figure out transformations of trigonometric functions. 17.1: Three Functions For each pair of graphs, be prepared to describe a transformation from the graph on top to the graph on bottom. 17.2: Info Gap: What's the Transformation? Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the data card: 1. Silently read the information on your card. 2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your 3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?” 4. Read the problem card, and solve the problem independently. 5. Share the data card, and discuss your reasoning. If your teacher gives you the problem card: 1. Silently read your card and think about what information you need to answer the question. 2. Ask your partner for the specific information that you need. 3. Explain to your partner how you are using the information to solve the problem. 4. When you have enough information, share the problem card with your partner, and solve the problem independently. 5. Read the data card, and discuss your reasoning. Suppose we considered the function $T$ which is the sum of $Q$ and $S$. Is $T$ also periodic? If yes, what is its period? If no, explain why not. 17.3: Match the Graph Here is the graph of $f(x)=\cos(x)$ and the graph of $g$, which is a transformation of $f$. 1. Identify a transformation that takes $f$ to $g$ and write an equation for $g$ in terms of $f$ matching the transformation. 2. Identify at least one other transformation that takes $f$ to $g$ and write an equation for $g$ in terms of $f$ matching the transformation. Here are graphs of two trigonometric functions: The function $f$ is given by $f(x) = \sin(x)$. How can we transform the graph of $f$ to look like the graph of $g$? Looking at the graph of $f$, we need to make the period and the amplitude smaller, translate the graph up, and translate the graph horizontally so it has a minimum at $x=0$. The amplitude of $g$ is $\frac{1}{2}$ and the period is $\frac{\pi}{2}$ so we can begin by changing $\sin(x)$ to $\frac{1}{2}\sin(4x)$. The midline of $g$ is 2.5 so we need a vertical translation of 2.5, giving us $\frac{1}{2}\sin(4x)+2.5$. The function $g$ has a minimum when $x = 0$ while $\frac{1}{2}\sin(4x)+2.5$ has a minimum when $x = \text-\frac{ \pi}{8}$. So a horizontal translation to the right by $\frac{\pi}{8}$ is needed. Putting all of this together, we have an expression for $g$: $g(x) = \frac{1}{2}\sin(4(x-\frac{\pi}{8}))+2.5$. Another way to think about the transformation is to first notice that $g$ has a minimum when $x$ is 0. If we translate $\sin(x)$ right by $\frac{\pi}{2}$, then $\sin(x-\frac{\pi}{2})$ also has a minimum at $x=0$. The period of $g$ is $\frac{\pi}{2}$, so we can write $\sin(4x-\frac{\pi}{2})$. The amplitude of $g$ is $\frac{1}{2}$ and it's midline is 2.5, so we end up with the expression $\frac{1}{2} \sin(4x-\frac{\pi}{2})+2.5$ for $g$. This is the same as $g(x) = \frac{1}{2}\sin(4(x-\frac{\pi}{8}))+2.5$, just thinking of the horizontal translation and scaling in different orders. • amplitude The maximum distance of the values of a periodic function above or below the midline. • midline The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose $y$-coordinate is that value.	{"url":"https://curriculum.illustrativemathematics.org/HS/students/3/6/17/index.html","timestamp":"2024-11-07T22:12:10Z","content_type":"text/html","content_length":"97661","record_id":"<urn:uuid:b44437eb-2c45-4d8d-b10f-91595e15dc72>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00470.warc.gz"}
Quadratic Equation Solver - Free Online Calculators Quadratic Equation Solver A “Quadratic Equation Solver” is a tool used to find the solutions of a quadratic equation in the form ax² + bx + c = 0. It simplifies solving by applying the quadratic formula, factoring, or completing the square, offering exact or approximate solutions. This tool is widely used in algebra to calculate the roots or “x” values where the equation equals zero. It’s helpful for students, teachers, and professionals solving math problems quickly. The solver saves time and reduces errors in complex calculations. Root 1 (x1): Root 2 (x2): The Quadratic Equation Solver: Unlocking the Solutions to Quadratic Equations Quadratic equations are a fundamental part of algebra, representing a wide range of real-world problems and mathematical concepts. These equations take the form of "ax^2 + bx + c = 0," where 'a,' 'b,' and 'c' are coefficients, and 'x' represents the variable. To find the values of 'x' that satisfy the equation, you can turn to a Quadratic Equation Solver, a valuable tool that simplifies the process of solving these equations. How Does the Quadratic Equation Solver Work? The Quadratic Equation Solver utilizes the quadratic formula, which is a well-known mathematical formula designed explicitly for solving quadratic equations: Here's how the Quadratic Equation Solver operates: 1. Coefficient Input: Users input the values of 'a,' 'b,' and 'c' from their quadratic equation into the solver. These coefficients determine the nature of the equation and its solutions. 2. Discriminant Calculation: The solver computes the discriminant, which is the value inside the square root of the quadratic formula: �2−4��b2−4ac. The discriminant provides essential information about the nature of the solutions. 3. Solution Computation: Based on the discriminant's value, the solver determines the type of solutions the quadratic equation has: □ If the discriminant is positive (�2−4��>0b2−4ac>0), the equation has two distinct real solutions. □ If the discriminant is zero (�2−4��=0b2−4ac=0), the equation has one real solution (a repeated root). □ If the discriminant is negative (�2−4��<0b2−4ac<0), the equation has no real solutions, but it has complex solutions. 4. Solution Presentation: The solver then presents the solutions ('x' values) to the user. In the case of complex solutions, it typically displays both the real and imaginary parts. Key Benefits of Using a Quadratic Equation Solver: 1. Accuracy: Quadratic equations can involve intricate calculations, and manual solving can be prone to errors. The solver ensures precise results. 2. Speed: Solving quadratic equations manually can be time-consuming, especially for equations with complex coefficients. The solver provides solutions almost instantly. 3. Understanding: Students can use the solver to verify their work, learn how to apply the quadratic formula, and gain a deeper understanding of the properties of quadratic equations. 4. Real-World Applications: Quadratic equations are prevalent in fields like physics, engineering, and economics. The solver is a valuable tool for professionals who need to solve such equations in their work. Considerations When Using a Quadratic Equation Solver: 1. Coefficient Accuracy: Ensure that you input the correct values for 'a,' 'b,' and 'c' to obtain accurate solutions. 2. Complex Solutions: Be prepared to interpret complex solutions if the discriminant indicates that they exist. Complex solutions involve both real and imaginary parts. 3. Multiple Solutions: Keep in mind that quadratic equations may have one, two, or no real solutions, depending on the discriminant's value. In conclusion, the Quadratic Equation Solver is a powerful tool that simplifies the process of solving quadratic equations. Whether you're a student learning algebraic concepts or a professional dealing with real-world problems, this solver provides accurate and efficient solutions to quadratic equations of all kinds. It's a valuable resource for anyone seeking to unlock the secrets hidden within these fundamental mathematical expressions.	{"url":"https://nowcalculator.com/quadratic-equation-solver/","timestamp":"2024-11-07T19:44:47Z","content_type":"text/html","content_length":"289718","record_id":"<urn:uuid:d131091b-b7fb-4064-9b3b-91460c23fb5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00816.warc.gz"}
Permutation puzzles August 26, 2024 This post builds further on God's number. Yesterday, I put together a small tool to explore permutation groups and how they are structured with respect to generating the entire group from some base elements. Here are some thoughts and First, I tried a series of simple group elements; swapping two adjacent numbers. For example, 1 0 2 3, 0 2 1 3 and 0 1 3 2. These generate S4S_4 in its entirety. God's number for this puzzle is 6, and, perhaps unsurprisingly, the single "hardest" element to reach is 3 2 1 0. This element is the only one that requires the full 6 moves. In general, it seems that this type of choice of base elements in SnS_n leads to God's number being g=12n(n−1)g = \tfrac12 n(n - 1) elements. Curiously, the amount of elements that need ii moves to be solved seems to be equal to the amount of elements that need g−ig - i. There is always one element the hardest to get, which is the identity written backwards. Quite satisfyingly, solving the group to the identity is just as hard as solving it to the reverse identity; the puzzle is completely symmetric. Initially, I thought; well, that kind of makes sense; the base elements we chose were entirely symmetric, so it's not surprising that the puzzle also is. But, as it turns out, that's not quite right. If we add the one element swapping the first and last number (in the example for S4S_4, 3 1 2 0) then the solution spaces are no longer symmetric, with elements needing a number of moves closer to God's number on average. Perhaps the reason the specific choice of base elements leads to a symmetric puzzle is due to how the solutions are composed. The optimal solution turns out to be moving the numbers in the permutation from out- to inwards; for example, this is the steps we'd use to solve 3 2 1 0: • 3 2 1 0 • 2 3 1 0 • 2 1 3 0 • 2 1 0 3 (now the puzzle is essentially reduced to the one in S3S_3) • 1 2 0 3 • 1 0 2 3 (reduced to S2S_2) • 0 1 2 3 It doesn't matter which side we are completing; the goal is to reduce the puzzle to a simpler one. Once the 3 is in place, the move 0 1 3 2 becomes usless; it doesn't do anything that the other permutations can't do in the same (or fewer) number of moves. This also explains that God's numbers for these types of puzzle groups are triangle numbers; we can show this with induction. We can define a "distance" d:G2→Nd: G^2\to\mathbb{N} between two elements in a puzzle as the least amount of moves needed to get from one to the other. Since we consider powers of each move as moves themselves, this means d(a,b)=d(b,a)d(a, b) = d(b, a). This must be true, since if qq represents a series of moves m1,m2,…,mim_1, m_2, \dots, m_i, then q−1=mi−1…m1−1q^{-1} = m_i^{-1} \dots m_1^{-1} also represents a series of moves, and an equally long one at that. That means if a⋅q=ba \cdot q = b then b⋅q−1=ab \cdot q^{-1} = a, proving that d(a,b)=d(b,a)d(a, b) = d(b, a). We also know that d(a,b)≤gd(a, b) \leq g (where gg is God's number). This must be true since, if a⋅m1…mi=ba \cdot m_1 \dots m_i = b, then b−1am1…mi=1b^{-1} a m_1 \dots m_i = 1. This shows that i≤gi\ leq g and therefore d(a,b)≤gd(a, b) \leq g for any a,b∈Ga,b\in G. The amount of moves needed to "solve" a specific state aa is then exactly d(a,1)d(a, 1). In general, d(a,b)=d(c⋅a,c⋅b)d(a, b) = d(c\cdot a, c\cdot b) since if qq is a solution to get from aa to bb, then a⋅q=ba\cdot q = b, and therefore c⋅a⋅q=c⋅bc\cdot a\cdot q = c\cdot b. This is not (necessarily) true for right multiplication, i.e. d(a,b)d(a, b) is not always equal to d(a⋅c,b⋅c)d(a \cdot c, b \cdot c). Lastly, like all well-behaving metric spaces, we find the triangle inequality d(a,b)+d(b,c)≥d(a,c)d(a, b) + d(b, c) \geq d(a, c) This is true since, if a⋅q=ba \cdot q = b and b⋅r=cb \cdot r = c, then definitely a⋅q⋅r=ca\cdot q\cdot r = c. The shortest distance to get from aa to cc is therefore at most as long as q⋅rq\cdot r, and potentially shorter. With this notion of distance, we can define a sort of "absolute value" to be the distance to the solved state. That is, ∣a∣:=d(a,1)\|a\| := d(a, 1). The triangle inequality then shows that ∣a∣+∣b∣≥∣b−1⋅a∣\|a\| + \|b\| \geq \|b^{-1}\cdot a\|. An interesting thing happens at some point down the line of generating solutions; a single state aa might have multiple distinct solutions. In formula, it means there are two different moves mm and kk such that both ∣a⋅m∣<∣a∣\|a\cdot m\| < \|a\| and simultaneously ∣a⋅k∣<∣a∣\|a\cdot k\| < \|a\|. For the Rubik's cube, this does not happen until ∣a∣=5\|a\| = 5. Specifically, we find: (L−1⋅U−1⋅L2⋅F−1⋅D−1)(L⋅D⋅L2⋅F⋅U)=1(L^{-1}\cdot U^{-1}\cdot L^2\cdot F^{-1}\cdot D^{-1}) (L\cdot D\cdot L^2\cdot F\cdot U) = 1 After the first 5 moves, we can actually choose whether we do DD or LL, and both solve the cube optimally. This type of product can actually be quite useful in solving some groups optimally, since we can first find a solution of any length (this is not particularly hard) and then find sub-prodocts in the solution that resemble part of the product above. If a series of 6 moves coincides with above, then we can remove them in favor of the four moves that are equivalent. I'm not sure if that is a viable strategy, though; for a Rubik's cube, I imagine you'd need a lot of these types of products to reduce a long solution down to an optimal one.	{"url":"https://vrugtehagel.nl/math/permutation-puzzles/","timestamp":"2024-11-09T19:20:37Z","content_type":"text/html","content_length":"20991","record_id":"<urn:uuid:dbb137c1-6af8-4474-93a8-7b835480aa81>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00519.warc.gz"}
Centre for Mathematics, University of Coimbra - Seminars On a quasi-linear elliptic equation depending on the gradient Speaker: Daniele Puglisi (University of Catania, Italy) Interpolations of a-Hölderian mappings between normed spaces Speaker: Amiran Gogatishvili (Czech Academy of Sciences, Prague) We present some well-known and new results for identifying some interpolation spaces on nonlinear interpolation of α-Hölderian mappings between normed spaces. We give applications of these results to obtain some regularity results on the gradient of the weak or... Speaker: Enrique Zuazua (Friedrich-Alexander-Univ. Erlangen-Nürnberg, Germany) Weighted sifted colimits Speaker: Jirí Adámek (Czech Technical Univ., Prague, Czech Republic) In ordinary categories both filtered colimits and reflexive coequalizers have the property that in every variety of algebras such colimits are formed on the level of the underlying sets. More generally, all sifted colimits have that property. A category $ \mathcal D $ is sifted if colimits... Speaker: Pedro Lopes (IST, Univ. Lisboa) Rectifiability in Carnot groups Speaker: Daniela Di Donato (University of Pavia, Italy) The mathematics of biomedical imaging: the past, the present and some open problems Speaker: F. Alberto Grunbaum (UC Berkeley, USA) Definability and full abstraction for algebraic effects with recursion Speaker: Norihiro Yamada (CMUC, Univ. Coimbra) In this talk, I present an overview of my recent work on an intersection between algebra, logic and topology: definability and full abstraction for algebraic theories in the sense of universal algebra combined with a well-known formal calculus for higher-order computation. The... Speaker: Lianet De la Cruz Toranzo (TU Bergakademie Freiberg, Germany) Speaker: Pedro Vaz (U.C. Louvain, Belgium)	{"url":"https://cmuc.mat.uc.pt/rdonweb/event/pplistseminar.do;jsessionid=8F3A3F907438F1CC46421E0CE9EDB411?menu=activities","timestamp":"2024-11-02T11:13:36Z","content_type":"text/html","content_length":"20311","record_id":"<urn:uuid:d1ed6f12-d5d2-42c4-b02b-2bb774a0e476>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00668.warc.gz"}