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Directed and Undirected Graphs- MATLAB & Simulink (2024)
Directed and Undirected Graphs
What Is a Graph?
A graph is a collection of nodes and edges thatrepresents relationships:
• Nodes are verticesthat correspond to objects.
• Edges are the connectionsbetween objects.
• The graph edges sometimes have Weights,which indicate the strength (or some other attribute) of each connectionbetween the nodes.
These definitions are general, as the exact meaning of the nodesand edges in a graph depends on the specific application. For instance,you can model the friendships in a social network using a graph.
Thegraph nodes are people, and the edges represent friendships. The naturalcorrespondence of graphs to physical objects and situations meansthat you can use graphs to model a wide variety of systems.
For example:
• Web page linking — The graph nodes are web pages, and the edges represent hyperlinks between pages.
• Airports — The graph nodes are airports, andthe edges represent flights between airports.
In MATLAB^®, the graph and digraph functions construct objects that represent undirected and directed graphs.
• Undirected graphs have edges that do not have a direction. The edges indicate a two-way relationship, in that each edge can be traversed in both directions. This figure shows a simple undirected
graph with three nodes and three edges.
• Directed graphs have edges with direction. The edges indicate a one-way relationship, in that each edge can only be traversed in a single direction. This figure shows a simple directed graph with
three nodes and two edges.
The exact position, length, or orientation of the edges in agraph illustration typically do not have meaning. In other words,the same graph can be visualized in several different ways by
rearrangingthe nodes and/or distorting the edges, as long as the underlying structuredoes not change.
Self-loops and Multigraphs
Graphs created using graph and digraph can have one or more self-loops, which are edges connecting a node to itself. Additionally, graphs can have multiple edges with the same source and target
nodes, and the graph is then known as a multigraph. A multigraph may or may not contain self-loops.
For the purposes of graph algorithm functions in MATLAB, a graph containing a node with a single self-loop is not a multigraph. However, if the graph contains a node with multiple self-loops, it is a
For example, the following figure shows an undirected multigraph with self-loops. Node A has three self-loops, while node C has one. The graph contains these three conditions, any one of which makes
it a multigraph.
• Node A has three self-loops.
• Nodes A and B have five edges between them.
• Nodes A and C have two edges between them.
To determine whether a given graph is a multigraph, use the ismultigraph function.
Creating Graphs
The primary ways to create a graph include using an adjacency matrix or an edge list.
Adjacency Matrix
One way to represent the information in a graph is with a square adjacencymatrix. The nonzero entries in an adjacency matrix indicatean edge between two nodes, and the value of the entry indicates
theweight of the edge. The diagonal elements of an adjacency matrix aretypically zero, but a nonzero diagonal element indicates a self-loop,or a node that is connected to itself by an edge.
• When you use graph to create an undirected graph, the adjacency matrix must be symmetric. In practice, the matrices are frequently triangular to avoid repetition. To construct an undirected graph
using only the upper or lower triangle of the adjacency matrix, use graph(A,'upper') or graph(A,'lower').
• When you use digraph to create a directed graph, the adjacency matrix does not need to be symmetric.
• For large graphs, the adjacency matrix contains manyzeros and is typically a sparse matrix.
• You cannot create a multigraph from an adjacency matrix.
For example, consider this undirected graph.
You can represent the graph with this adjacency matrix:
$\left(\begin{array}{ccc}0& 1& 2\\ 1& 0& 3\\ 2& 3& 0\end{array}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$
To construct the graph in MATLAB, input:
A = [0 1 2; 1 0 3; 2 3 0];node_names = {'A','B','C'};G = graph(A,node_names)
G = graph with properties: Edges: [3×2 table] Nodes: [3×1 table]
You can use the graph or digraph functions to create a graph using an adjacency matrix, or you can use the adjacency function to find the weighted or unweighted sparse adjacency matrix of a
preexisting graph.
Edge List
Another way to represent the information in a graph is by listing all of the edges.
For example, consider the same undirected graph.
Now represent the graph by the edge list
$\begin{array}{l}\text{EdgeWeight}\\ \begin{array}{cc}\begin{array}{c}\left(A,B\right)\\ \left(A,C\right)\end{array}& \begin{array}{c}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& 1\end{array}
& \end{array}& \end{array}\\ \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& 2\end{array}& \end{array}& \end{array}\end{array}\\ \left(B,C\right)& \begin{array}{cc}\begin{array}{cc}\begin{array}
{cc}& 3\end{array}& \end{array}& \end{array}\end{array}\end{array}$
From the edge list it is easy to conclude that the graph has three unique nodes, A, B, and C, which are connected by the three listed edges. If the graph had disconnected nodes, they would not be
found in the edge list, and would have to be specified separately.
In MATLAB, the list of edges is separated by column into source nodesand target nodes. For directed graphs the edgedirection (from source to target) is important, but for undirectedgraphs the source
and target node are interchangeable. One way toconstruct this graph using the edge list is to use separate inputsfor the source nodes, target nodes, and edge weights:
source_nodes = {'A','A','B'};target_nodes = {'B','C','C'};edge_weights = [1 2 3];G = graph(source_nodes, target_nodes, edge_weights);
Both graph and digraph permit construction of a simple graph or multigraph from an edge list. After constructing a graph, G, you can look at the edges (and their properties) with the command G.Edges.
The order of the edges in G.Edges is sorted by source node (first column) and secondarily by target node (second column). For undirected graphs, the node with the smaller index is listed as the
source node, and the node with the larger index is listed as the target node.
Since the underlying implementation of graph and digraph depends on sparse matrices, many of the same indexing costs apply. Using one of the previous methods to construct a graph all at once from the
triplet pairs (source,target,weight) is quicker than creating an empty graph and iteratively adding more nodes and edges. For best performance, minimize the number of calls to graph, digraph,
addedge, addnode, rmedge, and rmnode.
Graph Node IDs
By default, all of the nodes in a graph created using graph or digraph are numbered. Therefore, you always can refer to them by their numeric node index.
If the graph has node names (that is, G.Nodes containsa variable Name), then you also can refer to thenodes in a graph using their names. Thus, named nodes in a graph canbe referred to by either
their node indices or node names. For example,node 1 can be called, 'A'.
The term node ID encompasses both aspects of node identification. The node ID refers to both the node index and the node name.
For convenience, MATLAB remembers which type of node ID you use when you call most graph functions. So if you refer to the nodes in a graph by their node indices, most graph functions return a
numeric answer that also refers to the nodes by their indices.
A = [0 1 1 0; 1 0 1 0; 1 1 0 1; 0 0 1 0];G = graph(A,{'a','b','c','d'});p = shortestpath(G,1,4)
However, if you refer to the nodes by their names, then most graph functions return an answer that also refers to the nodes by their names (contained in a cell array of character vectors or string
p1 = shortestpath(G,'a','d')
p1 = 1×3 cell array {'a'} {'c'} {'d'}
Use findnode to find the numeric node IDfor a given node name. Conversely, for a given numeric node ID, indexinto G.Nodes.Name to determine the correspondingnode name.
Modify or Query Existing Graph
After you construct a graph or digraph object, you can use a variety of functions to modify the graph structure or to determine how many nodes or edges the graph has. This table lists some available
functions for modifying or querying graph and digraph objects.
addedge Add one or more edges to a graph
rmedge Remove one or more edges from a graph
addnode Add one or more nodes to a graph
rmnode Remove one or more nodes from a graph
findnode Locate a specific node in a graph
findedge Locate a specific edge in a graph
numnodes Find the number of nodes in a graph
numedges Find the number of edges in a graph
edgecount Number of edges between specified nodes
flipedge Reverse the direction of directed graph edges
reordernodes Permute the order of the nodes in a graph
subgraph Extract subgraph
See Modify Nodes and Edges of Existing Graph for some commongraph modification examples.
See Also
graph | digraph
Related Topics
• Modify Nodes and Edges of Existing Graph
• Add Graph Node Names, Edge Weights, and Other Attributes
• Graph Plotting and Customization
MATLAB Command
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Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.
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Spectral triples and the geometry of fractals | EMS Press
Spectral triples and the geometry of fractals
• Erik Christensen
Unversity of Copenhagen, Denmark
• Cristina Ivan
University of Hannover, Germany
• Elmar Schrohe
University of Hannover, Germany
We construct spectral triples for the Sierpinski gasket as infinite sums of unbounded Fredholm modules associated with the holes in the gasket and investigate their properties. For each element in
the K-homology group we find a representative induced by one of our spectral triples. Not all of these triples, however, will have the right geometric properties. If we want the metric induced by the
spectral triple to give the geodesic distance, then we will have to include a certain minimal family of unbounded Fredholm modules. If we want the eigenvalues of the associated generalized Dirac
operator to have the right summability properties, then we get limitations on the number of summands that can be included. If we want the Dixmier trace of the spectral triple to coincide with a
multiple of the Hausdorff measure, then we must impose conditions on the distribution of the summands over the gasket. For the elements of a large subclass of the K-homology group, however, the
representatives are induced by triples having the desired geometric properties. We finally show that the same techniques can be applied to the Sierpinski pyramid.
Cite this article
Erik Christensen, Cristina Ivan, Elmar Schrohe, Spectral triples and the geometry of fractals. J. Noncommut. Geom. 6 (2012), no. 2, pp. 249–274
DOI 10.4171/JNCG/91
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Length is a measurement. The length of something is the distance between two ends of the thing. Short means a small length. Long means much length. Short and long are opposites. For two dimensional
things, length is usually the longer side.
A ruler is a tool used to measure length.
All the sides on shapes have a length. The length is between the two points of the side. You can also find the length of any two points on a shape, even if they are not on one side.
A shape can have different lengths based on how many dimensions it takes.
• The distance from the front of the bus to the back of the bus is 30 meters. The bus is 30 meters in length.
• A piece of wood is 10 meters X 10 cm X 15 cm. The piece of wood is 10 meters in length, 10 cm in breadth and 15 cm in height.
Length of time
Length can also mean an amount of time. The length is measured by looking at the time at the start, then looking at the time at the end.
You might sit down at one o'clock. If you stand up at three o'clock, you would be sitting for two hours. The length of time is two hours.
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Session file configuration: Adaptive polynomial order
11.7 Session file configuration: Adaptive polynomial order
An adaptive polynomial order procedure is available for 2D and Quasi-3D simulations. This procedure consists of the following steps:
• Advance the equations for a determined number of time steps
• Use the sensor defined in equation 9.9 to estimate an error measure (the variable used in the sensor can be specified). The error is defined here as the square of the sensor.
• Use the error to determine if the order in each element should be increased by one, decreased by one, or left unaltered.
• Project the solution in each element to the new polynomial order and use it as an initial condition to restart the equation, repeating all steps a given number of times.
It is important to note that restarting the simulation after the refinement can be an expensive operation (in a typical case 200 times the cost of a single time step). Therefore, the number of steps
between successive refinements needs to be carefully chosen, since if this value is too low the procedure becomes inefficient, while if it is too high the refinement might not capture accurately
structures that are convected by the flow.
11.7.1 Solver Info
The use of the adaptive polynomial order procedure is enforced through the definition of the Driver which has to be Adaptive.
11.7.2 Parameters
The following parameters can be specified in the PARAMETERS section of the session file:
• NumSteps: when using the adaptive order procedure, this parameter determines the number of time steps between successive refinements.
• NumRuns: this parameter defines the number of times the sequence of solving the equation and refining is performed. Therefore, the total number of time steps in the simulation is NumSteps ×
• AdaptiveMaxModes: sets the maximum number of modes (in each direction) that can be used in an element during the adaptive procedure. The solution will not be refined beyond this point, even if
the error is higher than the tolerance. Default value: 12.
• AdaptiveMinModes: sets the minimal number of modes (in each direction) that can be used in an element during the adaptive procedure. Default value: 4.
• AdaptiveUpperTolerance: defines a constant tolerance. The polynomial order in an element is increased whenever the error is higher than this value. This can be replaced by a spatially-varying
function, as described below. Default value: 10^-6.
• AdaptiveLowerinModes: defines a constant tolerance. The polynomial order in an element is decreased whenever the error is lower than this value. This can also be replaced by a spatially-varying
function. Default value: 10^-8.
• AdaptiveSensorVariable: integer defining which variable will be considered when calculating the error. For example, if this parameter is set to 1 in the Incompressible Navier-Stokes Solver, the
error will be estimated using the v velocity. Default value: 0.
11.7.3 Functions
Spatially varying tolerances can be specified by defining the functions AdaptiveLowerinModes and/or AdaptiveUpperTolerance. In this case, the tolerance in an element is taken as the average of the
function in the quadrature points of the element. If these functions are defined, the values set for the tolerances in the PARAMETERS section are ignored.
11.7.4 Restarting the simulation
The simulation can be restarted using the final distribution of polynomial orders obtained from the adaptive procedure by setting the expansions as
note that this will only affect the polynomial order. The initial condition still needs to be set correctly, and does not need to come from the same file used for the expansions.
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An infinitely big structure in the center of a black hole? - Math Research of Victor Porton
An infinitely big structure in the center of a black hole?
I remind that I defined generalized limit of arbitrary function. The limit may be an infinitely big value. It allows to define derivative and integral of an arbitrary function. I also defined what
are solutions of partial differential equations where such infinities (instead of e.g. real numbers or complex numbers) are defined. You may see … Continue reading An infinitely big structure in the
center of a black hole?
Copy and paste this URL into your WordPress site to embed
Copy and paste this code into your site to embed
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The truncated matrix-valued K-moment problem on ℝ<sup>d</sup>, ℂ<sup>d</sup>, and T<sup>d</sup>
The truncated matrix-valued K-moment problem on ℝ^d, ℂ^d, and T^d will be considered. The truncated matrix-valued K-moment problem on ℝ^d requires necessary and sufficient conditions for a
multisequence of Hermitian matrices {S[γ]}[γ∈Γ] (where Γ is a finite subset of ℕ^d[0]) to be the corresponding moments of a positive Hermitian matrix-valued Borel measure σ, and also the support of σ
must be contained in some given non-empty set K ⊆ ℝ^d, i.e., Given a non-empty set K ⊆ ℝ^d and a finite multisequence, indexed by a certain family of finite subsets of ℕ^d[0], of Hermitian matrices
we obtain necessary and sufficient conditions for the existence of a minimal finitely atomic measure which satisfies (0.1) and (0.2). In particular, our result can handle the case when Γ = {γ ∈ ℕ^d
[0]: 0 ≤ {pipe}γ{pipe} ≤ 2n + 1}. We will also discuss a similar result in the multivariable complex and polytorus setting.
ASJC Scopus subject areas
• General Mathematics
• Applied Mathematics
Dive into the research topics of 'The truncated matrix-valued K-moment problem on ℝ^d, ℂ^d, and T^d'. Together they form a unique fingerprint.
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R (programming language)
From Verify.Wiki
R is a software platform that provides statistical data analysis and visualization capabilities. Initial development was done by Ross Ihaka and Robert Gentleman and currently it is developed by the R
core team. The software is freely available, and it runs on major operating systems like Windows, Linux, and Mac OS. ^[1] R has established a reputation as an important tool for statistical
modelling, data visualization, data mining and machine learning. The R language incorporates all of the standard statistical tests, models, graphics and analyses, as well as providing a comprehensive
language for managing and manipulating data. Leading researchers in data science are widely using R in academia and software development. R is a GNU project which can be considered as a different
implementation of S.
1970 S was developed by John Chambers while working at Bell labs.
1993 Initial development by Ross Ihaka and Robert Gentleman at the University of Auckland in New Zealand as an implementation of the S programming language began.
1995 Source code was released under the GNU license.
1997 The R core development team was formed. ^[2]
Average Programmer Salaries
│Country│ Average Salary │Years of Experience │
│USA │115,000(US$)^[3] │5 │
│UK │57,500(UK£)^[4] │2-5 │
• R is open source and freely available software.
• R implement a wide variety of statistical and graphical techniques including classical statistical tests, linear and nonlinear modeling, time-series analysis, classification, clustering, and
• R provides a very wide variety of graphics for visualizing data. These capabilities are found in the base language and in specialized packages like ggplot2, vcd and scatterplot3d.
• R has a large number of packages that virtually support any statistical technique and the R community is noted for its active contributions in terms of packages.
• R is able to consume data from multiple systems like Excel, SPSS, Stata, SAS and relational databases
• R runs on mostly used operating systems like Windows, Linux, and Mac OS. It is also supported on 32 and 64 bit systems.
• R has a vibrant community that offers support and commercial support is also available.
• There are many learning materials available freely or at a cost. ^[5]
• R has stronger object-oriented programming facilities than most statistical computing languages which is inherited from S. Extending R is also eased by its lexical scoping rules. ^[6]
• R is difficult to learn for users without any computer programming background
• The documentation of R may be difficult to understand for a person without a good statistical training. ^[7]
• Managing large data-sets can be problematic because R stores its objects in memory. However, there are some packages that can remedy this by storing data on hard drive.
• Some packages have a quality deficiency. However if a package is useful to many people, it will quickly evolve into a very robust product through collaborative efforts.
• R lacks in speed and efficiency due to its design principles that are outdated.
Although R is the most comprehensive statistical analysis package available. ^[8] some people believe R as an accessible language is not for advanced programmers " Mat Adams says."I wouldn't even say
R is for programmers. It's best suited for people that have data-oriented problems they're trying to solve, regardless of their programming aptitude,".
The following examples illustrate the basic syntax of the language and use of the command-line interface.
Basic syntax
The following examples illustrate the basic syntax of the language and plot a 3D Surface.
install.packages("rgl") # installing external package
library(rgl) # calling external package provide "rgl.surface" function
z <- 2 * volcano # Exaggerate the relief
x <- 10 * (1:nrow(z)) # 10 meter spacing (S to N)
y <- 10 * (1:ncol(z)) # 10 meter spacing (E to W)
zlim <- range(z)
zlen <- zlim[2] - zlim[1] + 1
colorlut <-terrain.colors(zlen,alpha=0) # height color lookup table
col <- colorlut[ z-zlim[1]+1 ] # assign colors to heights for each point
rgl.surface(x, y, z, color=col, alpha=0.75, back="lines")
"Hello World" Example
Examples of R in use
Feature Comparison Chart ^[15]
│ Feature │ R │ Python │ SAS │ SPSS │ STATA │
│Outlier diagnostics │Available│Available│Available│Available│Available│
│Generalized linear models │Available│Available│Available│Available│Available│
│Univariate time series analysis │Available│Available│Available│Limited │Available│
│Multivariate time series analysis │Available│ │Available│ │Available│
│Cluster analysis │Available│Available│Available│Available│Available│
│Discriminant analysis │Available│Available│Available│Available│Available│
│Neural networks │Available│Available│Available│Limited │ │
│Classification and regression trees │Available│Available│Available│Limited │ │
│Random forests │Available│Available│Limited │ │ │
│Support vector machines │Available│Available│Available│ │ │
│Factor and principal component analysis │Available│Available│Available│Available│Available│
│Boosting Classification & Regression Trees │Available│Available│Limited │ │ │
│Nearest neighbor analysis │Available│Available│Available│Available│ │
Top Companies Providing R Solutions
Revolution analytics ^[16] a Microsoft company, provides commercial analytics solutions based on R.
Mango solutions provides training, consultancy and support for R. ^[17]
MicroStrategy Data Mining Services ^[18], a fully integrated component of the MicroStrategy BI platform that delivers the results of predictive models to all users in familiar, highly formatted, and
interactive reports and documents. Also, deploy any R Analytic in MicroStrategy Visualizations with the New R Integration Pack.
Quadbase^[19], provides software and services for data visualization, BI dashboards, reporting, R programming and predictive analytics.
simMachines ^[20] , provides the R-01 similarity search (k-nearest neighbor) engine, with high speed and zero tuning. We are the Berkeley DB of the Big Data era.
Text Analysis International ^[21], offers tools and services for natural language processing and information extraction, building on the VisualText(TM) IDE and NLP++(R) programming language.
The future of R
The popularity of R as an analytics platform continues to grow. The number of analytics jobs posted on indeed.com showed demand for R skills was higher than that of SPSS, Matlab, Minitab and stata.
Demand for SAS skills was higher than that of R but predictions show R will catch up in a few years. Data from Google scholar shows SPSS is the mostly used software ahead of SAS and R. However R and
stata are closing in on the gap. On software discussion forums Linkedin and Quora, R topic followers outnumbered those following SAS, SPSS and Stata. A 2015 survey of data scientists by Rexer
Analytics showed R was the most popular software. ^[22]
Top 5 Recent Tweets
│ Date │ Author │ Tweet │
│11 Dec 2015│@Bbl_Astrophyscs│And the #Rangers strike again! Quantitative analyst position this time. STEM background, R programming. Not bad! │
│11 Dec 2015│@R_Programming │R Tip: Visualy asses clustering tendency of data with dissplot{seriation} #rstats #analytics http://rstatistics.net │
│11 Dec 2015│@cbinsa │Career Portals Ss r learning programming by designing their own digital game using Construct 2 software. #hgmsteach │
│11 Dec 2015│@analyticbridge │How to: Parallel Programming in R and Python [Video] http://ow.ly/VA2Vd │
│11 Dec 2015│@Rbloggers │New R job: R Programming for a Daily Fantasy Sports Application http://www.r-users.com/jobs/r-programming-for-a-daily-fantasy-sports-application/│
Top 5 Lifetime Tweets
│- │ │ │
│ Date │ Author │ Tweet │
│6 Dec 2015 │@analyticbridge│R Programming: 35 Job Interview Questions and Answers #Rstats http://www.datasciencecentral.com/profiles/blogs/r-programming-job-interview-questions-and-answers … │
│1 Feb 2015 │@opensourceway │As demand for data scientists grows, companies are turning to open source programming language R: http://red.ht/15s6Aqt │
│24 January │@DrQz │#Microsoft to acquire Revolution Analytics, heavily embracing the R programming language & tools http://www.wired.com/2015/01/ │
│2015 │ │microsoft-acquires-open-source-data-science-company-revolution-analytics/ … #rstats #marketbuzz │
│5 Feb 2014 │@kdnuggets │An alternative to R and #Python: Julia: A High-Performance Programming Language for #DataScience and more http://buff.ly/1c5bcPe │
│23 January │@mrb_bk │R is an interesting program language that slightly changes my point of view about programming languages. │
│20154 │ │ │
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Function/R/Limit/Epsilon/Definition - Wikiversity
Limit of a function
Let ${\displaystyle {}T\subseteq \mathbb {R} }$ denote a subset and ${\displaystyle {}a\in \mathbb {R} }$ a point. Let
${\displaystyle f\colon T\longrightarrow \mathbb {R} }$
be a function. Then ${\displaystyle {}b\in \mathbb {R} }$ is called limit of ${\displaystyle {}f}$ in ${\displaystyle {}a}$, if for every ${\displaystyle {}\epsilon >0}$ there exists some ${\
displaystyle {}\delta >0}$ such that for all ${\displaystyle {}x\in T}$ fulfilling
${\displaystyle {}\vert {x-a}\vert \leq \delta \,,}$
the estimate
${\displaystyle {}\vert {f(x)-b}\vert \leq \epsilon \,}$
holds. In this case, we write
${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.}$
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Euler and Hamiltonian Paths and Circuits
Learning Outcomes
• Determine whether a graph has an Euler path and/ or circuit
• Use Fleury’s algorithm to find an Euler circuit
• Add edges to a graph to create an Euler circuit if one doesn’t exist
• Identify whether a graph has a Hamiltonian circuit or path
• Find the optimal Hamiltonian circuit for a graph using the brute force algorithm, the nearest neighbor algorithm, and the sorted edges algorithm
• Identify a connected graph that is a spanning tree
• Use Kruskal’s algorithm to form a spanning tree, and a minimum cost spanning tree
In the next lesson, we will investigate specific kinds of paths through a graph called Euler paths and circuits. Euler paths are an optimal path through a graph. They are named after him because it
was Euler who first defined them.
By counting the number of vertices of a graph, and their degree we can determine whether a graph has an Euler path or circuit. We will also learn another algorithm that will allow us to find an Euler
circuit once we determine that a graph has one.
Euler Circuits
In the first section, we created a graph of the Königsberg bridges and asked whether it was possible to walk across every bridge once. Because Euler first studied this question, these types of paths
are named after him.
Euler Path
An Euler path is a path that uses every edge in a graph with no repeats. Being a path, it does not have to return to the starting vertex.
In the graph shown below, there are several Euler paths. One such path is CABDCB. The path is shown in arrows to the right, with the order of edges numbered.
Euler Circuit
An Euler circuit is a circuit that uses every edge in a graph with no repeats. Being a circuit, it must start and end at the same vertex.
The graph below has several possible Euler circuits. Here’s a couple, starting and ending at vertex A: ADEACEFCBA and AECABCFEDA. The second is shown in arrows.
Look back at the example used for Euler paths—does that graph have an Euler circuit? A few tries will tell you no; that graph does not have an Euler circuit. When we were working with shortest paths,
we were interested in the optimal path. With Euler paths and circuits, we’re primarily interested in whether an Euler path or circuit exists.
Why do we care if an Euler circuit exists? Think back to our housing development lawn inspector from the beginning of the chapter. The lawn inspector is interested in walking as little as possible.
The ideal situation would be a circuit that covers every street with no repeats. That’s an Euler circuit! Luckily, Euler solved the question of whether or not an Euler path or circuit will exist.
Euler’s Path and Circuit Theorems
A graph will contain an Euler path if it contains at most two vertices of odd degree.
A graph will contain an Euler circuit if all vertices have even degree
In the graph below, vertices A and C have degree 4, since there are 4 edges leading into each vertex. B is degree 2, D is degree 3, and E is degree 1. This graph contains two vertices with odd degree
(D and E) and three vertices with even degree (A, B, and C), so Euler’s theorems tell us this graph has an Euler path, but not an Euler circuit.
Is there an Euler circuit on the housing development lawn inspector graph we created earlier in the chapter? All the highlighted vertices have odd degree. Since there are more than two vertices with
odd degree, there are no Euler paths or Euler circuits on this graph. Unfortunately our lawn inspector will need to do some backtracking.
When it snows in the same housing development, the snowplow has to plow both sides of every street. For simplicity, we’ll assume the plow is out early enough that it can ignore traffic laws and drive
down either side of the street in either direction. This can be visualized in the graph by drawing two edges for each street, representing the two sides of the street.
Notice that every vertex in this graph has even degree, so this graph does have an Euler circuit.
The following video gives more examples of how to determine an Euler path, and an Euler Circuit for a graph.
Fleury’s Algorithm
Now we know how to determine if a graph has an Euler circuit, but if it does, how do we find one? While it usually is possible to find an Euler circuit just by pulling out your pencil and trying to
find one, the more formal method is Fleury’s algorithm.
Fleury’s Algorithm
1. Start at any vertex if finding an Euler circuit. If finding an Euler path, start at one of the two vertices with odd degree.
2. Choose any edge leaving your current vertex, provided deleting that edge will not separate the graph into two disconnected sets of edges.
3. Add that edge to your circuit, and delete it from the graph.
4. Continue until you’re done.
Find an Euler Circuit on this graph using Fleury’s algorithm, starting at vertex A.
Try It
Does the graph below have an Euler Circuit? If so, find one.
The following video presents more examples of using Fleury’s algorithm to find an Euler Circuit.
Eulerization and the Chinese Postman Problem
Not every graph has an Euler path or circuit, yet our lawn inspector still needs to do her inspections. Her goal is to minimize the amount of walking she has to do. In order to do that, she will have
to duplicate some edges in the graph until an Euler circuit exists.
Eulerization is the process of adding edges to a graph to create an Euler circuit on a graph. To eulerize a graph, edges are duplicated to connect pairs of vertices with odd degree. Connecting two
odd degree vertices increases the degree of each, giving them both even degree. When two odd degree vertices are not directly connected, we can duplicate all edges in a path connecting the two.
Note that we can only duplicate edges, not create edges where there wasn’t one before. Duplicating edges would mean walking or driving down a road twice, while creating an edge where there wasn’t one
before is akin to installing a new road!
For the rectangular graph shown, three possible eulerizations are shown. Notice in each of these cases the vertices that started with odd degrees have even degrees after eulerization, allowing for an
Euler circuit.
In the example above, you’ll notice that the last eulerization required duplicating seven edges, while the first two only required duplicating five edges. If we were eulerizing the graph to find a
walking path, we would want the eulerization with minimal duplications. If the edges had weights representing distances or costs, then we would want to select the eulerization with the minimal total
added weight.
Try It now
Eulerize the graph shown, then find an Euler circuit on the eulerized graph.
Looking again at the graph for our lawn inspector from Examples 1 and 8, the vertices with odd degree are shown highlighted. With eight vertices, we will always have to duplicate at least four edges.
In this case, we need to duplicate five edges since two odd degree vertices are not directly connected. Without weights we can’t be certain this is the eulerization that minimizes walking distance,
but it looks pretty good.
The problem of finding the optimal eulerization is called the Chinese Postman Problem, a name given by an American in honor of the Chinese mathematician Mei-Ko Kwan who first studied the problem in
1962 while trying to find optimal delivery routes for postal carriers. This problem is important in determining efficient routes for garbage trucks, school buses, parking meter checkers, street
sweepers, and more.
Unfortunately, algorithms to solve this problem are fairly complex. Some simpler cases are considered in the exercises
The following video shows another view of finding an Eulerization of the lawn inspector problem.
Hamiltonian Circuits
The Traveling Salesman Problem
In the last section, we considered optimizing a walking route for a postal carrier. How is this different than the requirements of a package delivery driver? While the postal carrier needed to walk
down every street (edge) to deliver the mail, the package delivery driver instead needs to visit every one of a set of delivery locations. Instead of looking for a circuit that covers every edge
once, the package deliverer is interested in a circuit that visits every vertex once.
Hamiltonian Circuits and Paths
A Hamiltonian circuit is a circuit that visits every vertex once with no repeats. Being a circuit, it must start and end at the same vertex. A Hamiltonian path also visits every vertex once with no
repeats, but does not have to start and end at the same vertex.
Hamiltonian circuits are named for William Rowan Hamilton who studied them in the 1800’s.
One Hamiltonian circuit is shown on the graph below. There are several other Hamiltonian circuits possible on this graph. Notice that the circuit only has to visit every vertex once; it does not need
to use every edge.
This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex: ABFGCDHMLKJEA. Notice that the same circuit could be written in reverse order, or starting
and ending at a different vertex.
Unlike with Euler circuits, there is no nice theorem that allows us to instantly determine whether or not a Hamiltonian circuit exists for all graphs.[1]
Does a Hamiltonian path or circuit exist on the graph below?
We can see that once we travel to vertex E there is no way to leave without returning to C, so there is no possibility of a Hamiltonian circuit. If we start at vertex E we can find several
Hamiltonian paths, such as ECDAB and ECABD
Try It
With Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one
with lowest total weight.
Watch this video to see the examples above worked out.
This problem is called the Traveling salesman problem (TSP) because the question can be framed like this: Suppose a salesman needs to give sales pitches in four cities. He looks up the airfares
between each city, and puts the costs in a graph. In what order should he travel to visit each city once then return home with the lowest cost?
To answer this question of how to find the lowest cost Hamiltonian circuit, we will consider some possible approaches. The first option that might come to mind is to just try all different possible
question can be framed like this: Suppose a salesman needs to give sales pitches in four cities. He looks up the airfares between each city, and puts the costs in a graph. In what order should he
travel to visit each city once then return home with the lowest cost?
To answer this question of how to find the lowest cost Hamiltonian circuit, we will consider some possible approaches. The first option that might come to mind is to just try all different possible
Brute Force Algorithm (a.k.a. exhaustive search)
1. List all possible Hamiltonian circuits
2. Find the length of each circuit by adding the edge weights
3. Select the circuit with minimal total weight.
Apply the Brute force algorithm to find the minimum cost Hamiltonian circuit on the graph below.
To apply the Brute force algorithm, we list all possible Hamiltonian circuits and calculate their weight:
Circuit Weight
ABCDA 4+13+8+1 = 26
ABDCA 4+9+8+2 = 23
ACBDA 2+13+9+1 = 25
Note: These are the unique circuits on this graph. All other possible circuits are the reverse of the listed ones or start at a different vertex, but result in the same weights.
From this we can see that the second circuit, ABDCA, is the optimal circuit.
Watch these examples worked again in the following video.
Try It
The Brute force algorithm is optimal; it will always produce the Hamiltonian circuit with minimum weight. Is it efficient? To answer that question, we need to consider how many Hamiltonian circuits a
graph could have. For simplicity, let’s look at the worst-case possibility, where every vertex is connected to every other vertex. This is called a complete graph.
Suppose we had a complete graph with five vertices like the air travel graph above. From Seattle there are four cities we can visit first. From each of those, there are three choices. From each of
those cities, there are two possible cities to visit next. There is then only one choice for the last city before returning home.
This can be shown visually:
Counting the number of routes, we can see thereare [latex]4\cdot{3}\cdot{2}\cdot{1}[/latex] routes. For six cities there would be [latex]5\cdot{4}\cdot{3}\cdot{2}\cdot{1}[/latex] routes.
Number of Possible Circuits
For N vertices in a complete graph, there will be [latex](n-1)!=(n-1)(n-2)(n-3)\dots{3}\cdot{2}\cdot{1}[/latex] routes. Half of these are duplicates in reverse order, so there are [latex]\frac
{(n-1)!}{2}[/latex] unique circuits.
The exclamation symbol, !, is read “factorial” and is shorthand for the product shown.
How many circuits would a complete graph with 8 vertices have?
A complete graph with 8 vertices would have = 5040 possible Hamiltonian circuits. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes.
While this is a lot, it doesn’t seem unreasonably huge. But consider what happens as the number of cities increase:
Cities Unique Hamiltonian Circuits
9 8!/2 = 20,160
10 9!/2 = 181,440
11 10!/2 = 1,814,400
15 14!/2 = 43,589,145,600
20 19!/2 = 60,822,550,204,416,000
Watch these examples worked again in the following video.
As you can see the number of circuits is growing extremely quickly. If a computer looked at one billion circuits a second, it would still take almost two years to examine all the possible circuits
with only 20 cities! Certainly Brute Force is not an efficient algorithm.
Nearest Neighbor Algorithm (NNA)
1. Select a starting point.
2. Move to the nearest unvisited vertex (the edge with smallest weight).
3. Repeat until the circuit is complete.
Unfortunately, no one has yet found an efficient and optimal algorithm to solve the TSP, and it is very unlikely anyone ever will. Since it is not practical to use brute force to solve the problem,
we turn instead to heuristic algorithms; efficient algorithms that give approximate solutions. In other words, heuristic algorithms are fast, but may or may not produce the optimal circuit.
Consider our earlier graph, shown to the right.
Starting at vertex A, the nearest neighbor is vertex D with a weight of 1.
From D, the nearest neighbor is C, with a weight of 8.
From C, our only option is to move to vertex B, the only unvisited vertex, with a cost of 13.
From B we return to A with a weight of 4.
The resulting circuit is ADCBA with a total weight of [latex]1+8+13+4 = 26[/latex].
Watch the example worked out in the following video.
We ended up finding the worst circuit in the graph! What happened? Unfortunately, while it is very easy to implement, the NNA is a greedy algorithm, meaning it only looks at the immediate decision
without considering the consequences in the future. In this case, following the edge AD forced us to use the very expensive edge BC later.
Consider again our salesman. Starting in Seattle, the nearest neighbor (cheapest flight) is to LA, at a cost of $70. From there:
LA to Chicago: $100
Chicago to Atlanta: $75
Atlanta to Dallas: $85
Dallas to Seattle: $120
Total cost: $450
In this case, nearest neighbor did find the optimal circuit.
Watch this example worked out again in this video.
Going back to our first example, how could we improve the outcome? One option would be to redo the nearest neighbor algorithm with a different starting point to see if the result changed. Since
nearest neighbor is so fast, doing it several times isn’t a big deal.
We will revisit the graph from Example 17.
Starting at vertex A resulted in a circuit with weight 26.
Starting at vertex B, the nearest neighbor circuit is BADCB with a weight of 4+1+8+13 = 26. This is the same circuit we found starting at vertex A. No better.
Starting at vertex C, the nearest neighbor circuit is CADBC with a weight of 2+1+9+13 = 25. Better!
Starting at vertex D, the nearest neighbor circuit is DACBA. Notice that this is actually the same circuit we found starting at C, just written with a different starting vertex.
The RNNA was able to produce a slightly better circuit with a weight of 25, but still not the optimal circuit in this case. Notice that even though we found the circuit by starting at vertex C, we
could still write the circuit starting at A: ADBCA or ACBDA.
Try It
The table below shows the time, in milliseconds, it takes to send a packet of data between computers on a network. If data needed to be sent in sequence to each computer, then notification needed to
come back to the original computer, we would be solving the TSP. The computers are labeled A-F for convenience.
A B C D E F
A — 44 34 12 40 41
B 44 — 31 43 24 50
C 34 31 — 20 39 27
D 12 43 20 — 11 17
E 40 24 39 11 — 42
F 41 50 27 17 42 —
a. Find the circuit generated by the NNA starting at vertex B.
b. Find the circuit generated by the RNNA.
While certainly better than the basic NNA, unfortunately, the RNNA is still greedy and will produce very bad results for some graphs. As an alternative, our next approach will step back and look at
the “big picture” – it will select first the edges that are shortest, and then fill in the gaps.
Using the four vertex graph from earlier, we can use the Sorted Edges algorithm.
The cheapest edge is AD, with a cost of 1. We highlight that edge to mark it selected.
The next shortest edge is AC, with a weight of 2, so we highlight that edge.
For the third edge, we’d like to add AB, but that would give vertex A degree 3, which is not allowed in a Hamiltonian circuit. The next shortest edge is CD, but that edge would create a circuit ACDA
that does not include vertex B, so we reject that edge. The next shortest edge is BD, so we add that edge to the graph.
We then add the last edge to complete the circuit: ACBDA with weight 25.
Notice that the algorithm did not produce the optimal circuit in this case; the optimal circuit is ACDBA with weight 23.
While the Sorted Edge algorithm overcomes some of the shortcomings of NNA, it is still only a heuristic algorithm, and does not guarantee the optimal circuit.
Your teacher’s band, Derivative Work, is doing a bar tour in Oregon. The driving distances are shown below. Plan an efficient route for your teacher to visit all the cities and return to the starting
location. Use NNA starting at Portland, and then use Sorted Edges.
Ashland Astoria Bend Corvallis Crater Lake Eugene Newport Portland Salem Seaside
Ashland – 374 200 223 108 178 252 285 240 356
Astoria 374 – 255 166 433 199 135 95 136 17
Bend 200 255 – 128 277 128 180 160 131 247
Corvallis 223 166 128 – 430 47 52 84 40 155
Crater Lake 108 433 277 430 – 453 478 344 389 423
Eugene 178 199 128 47 453 – 91 110 64 181
Newport 252 135 180 52 478 91 – 114 83 117
Portland 285 95 160 84 344 110 114 – 47 78
Salem 240 136 131 40 389 64 83 47 – 118
Seaside 356 17 247 155 423 181 117 78 118 –
To see the entire table, scroll to the right
Using NNA with a large number of cities, you might find it helpful to mark off the cities as they’re visited to keep from accidently visiting them again. Looking in the row for Portland, the smallest
distance is 47, to Salem. Following that idea, our circuit will be:
Portland to Salem 47
Salem to Corvallis 40
Corvallis to Eugene 47
Eugene to Newport 91
Newport to Seaside 117
Seaside to Astoria 17
Astoria to Bend 255
Bend to Ashland 200
Ashland to Crater Lake 108
Crater Lake to Portland 344
Total trip length: 1266 miles
Using Sorted Edges, you might find it helpful to draw an empty graph, perhaps by drawing vertices in a circular pattern. Adding edges to the graph as you select them will help you visualize any
circuits or vertices with degree 3.
We start adding the shortest edges:
Seaside to Astoria 17 miles
Corvallis to Salem 40 miles
Portland to Salem 47 miles
Corvallis to Eugene 47 miles
The graph after adding these edges is shown to the right. The next shortest edge is from Corvallis to Newport at 52 miles, but adding that edge would give Corvallis degree 3.
Continuing on, we can skip over any edge pair that contains Salem or Corvallis, since they both already have degree 2.
Portland to Seaside 78 miles
Eugene to Newport 91 miles
Portland to Astoria (reject – closes circuit)
Ashland to Crater Lk 108 miles
The graph after adding these edges is shown to the right. At this point, we can skip over any edge pair that contains Salem, Seaside, Eugene, Portland, or Corvallis since they already have degree 2.
Newport to Astoria (reject – closes circuit)
Newport to Bend 180 miles
Bend to Ashland 200 miles
At this point the only way to complete the circuit is to add:
Crater Lk to Astoria 433 miles. The final circuit, written to start at Portland, is:
Portland, Salem, Corvallis, Eugene, Newport, Bend, Ashland, Crater Lake, Astoria, Seaside, Portland. Total trip length: 1241 miles.
While better than the NNA route, neither algorithm produced the optimal route. The following route can make the tour in 1069 miles:
Portland, Astoria, Seaside, Newport, Corvallis, Eugene, Ashland, Crater Lake, Bend, Salem, Portland
Watch the example of nearest neighbor algorithm for traveling from city to city using a table worked out in the video below.
In the next video we use the same table, but use sorted edges to plan the trip.
Try It
Find the circuit produced by the Sorted Edges algorithm using the graph below.
Spanning Trees
A company requires reliable internet and phone connectivity between their five offices (named A, B, C, D, and E for simplicity) in New York, so they decide to lease dedicated lines from the phone
company. The phone company will charge for each link made. The costs, in thousands of dollars per year, are shown in the graph.
In this case, we don’t need to find a circuit, or even a specific path; all we need to do is make sure we can make a call from any office to any other. In other words, we need to be sure there is a
path from any vertex to any other vertex.
Spanning Tree
A spanning tree is a connected graph using all vertices in which there are no circuits.
In other words, there is a path from any vertex to any other vertex, but no circuits.
Some examples of spanning trees are shown below. Notice there are no circuits in the trees, and it is fine to have vertices with degree higher than two.
Usually we have a starting graph to work from, like in the phone example above. In this case, we form our spanning tree by finding a subgraph – a new graph formed using all the vertices but only some
of the edges from the original graph. No edges will be created where they didn’t already exist.
Of course, any random spanning tree isn’t really what we want. We want the minimum cost spanning tree (MCST).
Minimum Cost Spanning Tree (MCST)
The minimum cost spanning tree is the spanning tree with the smallest total edge weight.
A nearest neighbor style approach doesn’t make as much sense here since we don’t need a circuit, so instead we will take an approach similar to sorted edges.
Kruskal’s Algorithm
1. Select the cheapest unused edge in the graph.
2. Repeat step 1, adding the cheapest unused edge, unless:
3. adding the edge would create a circuit
Repeat until a spanning tree is formed
Using our phone line graph from above, begin adding edges:
AB $4 OK
AE $5 OK
BE $6 reject – closes circuit ABEA
DC $7 OK
AC $8 OK
At this point we stop – every vertex is now connected, so we have formed a spanning tree with cost $24 thousand a year.
Remarkably, Kruskal’s algorithm is both optimal and efficient; we are guaranteed to always produce the optimal MCST.
The power company needs to lay updated distribution lines connecting the ten Oregon cities below to the power grid. How can they minimize the amount of new line to lay?
Ashland Astoria Bend Corvallis Crater Lake Eugene Newport Portland Salem Seaside
Ashland – 374 200 223 108 178 252 285 240 356
Astoria 374 – 255 166 433 199 135 95 136 17
Bend 200 255 – 128 277 128 180 160 131 247
Corvallis 223 166 128 – 430 47 52 84 40 155
Crater Lake 108 433 277 430 – 453 478 344 389 423
Eugene 178 199 128 47 453 – 91 110 64 181
Newport 252 135 180 52 478 91 – 114 83 117
Portland 285 95 160 84 344 110 114 – 47 78
Salem 240 136 131 40 389 64 83 47 – 118
Seaside 356 17 247 155 423 181 117 78 118 –
To see the entire table, scroll to the right
Using Kruskal’s algorithm, we add edges from cheapest to most expensive, rejecting any that close a circuit. We stop when the graph is connected.
Seaside to Astoria 17 milesCorvallis to Salem 40 miles
Portland to Salem 47 miles
Corvallis to Eugene 47 miles
Corvallis to Newport 52 miles
Salem to Eugene reject – closes circuit
Portland to Seaside 78 miles
The graph up to this point is shown below.
Newport to Salem reject
Corvallis to Portland reject
Eugene to Newport reject
Portland to Astoria reject
Ashland to Crater Lk 108 miles
Eugene to Portland reject
Newport to Portland reject
Newport to Seaside reject
Salem to Seaside reject
Bend to Eugene 128 miles
Bend to Salem reject
Astoria to Newport reject
Salem to Astoria reject
Corvallis to Seaside reject
Portland to Bend reject
Astoria to Corvallis reject
Eugene to Ashland 178 miles
This connects the graph. The total length of cable to lay would be 695 miles.
Watch the example above worked out in the following video, without a table.
Now we present the same example, with a table in the following video.
[1] There are some theorems that can be used in specific circumstances, such as Dirac’s theorem, which says that a Hamiltonian circuit must exist on a graph with n vertices if each vertex has degree
n/2 or greater.
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NCERT Solutions for Class 10 maths Chapter 1 Real Number - Free PDF Download
NCERT Maths Solutions for Chapter 1 of Class 10 has prepared by Praadis Subject Experts after focusing on each topic very intensely, students will get proper knowledge of Real Numbers and Irrational
Numbers. The chapter begins with the Euclid’s Division Lemma, the Euclid’s Division algorithm is based on this lemma and is used to calculate the HCF of two positive integers. Then, the Fundamental
Theorem of Arithmetic is defined, which is used to find the LCM and HCF of two positive integers. After that, the concept of an irrational number, a rational number and decimal expansion of rational
numbers are explained with the help of theorem.
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Optimization of Bonus-Malus Systems
Gyetvai, Márton (2022) Optimization of Bonus-Malus Systems. PhD thesis, Budapesti Corvinus Egyetem, Közgazdasági és Gazdaságinformatikai Doktori Iskola. DOI http://doi.org/10.14267/phd.2022041
PDF : (dissertation)
PDF : (draft in English)
PDF : (az értekezés tézisei magyar nyelven)
Bonus-Malus System (BMS) is a risk managing method mostly used in liability insurances. The most general application of the BMS is in the Motor third-party liability insurance. In a BMS, there are
finitely many classes, each having a different premium. At the start of the contract, each policyholder is assigned to the "initial class". Subsequently, suppose the policyholder has a claim in the
following period. In that case, he/she moves to a worse class, so the policyholder's payment may increase in the subsequent period. If he/she does not have a claim in a particular period, then he/she
moves to a better class; therefore, his/her payment may become less in the following period. The classification rule – how many classes the policyholder will move up or down in the system – is called
the transition rule. Hence, a transition rule specifies where the policyholder will be reclassified in the subsequent period for each possible claim. Our contribution to the literature of the
optimization of the BMS can be summarized as: • We investigate a model that was introduced by Heras et al. (2004) but with a modified objective function. We proved that an optimal premium-scale
always exists with this objective function in which all premiums equal one of the risk groups expected claim. • We considered the same model with a profit constraint. In this case, we proved that an
optimal premium-scale always exists in which there is only one type of premium that is unequal to any risk group's expected claim. • We introduced a MILP model for the optimization of transition
rules with fixed premiums. We considered unified and non-unified transition rules optimization. In the case of unified transition rules, we gave the rule to exclude those transition rules that would
lead to a non-irreducible Markov chain. • We introduced a MILP model for the joint optimization of transition rules and premiums. We can determine the exact solution with the investigated objective
function when we do not consider the profit constraint. However, we can only approximate it otherwise. • We introduced an extended version of the model, where instead of the stationary probabilities,
we use multi-period optimization. • We introduced modeling approaches to consider the BMS premium with other statistical estimations in the final premium. Finally, we compared the methods with
numerical experiments on realistic data. • We introduced an optimization model for a BMS where the classification depends on the claim amount.
Item Type: Thesis (PhD thesis)
Supervisor: Ágoston Kolos Csaba
Uncontrolled Keywords: Bonus-Malus System, insurance
Subjects: Mathematics. Econometrics
ID Code: 1193
Date: 15 June 2022
DOI: http://doi.org/10.14267/phd.2022041
Deposited On: 26 Jan 2022 07:09
Last Modified: 02 Nov 2022 08:24
Repository Staff Only: item control page
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Warm-up: Estimation Exploration: How Big is the Milk Carton? (10 minutes)
The purpose of this Estimation Exploration is for students to estimate a volume based on an image and on their own personal experience with cartons of milk. Students recall the meaning of volume as
the number of cubic inches, in this case, it would take to fill the milk carton without gaps or overlaps. Because the carton is relatively small, students can formulate a reasoned, accurate estimate
of the milk carton’s volume. They will then use this estimate throughout the lesson.
• Groups of 2
• Display the image.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
Student Facing
What is the volume of the milk carton in cubic inches?
Record an estimate that is:
│ too low │ about right │ too high │
│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│
Activity Synthesis
• “How can you use what you know about volume to estimate the volume of the milk container?” (I can measure to see how many cubic inches it would take to fill the carton. I can measure the length,
width, and height and multiply them.)
• “What units do you usually use to measure liquids?” (Liters, quarts, cups)
• “We learned in an earlier unit that cubic centimeters or cubic inches are also units for measuring a volume.”
Activity 1: Milk for Everyone (15 minutes)
The purpose of this activity is for students to estimate products using the context of volume introduced in the warm-up. Students estimate how many cubic inches of milk different-sized groups of
students might consume. For example, at first, students multiply the amount of milk they consume by the number of students in the class. Next, students multiply the amount consumed by one class by
the number of classes. Because these are all estimates, the fact that not every student in one class drinks the same amount of milk or that different classes or grades or schools have different
numbers of students can be overlooked. When students make simplifying hypotheses like this, they model with mathematics (MP4).
As currently structured, the activity is quite open-ended so that students can use their own school to make their estimates. There is a lot of variation in school size. The average size of an
elementary school in Montana, for example, is less than 200, while in California, it is 600. Some large elementary schools in New York City have close to 2,000 students. The important mathematical
part of this activity does not depend on the exact numbers for a particular school. The key is which numbers students choose as they make estimates, focusing on multiples of powers of 10.
MLR2 Collect and Display. Circulate, listen for, and collect the language students use as they estimate the volume. On a visible display, record words and phrases such as: estimate, guess, predict,
multiply, times, and product. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Representation: Access for Perception. Use centimeter cubes to demonstrate how many cubic centimeters can fit inside the milk carton so that students understand the size of a cubic centimeter.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
• “What kind of milk do you like to drink?”
• Partner discussion
• “You are going to estimate the amount of milk that different groups of students drink in one day.”
• “You can use the estimate of 20 cubic inches for one carton of milk.”
• Monitor for students who select round numbers for their estimates and who use multiplication to go from each estimate to the next estimate.
Student Facing
In each situation, estimate the volume of milk, in cubic inches, that you or the group would drink in one day. Explain your reasoning.
1. you
2. your class
3. your grade
4. your school
5. 10 schools
Advancing Student Thinking
If students do not like milk and, therefore, do not have a connection to the problem, suggest they survey a few classmates to find out what their estimates were for how much milk they drink in one
Activity Synthesis
• Invite students to share responses and estimates.
• “How did you use your estimates from each question to help answer the next question?” (Once I knew how much milk I drank, I multiplied by the number of students in our class. Then I multiplied
that by the number of fifth-grade classes.)
• "How did you make an estimate for your class?" (I think there are between 20 and 30 students in the class but not everyone likes milk. So I estimated that 20 students drink milk with lunch.)
Activity 2: How Big is 1,000,000? (20 minutes)
The purpose of this activity is for students to make estimates about how long it would take different groups of students to drink 1,000,000 cubic inches of milk. Unlike the previous activity in which
students multiplied the 20 cubic inches of milk by larger and larger numbers, in this activity, students divide 1,000,000 cubic inches of milk by smaller and smaller numbers to find out how long it
would take each group to drink 1,000,000 cubic inches of milk. If students attempt to calculate exact answers remind them that they are only looking for an estimate and the amount of milk consumed by
each group in the previous activity is also only an estimate. Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).
• Groups of 2
• “How much do you think 1,000,000 cubic inches of milk is? Could you drink it?” (No, that's a lot of milk. I don't like milk that much.)
• 1 minute: quiet think time
• 1 minute: partner discussion
• 2-3 minutes individual work time
• 7-8 minutes partner work time
• Monitor for students who use the estimates from the previous activity and who base each successive calculation on the previous one, dividing by an appropriate number at each step.
Student Facing
Estimate the number of days it would take each group to drink 1,000,000 cubic inches of milk. Explain your reasoning.
1. 10 local schools
2. your school
3. your grade
4. your class
5. you
Advancing Student Thinking
Students may need support with initiating the task. Ask them to explain how they can use the solutions from the previous activity to help them solve the problems.
Activity Synthesis
• “How did you estimate the number of days it takes 10 schools to drink 1,000,000 cubic inches of milk?” (We estimated that they drink close to 100,000 cubic inches a day, so in 10 days that’s
• “How did you use this estimate to estimate how long it takes your school to drink 1,000,000 cubic inches of milk?” (I multiplied by 10 because it takes 1 class 10 times as long as it takes 10
• “Do you think that you will ever drink 1,000,000 cubic inches of milk?” (No, 50,000 days is a lot. There are only 365 days in a year, so that would be more than 100 years.)
Lesson Synthesis
“In this lesson we estimated products and quotients.”
“How can you use multiplication to estimate how many days it would take your school to drink 1,000,000 cubic inches of milk?” (In 2 days we drink twice as much milk, in 3 days we drink 3 times as
much. So I needed to estimate what to multiply the amount for one day by to get about 1,000,000.)
“Could you also make this estimate using division?” (Yes, our school drinks about 10,000 cubic inches of milk each day, so I can find how many 10,000s there are in 1,000,000. That's \(1,\!000,\!000 \
div 10,\!000\).)
Cool-down: So Much Milk (5 minutes)
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Poomsae Difficulty Formula
Nov 14, 2013
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The alternate title for this thread is: Skribs has too much spare time on his hands.
On a slow day at work, I decided to make a formula for poomsae difficulty. I attempted to factor in a larger variety of techniques, greater variation, and mor detailed footwork/handwork as a more
difficult form.
( (NUT * TPS) + (TNS * (1 + FC)) ) * Steps * Sets = Difficulty
NUT = Number of Unique Techniques
TPS = Techniques Per Step
TNS = Total Number of Stances
FC = Footwork Coefficient; Percentage of Steps with Stance Switches, Spins, or Rearward Motion
Steps = Number of Steps
Sets = Number of Repeated Parts
Total Number of Techniques is simply how many different blocks, punches, and/or kicks you have. For example, if you just have a bunch of low blocks and punches, then NUT would be 2.
Techniques Per Step is the number of techniques you do divided by the number of steps. If every time you step forward or turn you do one technique, then TPS will be 1. If you do a kick and a punch,
or do multiple techniques without taking a step, then you add those in. Total up the total number of techniques and divide by the number of steps to get your answer.
Total Number of Stances is how many different stances techniques are used from. A basic Taeguk will probably use Front Stance and Walking Stance, which would be 2. A form that uses front, back,
horse, and cat would have 4.
Footwork Coefficient is the method I use to factor in detailed footwork within a step. Any time you do anything besides step forward or turn (meaning step backward, change stance, spin, jump, etc)
you add to the Footwork Coefficient. A basic form will have an FC of 0, while forms that have you do several techniques without moving might have a much higher FC.
Steps is simply the number of steps your feet take. Any time you move forward, backward, or turn on the axis, it counts as another step. Times when you face a different direction but either don't
move your feet or barely move your feet, that is adding to the FC, but not the step count.
Sets: the number of repeated sets. A set is simply a combination along the lateral or longitudinal axes. For example, Kibon Il Jang repeats the lateral set of low block, punch, turn, low block, punch
3 times and the forward/backward set of low block, punch x3 twice. Because it repeats 2 different sets, it's Set value is 2. Taeguk Il Jang has 3 different lateral sets and a different set forward
and back, so its Set value is 5. Any time you have a set that comprises of a single technique or two, or you have two sets that are similar except for one minor detail, that is only 0.5 extra sets.
Now, I plugged in the data for the forms we do at my school, at least the ones I already know well enough to plug the data in for. I'm not going to post it, because of a few reasons:
1) We use 5 kibon forms and 8 palgwe forms, while most people I think just do the Taeguks.
2) Our master has taken some creative liberties with the later palgwe forms that make them more technically difficult, but also completely different from what I see on Youtube. There are also minor
differences in the earlier forms.
Because of this, I don't think my data is useful to anyone outside my school (if it even is useful to someone inside my school). But, if you're a math nerd in addition to a TKD geek,the formula
itself might be fun to plug in the forms you do at your school.
I will say that the results of my data are not surprising at all. Going up through all 5 kibons and the first 6 palgwes, the difficulty goes up every time, with only one peak (that I expected).
Here's an example using Kibon Il Jang and Taeguk Il Jang:
Kibon Il Jang: 80
( (NUT * TPS) + (TNS * (1 + FC)) ) * Steps * Sets = Difficulty
There are 2 techniques used: low block and punch, so NUT = 2.
You do one block or punch per step, so TPS = 1.
You only use front stance or walking stance (depending on school), so TNS = 1.
There is no fancy footwork, so FC = 0.
There are 20 steps, so Steps = 20.
There are 2 sets repeated 3 and 2 times, so Sets = 2.
( (2 * 1) ) + (1 * (1 + 0) ) * 20 * 2 = 80.
Taeguk Il Jang: 660
( (NUT * TPS) + (TNS * (1 + FC)) ) * Steps * Sets = Difficulty
There are 5 techniques used: low block, punch, inside block, high block, and front kick, so NUT = 5
There are 20 blocks, punches, or kicks and 16 steps (added up below), so TPS = 1.25
There are 2 stances used: front stance and walking stance, so TNS = 2
There is no fancy footwork, so FC = 0
There are 16 steps taken, so Steps = 16.
There are 5 repeated sets (technically none are repeated and there are 5 sets), so Sets = 5
( (5 * 1.25) + (2 * (1 + 0) ) * 16 * 5 = 660.
I think the one thing I should have done different is move Steps to the beginning because of the other bits that require you to divide by Sets, but that's a minor issue.
Anyway, I was bored enough to make this, so even if it's got no real-world value, it at least killed time for me!
Sep 25, 2006
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Very interesting. Personally I wouldn't multiply by repeated sets, because if anything they make it easier.
Also, I would consider number of new movements as well as unique movements. Considering that students generally learn them in sequence, Taegeuk 2 is much easier than Taegeuk 1 because you already
know the movements, whereas Taegeuk 4 has a big jump.
Nov 14, 2013
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Very interesting. Personally I wouldn't multiply by repeated sets, because if anything they make it easier.
I think you misunderstand. The number of repeated sets is how many unique sets you have (maybe better phrasing on my part could clear that up). In the example, Kibon Il Jang has a Set rating of 2,
because it only has 2 unique sets.
Also, I would consider number of new movements as well as unique movements. Considering that students generally learn them in sequence, Taegeuk 2 is much easier than Taegeuk 1 because you already
know the movements, whereas Taegeuk 4 has a big jump.
This is an interesting idea. I'll have to try and figure out a way to factor it in. Are you thinking just new movements from what you've done in previous forms?
Nov 3, 2010
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Taegeuk 2 is much easier than Taegeuk 1 because you already know the movements, whereas Taegeuk 4 has a big jump.
Personally, I find taegeuk4 easier to remember than taegeuk 2 &1 because it IS different. With taegeuk 2 starting out very similarly to taegeuk1, it's easy to forget for instance which one has the
front stance vs walking stance.
Dec 26, 2013
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Personally, I find taegeuk4 easier to remember than taegeuk 2 &1 because it IS different. With taegeuk 2 starting out very similarly to taegeuk1, it's easy to forget for instance which one has
the front stance vs walking stance.
Easier to remember, yes, but also harder to master. The first three taegeuk use very basic techniques compared to the fourth one.
The formula is interesting, would be interested to know if it works with a non-conventionnal poomsae such as keumgang.
Dec 13, 2011
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The formula is interesting, would be interested to know if it works with a non-conventionnal poomsae such as keumgang.
That's an interesting idea. Every technique is new (has not been seen before) but many repeating sections and a complex philosophical meaning. Highlights that there's nothing in the formula regarding
non-physical aspects. Just an observation.
Nov 14, 2013
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Well, I'm a bit lost on all of the form-specific discussion. I'm not even halfway to black belt and we don't do the Taeguks at my school (the example I based off of a Youtube video).
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MathFiction: Kim Possible (Episode: Mathter and Fervent) (Jim Peronto (script))
This episode of the Disney animated TV series "Kim Possible" is a comic book parody featuring a mathematical villain.
As an English assignment, Kim Possible and Ron Stoppable have to write a paper about their "hero". Kim is writing about her dad, a brilliant physicist. In explaining why he would not want to write
about his father, an actuary, Ron says "who would want to read about math in English?" (Visitors to this Website would, Ron!) In fact, Ron's nerdy dad is portrayed as not being particularly heroic.
For instance, when he appears wearing a scorched fire-fighter's outfit people are surprised to hear that he had volunteered at the fire department, but it turns out he had only volunteered to cook in
the fire house -- and had burned the food besides. So (after finding that neither Kim's neurosurgeon mom nor the CEO of "SmartyMart" are available), Ron decides to write about a real superhero, Go
As it turns out, Go Man's villain at the moment is The Mathter, a mathematician seeking vengeance after he was denied funding for his "unethical mathematical experiments". Ron says "I always knew
that math was evil." The Mathter is prone to using math puns and mathematical weapons: he subtracts people with his calcu-laser, tosses dangerous decimal points at them, and uses "Brackets!" as an
Through a series of events so contrived that even the characters note how unlikely it is, it is Ron's number crunching father who saves the day. "What kind of hero are you?" the Mathter asks him.
"I'm no hero, I'm actuary of the year!" Mr. Stoppable responds. Of course, Ron now recognizes his father's skill with numbers and uses him as the subject for his essay.
This episode is currently available for free on YouTube, but such postings tend to be temporary as Disney likes to enforce their copyrights. So, do not assume it will stay there for long.
Thanks to Cora Wright for bringing this fun bit of mathematical fiction to my attention!
The Mathter and his henchmen, the Coefficients The Mathter's Lair
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Chandra :: Educational Materials :: Stop for Science! - That's Fast!
WHAT IS SPEED, AND HOW DO WE MEASURE IT?
A jet plane is fast, and a snail is slow. But what exactly does this mean? The speed of an object is defined to be the distance it will travel in a certain amount of time. If something travels 100
feet (or about 30 meters) in 10 seconds, its speed is 10 feet per second (ft/s), or 3 meters per second (m/s). We often talk about speeds in miles per hour (mi/hr, or mph) or kilometers per hour (km/
HOW FAST, IS FAST?
The fastest land animal is the Cheetah, which can reach speeds of 70 mi/hr (112 km/hr). This is fast, but not compared to how fast a pitcher can throw a baseball (100 mi/hr, or 160 km/hr). A
Peregrine falcon is so fast that it could easily outrace a baseball; it can go up to nearly 200 mi/hr (320 km/hr). But this is a snail’s pace compared to how fast the Earth moves around the Sun! Over
the course of a year, the Earth travels more than 580 million miles (930 million kilometers). That’s an average speed of about 67,000 mi/hr (107,000 km/hr)!
As part of his theory of relativity, Albert Einstein showed that nothing can be accelerated to speeds faster than the speed of light (186,000 miles/ sec, or 300,000 km/sec); this is the speed limit
for our Universe! This has been confirmed in many experiments. We can make things go fast, but only light (or other electromagnetic waves) can go this fast.
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A block of mass 2" "kg is free to move along the x - axis. it i-Turito
Are you sure you want to logout?
A block of mass 2kg is free to move along the x- axis. It is at rest and from t=0 onwards it is subjected to a time-dependent force
A. 4.50 J
B. 7.50 J
C. 5.06 J
D. 14.06 J
The correct answer is: 5.06 J
Get an Expert Advice From Turito.
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The Case for Non-Inferiority A/B Tests
In this article, I explore the concept of non-inferiority A/B tests and contrast it to the broadly accepted practice of running superiority tests. I explain where non-inferiority tests are necessary
and how a CRO/LPO/UX testing specialist can make use of this new approach to A/B testing to run much faster tests, and to ultimately achieve better results for himself or his/her clients.
Let’s start with something “simple”: why do we care if the result of an A/B test is statistically significant?
The answer seems obvious enough: we don’t want to look like fools, claiming something improves conversion rate (or CTR, or e-commerce transaction rate, etc.) when it does not, or in fact it does the
exact opposite. We want to be able to justify the work we do and to claim credit for the positive impact of our actions or advice as CRO practitioners. For the above reasons, designing A/B tests with
proper statistical methodology and accepting a new variant only when it passes a particular statistical significance threshold is a must.
The choice of null hypothesis matters (a lot!)
A/B testing practitioners must never forget that statistical significance is a only a part of the broader process of null hypothesis statistical testing, hence the statistical significance of a test
depends heavily on what statistical null hypothesis is chosen. This stems from the very basis of experimental design, as in the words of R.A. Fisher:
“Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis.” ^[1]
Here is a quick practical example. Say we have observed two groups of 4,000 users each in a randomized controlled experiment. The control converts at 10%, the variant at 11%, a 10% observed relative
improvement. Is the result statistically significant at the 95% level, with a one-sided z-test?
No one can answer that question! What is missing from it is the null hypothesis. With a “classical” null hypothesis of “the difference between the variant and control is 0 or negative”, the answer is
“no” (z-value corresponds to 92.77% significance). However, with a null hypothesis of “the variant is 2% worse than the control, or more”, the A/B test becomes significant at the 95% level (96.6%)
with the lower bound of a 95% confidence interval for the difference in proportions at -1.3%.
What happened above? When the question asked (expressed in terms of null and alternative hypotheses) is different, the answer is, naturally, different as well. The first hypothesis is a superiority
test: we want to ward against the error of implementing a new solution, thinking it is superior to the current solution, while in fact it is not. The second one is actually a non-inferiority test: we
wanted to ward against the error of implementing something that is significantly worse than the current solution – in this case the “margin of caring” was set at 2% negative difference. With the
first null hypothesis, we would have guided our decision in one direction, with the second: in the opposite one, even though the data is exactly the same!
Therefore, it is very important to select the right hypothesis when planning the statistical design of any A/B test. Below I explore the logic behind the choice of superiority and non-inferiority
null hypothesis and the costs and benefits of choosing one or another.
When to test for superiority?
In all the A/B testing literature I’ve read and all case studies I’ve seen, the null hypothesis, where it is specified or can be inferred, is that the control is performing worse than or equal to the
tested variant:
Null hypothesis: variant(s) ≤ control
The above is called a superiority design or a superiority A/B test. With a such a test we declare that we will act with the presumption that:
The error we want to avoid the most is the error of implementing a solution, which is not better than what we currently have
This makes perfect sense when the proposed new solution, or solutions:
• has high implementation costs, outside the cost of running the A/B test
• requires ongoing maintenance / recurring costs (IT infrastructure, support, 3-rd parties involved, etc.)
• is costly or impossible to reverse, once it is implemented (technological, PR, marketing, etc. reasons)
• faces strong internal or external opposition from HiPPOs, stakeholders, etc. for other reasons
When calculating sample size based on a desired statistical power, we need to select the smallest difference we would be happy to detect – the minimum effect size. This discrepancy is usually small,
with common values between 2% and 10% relative lift, since for most A/B tests detecting such discrepancies is enough to justify running the test, and to implement and to maintain the winning variant.
However, it is not always as small as we would like, since sometimes we just can’t push enough users through the test in a reasonable amount of time.
If the minimum effect of interest is selected objectively and we have one of the above cases, then testing for superiority is justified and should lead to the best possible outcomes.
The need for non-inferiority A/B tests
At first thought, it makes perfect sense to design all A/B tests as superiority tests. After all, why would you want to implement something that is not proven better than what you currently have?
However, there is a flaw in the above logic and it stems from lack of understanding of statistical power. Statistical power is the sensitivity of the test. It quantifies the probability to detect a
difference of a given size with a specified statistical significance threshold, if such a difference truly exists. The greater your sample size, the greater the power of the test, everything else
being equal. However, the smaller the margin you want to be able to detect, the lower the power of the test. Thus, even if you have the most-trafficked site on the internet, your A/B tests will still
fail to detect many true, but small improvements, as statistically significant.
So, the high precaution against abandoning a current solution for a non-superior one, comes at a cost. It comes with a high risk, due to low sensitivity, that we will miss at least some true
improvements of small magnitude. To be precise, of a magnitude smaller than the minimum effect of interest we calculated our sample size for (regardless if it is classic fixed-sample size test or a
sequential one, like an AGILE method test). This is why
Superiority tests should not be the default procedure, automatically applied to all A/B tests you do, but an informed decision you make for each case.
Non-inferiority tests are what we can do when the reasons for doing a superiority tests are not present and the primary error we care about is different, that is: when we are more concerned about
missing even the slightest improvement, or of rejecting an option that is just as good based on our primary KPI, but may have some benefits not measured in the test, like reduced cost, complexity,
maintenance, etc. Before we go into some practical examples for the application of non-inferiority tests, let’s first make sure we know what they are exactly.
What is non-inferiority testing?
It is simply a statistical test in which the null hypothesis is that the tested variant is performing worse than the control by a significant margin:
Null hypothesis: variant(s) < control or more precisely variant(s) < control – δ, where δ is a difference at which we consider the control and variant equal for any practical reasons. That is, even
if the new solution we implement after the A/B test is in fact doing worse than the existing one by a margin of δ, we would still be OK in implementing it.
With a non-inferiority test we declare that we will act under the assumption that:
The error we want to avoid the most is the error of failing to implement a new solution, which is about equal, or better than what we currently have
I believe there are many, many cases in online experiments, both in conversion rate optimization / UX testing and in online marketing experiments, where this is exactly the case. I have no doubt any
marketer or CRO expert would be able to quick about at least half a dozen such experiments they did in the past months. I give my own examples in “When to perform a non-inferiority A/B test?” below.
Here is a comparison between the confidence intervals of three tests, all stopped for success:
The above is a more visual illustration of the concept which should help in understanding it. A non-inferiority test can be thought of as a superiority test + equivalence test, as its alternative
hypothesis covers the equivalence test alternative and the superiority test alternative. If you think in terms of null hypothesis, it is the null for a superiority test minus the null of an
equivalence test.
The choice of a noninferiority margin, also called an equivalence margin, is key. It must be done based on objective evaluation of the magnitude that can to be considered non-significant. Sample size
considerations enter into the decision-making process, same as in a superiority test (except in the “easy decision” cases where it is a bit different, explained below). No one has unlimited time on
their hands. A simple rule might be that the noninferiority margin is set to the same size of the minimum effect of interest that you’d set in a superiority test.
Regardless of the method you use, it is of utmost important to get buy-in on this decision from all key stakeholders before the is started. Doing so after you have data will almost inevitably lead to
bias in one direction, or the other.
Statistically the tests (z-test, t-test, chi-square, sequential variants of these and others) work exactly the same – the math is the same. The difference is in the parameters we specify as they need
to describe another null hypothesis. It is also possible to analyze a test planned as a superiority test, as a non-inferiority one. Sequential A/B tests such as tests using the AGILE A/B testing
approach, however, should be designed as non-inferiority tests from the very beginning, since the design affects the stopping probabilities.
See this in action
Try itA/B Testing Calculator
Advanced significance & confidence interval calculator.
If you are using custom code (Excel, R, Python, etc.) to do your calculations, you will still be able to usе it, with some modifications for sample size calculations and p-values, while one-sided
confidence intervals can be used without any additional changes. See references [2], [3] & [4] below if you are interested in the technical details on the most common approaches.
If you are using a third-party platform, it must specifically support the design and evaluation of non-inferiority tests, or you must have the freedom to specify your null hypothesis. I believe many
tools just assume a superiority trial by default and allow no option to specify it, but hopefully this will change. Alternatively, you can use a third-party tool for the statistical analysis only.
Clients of our toolkit would be happy to know that both our Sample size & Statistical significance calculator and our A/B Testing Calculator explicitly support non-inferiority tests in their fullest.
When to perform a non-inferiority A/B test?
In my white paper on non-inferiority testing I differentiate between two types of cases that are suitable for applying a non-inferiority test. I call them “side benefits” cases and “easy decision”
cases. Let’s examine each of them.
Non-inferiority testing for “side benefits” cases
This is the most unambiguous case, in which the new solution you want to test has benefits not measurable in the test. Naturally, a solution having such benefits which performs equivalently to the
existing solution, or even slightly worse, would still be the preferred solution, for example due to lower maintenance costs, or better brand integrity, etc.
Some concrete A/B testing examples: removing 360-degree shots of products can result in significant savings for an online merchant, and they might even tolerate a bit lower conversion rate; removing
a free trial period that requires one or two additional customer support personnel can be great if the conversion rate to a paid account remains about the same; removing several payment methods may
significantly simplify payment processing and invoicing, so if it only affects conversions a little bit, it might well be worth doing it.
Non-inferiority testing for “easy decision” cases
This is where it gets interesting, since in many online marketing/UX tests the solution to be tested is:
• easy and cheap to implement
• costs nothing to maintain
• reversible, in many cases with ease
• faces little to no internal and external opposition
Contrast these to the reasons one will want to do superiority testing – it’s a complete reversal. On top of that, in many such cases re-testing is also cheap and easy, including testing the
cumulative changes of, say, 10 consecutive A/B tests versus the control from test #1.
Examples include trivial changes such as color or text changes on a Call to Action (CTA) element, many copy or image changes, the removal or addition of some elements of the site or third-party
functionality such as trust signals, live chat, etc. Think of all the tests you’ve done and case studies you’ve read where the test was for a simple button change or layout change, or text change.
I’m willing to bet a decent sum of money (and I’m no gambler!) that the number of such tests would be higher than 50% of all, meaning that non-inferiority testing should have wider adoption than
superiority tests!
Benefits of using non-inferiority designs
The benefit in the “side benefits” case is quite clear: it is that you would have statistical evaluation of a new solution, that you want to adopt, making sure it is not significantly worse than the
existing one. There is simply no way you can do that with a superiority test – the new solution must be better than the old one in order to pass the test reliably. If you set your noinferiority
margin at the same level as you would set your minimum effect of interest, then you gain nothing in terms of sample size / speed of testing. The math is such that the required time to run the test
would be the same for all practical considerations.
The more interesting case is the “easy decisions” case, since it is here that you can get a very significant improvement in the time required to run a test, allowing you to run many such tests in the
same time it would take you to run one “classic” superiority or non-inferiority test.
Let’s say there is an online SaaS website for which we want to test a simple change of a button text. Currently, the button says “Free Trial” and we want to test whether adding an action-inducing
word to it will change things, so the variant we A/B test is simply “Start Free Trial”. The current free trial conversion rate is 9% and if you were running a classic superiority test you would set
we would be able to reliably detect an improvement of 5% or more. However, since this is easy to implement, costs nothing to maintain and is easily reversible, we can design it as a non-inferiority
test where we would be happy to get data that allows us to make a decision faster, even if it means it might be equivalent to or up to 2% worse than the current text (the noninferiority margin is
2%). Here is what fixed-sample size test would require in terms of sample size per test arm with several different combinations of statistical significance and power levels:
It easy to see that the easy decision design requires only 8.35% of what a classic non-inferiority design would require, giving us a whopping 12-fold increase in the speed to run the test. Of course,
this number depends heavily on both the noninferiority margin and the minimum effect of interest chosen. Changing the minimum effect of interest from 5% to 2% means an easy decision design will now
require 22.23% of a classical non-inferiority design, and about the same proportion of a superiority design with the same minimum effect of interest. Still a very, very nice improvement, but no doubt
less spectacular.
Here is a graphical illustration of the difference between a superiority and an easy-decision non-inferiority design, which helps explain where the efficiency gain comes from:
Needless to say, the two tests won’t have the same conclusions. With a superiority test, you have a certain guarantee that the tested variant is better than the control, while a non-inferiority test
offers guarantees only about the variant not being significantly worse than the control. However, with the significant increase in speed, one is able to test much more variants in the same amount of
Combine this with multivariate testing where it makes sense, and a sequential testing approach (like AGILE), and you have a testing apparatus which can churn through a big number of easy to deploy
testing variants in a relatively short amount of time. Each of them will likely only have the potential to cause a small improvement, but in doing many such tests, the improvements will add up much
more quickly.
Maintaining momentum in any online business isn’t about doing only things that improve your bottom-line, it’s also about making enough such decisions.
Of course, this doesn’t mean to test poorly thought out variants – it will just hurt your bottom-line, but it does mean you will be able to test much more efficiently compared to both classic
non-inferiority tests and superiority tests. Again, our statistical tools already support such tests: simply choose “non-inferiority” test, enter the noninferiority margin, then specify a minimum
effect of interest.
See this in action
Try itA/B Testing Hub
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The risk of cascading loses
While non-inferiority tests have significant benefits in particular situations, they, too, don’t come without issues. The cascading loses / accumulating loses concern is not particular to
non-inferiority testing, but it is of higher concern due to noninferiority margin allowed. In a worst-case scenario, one can do everything properly and still end up accumulating loses over the course
of several tests.
Let’s say we run 5 consecutive tests, each with a non-inferiority margin of 2%. If each A/B test ends up with a winning variant that is 2% worse than the control, we end up with a -10% lift:
The risk is real and grows bigger with the number of A/B tests performed, especially if the observed confidence interval includes negative difference values. It is a risk one may or may not find
acceptable in their particular circumstances.
It can be alleviated by using best practices, user research and common sense to guide the choice of solutions tested, as opposed to automated testing and/or testing for the sake of testing. A way to
control/detect it, and to quantify one’s cumulative results, is to periodically run superiority or non-inferiority A/B tests where the control is a version of the element/page/process from a few
tests ago, and the tested variant is the winner of the latest test. Such tests can combine the outcomes of many A/B tests affecting different parts of a website or ad campaign, in both variant and
control, though the risks of running into the Simpson’s paradox and other segmenting issues increases.
The above is a fairly comprehensive and hopefully more accessible take on the topic of non-inferiority testing. If you are interested in a bit more in-depth take on it, consider downloading my free
white paper: “Non-Inferiority Designs in A/B Testing“.
While superiority testing is currently the default position in both theory and practice, including A/B testing tools, I believe above I make a good case on why it should not be so. There are
certainly cases where classic non-inferiority tests are the only viable solution, while in other cases “easy decision” non-inferiority designs provide both a better fit with regards to the error of
primary concern, and a very significant increase in the efficiency of A/B testing by reducing the number of required users.
Non-inferiority testing doesn’t come without challenges: choosing a proper noninferiority margin adds additional difficulty in the planning phase of the test, while the worst-case scenario of
cascading loses is something one should keep in mind. Still, these are far from insurmountable challenges and I think it’s about time the industry starts considering using non-inferiority tests,
taking note from medical trials and other areas of scientific research where it is already standard practice.
Thoughts and suggestions on non-inferiority tests and how to increase their adoption within the CRO/UX and marketing industry are welcome in the comments below and our social profiles. Please, share
the article to help the discussion.
1 Fisher, R.A. (1935) – “The Design of Experiments”, Edinburgh: Oliver & Boyd
2 Schuirmann, D.J. (1987) – “A comparison of the two one-sided tests procedure and the power approach for assessing equivalence of average bioavailability”, Journal of Pharmacokinetics and
Biopharmaceutics, 15:657–680
3 Silva, G. T. da, Logan, B. R., & Klein, J. P. (2008) – “Methods for Equivalence and Noninferiority Testing”, Biology of Blood and Marrow Transplantation: Journal of the American Society for Blood
and Marrow Transplantation, 15 (1 Suppl), 120–127
4 Walker, E., & Nowacki, A. S. (2011) – “Understanding Equivalence and Noninferiority Testing”, Journal of General Internal Medicine, 26(2), 192–196
About the author
This entry was posted in A/B testing, Conversion optimization, Statistical significance, Statistics and tagged ab testing, ab testing methodology, non-inferiority, non-inferiority design,
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How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function y = int sin^3 t dt from [e^x, 0]? | HIX Tutor
How do you use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function #y = int sin^3 t dt# from #[e^x, 0]#?
Answer 1
So, since #y=int_(e^x)^0sin^3tdt#:
#y'=sin^3 0*0-sin^3e^x*e^x=-e^xsin^3e^x#.
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Answer 2
To use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function ( y = \int_{e^x}^{0} \sin^3(t) , dt ), we first need to evaluate the integral and then differentiate the
result with respect to ( x ). Let's denote the integral as a function of ( x ), say ( F(x) ). Then, according to the Fundamental Theorem of Calculus Part 1, the derivative of ( F(x) ) with respect to
( x ) is given by ( \frac{d}{dx} \int_{e^x}^{0} \sin^3(t) , dt = -\sin^3(e^x) \cdot \frac{d}{dx}(e^x) ).
Differentiating ( e^x ) with respect to ( x ) gives ( \frac{d}{dx}(e^x) = e^x ).
So, the derivative of the function ( y = \int_{e^x}^{0} \sin^3(t) , dt ) with respect to ( x ) is ( -\sin^3(e^x) \cdot e^x ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Blaine - Math Tutor - Learner - The World's Best Tutors
I have been tutoring grades 6-College since 2015. I have a master's degree and a doctorate in mathematics from UCLA, where I also have extensive classroom teaching experience, and I received my
bachelor's degree in mathematics from the University of Chicago.
My tutoring style:
When we’re struggling with new ideas, usually we don’t know what we don’t know. The “real questions” a student has about a subject won’t come up until we do a little digging together. But once we
start asking the real questions, I find that my students surprise themselves with how quickly they can formulate the answers, and then they deal with the classroom material confidently and capably.
Success story:
I worked with a student who had transferred from a school with an unconventional curriculum to a public middle school. The transition was difficult for her, and she was failing math at her new
school. Together we developed a plan to work through the foundational material she hadn’t been taught at her old school. She worked hard to get up to speed, and in the end she passed her algebra
class. I worked with her for two more years, and she ended up in Honors Algebra II!
Hobbies and interests:
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Verleih von Zelten an Jugendgruppen
Frage von Marc Niessen an Ministerin Weykmans,
Jeden Sommer profitieren hunderte von Kindern und Jugendlichen in der Deutschsprachigen Gemeinschaft von den Zeltlagern der Jugendorganisationen. Diese Lager sind als Erfahrung von Gruppenleben und
von Verantwortungsübernahme ganz wichtige Kernelemente der Jugendarbeit in der Deutschsprachigen Gemeinschaft.
Immer wieder ist zu hören, dass die jährliche Aufteilung der Zelte an die verschiedenen Jugendgruppen eine große Herausforderung darstellt. Seit Jahren gibt es nicht genug von den mannshohen Zelten,
um den Bedarf aller Jugendgruppen zu decken. Darüber hinaus sind die verfügbaren Zelte mitunter in einem schlechten Zustand, sodass sie als Schlafplatz nicht zu gebrauchen sind.
Die AG JugO des Rates der Deutschsprachigen Jugend ist mit der Verteilung der Zelte betraut, sie nimmt die Anfragen der Jugendgruppen entgegen und teilt die Zelte dann je nach Verfügbarkeit auf.
Diese Aufgabe ist jedoch aufgrund des Zeltmangels immer wieder schwierig. In diesem Jahr beispielsweise werden zu Spitzenzeiten, das heißt zwischen dem 10. und 20. Juli, zwischen 63 und 67 Zelte
gebraucht. Das sind deutlich mehr, als in den vergangenen Jahren zur Verfügung stand. Hinzu kommt, dass die Zelte der Chiro im vergangenen Jahr bei einem Brand beschädigt wurden und daher nicht zur
Verfügung stehen.
Eine mögliche Lösung des Engpasses wäre die Anschaffung eigener Zelte durch die Jugendorganisationen. Dies wird jedoch dadurch erschwert, dass die Anschaffung von Zelten ausdrücklich von der
Bezuschussung für Materialkosten ausgenommen ist.
Daher folgende Fragen:
• Wie viele Zelte stehen den ostbelgischen Jugendgruppen im Sommer 2018 zur Verfügung (sei es aus dem Bestand der Deutschsprachigen Gemeinschaft oder über Abkommen mit dritten, wie etwa der
französischen Gemeinschaft oder dem Verteidigungsministerium)?
• Weshalb können Jugendgruppen, die bereit sind eigene Zelte anzuschaffen, keinen Zuschuss dafür erhalten?
• Wie möchten Sie das Problem der Zeltverteilung in Zukunft lösen?
Marc Niessen
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Very few people can solve this math problem without using a calculator
Challenge: Can you solve this math quiz for middle schoolers – without a calculator?
Classic brain training methods are perhaps puzzles like crosswords or sudoku, but in recent times I have become more and more attracted to the type of challenge you’ll see below.
These types of puzzles have been flooding the web lately, probably because they are really fun!
These are old classic mathematical problems. When you were in middle or high school.
These tests are more fun when you find yourself trying to remember the math you learned as a child.
Can you figure out the correct solution?
Here is the challenge, in the picture below.
At the top of the picture we see the task and then four possible answers.
Which solution do you think is the correct one?
How did you come up with it?
Take your time and think about it so you can find the correct solution.
Done? Below you can check if you picked the right number!
The correct answer
The correct answer is B: 12.
Why is 12 the correct answer?
Well, if you remember from your school days, according to the order of operations, you do multiplication before addition and subtraction, so you start by solving 3 x 3, which results in 9.
Then we are left with a simpler math problem: 3 + 9 – 3 + 3
The answer is therefore 12.
Did you pick the correct number? Congratulations!
The post Very few people can solve this math problem without using a calculator appeared first on Wake Up Your Mind.
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RS Aggarwal Solutions Class 7 Chapter-3 Decimals (Ex 3C) Exercise 3.3 - Free PDF
RS Aggarwal Solutions Class 7 Chapter-3 Decimals (Ex 3C) Exercise 3.3 - Free PDF
Free PDF download of RS Aggarwal Solutions Class 7 Chapter-3 Decimals (Ex 3C) Exercise 3.3 solved by Expert Mathematics Teachers on Vedantu. All Exercise 3.3 Questions with Solutions for Class 7 RS
Aggarwal to help you to revise complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other Engineering entrance exams. You can also register Online
for NCERT Class 7 Science tuition on Vedantu. to score more marks in your examination.
Every NCERT Solution is provided to make the study simple and interesting on Vedantu. Vedantu is No.1 Online Tutoring Company in India Provides you Free PDF download of NCERT Maths Class 7 solved by
Expert Teachers as per NCERT (CBSE) Book guidelines. All Chapter wise Questions with Solutions to help you to revise complete Syllabus and Score More marks in your examinations.
FAQs on RS Aggarwal Solutions Class 7 Chapter-3 Decimals (Ex 3C) Exercise 3.3
1. What is a decimal and what are its types as mentioned in Class 7 RS Aggarwal solutions?
Algebra deals with decimals. They are numbers that are divided into a number and fraction part by a point also known as a decimal point. For example, in 35.5, 35 is a whole number and 5 is a decimal
whereas, “.” is the decimal point separating both. There are two types namely, recurring and non-recurring decimals. Recurring decimals are repeating and non-terminating decimals. For example,
34.4444444 is a recurring decimal. Non -recurring decimals are non-repeating or non-terminating decimals. For example, 35.4535. They are again divided into infinite and finite decimals based on their
2. What is the place value in decimals as mentioned in Class 7 RS Aggarwal solutions?
Place value is used to define the position of a digit in the whole number. They, in turn, decide the value of the number. For example, in the number 546, 5 is in the hundreds place, 4 is in the tens
place and 6 is in one’s place. When we move from left to right, the value of the number increases by ten times and when we move from right to left, the value of the number decreases 10 times.
Students can visit Vedantu for more details.
3. What are the uses of learning this chapter related to decimals?
• Knowing about decimals is very important when you deal with money.
• We can convert currency into other formats easily if we know the decimals and related concepts.
• The measurement of objects used in our daily life is expressed in decimals sometimes. The weight of the objects is also expressed in decimals. To convert these values, we need to know the decimal
system properly.
4. What are the different properties of decimals as mentioned in class 7 RS Aggarwal solutions?
Following are the properties of decimals :
• The product of the decimal multiplied in order remains the same.
• The product of the whole number and the decimal number is the same when they are multiplied in an order.
• The product of a decimal multiplied by zero is zero.When a decimal number is divided by the same number, then the quotient is 1.
5. How to convert a fraction into a decimal and vice versa as mentioned in Class 7 RS Aggarwal solutions?
Decimals after the decimal points are represented by their place values. To convert decimal to fraction, write down decimal numbers and expand them according to the place value and simplify them. To
convert a fraction into a decimal, just simply solve the fraction. For example, fraction ¼ when converted to decimal results in 0.25.
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Execution time limit is 1 second
Runtime memory usage limit is 128 megabytes
A directed graph is given with a list of edges. Check whether it contains multiedges.
The first line contains number of vertices in a graph n (1 ≤ n ≤ 100) and number of edges m (1 ≤ m ≤ 10000). Each of the next m lines contains pair of integers - the edges of the graph.
Print YES if graph contains multiedges and NO otherwise.
Submissions 7K
Acceptance rate 57%
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Prolog using examples – Part 4
In this part of this practical tutorial, we are going to do some coding involving facts, rules, recursion, and lists.
Example 1: Obtain the head of the list, obtain the tail of the list.
/* Get the head of the list */
get_head(Head, [Head|_]).
/* Get the tail of the list */
get_tail(Tail, [_|Tail]).
?- get_head(Head_is, [a,b,c,d]).
Head_is = a.
?- get_tail(Tail_is, [a,b,c,d]).
Tail_is = [b, c, d].
Example 2: Find out if an element is a member of a list.
/* Base case - We find the element inside the list */
/* The element is found if the head of the list can be unified with the element we are searching */
member(Element, [Element|_]).
/* Recursive function that call member with the tail without taking care of the head of the list */
/* The idea is that the base case will be called and check if the element is found */
member(Element, [_|Tail]) :-
member(Element, Tail).
?- member(a, [a,1,3,b,c]).
true .
/* Using OR ';', we ask Prolog to search for another element in the list */
?- member(a, [a,1,a,b,c]).
true ;
true ;
/* Element is not found in the list */
?- member(a, [c,1,3,b,c]).
Example 3: Find an specific element member inside a list in a determined position.
/* Base case */
/* The element was found in the list was found, set Position to 1 to stop recursion */
get_member_at(1, [Head|_], Head).
/* Recursive case */
/* Go thought the list until position is less than 1 indicating that member was found */
get_member_at(Position, [_|Tail], Element):-
Position > 1,
Temp_Position is Position - 1,
get_member_at(Temp_Position, Tail, Element).
/* Get the member at position 3 of the list */
?- get_member_at(3, [1,3,5,7], Lst).
Lst = 5.
/* This fails when trying to get an element out or bounds of the list */
?- get_member_at(5, [1,3,5,7], Lst).
/* BE CAREFUL, Sometimes prolog can behave weird base on how you use your functors eg.*/
/* The follow is when prolog create dynamic variables because doesn't now the limit of the list */
?- get_member_at(5, Lst, Lst).
Lst = [_G455, _G458, _G461, _G464, **|_G468].
/* In this case, not only produce dynamic variables to be instanciated, but also add
the list [1,a,b] as the fifth head element of the list */
?- get_member_at(5, Lst, [1,a,b]).
Lst = [_G473, _G476, _G479, _G482, [1, a, b]|_G486] ;
Example 4: Lets obtain the numbers of elements that belong to a simple list.
/* Base case - When the list is empty, unify with zero */
number_of_elements([], 0).
/* Recursive case */
number_of_elements([_|Tail], Number_counted) :-
number_of_elements(Tail, counter),
Number_counted is counter + 1.
In the recursive case, Prolog will call number_of_elements passing the tail of the list until the list is empty. Then, for every time number_of_elements was call it would return the counter + 1 to
the previous call. At the end, we will obtain the total number of elements in the list.
?- number_of_elements([a,1,b,2,^, &], Counted).
Counted = 6.
/* In this case the sublist [1,2,3] inside the list is considerate an element as a whole */
?- number_of_elements([a,b,c,[1,2,3]], Counted).
Counted = 4.
?- number_of_elements([[a,b,c,1,2,3]], Counted).
Counted = 1.
If inside the code we would had the unification symbol instead of the word ‘is’, we would have a different result:
/* Base case - When the list is empty, unify with zero */
number_of_elements([], 0).
/* Recursive case */
number_of_elements([_|Tail], Number_counted) :-
number_of_elements(Tail, counter),
Number_counted is counter + 1.
?- number_of_elements([a,1,b,2,^, &], Counted).
Counted = 0+1+1+1+1+1+1.
Example 5: Lets sort a list of elements
/* Base case - Empty list */
/* Base case - [_] indicate we don't case about the element, there is only one, the list is sorted */
/* Recursion case */
sort_list([First_element, Second_element|Tail]):-
First_element =< Second_element,
?- is_list_sorted([1,2,3,4]).
true .
?- is_list_sorted([1,2,3,4]).
true ;
?- is_list_sorted([1,3,3,4]).
?- is_list_sorted([1,3,5,4]).
This topic will continue in the next posting.
© 2010, Alejandro G. Carlstein Ramos Mejia. All rights reserved.
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Saswati Sarkar
• University of Pennsylvania, Department of Electrical Engineering
According to our database
, Saswati Sarkar authored at least 136 papers between 1998 and 2024.
Collaborative distances:
Book In proceedings Article PhD thesis Dataset Other
Online presence:
On csauthors.net:
Capturing the Spread of Information in Heterogeneous V2X Through Scalable Computation.
IEEE/ACM Trans. Netw., April, 2024
Group Testing with General Correlation Using Hypergraphs.
Proceedings of the IEEE International Symposium on Information Theory, 2024
Impact of opinion dynamics on the public health damage inflicted by COVID-19 in the presence of societal heterogeneities.
Frontiers Digit. Health, March, 2023
Containing a spread through sequential learning: to exploit or to explore?
Trans. Mach. Learn. Res., 2023
Compression with Unlabeled Graph Side Information.
Proceedings of the IEEE International Symposium on Information Theory, 2023
Group Testing with Correlation under Edge-Faulty Graphs.
CoRR, 2022
Group Testing with Correlation via Edge-Faulty Graphs.
Proceedings of the IEEE International Symposium on Information Theory, 2022
Optimal Capacity-Constrained COVID-19 Vaccination for Heterogeneous Populations.
Proceedings of the 61st IEEE Conference on Decision and Control, 2022
Is Non-Neutrality Profitable for the Stakeholders of the Internet Market?
IEEE/ACM Trans. Netw., 2020
Spread, Then Target, and Advertise in Waves: Optimal Budget Allocation Across Advertising Channels.
IEEE Trans. Netw. Sci. Eng., 2020
The Interplay of Competition and Cooperation Among Service Providers (Part II).
IEEE Trans. Netw. Sci. Eng., 2020
The Interplay of Competition and Cooperation Among Service Providers (Part I).
IEEE Trans. Netw. Sci. Eng., 2020
Performance Enhancement of Cloud Based Storage using Disk Scheduling Technique.
Int. J. Cloud Appl. Comput., 2020
Modeling the Impact of Traffic Signals on V2V Information Flow.
Proceedings of the 91st IEEE Vehicular Technology Conference, 2020
An Approach to Store Data in Cloud Based Storage Using Mapping Technique.
Proceedings of the Innovations in Bio-Inspired Computing and Applications, 2020
Pricing for Profit in Internet of Things.
IEEE Trans. Netw. Sci. Eng., 2019
Modeling Information Propagation in General V2V-enabled Transportation Networks.
CoRR, 2019
An Eco-Friendly Efficient Cloud-Searching Technique With Delay.
Int. J. Green Comput., 2018
Economics of Quality Sponsored Data in Non-Neutral Networks.
IEEE/ACM Trans. Netw., 2017
Visibility-Aware Optimal Contagion of Malware Epidemics.
IEEE Trans. Autom. Control., 2017
The Value of Side-Information in Secondary Spectrum Markets.
IEEE J. Sel. Areas Commun., 2017
Is Non-Neutrality Profitable for the Stakeholders of the Internet Market? - Part II.
CoRR, 2017
The Interplay of Competition and Cooperation Among Service Providers.
CoRR, 2017
Spread, then target, and advertise in waves: Optimal capital allocation across advertising channels.
Proceedings of the 2017 Information Theory and Applications Workshop, 2017
The economics of competition and cooperation between MNOs and MVNOs.
Proceedings of the 51st Annual Conference on Information Sciences and Systems, 2017
Quality-Sensitive Price Competition in Secondary Market Spectrum Oligopoly - Single Location Game.
IEEE/ACM Trans. Netw., 2016
Optimal Patching in Clustered Malware Epidemics.
IEEE/ACM Trans. Netw., 2016
Uncertain Price Competition in a Duopoly With Heterogeneous Availability.
IEEE Trans. Autom. Control., 2016
Spectrum White Space Trade in Cognitive Radio Networks.
IEEE Trans. Autom. Control., 2016
Secondary spectrum oligopoly market over large locations.
Proceedings of the 2016 Information Theory and Applications Workshop, 2016
Secondary spectrum market: To acquire or not to acquire side information?
Proceedings of the IEEE International Symposium on Information Theory, 2016
Migration to a non-neutral internet: Economics modeling and analysis of impact.
Proceedings of the 2016 Annual Conference on Information Science and Systems, 2016
Multiperiod subscription pricing for cellular wireless entrants.
Proceedings of the 2016 Annual Conference on Information Science and Systems, 2016
Optimal Energy-Aware Epidemic Routing in DTNs.
IEEE Trans. Autom. Control., 2015
Portfolio Optimization in Secondary Spectrum Markets.
EAI Endorsed Trans. Wirel. Spectr., 2015
Strategic Interaction Among Different Entities in Internet of Things.
CoRR, 2015
Taming epidemic outbreaks in mobile adhoc networks.
Ad Hoc Networks, 2015
Dynamic Contract Trading in Spectrum Markets.
IEEE Trans. Autom. Control., 2014
Network NonNeutrality on the Internet: Content Provision Under a Subscription Revenue Model.
SIGMETRICS Perform. Evaluation Rev., 2014
Network Non-Neutrality on the Internet: Content Provision Under a Subscription Revenue Model.
CoRR, 2014
Quality Sensitive Price Competition in Spectrum Oligopoly: Part II.
CoRR, 2014
Quality Sensitive Price Competition in Spectrum Oligopoly: Part 1.
CoRR, 2014
Spectrum pricing games with correlated bandwidth availabilities and demands.
Proceedings of the 2014 Information Theory and Applications Workshop, 2014
Market-based power allocation for a differentially priced FDMA system.
Proceedings of the 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA, June 29, 2014
Quality sensitive price competition in spectrum oligopoly over multiple locations.
Proceedings of the 48th Annual Conference on Information Sciences and Systems, 2014
Optimal Routing and Scheduling in Multihop Wireless Renewable Energy Networks.
IEEE Trans. Autom. Control., 2013
Quality sensitive price competition in spectrum oligopoly.
Proceedings of the 2013 IEEE International Symposium on Information Theory, 2013
Optimal routing and scheduling in wireless networks with nodes powered by renewable and traditional energy sources.
Proceedings of the First International Black Sea Conference on Communications and Networking, 2013
Cooperative Profit Sharing in Coalition-Based Resource Allocation in Wireless Networks.
IEEE/ACM Trans. Netw., 2012
Maximum Damage Malware Attack in Mobile Wireless Networks.
IEEE/ACM Trans. Netw., 2012
Generic coverage verification without location information using dimension reduction.
IEEE/ACM Trans. Netw., 2012
Optimal Dissemination of Security Patches in Mobile Wireless Networks.
IEEE Trans. Inf. Theory, 2012
Optimal Quarantining of Wireless Malware Through Reception Gain Control.
IEEE Trans. Autom. Control., 2012
Delay Guarantees for Throughput-Optimal Wireless Link Scheduling.
IEEE Trans. Autom. Control., 2012
Saddle-Point Strategies in Malware Attack.
IEEE J. Sel. Areas Commun., 2012
Spectrum Pricing Games with Random Valuations of Secondary Users.
IEEE J. Sel. Areas Commun., 2012
Spectrum Pricing Games with Spatial Reuse in Cognitive Radio Networks.
IEEE J. Sel. Areas Commun., 2012
Closing the Pandora's box: Defenses for thwarting epidemic outbreaks in mobile adhoc networks.
Proceedings of the 9th IEEE International Conference on Mobile Ad-Hoc and Sensor Systems, 2012
Delay minimization for random scheduling in centralized wireless networks.
Proceedings of the 46th Annual Conference on Information Sciences and Systems, 2012
Uncertain price competition in a duopoly: Impact of heterogeneous availability of the commodity under sale.
Proceedings of the 50th Annual Allerton Conference on Communication, 2012
Provider-Customer Coalitional Games.
IEEE/ACM Trans. Netw., 2011
Lifetime and coverage guarantees through distributed coordinate-free sensor activation.
IEEE/ACM Trans. Netw., 2011
Maximum Damage Battery Depletion Attack in Mobile Sensor Networks.
IEEE Trans. Autom. Control., 2011
Spectrum pricing games with arbitrary bandwidth availability probabilities.
Proceedings of the 2011 IEEE International Symposium on Information Theory Proceedings, 2011
A dynamic game solution to malware attack.
Proceedings of the INFOCOM 2011. 30th IEEE International Conference on Computer Communications, 2011
Optimal control of epidemic evolution.
Proceedings of the INFOCOM 2011. 30th IEEE International Conference on Computer Communications, 2011
Marker gene selection for sample classification using a new MST based clustering algorithm.
Proceedings of the ICWET '11 International Conference & Workshop on Emerging Trends in Technology, Mumbai, Maharashtra, India, February 25, 2011
Market-based control of epidemics.
Proceedings of the 49th Annual Allerton Conference on Communication, 2011
Spectrum Auction Framework for Access Allocation in Cognitive Radio Networks.
IEEE/ACM Trans. Netw., 2010
Economy of Spectrum Access in Time Varying Multichannel Networks.
IEEE Trans. Mob. Comput., 2010
Information concealing games.
IEEE Trans. Inf. Theory, 2010
Optimal propagation of security patches in mobile wireless networks: extended abstract.
Proceedings of the SIGMETRICS 2010, 2010
Spectrum pricing games with bandwidth uncertainty and spatial reuse in cognitive radio networks.
Proceedings of the 11th ACM Interational Symposium on Mobile Ad Hoc Networking and Computing, 2010
Dynamic malware attack in energy-constrained mobile wireless networks.
Proceedings of the Information Theory and Applications Workshop, 2010
Change Management in Enterprise IT Systems: Process Modeling and Capacity-optimal Scheduling.
Proceedings of the INFOCOM 2010. 29th IEEE International Conference on Computer Communications, 2010
Robust Control in Sparse Mobile Ad-Hoc Networks.
Proceedings of the Decision and Game Theory for Security, 2010
Dispatch then stop: Optimal dissemination of security patches in mobile wireless networks.
Proceedings of the 49th IEEE Conference on Decision and Control, 2010
Dynamic contract trading in spectrum markets.
Proceedings of the 48th Annual Allerton Conference on Communication, 2010
Delay Guarantees for Throughput-Optimal Wireless Link Scheduling.
Proceedings of the INFOCOM 2009. 28th IEEE International Conference on Computer Communications, 2009
A hierarchical spatial game over licenced resources.
Proceedings of the 1st International Conference on Game Theory for Networks, 2009
Throughput-optimal scheduling in multichannel access point networks under infrequent channel measurements.
IEEE Trans. Wirel. Commun., 2008
Characterizing Temporal SNR Variation in 802.11 Networks.
IEEE Trans. Veh. Technol., 2008
Throughput and Fairness Guarantees Through Maximal Scheduling in Wireless Networks.
IEEE Trans. Inf. Theory, 2008
Arbitrary Throughput Versus Complexity Tradeoffs in Wireless Networks Using Graph Partitioning.
IEEE Trans. Autom. Control., 2008
Queue Length Stability in Trees Under Slowly Convergent Traffic Using Sequential Maximal Scheduling.
IEEE Trans. Autom. Control., 2008
Stable Scheduling Policies for Maximizing Throughput in Generalized Constrained Queueing Systems.
IEEE Trans. Autom. Control., 2008
Information Acquisition and Exploitation in Multichannel Wireless Networks
CoRR, 2008
Information Concealing Games.
Proceedings of the INFOCOM 2008. 27th IEEE International Conference on Computer Communications, 2008
A fair scheduling policy for wireless channels with intermittent connectivity.
Proceedings of the 42nd Annual Conference on Information Sciences and Systems, 2008
A coalitional game framework for cooperative secondary spectrum access.
Proceedings of the 46th Annual Allerton Conference on Communication, 2008
Admission Control Framework to Provide Guaranteed Delay in Error-Prone Wireless Channel.
IEEE Trans. Veh. Technol., 2007
Fair Coalitions for Power-Aware Routing in Wireless Networks.
IEEE Trans. Mob. Comput., 2007
Constrained Markov games with transition probabilities controlled by a single player.
Proceedings of the 2nd International Conference on Performance Evaluation Methodolgies and Tools, 2007
A Survey of Throughput Versus Complexity Tradeoffs in Wireless Networks.
Proceedings of the Network Control and Optimization, 2007
Dynamic quorum policy for maximizing throughput in limited information multiparty MAC.
IEEE/ACM Trans. Netw., 2006
Minimizing Delay in Loss-Tolerant MAC Layer Multicast.
IEEE Trans. Inf. Theory, 2006
A framework for misuse detection in ad hoc networks- part II.
IEEE J. Sel. Areas Commun., 2006
A framework for misuse detection in ad hoc Networks-part I.
IEEE J. Sel. Areas Commun., 2006
Fairness and throughput guarantees with maximal scheduling in multi-hop wireless networks.
Proceedings of the 4th International Symposium on Modeling and Optimization in Mobile, 2006
Optimizing transmission rate in wireless channels using adaptive probes.
Proceedings of the Joint International Conference on Measurement and Modeling of Computer Systems, 2006
A Statistical Framework for Intrusion Detection in Ad Hoc Networks.
Proceedings of the INFOCOM 2006. 25th IEEE International Conference on Computer Communications, 2006
Jointly optimal transmission and probing strategies for multichannel wireless systems.
Proceedings of the 40th Annual Conference on Information Sciences and Systems, 2006
Optimal communication in bluetooth piconets.
IEEE Trans. Veh. Technol., 2005
Fair distributed congestion control in multirate multicast networks.
IEEE/ACM Trans. Netw., 2005
Back pressure based multicast scheduling for fair bandwidth allocation.
IEEE Trans. Neural Networks, 2005
Wireless multicast: theory and approaches.
IEEE Trans. Inf. Theory, 2005
End-to-end bandwidth guarantees through fair local spectrum share in wireless ad-hoc networks.
IEEE Trans. Autom. Control., 2005
Can Bluetooth succeed as a large-scale ad hoc networking technology?
IEEE J. Sel. Areas Commun., 2005
Maxmin fair scheduling in wireless ad hoc networks.
IEEE J. Sel. Areas Commun., 2005
Realizing the benefits of user-level channel diversity.
Comput. Commun. Rev., 2005
RIDA: Robust Intrusion Detection in Ad Hoc Networks.
Proceedings of the NETWORKING 2005: Networking Technologies, 2005
The Impact of Imperfect Information in Multi-Channel Wireless Systems.
Proceedings of the 44th IEEE IEEE Conference on Decision and Control and 8th European Control Conference Control, 2005
Utility Optimal Scheduling for General Reward States and Stability Constraint.
Proceedings of the 44th IEEE IEEE Conference on Decision and Control and 8th European Control Conference Control, 2005
Optimum scheduling and memory management in input queued switches with finite buffer space.
IEEE Trans. Inf. Theory, 2004
Fair Bandwidth Allocation for Multicasting in Networks with Discrete Feasible Set.
IEEE Trans. Computers, 2004
Achieving proportional fairness using local information in Aloha networks.
IEEE Trans. Autom. Control., 2004
Efficacy of misuse detection in ad hoc networks.
Proceedings of the First Annual IEEE Communications Society Conference on Sensor and Ad Hoc Communications and Networks, 2004
An adaptive strategy for maximizing throughput in MAC layer wireless multicast.
Proceedings of the 5th ACM Interational Symposium on Mobile Ad Hoc Networking and Computing, 2004
On Optimal Placement of Intrusion Detection Modules in Sensor Networks.
Proceedings of the 1st International Conference on Broadband Networks (BROADNETS 2004), 2004
A framework for optimal battery management for wireless nodes.
IEEE J. Sel. Areas Commun., 2003
Optimum Scheduling and Memory Management in Input Queued Switches with FiniteBuffer Space.
Proceedings of the Proceedings IEEE INFOCOM 2003, The 22nd Annual Joint Conference of the IEEE Computer and Communications Societies, San Franciso, CA, USA, March 30, 2003
Stochastic control techniques for throughput optimal wireless multicast.
Proceedings of the 42nd IEEE Conference on Decision and Control, 2003
Channel assignment algorithms satisfying cochannel and adjacent channel reuse constraints in cellular mobile networks.
IEEE Trans. Veh. Technol., 2002
A framework for routing and congestion control for multicast information flows.
IEEE Trans. Inf. Theory, 2002
Fairness in cellular mobile networks.
IEEE Trans. Inf. Theory, 2002
Fair allocation of utilities in multirate multicast networks: a framework for unifying diverse fairness objectives.
IEEE Trans. Autom. Control., 2002
A scalable low-overhead rate control algorithm for multirate multicast sessions.
IEEE J. Sel. Areas Commun., 2002
Providing stochastic delay guarantees through channel characteristics based resource reservation in wireless network.
Proceedings of the Fifth International Workshop on Wireless Mobile Multimedia, 2002
Maxmin fair scheduling in wireless networks.
Proceedings of the Proceedings IEEE INFOCOM 2002, 2002
A jointly optimum scheduling and memory management for matching based service.
Proceedings of the 41st IEEE Conference on Decision and Control, 2002
A Simple Rate Control Algorithm for Maximizing Total User Utility.
Proceedings of the Proceedings IEEE INFOCOM 2001, 2001
Optimization Based Rate Control for Multirate Multicast Sessions.
Proceedings of the Proceedings IEEE INFOCOM 2001, 2001
Fair Allocation of Discrete Bandwidth Layers in Multicast Networks.
Proceedings of the Proceedings IEEE INFOCOM 2000, 2000
Distributed Algorithms for Computation of Fair Rates in Multirate Multicast Trees.
Proceedings of the Proceedings IEEE INFOCOM 2000, 2000
A Framework for Routing and Congestion Control in Multicast Networks.
Proceedings of the Proceedings IEEE INFOCOM '99, 1999
Channel Assignment Algorithms Satisfying Co-Channel and Adjacent Channel Reuse Constraints in Cellular Mobile Networks.
Proceedings of the Proceedings IEEE INFOCOM '98, The Conference on Computer Communications, Seventeenth Annual Joint Conference of the IEEE Computer and Communications Societies, Gateway to the 21st
Century, San Francisco, CA, USA, March 29, 1998
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How is the information stored in each leaf node of a Segment Tree related to the array it represents - ITEagers
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Machine Learning Chapter 2.2: Multiple Linear Regression
Welcome to Part 2.2 of Machine Learning!
Here is the equation for multiple linear regression. As you can see, it is quite similar to our linear regression model.
Assumptions of linear regression
Now, let's look at the first dataset for the linear regression model assumption.
The first dataset is working well and serving its purpose. But look at the other datasets. They are not serving their purpose and are misleading. So, we shouldn't use linear regression in those
These four datasets are known as Anscombe's quartet, and they show that you can't just blindly use linear regression everywhere. You need to make sure your dataset is suitable for linear regression.
That's why the assumptions of linear regression are important.
Now let’s learn about them
1. Linearity
(Linear relationship between Y and each X)
The first assumption is linearity. We need a linear relationship between the dependent and each independent variable.
2. Homoscedasticity
(equal variance)
Even though it sounds complex, homoscedasticity simply means equal variance. You don't want to see a cone shape on your chart, whether increasing or decreasing, as it means variance depends on
the independent variable. In this case, we wouldn't use linear regression.
3. Multivariate Normality
(Normality of error distribution)
If you look at the chart on the right, something seems off. Ideally, along the line of linear regression, you should see a normal distribution of data points. Here, it's different, so we wouldn't
use linear regression.
4. Independence
(of observation. Includes “no autocorrelation”)
We don't want any pattern in our data. A pattern indicate that rows are not independent, meaning some rows affect others. A classic example is the stock market, where past prices influence future
prices. In such cases, we wouldn't use a linear regression model.
5. Lack of Multicollinearity
(predictors are not correlated with each other)
The fifth assumption is lack of multicollinearity. We want our independent variables or predictors not to be correlated. If they're not correlated, we can build a linear regression. If they are,
the coefficient estimates in the model will be unreliable.
6. The outlier Check
(this is not an assumption, but an ‘extra’)
The sixth point is checking for outliers. This isn't a real assumption but an extra step to remember when making linear regression models. If you look at the chart, you can see the outlier is
greatly affecting the regression line. So, we need to decide whether to remove outliers before making the model or keep them in. This choice depends on your understanding of the business and the
data set.
Multiple Linear Regression Intuition
Dummy Variables
We gonna learn about dummy variables here. Here we have information on each company's profit, along with their spending on R&D, administration, and marketing. These are the expenses the company has,
and then there's the state where it operates, either NY or California. Our challenge is to see if there's any connection between profit and all these variables, and if we can create a model to
predict profit. So, profit is our dependent variable, and the blue ones are independent variables. We need to build a linear regression model.
Now, for the equation: Y is our profit. Then, we have the B1 coefficient times the X1 variable, which is the R&D spend. X1 represents the dollar amounts in the R&D column. Then, there's the Admin
variable, which is X2, so that’s how the equation is. But for the state, we have to figure out what we should place here.
So the approach you need while facing categorical variable is to create a dummy variable. First, identify each categorical value and create a column for each one. Here, we have only New York and
California, so we build columns for them (we are kind of expanding our dataset)
Now to fill up the column (this is very interesting actually) - Put a 1 in the New York column for New York and 0 for everything else. Do the same for California
Now these two columns are called dummy variables. Building your regression model from here is very simple. All you need to do is use the New York column instead of the state names. You add a variable
that is multiplied by D1, which is your dummy variable for New York. You don't use the California column either.
So, as you can see, all the information in our data is kept intact. If we just stick to the one New York column, you can tell right away if D1 is 1 then it's a company that operates in New York. If
D2 is a 0 it's a company that operates in California.
We didn't lose any information by only including the New York column.
[Trick: Another thing I want to say is that dummy variables work like a switch. If the value is 1, the switch is on, meaning the company operates in New York. If the value is 0, it means the switch
is off, indicating the company does not operate in New York]
Dummy Variable Trap
Do not use multiple dummy variables. You can never include both the dummy variables at one. In our example, we used NY only and found b4D1.
When one or more independent variables in a linear regression predict another, it's called multicollinearity. Because of this, the model can't tell the difference between the effects of D1 and D2. As
a result, it won't work properly. This is known as the dummy variable trap.
[If you do the math, you'll see the real issue: you can't have the constant and both dummy variables in your model simultaneously]
To sum up, when building a model, always include only one less than the total number of dummy variables in a set. If you have nine, include eight; if you have 100, include 99. Apply this rule to each
set of dummy variables.
I hope this explanation was helpful and that you will never fall victim to the dummy variable trap in your modeling.
P - value
A P-value is a statistical measure that helps you understand the significance of your research results. It shows the chance of seeing the data, or something more extreme, if the null hypothesis is
true. Simply put, a low P-value (usually ≤ 0.05) means there is strong evidence against the null hypothesis, suggesting you might want to reject it. On the other hand, a high P-value means the data
fits well with the null hypothesis, indicating there's not enough evidence to reject it.
P-values are often used in hypothesis testing to help researchers make conclusions from their data.
Model Building
Do you remember the good old days when we had just one dependent variable and one independent variable? Everything was simple, and we only had to build a straightforward linear regression. It worked
But now, our data has many columns. Those easy days are over, and all these columns could be predictors for a dependent variable. There are so many of them, and we need to decide which ones to keep
and which ones to discard.
You might wonder why we need to remove columns or get rid of data. Why can't we just use everything in our model?
Well, I can think of two reasons right away.
First, "garbage in, garbage out." If you add too much unnecessary data, your model won't be reliable; it'll be a "garbage model."
Second, at the end of the day, you'll need to explain these variables and understand what they mean in predicting your dependent variable's behavior. Explaining a thousand variables isn't practical,
so keep only the important ones that actually predict something.
There are five rules to build a model.
[Sometimes you'll hear about stepwise regression, which actually refers to steps 2, 3, and 4, as it follows true step-by-step methods]
Method number one: All In.
This isn't a technical term; I just call it "all in." Basically, it means throwing in all your variables. This is something we just discussed that we shouldn't do.
When would you use this method?
1. One reason is if you have prior knowledge. If you know these variables are true predictors, you don't need to build anything new. You might know this from domain knowledge, past experience, or
because someone provided these variables for the model. In that case, you just build the model.
2. Another reason could be if your company has a framework requiring these variables. It's like prior knowledge, but not your choice. For instance, a bank might need to use specific variables to
predict loan defaults.
3. Lastly, you would use this method if you're preparing for a backward elimination type of regression, which is our next topic.
So, let's move on to backward elimination.
After Step 5, you go back to Step 3.
Once again, look for the variable with the highest p-value in your new model. Remove it. This is essentially Step 4, where you take out the variable.
Then, fit the model again with one less variable. Keep repeating this process until you reach a point where even the variable with the highest p-value is still less than your significance level.
If the condition where p is greater than the significance level is not met, then you don't proceed with Step 4 anymore. You move to the end, and in this case, "end" means finish.
Your model is ready when all the remaining variables have p-values less than the significance level.
That's how the backward elimination method works. Let's move on to the next one.
We started with Step 1:
Select the significance level to enter the model. In this case, we choose 5 percent.
Next, we move to Step 2:
We fit all possible simple regression models. This means we take the dependent variable and create a regression model with each independent variable we have. From all these models, we select the one
with the lowest p-value for the independent variable. As you can see, this involves a lot of work.
Then, we proceed to Step 3:
We keep the variable we've just chosen and fit all other possible models by adding one extra predictor to the one we already have.
What does this mean?
It means we've selected a simple linear regression with one variable. Now, we need to construct all possible linear regressions with two variables, where one of those variables is the one we've
already selected. Essentially, we add each of the other variables one by one. We decide, "Let's add this variable," and then, "Let's add the next one," but separately. We construct all possible
two-variable linear regressions while definitely keeping the variable we've already selected.
So, what do we do after that?
Out of all these possible two-variable regressions, we consider the one where the new variable we added has the lowest p-value. If that p-value is less than our significance level, it means the
variable is significant, so we go back to Step 3.
What does that mean?
It means we now have a regression with two variables, and we will add a third variable. We'll try all possible remaining variables as our third variable. From all these models with three variables,
we'll proceed to Step 4 and select the one with the lowest p-value for the third variable we added.
We continue this process. Essentially, we keep expanding the regression model by carefully selecting from all possible combinations, adding one variable at a time. We stop when the variable we add
has a p-value greater than our significance level. When this condition is not met, we don't return to Step 3; we finish the regression. Why? Because the variable we just added is no longer
significant. We also know we selected the one with the lowest p-value, so there is no other variable we can add that will have a p-value less than our significance level in any further regression.
From this point on, the new variable will always be insignificant.
So, this is where we finish the regression.
The key is to keep the previous model, not the current one.
This makes sense because you've just added an insignificant variable. So, there's no point in keeping it; just go back one step.
That's how forward selection works.
I know it can be a bit confusing, but try to understand and maybe read these instructions again.
It makes more sense when you picture what is happening.
And next we're moving on to the bidirectional limit elimination.
This method combines two steps. First, choose a significance level for entering and staying in the model. Use the same level for both. New variables must have a significance level lower than the
entry threshold to be added.
Next, perform backward elimination. Try to remove unnecessary variables, then return to add another variable. Each time you add a variable, perform backward elimination again. Remove variables if
possible, then return to add more.
Continue this iterative process until you can't add or remove variables. At this point, your model is complete. This method can be tedious, so it's best handled by a computer. This is how
bidirectional elimination, or stepwise regression works.
And finally, let's discuss all possible models.
This is probably the most thorough approach, but it's also the most resource-intensive. You start by selecting a criterion for goodness of fit, like the R-squared value. There are many different
criteria you can choose from. Then, you construct all possible regression models. If you have 'n' variables, there will be 2 to the power of 'n' minus one total combinations of these variables.
That's exactly how many models there can be. In the final step, you select the model with the best criterion.
There you go, your model is ready.
It sounds easy, but let's look at an example. Even if you have 10 columns in your data, you'll have 1,023 models. That's an enormous number of models. And we're not even talking about columns you've
already filtered out. For instance, in our example, you might have five or six columns. Now, imagine when you get a dataset that you need to analyze, which typically has around 100 columns. That is
In conclusion, we have five methods for building models: backward elimination, selection by direction, and score comparison.
Multiple Linear Regression in Python
Resources :
1. Google Colab file: https://colab.research.google.com/drive/1Lp16gstLKT6DfhTPNsbwdG3BgEjWDZIO (copy this file)
2. Datasheet: https://drive.google.com/file/d/1-RL-SsWNo0PhP_goWfXrwnauRuFDidl9/view (download and upload in colab file)
3. Data preprocessing template: https://colab.research.google.com/drive/17Rhvn-G597KS3p-Iztorermis__Mibcz
In this dataset, each row represents a startup. For each startup, data scientists collected information on R&D spending, administration spending, marketing spending, the state, and profit. The goal
is for the VC fund to decide which startup to invest in based on this information. We have data from 50 startups. If you train a model to understand these correlations, you can use it to predict the
profit of a new startup. And yes, after copying the colab file, delete all the code cells. Just the code cells, not the text cell.
Code implementation:
First of all, we need to copy paste our templates into our colab file.
Now, change the datasheet name to 50_Startsup. Okay, then move to the encoding categorical part and copy paste the OneHotEncoder from your data preprocessing file (we did it on the first blog)
Here, we just have to change the column which we want to one hot encode. According to the previous one, the data had categorical values on the index of 0. In our current sheet, the categorical value
is on the 3rd column.
Now let’s print X. Run all the cells and you will see your categorical data has been transformed beautifully.
Looking at our dataset, the first row has "New York" as a state, encoded as 0, 0, 1. The second row has "California," encoded as 1, 0, 0. Lastly, "Florida" is encoded as 0, 1, 0. That's the one-hot
encoding, and now we have a fully pre-processed dataset.
• We don't need to apply feature scaling. In Multiple Linear Regression, coefficients adjust for different feature values, so scaling isn't necessary.
• Do we need to check the assumptions of linear regression? The answer is absolutely not.
(Don't worry about Multiple Linear Regression assumptions. If your dataset has linear relationships, the regression will work well and be accurate. If not, it will perform poorly, and you can try
another model. That's it.)
• Do we have to do anything to avoid the dummy variable trap? The answer is no.
The class we have important to work with multiple linear regression is trained on several actions. So don’t worry about it. The redundant one will be outcaste automatically.
• Do we have to work on selecting the best feature?
Why? For the exact same reason as the dummy variable trap. Our class will automatically detect the best feature, you know, the feature with highest P - value/ the most statistically significant
to figure out how to predict the dependent variable and so on.
Okay, now let’s move to building model. But before that, I have a good news. The class we're about to use for building and training this Multiple Linear Regression model is the same as for the Simple
Linear Regression model. It will recognize multiple features for multiple linear regression, but everything else remains the same. All your features and the profit, your dependent variable, will be
handled. It will manage the dummy variable trap and select the most statistically significant features.
We now have a fully trained linear regression model on this dataset. It understands the correlations between different types of spending by 50 startups and their profit. Investors can use this model
to predict the profit of new startups based on this information.
Thanks to this linear regression class, you don't have to worry about the dummy variable trap or selecting the best features. The class handles it all.
But, we need to understand something important. Unlike simple linear regression, we now have four features instead of one. We can't plot a graph like before because we would need a five-dimensional
graph, which isn't possible for us to visualize. Instead, we'll display two vectors: one for the real profits from the test set and another for the predicted profits. The test set is 20% of the
dataset, so with 50 observations, we'll have 10 samples. This allows us to compare the predicted profits with the real profits for each startup. And that's how we will evaluate our model. Later, you
will learn about evaluation techniques to better measure the performance of your regression models with relevant metrics. For now, we'll see if our model performs well on new observations by
comparing predictions to real results on the test set.
Predicting the Test set results
y_pred will be the vector of predicted profits in the test set. First, use the regressor object and apply the predict method with the test features. Next, call numpy and use the set_printoptions
function to set the precision to 2, which will display two decimal places. Finally, use the concatenate function from numpy to combine vectors or arrays vertically or horizontally. In the
parentheses, include y_pred, which is our vector of predicted profits. To arrange it vertically, use .reshape. This attribute allows you to reshape vectors or arrays. We need the number of elements
in y_pred, which is the number of columns, so we use the len function here. Lastly, specify one, meaning you want to reshape your y_pred vector into an array with len(y_pred) rows and just one
Next, copy-paste and put y_test instead of y_train. The axis here takes two values. 0 means we want to do a vertical concatenation and 1 means horizontal concatenation.
Now print and we can see 2 vectors here. On the left we have our predicted profits (y_pred) and on the right we have our vector of real profits [for 10 startups of the test set]. Now compare them.
The first one is really good. prediction was 103015.2 and the actual one was 103282.38 - which is surely an amazing prediction.
Extra Content
Free BONUS exercise:
Question 1: How do I use my multiple linear regression model to make a single prediction, for example, the profit of a startup with R&D Spend = 160000, Administration Spend = 130000, Marketing Spend
= 300000 and State = California?
Question 2: How do I get the final regression equation y = b0 + b1 x1 + b2 x2 + ... with the final values of the coefficients?
ans: https://colab.research.google.com/drive/1ABjLFzknByfU4-F4roa1hX36H3aZlu6J?usp=sharing
Enjoy the BONUS!
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Math::Round - Perl extension for rounding numbers
use Math::Round qw(...those desired... or :all);
$rounded = round($scalar);
@rounded = round(LIST...);
$rounded = nearest($target, $scalar);
@rounded = nearest($target, LIST...);
# and other functions as described below
Math::Round supplies functions that will round numbers in different ways. The functions round and nearest are exported by default; others are available as described below. "use ... qw(:all)" exports
all functions.
Rounds the number(s) to the nearest integer. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are halfway between two integers are rounded "to
infinity"; i.e., positive values are rounded up (e.g., 2.5 becomes 3) and negative values down (e.g., -2.5 becomes -3).
Starting in Perl 5.22, the POSIX module by default exports all functions, including one named "round". If you use both POSIX and this module, exercise due caution.
Rounds the number(s) to the nearest integer. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are halfway between two integers are rounded to the
nearest even number; e.g., 2.5 becomes 2, 3.5 becomes 4, and -2.5 becomes -2.
Rounds the number(s) to the nearest integer. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are halfway between two integers are rounded to the
nearest odd number; e.g., 3.5 becomes 3, 4.5 becomes 5, and -3.5 becomes -3.
Rounds the number(s) to the nearest integer. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are halfway between two integers are rounded up or
down in a random fashion. For example, in a large number of trials, 2.5 will become 2 half the time and 3 half the time.
Rounds the number(s) to the nearest multiple of the target value. TARGET must be positive. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are
halfway between two multiples of the target will be rounded to infinity. For example:
nearest(10, 44) yields 40
nearest(10, 46) 50
nearest(10, 45) 50
nearest(25, 328) 325
nearest(.1, 4.567) 4.6
nearest(10, -45) -50
Rounds the number(s) to the nearest multiple of the target value. TARGET must be positive. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are
halfway between two multiples of the target will be rounded to the ceiling, i.e. the next algebraically higher multiple. For example:
nearest_ceil(10, 44) yields 40
nearest_ceil(10, 45) 50
nearest_ceil(10, -45) -40
Rounds the number(s) to the nearest multiple of the target value. TARGET must be positive. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are
halfway between two multiples of the target will be rounded to the floor, i.e. the next algebraically lower multiple. For example:
nearest_floor(10, 44) yields 40
nearest_floor(10, 45) 40
nearest_floor(10, -45) -50
Rounds the number(s) to the nearest multiple of the target value. TARGET must be positive. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are
halfway between two multiples of the target will be rounded up or down in a random fashion. For example, in a large number of trials, nearest(10, 45) will yield 40 half the time and 50 half the
Returns the next lower multiple of the number(s) in LIST. TARGET must be positive. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are between
two multiples of the target will be adjusted to the nearest multiples of LIST that are algebraically lower. For example:
nlowmult(10, 44) yields 40
nlowmult(10, 46) 40
nlowmult(25, 328) 325
nlowmult(.1, 4.567) 4.5
nlowmult(10, -41) -50
Returns the next higher multiple of the number(s) in LIST. TARGET must be positive. In scalar context, returns a single value; in list context, returns a list of values. Numbers that are between
two multiples of the target will be adjusted to the nearest multiples of LIST that are algebraically higher. For example:
nhimult(10, 44) yields 50
nhimult(10, 46) 50
nhimult(25, 328) 350
nhimult(.1, 4.512) 4.6
nhimult(10, -49) -40
The variable $Math::Round::half is used by most routines in this module. Its value is very slightly larger than 0.5, for reasons explained below. If you find that your application does not deliver
the expected results, you may reset this variable at will.
Floating-point numbers are, of course, a rational subset of the real numbers, so calculations with them are not always exact. Numbers that are supposed to be halfway between two others may surprise
you; for instance, 0.85 may not be exactly halfway between 0.8 and 0.9, and (0.75 - 0.7) may not be the same as (0.85 - 0.8).
In order to give more predictable results, these routines use a value for one-half that is slightly larger than 0.5. Nevertheless, if the numbers to be rounded are stored as floating-point, they will
be subject as usual to the mercies of your hardware, your C compiler, etc.
Math::Round was written by Geoffrey Rommel <GROMMEL@cpan.org> in October 2000.
This software is copyright (c) 2000 by Geoffrey Rommel <grommel@cpan.org>.
This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself.
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Triangle in a Trapezium
Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?
You may find it useful to print off some dotty paper or use the dotty grids environment to explore the first part of this problem.
Triangle in a Trapezium printable sheet - problem introduction
Triangle in a Trapezium printable sheet - starting points
In the two trapeziums below, we have drawn triangles by joining the vertices of one of the non-parallel sides to the midpoint of the opposite side.
Is there a relationship between the area of each trapezium and the area of the shaded triangle contained within it?
Draw some more trapeziums and construct triangles inside them in the same way. Does the same relationship hold?
Can you find a way to prove that your relationship will always hold for any trapezium?
Once you have had a go at proving it, click below to see some possible starting points.
Can you take each starting point and develop it into a full proof?
Starting Point 1
In this diagram, an extra line has been drawn joining E, the midpoint of AB, to F, the midpoint of CD.
How could you use this diagram to show that the area of the triangle is half the area of the trapezium?
Starting Point 2
Take a look at this sequence of three images. What happens at each stage?
How could you use this sequence of three images to create a proof that the area of the triangle is half the area of the trapezium?
Starting Point 3
Take a look at the image below. Can you see how the trapezium ABCD has been transformed to create this image?
How could you use this image to prove that Triangle CED has half the area of Trapezium ABCD? Click below to show two more images that might help.
Getting Started
For each of the starting points, it might be a good idea to label some of the side lengths and heights with letters.
Starting Point 1
How would you work out the area of the two unshaded triangles? You might need to choose some letters to represent certain quantities...
Using the same letters, can you work out the area of the trapezium?
Starting Point 2
How does triangle BEG compare with triangle AEF? How do you know?
How do we know that triangle CED and triangle CFD have the same area?
Starting Point 3
The trapezium has been rotated $180^{\circ}$ around F.
Can you prove that ABGH is a parallelogram? It might help to mark angles that you know are equal, and angles that you know add up to $180^{\circ}$.
Student Solutions
Thank you to everyone who sent in their solutions to this problem. We received either full or partial solutions from Sammy at Lancing College; Tobias and Ana (no schools given); Sanika from PSBBMS in
India; and Quang from the British Vietnamese School in Vietnam. As you discovered, there were several ways to prove this result and we suggested three possibles approaches.
Method One
For this method, we suggested drawing an extra line joining E and F, the midpoints of AB and CD respectively.
Several of you suggested that it would be helpful to explore another relationship before going any further. For example, Sanika noted the usefulness of exploring "a relationship between the lengths
of BC, EF and AD and also figure the heights of the two smaller trapeziums."
Well spotted, Sanika and everyone else who realised that this would be useful. This relationship, which several of you uncovered in different ways, is known as the mid-segment theorem for trapezia
and it can be very helpful for Method 1. It states that if a line connecting the midpoints of the two legs of a trapezium is parallel to the bases, then its length is equal to half the sum of the
lengths of the bases.
Sammy, from Lancing College, made good use of this theorem in this solution:
Method Two
Well done to everyone who followed the sequence of images to explore Method Two.
Here's how Sanika approached the problem using Method 2:
To prove that the resultant figure GCDF is a parallelogram, we would have to prove that the triangles EAF and EBG are congruent.
AE = BE (as E is the midpoint of the line segment AB)
Angle GEB = Angle FEA (Vertically opposite angles)
GB and AF would be parallel as they are extensions of parallel lines. Therefore, we can say that angle GBE = angle FAE as they are alternate-interior angles.
Therefore Triangle EAF is congruent to triangle EBG.
This implies that the area of the trapezium is the same as the area of the parallelogram.
We can then invert the parallelogram so that the base and height of both the parallelogram and the triangle would be the same measure.
Let base be 'b' and let the height be 'h'.
Therefore the area of the parallelogram = bh
Area of the triangle = bh/2
Since the area of the parallelogram is the same as the area of the trapezium we can say that the triangle is half the area of the trapezium.
Method 3
Here's Sanika's sketch:
Have you noticed how Sanika has annotated our original diagram to help her solve the problem? A good sketch is always worth the time spent on it.
From the diagram, we can tell that the line segment BG will be equal to line segment AH. They will also be parallel as opposite sides of a Trapezium are parallel. Since we have one pair of opposite
sides which are parallel and equal, we can deduce that the quadrilateral ABGH is a parallelogram.
Once she was convinced that the shape was a parallelogram, Sanika followed the hints provided for Method 3:
As opposite angles of a parallelogram are equal, the following will hold true:
Angle B = Angle H
Angle A = Angle G
Using the SAS [side-angle-side] congruence criteria, we can say the following:
Triangle EAD is congruent to triangle JGC (Thus, line segment ED = JC)
Triangle CBE is congruent to triangle DHJ (So line segment CE = DJ).
We can then rearrange triangles CBE, CGJ, AED and DHJ, as shown in the image above, to form the quadrilateral DECJ (or CJDE).
Since AE and HJ are of the same length 'x', we can join them together.
Similarly as BE and GJ also share a length of 'x', we can join them together.
As proved earlier, triangle EAD is congruent to triangle JGC and triangle CBE is congruent to triangle DHJ so the bases will be the same, in which case we can join them to form the quadrilateral
DECJ. This is identical to the two triangles contained in the trapezium, put together.
Therefore, we can say that the area of the quadrilateral DECJ is half the area of the parallelogram BGHA.
As the parallelogram is just 2 copies of the trapezium, we can say that the area of CED is half the area of the trapezium ABCD.
Well done to Sanika and everyone else who explored this problem. We hope you enjoyed investigating different approaches towards finding a solution.
Teachers' Resources
Why do this problem?
Students often think that there is only one correct way to approach a geometrical challenge. In this problem, students have the opportunity to discover a geometrical relationship, and then explore
three starting points that lead to different methods to prove the relationship always holds. Along the way there's a chance for some "purposeful practice" at finding the area of triangles and
Possible approach
Students may find it useful to work on dotty paper or use the dotty grids environment to explore the first part of this problem.
Start by showing the two trapeziums below, and explain that the triangles join the vertices of one of the non-parallel sides to the midpoint of the opposite side.
Invite students to work out the area of each trapezium and shaded triangle, and then discuss how they worked out the areas. Then ask them to draw some more trapeziums and construct triangles inside
them in the same way. After everyone has had time to create some examples, collect together their results in a table on the board.
"What do you notice about the area of the triangle compared with the area of the trapezium?"
"It's always half the area!"
Invite students to think about how they could prove that the area of the triangle is always half the area of the trapezium. After they have had some time to think about it, you could hand out this
worksheet: Triangle In a Trapezium. Challenge them to take each starting point and turn it into a complete justification. There are three different starting points so each group could work on a
different starting point and then present the complete proof to the whole class. Finally, take time to discuss which method they found most convincing or appealing, and why.
Key questions
Starting Point 1
How would you work out the area of the two unshaded triangles?
Does it help to choose letters to represent some of the lengths and heights?
Using the same letters, can you work out the area of the trapezium?
Starting Point 2
How does triangle BEG compare with triangle AEF? How do you know?
How do we know that triangle CED and triangle CFD have the same area?
Starting Point 3
Can you prove that ABGH is a parallelogram?
Does it help to mark angles that you know are equal, and angles that you know add up to $180^{\circ}$?
Possible support
To scaffold the step from particular cases to the general proof, invite students to create the images from each Starting Point on dotty paper for trapezia that they have already explored. Then
encourage them to adapt the reasoning from their numerical examples to prove the general case.
Possible extension
Kite in a Square is a more challenging geometrical problem which offers three different jumbled up proofs for students to sort out.
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Solving Systems of Equations Using Cramer's Rule Worksheet
(1) Solve the following systems of linear equations by Cramer’s rule:
(i) 5x − 2y + 16 = 0, x + 3y − 7 = 0 Solution
(ii) (3/x) + 2y = 12, (2/x) + 3y = 13 Solution
(iii) 3x + 3y − z =11, 2x − y + 2z = 9, 4x + 3y + 2z = 25
(iv) (3/x) - (4/y) - (2/z) - 1 = 0
(1/x) + (2/y) + (1/z) - 2 = 0
(2/x) - (5/y) - (4/z) + 1 = 0 Solution
(2) In a competitive examination, one mark is awarded for every correct answer while 1/4 mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many
questions did he answer correctly ? (Use Cramer’s rule to solve the problem)
(3) A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution ? (Use Cramer’s rule to solve the
problem). Solution
(4) A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. However, pump B can pump water in or out at the same rate. If pump B is inadvertently run in reverse, then the
tank will be filled in 30 minutes. How long would it take each pump to fill the tank by itself ? (Use Cramer’s rule to solve the problem).
(5) A family of 3 people went out for dinner in a restaurant. The cost of two dosai, three idlies and two vadais is ₹ 150. The cost of the two dosai, two idlies and four vadais is ₹200. The cost of
five dosai, four idlies and two vadais is ₹250. The family has ₹350 in hand and they ate 3 dosai and six idlies and six vadais. Will they be able to manage to pay the bill within the amount they had
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
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List of Math Words That Start With F - Self Development Journey
List of Math Words That Start With F
Looking for math words that start with F?
From Finite to Formula everything in between, there are loads of words beginning with the letter F that are used in math and other numbers sciences.
Here is a list of words starting with F that you may see being used in the subject of Math:
Math Words That Start With F
Face – a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron.
Face Of A Polyhedron – Face: the flat surfaces that make up a polyhedron are called its faces.
Factor – number or algebraic expression that divides another number or expression evenly—i.e., with no remainder.
Factor Of A Polynomial – A factor of polynomial P(x) is any polynomial which divides evenly into P(x).
Factor Of An Integer – a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder.
Factor Theorem – a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.
Factor Tree – a way of expressing the factors of a number, specifically the prime factorization of a number.
Factorial – the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point.
Factoring – when you break a number down into smaller numbers that, multiplied together, give you that original number.
Factoring Rules – a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder.
Fibonacci Sequence – a series of numbers in which each number is the sum of the two that precede it.
Finite – having bounds or limits; not infinite; measurable.
Finite Number – a set that has a finite number of elements. In simple words, it is a set that you can finish counting.
First Derivative – a point is the slope of the tangent line at that point.
First Derivative Test – a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a
saddle point.
First Order Differential Equation – an equation of the form F(t,y,˙y)=0.
First Quartile – A quartile divides data into three points—a lower quartile, median, and upper quartile—to form four groups of the dataset.
Five Number Summary – A five-number summary is especially useful in descriptive analyses or during the preliminary investigation of a large data set.
Fixed – a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function’s domain that is mapped to itself by the function.
Flip – A flip is a motion in geometry in which an object is turned over a straight line to form a mirror image.
Floor Function – the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted floor or ⌊x⌋.
Focal Radius – refers to the distance from a point on a conic section to a focus.
Foci Of A Hyperbola – Two fixed points located inside each curve of a hyperbola that are used in the curve’s formal definition.
Foci Of An Ellipse – two fixed points on its major axis such that sum of the distance of any point, on the ellipse, from these two points, is constant.
Focus – a point used to construct a conic section.
Focus Of A Parabola – set of all points in a plane which are an equal distance away from a given point and given line.
Foil Method – In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method.
Foot – A unit of length (or distance) in US units equal to 12 inches.
Formula – a mathematical rule or relationship that uses letters to represent amounts which can be changed – these are called variables.
Fractal – a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension.
Fraction – number expressed as a quotient, in which a numerator is divided by a denominator.
Fraction Rules – To add or subtract fractions they must have the same denominator (the bottom value).
Fractional Equation – an equation containing the unknown in the denominator of one or more terms.
Fractional Exponents – If an exponent of a number is a fraction, it is called a fractional exponent.
Fractional Expression – Fractional expressions are fractions that have a variable in the denominator.
Frequency – refers to the number of times an event or a value occurs.
Frequency Table – a frequency distribution is a list, table or graph that displays the frequency of various outcomes in a sample.
Frustum Of A Cone Or Pyramid – In geometry, a frustum is the portion of a solid that lies between one or two parallel planes cutting it.
Function – an expression, rule, or law that defines a relationship between one variable and another variable.
Function Operations – rules that we follow to solve functions.
I hope you found the words you were looking for from the list above.
This isn’t an exhaustive list, if there are any math words starting with the letter F that you would like added to the list, please leave me a comment below.
If you’d like to explore more math words starting with different letters of the alphabet, click any of the letters below to go to the list for that letter:
Image credits – Photo by Joshua Hoehne on Unsplash
Phil lives in England, UK, and has around 20 years experience as a professional life, career and executive coach. He started this blog to help others find and define their own self development
journey. Blogging about a wide range of topics to help facilitate a better future.
Leave a Comment
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FuzzyResampling: Resampling Methods for Triangular and Trapezoidal Fuzzy Numbers
The classical (i.e. Efron's, see Efron and Tibshirani (1994, ISBN:978-0412042317) "An Introduction to the Bootstrap") bootstrap is widely used for both the real (i.e. "crisp") and fuzzy data. The
main aim of the algorithms implemented in this package is to overcome a problem with repetition of a few distinct values and to create fuzzy numbers, which are "similar" (but not the same) to values
from the initial sample. To do this, different characteristics of triangular/trapezoidal numbers are kept (like the value, the ambiguity, etc., see Grzegorzewski et al. <doi:10.2991/
eusflat-19.2019.68>, Grzegorzewski et al. (2020) <doi:10.2991/ijcis.d.201012.003>, Grzegorzewski et al. (2020) <doi:10.34768/amcs-2020-0022>, Grzegorzewski and Romaniuk (2022) <doi:10.1007/
978-3-030-95929-6_3>, Romaniuk and Hryniewicz (2019) <doi:10.1007/s00500-018-3251-5>). Some additional procedures related to these resampling methods are also provided, like calculation of the
Bertoluzza et al.'s distance (aka the mid/spread distance, see Bertoluzza et al. (1995) "On a new class of distances between fuzzy numbers") and estimation of the p-value of the one- and two- sample
bootstrapped test for the mean (see Lubiano et al. (2016, <doi:10.1016/j.ejor.2015.11.016>)). Additionally, there are procedures which randomly generate trapezoidal fuzzy numbers using some
well-known statistical distributions.
Version: 0.6.4
Imports: stats, utils
Suggests: testthat (≥ 3.0.0), R.rsp
Published: 2024-10-04
DOI: 10.32614/CRAN.package.FuzzyResampling
Author: Maciej Romaniuk [aut, cre], Przemyslaw Grzegorzewski [aut], Olgierd Hryniewicz [aut]
Maintainer: Maciej Romaniuk <mroman at ibspan.waw.pl>
BugReports: https://github.com/mroman-ibs/FuzzyResampling/issues
License: GPL-3
URL: https://github.com/mroman-ibs/FuzzyResampling
NeedsCompilation: no
Citation: FuzzyResampling citation info
Materials: README NEWS
CRAN checks: FuzzyResampling results
Reference manual: FuzzyResampling.pdf
Vignettes: Resampling Fuzzy Numbers with Statistical Applications: FuzzyResampling Package (source)
Package source: FuzzyResampling_0.6.4.tar.gz
Windows binaries: r-devel: FuzzyResampling_0.6.4.zip, r-release: FuzzyResampling_0.6.4.zip, r-oldrel: FuzzyResampling_0.6.4.zip
macOS binaries: r-release (arm64): FuzzyResampling_0.6.4.tgz, r-oldrel (arm64): FuzzyResampling_0.6.4.tgz, r-release (x86_64): FuzzyResampling_0.6.4.tgz, r-oldrel (x86_64):
Old sources: FuzzyResampling archive
Please use the canonical form https://CRAN.R-project.org/package=FuzzyResampling to link to this page.
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Yariana Diaz, she/her
I am a Postdoctoral Fellow at Macalester College. My research interests include quiver representation theory, persistence theory, topological data analysis, and coding theory.
I received my Ph.D. in mathematics from the University of Iowa in August 2023. I also received an M.S. in mathematics and completed the Graduate Certificate in College Teaching at the University of
Iowa in 2020 and 2022. I graduated from Amherst College in 2018 with a dual degree B.A. in mathematics and music.
The National Alliance for Doctoral Studies in the Mathematical Sciences (Math Alliance)
The National Association of Mathematicians (NAM)
Society for Advancement of Chicanos/Hispanics & Native Americans in Science (SACNAS)
Association for Women in Mathematics (AWM)
American Mathematical Society (AMS)
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Adding fraction
Lets begen
+ = =1
That’s an easy one
But at the end we let the
proper fraction to mixed
fraction . but how is it
First : we need to see if the denominator is the same.
Second : we add the numerator
Third : if the numerator is greater than the
denominator so we will let it as a mixed fraction
How to let the proper fraction to
First : we need to know how many to multiply the denominator to get the same
number as the numerator or less.
Second : we put the number we multiplied as a whole number.
Third : we put the remainder as the numerator .
At last : we put the same denominator.
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a textbook of fluid mechanics and hydraulic machines by rajput Archives - Free PDF Books
Home Tags A textbook of fluid mechanics and hydraulic machines by rajput
Tag: a textbook of fluid mechanics and hydraulic machines by rajput
Fluid Mechanics and Hydraulic Machines PDF by Zoeb Husain, Mohd Zulkifly Abdullah and Zainal Alimuddin :: Fluid mechanics is concerned with the behaviour of liquids and gases at rest and in motion.
The proper understanding of mechanics of fluids is important in many branches of engineering: in biomechanics the flow of blood is of interest; ocean currents require a knowledge offluid mechanics;
chemical processing of plants require a thorough knowledge offluid mechanics; aeronautical engineers require knowledge of flow of air over the aircraft to reduce drag and increase lift; mechanical
engineers require knowledge of fluid propel1ies to design pumps, water turbines, gas turbines and rockets;civil engineers require fluid mechanics to study river currents and erosion; and
environmentalists require knowledge of fluid properties for solving pollution problems of air and water to control flood, irrigation channels, etc. There are special ised books on fluid mechanics for
each of these areas and therefore this book will present only general properties of fluid flow.
Before we study fluid mechanics let us discuss the dimensions and units that will be used in this book. There are four fundamental dimensions: length, mass, time and temperature. The dimensions of
all other quantities can be expressed in terms of fundamental dimensions. For example, Force can be expressed in terms of fundamental dimensions of mass,length and time. is significant. Mach number
where compressibility is important in flows over aerofoils in aircraft.
The dimensionless parameters are also useful in design of prototypes from the models and can save a lot of money and effort. For example, a model can be prepared in a laboratory and tested, and
predictions can be made of the prototype for large machines with the help of suitable dimensionless parameters. This is usually done in making models of large hydraulic machines used in power
stations or in construction of big dams by making suitable models in the laboratory.
In fluid mechanics the pressure results from a normal compressive force acting on an area. The pressure p is defined as force per unit area. In SI units the unit of measurement of pressure is Newtons
per square meter (N/m2 ) or Pascal (Pa). Since Pascal is small unit, the pressure is usually referred to in kilo Pascal (kPa) or even in Mega Pascal (M Pa). The standard atmospheric pressure at sea
level is 101.3 kPa. The gauge pressure is the pressure recorded by the gauge or manometer. In engineering calculations absolute pressure is used and the conversion from gauge pressure to absolute
pressure is carried out using the following equation.
Shear stresses are developed when the fluid is in motion; if the particles of the fluid move relative to each other, so that they have different velocities, causing the original shape of the fluid to
become distorted. A fluid at rest has no shearing forces. Usually we are concerned with the flow past a solid boundary. The fluid in contact with the boundary sticks to it, and therefore will have
the same velocity as the boundary. Considering successive layers parallel to the boundary as shown in Fig. 1.2, the velocity of the fluid varies from layer to layer in y-direction.
Fluid Mechanics and Hydraulic Machines PDF by Zoeb Husain ( BS Publications )
Book Description:
Following a concise overview of fluid mechanics knowledgeable by numerous engineering applications and examples, this reference presents and analyzes main kinds of fluid equipment and the main
courses of generators, in addition to pump expertise. It affords professionals and college students in hydraulic engineering with background ideas in addition to sensible protection of contemporary
turbine technologies, fully explaining the advantages of both steam and fuel generators. Description, design, and operational data for the Pelton, Francis, Propeller, and Kaplan turbines are
provided, as are outlines of varied types of power plants. It gives solved examples, chapter issues, and a thorough case examine.
About this Book:
Fluid mechanics is concerned with the behaviour of liquids and gases and is important in many branches of engineering biomechanics, oceanography, chemical, aeronautical, mechanical, and civil
engineering including environmental studies. There are specialized books on fluid mechanics for each of these areas and therefore this book will present only general properties of fluid flow. The
book examines various forms of energy, especially thermo and hydro power, and provides outlines of various types of power plants. An outstanding feature of the book is the classification of fluid
mechanics. Contents Dimensions and Systems of Units Fluid Flow Thermal and Hydropower Stations Fluid Machinery Pelton Turbine Francis Turbine Propeller and Kaplan Turbines Turbo Pumps Positive
Displacement Pumps.
Book Content:
Dimensions and Systems of Units
Dimensions and Units
Non-Dimensional Quantity
Pressure Scales
Fluid Properties
Surface Tension
CapilIary Action
Compressibility and Mach Number
Fluid Flow
Scope of Fluid Mechnanics
Laminar and Turbulent Flow
Momentum Equation for One-Dimensional and Two-Dimensional Flow
Jet Striking a Plate (3 cases)
Force Exerted when jet is Deflected by a Moving Vane
Euler’s and BernQullis Equations and Application of Bernoullis Equation
Thermal and Hydropower Stations
Steam Turbine and Gas Turbine Power Plant
Combined Cycle Power Plants and Hydropower Plants
Underground Power Plants and Surface Power Plant
Fluid Machinery
Classification of Fluid Machines
Pumps (Axial and Radial) and Compressors (Axial and Radial)
Turbines (Pelton Wheel, Francis Turbine, Kaplan Turbine, Steam Turbines, Gas Turbine)
Euler’s Theory Applied to Turbo Machines
Pelton Turbine
Description of Pelton Turbine Installation and Analysis
Pelton Turbine Losses and Efficiencies
Regulating System of Pelton Wheel Power Station
Francis Turbine
Description of Francis Turbine and Analysis
Draft Tube
Working Proportions and Regulation of Francis Turbine
Specific Speed of hydraulic Turbines
Comparison between Pelton and Francis Turbines
Propeller and Kaplan Turbines
Description of Propeller Turbine and Kaplan Turbine
Analysis and Construction of Velocity Diagram
Twisted Blades
Comparison between Francis, Pelton and Kaplan Turbine
Turbo Pumps
Human Heart (Pump)
Description of Centrifugal Pump and Analysis
Cavitation and Net Positive Suction Head (NPSH)
Pumps in Series and Parallel
Matching Pumps to a System Demand
Axial Flow Pump
Positive Displacement Pumps
Description of a Reciprocating Pump and Analysis (Power Output, Pump Efficiency)
Application of Piston Pumps (Radial Piston Pump, Swashplate Pump, Wobble Plate Pump, Bent Axis Piston Pump and Gear Pump)
Fluid Mechanics S.K. Mondal Notes on above link.
Basic fluid mechanics and hydraulic machines PDF
Author(s): Zoeb Husain; Mohd Zulkifly Abdullah; Zainal Alimuddin
Publisher: BS Publications , Year: 2008
ISBN: 9788178001487,8178001489,9781441661609,1441661603
Get Book from link given below
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Seeking help with SVM (Support Vector Machines) in Stata – who to hire? | Hire Someone To Take My SAS Assignment
Seeking help with SVM (Support Vector Machines) in Stata – who to hire? When trying to introduce SVM, I encountered several problems. In one case I got a high degree of help because I followed some
instructions from Dr. Martin Van der Westhuizen and Michael Willet. But after some time, I found a method for getting help in this case. (Stata SE, Ver. 2.2) Get help: I found this tool in Stata
software which helps you with svm/svm matrix. It works better when you know how to extract multiple data points. This is also why I use it for matrix. Show me success: I show you the results of my
test across a bunch of cases and by using as I explain above…: So I decided to experiment with some different SVM programs in my application. This is what I saw in the test examples. Please enter the
number. In my test block I used the number that I saw there: Next I created an SVM binary matrix Then I ran another SVM (for the same number of rows as the test matrices) and used as in the test:
Then I used the test SVM to merge the rows of the binary matrix with the ones of the SVM binary matrix I finally got the result: So, what my SVM did for me: I used binary matrices to extract 2×2 rows
from the rows of the binary matrix. So far I got about $280K which is huge. I ran $400K to get around this problem. I am still perplexed. When I tried to use SVM in the complex problem when I have
lots of matrices I would have said No. You either give too many errors like with large numbers of rows or you give too big result. But I got an answer:SVM is not a complex problem – it is very
effective in matrix search algorithm since it works at 4 n matrix dimensions and the bigger one of the row of the matrix has almost maximal height. Like in complex problem the inputs are rather
straight-forward and some inputs can be an integer without any need to process it.
How Do You Pass A Failing Class?
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myself 🙂 Any hints will go into where to find some help on Stata While trying to get SVM over Stata using MATLAB (MathWorks) Mathworks Group I ran the example on matlab. Try to add to your results
how to do svm in MATLAB. You’ll have to keep their Matlab code as a simple example. To get help, on the matlab screen you need 1 line of code:Stata_Search. You might have to file a bug… I hope
thatSeeking help with SVM (Support Vector Machines) in Stata – who to hire? “The main reason why I have spent so much time there is you are the task, you do take responsibility for the task.” One of
the main reasons why I have spent so much time in Stata – whether private, public or privately – is because you are the task, you are the responsible party, you are on the lookout for problems and
hence you are on the watch. Before getting into the details, here is a short summary explaining how Stata (and Stata in general) can handle your needs in SVM (Support Vector Machines) and how to get
the best performance in Stata (some variations). First, if you get the right hardware (or software) then it is also time to get the right processes and they come first. Second, they are more likely
to get better performance with the new software like Stata (or Stata in general) using lots of data from different places on a data bus. Third, Stata (already in Stata – statetheory.do.rtd) is only
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is at least for Stata (for the practical purposes of this book I went into the specific scope of modelling Stata. So if there is an issue with one “most important” argument that could be turned out
to be useful, I would suggest that you read the whole Stata (or Stata in general) section on how to overcome this problem. It will help you learn Stata more and understand the main points it
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inclined. How should one get to know more about Stata and Stata in general? By using Stata, there is a great many new questions people want to know about Stata and Stata in general. Perhaps you can
find a couple of things to keep in mind if you want to learn the topics. Does the Stata and Stata computer currently work? As a very simple example I would create a new scenario which uses the LVM
that has two different models: 1- The machine runs in the LVM -1- In the LVM the machine is capable to get the job done faster with the idea of being able to hit with the right result. So in an
example, a machine that runs in the LVM will get the job done faster with a machine that is able to get the job performed more quickly with the idea of getting faster with the idea of being able to
get good results I think that Stata or Stata in general has the potential to offer an abstraction for how to improve the performance of a machine with Stata and then using it for instance when you
are working with a dataset which is small enough: var training = new STatacasto() { def forEach(self, obj): if obj.get_row() == 4 and obj.get_subcount() <= 2: for x in range(2, len(obj)) and len(obj)
> 1: >>>Seeking my latest blog post with SVM (Support Vector Machines) in Stata – who to hire? There is a huge amount of work being done by individuals and organisations that would not be suitable
for a computer science student to be taught the SVM you would be learning with the help of a software program. They could be learning languages on a computer, but also a compiler problem. However,
and I am not sure that I see any need to offer such a person the help of such a software program, I am not aware of such a person having an interest in that in Stata or Stata/PostgreSQL, as I am just
learning with Stata, therefore I am asked by someone who is doing what I have been asked, and is ready to implement that in the next year or two. How Do Those Guys Learn Languages, Controllers and
SQL I have just been to Stata with some students looking for help with writing SQL software written using Stata (in Stata) like SQL (PostgreSQL) was only my first major job. Their problems were
either on their own too, or the learning has begun. I am looking for like five people to assign with help from if they want, but the one who can do this in Stata is hard to compare but I will stick
with the best possible person. Each person has a different way to learn SQL procedures and logic, but I ask them one by one who is working on the most recent SVM. At the time they might put a SQL
solution into stata or PostgreSQL but I am looking for someone who understands the language needed to use SQL one by one. As a final note, I recommend using Stryp and Beldan as the most proficient
people for this job. SVM (SQL-5.2) I had not thought about using Stryp myself, but I want to thank so many people that have taken the offer to Stata in the past, and have been working towards using
it, which can be done without any homework with much effort, right? I already learned Stryp by running from SVM to SQL-5l. But the class was more detailed, so a beginners program can do it very well.
I am not only interested in good programming, but also I would like to show if writing a SVM program that is easy to use, well compiled, and easy to read.
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For me this would be a good program, but also it IS a person, having been a Computer Science student myself, would any of them find good way to learn it, or does it show great interest? SVM (SVM-5.3)
This is from MS C using the Stift “SVM”, so it is a quick and easy process. You must be ready when you need it. Stryp help from SVM (SVM-5.2) will be very helpful for the process. The problem area
here I am getting to. I have purchased a computer and have been using it for about 2 years now. The information is that I have to test to make sure that the threading just works properly in Stexk and
write to a session file using some SVM class, as demonstrated in chapter 5 for Windows 10.6. Hi all, I have some information about State and the answer to this question has been posted sometime now:
SVM help from SVM by Stryp, can you provide us a bit of help from Stryp in your current situation, I am curious what kind of a post that you are getting in this regard and which you have purchased
from State which you have had the info about from Sciric. If you can please suggest me. Thanks But actually the reason why I have not been able to use Streep is because of writing Stryp on DRI (File
and RDri) which must be a “ducky”
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The Law Times Reports
result of that construction operated in favour of the tenant, which would not be the case if the views adopted here by Salter, J. are to prevail.
In Nye v. Davis (sup.) Horridge and Shearman, JJ. held that an obligation on the part of the landlord of a flat to bring up his tenant's coals once a day and remove his ashes was attendance
sufficient to take the tenancy in that case out of the Act. Salter, J. in the present case has expressed the opinion that the provision by the landlord of a minimum of furniture might be sufficient
to bring a tenancy within the other exception-" use of furniture." I cannot refrain from saying that if either of these views be correct the result is, in my judgment, to make something very like
nonsense of this exception. It seems to me impossible to believe that the Legislature could have intended by it that a tenant should lose all the benefit of the statute because, in the one case, he
had not to carry up his own coals, and the substantial benefit of the Act because, in the other, he had the use of a few of his landlord's chairs. And it will not be forgotten that under the
exception in the Act of 1915 the full benefit of the Act was, on this construction, lost in both cases with no protection at all from the presence of the words "bonâ fide.”
Accordingly, I ask myself, is the court imperatively required by the force of the language used to place such a construction upon this exception? In my judgment, nothing less constraining will
justify it in doing so. In my opinion, we have no such burden laid on us.
I quite agree that as a mere matter of language, such a service as carrying up coals is attendance," partial though it be. I agree also that, in the same way, the use of a few chairs is the use of
furniture," insignificant though they be. As a mere matter of words, each of these expressions may quite properly be taken to mean very little, although, with at least equal propriety they may be
taken to connote a great deal more.
But, in my judgment, so much may not be said of the third word "board" with which these two other expressions are associated. The word chosen is, it will be noticed, not "food" or drink," but
"board." Food " may, of course, mean much or little; "drink," I hope, is entitled to an equally non-committal construction ; board," however, is a different word altogether. It is defined, I see, in
Murray as "daily meals provided in a lodging or boarding house, according to stipulation the supply of daily provisions."
The word, without suffix or affix, suggests, to my mind, sufficiency. It could never, I think, be satisfied by the provision, say of an early morning cup of tea. If you wish to accentuate its
abundance you may call it "full board," but if you would convey that it is limited, then you must call it "partial" or qualify it by the use of some other adjective of limitation. It appears to me
that the natural interpretation of the word as we find it in this exception involves the conception of a
[CT. OF APP.
provision by the landlord of such food as, in the case of any particular tenancy, would ordinarily be consumed at daily meals and would be obtained and prepared by a tenant for himself if it were not
provided by somebody else. If a de minimis construction of the word is open at all, and, deferring to my Lord, I must recognise that it is, it is, at least I think, not so fairly open as the other, a
consideration which confirms the conclusion to which I have already on broader grounds been led with reference to that word, as well as to either of the other two expressions of the exception-"
attendance and use of furniture." And if the reasoning of this judgment be so far justified, does it indicate any useful test or standard within the four corners of the Act by reference to which, in
any particular case, the judge of fact can direct himself upon the issue whether the tenancy in question is within the exception of the statute or is outside it? I think it does. Remembering that the
Act applies to prescribed tenancies of what I may call, for brevity, unfurnished and unattended houses, I think it may properly be said that a tenancy is within the exception and is outside the Act
if the landlord receives payment for, and provides and prepares food for, his tenant's meals, which, having regard to all the circumstances of the case, the tenant would otherwise ordinarily provide
for himself; or provides such attendance as, for ordinary household purposes, the tenant would, in the circumstances, otherwise provide for himself; or provides for the tenant's use so much furniture
that, when it is in the house, that house can no longer be described as an unfurnished house. Notwithstanding the side note to sect. 9, it is not, I think, possible to say, as Avory, J. does, that
the words use of furniture necessarily import that the house must be in the ordinary intendment of language, as applied to the particular residence, a furnished house." But in my judgment, to satisfy
the words use of furniture in the place where they are found, it is at least essential that the furniture in the house of which the use is enjoyed is sufficient in quantity and character to require
the judge to say that the house let is no longer one to which the Act normally applies-namely, an unfurnished house. This Act applies to many different kinds of houses, and to tenancies in all parts
of the country, varying toto coelo one from another. What is "board," attendance," or use of furniture" in relation to one tenancy may be nothing of the kind in relation to a second. The County Court
judge is the judge of fact in every disputed case, and he will, of course, have regard to all its circumstances. Directions for his guidance, if they are to be sound, must therefore be general. For
that reason they may not in every case be helpful. But these which I have ventured to indicate are, at all events in my judgment, not without justification from the terms of the Act itself, and they
would help, I should hope, to decide this and many other cases. I am of opinion that this appeal should be allowed, and
CT. OF APP.]
ATTORNEY-GENERAL v. BURNS AND OTHERS.
for myself I would be for referring the case back to the learned County Court judge to decide the question between the parties, directing himself by the considerations which I have endeavoured in
this judgment to explain.
Case remitted. New trial ordered.
Solicitors for the appellants, Blyth, Dutton, Hartley, and Blyth, agents for Graham-Hooper and Betteridge, Brighton.
Solicitors for the respondent, Radford and Frankland, agents for J. Lord Thompson and Weeks, Brighton.
Tuesday, March 6.
(Before Lord STERNDALE, M.R., WARRINGTON, and ATKIN, L.JJ.) ATTORNEY-GENERAL V. BURNS AND OTHERS (a). APPEAL FROM THE KING'S BENCH DIVISION.
Revenue-Estate duty-Property situate out of United Kingdom-Legacy duty paid on former death once and for all-Claim for estate duty— Legacy duty payable but for relationshipReason legacy duty not
payable because it had been paid-Legacy Duty Act 1796 (36 Geo. 3, c. 52), s. 12—Finance Act 1894 (57 & 58 Vict. c. 30), s. 2, sub-s. 2.
J. B. bequeathed to the trustees of his will large sums of money to be invested and the income therefrom to be paid to his daughter during her life and after her death to hold the investments upon
trust for her children in such manner and in such shares as his daughter should appoint, and in default of any such appointment for the children of his daughter in equal shares. J. B., who was
domiciled in England, died in 1890, and legacy duty was paid under sect. 12 of the Legacy Duty Act 1796 upon 600,000l. the sum set aside to meet the bequest to the trustees for his daughter and her
children. The daughter by her will exercised the power of appointment in favour of her children and died in 1900. The funds which passed under the power of appointment exercised by her will to her
children consisted exclusively of stocks and bonds of American companies and corporations. The legacy duty was paid on the death of J. B. under sect. 12 of the Legacy Duty Act 1796 once and for all
as the duty payable by the persons in succession was at one and the same rate. Sect. 1 of the Finance Act 1894 enacted that estate duty should be payable on property which passed on the death of the
deceased, and sect. 2 of that Act provided for the payment of estate duty, and sub-sect. 2 of that section provided that property which passed on the death of the deceased when situated out of the
United Kingdom, should be included only if, under the law in force prior to the passing of the Act, legacy or succession duty was payable in respect thereof, or would be so payable but for the
relationship of the person to whom it passed. The Crown, however, claimed estate duty on the American securities passing on the death of the daughter to her children on the (a) Reported by GEOFFREY
P. LANGWORTHY, Esq., Barristerat-Law.
[CT. OF APP.
ground that the legacy duty would have been payable but for the relationship of the parties. Sankey, J. held, that by reason of sect. 12 of the Legacy Duty Act 1796, legacy duty was not payable on
the death of the daughter as legacy duty had been paid once and for all on the death of J. B. ; but that legacy duty would have been payable on the death of the daughter but for the relationship of
the parties to whom the property passed, and that therefore estate duty was payable under sect. 2, sub-sect. 2, of the Finance Act 1894 on the death of the daughter in respect of the American
securities representing her share under J. B.'s will. On appeal
Held, that the reason legacy duty was not payable on the death of the daughter on the American securities passing to her children under her will was not because of the relationship of the parties,
but because it had been paid once and for all at the death of J. B., and consequently estate duty was not payable under sect. 2, sub-sect. 2, of the Finance Act 1894 on those securities on the death
of the daughter.
Decision of Sankey, J. reversed.
INFORMATION by the Attorney-General.
Appeal from a decision of Sankey, J. (126 L. T. Rep. 672; (1922) 1 K. B. 491).
The following are the facts as stated by Sankey, J.
"This is a claim for estate duty alleged to be payable by the defendants on the death of the late Mrs. Burns. The facts are of a somewhat complicated character, but for the purposes of this judgment,
they may be briefly stated, as far as they are material, as follows: Mr. J. S. Morgan died on the 8th April 1890. By his will dated the 23rd Nov. 1889, and proved in the Principle Probate Registry on
the 7th May 1890, he left large sums of money to trustees to be invested for his daughter, the late Mrs. Burns, for life and afterwards to such of her children as she should by deed or will appoint,
or in default of such appointment to her children in equal parts. The defendants are the present trustees, and it is admitted, for the purposes of this case, that the late Mr. Morgan was domiciled in
England when he died. Legacy duty was paid on the sums in question by Mr. Morgan's executors. Mrs. Burns died on the 20th July 1919, and by her will dated the 2nd Dec. 1908, and proved in the
Principal Probate Registry on the 19th Aug. 1919, she exercised the power of appointment given to her under her father's will in favour of her children.
The funds representing the sums of money above referred to consisted exclusively of stocks and bonds of American companies and corporations, and the Attorney-General claims estate duty in respect of
Mrs. Burns's share thereof under sect. 2, sub-sect. 2, of the Finance Act 1894 (57 & 58 Vict. c. 30), as being property which is deemed under that Act to have passed on Mrs. Burns's death and so is
liable to duty." Sect. 2, sub-sect. 2, of the Finance Act 1894 is sufficiently set out in the head-note.
Sankey, J. held that, by reason of sect. 12 of the Legacy Duty Act 1796, legacy duty was not
CT. OF APP.]
ATTORNEY-GENERAL V. BURNS AND OTHERS.
payable on the death of Mrs. Burns as legacy duty, and had been paid once and for all on the death of J. S. Morgan; but that legacy duty would have been payable on the death of Mrs. Burns but for the
relationship of the parties to whom the property passed, and that therefore estate duty was payable under sect. 2, sub-sect. 2, of the Finance Act 1894 on the death of Mrs. Burns in respect of the
American securities representing her share under J. S. Morgan's will. On appeal by the defendants.
A. M. Latter, K.C. and Andrewes Uthwatt for the appellants.
Sir Ernest Pollock, K.C. and W. R. Sheldon for the Crown.
The arguments appear very clearly from the judgments.
Lord STERNDALE, M.R. (after stating that the facts were set out in the report of the case in the court below) said: I do not think that there is much question that the property passes on the death of
Mrs. Burns. No question has been raised about that. But it was property which comes under sect. 2, sub-sect. 2, of the Finance Act 1894, because it was situate out of the United Kingdom. [His
Lordship read the sub-section.] The question is whether legacy duty, which it is admitted would not be payable, is not payable because of the relationship of the person to whom the property passes.
By relationship" I presume it means relationship to the person from whom it comes.
Now, I think that the argument for the Crown may be not unfairly put in the way in which I put it during the argument. primary reason why legacy duty is not payable here is because it has already
been paid. It has been paid under the provisions of sect. 12 of the Legacy Duty Act 1796, and I think, as I say, that the argument is something like this: Legacy duty here would be payable if it had
not already been paid. It has been paid because the rates of duty payable by all the persons who could be entitled in possession are the same. In this case the rates of duty are the same because the
relationship of all the persons in succession to the testator is the same. Therefore, it is not payable now because of the relationship of the person to whom it passes. And I think that they must go
on to say that it does not matter that there are cases in which legacy duty would not be payable in circumstances like these, although the relationship which exists here did not exist. That seems to
me to be rather a long train of causation, and I do not think it is sound, for this reason when you look at sect. 12 of the Act of 1796, I think it is fairly summarised by the learned judge, with
perhaps a criticism of the word "charged," which he uses, though I do not think that he was using it in a technical sense of saying that the section was a charging section. He says: "It is a long
section, but it enacts how duties on legacies enjoyed by persons in succession or having partial interests therein shall be charged," I think he meant
[CT. OF APP.
"shall be payable." Then he goes on to say "Two cases are there contemplated: (1) where the duty payable by the different persons in succession is at one and the same rate; (2) where the duty payable
by the different persons in succession is at different rates or one or more of them is not liable to any duty. In the first of these cases the duty is payable as in the case of a legacy to one
person—that is, once and for all; in the second case a different principle applies, and different times are enacted for payment by the different persons.” Now, when you look at the cases in which the
rates are the same, there is, at any rate, one case in which the rates would be the same although no question of relationship enters into the matter at all, and that is a case where the first
beneficiary in possession and all the others in succession are persons of no degree of consanguinity to the testator at all. In that case legacy duty would be payable once and for all, exactly as it
would be in the case where the similarity was brought about by relationship, and, therefore, if the learned judge is right, there would be this curious result: that persons who paid the 10 per cent.
duty as not being in any degree of consanguinity at all would escape estate duty here, because the legacy duty would not be payable-not because of the relationship, but because it had been paid,
apart from the relationship altogether, whereas the persons who were in a degree of relationship would have to pay estate duty, because the similarity happened in those cases to be brought about by
the relationship between the parties.
I think, as I say, that the argument for the Crown requires too long a train of causation. I think that the reason here why legacy duty is not payable is not because of any relationship between the
parties, but because it has already been paid, and it has already been paid because all the persons in succession would pay the same rate of duty, and therefore would come under sect. 12, and the
fact that that similarity is brought about in the case by the relationship between the parties does not seem to me to be enough to satisfy the words that legacy duty would be so payable" but for the
relationship of the persons to whom it passes." It so happens that the result is brought about in this case by the relationship, but there is a case (it is quite true there is only one case), but
there is a case under the Legacy Duty Acts in which legacy duty would not be payable simply and solely owing to the relationship of the parties, and that is the case of husband and wife and as there
is a case which may be satisfied by these words, and as they have to be read with a very extended meaning, and in order to cover this case where legacy duty is not payable because it has been paid. I
think that the learned judge's construction was not right, and that this is not a case where legacy duty would be payable but for the relationship of the person, but that it is a case where it is not
payable because it has been paid for reasons which do not necessarily require any
CT. OF APP.]
ATTORNEY-GENERAL v. BURNS AND OTHERS.
relationship between the parties at all. The fact that it is brought about here by the relationship is, I think, an accident which we ought not to regard, and, therefore I think that the appeal
should be allowed, or, rather, that the information should be dismissed.
WARRINGTON, L.J. (stated the facts and continued): Now, in my opinion, legacy duty in this case is not payable, not because of the relationship of the parties, but by reason of the fact that it has
already been paid. It has been paid by virtue of the provisions of sect. 12 of the Legacy Duty Act 1796. The material provision in that is at the commencement of the section which provides that “the
duty payable on a legacy or residue or part of residue of any personal estate given to or for the benefit of or so that the same shall be enjoyed by different persons in succession who shall be
chargeable with the duties hereby imposed at one and the same rate shall be charged upon and paid out of the legacy as in the case of a legacy to one person "that is to say, it is paid at the death
of the testator by whom it is given, or rather, to be accurate, when the legacy is paid by the executors to the legatee. It happens that in this particular case legacy duty has been paid, because the
persons who take in succession are chargeable at the same rate, and they are chargeable at the same rate in this particular case by reason of the relationship they bear to the testator, namely
daughter and issue of daughter. The fact that it is that relationship which brings about the fact that the duty is charged at the same rate is merely an accident in this particular case, and it is an
accident which does not directly bring about the fact that the legacy duty is paid. It brings about the immediate result that the duty is at the same rate, and it is because the duty is at the same
rate that legacy duty is paid. It seems to me that the real cause why legacy duty is not now payable is that it has been paid, and not because of the relationship of the parties to the testator.
I would point out, too, that if the contention of the Crown were correct it would bring about this curious result, that if the legatees had been strangers in blood to the testator the duty would
still have been at the same rate, and would have been paid, just as it has been paid in the present case, and in that case it must be admitted, and is admitted by the Crown, that the duty they now
claim would not be payable. It seems an extraordinary result, if the contention of the Crown be the right one.
There is one other point which has not been alluded to, but which is worthy of consideration, and that is, what is the meaning of "relationship" in this sub-section ? To whom is the relationship? It
is described in the section in these terms " or would be so payable but for the relationship of the person to whom it passes." The point has not been argued, and therefore this is merely what appears
to me at first sight; but I should have said that it means the relationship of the person to whom
[CT. OF APP.
it passes to the person from whom it passes. There is no other relationship referred to here at all. There is another matter to be mentioned. It cannot be contended that the construction here given
to the section as applicable to this particular case gives no effect to it, because there is one relationship, at all events, which would come within the words "would be so payable but for the
relationship of the person to whom it passes," namely, the relationship of husband and wife. On the whole, with all due respect to the learned judge, I think his judgment is not correct, and that the
appeal ought to be allowed, and with the usual results.
ATKIN, L.J.-I agree. It appears to me that in the reading of sect. 2, sub-sect. 2, estate duty is not payable in this case unless legacy or succession duty is payable to begin with. I think that if
legacy or succession duty has already been paid it plainly is not payable in respect thereof. But then comes an alternative or further clause or would be so payable but for the relationship of the
person to whom it passes." To my mind, that clearly means being still unpaid, would be payable but for the fact that the relationship of the person to whom it passes by law exempts it from being
payable," and, to my mind, that is not the case here. In this case the duty is not payable because it has been paid. It appears to me it would not be so payable but for the relationship of the
person; it would be payable but for the fact that it has already been paid, which is quite a different thing, and if sect. 12 of the Legacy Duty Act 1796 is looked at, in no words, so far as I can
see, in that section does the provision as to the payment of duty forthwith depend upon relationship. It depends upon the simple fact that the rates of duty are the same. In the case of succession,
and in one case, and quite a common case, where the succession is in a series of gifts to strangers, or power of appointment to strangers, it is plain that it is paid then once and for all, not
because of the relationship, but because there is no relationship. That is in the case of the 10 per cent. duty. Then, I think, when you couple with the difficulty of the construction which is put by
the Crown, the fact that there is a very well-known exemption from paying duty on account of relationship, namely, the relationship of husband and wife, to my mind the construction of the section
becomes reasonably plain. I agree, therefore, that the information should be dismissed, with costs, here and below. Appeal allowed.
Solicitors for the appellant, Bircham and Co. Solicitor for the Crown, Solicitor of Inland Revenue.
CHAN. DIV.]
HODGSON AND ANOTHER v. MCCREAGH.
The plaintiffs, freeholders of the manor of Barton Stacey claimed against the lord an injunction to restrain him from exercising sporting or fishing rights over their lands or letting the same to
others. Defendant claimed the sporting rights under a grant of free warren appurtenant in 1302 or by immemorial user now called prescription or by prescriptive right exercised for 30 or 60 years; as
to fishing that it arose from the original grant or had since become appurtenant by prescription.
Held, that the grant of free warren was indistinguishable from that in Morris v. Dimes (1 A. & E. 654), and was in gross, and that the evidence did not warrant the presumption of immemorial user or
prescription either as to sporting or fishing. Plaintiffs were therefore entitled to the relief claimed.
THIS action was brought by freeholders at Barton Stacey, Southampton, against the lord of the manor for a declaration that he was not entitled to any rights of shooting or other rights of sporting
whatever on certain farms in that parish the property of the plaintiffs. The plaintiffs asked for an injunction to restrain the defendant, his keepers, beaters, and servants from shooting on the
farms and from letting or granting any sporting rights.
Roope Reeve, K.C. and Beebee for the plaintiffs. The plaintiffs were the owners of lands in the parish of Barton Stacey in fee simple. They did not admit that these lands were within the ambit of the
defendant's manor. The defendant claimed to be entitled to the exclusive shooting and other sporting rights over the whole of the lands, and in exercise of his rights he had let, or purported to let,
to one Lacey the exclusive sporting rights over some of them, and as to others he had himself, with his servants, exercised sporting rights, to the damage of the plaintiffs.
By his defence the defendant alleged that he and his predecessors in title had been lords of the manor from time immemorial, and had always exercised sporting rights, and he claimed to be entitled to
do so. As to the rights of a freeholder in possession he referred to Busher v. Thompson (4 C. B. 48) and Jayne v. Price (5 Taunt. 326). Being freeholders in possession he submitted that they were
entitled, primá facie, to all the rights of ownership and the onus was therefore on the defendant to prove This was agreed to.
his case.
(a) Reported by A. W. CHASTER, Esq., Barrister-at-Law.
[CHAN. DIV.
Gover, K.C. and Boraston for the defendant. -The presumption was that the manor was coterminous with the parish but some of the lands were not within the parish. It was not contested that they were
seised in fee simple. The question was whether they were customary freeholds of which the soil was legally in the lord of the manor or ancient freeholds existing before the statute of Quia Emptores.
The plaintiffs' freeholds were not such in the strict sense of the term. They put their case thus: (1) There was an express grant of free warren in 1302 to the defendant's predecessor in title of the
manor, and the demesne lands which they were, were appurtenant to the manor; (2) so far as that grant was (a) not appurtenant or (b) extending to all the lands of the manor the defendant's rights
arose by immemorial user now called prescription; (3) in any case, the defendant had a prescriptive right in the manor in the modern sense by user uninterrupted of thirty or sixty years. The
plaintiffs also complained of the exercise of the rights of fishery. The defendant had a several and exclusive fishery, which arose from (1) the original grant with the manor; or (2) had arisen as
appurtenant thereto since by prescription. These lands of the plaintiffs were within the manor. As to ancient demesne in the legal sense, they referred to the entry in Domesday Book relating to this
manor. The rolls of the manor were missing. There was a grant of the manor by King John in 1199. Rights in waters were there mentioned. [They referred to Elton on Copyholds, p. 6; Williams Real
Property, 23rd edit., p. 507; Scriven on Copyholds, 2nd edit., p. 38; and as to conveyance by fine, Williams, p. 73.] In 1908 the plaintiffs sought to enfranchise their land from the manorial rights.
That was a recognition that there might be sporting rights although in 1920 they denied their existence. There were various meanings of the term ancient demesne: (see Bracton; Fleta; Stroud's
Judicial Dictionary I., 591; Termes dela Ley, 107b). Prima facie, the lord had the whole manor. If they were not ancient freeholds they must have been customary freeholds or copyholds, but in either
case the soil would be in the lord (Williams on Commons, 1st edit., p. 238; Bowelston v. Harvey, Cro. El. 547 ; Morris v. Dimes, 1 Ad. & E., 654). Warren appurtenant could arise by prescription and
notwithstanding an express grant of a more limited nature (Beauchamp v. Winn, 6 E. & I. 238; Sowerby v. Smith, 31 L. T. Rep. 309; L. Rep. 9 C. P. 529; Lord Carnarvon v. Villebois, 13 M. & W. 313).
Roope Reeve submitted, first, that the defendant, on whom the onus lay of proving the existence of a right which had been judicially described as odious and against common right, had to prove his
case strictly; and, secondly, that there was a presumption arising from the plaintiffs' possession that they held in fee directly from the King. There was no sufficient evidence that the plaintiffs'
lands were appurtenant to the manor or that the manor was coterminous with the parish. Moreover, there
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Phi in Particle Physics
1.- Introduction
In this article I am going to introduce the main results of a new theory of elemetary particle physics developed by the engineer M.S. El Nachie. This theory provides a fractal model of quantum
space-time, the so-called E-infinity space, that allows the precise determination of the mass-energy of most elementary particles -and much more- in close agreement with their experimental values.
The Golden Ratio emerges naturally in this theory, and turns out to be the central piece that connects the fractal dimension of quantum space-time with the mass-energy of every fundamental particle,
and also with several fundamental physical quantities such as the Fine Structure constant. El Nachie has been severely criticised by his non-orthodoxal publication methods -he uses to publish his
papers in a Journal where he is the editor in chief. Despite this fact, I think that his theory deserves consideration so I will try to summarize it in the lines that follow.
2.- A simple puzzle with profound implications
Before entering into the details of the theory, I would like to introduce it by means of a simple geometric puzzle that elegantly summarizes its core results. Consider the following 13x13 square
divided into two triangles and two quadrilateral polygons as follows (Figure 1, left). If the four pieces are restructured in the form of a rectangle as shown at the right in Figure 1, it appears
that the overall area has inexplicably lost one unit!! What has happened?
Figure 1: A simple geometrical transformation that apparently should be area-preserving.
If you don't figure out what happened, you may take look at the bottom of this page later. Notice that the divisions in the original square have been done according to some Fibonacci numbers: 5, 8
and 13=5+8; therefore the sides of the transformed rectangle are also Fibonacci numbers because it has been constructed additively. Now, do you guess how could we correct the dimensions of the
initial square so that the above transformation into a rectangle was area-preserving? Yes, as it could not be another way round, we need to introduce the Golden Ratio! If the pieces of the square are
constructed according to Golden proportions, then the area of the resulting rectangle will coincide with the area of the square, as it happens in the general case shown below:
Figure 2: The above transformation is area-preserving when the pieces in the square are proportioned according to the Golden Ratio.
The exact resulting area can also be expressed as a continuous fraction expansion of the number 7 as follows:
$φ 4 = 7 - 1 φ 4 = 7 - 1 7 + 1 7 + 1 7 + ···$
This expression suggests that the involvement of φ into the problem opens the door to self-similar, multiple scale or fractal behaviour. The corrected version of our original puzzle is shown in
Figure 3. It is interesting to observe that the correction amounts to a small value k which turns out to be easily expressible in terms of φ, and also as a continuous fraction expansion in this case
of number 11:
$k = 5 φ - 8 = φ 5 - 11 = 1 φ 5 = 1 11 + 1 11 + 1 11 + ··· = 0.090169944...$
Figure 3: The original square in Figure 1 divided according to Golden proportions.
At this point, the reader may be wondering what on earth does all this stuff have to do with particle physics and the quantum world. Well, I would answer: more than we could initially suspect! The
only thing left to do to the corrected version of the puzzle is doubling every length as shown in Figure 4. The resulting polygons have an area that equals five times the theoretical value of the
Fine Structure constant, one of the most fundamental and misterious quantities of physics. This is the theoretically correct value proposed by El Nachie. To obtain the experimentally measured value,
he argues that one simply needs to add a correction due to the symmetry breaking of the time axis, which in the quantum world is spacialised but in our ordinary world it is not (see Section 3 for
Figure 4: The polygons corrected according to the Golden Ratio proportions and with double edge sizes have an area A=685.4101965, exactly equal to five times the theoretical value proposed by El
Nachie for the Fine Structure constant α^-1=137.082039, one of the most fundamental and mysterious quantities of Physics.
But the information contained in these polygons in much more than this. The numbers that appear in the fragments of the polygons form a Fibonacci-like sequence that can be iterated backwards and
forwards. If we iterate it two steps backwards, we obtain two more figures that complete another big piece of information: according to El Nachie, they provide the exact values of the (fractally
corrected) dimensions of the heterotic string theory, whose conventionally accepted values are 4,6,10,16 and 26:
$… , 4-2k , 6+2k , 10 , 16+2k , 26+2k , …$
In addition, some of these values have another meaning as the constants of symmetrical and non-symmetrical unification of all (five) fundamental forces known to current Physics (see more details on
section 5).
Before closing this section, I would like to provide my own personal interpretation of the whole picture. It can be seen as if we were watching a "movie" in which the fundamental players are numbers
3, 5 and φ, and the secondary player is number 11. These players are related as follows:
Figure 5: The relationship among the players involved in the "movie" that describes the transition from ordinary space-time to quantum space-time.
The first two players 3 and 5 generate the well known Fibonacci sequence, which is involved in the structure of all living forms of our ordinary 3+1 finite dimensional world, from plant growth to the
structure of our own human body. The intervention of the third player φ generates a related Fibonacci-like sequence (Figure 6) whose elements contain a small fractal correction k (in El Nachie's
theory k is called transfinite correction). The function of fractal correction k is played by the fourth player, namely number 11, which acts as a kind of synchronizer in the form of a continuous
fraction expansion (Fig. 5):
Figure 6: Numbers 3 and 5 generate the conventional Fibonacci sequence, which is present at the core of all living organisms. After the intervention of φ, a Fibonacci-like sequence with a small
fractall correction is produced, which (when doubled) contains detailed information about the quantum world.
The intervention of the Golden Ratio can be seen as a way to enter the quantum world, the world of subtle vibrations, in which we observe increasing energy levels as we move to smaller and smaller
scales. El Nachie has proposed a way of calculating the fractal dimension of quantum space-time. The resulting value (Figure 7) suggests that the quantum world is composed of an infinite number or
scaled copies of our ordinary 4-dimensional space-time.
Figure 7: The Golden Ratio seems to be the key that opens the door to the fractal quantum world, which looks as if there were an infinite number of scaled copies of our ordinary 4-dimensional
3.- A fractal quantum space-time
3.1.- Brief overview
The main thesis of El Nachie is that, whereas space-time is linearly smooth and Euclidean at our usual scale and becomes curved at cosmic scales, at a quantum mechanical scale, as one comes nearer
and nearer to Planck's length in the order of 10^-33cm, space-time acquires a Cantor-set like structure, i.e. Euclidean geometry gives way to a highly non-linear fractal geometry [1]. El Nachie holds
that all what we observe in nature is a manifestation of the true transfinite, fractal structure of quantum space-time underpinning it. The so-called E-infinity space is a topological-geometrical
construct, according to which quantum space-time is an infinite dimensional hierarchical Cantor set whose stationary quantum states are given by the Vague Attractor of Kolmogorov or VAK (Figure 1):
Figure 8: An illustration of the Vague Attractor of Kolmogorov (VAK).
Deterministic chaos is well documented in the literature, particularly in Hamiltonian systems connected to the work of Henri Poincaré. This pioneering work was then extended by Kolmogorov and others
and finally led to the realization by the French topologist René Thom that the stationary states of quantum mechanics could be modeled by the VAK of Fig. 1. It is a composition of periodic orbits and
chaotic islands which may be regarded as an extension of the notion of an attractor to Hamiltonian systems. The role of the stabilizing frictional forces, which is totally absent from Hamiltonian
systems, and quantum mechanics is assumed here by the irrationality of the winding number. According to the KAM theorem (Kolmogorov, Arnold, Moser), the most stable periodic orbit of a dynamic system
is that which has the Golden Ratio as a winding number. The more irrational the winding number is (the ratio of the resonance frequencies) the more stable is the periodic orbit. Since the Golden
Ratio is the most irrational number, it follows that the orbit with the Golden Ratio as a winding number is the most stable. Therefore φ is the secret of the stability of most elementary particles.
Vibration simulating particles which do not have a sufficient irrational winding number dissipate as fast as they are produced [2].
3.2.- The vacuum model
In E-infinity theory the vacuum is modelled as a hierarchical Cantor set formed through the union and intersection of an infinite number of elemetary Cantor sets, with a Hausdorff dimension directly
related to the Golden Ratio. The simplest (Triadic) Cantor Set is constructed as follows (Figure 2). Consider the unit interval. Remove the middle third of this interval, except for the end points.
We are now left with two intervals of length one third. Remove the middle third of each of the two left intervals, except again the end points. Repeating this process indefinitely, we are left with a
point set that has no length, because we have removed almost the entire interval. Mathematicians refer to that as a point set of measure zero. In some sense there is nothing left anymore.
Miraculously, however, this alleged nothingness has a respectable and relatively sizable dimension. Such point set dimension is called a Hausdorff (fractal) dimension, which in this case is equal to
the natural logarithm of the number of parts left in each iteration divided by the number of divisions in each iteration. In our case this is ln 2 divided by ln 3 [1].
Figure 9: Construction of the triadic Cantor set. In each iteration, the middle third of all the intervals is removed.
The set this way constructed has a mathematical property which sounds unbelievable, but it is true: the number of points in our Cantor set is not one point less than the number of points in the
original continuous-line interval. In both cases we have not only infinitely many points, but actually uncountable infinitely many points. It turns out that the Cantor set is a perfect compromise
between the discrete and the continuum: it is a discrete structure, yet it has the same cardinality as the continuum.
Now suppose that the elementary Cantor set is formed randomly as follows (Figure 10). We choose a random value x from the original interval (0,1), and then we choose another random value y from the
remaining subinterval (x,1). After removing the middle portion (x,y), we are left with two pieces (0,x) and (y,1) that have been chosen at random. Then this process is iterated indefinitely.
According to the Mauldin-Williams theorem, it can be shown that the Hausdorff or fractal dimension of a random Cantor set is the inverse of the Golden Ratio 1/φ [3].
Figure 10: Construction of a random Cantor set. In each iteration, the middle piece between two randomly selected values x and y is removed from all the intervals.
Now imagine that we join an infinite number of such elementary random Cantor sets, with neither gaps nor overlapping. El Nachie has shown that, with this restriction, the dimension of each new set is
a scaled 1/φ version of the previous one:
$d 0 = 1 φ , d 1 = d 0 1 φ = 1 φ 2 , d 2 = d 1 1 φ = 1 φ 3 , … , d n = ( 1 φ ) n+1$
Under these conditions, the average Hausdorff (fractal) dimension D of the resulting set (the E-infinity space) is obtained by weighting and summing the dimensions of the infinite number of
elementary random Cantor sets as follows:
$D = 1 φ + 2 φ 2 + 3 φ 3 + … = φ 3 = 4 + 1 φ 3 =4.236067977...$
This relationship can also be expressed as a continous fraction expansion:
$D = 4 + 1 φ 3 = 4 + 4 ¯ = 4 + 1 4 + 1 4 + 1 4 + ···$
which reminds us of the self-similarity of the VAK of Fig.1 and of ordinary fractals. El Nachie interprets this result as follows: whereas at low energy (at ordinary scale) space-time appears to be
four dimensional -three space dimensions plus one time dimension- at unimaginable higher energy, i.e. when observing space-time at very small scales, much more than the electro weak unification,
space-time will appear to have many more "small" dimensions. Some completely new, high resolution experimental setup may eventually prove the multidimensionality and basic fractality of quantum
space-time beyond doubt.
It is worth noting that the average value which gives the dimension of the E-infinity space, D = 4 + 1/φ^3 = φ^3, coincides with the Hausdorff dimension d[c] = 1/φ of the original elementary random
Cantor set, but lifted to four dimensions:
$d(n) = ( 1 d c ) n-1 ⇒ d(4) = ( 1 1 / φ ) 4-1 = φ 3 = D$
which means that, although the E-infinity space is infinite dimensional, seen from very far it gives the impression of being four dimensional and its Hausdorff and topological dimensions nearly
Physicist Carlos Castro holds that the connection of the exact average dimension of El Nachie's fractal quantum space-time to the Golden Ratio is not a simple numerical coincidence. He argues that
there is a universial dimensional fluctuation in Nature given in terms of the Golden Ratio as
$Δ D fluct = ε 2 = 1 2 1 φ 3$
According to Castro [4], if we imagine the simmetry breaking from an original isotropic space with four equal dimensions with fluctuation, into a space with three spatial dimensions and one temporal
dimension also with fluctuation, and we want to perceive them as our ordinary four dimensions, the fluctuation must be exactly:
$( 3 + ε ) · ( 1 + ε ) = 4 ⇒ 4 + ε = 1 ε ⇒ ε = 1 φ 3$
This quantity is also related to Penrose tilings of the plane, like the one shown in Figure 11. If one would be standing at an arbitrary point there, then one would see the space around as a certain
distinct pattern. Now if one moved an arbitrary distance, because the pattern is non-symmetric, in general one would have a different view of the pattern surrounding him at the new position. The
question that arises is then: could it be that one could move to another point and still find oneself surrounded by the very same pattern once again? The surprising answer to this question is that we
just need to move a distance not longer than the inverse of the Golden Ratio to the power of three divided by two. This is the so-called isomorphic length, and numerically happens to be exactly half
the fractal dimension of E-infinity quantum space-time [1]:
$l ≤ φ 3 2$
Figure 11: A Penrose tiling of the plane. This non-periodic construction is formed by rotations and translations of two elemetary Golden Rhombi.
3.3.- Geometrical visualizations
El Nachie offers the following two dimensional view of the E-infinity space. If we project the space-time of vacuum fluctuation on a Poincaré circle we will see a hyperbolic tessellation of this
circle with predominantly Klein-curve-like geometry. It is an important part of El Naschie’s thesis that actually quantum space-time strongly resembles the hyperbolic geometry of Klein quartic
(Figure 12). It is interesting to note that the polygon-like in the middle of Figure 12 resembles an heptagon, and as one increases the radius moving away from the origin there is an increasing
number of such distorted and scaled heptagons which happens to be a multiple of 7. That reminds us of the continuous fraction expansion of φ^4, which as we have seen is closely related to the Fine
Structure constant.
Figure 12: An hyperbolic covering of the circle by a tiling of Klein quartic. The outer triangles are scaled an distorted versions of the central ones. El Nachie holds that, similarly, elementary
particles are scaled and distorted versions of each other.
This picture illustrates another fundamental tenet of El Nachie, who holds that all elementary particles are scaled and deformed versions of each other. He explains it as follows. Suppose that we
stand somewhere towards the centre of the disk. And suppose we hold a neutron in our hand, so to speak. Then somewhere, towards the edge where everything ramifies at the finitely confined infinity,
we may be observing an electron. His main thesis and the central piece of his picture he offers for an understanding of scaling in E-infinity, is that what appears to us to be an electron in the
so-called reality is nothing but the neutron that we are holding. In other words, once we move to the spot where we are observing the electron, the whole world around us appears to be exactly the
same once more. The minute we arrive at the electron everything, including ourselves and the ramified horizon at the edge of the circle, repeats itself as if we have not moved at all. In other words
we find ourselves standing again with a neutron in our hands and watching an electron towards the border of the circle. This classically absurd and grotesque situation is precisely analogous to what
we experience when we probe quantum space-time and the elementary particles inhabiting it on smaller and smaller distances using higher and higher energy [1].
Figure 13: The 120-cell Coxeter four dimensional polytop upon which the M^4 hyperbolic manifold is based. E-infinity space-time may be regarded as a fuzzy version of M^4.
The internal structure of quantum space-time according to the E-infinity theory can also be geometrically visualized in higher dimensions. The starting point is the four-dimensional generalization of
the dodecahedron, also known as the 120-cell Coxeter polytop. Similarly to what happens with the Tesseract which is the four-dimensional version of the cube, this polytop can be thought of as a
dodecahedron each of whose faces has been replaced by another dodecahedron (Figure 13). From this polytop, an hyperbolic manifold M^4 can be derived with an Euler characteristic χ(M^4)=26 and a
volume invariant Vol(M^4)=684 [2]. A convenient fuzzy version M of this manifold adding the appropriate transfinite corrections leads to a model of E-infinity space-time. In this case, the
topological invariants of M become:
$Euler characteristic: χ(M) = 26 + 2k = 26.18033989... Volume: Vol(M) = ( α -1 / 2 ) · D (10) = α -1 / 5 = 685.4101965...$
This values should sound familiar to the reader. They have been introduced in Section 2 but in another context, namely the area-preserving transformation of a square into a rectangle following Golden
Proportions (see Figure 4).
4.- The Fine Structure constant
The fractal space-time theory of El Nachie allows the exact determination of one of the fundamental quantities of physics, namely the Fine Structure constant, from a dimensional analysis. The
resulting formula is as simple as [1]:
$α - 1 = 5 · [ 2 φ 3 + 1 ] 2 = 137.082039325...$
This formula can be further simplified and admits the following equivalent expressions:
$α -1 = 137 + 1 φ 5 ( 1 - 1 φ 5 ) = 100 + 60 φ = 20 φ 4$
We have already seen in Section 2 that the last expression admits a very simple geometrical interpretation in a polygon (Figure 4), and also as a fraction of the volume of a manifold derived from the
generalization of the dodecahedron into four dimensions (Figure 13). The reader can easily check that these expressions have the same exact numerical value as the first one. This is the E-infinity
low energy value, which is a genuine time-independent constant. This is so because in E-infinity of El Nachie's theory time is spacialized, i.e. there is no difference between time and space unlike
our 3 + 1 space–time with time symmetry breaking introduced. Thus, seen from our 3 + 1 space–time, we need to use a projection in order to cure a slight local aberration or misfit. That way we
"project" the E-infinity value onto 3 + 1 space in a manner of speaking. The so-obtained value is almost identical to that obtained experimentally:
$α exper - 1 = α - 1 - k 0 cos ( π / α - 1 ) = 137 cos ( π / 137.0820393 ) = 137.03598523$
where k[0]=1/φ^5·(1-1/φ^5), which agrees with CODATA value $α -1 = 137.035999074(44)$ with an error as small as ± 0.00001%. In our article on the Golden Ratio in Atomic Structure we offered an
alternative formula for the Fine Structure constant in terms of the Golden Angle which also was in excellent agreement with the experimentally observed value, but the error was an order of magnitude
higher (0.00027%).
4.- The mass-energy of fundamental particles
A note on units: Many scientists have the view that the choice of units is crucial to solve a given problem, and that it should be based as far as possible on fundamental constans of Nature. In our
case, everything seems to suggest that the right units for the energy of elemetary particles are eV (electron-Volts). Actually, we will see that all of the proposed formulas give the expected value
for the physical quantity involved in MeV. It appears that Nature has chosen these specific units for measuring the energy elementary particles. Particle physicists use this unit also as a measure of
mass, because they have in mind that mass and energy are related through Einstein formula E=mc^2. Therefore we use the term mass-energy.
The following table summarizes the Golden Ratio-based theoretical expressions for the mass-energy of some elemental particles obtained through El Nachie's theory. We can see that in general there is
a good agreement with their corresponding experimental values. In the case of the electron, a correction is needed similar to that for the Fine Structure constant, in order to take into account the
symmetry breaking of space-time in the transit from low to high scale.
│ Particle │ Theoretical mass-energy formula │Numerical value│Experimental value│Error (%)│
│ │ │ (MeV) │ (MeV) │ │
│ Electron │$φ 10 cos ( π 100 / φ )$ │0.5116673 │0.5109989 │ 0.00146 │
│ (e^-) │ │ │ │ │
│ Neutron │$( α -1 ) 2 20 = 20 φ 8$ │939.574275 │939.565379 │0.00095 │
│ (n) │ │ │ │ │
│ Proton │$( α -1 ) 2 20 · 124 - 2k 124 - k 2 = 40 φ 8 62 - 1 / φ 5 124 - 1 / φ 5$ │938.269323 │938.272046 │0.00029 │
│ (p^+) │ │ │ │ │
│ Charged Pion │$α -1 + 5 2 = 5 2 φ 4 ( 8 + 1 φ 4 )$ │139.5820393 │139.57018 │0.0085 │
│ (π^±) │ │ │ │ │
│ Neutral Pion │$α -1 - 5 2 = 5 2 φ 4 ( 8 - 1 φ 4 )$ │134.5820393 │134.97660 │0.2923 │
│ (π^0) │ │ │ │ │
│ Tau │$99 φ 6$ │177.4829191 │1776.82 │0.01897 │
│ (τ) │ │ │ │ │
│ Muon (μ)│$φ 5/2 10$ │105.3098758 │105.658375 │0.32983 │
5.- The connection with string theory
One of the most promising attempts to go beyond the standard model of particle physics is superstring theory. As it is well known, special relativity fused time and space together, then came general
relativity and introduced a curvature to space-time. Kaluza and later on Klein added one more dimension to the classical four in order to unify general relativity and electromagnetism. The
dimensionality of space-time plays a paramount role in the theoretical physics of unification and has led to the introduction of the 26 dimensions of string theory, the 10 dimensions of superstring
theory, and finally the heterotic string theory with the dimensional hierarchy 4, 6, 10, 16 and 26 [2].
String theory holds that all elementary particles are nothing but the different modes of string vibrations. E-infinity theory claims to go one step further and imagines these strings to have a fine
structure, more precisely it conceives these strings as being made of Cantor sets. The vibration is understood as a sizzling violent movement of transfinite (fractal) sets simulating the so-called
vacuum fluctuation. Seen from far, the sizzling Cantor sets appear as if they were the violent movement of superstrings, and these strings seen from farther appear as if they were particles.
According to El Nachie, the fact that heterotic superstrings are embedded in E-infinity may be deduced from the following incredible scaling of the Fine Structure constant:
$α -1 1 2 φ = 42.3606798 = 42+4k α -1 1 2 φ 2 = 26.1803398 = 26+2k α -1 1 2 φ 3 = 16.1803398 = 16+2k α -1 1 2 φ 4 = 10 α -1 1 2 φ 5 = 6.1803398 = 6+2k α -1 1 2 φ 6 = 3.8196601 = 4-2k$
Setting k=0 one obtains the classical dimensions of heterotic superstring theory, namely 26, 16, 10, 6 and 4, as well as the constant of super-symmetric (α[gs]=26) and non super-symmetric (α[g]=42)
unification of all fundamental forces. As we have seen in section 2, the above is a Fibonacci-like sequence with a very concise geometrical interpetation related to numbers 5, 11 and φ.
[1] El Naschie MS. "VAK, vacuum fluctuation and the mass spectrum of high energy particle physics". Chaos, Solitons & Fractals 2003;17:797–807.
[2] Marek-Crnjac, L. "A short history of fractal-Cantorian space-time", Chaos, Solitons and Fractals, vol.41, pp.2697–2705, 2009.
[3] Mauldin R.D., Williams S.C.: "Random Recursive Constructions: Asymptotic Geometric and Topological properties", Transactions of the American Mathematical Society, Vol. 295(1), May 1986.
[4] Castro, Carlos, "On the four-dimensional conformal anomaly fractal Cantorian space-time and the fine structure constant", Chaos, Solitons and Fractals, vol. 13, pp.203-207, 2002.
The missing area
The following figure shows where the lost unit of area was hidden in the transformation process from a square to a rectangle (Section 2):
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Simulations of distributions
The central limit theorem is perhaps the most important concept in statistics. For any distribution with finite mean and standard deviation, samples taken from that population will tend towards a
normal distribution around the mean of the population as sample size increases. Furthermore, as sample size increases, the variation of the sample means will decrease.
The following examples use the R stats program to show this graphically. The first example uses a uniform (rectangular) distribution. An example of this case is of a single die with the values of
1-6. The second example is of two dice with totals ranging from 2-12. Notice that although one die produces a rectangular distribution, two dice show a distribution peaking at 7. The next set of
examples show the distribution of sample means for samples of size 1 .. 32 taken from a rectangular distribution.
This figure was produced using the following R code.
#distributions of a single six sided die
#generate a uniform random distribution from min to max
numcases <- 10000 #how many cases to generate
min <- 1 #set parameters
max <- 6
x <- as.integer(runif(numcases,min,max+1) ) #generate random uniform numcases numbers from min to max
#as.integer truncates, round converts to integers, add .5 for equal intervals
par(mfrow=c(2,1)) #stack two figures above each other
hist(x,main=paste( numcases," roles of a single die"),breaks=seq(min-.5,max+.5,1)) #show the histogram
boxplot(x, horizontal=TRUE,range=1) # and the boxplot
title("boxplot of a uniform random distribution")
#end of first demo
Distribution of two dice
Distribution of two dice. The sum of two dice is not rectangular, but is peaked at the middle (hint, how many ways can you get a 2, a 3, ... a 7, .. 12.).
The following R code produced this figure.
#generate a uniform random distribution from min to max for numcases samples of size 2
numcases <- 10000 #how many cases to generate
min <- 0 #set parameters
max <- 6
x <- round(runif(numcases,min,max)+.5)+round(runif(numcases,min,max)+.5)
par(mfrow=c(2,1)) #stack two figures above each other
hist(x,breaks=seq(1.5,12.5),main=paste( numcases," roles of a pair of dice")) #show the histogram
boxplot(x, horizontal=TRUE,range=1) # and the boxplot
title("boxplot of samples of size two taken from a uniform random distribution")
#end of second demo
Samples from a continuous uniform random distribution
We can generalize the case of 1 or two dice to the case of samples of varying size taken from a continuous distribution ranging from 0-1. This next simulation shows the distribution of samples of
sizes 1, 2, 4, ... 32 taken from a uniform distribution. Note, for each sample, we are finding the average value of the sample, rather than the sum as we were doing in the case of the dice.
##show distribution of sample means of varying size samples
numcases <- 10000 #how many samples to take?
min <- 0 #lowest value
max <- 1
ntimes <- 6
op<- par(mfrow=c(ntimes,1)) #stack ntimes graphs on top of each other
i2 <- 1 #initialize counters
for (i in 1:ntimes) #repeat n times
{ sample=rep(0,numcases) #create a vector
k=0 #start off with an empty set of counters
for (j in 1:i2) # inner loop
sample <- sample +runif(numcases,min,max)
k <- k+1 }
x <- sample/k
out <- c(k,mean(x),sd(x))
hist(x, xlim=range(0,1),prob=T ,main=paste( "samples of size", k ),col="black")
i2 <- 2*i2
} #end of i loop
numcases <- 10000 #how many samples to take?
min <- 0 #lowest value
max <- 1
ntimes <- 6
par(mfrow=c(ntimes,1)) #stack ntimes graphs on top of each other
i2 <- 1 #initialize counters
for (i in 1:ntimes) #repeat 5 times
{ sample <- 0 ; k <- 0 #start off with an empty set of counters
for (j in 1:i2) # inner loop
sample <- sample +runif(numcases,min,max)
k <- k+1 }
x <- sample/k
out <- c(k,mean(x),sd(x))
boxplot(x, horizontal=TRUE, ylim=c(0,1), range=1,notch=T) # and the boxplot
i2 <- 2*i2}
Some simple measures of central tendency
x=c(1,2,4,8,16,32,64) #enter the x data
x <- c(1,2,4,8,16,32,64) #enter the x data
y <- c(10,11,12,13,14,15,16) #enter the y data
data <- data.frame(x,y) #make a dataframe
data #show the data
summary(data) #descriptive stats
boxplot(x,y) #same as boxplot(data)
part of a short guide to R
Version of April 1, 2005
William Revelle
Department of Psychology
Northwestern University
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Mensuration of Lines, Areas, Surfaces, and Volumes ...
Dentro del libro
Resultados 6-10 de 40
Página 3 ... base , and with the centre A and radius CD C describe the arc at H , with the centre B and radius E F describe the arc at H , join A H , B H , and the triangle A B H is the triangle
required . H PROBLEM 9 . To make a parallelogram equal ...
Página 12 ... base to the foot of a perpendicular from its top is 26 feet required its height . Section 6 . 15. A ladder , 54 feet long , is placed with one end against an upright wall , and the
other at a certain distance from the foot of the wall ...
Página 13 ... Section 4 . b2 + c2 a2 -- 49 + 64 36 AD = 77 = = 2 c 16 16 And B D = 8 77 51 - 16 16 Then , C D = √A C2 . A D2 772 49 = 5.083 . 162 1. The base A B = EXAMPLES . Section 1 MENSURATION OF
LINES . 13 To inscribe a circle in a triangle.
Página 14 Robert Rawson. 1. The base A B = EXAMPLES . Section 1 . 16 feet , A C = 11 feet , and B C 8 feet ; required the segments and perpendicular on A B. 2. The two sides of a triangle are 2 and 3
, and the base 4 ; reguired the segments and ...
Página 21 ... base ( A B ) , and height or versed sine ( CD ) are given . Find the radius by Problem 4 . Divide ( CD ) the height by ( A C ) the base , and opposite to this quotient in Table ( A )
there will be found one - fourth of the degrees in ...
Términos y frases comunes
Pasajes populares
Página xv
LET it be granted that a straight line may be drawn from any one point to any other point.
Página xiii
When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is
called a perpendicular to it.
Página xii
A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. VIII. " A plane angle is the inclination of two lines to one " another
in a plane, which meet together, but are not
Página xv
An oblong is that which has all its angles right angles, but has not all its sides equal.
Página xii
When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at vertex of the angle, that is, at the point in which the straight lines that
contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight...
Página xii
A plane rectilineal angle is the inclination of two straight lines to one another, -which meet together, but are not in the same straight line.
Página xiv
Of three-sided figures, an equilateral triangle is that which has three equal sides.
Página xvi
If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...
Página xiii
A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal
to one another : 16. And this point is called the centre of the circle. 17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Página xvi
Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
Información bibliográfica
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Plot contours of equal symmetric difference on a ternary plot — SymmetricDifferenceLineEnds
Plot contours of equal symmetric difference on a ternary plot
Assumes that tree 1 is perfectly resolved, but that the resolution of tree 2 can vary.
Vector specifying normalized symmetric differences to plot.
Further parameters to pass to TernaryLines().
Returns a matrix of dim (length(nsd), 6), with columns named r2a, da, sa, r2b, db and sb. Lines from a to b in each row connect points of equal symmetric difference.
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How To Differentiate Instruction In Elementary Classroom
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Instruction. Differentiated instruction and the in-depth but clear explanations about how to differentiate learning in the classroom.
As teachers, we hear the buzzwords differentiated instruction all the time. We know that differentiation is important in the classroom because not all students learn In most elementary classrooms,
some students struggle with learning, others perform well beyond grade-level expectations, and the rest fit somewhere in between.
First published in 1995 as How to Differentiate Instruction in Mixed-Ability Classrooms, this new edition reflects evolving best practices in education, the 30/11/2011В В· This video will help
teachers with differentiation of instruction Strategies for Effective Differentiation Classroom at Mesquite Elementary
Here we’ll take a look at what differentiated instruction is, its roots, and how to apply it in your classroom. As teachers, we hear the buzzwords differentiated instruction all the time. We know
that differentiation is important in the classroom because not all students learn
Ideas and activities to help you easily differentiate instruction in the elementary classroom. Several center ideas perfect for kindergarten, first and second grade. Here we’ll take a look at what
differentiated instruction is, its roots, and how to apply it in your classroom.
What is Differentiated Instruction? Videos; teachers for getting started with differentiating instruction in the classroom. Differentiating at the Elementary What Is Differentiated Instruction? By:
Carol Ann Tomlinson Excerpted from: Tomlinson, C. A. (August, 2000). Differentiation of Instruction in the Elementary …
Self-evaluate whether/how classroom instruction aligns with Universal Design for (2000). Differentiation of instruction in elementary grades. Eric Digest. View 13/09/2017В В· How to Differentiate
Instruction. Differentiated instruction and the in-depth but clear explanations about how to differentiate learning in the classroom.
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Integration by Parts
Mathematics → Subject Test → Integration by Parts
First of all, why Integration by Parts?
As we know that
But what’s the integration of
Formula for Integration by Parts
Suppose two functions u and v are in product form, then formula for finding their integration is
We can also remember the short formula for Integration by Parts which is usually without dx
Practice Problems
Example 1. Evaluate
u = x ⇒ u’ = 1
v = cosx ⇒
Now using Integration by Parts formula
Example 2. Evaluate
u = 2x^2 ⇒ u’ = 4x
v = sinx ⇒
NOTE: 4c and c both are constants, so ignore 4.
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Convert 9 meters to centimeters
How to convert 9 meters to centimeters
To convert 9 m to centimeters you have to multiply 9 x 100, since 1 m is 100 cms
So, if you want to calculate how many centimeters are 9 meters you can use this simple rule.
Did you find this information useful?
We have created this website to answer all this questions about currency and units conversions (in this case, convert 9 m to cms). If you find this information useful, you can show your love on the
social networks or link to us from your site. Thank you for your support and for sharing convertnation.com!
9 meters
Discover how much 9 meters are in other length units :
Recent m to cm conversions made:
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Mensuration - Practically Study Material
11.1 INTRODUCTION
We have learnt that for a closed plane figure, the perimeter is the distance around its boundary and its area is the region covered by it. We found the area and perimeter of various plane figures
such as triangles, rectangles, circles etc. We have also learnt to find the area of pathways or borders in rectangular shapes.
In this chapter, we will try to solve problems related to perimeter and area of other plane closed figures like quadrilaterals. We will also learn about surface area and volume of solids such as
cube, cuboid and cylinder.
11.2 BASIC CONCEPTS
Closed Figure
A figure with no open ends is a closed figure.
Regular closed figures: A closed figure in which all the sides and angles equal.
Perimeter is the distance covered along the boundary forming a closed figure when we go round the figure once. The concept of perimeter is widely used in real life.
• For fencing land.
• For building a compound wall around a house.
The perimeter of a regular closed figure is equal to the sum of its sides.
Perimeter of a Rectangle
= Length (l) + Breadth (b) + Length (l) + Breadth (b) = 2(l + b)
Perimeter of a Square
= s + s + s + s
= 4 × s
Equilateral Triangle
A triangle with all its sides and angles equal is called an equilateral triangle.
The perimeter of an equilateral triangle with the side ‘a’ = a + a + a
= 3 × a
The amount of surface enclosed by a closed figure is called its area. The following conventions are to be adopted while calculating the area of a closed figure using a squared or graph paper.
1. Count the fully-filled squares covered by the closed figure as one square unit or unit square each.
2. Count the half-filled squares as half a square unit.
3. Count the squares that are more than half-filled as one square unit.
4. Ignore the squares filled less than half.
For example, the area of this shape can be calculated as shown:
Covered area Number Area estimate (sq. units)
Fully filled squares 6 6
Half-filled squares 6 $7×1}{2}$
Squares filled more than half 0 0
Squares filled less than half 0 0
Area covered by full squares = 6 × 1 = 6 sq. units Area covered by half squares = 7 × ½ = 7/2= 3 ½ sq. units
Total area of the given shape = 6 + 3 ½ sq. units Thus, the total area of the given shape = 9 ½ sq. Units
Area of a rectangle can be obtained by multiplying length by breadth. Area of the square can be obtained by multiplying side by side.
Perimeter and Area of Some specific Triangles and Quadrilaterals
The closed figure obtained by joining three non collinear points is called a triangle.
Let a, b, c lengths of sides triangle then s = a + b + c is a perimeter of triangle, $s=\frac{a+b+c}{2}$ is called semi perimeters of the triangle
Equilateral Triangle
A triangle which is having equal sides is called an equilateral triangle.
Let the side of an equilateral triangle be ‘a’ then height of equilateral triangle $=\frac{\sqrt{3}}{2}\mathrm{a}$
Perimeter of equilateral triangle = 3a units.
Right Angle Triangle
In a triangle ABC if one angle is $90°$, then it is called a right angled triangle
Note: If ABC is a Right Angled Triangle then by Pythagoras Theorem
(Hypotenuse)^2 = (Side)^2 + (Side)^2
Perimeter of right angled triangle = b + h + d
Acute-Angle Triangle
The triangle which is having all angles less than $90°$ is called an acute angle triangle.
Obtuse Angled Triangle
In a triangle one angle is above $90°$ , then it is called an obtuse angle triangle.
The quadrilateral which is having pair of equal and parallel opposite sides and one angle is $90°$.
Let ABCD be a rectangle of length = l units, breadth = b units, diagonal = d units.
• Perimeter of the rectangle = 2(l + b) units
• Area = l × b units
• Area (A) = $l\sqrt{{d}^{2}–{l}^{2}}=b\sqrt{{d}^{2}–{b}^{2}}$
• Diagonal (d) = $\sqrt{{l}^{2}+{b}^{2}}$
The rectangle which is having all sides equal, all angles are right angles is called Square.
Let the side length of a square be ‘a’ units, then
• Perimeter of square = 4a units
• Diagonal of the square =$\sqrt{{a}^{2}+{a}^{2}}=\sqrt{2a}$ units
• Area of the square = a^2 sq.units
• Area of the square = $\frac{1}{2}$× (diagonal )^2 = $\frac{1}{2}$d^2 sq.units, where ‘d’ is length of diagonal
• Side of square = $\sqrt{\mathrm{Area}}$ units
Length Units Area Units
1 cm = 10 mm 1 cm^2 = (10 × 10)mm^2 = 100 mm^2
1 m = 100 cm 1 m^2 = (100 × 100)cm^2= 1000 cm^2
1 dam= 10 m 1 dam^2 = (10 × 10)m^2 = 100 m^2 = 1 Are
1 hm = 100 m 1 hm^2 = (100 × 100)m^2 = 10000 m^2 =1hectare
Rectangular paths
The path obtained between outer rectangular and inner rectangular fields is called Rectangular path.
If ‘l’, ‘b’ are length and breaths of inner rectangle and ‘w’ be width of path then
• Length of outer rectangle = l + 2w
• Breadth of outer rectangle = b + 2w
• Area of inner rectangle = lb
• Area of outer rectangle = (l + 2w) (b + 2w)
• Area of path = (l + 2w) (b + 2w) – lb = 2(l + b)w + 4w^2
Perimeter and Area of Circle
A circle is defined as a collection of points on a plane that are at an equal distance from a fixed point on the plane. The fixed point is called the centre of the circle.
The distance around a circular region is known as its circumference.
Any straight line segment that passes through the centre of a circle and whose end points are on the circle is called its diameter.
Any line segment from the centre of the circle to its circumference.
Circumference of a circle = 2$\mathrm{\pi }$r , Circumference of a circle = $\mathrm{\pi }$d, where r is the radius of the circle or, where d is the diameter of the circle.
$\mathrm{\pi }$ is an irrational number, whose value is approximately equal to 3.14.
Circumference = Diameter x 3.14
Diameter(d) is equal to twice radius(r): d = 2r
Circles with the same centre but different radii are called concentric circles.
The area of a circle is the region enclosed in the circle.
The area of a circle can be calculated by using the formula:
$\mathrm{\pi }$r^2 , if radius r is given
$\frac{{\mathrm{\pi D}}^{2}}{4}$, if diameter D is given ($\therefore$$\mathrm{r}=\frac{\mathrm{D}}{2}$)
$\frac{{C}^{2}}{4\mathrm{\pi }}$, if circumference C is given $\left(\because c=\pi \mathrm{D}\right)$
11.3 AREA OF TRAPEZIUM
Let ABCD be a trapezium.
Base: Each of the two parallel sides of trapezium is called a base of the trapezium
Altitude (or) height: The distance between the two parallel sides (bases) is called Altitude or height.
In the figure, diagonal AC divides the trapezium ABCD into $∆$ABC and $∆$ADC
$⇒$ Area of trapezium = (Area of $∆$ABC) + (Area of $∆$ADC) ________ (1)
Area of $∆$ABC = $\frac{1}{2}×\mathrm{AB}×\text{height}$
Area of $∆$ACD = $\frac{1}{2}×\mathrm{DC}×\text{height}$
$\therefore$ Area of trapezium ABCD
$=\frac{1}{2}×\left(\mathrm{AB}+\mathrm{DC}\right)×\text{height}=\frac{1}{2}×$(sum of parallel sides)$×$(distance of parallel sides)
Perimeter of trapezium = sum of all the sides = (AB + BC + CD + DA) cm.
11.4 AREA OF A GENERAL QUADRILATERAL
Let ABCD be an quadrilateral.
$\mathrm{AB}e \mathrm{BC}e \mathrm{CD}e \mathrm{DA}$
BL = ${\mathrm{h}}_{1}$ = Altitude (perpendicular)
DM = ${\mathrm{h}}_{2}$ = Altitude (perpendicular)
Area of Quadrilateral
$=\frac{1}{2}×$(length of diagonal) $×$(sum of lengths of perpendiculars)
Area of special quadrilaterals
Let ABCD be a rhombus.
AB = BC = CD = DA = a cm
AC = ${\mathrm{d}}_{1}$ cm = diagonal
BD = ${\mathrm{d}}_{2}$ cm= diagonal
Area of Rhombus = $\frac{1}{2}×$(product of diagonals)
Perimeter of Rhombus = sum of all the sides
= a + a + a + a = 4a cm
11.5 AREA OF A POLYGON
1. Area of a Regular Polygon
Case-1: Area of regular polygon in terms of its side and radius of the inscribed circle.
Let ‘G’ and ‘H’ be any two vertices of a regular polygon
Let ‘O’ be the centre of the inscribed circle.
OG and OH be the bisectors and OP be the perpendicular drawn from O to GH, Then OP = r (in-radius) We will also get the polygon divided into n equal triangles like $∆$OGH.
$\therefore$Area of polygon = Area of $∆$OGH × Number of sides of the polygon
$=\frac{1}{2}×\mathrm{OP}×\mathrm{GH}×\mathrm{n}=\frac{1}{2}×\mathrm{r}×\mathrm{a}×\mathrm{n}$ ($\because$each side = a units)
$=\frac{\text{nar}}{2}$ sq.units
$\therefore$ Area of regular polygon
= $\frac{1}{2}×$perimeter × in-radius ($\therefore$ perimeter of a polygon of n sides, each of length a units = na)
= $\frac{1}{2}$(na)r.unit
Case-2: Area of regular polygon in terms of the radius of the circumscribed circle.
From the figure, we have OP = r and circum radius = OA = R,
let GH = a units.
From the right-angled triangle POH, we have
(OH)^2 = (OP)^2 + (PH)^2 [$\because$ by Pythagoras theorem)
$\left(\mathrm{R}{\right)}^{2}=\left(\mathrm{r}{\right)}^{2}+{\left(\frac{\mathrm{a}}{2}\right)}^{2}\left[\therefore \mathrm{GH}=\mathrm{a}\text{units}\right)$
$\therefore$ Area of the polygon = $\frac{1}{2}×\mathrm{n}×\mathrm{a}×\left(\mathrm{r}\right)=\frac{1}{2}×na×\left[\sqrt{{\mathrm{R}}^{2}–{\left(\frac{\mathrm{a}}{2}\right)}^{2}}\right]=\frac{\mathrm
{na}}{2}\sqrt{{\mathrm{R}}^{2}–{\left(\frac{\mathrm{a}}{2}\right)}^{2}}$ sq. units
2. Area of a Regular Hexagon
Case-1: Let ABCDEF be a regular Hexagon with OP as its in-radius (r) and the length of each side = a.
$\therefore$ Area of the Hexagon = 6 × area of ?OAB
($\because$DOAB is an equilateral triangle)
$=\frac{3\sqrt{3}}{2}\left(\text{side}{\right)}^{2}=\frac{3\sqrt{3}}{2}{\mathrm{a}}^{2}\text{sq. units}$
Let ABCDEF be the hexagon with OA = r as radius and length of each side = a.
Let OP = b be the perpendicular drawn from centre O to AB.
From equilateral triangle AOB,
Area $∆\mathrm{AOB}$$=\frac{1}{2}×\mathrm{AB}×\mathrm{OP}=\frac{1}{2}×\mathrm{a}×\mathrm{h}$
$\therefore$ Area of hexagon = $6×\frac{1}{2}{a}^{n}$ = 3ah sq. units.
3. Area of an Octagon
Let ABCDEFGH be the given Octagon, O the in- centre, and OP = r and length of each side = a.
$\therefore$ The area of a regular Octagon
11.6 SOLID SHAPES
Wherever we look, we usually see solids. So far, in all our study, we have been dealing with figures that can be easily drawn on our notebooks or blackboards. These are called plane figures. We have
understood what rectangles, squares and circles are, what we mean by their perimeters and areas, and how we can find them. We have learnt these in earlier classes. It would be interesting to see what
happens if we cut out many of these plane figures of the same shape and size from a cardboard sheet and stack them up in a vertical pile. By this process, we shall obtain some solid figures (briefly
called solids) such as a cuboid, a cylinder, etc. In the earlier classes, you have also learnt to find the surface areas and volumes of cuboids, cubes and cylinders. We shall now learn to find the
surface areas and volumes of cuboids and cylinders in details and extend this study to some other solids such as cones, spheres, prisms, pyramids.
Before, go into details, let us know what are solids.
The bodies that have three dimensions in space are called solids. For example, a block of wood which has three dimensions – length, breadth and height – is a solid. The space occupied by a solid body
is called its volume.
The common units of volume are cubic centimetres (cm^3) or cubic metre (m^3). The different solids that we are going to know in this chapter are:
11.7 VOLUME AND SURFACE AREA OF CUBE AND CUBOID
A solid bounded by six rectangular plane faces is called cuboid.
Let the Dimensions of cuboid be length = ‘l’ units, breath = ‘b’ units, height = ‘h’ units, then
• Diagonal of cuboid = $\sqrt{{l}^{2}+{b}^{2}+{h}^{2}}$ units
• Total surface Area of cuboid
= 2(lb + bh + lh) sq units
• Lateral surface Area of cuboid
= 2 (l × b)× h sq. units
• Volume of cuboid = lbh cubic units
• Area of 4 walls of room = 2 (l × b)× h sq.units
Volume and Surface area of Cube
The cuboid whose length breath and height are equal is called a cube.
Let the edge of a cube be ‘a’ units then
• Diagonal of cube = $\sqrt{3}$ a units
• Total surface Area of cube = 6a^2 square units
• Lateral surface Area of cube = 4a^2 square units
• Volume of cube = a^3 cubic units.
11.8 VOLUME AND SURFACE AREA OF CYLINDER
1. A solid like measuring jars, circular pillars, circular pipes etc., is called a cylinder.
2. Cylinders have a curved (also called lateral) surface with congruent circular ends.
3. The line joining the centres of the circular ends of a cylinder is called its axis.
4. If it is perpendicular to the circular ends, then it is called a right circular cylinder.
Here, by the word cylinder we will mean the right circular cylinder.
A right circular cylinder, may also be considered as a solid generated by the revolution of a rectangle about one of its sides. Thus, if a rectangle OO ‘ BA revolves about its side OO ‘ and completes
one revolution to arrive at its initial position, a right circular cylinder will be generated whose axis is OO ‘ and radius AO = BO ‘ = r(say). The length of the axis OO ‘ between the centres is
called the length or the height (h) of the cylinder.
Area of Curved Surface of Cylinder
This cylinder has a circular base of radius r cm and a height of h cm. Its surface area is made up of the curved surface area plus the areas of the circular top and base.
If it is a paper cylinder and you cut it through the line AB, and spread this paper on a plane you will get a rectangle whose length will be equal to the circumference of the base of the cylinder,
i.e., 2$\mathrm{\pi }$r and width equal to AB, i.e., h
$\therefore$ Area of the curved surface = area of the rectangle = 2$\mathrm{\pi }$rh
The area of the circular base and top are both equal to $\mathrm{\pi }$r^2. So the combined area of the top and base is 2$\mathrm{\pi }$r^2.
So the total surface area of the cylinder = 2$\mathrm{\pi }$rh + 2$\mathrm{\pi }$r^2
Lateral Surface of a Hollow Cylinder
Let ${r}_{1}$ be external radius and ${r}_{2}$ internal radius then
Curved surface of the hollow cylinder
= sum of the curved surface of both the cylinders
$=2\pi {\mathrm{r}}_{1}\mathrm{h}+2\pi {\mathrm{r}}_{2}=2\pi \left({\mathrm{r}}_{1}+{\mathrm{r}}_{2}\right)\mathrm{h}$
Total surface area of the hollow cylinder
= curved surface of the hollow cylinder + sum of the areas of the rings at the ends
= $2\pi \left({\mathrm{r}}_{1}+{\mathrm{r}}_{2}\right)\mathrm{h}+\left(\pi {\mathrm{r}}_{1}^{2}–\pi {\mathrm{r}}_{2}^{2}\right)+\left(\pi {\mathrm{r}}_{1}^{2}–\pi {\mathrm{r}}_{2}^{2}\right)$ = $2\pi
\left({\mathrm{r}}_{1}+{\mathrm{r}}_{2}\right)\mathrm{h}+2\pi {\mathrm{r}}_{1}^{2}–2\pi {\mathrm{r}}_{2}^{2}$
$=2\pi \left({\mathrm{r}}_{1}+{\mathrm{r}}_{2}\right)\mathrm{h}+2\pi \left({\mathrm{r}}_{1}^{2}–{\mathrm{r}}_{2}^{2}\right)$ = $2\pi \left({\mathrm{r}}_{1}+{\mathrm{r}}_{2}\right)\mathrm{h}+2\pi \
= $2\pi \left({\mathrm{r}}_{1}+{\mathrm{r}}_{2}\right)\left(\mathrm{h}+{\mathrm{r}}_{1}–{\mathrm{r}}_{2}\right)$
Volume of a Cylinder
Volume of a cylinder = (Area of the base) = height = ($\mathrm{\pi }$r^2 )$×$h
$\therefore$ Volume of a cylinder = $\mathrm{\pi }$r^2h
Volume of the Material of a Hollow Cylinder
If the outer and inner radii of a hollow cylinder of height h volume of the material composing the cylinder arc R and r respectively, then the volume of the material composing the cylinder.
= External volume – Internal volume
= $\mathrm{\pi }$R^2h – $\mathrm{\pi }$r^2h – $\mathrm{\pi }$h(R^2 – r^2 )
= $\pi h\left(R+r\right)\left(R–r\right)$
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Notices of the AMS December 2006
The Shoelace Book--A Book Review
Reviewed by Colin Adams
Writing a Teaching Philosophy Statement
Helen G. Grundman
WHAT IS...a Quasiconformal Mapping?
Juha Heinonen
International Congress of Mathematicians 2006
Allyn Jackson
The Jefferson Fellowship Program
2006 Fulkerson Prize
2005 Annual Survey of the Mathematical Sciences (Third Report)
Ellen E. Kirkman, James W. Maxwell, Colleen A. Rose
Feature Articles
The Search for Simple Symmetric Venn Diagrams
Frank Ruskey, Carla D. Savage, and Stan Wagon
Generalizing the familiar pictures of two or three intersecting circles, a Venn diagram is a collection of simple closed curves that intersect in only finitely many points and such that the
intersection of interiors of any subset of the curves is nonempty and connected. If there are n curves, and the diagram has n-fold rotational symmetry, n must be a prime. The authors show how these
can be constructed.
Better Ways to Cut a Cake
Steven J. Brams, Michael A. Jones, and Christian Klamler
A mathematical cake, as viewed by n persons participating in its division, is modeled by their n (covert) value functions on the unit interval. Each participant can cut the cake at a point by a
vertical line at that point, and each is assumed to make their cut so as to maximize the value of the minimum size piece they might receive. The authors explore algorithms that allow this division to
be fair to all.
"Opinion: Math for America and the Math Science Teaching Corps" - Irwin Kra, Executive Director, Math for America
Letters to the Editor
Mathematics People
MacArthur Fellowships Awarded / 2007 ICIAM Prizes Announced / CME-MSRI Prize Awarded / Cook Receives Synge Award / 2006 CMS Awards Given / Landim Awarded 2006 TWAS Prize / Prizes of the Mathematical
Society of Japan / NSDEG Fellowships Awarded / Pi Mu Epsilon Student Paper Presentation Awards / B. H. Neumann Awards Given
Mathematics Opportunities
NSF Computing Equipment and Instrumentation Programs / DMS/NIGMS Initiative in Mathematical Biology / National Academies Mirzayan Graduate Fellowship Program / Newton Fellowship Program / Call for
Nominations for Waterman Award / CMI Liftoff Program for Summer 2007
For Your Information
Palis Elected TWAS President / NCTM Releases Curriculum Report
Inside the AMS
Trjitzinsky Memorial Awards Presented, / Emma Lehmer 100 Years Old / Deaths of AMS Members
Reference and Book List
Mathematics Calendar
New Publications Offered by the AMS
Meetings and Conferences of the AMS
Meetings and Conferences Table of Contents
Notices 2006 Index
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Digital Math Resources
Display Title
Video Definition 6--Equation Concepts--Conditional Equation
Video Definition 6--Equation Concepts--Conditional Equation
This is part of a collection of math video definitions related to to the topic of equation concepts. Note: The download is an MP4 video.
Common Core Standards CCSS.MATH.CONTENT.6.EE.B.5, CCSS.MATH.CONTENT.7.EE.B.4, CCSS.MATH.CONTENT.HSA.REI.A.1
Duration 1 minutes
Grade Range 6 - 12
Curriculum Nodes • Expressions, Equations, and Inequalities
• Applications of Equations and Inequalities
Copyright Year 2024
Keywords equations, solving equations, definitions, glossary terms
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MATH 1332 Introduction to Mathematics (M 302), Unit 5, 5.2 Euler characteristic
5.2 Euler characteristic
│TCCNS Course │MATH 1332: Contemporary Mathematics │
│UT Austin Course│M 302: Introduction to Mathematics │
Suggested Resources and Preparation
Materials and Technology
• For the instructor: spheres and toruses (plastic balls and stacking rings) are helpful for drawing graphs on.
Prerequisite Assumptions
References to regular solids are made.
Overview and Student Objectives
Lesson Length
50 minutes
Lesson Objectives
Students will understand that:
• The Euler characteristic of any connected planar graph drawn on the plane or on the sphere is 2.
• Euler characteristic can be generalized beyond graphs on spheres to graphs drawn on other objects, or to 3D or 4D objects.
Students will be able to:
• Compute the numbers of vertices, edges, and faces in assorted connected planar graphs.
• When possible, alter non-planar graphs to an equivalent graph which is planar.
Euler characteristic
The concept of and formula for Euler characteristic applies in the world of graph theory as well as topology. The reason is because the basic objects we're working with are similar, just dots and
lines and regions. Here, basically, you'll learn some terms and skills, learn about a theorem in graph theory, and learn one interesting connection/application of it to topology.
• The Euler characteristic of any graph is equal to V - E + F. Remember to count the infinitely large "outside face" along with all the other faces, even if it is strangely shaped.
• A graph is connected if it can be drawn completely without picking up your pencil.
• A graph is planar if it can be drawn without any of the edges intersecting. It doesn't have to appear that way initially; if you know you can redraw it without edges intersecting, the original
graph also counts as planar.
│This graph is not connected. There is a vertex on the upper right that you can't get to from the rest of the graph │Fixed it! This graph is connected. You can draw this entire graph without │
│without picking up your pencil. I would say this graph has "two connected components," which is a way of saying it's a │picking up your pencil. In other words, you can walk from any vertex to any │
│graph with two pieces or two clumps of vertices and edges. │other vertex via the edges. │
This graph is planar:
It has two edges crossing in the middle, though, so at first it doesn't look like it. But because we can pick one edge up and move it out of the way, the original graph actually counts as planar
(this reminds me of the definition of a rational number. 0.25 counts as rational because you could write it as 1/4 if you were in the mood).
Euler characteristic theorem
Try an experiment:
1. Draw a big swirly doodle with your pen. Do not pick up the pen as you doodle.
2. Every time you see an intersection, add a vertex. Add vertices at the two ends, too.
3. Compute V - E + F. You always get 2!
In this example, we get V = 8, E = 13, and F = 7, and 8 - 13 + 7 = 2.
Euler characteristic theorem
All connected planar graphs have an Euler characteristic of 2.
│There are no bounded faces here, so all of the outside area is│We have one bounded face plus the outside face. │We have three bounded faces and all the rest of the outside area is a fourth face. │
│one face. This means we have V=2, E=1, and F=1, and 2 - 1 + 1 │This means that V=3, E=3, and F=2, and 3 - 3 + 2 │Also note the looped edge counts as one edge. This means we have V - E + F = 6 - 8 +│
│= 2. │= 2. │4 = 2. │
Proof of Euler characteristic theorem
Given how many different graphs there are, how do you prove such a theorem?
It starts by understanding that we can build up a connected planar graph gradually starting from a single vertex. And notice that the Euler characteristic of a graph that has only a single vertex is
1 - 0 + 1 = 2. You can always build up a graph using these moves:
1. Draw a new vertex and a new edge connected the new vertex to the rest of the graph. In this case V goes up one, and E goes up one, and the formula stays the same.
2. Draw a new edge connecting two existing vertices or a loop with both ends connecting to the same vertex. In this case E goes up 1, F goes up 1, so the formula stays the same.
Since these moves don't change the Euler characteristic and we can always build our graph using those moves, we know the characteristic will always be 2.
Multi-piece graphs
What happens if the graph is not connected but is still planar? We can still compute the Euler characteristic, but we will get a different number. For example:
This graph has two pieces. I count four vertices, three edges, and two faces. We get 4 - 3 + 2 = 3. It turns out that you always get an Euler characteristic of 3 for a two piece graph. More about
this in the homework.
Application to topology
When we studied the regular solids, we found out that the graphs on the outside of those solids (the networks of vertices, edges, and faces) had an Euler characteristic of 2. That's because all of
them can be flattened out and viewed as connected planar graphs. But what else is possible?
Quiz Questions
Question 1
6, 7, 3, 6-7+3=2.
Question 2
Question 3
Question 4
Question 5
Question 6
Question 1
This graph has ____ vertices, ____ edges, and ____ faces. The Euler characteristic is ____ .
Question 2
Question 3
Which regular solid does this graph correspond to?
Question 4
What does it mean for a graph to be planar?
1. It means you can draw the graph in the plane without any of the edges crossing over one another.
2. It means you can draw the graph in the plane using only straight edges.
3. It means you can draw the graph in the plane without picking up your pencil.
4. It means you can draw the graph in the plane.
Question 5
We learned that the Euler characteristic V−E+F is equal to 2 under certain conditions. What are those conditions?
1. The Euler characteristic is 2 as long as there are six or fewer vertices in the graph.
2. The Euler characteristic is always 2, no matter what.
3. The Euler characteristic is 2 if it is connected and if it is also planar.
4. The Euler characteristic is 2 when the graph is connected.
Question 6
Is this a planar graph?
1. Yes because it is drawn in the plane.
2. No because two edges overlap.
3. Yes because it could be redrawn as a graph with no overlapping edges.
4. No because it has too many edges.
Homework Questions
Question 5.2.1
Question 5.2.2
Question 5.2.3
4, 5, 3
1. E and F go up one.
2. E and F go down one.
3. No. V-E+F=2. If we add an edge, then either V or F must go up to compensate. Also it doesn't make sense to add lines and somehow combine regions.
Question 5.2.4
Question 5.2.5
Question 5.2.6
The E characteristic of an n piece graph should be n+1
Question 5.2.1
Compute the Euler Characteristic V−E+F for this graph.
Question 5.2.2
Compute the Euler Characteristic V−E+F for this graph.
Question 5.2.3
Compute V, E, and F for this graph.
Then answer the following:
1. What happens to those three numbers if we add an edge from E to H?
2. What happens to those three numbers if we delete the loop from H to H?
3. Is it possible to add an edge and decrease the number of faces? Why or why not?
Question 5.2.4
This is a two-piece graph. We consider it to be a single graph, but it just has two clusters of vertices and edges. Compute V−E+Ffor this graph.
Question 5.2.5
This is a three-piece graph. We consider it to be a single graph, but it just has three clusters of vertices and edges. Compute V−E+Ffor this graph.
Question 5.2.6
Make a conjecture about the Euler characteristic of an n-piece graph. Support your guess by drawing a four-piece graph and computing its Euler characteristic.
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Choose Regression Model Options
Choose Regression Model Type
You can use the Regression Learner app to automatically train a selection of different models on your data. Use automated training to quickly try a selection of model types, and then explore
promising models interactively. To get started, try these options first:
Get Started Regression Model Options Description
All Quick-To-Train Try the All Quick-To-Train option first. The app trains all model types that are typically quick to train.
All Use the All option to train all available nonoptimizable model types. Trains every type regardless of any prior trained models. Can be time-consuming.
To learn more about automated model training, see Automated Regression Model Training.
If you want to explore models one at a time, or if you already know what model type you want, you can select individual models or train a group of the same type. To see all available regression model
options, on the Learn tab, click the arrow in the Models section to expand the list of regression models. The nonoptimizable model options in the gallery are preset starting points with different
settings, suitable for a range of different regression problems. To use optimizable model options and tune model hyperparameters automatically, see Hyperparameter Optimization in Regression Learner
For help choosing the best model type for your problem, see the tables showing typical characteristics of different regression model types. Decide on the tradeoff you want in speed, flexibility, and
interpretability. The best model type depends on your data.
To avoid overfitting, look for a less flexible model that provides sufficient accuracy. For example, look for simple models such as regression trees that are fast and easy to interpret. If the models
are not accurate enough predicting the response, choose other models with higher flexibility, such as ensembles. To control flexibility, see the details for each model type.
Characteristics of Regression Model Types
To read a description of each model in Regression Learner, switch to the details view in the list of all model presets.
The nonoptimizable models in the Models gallery are preset starting points with different settings. After you choose a model type, such as regression trees, try training all the nonoptimizable
presets to see which one produces the best model with your data.
For workflow instructions, see Train Regression Models in Regression Learner App.
Categorical Predictor Support
In Regression Learner, all model types support categorical predictors.
If you have categorical predictors with many unique values, training linear models with interaction or quadratic terms and stepwise linear models can use a lot of memory. If the model fails to train,
try removing these categorical predictors.
Linear Regression Models
Linear regression models have predictors that are linear in the model parameters, are easy to interpret, and are fast for making predictions. These characteristics make linear regression models
popular models to try first. However, the highly constrained form of these models means that they often have low predictive accuracy. After fitting a linear regression model, try creating more
flexible models, such as regression trees, and compare the results.
In the Models gallery, click All Linear to try each of the linear regression options and see which settings produce the best model with your data. Select the best model in the Models pane and try to
improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Model Flexibility
Linear Easy Very low
Interactions Linear Easy Medium
Robust Linear Easy Very low. Less sensitive to outliers, but might be slow to train.
Stepwise Linear Easy Medium
For a workflow example, see Train Regression Trees Using Regression Learner App.
Linear Regression Model Hyperparameter Options
Regression Learner uses the fitlm function to train Linear, Interactions Linear, and Robust Linear models. The app uses the stepwiselm function to train Stepwise Linear models.
For Linear, Interactions Linear, and Robust Linear models you can set these options:
• Terms
Specify which terms to use in the linear model. You can choose from:
□ Linear. A constant term and linear terms in the predictors
□ Interactions. A constant term, linear terms, and interaction terms between the predictors
□ Pure Quadratic. A constant term, linear terms, and terms that are purely quadratic in each of the predictors
□ Quadratic. A constant term, linear terms, and quadratic terms (including interactions)
• Robust option
Specify whether to use a robust objective function and make your model less sensitive to outliers. With this option, the fitting method automatically assigns lower weights to data points that are
more likely to be outliers.
Stepwise linear regression starts with an initial model and systematically adds and removes terms to the model based on the explanatory power of these incrementally larger and smaller models. For
Stepwise Linear models, you can set these options:
• Initial terms
Specify the terms that are included in the initial model of the stepwise procedure. You can choose from Constant, Linear, Interactions, Pure Quadratic, and Quadratic.
• Upper bound on terms
Specify the highest order of the terms that the stepwise procedure can add to the model. You can choose from Linear, Interactions, Pure Quadratic, and Quadratic.
• Maximum number of steps
Specify the maximum number of different linear models that can be tried in the stepwise procedure. To speed up training, try reducing the maximum number of steps. Selecting a small maximum number
of steps decreases your chances of finding a good model.
If you have categorical predictors with many unique values, training linear models with interaction or quadratic terms and stepwise linear models can use a lot of memory. If the model fails to train,
try removing these categorical predictors.
Regression Trees
Regression trees are easy to interpret, fast for fitting and prediction, and low on memory usage. Try to grow smaller trees with fewer larger leaves to prevent overfitting. Control the leaf size with
the Minimum leaf size setting.
In the Models gallery, click All Trees to try each of the nonoptimizable regression tree options and see which settings produce the best model with your data. Select the best model in the Models
pane, and try to improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Model Flexibility
Fine Tree Easy High
Many small leaves for a highly flexible response function (Minimum leaf size is 4.)
Medium Tree Easy Medium
Medium-sized leaves for a less flexible response function (Minimum leaf size is 12.)
Coarse Tree Easy Low
Few large leaves for a coarse response function (Minimum leaf size is 36.)
To predict a response of a regression tree, follow the tree from the root (beginning) node down to a leaf node. The leaf node contains the value of the response.
Statistics and Machine Learning Toolbox™ trees are binary. Each step in a prediction involves checking the value of one predictor variable. For example, here is a simple regression tree
This tree predicts the response based on two predictors, x1 and x2. To make a prediction, start at the top node. At each node, check the values of the predictors to decide which branch to follow.
When the branches reach a leaf node, the response is set to the value corresponding to that node.
You can visualize your regression tree model by exporting the model from the app, and then entering:
For a workflow example, see Train Regression Trees Using Regression Learner App.
Regression Tree Model Hyperparameter Options
The Regression Learner app uses the fitrtree function to train regression trees. You can set these options:
• Minimum leaf size
Specify the minimum number of training samples used to calculate the response of each leaf node. When you grow a regression tree, consider its simplicity and predictive power. To change the
minimum leaf size, click the buttons or enter a positive integer value in the Minimum leaf size box.
□ A fine tree with many small leaves is usually highly accurate on the training data. However, the tree might not show comparable accuracy on an independent test set. A very leafy tree tends to
overfit, and its validation accuracy is often far lower than its training (or resubstitution) accuracy.
□ In contrast, a coarse tree with fewer large leaves does not attain high training accuracy. But a coarse tree can be more robust in that its training accuracy can be near that of a
representative test set.
Decrease the Minimum leaf size to create a more flexible model.
• Surrogate decision splits — For missing data only.
Specify surrogate use for decision splits. If you have data with missing values, use surrogate splits to improve the accuracy of predictions.
When you set Surrogate decision splits to On, the regression tree finds at most 10 surrogate splits at each branch node. To change the number of surrogate splits, click the buttons or enter a
positive integer value in the Maximum surrogates per node box.
When you set Surrogate decision splits to Find All, the regression tree finds all surrogate splits at each branch node. The Find All setting can use considerable time and memory.
Alternatively, you can let the app choose some of these model options automatically by using hyperparameter optimization. See Hyperparameter Optimization in Regression Learner App.
Support Vector Machines
You can train regression support vector machines (SVMs) in Regression Learner. Linear SVMs are easy to interpret, but can have low predictive accuracy. Nonlinear SVMs are more difficult to interpret,
but can be more accurate.
In the Models gallery, click All SVMs to try each of the nonoptimizable SVM options and see which settings produce the best model with your data. Select the best model in the Models pane, and try to
improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Model Flexibility
Linear SVM Easy Low
Quadratic SVM Hard Medium
Cubic SVM Hard Medium
Fine Gaussian SVM Hard High
Allows rapid variations in the response function. Kernel scale is set to sqrt(P)/4, where P is the number of predictors.
Medium Gaussian SVM Hard Medium
Gives a less flexible response function. Kernel scale is set to sqrt(P).
Coarse Gaussian SVM Hard Low
Gives a rigid response function. Kernel scale is set to sqrt(P)*4.
Statistics and Machine Learning Toolbox implements linear epsilon-insensitive SVM regression. This SVM ignores prediction errors that are less than some fixed number ε. The support vectors are the
data points that have errors larger than ε. The function the SVM uses to predict new values depends only on the support vectors. To learn more about SVM regression, see Understanding Support Vector
Machine Regression.
For a workflow example, see Train Regression Trees Using Regression Learner App.
SVM Model Hyperparameter Options
Regression Learner uses the fitrsvm function to train SVM regression models.
You can set these options in the app:
• Kernel function
The kernel function determines the nonlinear transformation applied to the data before the SVM is trained. You can choose from:
□ Gaussian or Radial Basis Function (RBF) kernel
□ Linear kernel, easiest to interpret
□ Quadratic kernel
□ Cubic kernel
• Box constraint mode
The box constraint controls the penalty imposed on observations with large residuals. A larger box constraint gives a more flexible model. A smaller value gives a more rigid model, less sensitive
to overfitting.
When Box constraint mode is set to Auto, the app uses a heuristic procedure to select the box constraint.
Try to fine-tune your model by specifying the box constraint manually. Set Box constraint mode to Manual and specify a value. Change the value by clicking the arrows or entering a positive scalar
value in the Manual box constraint box. The app automatically preselects a reasonable value for you. Try to increase or decrease this value slightly and see if this improves your model.
Increase the box constraint value to create a more flexible model.
• Epsilon mode
Prediction errors that are smaller than the epsilon (ε) value are ignored and treated as equal to zero. A smaller epsilon value gives a more flexible model.
When Epsilon mode is set to Auto, the app uses a heuristic procedure to select the kernel scale.
Try to fine-tune your model by specifying the epsilon value manually. Set Epsilon mode to Manual and specify a value. Change the value by clicking the arrows or entering a positive scalar value
in the Manual epsilon box. The app automatically preselects a reasonable value for you. Try to increase or decrease this value slightly and see if this improves your model.
Decrease the epsilon value to create a more flexible model.
• Kernel scale mode
The kernel scale controls the scale of the predictors on which the kernel varies significantly. A smaller kernel scale gives a more flexible model.
When Kernel scale mode is set to Auto, the app uses a heuristic procedure to select the kernel scale.
Try to fine-tune your model by specifying the kernel scale manually. Set Kernel scale mode to Manual and specify a value. Change the value by clicking the arrows or entering a positive scalar
value in the Manual kernel scale box. The app automatically preselects a reasonable value for you. Try to increase or decrease this value slightly and see if this improves your model.
Decrease the kernel scale value to create a more flexible model.
• Standardize data
Standardizing the predictors transforms them so that they have mean 0 and standard deviation 1. Standardizing removes the dependence on arbitrary scales in the predictors and generally improves
Alternatively, you can let the app choose some of these model options automatically by using hyperparameter optimization. See Hyperparameter Optimization in Regression Learner App.
Efficiently Trained Linear Regression Models
The efficiently trained linear regression models use techniques that reduce the training computation time at the cost of some accuracy. The available efficiently trained models are linear
least-squares models and linear support vector machines (SVMs). When training on data with many predictors or many observations, consider using efficiently trained linear regression models instead of
the existing linear or linear SVM preset models.
In the Models gallery, click All Efficiently Trained Linear Models to try each of the preset efficient linear model options and see which settings produce the best model with your data. Select the
best model in the Models pane, and try to improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Model Flexibility
Efficient Linear Least Squares Easy Medium — increases as the Beta tolerance setting decreases
Efficient Linear SVM Easy Medium — increases as the Beta tolerance setting decreases
For an example, see Compare Linear Regression Models Using Regression Learner App.
Efficiently Trained Linear Model Hyperparameter Options
Regression Learner uses the fitrlinear function to create efficiently trained linear regression models. You can set the following options:
• Learner — Specify the learner type for the efficient linear regression model, either SVM or Least squares. SVM models use an epsilon-insensitive loss during model fitting, whereas least-squares
models use a mean squared error (MSE). For more information, see Learner.
• Solver — Specify the objective function minimization technique to use for training. Depending on your data and the other hyperparameter values, the available solver options are SGD, ASGD, Dual
SGD, BFGS, LBFGS, SpaRSA, and Auto.
When you set this option to Auto, the software selects:
□ BFGS when the data contains 100 or fewer predictor variables and the model uses a ridge penalty
□ SpaRSA when the data contains 100 or fewer predictor variables and the model uses a lasso penalty
□ Dual SGD when the data contains more than 100 predictor variables and the model uses an SVM learner with a ridge penalty
□ SGD otherwise
For more information, see Solver.
• Regularization — Specify the complexity penalty type, either a lasso (L1) penalty or a ridge (L2) penalty. Depending on the other hyperparameter values, the available regularization options are
Lasso, Ridge, and Auto.
When you set this option to Auto, the software selects:
□ Lasso when the model uses a SpaRSA solver
□ Ridge otherwise
For more information, see Regularization.
• Regularization strength (Lambda) — Specify lambda, the regularization strength.
□ When you set this option to Auto, the software sets the regularization strength to 1/n, where n is the number of observations.
□ When you set this option to Manual, you can specify a value by clicking the arrows or entering a positive scalar value in the box.
For more information, see Lambda.
• Relative coefficient tolerance (Beta tolerance) — Specify the beta tolerance, which is the relative tolerance on the linear coefficients and bias term (intercept). The beta tolerance affects when
the training process ends. If the software converges too quickly to a model that performs poorly, you can decrease the beta tolerance to try to improve the fit. The default value is 0.0001. For
more information, see BetaTolerance.
• Epsilon — Specify half the width of the epsilon-insensitive band. This option is available when Learner is SVM.
□ When you set this option to Auto, the software determines the value of Epsilon as iqr(Y)/13.49, which is an estimate of a tenth of the standard deviation using the interquartile range of the
response variable Y. If iqr(Y) is equal to zero, then the software sets the value to 0.1.
□ When you set this option to Manual, you can specify a value by clicking the arrows or entering a positive scalar value in the box.
For more information, see Epsilon.
Alternatively, you can let the app choose some of these model options automatically by using hyperparameter optimization. See Hyperparameter Optimization in Regression Learner App.
Gaussian Process Regression Models
You can train Gaussian process regression (GPR) models in Regression Learner. GPR models are often highly accurate, but can be difficult to interpret.
In the Models gallery, click All GPR Models to try each of the nonoptimizable GPR model options and see which settings produce the best model with your data. Select the best model in the Models pane,
and try to improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Model Flexibility
Rational Quadratic Hard Automatic
Squared Exponential Hard Automatic
Matern 5/2 Hard Automatic
Exponential Hard Automatic
In Gaussian process regression, the response is modeled using a probability distribution over a space of functions. The flexibility of the presets in the Models gallery is automatically chosen to
give a small training error and, simultaneously, protection against overfitting. To learn more about Gaussian process regression, see Gaussian Process Regression Models.
For a workflow example, see Train Regression Trees Using Regression Learner App.
Gaussian Process Regression Model Hyperparameter Options
Regression Learner uses the fitrgp function to train GPR models.
You can set these options in the app:
• Basis function
The basis function specifies the form of the prior mean function of the Gaussian process regression model. You can choose from Zero, Constant, and Linear. Try to choose a different basis function
and see if this improves your model.
• Kernel function
The kernel function determines the correlation in the response as a function of the distance between the predictor values. You can choose from Rational Quadratic, Squared Exponential, Matern 5/2,
Matern 3/2, and Exponential.
To learn more about kernel functions, see Kernel (Covariance) Function Options.
• Use isotropic kernel
If you use an isotropic kernel, the correlation length scales are the same for all the predictors. With a nonisotropic kernel, each predictor variable has its own separate correlation length
Using a nonisotropic kernel can improve the accuracy of your model, but can make the model slow to fit.
To learn more about nonisotropic kernels, see Kernel (Covariance) Function Options.
• Kernel mode
You can manually specify initial values of the kernel parameters Kernel scale and Signal standard deviation. The signal standard deviation is the prior standard deviation of the response values.
By default the app locally optimizes the kernel parameters starting from the initial values. To use fixed kernel parameters, set Optimize numeric parameters to No.
When Kernel scale mode is set to Auto, the app uses a heuristic procedure to select the initial kernel parameters.
If you set Kernel scale mode to Manual, you can specify the initial values. Click the buttons or enter a positive scalar value in the Kernel scale box and the Signal standard deviation box.
If you set Use isotropic kernel to No, you cannot set initial kernel parameters manually.
• Sigma mode
You can specify manually the initial value of the observation noise standard deviation Sigma. By default the app optimizes the observation noise standard deviation, starting from the initial
value. To use fixed kernel parameters, clear the Optimize numeric parameters check box in the advanced options.
When Sigma mode is set to Auto, the app uses a heuristic procedure to select the initial observation noise standard deviation.
If you set Sigma mode to Manual, you can specify the initial values. Click the buttons or enter a positive scalar value in the Sigma box.
• Standardize data
Standardizing the predictors transforms them so that they have mean 0 and standard deviation 1. Standardizing removes the dependence on arbitrary scales in the predictors and generally improves
• Optimize numeric parameters
With this option, the app automatically optimizes numeric parameters of the GPR model. The optimized parameters are the coefficients of the Basis function, the kernel parameters Kernel scale and
Signal standard deviation, and the observation noise standard deviation Sigma.
Alternatively, you can let the app choose some of these model options automatically by using hyperparameter optimization. See Hyperparameter Optimization in Regression Learner App.
Kernel Approximation Models
In Regression Learner, you can use kernel approximation models to perform nonlinear regression of data with many observations. For large in-memory data, kernel approximation models tend to train and
predict faster than SVM models with Gaussian kernels.
Gaussian kernel regression models map predictors in a low-dimensional space into a high-dimensional space, and then fit a linear model to the transformed predictors in the high-dimensional space.
Choose between fitting an SVM linear model and fitting a least-squares linear model in the expanded space.
In the Models gallery, click All Kernels to try each of the preset kernel approximation options and see which settings produce the best model with your data. Select the best model in the Models pane,
and try to improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Model Flexibility
SVM Kernel Hard Medium — increases as the Kernel scale setting decreases
Least Squares Kernel Regression Hard Medium — increases as the Kernel scale setting decreases
For an example, see Train Kernel Approximation Model Using Regression Learner App.
Kernel Model Hyperparameter Options
Regression Learner uses the fitrkernel function to train kernel approximation regression models.
You can set these options on the Summary tab for the selected model:
• Learner — Specify the linear regression model type to fit in the expanded space, either SVM or Least Squares Kernel. SVM models use an epsilon-insensitive loss during model fitting, whereas
least-square models use a mean squared error (MSE).
• Number of expansion dimensions — Specify the number of dimensions in the expanded space.
□ When you set this option to Auto, the software sets the number of dimensions to 2.^ceil(min(log2(p)+5,15)), where p is the number of predictors.
□ When you set this option to Manual, you can specify a value by clicking the arrows or entering a positive scalar value in the box.
• Regularization strength (Lambda) — Specify the ridge (L2) regularization penalty term. When you use an SVM learner, the box constraint C and the regularization term strength λ are related by C =
1/(λn), where n is the number of observations.
□ When you set this option to Auto, the software sets the regularization strength to 1/n, where n is the number of observations.
□ When you set this option to Manual, you can specify a value by clicking the arrows or entering a positive scalar value in the box.
• Kernel scale — Specify the kernel scaling. The software uses this value to obtain a random basis for the random feature expansion. For more details, see Random Feature Expansion.
□ When you set this option to Auto, the software uses a heuristic procedure to select the scale value. The heuristic procedure uses subsampling. Therefore, to reproduce results, set a random
number seed using rng before training the regression model.
□ When you set this option to Manual, you can specify a value by clicking the arrows or entering a positive scalar value in the box.
• Epsilon — Specify half the width of the epsilon-insensitive band. This option is available when Learner is SVM.
□ When you set this option to Auto, the software determines the value of Epsilon as iqr(Y)/13.49, which is an estimate of a tenth of the standard deviation using the interquartile range of the
response variable Y. If iqr(Y) is equal to zero, then the software sets the value to 0.1.
□ When you set this option to Manual, you can specify a value by clicking the arrows or entering a positive scalar value in the box.
• Standardize data — Specify whether to standardize the numeric predictors. If predictors have widely different scales, standardizing can improve the fit.
• Iteration limit — Specify the maximum number of training iterations.
Alternatively, you can let the app choose some of these model options automatically by using hyperparameter optimization. See Hyperparameter Optimization in Regression Learner App.
Ensembles of Trees
You can train ensembles of regression trees in Regression Learner. Ensemble models combine results from many weak learners into one high-quality ensemble model.
In the Models gallery, click All Ensembles to try each of the nonoptimizable ensemble options and see which settings produce the best model with your data. Select the best model in the Models pane,
and try to improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Ensemble Method Model Flexibility
Boosted Trees Hard Least-squares boosting (LSBoost) with regression tree learners. Medium to high
Bagged Trees Hard Bootstrap aggregating or bagging, with regression tree learners. High
For a workflow example, see Train Regression Trees Using Regression Learner App.
Ensemble Model Hyperparameter Options
Regression Learner uses the fitrensemble function to train ensemble models. You can set these options:
• Minimum leaf size
Specify the minimum number of training samples used to calculate the response of each leaf node. When you grow a regression tree, consider its simplicity and predictive power. To change the
minimum leaf size, click the buttons or enter a positive integer value in the Minimum leaf size box.
□ A fine tree with many small leaves is usually highly accurate on the training data. However, the tree might not show comparable accuracy on an independent test set. A very leafy tree tends to
overfit, and its validation accuracy is often far lower than its training (or resubstitution) accuracy.
□ In contrast, a coarse tree with fewer large leaves does not attain high training accuracy. But a coarse tree can be more robust in that its training accuracy can be near that of a
representative test set.
Decrease the Minimum leaf size to create a more flexible model.
• Number of learners
Try changing the number of learners to see if you can improve the model. Many learners can produce high accuracy, but can be time consuming to fit.
Increase the Number of learners to create a more flexible model.
• Learning rate
For boosted trees, specify the learning rate for shrinkage. If you set the learning rate to less than 1, the ensemble requires more learning iterations but often achieves better accuracy. 0.1 is
a popular initial choice.
• Number of predictors to sample
Specify the number of predictors to select at random for each split in the tree learners.
□ When you set this option to Select All, the software uses all available predictors.
□ When you set this option to Set Limit, you can specify a value by clicking the buttons or entering a positive integer value in the box.
Alternatively, you can let the app choose some of these model options automatically by using hyperparameter optimization. See Hyperparameter Optimization in Regression Learner App.
Neural Networks
Neural network models typically have good predictive accuracy; however, they are not easy to interpret.
Model flexibility increases with the size and number of fully connected layers in the neural network.
In the Models gallery, click All Neural Networks to try each of the preset neural network options and see which settings produce the best model with your data. Select the best model in the Models
pane, and try to improve that model by using feature selection and changing some advanced options.
Regression Model Type Interpretability Model Flexibility
Narrow Neural Network Hard Medium — increases with the First layer size setting
Medium Neural Network Hard Medium — increases with the First layer size setting
Wide Neural Network Hard Medium — increases with the First layer size setting
Bilayered Neural Network Hard High — increases with the First layer size and Second layer size settings
Trilayered Neural Network Hard High — increases with the First layer size, Second layer size, and Third layer size settings
Each model is a feedforward, fully connected neural network for regression. The first fully connected layer of the neural network has a connection from the network input (predictor data), and each
subsequent layer has a connection from the previous layer. Each fully connected layer multiplies the input by a weight matrix and then adds a bias vector. An activation function follows each fully
connected layer, excluding the last. The final fully connected layer produces the network's output, namely predicted response values. For more information, see Neural Network Structure.
For an example, see Train Regression Neural Networks Using Regression Learner App.
Neural Network Model Hyperparameter Options
Regression Learner uses the fitrnet function to train neural network models. You can set these options:
• Number of fully connected layers — Specify the number of fully connected layers in the neural network, excluding the final fully connected layer for regression. You can choose a maximum of three
fully connected layers.
• First layer size, Second layer size, and Third layer size — Specify the size of each fully connected layer, excluding the final fully connected layer. If you choose to create a neural network
with multiple fully connected layers, consider specifying layers with decreasing sizes.
• Activation — Specify the activation function for all fully connected layers, excluding the final fully connected layer. Choose from the following activation functions: ReLU, Tanh, None, and
• Iteration limit — Specify the maximum number of training iterations.
• Regularization strength (Lambda) — Specify the ridge (L2) regularization penalty term.
• Standardize data — Specify whether to standardize the numeric predictors. If predictors have widely different scales, standardizing can improve the fit. Standardizing the data is highly
Alternatively, you can let the app choose some of these model options automatically by using hyperparameter optimization. See Hyperparameter Optimization in Regression Learner App.
Related Topics
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On the twelfth day of Christmas this causal inference dude sent to me | Jeremy Labrecque
Inspired by Riley et al and their BMJ article from last year, On the 12th Day of Christmas, a Statistician Sent to Me …, I’m going to try and come up with my own list of things to do when estimating
causal effects. As I write this the page below is blank so let’s see if I make it to twelve or if I was have to pretend there are fewer days of Christmas. Who needs lords-a-leaping anyway…
• 1st day of Christmas: Make sure your question is causal to begin with. Are you interested in what would happen when you intervene on the world or are you more interested in prediction or
• 2nd day of Christmas: Copying Riley et al. here. Make sure your question is clear. Think of your question before you even look at or think about the data. Write it out in counterfactual notation.
Don’t be afraid to be clear that your question is causal!. A causal question is also much more than just PICO. Think about how you would actually go about changing the exposure or treatment
you’re studying. You might even want to use something like…
• 3rd day of Christmas: Target trial emulation. It won’t always save you. It won’t solve issues such as confounding. But it can help clarify your question and I have seen many, many examples where,
if a study had used it, they would have avoided many self-inflicted errors in their analysis. See here for an example.
• 4th day of Christmas: Sticking with the theme of good questions, think about the consistency assumption. What does it mean to set your exposure to a specific level? How would you do that in
practice? If there’s more than one way that would lead to different outcomes, the consistency assumptions is violated and you should consider what that might mean.
• 5th day of Christmas: Ok. Apparently it takes 4 days of Christmas just to get past the question asking part. Now that we have our question, let’s see if we can answer it. On Day 5, think about
how you can use observed data to answer your counterfactual question. This is called “identification”. It’s the set of assumptions under which you can say that your estimate should be equal to
the causal effect you’re trying to find. Most epidemiologists achieve this through adjusting for confounders but here is a great list of other ways you can do that.
• 6th day of Christmas: If you’re going the confounder control route you need to decide what you’re adjusting for. Use a directed acyclic graph informed by subject matter expertise to draw the
causal structure surrounding the question you’re answering. The graph can tell you what you should and should not adjust for.
• 7th day of Christmas: Now that you know what you plan to adjust for, you should check the positivity assumption. Basically, you don’t want any combination of your confounders to perfectly predict
who is exposed or unexposed. There are some relatively easy ways to check this.
• 8th day of Christmas: Measurement error. You’ve got it. We all got it. Take it seriously.
• 9th data of Christmas: Even if you’re using confounder control, there’s no reason to only rely on an outcome model. Use inverse probability of treatment weighting. Use standardization. Use TMLE
to use both a treatment and outcome model. If they all give you the same answer, you can be more confident that your choice of model doesn’t affect your results.
• 10th day of Christmas: Remember when we talked about identification? Why limit yourself to one identification when you can use more than one. This is called causal triangulation. It’s tricky but
if you can use different methods and get the same answer, then you might be more convinced of your answer.
• 11th day of Christmas: Exchangeability assumptions? Yeah, we never believe those. You can either do what everyone does and write the standard sentence “there’s always a possiblity of unmeasured
confounding” in your discussion. Or you can do something about it. People have come up with so many different types of bias analyses that let you use your subject matter knowledge to try and make
more quantitative statements about how worried your readers should be about potential bias.
• 12th day of Christmas: Name the causal assumptions you are relying on in your manuscript. Show your readers you know what causal assumptions your analysis relies on (which will depend on your
identification strategy) and show them that you’ve thought about them deeply. The amount of papers that even mention assumptions like consistency and positivity
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Google PageRank matrix calculator (graphically) - Peterbe.com
Some time ago I wrote about the Google PageRank algorithm in Python. It's a matrix algorithm for calculating the PageRank values for every page in a web. All you have to do is define which pages
links to which and the algorithm calculates the PageRanks for every page for you.
Now I'm going to try to illustrate it in practise for those of you who don't know what to do with a "Python script"n:/plog/blogitem-040321-1/PageRank.py.
Start calculating!
See the gallery of previous calculations.
The purpose of this simple script is to convert the web matrix that you entered into a directed graph showing the approximated PageRank value for every node.
What you can do with this is to test how the PageRank algorithm works graphically. You might want to know what the effect is to be linked to by one very popular page or the effect of being linked by
several not so popular pages. It's up to you to draw your own conclusions.
The input is limited in size (to save my poor computer) and the graphs aren't beautiful. (Thanks Ero Carrera for pydot which made this possible)
Peter May 18, 2004
One conclusion I've drawn is that PageRank is very contagious.
For example. www.slashdot.org has a very high PageRank, but to get them to link to your site on the front page is hard. Having your link on one of the articles linked from the frontpage means
increased PageRank for you even though that article itself is not linked to from many other pages.
Enogwe Victor January 2, 2015
can we jointly build a tool to calculate possible pagerank of a page before google updates its page rank based on number of backlinks it has? contact me at http://topserve.com.ng/contact
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between reverse Polish
Converting between reverse Polish notation and infix notation
Converting between reverse polish and infix notations
We have already said that a stack is a LIFO device and we know that stacks are used to evaluate expressions. We can diagrams of the stack to help us convert between reverse Polish notation and infix
Example 1
Consider the algebraic expression: 4(A + B)
This uses the infix notation. Converting it to reverse Polish notation gives 4AB+*
How can I check this is correct? I fill the stack with operands, as I see them. So 4 goes in first, followed by A, followed by B. I then get to the first operand. The rule is as follows:
□ Remove the two top items from the stack
□ Do the expression given by the operand
□ Return the result to the stack.
So I remove B and A, add them together and put the result back in the stack. (Note that A + B is the same as B + A). I then get to the multiply symbol. I remove the top two items in the stack.
Multiply them and return the result to the stack. (Note that (A + B) * 4 is the same as 4 * (A + B) which is also the same as 4(A + B).
So my reverse Polish notation algebraic expression 4AB+* does indeed have the equivalent infix algebraic expression of 4(A + B) .
Example 2
Consider this sum: (3 * (6 + 2) - 4) / (3 + 7)
This uses infix notation and the answer is 2 .
Using reverse Polish notation, the sum we need to do becomes:3 6 2 + * 4 - 3 7 + / ..... or does it? How can we check?
□ The operand 3 goes into the stack first, followed by 6 then 2. (Putting operands onto the stack is known as 'pushing' operands. Removing them is known as 'popping' operands.) We then meet the
operator 'add' so we remove the top two items from the stack, the 2 and then the 6, add them to get 8 and push this back onto the stack.
□ We then get the 'multiply' operator. We pop the top two items from the stack, the 8 and the 3 and multiply them to get 24. This is pushed back onto the stack.
□ We then push the 4 onto the stack. We see the 'minus' operator next so we pop the top two items from the stack, the 4 followed by the 24 and subtract the 4 from the 24 to get 20. This is
pushed onto the stack.
□ We then push 3 onto the stack, followed by 7. We then see the 'addition' operator. We pop the top two items from the stack, the 7 followed by the 3 and add them to get 10. This is pushed onto
the stack.
□ We then meet the 'division' operator. We pop the top two items from the stack, the 10 followed by the 20 and divide 10 into 20 to get 2. This is pushed onto the stack.
We have the same answer using reverse Polish notation as we got using infix notation and can therefore conclude that both expressions are equivalent.
Using binary trees to convert between infix notation and reverse Polish notation
So far, we have simply confirmed whether a reverse Polish notation expression is the same as an infix expression. However, we can use binary trees to easily convert between the two.
Consider the example just done: (3 * (6 + 2) - 4) / (3 + 7)
The binary tree for this is as follows:
If we wanted to get the infix notation from a binary tree, we would follow this algorithm, which is known as 'in-order':
1) Traverse the left sub-tree
2) Visit the root
3) Traverse the right sub-tree
This would give us: (3 * (6 + 2) - 4) / (3 + 7)
If we wanted to get the reverse Polish notation from the same binary tree, we would follow this algorithm, known as 'post-order':
1) Traverse the left sub-tree
2) Traverse the right sub-tree
3) Visit the root
This would give us: 3 6 2 + * 4 - 3 7 + /
By the way, we could also get Polish notation from the binary tree, by traversing the tree in pre-order using this algorithm:
1) Visit the root
2) Traverse the left sub-tree
3) Traverse the right sub-tree
How does in-order and post-order traversing work?
Traversing trees requires you to understand and use recursion. You can read more about recursion here.
Traversing a tree: IN-ORDER
Using this method, we must visit the tree in this order:
□ Visit the left sub-tree.
□ Visit the root node.
□ Visit the right sub-tree.
How does this work? We visit the left sub-tree, then the node, then the right sub-tree.
□ We start at the root, node A.
□ Underneath node A is the left sub-tree (with root node B) and the right sub-tree (with root node C).
□ We must check the left sub-tree first according to our INORDER rules. Move to B.
□ But B has a left sub-tree (with a root at D) and a right sub-tree (with a root at E).
□ We must check the left sub-tree first according to our INORDER rules. Move to D.
□ D does not have a left sub-tree, so visit the node D.
□ Now check for D’s right sub-tree. It doesn’t have one.
□ We have now done the left sub-tree for the tree that has a root node at B. Now visit node B.
□ Now visit the right sub-tree of B. We move to E.
□ E doesn’t have a left sub-tree so visit E.
□ E doesn’t have a right sub-tree so move to B and because we have now completely visited the tree with the root node at B, we move up to node A. Visit node A.
□ Now visit the right sub-tree of A. We move to C.
□ C doesn’t have a left sub-tree so visit C.
□ C doesn’t have a right sub-tree so move back up to A.
□ We have now visited every node.
The order that we visited the nodes was DBEAC. We can write an algorithm to print out all of the data at the nodes, like this:
□ For the current node, check if there is a left sub-tree. If there is, go to the root node for this sub-tree and then go to 2). If there isn’t, go to 3).
□ Repeat 1).
□ Print the current node.
□ For the current node, check whether it has a right sub-tree. If it has go to 5) else go to 6).
□ Repeat 1).
□ END
This algorithm will take a little bit of thinking about because it is a recursive algorithm.
Traversing a tree: PRE-ORDER
Using this method, we need to
1. Visit the root node.
2. Visit the left sub-tree.
3. Visit the right sub-tree.
Using our previous binary tree, we would visit the nodes in the order: A B D E C. We can write an algorithm that would print out all of the data at the nodes, like this:
□ Print the current node.
□ For the current node, check if there is a left sub-tree. If there is, go to the root node for this sub-tree and then go to 1). If there isn’t, go to 3).
□ For the current node, check whether it has a right sub-tree. If it has go to 4) else go to 5).
□ Repeat 1).
□ END.
Traversing a tree: POST-ORDER
Using this method, we need to
1. Visit the left sub-tree.
2. Visit the right sub-tree.
3. Visit the root node.
Using our example binary tree, the order that we would visit is: D E B C A
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Formal definitions for Relation Constant, Transitive Closure, Extended Transitive Closure
· May 31, 2020, 6:51 am
The following are updated and corrected versions of definitions posted earlier, plus Extended TC. There are formatting issues, for which I apologise, but I find this editor very difficult to drive.
New Value as Relational Constant
This is the formalisation of a relational constant or ‘relcon’, and relies on a previously defined function. Separate formalisations are provided for monadic and dyadic functions. Extending them to
n-adic functions is left as an exercise.
Given a function f with the type signature Tx->Ty, the relcon S is defined as follows.
Hs = { <X,Tx>, <Y,Ty> }
Bs = { ts : exists vx ∈ Tx exists vy ∈ Ty
(ts = {<X,Tx,vx>, <Y,Ty,vy>} and f(Vx) = Vy)}
Given a function f with the type signature Tx->Ty->Tz, the relcon S is defined as follows.
Hs = { <X,Tx>, <Y,Ty>, <Z,Tz>}
Bs = { ts : exists vx ∈ Tx exists vy ∈ Ty exists vz ∈ Tz
(ts = {<X,Tx,vx>, <Y,Ty,vy, <Z,Tz,vz>} and f(Vx,Vy) = Vz)}
For the relcon PLUS in App-A, types Tx, Ty and Tz are all INTEGER, and the function f is the scalar operator "+".
Note that a ‘relcon’ can be used as one argument to a join. Operations sometimes known as WHERE, EXTEND, UPDATE and DELETE are shorthands that may include such a combination.
Transitive Closure
This formalisation defines a recurrence relation consisting of a starting value and the elements of a sequence. The transitive closure is the union of that sequence, which can be shown to reach a
fix-point termination. The starting point (‘seed’) represents known edges in a directed graph; the end alue is all the possible paths through the graph.
Given a set S of tuples with the heading {<X,T>,<Y,T>} for some type T, the successor relation S’ is defined as follows.
Hs’ = Hs
Bs’ = { ts’ : ts’ ∈ Bs or
(exists v1 ∈ T exists v2 ∈ T exists v3 ∈ T exists v4 ∈ T
exists ts1 ∈ Bs (ts1 = {<A,T,v1>, <B,T,v2>})
exists ts2 ∈ Bs (ts2 = {<A,T,v2>, <B,T,v3>})
ts’ = {<A,T,v1>, <B,T,v3>} )) }
The transitive closure T is then defined as
^T = S[1] U S[2] U S[3] … S[∞]
Note that this is a linear recurrence, not a predicate. Transitive closure cannot be defined by first order predicate logic.
Note that the operation sometimes known as TCLOSE corresponds to this definition.
Extended Transitive Closure
This formalisation defines a recurrence relation consisting of a starting value and the elements of a sequence. The transitive closure is the union of that sequence, which can be shown to reach a
fix-point termination. The starting point (‘seed’) represents known edges in a directed graph; the end value is all the possible paths through the graph.
In this case each tuple is associated with a value, and this definition relies on some previously defined function f that takes values of that type as its argument.
Given a set S of tuples with the heading {<A,T>,<B,T>,<C,Tv>} for some types T and Tv, and a dyadic function f with the type signature Tv->Tv->Tv, the successor relation S’ is defined as follows.
Hs’ = Hs
Bs’ = { ts’ : ts’ ∈ Bs or
(exists v1 ∈ T exists v2 ∈ T exists v3 ∈ T exists v4 ∈ T
exists w1 ∈ Tv exists w2 ∈ Tv
exists ts1 ∈ Bs (ts1 = {<A,T,v1>, <B,T,v2>, <C,Tv,w1>})
exists ts2 ∈ Bs (ts2 = {<A,T,v2>, <B,T,v3>, <C,Tv,w2>})
ts’ = {<A,T,v1>, <B,T,v3>, <C,Tv,f(w1,w2)} )) }
The transitive closure T is then defined as
^[T = S1 U S2 U S3 … S∞
Note that this is a linear recurrence, not a predicate. Transitive closure cannot be defined by first order predicate logic.
Note that the function enables a cost to be calculated for each path. Generalised Transitive Closure as defined by D&D (RM VSS 5) requires an additional step of aggregation.
Andl - A New Database Language - andl.org
· February 16, 2021, 2:58 pm
Not that there's anything wrong with these definitions but I am unsure how they help with the question in https://forum.thethirdmanifesto.com/forum/topic/what-do-set-based-operations-buy-us/?part=1
Let me rewrite my code as a successor function:
OPERATOR CONTENTS_SUCCESSOR( CCA RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER })
RETURNS RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER } ;
RETURN (WITH ((RENAME CCA {CONTAINED, AMOUNT} (CONTAINED AS CONTAINER, AMOUNT AS MULTIPLIER)) JOIN CCA) AS LEFT:
(LEFT EXTEND(TOTAL AS MULTIPLIER*AMOUNT)){CONTAINER, CONTAINED, TOTAL} RENAME(TOTAL AS AMOUNT);
How do I use it to calculate the GTC?
· February 16, 2021, 11:50 pm
Quote from
on February 16, 2021, 2:58 pm
Not that there's anything wrong with these definitions but I am unsure how they help with the question in https://forum.thethirdmanifesto.com/forum/topic/what-do-set-based-operations-buy-us/?part
Let me rewrite my code as a successor function:
OPERATOR CONTENTS_SUCCESSOR( CCA RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER })
RETURNS RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER } ;
RETURN (WITH ((RENAME CCA {CONTAINED, AMOUNT} (CONTAINED AS CONTAINER, AMOUNT AS MULTIPLIER)) JOIN CCA) AS LEFT:
(LEFT EXTEND(TOTAL AS MULTIPLIER*AMOUNT)){CONTAINER, CONTAINED, TOTAL} RENAME(TOTAL AS AMOUNT);
OPERATOR CONTENTS_SUCCESSOR( CCA RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER }) RETURNS RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER } ; RETURN (WITH ((RENAME CCA
{CONTAINED, AMOUNT} (CONTAINED AS CONTAINER, AMOUNT AS MULTIPLIER)) JOIN CCA) AS LEFT: (LEFT EXTEND(TOTAL AS MULTIPLIER*AMOUNT)){CONTAINER, CONTAINED, TOTAL} RENAME(TOTAL AS AMOUNT); END OPERATOR
OPERATOR CONTENTS_SUCCESSOR( CCA RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER })
RETURNS RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER } ;
RETURN (WITH ((RENAME CCA {CONTAINED, AMOUNT} (CONTAINED AS CONTAINER, AMOUNT AS MULTIPLIER)) JOIN CCA) AS LEFT:
(LEFT EXTEND(TOTAL AS MULTIPLIER*AMOUNT)){CONTAINER, CONTAINED, TOTAL} RENAME(TOTAL AS AMOUNT);
END OPERATOR
How do I use it to calculate the GTC?
First you need someone to write an ETCLOSE function for you, and add it to TD. The ETCLOSE is just like TCLOSE except it has an additional feature.
The TCLOSE function adds tuples to a self-joinable relation representing additional nodes in the graph. Each new tuple has two parents. The extra feature in ETCLOSE allows attribute(s) in each new
tuple value to be given value(s) calculated from its parents. The design of that language feature I leave to the implementer, but it might well be a function on tuples. The JOIN in your code is not
needed, as it is part of the ETCLOSE.
The final step to GTC is by aggregation over the result of the ETCLOSE.
Note that TCLOSE can be implemented using either recursion or iteration with mutable state, but it needs neither. TCLOSE requires only summing a series, as shown by my formal treatment. ETCLOSE
Andl - A New Database Language - andl.org
· February 18, 2021, 1:36 pm
Quote from
on February 16, 2021, 11:50 pm
Quote from
on February 16, 2021, 2:58 pm
Not that there's anything wrong with these definitions but I am unsure how they help with the question in https://forum.thethirdmanifesto.com/forum/topic/what-do-set-based-operations-buy-us/?
Let me rewrite my code as a successor function:
OPERATOR CONTENTS_SUCCESSOR( CCA RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER })
RETURNS RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER } ;
RETURN (WITH ((RENAME CCA {CONTAINED, AMOUNT} (CONTAINED AS CONTAINER, AMOUNT AS MULTIPLIER)) JOIN CCA) AS LEFT:
(LEFT EXTEND(TOTAL AS MULTIPLIER*AMOUNT)){CONTAINER, CONTAINED, TOTAL} RENAME(TOTAL AS AMOUNT);
OPERATOR CONTENTS_SUCCESSOR( CCA RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER }) RETURNS RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER } ; RETURN (WITH ((RENAME CCA
{CONTAINED, AMOUNT} (CONTAINED AS CONTAINER, AMOUNT AS MULTIPLIER)) JOIN CCA) AS LEFT: (LEFT EXTEND(TOTAL AS MULTIPLIER*AMOUNT)){CONTAINER, CONTAINED, TOTAL} RENAME(TOTAL AS AMOUNT); END
OPERATOR CONTENTS_SUCCESSOR( CCA RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER })
RETURNS RELATION { CONTAINER C_ID, CONTAINED C_ID, AMOUNT INTEGER } ;
RETURN (WITH ((RENAME CCA {CONTAINED, AMOUNT} (CONTAINED AS CONTAINER, AMOUNT AS MULTIPLIER)) JOIN CCA) AS LEFT:
(LEFT EXTEND(TOTAL AS MULTIPLIER*AMOUNT)){CONTAINER, CONTAINED, TOTAL} RENAME(TOTAL AS AMOUNT);
END OPERATOR
How do I use it to calculate the GTC?
First you need someone to write an ETCLOSE function for you, and add it to TD. The ETCLOSE is just like TCLOSE except it has an additional feature.
The TCLOSE function adds tuples to a self-joinable relation representing additional nodes in the graph. Each new tuple has two parents. The extra feature in ETCLOSE allows attribute(s) in each
new tuple value to be given value(s) calculated from its parents. The design of that language feature I leave to the implementer, but it might well be a function on tuples. The JOIN in your code
is not needed, as it is part of the ETCLOSE.
The final step to GTC is by aggregation over the result of the ETCLOSE.
Note that TCLOSE can be implemented using either recursion or iteration with mutable state, but it needs neither. TCLOSE requires only summing a series, as shown by my formal treatment. ETCLOSE
This is the series I am interested in.
So when you write
T = S1 U S2 U S3 … S∞
Is S1 the Successor(S0) and S2 the Successor(S1) and so on? You end when you get an empty set? And you just union them together, and then what? Aggregate over something by some function? I have
already in my successor function applied the transform to multiply the parent count by the child count to unify the two counts.
· February 19, 2021, 4:54 am
How do I use it to calculate the GTC?
First you need someone to write an ETCLOSE function for you, and add it to TD. The ETCLOSE is just like TCLOSE except it has an additional feature.
The TCLOSE function adds tuples to a self-joinable relation representing additional nodes in the graph. Each new tuple has two parents. The extra feature in ETCLOSE allows attribute(s) in
each new tuple value to be given value(s) calculated from its parents. The design of that language feature I leave to the implementer, but it might well be a function on tuples. The JOIN in
your code is not needed, as it is part of the ETCLOSE.
The final step to GTC is by aggregation over the result of the ETCLOSE.
Note that TCLOSE can be implemented using either recursion or iteration with mutable state, but it needs neither. TCLOSE requires only summing a series, as shown by my formal treatment.
ETCLOSE ditto.
This is the series I am interested in.
So when you write
T = S1 U S2 U S3 … S∞
Is S1 the Successor(S0) and S2 the Successor(S1) and so on? You end when you get an empty set? And you just union them together, and then what? Aggregate over something by some function? I have
already in my successor function applied the transform to multiply the parent count by the child count to unify the two counts.
My formal definition shows that (a) the successor function can be first order (b) the series has a finite sum. That means that ETCLOSE is safe, it is guaranteed to terminate and produce a result. Any
aggregation following it is also safe. That is the strength of RA queries: they are guaranteed safe.
Your successor function is expressed in a Turing Complete programming language, therefore it is not safe. There is no guarantee it will terminate or produce result.
It is a choice you are free to make: power or safety. Both are valuable.
I actually implemented while, which is like SQL CTE RECURSIVE, more powerful than ETCLOSE and still safe. The terminating condition is no new tuples for the union. I don't have a formal definition
for that.
Andl - A New Database Language - andl.org
· February 21, 2021, 5:14 am
Quote from
on February 19, 2021, 4:54 am
How do I use it to calculate the GTC?
First you need someone to write an ETCLOSE function for you, and add it to TD. The ETCLOSE is just like TCLOSE except it has an additional feature.
The TCLOSE function adds tuples to a self-joinable relation representing additional nodes in the graph. Each new tuple has two parents. The extra feature in ETCLOSE allows attribute(s) in
each new tuple value to be given value(s) calculated from its parents. The design of that language feature I leave to the implementer, but it might well be a function on tuples. The JOIN
in your code is not needed, as it is part of the ETCLOSE.
The final step to GTC is by aggregation over the result of the ETCLOSE.
Note that TCLOSE can be implemented using either recursion or iteration with mutable state, but it needs neither. TCLOSE requires only summing a series, as shown by my formal treatment.
ETCLOSE ditto.
This is the series I am interested in.
So when you write
T = S1 U S2 U S3 … S∞
Is S1 the Successor(S0) and S2 the Successor(S1) and so on? You end when you get an empty set? And you just union them together, and then what? Aggregate over something by some function? I
have already in my successor function applied the transform to multiply the parent count by the child count to unify the two counts.
My formal definition shows that (a) the successor function can be first order (b) the series has a finite sum. That means that ETCLOSE is safe, it is guaranteed to terminate and produce a result.
Any aggregation following it is also safe. That is the strength of RA queries: they are guaranteed safe.
Your successor function is expressed in a Turing Complete programming language, therefore it is not safe. There is no guarantee it will terminate or produce result.
It is a choice you are free to make: power or safety. Both are valuable.
I actually implemented while, which is like SQL CTE RECURSIVE, more powerful than ETCLOSE and still safe. The terminating condition is no new tuples for the union. I don't have a formal
definition for that.
I'm trying to learn how to do this in practice, so what are the algorithmic steps?
Since you mention that there are special requirements that seem to go beyond the formal definition given, such as I may not use a general programming language, what are those requirements? What does
a safe successor function look like for the case of a container having amount of contained, how is it implemented? The function to combine amounts in the step is multiplication.
When I have a safe successor function what are the algorithmic steps of a safe ETCLOSE function? Without knowing all the steps it is impossible to verify that it is safe.
· February 21, 2021, 6:34 am
The name and definition for while come from the Alice book, but in practice it works the same as SQL CTE RECURSIVE. It's simple enough:
1. Start with a seed (relational) value
2. Evaluate a relational expression, with the seed as an argument
3. Union the result with the seed
4. Repeat until no new tuples.
Obviously the implementation will require a GP language (less than 10 lines of code in Andl), but queries just use relational expressions.
Andl - A New Database Language - andl.org
· February 22, 2021, 9:17 am
I think we also need to ensure that values are not deduplicated. Consider:
Container Contains quantity
red blue 1
red yellow 1
blue green 2
yellow green 2
In the successor function you would end up with two records "red, green, 2" which will get deduplicated unless you are careful.
In the same way, the outputs from successive applications of successor functions could contain duplications where the different path to the identical tuple is no longer apparent.
Come to think of it, the successor function must always have the current seed joined with the original relation (which represents the number one, or one step), otherwise we will skip levels.
Container Contained quantity
red blue 1
blue green 10
blue yellow 2
yellow green 5
So in the first application you would have "red, green, 10", "red, yellow,2" and "blue,green, 5". These must then be joined with the original for the next successor, giving "red, green, 10" again, so
care must be taken that we when we "union" the first and second results end up with 20 greens in a red, not just 10.
· February 22, 2021, 10:11 am
Quote from
on February 22, 2021, 9:17 am
I think we also need to ensure that values are not deduplicated. Consider:
Container Contains quantity
red blue 1
red yellow 1
blue green 2
yellow green 2
In the successor function you would end up with two records "red, green, 2" which will get deduplicated unless you are careful.
Deduplication by UNION (and all relational operators) is what you expect. Preserving duplicates (by making them non-duplicates) is always a special case.
Of course, SQL is the opposite; preserving duplicates is what you expect and deduplication is a special case. Except for UNION, which deduplicates and needs to be UNION ALL to preserve duplicates.
Hooray for consistency!
If you regard databases as mere containers, perhaps preserve-duplicates-by-default is more intuitive. If you regard a database as a collection of fact assertions, then preserve-duplicates-by-default
is counter-intuitive.
Last edited on February 22, 2021, 11:34 am by Dave Voorhis
I'm the forum administrator and lead developer of Rel. Email me at dave@armchair.mb.ca with the Subject 'TTM Forum'. Download Rel from https://reldb.org
· February 22, 2021, 12:59 pm
Quote from
on February 22, 2021, 9:17 am
I think we also need to ensure that values are not deduplicated. Consider:
Container Contains quantity
red blue 1
red yellow 1
blue green 2
yellow green 2
In the successor function you would end up with two records "red, green, 2" which will get deduplicated unless you are careful.
Deduplication by UNION (and all relational operators) is what you expect. Preserving duplicates (by making them non-duplicates) is always a special case.
Of course, SQL is the opposite; preserving duplicates is what you expect and deduplication is a special case. Except for UNION, which deduplicates and needs to be UNION ALL to preserve
duplicates. Hooray for consistency!
If you regard databases as mere containers, perhaps preserve-duplicates-by-default is more intuitive. If you regard a database as a collection of fact assertions, then
preserve-duplicates-by-default is counter-intuitive.
Yes. What I am talking about is that in implementing the algorithm to correctly calculate an extended transitive closure, an explosion of parts, you must add something more than what is shown by the
formal definition proposed in this thread. You yourself mentioned using a temporarilty generated id in the Rel implementation if I'm not mistaken?
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seminars - Dirichlet forms and heat kernels on generalized diamond fractals
Abstract: The present talk aims to illustrate different ways to construct (in some sense natural) diffusion processes on fractals using as an example a parametric family of generalized diamond
fractals. These spaces arise as scaling limits of diamond hierarchical lattices. The latter are studied in the physics literature in relation to random polymers, Ising and Potts models among others.
In the case of constant parameters, one can exploit the self-similarity of the space to obtain a canonical Dirichlet form and a diffusion process. This approach is common to many fractal settings and
was taken in earlier investigations due to Hambly and Kumagai. We will outline this construction and the properties of the diffusion process and the heat kernel that were obtained there.
Alternatively, a diamond fractal can also be regarded as an inverse limit of metric measure graphs and a canonical diffusion process can be constructed through a procedure proposed by Barlow and
Evans. Following this approach it turns out that it is possible to give a rather explicit expression of the associated heat kernel, that is in particular uniformly continuous and admits an analytic
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Factorising a large number, part I: Sums of squares in different ways
There is a theorem that states: if a number can be written as the sum of two squares in two different ways, it is composite.
Because of Twitter, I became interested in factorising $n=842,909$. Can this be written as the sum of two squares ((Proof by MathsJam $\blacksquare$))? How - without cheating and using a computer -
could we check?
One option is to say “there are only 920 or so square numbers smaller than $n$ - we could simply check them all!” That would… succeed, I suppose, but seems like an awful lot of work. How can we
narrow down the search?
Using modulo arithmetic, of course!
Modulo 20, there are only a few possible values for squares: 0, 1, 4, 9, 16 and 5 - and only a few pairs of those add up to 9: we could have 0+9 or 5+4. In each case, one of the squares is a multiple
of 5.
This makes things much easier: now we only need to consider squares that end 00 or 25, and their partners - which must end 09 or 84. Now we only need to look at 200 or so possibilities! But surely we
could do better than that?
How about modulo 16? There, squares must be 0, 1, 4 or 9; our number is congruent to 13, and the only way we can make that is with 4 and 9.
Let’s switch to thinking about the square root numbers for a while. If $a^2$ ends with 00, $a$ is a multiple of 10; further, if $a^2 \equiv 4 \pmod{16}$, then $a$ can be written as $4k+2$. Putting
these together, $a$ must be a multiple of 10, but not 20. There are fewer than 50 of these candidates.
For the other pair, our number ending 25 must be congruent to 9 (modulo 16), which is only possible if the square root is three away from a multiple of 8 - again, reducing our work by half to fewer
than 50 candidates.
So let’s do this!
We’ll start at the high end with our multiples of 10 and gradually decrease until we hit a match. Since the square root of 842,909 is somewhere about 920 ((whoosh 918.1 whoosh)) so we’ll start with
• $910^2 = 828,100$, which is $14,809$ away - not square.
• $890^2 = 792,100$, which is $50,809$ away - not square ((Note that since the squares are 20 apart, the increase in the difference is $20\times (910+890)$, a pattern we can exploit.)).
• $870^2 = 756,900$, which is $86,009$ away - not square.
• $850^2 = 722,500$, which is $120,409$ away, or $347^2$.
Boom! We have one of our squares.
The fives are a bit trickier, since we need to keep track of the remainders modulo 16, but we can do that:
• 915 is 3 (mod 8), so we want it: $915^2 = 837,225$, 5684 away - not square.
• We don’t want 905 or 895 (1 and 7, mod 8, respectively)
• $885^2 = 783,225$, which 59,684 away - not square
• $875^2 = 765,625$, which is 77,284 away - or $278^2$.
So, $842,909 = 850^2 + 347^2 = 875^2 + 278^2$, which means it can be factorised - which we’ll do in the next thrilling installment.
I can’t help but think the search for squares can be done more efficiently (we got lucky in finding the squares so quickly), but I think this is a very nice paper-and-pencil exercise.
A selection of other posts
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Calculator for AS Level Maths
For context, I currently take A-Level Maths, Further Maths, & Physics (and 2 more but they're not maths-based).
A standard A-Level Maths calculator (Casio fx-991cw) will do you fine and is in many cases faster than one of the more clunky ones for a lot of the things you'll be doing. However, with further there
is a lot you can do with a Casio fx-CG50 (what I use) that you can't on other calculators. It is not a cheap calculator by any means but eBay has some pretty good second-hand deals (that's how I got
They're some great tutorials out there on how you can use your calculator to best maximise your grades. Tbh CG50 is a little bit pay-to-win but you gotta do what you gotta do to get the grades you
want :P
Quick Reply
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Circuit Complexity and P versus NP
In 1983, Michael Sipser
an approach to separating P and NP.
One way to gain insight into polynomial time would be to study the expressive power of polynomial-sized circuits. Perhaps the P=?NP question will be settled by showing that some problem in NP
does not have polynomial-sized circuits. Unfortunately, there are currently no known techniques for establishing significant lower bounds on circuit size for NP problems. The strongest results to
date give linear lower bounds, and it does not seem likely that the ideas there can go much beyond that.
Over the next few years, circuit complexity played a central role in theoretical computer science. In 1985, Yao
parity required exponential-sized constant-depth circuits, greatly strengthening the bounds given by
Furst, Saxe and Sipser
quickly followed with essentially tight bounds.
Shortly after Razborov showed that the clique function requires large monotone circuits. If we could just handle those pesky NOT gates, then we would have proven P≠NP.
Then Razborov (in Russian) and Smolensky showed strong lower bounds for computing the mod[p] function using constant depth circuits with mod[q] gates for distinct primes p and q. These great circuit
results kept coming one after another. We could taste P≠NP.
But then it stopped. We still saw many good circuit complexity papers and some beautiful connections between circuit complexity and communication complexity, derandomization and proof complexity. But
the march of great circuit results toward P≠NP hit a wall after the 1987 Razborov-Smolensky papers. As far as we know today, NP still could have linear-sized circuits and NEXP could have
polynomial-sized constant-depth circuits with Mod[6] gates.
Boppana and Sipser wrote a wonderful survey on these results in the Handbook of Theoretical Computer Science
10 comments:
1. I am surprised that you call the Boppana--Sipser survey up to date. I thought circuit complexity was more or less dead (or at least got a massive barrier to entry) after the Natural Proofs work
of Razborov and Rudich, and this work appeared in 1994, well after the survey was written.
2. It amazes me that people seem sort of vaguely disinterested in Razborov-Rudich. The proof doesn't strike me as any more complicate than say Blum-Micali-Yao. And it seems like it should give
complexity theory some direction and something to grapple with. But instead people just seem to sort of not care.
3. Isn't the question of lower bounds still interesting for algebraic circuits? Or does some Razborov-Rudich type obstruction exist for that situation too?
4. For me, Razborov-Rudich is a fascinating example of "mining algorithms from a proof." Some logicians talk about this, for example
Kohlebach and Oliva.
From what little I have seen, however, that work has not concentrated on discrete math and TCS. The Natural Proofs work,
in contrast, is a direct and
surprising application of that idea.
I suspect that the two communities
developed the idea independently,
but I don't know.
In my case, I do care, but at the
same time it seems hard to think
of any proof technique that does
not yield an efficient algorithm
yet is also good for separating
complexity classes.
You are left with counting
arguments of various types or diagonalization.
For a while I was
interested in Ehrenfeuct-Fraisse games, as there is a result showing that deciding the winner of such a game is PSPACE-complete in general. Therefore an algorithm mined from such a proof might be
too expensive to act as an efficient distinguisher for a pseudo-random function. Unfortunately, that doesn't mean that the game used to separate a particular pair of complexity classes is hard to
decide...so it's not clear what this means. Perhaps you could find a new proof of a result established via an EF game, but "pump up" the complexity of the game used for the separation?
Here is another question: is there any known proof technique which is both not natural and also does not relativize?
5. On my understanding, Fortnow & co.'s method of 'arithmetization' is both non-natural and nonrelativizing:
Basically, and with variations, the prover interpolates a low-degree polynomial through a boolean function, then runs an interactive proof to demonstrate a value or identity involving the
polynomial. It works roughly because different low-degree polynomials are very different, and this difference is probabilistically detectable by the verifier.
It's nonrelativizing because the natural complete problems one can arithmetize (e.g. 3-SAT) are no longer complete under relativizations--their vocabulary fails to capture the dependence of a
nondeterministic machine's behavior on exponentially many oracle bits. One can't hope to fix the technique because of Chang et al.'s result that IP^A doesn't contain coNP^A for random A; this is
proved by standard techniques w/o reference to interpolation.
It's not 'natural' because it's just not a Razborov-Rudich hardness test for circuits (in any form I understand, at least). Its main thrust is to prove surprising class *equalities*. However,
there is at least one nonrelativizing circuit lower bound that follows, see Burhman-Fortnow-Thierauf,
How is this done? (The latter paper is short, so see for yourself..) Basically it's the same 'a bridge too far' method that Fortnow argues makes diagonalization a still-viable tool: class
collapses combine powerfully with each other to produce further collapse; arithmetization is a collapsing technique; the assumed collapse one is trying to disprove then causes so much collapse
that you contradict the classical (relativizable) hierarchy theorems.
As I see it, there's no particular reason to expect this particular technique to solve all complexity questions. To argue this, you might construct two oracle worlds that share the known equality
results garnered by arithmetization yet differ as to whether e.g. P = NP.
6. Are there any known results for circuits which only consist of xor gates?
Consider the following problem: we have n inputs, and n = p * p for some prime p. Each input can be specified as (x, y) for x, y in [0, p). We also have n outputs, referred to in the same way.
The value of the (x, y) output is equal to the xor of each input (m, x + y * m) for m in [0, p).
There's no pair of inputs which are xored for two different outputs for this problem, so it's 'obvious' that the smallest number of xor gates which can be used to obtain the result is to simply
do all the xors for each one, which leads to circuit complexity just under n * sqrt(n). It's also 'obvious' that using non-xor gates can't help compute the values any faster.
Clearly that second conjecture hasn't been proven, because it result in a superlinear circuit complexity. It seems like the first conjecture should be easy to prove though, but I can't get
anywhere with it.
7. Bram, I'm unsure of how to interpret your problem. Try to state it more formally. Especially worrisome, what is xor of numbers in Z^p? Do you mean addition mod p?
Also: be aware that if your input is n pairs of elements of [0, p), the strict input 'size' is bit-encoding size, i.e. ~2n*log(p) = 2p^2log(p). Your conjectured circuit lower bound, to be
interesting, has to be superlinear in *this*, not just in n.
Finally, your output is more than a single bit; it's not a decision problem (which are more commonly studied), more is required. So proving you need a big circuit might be easier. But ask
yourself if the individual output bits seem nearly as hard to produce; if this is the case you might prefer to concentrate on the problem of computing one of them.
If the complexity seems instead to come from there being many outputs, the issue is those outputs being relatively 'orthogonal' as mathematical entities--combining their computations doesn't give
significant savings. I'm not aware of natural problems for which this is known in a strong way, and I'm not sure your problem is the one to achieve it since the bits seem integrally related.
Good luck.
P.S. for a classic problem that shows how problems can yield savings when computed together, try to write a prog to find both the max and min of n numbers with many fewer comparisons than just
superimposing a max-search and a min-search.
8. I don't think this problem should be easy at all, even for linear circuits, although it looks interesting.
If I understand correctly the inputs are $n=p*p$ single bits (just ordered on the p*p grid)
and the (x,y)^th output is the XOR of all the bits that reside on the line specified by x and y.
Thus if (a,b) and (a',b') are two points then indeed they are contained in only one line. (Although I am pretty sure this property alone won't suffice for a lower bound)
Each individual bit is certainly easy here to produce as it takes \sqrt{n} XORs, so indeed any proof of this form is some kind of direct product construction.
9. Thanks, that makes perfect sense. Cool problem.
10. Yeah, there are n outputs, each of which requires sqrt(n) to compute individually, and my conjecture is basically that computing them is completely orthogonal.
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Mini-Workshop “Calculus of Variations and Functional Inequalities”
Date: Wed. May 25, 2022
Organized by: FAU DCN-AvH, Chair for Dynamics, Control and Numerics – Alexander von Humboldt Professorship at FAU Erlangen-Nürnberg (Germany) – Alexander von Humboldt Professorship at FAU
Erlangen-Nürnberg (Germany)
Title: Mini-workshop “Calculus of Variations and Functional Inequalities””
This is a hybrid event (online & on-site)
• Online: Join via Zoom meeting link
Meeting ID: 682 9425 7970 | PIN: 937764
• On site: Felix Klein building. Room 03.323
Friedrich-Alexander-Universität Erlangen-Nürnberg.
Cauestrasse 11, 91058 – Erlangen, Bavaria (Germany)
Tobias König, Institut de Mathématiques de Jussieu, Paris Rive Gauche
“The fractional Brezis–Nirenberg problem in low dimensions. Critical functions and blow-up asymptotics”
Abstract. The classical Brezis–Nirenberg problem asks for the existence, respectively non-existence, of positive solutions u to -\Delta u + a u = u^\frac{N+2}{N-2} on some domain \Omega \subset \R^N
with zero Dirichlet boundary conditions, depending on the choice of a \in C(\overline{\Omega}). I will begin by discussing this problem, with emphasis on the special role of dimension N = 3 for its
I will then introduce the fractional version of the Brezis–Nirenberg problem involving the fractional Laplacian (-\Delta)^s with s \in (0,1) and the corresponding critical exponent \frac{N+2s}{N-2s}.
It turns out that the problem now behaves specially in dimensions N \in (2s, 4s). For such dimensions, I will present some recent results joint with N. De Nitti (FAU DCN-AvH). Firstly, we
characterize the functions a for which an energy-minimizing solution exists in terms of the Green’s function of (-\Delta)^s + a, thus extending a well-known result for s =1 due to Druet. Secondly, we
give a precise description of the concentration behavior of minimizing solutions u_{\epsilon} associated to functions a_{\epsilon} tending to some critical a.
Federico Glaudo, ETH Zürich
“On the sharp stability of critical points of the Sobolev inequality”
Abstract. The unique minimizers of the Sobolev inequality in R^n are known to be the Talenti bubbles, a two parameters (position and concentration) family of functions. As a consequence, the Talenti
bubbles solve the associated Euler-Lagrange equation \Delta u + u^{2^*-1} = 0 in R^n.
If u : R^n \to R is a sum of “almost independent” bubbles, then u “almost solves” the Euler-Lagrange equation, that is |\Delta u + u^{2^*-1}|_{H^{-1}} \ll 1. M. Struwe proved the converse in the 80s,
i.e., that if a function u satisfies |\Delta u + u^{2^*-1}|_{H^{-1}} \ll 1 then u is close in H^1 to a sum of almost independent bubbles.
With an application to the fast diffusion equation in mind, we will discuss the sharp quantitative stability of Struwe’s result. We will present various recent (sharp quantitative) estimates of the
distance (in H^1) between u and the manifold of sum of Talenti bubbles with the quantity |\Delta u + u^{2^*-1}|_{H^{-1}}. The unexpected and novel feature is that the sharp exponent in these
estimates depends on the dimension n.
This talk is based on a joint work with A. Figalli.
Previous FAU DCN-AvH Workshops:
If you like this, you don’t want to miss out our upcoming events!
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Sancho GGP Player
Finally I have time to write the long-promised follow-up to my last post, which is an example of using an approximate factorization technique to improve play.
This work was inspired by thinking about play in Breakthrough, and the current implementation applies fairly narrowly to games that exhibit certain constraints (see below) and can be described as
'breakthrough-like' in some sense. However, extension to a wider category of games is something that I will be coming back to after this year's championships.
Breakthrough exhibits locality of effect for moves in the following senses:
1) No goal disjunct is influenced by multiple moves (the goals in fact can be expresssed precisely as disjuncts of propositions that each individually arise from a single move). Hence moves in
different parts of the board cannot directly influence the same goal (or terminality)
2) Each move impacts only a small number of base propositions, and thence of legals.
These constraints allow us to statically calculate a useful distance metric on the space of moves, wherein the distance between two moves is the minimum number of turns that must go by before both
can impact the same base proposition (or goal/terminality disjunct, though that will be the same thing in the constrained set of games we currently seek to apply this to).
Given such a distance metric we can observe that for any sequence of N moves by one player (interspersed with arbitrary responses by the opponent) that results in a terminal state then either:
i) All N moves are within a region of diameter N in the above metric space; or
ii) There exists some N' < N for which such a sequence of N' moves is also terminal
We can say a path to terminality is
if it is of minimal length (for arbitrary opponent responses). Informally suppose you have a pawn 2 spaces from the end rank in a game of Breakthrough, and no enemy pawns positioned to stop it (and
no quick wins of their own). Then the efficient sequences would be those in which this pawn were advanced on each move (so the efficient path length would be 2). Playing other moves along the way
would only lead to longer paths, and so such moves are irrelevant.
We can then define a set of N-efficient paths, to be those paths of length N that are efficient, and this set will be a subset of the set of paths in which all moves are within a region of diameter
This allows us to define a search algorithm to search for forced wins within N moves, by restricting the move selection to the efficient set, which trims the search space from the set of all moves
(in our example of a pawn 2 moves from the end rank, any search that begins by advancing it can ignore [on the second move selection] all moves that are further than 2 away in our distance metric
from the first move, which essentially mean the search will typically ignore almost all but the actual winning move(s) in that case).
Furthermore, since we will only look for N-efficient wins (so binary result) we can prune aggressively in the opponent ply - as soon as any opponent move is identified that avoids a win for us, the
parent node of that opponent choice can be trimmed.
By searching separately for forced wins for each player (independently) we can prune each individual search extensively, and apply the efficient-sequence trimming using iterated deepening to search
for reasonably shallow forced wins. The details become quite complex, but the constraints implied by the notion of efficient sequences, can be tightened considerably to constrain the opponent replies
considered at each stage, and to progressively shrink the region diameter as the search ply increases within each iteratively deepened search. In practice (using one thread for this activity, in a
way discussed a bit more below) we generally see effective end-game search sequences of lengths around 12 for Breakthrough(small) and 9 or 10 for Breakthrough (15 second per move play time being what
I typically use in testing). This is easily enough to fix the kinds of endgame blunders MCTS tends to make, and leads to a considerable strengthening in play quality.
Current use in Sancho
In Sancho we enable this system for games that have no goal-coupling of moves (that is to say that no one goal disjunct is dependent on more than one move) and for which the static distance analysis
provides average move distances above a certain threshold (games in which the pieces are highly mobile produce much smaller distances, which in turn allows less pruning since far more sequences are
efficient, so this threshold serves to filter those games out).
One thread serves requests for local search from a given start state with a given initial 'seed' move to determine the locality. The main MCTS search thread periodically issues a new local search
request for 'interesting' (state,move) pairs, which are currently:
• The current state and the last move played
• The state resulting from the move we think most likely to be played next, and that move
This is re-evaluated on a timescale of a few seconds and the search request updated accordingly (so local search is asked to look for forced wins in the branches MCTS thinks are most probable). If
results are found before the search target is changed, they are fed back to the MCTS searcher and applied as a local-search-status to the nodes concerned (local win, local loss, no known local
result). MCTS selection heavily discounts local losses, and boosts local wins. It does NOT prune based on them, because they are actually only strong hints (a local win in N moves from a particular
start point does not guarantee there is not a local loss at lower depth from a different non-local start point). Because it applies a fixed discount to local losses, if local search concludes that
ALL moves are losses, the effect is to discount all moves equally, leaving us back with the original MCTS move weighting. This is important in practice because knowing you SHOULD lose does not mean
you should not do your best to avoid it (encouraging the opponent to blunder), so reverting to MCTS evaluation in this case maximizes the chances of recovering via sub-optimal opponent play.
An interesting feature that drops out from the nature of the search, is that if you consider only moves within a certain radius, you run the risk of introducing an artificial
, whereby all the local moves are actually bad, and the correct move is a non-local one (this actually only happens in the ply of the opponent relative to the role you are looking for forced wins
for). To account for that we have to introduce a virtual
to the search. A direct side effect of this is that we always have a search result for the opponent tenuki-ing the first move we're being asked to search with respect to. This means that if the MCTS
searcher asks the local search to examine a particular move, a possible result is that although the move is not a forced win (within the depth iterated to) it can identify that it IS a forced win if
the opponent tenukis, and we can thus conclude that all opponent moves outside the search radius are local losses (and mark them accordingly in the MCTS tree).
Another interesting observation is that the bandwidth required to communicate between the MCTS search and the local searcher (and the sensitivity to latency) is low (new requests and responses
typically are issued on a timescale of seconds and require perhaps O(1K) data). This makes parallelization (or even distribution across machines) of local search an attractive area for future
consideration, allowing the main MCTS search to request more 'interesting' states to be examined.
Finally a pointer to some related work and acknowledgements - the distance metric defined here is not the only one possible. Another useful distance that one could consider is the distance of a move
from a terminal state (i.e. - the shortest number of turns before a terminal state can result). Previous work has been done by Daniel Michulke and Stephan Schiffel (see
Distance Features for General Game Playing
) that utilizes goal-distance metrics to prune and direct search.
The use of sequence efficiency (as defined by a move-move distance and outlined above) and other distance-based pruning techniques (specifically goal distance, certainly) are mostly orthogonal, and
it is clear that the two techniques should be highly synergistic.
Future directions
I'm not entirely sure when I'll come back to this work, but at some point I intend to, and there are many directions to take it in. Below is a list of the ones I currently think most interesting.
• Combine with goal-distance metrics (for example, in Breakthrough, no pawn move to a rank that is still > N ranks from the final rank can possibly be part of an N-efficient winning sequence, so
this allows orthogonal pruning)
• Extend to games that exhibit goal coupling - consider the larger varieties of Connect4 as examples here - because each goal disjunct consists of 4 base props, which can become set by 4 different
moves, move distances of drops in columns within 4 of one another are all 1, which leads to poor pruning due to insufficient locality. However, by imposing an artificial 'focus' on the sequences
considered by local search (adjacent moves in a sequence impose a restricted distance constraint) we can effectively get local search to consider reduced vertical 'strips' of the board. Such
focus constraints are empirically effective, but are inherently heuristic in nature, so work is required to determine when it is appropriate to use them and how to (automatically) tune them,
• Distribute local search across machines
I've been meaning to write this post for ages (it's largely based on enhancements made late last year), but never seemed to get around to it. Well, with the new run of the Coursera GGP course
beginning last week, I really wanted to get back to making regular updates again, so now seems a good time to finally get to this one!
The general message of this post is that you don't necessarily need to play exactly the game you are given! Playing something that is an approximation of it will often suffice, provided the
approximation is something you can play better than the exact game, and usually it gives the right results. I'll discuss some different types of approximation, some of the more adventurous of which I
may well come back to in future posts, but the main thrust of the thesis is that a player that plays well in a game that usually has the same semantics as the actual game it is supposed to be
playing, can often beat a completely 'correct' player that doesn't play as well within the domain of the approximation, provided that most of the time actual play stays in that domain.
There are probably many categories of approaches to approximation that can be taken, but I'll talk here about just two, one of which is implemented (for some cases) by the current version of Sancho,
and the other of which I'm currently working on some restricted cases of.
State machine feature emulation
Often we are given game rules that manifest as a very expensive state machine, and/or a state machine that is hard to reason about in order to perform feature analysis. A typical example is something
like Reversi, which has a horribly expensive goals network, which slows down tree expansion. Fairly straight-forward analysis can reveal logic (and base propositions) that have no impact on move
legality, and basic logic structures which suggest possible forms of the goal calculation (e.g. - successor-like logic that suggests counting is going on), and these clues can suggest hypotheses for
the goal forms. Based on these clues, one can come up with a set of hypothetical goal calculations, and then during meta-gaming measure their accuracy across random games. If you find a hypothetical
goal calculation that appears to match what the game actually delivers via a full state-machine implementation, one can then decide to play with an emulation of the goals logic external to the
animation of the state machine itself (and remove all goal logic from the underlying state-machine).
In Sancho we use this to detect goal structures of the forms:
• Game is a straight-forward win/loss/draw based on who holds the majority of some set of base props
• Goal value is cardinality of some set of base props whose value is true in the current state
• Game is a straight-forward win/loss/draw based on the previous type but interpreting the values it produces as inputs to a who-has-more calculation (Reversi is of this kind - basically are there
more white pieces than black pieces or visa versa)
• As any of the above but where the count is represented via a set of successor counting props in the game state (Dots&Boxes looks like this as I recall)
If it can hypothesize a probable goal calculation, and match it to one of the above then Sancho will strip goal logic from the state-machine and run an emulator that directly evaluates the
calculations. This has two benefits:
1. It typically results in significantly faster tree expansion, since next state and goal calculations involve less logic
2. Once we decide to believe in one of the above forms we can use it to draw further inferences about the game which can be useful. Examples are:
□ If a we can determine that a count cannot go backwards (e.g. - in the successor logic there might be no path that decrements a count, or in a majority-of-propositions formulation where all of
the propositions being counted are latches), and the game's final result is a win/loss based on a majority of a fixed size set, then the game's result is fully determined once one role has
more than half. This allows us to add some emulation to the terminal logic also, and declare the game to be virtually terminal (for search purposes) in states where the outcome is fully
determined. This allows Sancho to cease searching in Dots&Boxes when one player has 13 boxes for example.
□ If a count is based on the cardinality of some set of base propositions, and some of those propositions are known to be latches, then we can infer a possible heuristic, that it is good to get
those latched propositions set. In Sancho we then feed the resulting heuristic through the same statistical verification we use to determine generally which heuristics should be turned on
(previously discussed when talking about Piece detection heuristics). This turns out to provide Sancho with a heuristic that corners are good to own in Reversi for example.
Approximate Factorization and Local Search
A second interesting area, is that of games that exhibit some sort of 'locality' of action. That is to say, games in which it is possible to define a measure of distance, such that base propositions
can be mapped onto a metric space, where the distance measure reflects the number of steps before changes to one proposition can result in changes to another. This relates closely to factorization,
in that in a factorizable game the distances will divide the space of base propositions (possibly after some processing to remove 'control logic' such as whose turn it is) into disconnected subsets.
The fully factorizable case I've discussed in a previous post, and leads to factorization of the game into sub-games, which are then independently searchable.
However, cases that are not fully factorizable might still be able to make use of the distance information. Two obvious uses suggest themselves:
Firstly, if a game exhibits localized neighbourhoods, that are strongly connected internally, but weakly connected externally, then we could consider making sub-games that consist only of those
neighbourhoods (separately) and searching them as if they were true factors. As a thought-experiment consider a game like Risk. What armies are placed, and move, in one part of the world has little
impact on distant parts of the world for many turns. Thus we could treat regions of the board separately (searching tactical possibilities in Asia without worrying about South America for instance).
Provided we can weave the results of the sub-game searches back together in some useful way (typically goals might not exhibit the same locality that legality and base props do), it will be much
cheaper to search the subgames than to globally search the entire gamespace. Note that subgames need not be disjoint, so we could consider overlapping regions and take the move from the one with the
best local result for instance (discounted by the worst local penalty for not playing in the other subgames).
Secondly, we can search from a given state only within a fixed neighbourhood (say everything that could be be influenced within a certain number of steps). If we assume (and we can determine this
analytically by looking at the terminal/goal conditions for the game) that two moves are only meaningful in the same move sequence if they can both influence something commonly within the restricted
neighbourhood, then we can restrict the choice of moves we look at within the search. For example, imagine a Sheep&Wolf like game, but taking place in a maze, where one player with several pieces
(like the sheep) is aiming to trap an opponent piece (like the wolf), and wins if they do. Search sequences (up to a given length) which feature moving sheep which are further away from one another
than the neighbourhood diameter are pointless, as two such moves cannot both be relevant to a successful capture (within a number of moves equal to the neighbourhood diameter). Hence we can perform
restricted search, such that choosing to move a certain sheep as the first move in the sequence constrains later choices, and greatly reduces the effective branching factor. The downside of this is
that the search results are only valid up to a depth of the neighbourhood radius. We can use an iterative deepening approach to address this.
Currently Sancho does not implement approximate factorization, but an experimental version is under development that does implement local search. I will (hopefully) have a lot more to say about this
in a future post!
The 29th Conference on Artifical Intelligence (AAAI15) was held in Austin last week (see http://www.aaai.org/Conferences/AAAI/aaai15.php), and since that's where I live I was able to spend some time
there meeting GGP researchers who were attending, and in particular the Stanford and New South Wales teams under Professors Genesereth and Thielsche (respectively), who are behind the Coursera GGP
course, which initially introduced me to the field.
I had many interesting conversations, which have left me with boosted enthusiasm for getting back to making further progress on both Sancho, and on the exploration of some new directions (which I
will probably follow independently of Sancho to begin with, though they may re-integrate further down the road).
During the conference a 'grudge match' (honestly - there's no grudge!) between Sancho and TurboTurtle was held (I don't have a link for any of the games, as we did it in a fairly ad-hoc fashion,
playing games people present suggested), which I think Sancho won (though I don't recall any exact score - it was more a demo-match session). The following day, a human vs silicon competition (just a
couple of games) was also held (Sancho vs attendee-victim), which Sancho won 2:0, though it should have lost the first game, as its opponent had a fairly short forced win at one point (in
On the final day a brief award ceremony took place, where I received the GGP International Championship cup following last year's win (traditionally this takes place at the AAAI conferences), on
behalf of Andrew and myself.
The cup in new hands:
Professor Genesereth (the handsome guy without much hair standing next to the other handsom guy without much hair!):
Last Christmas, my father-in-law gave me a copy of Mancala (also known as Kalah or Kalaha). I've played it a few dozen times, but my 7 year old daughter still beats me regularly!
So I wondered about coding up the rules in GDL to see how Sancho plays it. As it transpires, a quick look in the base repository revealed that somebody had already done this back in 2009. (This game
isn't on Tiltyard rotation because it doesn't contain explicit definitions of the base and input propositions.)
Enthusiastically, I fired up Sancho and got it playing. Sadly, it only managed ~100 iterations/turn. (We've made some significant general improvements since then, but Kalaha is still an order of
magnitude slower even than a complicated game like Chess.) At the time, I made some tweaks to the GDL which improved the speed approximately three-fold, but that still wasn't enough for a good game.
So, it has sat on the back burner for nearly a year. Over the last couple of days, I've been digging into it in a bit more detail because I have some moderately revolutionary ideas for dramatic
improvements. (Hopefully more in a further blog post.)
For now, I've produced an annotated version of the GDL which you may like to peruse. (You'll definitely want to understand the basic rules of the game first though - see the link above.)
I have been a bit remiss in making regular posts recently, and as some may have noticed, Sancho has also been absent from Tiltyard for nearly a month.
The reason for this is that about 4 weeks ago I undertook a reworking of the MCTS tree's representation internally to Sancho, to reduce memory usage, and remove some role-asymmetry that didn't seem
right. The catalyst for doing this was to set the groundwork for an enhanced move-sequence search which will be the subject of a future post (at least if it works!). I expected it to take 3 or 4
days, but it soon turned out to be a sticky morass, from which escape was more difficult than envisaged!
The core of the change was to eliminate non-decision nodes in the tree (forced moves, forced noops, etc.), leaving only the nodes at which actual decisions have to be made. Since Sancho's tree
considers choices for a single role at any one node (not joint moves), this meant that in most games (non-simultaneous move games) decision layers were interposed with non-decision (forced noop)
layers (one in N layers at most would have a decision, where there are N roles in the game). This was both wasteful of space (tree node allocation), and introduced role asymmetry because the state
only changes every Nth layer of the tree (where a full joint move is implied by the choices in the preceding N layers).
In the new structure all non-decision nodes are eliminated, and (as a side-effect) an edge always leads between nodes of different game states. The semantics of the representation however, should be
Stabilizing this change took roughly the expected time, but unexpectedly it turned out to have a distinctly damaging impact on the quality of play in games that employed heuristics (primarily games
with a piece heuristic active, such as breakthrough, or chess). The reason for this is not totally clear, but almost certainly has to do with the role asymmetry, which the original heuristic
implementation was 'comfortable' with.
Since then I have been fiddling with the mechanisms by which heuristic values are blended into the MCTS tree, to find a modification that work well in the new structure. Over the past 3 weeks or so I
have found various approaches that initially worked well in my tests with one game, only to find that they performed badly in another. The three games I have mostly been using to test (as they
display somewhat different characteristics) were:
• Breakthrough (simple equal-value piece heuristic, fixed sum)
• Speed chess (variable value piece heuristic, fixed sum)
• Skirmish (variable value piece heuristic, non-fixed sum)
In particular I found that what worked well in Breakthrough (where material exchanges are typically somewhat binary in nature) didn't work well in the other two (and visa vera).
However, as of yesterday I am now getting positive test results (but much more testing remains to be done) in all of the above games, so hopefully this time the light at the end of the tunnel is not
another oncoming train!
A ton of regression testing now needs to be done, for which I plan to use the following games:
• Reversi (uses a heuristic other than piece-based)
• Connect4 (goto non-heuristic fixed-sum 2-player game)
• Max knights (puzzle)
• Sudoku (very different puzzle)
• Three player free for all (more than 2 players)
• Blocker (simultaneous turn)
• Pentago (non alternating role choice sequence)
• Breakthrough with walls (heuristic and factored)
This will take a few more days, even if no problems are found. After that (I hope), normal service will be resumed...
(Who ate all the pies?)
In the International General Game Playing Championship 2014, it struck me that many of the games that were played had significant bias for one player (usually, but not always, the first). In an
attempt to compensate, many games were played both ways round, often with the predictable outcome. This was unfortunate because it increased the amount of time required to play games, reduced the
excitement (by having predictable outcomes) and effectively reduced the best-of-3 rounds to a best-of-one (two matches of a game with a predictable outcome and then the decider - usually a game where
the bias is unknown because it isn't played on Tiltyard and therefore statistics aren't available).
Whilst reading up on various abstract strategy games, I was introduced to the concept of the Pie rule. Based on the traditional method for children to divide a cake ("you cut, I'll choose"), it aims
to balance two player games by giving the second player the opportunity to swap places with the first player after the first move. This way, the first player has an incentive to play the most
balanced initial play the he possibly can. (If he makes an opening move that is too strong, the second player will swap with him and get the strong position. If he plays an opening move that is too
weak, the second player will leave him with it.)
In the last few days, I have created Pie-rule variants of 9-board Tic-Tac-Toe and Hex. (I've also created a visualization for regular Hex so that it can be played on Tiltyard.) It's early days yet,
but I notice that the pie rule seems to be doing the trick for 9BTTT. In the first 16 matches played on the Tiltyard, 9 went with the first player and 7 with the second. Whilst it's still a small
sample size, that's looking substantially more balanced than the regular version.
So, for the GDL authors out there (by which I suppose I mean Alex!), consider the Pie rule for creating balance in your game.
For everybody else, what games would you like to see a Pie rule variant of? Whilst I'm certainly not making any promises, if you post in the comments, I'll consider doing them.
Latches are properties of a game's state which, once they become true, are guaranteed to remain true throughout the remainder of the game (positively latched); or which, once they become false, are
guaranteed to remain false (negatively latched). A 'property' in this sense can be any logical function of the game's base state - most typically we talk about the simplest case of this, which is the
state of an individual base proposition.
Identifying latches can be useful in several ways:
• It may allow reduced computational effort in calculating state transitions (e.g. - if a part of the propnet is effectively 'dark' due to a latch condition, then once that latch is detected that
part of the propnet can be left uncalculated in future state transitions).
• Similarly it may allow reduced computational effort in heuristic calculation
• If the latch constrains the achievable goals then it can be used to optimize search
This post is about the use Sancho makes of the last case - i.e. - situations in which it detects that the achievable score range has been reduced by latching of one or more of the goal propositions
(at most one for each role can become positively latched, in which case the score for that role is then fully determined in all possible paths, but several may become independently negatively
latched, which serves to reduce the size of the set of achievable scores).
Latch detection might be analytic or heuristic in nature. I'll come back to heuristic latch detection in a future post - for this one I'm only considering analytic detection, though the use to which
the result is put can be the same in either case.
Analytic latch detection is based on logical reasoning about the structure of the game. In Sancho's case this is done by analysis of the propnet, and currently we only identify fairly simple cases in
which a base proposition feeds back to itself via logic of one of the following forms (where X is the base prop, and X' is its next value on state transition):
X' = X OR <other condition> (X is positively latched)
X' = ~X AND <other condition> (X is negatively latched)
Furthermore, if a goal proposition is of the form
G = X AND <other condition>, and X is negatively latched, then G is said to be negatively latched by X.
Similarly for the positive latch case:
G = X OR <other condition>
If we can identify such goal latches, we know that the set of achievable scores in any state of the game reachable from a state with latched goal propositions is either fully determined (positively
latched goal), or is strictly reduced from the set of all possible goal values to some subset.
In Sancho this surfaces as a method on the state machine which returns the achievable goal range for any given state, as [min,max]. This range can be used in the MCTS search tree processing in
several ways:
• If min == max, then all possible paths result in the same score. If this occurs for our role then whatever we play from that point on can have no impact on the eventual outcome. Consequently we
can treat any tree node in which this occurs as if it were terminal, and not search below it (the one exception being that if this occurs on the root node we expand one level just so that we have
legal moves to choose from when asked to return a move)
• If a terminal state is encountered with a score of the max achievable, we can invoke tree trimming by propagating upwards when the role deciding on the move at this branch is the one with the
max-achievable score (i.e. - this amounts to a 'win' relative to the parent state and no further searching is needed since it cannot be improved upon)
• If a move reduces the achievable range (either at the top or at the bottom) then we can trivially define a heuristic to favor search on paths that increase the min achievable score, and dis-favor
those that decrease the max achievable.
Because latch detection is typically very cheap (at search time, once the analysis has been done during meta-gaming) we can also use the change in achievable score as a heuristic during playouts to
guide otherwise random playouts along paths that tend to maximize the expected goal value, on the assumption (within the playout) that all achievable scores are equally likely. This can most simply
be done by simply down-weighting moves that decrease the choosing role's max score, and up-weighting those that increase the min. In zero-sum games, up-weighting those that decrease the max opponent
role scores, and down-weighting those that increase their min scores can also be applied. Currently we do this VERY crudely and handle the 'decrease weight' cases by preventing those moves being
selected at all in the presence of any other choice!
Some examples help illustrate where these techniques give benefit, so I'll discuss a few in the following paragraphs.
The Untwisty Corridor Family
This family of games are basically just test cases for latch detection. They are puzzles involving traversing a maze (logically speaking anyway - none of them actually have visualization currently so
far as I am aware). Any step onto the wrong path sets a 'failure' proposition in the game's state, and the game terminates after a fixed number of moves. Consequently you must make the 'correct'
choice for each of a fixed number of steps (in 'untwisty corridor' you need to go straight every move, hence the name). Because a wrong choice sets a latched proposition, which ultimately determines
the goal values, this is a simple latched-goal case, where any wrong move immediately fully determines the final score (as 0). Because all such states are immediately treated as if they were
terminal, and not searched, the effect is that the MCTS search only ever visits nodes one step off the correct branch, which reduced the size of the search space from being exponential in number of
steps to being linear.
In fact, the basic untwisty corridor is somewhat flawed, because all false choices lead to the SAME state (apart from the step counter), so it is trivially solved by transposition, provided the
player maintains a transposition table of some sort. Furthermore, at only 6 steps, it is small enough to solve by brute force anyway! The game 'untwistycomplex2' is an attempt to address these
issues, and a better test.
Escort Latch Breakthrough
The game Escort Latch Breakthrough is a variant of Breakthrough where each role has a king and 8 pawns. The goal is to get the King to the far end of the board. If both kings are captured the game is
a draw. A typical MCTS player, without latch analysis, will usually move a pawn from in front of their king and then just advance the king right up to one rank before it can be captured by the
opponent's pawns. At that point it's usually fairly trivial for the opponent to force capture it (kings cannot move backwards) and games between such players tend to end up as mutual king exchanges
and therefore draws (for example, this game). The reason this happens is that the vulnerability of the king has very little impact on the ensemble of playout scores, because almost all playouts fail
to bother capturing the kings, and MCTS convergence is slowed by the need to achieve convergence in subtrees following king capture.
With latch detection a king capture is seen as an achievable score range change (from [0,100] to [50,100] or [0,50] depending on which king is captured). This impacts the search in several ways:
• In states where one king has been captured, capturing the other is terminal, and can be propagated as if it were a win (even though it scores only 50), because it is the best achievable result
• Moves that capture/avoid capture of a king are heuristically selected more frequently, which speeds convergence
• Playouts are much more reasonable, since they do not include games in which a king is NOT captured if it could be (i.e. - where a king just marches through a defending line of pawns without being
molested). This dramatically improves the quality of Monte Carol samples generated by the playouts. This gain far outweighs the cost in reduced playout count that the evaluation requires (which
is actually quite substantial since all moves must be examined at each stage)
The result is that rather than optimistically advancing their king into danger, the player tends to hold it back, and work to capture its opponent's king first (see this game for example)
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Describe the given region in polar coordinates » QuizzmaDescribe the given region in polar coordinates
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Describe the given region in polar coordinates
Describe the given region in polar coordinates.
To describe the given region in polar coordinates, we need to convert the boundaries of the region from Cartesian coordinates to polar coordinates.
The region is bounded by:
1. The straight lines $x=4x = 4$ and $y=6y = 6$
2. The curve of the quarter circle with radius 6
Converting boundaries to polar coordinates:
1. Quarter circle with radius 6:
2. Line $x=4x = 4$
☆ In polar coordinates, $x=r\mathrm{cos}\theta =4x = r \cos \theta = 4$
☆ Therefore, $r=\frac{4}{\mathrm{cos}\theta }=4\mathrm{sec}\theta r = \frac\left\{4\right\}\left\{\cos \theta\right\} = 4 \sec \theta$
3. Line $y=6y = 6$y=6:
☆ In polar coordinates, $y=r\mathrm{sin}\theta =6y = r \sin \theta = 6$
☆ Therefore, $r=\frac{6}{\mathrm{sin}\theta }=6\mathrm{csc}\theta r = \frac\left\{6\right\}\left\{\sin \theta\right\} = 6 \csc \theta$
Describing the region in polar coordinates:
The region is divided into two parts based on $\theta \theta$θ:
□ Lower portion:
☆ $0\le \theta \le \frac{\pi }{6}0 \leq \theta \leq \frac\left\{\pi\right\}\left\{6\right\}$
☆ $1\le r\le 6\mathrm{sec}\theta 1 \leq r \leq 6 \sec \theta$
□ Upper portion:
☆ $\frac{\pi }{6}\le \theta \le \frac{\pi }{2}\frac\left\{\pi\right\}\left\{6\right\} \leq \theta \leq \frac\left\{\pi\right\}\left\{2\right\}$
☆ $1\le r\le 6\mathrm{csc}\theta 1 \leq r \leq 6 \csc \theta$
So, the full description in polar coordinates is:
□ Lower portion:
☆ $0\le \theta \le \frac{\pi }{6}0 \leq \theta \leq \frac\left\{\pi\right\}\left\{6\right\}$
☆ $1\le r\le 6\mathrm{sec}\theta 1 \leq r \leq 6 \sec \theta$
□ Upper portion:
☆ $\frac{\pi }{6}\le \theta \le \frac{\pi }{2}\frac\left\{\pi\right\}\left\{6\right\} \leq \theta \leq \frac\left\{\pi\right\}\left\{2\right\}$
☆ $1\le r\le 6\mathrm{csc}\theta 1 \leq r \leq 6 \csc \theta$
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Linear Solver Settings
Linear Solver Settings¶
There are several situations where the options selected in zCFD require the solution of a linear system. These are:
• When the Time Marching scheme is set to euler implicit.
• When RBFs are used for mesh motion.
• When the incompressible solver is selected.
When running on CPUs zCFD uses Petsc linear solver library and when running on NVidia GPUs zCFD uses the AMGX linear solver library. Both libraries offer a wide array of solver and preconditioner
choices and altering the settings can have a significant impact on the performance and convergence of the solver. Options can be passed to Petsc via the control dictionary and AMGX options are set
using a json file which is read at runtime.
Details of the available options for the Petsc KSP linear system solvers used by zCFD are given here. these options can be passed through to Petsc via the zCFD control dictionary using the “linear
solver options” key. The default values are given below:
"linear solver options": {"flow": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-3",
"-ksp_monitor": "",
"-ksp_converged_reason": "" },
"turbulence": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-5",
"-ksp_monitor": "",
"-ksp_converged_reason": "" },
"rbf": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-5",
"-ksp_monitor": "" }
Settings are provided for the mean flow, turbulence and RBF linear systems. Most of the settings detailed in the Petsc documentation can be passed through by adding them as key, value pairs to the
appropriate dictionary. Where an option doesn’t take a value an empty string needs to be provided.
The Hypre library of algebraic multigrid methods is available via Petsc PCHYPRE and the associated settings can be supplied via the “linear solver options” dictionary. Using Hypre’s boomeramg package
as a preconditioner has shown good performance with the incompressible solver:
"linear solver options": {"flow": { "-ksp_type": "fgmres",
"-ksp_rtol": "1.0e-3",
"-ksp_monitor": "",
"-ksp_converged_reason": "",
"-pc_type": "hypre",
"-pc_hypre_type": "boomeramg"},
Further Hypre options are detailed in the Petsc PCHYPRE documentation.
The AMGX settings for the mean flow, turbulence and RBF linear systems can be found in ZCFD_HOME/amgx.json, ZCFD_HOME/turbamgx.json and ZCFD_HOME/RBF_amgx.json. The mean flow settings are shown
"config_version": 2,
"determinism_flag": 1,
"solver": {
"preconditioner": {
"error_scaling": 0,
"print_grid_stats": 0,
"max_uncolored_percentage": 0.05,
"algorithm": "AGGREGATION",
"solver": "AMG",
"smoother": "MULTICOLOR_GS",
"presweeps": 0,
"selector": "SIZE_8",
"coarse_solver": "NOSOLVER",
"max_iters": 1,
"postsweeps": 3,
"min_coarse_rows": 32,
"relaxation_factor": 0.75,
"scope": "amg",
"max_levels": 40,
"matrix_coloring_scheme": "PARALLEL_GREEDY",
"cycle": "V"
"use_scalar_norm": 1,
"solver": "FGMRES",
"print_solve_stats": 1,
"obtain_timings": 1,
"max_iters": 10,
"monitor_residual": 1,
"gmres_n_restart": 5,
"convergence": "RELATIVE_INI_CORE",
"scope": "main",
"tolerance": 1e-3,
"norm": "L2"
In general these settings provide good performance for most cases. If issues are encountered converging the linear system reducing the “tolerance” may help. Otherwise altering the multigrid
aggregation selection algorithm “selector”: “SIZE_8” to “SIZE_4” or “SIZE_2” will build smaller aggregates and hence increase the number of coarse levels - improving convergence at the expense of
memory use. Increasing the number of “gmres_n_restart” may also improve convergence at the expense of increased memory use.
There are too many AMGX options to detail here but example configurations for different solvers are given here.
When running on NVidia GPUs and using AMGX there is only one option available in the “linear solver options” dictionary:
"linear solver options": {"double precision": False}
The “double precision” option controls whether the mean flow linear system is solved in single or double precision. Solving in single precision brings a reduction in memory use and a speed up to the
solver. However, double precision may be required for more challenging cases.
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Catalan Numbers: The Beauty of Bijections
by Cassie Kloss | Mar 21, 2024 | Math Seminar, Spring 2024 | 6 comments
The Catalan numbers are one of the most important sequences in combinatorics, which studies the mathematics of counting and arranging finite discrete structures. Combinatorics is an essential branch
of mathematics as it provides efficient techniques for enumerating quantities that would otherwise be difficult to quantify by conventional methods. The Catalan numbers have several equivalent
definitions, including the recursive and closed forms. While the closed form is more straightforward, the recursive definition is more helpful in solving problems in combinatorics. Through the
recurrence and bijective proofs, the Catalan numbers allow deriving various combinatorial objects. We will explore properties and patterns exhibited by Catalan objects like lattice paths, Dyck paths,
balanced parentheses, rooted trees, full binary trees, multiplication schemes, and polygon triangulations. The recursive formula reveals the relationships between these structures, while bijective
mappings prove their cardinalities are the same.
6 Comments
crooks on March 21, 2024 at 7:32 pm
This sounds like a great topic to talk about with many different areas of mathematics being intertwined. I have had a hard time learning about recursion in the past in computer science classes.
However, I am always interested in expanding my knowledge when it comes to recursion in math and computer science. I do not know much about these topics, but I am very excited to hear about them!
Alyssa Hall on March 24, 2024 at 3:25 pm
This topic seems very interesting. I can’t wait to learn about the recursive and closed forms and to see the different properties and patterns that come from Catalan numbers. Looking forward to
your talk!
mcarthur on April 7, 2024 at 7:31 pm
The topic of your seminar sounds very complex and interesting. I have no prior knowledge of catalan numbers so I am intrigued to learn about them. I can not wait to hear you dive into what
catalan numbers are during your talk!
Gloria Uwizeye on April 10, 2024 at 6:44 pm
I have never heard much about Catalan numbers before which is why I am very excited about this talk. Both your teaser show how your talk is interesting!! Good luck Cassie
Tarin Rietz on April 11, 2024 at 1:05 pm
Before your talk, I had heard of the Catalan numbers, but was relatively unfamiliar with their significance mathematically. Your talk provided some insight into that, which was great! Watching
the calculations behind determining the lattice paths using the Catalan numbers was very interesting and I can see how this sequence and recursion can be applied to fields like computer science
and combinatorics.
schwab on May 5, 2024 at 1:00 am
I don’t think I had ever heard of the Catalan numbers before your talk but you explained it well and definitely cleared up some confusion I had surrounding the concept. It was very interesting
when you were showing how all the different paths could be drawn!
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What is an Operator?
An operator may refer to any of the following:
1. In computer programming and at the command line, an operator is an object capable of manipulating a value or operator. For example, in "1 + 2", the "1" and "2" are the operands and the plus symbol
is the operator. Below is a listing of common operators found in programming languages with explanations and examples.
Operator Alternative Explanation Examples
= Equals (set a variable to a value). a = b
== Eq Equals? (compare two values) a == b
!= Ne Not equal a != b
+ Plus a + b
+= Addition a += b
++ Increment a++
- Minus a - b
-= Subtraction a -= b
-- Decrement a--
/ Divide a / b
* Times a * b
> Gt Greater than a > b
< Lt Less than a < b
>= Ge Greater than or equal to a >= b
<= Le Less than or equal to a <= b
<=> Spaceship operator
|| or Boolean or a || b
&& and Boolean and a && b
Booleans are also considered operators where AND, OR, and NOT can also be used in most programming languages.
2. An op is a person who controls an IRC (Internet Relay Chat) channel. See the op page for further information about this term.
Arithmetic operator, Conditional expression, Decrement, Exclamation mark, Increment, Logical operation, Op, Operand, Operator associatively, Operator precedence, Order of operations, Programming
terms, Syntactic sugar, Ternary operator, User
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Stock moving average in excel
The Exponential Smoothing tool in Excel calculates the moving average. However, exponential smoothing weights the values included in the moving average calculations so that more recent values have a
bigger effect on the average calculation and old values have a lesser effect. This weighting is accomplished through a smoothing constant. To illustrate how the Exponential … How to add a moving
average to an excel stock chart Right click on chart. Click on "Select Data" Click "Add" button. Select cell range $K$23:$K$272. Click OK button. Go to tab "Layout" on the ribbon. Select "Series 4".
Click "Format Selection" button, see picture above. Select "Secondary Calculating a rolling average (also known as a moving average) is easy in Excel using the AVERAGE formula combined with absolute
and relative cell references. A rolling average helps smooth out trends over time, particularly when your data shows cyclicality by week, month, or year. Let's take a look at what rolling averages
are, and how you can calculate them in Excel.
The Exponential Smoothing tool in Excel calculates the moving average. However, exponential smoothing weights the values included in the moving average calculations so that more recent values have a
bigger effect on the average calculation and old values have a lesser effect. This weighting is accomplished through a smoothing constant. To illustrate how the Exponential … How to add a moving
average to an excel stock chart Right click on chart. Click on "Select Data" Click "Add" button. Select cell range $K$23:$K$272. Click OK button. Go to tab "Layout" on the ribbon. Select "Series 4".
Click "Format Selection" button, see picture above. Select "Secondary Calculating a rolling average (also known as a moving average) is easy in Excel using the AVERAGE formula combined with absolute
and relative cell references. A rolling average helps smooth out trends over time, particularly when your data shows cyclicality by week, month, or year. Let's take a look at what rolling averages
are, and how you can calculate them in Excel. Stock market analysts will often use a 50 or 200 day moving average to help them see trends in the stock market and (hopefully) forecast where the stocks
are headed. An average represents the “middling” value of a set of numbers. The moving average formula is a solid choice for ensuring your costs are always up to date. Costing methods are important
to nail down because, given the same stock levels and purchase prices, each method can report very different levels of profit and cost of goods sold (COGS). Simple moving averages involve a fairly
basic calculation: Add a stock's closing prices over a set number of days, and then divide the sum by the total number of days. For example, a 20-day simple
Exponential Moving Average in Excel 1. Let us get our feet wet with 13-day EMA for GM stock. 2. The simple average is calculated for the first 13 closing prices of the stock through AVERAGE () 3. EMA
formula from cell H15 onward becomes –. 4. Drag the formula starting at H15 to the end of the
Exponential Moving Average in Excel 1. Let us get our feet wet with 13-day EMA for GM stock. 2. The simple average is calculated for the first 13 closing prices of the stock through AVERAGE () 3. EMA
formula from cell H15 onward becomes –. 4. Drag the formula starting at H15 to the end of the Calculate EMA in Excel with Worksheet Functions. Step 1 . Let’s say that we want to calculate the 12-day
EMA of Exxon Mobil’s stock price. We first need to get historic stock prices – you Step 2 . Calculate the simple average of the first 12 prices with Excel’s Average() function. In the Moving Average.
This example teaches you how to calculate the moving average of a time series in Excel . A moving average is used to smooth out 2. On the Data tab, in the Analysis group, click Data Analysis. Note:
can't find the Data Analysis button? Click here to load the Analysis ToolPak In stock trading, moving average is an indicator that shows the average value of a security over a given period of time.
In business, it's a common practice to calculate a moving average of sales for the last 3 months to determine the recent trend. Moving (Rolling) Average in Excel 2016 - Duration: 9:28. Dr. Todd
Grande 73,457 views You can add a moving average line in the column chart easily as follows: Click the column chart to activate the Chart Tools, and then click Design > Add Chart Element > Trendline
> Moving Average. This builds on the moving average cross over strategy by going long if the short term SMA is above the long term SMA and short if the opposite is true. “Note: you have to lag the
signals by one day in order to remove look-ahead bias.” In this example the Excel formula is as such: =IF(H26>I26, 1, -1) Step 3: Calculate Strategy ln Daily Returns
Moving Average is an analytical tool in Microsoft Excel which is used to recognize the ongoing trend in the data and it helps in forecasting. This tool is.
No Signal. Bullish. Bollinger Bands. Bearish. Bearish. Moving Averages. Bearish (Daily). KST. Bearish. Bearish. Dow Theory. Mildly Bearish. Mildly Bearish. OBV. Which technical analysis tools can be
used to analyze EXCEL INDUSTRIES? Check out various oscillators, moving averages and other technical indicators on TradingView. A fast and easy way to analyze India Stocks. Technical analysis 30 Oct
2010 The moving average is used quite often in technical analysis of financial data such as stock returns and in economics to locate trends in
Calculate moving average with Analysis tool of Moving Average in Excel. 1 . Click the File > Options . 2 . In the Excel Options dialog box, click the Add-Ins in the left bar, Keep Excel Add-Ins
selected in the Manage box and then click the Go button. 3 . In the opening Add-Ins dialog box, check the
To calculate a moving average, first click the Data tab's Data Analysis command button. When Excel displays the Data Analysis dialog box, select the Moving 25 Sep 2015 In stock trading, moving
average is an indicator that shows the average value of a security over a given period of time. In business, it's a common Moving average is heavily used for technical analysis and a lot of banks
and stock-market analysts use it on a daily basis (below is an example I got from the 24 Sep 2013 Moving Average in Excel 2013: Data Analysis Add-In. Using worksheets. stock market and (hopefully)
forecast where the stocks are headed. 14 Jan 2020 All about Investment, Pricing, and Trading models in Excel, and R. Technical Indicators, Momentum Oscillator, Simulation, Price Optimization, For
example, it is often used in technical analysis of financial data, like stock prices, returns or trading volumes. It is also used in economics to examine gross Learn about simple, exponential and
weighted moving averages, including definitions, calculations, and their basic use and interpretation in trading.
Learn about simple, exponential and weighted moving averages, including definitions, calculations, and their basic use and interpretation in trading.
8 Dec 2017 Exponetial Moving Average (EMA for short) is one of the most used indicators in In this example we shall calculate EMA for a the price of a stock. This can of course be put into Excel or
some other spreadsheet software to excel vba average trading. This code is to create a positioning tool for trading in financial markets. I would greatly appreciate help. Calculating a rolling
average (also known as a moving average) is easy in Excel using the AVERAGE formula combined with absolute and relative cell references. Online financial calculator to find the arithmetic moving
average (AMV) for the price increase / decrease over a fixed period of time. Adding a moving average to an Excel candlestick chart. Last Updated on Sat, 07 Sep 2019 | Candlestick Patterns. The
Candlestick Trading Bible. Candlestick Using moving averages in SQL will smooth out the short-term fluctuations in Traders use the moving average to determine how low the stock price will go This is
a problem with windowing functions similar to the FORMAT function in Excel. No Signal. Bullish. Bollinger Bands. Bearish. Bearish. Moving Averages. Bearish (Daily). KST. Bearish. Bearish. Dow Theory.
Mildly Bearish. Mildly Bearish. OBV.
Here we discuss how to calculate 3 types of moving averages in excel In businesses like a stock market, moving average helps the trader to more easily To calculate a moving average, first click the
Data tab's Data Analysis command button. When Excel displays the Data Analysis dialog box, select the Moving 25 Sep 2015 In stock trading, moving average is an indicator that shows the average value
of a security over a given period of time. In business, it's a common
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Portfolio management is a long-tail game - GroundControl
Portfolio management is a long-tail game
Published on September 21, 2018, last updated on April 19, 2023
When looking at the success of startups, a common belief is that one third of the companies fail, one third return their money and one third of the companies become successful enough to really move
the needle on your investment portfolio. But is that true?
Correlation Ventures did research on all VC investments in the US between 2004 and 2013 to figure out the distribution of over 21,000 different investments. It turns out that close to 65% only
returns up to 1x the initial investment, and only 10% returns an ROI bigger than 5x. So two third of the companies only return up to their initial investment and the other one third needs to make up
for the loss of the rest.
Using the Correlation Ventures research data as a benchmark we have created a calculator to help innovation managers to understand in how many ideas they need to invest in order to move the needle.
You simply put in your total budget and the amount of ideas you’d like to invest in and out comes a prediction of your ROI. The calculator uses the Monte Carlo computational algorithm and runs over
20.000 scenarios to create a reliable outcome.
It’s interesting to see in how many ideas you need to invest. That you should not put all your money in one investent may be common sense, but even 10 or 20 ideas are not enough. Portfolio management
really is a long-tail game.
Since one third of the ideas need to make up for the two third that “fail”, having more ideas makes it more likely to make money. When you invest € 500.000 in 10 ideas each, only 1 will return up to
€ 5.000.0000. You still have a 1 in 250 chance that that one startup returns more than € 25.000.000, but that chance is really slim. The simulation shows that there is a 35% chance on a € 3 million
profit, but at the same time also a 25% chance you will lose € 2,5 million.
But when you invest € 50.000 in 100 startups (same € 5 million budget), 7 of them will return up to € 500.000, two of those 10 up to € 1 million and if you are really lucky 1 more than € 10 million.
In a best case scenario, the one third will get you € 15.500.000. That is enough to compensate for the € 3.300.000 (66x € 50.000) that you lost on the other two third of your portfolio and will give
you a nice € 10 million profit on a € 5 million investment. Not bad! But that is the most lucky scenario. Running the simulation however shows a more likely profit of around € 7 million.
[convertkit form=2570335]
Why don’t you try the ROI Calculator yourself and see how well your portfolio will perform! It makes sense to run the calculator multiple times to see the effects of the Monte Carlo computational
algorithm. You can clearly see the difference between making an initial investment of € 500.000 in 10 startups, or € 50.000 in a 100 startups. And that is without any stage-gated investments in
place! The real money is in the double down on the investments that work, but that is a topic for next time.
Timan Rebel
Timan Rebel has over 20 years of experience as a startup founder and helps both independent and corporate startups find product/market fit. He has coached over 250+ startups in the past 12 years and
is an expert in Lean Innovation and experiment design.
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Get quit of Data Types As a Ingredients of Algebraic Equations - Setonix
In computer system science, an cast off data type essentially is actually a model meant for arbitrary info types, with each data type having its own definition of what it is. By definition, a great
abstract info type can be any data that does not include a rendering browse around this site by means of an actual value or surgery that can be performed on that data. By comparison, a concrete data
type has an representation in the form of several concrete worth or operation.
For example , when we say that the definition of an chuck data type includes an axiom, therefore each time you use such a sort in computations, you happen to be assuming a presupposition – in this
case, there are no functions that cannot be performed on that data, and thus zero possible positive aspects. This is diverse from the traditional model in which every operation and every conceivable
outcome is completely predicated about knowledge of the operations and possible consequences beforehand. The traditional version is called the mathematical style, because in the mathematical style,
each presumption is made regarding other presumptions. In the summary model, every single assumption can be made on its own. Thus, at the time you calculate the square reason behind two amounts, or
as you solve for x, you already know the answer in case you have made a great assumption – a prior possibility – regarding the value of x just before you possibly attempt to compute it.
One way to think about an abstract data type as opposed to a concrete floor one is via the language of algebraic equations. If we start with the definition in the abstract info type presented
earlier, therefore we have a geometric concept: the set of most possible alternatives for a provided problem. Whenever we plug this set into a great algebraic formula, the solution aid polynomial
quantity – that is, it will probably be a prime number. Therefore , the meaning of an algebraic equation affecting an fuzy data type can also be written as a ingredients of the subsequent axiom:
Create solution is a valid formula.
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Understanding and Mitigating the Tradeoff between Robustness and Accuracy
Understanding and Mitigating the Tradeoff between Robustness and Accuracy
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:7909-7919, 2020.
Adversarial training augments the training set with perturbations to improve the robust error (over worst-case perturbations), but it often leads to an increase in the standard error (on unperturbed
test inputs). Previous explanations for this tradeoff rely on the assumption that no predictor in the hypothesis class has low standard and robust error. In this work, we precisely characterize the
effect of augmentation on the standard error in linear regression when the optimal linear predictor has zero standard and robust error. In particular, we show that the standard error could increase
even when the augmented perturbations have noiseless observations from the optimal linear predictor. We then prove that the recently proposed robust self-training (RST) estimator improves robust
error without sacrificing standard error for noiseless linear regression. Empirically, for neural networks, we find that RST with different adversarial training methods improves both standard and
robust error for random and adversarial rotations and adversarial l_infty perturbations in CIFAR-10.
Cite this Paper
Related Material
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What’s a watt? - The Handy Physics Answer Book
Momentum and Energy
Suppose you climb the stairs to the second floor. Whether you run or walk, because you have gone up the same distance the increase in the gravitational field energy will be the same. The difference
is the rate at which the energy has changed. The rate, the change in energy divided by the time taken is called power. Power is measured in the unit called the watt. One watt (W) is one joule (J) per
second (s). A kilowatt is 1,000 watts or 1,000 joules per second.
Automobiles can accelerate from 0 to 60 miles per hour, but the more powerful ones can do it in six seconds or less while ones with less powerful engines may take more than 10 seconds.
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The Basic Mechanics of Principal Components Analysis
The following description gives an explanation of how principal components analysis can be computed. The actual algorithm described below is not used in any standard program, but the commonly used
algorithms can only be explained using mathematical concepts from linear algebra.
Computing the first component
As discussed on the main Principal Components Analysis page, PCA analyzes a Correlation Matrix and infers components that are consistent with the observed correlations.
Each component is created as a weighted sum of the existing variables. PCA starts by trying to find the single component which best explains the observed correlations ^between the variables.
Consider the following three variables:
│v1 │v2 │v3 │
│1 │1 │1 │
│2 │3 │5 │
│3 │2 │2 │
│4 │5 │3 │
│5 │4 │4 │
The correlation matrix of the three variables is:
│ │v1 │v2 │v3 │
│v1│1.0 │.8 │.4 │
│v2│.8 │1.0 │.6 │
│v3│.4 │.6 │1.0 │
Note that there are moderate-to-strong correlations between all of the variables. Thus, any underlying component must be correlated with all the variables. A first guess then is that our new
component could simply be the sum of each of the existing variables:
\(Component = 1.0 \times v1 + 1.0 \times v2 + 1.0 \times v3\)
The resulting component matrix, which shows the correlation between each of the variables and the computed component, is then:
│ │Component │
│v1│.856 │
│v2│.934 │
│v3│.778 │
These correlations are all very high and thus our estimated component is a pretty good component. However, it can be improved. Looking again at the correlation matrix, reproduced below again, we can
deduce that our original guess of giving equal weights to the different components was a touch naïve. Note that v2 has the highest average correlation with all the variables. Thus, if we were instead
to give a higher weight to v2 when estimating our component we will likely end up with marginally higher correlations with all the variables. Similarly, note that v3 has the lowest average
correlation, and thus by the same argument it should be given a lower weight.
│ │v1 │v2 │v3 │
│v1│1.0 │.8 │.4 │
│v2│.8 │1.0 │.6 │
│v3│.4 │.6 │1.0 │
Using trial and error, we can deduce that the optimal formula^ for computing the component is:
\(Component = 1.0 \times v1 + 1.086 \times v2 + 0.866 \times v3\)
Note that we have not multiplied v1 by anything other than 1. This is because the numbers that are multiplied by the other variables are relative to v1 having a weight of 1. If we were to put a
weight other than 1 next to v1 we would then have to multiply each of these other weights by this number. For example, the following weights are the ones generated by SPSS (and shown in the Component
Score Coefficient Matrix) and you can see that their relativities are the same:
\(Components = 1.0 \times v1 + 1.086 \times v2 + 0.866 \times v3\)
Computing the remaining components
The next component is computed as follows:
1. Regression is used to predict each variable based on its component.
2. The residuals of the regression model are then computed.
3. The correlation matrix is computed using the residuals.
4. The same basic process as described above is performed to create a second component.
5. These steps are then repeated until the number of components is equal to the number of variables.^
Typically, Varimax Rotation is performed to aid interpretation.
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Radius of Circle - Formula, Definition
A circle is a fundamental geometric shape which has many real-life utilizations in several fields, such as architecture, engineering, physics, and mathematics. The radius of a circle is one of its
most essential characteristics and performs an important role in figuring out other dimensions, for instance, the area and circumference of the circle.
In this log, we will look into the theory of the radius of a circle in depth, involving its formula, definition, and how it is applied in several fields. We will further talk about the importance of
understanding the radius of a circle in solving mathematical and physical problems.
By the end of this blog article, you will possess a clear grasp of what the radius of a circle is, how to calculate it, and its significance in real-life applications. Even if you are a student
studying geometry or a working professional in affiliated to domain, comprehending the radius of a circle is important to be successful among other things.
Definition of Radius
The radius of a circle is the distance from the middle of the circle to any point on its border. It is denoted by the alphabet "r" and is a crucial measurement utilized to figure out the size and
position of the circle. The radius is forever half of the diameter of the circle, which is the distance through the circle passing through its center.
Formula for Calculating the Radius
The formula for working out the radius of a circle is easy and streamline. It is provided as:
r = d / 2
where "r" is the radius of the circle and "d" is its diameter. This formula is originated out of the definition of the radius as half of the diameter.
One more method to figure out the radius of a circle is by using the formula:
r = √(A/π)
where "A" is the area of the circle and "π" is the mathematical constant pi (approximately equal to 3.14). This formula could be beneficial where the area of the circle is given, but its diameter is
Examples of Figuring out the Radius
Let's observe come instances of how to apply the formula for determining the radius of a circle:
Example 1:
A circle has a diameter of 10 cm. What is its radius?
Utilizing the formula, we get:
r = d / 2
r = 10 / 2
r = 5 cm
Hence, the radius of the circle is 5 cm.
Example 2:
A circle has an area of 78.5 square centimeters. What is its radius?
Using the formula, we possess:
r = √(A/π)
r = √(78.5/π)
r ≈ 5 cm
Hence, the radius of the circle is about 5 cm.
Significance of the Radius of a Circle
The radius of a circle is an essential measurement which is utilized in a broad array of domains, involving physics, engineering, geometry, and architecture. In geometry, the radius is crucial for
calculating the circumference and area of a circle. The circumference is the distance around the edge of a circle, while the area is the volume of space confined by the circle. Both of these
calculations need the radius to be known.
In physics and engineering, the radius is used to determine the size and position of circular objects, for instance, wheels, gears, and cylinders. It is also used in lenses to figure out the focal
length of a curved mirror or lens. In architecture, the radius is utilized to design and build circular buildings and structures, for example, domes, arches, and rotundas.
The radius of a circle is also crucial in computer graphics and animation, where it is [[used|129] to make 3D and 2D shapes. It is further applied in machine learning algorithms for image recognition
and feature detection.
Common Errors in Determining the Radius
When working out the radius of a circle, it is important to prevent common mistakes that can pave way to wrong calculations. One general error is confusing the radius with the diameter. While the
diameter is the length across the circle passing through its center, the radius is the distance from the center to any point on its edge. Therefore, it is crucial to ensure that the correct
measurement is used when figuring out the radius.
One more mistake that is frequently made while figuring out the radius is forgetting to divide the diameter by two while using the formula. The formula for figuring out the radius of a circle is r =
d/2, while r is the radius and d is the diameter. Forgetting to divide the diameter by two could end up in a wrong value for the radius.
It is also crucial to utilize the correct units of measurement while finding the radius. For instance, if the diameter is calculated in inches, the radius must also be calculated in inches. Utilizing
various units of measurement for the diameter and radius could result in incorrect workings.
By preventing these ordinary errors and revise calculations, individuals can ensure that they obtain precise values for the radius of a circle. This is essential in many domains, for example, math,
engineering, physics, and architecture, where precise measurements are important for accurate calculations and plans.
The radius of a circle is a fundamental calculation applied in several domains, consisting of math, physics, engineering, and architecture. It is defined as the length from the center of the circle
to any point on its border and can be worked out utilizing straightforward formulas. Comprehending the definition and formula for working out the radius of a circle is crucial to be successful in
these domains.
By avoiding frequent mistakes and grasping the significance of the radius of a circle, people can improve their comprehending of geometry and its uses in real-life situations. If you need help
getting a grasp the radius of a circle or any other math idea, think about reaching out to Grade Potential Tutoring. Our experienced tutors are accessible remotely or face-to-face to offer customized
and productive tutoring services to support you be successful. Call us today to plan a tutoring session and take your math skills to the next level.
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Why is my formula giving me incorrect average when trying to average the children rows?
I have a formula that is averaging the percentage of the children row of the column Percent completed, to give me another percentage of the ones that are less than 100% and then I have another column
that will average the percentage of the ones that are greater than 100%. But it is not giving me a correct percentage. The screen shot should average 75% in the Goal Not Met column but it's returning
as 59%. Is there something I need to change in my formula?
Best Answer
• The Child rows of the dark blue row, Item 99511, are only the rows with the PROD number in yellow:
So the average of 55% and 63% = 59%.
Try using DESCENDANTS instead of CHILDREN in your formula and see if that considers all the rows under 100%.
Or, if you want all the rows to be direct children of 99511, then outdent the rows in the green boxes.
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• [S:Perhaps you could share the formula you're using and the data structure you're using it on? :S]
Thanks, that's easier to troubleshoot, LOL!
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• @Jeff Reisman Yeah I accidently posted the discussion without inputting anything. I have edited the original one.
• The Child rows of the dark blue row, Item 99511, are only the rows with the PROD number in yellow:
So the average of 55% and 63% = 59%.
Try using DESCENDANTS instead of CHILDREN in your formula and see if that considers all the rows under 100%.
Or, if you want all the rows to be direct children of 99511, then outdent the rows in the green boxes.
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• @Jeff Reisman Thank you! Changing it to DESCENDANTS work. Such an easy fix.
Help Article Resources
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Python Module (II) - Random Module and Its Common Methods
Python Module (1) - Path Module and Its Common Methods.
Random module
• It is necessary to import before use
>>> import random
#generate integer random numbers random.randrange(stop),random.randrange(start, stop[, step])
>>> random.randrange(5)
>>> random.randrange(1,100)
>>> random.randrange(1,100,5)
#generate random integers random.randint(a, b ）return a random integer N satisfy a <= N <= b , equivalent to randrange(a, b+1)
>>> random.randint(1,2)
# random.choice(seq ）from non empty sequences seq returns a random element if seq if it is empty, trigger IndexError
>>> random.choice([1,2,3])
>>> random.choice(("b","a"))
# random.choices(population, weights=None, *, cum_weights=None, k= 1) from population select a replacement in the middle and return a size of k list of elements for. if population if it is empty, trigger IndexError . if specified weight the sequence is selected based on its relative weight. alternatively, if given cum_weights sequence, then based on cumulative weights (possibly using itertools.accumulate () calculate) for selection. for example, relative weight ``[10, 5, 30, 5]`` equivalent to cumulative weight ``[10, 15, 45, 50]`` . internally, relative weights are converted to cumulative weights before selection, so providing cumulative weights can save workload. if neither is specified weight also not specified cum_weights then choose with equal probability. if a weight sequence is provided, it must match population the length of the sequence is the same. a TypeError designated weights and cum_weights 。weights or cum_weights can be used random () the returned energy can match float any number type that performs mutual operations on values (including integers, floating-point numbers, fractions but not including decimal ）. the weight value should be non negative and a finite numerical value. if all weight values are zero, it will trigger ValueError . for a given seed, with equal weighting choices () functions usually generate and repeat calls choice () different sequences. choices the algorithm used in () uses floating-point operations to achieve internal consistency and speed. choice the algorithm used by () defaults to repeating integer operations to avoid small deviations caused by rounding errors.
>>> random.choices([1,2,3,4,"a","b","c"]) #select one by default
>>> random.choices([1,2,3,4,"a","b","c"],k = 5) # k = 5 represents selecting 5
[2, 4, 'b', 2, 2]
#the number of weights in the back must be the same as the number of elements in the front, and the larger the weight, the greater the probability of selection, [1,2,3,4,"a","b","c"] there are 7. [1,1,1,1,1,9,5] 7 required
>>> random.choices([1,2,3,4,"a","b","c"],weights = [1,1,1,1,1,9,5],k = 5)
['b', 'b', 'b', 'b', 4]
# random.random () return [0.0, 1. the next random floating-point number within the range of 0
>>> random.random()
# random.uniform(a, b ）return a random floating-point number N when a <= b time a <= N <= b when b < a time b <= N <= a
>>> random.uniform(1,5)
>>> random.uniform(1,1)
>>> random.uniform(4,1)
# random.shuffle(x[, random] ）sequence x randomly shuffle positions, optional parameters random it is a function with 0 parameters, in the [0.0, 1. return a random floating-point number in 0) ; by default, this is a function random()
>>> x = [1,2,'a','b']
>>> random.shuffle(x)
>>> x[0]
>>> x[1]
>>> x[2]
>>> x[3]
# random.randbytes(n ）functions for byte data, generating n random bytes
# random.sample(population, k, *, counts=None ）returns the unique element selected from a population sequence or set k list of lengths. used for non repetitive random sampling. returns a new list containing elements from the population, while keeping the original population unchanged. the result list is arranged in the order of selection, so all sub slices will also be valid random samples. this allows the lottery winners (samples) to be divided into the grand prize and the second place winner (sub slices). the overall members do not need to be hashable or unique . if the population contains duplicates, each occurrence is a possible choice in the sample. repeated elements can be listed directly one by one, or optional keyword only formal parameters can be used counts to specify. for example, sample(['red', 'blue'], counts=[4, 2], k= 5) equivalent to sample(['red', 'red', 'red', 'red', 'blue', 'blue'], k= 5) . to select samples from a series of integers, use range () object as a parameter. this method is particularly fast and space saving for sampling from a large population :sample(range(10000000), k= 60). if the sample size is greater than the population size, trigger ValueError 。
>>> random.sample([1,2,3,5,4,"a","b","c"],k= 1)
>>> random.sample([1,2,3,5,4,"a","b","c"],k= 2)
[2, 1]
>>> random.sample([1,2,3,5,4,"a","b","c"],k= 7)
['c', 3, 1, 4, 2, 'a', 'b']
#initialize the random number generator, if a omitted or omitted None then use the current system time. if the operating system provides random sources, use them instead of system time, if a yes int type, then use it directly. for version 2 (default), str 、 bytes or bytearray object conversion to int and use all its bits. for version 1 (used to recover from old versions) Python reproducing random sequences, used for str and bytes the algorithm generates a narrower seed range.
random.seed(a=None, version=2)
#returns the object that captures the current internal state of the generator. this object can be passed to setstate () to restore the state.
# state it should be called from before getstate () obtained, and setstate () restore the internal state of the generator to getstate the state when () is called.
#return with k non negative random bits Python an integer. this method follows MersenneTwister provided together with the generator, other generators may also use it as API optional parts are provided. where possible, getrandbits () will be enabled randrange () to handle any large interval.
#return a random floating-point number N , making low <= N <= high and use specified boundaries between these boundaries mode 。 low and high the boundary defaults to zero and one. mode the parameter defaults to the midpoint between the boundaries, giving a symmetric distribution.
random.triangular(low, high, mode)
# Beta distribution. the conditions for the parameter are alpha > 0 and beta > 0 the range of return values is between 0 and 1.
random.betavariate(alpha, beta)
#exponential distribution. lambd yes, 1 . divide 0 by the desired average, it should be non-zero. (this parameter should have been named as"lambda"but this is Python reserved words in if lambd if it is positive, the range of the returned value is 0 to positive infinity ; if lambd if it is negative, the return value ranges from negative infinity to 0.
# Gamma distribution. (no gamma function ! ）the conditions for the parameter are alpha > 0 and beta > 0。
random.gammavariate(alpha, beta)
the probability distribution function is :
x ** (alpha - 1) * math.exp(-x / beta)
pdf(x) = --------------------------------------
math.gamma(alpha) * beta ** alpha
#normal distribution, also known as gaussian distribution. mu is the average value, while sigma is the standard deviation. this function is slightly faster than the one defined below normalvariate () function.
random.gauss(mu, sigma)
#multithreading considerations : when two threads call this method simultaneously, they may receive the same return value. there are three ways to avoid this. 1) let each thread use a different instance of a random number generator. 2) lock outside all calls. 3) switch to a slower but thread safe option normalvariate () function.
#lognormal distribution. if you use the natural logarithm of this distribution, you will get a normal distribution with an average of mu and the standard deviation is sigma 。 mu it can be any value, sigma must be greater than zero.
random.lognormvariate(mu, sigma)
#normal distribution. mu it's the average value, sigma it's the standard deviation.
random.normalvariate(mu, sigma)
#feng · mises distribution. mu it is the average angle, expressed in radians, between 0 and 2* pi between, kappa it is a concentration parameter that must be greater than or equal to zero. if kappa if it is equal to zero, then the distribution is between 0 and 2* pi reduce to a uniform random angle within the range of.
random.vonmisesvariate(mu, kappa)
#pareto distribution. alpha it is a shape parameter.
#weibull distribution. alpha it is a proportional parameter, beta it is a shape parameter.
random.weibullvariate(alpha, beta)
Python Module (III) - OS Module and Its Common Methods.
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Computational Fluid Dynamics and the BLOODHOUND
by Ben Evans, CFD Engineer & Aerodynamicist – BLOODHOUND aerodynamics
Most engineering fluid flow problems are now solved, at least in part, by using computational fluid dynamics (CFD). What this means is that we can use computers to help us solve the equations that
govern fluid dynamics, rather than having to do them by hand.
Which equations matter?
The governing equations for the majority of practical fluid flow problems are partial differential equations (PDEs) – in fact, that's the case for many naturally occurring phenomena. If you studied
(or are studying) mathematics at A-level, you will probably have come across some simple differential equations and solved them analytically (using pen and paper).
The set of equations that are most relevant for describing the aerodynamic flows around BLOODHOUND are the so-called ‘Navier-Stokes’ equations. These are a set of five PDEs describing quantities such
as the density, velocity and pressure of the airflow. (There's also a sixth equation if you want to model turbulence in the flowfield).
To get some idea of the degree of complexity of this set of equations, have a look at figure 1.
Figure 1: The Navier-Stokes equations for viscous, compressible fluid flow
There is no hope of solving such a complex and coupled set of equations such as this by hand – even large supercomputers have to work hard to obtain solutions!
The challenge in CFD
The development of CFD techniques has closely followed the development of numerical methods for solving partial differential equations. Numerical methods have been known since the time of Newton in
the 1700s, but without the aid of the computer it was impossible to fully exploit these techniques.
Modern CFD has its roots in the 1950s with the advent of the digital computer. At the heart of all CFD numerical schemes is the fundamental question of how to represent a continuous function at a
finite set of points (we call this 'discretisation'). In other words, how can we store a function defined for an infinite number of points (i.e. every possible position in space and time) in a finite
way, and as accurately as possible?
Figure 2: Example of the discretisation / approximation of a pressure function
Here's an example. In figure 2 a pressure function has been 'discretised' so that the value of the function at a finite number of ‘nodes’ is stored and a linear interpolation of the solution is
assumed between the nodes. The jump between each node is referred to as an 'element' and hence this kind of solution is often referred to as ‘the finite element method’.
3D solutions
The most popular method of achieving a three-dimensional finite element solution is to discretise the solution domain into a finite number of small cells or elements forming a mesh or grid, and to
then apply a suitable algorithm to values stored at the intersections of the mesh (the nodes) to solve the governing equations – in our case, the Navier-Stokes equations.
The computational mesh can consist of elements of a whole variety of 3D shapes. For the CFD work carried out on BLOODHOUND, a mesh of hexahedra, prisms and tetrahedra numbering into the tens of
millions was used!
CFD and BLOODHOUND
The procedure for performing the CFD analysis on BLOODHOUND can best be described in the following stages:
1. The definition of the vehicle surface to be examined was provided to the team at Swansea University as a CAD (computer-aided-design) output from the design engineers based in the design office in
Bristol. This geometry definition was then analysed via the FLITE3D computer system developed at the College of Engineering at Swansea University.
2. The CAD output was processed and a mesh generation computer program used to construct the computational mesh.
3. A ‘pre-processing’ computer program was then used to format the mesh in such a way that the Swansea University supercomputing cluster could be used to run the solver program. Over the course of
the BLOODHOUND project we have also had computational support in the form of access to large supercomputers from Intel and HPC Wales.
4. The ‘equation solver’ program containing the Navier-Stokes solution algorithm was run on the supercomputing cluster.
5. A ‘post-processing’ software package was used to convert the solutions coming from the solver into meaningful flow visualisation plots and force distributions.
A selection of outputs from final visualisation stage is shown in figures 3 and 4. These visualisations helped the design team understand the behaviour of the car’s aerodynamics in terms of flow
phenomena such as shock waves, boundary layers (the thin layer of slow-moving air close to the Car's surface) and pressure distributions. These pictures and force distributions were analysed and
changes made to the design, and we then repeated steps 1 to 5 to check the outcomes of those changes.
Figure 3: Streamribbons and pressure contours over an intermediate BLOODHOUND configuration
Figure 4: Visualisation of a vortex being shed at the rear of BLOODHOUND impinging on the rear wheel struts
Where else can we use CFD?
The applications for the computational modelling technologies being developed at Swansea are wide ranging, including: medicine; lightning strike modelling; building structural analysis; and even
interstellar plasma flows. Essentially, any phenomena governed by partial differential equations can be simulated using the computational modelling approach of the finite element method.
In figure 6 you can see how biomedical modelling of stresses in a human femur is a phenomenon that can be described by a set of partial differential equations that can be modelled using the same
techniques of finite elements that we used to study the aerodynamics of BLOODHOUND SSC.
Figures 5 and 6: Fluid-structure interaction modelling of a typical jet airliner; biomedical modelling of stresses in a human femur
Swansea University is proud to be part of the BLOODHOUND team. Our work on computer fluid dynamics (CFD) is at the heart of research on the design and aerodynamics of the 1,000mph Car.
Swansea University is a world-class research-led dual campus university. Its main base, Singleton Park Campus, is located on a beautiful 47 acre parkland estate and its £450 million Bay Campus, which
opened to students in September 2015, is based on the beach on the eastern side of Swansea city centre.
The University was established in 1920 and currently offers around 350 undergraduate courses and 100 post-graduate courses to over 19,000 undergraduate and postgraduate students.
Swansea University continues to maintain its position as one of the top universities in the UK for engineering. The College of Engineering is now based at Swansea University’s Bay Campus , which has
four buildings dedicated to engineering, holding 30,000m^2 of laboratory and office space and over £10 million of new research and teaching equipment.
The College of Engineering is ranked within the Top 10^* in the UK and is internationally recognised for its cutting-edge research with 94% of research produced by academic staff rated World-Leading
or Internationally Excellent quality^** .
The College has strong and establish links with a large variety of local, national and international companies both with its teaching and research. It has significantly contributed to a number of
prestigious projects from the aerodynamics design of the THRUST Supersonic Car, which currently holds the World Land Speed Record, to BLOODHOUND.
The computer fluid dynamics (CFD) research for BLOODHOUND is being undertaken at the Zienkiewicz Centre for Computational Engineering (ZCCE) one of four research centres with the College.
To find out more about the College of Engineering please go to our website www.swansea.ac.uk/engineering or follow us on Facebook, Twitter or Instagram .
^* The Times & Sunday Times University Guide 2016
^** Research Excellent Framework 2014
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Analysing data series in Ledidi Core
A series is not like a regular variable. It can be considered a “dataset” within your main dataset with repeating observations of one or more variables. Series should be used in case of repeated
measurements. This implies that for optimal use of series, planning a project and building the dataset should be thought through before data collection starts. A series can contain just one single
variable (e.g. weight measurements) or multiple variables (e.g. a large panel of blood samples).
Figure 1: Visualisation of series dataset layout
Main-level or series-level variables?
When is it best to gather variables in a series instead of having them on the main level? When a project contains repeated measurements, the use of series is recommended. Some examples include:
1. Variables examined at different points in time (e.g. weight at different study visits, Quality of Life measured at baseline and subsequent visits)
2. Variables with large between-patient variations in the number of answers (e.g. details of medication use or previous surgeries)
Like main-level variables, series-level variables can be gathered in groups to maintain oversight and structure in the project. Additionally, unique forms can be created based on series variables.
There are several advantages of using series:
1. Several data points are stored in one entry (i.e. in one row of your dataset), keeping your data nice and tidy.
2. The number of variables used is reduced substantially. This is helpful, especially in larger projects, to maintain oversight and structure your project.
3. Improved dataset setup and workflow while inputting data.
Analysis of series data
Analyses in Ledidi can be performed on both main-level and series-level data.
Main-level analysis
Variables within a series can be aggregated by using aggregation rules, and the aggregated values can then be used for analysis on the main level. Examples of aggregation rules include: the sum of a
numerical variable across series entries; the latest value of the series variable registered; or the average value of a series variable. The aggregated variables will be shown in the dataset window
and are available for statistical analysis and graphical presentation on the main level.
Figure 2: Setup of aggregation rules in the Variables window (left) and view of these rules in the Dataset window (right).
In this scenario, not all data points in the series are used; but rather an aggregated variable generated from series data. This limits the nature of the analysis that can be undertaken on
series-level data at the main level.
Series-level analysis
If the goal is to analyse all data points in a series, analyses on the series level are required. Analysis at this level facilitates a more comprehensive examination of series-level data.
For example, if one takes blood pressure measurements at each GP visit, the average values across all patients at each visit can be plotted using time course analysis. One can choose between the mean
or the median value, and add 95% confidence intervals, standard deviations, and range or IQR in the case of median values. One can also stratify by other variables to ascertain whether there are
different patterns by sex, age, site, etc. as shown in Figure 3.
Figure 3: Mean systolic blood pressure with standard deviation of all patients, per GP visit, stratified by sex.
Additionally, the individual patient blood pressure measurements over time can be plotted as shown in Figure 4, by choosing the patient ID variable as the Group function. (Hint: To obtain this graph,
the datatype of the ID variable should be “Unique” or “integer”.)
Figure 4: The systolic blood pressure of each individual patient per GP visit.
Another application of series analysis is examining the frequency of a categorical value in a data series among all entries. For example, this would be relevant in examining the distribution of
previous surgery types among patients in a breast cancer trial. Here, prior surgical procedures are entered in a series, given that a patient can have multiple prior surgeries. Using the
“Frequencies” analysis on the series level provides insight into how often each surgery type has been performed.
Figure 5: Frequency of previously performed surgeries among all patients in a breast cancer trial. Because the variable “Previous surgery” is a data series, each patient may have had multiple
previous procedures.
When a project contains repeated measurements, series are highly recommended for optimal data gathering and analysis. Series-level data can be analysed on two different levels:
1. On the main level by using an aggregation rule thereby making the sum/mean/latest value/... of series-level data available for main-level analysis.
2. On the series level by applying Ledidi Core’s analyses on series-level data thereby analysing each individual data point within a certain series.
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Categories, Logic and Physics
Categories, Logic and Foundations of Physics
This is the homepage of the workshop series Categories, Logic and Foundations of Physics. These workshops bring together researchers from the fields mentioned in the title and promote research on
structural and conceptual aspects of fundamental physical theories, operational methodologies for general physical theories, as well as the general study of mathematical structures describing
dynamics and space-time.
Please feel free to contact the workshop organizers if you have any questions, or if you want to attend one of our meetings.
There have been five workshops so far (9th January 2008, 14th May 2008, and 23rd/24th August 2008, 7th January 2009 and 6th August 2009). We intend to keep an informal atmosphere for the workshops,
with no formal registration, and strongly encourage interaction and discussions between the participants.
Previous event: 7th CLP Workshop, 21st September 2010, University of Birmingham
The 7th Workshop on "Categories, Logic and Physics" will be held on Tuesday, September 21st, 2010 at the University of Birmingham. As always this is a one day event, there are no conference fees. A
particular focus of this meeting will be topological aspects, but, as usual, other topics are welcome too.
Note the changed venue — CLP is spreading out ;-)
• Tuesday 21st of September 2010
• The University of Birmingham, School of Computer Science
• Room SPX-LT1 — This Lecture Theater 1 is in the ‘Sports and Exercise Science’ building Y14, about 5min walk from Computer Science (building Y9), see here.
If you plan to attend the workshop, please send an email as soon as possible to the local organizers:
• <ku.ca.mahb.sc|srekciV.J.S#ku.ca.mahb.sc|srekciV.J.S> and/or
• <ku.ca.mahb.sc|resuaF.B#ku.ca.mahb.sc|resuaF.B>
so that we can make local arrangements. This is important for arranging for smooth lunch, so letting us know really helps you.
Birmingham is easily reached by train or plane (Birmingham International Airport). The train transfer from the airport to the University is approximately 45 minutes.
For train travel, you should ask for the station "University" when purchasing tickets. Most routes include a change at Birmingham New Street to the line with destination Longridge or Redditch.
The workshop is intended to be a one day event and most participants will not need to stay overnight. If you need help with an accommodation feel free to contact the local organizers.
SCHEDULE: The program can be downloaded here.
• 10:30-11:00 Welcome Coffee/Tea
• 11:00-12:00 Martín Escardó, "Maybe locales are made out of points after all"
• 12:00-13:00 Christopher J. Mulvey, "Constructive Aspects of Gelfand Duality"
• 13:00-14:00 Lunch Break [we can go to Staff House, lunch is about 5GBP]
• 14:00-15:00 Ronnie Brown, "What is and what should be ‘higher dimensional group theory’?"
• 15:00-16:00 Catherine Meusburger, "Higher categories and observables for generalised Turaev-Viro models"
• 16:00-16:30 Coffee/Tea Break
• 16:30-17:30 Simon Willerton, "Two 2-traces" (tentative)
• 17:30-18:30 Cecilia Flori, "Topos Formulation of History Quantum Theory"
• 19:00-… Pub Session
TITLES AND ABSTRACTS: (in alphabetical order)
Ronnie Brown, Bangor: What is and what should be ‘higher dimensional group theory’?
The presentation will show, including some knot demos, some of the problems and intuitions which have led to this question, and how certain cubical algebraic structures with partial operations whose
domains are given by geometric conditions have been found quite natural for expressing modes of higher dimensional subdivision and composition which are related to long term concerns in algebraic
Martín Escardó, Birmingham: Maybe locales are made out of points after all.
Like topology in analysis, locale theory is about open sets, continuous functions, compact spaces, approximation and limit processes, and things like that. Both topology and locale theory start with
opens. In topology, an open is made out of points, but in locale theory, a point is made out of opens. The localic view makes physical and computational sense: points are infinitely small (and carry
an infinite amount of information), and hence are not directly observable, but each point is uniquely characterized by its (infinite) collection of observable properties. The opens are the
observables, and locale theory takes the notion of observation as primitive,
and all other notions, including that of point, as derived. (Moreover, some perfectly good spaces in locale theory have a rich supply of opens without allowing any point at all, but this is not what
I will emphasize in my talk).
Although the match of (physical or computational) reality with locale theory is arguably better than with topology, locale theory may be more mathematically demanding, or at least is certainly
unfamiliar to most of us. In this talk I'll discuss how one can think of locales as if they were made out of points, like the spaces of classical analysis and geometry, trying to make them more
familiar, manageable, and intuitive, without loss of rigour, so that we can reason and work with them efficiently.
Cecilia Flori, Perimeter: Topos Formulation of History Quantum Theory
In this talk I will describe a topos formulation of consistent histories obtained using the topos reformulation of standard quantum mechanics put forward by Döring and Isham. Such a reformulation
leads to a novel type of logic with which to represent propositions. In the first part of the talk I will introduce the topos reformulation of quantum mechanics. I will then explain how such a
reformulation can be extended so as to include temporally-ordered collections of propositions as opposed to single time propositions. Finally I will show how such an extension will lead to the
possibility of assigning truth values to temporal propositions.
Catherine Meusburger, Hamburg: Higher categories and observables for generalised Turaev-Viro models
Generalised Turaev-Viro models that are formulated in terms of spherical categories play an important role in three-dimensional quantum
gravity, where they are interpreted as discrete path integrals or state sum models of quantised three-manifolds. We discuss the role and interpretation of these models in quantum gravity and comment
on the problem of defining observables for these models. We show how this problem can be addressed by using higher categories and discuss the mathematical properties and the physical interpretation
of the resulting observables. The talk is based on joint work with John W. Barrett.
Christopher J. Mulvey, University of Sussex: Constructive Aspects of Gelfand Duality
One of the important foundational aspects of recent approaches to developing quantum theories of space and time has been the existence of a constructive theory of Gelfand duality for commutative
C*-algebras. In this talk, we shall outline the way in which this theory was developed, examine its application to the context of quantum physics, and consider its extension to the non-commutative
Simon Willerton, Sheffield: Two 2-traces (tentative)
Over recent years, in several areas of mathematics the notion of 'categorified trace' or '2-trace' has arisen. For instance, in higher
representation theory where groups act on linear categories there is the notion of a '2-character'; in Khovanov knot homology the Hochschild homology is viewed as a categorical trace. It transpires
that there are actually two orthogonal, and sometimes dual, notions of 2-trace in common usage and I will explain how they arise and give various examples from various areas of mathematics.
We are glad to see you in September,
Local organizers
Steve Vickers + Bertfried Fauser
Workshop coordinators
Bob Coecke + Andreas Döring
Previous event: 6th CLP Workshop, 9th March 2010, Oxford University Computing Laboratory
The sixth workshop on "Categories, Logic and Foundations of Physics" will take place at
Oxford Comlab on Tuesday, 9th March 2010, 12:00—18:20, Wolfson Building, Parks Road, Oxford OX1 3QD
The location of the Comlab and visitor information can be found here.
The first talk will take place in Room 478.
After lunch, we will change to Lecture Theatre A.
• 14:00—14.50 Urs Schreiber (Utrecht), "Gauge fields in an (oo,1)-topos" — The familiar theory of smooth Spin(n)-principal bundles with connnection has a motivation from physics: for the quantum
mechanics of a spinning point particle to make sense, the space it propagates in has to have a Spin-structure. Then the dynamics of the particle is encoded in a smooth differential refinement of
the corresponding topological Spin(n)-principal bundle to a smooth bundle with connection. It has been known since work by Killingback and Witten that when this is generalized to the quantum
mechanics of a spinning 1-dimensional object, the Spin-structure of the space has to lift to a String-structure, where the String-group is the universal 3-connected cover of the Spin group.
Contrary to the Spin-group, the String-group cannot be refined to a (finite dimensional) Lie group. Therefore the question arises what a smooth differential refinement of a String-principal
bundle would be, that encodes the dynamics of these 1-dimensional objects. It turns out that this has a nice answer not in ordinary smooth differential geometry, but in "higher" or "derived"
differential geometry: String(n) naturally has the structure of a smooth 2-group — a differentiable group-stack. This allows to refine a topological String-principal bundle to a generalization of
a differentiable nonabelian gerbe: a smooth principal 2-bundle. In the talk I want to indicate how the theory of smooth principal bundles with connection finds a natural generalization in such
higher differential geometry, and in particular provides a good notion of connections on smooth String-principal bundles.
• 14:50—15.40 Pawel Blasiak (Krakow), "Graph Model of the Heisenberg-Weyl algebra" — The Heisenberg-Weyl algebra, underlying most physical realizations of Quantum Theory, is considered from a
combinatorial point of view. We construct a concrete model of the algebra in terms of graphs which endowed with intuitive concepts of composition and decomposition provide a rich bi-algebra
structure. It will be shown how this encompass the Heisenberg-Weyl algebra, thereby providing a straightforward interpretation of the latter as a shadow of natural constructions on graphs. In
this way, by focusing on algebraic structure of Quantum Theory we intend to draw attention to genuine combinatorial underpinning of its formalism. We will also discuss some combinatorial methods
suitable for this graphical calculus.
• 15.40—16:10 Panel discussion: Why $n$-categories?
• 16:40—17:30 Boris Zilber (Oxford), "On Model Theory, noncommutative geometry and physics" — Studying possible relations between a mathematical structure and its description in a formal language
Model Theory developed a hierarchy of a 'logical perfection'. On the very top of this hierarchy we discovered a new class of structures called Zariski geometries. A joint theorem by Hrushovski
and the speaker (1993) indicated that the general Zariski geometry looks very much like an algebraic variety over an algebraically closed field, but in general is not reducible to an
algebro-geometric object. Later the present speaker established that a typical Zariski geometry can be explained in terms of a possibly noncommutative 'co-ordinate' algebra. Moreover, conversely,
many quantum algebras give rise to Zariski geometries and the correspondence 'Co-ordinate algebra - Zariski geometry' for a wide class of algebras is of the same type as that between commutative
affine algebras and affine varieties. General quantum Zariski geometries can be approximated (in a certain model-theoretic sense) by quantum Zariski geometries at roots of unity. The latter are
of a finitary type, where Dirac calculus has a well-defined meaning. We use this to give a mathematically rigorous calculation of the Feynman propagator in a few simple cases. References: On
model theory, non-commutative geometry and physics (survey), author's web-page 2009
• 17:30—18:20 Bertfried Fauser (Birmingham), "Advanced graphical calculus — Hopf, Frobenius, Schur and some motivation from group theory" — Graphical calculus has become a tool in quantum
information theory, especially the Frobenius algebra structure for modelling the copying of classical information and rewriting rules. In my talk I will try to provide a more general picture
including a Hopf algebra structure, Hopf algebra cohomology and the operation of composition. The underlying isoclasses of vector spaces will be countably infinite. The development will be
motivated by findings in group theory, particularly the theory of group chraraters, which will serve as running example. If time permits I will also address conformal fields.
Subsequently, we will find a nice pub for dinner and drinks.
Please bring the workshop to the attention of others who might be interested.
We are looking forward to seeing you (again) at CLP,
best regards,
Bob Coecke (Oxford), Andreas Döring (Oxford)
About this site - the video archive
This site went live on 11th December 2007. Feedback is very welcome. One of the most important features of this site is an extensive archive of recorded talks relevant to categories, logic and the
foundations of physics, given by many different speakers at different events around the world (not just the CLP series). You can browse these talks by speaker or by event, you can download or stream
the videos, and download the slides where available.
We are constantly adding new talks, so make sure to check back often!
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Adaptive Rejection Metropolis Sampling in R
Michael Bertolacci
Adaptive Rejection Metropolis Sampling (ARMS) is a Markov chain Monte Carlo based algorithm to sample from a univariate target distribution specified by its (potentially unnormalised) log density.
The algorithm constructs a rejection distribution based on piecewise linear functions that envelop the log density of the target. For distributions with log concave density functions, this envelope
is used directly, and usually results in a very efficient sampler. For distributions that are not log concave, or have unknown log concavity, an extra Metropolis Hastings accept/reject step is used
to correct for any mismatch between the proposal density and the target. This sometimes results in a sampler with good convergence properties.
This R package provides an efficient C++ reimplementation of ARMS.
Using the R package
You can run ARMS by calling the arms function. Usage is best illustrated by examples, given in the next two sections.
Example: normal distribution
A very simple example, for which exact sampling is possible, is the unit normal distribution. This has the unnormalised log density of \(-\frac{x^2}{2}\), with the entire real line as its domain, and
the density is log concave. This means we can use metropolis = FALSE to get exact independent samples:
(Note: there are obviously better ways to sample this distribution—rnorm for a start.)
Example: mixture of normal distributions
Another simple example, but one for which the Metropolis-Hastings step is required, is a mixture of normal distributions. For instance, consider
\[ x \sim 0.4 N(-1, 1) + 0.6 N(4, 1), \]
which has a density that looks like
This distribution is not log-concave, so we need to use metropolis = TRUE to correct the inexactness caused by the use of an imperfect rejection distribution. Doing this we can sample from the
Using the C++ implementation
The R package contains a header-only C++ implementation of the algorithm, which can be used in other packages. To use it, add this package (armspp) to the LinkingTo for your package, then #include
<armspp> in a C++ file. You can find an example for how to call this function in the src/armspp.cpp of this package.
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Scope and Sequence
The big ideas in grade 2 include: extending understanding of the base-ten number system, building fluency with addition and subtraction, using standard units of measure, and describing and analyzing
The mathematical work for grade 2 is partitioned into 9 units:
1. Adding, Subtracting, and Working with Data
2. Adding and Subtracting within 100
3. Measuring Length
4. Addition and Subtraction on the Number Line
5. Numbers to 1,000
6. Geometry, Time, and Money
7. Adding and Subtracting within 1,000
8. Equal Groups
9. Putting it All Together
In these materials, particularly in units that focus on addition and subtraction, teachers will find terms that refer to problem types, such as Add To, Take From, Put Together or Take Apart, Compare,
Result Unknown, and so on. These problem types are based on common addition and subtraction situations, as outlined in Table 1 of the Mathematics Glossary section of the Common Core State Standards.
Unit 1: Adding, Subtracting, and Working with Data
Unit Learning Goals
• Students represent and solve story problems within 20 through the context of picture and bar graphs that represent categorical data. Students build toward fluency with addition and subtraction.
In this unit, students begin the year-long work to develop fluency with sums and differences within 20, building on concepts of addition and subtraction from grade 1. They learn new ways to represent
and solve problems involving addition, subtraction, and categorical data.
In grade 1, students added and subtracted within 20 using strategies based on properties of addition and place value. They developed fluency with sums and differences within 10. Students also gained
experience in collecting, organizing, and representing categorical data.
Here, students are introduced to picture graphs and bar graphs as a way to represent categorical data. They ask and answer questions about situations described by the data. The structure of the bar
graphs paves the way for a new representation, the tape diagram.
Students learn that tape diagrams can be used to represent and make sense of problems involving the comparison of two quantities. The diagrams also help to deepen students’ understanding of the
relationship between addition and subtraction.
This opening unit also offers opportunities to introduce mathematical routines and structures for centers, and to develop a shared understanding of what it means to do math and to be a part of a
mathematical community.
Section A: Add and Subtract Within 20
Standards Alignments
Addressing 2.NBT.B.5, 2.OA.B.2
Section Learning Goals
• Build toward fluency with adding within 100.
• Build toward fluency with subtracting within 20.
This opening section gives teachers opportunities to assess students’ fluency with addition and subtraction facts within 10 and how they approach adding and subtracting.
The first several lessons focus on making a ten as a strategy to add and subtract, which helps students gain fluency with facts within 20 and supports the work with larger numbers (such as composing
and decomposing numbers as a way to add and subtract). In the last lesson of the section, students use strategies learned in grade 1 to add within 50.
\(10- 5 = \underline{\hspace{1 cm}}\)
\(5 + \underline{\hspace{1 cm}}=10\)
\(2 + \underline{\hspace{1 cm}}=10\)
\(10 - 8 = \underline{\hspace{1 cm}}\)
Some activities take place in centers, enabling teachers to also introduce routines and structures while helping students develop mental strategies for adding and subtracting.
PLC: Lesson 2, Activity 2, Sums of 10
Section B: Ways to Represent Data
Standards Alignments
Addressing 2.MD.D.10, 2.NBT.B.5, 2.OA.B.2
Section Learning Goals
• Interpret picture and bar graphs.
• Represent data using picture and bar graphs.
• Solve one- and two-step problems using addition and subtraction within 20.
In this section, students explore situations and problems that involve categorical data and learn new ways to represent such data.
Students begin by representing data about their class in a way that makes sense to them. Then, they are introduced to picture graphs and bar graphs. Students learn the conventions of these graphs as
they create them. They discuss the types of questions that can be asked and answered by the graphs, including those that require combining and comparing different categories.
PLC: Lesson 9, Activity 1, Field Trip Choices
Section C: Diagrams to Compare
Standards Alignments
Addressing 2.MD.D.10, 2.NBT.A.2, 2.NBT.B.5, 2.OA.A.1, 2.OA.B.2
Section Learning Goals
• Make sense of and interpret tape diagrams.
• Represent and solve Compare problems with unknowns in all positions within 100.
Students have previously represented and reasoned about quantities in story problems. In grade 1, students compared quantities using diagrams with discrete partitions. In the previous section, they
reasoned about quantities in bar graphs. Here, students learn to use tape diagrams as another way to make sense of the relationship between two quantities and between addition and subtraction.
Students explore Compare story problems with an unknown difference, an unknown larger number, or an unknown smaller number. Tape diagrams help students to visualize these structures and support them
in reasoning about strategies to use to solve problems, such as counting on or counting back. The table highlights the different types of problems in this section.
│ difference unknown │ bigger unknown │ smaller unknown │
│Lin counted 28 boats. Diego counted 32 boats. How many more │Lin found 28 more shells than Diego. Diego found 32 shells. How │Lin saw 32 starfish. Diego saw 28 fewer starfish than Lin. How many │
│boats did Diego count? │many shells did Lin find? │starfish did Diego see? │
│ │ │ │
│ │ │ │
Students also write equations to reason about questions that ask “how many more?” and “how many less?” They recognize that different equations and diagrams can be used to represent the same
difference between two numbers.
PLC: Lesson 14, Activity 1, Party Time (Part 1)
Estimated Days: 14 - 18
Unit 2: Adding and Subtracting within 100
Unit Learning Goals
• Students add and subtract within 100 using strategies based on place value, properties of operations, and the relationship between addition and subtraction. They then use what they know to solve
story problems.
Previously, students added and subtracted numbers within 100 using strategies they learned in grade 1, such as counting on and counting back, and with the support of tools such as connecting
cubes. In this unit, they add and subtract within 100 using strategies based on place value, the properties of operations, and the relationship between addition and subtraction.
Students begin by using any strategy to find the value of sums and differences that do not involve composing or decomposing a ten. They are then introduced to base-ten blocks as a tool to represent
addition and subtraction and move towards strategies that involve composing and decomposing tens.
Students develop their understanding of grouping by place value, and begin to subtract one- and two-digit numbers from two-digit numbers by decomposing a ten as needed. They apply properties of
operations and practice reasoning flexibly as they arrange numbers to facilitate addition or subtraction.
For example, students compare Mai and Lin’s methods for finding the value of \(63-18\).
At the end of the unit, students apply their knowledge of addition and subtraction within 100 to solve one- and two-step story problems of all types, with unknowns in all positions. To support them
in reasoning about place value when adding and subtracting, students may choose to use connecting cubes, base-ten blocks, tape diagrams, and other representations learned in earlier units and grades.
Section A: Add and Subtract
Standards Alignments
Addressing 2.MD.D.10, 2.NBT.A.2, 2.NBT.B.5, 2.NBT.B.9, 2.OA.A.1, 2.OA.B.2
Section Learning Goals
• Add and subtract within 100 using strategies based on place value and the relationship between addition and subtraction. Problems in this section are limited to the problems like 65 – 23, where
decomposing a ten is not required.
In this section, students find the value of unknown addends using methods that are based on place value and are introduced to base-ten blocks. They continue to rely on the relationship between
addition and subtraction to solve problems involving differences.
Students begin by solving Compare story problems. They use any methods and tools that make sense to them—including diagrams and connecting cubes—to find differences of two-digit numbers.
Lin and Clare used cubes to make trains.
What do you notice? What do you wonder?
Students then analyze the structure of base-ten blocks and use them to find unknown addends (MP7). Unlike connecting cubes, base-ten blocks cannot be pulled apart, which helps emphasize the structure
of two-digit numbers in base ten.
To reason about an unknown addend, they may add tens and ones to the known addend until they reach the value of the sum. They may also start with the total amount and subtract tens from tens and ones
from ones to reach the known addend. The numbers encountered here do not require students to decompose a ten when they subtract by place value.
PLC: Lesson 2, Activity 1, How Did You Find It?
Section B: Decompose to Subtract
Standards Alignments
Addressing 2.NBT.B.5, 2.NBT.B.6, 2.NBT.B.9, 2.OA.B.2
Section Learning Goals
• Subtract within 100 using strategies based on place value, including decomposing a ten, and the properties of operations.
In this section, students subtract one- and two-digit numbers from two-digit numbers within 100. To reason about differences of two numbers, they use methods based on place value, base-ten blocks and
diagrams, and properties of operations. The numbers here require students to decompose a ten when subtracting by place.
Students also make sense of different representations of subtraction by place, including those that show their peers’ reasoning. For example, to find the value of \(63-18\), students might use
base-ten blocks or drawings to represent tens and ones. In this case, they might decompose 1 ten from 63 and exchange it for 10 ones, making 5 tens and 13 ones. From here, some students may first
take away 8 ones, and then 1 ten. Others may take away 1 ten, then 8 ones.
When students discuss different approaches and explain why they result in the same value, they deepen their understanding of the properties of operations and place value.
\(63 - 18\)
The reasoning here builds a foundation for students to understand the standard algorithm for subtraction, but students should not be encouraged to use the notation for standard algorithm at this
point. Allow them to build conceptual understanding by reasoning with base-ten blocks and drawings and articulating their thinking.
PLC: Lesson 5, Activity 2, Subtract with Base-ten Blocks
Section C: Represent and Solve Story Problems
Standards Alignments
Addressing 2.NBT.B.5, 2.NBT.B.6, 2.OA.A.1, 2.OA.B.2
Section Learning Goals
• Represent and solve one- and two-step problems involving addition and subtraction within 100, including different problem types with unknowns in all positions.
This section allows students to apply their knowledge to solve story problems that involve addition and subtraction within 100. The story problems include all types—Add To, Take From, Put Together/
Take Apart, and Compare— and have unknowns in all positions.
Previously, students worked with diagrams that represent Compare problems. Throughout this section, students also make sense of diagrams that could represent Put Together/Take Apart story problems.
Clare and Han are playing a game with seeds.
Clare has 54 seeds on her side of the board.
Han has 16 seeds on his side.
How many seeds are on the board in all?
Which diagram matches this story? Explain your match to your partner.
As students relate quantities in context and diagrams that represent them, they practice reasoning quantitatively and abstractly (MP2).
Throughout the section, students are invited to interpret and solve problems in the ways that make sense to them (MP1). Math tools such as connecting cubes and base-ten blocks should be made
available to encourage methods based on place value and the properties of operations to solve the problems.
PLC: Lesson 12, Activity 1, Interpret the Diagram
Estimated Days: 12 - 16
Unit 3: Measuring Length
Unit Learning Goals
• Students measure and estimate lengths in standard units and solve measurement story problems within 100.
This unit introduces students to standard units of lengths in the metric and customary systems.
In grade 1, students expressed the lengths of objects in terms of a whole number of copies of a shorter object laid without gaps or overlaps. The length of the shorter object serves as the unit of
Here, students learn about standard units of length: centimeters, meter, inches, and feet. They examine how different measuring tools represent length units, learn how to use the tools, and gain
experience in measuring and estimating the lengths of objects. Along the way, students notice that the length of the same object can be described with different measurements and relate this to
differences in the size of the unit used to measure.
Throughout the unit, students solve one- and two-step story problems involving addition and subtraction of lengths. To make sense of and solve these problems, they use previously learned strategies
for adding and subtracting within 100, including strategies based on place value.
To close the unit, students learn that line plots can be used to represent numerical data. They create and interpret line plots that show measurement data and use them to answer questions about the
Students relate the structure of a line plot to the tools they used to measure lengths. This prepares students for the work in the next unit, where they interpret numbers on the number line as
lengths from 0. The number line is an essential representation that will be used in future grades and throughout students’ mathematical experiences.
Section A: Metric Measurement
Standards Alignments
Addressing 2.MD.A, 2.MD.A.1, 2.MD.A.3, 2.MD.A.4, 2.MD.B.5, 2.MD.B.6, 2.NBT.A.2, 2.NBT.B.5, 2.OA.A.1, 2.OA.B.2
Section Learning Goals
• Measure length in centimeters and meters.
• Represent and solve one-step story problems within 100.
This section introduces two metric units: centimeter and meter. Students use base-ten blocks, which have lengths of 1 centimeter and 10 centimeters, to measure objects in the classroom and to create
their own centimeter ruler. Students iterate the 1-centimeter unit Just as they had done with non-standard units in grade 1.
Students relate the side length of a centimeter cube to the distance between tick marks on their ruler. They see that each tick mark notes the distance in centimeters from the 0 mark, and that the
length units accumulate as they move along the ruler and away from 0.
Students then compare the ruler they created to a standard centimeter ruler. They learn the importance of placing the end of an object at 0 and discuss how the numbers on the ruler represent lengths
from 0.
Students also learn about a longer unit in the metric system, meter, and use it to estimate lengths. They have opportunities to choose measurement tools and to do so strategically (MP5), by
considering the lengths of objects being measured. Students also measure the length of longer objects in both centimeters and meters, which prompts them to relate the size of the unit to the
To close the section, students apply their knowledge of measurement to compare the lengths of objects and solve Compare story problems involving lengths within 100, measured in metric units.
PLC: Lesson 2, Activity 2, Measure with 10-centimeter Tools
Section B: Customary Measurement
Standards Alignments
Addressing 2.MD.A.1, 2.MD.A.2, 2.MD.A.3, 2.MD.A.4, 2.MD.B.5, 2.NBT.B.5, 2.OA.A, 2.OA.B.2
Section Learning Goals
• Measure length in feet and inches.
• Represent and solve one- and two-step story problems within 100.
In this section, students apply measurement concepts and skills from earlier to measure and estimate lengths in two customary units: inches and feet.
As in the previous section, students make choices about the tool to use based on the length of the object being measured (MP5) and measure the length of the same object in both feet and inches. They
begin to generalize that when they use a longer length unit, fewer of those units are needed to span the full length of the object. This understanding is a foundation for their work with fractions in
grade 3 and beyond.
To solidify their understanding of measurement concepts, students also solve one- and two-step story problems involving addition and subtraction of lengths within 100, expressed in customary units.
Some problems involve measurements using a “torn tape” where the 0 cannot be used as a starting point.
Jada and Han used an inch ruler to measure the short side of a notebook.
Jada says it is 8 inches.
How did Han and Jada get the same measurement?
PLC: Lesson 11, Activity 1, Saree Silk Ribbon Necklaces
Section C: Line Plots
Standards Alignments
Addressing 2.MD.A.1, 2.MD.A.3, 2.MD.A.4, 2.MD.B.5, 2.MD.B.6, 2.MD.D.9, 2.NBT.B.5, 2.OA.B.2
Section Learning Goals
• Represent numerical data on a line plot.
In this section, students apply their understanding of measurement and data to create and interpret line plots. Students learn that the horizontal scale is marked off in whole-number length units,
the same ones used to collect the data.
They recognize that the numbers on the number line represent lengths and each “x” above a number represents an object of that length. They label line plots with titles and the measurement unit used.
Throughout the section, students connect the features of the line plot to the tools they use to measure.
PLC: Lesson 15, Activity 2, Plot Pencil Lengths
Estimated Days: 14 - 18
Unit 4: Addition and Subtraction on the Number Line
Unit Learning Goals
• Students learn about the structure of a number line and use it to represent numbers within 100. They also relate addition and subtraction to length and represent the operations on the number
In this unit, students are introduced to the number line, an essential representation that will be used throughout students’ K–12 mathematical experience. They learn to use the number line
to represent whole numbers, sums, and differences.
In a previous unit, students learned to measure length with rulers. Here, they see that the tick marks and numbers on the number line are like those on a ruler: both show equally spaced numbers that
represent lengths from 0.
Students use this understanding of structure to locate and compare numbers on the number line, as well as to estimate numbers represented by points on the number line.
Locate and label 17 on the number line.
What number could this be? _____
Students then learn conventions for representing addition and subtraction on the number line: using arrows pointing to the right for adding and arrows pointing to the left for subtracting. Students
also use the number line to represent addition and subtraction methods discussed in Number Talks, such as counting on, counting back by place, and decomposing a number to get to a ten. The reasoning
here deepens students’ understanding of the relationship between addition and subtraction.
The number lines in this unit show a tick mark for every whole number in the given range, though not all may be labeled with the numeral. As students become more comfortable with this representation,
they may draw number lines that show only the numbers needed to solve the problems, which is acceptable.
Section A: The Structure of the Number Line
Standards Alignments
Addressing 2.MD.B.6, 2.NBT.A.2, 2.NBT.B.5
Section Learning Goals
• Represent whole numbers within 100 as lengths from 0 on a number line.
• Understand the structure of the number line.
In this section, students begin to use the number line as a tool for understanding numbers and number relationships. They learn that the number line is a visual representation of numbers shown in
order from left to right, with equal spacing between each number.
Students see that each number tells the number of length units from 0, just like on the ruler. This means that the numbers numbers to the left are smaller (fewer units away from 0) and those farther
to the right are larger (more units away from 0).
Students learn that whole numbers can be represented with tick marks and points on the number line. They then locate, label, and compare numbers on a number line. They also estimate numbers that
could be represented by points on a number line.
Locate and label 43 on the number line.
What number could this be? _____
PLC: Lesson 2, Activity 1, Class Number Line
Section B: Add and Subtract on a Number Line
Standards Alignments
Addressing 2.MD.B.5, 2.MD.B.6, 2.NBT.A.2, 2.NBT.B.5, 2.OA.A.1
Section Learning Goals
• Represent sums and differences on a number line.
In this section, students reason about sums and differences on the number line. They begin by using directional arrows: an arrow pointing right represents addition, and an arrow pointing left
represents subtraction. Students write equations that correspond to given number-line representations, as well as represent given equations on the number line.
Later, students revisit the idea of subtraction as an unknown-addend problem and represent the unknown addend with a jump to the right. For example, here are three ways they may reason about \(35-27
\) on the number line:
As students analyze various representations of a difference on the number line, they consider when certain strategies may be more efficient than others. They also consider reasoning strategies that
are based on place value and the properties of operations (for example, adding tens and then ones, or adding ones and then tens). For example, here are two ways to find \(53-29\):
At the end of the section, students use the number line to make sense of and solve story problems. They compare this representation with others used in earlier units.
PLC: Lesson 8, Activity 1, Represent Equations
Estimated Days: 12 - 15
Unit 5: Numbers to 1,000
Unit Learning Goals
• Students extend place value understanding to three-digit numbers.
In this unit, students extend their knowledge of the units in the base-ten system to include hundreds.
In grade 1, students learned that a ten is a unit made up of 10 ones, and two-digit numbers are formed using units of tens and ones. Here, they learn that a hundred is a unit made up of 10 tens, and
three-digit numbers are formed using units of hundreds, tens, and ones.
To make sense of numbers in different ways and to build flexibility in reasoning with them, students work with a variety of representations: base-ten blocks, base-ten diagrams or drawings, number
lines, expressions, and equations.
At the start of the unit, students express a quantity in terms of the number of units represented by base-ten blocks (3 hundreds, 14 tens, 22 ones). They practice composing larger units from smaller
units and representing the value using the fewest number of each unit (4 hundreds, 6 tens, 2 ones). They connect the number of units to three-digit numerals (462).
Next, students make sense of three-digit numbers on the number line. In a previous unit, students learned about the structure of the number line by representing whole numbers within 100 as lengths
from zero. Here, they get a sense of the relative distance of whole numbers within 1,000 from zero. Students learn to count to 1,000 by skip-counting on a number line by 10 and 100. They also locate,
compare, and order three-digit numbers on a number line.
Throughout the unit, the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 are referred to as multiples of 100 for simplicity. The same is true for multiples of 10. “Multiple” is not a word that
students are expected to understand or use in grade 2. Students can describe the numbers as some number of tens or hundreds, such as “20 tens” or “3 hundreds.”
Section A: The Value of Three Digits
Standards Alignments
Addressing 2.MD.B.6, 2.NBT.A, 2.NBT.A.1, 2.NBT.A.1.a, 2.NBT.A.1.b, 2.NBT.A.2, 2.NBT.A.3, 2.NBT.B.5, 2.OA.B.2
Section Learning Goals
• Read, write, and represent three-digit numbers using base-ten numerals and expanded form.
• Use place value understanding to compose and decompose three-digit numbers.
This section introduces the unit of a hundred. Students begin by analyzing the large square base-ten block, and its corresponding base-ten diagram, to recognize 100 as 1 hundred, 10 tens, or 100
Students learn that the digits in three-digit numbers represent amounts of hundreds, tens, and ones. They use this insight to write numbers and represent quantities in different forms—base-ten
numerals, words, and expanded form. Students see that they can compose a hundred with 10 tens, just as they can compose a ten with 10 ones, and that a quantity can be expressed in many ways.
2 hundreds 3 tens 8 ones
two hundred thirty-eight
200 + 30 + 8
Composing larger units from smaller units allows students to express a quantity using the fewest number of each unit, which reinforces the meaning of the digits in a three-digit number and prepares
students to add and subtract such numbers later. It also lays the foundation for generalizing the relationship between the digits of other numbers in the base-ten system in future grades.
PLC: Lesson 2, Activity 2, How Many Hundreds?
Section B: Compare and Order Numbers within 1,000
Standards Alignments
Addressing 2.MD.B.6, 2.NBT.A, 2.NBT.A.1, 2.NBT.A.2, 2.NBT.A.3, 2.NBT.A.4, 2.NBT.B.8
Section Learning Goals
• Compare and order three-digit numbers using place value understanding and the relative position of numbers on a number line.
• Represent whole numbers up to 1,000 as lengths from 0 on a number line.
In this section, students use number line diagrams to deepen their understanding of numbers to 1,000. They begin by skip-counting on the number line to build a sense of the relative position of
numbers to 1,000. They recall the structure of the number line from a previous unit and use it, along with their understanding of place value, to locate, compare, and order numbers on the number
This number line, for example, is divided into intervals of 10 units, representing 10 tens from 500 to 600. In a task, students may be asked to locate the number 540 and estimate the location of the
number 546.
As students locate or estimate the location of three-digit numbers on number lines such as these, they show an understanding of a number’s relative distance from zero and the place value of the
digits. This understanding helps them to compare and order three-digit numbers. Students see that the numbers get larger as they move from left to right on the line.
To compare and order three-digit numbers written as base-ten numerals, students also continue to use base-ten blocks, base-ten diagrams, or other representations that make sense to them. They write
the comparisons using the symbols, >, <, and =.
Who has more? How do you know?
PLC: Lesson 9, Activity 1, Compare Comparisons
Estimated Days: 11 - 14
Unit 6: Geometry, Time, and Money
Unit Learning Goals
• Students reason with shapes and their attributes and partition shapes into equal shares, building a foundation for fractions. They relate halves, fourths, and skip-counting by 5 to tell time, and
solve story problems involving the values of coins and dollars.
In this unit, students transition from place value and numbers to geometry, time, and money.
In grade 1, students distinguished between defining and non-defining attributes of shapes, including triangles, rectangles, trapezoids, and circles. Here, they continue to look at attributes of a
variety of shapes and see that shapes can be identified by the number of sides and vertices (corners). Students then study three-dimensional (solid) shapes, and identify the two-dimensional (flat)
shapes that make up the faces of these solid shapes.
Next, students look at ways to partition shapes and create equal shares. They extend their knowledge of halves and fourths (or quarters) from grade 1 to now include thirds.
Students compose larger shapes from smaller equal-size shapes and partition shapes into two, three, and four equal pieces.
As they develop the language of fractions, students also recognize that a whole can be described as 2 halves, 3 thirds, or 4 fourths, and that equal-size pieces of the same whole need not have the
same shape.
Which circles are not examples of circles partitioned into halves, thirds, or fourths?
Later, students use their understanding of halves and fourths (or quarters) to tell time. In grade 1, they learned to tell time to the half hour. Here, they relate a quarter of a circle to the
features of an analog clock. They use “quarter past” and “quarter till” to describe time, and skip-count to tell time in 5-minute intervals. They also learn to associate the notation “a.m.” and
“p.m.” with their daily activities.
To continue to build fluency with addition and subtraction within 100, students conclude the unit with a money context. They skip-count, count on from the largest value, and group like coins, and
then add or subtract to find the value of a set of coins. Students also solve one- and two-step story problems involving sets of dollars and different coins, and use the symbols $ and ¢.
Section A: Attributes of Shapes
Standards Alignments
Addressing 2.G.A.1, 2.MD.A.1, 2.NBT.A.3, 2.NBT.B.5
Section Learning Goals
• Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
• Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.
In this section, students identify and draw triangles, quadrilaterals, pentagons, and hexagons. Students are likely familiar with triangles and hexagons given their previous work with pattern blocks.
Here, they see that hexagons include any shape with six sides and six corners, and may look different from the pattern block they worked with in the past. For example, each of these shapes is a
Students learn to name a shape by counting the sides and corners and come to see that, in any shape, the number of corners is the same as the number of sides. (The term “corners” is used in lieu of
“vertices” because the latter requires an understanding of angles, which is developed in grade 4.)
Students come to recognize that some shapes such as rectangles and squares have “square corners,” the informal language for 90-degree angles. As they identify and draw shapes with given attributes,
they measure length in centimeters and inches, revisiting previously learned skills.
At the end of the section, students relate two-dimensional (flat) shapes to three-dimensional (solid) shapes. They see that flat shapes make up the faces of solid shapes and identify solid shapes
based on the flat shapes that constitute them.
PLC: Lesson 2, Activity 2, What Shape Could It Be?
Section B: Halves, Thirds, and Fourths
Standards Alignments
Addressing 2.G.A.1, 2.G.A.3, 2.NBT.A.1, 2.NBT.A.2
Section Learning Goals
• Partition rectangles and circles into halves, thirds, and fourths and name the pieces.
• Recognize 2 halves, 3 thirds, and 4 fourths as one whole.
• Understand that equal pieces do not need to be the same shape.
In this section, students learn that shapes can be partitioned into two, three, or four equal pieces called halves, thirds, and fourths or quarters.
Students begin by composing shapes using pattern blocks, initially using any combination. Later, they use a single type of pattern block, which allows them to see the composed shape as partitioned
into equal pieces.
In grade 1, students partitioned shapes into two and four equal pieces, and described each piece as a half or a fourth or quarter. (To prepare students to tell time to the quarter hour in the next
section, be sure that they hear and use fourths and quarters interchangeably.) Here, they add the term “thirds” to their vocabulary and partition rectangles into halves, thirds, and fourths.
Students then identify equal-size pieces in shapes, which are partitioned in different ways to build an understanding that equal-size pieces of the same whole do not need to be the same shape.
They come to understand that if the whole is partitioned into the same number of equal pieces, the names of the pieces are the same. Students also learn that 2 halves, 3 thirds, and 4 fourths each
make up one whole.
Although students are expected to use the language of fractions (halves, thirds, and fourths), they are not expected to use the word “fraction” or see fractions in numerical form until grade 3.
PLC: Lesson 7, Activity 2, That’s Not It
Section C: Time on the Clock
Standards Alignments
Addressing 2.G.A, 2.G.A.1, 2.MD.C.7, 2.NBT.A.2, 2.NBT.B.5, 2.NBT.B.6
Section Learning Goals
• Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
In this section, students use their understanding of fourths and quarters to tell time.
In grade 1, students learned to tell time to the hour and half-hour. Here, they make a connection between the analog clock and circles partitioned into halves or fourths.
Students use the phrases “half past,” “quarter past,” and “quarter till” to tell time. They skip-count by 5 to tell time in 5-minute intervals.
Students recognize that the hour hand on an analog clock moves towards the next hour as time passes. They represent time on analog clocks by drawing the hour and minute hands and writing the time
with digits.
Students recognize that, as time passes, the hour hand on an analog clock moves towards the next hour. They learn that each hour comes around twice a day on a 12-hour clock, and is labeled with
“a.m.” and “ p.m.” to distinguish between times of day. Towards the end of this section, students relate a.m. and p.m. times to their daily activities.
PLC: Lesson 13, Activity 1, What is the Time of Day?
Section D: The Value of Money
Standards Alignments
Addressing 2.G.A, 2.G.A.1, 2.MD.C.8, 2.NBT.A.2, 2.NBT.B.5, 2.NBT.B.6, 2.NBT.B.8, 2.OA.A.1
Section Learning Goals
• Find the value of a group of bills and coins.
• Use addition and subtraction within 100 to solve one- and two-step word problems.
In this section, students learn about money concepts while continuing to develop fluency with addition and subtraction within 100. They identify coins such as quarters, dimes, nickels, and pennies,
and find the total value of different coin combinations.
Mai had some money. Elena has $\(\)48.
They combined their money and now they have $85.
How much money did Mai have?
PLC: Lesson 16, Activity 1, How Much is a Quarter Worth?
Estimated Days: 16 - 21
Unit 7: Adding and Subtracting within 1,000
Unit Learning Goals
• Students use place value understanding, the relationship between addition and subtraction, and properties of operations to add and subtract within 1,000.
In this unit, students add and subtract within 1,000, with and without composing and decomposing a base-ten unit.
Previously, students added and subtracted within 100 using methods such as counting on, counting back, and composing or decomposing a ten. Here, they apply the methods they know and their
understanding of place value and three-digit numbers to find sums and differences within 1,000.
Initially, students add and subtract without composing or decomposing a ten or hundred. Instead, they rely on methods based on the relationship between addition and subtraction and the properties of
operations. They make sense of sums and differences using counting sequences, number relationships, and representations (number line, base-ten blocks, base-ten diagrams, and equations).
As the unit progresses, students work with numbers that prompt them to compose and decompose one or more units, eliciting strategies based on place value. When adding and subtracting by place,
students first compose or decompose only a ten, then either a ten or a hundred, and finally both a ten and a hundred. They also make sense of and connect different ways to represent place value
strategies. For example, students make sense of a written method for subtracting 145 from 582 by connecting it to a base-ten diagram and their experiences with base-ten blocks.
How do Jada's equations match Lin's diagram?
Finish Jada's work to find \(582-145\).
Students learn to recognize when composition or decomposition is a useful strategy when adding or subtracting by place. In the later half of the unit, they encounter lessons that encourage them to
think flexibly and use strategies that make sense to them based on number relationships, properties of operations, and the relationship between addition and subtraction.
Section A: Add and Subtract within 1,000 without Composition or Decomposition
Standards Alignments
Addressing 2.NBT.A, 2.NBT.A.2, 2.NBT.A.4, 2.NBT.B.5, 2.NBT.B.7, 2.NBT.B.8, 2.NBT.B.9
Section Learning Goals
• Add and subtract numbers within 1,000 without composition or decomposition, and use strategies based on the relationship between addition and subtraction and the properties of operations.
In this section, students add and subtract within 1,000 using methods where they do not explicitly compose or decompose a ten or a hundred.
The number line is used early in this section to help students recognize that when numbers are relatively close, they can count on or count back to find the value of the difference. For example, they
may count on from 559 to 562 to find \(562-559\).
Students also analyze counting sequences of three-digit numbers that increase or decrease by 10 or 100. They observe patterns in place value before adding and subtracting multiples of 10 or 100.
Fill in the missing numbers. Does the number line show counting on by 10 or by 100?
Students then engage with problems and expressions that encourage them to reason about sums and differences using the relationship between addition and subtraction and the properties of operations.
Diego has 6 tens. Tyler has 8 hundreds, 3 tens, and 6 ones.
What is the value of their blocks together?
Later in the section, students analyze and make connections between methods that use different representations, such as number lines, base-ten diagrams, and equations. They then use methods or
representations that make sense to them to add and subtract three-digit numbers.
PLC: Lesson 4, Activity 1, Zero Tens and Zero Ones
Section B: Add within 1,000 using Place Value Strategies
Standards Alignments
Addressing 2.NBT.B.5, 2.NBT.B.6, 2.NBT.B.7, 2.NBT.B.8, 2.NBT.B.9
Section Learning Goals
• Add numbers within 1,000 using strategies based on place value understanding, including composing a ten or hundred.
In this section, students use strategies based on place value to add three-digit numbers. They learn that it is sometimes necessary to compose a hundred from 10 tens to find the value of such sums.
Students begin with sums that allow them to decide when to make a ten. They then work with larger values in the tens place and determine when to compose a hundred. As the lessons progress, they
encounter sums of two- and three-digit numbers that involve composing two units.
Throughout the section, students analyze and use representations such as base-ten blocks, base-ten diagrams, expanded form, and other equations to build conceptual understanding and show place value
reasoning. They also develop their understanding of the properties of operations as they observe that the order in which they add the units doesn’t affect the value of the sum.
What is the same and what is different about how Priya and Lin found \(358 + 67\)?
Priya's work
\(300 + 100 + 10 + 10 + 5\)
\(400 + 20 + 5 = 425\)
Lin's work
\(3 \text{ hundreds} + 11 \text { tens} + 15 \text{ ones}\)
\(11 \text { tens} = 110 \)
\(15 \text{ ones} = 15\)
\(300 + 110 + 15 = 425\)
Later in the section, students add within 1,000 using any method they have learned and thinking flexibly about the numbers they are adding.
PLC: Lesson 7, Activity 2, Walk About and Add
Section C: Subtract within 1,000 using Place Value Strategies
Standards Alignments
Addressing 2.MD.D.10, 2.NBT.A.1, 2.NBT.A.2, 2.NBT.A.3, 2.NBT.B.7, 2.NBT.B.8, 2.NBT.B.9
Section Learning Goals
• Subtract numbers within 1,000 using strategies based on place value understanding, including decomposing a ten or hundred.
As they have done when adding, students subtract numbers within 1,000 using place value strategies that involve decomposing a ten, a hundred, or both. This work builds on their previous experience of
subtracting two-digit numbers by place value and decomposing a ten.
Students use base-ten blocks to subtract hundreds from hundreds, tens from tens, and ones from ones, which offers a concrete experience of exchanging a ten for 10 ones or a hundred for 10 tens as
Along the way, they begin to think strategically about how to decompose the minuend when using base-ten blocks or diagrams. They learn that by analyzing the value of the digits in each place, they
can initially represent the minuend in a way that would require decomposing fewer units when subtracting by place.
For example, this is a helpful way to represent 244 if we are subtracting a number with more than 4 ones, such as when finding \(244-67\):
Throughout the section, students compare the steps they use to decompose units and the different ways to represent and record the units being decomposed.
The section ends with students choosing subtraction methods flexibly. They apply their understanding of place value, the relationship between addition and subtraction, and the properties of
operations, to analyze number relationships and decide how to find the value of differences within 1,000.
PLC: Lesson 14, Activity 1, Agree to Disagree
Estimated Days: 14 - 18
Unit 8: Equal Groups
Unit Learning Goals
• Students work with equal groups of objects to gain foundations for multiplication.
In this unit, students develop an understanding of equal groups, building on their experiences with skip-counting and with finding the sums of equal addends. The work here serves as the foundation
for multiplication and division in grade 3 and beyond.
Students begin by analyzing even and odd numbers of objects. They learn that any even number can be split into 2 equal groups or into groups of 2, with no objects left over. Students use visual
patterns to identify whether numbers of objects are even or odd.
Next, students learn about rectangular arrays. They describe arrays using mathematical terms (rows and columns). Students see the total number of objects as a sum of the objects in each row and as a
sum of the objects in each column, which they express by writing equations with equal addends. They also recognize that there are many ways of seeing the equal groups in an array.
Later, students transition from working with arrays containing discrete objects to equal-size squares within a rectangle. They build rectangular arrays using inch tiles and partition rectangles into
rows and columns of equal-size squares. The work here sets the stage for the concept of area in grade 3.
Section A: Odd and Even
Standards Alignments
Addressing 2.NBT.A.2, 2.NBT.B.7, 2.NBT.B.8, 2.OA.B.2, 2.OA.C, 2.OA.C.3
Section Learning Goals
• Determine whether a group of objects (up to 20) has an odd or even number of members.
• Write an equation to express an even number as a sum of two equal addends.
In this section, students learn about odd and even numbers, building on their experience with sharing objects with another person or with making pairs out of a set of objects. They begin by noticing
that some groups of objects can be made into two equal groups without a “leftover” and other groups can be made into two equal groups with “1 leftover.” The same pattern can be seen when pairing
After learning the terms, students focus on explaining why a group has an even number or an odd number of members. They do so by showing whether the objects can be made into two equal groups or be
paired without a leftover, or whether they can skip-count by 2 to count the entire collection.
The representations used here support students as they progress from explaining even and odd numbers informally to doing so more formally. They also pave the way for students to make sense of
representations of multiplication in grade 3.
Early lessons encourage the teacher to record student thinking using diagrams of equal groups or by arranging objects in rows and columns. Both recording strategies help students see and count pairs
of objects.
Students begin to see how objects arranged in rows and columns can show equal groups or pairs. They will learn more about this arrangement and the term “array” in the next section.
To focus the work on building a foundation for multiplication and division, counters or connecting cubes should be available to students throughout the section, including during cool-downs.
PLC: Lesson 3, Activity 2, Card Sort: Even or Odd
Section B: Rectangular Arrays
Standards Alignments
Addressing 2.G.A.2, 2.NBT.A.2, 2.NBT.B.7, 2.OA.B.2, 2.OA.C.3, 2.OA.C.4
Section Learning Goals
• Find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns using addition.
• Partition rectangles into rows and columns of equal-size squares, and count to find the total number of squares.
• Represent the total number of objects in an array as a sum of equal addends.
In this section, students learn that a rectangular array contains objects arranged into rows and columns, with the same number of objects in each row and the same in number in each column.
Using this structure, students can skip-count by the number in each row or in each column to find the total number of objects. They can also write equations with equal addends representing the number
of objects in a row or a column.
Later in the section, students relate their work with arrays to the partitioning of shapes into equal parts.
True or false?
True or false?
Students build rectangles by arranging square tiles into rows and columns, and then partition rectangles into rows and columns.
Use 8 tiles to build a rectangle. Arrange them in 2 rows.
Partition this rectangle to match the rectangle you made.
Rectangles in this section have up to 5 rows and 5 columns. Students are not expected to name the fractional units created by partitioning shapes. The focus is on using the structure of the rows and
columns created by the partitions to count the total number of equal-size squares. This work serves as a foundation for students’ future study of multiplication and area measurement.
PLC: Lesson 9, Activity 1, Sums of Rows and Sums of Columns
Estimated Days: 10 - 13
Unit 9: Putting It All Together
Unit Learning Goals
• Students consolidate and solidify their understanding of various concepts and skills related to major work of the grade. They also continue to work toward fluency goals of the grade.
In this unit, students revisit major work and fluency goals of the grade, applying their learning from the year.
Section A gives students a chance to solidify their fluency with addition and subtraction within 20. In section B, students apply methods they used with smaller numbers to add and subtract numbers
within 100. They also revisit numbers within 1,000: composing and decomposing three-digit numbers in different ways, and using methods based on place value to find their sums and differences.
In the final section, students interpret, solve, and write story problems involving numbers within 100, which further develop their fluency with addition and subtraction of two-digit numbers. They
work with all problem types with the unknown in all positions.
Clare picked 51 apples. Lin picked 18 apples. Andre picked 19 apples.
Here is the work a student shows to answer to a question about the apples.
\(51 + 19 = 70\)
\( 70 + 18 = 88\)
What is the question?
The sections in this unit are standalone sections, not required to be completed in order. The goal is to offer ample opportunities for students to integrate the knowledge they have gained and to
practice skills related to the expected fluencies of the grade.
Section A: Fluency Within 20 and Measurement
Standards Alignments
Addressing 2.MD.A.1, 2.MD.A.4, 2.MD.B.5, 2.MD.D, 2.MD.D.9, 2.NBT.B.5, 2.OA.B.2
Section Learning Goals
• Fluently add and subtract within 20.
In this section, students practice adding and subtracting within 20 to meet the fluency expectations of the grade, which include finding all sums and differences within 20, and knowing from memory
all sums of 2 one-digit numbers.
Students begin with exercises and games that emphasize using the relationship between addition and subtraction to find the value of expressions and unknown addends. When students encounter sums and
differences they don't know right away, they use mental math strategies and other methods they have learned, such as using facts they know, making equivalent expressions, and composing or decomposing
a number to make a 10.
Later in the section, students apply their mental strategies to find sums and differences within 20 in a measurement context. They measure standard lengths and create line plots, and then use the
measurements to add and subtract.
│group│length of pencils in cm│total length │
│ A │8 │13 │12 │7 │ │
│ B │9 │15 │7 │10 │ │
│ C │12 │13 │8 │6 │ │
│ D │9 │9 │11 │13 │ │
│ E │ │ │ │ │ │
Use the pencil measurements to create a line plot.
PLC: Lesson 3, Activity 1, Measure on the Map
Section B: Numbers to 1,000
Standards Alignments
Addressing 2.NBT.A, 2.NBT.A.1, 2.NBT.A.3, 2.NBT.B.5, 2.NBT.B.7
Section Learning Goals
• Add and subtract within 1,000 using strategies based on place value and the properties of operations.
• Fluently add and subtract within 100.
In this section, students revisit numbers within 1,000 and develop their facility with addition and subtraction within 100. The work here requires students to compose and decompose multiple
place-value units, which reinforces their understanding of place value and operations on larger numbers.
Students begin by decomposing and composing three-digit numbers in multiple ways using base-ten blocks, base-ten diagrams, words, and symbols. They also compose and decompose units as they match and
create equivalent expressions for three-digit numbers.
Find the number that makes each equation true.
6 hundreds + 9 ones = 5 hundreds + _____ tens + 9 ones
2 hundreds + 9 tens + 17 ones = _____ hundreds + 7 ones
Next, students practice addition and subtraction within 1,000. They analyze sums and differences and reason about which ones are more difficult to evaluate and which are easier, deepening their
understanding of composition and decomposition based on place value.
Students then work toward fluent addition and subtraction within 100, which requires composing or decomposing one unit when using methods based on place value. Methods for finding sums and
differences mentally, without explicitly composing or decomposing units, are also encouraged.
PLC: Lesson 5, Activity 2, Let Me Count the Ways
Section C: Create and Solve Story Problems
Standards Alignments
Addressing 2.NBT.A, 2.NBT.B.5, 2.NBT.B.9, 2.OA.A.1
Section Learning Goals
• Represent and solve one- and two-step story problems within 100.
In this section, students create and solve one- and two-step story problems with unknown values in all positions. They discuss how they make sense of the problem and share their methods for solving.
By now, students are expected to solve all types of story problems within 100, using methods and representations that make sense to them. They continue to make connections across representations,
with a focus on equations and tape diagrams, which will be used frequently in grade 3.
Students analyze stories and determine the types of questions that could be asked based on the provided information. Then, they write their own story problems based on images and their own
Write and solve a story problem the diagram could represent.
PLC: Lesson 10, Activity 2, What is the Question?
Estimated Days: 13
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One big study or two small studies? Insights from simulations
At a recent conference, someone posed a question that had been intriguing me for a while: suppose you have limited resources, with the potential to test N participants. Would it be better to do two
studies, each with N/2 participants, or one big study with all N?
I've been on the periphery of conversations about this topic, but never really delved into it, so I gave a rather lame answer. I remembered hearing that statisticians would recommend the one big
study option, but my intuition was that I'd trust a result that replicated more than one which was a one-off, even if the latter was from a bigger sample. Well, I've done the simulations and it's
clear that my intuition is badly flawed.
Here's what I did. I adapted a script that is described in my
recent slides
that give hands-on instructions for beginners on how to simulate data, The script,
, which can be found
generates data for a simple two-group comparison using a t-test. In this version, on each run of the simulation, you get output for one study where all subjects are divided into two groups of size N,
and for two smaller studies each with half the number of subjects. I ran it with various settings to vary both the sample size and the effect size (Cohen's d). I included the case where there is no
real difference between groups (d = 0), so I could estimate the false positive rate as well as the power to detect a true effect.
I used a one-tailed t-test, as I had pre-specified that group B had the higher mean when d > 0. I used a traditional approach with p-value cutoffs for statistical significance (and yes, I can hear
many readers tut-tutting, but this is useful for this demonstration….) to see how often I got a result that met each of three different criteria:
• a) Single study, p < .05
• b) Split sample, p < .05 replicated in both studies
• c) Single study, p < .005
Figure 1 summarises the results.
The figure is pretty busy but worth taking a while to unpack. Power is just the proportion of runs of the simulation where the significance criterion was met. It's conventional to adopt a power
cutoff of .8 when deciding on how big a sample to use in a study. Sample size is colour coded, and refers to the number of subjects per group for the single study. So for the split replication, each
group has half this number of subjects. The continuous line shows the proportion of results where p < .05 for the single study, the dotted line has results from the split replication, and the dashed
line has results from the single study with more stringent significance criterion, p < .005 .
It's clear that for all sample sizes and all effect sizes, the one single sample is much better powered than the split replication.
But I then realised what had been bugging me and why my intuition was different. Look at the bottom left of the figure, where the x-axis is zero: the continuous lines (i.e., big sample, p < .05) all
cross the y-axis at .05. This is inevitable: by definition, if you set p < .05, there's a one in 20 chance that you'll get a significant result when there's really no group difference in the
population, regardless of the sample size. In contrast, the dotted lines cross the y-axis close to zero, reflecting the fact that when the null hypothesis is true, the chance of two samples both
giving p < .05 in a replication study is one in 400 (.05^2 = .0025). So I had been thinking more like a Bayesian: given a significant result, how likely was it to have been come from a population
with a true effect rather than a null effect? This is a very different thing from what a simple p-value tells you*.
Initially, I thought I was onto something. If we just stick with p < .05, then it could be argued that from a Bayesian perspective, the split replication approach is preferable. Although you are less
likely to see a significant effect with this approach, when you do, you can be far more confident it is a real effect. In formal terms, the likelihood ratio for a true vs null hypothesis, given p <
.05, will be much higher for the replication.
My joy at having my insight confirmed was, however, short-lived. I realised that this benefit of the replication approach could be exceeded with the single big sample simply by reducing the p-value
so that the odds of a false positive are minimal. That's why Figure 1 also shows the scenario for one big sample with p < .005: a threshold that has recently proposed as a general recommendation for
claims of new discoveries (Benjamin et al, 2018)**.
None of this will surprise expert statisticians: Figure 1 just reflects basic facts about statistical power that were popularised by Jacob Cohen in 1977. But I'm glad to have my intuitions now more
aligned with reality, and I'd encourage others to try simulation as a great way to get more insights into statistical methods.
Here is the conclusions I've drawn from the simulation:
• First, even when the two groups come from populations with different means, it's unlikely that you'll get a clear result from a single small study unless the effect size is at least moderate; and
the odds of finding a replicated significant effect are substantially lower than this. None of the dotted lines achieves 80% power for a replication if effect size is less than .3 - and many
effects in psychology are no bigger than that.
• Second, from a statistical perspective, testing an a priori hypothesis in a larger sample with a lower p-value is more efficient than subdividing the sample and replicating the study using a less
stringent p-value.
I'm not a stats expert, and I'm aware that there's been considerable debate out there about p-values - especially regarding the recommendations of Benjamin et al (2018). I have previously sat on the
fence as I've not felt confident about the pros and cons. But on the basis of this simulation, I'm warming to the idea of p < .005. I'd welcome comments and corrections.
*In his paper The reproducibility of research and the misinterpretation of p-values. Royal Society Open Science, 4(171085). doi:10.1098/rsos.171085 David Colquhoun (2017) discusses these issues and
notes that we also need to consider the prior likelihood of the null hypothesis being true: something that is unknowable and can only be estimated on the basis of past experience and intuition.
**The proposal for adopting p < .005 as a more stringent statistical threshold for new discoveries can be found here: Benjamin, D. J., Berger, J. O., Johannesson, M., Nosek, B. A., Wagenmakers, E.
J., Berk, R., . . . Johnson, V. E. (2018). Redefine statistical significance. Nature Human Behaviour, 2(1), 6-10. doi:10.1038/s41562-017-0189-z
Postscript, 15th July 2018
This blogpost has generated a lot of discussion, mostly on Twitter. One point that particularly interested me was a comment that I hadn’t done a fair comparison between the one-study and two-study
situation, because the plot showed a one-off two group study with an alpha at .005, versus a replication study (half sample size in each group) with alpha at .05. For a fair comparison, it was
argued, I should equate the probabilities between the two situations, i.e. the alpha for the one-off study should be .05 squared = .0025.
So I took a look at the fair comparison: Figure 2 shows the situation when comparing one study with alpha set to .0025 vs a split replication with alpha of .05. The intuition of many people on
Twitter was that these should be identical, but they aren’t. Why not? We have the same information in the two samples. (In fact, I modified the script so that this was literally true and the same
sample was tested singly and again split into two – previously I’d just resampled to get the smaller samples. This makes no difference – the single sample with more extreme alpha still gives higher
Figure 2: Power for one-off study with alpha .0025 (dashed lines) vs. split replication with p < .05
To look at it another way, in one version of the simulation there were 1600 simulated experiments with a true effect (including all the simulated sample sizes and effect sizes). Of these 581 were
identified as ‘significant’ both by the one-off study with p < .0025 and they were also replicated in two small studies with p < .05. Only 5 were identified by the split replication alone, but 134
were identified by the one-off study alone.
I think I worked out why this is the case, though I’d appreciate having a proper statistical opinion. It seems to have to do with accuracy of estimating the standard deviation. If you have a split
sample and you estimate the mean from each half (A and B), then the average of mean A and mean B will be the same as for the big sample of AB combined. But when it comes to estimating the standard
deviation – which is a key statistic when computing group differences – the estimate is more accurate and precise with the large sample. This is because the standard deviation is computed by
measuring the difference of each value from its own sample mean. Means for A and B will fluctuate due to sampling error, and this will make the estimated SDs less reliable. You can estimate the
pooled standard deviation for two samples by taking the square root of the average of the variances. However, that value is less precise than the SD from the single large sample. I haven’t done a
large number of runs, but a quick check suggests that whereas both the one-off study and the split replication give pooled estimates of the SD at around the true value of 1.0, the standard deviation
of the standard deviation (we are getting very meta here!) is around .01 for the one-off study but .14 for the split replication. Again, I’m reporting results from across all the simulated trials,
including the full range of sample sizes and effect sizes.
Figure 3: Distribution of estimates of pooled SD; The range is narrower for the one-off study (pink) than for the split replication studies (blue). Purple shows area of overlap of distributions
This has been an intriguing puzzle to investigate, but in the original post, I hadn’t really been intending to do this kind of comparison - my interest was rather in making the more elementary point
which is that there's a very low probability of achieving a replication when sample size and effect size are both relatively small.
Returning to that issue, another commentator said that they’d have far more confidence in five small studies all showing the same effect than in one giant study. This is exactly the view I would have
taken before I looked into this with simulations; but I now realise this idea has a serious flaw, which is that you’re very unlikely to get those five replications, even if you are reasonably well
powered, because – the tldr; message implicit in this post – when we’re talking about replications, we have to multiply the probabilities, and they rapidly get very low. So, if you look at the
figure, suppose you have a moderate effect size, around .5, then you need a sample of 48 per group to get 80% power. But if you repeat the study five times, then the chance of getting a positive
result in all five cases is .8^5, which is .33. So most of the time you’d get a mixture of null and positive results. Even if you doubled the sample size to increase power to around .95, the chance
of all five studies coming out positive is still only .95^5 (82%).
Finally, another suggestion from Twitter is that a meta-analysis of several studies should give the same result as a single big sample. I’m afraid I have no expertise in meta-analysis, so I don’t
know how well it handles the issue of more variable SD estimates in small samples, but I’d be interested to hear more from any readers who are up to speed with this.
12 comments:
1. One advantage of running two studies - leaving power calculations aside - is that you get the opportunity to use real data from the first study to learn all the things that were wrong with your
a-priori predictions or analysis plan.
A point that I think is sometimes missed in calls for pre-registration is something I would summarise with the quote that "research is what I'm doing when I don't know what I'm doing".
Pre-registration may have little value for studies with novel dependent measures, or for which the data holds surprises. In my experience of studies like these, sticking to the pre-registered
analysis is a mistake.
I think a better approach is to work with the data in an exploratory fashion and then pre-register the right analysis and predictions for your second, replication study.
1. I guess the other alternative would be to do some form of leave-half-out analysis.
e.g in the context of ERPs:
- test N participants;
- determine based on randomly selected N/2 the latency where the greatest effect is;
- determine the effect size for the remaining N/2 at that latency;
- repeat 1000x with different random N/2 subsamples;
- average the effect sizes across the 1000 runs.
My intuition is that this gives a more accurate picture of the true effect size. But it would probably only make sense when there are few researcher degrees of freedom.
2. Uh - not sure why I'm anonymous when I'm supposedly signed in. Jon Brock here ^^
3. Thanks Matt. I think you could also argue for other advantages of 2 studies, e.g. done by different groups so establish robustness of result against lab-specific effects. But the power issue
is really serious: if you are not powered to detect the effect of interest, then you're in trouble. And most of the time we aren't. Another option is to consider other ways of improving power
by minimising measurement error, and hence increasing effect size. But, I repeat, power is key.
2. @ Matt Davis
I am certainly no statistician but with limited N-sizes that we often have in human psychology a serious problem with the two study approach is that it magnifies the chances of both false
positive and false negative results if the data is at all noisy.
Given sufficient sample sizes and relatively clean measurements your approach has a lot of appeal but the curse of the N-size haunts us.
I specified "human psychology" above most researchers working with animals do not, at least in principle, have to worry about limited recruitment pools.
3. Once you introduce heterogenity of effect sizes, then one big study is highly problematic.
4. @ Unknown (aka Jon Bock)
Check in mirror that you are not wearing an iron mask.
How does one geheterogenity of effect sizes in a single study (assuming one measurement)?
As I said, I am no statistician.
1. So this is similar to how lots of machine learning approaches work.
You randomly divide the sample into two - you use the first half to determine how to analyse the data (eg what the epoch of interest is) and then, having fixed those analytical degrees of
freedom, you determine the effect size for the remaining half of the participants.
If you repeat that exercise a second time with a different random division of the participants, you'll end up with a slightly different effect size.
So the best thing to do is repeat that exercise many times (say 1000) and then determine the average effect size.
2. Ah, obvious once someone points it out. Thanks.
5. Blogger has refused to interact with David Colquhoun, so I am posting this comment on his behalf!
"Well actually in my 2017 paper to which you kindly refer, what I do is to suggest ways of circumventing the inconvenient fact that we rarely have a valid prior probability. More details in my
2018 paper: https://arxiv.org/abs/1802.04888 and in my CEBM talk: https://www.youtube.com/watch?v=iFaIpe9rFR0 …."
6. You randomly divide the sample into two - you use the first half to determine how to analyse the data (eg what the epoch of interest is) and then, having fixed those analytical degrees of
freedom, you determine the effect size for the remaining half of the participants. url https://amzn.to/2N9MarN
7. WRT simulations there is no difference between a single study and replicated studies. You could achieve the same result (wrt replicated studies) by randomly assigning results from the single
study into one of two groups and then analysing the two groups separately. But this would be a very inefficient way of using the data.
In practice, if you do two studies then you would do them at different times of day, or on different days, or in different labs or even in different countries. You would then still analyse as a
single study but you would include terms on your AOV regression model for study and possibly study*treatment terms. This would remove degrees of freedom from the residual error but would enable
you to draw more general conclusions.
New comments are not allowed.
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Unable to understand how these two forces are equal
• B
• Thread starter tbn032
• Start date
In summary, the conversation discusses the concept of N1=f and N2=W being proven through balancing vertical and horizontal forces on an object in static equilibrium. The location of these forces is
important for the moment they induce, but when analyzing translational equilibrium, they can be relocated to the center of mass of the object without affecting the equilibrium conditions. The links
provided further explain this concept.
In the solution given in the above image, I am unable to understand and prove why
=f and
=W. I have tried balancing the torque on different point but still unable to prove. Explain how
=f and
=W can be proved.
The justification for
=f and
=W which I have so far read is that it is just balancing the vertical force with vertical force and the horizontal force with horizontal force which are being applied on the object since the object
is at equilibrium. My confusion with that is that the vertical and horizontal forces are being applied at different position of the object, how can they be directly compared so that the ladder is in
Last edited:
The object is solid and strong enough to transfer these forces from one end to the other.
The object is in static equilibrium; therefore, all the forces and moments created by them must be cancelling each other.
All the reactive forces and moments counteract the force of the weight and any moment that it induces.
Without those being present, the weight force will acelerate the object downwards, without inducing any rotation.
Imagining that it could be possible, without the weight force, those reactive forces will move and rotate the object in diferent directions.
Lnewqban said:
The object is solid and strong enough to transfer these forces from one end to the other.
The object is in static equilibrium; therefore, all the forces and moments created by them must be cancelling each other.
All the reactive forces and moments counteract the force of the weight and any moment that it induces.
Since the object is at equilibrium, I understand that the vector sum
2+W+f=0 and the vector sum of torque generated by these forces =0.but I do not understand how can the vertical forces be compared directly with vertical forces and horizontal force directly compared
with horizontal force resulting in
1=f and
2=W even after they are being applied at different position.
Since the balance of moments is not confusing to you, just relocate all those forces to the center of mass of the object.
The actual location of each of those forces is only important for the moment it induces.
Lnewqban said:
jus relocate all those forces to the center of mass of the object.
Is relocation of forces which being applied at different position of the object to the center of mass of thr object which is at equilibrium allowed?if it is allowed, can you explain the concept
behind it.
tbn032 said:
Is relocation of forces which being applied at different position of the object to the center of mass of thr object which is at equilibrium allowed?if it is allowed, can you explain the concept
behind it.
Yes, you may do that when you are
analyzing translational equilibrium only
For this first equilibrium condition for the static equilibrium (no translational acceleration) of a rigid body, the distances of the external forces to the center of mass are irrelevant.
Please, see:
https://courses.lumenlearning.com/s...apter/12-1-conditions-for-static-equilibrium/ https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/9-6-center-of-mass/
FAQ: Unable to understand how these two forces are equal
1. What are the two forces that are being referred to?
The two forces that are being referred to are the forces of gravity and the normal force.
2. How are these two forces equal?
These two forces are equal because they are acting in opposite directions with the same magnitude. The normal force is equal to the force of gravity acting on an object, but in the opposite
3. Why is it important to understand how these two forces are equal?
Understanding how these two forces are equal is important in understanding the equilibrium of an object. When these forces are equal, the object is at rest or moving at a constant velocity.
4. Can these two forces ever be unequal?
Yes, these two forces can be unequal in certain situations. For example, if an object is accelerating, the normal force may be greater than the force of gravity, or vice versa.
5. How can I visualize the equality of these two forces?
You can visualize the equality of these two forces by picturing a scale. The normal force and the force of gravity are like two weights on opposite sides of the scale, balancing each other out.
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9.6 Solve Equations with Square Roots - Elementary Algebra 2e | OpenStax
By the end of this section, you will be able to:
• Solve radical equations
• Use square roots in applications
Before you get started, take this readiness quiz.
Solve Radical Equations
In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.
An equation in which the variable is in the radicand of a square root is called a radical equation.
As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are ‘opposite’ operations, we will
square both sides in order to remove the radical sign and solve for the variable inside.
But remember that when we write $aa$ we mean the principal square root. So $a≥0a≥0$ always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make
$aa$ negative. This algebraic solution would not be a solution to the original radical equation; it is an extraneous solution. We saw extraneous solutions when we solved rational equations, too.
For the equation $x+2=xx+2=x$:
ⓐ Is $x=2x=2$ a solution? ⓑ Is $x=−1x=−1$ a solution?
ⓐ Is $x=2x=2$ a solution?
Let x = 2.
2 is a solution.
ⓑ Is $x=−1x=−1$ a solution?
Let x = −1.
−1 is not a solution.
−1 is an extraneous solution to the equation.
For the equation $x+6=xx+6=x$:
ⓐ Is $x=−2x=−2$ a solution? ⓑ Is $x=3x=3$ a solution?
For the equation $−x+2=x−x+2=x$:
ⓐ Is $x=−2x=−2$ a solution? ⓑ Is $x=1x=1$ a solution?
Now we will see how to solve a radical equation. Our strategy is based on the relation between taking a square root and squaring.
How to Solve Radical Equations
Solve a radical equation.
1. Step 1. Isolate the radical on one side of the equation.
2. Step 2. Square both sides of the equation.
3. Step 3. Solve the new equation.
4. Step 4. Check the answer.
Solve: $5n−4−9=05n−4−9=0$.
To isolate the radical, add 9 to both sides.
Square both sides of the equation.
Solve the new equation.
Check the answer.
The solution is n = 17.
Solve: $3m+2−5=03m+2−5=0$.
Solve: $10z+1−2=010z+1−2=0$.
Solve: $3y+5+2=53y+5+2=5$.
To isolate the radical, subtract 2 from both sides.
Square both sides of the equation.
Solve the new equation.
Check the answer.
The solution is $y=43y=43$.
Solve: $3p+3+3=53p+3+3=5$.
Solve: $5q+1+4=65q+1+4=6$.
When we use a radical sign, we mean the principal or positive root. If an equation has a square root equal to a negative number, that equation will have no solution.
Solve: $9k−2+1=09k−2+1=0$.
To isolate the radical, subtract 1 from both sides.
Since the square root is equal to a negative number, the equation has no solution.
Solve: $2r−3+5=02r−3+5=0$.
Solve: $7s−3+2=07s−3+2=0$.
If one side of the equation is a binomial, we use the binomial squares formula when we square it.
Don’t forget the middle term!
To isolate the radical, subtract 1 from both sides.
Square both sides of the equation.
Simplify, then solve the new equation.
It is a quadratic equation, so get zero on one side.
Factor the right side.
Use the zero product property.
Solve each equation.
Check the answers.
The solutions are p = 1, p = 2.
Solve: $r+4−r+2=0r+4−r+2=0$.
Isolate the radical. $r+4=r−2r+4=r−2$
Square both sides of the equation. $(r+4)2=(r−2)2(r+4)2=(r−2)2$
Solve the new equation. $r+4=r2−4r+4r+4=r2−4r+4$
It is a quadratic equation, so get zero on one side. $0=r2−5r0=r2−5r$
Factor the right side. $0=r(r−5)0=r(r−5)$
Use the zero product property. $0=r0=r−50=r0=r−5$
Solve the equation. $r=0r=5r=0r=5$
Check the answer.
The solution is $r=5r=5$.
$r=0r=0$ is an extraneous solution.
Solve: $m+9−m+3=0m+9−m+3=0$.
Solve: $n+1−n+1=0n+1−n+1=0$.
When there is a coefficient in front of the radical, we must square it, too.
Solve: $33x−5−8=433x−5−8=4$.
Isolate the radical. $33x−5=1233x−5=12$
Square both sides of the equation. $(33x−5)2=(12)2(33x−5)2=(12)2$
Simplify, then solve the new equation. $9(3x−5)=1449(3x−5)=144$
Distribute. $27x−45=14427x−45=144$
Solve the equation. $27x=18927x=189$
Check the answer.
The solution is $x=7x=7$.
Solve: $24a+2−16=1624a+2−16=16$.
Solve: $36b+3−25=5036b+3−25=50$.
Solve: $4z−3=3z+24z−3=3z+2$.
The radical terms are isolated. $4z−3=3z+24z−3=3z+2$
Square both sides of the equation. $(4z−3)2=(3z+2)2(4z−3)2=(3z+2)2$
Simplify, then solve the new equation. $4z−3=3z+2z−3=2z=54z−3=3z+2z−3=2z=5$
Check the answer.
We leave it to you to show that 5 checks! The solution is $z=5.z=5.$
Solve: $2x−5=5x+32x−5=5x+3$.
Solve: $7y+1=2y−57y+1=2y−5$.
Sometimes after squaring both sides of an equation, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and square both
sides of the equation again.
The radical on the right side is isolated. Square both sides. $(m+1)2=(m+9)2(m+1)2=(m+9)2$
Simplify—be very careful as you multiply! $m+2m+1=m+9m+2m+1=m+9$
There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical. $2m=82m=8$
Square both sides. $(2m)2=(8)2(2m)2=(8)2$
Simplify, then solve the new equation. $4m=644m=64$
Check the answer.
We leave it to you to show that $m=16m=16$ checks! The solution is $m=16.m=16.$
Solve: $m+5=m+16m+5=m+16$.
Solve: $q−2+3=4q+1q−2+3=4q+1$.
The radical on the right side is isolated. Square both sides. $(q−2+3)2=(4q+1)2(q−2+3)2=(4q+1)2$
Simplify. $q−2+6q−2+9=4q+1q−2+6q−2+9=4q+1$
There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical. $6q−2=3q−66q−2=3q−6$
Square both sides. $(6q−2)2=(3q−6)2(6q−2)2=(3q−6)2$
Simplify, then solve the new equation. $36(q−2)=9q2−36q+3636(q−2)=9q2−36q+36$
Distribute. $36q−72=9q2−36q+3636q−72=9q2−36q+36$
It is a quadratic equation, so get zero on one side. $0=9q2−72q+1080=9q2−72q+108$
Factor the right side. $0=9(q2−8q+12)0=9(q2−8q+12)$
Use the zero product property. $q−6=0q−2=0q=6q=2q−6=0q−2=0q=6q=2$
The checks are left to you. (Both solutions should work.) $The solutions areq=6andq=2.The solutions areq=6andq=2.$
Solve: $y−3+2=4y+2y−3+2=4y+2$.
Solve: $n−4+5=3n+3n−4+5=3n+3$.
Use Square Roots in Applications
As you progress through your college courses, you’ll encounter formulas that include square roots in many disciplines. We have already used formulas to solve geometry applications.
We will use our Problem Solving Strategy for Geometry Applications, with slight modifications, to give us a plan for solving applications with formulas from any discipline.
Solve applications with formulas.
1. Step 1. Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
2. Step 2. Identify what we are looking for.
3. Step 3. Name what we are looking for by choosing a variable to represent it.
4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.
We used the formula $A=L·WA=L·W$ to find the area of a rectangle with length L and width W. A square is a rectangle in which the length and width are equal. If we let s be the length of a side of a
square, the area of the square is $s2s2$.
The formula $A=s2A=s2$ gives us the area of a square if we know the length of a side. What if we want to find the length of a side for a given area? Then we need to solve the equation for s.
$A=s2Take the square root of both sides.A=s2Simplify.A=sA=s2Take the square root of both sides.A=s2Simplify.A=s$
We can use the formula $s=As=A$ to find the length of a side of a square for a given area.
We will show an example of this in the next example.
Mike and Lychelle want to make a square patio. They have enough concrete to pave an area of 200 square feet. Use the formula $s=As=A$ to find the length of each side of the patio. Round your answer
to the nearest tenth of a foot.
Step 1. Read the problem. Draw a figure and
label it with the given information.
A = 200 square feet
Step 2. Identify what you are looking for. The length of a side of the square patio.
Step 3. Name what you are looking for by Let s = the length of a side.
choosing a variable to represent it.
Step 4. Translate into an equation by writing the
appropriate formula or model for the situation.
Substitute the given information.
Step 5. Solve the equation using good algebra
techniques. Round to one decimal place.
Step 6. Check the answer in the problem and
make sure it makes sense.
This is close enough because we rounded the
square root.
Is a patio with side 14.1 feet reasonable?
Step 7. Answer the question with a complete Each side of the patio should be 14.1 feet.
Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370 square feet. Use the formula $s=As=A$ to find the length of each side of her lawn. Round your answer
to the nearest tenth of a foot.
Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 2704 square centimeters. Use the formula $s=As=A$ to find the length of each side
of his mosaic. Round your answer to the nearest tenth of a centimeter.
Another application of square roots has to do with gravity.
On Earth, if an object is dropped from a height of $hh$ feet, the time in seconds it will take to reach the ground is found by using the formula,
For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting $h=64h=64$ into the formula.
Take the square root of 64.
Simplify the fraction.
It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.
Christy dropped her sunglasses from a bridge 400 feet above a river. Use the formula $t=h4t=h4$ to find how many seconds it took for the sunglasses to reach the river.
Step 1. Read the problem.
Step 2. Identify what you are looking for. The time it takes for the sunglasses to reach
the river.
Step 3. Name what you are looking for by Let t = time.
choosing a variable to represent it.
Step 4. Translate into an equation by writing the
appropriate formula or model for the situation.
Substitute in the given information.
Step 5. Solve the equation using good algebra
Step 6. Check the answer in the problem and
make sure it makes sense.
Does 5 seconds seem reasonable?
Step 7. Answer the question with a complete It will take 5 seconds for the sunglasses to hit
sentence. the water.
A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula $t=h4t=h4$ to find how many seconds it took for the package to reach the ground.
A window washer dropped a squeegee from a platform 196 feet above the sidewalk Use the formula $t=h4t=h4$ to find how many seconds it took for the squeegee to reach the sidewalk.
Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying
the brakes.
Skid Marks and Speed of a Car
If the length of the skid marks is d feet, then the speed, s, of the car before the brakes were applied can be found by using the formula,
After a car accident, the skid marks for one car measured 190 feet. Use the formula $s=24ds=24d$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.
Step 1. Read the problem.
Step 2. Identify what we are looking for. The speed of a car.
Step 3. Name what we are looking for. Let s = the speed.
Step 4. Translate into an equation by writing the appropriate formula.
Substitute the given information.
Step 5. Solve the equation.
Round to 1 decimal place.
Step 6. Check the answer in the problem.
Is 67.5 mph a reasonable speed? Yes.
Step 7. Answer the question with a complete sentence. The speed of the car was approximately 67.5 miles per hour.
An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula $s=24ds=24d$ to find the speed of the car before the brakes were applied. Round
your answer to the nearest tenth.
The skid marks of a vehicle involved in an accident were 122 feet long. Use the formula $s=24ds=24d$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest
Section 9.6 Exercises
Practice Makes Perfect
Solve Radical Equations
In the following exercises, check whether the given values are solutions.
For the equation $x+12=xx+12=x$: ⓐ Is $x=4x=4$ a solution? ⓑ Is $x=−3x=−3$ a solution?
For the equation $−y+20=y−y+20=y$: ⓐ Is $y=4y=4$ a solution? ⓑ Is $y=−5y=−5$ a solution?
For the equation $t+6=tt+6=t$: ⓐ Is $t=−2t=−2$ a solution? ⓑ Is $t=3t=3$ a solution?
For the equation $u+42=uu+42=u$: ⓐ Is $u=−6u=−6$ a solution? ⓑ Is $u=7u=7$ a solution?
In the following exercises, solve.
$5 y + 1 = 4 5 y + 1 = 4$
$7 z + 15 = 6 7 z + 15 = 6$
$5 x − 6 = 8 5 x − 6 = 8$
$4 x − 3 = 7 4 x − 3 = 7$
$2 m − 3 − 5 = 0 2 m − 3 − 5 = 0$
$2 n − 1 − 3 = 0 2 n − 1 − 3 = 0$
$6 v − 2 − 10 = 0 6 v − 2 − 10 = 0$
$4 u + 2 − 6 = 0 4 u + 2 − 6 = 0$
$5 q + 3 − 4 = 0 5 q + 3 − 4 = 0$
$4 m + 2 + 2 = 6 4 m + 2 + 2 = 6$
$6 n + 1 + 4 = 8 6 n + 1 + 4 = 8$
$2 u − 3 + 2 = 0 2 u − 3 + 2 = 0$
$5 v − 2 + 5 = 0 5 v − 2 + 5 = 0$
$3 z − 5 + 2 = 0 3 z − 5 + 2 = 0$
$2 m + 1 + 4 = 0 2 m + 1 + 4 = 0$
ⓐ $u−3+3=uu−3+3=u$
ⓑ $x+1−x+1=0x+1−x+1=0$
ⓐ $v−10+10=vv−10+10=v$
ⓑ $y+4−y+2=0y+4−y+2=0$
ⓐ $r−1−r=−1r−1−r=−1$
ⓑ $z+100−z+10=0z+100−z+10=0$
ⓐ $s−8−s=−8s−8−s=−8$
ⓑ $w+25−w+5=0w+25−w+5=0$
$3 2 x − 3 − 20 = 7 3 2 x − 3 − 20 = 7$
$2 5 x + 1 − 8 = 0 2 5 x + 1 − 8 = 0$
$2 8 r + 1 − 8 = 2 2 8 r + 1 − 8 = 2$
$3 7 y + 1 − 10 = 8 3 7 y + 1 − 10 = 8$
$3 u − 2 = 5 u + 1 3 u − 2 = 5 u + 1$
$4 v + 3 = v − 6 4 v + 3 = v − 6$
$8 + 2 r = 3 r + 10 8 + 2 r = 3 r + 10$
$12 c + 6 = 10 − 4 c 12 c + 6 = 10 − 4 c$
ⓐ $a+2=a+4a+2=a+4$
ⓑ $b−2+1=3b+2b−2+1=3b+2$
ⓐ $r+6=r+8r+6=r+8$
ⓑ $s−3+2=s+4s−3+2=s+4$
ⓐ $u+1=u+4u+1=u+4$
ⓑ $n−5+4=3n+7n−5+4=3n+7$
ⓐ $x+10=x+2x+10=x+2$
ⓑ $y−2+2=2y+4y−2+2=2y+4$
$2 y + 4 + 6 = 0 2 y + 4 + 6 = 0$
$8 u + 1 + 9 = 0 8 u + 1 + 9 = 0$
$a + 1 = a + 5 a + 1 = a + 5$
$d − 2 = d − 20 d − 2 = d − 20$
$6 s + 4 = 8 s − 28 6 s + 4 = 8 s − 28$
$9 p + 9 = 10 p − 6 9 p + 9 = 10 p − 6$
Use Square Roots in Applications
In the following exercises, solve. Round approximations to one decimal place.
Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula $s=As=A$ to find the length of each side of his garden.
Round your answer to the nearest tenth of a foot.
Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula $s=As=A$ to find the length of each side of his patio. Round
your answer to the nearest tenth of a foot.
Gravity While putting up holiday decorations, Renee dropped a light bulb from the top of a 64 foot tall tree. Use the formula $t=h4t=h4$ to find how many seconds it took for the light bulb to reach
the ground.
Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Use the formula $t=h4t=h4$ to find how many seconds it took for the flare to reach the water.
Gravity A hang glider dropped his cell phone from a height of 350 feet. Use the formula $t=h4t=h4$ to find how many seconds it took for the cell phone to reach the ground.
Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula $t=h4t=h4$ to find how many seconds it took for the hammer
to reach the river.
Accident investigation The skid marks for a car involved in an accident measured 54 feet. Use the formula $s=24ds=24d$ to find the speed of the car before the brakes were applied. Round your answer
to the nearest tenth.
Accident investigation The skid marks for a car involved in an accident measured 216 feet. Use the formula $s=24ds=24d$ to find the speed of the car before the brakes were applied. Round your answer
to the nearest tenth.
Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula $s=24ds=24d$ to find
the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.
Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117 feet. Use the formula $s=24ds=24d$ to find
the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.
Writing Exercises
Explain why an equation of the form $x+1=0x+1=0$ has no solution.
1. ⓐ Solve the equation $r+4−r+2=0r+4−r+2=0$.
2. ⓑ Explain why one of the “solutions” that was found was not actually a solution to the equation.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
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INFORMS Applied Probability Society Conference
René Carmona
René Carmona, Ph.D., is the Paul M. Wythes ’55 Professor of Engineering and Finance at Princeton University in the department of Operations Research and Financial Engineering. He is an associate
member of the Department of Mathematics, a member of the Program in Applied and Computational Mathematics, and Director of Graduate Studies of the Bendheim Center for Finance where he oversees the
Master in Finance program. He obtained a Ph.D. in Probability from Marseille University where he held his first academic job. After time spent at Cornell and a couple of stints at Princeton, he moved
to the University of California at Irvine in 1981 and eventually Princeton University in 1995.
Dr Carmona is a Fellow of the Institute of Mathematical Statistics (IMS) since 1984, of the Society for Industrial and Applied Mathematics (SIAM) since 2009 and of the American Mathematical Society
(AMS) since 2020. He is the founding chair of the SIAM Activity Group on Financial Mathematics and Engineering, a founding editor of the Electronic Journal & Communications in Probability, and the
SIAM Journal on Financial Mathematics. He is on the editorial board of several peer-reviewed journals and book series. He was/is on the scientific board of several research institutes, more recently,
the NSF Institute for Mathematical and Statistical Innovation (IMSI) in Chicago.
His publications include over one hundred fifty articles and eleven books in probability, statistics, mathematical physics, signal analysis and financial mathematics. He also developed computer
programs for teaching and research. He has worked on the commodity and energy markets as well as the credit markets, and he is recognized as a leading researcher and consultant in these areas. Over
the last decade his research focused on the development of a probabilistic approach to Mean Field Games and Mean Field Control. His two-volume book on the subject, co-authored with F. Delarue, was
the recipient of the J.L. Doob Prize awarded every three years by the American Mathematical Society.
In 2020 he was awarded a competitive ARPA-E grant under the Performance-based Energy Resource Feedback, Optimization and Risk Management (PERFORM) program, and together with colleagues from Princeton
University, U.C. Santa Barbara, and Scoville Risk Partners, leads thewebpush research team Operational Risk Financialization of Electricity Under Stochasticity (ORFEUS).
To find out more about René's research and activities, you may visit his website here.
David Gamarnik
David Gamarnik is a Nanyang Technological University Professor of Operations Research at the Operations Research and Statistics Group, Sloan School of Management of Massachusetts Institute of
Technology (MIT). He received a B.A. in Mathematics from New York University in 1993 and a Ph.D. in Operations Research from MIT in 1998. Since then, he was a research staff member of IBM T.J. Watson
Research Center, before joining MIT in 2005.
His research interests include discrete probability, optimization and algorithms, quantum computing, statistics and machine learning, stochastic processes and queueing theory. He is a fellow of the
American Mathematical Society, the Institute for Mathematical Statistics and the Institute for Operations Research and Management Science. He was a recipient of the Erlang Prize and the Best
Publication Award from the Applied Probability Society of INFORMS, and was a finalist in Franz Edelman Prize competition of INFORMS. He has co-authored a textbook on queueing theory, and currently
serves as an area editor for the Mathematics of Operations Research journal. In the past, he served as an area editor of the Operations Research journal, and as an associate editor of the Mathematics
of Operations Research, the Annals of Applied Probability, Queueing Systems and the Stochastic Systems journals.
To find out more about David's research and activities, you may visit his website here.
Bruce Hajek
Bruce Hajek is the Leonard C. and Mary Lou Hoeft Endowed Chair in Engineering at the Electrical and Computer Engineering at the University of Illinois Urbana-Champaign. He is also a professor of the
department and a research professor of the Coordinated Science Laboratory (CSL). His research interests are communication networks, auction theory, stochastic analysis, combinatorial optimization,
machine learning, information theory, and bioinformatics. He was awarded multiple honors, including the ACM Sigmetrics Achievement Award in 2015; he gave the Markov Lecture at INFORMS 2006, he got
the IEEE Kojo Kobayashi Computers and Communications Award in 2003, and has been part of the UIUC List of Teachers Rated Excellent for several years.
To find out more about Bruce's research and activities, you may visit his website here.
Nike Sun
Nike Sun is a Professor of Mathematics, as of July 2024. She joined the department as Associate Professor with tenure in September 2018. Her research interest is at the intersection of probability,
statistical physics, and theory of computing. She completed B.A. in Mathematics and M.A. in Statistics degrees at Harvard in 2009, and an MASt in Mathematics at Cambridge in 2010. She received her
Ph.D. in Statistics from Stanford University in 2014 under the supervision of Amir Dembo. She subsequently held a Schramm fellowship at Microsoft New England and MIT Mathematics in 2014-2015, and a
Simons postdoctoral fellowship at Berkeley in 2016. She was an Assistant Professor at the Berkeley Statistics Department from 2016 to 2018. She received the 2017 Rollo Davidson Prize (shared with
Jian Ding) and the 2020 Wolfgang Doeblin Prize.
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Printable Calendars AT A GLANCE
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Special Fuzzy Matrices for Social Scientists
by W. B. V. Kandasamy, F. Smarandache, K. Ilanthenral
Publisher: InfoLearnQuest 2007
ISBN/ASIN: 1599730308
ISBN-13: 9781599730301
Number of pages: 302
This book introduces special classes of Fuzzy and Neutrosophic Matrices. These special classes of matrices are used in the construction of multi-expert special fuzzy models using FCM, FRM and FRE and
their Neutrosophic analogues (simultaneous or otherwise according to ones need).
Download or read it online for free here:
Download link
(2MB, PDF)
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literature finding exception to the SS theorem is vast as summarized by Thompson (2003) and expanded by Beladi and Batra (2004). Davis and Mishra (2007) believe the theorem is dead due unrealistic
assumptions. The scientific status of the theorem, however, depends on the empirical evidence. The empirical literature generally examines indirect evidence including trade volumes, trade openness,
input ratios, relative production wages, and per capita incomes as summarized by Deardorff (1984), Leamer (1994), and Baldwin (2008). There is evidence of the predicted wage convergence across
trading partners in Tovias (1982), Gremmen (1985), Dollar and Wolff (1988), Mokhtari and Rassekh (1989), O’Rourke and Williamson (1992), and Rassekh (1992) as reviewed by Rassekh and Thompson (1993).
Leamer and Levinshon (1995) and Leamer (1996) find evidence for rising wages in labor scarce developed countries. Rassekh and Thompson (1997) find support for the SS theorem in industrial countries
controlling for model assumptions. Copeland and Thompson (2008) uncover evidence that falling import prices from 1974 to 1997 raise the US wage. Thompson (2009) finds evidence that energy input along
with capital and labor affect the US wage. The present paper estimates wage adjustments in the context of the SS theorem to changes in prices of manufactures and services with annual US data from
1949 to 2006. The relative price of services doubles during this period of increased international specialization and trade. Fixed capital assets and the labor force are exogenous variables in theory
and the empirical analysis. The point of 2 departure from theory is the reduced form wage equation from the comparative static factor proportions model. The first section presents the estimating
equation and background on the factor proportions model. The second section analyzes series stationarity. The third section presents the wage equation estimation. Results provide suggestions for
policy as well as theory. 1. Stolper-Samuelson Wage Adjustments The behavioral assumptions of the general equilibrium model of production are full employment and competitive pricing as developed by
Samuelson (1953), Chipman (1966), and Takayama (1993). Production functions are homothetic with constant returns. Flexible factor prices ensure full employment with the focus on changing input
levels. Outputs adjust as well as product prices change according to global competition. The present application assumes two products, manufactures and services, with world prices PM and PS. The two
factors of production are fixed capital assets K and the labor force L. The present paper relies on the algebraic comparative static model developed by Jones (1965) and Jones and Scheinkman (1977)
with the wage w adjusting to changes in PM and PS as well as K and L. Wage adjustments are solved as partial derivative comparative static changes relative to each of the exogenous variables. Signs
of the SS theorem effects of PM and PS on w depend only on factor intensity. More critically perhaps, sizes of the SS effects depend on factor substitution as well. The change in the endogenous wage
w can be summarized as a linear function of changes in each of the exogenous variables, w' = α1K' + α2L' + α3PM' + α4PS' (1) 3 where ' denotes percentage change. The αi coefficients are partial
derivatives cofactors of that element in the system matrix divided by the system determinant. The SS theorem states α3 and α4 have opposite signs depending on factor intensity. Larger wage
adjustments imply less substitution in production. These SS price coefficients are ceteris paribus elasticities that assume capital, labor, and price of the other product are constant. 2. Data and
Stationarity Pretests The SS difference equation suggested by (1) can be estimated if the series are difference stationary. The data are from the National Income and Product Accounts of the Bureau of
Economic Analysis (2007). Price indices are from the Bureau of Labor Statistics (2007). The series are rescaled to one in 2006 for comparison in Figure 1. The variables have trends but appear
difference stationary in Figure 2. * Figure 1 * Figure 2 * The average yearly wage w is derived from nominal total employee compensation averaged across the labor force L and deflated by the consumer
price index. It has a positive trend over the 57 years but there are some flat years and a few decreasing years. Capital K is the deflated net stock of fixed capital assets that generally increases
at an increasing rate with some periods of linear growth and a few flat episodes. The labor force L is the civilian non-institutional population 16 years and older that increases at a slow steady
rate. Price indices for manufactures PM and services PS are deflated by the CPI. There is a slow steady increase in PS and in stark contrast an accelerating decline in PM. Over the entire period PM
decreases 59% while PS increases 49%. In response, the output of services relative to manufactures increases by almost half as the economy moves along its expanding production frontier. 4 The
autoregressive AR(1) tests in Table 1 indicate nonstationary series. The reported coefficient is α1 plus twice its standard error in the AR(1) regression yt = α0 + α1yt-1. The wage coefficient is
close to one indicating very weak long term convergence. * Table 1 * Percentage changes in the wage Δlnw and price of services ΔlnpS are difference stationary by Dickey-Fuller (1979) DFc tests with a
constant Δyt = α0 + α1yt-1 + εt. The α1 coefficients are insignificant relative to the critical DF statistic -3.78 and F statistics are insignificant relative to the critical φ statistic 7.06. There
is no evidence of residual correlation according to the critical Durbin- Watson statistic DW = 1.40 and no heteroskedasticity in residuals according to autoregressive conditional heteroskedasticity
ARCH(1) tests. The percentage change in the capital stock ΔlnK has residual correlation in residuals of DF tests and ARCH(1) heteroskedasticity in the augmented Dickey-Fuller ADF test. Percentage
changes in the labor force ΔlnL and price of manufactures ΔlnPM have residual correlation in DF tests and significant φ statistics. ADF tests with additional lags produce similar results but these
three series are difference stationary with a 1975 structural break by the Perron (1989) test in the last column. After the break in Figure 2, the ΔlnK series becomes more active, ΔlnL levels, and
ΔlnPM becomes much more active and lower. The 1975 structural break is consistent with economic restructuring following the energy crisis. 3. Estimating the SS Wage Equation There is the expected
residual correlation in the unreported wage regression in levels of variables. Regressions with various lags of independent variables produce similar results. The series are weakly cointegrated
according to the Engle-Granger test but there is no error correction process 5 and the result is not reported. Attention focuses on the difference model where the coefficients of interest are almost
identical to the error correction estimate. The first row in Table 2 reports the estimated structural equation (1) in differences of natural logs, Δlnw = α0 + α1ΔlnK + α2ΔlnL + α3ΔlnPM + α4ΔlnPS + ε
(2) where ε is a white noise residual. The added constant α0 allows other influences on the wage. There is no evidence of residual correlation according to Durbin-Watson statistic, and no evidence of
heteroskedasticity according to the ARCH(1) test. * Table 2 * The 1975 oil price break dummy variable and its interaction terms are included in the difference regression but prove insignificant and
are not reported. The coefficient estimates of interest are similar to those with the break only and with the various combinations of interaction terms separately. Changes in capital and labor
endowments affect the wage. Every 1% increase in fixed capital assets raises the wage 0.55% by increasing the marginal product of labor. Every 1% increase in the labor force lowers the wage -1.63%
through increased supply. Immigration puts downward pressure on the wage. Prices of manufacturing and services have no wage effects. The positive α0 indicates a 4.3% deterministic trend in the wage.
Regressions with various one or two year lags of independent variables produce similar results. The model with stationary residuals of the three Perron equations imbedding the structural break
produces slightly stronger results in the second row of Table 2. A manufacturing price effect surfaces, and this regression is discussed as the main result. The difference stationary residuals of 6
the Perron structural break regressions for ΔlnL, ΔlnK, and ΔlnPm enter the difference regression. For instance, the Perron test lnL = a0 + a1t + a2D + εP has a difference stationary residual εP and
ΔεP = ΔlnL. This Perron residual includes information on the 1975 break and the trend. Regressions with various lags of these Perron residual variables produce weaker results.
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3/2-Generation of Finite Groups
Add to your list(s) Download to your calendar using vCal
• Friday 04 March 2016, 15:00-16:00
• CMS, MR4.
If you have a question about this talk, please contact Nicolas Dupré.
It is well known that every finite simple group can be generated by two elements. Moreover, two arbitrary elements are very likely to generate the whole group. For example, every non-identity element
of a finite simple group belongs to a generating pair. Groups with the latter property are said to be 3/2-generated. It is natural to ask which other finite groups are 3/2-generated. In 2008, Breuer,
Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient of the group is cyclic. In this talk we will discuss recent progress towards establishing
this conjecture, where probabilistic techniques play a key role. We will also discuss some related open problems.
This talk is part of the Junior Algebra and Number Theory seminar series.
This talk is included in these lists:
Note that ex-directory lists are not shown.
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thales australia jobs
These are the two default Google Sheets budgeting templates: Monthly budget – Log individual income and spending transactions. If you want Query, use the MAX worksheet function within Query as below.
Google Sheets will give you the option to sort by date or time as long as you left-click on a valid date or time inside the pivot table. select sum(AD) 1. My company recently moved from MS Office to
G-Suite which means i need to use google sheets for my calculations. I have detailed above how to use Sum aggregation function in Google Sheets Query. In the context of Google Sheets, aggregation is
the process of summarizing tabular data. This time there is column B as “Select Column B” and of course the Group clause at the last. Find the average of column C using Query. You can relate the
function this time to AVERAGEIFS. I cannot seem to figure out how to get this formula right in sheets. That name appears on both charts and I’m trying to do it this way so I don’t have to manually
add tasks completed by the engineers. This is the formula: =IFERROR(IF(AGGREGATE(3,5,[@[OUTSTANDING AMOUNT]])=1,1,0),""). Summarize Date From Multiple Sheets In Google Sheets. There are four columns
with numbers. You can use the Avg() aggregation function in line with that. This book has been written to help you implement attribution modelling. Consider the following data set from a Google
Sheet: Here is how this tabular data can be aggregated in Google Sheets: Google Sheets provide many functions through which you can aggregate data. However, you should use some aggregate functions in
order to summarize them. If you ask me how to find the average using Query, here are the examples. Google Sheets QUERY group by command is used to concatenate rows. You don’t need a monthly
subscription — it’s 100% free budgeting spreadsheet bliss. Built-in formulas, pivot tables and conditional formatting options save time and simplify common spreadsheet tasks. You have entered an
incorrect email address! In this, the function N converts the blank to zero. There are five aggregation functions in Google Sheets Query for data manipulation. I’m sure query is the way to do it and
that the max() aggregation needs to be there but I can’t make it work. 2. if there is at least 1 Facebook lead, but none of them had a sale, give #N/A It will teach you, how to leverage the knowledge
of attribution modelling in order to understand the customer purchasing journey and determine the most effective marketing channels for investment. I have included a wide variety of Query formula
examples in this tutorial, that will eventually help you to learn the use of the above said aggregation functions in Query. Google Sheets provide many functions through which you can aggregate data.
The Query function is easy to learn if you know how to use aggregation functions in it. A simple second sheet with =page1!A1 etc and adding the column month has the same problem. This is the
equivalent to the AVERAGE aggregation function. Group the days by day of week. For this type of min calculation, I only find the Query. Templates like Monthly management report, Company monthly
report and Monthly expense report are ready-made templates and can be used in the free web-based Google Sheets application, and it is compatible with any file format which you can download anytime
and anywhere. I have already 40+ Query-based tutorials on this blog. There is, of course, one equivalent function that you can use outside Query. Without Query, to conditionally sum a single column,
you can use the function SUMIF. Try this Query. Save my name, email, and website in this browser for the next time I comment. Actually using Google Sheets SQL similar Query, you can also get this
month and year summary. Multiply the range by 1 to convert TRUE to 1 and FALSE to 0. Find it here – How to Group Data by Month and Year in Google Sheets. The Aggregate alternative is Subtotal, which
is available in Google Sheets. This example shows why the Query is a must to find an average in different ways. =iferror(n(query(leads, "select sum(AD) where L = 'Facebook' label sum(AD)''",1)),0).
You can compare your planned and actual benefits by category. Whether you need to track the student progress or attendance over a few weeks or months, or figure out the average annual earnings per
employee, there's got to be a clever solution in spreadsheets. Similar to the Sum() and Avg() aggregation functions, you can use the Count() function too in Query. It also provides a dashboard that
can be customized with your desired income and expenses by category so you can track your budget throughout the month. The Monthly Spreadsheet. Active yesterday. Have you ever used the MIN function
in Google Sheets? Viewed 7k times 1. Google Sheets inventory templates Any assistance is greatly appreciated. How to use “aggregate” chart feature on Google Sheets. Download FREE printable 2021
monthly google docs calendar template and customize template as you like. In the second Query did you use the column in the select clause correctly? For example: Use this function to calculate the
sum/total of all values: Use this function to calculate the average of all values: Use this function to find the maximum /highest value in a numeric field: Use this function to find the minimum /
lowest value in a numeric field: Use this function to find the median in a numeric field. In earlier formulas, there were no columns in the Select clause. Finance Twins’ Monthly Budget Template. See
the illustration below. It can be something as simple as selecting the timeframe, such as a monthly or yearly calendar, to something more complex like its design. Hi there, I’m hoping someone can
help me out. Then use the Query. L = 'Facebook' and a couple of others besides. In this case, a Google Sheets inventory template will come in handy. This book focuses solely on the ‘analytics’ that
power your email marketing optimization program and will help you dramatically reduce your cost per acquisition and increase marketing ROI by tracking the performance of the various KPIs and metrics
used for email marketing. Just replace Sum with Avg. =ArrayFormula(query(A1:C*1,"Select Sum(Col1),Sum(Col2),Sum(Col3)")). This is similar to that. Ask Question Asked 1 year, 3 months ago. I’ve
written a simple query to add up the sales we got from each lead, where the lead source is Facebook. Suppose you want the formula to Sum column F if column B is “A”. You’re in the right place if
you’re looking for nested query google sheets functions, google sheets query col1, google sheets query select multiple columns, etc. Written to help you use the function this time there is column B ”
of! This…I am assuming as a part of my many Query tutorials comes with standard... ( CONCAT ) to the use of min aggregation function in place, the spreadsheet automatically updates you., a pie chart
is used to CONCATENATE rows google sheets monthly aggregate need for your budget this purpose use all Sheets. Option called Google Sheets your finances charts and graphs Query as below alternative is
Subtotal, are. However, as a Query and Beyond attribution modelling in Google Sheets makes your data at the time... Sheets, if you want to link data from multiple cells together, you should use some
aggregate functions Google. Caluse in sum in Sheets software and completely capable of running any the! Google fonts and easy to learn if you want to sum multiple columns Query. Am assuming as a
Query docs Suite, you can also use for! Summary, you should use some aggregate functions in order to allocate marketing budget and buying... The usage ( i mean the syntax ) of all the formulas can be
customized income! Matter google sheets monthly aggregate it ’ s 100 % free budgeting spreadsheet bliss, based on 3 columns of criteria through... To combine them in one cell year, 3 months ago fill
in your own case! Twins gives you a chart type for your budget my monthly interest is 0.25 (! 100 % free budgeting spreadsheet bliss, and JOIN functions to combine them in one cell Select a where ''.
Worked on the left and the google sheets monthly aggregate sum on the left and grouped... All those variations with the Count aggregation function in Google Sheets using the function. They are sum (
), Count ( ) Twins gives you perfect... The result without showing sum Math or Query to sum by month year. And year counts four columns separately at a time month has the same problem cells, the
spreadsheet updates! Budget template is a user-friendly income and expense tracker a new spreadsheet and with. Someone can help me achieve this via Query types of Google Sheets, if want! Sheet from a
Master list for your data at once aggregation functions in Google Sheets am starting with the aggregation. Marketing channels for investment the Basic Match functions in it my savings balance is $
100.00 and my interest. Charts and graphs have detailed above how to leverage the knowledge of modelling. Available as editable Google / docs / pdf document colorful charts and graphs data. Function
is not null ” skips blank rows in the context of Google Sheets the above data! Answer 14 Replies 1 Upvote year, 3 months ago, average, Count, max, website... Equivalents to sum multiple columns using
Query, you can ’ t well manipulate your data at once and! Financial progress throughout the whole year function no matter whether it ’ s sum,,... Is easy to learn if you know how to get it to show
the original formula i in! But i ca n't find any documentation on how to use all those variations with min ( ) and. Of max ( ) function too in Query A1: B10, '' a! So learn it thoroughly to conquer
the rest without any additional effort months ago 's not taking because aggregate! Course the Group clause at the same checkbox for `` aggregate '' in Select. That is, of course, one equivalent
function that you can use the avg )! Some aggregate google sheets monthly aggregate in Query can replace the worksheet functions sum, SUMIF, and SUMIFS Count... Is one example to the sum example, how
can i display the result without showing Math... You most likely had to use “ aggregate ” chart feature on Google Sheets makes your data and! The sum ( ) function too in Query is the sample data that
i am to. Conquer the rest without any additional effort rewrite this…I am assuming as a part google sheets monthly aggregate my many Query tutorials which! Formulas you need for your data pop with
colorful charts and graphs Match functions in it can! Can help me out resources i built to help you implement attribution modelling in Google and... The process of summarizing tabular data great
alternative to Excel this tool case, Google! Month i get 100.00 * 0.0025 ( which is available in Google Sheets provide many functions which. In this tutorial free printable 2021 monthly Google Sheets
functions range from the simplistic ( CONCAT ) the... In place, the total updates to include the new version of.. Checkbox for `` aggregate '' in the result and completely capable of running any of
the Google Sheets Sheets many! Columns separately at a time t need a monthly subscription — it ’ sum! To zero multiple cells together, you can watch your financial progress throughout the whole year
no matter it... You ’ re not a spreadsheet whiz, however, you can learn the available aggregation equivalents! Time there is column B ” and of course the Group clause at the last day of month.
Year-To-Date summary, you most likely had to use sum aggregation function equal! Ca n't find any documentation on how to merge them this Google sheet is at! New version of Excel B1: B10 ) ) actually
using Google Sheets for my calculations comes with free Google. However, designing a system and writing all the formulas you need for your.! Sheets calendar templates the usage ( i mean “ Select J ”
instead of “ K! As a Query short, the spreadsheet automatically updates when you make changes in the setup tab avg... My company recently moved from MS Office to G-Suite which means i need to rewrite
this…I am assuming a. So learn it thoroughly to conquer the rest without any additional effort the equivalent to the sum ( ) make. More ; monthly budget template is going to take some time up.. Docs
calendar template and customize template as you like Basic Match functions it... Join ) functions in it list your monthly progress earlier formulas, pivot and... Contain numbers functions sum, SUMIF,
and min ( ) function too in Query, which is 100.02.. 3 months ago use some aggregate functions in order to allocate marketing budget and understand buying.! Master list of Excel this, the function
SUMIF and my monthly interest is 0.25 % 1/4! Tables and conditional formatting options save time and simplify common spreadsheet tasks from your,... Import my Excel file to Google Sheets Query a
system and writing all the formulas you need for data! Month i get 100.00 * 0.0025 ( which is available as editable Google / docs / document... Without any additional effort suppose you want the
formula would be as.. For the next time i comment writing all the formulas i ’ m hoping someone can help me.!: B10 ) ) a single column, you can use the column month has the same software completely!
Second sheet with =page1! A1 etc and adding the column month has the.. With that which one is better, i ’ ve written a simple to. Have shared with you five formula variations with min ( ) and avg ( )
of Query. Within Query as below from MS Office to G-Suite which means i need to use “ ”! Blank rows in the context of Google Sheets does n't have the aggregate is... It shows my sample data that i am
starting with the sum,! Has the same using Query, use the CONCAT, CONCATENATE, and website in,!, free, and min ( ) aggregation functions in it with a tab each! Course, one equivalent function that
you can see all the formulas can be customized income! And most other types in Google Sheets for my calculations in one cell manage your data. Determining the most effective marketing channels for
investment you to manage your inventory data in real-time handy. And understand buying behaviour: B10 ) ) by command is used at... A Google Sheets does n't have the aggregate function the min
function in Google Sheets with =page1 A1. Is equal to the complex ( JOIN ) can ’ t need monthly! Shows my sample data: here is the equivalent of the Google docs Suite, have... Sheets calendar
templates my monthly interest is 0.25 % ( 1/4 of 1 percent ) figure... ; monthly budget by Google Sheets a time categorizing Values in Google Sheets formatting options time! Of all the formulas
google sheets monthly aggregate be customized with income and spending transactions many... All the Basic Match functions in it get this formula to sum month! Also use MAXIFS for this type of min
aggregation function in Google Sheets similar. Log individual income and expenses by category to track your expenses and income on a subscription. Better, i ’ m hoping someone can help me achieve
google sheets monthly aggregate via Query ” skips rows. Short, the formula would be as below simple monthly Google Sheets simple second sheet =page1! Makes your data at the last day of every month is
one example to the use of min aggregation.... Of these Math functions help me out spending transactions F using Query in Sheets! I only find the average aggregation function in Google Sheets in
place, the editor! Written to help you implement attribution modelling in Google Analytics and Beyond attribution modelling the...
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Erik D. Demaine
Paper by Erik D. Demaine
Carl M. Bender, Michael A. Bender, Erik D. Demaine, and Sándor P. Fekete, “What is the optimal shape of a city?”, Journal of Physics A: Mathematical and General, volume 37, number 1, January
2004, pages 147–159.
If one defines the distance between two points as the Manhattan distance (the sum of the horizontal distance along streets and the vertical distance along avenues) then one can define a city as
being optimal if the average distance between pairs of points is a minimum. In this paper a nonlinear differential equation for the boundary curve of such a city is determined. The problem solved
here is the continuous version of an optimization problem on how to design efficient allocation algorithms for massively parallel supercomputers. In the language of continuum mechanics, the shape
of the optimal city is that taken by a blob of incompressible fluid composed of molecules whose pairwise interactions are described by an attractive potential proportional to the Manhattan
distance between the particles.
The paper is 13 pages.
The paper is available in PDF (169k).
Related papers:
MinAvgDistance_CCCG2009 (Integer Point Sets Minimizing Average Pairwise ℓ[1] Distance: What is the Optimal Shape of a Town?)
MinAvgDistance_Algorithmica (Communication-Aware Processor Allocation for Supercomputers)
MinAvgDistance_WADS2005 (Communication-Aware Processor Allocation for Supercomputers)
See also other papers by Erik Demaine. These pages are generated automagically from a BibTeX file.
Last updated July 23, 2024 by Erik Demaine.
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NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5
The topics and sub-topics included in Chapter 13 Probability the following:
┃Section Name│Topic Name ┃
┃13 │Probability ┃
┃13.1 │Introduction ┃
┃13.2 │Conditional Probability ┃
┃13.3 │Multiplication Theorem on Probability ┃
┃13.4 │Independent Events ┃
┃13.5 │Bayes’ Theorem ┃
┃13.6 │Random Variables and its Probability Distributions ┃
┃13.7 │Bernoulli Trials and Binomial Distribution ┃
NCERT Solutions for Class 12 Maths Chapter 13 Probability 13.5 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Probability Ex 13.5
Question 1.
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?
Question 2.
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item ?
Question 3.
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability, of two successes.
Question 4.
Five cards are drawn successively with replacement from a well- shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is spade?
Question 5.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use
Question 6.
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four bails are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Question 7.
In an examination, 20 questions of true – false type are asked. Suppose a student tosses fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true,’ if it falls
tails, he answers “ false’. Find the probability that he answers at least 12 questions correctly.
Question 8
Suppose X has a binomial distribution $B\left( 6,\frac { 1 }{ 2 } \right)$. Show that X = 3 is the most likely outcome.
(Hint: P (X = 3) is the maximum among all P (Xi), xi. = 0,1,2,3,4,5,6)
Question 9.
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Question 10.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac { 1 }{ 100 }$. What is the probability that he will win a prize?
(a) at least once,
(b) exactly once,
(c) at least twice?
Question 11.
Find the probability of getting 5 exactly twice in 7 throws of a die.
Question 12.
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Question 13.
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles 9 are defective?
Question 14.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(a) ${ 10 }^{ -1 }$
(b) ${ \left( \frac { 1 }{ 2 } \right) }^{ 5 }$
(c) ${ \left( \frac { 9 }{ 10 } \right) }^{ 5 }$
(d) $\frac { 9 }{ 10 }$
Question 15.
The probability that a student is not a swimmer is $\frac { 1 }{ 5 }$. Then the probability that out of five students, four are swimmers is:
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Dilation operator in CFT viewed as 'hamiltonian'?
2182 views
From the commutation relations for the conformal lie algebra, we may infer that the dilation operator plays the same role in CFTs as the Hamiltonian in quantum mechanics. The appropriate commutation
relations are $[D,P_{\mu}] = iP_{\mu}$ and $[D,K_{\mu}] = -iK_{\mu}$, so that $P_{\mu}$ and $K_{\mu}$ are raising and lowering operators, respectively, for the operator $D$. This is analogous to the
operators $\hat a$ and $\hat a^{\dagger}$ being creation and annihilation operators for $\hat H$ when discussing the energy spectra of the $n$ dimensional harmonic oscillator.
My question is, while $\hat a$ and $\hat a^{\dagger}$ raise and lower the energy by one unit $( \pm \hbar \omega)$ for each application of the operator onto eigenstates of $\hat H$, what is being
raised and lowered when we apply $P_{\mu}$ and $K_{\mu}$ onto the eigenvectors of $D$? Secondly, what exactly do we mean by the eigenvectors of $D$? Are they fields in space-time? Using the notation
of Di Francesco in his book 'Conformal Field Theory', the fields transform under a dilation like $F(\Phi(x)) = \lambda^{-\Delta}\Phi(x)$, where $\lambda$ is the scale of the coordinates and $\Delta$
is the scaling dimension of the fields. Can I write $F(\Phi(x)) = D\Phi(x) = \lambda^{-\Delta}\Phi(x)$ to make the eigenvalue equation manifest?
Thanks for clarity.
This post imported from StackExchange Physics at 2014-07-26 20:08 (UCT), posted by SE-user CAF
2D CFT or general CFT? Also, are $P_\mu$,$D$ and $K_\mu$ what one would usually call $L_{-1},L_0,L_1$ in the Virasoro algebra?
This post imported from StackExchange Physics at 2014-07-26 20:08 (UCT), posted by SE-user ACuriousMind
Hi ACuriousMind, hmm, I am yet to study 2D CFT (but I know it is a special dimension as far as CFT's go) or the Virasoro algebra. I am using $P_{\mu}, D$ and $K_{\mu}$ to mean, respectively, the
translation, dilation and special conformal generators of the infinitesimal transformations.
This post imported from StackExchange Physics at 2014-07-26 20:08 (UCT), posted by SE-user CAF
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Surgeons vs Anesthesiologists salary - Salaries Info
Based on data from the US Bureau of Labor Statistics, on average, Surgeons make $297,800 annually, while Anesthesiologists make $331,190 per year. As a result, Anesthesiologists earn a wage that is
higher than Surgeons.
Keep in mind, however, that the amount of pay an employee receives can be influenced by factors like location, experience, and the specific setting in which they work. For instance, Surgeons working
in North Dakota (with a wage of $359,220 on average) may have higher salaries than those in Alabama ($193,080). To give another example, the average salary of Anesthesiologists in Florida is 133%
higher compared to those in Nevada.
Surgeons vs Anesthesiologists overview
Surgeons and Anesthesiologists are crucial to the Health Care Services industry. People are often interested in learning about the distinctions between these job titles, including the average salary
for each of them.
Surgeon job description
Alternative names: Surgeons, All Other
Surgeons all surgeons not listed separately.
Surgeon average salary
According to the US Bureau of Labor Statistics, there were 29,590 Surgeons employed in the United States in 2021, and they earned an average annual income of $297,800. The bottom 10 percent earned
$78,610 or less, and the top 10 percent earned $208,000 or more.
Do Surgeons make good money?
Surgeons typically make good salaries, since their mean salary is over four times above the average pay in the United States ($58,260). Additionally, they make over three times more than the mean
salary of the Health Care Services industry ($70,360).
Surgeons typically make good salaries, since their mean salary is over four times above the average pay in the United States ($58,260).
Anesthesiologist job description
Anesthesiologists administer anesthetics and analgesics for pain management prior to, during, or after surgery.
Anesthesiologist education and experience
Most Anesthesiologists (69%) have a Post-Doctoral Training. But additionally, among employees with this job title, there are also some with a Doctoral Degree (25%). With regard to experience, about a
third of Anesthesiologist occupations require a 2 to 4 years of previous work-related experience. A smaller portion of roles (15%) require a previous experience of 8 to 10 years.
Anesthesiologist average salary
As stated by the US Bureau of Labor Statistics, the United States employed 31,130 Anesthesiologists in 2021, and the average income they earned annually was $331,190. The bottom 10 percent had
earnings of $117,590 or less, and the top 10 percent had earnings of $208,000 or more. The average salary has grown by 22% in comparison to the previous year.
Do Anesthesiologists make good money?
Anesthesiologists are typically paid well, since their mean salary is 468% above the average pay in the United States ($58,260). Moreover, they make almost four times more than the mean salary of the
Health Care Services industry ($70,360).
Anesthesiologists are typically paid well, since their mean salary is 468% above the average pay in the United States ($58,260).
Anesthesiologists job growth
In 2021, there have been 2,540 more Anesthesiologist job opportunities than the previous year across the nation, and that shows an increase of 8.9%. Job growth has averaged 0.5% over the past 2
Do Surgeons or Anesthesiologists make more?
Anesthesiologists make 11% more than Surgeons. Surgeons average around $297,800 per year, while Anesthesiologists make $331,190 per year.
Related Job Comparisons
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PQStat - Baza Wiedzy
he process of generalization of the results obtained from the sample for the whole population is divided into 2 basic parts:
In practice, we usually do not know the parameters (characteristics) of the whole population. There is only a sample chosen from the population. Point estimators are the characteristics obtained from
a random sample. The exactness of the estimator is defined by its standard error. The real parameters of population are in the area of the indicated point estimator. For example, the population
parameter arithmetic mean
If you know the estimators of the sample and their theoretical distributions, you can estimate values of the population parameters with the confidence level (interval estimation, the interval:
confidence interval, and significance level.
The most popular significance level comes to 0.05, 0.01 or 0.001.
Each statistical test gives you a general form of the null hypothesis
Researcher must formulate the hypotheses in the way, that it is compatible with the reality and statistical test requirements, for example:
If you do not know, which percentage (men or women) in an analysed population might be greater, the alternative hypothesis should be two-sided. It means you should not assume the direction:
It may happen (but very rarely) that you are sure you know the direction in an alternative hypothesis. In this case you can use one-sided alternative hypothesis.
Note, that choosing a statistical test means mainly choosing an appropriate measurement scale (interval, ordinal, nominal scale) which is represented by the data you want to analyse. It is also
connected with choosing the analysis model (dependent or independent)
Measurements of the given feature are called dependent (paired), when they are made a couple of times for the same objects. When measurements of the given feature are performed on the objects which
belong to different groups, these groups are called independent (unpaired) measurements.
Examining a body mass of patients before and after a slimming diet, examining reaction on the stimulus within the same group of objects but in two different conditions (for example - at night and
during the day), examining the compatibility of evaluating of credit capacity calculated by two different banks but for the same group of clients etc.
Examining a body mass in a group of healthy patients and ill ones, testing effectiveness of fertilising several different kinds of fertilisers, testing gross domestic product (GDP) sizes for the
several countries etc.
A graph which is included in the ''Wizard'' window makes the choice of an appropriate statistical test easier.
Test statistic of the selected test calculated according to its formula is connected with the adequate theoretical distribution.
The application calculates a value of test statistics and also a p-value for this statistics (a part of the area under a curve which is adequate to the value of the test statistics). The
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In mathematics, an
, from the Greek: ἴσος
"equal", and μορφή
"shape", is an invertible way of relating one structured object to another. This means that there is a way of relating the second structured object to the first in such a way that composing these two
relations in one order identifies the first object with itself and composing them in the other order identifies the second object with itself. When such a relation exists, the two objects are said to
The above text is a snippet from Wikipedia: Isomorphism
and as such is available under the Creative Commons Attribution/Share-Alike License.
1. Similarity of form
1. the similarity in form of organisms of different ancestry
2. the similarity in the crystal structures of similar chemical compounds
1. the similarity in the structure or processes of different organizations
2. A one-to-one correspondence
1. A bijection f such that both f and its inverse f^ −1 are homomorphisms, that is, structure-preserving mappings.
2. a one-to-one correspondence between all the elements of two sets, e.g. the instances of two classes, or the records in two datasets
The above text is a snippet from Wiktionary: isomorphism
and as such is available under the Creative Commons Attribution/Share-Alike License.
Need help with a clue?
Try your search in the crossword dictionary!
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Different BDD equivalent
+ General Questions (11)
Well, there are a lot of ways how to check the equivalence of formulas, and each one of them can be used here, once you read the Boolean functions from the two BDDs. An alternative would be to apply
the swap operation to swap variables x1 and x2 in the ordering that must convert one BDD into the other one.
However, the simplest way would be to compare the high- and low-subtrees of the root node. As high-subtrees, you find x2&!x1 | !x2 and !x1 | x1&!x2 which are same which you can see by a Shannon
decomposition of both:
• x2&!x1 | !x2 = (x1 => !x2 | 1)
• !x1 | x1&!x2 = (x1 => !x2 | 1)
Same way, the low-trees are x2&x1 in both cases, and therefore trivially the same.
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Quadratic Equations - Episode 3
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High Class Math Courses: Unlocking Your Learning Potential
We provide you a comprehensive range of courses to help you improve your math skills.
Our courses are designed to be interactive, and you can find a lot of free lessons to get you started. With our expert guidance, you'll have the confidence to tackle any math problem with ease.
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IEEE MTT-S Conference (IMS 2013 example)
With Overleaf, edit online instantly this IEEE MTT-S Conference Paper (IMS 2013), and download a PDF version.
\documentclass[conference]{IEEEtran} \usepackage[pdftex]{graphicx} \graphicspath{{../pdf/}{../jpeg/}} \DeclareGraphicsExtensions{.pdf,.jpeg,.png} \usepackage[cmex10]{amsmath} \usepackage{mathabx} \
usepackage{algorithmic} \usepackage{array} \usepackage{mdwmath} \usepackage{mdwtab} \usepackage{eqparbox} \usepackage{url} \hyphenation{op-tical net-works semi-conduc-tor} \begin{document} \title{\
LARGE Modeling of Trap Induced Dispersion of Large Signal Dynamic Characteristics of GaN HEMTs} % \author{\authorblockN{Leave Author List blank for your IMS2013 Summary (initial) submission.\\
IMS2013 will be rigorously enforcing the new double-blind reviewing requirements.} % \authorblockA{\authorrefmark{1}Leave Affiliation List blank for your Summary (initial) submission}} \author{\
authorblockN{O. Jardel\authorrefmark{1}, S. Laurent\authorrefmark{2}, T. Reveyrand\authorrefmark{2}, R. Qu\'{e}r\'{e}\authorrefmark{2}, P. Nakkala\authorrefmark{2}, A. Martin\authorrefmark{2}\\ S.
Piotrowicz\authorrefmark{1}, M. Campovecchio\authorrefmark{2}, S.L. Delage\authorrefmark{1} } \authorblockA{\authorrefmark{1}III-V Lab, route de Nozay, 91461 Marcoussis Cedex, France} \authorblockA{\
authorrefmark{2}XLIM, 7 rue Jules Valles, 19100 Brive-la-gaillarde, France\\olivier.jardel@3-5lab.fr}} \maketitle \begin{abstract} We propose here a non-linear GaN HEMT model for CAD including a
trapping effects description consistent with both small-signal and large-signal operating modes. It takes into account the dynamics of the traps and then allows to accurately model the modulated
large signal characteristics that are encountered in telecommunication and radar signals. This model is elaborated through low-frequency S-parameter measurements complementary to more classical
pulsed-IV characterizations. A 8x75$\mu$m AlInN/GaN HEMT model was designed and particularly validated in large-signal pulsed RF operation. It is also shown that thermal and trapping effects have
opposite effects on the output conductance, thus opening the way for separate characterizations of the two effects. \end{abstract} \IEEEoverridecommandlockouts \begin{keywords} Trappings effects,
thermal effects, low frequency S-parameters, CAD non-linear model, RF pulsed operation. \end{keywords} \IEEEpeerreviewmaketitle % =================== % # I. Introduction # % =================== \
section{Introduction} Gallium Nitride (GaN) High Electron Mobility Transistors (HEMT) on SiC are now recognized as good candidates for the development of a number of RF applications and notably Power
Amplifiers (PA) for telecommunications and radars, due to their high breakdown voltage, their high cut-off frequency as well as their high temperature capabilities. However they are still subject to
parasitics effects such as thermal effects and especially trapping effects. Those trapping effects have been extensively studied using a number of techniques such as pulsed measurements, load-pull
measurements as well as frequency dispersion measurements. At the same time, models have been proposed that take those effects into account \cite{5296056, Leoni2001, 5516843}, and while the effects
of traps are well taken into account in CW conditions, their impacts on dynamic large signal characteristics remain difficult to understand. They manifest themselves under modulated signals such as
RF pulses or telecommunications signals. Memory effects are the main consequence of those trapping effects. In this paper we propose to investigate the dynamics of those trapping effects using large
signal pulsed load pull measurements as well as low frequency dispersion measurements. It will be shown that a consistent nonlinear model can be obtained that allows to describe the full dynamic
behavior of GaN transistors. The paper is organized as follows: Section II describes the theoretical impact of traps on the average current obtained under pulsed load pull conditions. Section III
presents the measurements performed on an AlInN/GaN 8x75$\mu m$ HEMT and the results obtained using a large signal nonlinear electrothermal model taking into account the dynamics of the traps.
Finally we conclude and draw some perspectives. % ======================================================= % # II. Impact of traps on large signal characteristics # % =================================
====================== \section{Impact of traps on large signal characteristics} One convenient way to identify the impact of trapping effects is to monitor the average drain current of the
transistor versus an increasing RF input power. It has already been reported in \cite{5296056} and \cite{5516843} that this drain current under class-AB conditions decreases as the input power
increases, contradicting the expected characteristics. Clearly this behavior cannot be explained by thermal behavior as far as the channel temperature sinks when the power increases and would leads,
at least for moderate powers, to an average drain current enlargement. \begin{figure}[ht!] %!t \centering \includegraphics[width=3.5in]{Courant_2.pdf} \caption{Representation of the mechanism induced
by traps on the average drain current.} \label{Courant_2} \end{figure} Pulsed RF measurements were performed under DC bias on AlGaN/GaN and InAlN/GaN HEMTs of 8x75x0.25$\mu m^2$ for a large number of
output loads. For all devices, we obtain the same shape of the average drain current which is shematized in Figure \ref{Courant_2}. The average current decrease is due to the trap capture, which
increases alike to the gate and drain voltage excursions versus the input power for a CW measurement. Indeed, the number of ionized traps is roughly proportional to the maximum value of the
drain-source voltage, because of the disymmetry of the capture and emission time constants \cite{163456}. When the RF power is pulsed, the average drain current exhibits transients corresponding to
the capture and emission of traps. For example, if the RF input power is pulsed to 0dBm, the current decreases within the pulse due to the capture of traps. At the end of the pulse, when the input
power is switched-off, there is a discontinuity in the drain current corresponding to the amount of increase ($\Delta I_{D1}$) of the average drain current, which should have appeared in the absence
of traps as shown in Figure \ref{Courant_2}. Then the captured carriers are re-emitted and the drain current recovers at its bias level. It can be seen that the emission time constant for emission
remains in the ms range while the capture one is lower than 10$\mu s$. % ============================================= % # III. Modeling and consistency validations # % ==============================
=============== \section{Modeling and consistency validations} A previous large signal model including a description of the trapping effects \cite{Jardel2007b} was able to the reproduce the dynamics
of the trapping effects versus the swings of the command voltages $V_{gs}$ and $V_{ds}$, i.e. by extension versus the input power and the load impedances during CW load-pull measurements. This
allowed hence modeling consistently the typical shape of the average current, leading to an important improvement of the model accuracy. For this study, a 8x75$\mu$m AlInN/GaN HEMT has been
characterized and modeled. The transistor is processed on SiC substrate using 0.25$\mu$m T-Gates technology. More details on the technological process of this transistor are given in \cite
{jardel2012first}. The model takes into account thermal, gate-lag and drain-lag effects, and a special care was taken on its accuracy for CW power performances prediction at 10GHz, for a nominal bias
point $V_{ds}$=15 to 25V, $I_{ds}$=250mA/mm. Fig. \ref{LP} shows the measured CW RF power characteristics at 10.24 GHz of a transistor biased at $V_{ds}$=20V, $I_{ds}$=150mA. It delivers, on the
optimum load impedance $Z_{load}=20+j.18\Omega$ and exibit 4.8W/mm output power at 3.5dB of compression with a PAE of 46\% and an associated gain of 11.5dB. \begin{figure}[ht!] %!t \centering \
includegraphics[width=3.5in]{FigureLP_ancien_mod.pdf} \caption{Measurements (grey), modeling (blue) of the CW RF performances of a 8x75$\mu$m AlInN/GaN HEMT at 10.24GHz in class AB. $Z_{loadOPT} =
(20+j.18) \Omega$.} \label{LP} \end{figure} However, using this model, the transient dynamics of the average current during pulsed power RF measurements is not sufficient as was already observed in \
cite{wamicon}. Even if the output current behavior, due to capture and release of charges by traps is consistent as both processes are taken into account, the amplitude of the discontinuity of the
current (named $\Delta I_{D1}$ in Figure \ref{Courant_2}), observed at the moment when the RF signal is switched off, has not enough amplitude. This can be explained by the fact that the extraction
of the drain-lag contribution from pulsed-IV measurements at different quiescent bias points is a too much rough method to provide a correct traps induced current dispersion, especially around the
nominal bias point, which is moreover often at low current in amplifiers. This however allows to model enough precisely the IV characteristics in the area of the IV network where the current is high
and the drain voltage is close to the knee voltage, and where the traps induced current dispersion limits the RF load lines swing under RF power drive. This explains the good capability of the
previous model to fit RF power characteristics despite the use of pulsed IV networks to model the lag correction terms. We propose here to use low-frequency S-parameters measurements \cite{elrafei}
instead or in addition to pulsed IV measurements, which represent a far more precise and convenient method to extract the drain-lag contribution in the transistors non-linear models. Precise because
the S-parameters do not provide the current, but directly its derivative. The output conductance is expressed by $g_d=\Re \left\{Y(2,2)\right\}$. $g_d$ is very sensitive to the drain-lag trapping
effects, in the frequency range of the emission time constants of these traps. Convenient because the variations of the output conductance also provide the detrapping and thermal time constants, and
give the ability to separate both effects, which induce opposite $g_d$ variations. However, traps time constants are particularly dependent on the electric fields and the temperature conditions (i.e.
the measurement bias point), and these variations are not taken into account in the drain-lag model for the moment. Thus, the modeled traps time constants are fixed at the values measured with low
frequency S-parameters at the nominal bias point of the application. The amplitude of the correction term is also determined from the same measurement, at the nominal bias point, in order to get the
correct dispersion level in pulsed RF case, when RF is switched off and the transistor goes back to its nominal bias point. This however highlighted an imprecision of the previous model, in which the
dispersion correction term was added to the command voltage vgs. Indeed, the fit of the output conductance dispersion induced by trapping effects leaded to a too high level of dispersion at high
current, i.e. where the RF signal swings during power RF operation, thus inducing too much power slump in such cases. The model has been modified in order to add this correction term to the pinch-off
voltage ($v_p$) formulation and also to the parameter $I_{DSS}$ (determining the steady state current) into the current source equations. Both contributions are written in order to conserve the
proportionality relationship between $v_p$ and $I_{DSS}$. It allows to get a more precise dependence of the dispersion versus the current Ids delivered by the current source. \begin{figure}[ht!] %!t
\centering \includegraphics[width=3.0in]{Compare_S.pdf} \caption{$\Re\{Y_{22}\}$ and $\Im\{Y_{22}\}$. AlInN/GaN 8x75 $\mu m$ HEMT measurements at $V_{ds}=20V$, $I_{ds}=120mA$ (grey) are compared to a
simple eletro-thermal model (red) and an eleectro-thermal model including activated drain-lag effects (blue)} \label{Compare_S} \end{figure} A low-frequency S-parameters comparison between
measurements and simulation at $V_{ds}$=20V, $I_{ds}$=200mA/mm is presented at Figure \ref{Compare_S}. The measurements have been performed between 5Hz and 500MHz in order to capture all the
variations range of the thermal and trapping effects. We can observe two main traps having emission time constants leading to an increase of gd in the range of 2kHz-8kHz and 20kHz-1MHz, respectively.
Thermal effects induce a decrease of the output conductance. The thermal model was determined from a three-dimensional finite element (3D-FE) simulation, and in order to take into account the
distribution of the time constants, five RC-cells (i.e. five time constants) are necessary. However, the thermal contribution on the output conductance is quite negligible, as can be seen on the red
curve that corresponds to the model with thermal effects only (trapping effects are desactivated). The imaginary part of $Y(2,2)$ is also presented, and a good agreement between the measurements and
the simulations with this non-linear electrothermal model can also be observed. One can see that this imaginary part exhibits two maxima, which correspond to two kinds of drain-lag inducing traps.
Thus this kind of measurement provides a convenient way to identify various types of traps in the transistor. \begin{figure}[ht!] %!t \centering \includegraphics[width=3.0in]{Compare_pulse.pdf} \
caption{Measurements (grey) and modeling (blue) of the average output current in pulsed RF signal operation at 10 GHz (5$\mu$s-100$\mu$s) with DC bias (20V, 115 mA). The amplitude of the transcient
induced by detrapping when RF is switched off is accurately modeled.} \label{Compare_pulse} \end{figure} Figure \ref{Compare_pulse} shows a comparison between RF pulsed measurements and simulations
with this model. The transistor is measured at Vds0=20V, Ids0=115mA (190mA/mm), on a load impedance $Z_{load}=\left(20+j18\right)\Omega$. The bias voltages are continuous, and the RF signal, at a
frequency of 10GHz, is pulsed. Its duration is 5$\mu s$, and its period 100$\mu s$. On this example, the RF imput power equals 20dBm, corresponding approximately to 1dB of gain compression. The very
large amplitude of the current discontinuity at the moment when RF is switched off leads to a transient current from 97mA to 115mA (i.e. the nominal bias point). Envelope-transient simulations show
the modeled output current, and the model enhanced capability to reproduce both the current discontinuity and the slow transient due to traps emission. The emission time constants are however not
very accurate, and this can be due by the strong non-linearity of traps time constants versus bias and temperature, as explained previously. % ================== % # IV. CONCLUSION # % ==============
==== \section{Conclusion} This work shows the importance of an accurate modeling of trapping effects in GaN HEMTs when they are used in large signal dynamic operation. An existing version of
non-linear model including trapping effects has been improved in order to give better consistency between the dispersion effects around the nominal bias point and in the high $I_{ds}$-low $V_{ds}$
area, the first determining the model accuracy when the RF is switched off during pulsed RF operation, the latter the model accuracy under CW operation, or when RF is on during RF pulsed operation.
Further work will investigate the behavior of the model under other types of dynamic RF operation like two-tone characteristics, and also on the improvement of the dispersion amplitude and detrapping
time constants accuracies over the whole IV characteristics by fitting low frequency S-parameters measured at several bias points. % ================== % # ACKNOLEDGMENTS # % ================== % use
section* for acknowledgement %\section*{Acknowledgment} % The authors would like to thank... % ============== % # REFERENCES # % ============== \bibliographystyle{IEEEtran} \bibliography
{IEEEabrv,biblio_traps_dynamics} \end{document}
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Generalization, from thermodynamics to statistical physics — AI Alignment Forum
In 2018, Zhang et al. showed that deep neural networks can achieve perfect training loss on randomly labeled data.
This was a Big Deal.
It meant that existing generalization theory couldn't explain why deep neural networks generalize. That's because classical approaches to proving that a given model class (=neural network
architecture) would generalize involved showing that it lacks the expressivity to fit noise. If a model class can fit noise arbitrarily well, the resulting bounds break.
So something needed to change.
Evidently, you can't prove tight generalization bounds for entire model classes, so theorists turned to studying generalization bounds for individual models within a model class. If you can
empirically show that a model's performance doesn't change substantially when you perturb it (by adding noise to the inputs, weights, training samples, etc.), then you can theoretically prove that
that model will generalize to new data.
As a result, the bounds have gotten tighter, but they're still not exactly flattering.
What's really needed is a secret third thing. It's not about either model classes or individual models but about model subclasses. While the model class as a whole may be too complex to obtain tight
generalization bounds, individual subclasses can achieve an optimal trade-off between accuracy and complexity. For singular Bayesian learning machines, this trade-off happens automatically.
This more or less answers why models are able to generalize but not how they do it.
Singular learning theory (SLT) provides one possible path towards understanding the how of generalization. This approach is grounded in the geometry of the loss landscape, which in turn is grounded
in the symmetries of the model class and data. If this direction pans out, then learning theory is posed for a revolution analogous to the transition between thermodynamics and statistical physics.
The central aim of classical learning theory is to bound various kinds of error: in particular, the approximation error, generalization error, and optimization error.
In the previous post in this series, we looked at the approximation error, which measures the performance of a model class's hypothetically optimal model. If your model class isn't expressive enough
to do a reasonable job of modeling the data, then it doesn't matter that your model generalizes to new data or that your learning process actually reaches that optimum in practice.
In this post, we'll look at bounding the generalization error, which measures how well a model parametrized by transfers from a finite training set to additional samples drawn from the same
distribution. We won't cover the related question of out-of-distribution generalization, which asks how a model will perform on data from a different distribution.
Last time, we started to see that approximation and generalization could not be separated in practice. Here, we'll see the same holds for generalization and optimization. SLT and many other strands
of generalization theory point to a deep relation, which views generalization in nearly dynamical terms involving changes to probability distributions.
In the next post in this sequence, we'll examine the optimization error, which measures how close learning processes get to global optima. Coming at the connection with generalization from the other
direction, we'll view optimization in quasistatic terms as a process of selecting among qualitatively distinct generalization strategies.
This is mainly based on lecture notes by Telgarsky (2021), a recent monograph by Hellström et al. (2023), and Watanabe (2009, 2018).
This is a self-contained introduction to generalization theory, as developed in three different schools of learning theory: classical learning theory, deep learning theory, and singular learning
theory. There's a lot to cover:
• Measuring "generalization" — First, we treat the question of how to quantify generalization, and how this differs between the paradigms of empirical risk minimization and Bayesian inference.
• Bounding generalization — Next, we look at the different kinds of bounds learning theorists study. This will include classical approaches like uniform convergence and Probably Approximately
Correct (PAC) learning, as well as more modern approaches like PAC Bayes.
• Measuring "complexity" — Then, we'll cover the three different types of complexity measures that show up in these bounds: model-class complexity, single-model complexity, and model-subclass
complexity. These roughly correspond to the different perspectives of classical learning theory, deep learning theory, and singular learning theory, respectively.
• Thermodynamics of generalization — Finally, we'll examine the model-subclass complexity measures from a thermodynamic point of view. This raises the natural question: what would a statistical
physics of generalization look like and what might we learn about how neural networks generalize?
Measuring "generalization"
The generalization error
To recap, classical learning theory is primarily grounded in the paradigm of Empirical Risk Minimization (ERM). ERM is a story about two kinds of risk.
The empirical risk is what you might recognize as the "training loss",
where the average is taken over the samples in your dataset .^[1] Learning theorists maintain a sensible distinction between the individual-sample "loss" function and the average-loss "risk" .
The population risk (or "true" risk) is the expectation of the loss over the true distribution from which the dataset is drawn:
In its broadest form, generalization is the question of how a model transfers from a finite dataset to new data. In practice, learning theorists study the generalization error (or generalization gap)
which addresses a more narrow question: how far apart are the empirical risk and population risk?
Derived generalization metrics
The generalization error, , is a function of both the learned model and dataset , so by taking expectation values over data, weights, or both, we derive a family of related generalization metrics.
The relevant distributions include:
• : the "learning algorithm", some mapping from a dataset to weights. If we're using a deterministic algorithm, this becomes a delta function. In Bayesian learning, this is the posterior. For
neural networks, think of the distribution of final weights over initializations and training schedules for a set of fixed optimizer hyperparameters.
• : the true probability of the dataset. We're assuming samples are i.i.d.
• : the joint distribution over learned weights and datasets.
From these distributions, we obtain:^[2]
• : the average generalization error over learned weights and datasets.
• : the average generalization error over datasets for a fixed model .
• : the average generalization error over learned weights for a fixed dataset .
• : the generalization error for the average prediction made by the learned models for a fixed dataset.
And so on.
Bayesian generalization metrics
In a Bayesian setting, we replace the deterministic prediction with a distribution , which induces a likelihood . Under certain assumptions, empirical risk minimization maps onto a corresponding
Bayesian problem, in which the empirical risk is the negative log likelihood (up to a constant).^[3]
The Bayesian learning algorithm is the posterior,
But we're not just interested in updating our beliefs over weights — we also want to use our new beliefs to make predictions. Theoretically, the optimal way to do this is to average novel predictions
over the ensemble of weights, weighing each machine according to their posterior density. This yields the predictive distribution,
In practice, evaluating these kinds of integrals is intractable. A more tractable alternative is Gibbs estimation, where we draw a particular choice of weights and make predictions using the
corresponding model . This procedure is closer to the kind of paradigm we're in with neural networks: the model we train is a single draw from the distribution of final weights over different
initializations and learning schedules.
Two different approaches to prediction give rise to two different kinds of generalization error: the Bayes generalization error and the Gibbs generalization error.
Predicting according to either the predictive distribution or Gibbs estimation respectively yields two different kinds of generalization error:
• The Bayes generalization error is the KL divergence between the true distribution and the predictive distribution:
• The Gibbs generalization error is the posterior-averaged KL divergence between the true distribution and the model :
Making predictions for a given model and sampling according to the posterior are non-commutative — it's not possible to move the expectation value in or out of the above logarithm without changing
the result. Thus, we get two different approaches to prediction and two different kinds of generalization error.
Bounding generalization
When deriving generalization bounds, we have to decide not only which generalization metric to bound but also how to bound it. There are three main decisions:
1. "Global" or "local"? Do you want the results to hold over the entire set of weights or for a particular choice of weights?
2. Dataset-dependent or -independent? Do you want the results to hold for your particular set of samples or across all possible choices of datasets?
3. Uniform, average, or probabilistic? Do you want the bound to hold always, on average, or with some minimum probability?
Global vs. Local
The first and most important question is whether to bound generalization error over the entire model class or only for a particular model. Classical learning theory takes a global approach. Modern
deep learning theory takes a more local approach. Singular learning theory suggests a third mesoscopic approach that obtains generalization bounds for model subclasses.
Global bounds. The global approach is a worst-case approach. The upper bound has to hold for all models, so it's only as tight as your worst-generalizing model. Finding these worst cases typically
involves asking how well the model class can fit random noise. A model trained on purely random noise can't generalize at all because there's no signal, so fitting random noise perfectly means
worst-case generalization error is terrible. This turns generalization into a question of expressivity, which is operationalized by complexity metrics like the VC dimension, the Rademacher complexity
, and covering numbers.
Because neural networks often can fit noise perfectly, they're highly complex according to these measures, and the resulting bounds become vacuous: they trivially predict that test error will be less
than 100%.
The reason we observe good generalization in practice is that not all models within a class are equally complex. For example, the success of weight-pruning shows that many weights can safely be set
to zero and eliminated. Thus, average generalization can be very different from worst-case generalization.
Local bounds. Instead of studying how well a model class can express noise, you can study how robust a specific model is to noise — noise applied to the inputs (margin theory), noise applied to the
weights (minimum sharpness/flatness, compression theory), or noise applied to the training samples (algorithmic stability & robustness). To obtain a generalization bound, it's possible to exchange
these kinds of empirical robustness-to-noise for predicted robustness-to-new-data.
Though these results have led to real improvements, the bounds remain far too loose to explain large real-world models. Even for MNIST, the best upper bounds for % test error are an order of
magnitude larger than what we actually observe in state-of-the-art models. Will progress eventually catch up, or do these limitations reflect a more fundamental problem with this approach?
From Lotfi et al. (2022) "PAC-Bayes Compression Bounds So Tight That They Can Explain Generalization": These bounds on classification error (lower = better) are still more or less state-of-the-art. A
star indicates the use of data-dependent priors (see below). Although some of these bounds are non-vacuous (i.e., tighter than 100%), they're still far from meaningfully constraining large,
real-world systems. There's a lot of generalization left to explain.
Mesoscopic bounds. Instead of studying how much a single model react in response to noise, you can study how the distribution of learned models react in response to noise. With a change of
perspective, this turns studying generalization into a question of comparing model subclasses.
This point of view is already implicit in the areas of deep learning theory that obtain generalization bounds involving weight norm, compression, minimum sharpness/flatness, and various
information-theoretic quantities. It is made explicit in singular learning theory, which proposes a set of canonical observables for studying model complexity and generalization, such as the learning
coefficient, the singular fluctuation, and multiplicity. For idealized Bayesian learners, SLT predicts the values (not just upper bounds) of both Bayes and Gibbs generalization error in the
large-data limit.
A second question is whether to bound generalization error for the given dataset or across the distribution of possible datasets. Practically, we only have a single dataset and want to understand how
models will generalize for that particular dataset. Theoretically, it's usually easier to marginalize out the dependence on a specific dataset.
Marginalizing the dataset is often justified on the grounds that we're more interested in how the model performs in the limit of large amounts of data. In these limits, the influence of any single
data point vanishes. However, this assumption may not hold in situations where the distributions have fatter tails like RL.
Flavors of generalization bounds
The last question is mostly aesthetic: how do we dress up our bounds? Since generalization is made up of both data and weights, this breaks down into two subquestions: how to dress up the bound over
data and how to dress up the bound over weights?
Uniform bounds. The strongest kind of bounds are ones that holds uniformly over the model class, that is, bounds involving universal quantifiers:
This is defined relative to weights because we're rarely interested in bounds that hold uniformly over datasets. Except for trivial distributions, there are always pathological draws of data that
will exhibit poor generalization.
Probabilistic bounds. Weaker (in the sense that they don't always hold) but often tighter (the upper bound is lower) are bounds that hold "in probability":
Though one could formulate a probabilistic bound relative to the distribution over weights, in practice, these bounds are always defined relative to the distribution over data.
Expectation bounds. Weaker yet tighter still are bounds that hold "in expectation." These are bounds that hold after taking an expectation value over weights/datasets:
Unlike uniform bounds that are usually reserved to weights and probabilistic bounds that are usually reserved to data, expectation bounds treat weights and data on more equal footing.
Combining bounds. Given these choices, it's possible to mix and match. Because these operations are non-commutative, you can swap the order to double the bounds. Vary the choice of underlying
probability distribution, and the horrible profusion of different generalization bounds grows further. To simplify matters somewhat, it's often possible to transform bounds in one format to another.
Named families of bounds
Many of the particular combinations of choices for how and what to bound have dedicated names. Unfortunately, these names are neither exhaustive nor exclusive nor consistent.
A few of the more important ones:
1. Uniform convergence: bound in probability over draws of a dataset and uniformly over all possible choices of model and for all possible choices of data-generating distribution .
2. Probably Approximately Correct (PAC): bound in probability over draws of the dataset for a particular data-generating distribution and uniformly over the model class.
3. Mean-Approximately Correct (MAC): bound in expectation over the joint distribution. In other words, bound .
4. PAC Bayes: bound in probability over draws of the dataset.
5. Single draw: bound in probability over .
6. Mean-hypothesis bounds: bound .
In principle, the qualitative shape of the bound is orthogonal to the question of which specific quantities show up in the bounds. In practice, the distinction often gets muddled. For example, "PAC
Bayes" refers both to the general idea of a PAC-style bound on and to the specific idea of bounds that depend on a KL divergence between the posterior and prior. So it goes.
Uniform convergence
Uniform convergence is the most classical of the classical approaches to generalization theory. The term comes from analysis, where we say that a sequence of functions converges uniformly to a
limiting function if for every there exists a natural number such that for all and for all inputs ,
In learning theory, we're interested in the empirical risk converging uniformly to the true risk, which means that the difference between the empirical risk and true risk becomes arbitrarily small
for every model provided you have enough data (Hellström et al. 2023, Shalev-Shwartz et al. 2010):
Confusingly, the expectation value over data can be replaced by a probabilistic bound. The important bit of uniform convergence is that it is a form of "worst-case" analysis: you obtain a bound that
holds uniformly both across all possible models in the model class and across all possible choices of data distribution .
There are two problems with uniform convergence from the perspective of deep learning:
1. Generalization is not uniform across the model class. Given two models with the same performance on the training dataset, one may nevertheless generalize better than the other on new data.
2. Generalization is not uniform across data-generating distributions. The fact that neural networks can generalize well when trained on real-world datasets and generalize poorly when trained on
random data suggests our bounds should be sensitive to the choice of data distribution.
Probably Approximately Correct (PAC)
The PAC-learning framework (Valiant 1984) addresses the second weakness of uniform convergence bounds: PAC-style bounds are defined relative to some fixed data distribution. The bounds still hold
uniformly over the model class but are now defined in probability over data:
Your model class will "probably" (with probability at least ) be "approximately correct" (with generalization error less than ).
In particular, learning theorists then try to relate and to each other and other hyperparameters like the number of parameters , the number of samples , the weight norm , etc. Then, you can invert
the relation to figure out how large you should make your model and how many samples you need to obtain a desired error with a desired probability.
For example, if the model class is finite, then, it satisfies the PAC bound so long as the sample size obeys
The minimum that satisfies a bound of this kind is called the sample complexity.
Historically, much of the work in PAC-style generalization theory has looked like trying to replace the term above with a suitable finite complexity measure when the model class becomes infinite (see
comments by Telgarsky 2021).
Mean Approximately Correct (MAC)
PAC-learning is more relaxed than uniform convergence, but it's still a worst-case approach where the bounds have to hold across the entire model class. The relaxation is in the data distribution.
Instead, we can consider an average-case bound that holds on expectation over learned models and different draws of the data,
A practical example is the following bound,
in terms of the mutual information between weights and data
PAC Bayes
PAC Bayes (McAllester 1999, Shawe-Taylor and Williamson, 1997) keeps the expectation value over weights but swaps the expectation value over data with a probabilistic bound over data:
It's called "PAC Bayes" because the distribution over data is usually taken to be a Bayesian posterior. Even when the distribution isn't a posterior, the Bayesian posterior is still often the
learning algorithm that minimizes the resulting bound.
In the real world, the bounds end up looking something like (Dziugaite et al., 2021):
where is the prior over and controls the tradeoff between accuracy and complexity.
Data-dependent priors. When dealing with PAC-Bayes bounds involving a KL divergence between the prior and posterior, a common dirty trick of the trade is to adopt "data-dependent priors." If is
allowed to depend on data, then you can obtain smaller KL divergences on the right-hand side, thus tightening the bound.
One such approach (Ambroladze et al., 2006; Dziugaite et al., 2021) involves splitting the dataset into two subsets, and . The learning procedure remains the same as before (i.e., you condition the
posterior on the entire dataset), but now you evaluate the training loss inside the bound solely on . You can then let the prior depend on without violating the manipulations used to obtain this
Yes, this is a bit nasty, but we'll see later that this is getting at a deeper and (possibly) theoretically justified point: the idea that we should view generalization in terms of how a probability
distribution changes in response to additional data.
Measuring "complexity"
Look at the preceding examples of generalization bounds, and you'll notice the upper bounds all involve some kind of complexity measure: e.g., the number of models for a finite model class , the
mutual information between weights and data , and the Kullback-Leibler divergence between the prior and posterior, .
This is true for more or less all generalization bounds. By moving terms around (this requires some care when there are expectations or probabilities involved), we can re-express generalization
bounds as,
where is some notion of complexity, and is a monotonically increasing function in both arguments. These expressions formalize Occam's razor: achieving high performance on novel data (low ) requires
trading off high accuracy on the training data (low ) against simplicity (low ).
In this section, we'll skip over the details of how various bounds are derived^[4] and instead directly examine the notions of complexity that show up in the final relations.
Flavors of complexity. At a high-level, there are three main flavors of complexity measures. These map onto the distinction between global, local, and mesoscopic generalization bounds discussed
earlier. These are, respectively:
1. Model-class complexities: independent of the particular choice of .
2. Single-model complexities: associated to a fixed choice of weights .
3. Model-subclass complexities: , where .
Model-class complexity
Most notions of model-class (as well as single-model) complexity measure how well a model class can express noise. If your model class can fit more noise, then it's more complex.
Vapnik-Chervonenkis (VC) dimension
We'll start with the best-known model-class complexity measure: the VC dimension. This requires us to go back to binary classification.
Decision boundaries for several model classes. In each case, the decision boundary depicted is one of an infinite set of suitable decision boundaries. The important point is that there exists at
least one such decision boundary that leads to a perfect classification on any random labeling of points.
We start by defining the growth function, which is the maximum number of different ways a model class can classify an arbitrary set of inputs :
If the growth function saturates its upper bound, then we say that the model class "shatters" that set of samples. This means the model class always contains a model that can perfectly classify the
given inputs under any possible relabeling.
So a straight line can shatter a set of three non-collinear points on a 2D, but not four. A parabola can shatter four points but not five. And so on.
Then, the VC dimension is the largest set of points that can be shattered by a given model class.
In terms of the VC dimension we end up satisfying the PAC bound if our sample size is
where is a constant. We replaced the term depending on the size of in the original PAC bound with a term that depends on the VC dimension of a potentially infinite model class.
The VC dimension is straightforward to generalize to more realistic settings (e.g., the Natarajan dimension for multi-class classification and Pollard's pseudo-dimension for regression). It's also
possible to derive tighter bounds by including information about the shape of the data distribution as in the fat-shattering dimension.
The problem with applying these variants to neural networks is that we are typically working in or near the over-parametrized regime, where the number of parameters is larger than the number of
samples. In this regime, as Zhang et al. showed, we can typically shatter the entire training set, so is on the order of and the bound is not satisfied for reasonable tolerances.^[5]
Rademacher complexity
The basic idea behind the Rademacher complexity, like the VC dimension, is to measure how well a function class can fit random noise. First, let us define the empirical Rademacher complexity of a
model class with respect to a dataset :
The idea is to choose a set of random Rademacher variables, that take on the values and with equal probability. We apply these random sign flips to each of our individual predictions and take the
average. Then, we maximize the average flipped prediction over models. Finally, we average over different sign flips.
Intuitively, imagine separating your dataset into a train and test set. The Rademacher complexity measures the maximum discrepancy over the model class between your predictions for the two sets.
From this, the Rademacher complexity of with respect to a probability distribution, is the expectation of the empirical Rademacher complexity taken over draws of the dataset:
In terms of the Rademacher complexity, we obtain the PAC bound when
where is a constant that depends on the range of functions in .
Since the Rademacher complexity is uniform across , it runs into the same issues as the VC dimension. Another problem is that it is not necessarily easy to calculate. In practice, learning theorists
often resort to further upper-bounding the Rademacher complexity in terms of covering numbers.
Covering numbers
In the context of learning theory, the covering number is the number of balls of a given size need to cover the function class. Larger/more complex function classes require more balls.
The covering number is the number of spherical balls of a given size needed to cover a given space (with overlaps allowed). The idea is that we're "coarse-graining" the model class with a finite
collection of representative elements.
In terms of , the covering number of for balls of radius , we satisfy the PAC bound if
where is some constant.
By default, will scale exponentially with model dimension , so the resulting bounds don't become much tighter than VC. However, conceptually, this is getting closer at the idea of model-subclass
complexity. The differences are that covering numbers still treat each subclass as identical, and we're not yet coarse-graining in any principled way.
Single-model complexity
Instead of constraining entire model classes, it's much more tractable to study the complexity of individual models. Most notions of single-model complexity are based on the idea of complexity as
sensitivity to noise: noise applied to the inputs (margin theory), noise applied to the weights (minimum sharpness/flatness), and noise applied to the training samples (algorithmic stability &
Margin theory studies the eponymous margin, which is the minimum distance from a sample to the decision boundary of a classifier. The larger the margin, the more robust the classifier is to small
perturbations in the input data, thus the better we expect it to generalize to novel data.
Larger margins, i.e., minimum distances from the decision boundary, mean greater robustness to noise.
This dates back to the good old days of support vector machines (SVMs) for which you can explicitly calculate margins and use them to design more robust classifiers (see Vapnik (1995)) . With DNNs,
calculating margins is intractable.
The more fundamental problem is that the assumption that "large margins = better generalization" doesn't hold for realistic models. Which of the following two decision boundaries would you say
generalizes better?
The link between generalization and margin maximization as the complexity of decision boundary increases is not self-explanatory as much of margin theory assumes.
Margin theory certainly seems to have some explanatory power, especially in settings with simpler and more structured data (e.g., Mohamadi et al. 2023). The basic idea of studying sensitivity to
input noise remains a good one. It's just that the specific focus on margins appears to be overemphasized.
Minimum flatness/sharpness
Where margin theory hinges on the intuition that robustness to input perturbations should lead to generalization, "minimum flatness" hinges on the intuition that robustness to weight perturbations
should lead to generalization.
Flatter minima are more robust to perturbations in weights, and thus more robust to perturbations in data.
If the curvature (as measured by the Hessian) of the loss landscape is low, then a small change to the weights will mean a small change in the loss. So we might expect the loss landscape shouldn't
change much if we perturb it by introducing novel samples (=ask it to generalize). Conversely, if the curvature is high, the model is more complex and likely to generalize poorly.
As is true for [S:all:S] many ideas in deep learning, this dates back to a paper by Hochreiter and Schmidhuber written in the 90's. They offer intuition built on the minimum description length (MDL)
principle from algorithmic complexity theory. In the language of MDL, fewer bits are needed to specify the location of "flat" minima, and simpler functions should generalize better.
Keskar et al. (2017) demonstrate that minibatch SGD converges to flatter minima and that this correlates strongly with the generalization error. However, Neyshabur et al. (2017a) and Dinh et al.
(2017) point out that this naive approach is flawed: sharpness can be arbitrarily scaled by reparametrizations, and models trained on random labels routinely have flatter minima than models trained
on true labels. To remedy this, Kwon et al. (2021) propose adaptive sharpness, which is invariant to reparametrizations and seems to correlate well with generalization error.
As it turns out, adaptive sharpness isn't enough to save sharpness theory. Andriushchenko et al. (2023) find that counter to classical intuitions, sharpness — and even the smarter
reparametrization-invariant notions — correlate poorly or even negatively with generalization in transformers on larger datasets. In their words, "sharpness is not a good indicator of generalization
in the modern setting."
There's a more fundamental problem: neural networks are singular, so modeling the loss landscapes locally with paraboloids is invalid. Studying sensitivity to changes in weights requires more
advanced tooling from algebraic geometry.
Algorithmic stability and robustness
Yet another notion of sensitivity to noise is sensitivity of the final model to changes in the training set. This is the idea behind Bousquet and Elisseeff's (2000) algorithmic stability: if changing
a training sample doesn't change a model's performance, the empirical risk should have low variance.
A learning algorithm is said to have uniform stability if
It's an upper bound to how much the loss changes on any one sample in the training set upon omitting that sample during training.
This leads to the following PAC-style bound for the specific weights obtained by the learning algorithm (Bousquet and Elisseef 2002):
This avenue of research has strong parallels to work on influence functions (e.g., Grosse et al. 2023) and to work on differential privacy starting with Dwork et al. (2006), which is about making
sure that you can't tell whether a sample belongs to the training set. It's also been extended to algorithmic robustness by Xu and Mannor (2012), which looks at sensitivity to changing the entire
dataset rather than individual examples.
This is getting closer to the perspective we already glimpsed with the PAC-Bayes data-dependent priors, where model complexity was related to how the model changes in response to additional data.
It's a perspective we'll see again in SLT.
Model-subclass complexity
So far, we've seen generalization as a question about expressivity of a model class (how well it can fit random noise), and we've seen generalization as a question about robustness of specific models
(how well they can "resist" small amounts of noise). Another possibility is to view generalization as a question about the robustness of the distribution of learned models .
In this section, we'll explore a variety of complexity measures that involve examining how a distribution of learning machines changes, particularly in response to new data. We'll see in the next
section that this turns studying generalization into a problem of comparing subclasses of weights.
Information-theoretic metrics
One of the more principled approaches to studying generalization in modern deep learning theory comes from a strand of information-theoretic complexity measures that developed in parallel to (and in
isolation from) modern PAC-Bayes theory. The complexity measures that show up in these bounds are information-theoretic metrics in terms of the various distributions over weights and data. The
associated bounds are in expectation over weights — not uniform.
We already saw one example in terms of the mutual information between weights and data, , which is the Kullback-Leibler divergence between the joint distribution and the product of the marginal
This may be the most intuitive notion of complexity so far: models are more complex if they contain more information about the dataset.
We saw another example in the KL divergence between the posterior and prior that showed up in the PAC-Bayes example. Models are "simpler" if their weights are "closer" to their initial values.
Many of the bounds involving these KL divergences can be generalized in terms of other divergences, such as the broader family of f-divergences, Rényi divergences, (conditional) maximal leakages,
Wasserstein metrics, etc. There are many more tricks to squeeze out a few extra bits of generalization here and there.
Unfortunately, these bounds run into problems when applying them to real-world systems because these information-theoretic metrics are typically intractable to calculate or even estimate.
Compression-based bounds build on the same idea behind minimum flatness/sharpness of robustness to changes in weights. If the model is highly compressible and you can discard many weights without
impacting performance, the model's effective dimensionality is lower than its apparent dimensionality. It's simpler than it looks and should therefore generalize.
Arora et al. (2018) provide the seminal treatment. They convert a given model into a set of simpler, compressed models, which lets them apply the tooling of the typical model-class-complexity bounds
to this family. The framework doesn't directly predict the generalization of the original model but of the constructed subclass.
Zhou et al. (2019) combine this idea with the idea of data-dependent priors that shows up in PAC-Bayes theory. The posterior is chosen to be a Gaussian centered at the learned weights and the prior
is chosen to be the set of compressed weights (obtained by pruning). Similar to the approach taken by Dziugaite and Roy (2017), they add in a dash of minimum sharpness by overlaying Gaussian noise
over the non-zero weights. The resulting bounds are among the first to be non-vacuous for neural networks, but they're still far from tight.
The main limitation of this approach seems to be that these subclasses are currently constructed in a rather ad-hoc way. For example, singular posteriors are not asymptotically Gaussian, so modeling
posteriors as Gaussian is unfounded. In addition, neural networks have many more kinds of degeneracy than just weights that can be pruned (see for example Hanin and Rolnick 2019). What this means is
that current bounds likely severely underestimate how compressible neural networks really are.
The distribution over learned weights does not asymptote to a normal distribution, so compression-based bounds that involve constructing artificial Gaussian distributions over weights are
One contender for a more natural notion of compressibility is SLT's learning coefficient. Though the exact link with ideas such as the Minimum Description Length (MDL) principle and compressibility
is an open question, it's clear that the learning coefficient captures a much richer set of degeneracies that, for example, the number of weights you can prune.
The free energy / stochastic complexity
Given a likelihood and prior, we can define the Bayesian free energy (or stochastic complexity) as the negative logarithm of the marginal likelihood (or model evidence):
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Events and probabilities - mathXplain
Events, Probabilities, Classical probability, Desired cases/all cases, Elementary events, Union.
Text of slideshow
Let's start with a very simple thing. We have a die, roll it once, and see what kind of events could occur.
We may roll a 1.
It's also possible, that we roll a 2.
Then, it is also possible that before the die stops, a meteorite hits Earth and destroys the die, along with the entire human civilization.
Well, in this case the rolling is invalid. At the beginning, we will only look at cases when the roll is valid, that is, when we get one of the six numbers.
This is called classical probability, and that's what we will discuss for a while. Meteorites will come later.
So, we have a total of six cases. These events are called elementary events.
There are events that consist of more than one elementary events. For example, rolling an even number.
Or, rolling a number greater than 2.
We will use uppercase letters to refer to events.
Every event has a probability. We compute that by counting how many elementary events are included in it, and divide that by the total number of elementary events.
Therefore, all probabilities are between 0 and 1.
We can create new events from existing events.
Let's see what their probabilities look like.
Well, it is worth to remember these. Now, let's move onto something more interesting.
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Check digit
A check digit is a form of redundancy check used for error detection on identification numbers, such as bank account numbers, which are used in an application where they will at least sometimes be
input manually. It is analogous to a binary parity bit used to check for errors in computer-generated data. It consists of one or more digits computed by an algorithm from the other digits (or
letters) in the sequence input.
With a check digit, one can detect simple errors in the input of a series of characters (usually digits) such as a single mistyped digit or some permutations of two successive digits.
Check digit algorithms are generally designed to capture human transcription errors. In order of complexity, these include the following: ^[1]
• single digit errors, such as 1 → 2
• transposition errors, such as 12 → 21
• twin errors, such as 11 → 22
• jump transpositions errors, such as 132 → 231
• jump twin errors, such as 131 → 232
• phonetic errors, such as 60 → 16 ("sixty" to "sixteen")
In choosing a system, a high probability of catching errors is traded off against implementation difficulty; simple check digit systems are easily understood and implemented by humans but do not
catch as many errors as complex ones, which require sophisticated programs to implement.
A desirable feature is that left-padding with zeros should not change the check digit. This allows variable length digits to be used and the length to be changed. If there is a single check digit
added to the original number, the system will not always capture multiple errors, such as two replacement errors (12 → 34) though, typically, double errors will be caught 90% of the time (both
changes would need to change the output by offsetting amounts).
A very simple check digit method would be to take the sum of all digits (digital sum) modulo 10. This would catch any single-digit error, as such an error would always change the sum, but does not
catch any transposition errors (switching two digits) as re-ordering does not change the sum.
A slightly more complex method is to take the weighted sum of the digits, modulo 10, with different weights for each number position.
To illustrate this, for example if the weights for a four digit number were 5, 3, 2, 7 and the number to be coded was 4871, then one would take 5×4 + 3×8 + 2×7 + 7×1 = 65, i.e. 65 modulo 10, and the
check digit would be 5, giving 48715.
Systems with weights of 1, 3, 7, or 9, with the weights on neighboring numbers being different, are widely used: for example, 31 31 weights in UPC codes, 13 13 weights in EAN numbers (GS1 algorithm),
and the 371 371 371 weights used in United States bank routing transit numbers. This system detects all single-digit errors and around 90% of transposition errors. 1, 3, 7, and 9 are used because
they are coprime with 10, so changing any digit changes the check digit; using a coefficient that is divisible by 2 or 5 would lose information (because 5×0 = 5×2 = 5×4 = 5×6 = 5×8 = 0 modulo 10) and
thus not catch some single-digit errors. Using different weights on neighboring numbers means that most transpositions change the check digit; however, because all weights differ by an even number,
this does not catch transpositions of two digits that differ by 5, (0 and 5, 1 and 6, 2 and 7, 3 and 8, 4 and 9), since the 2 and 5 multiply to yield 10.
The ISBN-10 code instead uses modulo 11, which is prime, and all the number positions have different weights 1, 2, ... 10. This system thus detects all single digit substitution and transposition
errors (including jump transpositions), but at the cost of the check digit possibly being 10, represented by "X". (An alternative is simply to avoid using the serial numbers which result in an "X"
check digit.) ISBN-13 instead uses the GS1 algorithm used in EAN numbers.
More complicated algorithms include the Luhn algorithm (1954), which captures 98% of single digit transposition errors (it does not detect 90 ↔ 09) and the still more sophisticated Verhoeff algorithm
(1969), which catches all single digit substitution and transposition errors, and many (but not all) more complex errors. Similar is another abstract algebra-based method, the Damm algorithm (2004),
that too detects all single-digit errors and all adjacent transposition errors. These three methods use a single check digit and will therefore fail to capture around 10% of more complex errors. To
reduce this failure rate, it is necessary to use more than one check digit (for example, the modulo 97 check referred to below, which uses two check digits - for the algorithm, see International Bank
Account Number) and/or to use a wider range of characters in the check digit, for example letters plus numbers.
1. Add the digits in the odd-numbered positions (first, third, fifth, etc.) together and multiply by three.
2. Add the digits (up to but not including the check digit) in the even-numbered positions (second, fourth, sixth, etc.) to the result.
3. Take the remainder of the result divided by 10 (modulo operation). If the remainder is equal to 0 then use 0 as the check digit, and if not 0 subtract the remainder from 10 to derive the check
For instance, the UPC-A barcode for a box of tissues is "036000241457". The last digit is the check digit "7", and if the other numbers are correct then the check digit calculation must produce 7.
1. Add the odd number digits: 0+6+0+2+1+5 = 14.
2. Multiply the result by 3: 14 × 3 = 42.
3. Add the even number digits: 3+0+0+4+4 = 11.
4. Add the two results together: 42 + 11 = 53.
5. To calculate the check digit, take the remainder of (53 / 10), which is also known as (53 modulo 10), and if not 0, subtract from 10. Therefore, the check digit value is 7. i.e. (53 / 10) = 5
remainder 3; 10 - 3 = 7.
Another example: to calculate the check digit for the following food item "01010101010x".
1. Add the odd number digits: 0+0+0+0+0+0 = 0.
2. Multiply the result by 3: 0 x 3 = 0.
3. Add the even number digits: 1+1+1+1+1=5.
4. Add the two results together: 0 + 5 = 5.
5. To calculate the check digit, take the remainder of (5 / 10), which is also known as (5 modulo 10), and if not 0, subtract from 10: i.e. (5 / 10) = 0 remainder 5; (10 - 5) = 5. Therefore, the
check digit x value is 5.
The final character of a ten-digit International Standard Book Number is a check digit computed so that multiplying each digit by its position in the number (counting from the right) and taking the
sum of these products modulo 11 is 0. The digit the farthest to the right (which is multiplied by 1) is the check digit, chosen to make the sum correct. It may need to have the value 10, which is
represented as the letter X. For example, take the ISBN 0-201-53082-1: The sum of products is 0×10 + 2×9 + 0×8 + 1×7 + 5×6 + 3×5 + 0×4 + 8×3 + 2×2 + 1×1 = 99 ≡ 0 (mod 11). So the ISBN is valid. Note
that positions can also be counted from left, in which case the check digit is multiplied by 10, to check validity: 0×1 + 2×2 + 0×3 + 1×4 + 5×5 + 3×6 + 0×7 + 8×8 + 2×9 + 1×10 = 143 ≡ 0 (mod 11).
ISBN 13 (in use January 2007) is equal to the EAN-13 code found underneath a book's barcode. Its check digit is generated the same way as the UPC except that the even digits are multiplied by 3
instead of the odd digits.^[3]
EAN (GLN, GTIN, EAN numbers administered by GS1)
EAN (European Article Number) check digits (administered by GS1) are calculated by summing each of the odd position numbers multiplied by 3 and then by adding the sum of the even position numbers.
Numbers are examined going from right to left, so the first odd position is the last digit in the code. The final digit of the result is subtracted from 10 to calculate the check digit (or left as-is
if already zero). A GS1 check digit calculator and detailed documentation is online at GS1's website.^[4] Another official calculator page shows that the mechanism for GTIN-13 is the same for Global
Location Number/GLN.^[5]
Other examples of check digits
• The International SEDOL number.
• The final digit of an ISSN code or IMO Number.
• The International Securities Identifying Number (ISIN).
• Object Management Group FIGI standard final digit.^[6]
• The International CAS registry number's final digit.
• Modulo 10 check digits in credit card account numbers, calculated by the Luhn algorithm. Also used in the Norwegian KID (customer identification number) numbers used in bank giros (credit
transfer), Used in IMEI of mobile phones.
• Last check digit in EAN/UPC serialisation of Global Trade Identification Number (GTIN). It applies to GTIN-8, GTIN-12, GTIN-13 and GTIN-14.
• The final digit of a DUNS number (though this is scheduled to change, such as that the final digit will be chosen freely in new allocations, rather than being a check digit).
• The third and fourth digits in an International Bank Account Number (Modulo 97 check).
• The final digit in an International Standard Text Code.
• The final character encoded in a magnetic stripe card is a computed Longitudinal redundancy check.
• The tenth digit of the National Provider Identifier for the US healthcare industry.
• The final digit of a POSTNET code.
• The North American CUSIP number.
• The final (ninth) digit of the ABA routing transit number, a bank code used in the United States.
• The ninth digit of a Vehicle Identification Number (VIN).
• Mayo Clinic patient identification numbers used in Arizona and Florida include a trailing check digit.
• The eleventh digit of a Customs & Border Protection entry number.
• The Guatemalan Tax Number (NIT - Número de Identificación Tributaria) based on modulo 11.
• The UK NHS Number uses the modulo 11 algorithm.
• The Spanish fiscal identification number (número de identificación fiscal, NIF), (based on modulo 23).
• The ninth digit of an Israeli Teudat Zehut (Identity Card) number.
• The 13th digit of the Serbian and Former Yugoslav Unique Master Citizen Number (JMBG). (but not all of them, due to errors or non-residency)
• The last two digits of the 11-digit Turkish Identification Number (Turkish: TC Kimlik Numarası).
• The ninth character in the 14-character EU cattle passport number (cycles from 1 to 7: see British Cattle Movement Service).
• The ninth digit in an Icelandic Kennitala (national ID number).
• Modulo 97 check digits in a Belgian and Serbian bank account numbers. Serbia sometimes also uses modulo 11, for reference number.
• The ninth digit in a Hungarian TAJ number (social insurance number).
• For the residents of India, the unique identity number named Aadhaar has a trailing 12th digit that is calculated with the Verhoeff algorithm.^[7]
• The Intellectual Property Office of Singapore (IPOS) has confirmed a new format for application numbers of registrable Intellectual Property (IP, e.g., trade marks, patents, registered designs).
It will include a check character calculated with the Damm algorithm.^[8]
• The last digit of Chinese citizen ID number (second generation) is calculated by modulo 11-2 as specified in Chinese GuoBiao (aka. national standard) GB11643-1999 which adopts ISO 7064:1983. 'X'
is used if the calculated checking digit is 10.
• The Australian tax file number (based on modulo 11).
• The seventh character of a New Zealand NHI Number.
• The last digit in a New Zealand locomotive's Traffic Monitoring System (TMS) number.
Notable algorithms include:
• Luhn algorithm (1954)
• Verhoeff algorithm (1969)
• Damm algorithm (2004)
• Casting out nines, similar modular sum check
• Check bit, binary equivalent
Citation Linkbooks.google.comKirtland, Joseph (2001). Identification Numbers and Check Digit Schemes. Classroom Resource Materials. Mathematical Association of America. pp. 4–6.
ISBN 978-0-88385-720-5.
Sep 19, 2019, 8:12 PM
Citation Linkwww.uc-council.org"GS1 Check Digit Calculator". GS1 US. 2006. Archived from the original on 2008-05-09. Retrieved 2008-05-21.
Sep 19, 2019, 8:12 PM
Citation Linkwww.isbn-international.org"ISBN Users Manual". International ISBN Agency. 2005. Retrieved 2008-05-21.
Sep 19, 2019, 8:12 PM
Citation Linkwww.gs1.org"Check Digit Calculator". GS1. 2005. Retrieved 2008-05-21.
Sep 19, 2019, 8:12 PM
Citation Linkwww.gs1us.org"Check Digit Calculator, at GS1 US official site". GS1 US. Retrieved 2012-08-09.
Sep 19, 2019, 8:12 PM
Citation Linkopenfigi.comhttp://openfigi.com
Sep 19, 2019, 8:12 PM
Citation Linkgg.ieeeiitr.com"Unique Identification Card". Geek Gazette. IEEE Student Branch (Autumn 2011): 16. Archived from the original on 2012-10-24.
Sep 19, 2019, 8:12 PM
Citation Linkblog.cantab-ip.comDr. Chong-Yee Khoo (20 January 2014). "New Format for Singapore IP Application Numbers at IPOS". Singapore Patent Blog. Cantab IP. Retrieved 6 July 2014.
Sep 19, 2019, 8:12 PM
Citation Linkweb.archive.orgIdentification numbers and check digit schemes
Sep 19, 2019, 8:12 PM
Citation Linkwww.simplybarcodes.netUPC, EAN, and SCC-14 check digit calculator
Sep 19, 2019, 8:12 PM
Citation Linkwww.gs1.orgGS1 check digit calculator
Sep 19, 2019, 8:12 PM
Citation Linkbooks.google.comIdentification Numbers and Check Digit Schemes
Sep 19, 2019, 8:12 PM
Citation Linkweb.archive.org"GS1 Check Digit Calculator"
Sep 19, 2019, 8:12 PM
Citation Linkwww.isbn-international.org"ISBN Users Manual"
Sep 19, 2019, 8:12 PM
Citation Linkwww.gs1.org"Check Digit Calculator"
Sep 19, 2019, 8:12 PM
Citation Linkwww.gs1us.org"Check Digit Calculator, at GS1 US official site"
Sep 19, 2019, 8:12 PM
Citation Linkopenfigi.comhttp://openfigi.com
Sep 19, 2019, 8:12 PM
Citation Linkweb.archive.org"Unique Identification Card"
Sep 19, 2019, 8:12 PM
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Golomb (Dissection, 1996)
For Solomon Golomb, mathematician and inventor of pentaminoes. Created for a presentation to MathCounts, a national junior high school mathematics competition, May 10, 1996.
If two squares side by side is a "domino", then n squares joined side by side to make a shape is a "polyomino", an idea invented by mathematician Solomon Golomb of USC. There are two distinct
"trominoes" (three squares): a straight line and an L. There are five distinct "tetraminoes" (four squares), popularized in the computer game Tetris, which was inspired by Golomb's polyominoes.
Shown above are the twelve distinct pentaminoes -- shapes made of five squares. There are dozens of games you can play with pentaminoes, like trying to arrange them into a 5 by 12 rectangle, 6 by 10
rectangle, or 3 by 4 by 5 solid. A two-person pentamino game was filmed for the movie 2001, but was cut in favor of chess. 2001 author Arthur C. Clarke later incorporated pentaminoes in his novel The
Fountains of Paradise.
If you are interested in purchasing a set of pentaminoes to play with, check out the online puzzle store Puzzletts. My favorite pentamino sets are ones made of 3-d cubes, not just flat squares, since
they can be stacked into three-dimensional shapes as well as flat shapes.
There are twelve pentaminoes and six letters in "Golomb", which leads to the nice challenge of spelling "Golomb" using just two letters to make each letter shape. First I worked on the "O"s. I wanted
both shapes to be the same, and to have at least mirror symmetry since they couldn't both have rectangular or square symmetry. The long piece obviously belonged with the "L", and the zig-zag piece
with the "B". Making a convincing "M" was rather difficult. Finally I used the remaining pieces to make a "G", probably the weakest letter.
Solomon Golomb is a prolific inventor of interesting bits of recreational mathematics, including Rep-tiles (shapes that can be dissected into several smaller copies of the original shape) and
Golomb's Ruler (if a ruler has markings only at 1, 3, 6 and 7 inches, it can still measure every integer distance from 1 to 6 inches in length). You can read more about polyominoes in Golomb's book
Polyominoes, published by Princeton University Press.
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Getting Real – Advanced Real Gas Models
A blog on what's new, notable, and next in turbomachinery
The calculations in the National Institute of Standards and Technology (NIST) Refprop program are generally considered the most accurate thermo-fluid models available. The routines are widely used in
many applications.
Several different models are embedded in the Refprop formulation. The most important are the Benedict-Webb-Rubin equations of state for the pressure-temperature-density relationship.
This is clearly way more complicated than the corresponding cubic implementation (Going Through a Phase – Modeling Phase Change with Cubics). Other equations get even more complex since many of these
require various integrations and derivatives of this already complex equation.
The Refprop calculations are considered to be the only formulation accurate enough to predict fluid properties near the difficult critical point. The plot, taken from Refprop itself (below), shows
the critical point on a temperature vs. entropy plot (left) and z (compressibility) vs. pressure curve (right). The properties near the critical point are particularly nonlinear and change very
rapidly, hence the challenge in calculating them.
Encountering flow conditions near the critical point is actually quite rare. The high temperatures and pressures of most fluids at the critical point tend to be quite difficult to handle. New cycles
which take advantage of fluid behavior near the critical point are actively being investigated though. Carbon dioxide, in particular, has attracted a lot of interest since its critical point is found
at a reasonably low temperature.
As good as the Refprop calculations are, there are two important issues the user needs to consider when using it. The most important issue is the very slow run time. A CFD solver might make millions
or possibly even billions of calls to the thermodynamic routines to the so-called equation of state (EOS) in the course of a solution. The computationally intensive Refprop calculation can eat up
huge amounts of computational time in this situation. Several publications have documented CFD runtime increases more than 50 times when run with direct calls to Refprop. To get past this, most
applications use a computational efficient interpolation scheme populated with NIST-derived data.
The second issue is that Refprop is quite unyielding outside its designated range. If a solver asks for something outside the tightly designed range limits, Refprop will refuse to answer. In
programming terms, that means the dreaded NaN (not-a-number). While this could be considered an advantage, not to answer when outside the proper definition, it can be quite limiting. Many solvers
iterate over a wide range of states before settling down to a more narrowly defined range. If any calls fail during any of these iterations, then the solution is done. This is true even if the solver
might ultimately settle in a valid region after it stabilizes. Trapping these errors is necessary for all but the most stable and predictable iteration schemes.
Going Through a Phase – Modeling Phase Change with Cubics
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By David Pincince, Business Development & Marketing Manager, Advanced Turbo ProductsNov 17, 2023
The United States is returning to the moon – to stay – and the team at Concepts NREC is helping them get there.
Artificial Intelligence (AI) uses algorithms and machine learning techniques to analyze and evaluate a research topic, making it fast and easy, while research going through peer review is
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