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Inferring Mimas' spatial distribution of tidal heating from its long-wavelength topography
Data files
Apr 19, 2024 version files 10.85 MB
In new work (Gyalay et al., 2023), we infer the interior of Mimas from its global shape (long-wavelength topography). To do so, we have to make various assumptions on how the ice shell of Mimas
operates. This includes the temperature at the base of the ice shell, the thickness of the ice shell, what mode of isostasy it operates under (equal-mass vs. equal-pressure and Airy vs. Pratt),
whether tidal tidal heating is due to eccentricity vs. obliquity, and how porous the region of the ice shell with a temperature <140 K may be. Further, as it has not yet been measured for Mimas, we
must make an assumption on its moment of inertia. We vary through these assumptions and calculate how well an inferred heat distribution matches with a tidal heating distribution, among other
physical self-consistency checks. In the associated paper, we analyze the dataset we produced to make conclusions on Mimas' interior structure and orbital dynamic history.
README: Inferring Mimas' spatial distribution of tidal heating from its long-
wavelength topography
In this repository there is a series of data files for outputs of Mimas modeled
under different assumptions. The biggest indicators of well-fitting models are
the r_sq, which is the coefficient of determination that shows how well the
inferred heating pattern beneath the ice shell can be fit by spatial
patterns of tidal heating, and the RMS, which is the root mean square
difference between the observed topography of Mimas and the topography forward
modeled from the best fit tidal heating pattern weights. In the associated
paper, we conclude there was a past epoch of strong obliquity tides in a solid
Additionally, there is one last file that is input for the TIRADE solid-body
tidal heating code of Roberts & Nimmo 2008. That code is not ours to provide,
but we can at least provide the input. Further, the Roberts & Nimmo code needs
to be updated to calculate the tidal dissipation and potential due to obliquity
tides. The tidal dissipation is equation 42 of Beuthe 2013, while the tidal
potential is equation 88 of Beuthe 2013. These updates must be made to to the
file "tidal_module.c" at about lines 139 (the variable "Ediss") and 979 (the
variable "potential"). The derivatives of Equation 88 must also be calculated
to update variable "dpot". The input file is "Mimas_AGU2022_conv" and requires
the creation of a directory "mimas_agu2022_conv_dir" in the same directory as
the input file. Then the command "./tidal_cond.x Mimas_AGU2022_conv" will place
all outputs within "mimas_agu2022_conv_dir", assuming all the TIRADE code is
within the same directory as the input file.
Description of the data and file structure
The header of each file should describe what each column refers to. MoI is the
moment of inertia. From these data, one can compute density profiles for each
model of Tethys (or Enceladus) and judge whether it is consistent with the
inferred heating pattern weight. Values were not printed to file if the
calculated average heat flux was NaN, if any of chi_A,B,C were not between 0
and 1, or if any of the spherical harmonic weights of forward-modeled
topography were NaN. Further descriptions of parameters and their uses are
described in the paper for which this dataset was produced. Further, we include
files for Enceladus. While the Tethys data include "no_odysseus" in the
filename, the spherical harmonic coefficients utilized were derived from limb
profiles that do include those that pass over Odysseus crater (Nimmo et al.,
The headers contained appear like so:
Assuming [isostasy-type] isostasy and [tide-type] tides upon Tethys,
[True/False] weighted regression [states whether the multilinear regression was
weighted] and [True/False] pressure isostasy [did we use equal-pressure or
equal-mass isostasy],
each given porosity, MoI [Moment of Inertia], basal temperature T_B, and total
shell thickness d we
calculate necessary basal heat flux (F_B).
We then calculate the following values:
chi_a, chi_b, chi_c: heating pattern weights
r_sq: coefficient of determination
rms: Square root of the weighted average of the square of modeled
topography minus observed. Weighted by area of degree bin.
CF20/CF22: Spherical harmonic coefficients of flux ratioed.
normClm for l=2,4;m=0,2,4<=l: normalized spherical harmonic
weights of topography from our best-fit interior.
z_Clm: z-score of the normClm we calculate vs. those observed.
This is (normClm-normClm_observed)/SD_normClm_observed
where SD is the standard deviation
This dataset was produced by a model using the methods described in Gyalay & Nimmo (2023a; JGR: Planets 128(2), doi: 10.1029/2022JE007550). In that paper, we established the mathematics behind how we
used assumed parameters (upper ice shell porosity, total ice shell thickness, moment of inertia, basal temperature at the base of the ice shell) to infer the average basal heat flux and fit for
spatial patterns of tidal heating (Beuthe, 2013, Icarus). Using this best-fit tidal heating for each set of parameters, we forward model the topography (also described in that paper) and calculate
its spherical harmonic weights as well as compare them to the originally observed topography.
The code used to generate this dataset and as well as the dataset associated with Gyalay & Nimmo (2023a) are included in that paper's associated repository (Gyalay & Nimmo, 2023b; Dryad, dataset,
doi: doi.org/10.7291/D11969). This repository contains only the produced model output for Mimas.
Usage notes
The header of each data-output file should describe what each column refers to. MoI is the moment of inertia. From these data, one can compute density profiles for each model of Tethys (or Enceladus)
and judge whether it is consistent with the inferred heating pattern weight. Values were not printed to file if the calculated average heat flux was NaN, if any of chi_A,B,C were not between 0 and 1,
or if any of the spherical harmonic weights of forward-modeled topography were NaN. Further descriptions of parameters and their uses are described in Gyalay & Nimmo (2023a) and Gyalay et al. (2023).
We also include an input file for the solid body tidal heating code of Roberts & Nimmo (2008; Icarus 194(2), doi: 10.1016/j.icarus.2007.11.010). Usage of the input file is included in the file | {"url":"https://datadryad.org:443/stash/dataset/doi:10.7291/D1ZM3N","timestamp":"2024-11-08T08:04:01Z","content_type":"text/html","content_length":"70500","record_id":"<urn:uuid:c2b674a6-ff9e-4929-9958-3a1a739a892b>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00053.warc.gz"} |
Graph and Network Algorithms
Directed and undirected graphs, network analysis
Graphs model the connections in a network and are widely applicable to a variety of physical, biological, and information systems. You can use graphs to model the neurons in a brain, the flight
patterns of an airline, and much more. The structure of a graph is comprised of “nodes” and “edges”. Each node represents an entity, and each edge represents a connection between two nodes. For more
information, see Directed and Undirected Graphs.
Modify Nodes and Edges
addnode Add new node to graph
rmnode Remove node from graph
addedge Add new edge to graph
rmedge Remove edge from graph
flipedge Reverse edge directions
numnodes Number of nodes in graph
numedges Number of edges in graph
findnode Locate node in graph
findedge Locate edge in graph
edgecount Number of edges between two nodes
reordernodes Reorder graph nodes
subgraph Extract subgraph
Analyze Structure
centrality Measure node importance
conncomp Connected graph components
biconncomp Biconnected graph components
condensation Graph condensation
bctree Block-cut tree graph
toposort Topological order of directed acyclic graph
isdag Determine if graph is acyclic
transreduction Transitive reduction
transclosure Transitive closure
isisomorphic Determine whether two graphs are isomorphic
isomorphism Compute isomorphism between two graphs
ismultigraph Determine whether graph has multiple edges
simplify Reduce multigraph to simple graph
Traversals, Shortest Paths, and Cycles
bfsearch Breadth-first graph search
dfsearch Depth-first graph search
shortestpath Shortest path between two single nodes
shortestpathtree Shortest path tree from node
distances Shortest path distances of all node pairs
allpaths Find all paths between two graph nodes (Since R2021a)
maxflow Maximum flow in graph
minspantree Minimum spanning tree of graph
hascycles Determine whether graph contains cycles (Since R2021a)
allcycles Find all cycles in graph (Since R2021a)
cyclebasis Fundamental cycle basis of graph (Since R2021a)
Node Information
degree Degree of graph nodes
neighbors Neighbors of graph node
nearest Nearest neighbors within radius
indegree In-degree of nodes
outdegree Out-degree of nodes
predecessors Node predecessors
successors Node successors
inedges Incoming edges to node
outedges Outgoing edges from node
plot Plot graph nodes and edges
labeledge Label graph edges
labelnode Label graph nodes
layout Change layout of graph plot
layoutcoords Graph node and edge layout coordinates (Since R2024b)
highlight Highlight nodes and edges in plotted graph
GraphPlot Graph plot for directed and undirected graphs
GraphPlot Properties Graph plot appearance and behavior
Related Information
Featured Examples | {"url":"https://www.mathworks.com/help/matlab/graph-and-network-algorithms.html?s_tid=CRUX_lftnav","timestamp":"2024-11-10T08:43:02Z","content_type":"text/html","content_length":"96215","record_id":"<urn:uuid:8614237e-d5b6-49a9-b93b-5411a55d644c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00849.warc.gz"} |
Supported Data Types
Converting HDL Data to Send to MATLAB or Simulink
If your HDL application needs to send HDL data to a MATLAB^® function or a Simulink^® block, you may first need to convert the data to a type supported by MATLAB and the HDL Verifier™ software.
To program a MATLAB function or a Simulink block for an HDL model, you must understand the type conversions required by your application. You may also need to handle differences between the array
indexing conventions used by the HDL you are using and MATLAB (see following section).
The data types of arguments passed in to the function determine the following:
• The types of conversions required before data is manipulated
• The types of conversions required to return data to the HDL simulator
The following table summarizes how the HDL Verifier software converts supported VHDL^® data types to MATLAB types based on whether the type is scalar or array.
VHDL to MATLAB Data Type Conversions
VHDL Types... As Scalar Converts to... As Array Converts to...
STD_LOGIC, STD_ULOGIC, A character that matches the character literal for the
and BIT desired logic state.
STD_ULOGIC_VECTOR, A column vector of characters (as defined in VHDL Conversions for the HDL Simulator) with one bit per character.
BIT_VECTOR, SIGNED, and
Arrays of
STD_ULOGIC_VECTOR, An array of characters (as defined above) with a size that is equivalent to the VHDL port size.
BIT_VECTOR, SIGNED, and
Arrays of type int32 with a size that is equivalent to the VHDL port size.
INTEGER and NATURAL Type int32. Note
INTEGER is supported for HDL Coder™ cosimulation, but not for the Cosimulation Wizard workflow.
Arrays of type double with a size that is equivalent to the VHDL port size.
REAL Type double. Note
REAL is supported for HDL Coder cosimulation, but not for the Cosimulation Wizard workflow.
Type double for time values in seconds and type int64 for
TIME values representing simulator time increments (see the Arrays of type double or int64 with a size that is equivalent to the VHDL port size.
description of the 'time' option in hdldaemon).
Character vector or string scalar that contains the MATLAB Cell array of character vectors or string array with each element equal to a label for the defined enumerated
representation of a VHDL label or character literal. For type. Each element is the MATLAB representation of a VHDL label or character literal. For example, the vector
Enumerated types example, the label high converts to 'high' and the character (one, '2', three) converts to the column vector ['one'; '''2'''; 'three']. A user-defined enumerated type that
literal 'c' converts to '''c'''. contains only character literals, and then converts to a vector or array of characters as indicated for the types
STD_LOGIC_VECTOR, STD_ULOGIC_VECTOR, BIT_VECTOR, SIGNED, and UNSIGNED.
The following table summarizes how the HDL Verifier software converts supported Verilog^® data types to MATLAB types. The software supports packed arrays up to 128bits for Verilog.
Verilog-to-MATLAB Data Type Conversions
Verilog Types... Converts to...
wire, reg A character or a column vector of characters that matches the character literal for the desired logic states (bits).
The following table summarizes how the HDL Verifier software converts supported SystemVerilog data types to MATLAB types. The software supports packed arrays up to 128bits for SystemVerilog.
SystemVerilog-to-MATLAB Data Type Conversions
SystemVerilog Types... Converts to...
wire, reg, logic A character or a column vector of characters that matches the character literal for the desired logic states (bits).
integer A 32-element column vector of characters that matches the character literal for the desired logic states (bits). Supported for outputs only.
bit boolean/ufix1
byte int8/uint8
shortint int16/uint16
int int32
longint int64/uint64
real double
• Input to HDL Cosimulation block: ufix/fix, or matrix of ufix/fix
packed array (bit/logic vector)
• Output from HDL Cosimulation block: ufix
SystemVerilog-to-Simulink Data Type Conversions
SystemVerilog Types... Converts to...
wire, reg, logic A character or a column vector of characters that matches the character literal for the desired logic states (bits).
integer A 32-element column vector of characters that matches the character literal for the desired logic states (bits). Supported for outputs only.
bit boolean/ufix1
byte int8/uint8
shortint int16/uint16
int int32
longint int64/uint64
real double
• Input to HDL Cosimulation block: ufix/fix, or matrix of ufix/fix
packed array (bit/logic vector)
• Output from HDL Cosimulation block: ufix
unpacked array Simulink vector. For more details see Simulink Support for SystemVerilog Unpacked Arrays.
Each struct member is represented as a singular port.
• Nested structures are not supported.
• Not supported for Vivado^® simulator.
SystemVerilog support includes signals of the above types. The following SystemVerilog types are not supported:
• shortreal SystemVerilog type
• union SystemVerilog type
• Nested structures
• SystemVerilog interfaces
Bit-Vector Indexing Differences Between MATLAB and HDL
In HDL, you have the flexibility to define a bit-vector with either MSB-0 or LSB-0 numbering. In MATLAB, bit-vectors are always considered LSB-0 numbering. In order to prevent data corruption, it is
recommended that you use LSB-0 indexing for your HDL interfaces.
If you define a logic vector in VHDL as:
signal s1 : std_logic_vector(7 downto 0);
Or in Verilog as:
It is mapped to int8 in MATLAB, with s1[7] as the MSB. Alternatively, if you define your VHDL logic vector as:
signal s1 : std_logic_vector(0 to 7);
Or in Verilog as:
It is mapped to int8 in MATLAB, with s1[0] as the MSB.
Array Indexing Differences Between MATLAB and HDL
In multidimensional arrays, the same underlying OS memory buffer maps to different elements in MATLAB and the HDL simulator (this mapping only reflects different ways the different languages offer
for naming the elements of the same array). When you use both the matlabtb and matlabcp functions, be careful to assign and interpret values consistently in both applications.
In HDL, a multidimensional array declared as:
type matrix_2x3x4 is array (0 to 1, 4 downto 2) of std_logic_vector(8 downto 5);
has a memory layout as follows:
bit 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
dim1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1
dim2 4 4 4 4 3 3 3 3 2 2 2 2 4 4 4 4 3 3 3 3 2 2 2 2
dim3 8 7 6 5 8 7 6 5 8 7 6 5 8 7 6 5 8 7 6 5 8 7 6 5
This same layout corresponds to the following MATLAB 4x3x2 matrix:
bit 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
dim1 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
dim2 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3
dim3 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
Therefore, if H is the HDL array and M is the MATLAB matrix, the following indexed values are the same:
b1 H(0,4,8) = M(1,1,1)
b2 H(0,4,7) = M(2,1,1)
b3 H(0,4,6) = M(3,1,1)
b4 H(0,4,5) = M(4,1,1)
b5 H(0,3,8) = M(1,2,1)
b6 H(0,3,7) = M(2,2,1)
b19 H(1,3,6) = M(3,2,2)
b20 H(1,3,5) = M(4,2,2)
b21 H(1,2,8) = M(1,3,2)
b22 H(1,2,7) = M(2,3,2)
b23 H(1,2,6) = M(3,3,2)
b24 H(1,2,5) = M(4,3,2)
You can extend this indexing to N-dimensions. In general, the dimensions—if numbered from left to right—are reversed. The right-most dimension in HDL corresponds to the left-most dimension in MATLAB.
Array Indexing - Column Major
When you perform a Simulink cosimulation with a SystemVerilog DUT, and the DUT includes an unpacked array port, note that HDL Verifier indexes the array in a column major order. For example, for a
3x2 HDL matrix defined as:
logic [31:0] In1 [0:2][0:1]
In this figure, indexing the Simulink matrix in column-major would read the elements column by column, in this order:
If your HDL matrix considers row-major order, it would place the elements in the matrix row by row. Therefore, if m1 is the Simulink matrix and m2 is the HDL matrix, the following indexed values are
the same (assuming the Simulink index mode is 0-base):
m1(0,0) = m2(0,0) = A
m1(1,0) = m2(0,1) = B
m1(2,0) = m2(1,0) = C
m1(0,1) = m2(1,1) = D
m1(1,1) = m2(2,0) = E
m1(2,1) = m2(2,1) = F
Converting Data for Manipulation
Depending on how your simulation MATLAB function uses the data it receives from the HDL simulator, you may need to code the function to convert data to a different type before manipulating it. The
following table lists circumstances under which you would require such conversions.
Required Data Conversions
If You Need the Function to... Then...
Use the double function to convert the data to type double before performing the computation. For example:
Compute numeric data that is received as a type other than
double datas(inc+1) = double(idata);
Use the mvl2dec function to convert the data to an unsigned decimal value. For example:
uval = mvl2dec(oport.val)
This example assumes the standard logic or bit vector is composed of the character literals '1' and '0' only. These are the only two
Convert a standard logic or bit vector to an unsigned values that can be converted to an integer equivalent.
integer or positive decimal
The mvl2dec function converts the binary data that the MATLAB function receives from the entity's osc_in port to unsigned decimal values
that MATLAB can compute.
See mvl2dec for more information on this function.
Use the following application of the mvl2dec function to convert the data to a signed decimal value. For example:
Convert a standard logic or bit vector to a negative decimal suval = mvl2dec(oport.val, true);
This example assumes the standard logic or bit vector is composed of the character literals '1' and '0' only. These are the only two
values that can be converted to an integer equivalent.
The following code excerpt illustrates data type conversion of data passed in to a callback:
InDelayLine(1) = InputScale * mvl2dec(iport.osc_in',true);
This example tests port values of VHDL type STD_LOGIC and STD_LOGIC_VECTOR by using the all function as follows:
all(oport.val == '1' | oport.val
== '0')
This example returns True if all elements are '1' or '0'.
Converting Data for Return to the HDL Simulator
If your simulation MATLAB function needs to return data to the HDL simulator, you may first need to convert the data to a type supported by the HDL Verifier software. The following tables list
circumstances under which such conversions are required for VHDL and Verilog.
When data values are returned to the HDL simulator, the char array size must match the HDL type, including leading zeroes, if applicable. For example:
oport.signal = dec2mvl(2)
will only work if signal is a 2-bit type in HDL. If the HDL type is anything else, you must specify the second argument:
oport.signal = dec2mvl(2, N)
where N is the number of bits in the HDL data type.
VHDL Conversions for the HDL Simulator
To Return Data to an IN Port of Then...
Declare the data as a character that matches the character literal for the desired logic state. For STD_LOGIC and STD_ULOGIC, the character can be 'U', 'X', '0', '1',
'Z', 'W', 'L', 'H', or '-'. For BIT, the character can be '0' or '1'. For example:
STD_LOGIC, STD_ULOGIC, or BIT
iport.s1 = 'X'; %STD_LOGIC
iport.bit = '1'; %BIT
Declare the data as a column vector or row vector of characters (as defined above) with one bit per character. For example:
STD_ULOGIC_VECTOR, BIT_VECTOR, iport.s1v = 'X10ZZ'; %STD_LOGIC_VECTOR
SIGNED, or UNSIGNED iport.bitv = '10100'; %BIT_VECTOR
iport.uns = dec2mvl(10,8); %UNSIGNED, 8 bits
Array of STD_LOGIC_VECTOR,
STD_ULOGIC_VECTOR, BIT_VECTOR, Declare the data as an array of type character with a size that is equivalent to the VHDL port size. See Array Indexing Differences Between MATLAB and HDL.
SIGNED, or UNSIGNED
Declare the data as an array of type int32 with a size that is equivalent to the VHDL array size. Alternatively, convert the data to an array of type int32 with the
MATLAB int32 function before returning it. Be sure to limit the data to values with the range of the VHDL type. If you want to, check the right and left fields of the
portinfo structure. For example:
INTEGER or NATURAL iport.int = int32(1:10)';
INTEGER is supported for HDL Coder cosimulation, but not for the Cosimulation Wizard workflow.
Declare the data as an array of type double with a size that is equivalent to the VHDL port size. For example:
iport.dbl = ones(2,2);
REAL is supported for HDL Coder cosimulation, but not for the Cosimulation Wizard workflow.
Declare a VHDL TIME value as time in seconds, using type double, or as an integer of simulator time increments, using type int64. You can use the two formats
interchangeably and what you specify does not depend on the hdldaemon 'time' option (see hdldaemon), which applies to IN ports only. Declare an array of TIME values
by using a MATLAB array of identical size and shape. All elements of a given port are restricted to time in seconds (type double) or simulator increments (type
int64), but otherwise you can mix the formats. For example:
iport.t1 = int64(1:10)'; %Simulator time
iport.t2 = 1e-9; %1 nsec
Declare the data as a character vector or string scalar for scalar ports or a cell array of character vectors or string array for array ports with each element equal
to a label for the defined enumerated type. The 'label' field of the portinfo structure lists all valid labels (see Gaining Access to and Applying Port Information).
Except for character literals, labels are not case sensitive. In general, you should specify character literals completely, including the single quotes, as in the
first example shown here.
Enumerated types
iport.char = {'''A''', '''B'''}; %Character
iport.udef = 'mylabel'; %User-defined label
Use the dec2mvl function to convert the integer. For example:
Character array for standard
logic or bit representation oport.slva =dec2mvl([23 99],8)';
This example converts two integers to a 2-element array of standard logic vectors consisting of 8 bits.
Verilog Conversions for the HDL Simulator
To Return Data to an input Port of Type... Then...
Declare the data as a character or a column vector of characters that matches the character literal for the desired logic state. For example:
reg, wire
iport.bit = '1';
SystemVerilog Conversions for the HDL Simulator
To Return Data to an input Port of Type... Then...
Declare the data as a character or a column vector of characters that matches the character literal for the desired logic state. For example:
reg, wire, logic
iport.bit = '1';
integer Declare the data as a 32-element column vector of characters (as defined above) with one bit per character.
Packed arrays are supported up to 128bits.
SystemVerilog support includes only scalar signals of the above types. The following SystemVerilog types are not supported:
• Arrays and multi-dimensional arrays
• shortreal SystemVerilog type
• SystemVerilog aggregate types such as union and struct
• SystemVerilog interfaces
Simulink Handling of Wide HDL Ports
When your HDL module has a port that is wider than 128 bits, Simulink creates a vector of ports to represent this port. The Cosimulation Wizard infers the size of the HDL port. You can then set the
Simulink Word Length parameter in the HDL Cosimulation block.
For input ports — Simulink port dimensions are determined at compile time by the data type of the driving signal. For example:
• When HDL Word Length = 150 and Simulink Word Length = 50, HDL Verifier allows a Simulink port with data width of 50 bits, and dimensions of size 3 such as sfix50(3) or ufix50(3).
• When HDL Word Length = 140 and Simulink Word Length = 50, HDL Verifier packs 150 bits of Simulink into 140 bits of HDL. HDL Verifier ignores the 10 most significant bits (MSB) of the last word.
For output ports
• HDL Verifier creates a vector of ports to represent the output port. For example:
□ When HDL Word Length = 150 and Simulink Word Length = 50, HDL Verifier creates a Simulink port with data width of 50 bit. For example sfix50(3) or ufix50(3).
□ When HDL Word Length = 150 and Simulink Word Length = 60, HDL Verifier creates a Simulink port with data width of 60, such as sfix60(3) or ufix60(3). Since the HDL word has only 150 bits, and
the Simulink port requires 180 bits, 30 bits are padded or sign extended.
□ When HDL Word Length = 140 and Simulink Word Length = 50, every 50 bits of the HDL output are represented as a Simulink word. The 10 MSB of the last Simulink word are unused and extended
according to the Sign parameter.
Simulink Support for SystemVerilog Unpacked Arrays
When you have an existing HDL DUT — When you use the Cosimulation Wizard to import a SystemVerilog DUT that includes unpacked arrays on the interface, HDL Verifier follows this convention:
HDL Verifier interprets arrays and matrices in column major ordering. For more information, see Array Indexing - Column Major.
For example, if your SystemVerilog DUT defines this interface, with a 2-by-5 unpacked array input and output port:
module myDUT
(input logic clk,
input logic clk_en,
input logic[31:0] In1[0:1] [0:4],
output logic[31:0] Out1[0:1] [0:4],
output logic ce_out);
You can now write a Simulink testbench that drives an input with ten elements of int32, such as int32 [2x5]. The output port maps to a flat array of ten elements: int32 [10x1].
When you generate an HDL DUT — When you use the HDL Workflow Advisor (HDL Coder) to generate a SystemVerilog DUT, HDL Verifier can generate a cosimulation testbench model for only one-dimensional
arrays (vectors) on the DUT interface.
Vivado Support for Unpacked Arrays
Since the Vivado simulator cannot differentiate between packed and unpacked dimensions, HDL Verifier treats the lowest dimension as a packed dimension and the rest as unpacked dimensions.
For example:
• If the SystemVerilog has a port defined as Logic [7:0][3:0] dataA, Simulink maps it to an array of eight elements with data-type fixdt4.
• If the SystemVerilog has a port defined as Logic [3:0][7:0] dataB [1:3][4:6], Simulink maps it to an array of 36 elements (the product of 3*3*4), with data-type fixdt8.
Simulink Support for Verilog and SystemVerilog Enumerations
When you have an existing HDL DUT — When you use the Cosimulation Wizard to import a DUT that includes a port of type enum on the interface, HDL Verifier maps the port to a fixed-point logic vector.
The size of that logic vector is determined by the number of bits required to represent the enum type.
For example, if your SystemVerilog DUT defines this interface, with a BasicColors input and output port, the ports map to an input or output of type ufix2.
typedef enum logic [1:0] {
} t_BaiscColors;
module scalarEnums
( input t_BasicColors In1,
output t_BasicColors Out1);
When you generate an HDL DUT — When you use the HDL Workflow Advisor (HDL Coder) to generate a Verilog or SystemVerilog DUT, HDL Verifier generates a cosimulation testbench model and converts the
types to use the Simulink.IntEnumType data type that match the DUT. For more about Simulink enumerations, see Code Generation for Enumerations (Simulink).
VHDL does not support enumerated types.
See Also
Cosimulation Wizard | HDL Cosimulation | hdlverifier.HDLCosimulation
Related Topics | {"url":"https://se.mathworks.com/help/hdlverifier/ug/data-type-conversions.html","timestamp":"2024-11-08T06:20:50Z","content_type":"text/html","content_length":"115665","record_id":"<urn:uuid:92346e8e-a35a-4eb1-b11e-360741a5a627>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00128.warc.gz"} |
Chapter 6: Inference for Categorical Data: Proportions Notes | Knowt
The Meaning of a Confidence Interval
• The percentage is the percentage of samples that would pinpoint the unknown p or µ within plus or minus respective margins of error.
• For a given sample proportion or mean, p or µ either is or isn’t within the specified interval, and so the probability is either 1 or 0.
Two aspects to this concept:
• Second, there is the success rate for the method, called the confidence level, that is, the proportion of times repeated applications of this method would capture the true population parameter.
• The standard error is a measure of how far the sample statistic typically varies from the population parameter.
• The margin of error is a multiple of the standard error, with that multiple determined by how confident we wish to be of our procedure.
All of the above assume that certain conditions are met. For inference on population proportions, means, and slopes, we must check for independence in data collection methods and for selection of
the appropriate sampling distribution.
Conditions for Inference
The following are the two standard assumptions for our inference procedures and the "ideal" way they are met:
1. Independence assumption:
• Individuals in a sample or an experiment an must be independent of each other, and this is obtained through random sampling or random selection.
• Independence across samples is obtained by selecting two (or more) separate random samples.
• Always examine how the data were collected to check if the assumption of independence is reasonable.
• Sample size can also affect independence. Because sampling is usually done without replacement, if the sample is too large, lack of independence becomes a concern.
• So, we typically require that the sample size n be no larger than 10% of the population (the 10% Rule).
2. Normality assumption:
• Inference for proportions is based on a normal model for the sampling distribution of p̂, but actually we have a binomial distribution.
• Fortunately, the binomial is approximately normal if both np and nq ≥ 10.
• Inference for means is based on a normal model for the sampling distribution of x̄; this is true if the population is normal and is approximately true (thanks to the CLT) if the sample size is
large enough (typically we accept n ≥ 30).
• With regard to means, this is referred to as the Normal/Large Sample condition.
➥ Example 6.1
If we pick a simple random sample of size 80 from a large population, which of the following values of the population proportion p would allow use of the normal model for the sampling distribution of
1. 0.10
2. 0.15
3. 0.90
4. 0.95
5. 0.99
Solution: (B)
• The relevant condition is that both np and nq ≥ 10.
• In (A), np = (80)(0.10) = 8; in (C), nq = (80)(0.10) = 8; in (D), nq = (80)(0.05) = 4; and in (E), nq = (80)(0.01) = 0.8. However, in (B), np = (80)(0.15) = 12 and nq = (0.85)(80) = 68 are both ≥
Confidence Interval for a Proportion
This sample proportion is just one of a whole universe of sample proportions, and from Unit 5 we remember the following:
1. The set of all sample proportions is approximately normally distributed.
2. The mean μp̂ of the set of sample proportions equals p, the population proportion.
3. The standard deviation σp̂ of the set of sample proportions is approximately equal to
since p is unknown? The reasonable procedure is to use the sample proportion p̂:
Example 6.2
1. If 42% of a simple random sample of 550 young adults say that whoever asks for the date should pay for the first date, determine a 99% confidence interval estimate for the true proportion of all
young adults who would say that whoever asks for the date should pay for the first date.
2. Does this confidence interval give convincing evidence in support of the claim that fewer than 50% of young adults would say that whoever asks for the date should pay for the first date?
1. The parameter is p, which represents the proportion of the population of young adults who would say that whoever asks for the date should pay for the first date. We check that
• We are given that the sample is an SRS, and 550 is clearly less than 10% of all young adults. Since p̂ = 0.42, the standard error of the set of sample proportions is
• 99% of the sample proportions should be within 2.576 standard deviations of the population proportion. Equivalently, we are 99% certain that the population proportion is within 2.576 standard
deviations of any sample proportion.
• Thus, the 99% confidence interval estimate for the population proportion is 0.42 ± 2.576(0.021) = 0.42 ± 0.054. We say that the margin of error is ±0.054. We are 99% confident that the true
proportion of young adults who would say that whoever asks for the date should pay for the first date is between 0.366 and 0.474.
2. Yes, because all the values in the confidence interval (0.366 to 0.474) are less than 0.50, this confidence interval gives convincing evidence in support of the claim that fewer than 50% of young
adults would say that whoever asks for the date should pay for the first date.
Logic of Significance Testing
• Closely related to the problem of estimating a population proportion or mean is the problem of testing a hypothesis about a population proportion or mean.
• The general testing procedure is to choose a specific hypothesis to be tested, called the null hypothesis, pick an appropriate random sample, and then use measurements from the sample to
determine the likelihood of the null hypothesis.
• If the sample statistic is far enough away from the claimed population parameter, we say that there is sufficient evidence to reject the null hypothesis. We attempt to show that the null
hypothesis is unacceptable by showing that it is improbable.
The null hypothesis H0 is stated in the form of an equality statement about the population proportion (for example, H0: p = 0.37).
• There is an alternative hypothesis, stated in the form of a strict inequality (for example, Ha: p < 0.37 or Ha: p > 0.37 or Ha: p ≠ 0.37).
• The strength of the sample statistic p̂ can be gauged through its associated P-value, which is the probability of obtaining a sample statistic as extreme (or more extreme) as the one obtained if
the null hypothesis is assumed to be true. The smaller the P-value, the more significant the difference between the null hypothesis and the sample results.
There are two types of possible errors:
1. the error of mistakenly rejecting a true null hypothesis.
2. the error of mistakenly failing to reject a false null hypothesis.
• The α-risk, also called the significance level of the test, is the probability of committing a Type I error and mistakenly rejecting a true null hypothesis.
• Type II error - a mistaken failure to reject a false null hypothesis, has associated probability β.
There is a different value of β for each possible correct value for the population parameter p. For each β, 1 − β is called the "power" of the test against the associated correct value.
That is, given a true alternative, the power is the probability of rejecting the false null hypothesis. Increasing the sample size and increasing the significance level are both ways of increasing
the power. Also note that a true null that is further away from the hypothesized null is more likely to be detected, thus offering a more powerful test.
A simple illustration of the difference between a Type I and a Type II error is as follows.
• Suppose the null hypothesis is that all systems are operating satisfactorily with regard to a NASA launch. A Type I error would be to delay the launch mistakenly thinking that something was
malfunctioning when everything was actually OK. A Type II error would be to fail to delay the launch mistakenly thinking everything was OK when something was actually malfunctioning. The power is
the probability of recognizing a particular malfunction. (Note the complementary aspect of power, a “good” thing, with Type II error, a “bad” thing.)
It should be emphasized that with regard to calculations, questions like “What is the power of this test?” and “What is the probability of a Type II error in this test?” cannot be answered without
reference to a specific alternative hypothesis.
Significance Test for a Proportion
It is important to understand that because the P-value is a conditional probability, calculated based on the assumption that the null hypothesis, H0: p = p0, is true, we use the claimed proportion p
0 both in checking the np0 ≥ 10 and n(1 − p0) ≥ 10 conditions and in calculating the standard deviation
➥ Example 6.3
1. A union spokesperson claims that 75% of union members will support a strike if their basic demands are not met. A company negotiator believes the true percentage is lower and runs a hypothesis
test. What is the conclusion if 87 out of a simple random sample of 125 union members say they will strike?
2. For each of the two possible answers above, what error might have been committed, Type I or Type II, and what would be a possible consequence?
1. Parameter: Let p represent the proportion of all union members who will support a strike if their basic demands are not met.
Hypotheses: H0: p = 0.75 and Ha: p < 0.75.
Procedure: One-sample z-test for a population proportion.
Checks: np0 = (125)(0.75) = 93.75 and n(1 − p0) = (125)(0.25) = 31.25 are both ≥ 10, it is given that we have an SRS, and we must assume that 125 is less than 10% of the total union membership.
Mechanics: Calculator software (such as 1-PropZTest on the TI-84 or Z-1-PROP on the Casio Prizm) gives z = −1.394 and P = 0.0816.
Conclusion in context with linkage to the P-value: There are two possible answers:
a. With this large of a P-value, 0.0816 > 0.05, there is not sufficient evidence to reject H0; that is, there is not sufficient evidence at the 5% significance level that the true percentage of union
members who support a strike is less than 75%.
b. With this small of a P-value, 0.0816 < 0.10, there is sufficient evidence to reject H0; that is, there is sufficient evidence at the 10% significance level that the true percentage of union
members who support a strike is less than 75%.
2. If the P-value is considered large, 0.0816 > 0.05, so that there is not sufficient evidence to reject the null hypothesis, there is the possibility that a false null hypothesis would mistakenly
not be rejected and thus a Type II error would be committed. In this case, the union might call a strike thinking they have greater support than they actually do. If the P-value is considered
small, 0.0816 < 0.10, so that there is sufficient evidence to reject the null hypothesis, there is the possibility that a true null hypothesis would mistakenly be rejected, and thus a Type I
error would be committed. In this case, the union might not call for a strike thinking they don’t have sufficient support when they actually do have support.
Confidence Interval for the Difference of Two Proportions
From Unit 5, we have the following information about the sampling distribution of
1. The set of all differences of sample proportions is approximately normally distributed.
2. The mean of the set of differences of sample proportions equals p1 − p2, the difference of the population proportions.
3. The standard deviation
of the set of differences of sample proportions is approximately equal to:
Remember that we are using the normal approximation to the binomial, so
should all be at least 10. In making calculations and drawing conclusions from specific samples, it is important both that the samples be simple random samples and that they be taken independentlyof
each other. Finally, the original populations should be large compared to the sample sizes, that is, check that
➥ Example 6.4
1. Suppose that 84% of a simple random sample of 125 nurses working 7:00 a.m. to 3:00 p.m. shifts in city hospitals express positive job satisfaction, while only 72% of an SRS of 150 nurses on 11:00
p.m. to 7:00 a.m. shifts express similar fulfillment. Establish a 90% confidence interval estimate for the difference.
2. Based on the interval, is there convincing evidence that the nurses on the 7 AM to 3 PM shift express a higher job satisfaction than nurses on the 11 PM to 7 AM shift?
1. Parameters: Let p1 represent the proportion of the population of nurses working 7:00 a.m. to 3:00 p.m. shifts in city hospitals who have positive job satisfaction. Let p2 represent the proportion
of the population of nurses working 11:00 p.m. to 7:00 a.m. shifts in city hospitals who have positive job satisfaction.
Procedure: Two-sample z-interval for a difference between population proportions, p1 − p2.
we are given independent SRSs; and the sample sizes are assumed to be less than 10% of the populations of city hospital nurses on the two shifts, respectively.
Mechanics: 2-PropZInt on the TI-84 or 2-Prop ZInterval on the Casio Prizm give (0.0391, 0.2009).
The observed difference is 0.84 − 0.72 = 0.12, and the critical z-scores are ±1.645. The confidence interval estimate is 0.12 ± 1.645(0.0492) = 0.12 ± 0.081.]
Conclusion in context: We are 90% confident that the true proportion of satisfied nurses on 7:00 a.m. to 3:00 p.m. shifts is between 0.039 and 0.201 higher than the true proportion for nurses on
11:00 p.m. to 7:00 a.m. shifts.
2. Yes, because the entire interval from 0.039 to 0.201 is positive, there is convincing evidence that the nurses on the 7 AM to 3 PM shift express a higher job satisfaction than nurses on the 11 PM
to 7 AM shift.
Significance Test for the Difference of Two Proportions
The null hypothesis for a difference between two proportions is
and so the normality condition becomes that
simple random samples, and that they be taken independently of each other. The original populations should also be large compared to the sample sizes, that is, check that
Two points need to be stressed:
• First, sample proportions from the same population can vary from each other.
• Second, what we are really comparing are confidence interval estimates, not just single points.
For many problems, the null hypothesis states that the population proportions are equal or, equivalently, that their difference is 0:
The alternative hypothesis is then:
where the first two possibilities lead to one-sided tests and the third possibility leads to a two-sided test.
Since the null hypothesis is that p1 = p2, we call this common value pc and use this pooled value in calculating σd:
In practice, if
we use
as an estimate of pc in calculating σd.
➥ Example 6.5
1. In a random sample of 1500 First Nations children in Canada, 162 were in child welfare care, while in an independent random sample of 1600 non-Aboriginal children, 23 were in child welfare care.
Many people believe that the large proportion of indigenous children in government care is a humanitarian crisis. Do the above data give significant evidence that a greater proportion of First
Nations children in Canada are in child welfare care than the proportion of non-Aboriginal children in child welfare care?
2. Does a 95% confidence interval for the difference in proportions give a result consistent with the above conclusion?
1. Parameters: Let p1 represent the proportion of the population of First Nations children in Canada who are in child welfare care. Let p2 represent the proportion of the population of
non-Aboriginal children in Canada who are in child welfare care.
Hypotheses: H0: p1 − p2 = 0 or H0: p1 = p2 and Ha: p1 – p2 > 0 or Ha: p1 > p2.
Procedure: Two-sample z-test for a difference of two population proportions.
are all at least 10; the samples are random and independent by design; and it is reasonable to assume the sample sizes are less than 10% of the populations.
Mechanics: Calculator software (such as 2-PropZTest) gives z = 11.0 and P = 0.000.
Conclusion in context with linkage to the P-value: With this small of a P-value, 0.000 < 0.05, there is sufficient evidence to reject H0; that is, there is convincing evidence that that the true
proportion of all First Nations children in Canada in child welfare care is greater than the true proportion of all non-Aboriginal children in Canada in child welfare care.
2. Calculator software (such as 2-PropZInt) gives that we are 95% confident that the true difference in proportions (true proportion of all First Nations children in Canada in child welfare care
minus the true proportion of all non-Aboriginal children in Canada in child welfare care) is between 0.077 and 0.110. Since this interval is entirely positive, it is consistent with the
conclusion from the hypothesis test. | {"url":"https://knowt.com/note/886406e4-6099-4c24-aba7-41f5acf9e454/Chapter-6-Inference-for-Categorical-Dat","timestamp":"2024-11-07T14:19:50Z","content_type":"text/html","content_length":"241726","record_id":"<urn:uuid:d238226b-18a6-4063-abea-aa375cb408c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00644.warc.gz"} |
Dropkick Math can help students overcome math anxiety.
Mathematics is a skill that people use throughout their lives, so children must learn this skill at school. Unfortunately, both children and adults can feel stressed and anxious when doing math.
People who experience these feelings of stress when faced with math-related situations may be experiencing what is called “math anxiety.”
Math anxiety can affect anyone at any stage in life because it is related to poor math ability in school and later adulthood. So, if you have ever felt stressed or anxious when dealing with a
math-related situation or have seen your child becoming stressed when doing math homework, it may be math anxiety.
You Are Not Alone
You are not on your own if you have ever experienced stress or anxiety when dealing with math. Many people can feel extremely nervous and overwhelmed when faced with a situation that requires
mathematics. But math anxiety is more than just a feeling of nervousness when facing problems. Nervousness is a sensible reaction to a problem that is actually scary or poses a danger. However,
anxiety does not make sense when dealing with math. This would mean that a person may feel anxious even though he or she knows there is no real reason to feel threatened or in danger.
Anxiety can cause physical symptoms such as racing heart or sweating. With such physical reactions, many people who have math anxiety tend to avoid situations in which they have to do math. Children
with math anxiety will often have poor math skills because their first instinct is to avoid the problem. Adults with math anxiety are less likely to succeed in careers relating to science,
technology, engineering, and mathematics.
Understanding Math Anxiety
It is essential to understand how math anxiety first appears especially when diagnosing a child. It is important to understand what is happening in the brain when a child feels anxious about math so
a parent can best help their child with math anxiety.
Until recently, educators thought that math anxiety first appeared when children learned complicated mathematics (such as algebra). So, this would mean that young children who do not yet do
complicated math would not experience math anxiety. However, recent research shows that some children as young as six years old say that they feel anxious about math.
A recent study examined 154 children in grades 1 and 2 who were asked questions such as,”How do you feel when taking a big test in your math class?” The children were required to indicate how nervous
they felt by pointing to a position on a scale ranging from very nervous to calm. After answering these questions, children took a math test that measured their math abilities. It was found that
almost half of the children who participated in this study reported that they were at least somewhat nervous about doing math, and the children with higher math anxiety got worse scores on the math
test. This research can show that math anxiety and the relationship between math anxiety and math ability can develop when children are very young.
How It Develops
Although research has found that math anxiety and math abilities are related, no study so far has been able to tell which comes first. In other words, it is not yet known if poor math skills cause
anxiety or if having math anxiety makes people worse at math.
Educators do have two ideas about how math anxiety may develop. The first is that children who have difficulty with learning numbers when they are very young are more likely to develop math anxiety
when they start going to school. The other idea is that math anxiety develops in children who experience certain social situations that can influence the child’s thoughts or feelings. This means the
child’s emotions, behaviours, or opinions are affected by things that other people say or do. One small study has shown that teachers with high math anxiety are more likely to have students with
poorer math achievements at the end of the school year. This study helps to show that the way the teacher acted somehow affected the students’ math ability.
Changes In The Brain
To better help a child suffering from math anxiety, a parent must understand the changes in the brain while doing math. Researchers believe that the human brain can only process a certain amount of
information at a time. Working memory, the system in the brain that allows us to process information, is part of the human memory system that will enable us to remember and think about several things
simultaneously. This skill is critical for doing math. For example, when a teacher presents a math problem, students must hold all the numbers in their minds, consider the steps needed to solve the
problem and write out the answer simultaneously. Researchers believe that when people feel anxious, the math anxiety they feel is using up some of their working memory, so there is not as much
leftover to help solve the math problem. If these people did not feel so anxious, they might have more working memory to solve the math problem.
Various studies have supported the idea that math anxiety uses working memory. Researchers have reported that students who have a high level of working memory perform better on math tests compared to
those with a low level of working memory.
A separate study analyzed children with and without math anxiety while they were in a device called a magnetic resonance imaging (MRI). The MRI scanner was able to measure how hard each region of the
brain was working during a specific task. This measurement, called “brain activation,” is counted when a brain region is working hard. Researchers found that a part of the brain called the amygdala
is more activated in children with high math anxiety compared to children with low math anxiety. Overall, this study suggested that when children solve math problems, those with high math anxiety
activate brain regions involved in anxiety. In contrast, those with low math anxiety activate brain regions involved in solving math problems.
How To Help A Child With Math Anxiety
While there is no treatment for math anxiety, educators believe a few tools and actions can help children overcome the condition. The tools that have been created to help people with math anxiety are
called “interventions.” For example, educators have made interventions based on research showing that writing down feelings and thoughts beforehand can make children feel less nervous when taking a
test. They believe that when children write down their thoughts and feelings, they would no longer occupy working memory while completing a math test. Breathing exercises have also been suggested to
help students calm down before a math test. Students have indicated that they feel calmer before a test, and their scores have shown improvements. Together these intervention studies can provide ways
to help students with math anxiety.
How Dropkick Math Can Help
Along with interventions, Dropkick Math offers programs that can help a child improve their math skills. When a child becomes more confident in mathematics, their level of math anxiety decreases.
With our fun and engaging programs, children will learn to become more at ease with math problems.
By understanding the fundamentals of the four pillars of math, students can reduce their math anxiety and acquire new skills that will set them up for a future of success. To help your child overcome
their math anxiety, start by learning more about our programs. | {"url":"https://dropkickmath.com/blog/what-is-math-anxiety/","timestamp":"2024-11-04T20:49:04Z","content_type":"text/html","content_length":"82770","record_id":"<urn:uuid:33b06896-3034-487c-9f75-ad19f32a6e6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00452.warc.gz"} |
Unitary Flow
Monday, October 6, 2014
Wednesday, October 1, 2014
Vectors are present in all domains of fundamental physics, so if you want to understand physics, you will need them. You may think you know them, but the truth is that they appear in so many guises,
that nobody really knows everything about them. But vectors are a gate that allows you to enter the Cathedral of physics, and once you are inside, they can guide you in all places. That is, special
and general relativity, quantum mechanics, particle physics, gauge theory... all these places need vectors, and once you master the vectors, they become much simpler (if you don't know them and are
interested, read this post).
The Cathedral has many gates, and vectors are just one of them. You can enter through groups, sets and relations, functions, categories, through all sorts of objects or structures from algebra,
geometry, even logic. I decided to show you now the way of vectors, because I think is fast and deep in the same time, but remember, this is a matter of choice. And vectors will lead us, inevitably,
to the other gates too.
I will explain some elementary and not so elementary things about vectors, but you have to read and practice, because here I just give some guidelines, a big picture. The reason I am doing this is
that when you study, you may get lost in details and miss the essential.
Very basic things
A vector can be understood in many ways. One way is to see it as a way to specify how to move from one point to another. A vector is like an arrow, and if you place the arrow in that point, you find
the destination point. To find the new position for any point, just place the vector in that point, and the tip of the vector will show you the new position. You can compose more such arrows, and
what you'll get is another vector, their sum. You can also subtract them, just place their origins in the same point, and the difference is the vector obtained by joining their tips with another
Once you fix a reference position, an
, you can specify any position, by the vector that tells you how to move from origin to that position. You can see that vector as being the difference between the destination, and the starting
You can add and subtract vectors. You can multiply them with numbers. Those numbers are from a field $\mathbb{K}$, and we can take for example $\mathbb{K}=\mathbb{R}$, or $\mathbb{K}=\mathbb{C}$, and
are called
. A
vector space
is a set of vectors, so that no matter how you add them and scale them, the result is from the same set. The vector space is real (complex), if the scalars are real (complex) numbers. A sum of
rescaled vectors is named
linear combination
. You can always pick a
, or a
, a set of vectors so that any vector can be written as a linear combination of the basis vectors, in a unique way.
Vectors and functions
Consider a vector $v$ in an $n$-dimensional space $V$, and suppose its components in a given basis are $(v^1,\ldots,v^n)$. You can represent any vector $v$ as a function $f:\{1,\ldots,n\}\to\mathbb
{K}$ given by $f(i)=v^i$. Conversely, any such function defines a unique vector. In general, if $S$ is a set, then the set of the functions $f:S\to\mathbb{K}$ form a vector space, which we will
denote by $\mathbb{K}^S$. The cardinal of $S$ gives the dimension of the vector space, so $\mathbb{K}^{\{1,\ldots,n\}}\cong\mathbb{K}^n$. So, if $S$ is an infinite set, we will have an infinite
dimensional vector space. For example, the scalar fields on a three dimensional space, that is, the functions $f:\mathbb{R}^3\to \mathbb{R}$, form an infinite dimensional vector space. Not only the
vector spaces are not limited to $2$ or $3$ dimensions, but infinite dimensional spaces are very natural too.
Dual vectors
If $V$ is a $\mathbb{K}$-vector space, a linear functions $f:V\to\mathbb{K}$ is a function satisfying $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u,v\in V,\alpha\in\mathbb{K}$. The
linear functions $f:V\to\mathbb{K}$ form a vector space $V^*$ named the
dual space
of $V$.
Consider now two sets, $S$ and $S'$, and a field $\mathbb{K}$. The
Cartesian product
$S\times S'$ is defined as the set of pairs $(s,s')$, where $s\in S$ and $s'\in S'$. The functions defined on the Cartesian product, $f:S\times S'\to\mathbb{K}$, form a vector space $\mathbb{K}^{S\
times S'}$, named the
tensor product
of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, $\mathbb{K}^{S\times S'}=\mathbb{K}^{S}\otimes\mathbb{K}^{S'}$. If $(e_i)$ and $(e'_j)$ are bases of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, then $(e_ie'j)
$, where $e_ie'_j(s,s')=e_i(s)e'_j(s')$, is a basis of $\mathbb{K}^{S\times S'}$. Any vector $v\in\mathbb{K}^{S_1\times S_2}$ can be uniquely written as $v=\sum_i\sum_j \alpha_{ij} e_ie'j$.
Also, the set of functions $f:S\to\mathbb{K}^{S'}$ is a vector space, which can be identified with the tensor product $\mathbb{K}^{S}\otimes(\mathbb{K}^{S'})^*$.
The vectors that belong to tensor products of vector spaces are named
. So, tensors are vectors with some extra structure.
The tensor product can be defined easily for any kind of vector spaces, because any vector space can be thought of as a space of functions. The tensor product is associative, so we can define it
between multiple vector spaces. We denote the tensor product of $n>1$ copies of $V$ by $V^{\otimes n}$. We can check that for $m,n>1$, $V^{\otimes (m+n)}=V^{\otimes {m}}\otimes V^{\otimes {n}}$. This
can work also for $m,n\geq 0$, if we define $V^1=V$, $V^0=\mathbb{K}$. So, vectors and scalars are just tensors.
Let $U$, $V$ be $\mathbb{K}$-vector spaces. A
linear operator
is a function $f:U\to V$ which satisfies $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u\in U,v\in V,\alpha\in\mathbb{K}$. The operator $f:U\to V$ is in fact a tensor from $U^*\otimes
Inner products
Given a basis, any vector can be expressed as a set of numbers, the
of the vector. But the vector is independent of this numerical representation. The basis can be chosen in many ways, and in fact, any non-zero vector can have any components (provided not all are
zero) in a well chosen basis. This shows that
any two non-zero vectors play identical roles
, which may be a surprise. This is a key point, since a common misconception when talking about vectors is that they have definite intrinsic sizes and orientations, or that they can make an angle.
But in fact the sizes and orientations are relative to the frame, or to the other vectors. Moreover, you can say that from two vectors, one is larger than the other, only if they are collinear.
Otherwise, no matter how small is one of them, we can easily find a basis in which it becomes larger than the other.
It makes no sense to speak about the size, or magnitude, or length of a vector, as an intrinsic property.
But wait, one may say, there is a way to define the size of a vector! Consider a basis in a two-dimensional vector space, and a vector $v=(v^1,v^2)$. Then, the size of the vector is given by
Pythagoras's theorem, by $\sqrt{(v^1)^2+(v^2)^2}$. The problem with this definition is that, if you change the basis, you will obtain different components, and different size of the vector. To make
sure that you obtain the same size, you should allow only certain bases. To speak about the size of a vector, and about the angle between two vectors, you need an additional object, which is called
inner product
, or
scalar product
. Sometimes, for example in geometry and in relativity, it is called
Choosing a basis gives a default inner product. But the best way is to define the inner product, and not to pick a special basis. Once you have the inner product, you can define angles between
vectors too. But size and angles are not intrinsic properties of vectors, they depend on the scalar product too.
The inner product between two vectors $u$ and $v$, defined by a basis, is $u\cdot v = u^1 v^1 + u^2 v^2 + \ldots + u^n v^n$. But in a different basis, it will have a general form $u\cdot v=\sum_i\
sum_j g_{ij} u^i v^j$, where $g_{ij}=g_{ji}$ can be seen as the components of a symmetric matrix. These components change when we change the basis, they form the components of a tensor from $V^*\
otimes V^*$. Einstein had the brilliant idea to omit the sum signs, so the inner product looks like $u\cdot v=g_{ij} u^i v^j$, where you know that since $i$ and $j$ appear both in upper and in lower
positions, we make them run from $1$ to $n$ and sum. This is a thing that many geometers hate, but physicists find it very useful and compact in calculations, because the same summation convention
appears in many different situations, which to geometers appear to be different, but in fact are very similar.
Given a basis, we can define the inner product by choosing the coefficients $g_{ij}$. And we can always find another basis, in which $g_{ij}$ is diagonal, that is, it vanishes unless $i=j$. And we
can rescale the basis so that $g_{ii}$ are equal to $-1$, $1$, or $0$. Only if $g_{ii}$ are all $1$ in some basis, the size of the vector is given by the usual Pythagoras's theorem, otherwise, there
will be some minus signs there, and even some terms will be omitted (corresponding to $g_{ii}=0$).
Quantum mechanics
Quantum particles are described by Schrödinger's equation. Its solutions are, for a single elementary particle, complex functions $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}$, or more general, $|\psi\
rangle:\mathbb{R}^3\to\mathbb{C}^k$, named
. They describe completely the states of the quantum particle. They form a vector space $H$ which also has a
hermitian product
(a complex scalar product so that $h_{ij}=\overline{h_{ji}}$), and is named the
Hilbert space
(because in the infinite dimensional case also satisfies an additional property which we don't need here), or the
state space
. Linear transformations of $H$ which preserve the complex scalar product are named
unitary transformations
, and they are the complex analogous of rotations.
The wavefunctions are represented in a basis as functions of positions, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$. The element of the position basis represent
point particles
. But we can make a unitary transformation and obtain another basis, made of functions of the form $e^{i (k_x x + k_y y + k_z z)}$, which represent pure
. Some observations use one of the bases, some the other, and here is why there is a duality between waves and point particles.
For more elementary particles, the state space is the tensor product of the state spaces of the individual particles. A tensor product of the form $|\psi\rangle\otimes|\psi'\rangle$ represents
separable states
, which can be observed independently. If the system can't be written like this, but only as a sum, the particles are
. When we measure them, the outcomes are correlated.
The evolution of a quantum system is described by Schrödinger's equation. Basically, the state rotates, by a unitary transformation. Only such transformations conserve the probabilities associated to
the wavefunction.
When you
the quantum systems, you need an observable. One can see an
as defining a decomposition of the state space, in perpendicular subspaces. After the observation, the state is found to be in one of the subspaces. We can only know the subspace, but not the actual
state vector. This is strange, because the system can, in principle, be in any possible state, but the measurement finds it to be only in one of these subspaces (we say it
). This is the
measurement problem
. The things become even stranger, if we realize that if we measure another property, the corresponding decomposition of the state space is different. In other words, if you look for a point
particle, you find a point particle, and if you look for a wave, you find a wave. This seems as if the unitary evolution given by the Schrödinger's equation is broken during observations. Perhaps the
wavefunction remains intact, but to us, only one of the components continues to exist, corresponding to the subspace we obtained after the measurement. In the
many worlds interpretation
the universes splits, and all outcomes continue to exist, in new created universes. So, not only the state vector contains the universe, but it actually contains many universes.
I have a proposed explanation for some strange quantum features, in [
], and in these videos:
Special relativity
An example when there is a minus signs in the Pythagoras's theorem is given by the
theory of relativity
, where the squared size of a vector is $v\cdot v=-(v^t)^2+(v^x)^2+(v^y)^2+(v^z)^2$.
This inner product is named the
Lorentz metric
. Special relativity takes place in the
Minkowski spacetime
, which has four dimensions. A vector $v$ is named
if $v\cdot v < 0$,
if $v\cdot v > 0$, and
if $v\cdot v = 0$. A particle moving with the speed of light is described by a lightlike vector, and one moving with an inferior speed, by a timelike vector. Spacelike vectors would describe faster
than light particles, if they exist. Points in spacetime are named
. Events can be simultaneous, but this depends on the frame. Anyway, to be simultaneous in a frame, two events have to be separated by a spacelike interval. If they are separated by a lightlike or
timelike interval, they can be connected causally, or joined by a particle with a speed equal to, respectively smaller than the speed of light.
In Newtonian mechanics, the laws remain unchanged to translations and rotations in space, translations in time, and inertial movements of the frame - together they form the
Galilei transformations
. However, electromagnetism disobeyed. In fact, this was the motivation of the research of Einstein, Poincaré, Lorentz, and FitzGerald. Their work led to the discovery of special relativity,
according to which the correct transformations are not those of Galilei, but those of Poincaré, which preserve the distances given by the Lorentz metric.
Curvilinear coordinates
A basis or a frame of vectors in the Minkowski spacetime allows us to construct Cartesian coordinates. However, if the observer's motion is accelerated (hence the observer is
), her frame will rotate in time, so Cartesian coordinates will have to be replaced with curved coordinates. In curved coordinates, the coefficients $g_{ij}$ depend on the position. But in special
relativity they have to satisfy a flatness condition, otherwise spacetime will be curved, and this didn't make much sense back in 1905, when special relativity was discovered.
General relativity
Einstein remarked that to a non-inertial observer, inertia looks similar to gravity. So he imagined that a proper choice of the metric $g_{ij}$ may generate gravity. This turned out indeed to be
true, but the choice of $g_{ij}$ corresponds to a curved spacetime, and not a flat one.
One of the problems of general relativity is that it has
. Singularities are places where some of the components of $g_{ij}$ become infinite, or where $g_{ij}$ has, when diagonalized, some zero entries on the diagonal. For this reason, many physicist
believe that this problem indicates that general relativity should be replaced with some other theory, to be discovered. Maybe it will be solved when we will replace it with a theory of quantum
gravity, like string theory or loop quantum gravity. But until we will know what is the right theory of quantum gravity, general relativity can actually deal with its own singularities (while the
ones mentioned above did not solve this problem). I will not describe this here, but you can read
my articles about this
, and also
this essay
, and these posts about the
black hole information paradox
]. And watch this video
Vector bundles and forces
We call
the functions defined on the space or the spacetime. We have seen that fields valued in vector spaces are actually vector spaces. On a flat space $M$ which looks like a vector space, the fields
valued in vector spaces can be thought of as being valued in the same vector space, for example $f:M\to V$. But if the space is curved, or if it has nontrivial topology, we are forced to consider
that at each point there is another copy of $V$. So, such a field will be more like $f(x)\in V_x$, where $V_x$ is the copy of the vector space $V$ at the point $x$. Such fields still form a vector
space. The union of all $V_x$ is called a
vector bundle
. The fields are also called
, and $V_x$ is called the
at $x$.
Now, since $V_x$ are copies of $V$ at each point, there is no invariant way to identify each $V_x$ with $V$. In other words, $V_x$ and $V$ can be identified, for each $x$, up to a linear
transformation of $V$. We need a way to move from $V_x$ to a neighboring $V_{x+d x}$. This can be done with a
. Also, moving a vector from $V_x$ along a closed curve reveals that, when returning to $V_x$, the vector is rotated. This is explained by the presence of a
, which can be obtained easily from the connection.
Connections behave like
of force fields. And a
force field
corresponds to the curvature of the connection. This makes very natural to use vector bundles to describe forces, and this is what
gauge theory
Forces in the
standard model
of particles are described as follows. We assume that there is a typical complex vector space $V$ of dimension $n$, endowed with a hermitian scalar product. The connection is required to preserve
this hermitian product when moving among the copies $V_x$. The set of linear transformations that preserve the scalar product is named
unitary group
, and is denoted by $U(n)$. The subset of transformations having the determinant equal to $1$ is named the
special unitary group
, $SU(n)$. The
electromagnetic force
corresponds to $U(1)$, the
weak force
to $SU(2)$, and the
strong force
to $SU(3)$. Moreover, all particles turn out to correspond to vectors that appear in the representations of the gauge groups on vector spaces.
What's next?
Vectors are present everywhere in physics. We see that they help us understand quantum mechanics, special and general relativity, and the particles and forces. They seem to offer a unitary view of
fundamental physics.
However, up to this point,
we don't know how to unify
• unitary evolution and the collapse of the wavefunction
• the quantum level with the mundane classical level
• quantum mechanics and general relativity
• the electroweak and strong forces (we know though how to combine the electromagnetic and weak forces, in the unitary group $U(2)$)
• the standard model forces and gravity | {"url":"http://www.unitaryflow.com/2014/10/","timestamp":"2024-11-08T09:02:45Z","content_type":"application/xhtml+xml","content_length":"100897","record_id":"<urn:uuid:7f342b77-e32c-42a5-9bb9-5657dfd99157>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00820.warc.gz"} |
Slope angle determination method
1. The height can vary when a heap or powder shape crosses the height of the funnel, it can be determined relative to the base.
2. The base on which the structure of the pile shapes can be fixed in diameter or the diameter of the powder cone can be allowed to become pile shapes.
As the size of the granules decreases, the flow of the granules is less free. Particle size distribution affects grain separation and internal flow. With particles of relatively small size, the flow
of particles through the hole is restricted, because the forces attached between the particles are of a magnitude equal to the gravitational force. Since the gravitational force is the work of large
diameter to the power of 3, it is important that as the size of the particle increases, the flow is facilitated. Through the flow of the hopper, the pellets show the internal flow and demixing. Due
to the friction force, the flow of the granules is interrupted. Therefore, the digression from the angle of repose articulates as the coefficient of friction μ. | {"url":"https://whatishplc.com/chemistry/slope-angle-determination-method/","timestamp":"2024-11-06T23:13:58Z","content_type":"text/html","content_length":"71417","record_id":"<urn:uuid:2f5dfbdb-3c8f-4c39-b035-7217ea70f1e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00838.warc.gz"} |
[Solved] ImportError: numpy.core.multiarray Failed to Import
In this article, we will take a look at how to solve “ImportError: numpy.core.multiarray failed to import “. We will understand why these kinds of errors occur and what are the possible solutions for
them. Let’s get started,
What is Numpy?
NumPy is the fundamental package for scientific computing in Python. It is a Python library that provides a multidimensional array object, various derived objects (such as masked arrays and
matrices), and an assortment of routines for fast operations on arrays, including mathematical, logical, shape manipulation, sorting, selecting, I/O, discrete Fourier transforms introductory linear
algebra, basic statistical operations, random simulation and much more.
It is vastly used in data pre-processing and later implemented on ML and DL algorithms. Hence, we can say that it is a much-needed library in the field of data science and also provides bases to
several other libraries like pandas, matplotlib e.t.c.
You can refer here to learn more about numpy.
Understanding the error
Understanding the error is one of the significant aspects of programming, as there may be one or more reasons for the error. So, it is essential to know the exact reason for the error and then fix
We will discuss significant reasons why this error occurs:
Incompatible Numpy Version
The main reason for occurrence is that we try to import an incompatible version of numpy to build our program. However, there will be several cases in that also, and we will discuss them one by one.
ImportError: numpy.core.multiarray failed to import in cv2 / matplotlib / pyinstaller / pytorch
Let’s try to understand this. Most machine learning and deep learning python libraries like cv2, matplotlib, pyinstaller, PyTorch, etc. Uses numpy for several operations they perform. More often, we
need more than one of them to build our program. To meet that requirement, sometimes, we install one module after the other. Subsequently, each module alters the numpy version according to its
Now, when we use two or more modules, it might be the case that both the modules use different numpy versions. The system fails to meet the dependable requirement and then raises the “ImportError:
numpy.core.multiarray failed to import” error.
Solving issue for cv2 / matplotlib / pyinstaller / pytorch
In this case, we can fix the error by updating the existing numpy and other modules version to the latest version. To do that open the command-line interface in your system and run the following
python --version
If your python version is python 3.x, then run the following command to uninstall numpy and OpenCV :
pip3 uninstall numpy
pip3 uninstall opencv-python
pip3 uninstall matplotlib
pip3 uninstall pyinstaller
pip3 uninstall torch #run this command twice
Then run the following command to install it again:
pip3 install numpy
pip3 install opencv-python
pip3 install pyinstaller
pip3 install torch==1.3.0+cpu torchvision==0.4.1+cpu -f https://download.pytorch.org/whl/torch_stable.html
If your python version is python 2.x, then run the following command to uninstall numpy and OpenCV :
pip uninstall numpy
pip uninstall opencv-python
pip uninstall matplotlib
pip uninstall pyinstaller
pip uninstall torch #run this command twice
Then run the following command to install it again:
pip install numpy
pip install opencv-python
pip install pyinstaller
pip install torch==1.3.0+cpu torchvision==0.4.1+cpu -f https://download.pytorch.org/whl/torch_stable.html
Now, You can check the numpy, OpenCV, and other versions from python shell.
ImportError: numpy.core.multiarray failed to import in Raspberry Pi
Several times, it happens that we will able to successfully build our model on our local computer or google collab. But when we try to download that model on our Raspberry Pi model, it cannot compile
them. This might happen due to the different version of numpy we are using on our local computer or google collab than one installed on the raspberry pi model.
Solving issue for Raspberry pi
To solve this issue, we need to reinstall numpy with the compatible version of the raspberry pi model. To do that, we have to use the following command in CLI of the raspberry pi model:
pip uninstall numpy
pip install numpy==<version>
ImportError: numpy.core.multiarray failed to import in Docker / Shap
In this scenario again, the reason for getting the error is the same, i.e., the numpy version is not compatible with the docker container version or the shap version. This incompatibility is the main
reason for the occurrence of error.
There are some data points, according to which:
1.) Python 3.6 works with numpy 1.19 and shap 0.39.
2.) Python 3.9 fails with numpy 1.19 and shap 0.39
3.) Python 3.9 works with numpy 1.20 and shap 0.39
4.)Python 3.9 works with numpy 1.19 and shap 0.38
Solving issue for Docker / Shap
To solve the issue, numpy and the shap version must be compatible with the python version. To do that, first, check the python version using:
python --version
After that, we need to install the numpy version and the shap version according to need. To do that, run the following command in CLI:
pip install numpy==<version>
pip install shap==<version>
pip install -r requirements.txt
After completing, run the following command in python shell to check shap and numpy version:
import numpy
import shap
So, before building any program, we should be aware of the libraries and our versions. We should also check the required dependency for the project. I always prefer to work on virtual environments
and manage different virtual environments according to the need of the project.
If we are deploying our model to raspberry pi or some other devices, we should ensure the dependencies are successfully installed there and take care of the versions installed.
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Unofficial Quake 3 Map Specs
This document describes the Quake 3 BSP file format as the author understands it. While every effort has been made to ensure that the contents of this document are accurate, the author does not
guarantee that any portion of this document is actually correct. In addition, the author cannot be held responsible the consequences of the any use or misuse of the information contained in this
Index Lump Name Description
0 Entities Game-related object descriptions.
1 Textures Surface descriptions.
2 Planes Planes used by map geometry.
3 Nodes BSP tree nodes.
4 Leafs BSP tree leaves.
5 Leaffaces Lists of face indices, one list per leaf.
6 Leafbrushes Lists of brush indices, one list per leaf.
7 Models Descriptions of rigid world geometry in map.
8 Brushes Convex polyhedra used to describe solid space.
9 Brushsides Brush surfaces.
10 Vertexes Vertices used to describe faces.
11 Meshverts Lists of offsets, one list per mesh.
12 Effects List of special map effects.
13 Faces Surface geometry.
14 Lightmaps Packed lightmap data.
15 Lightvols Local illumination data.
16 Visdata Cluster-cluster visibility data.
The faces lump stores information used to render the surfaces of the map. There are a total of length / sizeof(faces) records in the lump, where length is the size of the lump itself, as specified in
the lump directory.
int texture Texture index.
int effect Index into lump 12 (Effects), or -1.
int type Face type. 1=polygon, 2=patch, 3=mesh, 4=billboard
int vertex Index of first vertex.
int n_vertexes Number of vertices.
int meshvert Index of first meshvert.
int n_meshverts Number of meshverts.
int lm_index Lightmap index.
int[2] lm_start Corner of this face's lightmap image in lightmap.
int[2] lm_size Size of this face's lightmap image in lightmap.
float[3] lm_origin World space origin of lightmap.
float[2][3] lm_vecs World space lightmap s and t unit vectors.
float[3] normal Surface normal.
int[2] size Patch dimensions.
There are four types of faces: polygons, patches, meshes, and billboards.
Several components have different meanings depending on the face type.
For type 1 faces (polygons), vertex and n_vertexes describe a set of vertices that form a polygon. The set always contains a loop of vertices, and sometimes also includes an additional vertex near
the center of the polygon. For these faces, meshvert and n_meshverts describe a valid polygon triangulation. Every three meshverts describe a triangle. Each meshvert is an offset from the first
vertex of the face, given by vertex.
For type 2 faces (patches), vertex and n_vertexes describe a 2D rectangular grid of control vertices with dimensions given by size. Within this rectangular grid, regions of 3×3 vertices represent
biquadratic Bezier patches. Adjacent patches share a line of three vertices. There are a total of (size[0] - 1) / 2 by (size[1] - 1) / 2 patches. Patches in the grid start at (i, j) given by:
i = 2n, n in [ 0 .. (size[0] - 1) / 2 ), and
j = 2m, m in [ 0 .. (size[1] - 1) / 2 ).
For type 3 faces (meshes), meshvert and n_meshverts are used to describe the independent triangles that form the mesh. As with type 1 faces, every three meshverts describe a triangle, and each
meshvert is an offset from the first vertex of the face, given by vertex.
For type 4 faces (billboards), vertex describes the single vertex that determines the location of the billboard. Billboards are used for effects such as flares. Exactly how each billboard vertex is
to be interpreted has not been investigated.
The lm_ variables are primarily used to deal with lightmap data. A face that has a lightmap has a non-negative lm_index. For such a face, lm_index is the index of the image in the lightmaps lump that
contains the lighting data for the face. The data in the lightmap image can be located using the rectangle specified by lm_start and lm_size.
For type 1 faces (polygons) only, lm_origin and lm_vecs can be used to compute the world-space positions corresponding to lightmap samples. These positions can in turn be used to compute dynamic
lighting across the face.
Feel free to ask for clarification, but please accept my apologies if I can't find the time to answer.
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Review of Short Phrases and Links
This Review contains major "Cryptography"- related terms, short phrases and links grouped together in the form of Encyclopedia article.
1. Cryptography is the science of secret codes, enabling the confidentiality of communication through an insecure channel.
2. Cryptography is a branch of mathematics concerned with obscuring information and then controlling who can retrieve the information. (Web site)
3. Cryptography is a very important domain in computer science with many applications.
4. Cryptography is an evolving field of mathematics that demonstrates the correlation between real-world applications and math. (Web site)
5. Cryptography is an interdisciplinary subject, drawing from several fields.
1. These export restrictions and some of the solutions for strong cryptography, will be discussed later in the section on Legal issues.
2. Legal issues involving cryptography Prohibitions Cryptography has long been of interest to intelligence gathering agencies and law enforcement agencies. (Web site)
1. Surprisingly, it is proven that cryptography has only one secure cipher: the one-time pad. (Web site)
2. Also see: cryptography, block cipher, stream cipher, a cipher taxonomy, and substitution.
3. In classical cryptography, a transposition cipher changes one character from the plaintext to another (to decrypt the reverse is done). (Web site)
4. Image:SAFER.png The study of modern symmetric-key cryptography relates mainly to the study of block ciphers and stream ciphers and their applications. (Web site)
1. In addition to encryption, public-key cryptography can be used to implement digital signature schemes. (Web site)
2. Elliptic curve cryptography is a type of public-key algorithm that may offer efficiency gains over other schemes. (Web site)
3. Ellis's discoveries are kept secret within the British government's cryptography organization. (Web site)
4. Furthermore, public-key cryptography can be used for authentication (digital signatures) as well as for privacy (encryption).
1. Cryptography and cryptanalysis are sometimes grouped together under the umbrella term cryptology, encompassing the entire subject.
2. The study of how to circumvent the use of cryptography is called cryptanalysis, or codebreaking.
1. Most of the interesting new research and applications in cryptography lie in other areas, such as public key cryptography.
2. One solution is based on mathematics, public key cryptography.
1. Also see: cryptography, block cipher, stream cipher, a cipher taxonomy, and substitution.
2. More simply, cryptography is a process associated with scrambling plaintext.
3. In cryptography, a key is a variable value that is applied using an algorithm to a string or block of unencrypted text to produce encrypted text.
1. In cryptography, the Rip van Winkle cipher is a provably secure cipher with a finite key, assuming the attacker has only finite storage. (Web site)
2. Quantum cryptography is still vulnerable to a type of MITM where the interceptor (Eve) establishes herself as "Alice" to Bob, and as "Bob" to Alice.
3. Public-key cryptography may be vulnerable to impersonation, even if users' private keys are not available. (Web site)
4. In its broadest sense, cryptography includes the use of concealed messages, ciphers, and codes.
5. Primality tests are important in cryptography and in particular in the RSA public key algorihtm.
1. The advantage of steganography over cryptography alone is that messages do not attract attention to themselves, to messengers, or to recipients.
2. Later, Johannes Trithemius published the book Steganographia, a treatise on cryptography and steganography disguised as a grimoire.
1. Peter Wayner's website - sample implementations of steganographic techniques, by the author of Disappearing Cryptography. (Web site)
2. The Cryptography FAQ is posted to the newsgroups sci.crypt, talk.politics.crypto, sci.answers, and news.answers every 21 days.
3. This method is known as secret-key cryptography.
4. It is used extensively in a wide variety of cryptographic systems, and in fact, most implementations of public-key cryptography include DES at some level.
1. Linear and differential cryptanalysis are general methods for symmetric key cryptography.
2. Kerberos is based on symmetric cryptography with a trusted key distribution center (KDC) for each realm.
3. Other cryptographic primitives are sometimes classified as symmetric cryptography: Cryptographic hash functions produce a hash of a message.
1. Linear and differential cryptanalysis are general methods for symmetric key cryptography.
2. Conversely, secret key cryptography, also known as symmetric cryptography uses a single secret key for both encryption and decryption.
3. Books on cryptography — an annotated list of suggested readings. (Web site)
4. One advantage is that Category:Cryptography stubs gives a list of all the cryptography-related stubs. (Web site)
5. Importance: If one way functions exist then public key cryptography is secure.
1. Cryptography can be used to implement digital rights management. (Web site)
2. Cryptography World - A very basic guide to cryptography and key management. (Web site)
3. Secret-key cryptography often has difficulty providing secure key management.
1. Since then, cryptography has become a widely used tool in communications, computer networks, and computer security generally. (Web site)
2. By contrast, cryptography obscures the meaning of a message, but it does not conceal the fact that there is a message.
3. Cryptography is central to the techniques used in computer and network security for such things as access control and information confidentiality.
4. Symmetric-key cryptography encompasses problems other than encryption, mainly those that can be accomplished with block ciphers. (Web site)
5. This means that users of weak cryptography (in countries where strong cryptography is forbidden) can safely hide OTP messages in their session keys.
1. Cryptography has a cipher with a strong proof of security: the one-time pad.
2. Introduction to theory of cryptography, stressing rigorous definitions and proofs of security.
1. Quantum cryptography provides the means for two parties to exchange an enciphering key over a private channel.
2. There is also active research examining the relationship between cryptographic problems and quantum physics (see quantum cryptography and quantum computing). (Web site)
3. Quantum cryptography uses the laws of quantum mechanics to encode data securely.
4. For example, some quantum cryptography vendors offer systems that change AES keys 100 times a second.
1. This is usually inconvenient, and public-key (or asymmetric) cryptography provides an alternative. (Web site)
2. Other topics See also: Topics in cryptography The security of all practical encryption schemes remains unproven, both for symmetric and asymmetric schemes.
1. The study of how to circumvent the use of cryptography is called cryptanalysis, or codebreaking.
2. Before the modern era, cryptography was concerned solely with message confidentiality (i.e. (Web site)
3. Generally, the earliest forms of secret writing (now collectively termed classical cryptography) required only pen and paper.
4. As sensitive information is often encrypted, SIGINT often involves the use of cryptography.
5. In cryptography, plaintext is information used as input to an encryption algorithm; the output is termed ciphertext. (Web site)
1. One particularly important issue has been the export of cryptography and cryptographic software and hardware. (Web site)
2. In the United States, cryptography is legal for domestic use, but there has been much conflict over legal issues related to cryptography. (Web site)
3. Another contentious issue in cryptography in the United States was the National Security Agency and its involvement in high quality cipher development.
1. In addition to encryption, public-key cryptography can be used to implement digital signature schemes. (Web site)
2. In addition to encryption, public-key cryptography encompasses digital signatures. (Web site)
3. The key lengths used in public-key cryptography are usually much longer than those used in symmetric ciphers.
4. Cryptography Applies results from complexity, probability and number theory to invent and break codes.
5. Diffie and Hellman showed that public-key cryptography was possible by presenting the Diffie-Hellman key exchange protocol. (Web site)
1. An introduction to number theory and its applications to cryptography.
2. These, in addition to his other works on information and communication theory established a solid theoretical basis for cryptography and for cryptanalysis. (Web site)
1. Quantum cryptography: Public key distribution and coin tossing.
2. In cryptography, the Rip van Winkle cipher is a provably secure cipher with a finite key, assuming the attacker has only finite storage. (Web site)
3. In the cryptography literature this is referred to as the key distribution problem.
4. Quantum cryptography is an effort to allow two users of a common communication channel to create a body of shared and secret information.
1. Cryptographer Someone who creates ciphers using cryptography.
2. The Encyclopedia of Cryptography and Security 684 pages, (Aug 10, 2005), Springer () [1], [2] has some notable cryptographers contributing (e.g.
1. American Revolutionary leaders used various methods of cryptography to conceal diplomatic, military, and personal messages.
2. In practice, "cryptography" is also often used to refer to the field as a whole; crypto is an informal abbreviation.
3. The two complementary properties that are often used in quantum cryptography, are two types of photon---s polarization, e.g.
4. In colloquial parlance, the term " code " is often used synonymously with " cipher." In cryptography, however, "code" traditionally had a specific meaning. (Web site)
1. Cryptography can be used to implement various protocols: zero-knowledge proof, secure multiparty computation and secret sharing, for example.
2. Quantum cryptography protocols achieve something that ordinary classical cryptography cannot.
1. In cryptography, RSA is an algorithm for public key encryption. (Web site)
2. The ElGamal algorithm is an asymmetric-key encryption algorithm for public-key Cryptography which is based on Diffie-Hellman key agreement. (Web site)
3. In cryptography, plaintext is information used as input to an encryption algorithm; the output is termed ciphertext. (Web site)
4. What is certain though, is that modern cryptography does involve more than secret writing, encryption and decryption.
1. This is usually inconvenient, and public-key (or asymmetric) cryptography provides an alternative. (Web site)
2. Diffie and Hellman showed that public-key cryptography was possible by presenting the Diffie-Hellman key exchange protocol.
3. Cryptography can be used to implement some remarkable protocols: zero-knowledge proof, secure multiparty computation and secret sharing, for example. (Web site)
4. Asymmetric cryptography also provides the foundation for password-authenticated key agreement and zero-knowledge password proof techniques. (Web site)
5. The distinction between theory and practice is pronounced in cryptography. (Web site)
1. In practice, "cryptography" is also often used to refer to the field as a whole; crypto is an informal abbreviation.
2. Generally, the earliest forms of secret writing (now collectively termed classical cryptography) required only pen and paper.
3. The two complementary properties that are often used in quantum cryptography, are two types of photon-s polarization, e.g.
1. We might wish to tag the Talk: pages of articles which are directly about cryptography with a notice about this WikiProject.
2. Category:Cryptography --- Should contain all cryptography articles (some of course in subcategories).
3. This article is part of WikiProject Cryptography, an attempt to build a comprehensive and detailed guide to cryptography on Wikipedia.
1. Cryptography and cryptanalysis are sometimes grouped together under the umbrella term cryptology, encompassing the entire subject.
2. In the past, cryptography helped ensure secrecy in important communications, such as those of spies, military leaders, and diplomats.
3. Elliptic curve cryptography is a type of public-key algorithm that may offer efficiency gains over other schemes. (Web site)
4. The modern field of cryptography can be broken down into several areas of study. (Web site)
5. A disadvantage of using public-key cryptography for encryption is speed. (Web site)
1. The Cryptography FAQ is posted to the newsgroups sci.crypt, talk.politics.crypto, sci.answers, and news.answers every 21 days. (Web site)
2. Table of Contents and introduction to the cryptography FAQ. The part of the FAQ discussing the RSA encryption scheme (the one related to factoring). (Web site)
1. In the 1990s, several challenges were launched against US export regulations of cryptography.
2. Cryptography may also be seen as a zero-sum game, where a cryptographer competes against a cryptanalyst.
3. Export Controls Main article: Export of cryptography In the 1990s, there were several challenges to US export regulations of cryptography. (Web site)
4. History of cryptography Main article: History of cryptography The Ancient Greek scytale may have been one of the earliest devices used to implement a cipher.
1. It gives the history of cryptography from ancient times up to recent events.
2. History of cryptography Main article: History of cryptography The Ancient Greek scytale may have been one of the earliest devices used to implement a cipher.
1. One particularly important issue has been the export of cryptography and cryptographic software and hardware. (Web site)
2. Cryptography Software Code in Visual Basic and C Using keys in cryptography a brief introduction to keys and passwords in cryptography. (Web site)
1. The Codebreakers by David Kahn, a comprehensive history of classical (pre-WW2) cryptography.
2. Our Cryptography tutorial and the history of Cryptography, are very interesting.
1. Many countries have tight restrictions on the use of cryptography. (Web site)
2. In some countries, even the domestic use of cryptography is, or has been, restricted. (Web site)
3. Importance If one-way functions do not exist then public key cryptography is impossible.
1. Quantum cryptography was discovered independently in the US and Europe.
2. In particular, a digitally signed contract may be questioned when a new attack on the cryptography underlying the signature is discovered.
1. Abstract: This article is intended as an introduction for non-specialists to Cryptography.
2. Learning About Cryptography - by Terry Ritter (2002); This is a good introduction to cryptography.
1. Cryptography can be used to implement digital rights management. (Web site)
2. The study of how best to implement and integrate cryptography is a field in itself, see: cryptographic engineering, security engineering and cryptosystem. (Web site)
3. Open problems in cryptography List of cryptography topics — an alphabetical list of cryptography articles. (Web site)
4. Cryptonomicon — a novel by Neal Stephenson in which cryptography plays an important role. (Web site)
5. The Codebreakers by David Kahn, a comprehensive history of classical (pre-WW2) cryptography.
1. Since then, cryptography has become a widely used tool in communications, computer networks, and computer security generally. (Web site)
2. In the past, cryptography helped ensure secrecy in important communications, such as those of spies, military leaders, and diplomats.
3. Secure communications See also: Information security Cryptography is commonly used for securing communications.
1. Prior to the early 20th century, cryptography was chiefly concerned with linguistic patterns. (Web site)
2. Before the modern era, cryptography was concerned solely with message confidentiality (i.e. (Web site)
1. Singh, Simon, The Code Book ( ISBN 1-85702-889-9): an anecdotal introduction to the history of cryptography.
2. Alvin's Secret Code by Clifford B. Hicks (children's novel that introduces some basic cryptography and cryptanalysis). (Web site)
3. The invention of radio thus created a new importance for cryptography, the art and science of making secret codes.
4. Cryptography Management Kit includes basic sample source code for the MD5 algorithm. (Web site)
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Question ID - 51185 | SaraNextGen Top Answer
In a simple pendulum experiment for determination of acceleration due to gravity (g), time taken for 20 oscillations is measured by using a watch of 1 second least count. The mean value of time taken
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Non-Gaussianity and cross-scale coupling in interplanetary magnetic field turbulence during a rope–rope magnetic reconnection event
Articles | Volume 36, issue 2
© Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License.
Non-Gaussianity and cross-scale coupling in interplanetary magnetic field turbulence during a rope–rope magnetic reconnection event
In a recent paper (Chian et al., 2016) it was shown that magnetic reconnection at the interface region between two magnetic flux ropes is responsible for the genesis of interplanetary intermittent
turbulence. The normalized third-order moment (skewness) and the normalized fourth-order moment (kurtosis) display a quadratic relation with a parabolic shape that is commonly observed in
observational data from turbulence in fluids and plasmas, and is linked to non-Gaussian fluctuations due to coherent structures. In this paper we perform a detailed study of the relation between the
skewness and the kurtosis of the modulus of the magnetic field $|\mathbit{B}|$ during a triple interplanetary magnetic flux rope event. In addition, we investigate the skewness–kurtosis relation of
two-point differences of $|\mathbit{B}|$ for the same event. The parabolic relation displays scale dependence and is found to be enhanced during magnetic reconnection, rendering support for the
generation of non-Gaussian coherent structures via rope–rope magnetic reconnection. Our results also indicate that a direct coupling between the scales of magnetic flux ropes and the scales within
the inertial subrange occurs in the solar wind.
Keywords.Space plasma physics (turbulence)
Received: 25 Aug 2017 – Revised: 14 Dec 2017 – Accepted: 29 Jan 2018 – Published: 23 Mar 2018
The solar wind can be regarded as a network of entangled magnetic flux tubes and Alfvénic fluctuations propagating within each flux tube (Bruno et al., 2001; Borovsky, 2008). Flux tubes can emerge
locally in the solar wind as a consequence of the magnetohydrodynamic turbulent cascade (Matthaeus and Montgomery, 1980; Veltri, 1999; Greco et al., 2008, 2009; Telloni et al., 2016). An alternative
view describes coherent structures as “fossile” structures that emanate from the solar surface and are advected by the solar wind (Borovsky, 2008; Bruno et al., 2001).
The probability distribution functions (PDFs) of turbulent space plasmas display sharp peaks and fat tails on small scales within the inertial subrange (Sorriso-Valvo et al., 2001; Bruno et al., 2001
; Koga et al., 2007; Chian and Miranda, 2009), as well as departures from self-similarity and monofractality (Bershadskii and Sreenivasan, 2004; Bruno et al., 2007; Miranda et al., 2013). These
features are due to the presence of rare, large-amplitude coherent structures which dominate the statistics of fluctuations on small scales and can be quantified by the computation of statistical
A robust parabolic dependence between the normalized third-order moment (skewness) and the normalized fourth-order moment (kurtosis) has been found in local concentrations of contaminants in
atmospheric turbulence as found by Mole and Clarke (1995). Sura and Sardeshmukh (2007) also found a similar skewness–kurtosis parabolic relation using global data of sea-surface temperature
fluctuations. Labit et al. (2007) reported a similar skewness–kurtosis dependence in electron density fluctuations in plasma confinement experiments. Medina and Díaz (2016) obtained
a skewness–kurtosis parabolic relation for datasets of human reaction times for visual stimuli. Since then, the presence of a skewness–kurtosis relation in different physical scenarios has attracted
much attention (Krommes, 2008; Sattin et al., 2009; Sandberg et al., 2009; Guszejnov et al., 2013; Bergsaker et al., 2015) and has been associated with the presence of non-Gaussian fluctuations due
to coherent structures (Labit et al., 2007; Sandberg et al., 2009; Guszejnov et al., 2013; Bergsaker et al., 2015). The skewness–kurtosis parabolic relation was also found in time series of two-point
differences of the modulus of the magnetic field by Vörös et al. (2006). They demonstrated that the parabolic relation is due to nonlocal interaction between large-scale structures and small-scale
In this paper we investigate the skewness–kurtosis relation during a triple interplanetary magnetic flux rope (IMFR) event detected by Cluster-1 in the solar wind. This event was recently
characterized by Chian et al. (2016). They demonstrated the occurrence of magnetic reconnection at the interface region of two IMFRs and that this reconnection can be the origin of interplanetary
intermittent turbulence. Our results show that the skewness–kurtosis parabolic relation is enhanced during the reconnection between flux ropes, and that is a natural consequence of the interaction
between flux ropes.
This paper is organized as follows. Section 2 presents the statistical tools employed for the data analysis, including the equations to compute the skewness and the kurtosis. Section 3 describes the
triple-IMFR event. The skewness–kurtosis relation is analyzed in detail in Sect. 4. The interpretations of these results are presented in Sect. 5. Finally, we conclude in Sect. 6.
Let θ[i], $i=\mathrm{1},\mathrm{\dots },N$ be the time series of a quantity of interest (e.g., the modulus of the magnetic field $|\mathbit{B}|$). The skewness of θ[i] can be computed as follows:
$\begin{array}{}\text{(1)}& S=\frac{\mathrm{1}}{N}\sum _{i=\mathrm{1}}^{N}{\left(\frac{{\mathit{\theta }}_{i}-〈{\mathit{\theta }}_{i}〉}{\mathit{\sigma }}\right)}^{\mathrm{3}},\end{array}$
where 〈θ[i]〉 represents the average of θ[i], N represents the number of data points, and σ is the SD of θ[i]. The flatness of θ[i] is given by
$\begin{array}{}\text{(2)}& F=\frac{\mathrm{1}}{N}\sum _{i=\mathrm{1}}^{N}{\left(\frac{{\mathit{\theta }}_{i}-〈{\mathit{\theta }}_{i}〉}{\mathit{\sigma }}\right)}^{\mathrm{4}},\end{array}$
from which the kurtosis can be obtained by
$\begin{array}{}\text{(3)}& K=F-\mathrm{3}.\end{array}$
For a Gaussian function $S=K=\mathrm{0}$. The skewness quantifies the degree of asymmetry of the PDF of θ[i], whereas the kurtosis quantifies the departure of the flatness of the PDF of θ[i] from the
flatness of a Gaussian distribution which is equal to 3. The definition of kurtosis in Eq. (3) is sometimes called “excess kurtosis” (Sattin et al., 2009).
A common way to characterize asymmetry and non-Gaussianity of θ[i] as a function of scale τ is through the time series of two-point differences:
$\mathit{\delta }{\mathit{\theta }}_{i}\left(\mathit{\tau }\right)={\mathit{\theta }}_{i+\mathit{\tau }}-{\mathit{\theta }}_{i}.$
The skewness of θ[i] on scale τ is then
$\begin{array}{}\text{(4)}& S\left(\mathit{\tau }\right)=\frac{\mathrm{1}}{N}\sum _{i=\mathrm{1}}^{N}{\left(\frac{\mathit{\delta }{\mathit{\theta }}_{i}-〈\mathit{\delta }{\mathit{\theta }}_{i}〉}{{\
mathit{\sigma }}_{\mathit{\tau }}}\right)}^{\mathrm{3}},\end{array}$
and the flatness is
$\begin{array}{}\text{(5)}& F\left(\mathit{\tau }\right)=\frac{\mathrm{1}}{N}\sum _{i=\mathrm{1}}^{N}{\left(\frac{\mathit{\delta }{\mathit{\theta }}_{i}-〈\mathit{\delta }{\mathit{\theta }}_{i}〉}{{\
mathit{\sigma }}_{\mathit{\tau }}}\right)}^{\mathrm{4}},\end{array}$
where σ[τ] is the SD of δθ[i](τ). From Eq. (5) the kurtosis as a function of scale is obtained by
$\begin{array}{}\text{(6)}& K\left(\mathit{\tau }\right)=F\left(\mathit{\tau }\right)-\mathrm{3}.\end{array}$
A functional relation between the skewness and the kurtosis of θ[i] as defined by Eqs. (1)–(3) has been observed in a variety of scenarios (e.g., Mole and Clarke, 1995; Labit et al., 2007; Medina and
Díaz, 2016). This relation is given by
$\begin{array}{}\text{(7)}& K=\mathit{\alpha }{S}^{\mathrm{2}}+\mathit{\beta },\end{array}$
where α and β are the coefficients that characterize a parabolic curve.
We compute the α and β coefficients by applying a least-square fit between (S,K) values obtained from the observational data and Eq. (7) following the Levenberg–Marquardt algorithm (Levenberg, 1944;
Marquardt, 1963; Bard, 1974), which is a popular method to fit a dataset into nonlinear equations. In order to quantify how well the computed (S,K) values are fitted into Eq. (7) we employ the
correlation index r which measures the correlation between two datasets X[i] and Y[i], $i=\mathrm{1},\mathrm{\dots },N$:
$\begin{array}{}\text{(8)}& r=\frac{\mathrm{1}}{{\mathit{\sigma }}_{X}{\mathit{\sigma }}_{Y}}\sum _{i=\mathrm{1}}^{N}\frac{\left({X}_{i}-〈{X}_{i}〉\right)\left({Y}_{i}-〈{Y}_{i}〉\right)}{N},\end
where σ[X] and σ[Y] represent the SD of X[i] and Y[i], respectively. The correlation index $r\in \left[-\mathrm{1},\mathrm{1}\right]$. If r=1 there is complete correlation between X[i] and Y[i],
whereas $r=-\mathrm{1}$ indicates anticorrelation. The value r=0 represents absence of correlation.
In summary, the analysis is described by the following steps:
We repeat these steps for S(τ) and K(τ) of two-point differences using Eqs. (4) and (6) in the first step. There are several computational programs for data analysis that implement the
Levenberg–Marquardt algorithm. Here we use the implementation available in the GNU Octave program (Eaton, 2012; Eaton et al., 2014).
We note that several papers regarding the relation between skewness and kurtosis have employed the definition of what we refer to as flatness (Eq. 2). Throughout this paper we will focus on the
kurtosis defined by Eq. (3).
Figure 1a shows the time series of the modulus of magnetic field $|\mathbit{B}|$ obtained by the FGM instrument onboard Cluster-1 (Balogh et al., 2001) from 00:00 to 12:00UT on 2 February 2002.
During this interval Cluster-1 was in the solar wind upstream of the Earth's bow shock (Chian and Miranda, 2009). The magnetic field data are collected by Cluster-1 at a resolution of 22Hz (Balogh
et al., 2001). Figure 1 also presents an overview of other in situ plasma parameters for the selected interval, namely, the three components of B in the GSE coordinates, the angles Φ[B] and Θ[B] of
the solar wind magnetic field B relative to the Sun–Earth x axis in the ecliptic plane, and out of the ecliptic, respectively, in the polar GSE coordinates; the modulus of the ion bulk flow velocity
$|{\mathbf{V}}_{\mathrm{i}}|$, the ion number density n[i], the ion temperature perpendicular to the magnetic field T[i] and the ion plasma β[i], which is the ratio between plasma kinetic pressure
and magnetic pressure. The Cluster-1 plasma measurements are given by the ion spectrometry experiment CIS (Rème et al., 2001).
This event is characterized by the presence of three interplanetary magnetic flux ropes. Magnetic flux ropes are magnetic structures described as bundles of twisted, current-carrying magnetic field
lines bent into a tube-like shape, spiralling around a common axis (Russell and Elphic, 1979; Telloni et al., 2016; Chian et al., 2016). During this event three IMFRs were identified by Chian et al.
(2016) using a combination of criteria for large-scale magnetic cloud boundary layers (Lepping et al., 1997; Wei et al., 2003) and small-scale IMFRs (Moldwin et al., 2000; Feng et al., 2007). The
interval of each IMFR is indicated by horizontal arrows in Fig. 1a, and their timings are shown in Table 1.
4 Skewness–kurtosis relation
4.1 Time series of $|\mathbit{B}|$
Figure 2a shows the time series of $|\mathbit{B}|$ detected by Cluster-1 on 2 February 2002 (Julian day 32) from 00:32 to 03:18UT. Five regions were defined during this interval and are indicated
using arrows. These regions represent the interior region of IMFR-1 (R[1]), the interface of IMFR-1 and IMFR-2 (I[12]), the interior of IMFR-2 (R[2]), the interface of IMFR-2 and IMFR-3 (I[23]), and
the interior of IMFR-3 (R[3]). Their timings are indicated in Table 2. Each region has a duration of 30min, which gives 40358 data points. During this event current sheets were detected at the
front boundary layer of IMFR-1 and at the interface region between IMFR-2 and IMFR-3. This interface region was identified as a source of intermittent turbulence by Chian et al. (2016). A current
sheet was detected at the leading edge of IMFR-1 using data from ACE and Cluster-1, and a current sheet was detected at the interface region between IMFR-2 and IMFR-3 using data from Cluster-1, ACE
and Wind (Chian et al., 2016).
The S–K parabolic relation described by Eq. (7) can be verified by computing S and K from a number of datasets corresponding to different realizations of an experiment. In the case of a time series,
the parabolic relation can be tested by computing S and K using datasets extracted from the time series with sliding windows. The size of the sliding window is a critical parameter for this type of
analysis. Since S and K are higher statistical moments, the number of data points inside the window should be large enough to guarantee the robust estimation of S and K. However, if the time series
is divided into sliding windows with a large number of data points, then the number of (S, K) values may be insufficient to verify the parabolic relation of Eq. (7). This can be solved by defining
overlapping windows; nevertheless, the overlapping cannot be too large in order to obtain a set of independent (S, K) values. To determine the optimal window size, we applied a procedure to estimate
the maximum order of the statistical moment in a time series (de Wit, 2004; Miranda et al., 2013). We computed the maximum statistical order in each sliding window of size 5000 data points across the
time series of Fig. 2, and a window shift of 400 data points. Then, we increased the size of the window by 1000 data points (keeping the same window shift), computed the maximum order in each window
and then repeated the procedure. We found that a sliding window of size 10000 data points is large enough for a robust estimation of moments up to the sixth order in all windows and at the same time
allows a sufficient number of estimations of S an K to be obtained to test the parabolic relation of Eq. (7). Figure 2b and c show the resulting time series of S and K, respectively. The SD gives an
estimation of the uncertainty of the computed S and K inside each window, and is represented using a gray area. From this figure we observe that from 02:26 to 02:35UT the uncertainty of S and K
increases due to the large variation in $|\mathbit{B}|$ at the interface between IMFR-2 and IMFR-3. A similar behavior was observed in magnetic field data during an interplanetary shock event by
Vörös et al. (2006). The uncertainty within sliding windows that contain the large variations in $|\mathbit{B}|$ increases due to nonstationarity. Following Vörös et al. (2006), we exclude these
windows from further analysis.
Figure 3 shows K as a function of S for the five regions previously defined. A least-square fit with Eq. (7) is displayed as a dashed line. Table 3 shows the resulting fit for each region, as well as
the correlation index r between the points in the scatter plot and the fitted parabolic function computed using Eq. (8). Since the interpretation of α and β is under debate (see the discussion in
Sect. 5) we will focus on the computed value of r.
The correlation index r shown in the last column of Table 3 measures how well the data points can be adjusted by the parabolic function given by Eq. (7). All regions display r>0.5. The lowest
correlation is obtained for the interval corresponding to R[2], in agreement with a visual inspection of Fig. 3c. For this interval, most of the points in Fig. 3c tend to accumulate around $\left(S,K
\right)=\left(\mathrm{0},\mathrm{0}\right)$, which is the value obtained for a Gaussian distribution (i.e., in the absence of coherent structures). Therefore, the interior of IMFR-2 is characterized
by a low degree of non-Gaussianity and intermittency in comparison with the other intervals.
The highest value of the correlation is obtained during I[23] (see Table 3). Figure 3d shows that points spread near the fitted parabola and far from the (0, 0) Gaussian point. This indicates that
this interval is characterized by a higher degree of non-Gaussianity. These results are in agreement with the results of Chian et al. (2016), which found that the interior of IMFR-2 has lower degrees
of non-Gaussianity and phase coherence, and a nearly monofractal scaling when compared with other intervals. For the interface of IMFR-2 and IMFR-3 they observed higher degrees of non-Gaussianity and
phase synchronization, and a strong departure from monofractality.
4.2 Time series of $\mathit{\delta }|\mathbit{B}|$
Next, we investigate the S–K parabolic relation as a function of scale within the inertial subrange. The left side of Fig. 4a shows the power spectral density (PSD) as a function of frequency f of
the time series of $|\mathbit{B}|$ from the beginning of IMFR-1 at 00:32UT until the end of IMFR-3 at 08:40UT. The right side of Fig. 4a shows the compensated PSD which is the original PSD
multiplied by f^5∕3 (Biskamp et al., 1999). The inertial subrange should appear as a frequency range in which the compensated PSD is almost horizontal. The following panels in Fig. 4 show the PSD and
the compensated PSD for R[1], I[12], R[2], I[23] and R[3]. A common frequency range in which the compensated PSD is almost horizontal for all regions is indicated by two vertical dashed lines. From
Fig. 4, the inertial subrange starts at f=0.01Hz and ends at f=0.1Hz, which correspond to scales τ=100s and τ=10s, respectively.
The intermittent aspect of interplanetary magnetic field turbulence can be demonstrated by constructing the PDF of the normalized magnetic-field differences
$\mathrm{\Delta }B\left(\mathit{\tau }\right)=\frac{\mathit{\delta }B\left(\mathit{\tau }\right)-〈\mathit{\delta }B〉}{{\mathit{\sigma }}_{B}},$
where $\mathit{\delta }B\left(\mathit{\tau }\right)=|\mathbit{B}\left(t+\mathit{\tau }\right)|-|\mathbit{B}\left(t\right)|$, and the brackets denote the average value. Figure 5 shows the PDFs of ΔB
constructed from the magnetic field fluctuations of the five regions, for τ=10s and τ=100s. From this figure it is clear that the PDFs are closer to a Gaussian distribution (represented by the gray
area in Fig. 5) at τ=100s (large scale), and become non-Gaussian at τ=10s (small scale), exhibiting sharp peaks and fat tails. This figure demonstrates that magnetic field fluctuations become more
intermittent as the scale τ becomes smaller.
Next, we analyze the S–K relation of ΔB(τ) at τ=10s and τ=100s. Figure 6a shows the time series of ΔB(τ=100s). Figure 6b and c show the time series of S and K computed using a sliding overlapping
window as in Sect. 4.1. The gray area indicates the uncertainty of the S and K values. As in Fig. 2, we observe a large uncertainty from 02:26 to 02:35UT due to the interface between IMFR-2 and
IMFR-3; therefore these S and K values are excluded from further analysis.
Figures 7 and 8 show the S–K scatter plots for τ=100s and τ=10s, respectively. From these figures, we note that R[2] does not display a parabolic shape on the two selected scales. The low value of
the correlation index of R[2] shown in Tables 4 and 5 confirms that the data points fit the parabolic shape poorly. This indicates that magnetic field fluctuations during R[2] are nearly Gaussian
even on the smallest scale.
Except for R[2], all other regions show a parabolic shape at τ=100s that is enhanced at τ=10s, in agreement with the intermittent nature of magnetic field turbulence. Magnetic field fluctuations in
the solar wind turbulence display a scale dependence in which they become intermittent as the scale becomes smaller, within the inertial subrange, due to rare, large-amplitude coherent structures. As
a consequence, statistics of magnetic field fluctuations such as the PDFs of the ΔB (Fig. 5) departure from Gaussian statistics as τ decreases. By comparing the values of the correlation index shown
in Table 4 for τ=100s with those of Table 5 for τ=10s we note that, for each region, the correlation r increases on the smallest scale, confirming that the S–K parabolic relation displays scale
dependence within the inertial subrange.
The highest correlation value for τ=100s (Table 4) corresponds to I[23]. This indicates that the ongoing magnetic reconnection occurring in this region can act as a source of non-Gaussianity and
intermittent turbulence even on the largest scale. At τ=10s, Table 5 shows that r=0.99 at I[23] and r=0.98 at R[1]. Small-scale current sheets were detected in these two intervals by Chian et al. (
2016) and are responsible for intermittency and non-Gaussian fluctuations. Our result demonstrates that they are also responsible for the enhancement of the S–K parabolic relation. Note that there
are points in Fig. 8d that are further away from the (0, 0) Gaussian point, compared to Fig. 8a. This means that while the scatter plots of R[1] and I[23] are highly correlated with Eq. (7), the
numerical values of S and K, which measure the degree of asymmetry and non-Gaussianity respectively, can be higher at I[23].
A theoretical explanation of the parabolic relation between the skewness and kurtosis of turbulent fluids and plasmas is still an open question. Sura and Sardeshmukh (2007) proposed a nonlinear
Langevin equation with external forcing that can account for the parabolic relation between S and K. Krommes (2008) extended this model to include self-generated internal instabilities in plasmas.
Sattin et al. (2009) argued that a parabolic relation can be obtained as a natural consequence of a number of constraints expected to be met for most physical systems. Guszejnov et al. (2013)
proposed a simplified model of a synthetic intermittent time series, constructed from a random number of coherent structures with random amplitudes embedded in a background Gaussian noise, and
demonstrated that their model can predict a S–K parabolic relation. A similar study was performed by Bergsaker et al. (2015) using a model of coherent plasma flux events.
Although a theoretical explanation of the S–K relation is still unclear, there is a consensus that the parabolic shape is due to non-Gaussianity related to coherent structures, whereas points near $\
left(S,K\right)=\left(\mathrm{0},\mathrm{0}\right)$ correspond to Gaussian fluctuations. This is confirmed by models of synthetic time series. For example, Sandberg et al. (2009) proposed a model of
intermittent time series which consists of a superposition of Gaussian and non-Gaussian random fluctuations. Their model includes a parameter that measures the deviation from Gaussianity. The
resulting PDF derived from their model displays asymmetric long tails that reproduce measured distributions of plasma density fluctuations in plasma magnetic confinement devices (Antar et al., 2001,
2003) as well as distributions of X-ray emissions detected from accretion disks (Sandberg et al., 2009). Their model also leads to a parabolic relation between S and K. Bergsaker et al. (2015)
observed a transition from a parabolic shape to the $\left(S,K\right)=\left(\mathrm{0},\mathrm{0}\right)$ point by increasing the intensity of the Gaussian noise in their model of synthetic time
series, constructed by adding deterministic fluctuations and Gaussian noise. However, a quantification of the parabolic shape is needed for an objective comparison between different datasets. We have
found that the computation of the correlation index r allows time series dominated by either Gaussian and non-Gaussian fluctuations to be clearly distinguished. Despite the simplicity of this
approach, it represents an alternative way to compare the degree of non-Gaussianity due to asymmetry and fat tails in the PDFs of different datasets, and can be applied to observational data and
results from numerical simulations.
The stochastic model of a time series proposed by Sandberg et al. (2009) assumes that the non-Gaussian fluctuations arise from a quadratic nonlinear term. By increasing the degree of non-Gaussianity
the skewness and the kurtosis converge to extreme values: $S=±\mathrm{2}\sqrt{\mathrm{2}}$ and K=12. This means that experimental data governed by nonlinear processes of quadratic order should lead
to S–K scatter plots with $S\in \left[-\mathrm{2}\sqrt{\mathrm{2}},\mathrm{2}\sqrt{\mathrm{2}}\right]$ and K<12. The scatter plots shown in Fig. 3a, c and e seem to agree with these limits; however,
in Fig. 3d there are some points in which $S<-\mathrm{2}\sqrt{\mathrm{2}}$. Sandberg et al. (2009) also propose that processes described by higher-order nonlinearities can result in S–K parabolic
shapes with S outside the interval $\left[-\mathrm{2}\sqrt{\mathrm{2}},\mathrm{2}\sqrt{\mathrm{2}}\right]$, which can explain the behavior of $|\mathbit{B}|$ during the magnetic reconnection
occurring in the I[23] interval.
In the previous sections we showed and discussed the value of the correlation index measuring how well the S–K scatter plots fit with a parabola. As mentioned before, there is no agreement on the
interpretation of the coefficients α and β in Eq. (7). Sattin et al. (2009) argues that the coefficients are not likely to offer relevant information about the underlying process. However, Guszejnov
et al. (2013) discussed an interpretation of the α and β coefficients based on their model of a synthetic time series. The value of the α coefficient depends on the statistics of the fluctuations due
to coherent structures and is not necessarily constant in time. For the β coefficient, if the number of coherent structures in a time series can be represented as random independent variables that
follow a Poisson distribution function (which models the occurrence of rare events), then β=3. Deviations from this value can be interpreted as a departure from the independence assumption, which
means that there is interaction among coherent structures (Guszejnov et al., 2013). Since we define kurtosis to be the flatness minus three, the previous statement is equivalent to say that
deviations from β=0 are due to interacting coherent structures. From Table 3, we note that all intervals have nonzero values of β. Recall that this event is characterized by a rope–rope magnetic
reconnection involving IMFR-2 and IMFR-3, with formation of a bifurcated current sheet acting as a source of intermittent turbulence (Chian et al., 2016). The interaction between the small-scale
IMFR-2 and the medium-scale IMFR-3 occurring during this event gives support for the interpretation of the β parameter by Guszejnov et al. (2013).
Vörös et al. (2006) demonstrated that the S–K parabolic relation is also observed for time series of two-point differences of $|\mathbit{B}|$ in the solar wind. They showed that this relation is
enhanced in the presence of large-scale events such as interplanetary shocks, whereas for nonshock intervals, the parabolic relation is not observed. In this case the S–K parabolic relation
represents a signature of direct coupling between large-scale structures (interplanetary shocks) and small-scale intermittency. Our results indicate that the S–K parabolic relation is present during
reconnection between a small-scale IMFR with a duration of ∼60min and a medium-scale IMFR with a duration of ∼7h (see Table 1). The only region in which the parabolic relation is not observed is in
the interior of IMFR-2. This region was found to have a low degree of intermittency and nearly monofractal scaling. Therefore, our results are in accordance with cross-scale coupling between IMFR
scales and scales within the inertial subrange.
In this paper we investigated the relation between the skewness and the kurtosis during a triple-IMFR event on 2 February 2002. This event was divided into five regions, namely, the interior of
IMFR-1, the interface of IMFR-1 and IMFR-2, the interior of IMFR-2, the interface of IMFR-2 and IMFR-3, and the interior of IMFR-3. We then computed the skewness S and the kurtosis K of $|\mathbit{B}
|$ using a sliding window, and showed that the scatter plots of K as a function of S display a parabolic shape for all regions. The highest value of the correlation index computed by a least-square
fit between the (S,K) values and Eq. (7) occurs at the interface of IMFR-2 and IMFR-3. This region was found to be the source of intermittent turbulence due to a magnetic reconnection between the
small-size IMFR-2 and the medium-size IMFR-3 (Chian et al., 2016). Therefore, the enhanced S–K parabolic relation is related to non-Gaussian fluctuations due to coherent structures emerging from
intermittent turbulence generated via magnetic reconnection. The lowest value of the correlation index was obtained at the interior of IMFR-2, in agreement with the results of Chian et al. (2016),
who found that this region is characterized by a low degree of non-Gaussianity and phase synchronization, and nearly monofractal scaling.
We also analyzed the S–K relation using two-point differences of $|\mathbit{B}|$ on two different scales within the inertial subrange. By computing the compensated PSD we selected an interval of
frequencies in which all regions exhibit $-\mathrm{5}/\mathrm{3}$ scaling corresponding to the inertial subrange and selected two timescales representing the largest scale (τ=100s) and the smallest
scale (τ=10s) within the inertial subrange. We found that the scatter plot of IMFR-2 on the largest scale (τ=100s) and on the smallest scale (τ=10s) accumulate around the $\left(S,K\right)=\left(\
mathrm{0},\mathrm{0}\right)$ point. The least-square fit with Eq. (7) results in a low correlation index, which confirms that magnetic field fluctuations in this region are nearly Gaussian. All other
regions displayed parabolic shapes. At τ=100s, the correlation index is high for the interface of IMFR-2 and IMFR-3, indicating that the magnetic reconnection that occurs in this region can generate
non-Gaussian fluctuations on the largest scale. On the smallest scale, the correlation index is higher for two regions, namely, the interior of IMFR-1 and the interface of IMFR-2 and IMFR-3. This
result can be due to non-Gaussian fluctuations resulting from small-scale current sheets detected within these regions (Chian et al., 2016). Our analysis indicates that the S–K parabolic relation
observed in interplanetary magnetic field turbulence is enhanced on small scales within the inertial subrange.
Our findings give support to the conclusion by Chian et al. (2016) that rope–rope magnetic reconnection acts as a source of interplanetary intermittent turbulence and suggest that magnetic
reconnection is responsible for non-Gaussian PDFs with asymmetric shapes and fat tails. The results are also in agreement with the results of Vörös et al. (2006) in that the S–K parabolic relation is
a signature of direct coupling between IMFR scales and small-scale intermittency.
Code and data availability
All data analyzed in this paper are publicly available via the Cluster Science Archive at http://www.cosmos.esa.int/web/csa (ESA, 2018). Numerical codes are also freely available at https://
github.com/rmiracer (Miranda, 2018).
The authors declare that they have no conflict of interest.
This article is part of the special issue “Space weather connections to near-Earth space and the atmosphere”. It is a result of the 6^∘ Simpósio Brasileiro de Geofísica Espacial e Aeronomia (SBGEA),
Jataí, Brazil, 26–30 September 2016.
The authors are grateful to the reviewer for valuable comments. The authors would like to thank Heng Qiang Feng for providing the estimated times of the boundary layers for the three IMFRs observed
by Cluster-1. Rodrigo A. Miranda acknowledges support from FAPDF (Brazil) under grant 0193.000984/2015. Adriane B. Schelin acknowledges support from FAPDF under grant 0193.000.884/2015.
Abraham C.-L. Chian acknowledges the award of a PVE Distinguished Visiting Professor Fellowship by CAPES (grant no. 88881.068051/2014-01) and the hospitality of Erico Rempel of ITA. José L. Ferreira
acknowledges support from the UNIESPAÇO program of the Brazilian Space Agency (AEB), the National Council of Technological and Scientific Development (CNPq), and FAPDF.
The topical editor, Alisson Dal Lago, thanks the two anonymous referees for help in evaluating this paper.
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P., Parks, G. K., Phan, T., Formisano, V., Amata, E., Bavassano-Cattaneo, M. B., Baldetti, P., Bruno, R., Chionchio, G., Di Lellis, A., Marcucci, M. F., Pallocchia, G., Korth, A., Daly, P. W.,
Graeve, B., Rosenbauer, H., Vasyliunas, V., McCarthy, M., Wilber, M., Eliasson, L., Lundin, R., Olsen, S., Shelley, E. G., Fuselier, S., Ghielmetti, A. G., Lennartsson, W., Escoubet, C. P., Balsiger,
H., Friedel, R., Cao, J.-B., Kovrazhkin, R. A., Papamastorakis, I., Pellat, R., Scudder, J., and Sonnerup, B.: First multispacecraft ion measurements in and near the Earth's magnetosphere with the
identical Cluster ion spectrometry (CIS) experiment, Ann. Geophys., 19, 1303–1354, https://doi.org/10.5194/angeo-19-1303-2001, 2001. a
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Sandberg, I., Benkadda, S., Garbet, X., Ropokis, G., Hizanidis, K., and del-Castillo-Negrete, D.: Universal probability distribution function for bursty transport in plasma turbulence, Phys. Rev.
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Phys. Scripta, 79, 045006, https://doi.org/10.1088/0031-8949/79/04/045006, 2009.a, b, c, d
Sorriso-Valvo, L., Carbone, V., Giuliani, P., Veltri, P., Bruno, R., Antoni, V., and Martines, E.: Intermittency in plasma turbulence, Planet. Space Sci., 49, 1193–1200, 2001.a
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, 2016.a, b
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Geophys., 25, 13–17, https://doi.org/10.5194/angeo-25-13-2007, 2007.
Wei, F. S., Liu, R., Fan, Q., and Feng, X. S.: Identification of the magnetic cloud boundary layers, J. Geophys. Res., 108, A1263, https://doi.org/10.1029/2002JA009511, 2003.a | {"url":"https://angeo.copernicus.org/articles/36/497/2018/","timestamp":"2024-11-09T10:37:38Z","content_type":"text/html","content_length":"322830","record_id":"<urn:uuid:8d124be1-cb12-4177-aab5-e1047539de17>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00429.warc.gz"} |
Which word best describes a rectangle with diagonals that are perpendicular? - Our Planet Today
on April 26, 2022
Which word best describes a rectangle with diagonals that are perpendicular?
Space and Astronomy
What best describes a rectangle with diagonals that are perpendicular?
If the diagonals of a quadrilateral are perpendicular, then it is a rhombus. False – diagonals don’t have to be congruent or bisect each other. The diagonals of a rectangle bisect its angles.
What is it called when diagonals are perpendicular?
Basic properties
Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.
Is a rectangle a perpendicular diagonals?
The diagonals of a rectangle are perpendicular to each other.
Which describes the diagonals of rectangles?
A rectangle is a parallelogram, so its opposite sides are equal. The diagonals of a rectangle are equal and bisect each other.
Are rectangles perpendicular?
Like a square, a rectangle has lines that are perpendicular to two other lines. This means it also has four right angles.
Are the diagonals of a square perpendicular to each other?
The diagonals of a square are perpendicular bisectors of one another. As a result, Their intersection forms four right angles, and each diagonal is split into two congruent pieces.
Are the diagonals of a rectangle perpendicular bisectors of each other?
<br> Diagonals of a rectangle are perpendicular bisector of each other. <br> Diagonals of a rhombus are perpendicular bisectors of each other.
Are diagonals of a rectangle congruent?
Rectangle: A parallelogram with a right angle. When the diagonals are drawn, they bisect each other. The diagonals themselves are congruent, so each bisected line segment is also congruent. The four
triangles formed from the diagonals will be congruent to the one opposite itself.
What is perpendicular in square?
In a square or other rectangle, all pairs of adjacent sides are perpendicular. A right trapezoid is a trapezoid that has two pairs of adjacent sides that are perpendicular. Each of the four
maltitudes of a quadrilateral is a perpendicular to a side through the midpoint of the opposite side.
Are diagonals perpendicular?
The diagonals are perpendicular to and bisect each other. A square is a special type of parallelogram whose all angles and sides are equal. Also, a parallelogram becomes a square when the diagonals
are equal and right bisectors of each other.
How many perpendicular lines are in a rectangle?
How many perpendicular lines does a rectangle have? There are 4 right angles, but 2 pairs of perpendicular lines.
What quadrilaterals have diagonals that are perpendicular?
A B
in these quadrilaterals, the diagonals are perpendicular rhombus, square
a rhombus is always a… parallelogram
a square is always a… parallelogram, rhombus, and rectangle
a rectangle is always a… parallelogram
Which shape has perpendicular lines?
Some shapes which have perpendicular lines are: Square. Right-angled triangle. Rectangle.
What is a perpendicular shape?
A perpendicular shape is a shape that has at least two sides that come together at a 90-degree angle. The box symbol where two lines or sides meet verifies that they are perpendicular. A right
triangle has one right angle and two perpendicular lines.
What is perpendicular in geography?
1. perpendicular – a straight line at right angles to another line. straight line – a line traced by a point traveling in a constant direction; a line of zero curvature; “the shortest distance
between two points is a straight line”
Can a square be a rectangle?
FAQs on Is Square a Rectangle
A square is a rectangle because it possesses all the properties of a rectangle. These properties are: Interior angles measure 90^∘ each. Opposite sides that are parallel and equal.
What is perpendicular example?
Perpendicular – Definition with Examples
Two distinct lines intersecting each other at 90° or a right angle are called perpendicular lines. Here, AB is perpendicular to XY because AB and XY intersect each other at 90°. The two lines are
parallel and do not intersect each other.
Which best describes perpendicular lines?
In geometry, a branch of mathematics, perpendicular lines are defined as two lines that meet or intersect each other at right angles (90°).
Does perpendicular mean vertical?
A vertical line and a horizontal line are perpendicular. For lines that are neither vertical nor horizontal, they are perpendicular if and only if the slope of one is the negative reciprocal of the
slope of the other.
Which statement best describes perpendicular lines?
Perpendicular lines are lines that intersect at a right (90 degrees) angle.
Which of the following best describes a perpendicular bisector of a segments?
A perpendicular bisector is defined as a line or a line segment that divides a given line segment into two parts of equal measurement. ‘Bisect’ is the term used to describe dividing equally.
Perpendicular bisectors intersect the line segment that they bisect and make four angles of 90° each on both sides.
How do you find perpendicular?
Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2x + b and solving for b, we get b = 6. Thus,
the equation of the line is y = ½x + 6. Rearranged, it is –x/2 + y = 6. | {"url":"https://geoscience.blog/which-word-best-describes-a-rectangle-with-diagonals-that-are-perpendicular/","timestamp":"2024-11-10T02:06:39Z","content_type":"text/html","content_length":"191782","record_id":"<urn:uuid:11e883df-0607-48c1-aa71-bd4b805497f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00837.warc.gz"} |
Algebraic Identities For Class 8
Algebraic identities for class 8, will teach you the standard identities list, we use to solve the mathematical expressions, which are based on these formulas and identities. Also, students of the
8th standard will learn to prove these identities using distributive law and multiplications techniques.
As we know, identity is equality which is true for all values of the variable. These identities are the algebraic expressions, which defines that the Left-Hand Side(LHS) and the Right-Hand Side(RHS)
of the equation is equal for all the values of the variable.
A variable is a term that can take any value. It can take any position in the number line, which has an infinite number of points. The value of algebraic expression changes with the changed value of
the variable contained in it. The algebraic identities for class 8 also follow the same perception.
The algebraic expressions are usually expressed as monomials, binomials and trinomials based on one, two or three terms present in it. In fact, the expression which has one or more than one terms
present in it is called a polynomial. The number attached to the term of an algebraic expression is called a coefficient.
The algebraic identities for class 8 consist of three major identities, which consist of algebraic expressions and is true for identity definition. The algebraic formulas for class 8 are also derived
using these identities. These identities and formulas will be used to solve algebraic equations. Also, with the help of these identities, we can easily express any given equation which is relevant to
the algebraic identities in the simpler form.
Standard Algebraic Identities List
(1) (a + b)^2 = a^2 + 2ab + b^2
(2) (a – b)^2 = a^2 – 2ab + b^2
(3) (a + b) (a – b) = a^2 – b^2
These are the general algebraic identities. If we put the values for a and b, in any of the above three expressions, the left-hand side of the equation will be equal to the right-hand side.
Therefore, these expressions are called as identities.
Based on these identities, there are a number of algebraic formulas created. These formulas are used to solve algebraic problems. For class 8 and class 9 standard, these algebraic identities and
formulas are commonly used. So, this article will be helpful for the students who are appearing for class 8 and class 9 exams.
As we already discussed algebraic identities, let us now discuss how to prove that these algebraic expressions are actually identities. These proofs will help you to solve many problems of algebraic
questions for class 8 and class 9.
Proof of Standard Algebraic Identities
Hence, with this, all three identities are proved. Now let us solve some problems based on these identities.
Algebra Identities Examples
Example 1: Solve (2x + 3) (2x – 3) using algebraic identities.
Solution: By the algebraic identity number 3, we can write the given expression as;
(2x + 3) (2x – 3) = (2x)^2 – (3)^2 = 4x^2 – 9
Example 2: Solve (3x + 5)^2 using algebraic identities.
Solution: We know, by algebraic identity number 1,we can write the given expression as;
(3x + 5)^2 = (3x)^2 + 2*3x*5 + 5^2
(3x + 5)^2 = 9x^2 + 30x + 25
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How to Make a Computer That Sees the FutureHow to Make a Computer That Sees the Future
A computer that sees the future is the key to time travel … at least in the science fiction series, The Shadows of Time,
But it’s not as crazy as it sounds.
In a way, a forecasting computer that sees the future already exists.
A close-up view of an IBM quantum computer.
Superposition—Collapsing Probabilities
Before we look at this forecasting computer, we must understand a little about superposition—at least in its definition for modern and not classical physics. A good explanation for quantum
superposition goes way beyond the scope of this blog. To keep it simple, let’s just say the state of a subatomic particle is not set until it’s measured. The measurement itself sets the state.
Until the particle is measured, the particle is in an indeterminate state we call a wave function, a set of possibilities that collapses when observed.
For a better explanation of it in layman’s terms, check out Quantum Superposition, Explained Without Woo Woo – Science Asylum, Nick Lucid.
The Future
In the science fiction series, The Shadows of Time, the future is viewed in the same way as a set of possibilities that collapses when observed. Until the future is observed in the present, we can
view it as a wave function. Well, not us. I mean, we’re not smart enough to create a wave function in real time. That would take a computer so advanced that only pan-dimensional beings with whiskers
and tails could create it.
The future is a set of probabilities that collapses when observed.
Seeing the Future
So, how would we create a forecasting computer?
In quantum computing, the state of a qubit exists as a superposition of all possible states. The computer assigns each state with a probability of it being of 0 or 1. Until the qubit is measured, it
is both 0 and 1, but with a certain likelihood of collapsing to 0 versus 1. The act of measuring the qubit causes the quantum superposition to collapse.
In a study published in Nature Communications, researchers discussed how they accurately predicted the decay of a quantum system. They even prevented this breakdown from occurring.
What’s the big deal about a computer than can tell you the position, direction, and speed of a particle at the same time? I want the winning lottery numbers.
I’m glad you asked. In The Shadows of Time series, you can use this computer to:
• Seed a wormhole,
• Hold it open long enough for your safe passage, and
• Make sure you end up in the right time and place.
In my novels, they call that forecasting computer the Ox Shalay.
A computer that sees the future
is the key to time travel.
Telling the Future
Great. Your Ox Shalay computer is a fancy flux-capacitor. It makes time travel possible. Can I get it to do one more thing for me? Can it tell me the winning lottery numbers? I want the 2052 edition
of the Grays Sports Almanac without running into Biff.
How would this super intelligent computer tell me the future if all it sees is a bunch of collapsing probabilities?
The future is not written in stone, at least from our perspective. So, if all the Ox Shalay sees is a bunch of possibilities that collapse in the present, how would it communicate those
possibilities? One way is through poetic verse that adds uncertainty to the same degree of the probability rating.
In other words, if there is a 100% chance the light is on in a room I’m going into, then the Ox Shalay would report that the room is bright. If there’s a 50% chance the light will be on, the Ox
Shalay will report something like, “the way before you will be unclear.” Whoever reads the Ox Shalay’s predictions will have the same level of uncertainty that the computer has.
Either way, I’m hoping there’s a computer like the Ox Shalay wrapped under a tree with my name on it this Christmas.
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How to Create Superscripts in LaTeX: An In-Depth Guide – TheLinuxCode
How to Create Superscripts in LaTeX: An In-Depth Guide
Superscripts are an indispensable tool for technical writing and formatting mathematical notation. But learning how to make superscripts in LaTeX can be tricky for beginners.
This comprehensive 2500+ word guide will teach you everything you need to know about creating superscripts elegantly and effortlessly in LaTeX documents. We‘ll cover:
• Key use cases and conventions for superscripts
• Basic and advanced superscript usage
• Proper formatting in text and math mode
• Troubleshooting common errors
• Automating superscripts in documents
• And more, with plenty of examples throughout!
Follow along step-by-step to become a LaTeX supercript expert!
Introduction to Superscripts
Before we dive into LaTeX methods, let‘s briefly overview what superscripts are and when to use them.
A superscript is text set slightly above the normal line height and reduced in font size from the body text. Superscripts are commonly used for:
• Exponents and indices in mathematical or scientific formulas
• Annotating citations and footnotes with numbering
• Numbering chapter/section headings (e.g. "3.2 Research Methodology")
• Typesetting ordinals like 1st, 2nd, and 3rd
• Marking trademarks, copyright symbols, and special characters
You‘ve surely seen superscripts in technical documents, research papers, physics textbooks, and more. But what are the conventions for properly using them?
The Chicago Manual of Style provides some guidance:
• Use superscripts for citations and footnotes/endnotes
• Format ordinal numbers as superscripts (1st, 2nd, etc.)
• Mark foot measure symbols ‘ and " as superscripts
• Use superscripts for exponents in simple mathematical equations
Importantly, avoid using superscripts for:
• Complex multiline equations – use regular script size
• Ordinal numbers in headings – use normal text (e.g. First Chapter)
• Cross-referencing figures or tables – use plain text callouts
Adhering to these standards improves the readability of documents. Now let‘s see how to implement superscripts in LaTeX.
Basic Superscript Usage
The most common way to create a superscript in LaTeX is using the ^ operator:
This will set "2" as a superscript above the x.
You can use ^ to superscript any symbol, number, or character:
H^2^O, n^th^, x^10^
For superscripting text strings, enclose the full phrase in braces:
The \textsuperscript{1st} time
Note that text set with \textsuperscript will have spacing and font size adjusted automatically to suit a superscript.
You can also stack multiple superscripts vertically by nesting braces:
Now you know the basics of making superscripts with the ^ operator and \textsuperscript command. But there‘s much more we can do to customize and advanced superscripts in LaTeX.
Advanced Superscript Usage
LaTeX provides finer control over superscript positioning and formatting for optimal results.
Adjusting Superscript Height
By default, \textsuperscript raises text approximately 3.25pt above the baseline. You can tweak this height by passing an optional argument of a length value:
\textsuperscript[1.5pt]{Hi} % Raise 1.5pt
This gives you flexibility when the default supscript position looks off.
Superscripts in Math Mode
For quality typesetting of mathematical and scientific notation, LaTeX‘s math mode is essential.
Math mode handles superscript spacing, sizing, and positioning seamlessly, even for complex nested formulas:
$\mathbf{F} = G\frac{m_1m_2}{r^2}\hat{\mathbf{r}}$
The superscript becomes a natural part of the expression.
Math mode also adapts spacing around superscripts appropriately:
$x^{2}y$ versus $x^2y$
The extra {} around 2 avoids cramped spacing.
For advanced usage, explore packages like physics for specialized math typesetting.
Superscripting Entire Expressions
At times you may want to superscript an entire phrase or expression as one unit:
$z = x\textsuperscript{2 + sin($\theta$)} + y$
This is useful in complex equations where you need to group terms visually.
Stacked Superscripts
LaTeX handles stacked/nested superscripts well, but brace grouping can help:
• $((g^2)^3)$ – Clear, but braces optional
• $(g^2^3)$ – Ambiguous, needs {g^(2^3)}
Always check that stacked superscripts render clearly and unambiguously. Add {} grouping if needed.
Superscripting Symbols
Certain symbols carry semantic meaning as superscripts, like:
• Degrees °
• Primes ‘ and " for arcminutes/seconds
• Registered trademark ®
To make these symbols superscripts:
60‘‘\textsuperscript{‘‘} = 1‘
Escape special characters and check for font support.
With these advanced techniques, you can fine-tune superscripts for any situation.
Superscripts in Math Formulas
One of the most common uses for superscripts is typesetting mathematical formulas, especially:
$F = G\frac{m_1m_2}{r^2}$
Indices for summation/products
$\prod\limits_{i=1}^{n} x_i$
Subatomic particles
Ionic compounds
Foot marks for arcminutes/seconds
Superscripts lend semantic meaning to variables, terms, annotations, and units.
In a study published in the ACS Journal of Chemical Education, researchers found that using proper superscripts and subscripts "enhanced the readability of math formulas for easier comprehension"
(Wang, 2022).
Follow these best practices for superscripts in formulas:
• Use math mode for chemical compounds, equations, etc.
• Add {} grouping when needed for clear nesting
• Leave ample space around superscripts
• Break long expressions over multiple lines
• Use roman font for ionic charges like Na^+
• Make foot marks ‘ and " superscript symbols
With correct superscript usage, you can make complex formulas far more readable.
Troubleshooting Superscripts
Here are some common superscript issues and how to fix them:
Spacing looks too tight
• Add spacing manually with \, or \
• Use math mode for more robust spacing
Superscripts too high/low
• Adjust raise amount for \textsuperscript
• Check math mode vs. text mode position
Glyph or symbol not available
• Use a font with large math support like Latin Modern or STIX
• Consult LaTeX glyph/symbol docs to escape properly
Ambiguous nested superscripts
• Add {} grouping to clarify, like {(x^2)^y}
Package conflicts
• Isolate packages to identify issue
• Import only needed superscript functions
Compilation errors
• Check for missing braces or escaping
• Enable messages to see where issue occurs
Following best practices for spacing, indentation, line breaking, and packages will prevent many errors. Always check compiler messages and warnings for clues!
Automating Superscripts in Documents
Manually formatting superscripts everywhere can be tedious. Luckily, LaTeX provides ways to automate them!
Section Numbering
Auto-number section headings with \section:
\section{Chapter III\textsuperscript{4}}
You can customize the format through your document class.
Tables and Figures
Automatically number tables and figures with packages like caption:
\caption{Figure \textsuperscript{6}: Results...}
This way you don‘t need to manually update when adding content.
Table of Contents
The tocloft package gives you advanced control over the table of contents.
Format chapter numbers, indents, leaders, and more:
\setlength{\cftchapnumwidth}{2em} % Number width
\renewcommand{\cftchappresnum}{Chapter } % Label
\setlength{\cftchapnumwidth}{2em} % Number width
Consult the tocloft documentation for detailed configuration options.
Automating recurring superscripts makes your document source much cleaner and more maintainable.
Use Cases for Superscripts
Let‘s examine some common scenarios where superscripts are conventionally used:
Academic Writing
In research papers and reports, superscripts mark:
• Citations [^1]
• Footnotes[^see] and endnotes
• Equation terms like exponents
This allows readers to quickly check sources and annotations without losing their place.
In fact, a analysis of over 350,000 research articles found that over 92% used superscripts for citations (Smith et al, 2019). Following accepted scholarly conventions makes your academic writing
more readable.
Math and Physics Notation
As discussed earlier, superscripts are ubiquitous in scientific formulas and equations across disciplines like:
• Physics – for quantities like force, work, velocity
• Chemistry – ionic compounds, molecular formulas
• Mathematics – expressions, exponents, factorial notation
Superscripts compactly convey meaning without interrupting the flow of complex notation.
Superscripts are an elegant way to indicate versions, releases, and revisions:
Our newest product - AcmePlus\textsuperscript{4.2}
Tech products and documentation often use superscript versioning.
Ordinal Numbers
The Chicago Manual of Style prescribes formatting ordinal numbers like 1st, 2nd, and 3rd with superscript:
On the 1\textsuperscript{st} day of the month...
This applies for formal writing in books, legal documents, and other publications.
Trademarks and Copyrights
To properly annotate trademarks, registered symbols ® should be set in superscript:
LEGO\textsuperscript{®} is a trademark...
The same goes for copyright © and service marks TM symbols.
Superscripts help identify protected IP.
We‘ve covered a lot of ground here! To recap:
• Use superscripts for citations, exponents, ordinals, and other semantic formatting
• LaTeX offers easy creation with ^ and \textsuperscript
• Math mode provides robust superscript typesetting
• Format trademarks, versioning, and formulas with superscripts
• Automate recurring superscripts in sectioning, ToC, captions
• Fix spacing, nesting, and glyph errors through troubleshooting
LaTeX empowers you to leverage superscripts effectively with semantic meaning. Superscripts are a tool, so be sure to use them only when appropriate according to conventions.
For more help, refer to:
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Hash tables and bit buckets
So I was working on homework for my database II class this morning (very early this morning, I might add) when I came across the final problem asking what a hashing table with the following hashing
formula with various values of B plugged into it would be useful for:
H = n^2 mod B
Now, using the value of 10 for B doesn’t really help all that much, as you leave four of the buckets empty at all times and four of the other buckets will get double filled, assuming an equal
distribution of values for n (0,1,2…) But there is a reason to use this! What happens if you set B=2, and start plugging in negative values in for n? Well….
• 10^2 mod 2 = 0
• 11^2 mod 2 = 1
• 12^2 mod 2 = 0
• etc.
We get alternating values of 0 and 1. But we could do this without going the extra step of squaring the value and get the same result, right? Well, yes and no… what happens if we input a negative
number for n?
• (-10) mod 2 = 0
• (-11) mod 2 = -1
Uh-oh! Now we have a situation where we are getting a value that does not fall into a bit bucket that exists! What to do, what to do? Well, if we square the value of n in this case, all of the even
values of n, whether they are positive or negative, will fall into the 0 bucket, and all of the odd values of n, positive or negative, will fall into the 1 bucket:
• ( -10)^2 mod 2 = 0
• (-11)^2 mod 2 = 1
Thus, problem solved! Now I just hope that I got this right and I get the nice extra credit that I so richly deserve for being a fucking genius. 🙂 | {"url":"https://blog.inthewings.net/?p=7","timestamp":"2024-11-05T22:39:40Z","content_type":"text/html","content_length":"223072","record_id":"<urn:uuid:b39bfc6a-bda2-4bdb-ae61-3b07a6d31733>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00159.warc.gz"} |
Hydromount Damping Block
This block stores a set of coefficients for the hydromount formulation. The block name is a concatenation of HYDROMOUNT_DAMPING_ and the force or moment direction. For instance, if the direction
block selects the hydromount formulation for the FZ direction, then the property file should contain a [HYDROMOUNT_DAMPING_FZ] block.
The data block contains a scalar value that determines the number of preload sub-blocks. For each preload there must be a corresponding data sub-block defined as: (PRELOAD_N) where N is a number from
1 to M with M being the total number of preloads.
Each Hydromount Damping block in the property file must contain all twenty-two (22) hydromount damping parameters.
The following tables shows the names, type and dimension of the parameters. Abbreviations are: [A] = angle dimension, [F] = force, [L] = length, [T] = time.
Table 1. Parameters Associated with
Parameter Name Type Units FX, FY, FZ
R real [T^-1]
K0 real [F][L^-1]
K1 real [F][L^-1]
K2 real [F][L^-1]
C0 real [F][T][L^-1]
C1 real [F][T][L^-1]
C2 real [F][T][L^-1]
P0 real No-Units
P1 real [L^-P2]
P2 real No-Units
Q0 real No-Units
Q1 real [L^-Q2][T^Q2]
Q2 real No-Units
Table 2. Parameters Associated with
the Fluid Model
Parameter Name Type Units FX, FY, FZ
MN real [M]
KH real [F][L^-1]
CH1 real [F][T][L^-1]
CH2 real [F][T][L^-1]
J0 real No-Units
J1 real [L^-P2]
J2 real No-Units
L0 real No-Units
L1 real [[L^-Q2][T^Q2]
L2 real No-Units
Table 3. Parameters Associated with
Transition from Rubber to Full
Parameter Name Type Units FX, FY, FZ
XR real [L]
XH real [L]
In a property file, a Hydromount Damping block has this form:
SPD_FILE_NAME = 'C:\Users\rajivr\Desktop\input_50289_fz.spd'
NPRELOADS = 1
PRELOAD = 0.0
R = 0.001
C0 = 308.74
K0 = 275.345
C1 = 111.554
K1 = 1659.511332
C2 = 0.266419
K2 = 4.01626
P0 = 1.0
P1 = 0.0
P2 = 1.0
Q0 = 1.0
Q1 = 0.0
Q2 = 1.0
Mh = 0.01
Kh = 90.0
Ch1 = 0.10
Ch2 = 0.50
J0 = 1.0
J1 = 0.0
J2 = 1.0
L0 = 1.0
L1 = 0.0
L2 = 1.0
XR = -1.0
XH = 0.0 | {"url":"https://help.altair.com/hwsolvers/altair_help/topics/tools/bushing_model_property_file_hydromount_damping_block_r.htm","timestamp":"2024-11-11T17:15:43Z","content_type":"application/xhtml+xml","content_length":"73242","record_id":"<urn:uuid:19bc2a55-4a1a-43a4-8dcc-7aa981e96e4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00027.warc.gz"} |
What is an Option Spread? - Definition | Meaning | Example
Definition: An option spread is an options strategy that requires the opening two opposite positions to hedge against risk. With an options spread strategy, investors buy and sell the same number of
options on an underlying asset, but at a different strike price and maturity.
What Does Options Spread Mean?
What is the definition of options spread? An options spread is defined based upon the relationship between the strike price and maturity. There are a few different types of spreads. Here are the main
The horizontal spreads are option contracts on an underlying asset with the same strike prices, but different maturity.
The vertical spreads are option contracts on an underlying asset with the same maturity, but different strike prices.
The diagonals spread are option contracts on an underlying asset with different strike prices and maturity.
Furthermore, with a bull spread investors earn a profit if the stock price rises above the strike price, whereas, with a bear spread, investors earn a profit if the stock price falls below the strike
Finally, with a credit spread, investors earn a profit if the premium of the sold options is higher than the premium of the purchased options, whereas, with a debit spread, investors earn a profit if
the premium of the sold options is lower than the premium of the purchased options. All options spread strategies can be constructed using calls or puts.
Let’s look at an example.
Kim is bullish on a technology stock that trades at $120. Because it is too expensive to buy 100 shares of the stock, she decides to buy a bull call spread on the stock to hedge the risk and acquire
the stock at a lower price. Therefore, she buys a bull call spread for $2.20, paying $220.
How can Kim profit from the bull call spread?
Kim believes that the stock price will rise to $125 before maturity. So, she buys a call option at a strike price of $120 and a call option at a strike price of $125. The $120 call gives Kim the
right to buy the underlying asset at the strike price of $120 and the $125 obligates Kim to sell the underlying asset at the strike price of $125. By buying one call option and selling the other call
option, Kim is hedging the risk.
If the stock price rises, above $120, both legs of the bull spread will rise as well. Kim will make a profit from buying the underlying asset at the strike price of $120 and selling it at the strike
price of $125, thus realizing a profit of $5 x 100 shares = $500 – $220 = $280.
Summary Definition
Define Options Spread: An option spread is an investment strategy used to mitigate risk by purchasing options at different strike prices with the spread being the range of potential earnings.
Accounting & CPA Exam Expert
Shaun Conrad is a Certified Public Accountant and CPA exam expert with a passion for teaching. After almost a decade of experience in public accounting, he created MyAccountingCourse.com to help
people learn accounting & finance, pass the CPA exam, and start their career. | {"url":"https://www.myaccountingcourse.com/accounting-dictionary/option-spread","timestamp":"2024-11-02T21:06:37Z","content_type":"text/html","content_length":"154280","record_id":"<urn:uuid:b680455b-58ef-429e-8c1d-a27d5279dc56>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00893.warc.gz"} |
Anything worth doing is worth over doing, right? This time we have another two problems from Programming Praxis, aptly title “Turtle Graphics instead of just printing the characters. 😄
As always if you would like to download the entire source code, you can do so here. Granted, you’ll almost surely need source code.
Now the big question is how do you actually draw those? It turns out, it’s not so bad. Here’s the first function which takes a vector of point lists (like digits above) and draws the list at a given
point with the given turtle:
; using turtle t, draw the digit/letter with index i from table chars
; with the top left at r, c
; reset the turtle to wherever it was after that
(define (draw-thing chars t r c i)
(block t
(let ([ps (vector-ref chars i)])
(lift-pen! t)
(move-to! t (+ c (cadar ps)) (+ r (caar ps)))
(drop-pen! t)
(let loop ([ps (cdr ps)])
(unless (null? ps)
(move-to! t (+ c (cadar ps)) (+ r (caar ps)))
(loop (cdr ps)))))))
Pretty straight forward. Save the state, lift the pen to move to the first point, and then recursively draw lines from point to point. Now what if we want to use that to draw a number? Still pretty
straight forward. Just loop through the digits, drawing them one at a time. The hardest part was actually formatting them from the right so that the standard mod/div method for extracting digits
would work.
; using turtle t, draw a number n with the top left at r, c
; reset the turtle to whever it was after that
(define (draw-number t r c n)
(block t
(if (= n 0)
(draw-thing digits t r (+ c 10) 0)
(let ([? (< n 0)])
(let loop ([c (+ c (* 10 (digits-in n)) 1)]
[n (abs n)])
(when (> n 0)
(draw-thing digits t r c (mod n 10))
(loop (- c 10) (div n 10)))
(when (and ? (= n 0))
(draw-thing digits t r c 10)))))))
In case you were wondering, here’s the function that will tell me how many characters are in a number, including an extra one for the - in negative numbers:
; helper to calculate how wide a number is (add one for negative numbers)
(define (digits-in n)
(if (= n 0)
(+ (if (< n 0) 1 0)
(inexact->exact (+ 1 (floor (log (abs n) 10)))))))
After getting numbers working, drawing strings was much easier. Particularly because you can directly access the letters from left to right with string-ref:
; using turtle t, draw a string s with the top left at r, c
; reset the turtle to whever it was after that
(define (draw-string t r c s)
(block t
(lambda (i)
(unless (eq? #\space (string-ref s i))
(draw-thing letters t r (+ c (* i 10))
(- (char->integer (char-upcase (string-ref s i))) 65))))
(iota (string-length s)))))
And that’s it. Well, except for the minor fact that we haven’t actually solved the problem yet. 😄
; draw a temperature table
; (draw-image (make-temperature-table))
(define (make-temperature-table)
(let ([t (hatch)])
(draw-string t 15 0 " F")
(draw-string t 15 50 " C")
(lambda (row)
(let* ([f (* row 20)]
[c (inexact->exact (round (/ (* (- f 32) 5) 9)))])
(draw-number t (* (- row) 15) 0 f)
(draw-number t (* (- row) 15) 50 c)))
(iota 16))
(turtle->image t)))
; draw hello world
; (draw-image (hello-world))
(define (hello-world)
(let ([t (hatch)])
(draw-string t 0 0 "Hello World")
(turtle->image t)))
Much better.
That was actually really fun to do. Perhaps I’ll see what other [DEL:trouble:DEL] fun I can get into with it.
If you would like to download the entire source code, you can do so Turtle Graphics).
As a random side note, it’s amusing to watch the turtles actually going about the drawing. Try turning on (live-display #t). | {"url":"https://blog.jverkamp.com/2012/09/08/the-first-two-problems/","timestamp":"2024-11-07T03:03:00Z","content_type":"text/html","content_length":"21002","record_id":"<urn:uuid:1e3116a8-3125-4534-ac9f-d5cec1be250b>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00486.warc.gz"} |
How an abacus work
The abacus is an old Asian tool for calculations that is still used in schools today, as it has proven itself a very useful tool when developing your math skills. Its based on a top section and a
bottom section with columns of beads. The top beads in the top section has a value of 5, 50, 500 and so on for each column, going from right to left. The beads in the bottom section have values of 1,
10, 100, etc.
The user of the abacus can easily perform addition and subtraction of large numbers, with different types of methods. And with some practice the user will eventually not need the abacus itself at all
as the calculations are done on a mental abacus.
With further training multiplication, division and even methods of solving cube roots are possible with the abacus. And children that are taught the abacus get extraordinary math skills at a very
young age, which can be seen in this film clip:
The abacus also comes with two different layouts: classical (Suanpan) or modern (Soroban).
2 thoughts on “How an abacus work”
1. I want to learn abacus do we have any videos
1. There is a great youtube tutorial series to get started with the Abacus: | {"url":"https://realglitch.com/2013/04/how-an-abacus-work/","timestamp":"2024-11-03T18:52:43Z","content_type":"text/html","content_length":"28290","record_id":"<urn:uuid:b2bda43b-015e-487e-b51c-aec3ff6285c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00368.warc.gz"} |
Precision prediction model in FDM by the combination of genetic algorithm and BP neural network algorithm
The accuracy of fused deposition modeling (FDM) prototype is affected by many factors, which process parameters are the most important factor. It is difficult to establish mathematical model
accurately; the reason is that process parameters in FDM are coupled and the forming process is nonlinear. In order to define the effect of various process parameters on the forming precision and
improve the precision of FDM printing, this paper established the precision prediction model based on process parameters by genetic algorithm optimizing the BP neural network’s weight and threshold.
Compared with BP prediction model, the result has shown that the precision of the prediction model is better than those of BP prediction model.
1. Introduction
Addictive manufacturing technology is the advanced manufacturing technology which is the multidiscipline combination of manufacturing technology, information technology and new materials technology
in recent 20 years [1]. The parts are produced by accumulating material layer by layer relatively to the traditional material removal (cutting machining) method. FDM is one of the most mature
addictive manufacturing technology which was put forward by Scott Crump in 1988. It has been successfully applied in many fields in the conceptual model. As shown in Fig. 1, FDM rapid prototyping
machine mainly consists of nozzle device, feeding device and workbench, et al. The nozzle is moved in horizontal $x$ and $y$ planes, the workbench moves vertically along the $Z$ direction. The layers
are formed by extrusion of a plastic filament (ABS, PLA) that is melted in the heating device. FDM in rapid prototyping technology is widely applied in industry because of its simple molding
equipment, low expense of equipment and high reliability, but FDM prototype parts’ precision is low and its surface has the obvious texture, which has seriously restricted and hindered the further
application of FDM [2]. In process, the precision of prototype parts is affected by multiple process parameters which are chosen based on experience and experiment. So, it is unfavorable to FDM
users. Therefore, how to choose the process parameters scientifically and reasonably to improve the precision becomes an urgent demand to solve.
Fig. 1Principle diagram of FDM technology
2. Literature review
Some experiments have been made to improve dimensional accuracy of FDM parts by optimizing process parameters. Zou et al. [3] measured the various characteristics of FDM prototype parts by coordinate
measuring machine and the surface roughness, then they get the relationship between the prototype parts quality and the process parameters by MATLAB. Luo et al. [4]. studied the important process
parameters in FDM process. They discussed the effect of the FDM process control technology on the optimum choice of the process parameters, which provide rational selection for users.
The above literature review proved that the quality of FDM mechanical parts can be improved by the proper choice of optimum process parameters. However, it is difficult to establish functional
relationship between process parameters and dimensional accuracy by the traditional method. In recent years, it has got a good predictive effect by combining the artificial neural network (ANN) with
wavelet transform, fuzzy theory, simulated annealing algorithm and support vector machine method [5]. Gao et al. [6] diagnosed the degree of damage within a certain range of train speed wheel, which
combined genetic algorithm with wavelet neural network, the results show that the algorithm is highly accurate and valid. Mei et al. [7] improved the function of the learning and inference of expert
system through the sample analysis of the artificial neural network, the learning and reasoning functionalizes are enhanced and proved effective in terms of fault diagnosis on multilevel planetary
gear increasers and reducers. D. Bellante et al. [8] developed forecasting model based on the design characteristics of homemade parts, they combined it with the neural network optimization
algorithm, which determined the CAD model with the optimal value. Ji et al. [9] studied the wavelet neural network prediction model of product precision, which was built by using the MATLAB software.
Simulation results indicate that the prediction model has sufficient accuracy. Based on the above research, this paper established a hybrid prediction precision algorithm, which is GA-BP model by
combining genetic algorithm with BP neural network.
3. Methodology
In this paper, five process parameters (cable width offset, layer thickness, filling speed, extrusion speed and the fallback speed) are discussed. Table 1 shows the five process parameters and their
levels, other FDM parameters are controlled at their fixed level.
Table 1Process parameters and their levels
Process parameters Symbols
${V}_{1}$ ${V}_{2}$ ${V}_{3}$ ${V}_{4}$
Cable width offset (mm) A 0.15 0.2 0.25 0.3
Lay thickness (mm) B 0.1 0.15 0.2 0.25
Filling speed (mm/s) C 30 40 50 60
Extrusion speed (mm/s) D 25 30 35 40
The fallback speed (mm/s) E 30 45 60 75
3.1. Acquisition of experimental samples
In order to assess the dimensional precision of parts produced by FDM printing machine, according to the research of H. S. Cho [10], this paper select the standard parts “letter-H” geometry. The
shape of the “letter-H” is so sample that it can be easily measured and analyzed. It can be used to reflect on the material contraction error but also reflect on the warp deformation error. For the
standard part, five dimensional parameters are required to measure, as shown in Fig. 2. Dimensional parameters $a$, $b$ and $c$, $d$ are correspond to the $x$ and $y$ directions respectively, and
dimension $e$ is corresponding to the $z$ direction.
Fig. 3The Corexy structure machine of DIY 3D printer
For the sake of improving the accuracy and efficiency of the experiment, this paper got 24 sets of process parameters with orthogonal test firstly, and then an additional 8 sets of data were obtained
using the interpolation method. Finally, the parts were manufactured by the FDM printing machine based on above 32 sets of process parameters. To get a set of experimental samples, this article
measures the standard parts one by one [11]. This experiment chooses the corexy structure machine of DIY 3D printer, as shown in Fig. 3. PLA is selected as experimental material. In this experiment,
the temperature of extruder nozzle is kept at 210 degrees Celsius, the temperature of hot bed reaches 50 degrees Celsius and the environmental temperature is set to 25 degrees Celsius. The material
of extruder nozzle is brass, and the inner diameter is 0.4 mm. The dimensional parameters of standard part can be measured with micrometer. For the dimension, each value should be measured by three
times. Dimension errors can be obtained by calculating the difference between the actual size and the theoretical size, which are expressed as ${∆}_{a}$, ${∆}_{b}$, ${∆}_{c}$, ${∆}_{d}$, ${∆}_{e}$
.The final experimental samples can be seen in Table 2, 6 groups experimental data (No.4, 7, 11, 15, 20, 31) are chosen as test samples from 32 groups experimental data, others are established as the
training samples.
3.2. Training and simulation
For BP neural network, a set of weights are selected randomly, the given target output can be established as linear equation algebra directly and then come to power [12]. In the actual neural
network, BP neural network and its various forms of transform occupy 80-90 % of the artificial neural network, but BP neural network also has many shortcomings, including slow convergence rate, low
learning rate and easy to fall into local minima, etc. The genetic algorithm (GA) is a kind of natural selection and population genetics random optimization algorithm [13]. BP neural network’s
weights and thresholds are random numbers within the range of –0.5-0.5. The initialization of these parameters has great influence on the network training, but they can’t be obtained accurately.
Therefore, it is necessary to optimize the weights and thresholds by the combination of genetic algorithms and BP neural network to improve the prediction precision [14].
Table 2Experiment samples
Serial A (mm) B (mm) C (mm/s) D (mm/s) E (mm/s) ${∆}_{a}$ (mm) ${∆}_{b}$ (mm) ${∆}_{c}$ (mm) ${∆}_{d}$ (mm) ${∆}_{e}$ (mm)
1 0.15 0.1 30 35 60 –0.433 –0.17 –0.332 –0.158 –0.149
2 0.15 0.1 60 55 30 –0.373 –0.2 –0.412 –0.179 –0.007
3 0.15 0.15 30 45 75 –0.391 –0.138 –0.32 –0.106 –0.007
4 0.15 0.15 60 25 30 –0.365 –0.162 –0.339 –0.185 0.024
5 0.15 0.2 40 35 60 –0.399 –0.145 –0.356 –0.108 –0.021
6 0.15 0.2 50 55 30 –0.376 –0.131 –0.365 –0.126 –0.243
7 0.15 0.25 40 45 75 –0.418 –0.129 –0.374 –0.075 0.083
8 0.15 0.25 50 25 30 –0.38 –0.138 –0.37 –0.124 0.088
9 0.2 0.1 30 25 60 –0.466 –0.189 –0.355 –0.17 –0.013
10 0.2 0.1 60 45 45 –0.455 –0.134 –0.279 –0.083 0.069
11 0.2 0.15 30 55 75 –0.429 –0.119 –0.434 –0.136 0.038
12 0.2 0.15 60 35 30 –0.373 –0.176 –0.349 –0.173 0.092
13 0.2 0.2 40 25 60 –0.434 –0.189 –0.357 –0.169 –0.067
14 0.2 0.2 50 45 45 –0.448 –0.149 –0.37 –0.142 -0.042
15 0.2 0.25 40 55 75 –0.508 –0.127 –0.393 –0.093 0.053
16 0.2 0.25 50 35 30 –0.55 –0.131 –0.427 –0.124 0.053
17 0.25 0.1 40 55 30 –0.746 –0.214 –0.399 –0.277 0.002
18 0.25 0.1 50 35 75 –0.406 –0.22 –0.406 –0.211 0.001
19 0.25 0.15 40 25 45 –0.417 –0.193 –0.381 –0.164 0.043
20 0.25 0.15 50 45 60 –0.46 –0.168 –0.349 –0.105 0.022
21 0.25 0.2 30 55 30 –0.428 –0.167 –0.379 –0.13 0.034
22 0.25 0.2 60 35 75 –0.52 –0.157 –0.398 –0.151 0.053
23 0.25 0.25 30 25 45 –0.43 –0.134 –0.332 –0.089 –0.08
24 0.25 0.25 60 45 60 –0.562 –0.178 –0.459 –0.167 0.221
25 0.3 0.1 40 45 30 –0.444 –0.224 –0.421 –0.224 0.01
26 0.3 0.1 50 25 75 –0.436 –0.189 –0.384 –0.244 –0.079
27 0.3 0.15 40 35 45 –0.423 –0.201 –0.355 –0.155 –0.115
28 0.3 0.15 50 55 60 –0.402 –0.171 –0.373 –0.168 –0.15
29 0.3 0.2 30 45 30 –0.363 –0.15 –0.336 –0.128 –0.109
30 0.3 0.2 60 25 75 –0.449 –0.151 –0.405 –0.145 0.113
31 0.3 0.25 30 35 60 –0.375 –0.141 –0.374 –0.073 –0.202
32 0.3 0.25 60 55 75 –0.676 –0.185 –0.49 –0.179 –0.021
In this study, the hierarchy of chosen neural network is made up of input layer, hidden layer and output layer, as shown in Fig. 4. These dimensional errors (${∆}_{b}$, ${∆}_{d}$, ${∆}_{e}$) are
chosen as simulation samples. Therefore, there are three input parameters and three output parameters in this prediction precision model. The number of nodes in the hidden layer is related to the
network learning time and the size of the error, thus, the selection of the number of hidden neurons has a great influence on the prediction of the whole neural network. According to the study of
Shen et al. [15], the optimal number of hidden neurons is ${n}_{1}$ (refer to Eq. (1)):
where $m$ is the number of output neurons, $n$ is the number of input neurons, and a is a constant between 1 and 10. Thus, the number of optimal hidden neurons should be between 4 and 13, The
training error is shown in Table 3, This paper chooses the number of hidden neurons that minimize the training error (the value is 10), therefore, a 3-10-3 neural network is built in this model.
Table 3The training error of different number of hidden neurons
The number of hidden neurons 4 5 6 7 8
Training error 9.0E-9 2.43E-9 8.92E-10 4.23E-10 9.87E-11
The number of hidden neurons 9 10 11 12 13
Training error 4.65E-11 3.39E-11 6.85E-11 1.20E-10 3.52E-10
The training of neural network is a process of optimizing the weights and thresholds which can make the network output errors small constantly. The BP neural network’s training function is “trainlm”,
which use Levenberg-Marquardt to train network. The error of proposed network target is 0.00001, the choice of function for hidden layer and output layer has large effect on the prediction accuracy
of BP neural network. In this simulation, the transfer functions of hidden layer neuron and output layer neuron are “logsig” and “purelin” respectively. The crossover probability and mutation
probability of genetic algorithm paper are set 0.9 and 0.1 respectively [16], then the network can be trained after the determination of the network structure and corresponding parameters.
Fig. 4Structure of neural network
4. Analysis
Fig. 5 shows the change process of network error. After training for six times, the application of the optimization of the BP neural network by genetic algorithm can reach high precision. The
accuracy of trained model reaches 3.3902e-11, which satisfies the required accuracy for 1.0e-5. Therefore, the optimal weight and threshold is chosen by training network for six times.
Fig. 6 shows the prediction results of dimension errors, ${∆}_{b}$, ${∆}_{d}$, ${∆}_{e}$, which are dimension errors of the $x$, $y$ and $z$ direction respectively. Fig. 6(a) is the dimensional error
of ${∆}_{b}$, Fig. 6(b) is the dimensional error of ${∆}_{d}$, Fig. 6(c) is the dimensional error of ${∆}_{e}$. Among them, the black lines are the actual dimension errors, the blue lines are the
predictive error of BP model and the red lines are predictive error of GA-BP model. In addition to a few cases, the figures show that the precision of GA-BP model is considerably higher than the
prediction precision of the BP model. The combination of genetic algorithm with neural network can significantly reduce the possibility of local optimum. This is because the genetic algorithm can
optimize the weight and threshold individually, which can reduce the possibility of BP neural network’s divergence and vibration. In short, the intrinsic mechanism of GA-BP model determines all kinds
of training and prediction performance, which indicates that the method is feasible and valid in the evaluation of prediction precision and adaptive ability.
Fig. 6The prediction result of dimensional errors
a) Dimensional error of ${∆}_{b}$
b) Dimensional error of ${∆}_{d}$
c) Dimensional error of ${∆}_{e}$
Fig. 7 shows the dimension errors in the $x$, $y$ and $z$ directions. The red line is the actual dimension error ${∆}_{b}$, for which the absolute average error is 0.165 mm. The blue lines are the
actual dimension errors ${∆}_{d}$, for which the absolute average error is 0.149 mm. And the green line is the actual dimension error ${∆}_{e}$, for which the absolute average error is 0.072 mm. In
this figure, in most cases, the error in $z$ direction is less than errors in $x$ and $y$ direction. It is concerned with the principle of FDM, which is accumulating layer by layer. Therefore, the
deformation is focused on the $x$ and $y$ directions. Besides, the dimension error in $y$ direction is less than the error in $x$ direction. The error is largely due to scan of machine, the gliding
in the $x$ direction is in single guideway while the gliding in the $y$ direction is in double guideway, as shown in Fig. 8. Therefore, the motion of nozzle in $y$ direction is smoother than the
motion in the $x$ direction which causes the error in the $y$ direction less than the error in the $x$ direction.
Fig. 7The contrast figure in the x, y and z directions
Fig. 8The figure of guideway: 1 – The guideway of x direction, 2 – The left guideway of y direction, 3 – The right guideway of y direction
5. Conclusions
This study has proposed an effective prediction method of FDM process parameters by using GA-BP model. The work presented the successful application of GA-BP model in FDM process parameter prediction
method and solved the difficult problem (It is difficult to build accurate mathematical model). By the comparison on the same test data, this algorithm has a higher prediction accuracy than the BP
neural network algorithm, which indicates that the GA-BP model is feasible and valid in the evaluation of prediction precision and adaptive ability.
Based on the data comparison of different direction and the same size, the error in $z$ direction is less than the error in $x$ and $y$ directions. This is concerned with the principle of FDM, which
is accumulating layer by layer. The dimension error of $y$ direction is less than the error in $x$ direction. The error is largely due to scan of machine, the gliding in the $x$ direction is in
single guideway while the gliding in the $y$ direction is in double guideway, which causes the motion of nozzle in $y$ direction is smoother than the motion in the $x$ direction.
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Manufacturing, Vol. 17, Issue 12, 2016, p. 1589-1603.
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• Luo Jin,Ye Chunsheng, Huang Shuhuai The study on the important technologic parameters and their control of FDM system. China Metal Forming Equipment and Manufacturing Technology, Vol. 40, Issue
6, 2005, p. 77-80.
• Tian Jingwen, Gao Meijuan Artificial Neural Network Algorithms and Application. Beijing Institute of Technology Press, Beijing, 2006.
• Gao Ruipeng, Shang Chunyang, Jiang Hang A fault detection strategy for wheel flat scars with wavelet neural network and genetic algorithm. Journal of Xi’an Jiaotong University, Vol. 47, Issue 9,
2013, p. 88-91.
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9, Issue 1, 2011, p. 117-121.
• Bellante D. Dimensional accuracy improvement of FDM square cross-section parts using artificial neural networks and an optimization algorithm. The International Journal of Advanced Manufacturing
Technology, Vol. 69, Issue 9, 2013, p. 2301-2313.
• Ji Liangbo Precision prediction model in fused deposition modeling of three-dimensional printing based on wavelet neural network. Journal of Shanghai Jiaotong University, Vol. 49, Issue 3, 2015,
p. 375-378.
• Cho H. S., Park W. S., Choi B. W., Leu M. C. Determining optimal parameters for stereolithography processes via genetic algorithm. Journal of Manufacturing Systems, Vol. 19, Issue 1, 2000, p.
• Fang Kaitai, Ma Changxing Orthogonal and Uniform Experimental Design. Science Press Co. Ltd, Beijing, 2001.
• Xu Liming, Wang Qing Chen Jianping, Pan Yuzhen Forecast for average velocity of debris flow based on BP neural network. Journal of Jilin University, Earth Science Edition, Vol. 43, Issue 1, 2013,
p. 186-191.
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Issue 1, 2003, p. 97-99.
About this article
fused deposition modeling
process parameters
genetic algorithm
neural network
This work was supported by the Fundamental Research Funds for the Central Universities (2017MS150).
Copyright © 2017 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
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Applied Gyrodynamics - A 1932 book on gyroscopes and gyroscope based apperatus
any other axis through the same point if the length of the cylinder is greater than the diameter. The moment of inertia of the cylinder about an axis through the center of mass and in the direction
of the length is less than that about any other axis. In the case of most bodies, the values of the moments of inertia relative to their mutually perpendicular axes are unequal.
It can be shown that the axis of greatest and that of least moment of inertia are perpendicular to one another. The perpendicular axis through the center of mass of a body about which the moment of
inertia is maximum, that about which it is minimum, and the other axis perpendicular to these two, are called the principal momental axes of the body. The moments of inertia about the principal
momental axes are called the principal moments of inertia of the body. If the principal moments of inertia of one body or system are equal, respectively, to the principal moments of inertia of
another body or system, the two bodies or systems are said to be equimomental.
For any rigid body there can be constructed an equimomental system consisting of three slender and uniform rigid rods bisecting each other at right angles, and coinciding in direction with the
principal momental axes of the given body.
14. Centripetal Forces Acting upon an Unsymmetrical Pendulum Bob. - Consider a pendulum having a bob that is capable of rotation with negligible friction about the axis of the pendulum rod and that
is unsymmetrical with respect to the axis of the pendulum rod. In Fig. 13, the pendulum bob is a rectangular bar with the long axis perpendicular to the pendulum rod. The long axis of the bob is
inclined at an angle 0 to the plane of the knifeedge and the pendulum rod. Consider A and A' at the centers of the two ends of the bob at the instant when the pendulum is passing through the
equilibrium position. From A and A' draw lines A B and A'B' perpendicular to the line CC' through the center of the bob parallel to the knife-edge. From B and B' draw lines BD and B'D' perpendicular
to the knife-edge. The points D and D' are the points about which oscillate the two particles at A and A'.
In order that the particles at A and A' may rotate about DD', the particles must be acted upon by centripetal forces F, and F,' directed toward D and D', respectively. Each of the centripetal forces
F,, etc., can be resolved into three components, one vertical, one parallel to the axis of the bob, and one horizontal and per | {"url":"http://gyros.biz/index.asp?pageon=16&pdf=False","timestamp":"2024-11-08T01:11:28Z","content_type":"text/html","content_length":"21030","record_id":"<urn:uuid:f825d7d6-6dc3-443f-bc4c-71a805e6e689>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00870.warc.gz"} |
Free Cash Flow Models: FCFF and FCFE in Valuation
B.2.2 Free Cash Flow Models
In the realm of financial analysis and investment, understanding the concept of free cash flow (FCF) is crucial for assessing a company’s financial health and its potential as an investment
opportunity. Free cash flow models, particularly Free Cash Flow to Firm (FCFF) and Free Cash Flow to Equity (FCFE), provide a comprehensive approach to valuing companies, especially those that do not
pay dividends or have fluctuating dividend policies. This section delves into the definitions, calculations, and applications of FCFF and FCFE, equipping you with the knowledge to make informed
investment decisions.
Understanding Free Cash Flow
Free Cash Flow (FCF) represents the cash generated by a company after accounting for capital expenditures necessary to maintain or expand its asset base. It is a critical measure of a company’s
ability to generate cash and is often used by investors to assess the company’s financial performance and potential for growth.
Free Cash Flow to Firm (FCFF)
FCFF is the cash flow available to all investors, including both debt and equity holders. It reflects the company’s ability to generate cash from its operations, which can be used to pay interest,
dividends, or reinvest in the business. The formula for calculating FCFF is as follows:
$$ FCFF = EBIT \times (1 - \text{Tax Rate}) + \text{Depreciation} - \text{Capital Expenditures} - \text{Increase in Net Working Capital} $$
Components of FCFF:
• EBIT (Earnings Before Interest and Taxes): Represents the company’s operating income before accounting for interest and taxes.
• Tax Rate: The corporate tax rate applicable to the company.
• Depreciation: A non-cash expense that reflects the wear and tear of the company’s assets.
• Capital Expenditures (CapEx): Investments made by the company to acquire or upgrade physical assets such as property, industrial buildings, or equipment.
• Increase in Net Working Capital (NWC): The change in current assets minus current liabilities, indicating the additional capital required to support operations.
Free Cash Flow to Equity (FCFE)
FCFE is the cash flow available to equity holders after accounting for all expenses, reinvestments, and debt repayments. It represents the cash that can be distributed to shareholders in the form of
dividends or stock buybacks. The formula for calculating FCFE is:
$$ FCFE = FCFF - \text{Interest} \times (1 - \text{Tax Rate}) + \text{Net Borrowing} $$
Components of FCFE:
• Interest: The cost of debt financing, which is tax-deductible.
• Net Borrowing: The net amount of new debt raised minus debt repayments during the period.
Calculating FCFF and FCFE
To effectively calculate FCFF and FCFE, follow these steps:
1. Gather Financial Data:
□ Obtain the company’s income statement and balance sheet.
□ Identify key figures such as EBIT, depreciation, capital expenditures, and changes in net working capital.
2. Adjust for Non-Cash Expenses:
□ Add back non-cash expenses like depreciation to EBIT, as they do not involve actual cash outflows.
3. Account for Changes in Working Capital:
□ Calculate the increase in net working capital by subtracting the previous period’s NWC from the current period’s NWC.
4. Calculate FCFF:
□ Use the FCFF formula to determine the cash flow available to all investors.
5. Calculate FCFE:
□ Adjust FCFF for interest expenses and net borrowing to find the cash flow available to equity holders.
Example Calculation:
Consider a company with the following financial data:
• EBIT: $500,000
• Tax Rate: 30%
• Depreciation: $50,000
• Capital Expenditures: $100,000
• Increase in NWC: $20,000
Calculate FCFF:
$$ FCFF = \$500,000 \times (1 - 0.30) + \$50,000 - \$100,000 - \$20,000 = \$280,000 $$
Assuming interest expenses of $30,000 and net borrowing of $10,000, calculate FCFE:
$$ FCFE = \$280,000 - \$30,000 \times (1 - 0.30) + \$10,000 = \$261,000 $$
Discounted Cash Flow (DCF) Valuation Using Free Cash Flows
DCF valuation is a method used to estimate the value of an investment based on its expected future cash flows. When applying DCF to free cash flows, the process involves forecasting future FCFF or
FCFE and discounting them back to their present value.
Steps in DCF Valuation:
1. Forecast Future Free Cash Flows:
□ Project the company’s FCFF or FCFE over a specific period, typically 5 to 10 years.
□ Consider factors such as revenue growth, operating margins, and capital expenditure requirements.
2. Determine the Discount Rate:
□ For FCFF, use the Weighted Average Cost of Capital (WACC) as the discount rate, reflecting the average rate of return required by all investors.
□ For FCFE, use the cost of equity, representing the return required by equity investors.
3. Calculate the Present Value:
□ Discount the forecasted free cash flows to their present value using the chosen discount rate.
4. Estimate Terminal Value:
□ Calculate the terminal value, representing the value of the company beyond the forecast period, using a perpetuity growth model or exit multiple approach.
5. Sum the Present Values:
□ Add the present values of the forecasted cash flows and the terminal value to determine the total enterprise value (for FCFF) or equity value (for FCFE).
When to Use Free Cash Flow Models
Free cash flow models are particularly useful in the following scenarios:
• Non-Dividend Paying Companies: For companies that do not pay dividends, free cash flow models provide a more accurate reflection of their financial performance and value.
• Significant Capital Expenditures: Companies with substantial capital expenditure requirements benefit from free cash flow models, as they account for the cash needed to maintain and grow the
• Changing Dividend Policies: When a company’s dividend policy is inconsistent or expected to change, free cash flow models offer a stable basis for valuation.
Interpreting Valuation Results
The results of a free cash flow valuation provide insights into a company’s financial health and investment potential. A positive free cash flow indicates that the company generates more cash than it
needs to fund its operations and investments, which can be used to pay dividends, reduce debt, or reinvest in the business. Conversely, a negative free cash flow suggests that the company may need to
raise additional capital to fund its operations.
Free cash flow models, including FCFF and FCFE, offer a comprehensive approach to valuing companies, particularly those with significant capital expenditures or fluctuating dividend policies. By
understanding and applying these models, investors can gain valuable insights into a company’s ability to generate cash and its potential as an investment opportunity. Whether used in conjunction
with other valuation methods or as a standalone analysis, free cash flow models are an essential tool in the investor’s toolkit.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What does FCFF represent? - [x] Cash flow available to all investors, including debt and equity holders. - [ ] Cash flow available only to equity holders. - [ ] Cash flow after dividends are
paid. - [ ] Cash flow before accounting for capital expenditures. > **Explanation:** FCFF represents the cash flow available to all investors, including both debt and equity holders, after accounting
for operating expenses and capital expenditures. ### How is FCFE calculated? - [x] FCFE = FCFF - Interest × (1 - Tax Rate) + Net Borrowing - [ ] FCFE = FCFF + Interest × (1 - Tax Rate) - Net
Borrowing - [ ] FCFE = EBIT × (1 - Tax Rate) + Depreciation - Capital Expenditures - [ ] FCFE = Net Income + Depreciation - Capital Expenditures > **Explanation:** FCFE is calculated by adjusting
FCFF for interest expenses and net borrowing, reflecting the cash flow available to equity holders. ### What is the primary use of free cash flow models? - [x] Valuing companies that do not pay
dividends. - [ ] Calculating dividend payout ratios. - [ ] Estimating future stock prices. - [ ] Determining tax liabilities. > **Explanation:** Free cash flow models are primarily used for valuing
companies that do not pay dividends, providing a more accurate reflection of their financial performance. ### Which discount rate is used for FCFF in DCF valuation? - [x] Weighted Average Cost of
Capital (WACC) - [ ] Cost of Equity - [ ] Risk-Free Rate - [ ] Dividend Yield > **Explanation:** The Weighted Average Cost of Capital (WACC) is used as the discount rate for FCFF in DCF valuation,
reflecting the average rate of return required by all investors. ### What does a positive free cash flow indicate? - [x] The company generates more cash than needed for operations and investments. -
[ ] The company is in financial distress. - [ ] The company has a high dividend payout ratio. - [ ] The company needs to raise additional capital. > **Explanation:** A positive free cash flow
indicates that the company generates more cash than needed to fund its operations and investments, which can be used for dividends, debt reduction, or reinvestment. ### Which component is not part of
the FCFF calculation? - [ ] EBIT - [ ] Depreciation - [x] Dividends - [ ] Capital Expenditures > **Explanation:** Dividends are not part of the FCFF calculation, as FCFF focuses on cash flow
available to all investors before any distributions. ### When is FCFE preferred over FCFF? - [x] When evaluating cash flow available to equity holders. - [ ] When assessing overall company
performance. - [ ] When calculating tax liabilities. - [ ] When estimating future capital expenditures. > **Explanation:** FCFE is preferred when evaluating the cash flow available specifically to
equity holders, after accounting for debt servicing. ### What is the terminal value in DCF valuation? - [x] The estimated value of a company beyond the forecast period. - [ ] The initial investment
cost. - [ ] The present value of future dividends. - [ ] The book value of assets. > **Explanation:** The terminal value in DCF valuation represents the estimated value of a company beyond the
forecast period, often calculated using a perpetuity growth model or exit multiple approach. ### Which factor is considered in FCFE but not in FCFF? - [x] Net Borrowing - [ ] Depreciation - [ ]
Capital Expenditures - [ ] Tax Rate > **Explanation:** Net Borrowing is considered in FCFE to adjust for changes in debt, reflecting the cash flow available to equity holders. ### True or False: Free
cash flow models are only applicable to large companies. - [ ] True - [x] False > **Explanation:** False. Free cash flow models are applicable to companies of all sizes, as they provide insights into
a company's ability to generate cash and its potential as an investment opportunity. | {"url":"https://csccourse.ca/32/4/2/","timestamp":"2024-11-09T13:22:02Z","content_type":"text/html","content_length":"119863","record_id":"<urn:uuid:eea4b779-76e0-4094-86b1-3878ff7c4c7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00291.warc.gz"} |
CBSE Class 6 Maths - MCQ and Online Tests - Unit 13 - Symmetry
CBSE Class 6 Maths – MCQ and Online Tests – Unit 13 – Symmetry
Every year CBSE students attend Annual Assessment exams for 6,7,8,9,11th standards. These exams are very competitive to all the students. So our website provides online tests for all the 6,7,8,9,11th
standards’ subjects. These tests are also very effective and useful for those who preparing for any competitive exams like Olympiad etc. It can boost their preparation level and confidence level by
attempting these chapter wise online tests.
These online tests are based on latest CBSE syllabus. While attempting these, our students can identify their weak lessons and continuously practice those lessons for attaining high marks. It also
helps to revise the NCERT textbooks thoroughly.
CBSE Class 6 Maths – MCQ and Online Tests – Unit 13 – Symmetry
Question 1.
Which of the following letters has horizontal line of symmetry?
(a) C
(b) A
(c) J
(d) L.
Answer: (a)
Question 2.
How many lines of symmetry does the figure have ?
(a) 1
(b) 2
(c) 3
(d) no line of symmetry
Answer: (d)
Question 3.
Which of the following letters has horizontal line of symmetry?
(a) Z
(b) V
(c) U
(d) E.
Answer: (d)
Question 4.
How many lines of symmetry does the figure have?
(a) 1
(b) 2
(c) 3
(d) Countless.
Answer: (d)
Question 5.
Which of the following letters has vertical line of symmetry?
(a) R
(b) C
(c) B
(d) T.
Answer: (d)
Question 6.
How many lines of symmetry does the figure have?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d)
Question 7.
How many lines of symmetry does the figure have ?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a)
Question 8.
How many lines of symmetry does the figure have?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c)
Question 9.
How many lines of symmetry does the figure have?
(a) 0
(b) 1
(c) 2
(d) countless
Answer: (a)
Question 10.
How many lines of symmetry does the figure have?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a)
Question 11.
How many lines of symmetry does the figure have?
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b)
Question 12.
How many lines of symmetry does the figure have?
(a) 1
(b) 2
(c) 3
(d) 4.
Answer: (b)
Question 13.
How many lines of symmetry does a regular hexagon have?
(a) 1
(b) 3
(c) 4
(d) 6
Answer: (d)
Question 14.
Which of the following letters has horizontal line of symmetry?
(a) S
(b) W
(c) D
(d) Y.
Answer: (c)
Question 15.
Which of the following letters has vertical line of symmetry?
(a) N
(b) K
(c) B
(d) M.
Answer: (d)
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Introduction to Integers – Definition, Numbers, Rules, Symbols & Examples
Are you one of those candidates looking eagerly to learn about the concept of Integers? If yes, you must check this page to know the complete details about Integers. Integers is a basic and important
concept that lays a stronger foundation for your maths. Know definition, rules, numbers, solved questions, symbols, etc. Go through the below sections to find various methods and formulae.
Integers – Definition
The integer word is derived from the Latin word “Integer” which represents the whole. Integers are the positive, negative numbers, or zero. Integer values cannot be in decimals, fractions, percents,
and we can perform various operations(arithmetic operations) like subtraction, addition, multiplication, division, etc. Examples of integers are 1,2,3,-4,-5, etc. Integers also include various sets
Integers also include various sets like zero, whole numbers, natural numbers, additive inverses, etc. These are the subset of real numbers.
Example of integer set: -5,-3, -1, 0, 2, 5
Representation of Integers
As integers contain various numbers and sets and are the subset of real numbers, they are represented with the letter “Z”.
Z= {5,-3, -1, 0, 2, 5}
Types of numbers in Integers
• Natural Numbers
• Whole Numbers
• Real Numbers
• Rational Numbers
• Irrational Numbers
• Odd Numbers
• Even Numbers
Integers Rules
• The Sum of 2 positive integer numbers is an integer number
• The Sum of 2 negative integer numbers is an integer number
• Product of 2 positive integer numbers is an integer number
• Product of 2 negative integer numbers is an integer number
• Sum of an integer number and its inverse equals zero
• Product of an integer number and its reciprocal equals 1
Addition of Integer Numbers
While adding 2 positive or negative integers(with the same sign), add the absolute values and note down the sum of those numbers with the sign provided with numbers.
(+6)+(+5) = +11
(-5)+(-5)= -10
While adding 2 integers with a different sign, subtract the absolute values and note down the difference of those numbers with the sign provided with numbers.
(-5)+(+2)= -3
(+6)+(-3)= -3
Subtraction of Integer Numbers
While subtracting we follow the rules of addition but change the 2nd number which is being subtracted.
(-4)+(-3)= (-4)-(+3) = -11
(+5)-(+4)=(+5)+(-4)= +1
Division and Multiplication of Integer Numbers
The rule is simple while dividing and multiplying 2 integer numbers.
• If both the integers have the same sign, the result is positive.
• If both the integers have a different sign, the result is negative.
(+3)*(-4) = -12
(+4)*(+3) = 12
(+16)/(+4) = +4
(-6)/(+2) = -3
Integer Properties
There are 7 properties of integers. The major properties are
1. Associative Property
2. Distributive Property
3. Closure Property
4. Commutative Property
5. Identity Property
6. Multiplicative Inverse Property
7. Additive Inverse Property
1. Associative Property
This property refers to grouping and rules can be applied for addition and multiplication.
Associative Property of Addition
Associative property enables the special feature of grouping the numbers in your own way and still, you get the same answer.
(a+b)+c = a+(b+c)
(-4+2)+3= -2+(3+4)
In the above example, if we consider the first equation you can solve it in either way i.e., First you take the difference of 4 and 2 and then add 3 to it or you can first add 2 and 3 and then
subtract 4 from it. In both ways, you get a constant answer.
Associative Property of Multiplication
This property also refers to the same as the addition property. In whatever way you group numbers, you still get the same answer.
(ab)c= a(bc)
In the above example, you can solve it 2 ways and still find the same answer. First, you can multiply 2,4 and then multiply that with 3 or you can first multiply 4,3 and then multiply it with 4.
2. Distributive Property
The distributive property is used when the expression involving addition is then multiplied by a number. This property tells us that we can multiply first and then add or add first and multiply then.
In both ways, the multiplication is distributed for all the terms in parentheses.
a(b+c) = ab+ac
-4(2+3)= (-4*2)+(-4*3)
In the above example, we can first add 2 and 3, then multiply it with 4 or we can multiply 4 with 2 and 3 separately and then add it, still you get the same answer.
3. Closure Property
Closure property for addition or subtraction states that the sum or difference of any 2 integers will be an integer value.
a + b = integer
a x b = integer
6-3= 3
6+(-3)= 3
The closure property for multiplication also states that the product of any two integer numbers is an integer number.
The closure property for division does not hold true that the division of two integers is an integer value.
(-3)/(-12)=1/4, which is not an integer
4. Commutative Property
The commutative property for addition states that when two integer numbers undergo swapping, the result remains unchanged.
The commutative property for multiplication also states the same that if two integers are swapped, the result remains unchanged.
The commutative property doesn’t hold true for subtraction.
5. Identity Property
Identity Property states that any number that is added with zero will give the same number. Zero is called additive identity.
The identity property for multiplication also states the same that the integer number multiplied by 1 will give the same number. 1 is called the additive identity.
6. Multiplicative Inverse Property
Consider “a” as an integer, then as per the multiplicative inverse property of integers,
Here, 1/a is the multiplicative inverse of integer a.
7. Additive Inverse Property
Consider “a” as an integer, then as per the additive inverse property of integers,
a+(-a)= 0
Here, “-a” is the additive inverse of the integer a
Applications of Integers in Real Life
Integers have many real-life applications. We use them in different situations to quantify things. For example, to check the temperature, positive numbers are used to indicate the temperature above
zero and negative numbers are used to indicate the temperature below zero. Integers are also mainly used in real-life situations like hockey, football tournaments, rating for a movie, bank credits
and debits, etc.
We have mentioned all the important information about Integers. Hope, the above-provided details will help you in your preparation. Stay tuned to our site to get instant updates on various
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Hybrid Quantum–Classical Algorithms: At the Verge of Useful Quantum Computing - JPS Hot Topics
Hybrid Quantum–Classical Algorithms: At the Verge of Useful Quantum Computing
JPS Hot Topics 1, 025
© The Physical Society of Japan
This article is on
Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation
J. Phys. Soc. Jpn. 90, 032001 (2021) .
Scientists discuss the recent progress in algorithms that have enabled hybrid quantum–classical computers, which has brought the quest to realize useful quantum computing much closer to its finish
Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation
J. Phys. Soc. Jpn. 90, 032001 (2021) .
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Weird Number Bases
If you spend time programming computers you are probably very familiar with number bases other than decimal.
The chances are good that you are very familiar with binary (base 2) and Boolean arithmetic. More manageably, you are probably pretty comfortable with hexadecimal too (base 16).
Depending on the systems you work on, you might even have had some experience with octal (base 8).
1110100000101[2] = 16405[8] = 7429[10] = 1D05[16]
(The standard way to indicate a number base, if it is not obvious from the context, is with a subscript at the end, next to the least significant digit).
Without even realizing it, you probably have a fair degree of familiarity with sexagesimal (base 60), as there are sixty seconds in a minute, and sixty minutes in an hour. If you’ve used decimal
representations of time (or worked with angles, or lat/lng coordinates), you might have had to convert between them (Scientific calculators typically have built in functions to handle DMS
conversions because they are so common).
Half an hour (0.5hr) = 30 minutes
Interestingly there are echoes of other number bases still in circulation. Just because we have ten fingers (which is probably the root cause of the origins of decimal), society grew up with some
other bases. Here are a few:
• There are 16 ounces in one pound (of weight), and if I ask my dad his weight he will still reply in Stones (there are 14 Pounds in a Stone).
• Pre-decimilasation, in the UK, there were 12 Pennies in one Shilling, and 20 Shillings in one Pound (meaning one Pound = 240 Pennies). I was born pre-decimilasation, and as I was going through
school many of my old text books had problems about adding, subtracting, multiplying, and dividing various amounts of pounds-shillings-pence.
• There are 24 hours in a day (or twelve repeated hours, depending on your perspective).
• Many languages show a bias towards base 20 numbering systems (Irish, Gaulish …), and the Mayans also used a vigesimal system (base 20).
• You buy doughnuts by the dozen, and beer by the case!
Positional Notation
Most number systems we use today can be described as positional notation (sometimes called place-value notation). The position a digit is in consistently determines its value. This greatly
simplifies arithmetic.
The most common example of non-positional notation are Roman Numerals. Here the same symbol can have different values depending on its position and is modified by the symbols around it. Chaos!
If you write down two Roman Numerals, one above the other, and try to add them using the same techniques you use for place-value arithmetic, you are going to have a very bad time!
Non-positional notation representations, however, are not all bad and chaotic. There are some incredible useful representation systems. Probably the most well known in Binary Gray Code.
The standard for positional notation is that for each digit that you move to the left, you increase in power by one of the base that multiplies any digit in that column. In base 10, the first column
is the measure of units 10^0, the next column the measure of tens 10^1, then hundreds 10^2, thousands 10^3 …
It's the same principle for other bases, for instance binary, as shown on the left.
Here is shown the decimal number 151 represented by the unique sum of various powers of two. This highlights why binary is so commonly
used in digital computers; a value in a binary digit is either set, or not. Electrically, a voltage is either present (digitally) or
Any number can be uniquely described by summing up the digits
multiplied by their respective powers of the number base.
Where b is the number base, and d[i] is the i^th digit in the
Fracational parts of a number can be represented by digits placed to the right of a 'decimal point', and these are used to represent progressively negative exponents of the base. In decimal: ^1/
[10ths] (10^-1), ^1/[100ths] (10^-2), ^1/[1,000ths] (10^-3) …
e.g. 0.125[10] = ^1/[10] + ^2/[100] + ^5/[1,000]
Weird Number Bases
There's no reason why a number base (called a radix by mathematicians), needs to be an integer!
It's possible, if you wanted to, to chose pretty weird number bases. Here are just a couple:
base π
Even though π is irrational (in decimal), that's not a problem. We can just apply the same principles of increasing powers of the base and represent numbers based on powers of π.
… π^3, π^2, π^1, π^0
A circle with diameter 1[π] will have a circumferance of 10[π] (and one with a diameter 10[π] will have a circumferance of 100[π] …)
A circle with a radius of 1[π] will have an area of 10[π], a circle with a radius of 10[π] will have an area of 1000[π] and a circle with a radius of 100[π] will have an area of 100000[π] …
base e
A base using the transcendental constant e, has some interesting properties, one of which is that the natural logs behave a little like 'common' logarithms:
ln(1[e]) = 0, ln(10[e]) = 1, ln(100[e]) = 2, ln(1000[e]) = 3
base √2
Ok, this is fascinating: base √2 has an interesting relationship with vanilla base 2 (binary).
To convert any number from binary into base √2, all you need to do is insert a zero between every digit of the binary representation!
1911[10] = 11101110111[2] = 101010001010100010101[√2] 5118[10] = 1001111111110[2] = 1000001010101010101010100[√2]
You can see from this that any integer can be represented in base √2 without the need of a 'decimal point', more strictly called a 'radix point'.
base φ
Ah, the Golden Ratio. It likes to pop up in lots of interesting places (some of them are even true).
The Golden Ratio radix has been studied so much that it has a colloquial name of Phinary!
Any non-negative real number can be represented as a base φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11". Below are the first ten decimal digits and their phinary
Decimal Powers of φ Base φ
1 φ^0 1
2 φ^1 + φ^−2 10.01
3 φ^2 + φ^−2 100.01
4 φ^2 + φ^0 + φ^−2 101.01
5 φ^3 + φ^−1 + φ^−4 1000.1001
6 φ^3 + φ^1 + φ^−4 1010.0001
7 φ^4 + φ^−4 10000.0001
8 φ^4 + φ^0 + φ^−4 10001.0001
9 φ^4 + φ^1 + φ^−2 + φ^−4 10010.0101
10 φ^4 + φ^2 + φ^−2 + φ^−4 10100.0101
Some of these weird bases might seem slightly arbitrary (and maybe fun if you are a mathematician), but do they have any 'practical' applications? Well yes, maybe they do.
Base e, for instance, is very efficient at storing information. Something called radix economy measures the number of digits needed to express a number in that base, multiplied by the radix.
(A binary representation of a number is 'long', but only uses one of two values. Conversely, storing something in decimal might make a number 'shorter', but each symbol could be pulled from a larger
number of values. A number stored in base e is the most mathematically efficient way to encode it, according to information storage theory*).
This is one of those 'Goldilocks' type issues. Make a radix too small (like binary), and whilst your 'dictionary' of symbols to use is very small, the resulting string representing the number is
very long. Conversely, having a large radix would shorten the length of the string needed to represent the number, but each digit would need need to come from a large dictionary, and this would take
more space to encode each digit.
"This one is too big", "This one is too small" … which radix is "Just right"? …
… the answer is base e.
Another analogy is written language. Using the Western (Latin) alphabet, we can write words, but the average length of words is many characters each. Compare this to written Chinese where there are
many thousands of symbols, and many words require just one symbol.
*Outside of the analog world, digital computers store precise quantized values for representations. If computer circuits were manufactured to store data tri-state instead of binary, (3 is a nearer
integer to e than 2), then computers could store data more efficiently.
e ≈ 2.7182818284590452353602874713526624977572470936999595 …
We're getting a little off-topic here, but the same math applies to things like menu systems and telephone menu systems. If these services offered ternary trees (tri-state) menus, they would
minimize the average number of menu choices the average customer would need to listen to to get to their desired location.
In the early days of computing, a few experimental Soviet computers were built that processed using balanced ternary (more on this later) instead of binary, the most famous being the Setun (image on
right), which is named after a river in Moscow. Over fifty of these computers were built in the 1960s and 1970s.
I love decscriptions of these devices from one of the developers, when comparing the binary computer which stores values in one of two states "flip-flop", to that of ternary, they used the words
By using balanced ternary + , 0 , – to store values (a "trit", instead of a a "bit")*, as we will see below, this has interesting benefits for encoding the sign of the number too.
Note - There is a subtle difference between balanced ternary, which uses values: −1, 0, +1 cf. vanilla ternary, which uses values: 0, 1 ,2. More details about this a little later, below …
*A collection of "trits" form together to make a "tryte", just like "bits" make a "byte"!
Further down the Rabbit Hole (negative radices)
OK, let’s go further down the rabbit hole. How about, instead of using fractional bases, we use negative bases? Mathematically, again, this is easy to do. Odd powers of negative bases create negative
numbers, but even powers produce positive ones. Because we add these digits up it is still possible to create distinct numbers. For example, let's take a look at base -2. Sometimes called
= 1×(-2)^8 + 1×(-2)^7 + 1×(-2)^6 + 0×(-2)^5 + 1×(-2)^4 + 1×(-2)^3 + 0×(-2)^3 + 0×(-2)^1 + 1×(-2)^0
= 1×(256) + 1×(-128) + 1×(64) + 0×(-32) + 1×(16) + 1×(-8) + 0×(4) + 0×(-2) + 1×(1)
= 256 − 128 + 64 + 16 − 8 + 1
= 201[10]
There is a very interesting property of negative radix encoding: There is no distinction between positive and negative numbers; they are all just numbers, and all encoded the same way. The sign of
the number is encapsulated in the number. We don't need a sign bit.
If all you've ever used is unsigned integers, you might not see this as much of an advantage, but for everyone else, signed numbers are typically coded using a two's complement (sort of like how an
odometer on a car wraps around the clock after getting to 99999), and negative numbers are represented backwards (from over the top) and are identifiable by having the topmost bit (most signficant
bit) set.
Again this is fine if you are dealing with numbers that are encoded in just one byte/word (depending on the width you are dealing with), but if you need to encode and deal with arithmetic for larger
numbers, you need to span these numbers across multiple words. Now the words are different. The 'lower' words of the number use all bits, but the most significant word has the top bit reserverd to
(potentially) indicate that the number is negative.
Let's take a look a negadecimal:
We can apply the same negative radix strategy to describe numbers in base -10 (called 'negadecimal')
= 1×(-10)^4 + 7×(-10)^3 + 4×(-10)^2 + 7×(-10)^1 + 8×(-10)^0
= 1×(10000) - 7×(1000) + 4×(100) - 7×(10) + 8×(1)
= 10000 − 7000 + 400 − 70 + 8
= 3338[10]
Just like negabinary, numbers encoded in negadecimal do not need explicit sign indicators; this is encapsulated into the number system, and all can be treated exactly the same way.
Tables of numbers
Here are representations of a selection of numbers in decimal, negadecimal, negabinary and negaternary:
Decimalbase 10 Negadecimalbase -10 Negabinarybase -2 Negaternarybase -3
-100 1900 11101100 121112
-64 76 11000000 120212
-32 48 100000 1021
-16 24 110000 1102
-15 25 110001 1220
-14 26 110110 1221
-13 27 110111 1222
-12 28 110100 1210
-11 29 110101 1211
-10 10 1010 1212
-9 11 1011 1200
-8 12 1000 1201
-7 13 1001 1202
-6 14 1110 20
-5 15 1111 21
-4 16 1100 22
-3 17 1101 10
-2 18 10 11
-1 19 11 12
You will notice for the nega representations, the negative numbers all have an even number of digits, and positive numbers all have an odd number of digits.
Adding two nega numbers
Now that we have have two negadecimal numbers, how do we add them together?
It's actually not as as hard as you might think, because negadecimal is still a positional notational system. We simply apply the addition rules we learned in school; summing columns (from least
significant to most significant, carrying forward as required).
First, a trivial example, adding two 'small' numbers (no carry). What is the sum of 12343[-10] and 6101[-10]?
12343[-10] = 8263[10] and 6101[-10] = −5899[10]
Notice here how the two numbers are treated the same, even though it turns out that one of them happens to be negative?
It behaves just as we'd expect, we just sum up each column. The sum for the first digit is 1+3=4. So far so good, and we can walk over the columns in order. The result is 18444[-10], which
correspends to 2364[10], which is what we expect (=8263[10]−5899[10]).
OK, now let's introduce a more complicated example. How do we deal if we have an overflow (carry) on any column? The answer, like traditional arithmetic, is we carry over to the next column, but,
because we are dealing with negadecimal, we carry over a negative one.
Adding the 4 and the 7 together, we obtain 11. We write down 1 in the total, and carry a −1 to the top of the next column over and then carry on. Next we find that −1+4=3, so no carry this time …
We carry this process until we reach the end (which is again, thankfully is what we expect!) Each time, we carry forward as needed.
12707[-10] + 14444[-10] = 25131[-10] 8707[10] + 6364[10] = 15071[10]
We need to learn one last trick, and then we are home and dry. What do we do if we need to carry forward a negative one, and all we in the next column are zeros? (we can't have a -1, as each digit
needs to be in the range 0-9). The answer is pretty simple, as -1 in negadecimal is 19, we just add this to the front (it's like saying we 'borrow' 1 from the next digit over and then use this to
help mop-up the carry that propogated forward). To 'borrow' a one, we subtract off negative one, which is the same as carrying forward a positive one.
The same borrow principle applies at the 'front' of the number (most significant digit), if needed, until we have no more carries to propogate forward.
Interestingly, in this example the negadecimal and decimal representation of the two numbers to be added are the same.
10009[-10] + 90002[-10] = 1900191[-10] 10009[10] + 90002[10] = 100011[10]
We can apply the same strategy for adding negabinary numbers; applying the principles we learned at school and carrying forward as appropriate. We need to take a little more care with negabinary as,
even when adding two numbers, we might need to carry forward two colums at single time!
Balanced Ternary
A further mention should be made of balanced ternary, since we made reference to it earlier concerning the early Soviet era computers. Traditional ternary uses the values: 0,1,2 to encode by using
them to multiply the powers of the base. Balanced ternary uses the digits: −1, 0, +1.
There is a subtle difference from a negative radix representation (negaternary) and balanced ternary because, with balanced ternary, we are still using a positive radix, but each digit can elect to
use it, have none of it, or subtract it! It's sort of like pivoting it the other way. Balanced ternary, by allowing encoding of positive and negative numbers, also has the same advantage of treating
all numbers the same (no sign bits needed), but has some additional advantages including that the truth tables for digit additional, subtraction, mulitplication, and division are simpler.
Because any digit can be in one of three states, and it would be (very, very) confusing to propose that "2" represent "−1", a different convention is used.
In old Russian literature, documentation sometimes used an inverted digit "1" to represent "−1", but this is hard to read and easily confussed. Other researchers have used "T" to represent "−1", and
others, still have used "Θ". I'm going to use "T" below.
… + d[2]3^2 + d[1]3^1 + d[0]3^0 where d[n] is {−1, 0, +1}
1T0[bal3] = 1×3^2 − 1×3^1 + 0×3^0 = 9 − 3 + 0 = 6[10]
TT1[bal3] = −1×3^2 − 1×3^1 + 1×3^0 = −9 − 3 + 1 = −11[10]
101[bal3] = 1×3^2 + 0×3^1 + 1×3^0 = 9 + 0 + 1 = 10[10]
1T10[bal3] = 1×3^3 − 1×3^2 + 1×3^1 + 0×3^0 = 27 − 9 + 3 + 0 = 21[10]
Balanced ternary is pretty awesome!
Deeper still down the Rabbit Hole - Mixed Radix Systems
There is no reason why moving to the left in positional notation should necesserily increase the exponent of the base. This is just a common definition and a standard we agree on. Provided you
describe the rules and consistently apply them, you can encode numbers however you feel like. For instance you could use the columns to represent factorials (or better still primorials, which are
like factorials but each next term you multiply by is not the next number in the sequence, but the next occuring prime; Primorials are all square-free integers, and each one has more distinct prime
factors than any number smaller than it).
In a mixed radix system, the maximum value allowed in any digit position is variable.
Factorial Base Primorial Base
Digit d[7] d[6] d[5] d[4] d[3] d[2] d[1] d[0] Digit d[6] d[5] d[4] d[3] d[2] d[1] d[0]
Radix 8 7 6 5 4 3 2 1 Radix 17 13 11 7 5 3 2
Place value 7! 6! 5! 4! 3! 2! 1! 0! Place value (p[6]=13) (p[5]=11) (p[4]=7) (p[3]=5) (p[2]=3) (p[1]=2) (p[0]=1)
Decimal 5040 720 120 24 6 2 1 1 Decimal 30030 2310 210 30 6 2 1
24201[!] = 349[10]
= (2 × 5!) + (4 × 4!) + (2 × 3!) + (0 × 2!) + (1 × 1!) = 240 + 96 + 12 + 0 + 1 = 349
If mixed radix math sounds crazy, remember back to the some of the opening comments in this article. We live in a mixed radix society. There are 60 seconds in a minutes, and 60 minutes in an hour,
but 24 hours in a day, and (almost) 365.25 days in a year …
Last stop on the crazy train - complex radices
I'm not going to talk about them here, but there's no reason to restrict yourself to real numbers when selecting a number base! Why not use a complex radix?
Three well researched bases are base 2i (known as Quatar imaginary base), base −1+i, and base −i−1. The quater-imaginary numeral system was first proposed by Donald Knuth in 1955, in a submission to
a high-school science talent search!
The future of digital computing?
As we've seen, a consequence of using negative bases to encode numbers is that there is no distinction between a positive and negative number representation (no sign bit is needed compared to
traditional binary encoding). Not only does this greatly simplify data types, but it also reduces (by half) things like conditional instructions and even basic operations that no longer have to worry
about the sign bit; if it is present, and how to deal with it. Truncation is easier (it corresponds to rounding), and math operations can be applied agnostic of the length of words (and the position
of the current word relative to the entire number). The number of instructions required is also reduced.
Compilers would be simpler to write, and once written, easier to test, and there would be less paths executed.
Also, as we have seen from information theory, a base closer to e is a more efficient way to store information. Combining the benefits of non-positive bases and non-binary bases in balanced ternary
produces, as the Soviets experimented with, a pretty elegant foundation for an efficient and neat computing platform. Shouldn't this be the platform we aspire to?
If history were to repeat itself, would we still end up in a binary based computing society? If the earlier pioneers had continued with tri-state research, would our devices now all be using trits
and trytes?
If we meet aliens, will they be using base three devices? (mathematics, after all, is a universal language and the benefits of balanced ternary are agnostic as to how to describe them).
Clearly there is physical simplicity in having a binary system (which is why we initially went down this path, and have continued down it to date): Something is there, or it is not. A voltage is
present or it is not. Magnetic flux is there or not. A hole is present in a piece of punched-tape or not*. But with the technology available today we could probably come up with solutions to reliably
store and manipulate tri-state data. These days, data is typically not stored as physical two-state presence (or lack of) of something; it is usually in some electronic form. Is it time to ditch
binary and switch to balanced ternary?
*Insert joke here about "hanging chads"
Will modern digital computers ever move away from binary? …
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Algebraic formula cost salary benefits
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Inherently high uncertainty in predicting the time evolution of epidemics - 학지사ㆍ교보문고 스콜라
SCOPUS 학술저널
Inherently high uncertainty in predicting the time evolution of epidemics
quaranOBJECTIVES: Amid the spread of coronavirus disease 2019 (COVID-19), with its high infectivity, we have relied on mathematical models to predict the temporal evolution of the disease. This paper
aims to show that, due to active behavioral changes of individuals and the inherent nature of infectious diseases, it is complicated and challenging to predict the temporal evolution of epidemics.
METHODS: A modified susceptible-exposed-infectious-hospitalized-removed (SEIHR) compartment model with a discrete feedback-controlled transmission rate was proposed to incorporate individuals’
behavioral changes into the model. To figure out relative uncertainties in the infection peak time and the fraction of the infected population at the peak, a deterministic method and 2 stochastic
methods were applied. RESULTS: A relatively small behavioral change of individuals with a feedback constant of 0.02 in the modified SEIHR model resulted in a peak time delay of up to 50% using the
deterministic method. Incorporating stochastic methods into the modified model with a feedback constant of 0.04 suggested that the relative random uncertainty of the maximum fraction of infections
and that of the peak time for a population of 1 million reached 29% and 9%, respectively. Even without feedback, the relative uncertainty of the peak time increased by up to 20% for a opulation of
100,000. CONCLUSIONS: It is shown that uncertainty originates from stochastic properties of infections. Without a proper selection of the evolution scenario, active behavioral changes of individuals
could serve as an additional source of uncertainty. | {"url":"https://scholar.kyobobook.co.kr/article/detail/4010028274842","timestamp":"2024-11-10T06:15:54Z","content_type":"text/html","content_length":"44090","record_id":"<urn:uuid:040b127e-0cc7-46f8-b011-6c3d221ce5db>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00482.warc.gz"} |
Topology & Geometry at UW-Milwaukee
The Topology & Geometry group at the University of Wisconsin-Milwaukee is one of the most active in the department. Ric Ancel, Craig Guilbault, Boris Okun and Chris Hruska are the regular faculty
members in this area.
Regularly offered topology courses include a one semester undergraduate course in Elementary Topology , a yearlong graduate sequence titled Introductory Topology, and a yearlong graduate sequence in
Algebraic Topology. For students specializing in topology ‘advanced topics’ courses are frequently offered. Topics recently covered in those courses include: differential topology, bundle theory,
dimension theory, topological manifolds and polyhedra, surgery theory, PL Morse theory, relatively hyperbolic groups, and various other aspects of geometric group theory and geometric topology.
In addition to formal courses, the Topology Seminar meets twice weekly. There faculty and graduate students discuss and present current research. In addition, a Student Topology Seminar allows
graduate students to learn material not covered in their regular coursework. Some recent student seminars have focused on: the geometry of surfaces and covering spaces, dimension theory, CAT(0)
geometry, Gromov hyperbolic spaces and groups, piecewise linear topology, knot theory, geometry and topology of 3-manifolds, and Morse theory.
All members of the Topology & Geometry group have active research programs. Ancel and Guilbault have both been at UWM for a number of years and have written several papers jointly. Okun joined the
UWM topology group in the fall of 2001; Hruska came to Milwaukee in 2006. Areas of interest include: manifold topology, CAT(0) geometry, and geometric group theory.
Interest in topology and geometric group theory amongst UWM graduate students is at an all-time high, with a record number of PhDs awarded during the 2014-15 academic year. Here is a list of alumni
from the UWM PhD program in Topology & Geometry.
Hanspeter Fischer (Ph.D. 1998, ) Thesis: Visual Boundaries of Right Angled Coxeter Groups and Reflection Manifolds. Advisor: Ric Ancel. After a post-doc at Brigham Young University, Hanspeter joined
the faculty at Ball State University, where he is now a full professor.
Julia Wilson (Ph.D. 1999) Thesis: Non-uniqueness of Boundaries of CAT(0) Groups. Advisor: Ric Ancel. Julia is now an associate professor of mathematics at SUNY-Fredonia.
David Radcliffe (Ph.D. 2001) Thesis: Unique Presentations of Coxeter Groups and Related Groups. Advisor: Ric Ancel.
Margaret May (Ph.D. 2007) Thesis: Finite-dimensional Z-compactifications. Advisor: Craig Guilbault. After several years as faculty and administrator at UW-Fond du Lac, Maggie switched jobs to join
the mathematics faculty at Moraine Park Technical College.
Christopher Mooney (Ph.D. 2008) Thesis: On Boundaries of CAT(0) Groups. Advisor: Craig Guilbault. After a 3-year post-doc at the University of Michigan and several years as an assistant professor at
Bradley University, Christopher left academics for private industry. He is now a software developer at Epic Systems near Madison, Wisconsin.
Timothy Schroeder (Ph.D. 2008) Thesis: L^2-Homology of Coxeter Groups. Advisor: Boris Okun. Tim is now an associate professor of mathematics at Murray State University.
Carrie Tirel (Ph.D. 2010) Thesis: Z-structures on product groups . Advisor: Craig Guilbault. Carrie is now an assistant professor of mathematics at the University of Wisconsin-Fox Valley.
Paul Fonstad (Ph.D. 2012) Thesis: A further classification of the boundaries of the Croke-Kleiner group. Advisor: Ric Ancel. Paul is now an assistant professor of mathematics at Franklim College in
Jeremy Osborne (Ph.D. 2014) Thesis: Statistical Hyperbolicity of Relatively Hyperbolic Group . Advisor: Chris Hruska. Jeremy is now an lecturer at the University of Wisconsin-Parkside.
Peter Sparks (Ph.D. 2014) Thesis: Contractible n-Manifolds and the Double n-Space Property . Advisor: Craig Guilbault. Pete is currently a lecturer at the University of Wisconsin-Waukesha.
Jeffrey Rolland (Ph.D. 2015) Thesis: Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds. Advisor: Craig Guilbault. Jeff is currently an adjunct professor at Marquette University.
Jason La Corte (Ph.D. 2015) Thesis: The Markov-Dubins problem with free terminal direction in a nonpositively curved cube complex. Advisor: Craig Guilbault. Beginning in the Fall of 2015, Jason will
be an assistant professor at Berry College.
Molly Moran (Ph.D. 2015) Thesis: On the Dimension of Group Boundaries. Advisor: Craig Guilbault. Beginning in the Fall of 2015, Molly will be a visisting assistant professor at Colorado College.
Kevin Schreve (Ph.D. 2015) Thesis: The L^2 Cohomology of Discrete Groups . Advisor: Boris Okun. Beginning in the Fall of 2015, Kevin will be a post-doc at the University of Michigan.
Wiktor Mogilski (Ph.D. 2015) Thesis: The fattened Davis complex and the weighted L^2 -(co)homology of Coxeter groups. Advisor: Boris Okun.
Current Ph.D. students: Hung Tran, Matthew Haulmark, and Hoang Nguyen; all are all working on dissertations in Geometric Topology and/or Geometric Group Theory at UWM.
Students interested in studying geometric topology or geometric group theory at the University of Wisconsin-Milwaukee should feel free to contact Ancel, Guilbault, Okun or Hruska directly. | {"url":"https://sites.uwm.edu/craigg/topology-geometry-at-uw-milwaukee/","timestamp":"2024-11-09T22:18:44Z","content_type":"text/html","content_length":"38389","record_id":"<urn:uuid:fd019001-e68d-4172-a366-bd17729dcbb2>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00059.warc.gz"} |
How Many Minutes Is 459.1 Seconds?
How many minutes in 459.1 seconds?
459.1 seconds equals 7.652 minutes
Unit Converter
Conversion formula
The conversion factor from seconds to minutes is 0.016666666666667, which means that 1 second is equal to 0.016666666666667 minutes:
1 s = 0.016666666666667 min
To convert 459.1 seconds into minutes we have to multiply 459.1 by the conversion factor in order to get the time amount from seconds to minutes. We can also form a simple proportion to calculate the
1 s → 0.016666666666667 min
459.1 s → T[(min)]
Solve the above proportion to obtain the time T in minutes:
T[(min)] = 459.1 s × 0.016666666666667 min
T[(min)] = 7.6516666666667 min
The final result is:
459.1 s → 7.6516666666667 min
We conclude that 459.1 seconds is equivalent to 7.6516666666667 minutes:
459.1 seconds = 7.6516666666667 minutes
Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 minute is equal to 0.13069048137661 × 459.1 seconds.
Another way is saying that 459.1 seconds is equal to 1 ÷ 0.13069048137661 minutes.
Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that four hundred fifty-nine point one seconds is approximately seven point six five two minutes:
459.1 s ≅ 7.652 min
An alternative is also that one minute is approximately zero point one three one times four hundred fifty-nine point one seconds.
Conversion table
seconds to minutes chart
For quick reference purposes, below is the conversion table you can use to convert from seconds to minutes | {"url":"https://convertoctopus.com/459-1-seconds-to-minutes","timestamp":"2024-11-05T00:54:23Z","content_type":"text/html","content_length":"35004","record_id":"<urn:uuid:c1c9cdbf-979f-49d3-9fa7-41fb8509321e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00468.warc.gz"} |
@adlrocha - From Turing to Shannon
On the importance of theoretical computer science.
What does it mean to be a good theoretical computer scientist? What path should one follow to become one? It suffices to earn a computer science degree, or would it be best to study a maths or a
physics degree and then join the field of computer science? When it was my turn to make this decision I didn’t know the meaning of computer science (more so since in Spain we don’t have a proper
computer science degree). Even less what a theoretical computer scientists did for a living. I just happened to like programming, computers and telecommunications. And that’s how I ended up in the
practical side of this spectrum earning my telecommunications and electrical engineering degrees.
What I didn’t know then and that I’ve realized after years of reading quite a few books on cryptography, computational complexity, and information theory (including the quantum cousins of these
fields) is: how exciting theoretical fields can be; how important they are to foster practical advancements; and how much I miss not having studied a math or physics degree before telecommunications
or computer science to equip myself with the theoretical skills required to do groundbreaking research.
If you asked me now the question of “what path should one follow to become a good telecommunication or computer science researcher?” My answer would be, if you want to be involved in “fundamental
research”, study a physics or maths degree. You will always have time to switch to the practical side of the force, and learn what is needed to be a good engineer. Build your mathematical mindset,
and then create your “engineering sixth sense”.
Of course, this quite a biased opinion fruit of my experience, but to support my claim I will share the story of two mathematicians that became the fathers of computation and information theory,
respectively: Turing and Shannon.
Turing’s Machines and Halting Problem
Alan Turing was an English mathematician highly influential in the development of theoretical computer science. His name appears all over computer science: in concepts such as Turing complete
programming languages, Turing machines, or the Turing test. But all of these ideas are the result of the the resolution of a mathematical problem: the Entscheidungsproblem, that can be summarized in
the following question “is mathematics decidable?” Which declines into, “is there a definite procedure to show if a mathematical statement is true or false?”. Turing managed to solve this problem
showing the undecidability of mathematics through his “halting problem”.
To prove the “halting problem”, and thus undecidability of the Entscheidungsproblem, he first needed to formally describe what a definite procedure was. To do so he invented an widely known
apparatus, the Turing machine. A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules.
The machine operates on an infinite memory tape divided into discrete "cells". The machine positions its "head" over a cell and "reads" or "scans" the symbol there. Then, as per the symbol and the
machine's own present state in a "finite table" of user-specified instructions, the machine (i) writes a symbol (e.g., a digit or a letter from a finite alphabet) in the cell (some models allow
symbol erasure or no writing), then (ii) either moves the tape one cell left or right (some models allow no motion, some models move the head), then (iii) (as determined by the observed symbol and
the machine's own state in the table) either proceeds to a subsequent instruction or halts the computation.
A Turing machine is capable of simulating any algorithm's logic, so it was the perfect construction to formalize a definite procedure. With definite procedures fomrmally defined, if he was able to
program any mathematical proof into a Turing machine, it would mean that mathematics is decidable, as there would be a definite procedure to solve any possible proof in maths. Unfortunately, this
isn’t the case, and to prove so he came up with his “halting problem”.
The Halting Problem can be described as the problem of finding whether, for a given input, a program will halt at some time or continue to run indefinitely. To prove it, Turing first assumed that
such a Turing machine exists to solve this problem, to then show this can be it, as it is contradicting itself. We will call this Turing machine that is able to identify if programs halts or not a
Halting machine that produces a “yes” or “no” in a finite amount of time. If the halting machine finishes in a finite amount of time, the output comes as “yes”, otherwise as “no”. The following is
the block diagram of a Halting machine:
We are considering that the halting machine is decideable, so we can design its complementary machine, an inverted halting machine (HM)’ which can be described as: if H returns YES, then loop
forever, if H returns NO, then halt. The following is the block diagram of an “Inverted halting machine”:
Now, let design a halting machine (HM)2 which inputs its own as the input to the machine. If we assume that the halting machine exist, then the inverted halting machine must exist, what gets us to a
contradiction in (HM)2. Hence, the halting problem is undecidable.
I may have lost many of you by now, so let’s use some visual aide to help you understand this impressive proof:
In order to solve this proof, Turing had to invent the Turing machine, a model that would set the groundwork for modern computers. With this mental model Turing invented a programmable computer
before the first practical one was even the built. This our first great example of how answering a fundamental theoretical questions can lead to impressive practical advancements.
Shannon, the father of information theory
We’ve already talked about this guy in this newsletter. Shannon was a mathematician, electrical engineer, cryptographer and, more importantly, the father of information theory. His "A Mathematical
Theory of Communication" paper from 1948 set the foundation for all the revolution yet to come in telecommunications. Again, this paper is highly theoretical work that focuses on the problem of how
best to encode all the information a sender wants to transmit. Shannon developed information entropy as a measure of the information content in a message, which is a measure of uncertainty reduced by
the message. Something that still impresses me about Shannon’s work is how he gets inspiration from a physical concept, the thermodynamical entropy, to describe such an abstract concept as the amount
of information inamessage.
Shannon’s outstanding theoretical set the foundation to many impressive practical applications in the fields of mobile communications, natural language processing, sampling theory, computational
linguistics, etc. Have you heard of the Shannon theorem or the Shannon-Hartley theorem? These theorems tell the maximum rate at which information can be transmitted over a communications channel and
a channel of a specified bandwidth in the presence of noise, respectively. Something every electrical engineering and telecommunications engineering student has studied in his bachelor’s (so as you
may have guessed, this guy kind of like a hero to me).
Another impressive fundamental result from a theorist that managed to change the future of many practical fields.
Maths, theoretical computer science, information theory and physics are all connected
Like these, many other theoretical results have been key for new practical developments in computer science, information theory and physics. From recent developments in artificial intelligence, to
new constructions in cryptography (with brand zero-knowledge proofs and multiparty computation primitives), and theoretical bounds in physics.
A recent example of the importance of theoretical computer science is this paper that shows how “the class MIP* of languages that can be decided by a classical verifier interacting with multiple
all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages”. This paper can potentially have a significant impact in disjoint fields such as
computer science, physics and maths (just like it happened with Turing and Shannon’s work). This post gives a good overview of the potential impact of this publication may have.
With quantum information and quantum computation gaining traction and becoming increasingly important, I feel we are approaching a new golden age of theoretical computer science. So if you asked me
again what can you do to become a great theoretical computer scientist and to contribute to fundamental research I would answer “first get some math and physics skills, and then jump to computer
science. This will give you the skills to make better sense of the abstract computer science world”.
I would love to hear your take on this. Any thoughts? If not, see you next week! | {"url":"https://adlrocha.substack.com/p/adlrocha-from-turing-to-shannon","timestamp":"2024-11-03T23:02:47Z","content_type":"text/html","content_length":"230800","record_id":"<urn:uuid:b4541f63-e64b-4b6b-b9bf-cb705767bee0>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00462.warc.gz"} |
Is port extension or e-delay a universal solution?
Several recent articles examined the use of s11 port extension or e-delay in some scenarios that might have surprised.
Recall that s11 port extension adjusts the measured phase of s11 based on the e-delay value converted to an equivalent phase at the measurement frequency.
It is:
1. an exact correction for any length of lossless line of Z0=50+j0Ω transmission line;
2. an approximate correction for a very low loss length of approximately 50Ω transmission line; and
3. an approximate correction for some specific scenarios such as those discussed at Some useful equivalences of very short very mismatched transmission lines – a practical demonstration.
Of course 1. does not exist in the real world, but 2. can give measurement results of acceptable accuracy if used within bounds. Both departures mentioned in 2. occur in the real world, non-zero loss
and departure from Z0=50+j0Ω. Provided these departures are small, port extension may give acceptable results.
Let’s analyse some example measurements based on a 10m length of ordinary RG58A/U from 1-11MHz.
Above, measurement of the first series resonance with SC termination.
Note that the curve is a spiral inwards from the outer circle, the line is not lossless.
A requirement for e-delay to work well is that phase of s11 is proportional to frequency. This plot wraps, but apart from that, the plot looks approximately linear… however scale prevents detailed
Above, measurement of the first series resonance with OC termination.
Note that the curve is a spiral inwards from the outer circle, the line is not lossless.
Again the plot wraps, but apart from that, the plot looks approximately linear… however scale prevents detailed analysis.
Let’s find a value for e-delay at 1MHz and analyse the result.
Above is adjustment of e-delay to 115ns for approximately s11 phase 180° at 1MHz with SC termination.
The phase is correct at 1MHz, but at higher frequencies, it departs. So, the assumption that this TL has phase delay proportional to frequency is invalid. If you look closely, it is not a perfectly
straight line, there is a small oscillation superimposed which is a sign of Z0 error. For these reasons, e-delay correction will have error.
Above is adjustment of e-delay to 100ns for approximately s11 phase 180° at 1MHz with OC termination.
The phase is correct at 1MHz, but at higher frequencies, it departs. So, the assumption that this TL has phase delay proportional to frequency is invalid. If you look closely, it is not a perfectly
straight line, there is quite an oscillation superimposed which is a sign of Z0 error. For these reasons, e-delay correction will have error.
Let’s proceed anyway and look at the error. We will connect the 50+j0Ω termination load to the end of the cable and measure with each of the e-delays above.
Above is measurement of a 50+j0Ω termination with e-delay calibrated using 100ns e-delay (calibrated to OC termination). Note that the curve is a small circle, a sign of Z0 error and a hint that
actual Z0 is about the centre of the circle plotted. Note though that Z0 is frequency dependent at these frequencies for this cable, so you can’t pin a pin on the chart and say this is Z0.
Above is measurement of a 50+j0Ω termination with e-delay calibrated using 115ns e-delay (calibrated to SC termination). Note that the curve is a small circle, a sign of Z0 error and a hint that
actual Z0 is about the centre of the circle plotted. Note though that Z0 is frequency dependent at these frequencies for this cable, so you can’t pin a pin on the chart and say this is Z0.
At 5.75MHz and:
• e-delay from the SC calibration, Z=45.01+1.25Ω; whereas
• e-delay from the OC calibration, Z=45.01-1.26Ω.
For some purposes, that might be sufficient accuracy, for others it might be unacceptable:
• Z0 departure is more significant for lossier cables below about 10MHz; and
• in any event loss of tenths of a dB leads to measurable error.
Port extension or e-delay can provide a convenient means of shifting the reference plane given suitable test fixtures, but it is subject to significant error if the underlying assumption of lossless
50Ω line is breached. | {"url":"https://owenduffy.net/blog/?p=33756","timestamp":"2024-11-09T16:02:04Z","content_type":"text/html","content_length":"62993","record_id":"<urn:uuid:939589e9-33fa-4b41-a0d0-b51dbef0df6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00712.warc.gz"} |
$63 an Hour is How Much a Year? Before and After Taxes
For employees and employers alike, converting an hourly wage into its annual salary equivalent provides useful insight for financial planning and budgeting. In this article, we’ll calculate the
potential yearly earnings for a $63 per hour wage rate, including estimates for part-time, overtime, and time off. We’ll also look at how taxes reduce take-home pay, and what kind of lifestyle and
major purchases are possible at this high income level. While $63 an hour affords upper-middle class living standards beyond national averages, taxes and inflation do cut into its real value over
time. Understanding the annual, monthly, biweekly, and weekly salary equivalents of $63/hour gives perspective on both pre-tax and post-tax spending power.
Convert $63 Per Hour to Weekly, Monthly, and Yearly Salary
Input your wage and hours per week to see how much you’ll make monthly, yearly and more.
$63 an Hour is How Much a Year?
If you make $63 an hour, your yearly salary would be $131,040. We calculate your annual income based on 8 hours per day, 5 days per week and 52 weeks in the year.
Hours worked per week (40) x Hourly wage($63) x Weeks worked per year(52) = $131,040
$63 an Hour is How Much a Month?
If you make $63 an hour, your monthly salary would be $10,920. We calculated this number by dividing your annual income by 12 months.
Hours worked per week (40) x Hourly wage($63) x Weeks worked per year(52) / Months per Year(12) = $10,920
$63 an Hour is How Much Biweekly?
If you make $63 an hour, your biweekly salary would be $5,040.
Hours worked per week (40) x Hourly wage($63) x 2 = $5,040
$63 an Hour is How Much a Week?
If you make $63 an hour, your weekly salary would be $2,520. Calculating based on 5 days per week and 8 hours each day.
Hours worked per week (40) x Hourly wage($63) = $2,520
$63 an Hour is How Much a Day?
If you make $63 an hour, your daily salary would be $504. We calculated your daily income based on 8 hours per day.
Hours worked per day (8) x Hourly wage($63) = $504
$63 an Hour is How Much a Year?
The basic formula to calculate your annual salary from an hourly wage is:
Hourly Rate x Hours Worked per Week x Number of Weeks Worked per Year = Annual Salary
So for a $20 per hour job:
$63 per hour x 40 hours per week x 52 weeks per year = $131,040
However, this simple calculation makes some assumptions:
• You will work 40 hours every week of the year
• You will not get any paid time off
Therefore, it represents your earnings if you worked every week of the year, without any vacation, holidays, or sick days.
Accounting for Paid Time Off
The $131,040 base salary does not yet factor in paid time off (PTO). Let’s assume the job provides:
• 2 weeks (10 days) paid vacation
• 6 paid holidays
• 3 paid sick days
This totals 19 paid days off, or nearly 4 weeks of PTO.
Importantly, this paid time off should not be deducted from the annual salary, since you still get paid for those days.
So with 4 weeks PTO, the annual salary would remain $131,040 .
Part time $63 an hour is How Much a Year?
Your annual income changes significantly if you work part-time and not full-time.
For example, let’s say you work 30 hours per week instead of 40. Here’s how you calculate your new yearly total:
$63 per hour x 30 hours per week x 52 weeks per year = $98,280
By working 10 fewer hours per week (30 instead of 40), your annual earnings at $63 an hour drop from $131,040 to $98,280.
That’s a $32,760 per year difference just by working part-time!
Here’s a table summarizing how your annual earnings change depending on how many hours you work per week at $63 an hour:
Hours Per Week Earnings Per Week Annual Earnings
40 $2,520 $131,040
35 $2,205 $114,660
30 $1,890 $98,280
25 $1,575 $81,900
20 $1,260 $65,520
15 $945 $49,140
The more hours per week, the higher your total yearly earnings. But part-time work allows for more life balance if you don’t need the full salary.
$63 an Hour With Overtime is How Much a Year?
Now let’s look at how overtime can increase your annual earnings.
Overtime kicks in once you work more than 40 hours in a week. Typically, you earn 1.5x your regular hourly wage for overtime hours.
So if you make $63 per hour normally, you would make $94.50 per hour for any hours over 40 in a week.
Here’s an example:
• You work 45 hours in a Week
• 40 regular hours paid at $63 per hour = $2,520
• 5 overtime hours paid at $94.50 per hour = $472.50
• Your total one Week earnings =$2,520 + $472.50 = $2,992.50
If you worked 45 hours each week for 52 weeks, here’s how your annual earnings increase thanks to overtime pay:
$2,992.50 per week x 52 weeks per year = $155,610
That’s $24,570 more than you’d earn working just 40 hours per week at $63 an hour.
Overtime can add up! But also consider taxes and work-life balance when deciding on extra hours.
Here’s a table summarizing how your annual earnings change depending on how many hours you work per week at $63 an hour:
Overtime hours per work day Hours Per Week Earnings Per Week Annual Earnings
0 40 $2,520 $131,040
1 45 $2,992.50 $155,610
2 50 $3,465 $180,180
3 55 $3,937.50 $204,750
4 60 $4,410 $229,320
5 65 $4,882.50 $253,890
6 70 $5,355 $278,460
7 75 $5,827.50 $303,030
How Unpaid Time Off Impacts $63/Hour Yearly Earnings
So far we’ve assumed you work 52 paid weeks per year. Any unpaid time off will reduce your total income.
For example, let’s say you take 2 weeks of unpaid leave. That brings your paid weeks down to 50:
Hours worked per week (40) x Hourly wage($63) x Weeks worked per year(50) = $126,000 annual salary
With 2 weeks unpaid time off, your annual earnings at $63/hour would drop by $5,040.
The table below summarizes how your annual income changes depending on the number of weeks of unpaid leave.
Weeks of unpaid leave Paid weeks per year Earnings Per Week Annual Earnings
0 52 $2,520 $131,040
1 51 $2,520 $128,520
2 50 $2,520 $126,000
3 49 $2,520 $123,480
4 48 $2,520 $120,960
5 47 $2,520 $118,440
6 46 $2,520 $115,920
7 45 $2,520 $113,400
Key Takeaways for $63 Hourly Wage
In summary, here are some key points on annual earnings when making $63 per hour:
• At 40 hours per week, you’ll earn $131,040 per year.
• Part-time of 30 hours/week results in $98,280 annual salary.
• Overtime pay can boost yearly earnings, e.g. $24,570 extra at 45 hours/week.
• Unpaid time off reduces your total income, around $5,040 less per 2 weeks off.
• Your specific situation and location impacts taxes and PTO.
Knowing your approximate annual salary and factors impacting it makes it easier to budget and plan your finances. The next step is calculating take-home pay after deductions like taxes.
$63 An Hour Is How Much A Year After Taxes
Figuring out your actual annual earnings based on an hourly wage can be complicated once taxes are taken into account. In addition to federal, state, and local income taxes, 7.65% of your gross pay
also goes to Social Security and Medicare through FICA payroll taxes. So how much does $63 an hour equal per year after FICA and income taxes are deducted from your gross pay?
Below we’ll walk through the steps to calculate your annual net take home pay if you make $63 per hour. This will factor in estimated federal, FICA, state, and local taxes so you know exactly what to
Factoring in Federal Income Tax
Your federal income tax will be a big chunk out of your gross pay. Federal tax rates range from 10% to 37%, depending on your tax bracket.
To estimate your federal income tax rate and liability:
Look up your federal income tax bracket based on your gross pay.
2023 tax brackets: single filers
Tax rate Taxable income bracket Tax owed
10% $0 to $11,000. 10% of taxable income.
12% $11,001 to $44,725. $1,100 plus 12% of the amount over $11,000.
22% $44,726 to $95,375. $5,147 plus 22% of the amount over $44,725.
24% $95,376 to $182,100. $16,290 plus 24% of the amount over $95,375.
32% $182,101 to $231,250. $37,104 plus 32% of the amount over $182,100.
35% $231,251 to $578,125. $52,832 plus 35% of the amount over $231,250.
37% $578,126 or more. $174,238.25 plus 37% of the amount over $578,125.
For example, if you are single with $131,040 gross annual pay, your federal tax bracket is 24%.
Your estimated federal tax would be:
$16,290 + ($131,040 – $95,376) x 24% = $24,849.36
So at $63/hour with $131,040 gross pay, you would owe about $24,849.36 in federal income taxes.
Considering State Income Tax
In addition to federal tax, most states also charge a state income tax. State income tax rates range from about 1% to 13%, with most falling between 4% and 6%.
Key Takeaways
□ California, Hawaii, New York, New Jersey, and Oregon have some of the highest state income tax rates.
□ Alaska, Florida, Nevada, South Dakota, Tennessee, Texas, Washington, and Wyoming don’t impose an income tax at all.
□ Another 10 U.S states have a flat tax rate—everyone pays the same percentage regardless of how much they earn.
A State-by-State Comparison of Income Tax Rates
STATE TAX RATES LOWEST AND HIGHEST INCOME BRACKETS
Alaska 0% None
Florida 0% None
Nevada 0% None
South Dakota 0% None
Tennessee 0% None
Texas 0% None
Washington 0% None
Wyoming 0% None
Colorado 4.55% Flat rate applies to all incomes
Illinois 4.95% Flat rate applies to all incomes
Indiana 3.23% Flat rate applies to all incomes
Kentucky 5% Flat rate applies to all incomes
Massachusetts 5% Flat rate applies to all incomes
New Hampshire 5% Flat rate on interest and dividend income only
North Carolina 4.99% Flat rate applies to all incomes
Pennsylvania 3.07% Flat rate applies to all incomes
Utah 4.95% Flat rate applies to all incomes
Michigan 4.25% Flat rate applies to all incomes
Arizona 2.59% to 4.5% $27,806 and $166,843
Arkansas 2% to 5.5% $4,300 and $8,501
California 1% to 13.3% $9,325 and $1 million
Connecticut 3% to 6.99% $10,000 and $500,000
Delaware 0% to 6.6% $2,000 and $60,001
Alabama 2% to 5% $500 and $3,001
Georgia 1% to 5.75% $750 and $7,001
Hawaii 1.4% to 11% $2,400 and $200,000
Idaho 1.125% to 6.5% $1,568 and $7,939
Iowa 0.33% to 8.53% $1,743 and $78,435
Kansas 3.1% to 5.7% $15,000 and $30,000
Louisiana 1.85% to 4.25% $12,500 and $50,001
Maine 5.8% to 7.15% $23,000 and $54,450
Maryland 2% to 5.75% $1,000 and $250,000
Minnesota 5.35% to 9.85% $28,080 and $171,221
Mississippi 0% to 5% $5,000 and $10,001
Missouri 1.5% to 5.3% $1,121 and $8,968
Montana 1% to 6.75% $2,900and $17,400
Nebraska 2.46% to 6.84% $3,340 and $32,210
New Jersey 1.4% to 10.75% $20,000 and $1 million
New Mexico 1.7% to 5.9% $5,500 and $210,000
New York 4% to 10.9% $8,500 and $25 million
North Dakota 1.1% to 2.9% $41,775 and $458,350
Ohio 0% to 3.99% $25,000 and $110,650
Oklahoma 0.25% to 4.75% $1,000 and $7,200
Oregon 4.75% to 9.9% $3,750 and $125,000
Rhode Island 3.75% to 5.99% $68,200 and $155,050
South Carolina 0% to 7% $3,110 and $15,560
Vermont 3.35% to 8.75% $42,150 and $213,150
Virginia 2% to 5.75% $3,000 and $17,001
Washington, D.C. 4% to 9.75% $10,000 and $1 million
West Virginia 3% to 6.5% $10,000 and $60,000
Wisconsin 3.54% to 7.65% $12,760 and $280,950
To estimate your state income tax:
Look up your state income tax rate based on your gross pay and filing status.
Multiply your gross annual pay by the state tax rate.
For example, if you live in Pennsylvania which has a flat 3.07% tax rate, your estimated state tax would be:
$131,040 gross pay x 3.07% PA tax rate = $4,022.93 estimated state income tax
So with $131,040 gross annual income, you would owe around in $4,022.93 Pennsylvania state income tax. Verify your specific state’s income tax rates.
Factoring in Local Taxes
Some cities and counties levy local income taxes ranging from 1-3% of taxable income.
To estimate potential local taxes you may owe:
• Check if your city or county charges a local income tax.
• If yes, look up the local income tax rate.
• Multiply your gross annual pay by the local tax rate.
For example, say you live in Columbus, OH which has a 2.5% local income tax. Your estimated local tax would be:
$131,040 gross pay x 2.5% local tax rate = $3,276 estimated local tax
So with $131,040 in gross earnings, you may owe around $3,276 in Columbus local income taxes. Verify rates for your own city/county.
Accounting for FICA Taxes (Social Security & Medicare)
FICA taxes are a combination of Social Security and Medicare taxes that equal 15.3% of your earnings. You are responsible for half of the total bill (7.65%), which includes a 6.2% Social Security tax
and 1.45% Medicare tax on your earnings.
In 2023, only the first $160,200 of your earnings are subject to the Social Security tax
There is an additional 0.9% surtax on top of the standard 1.45% Medicare tax for those who earn over $200,000 (single filers) or $250,000 (joint filers).
To estimate your FICA tax payment:
$131,040 x 6.2% + $131,040 x 1.45% = $10,024.56
So you can expect to pay about $10,024.56 in Social Security and Medicare taxes out of your gross $131,040 in earnings.
Total Estimated Tax Payments
Based on the examples above, your total estimated tax payments would be:
Federal tax: $24,849.36
State tax: $4,022.93
Local tax: $3,276
FICA tax: $10,024.56
Total Estimated Tax: $42,172.85
Calculating Your Take Home Pay
To calculate your annual take home pay at $63 /hour:
1. Take your gross pay
2. Subtract your estimated total tax payments
$131,040 gross pay – $42,172.85 Total Estimated Tax = $88,867.15 Your Take Home Pay
n summary, if you make $63 per hour and work full-time, you would take home around $88,867.15 per year after federal, state, local , FICA taxes.
Your actual net income may vary depending on your specific tax situation. But this gives you a general idea of what to expect.
Convert $63 Per Hour to Yearly, Monthly, Biweekly, and Weekly Salary After Taxes
If you make $63 an hour and work full-time (40 hours per week), your estimated yearly salary would be $131,040 .
The $131,040 per year salary does not account for taxes. Federal, state, and local taxes will reduce your take-home pay. The amount withheld depends on your location, filing status, dependents, and
other factors.
Just now during our calculation of $63 An Hour Is How Much A Year After Taxes, we assumed the following conditions:
• You are single with $131,040 gross annual pay, your federal tax bracket is 24 %.
• You live in Pennsylvania which has a flat 3.07% tax rate
• You live in Columbus, OH which has a 2.5% local income tax.
In the end, we calculated your Total Estimated Tax is $42,172.85 , Your Take Home Pay is $88,867.15 , Total tax rate is 32.18%.
So next we’ll use 32.18% as the estimated tax rate to calculate your weekly, biweekly, and monthly after-tax income.
$63 Per Hour to Yearly, Monthly, Biweekly, Weekly,and Week Salary After Taxes Table
Income before taxes Estimated Tax Rate Income Taxes After Tax Income
Yearly Salary $131,040 32.18% $42,172.85 $88,867.15
Monthly Salary $10,920 32.18% $3,514.40 $7,405.60
BiWeekly Salary $5,040 32.18% $1,622.03 $3,417.97
Weekly Salary $2,520 32.18% $811.02 $1,708.98
$63 an hour is how much a year after taxes
Here is the adjusted yearly salary after a 32.18% tax reduction:
□ Yearly salary before taxes: $131,040
□ Estimated tax rate: 32.18%
□ Taxes owed (32.18% * $131,040 )= $42,172.85
□ Yearly salary after taxes: $88,867.15
Hourly Wage Hours Worked Per Week Weeks Worked Per Year Total Yearly Salary Estimated Tax Rate Taxes Owed After-Tax Yearly Salary
$63 40 52 $131,040 32.18% $42,172.85 $88,867.15
$63 an hour is how much a month after taxes
To calculate the monthly salary based on an hourly wage, you first need the yearly salary amount. Then divide by 12 months.
☆ Yearly salary before taxes at $63 per hour: $131,040
☆ Divided by 12 months per year: $131,040 / 12 = $10,920 per month
The monthly salary based on a 40 hour work week at $63 per hour is $10,920 before taxes.
After applying the estimated 32.18% tax rate, the monthly after-tax salary would be:
□ Monthly before-tax salary: $10,920
□ Estimated tax rate: 32.18%
□ Taxes owed (32.18% * $10,920 )= $3,514.40
• Monthly after-tax salary: $7,405.60
Monthly Salary Based on $63 Per Hour
Hourly Wage Yearly Salary Months Per Year Before-Tax Monthly Salary Estimated Tax Rate Taxes Owed After-Tax Monthly Salary
$63 $131,040 12 $10,920 32.18% $3,514.40 $7,405.60
$63 an hour is how much biweekly after taxes
Many people are paid biweekly, meaning every other week. To calculate the biweekly pay at $63 per hour:
• Hourly wage: $63
• Hours worked per week: 40
• Weeks per biweekly pay period: 2
• $63 * 40 hours * 2 weeks = $5,040 biweekly
Applying the 32.18%estimated tax rate:
• Biweekly before-tax salary: $5,040
• Estimated tax rate: 32.18%
• Taxes owed (32.18% * $5,040 )= $1,622.03
• Biweekly after-tax salary: $3,417.97
Biweekly Salary at $63 Per Hour
Hourly Wage Hours Worked Per Week Weeks Per Pay Period Before-Tax Biweekly Salary Estimated Tax Rate Taxes Owed After-Tax Biweekly Salary
$63 40 2 $5,040 32.18% $1,622.03 $3,417.97
$63 an hour is how much weekly after taxes
To find the weekly salary based on an hourly wage, you need to know the number of hours worked per week. At 40 hours per week, the calculation is:
• Hourly wage: $63
• Hours worked per week: 40
• $63 * 40 hours = $2,520 per week
Accounting for the estimated 32.18% tax rate:
• Weekly before-tax salary: $2,520
• Estimated tax rate: 32.18%
• Taxes owed (32.18% * $2,520 )= $811.02
• Weekly after-tax salary: $1,708.98
Weekly Salary at $63 Per Hour
Hourly Wage Hours Worked Per Week Before-Tax Weekly Salary Estimated Tax Rate Taxes Owed After-Tax Weekly Salary
$63 40 $2,520 32.18% $811.02 $1,708.98
Key Takeaways
• An hourly wage of $63 per hour equals a yearly salary of $131,040 before taxes, assuming a 40 hour work week.
• After accounting for an estimated 32.18% tax rate, the yearly after-tax salary is approximately $88,867.15 .
• On a monthly basis before taxes, $63 per hour equals $10,920 per month. After estimated taxes, the monthly take-home pay is about $7,405.60 .
• The before-tax weekly salary at $63 per hour is $2,520 . After taxes, the weekly take-home pay is approximately $1,708.98 .
• For biweekly pay, the pre-tax salary at $63 per hour is $5,040 . After estimated taxes, the biweekly take-home pay is around $3,417.97 .
Understanding annual, monthly, weekly, and biweekly salary equivalents based on an hourly wage is useful when budgeting and financial planning. Taxes make a significant difference in take-home pay,
so be sure to account for them when making income conversions. Use this guide as a reference when making salary calculations.
What Is the Average Hourly Wage in the US?
Last Updated: Sep 1 2023
US Average Hourly Earnings is at a current level of $33.82, up from 33.74 last month and up from 32.43 one year ago. This is a change of 0.24% from last month and 4.29% from one year ago.
Average Hourly Earnings is the average dollars that a private employee makes per hour in the US. This metric is a part of one of the most important releases every month which includes unemployment
numbers as well. This is normally released on the first Friday of every month. This metric is released by the Bureau of Labor Statistics (BLS).
What is the average salary in the U.S.?
Last Updated: July 18, 2023
The U.S. Bureau of Labor Statistics uses median salary data rather than averages to avoid skewed numbers from outlying high and low numbers. Median weekly earnings of the nation's 121.5 million
full-time wage and salary workers were $1,100 in the second quarter of 2023, the U.S.
If a person works 52 weeks in the year, then this represents a national annual salary of $57,200.
Is $63 an Hour a Good Salary?
Whether $63 an hour is considered a good wage depends heavily on where you live and your household size and expenses. In moderate cost-of-living areas, $63/hour provides a reasonably comfortable
middle-class lifestyle for many. But in high cost cities, it may only be lower-middle class earnings.
To put $63/hour in context, it equates to an annual salary of around $131,040 if you worked full-time year-round (40 hours per week). That places you in the top 25-30% of all personal incomes
However, after federal and state taxes, your annual take-home pay could be around $95,000 or less. And without employer benefits, you’d have to pay for your own healthcare, insurance, retirement and
other costs.
So while a healthy income for many people, $63/hour does not guarantee economic security, especially for larger families. You’d likely enjoy a moderate middle-class lifestyle, but not necessarily an
abundance of discretionary spending power. Location and budgeting are key.
Jobs that pay $63 an hour
Here are some of the most common professions paying around $63 per hour:
• Electrical engineers – Experienced engineers make $55-70 per hour in many industries.
• Technical writers – Technical writers with 5-10 years of experience earn $55-75 per hour.
• Mechanical engineers – Engineers in the auto, aerospace and manufacturing fields earn $55-70 per hour once established.
• Physician assistants – Experienced PAs in specialty fields make $60-75 per hour or more.
• Web developers – Senior web developers at tech companies earn $55-70 per hour in total compensation.
• Project managers – IT, construction and product development PMs earn $55-70 hourly when fully trained.
• Radiologic technologists – Senior radiology techs make $50-65 per hour in hospital settings.
• Clinical laboratory technologists – Experienced lab techs can make $50-70 per hour in medical labs.
• Occupational therapists – Therapists bill $60-75 per hour in clinical settings once fully licensed.
Reaching a $63 per hour income level typically requires specialized skills, education and certifications. Entry-level and unskilled positions are very unlikely to pay this hourly rate.
Can You Live Off $63 An Hour?
Can the average household live comfortably on $63 per hour? Here’s an overview of typical monthly expenses:
• Housing – Rent or mortgage for a modest 2-3 bedroom home, around $1,500-$2,500 depending on location.
• Transportation – Car payments plus gas, insurance and repairs: $500-$1,000.
• Food – Groceries, dining out for a family: $800-$1,200.
• Utilities – Electric, gas, water, trash service: $300-$500.
• Phone/internet – Cell phone plans, cable/internet: $200-$400.
• Insurance – Health, dental, vision, life, disability: $500-$1,000.
• Entertainment – Activities, streaming services, other subscriptions: $300-$600.
• Clothing – New clothes, dry cleaning: $150-$300.
• Travel – Occasional vacations, trips: $200-$500.
• Miscellaneous – Everything else including pets, hobbies etc: $300-$500.
• Debt payments – Student loans, credit cards, personal loans: $500-$1,500.
• Savings & Investments – Ideally 10-20% of net income: $1,000-$2,000.
• Taxes – Federal and state taxes eat up 25% or more of gross.
Total: Around $6,000-$10,000 per month for a moderately comfortable, middle-class life in most areas.
This rough monthly budget shows that $63/hour provides a reasonable standard of living for many middle-class households able to budget wisely. Strict budgeting in areas like housing, transportation
and discretionary costs would be required to save substantially and pay down debt though.
The impact of inflation on the value of $63 an hour
While $63 an hour provides a comfortable wage today, inflation will steadily erode its real value over time. According to the U.S. Bureau of Labor Statistics, prices for housing, food,
transportation, medical care, and other common expenses have risen over 3% per year on average during the past 20 years.
Assuming a 3.5% annual inflation rate, $63 today will only have the purchasing power of around $57 five years from now and $51 ten years from now. Your costs slowly claim more of your monthly budget.
This means wage earners have to seek regular raises just to maintain their current lifestyle. Those on fixed incomes see their purchasing power diminish each year as prices outpace earnings.
Coping with inflation often means changing jobs strategically for higher pay. On a flat $63 wage, building savings and wealth becomes more difficult over decades. Careful budgeting and smart
investments become essential.
5 Ways To Increase Your Hourly Wage
Here are some potential strategies for moving beyond $63 per hour:
1. Ask for raises and negotiate higher pay based on your contributions and experience. Come armed with market research.
2. Pursue promotions or higher job titles that come with increased compensation.
3. Gain new skills/certifications to qualify for higher-paying roles.
4. Change employers to one that pays better for your experience level.
5. Start a side business that allows you to earn supplemental income.
6. Work overtime when possible, especially at time-and-a-half rates.
7. Enroll in higher education like a Master’s program to access more advanced roles.
Depending on your career path, a combination of adding skills, seeking promotions, strategic job changes and taking on side work can potentially help you surpass an hourly income of $63. Consistent
effort is key.
Buying a car on $63 an hour
Is buying a car affordable if you make $63 per hour? Here’s an overview:
• The average new car transaction price is around $48,000 today.
• Even with a sizable downpayment, loan payments may exceed $600 per month over 5 years.
• Plus insurance, gas and repairs of $200-$400 monthly.
• That’s around $800+ per month for basic transportation.
• On a $63 hourly income, that’s likely unaffordable for most budgets.
However, buying a used car for $15,000-25,000 is much more viable. This keeps total monthly costs (loan, insurance, gas) under $500 comfortably.
You may also be able to afford a new compact model around $25,000 with strong credit and a large down payment. But overall, buying a car on $63 an hour requires sticking to cheaper vehicles while
limiting loan amounts.
Can You Buy a House on $63 An Hour?
Next, let’s assess whether home ownership is feasible on an income of $63 per hour:
• The median home list price is currently around $325,000 nationwide.
• With a 10% down payment of $32,500, your mortgage principal will be $292,500.
• At a 5% fixed rate over 30 years, the monthly mortgage payment is approximately $1,550.
• On a $63 hourly income of $131,040 gross pay yearly, that $1,550 payment is 14% of your gross annual earnings.
• Experts recommend total housing costs stay under 28% of gross income.
• 28% of your gross income is $36,691 annually or $3,058 monthly.
Based on this, buying a median priced home may be challenging but possible on a $63 hourly wage if you budget diligently. You’d want to aim for the lower end of your price range while keeping total
monthly housing costs under $2,500.
However, in high cost areas where the median home is $600,000+, ownership may be out of reach on this income level. Renting a more modest home or apartment may be the better option financially.
In summary, home ownership on $63 an hour is feasible but requires careful budgeting, especially in higher cost regions. Staying well below median prices is key.
Example Budget For $63 Per Hour
Here is a realistic monthly budget for someone earning $63 per hour or $131,040 per year:
Monthly Net Income:
• Gross income: $131,040
• Taxes and deductions (30%): $39,312
• Take Home Pay: $91,728 annually / $7,644 monthly
Monthly Expenses:
• Rent: $1,500 (2-bedroom apartment)
• Used Car Payment: $300
• Car Insurance: $120
• Gas: $120
• Groceries and Dining Out: $650
• Utilities: $300
• Cable/Internet: $100
• Cell Phone: $100
• Health Insurance: $400
• Entertainment: $200
• Clothing: $150
• Gifts and Miscellaneous: $200
• Retirement Savings: $500
• Emergency Fund Savings: $300
• Total Expenses: $4,940
Remaining Income: $2,704
This sample budget shows $63 an hour providing a moderately comfortable middle-class lifestyle. There’s some room for additional discretionary spending and financial goals after basic needs are met.
But budgets would need to be monitored closely.
In Summary
• An income of $63 per hour translates to around $131,040 per year assuming full-time employment.
• This salary can support a moderately comfortable middle-class lifestyle in most regions.
• Education, certifications and experience can potentially help you earn $63 per hour.
• Buying an average-priced home may be challenging but possible at this income level with prudent budgeting.
• Inflation steadily diminishes the real value of a fixed $63 wage over decades.
• While covering basic needs, $63 an hour requires budget tradeoffs and limits discretionary spending capability.
• Home ownership is feasible in many markets with realistic expectations, but rents may be preferable in high cost areas.
• With careful money management and supplemental income, a $63 hourly wage can open the door to moderate wealth building over time.
In summary, an income of $63 an hour affords a reasonably comfortable but not lavish lifestyle for middle-class households and individuals willing to budget wisely. Ongoing career development and
prudent financial habits remain important for long-term wealth creation. | {"url":"https://timehackhero.com/63-an-hour-is-how-much-a-year/","timestamp":"2024-11-05T19:58:16Z","content_type":"text/html","content_length":"230572","record_id":"<urn:uuid:eb18bbfb-287e-4ba4-9bfe-4f7ea76ae7c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00309.warc.gz"} |
Counterfactual Physics: Lorentz Variance?
The theory of relativity takes its name from a very simple and appealing idea: that the laws of physics should look the same to moving observers as to stationary ones. "Laws of physics" here includes
Maxwell's equations for electricity and magnetism, which necessarily means that moving observers must see the same speed of light as stationary observers (Einstein included the constancy of the speed
of light as a second postulate in his original relativity paper, but it's redundant-- the constancy of the speed of light is a direct consequence of the principle of relativity). This leads directly
to all the observed weirdness of relativity-- clocks running at different rates for different observers, moving objects shrinking, disagreements about the simultaneity of events, etc.
Of course, the notion of a single universal time also has a certain aesthetic appeal, which is part of why it was the default assumption of physics from the days of Galileo and Newton through to
1905. Having every clock in the universe, moving or not, tick at exactly the same rate would be simple and elegant in a manner similar to that of relativity. It would, however, require some drastic
revisions of the laws of physics as we understand them.
The question is, what would need to change, and what would the consequences be?
You would clearly need to change the structure of Maxwell's equations to accommodate a variable speed of light, but what would that do to, say, atoms and molecules, that are held together by
electromagnetic forces? The speed of light shows up in things like the Rydberg constant that gives you the energy of atomic energy levels, so presumably these would get pushed around as the speed of
an object changed. Would that mean, for example, that objects moving at too high a speed would literally fall apart as the energy levels determining the bonds between their component atoms shifted
into a new configuration?
(I'm imagining a world that's as similar to ours as possible, but has a variable speed of light, so things like the Michelson-Morley experiment would not give the negative result that they do in our
world. I'm not looking for aether drag theories that use some baroque method to match those observations, but a version of reality in which those observations are consistent with a variable speed of
I don't expect that anybody has put all that much thought into this-- it would take a good deal of mathematical effort to reformulate E&M in a way that wasn't consistent with relativity, and why
would you do that? (Then again, there's some amazingly abstract stuff on the arxiv, so who knows?) It's kind of an interesting topic for idle speculation, though. So consider the comments section an
open thread for idle speculation, or the posting of actual relevant information, should any exist.
(This post brought to you by Emmy, who asked "Would having all observers see the same time really be all that bad?" in chapter 3 of the book-in-progress.)
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"the constancy of the speed of light is a direct consequence of the principle of relativity)"
Careless statement there, what about galilean relativity. It certainly doesn't imply the constancy of the speed of light. I know you are thinking of Lorentz invariance when saying that, but still.
The context of that sentence was a discussion of Einstein's theory, so I thought it was clear that "principle of relativity" referred to the first principle in Einstein's paper.
Of course, I would say that the only real difference between Galileian and Einsteinian relativity is that physicists had learned about electricity and magnetism in the intervening 200-odd years. They
have the same central idea-- physics looks the same to an observer moving at constant speed-- it's just that the meaning of "physics" expanded to include Maxwell's equations.
You would clearly need to change the structure of Maxwell's equations to accommodate a variable speed of light
There is an entire branch of physics called plasma physics which deals with this problem. The difference is that plasma physics deals with the speed of electromagnetic waves in a medium and how that
speed changes as a function of frequency. The constancy of the speed of light in vacuum is taken for granted. Since it's the speed of light in vacuum that has to be constant according to Lorenz, one
would expect that the speed of an electromagnetic wave in the actual medium might vary according to the observer's relative velocity.
How you get from this formulation to one where the speed of light in vacuum might depend on the observer's frame of reference is not obvious to me, but I suspect that if it were possible to devise
such a formulation self-consistently it would look quite a bit like plasma physics. You would be likely to run into lots of phenomena like wave dispersion in unexpected contexts.
I think what Chad means to say is:
"Given that we know that the Maxwell equations are true empirically, the constancy of the speed of light is a direct consequence of the principle of relativity."
Whilst this is clearly true, I think it is the wrong way of thinking about it. I think we should say that the Maxwell equations are the way they are because of special relativity (in addition to
other symmetry principles like U(1) gauge invariance) rather than saying that relativity is the way it is because of Maxwell's equations. This is how we think about determining field theories these
days, i.e. there is a list of principles that have to be satisfied, including Lorentz invariance, gauge symmetries and renormalizability. Taken together, these often uniquely determine the Lagrangian
of the theory.
The empirical correctness of Maxwell's equations obviously provided the main motivation for taking special relativity seriously in the first place. Therefore, I can see why Chad (the experimentalist)
might therefore want to take Maxwell's equations as a given in understanding relativity. However, this is a historical accident and the existence of an invariant speed seems to be the more
fundamental principle from which we should derive all of the more specific theories like Maxwell, Yang-Mills, etc.
Regarding the main question of the post, you might try doing a literature search for "luminiferous aether". However, I don't think you will find that much of the literature on the subject has been
digitized :)
I think what Chad means to say is:
"Given that we know that the Maxwell equations are true empirically, the constancy of the speed of light is a direct consequence of the principle of relativity."
That's another way of putting it, yes. It's close to the historical trajectory, too.
Whilst this is clearly true, I think it is the wrong way of thinking about it. I think we should say that the Maxwell equations are the way they are because of special relativity (in addition to
other symmetry principles like U(1) gauge invariance) rather than saying that relativity is the way it is because of Maxwell's equations. This is how we think about determining field theories these
days, i.e. there is a list of principles that have to be satisfied, including Lorentz invariance, gauge symmetries and renormalizability.
Sure, if you want to be all theorist-y about it...
Even phrased that way, though, you could imagine some different symmetry that would lead to different rules. I'm not sure exactly what that would be, but there's presumably some symmetry you could
use that would give time as a universal quantity, and imposing that symmetry would give you different rules for E&M. The question is, what would those rules look like, and what would that do to the
rest of physics?
there is a list of principles that have to be satisfied, including Lorentz invariance, gauge symmetries and renormalizability
Most of us wouldn't include renormalizability in that list.
The empirical correctness of Maxwell's equations obviously provided the main motivation for taking special relativity seriously in the first place. Therefore, I can see why Chad (the experimentalist)
might therefore want to take Maxwell's equations as a given in understanding relativity.
I will disagree here and say that Chad is entirely correct to do things in this order. Relativity as a concept goes all the way back to Galileo, as Chad @2 points out. Classical mechanics is quite
simple under Galilean relativity, but Maxwell's equations were found to be extremely messy under Galilean transforms. Thus it was necessary to come up with a different conception of relativity, and
this was an active area of research in the late 19th and early 20th centuries. The correct transform bears Lorentz's name because he was the first to show that Maxwell's equations were indeed
invariant under such transforms. Fixing classical mechanics to work under Lorentz transforms turned out to be a good deal easier than fixing Maxwell's equations to work under Galilean transforms
(propagation of EM waves in media was IIRC not understood at the time, and the same mathematical tools needed to treat that problem would be needed to make a Galilean invariant version of Maxwell's
I suppose there might be pedagogical reasons for taking c to be a frame-independent constant, but there are other pedagogical reasons for deriving it as a consequence of Lorentz invariance. I took my
Jackson course from a high-powered theorist who nonetheless adopted the approach Chad takes.
"Given that we know that the Maxwell equations are true empirically, the constancy of the speed of light is a direct consequence of the principle of relativity."
Yeah, that is about as deep as saying: given that the speed of light is a constant empirically, the constancy of the speed light is a direct consequence of the principle of relativity.
For dog's sake, Maxwell's equations entail the constancy of the speed of light. They are Lorentz invariant. This was known long before Einstein came along. Poincaré knew it, Lorentz and Fitzgerald
know it, Voigt knew it.
The question is, what would those rules look like, and what would that do to the rest of physics?
It sounds like you're looking for a Galilean-invariant gauge theory, but I think the main difficulty is that the concept of a massless particle doesn't make sense in Galilean-invariant field theory.
So it's tempting to say that E&M would be forced to be short-range, but maybe there's some sort of loophole. Probably this is discussed in the literature in various places....
"you might try doing a literature search for "luminiferous aether". However, I don't think you will find that much of the literature on the subject has been digitized :)"
You might have some good luck here at the OU History of Science collections. http://libraries.ou.edu/locations/?id=20
(we're trying to organize a tour of the collections with the HoS department here for the Midwest Solid State Conference, if anyone here is going. Confirmed; they do have some stuff on "luminiferous
aether" e.g. "On the relative motion of the earth and the luminiferous Åther microform" by Albert A. Michelson and Edward W. Morley, he London, Edinburgh and Dublin Philosophical Magazine and Journal
of Science. 5th Series, v. 24, no. 151 (Dec. 1887) If you want older, they have some cool stuff from e.g. Galileo and Darwin (some original Darwin notes, and books with e.g. notes in the margins in
their hand and notes for future revisions of the book! My wife is in the HoS department and was surprised to find that the slightly singed book she had set in front of her one day was singed because
it was saved from the Great Fire of London in 1666....)
On second thought I'm not satisfied with the "no massless particles" argument, since gapless excitations exist in condensed matter systems. It's a fun exercise to think about what the closest
Galilean analogue of relativistic E&M could be, I'll give it more thought....
Count me among those who don't think the constancy of the speed of light is implied by the principle of relativity. After all, the speed of sound is not the same to every observer, but nobody thinks
that's a violation of relativity. It's just a reflection of the fact that sound moves through a medium, and any particular example of that medium has a rest frame. Pre-Einstein, that's exactly how
people thought E&M worked; the medium was the aether. Einstein's great breakthrough was to show how you could get a consistent reconciliation of Maxwell's equations and the principle of relativity by
replacing aether with a constant speed of light.
Freeman Dyson gave a nice picture of what groups (including Lorentzian or Newtonian) could accommodate reasonable physics in his Missed Opportunities Lecture .
In general, I think of c not as the speed of light, but as the speed of massless particles--this seems like a more likely fundamental basis for a universe than the particulars of any individual gauge
interaction. Photons in vacuum happen to be massless. You generally recover the Newtonian limit if you let c go to infinity.
For instance, you can write the Rydberg formula in terms of the coupling constant alpha (presumably unaffected) and c, and if you do so the Rydberg energy goes to infinity. Note that if you write the
binding energy in terms of the electron charge c appears downstairs, but the apparent electron charge depends on the speed of light (sqrt(alpha*hbar*c)) whereas alpha is fixed. I think.
I just found this:
I don't like how they speak about nonrelativistic limit to designate the limit c to infinity. It's just confusing the issue of the difference between the principle of relativity and the invariance of
lightspeed again. Anyway, there you have it, electromagnetism with galilean relativity.
PS: Doesn't surprise Lévy-Leblond is involved. He's also the guy who deduced a spin 1/2 equation from galilean invariance.
Very quickly and naively (which is all I have time for), I'd think that a world where the electric field exists, but the magnetic field doesn't, is an example of a world that obeys the principle of
Gallilean relativity without being special relativistic. There may be other, less simple minded, limits.
If true, I don't think energy levels in the hydrogen atom would change all that much in such a world (though you may need to change units in order to keep them finite. Incidentally, these sort of
theoretical speculations is the place where it is cleanest to talk about dimensionless quantities, such as energy ratios, i.e the pattern of energy levels as opposed to their absolute normalization).
This may be a cop-out answer, but if you take a universe where the speed of light is infinite, you get synchronized clocks.
So just take that limit in all your physical equations.
Does this limit results in no magnetic fields as in Moshe's (#16) suggested Galilean relativity world? I suspect so. Is it equivalent? I dunno. Like Moshe's suggestion, I think this limit would keep
the gross features of atomic structure intact, although the fine structure details would get all weird.
Dear Chad,
you are looking for either Hertz (see P. Moon et al. Physics Essays 7, 28 (1994) for instance) or Weber electrodynamics (see any of Assis' papers listed here: http://www.ifi.unicamp.br/~assis/
wpapers.htm). In both theories you will get Galilean invariance (universal time).
In Hertz electrodynamics, there is no magnetic forces. Instead, a magnetic field creates an electric field that acts on the charged particles. Curiously enough, speed of light is still a constant...
In Weber electrodynamics, there is really no field, just forces between particles.
Good luck,
Reading Dyson's lecture on missed opportunities was fascinating. Some of the ideas are really far out, like a universe with absolute space instead of absolute time, except, I just read an article
proposing something an awful lot like that just a few weeks ago.
In the 19th century, the physicists were well ahead of the mathematicians, but now it seems that the string theorists are mathematicians ahead of the physicists.
Speaking of counterfactuals, I looked up luminiferous aether in the journal Science and came across a fascinating paper on a repeat of the Michelson Morley experiment done in 1925, except that this
experiment WAS able to measure the absolute motion of the earth and solar system! There is aether and there is aether drag, but there is also absolute motion. I'll have to read it again, but it sure
doesn't correspond to conventional wisdom. I assume there was a follow up paper or two, or I may have accidentally found an internet gateway to an alternate universe.
I've put a copy of the paper at: http://www.kaleberg.com/misc/1925experiments.pdf
Please, tell me I misread it. | {"url":"https://scienceblogs.com/principles/2010/09/08/counterfactual-physics-lorentz","timestamp":"2024-11-12T00:42:26Z","content_type":"text/html","content_length":"79068","record_id":"<urn:uuid:c5bb0e03-5015-4171-a709-b97e001dab6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00789.warc.gz"} |
Why Markov chains, Brownian motion and the Metropolis algorithm
• We want to study a physical system which evolves towards equilibrium, from given initial conditions.
• We start with a PDF $w(x_0,t_0)$ and we want to understand how the system evolves with time.
• We want to reach a situation where after a given number of time steps we obtain a steady state. This means that the system reaches its most likely state (equilibrium situation)
• Our PDF is normally a multidimensional object whose normalization constant is impossible to find.
• Analytical calculations from $w(x,t)$ are not possible.
• To sample directly from from $w(x,t)$ is not possible/difficult.
• The transition probability $W$ is also not known.
• How can we establish that we have reached a steady state? Sounds impossible!
Use Markov chain Monte Carlo
Brownian motion and Markov processes
A Markov process is a random walk with a selected probability for making a move. The new move is independent of the previous history of the system.
The Markov process is used repeatedly in Monte Carlo simulations in order to generate new random states.
The reason for choosing a Markov process is that when it is run for a long enough time starting with a random state, we will eventually reach the most likely state of the system.
In thermodynamics, this means that after a certain number of Markov processes we reach an equilibrium distribution.
This mimicks the way a real system reaches its most likely state at a given temperature of the surroundings.
Brownian motion and Markov processes, Ergodicity and Detailed balance
To reach this distribution, the Markov process needs to obey two important conditions, that of ergodicity and detailed balance. These conditions impose then constraints on our algorithms for
accepting or rejecting new random states.
The Metropolis algorithm discussed here abides to both these constraints.
The Metropolis algorithm is widely used in Monte Carlo simulations and the understanding of it rests within the interpretation of random walks and Markov processes.
Brownian motion and Markov processes, jargon
In a random walk one defines a mathematical entity called a walker, whose attributes completely define the state of the system in question.
The state of the system can refer to any physical quantities, from the vibrational state of a molecule specified by a set of quantum numbers, to the brands of coffee in your favourite supermarket.
The walker moves in an appropriate state space by a combination of deterministic and random displacements from its previous position.
This sequence of steps forms a chain.
Brownian motion and Markov processes, sequence of ingredients
• We want to study a physical system which evolves towards equilibrium, from given initial conditions.
• Markov chains are intimately linked with the physical process of diffusion.
• From a Markov chain we can then derive the conditions for detailed balance and ergodicity. These are the conditions needed for obtaining a steady state.
• The widely used algorithm for doing this is the so-called Metropolis algorithm, in its refined form the Metropolis-Hastings algorithm.
Applications: almost every field in science
• Financial engineering, see for example Patriarca et al, Physica 340, page 334 (2004).
• Neuroscience, see for example Lipinski, Physics Medical Biology 35, page 441 (1990) or Farnell and Gibson, Journal of Computational Physics 208, page 253 (2005)
• Tons of applications in physics
• and chemistry
• and biology, medicine
• Nobel prize in economy to Black and Scholes | {"url":"https://notebook.community/CompPhysics/ComputationalPhysics/doc/pub/rw/ipynb/.ipynb_checkpoints/rw-checkpoint","timestamp":"2024-11-10T14:45:32Z","content_type":"text/html","content_length":"144434","record_id":"<urn:uuid:316d8c3b-b1f5-40d7-9a78-157657ade31a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00065.warc.gz"} |
How to Find the Cube Root of 64?
Cube Root of 64
The cube root of 64, denoted as 364. The Cube root of 64 is the number which multiplied by itself three times gives the product as 64. Since 64 can be expressed as 2*2*2*2*2*2. Therefore, the cube
root of 64 = 3 2 2 2 2 2 2 = 4
What is Cube Root ?
The cube root of a number is a value that represents the number that, when multiplied by itself three times, gives the original number. In other words, it's a number that, when raised to the power of
3, gives the original number. For example, the cube root of 27 is 3, since 3 3 3 = 27. The cube root of a number is denoted using the symbol.
Cube Root Symbol
The symbol for the cube root of a number is 3 . For example, to indicate that x is the cube root of a number n, we write x=3n .
Perfect cube
A perfect cube is a number that is the result of multiplying an integer by itself three times. In mathematical terms, a number n is a perfect cube if there exists an integer x such that x3=n .
For example, the first few perfect cubes are:
and so on.
Non Perfect Cube
A non-perfect cube is a number that is not a perfect cube, meaning it cannot be represented as the result of an integer raised to the power of 3. For example, 7 is a non-perfect cube because there is
no integer value of x such that x3=7 .
The cube root of a non-perfect cube number is not an integer, and can be either a rational or an irrational number. To find the cube root of a non-perfect cube, you can use the estimation and
refinement method, However, it's important to note that finding an exact value for the cube root of a non-perfect cube is not always possible, and the answer will typically be an approximation to a
certain number of decimal places.
How to Find Cube Root?
There are different methods to find the cube root of a number, but one common method is the estimation and refinement method:
1. Estimate the value: Find the nearest perfect cube to the number and use that as the estimate for the cube root.
2. Refine the estimate: Use the formula, (x+n/x2)/3 where x is the estimate and n is the original number.
3. Repeat step 2: Keep refining the estimate until you have an answer that is accurate to the desired level of precision.
Prime Factorization Method
The prime factorization method is a way to find the cube root of a perfect cube number by factoring the number into its prime components. This method is particularly useful for finding the cube root
of perfect cube numbers that are difficult to estimate, such as large numbers or irrational numbers.
Here's how the prime factorization method works:
1. Factor the perfect cube number into its prime components.
2. Take the cube root of each prime component.
3. Multiply the cube roots of the prime components together to find the cube root of the original number.
For example, to find the cube root of 8 (which is 2), we can factor
8 into 2*2*2 and then take the cube root of each component, which is 2. So, the Cube root of 8 is 2.
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The Outer Connected Detour Monophonic Number of a Graph
For a connected graph ???? = (????, ????) of order a set is called a monophonic set of ????if every vertex of ????is contained in a monophonic path joining some pair of vertices in ????. The
monophonic number (????) of is the minimum cardinality of its monophonic sets. If or the subgraph is connected, then a detour monophonic set of a connected graph is said to be an outer connected
detour monophonic setof .The outer connecteddetourmonophonic number of , indicated by the symbol , is the minimum cardinality of an outer connected detour monophonic set of . The outer connected
detour monophonic number of some standard graphs are determined. It is shown that for positive integers , and ???? ≥ 2 with ,there exists a connected graph ????with????????????[????]???? = [,
]????????????m[????]???? = and = ????. Also, it is shown that for every pair of integers ????and b with 2 ≤ ???? ≤ ????, there exists a connected graph with and .
chord, monophonic path, monophonic number, detour monophonic path, detour monophonic number, outer connected detour monophonic number
T. W. Haynes, S. T. Hedetniemi and P. J, Slater, Fundamentals Of Domination In Graphs, Marcel Dekker, New York, (1998).
J. John, The Forcing Monophonic And The Forcing Geodetic Numbers Of A Graph, Indonesian Journal of Combinatorics 4(2), (2020), 114-125.
J. John and S. Panchali 2, The upper monophonic number of a graph, Int. J. Math.Combin. 4, (2010),46 – 52.
P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of A Graph. J. Combin. Math. Combin. Comput., (84), (2013),179-188
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Copyright (c) 2022 N.E Johnwin Beaula, S Joseph Robin
This work is licensed under a
Creative Commons Attribution 4.0 International License
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Runtime Performance
• GraalVM for JDK 23 (Latest)
• GraalVM for JDK 24 (Early Access)
• GraalVM for JDK 21
• GraalVM for JDK 17
• Archives
• Dev Build
Runtime Performance
GraalVM optimizes R code that runs for extended periods of time. The speculative optimizations based on the runtime behaviour of the R code and dynamic compilation employed by the GraalVM runtime are
capable of removing most of the abstraction penalties incurred by the dynamism and complexity of the R language.
Examine the algorithm in the following example which calculates the mutual information of a large matrix:
x <- matrix(runif(1000000), 1000, 1000)
mutual_R <- function(joint_dist) {
joint_dist <- joint_dist/sum(joint_dist)
mutual_information <- 0
num_rows <- nrow(joint_dist)
num_cols <- ncol(joint_dist)
colsums <- colSums(joint_dist)
rowsums <- rowSums(joint_dist)
for(i in seq_along(1:num_rows)){
for(j in seq_along(1:num_cols)){
temp <- log((joint_dist[i,j]/(colsums[j]*rowsums[i])))
temp = 0
mutual_information <-
mutual_information + joint_dist[i,j] * temp
# user system elapsed
# 1.321 0.010 1.279
Algorithms such as this one usually require C/C++ code to run efficiently:^1
if (!require('RcppArmadillo')) {
x <- matrix(runif(1000000), 1000, 1000)
# user system elapsed
# 0.037 0.003 0.040
(Uses r_mutual.cpp.)
However, after a few iterations, GraalVM runs the R code efficiently enough to make the performance advantage of C/C++ negligible:
# user system elapsed
# 0.063 0.001 0.077
The GraalVM R runtime is primarily aimed at long-running applications. Therefore, the peak performance is usually only achieved after a warmup period. While startup time is currently slower than GNU
R’s, due to the overhead from Java class loading and compilation, future releases will contain a native image of R with improved startup.
^1 When this example is run for the first time, it installs the RcppArmadillo package,which may take a few minutes. Note that this example can be run in both GraalVM’s R runtime and GNU R. | {"url":"https://www.graalvm.org/dev/reference-manual/r/Performance/","timestamp":"2024-11-10T09:36:24Z","content_type":"text/html","content_length":"30630","record_id":"<urn:uuid:7e34a577-4898-48e3-8778-e49e29afccb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00325.warc.gz"} |
22 inches to feet (Inches to Feet)
By Kshitij Singh / Under Inches To Feet / Published on
Convert 22 inches to feet with this simple guide. Learn the equation and practical examples for converting 22 inches to feet.
Let us understand the process of converting 22 inches to feet
22 inches is equal to approximately 1.83333 feet.
Converting inches to feet is a common requirement in both personal and professional contexts. Whether you are working on a home improvement project, designing a piece of furniture, or calculating the
dimensions of a plot of land, knowing how to convert inches to feet can be very useful.
To perform this conversion, you simply need to understand the basic mathematical relationship between inches and feet. One foot is equivalent to 12 inches. Therefore, to convert inches to feet, you
divide the number of inches by 12. For example, 22 inches divided by 12 equals approximately 1.83333 feet.
Importance of Converting Inches to Feet
Understanding conversions between units of measurement is vital in a variety of fields. From construction and architecture to tailoring and interior design, precise measurement conversions are
crucial. By mastering this simple conversion, you can ensure the accuracy of your projects and plans.
Practical Application of Converting 22 Inches to Feet
When you think about everyday situations where this conversion might be necessary, picture a scenario where you’re setting up a new television. Most TVs are measured diagonally in inches. A TV with a
22-inch screen would therefore have a diagonal size of about 1.83333 feet. Similarly, if you are selecting a computer monitor or even a particular size of fabric, knowing how to convert inches to
feet can save you a lot of time and hassle.
Conversion Formula
To convert inches to feet, use the following formula:
[ \text{feet} = \frac{\text{inches}}{12} ]
Here is a step-by-step example:
1. Take the measurement in inches: 22 inches.
2. Divide by 12: ( \frac{22}{12} \approx 1.83333 ) feet.
This formula is universally applicable, making it easy to convert any measurement from inches to feet.
Why This Conversion Matters
Consider statistics showing that 60% of DIY enthusiasts encounter measurement errors in their projects. Knowing simple conversions like inches to feet can help mitigate these errors significantly.
According to a survey, proper measurement understanding reduces rework by up to 30%, leading to greater efficiency and cost savings.
External Resources
For further information and additional conversion tools, visit Measurement Conversion.
How many feet is 22 inches?
22 inches is approximately 1.83333 feet. This conversion is useful in various practical scenarios such as setting up a TV or measuring fabric.
Why is converting inches to feet useful?
Converting inches to feet is useful in many fields, including construction, interior design, and everyday tasks like selecting the correct size for a TV or monitor. It ensures accuracy in
measurements and helps avoid errors.
What is the formula to convert inches to feet?
The formula to convert inches to feet is: feet = inches ÷ 12. For example, ( \frac{22}{12} \approx 1.83333 ) feet.
How do you visualize 1.83333 feet?
Visualizing 1.83333 feet can be easier if you think about it as almost 2 feet, which is roughly the length of a standard-sized dining chair.
Mastering the conversion from inches to feet is an essential math skill that can greatly improve the accuracy of your measurements. By understanding these simple conversions, you can confidently plan
and execute projects without the worry of measurement inaccuracies. | {"url":"https://unlearningmath.com/22-inches-to-feet/","timestamp":"2024-11-07T23:12:21Z","content_type":"text/html","content_length":"53393","record_id":"<urn:uuid:68fce502-1fa2-4e84-a4fb-6f6a013e0706>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00832.warc.gz"} |
divided power algebra
Algebraic theories
Algebras and modules
Higher algebras
Model category presentations
Geometry on formal duals of algebras
A divided power algebra is a commutative ring $A$ together with an ideal $I$ and a collection of operations $\{\gamma_{n}\colon I\to A\}_{n\in\mathbb{N}}$ which behave like operations of taking
divided powers $x\mapsto x^{n}/n!$ in power series.
A divided power algebra is a triple $(A,I,\gamma)$ with
where we additionally adopt the convention $\gamma_0(x) = 1$ (which is usually not in $I$), and this data is required to satisfy the following conditions:
1. For each $x\in I$, we have $\gamma_{1}(x)=x$.
2. For each $x,y\in I$ and $n\geq 0$, we have
3. For each $\lambda\in A$, each $x\in I$ and $n\geq 0$, we have
$\gamma_{n}(\lambda x)=\lambda^{n}\gamma_{n}(x).$
4. For each $x\in I$ and each $m,n\geq 0$, we have
5. For each $x\in I$ and each $m\geq 0$, $n\geq 1$, we have
$\gamma_{m}(\gamma_{n}(x))=\frac{(n m)!}{(n!)^{m}m!}\gamma_{n m}(x).$
For a given $(A,I)$, a divided power structure on $(A,I)$ is a $\gamma$ making $(A, I, \gamma)$ a divided power algebra.
If $A$ is an $R$-algebra for a ring $R$, we call it a divided power $R$-algebra or PD-$R$-algebra.
Genuine powers can be constructed in the expected way from the divided powers, and when $A$ is torsion free, the reverse is true:
If $(A,I,\gamma)$ is a divided power algebra, then $n! \gamma_n(x) = x^n$ for every $x \in I$ and $n \geq 0$ (taking $x^0:=1$).
It is true for $n=0$ and $n=1$ by definition. For $n \geq 2$, this follows by induction, since $n! \gamma_n(x) = (n-1)! \gamma_{n-1}(x) \cdot 1! \gamma_1(x) = x^{n-1} \cdot x$.
If $A$ is a commutative, torsion free ring with an ideal $I$ such that $x^n$ is an $(n!)$-th multiple for every $x \in I$ and $n \geq 0$, then $(A,I)$ has a unique divided power structure, and it is
given by $\gamma_n(x) = x^n / n!$.
The hypotheses imply the quotients $x^n / n!$ are unique and well-defined, and any divided power structure on $(A,I)$ must be given by that formula. It’s straightforward to check the definition does
give a divided power algebra.
So in the torsion free case, the divided power algebras are precisely of the motivating form. In positive characteristic, though, examples can be somewhat more exotic.
We can define the concept of divided power in any symmetric monoidal category.
Let $\mathcal{C}$ be a symmetric monoidal category.
The $n^{th}$ divided power of an object $A$ is then defined as the equalizer of the following diagram:
where there is one arrow for every $\sigma \in \mathfrak{S}_{n}$. We write $\sigma$ for the natural transformation associated to $\sigma$, which is defined in the entry symmetric monoidal category.
The $n-th$ divided power $\Gamma_{n}(A)$ is thus described by the following equalizer diagram:
In the category of modules over some commutative ring, the $n-th$ divided power of a module $A$ is equivalently descibed as the space $(A^{\otimes n})^{\mathfrak{S}_{n}}$ of invariants under the
action of $\mathfrak{S}_{n}$ on $A^{\otimes n}$ by permutation of the factors. Note that the $n-th$ symmetric power $S_{n}(A)$ of an object $A$ in a symmetric monoidal category is described as the
coequalizer of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$. In a category of modules, it is equivalently described as the space $(A^{\otimes n})_{\mathfrak{S}_{n}}$ of
coinvariants by the action of $\mathfrak{S}_{n}$. We then have the relation $(S_{n}(A^{*}))^{*} = \Gamma_{n}(A)$ which probably means that these powers can be interpreted as graded exponential
modalities in a kind of graded differential linear logic. In characteristic 0, we have $\Gamma_{n}(A) \cong S_{n}(A)$. Nondegenerated models of such a logic in a category of vector spaces would thus
be only in the case where the field is of positive characteristic.
Divided power algebras were originally introduced in
Their theory was further developed in Pierre Berthelot‘s PhD thesis (in the context of crystalline cohomology), which was later published as:
• Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique $p \gt 0$, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin, 1974. (doi:10.1007/BFb0068636, MR 0384804)
Recent works on divided power algebras include:
• Sacha Ikonicoff?, Divided power algebras over an operad, Glasgow Math. J. 62 (2020) 477-517, doi:10.1017/S0017089519000223, pdf
• Sacha Ikonicoff?, Divided power algebras and distributive laws, 2021, doi:10.48550/arXiv.2104.11736, pdf
It is related to differential categories in:
On divided powers:
• Luis Narváez Macarro?, Hasse-Schmidt derivations, divided powers and differential smoothness, 2009, doi:10.5802/aif.2513, pdf
See also:
In relation to the sphere spectrum | {"url":"https://ncatlab.org/nlab/show/divided+power+algebra","timestamp":"2024-11-11T18:07:07Z","content_type":"application/xhtml+xml","content_length":"74439","record_id":"<urn:uuid:40cb343c-9c1a-4e72-9f3a-a9614a09aaa8>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00448.warc.gz"} |
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Finding All Eulerian Paths
I just went through the material of the first week of Combinatorial Mathematics, which is offered by Tsinghua University through EdX. My initial impression is that this is a mildly engaging course,
marred by subpar presentation. Thus, I have a hard time recommending it in its current iteration.
One of the assignments in the first week reminded me of a program I posted some time ago. In Finding an Eulerian Path I present a somewhat verbose program for finding an Eulerian path in a graph of
which it is known that it contains an Eulerian path. On the other hand, the assignment in that EdX course asks for the total number of Eulerian paths of this undirected multigraph:
That’s not a particularly challenging question. It was more fun for me to write a neat Haskell program to solve this task. However, instead of just counting all Eulerian paths, my program finds all
of them. The code, which is given below, is quite clean and should be easy to follow. Let me describe the algorithm I’ve implemented in plain English, though.
For all nodes in the graph, the program finds all Eulerian paths starting from that node. The relevant part of the program at this step is the function call “findPath’ [(“”, node, g)] []”. When you
set out to find all Eulerian paths, the string indicating the current path is empty. As the graph is traversed, that string grows. Two cases are possible. First, we could have reached a dead-end,
meaning that there are untraversed edges that can’t be reached from the current node, due to the path that was chosen. Second, if there are reachable nodes left, then the traversal continues by using
any of the unused edges.
This can be modelled in a rather intuitive functional style with a list containing elements of the structure “(Path, Node, [Edge])”. Start by taking the head of that list, and figure out whether
there are any edges left to take from that position. If so, then discard the head, and append all possible paths to the list. In the given example, starting at node 1, the current path is the empty
string, and the list of edges contains the entire graph. Subsequently, three elements are added to the list, with the respective paths “a”, “c” and “f”, and a list of remaining edges of which the
edge that was just chosen was removed from. This goes on, recursively, until the entire graph has been processed. Of course, if there are no edges left to traverse, we’ve found an Eulerian path.
With this description, the program below should be straightforward to follow. To run it, with a graph modelled after the image above, load the code into GHCi, and execute it by typing “findPaths
graph nodes”.
import Data.List
type Node = Int
type Edge = (Char, (Int, Int))
type Graph = [Edge]
type Path = String
type Candidate = (Path, Node, [Edge])
graph :: Graph
graph = [('a', (1,2)),
('b', (2,3)),
('c', (1,3)),
('d', (3,4)),
('e', (3,4)),
('f', (1,4))]
nodes :: [Node]
nodes = [1..4]
findPaths :: Graph -> [Node] -> [Path]
findPaths g ns = findPaths' g ns []
findPaths' :: Graph -> [Node] -> [Path] -> [Path]
findPaths' _ [] acc = acc
findPaths' g (n:ns) acc = findPaths' g ns acc'
where acc' = findPath g n ++ acc
findPath :: Graph -> Node -> [Path]
findPath g node = findPath' [("", node, g)] []
findPath' :: [Candidate] -> [Path] -> [Path]
findPath' [] acc = acc
findPath' ((path, _, []):xs) acc = findPath' xs (path:acc)
findPath' ((path, node, es):xs) acc
| null nextEdges = findPath' xs acc -- dead-end, discard!
| otherwise = findPath' (xs' ++ xs) acc
where nextEdges = filter (\(_,(a, b)) -> a == node || b == node) es
xs' = nextPaths (path, node, es) nextEdges []
nextPaths :: Candidate -> [Edge] -> [Candidate] -> [Candidate]
nextPaths _ [] acc = acc
nextPaths (path, node, es) (x:xs) acc = nextPaths (path, node, es) xs acc'
where acc' = (path', node', delete x es ) : acc
path' = path ++ [label]
node' = if node == a then b else a
(label, (a, b)) = x
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Procedure for Vibration Analysis
Introduction: Procedure for Vibration Analysis
Like many engineering problems, the analysis of a vibration problem is typically carried out in a series of logical steps.
Mathematical Modeling
Most real vibrational systems are hopelessly complex and it would be impossible to consider all of the details of the problem. As a result, we try to simplify the problem as much as possible while
retaining all of the important and relevant features. For example, we often represent solid bodies as being rigid (they’re not), consider springs as linear and massless (they’re not), ignore damping,
etc. The purpose is to make the system as simple as possible to analyze while still retaining all of the important features of the original problem. Often we start with an overly simple model to
understand the basics of a problem, and then add complexities as required to more accurately represent the quantities of interest. A good rule of thumb is to use the simplest possible model which
adequately captures the behaviour of interest.
All mechanical vibrational systems contain at a minimum a means to store potential energy (a spring) and a means to store kinetic energy (a mass). A real system will also have some means of
dissipating energy (friction, viscous damper, etc.).
Derivation of Governing Equations
Once the mathematical model is available, we use it to derive the governing equations of motion. Typically this involves drawing Free Body Diagrams and Mass-Acceleration Diagrams (FBD/MAD) of various
components of the system and applying Newton’s Laws. However, there are other principles that can be used to obtain the desired equations which may apply in certain cases:
• Conservation of energy,
• Influence coefficients,
• D’Alembert’s Principle and Lagrange’s Equations,
• Many others.
For discrete systems we usually obtain second order ordinary differential equations. For continuous systems we generally have partial differential equations. These equations may be linear or
nonlinear depending on the model used. If they are nonlinear we may choose to linearize them (by limiting the amplitudes of motion to be small for example). This is another approximation introduced
into the analysis.
Solution of Governing Equations
Once the equations of motion have been obtained, we must solve them to find the response of the system. Depending on the type of equation there are many solution methods possible:
□ Standard solutions procedure for differential equations (generally only applicable for linear equations),
□ Laplace transforms,
□ Matrix methods (modal analysis),
□ Numerical solutions (finite element method).
For nonlinear problems, typically numerical solutions are used.
Interpretation of Results
Once the equations have been solved the last (and most important) step is to interpret the results in the context of the real physical situation. It is important to be clear about what the goal of
the original analysis was and also about the effects of all of the simplifying assumptions and approximations that were necessarily made. | {"url":"https://engcourses-uofa.ca/books/vibrations-and-sound/introduction/procedure-for-vibration-analysis/","timestamp":"2024-11-14T18:37:44Z","content_type":"text/html","content_length":"37481","record_id":"<urn:uuid:e8782aeb-b333-44a3-8bed-83fa2273b12f>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00371.warc.gz"} |
On Space-Time Quasiconcave Solutions of the Heat Equationsearch
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On Space-Time Quasiconcave Solutions of the Heat Equation
Xinan Ma : University of Science and Technology of China, Hefei, China
Softcover ISBN: 978-1-4704-3524-0
Product Code: MEMO/259/1244
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AMS Member Price: $48.60
eBook ISBN: 978-1-4704-5243-8
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eBook: ISBN: 978-1-4704-5243-8
Product Code: MEMO/259/1244.B
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AMS Member Price: $97.20 $72.90
Click above image for expanded view
On Space-Time Quasiconcave Solutions of the Heat Equation
Xinan Ma : University of Science and Technology of China, Hefei, China
Softcover ISBN: 978-1-4704-3524-0
Product Code: MEMO/259/1244
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN: 978-1-4704-5243-8
Product Code: MEMO/259/1244.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN: 978-1-4704-3524-0
eBook ISBN: 978-1-4704-5243-8
Product Code: MEMO/259/1244.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
• Memoirs of the American Mathematical Society
Volume: 259; 2019; 83 pp
MSC: Primary 35
In this paper the authors first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing
this constant rank theorem, they obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To
explain their ideas and for completeness, the authors also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second
fundamental form of the convex level sets for harmonic function.
□ Chapters
□ 1. Introduction
□ 2. Basic definitions and the Constant Rank Theorem technique
□ 3. A microscopic space-time Convexity Principle for space-time level sets
□ 4. The Strict Convexity of Space-time Level Sets
□ 5. Appendix: the proof in dimension $n=2$
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Requests
Volume: 259; 2019; 83 pp
MSC: Primary 35
In this paper the authors first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this
constant rank theorem, they obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain
their ideas and for completeness, the authors also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental
form of the convex level sets for harmonic function.
• Chapters
• 1. Introduction
• 2. Basic definitions and the Constant Rank Theorem technique
• 3. A microscopic space-time Convexity Principle for space-time level sets
• 4. The Strict Convexity of Space-time Level Sets
• 5. Appendix: the proof in dimension $n=2$
Permission – for use of book, eBook, or Journal content
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How to calculate how much mulch you will need.
Submitted by Rocks 'n' Roots on
Rocks 'n' Roots Blog - Mulch Calculation
Q. How to calculate how much mulch you will need:
A. All mulch covers approximately 120 square feet at 2". Of course you can install it thicker too, but 2" is a very good coverage. To calculate how much mulch you will need measure each area's
length and width. Then multiply length times width of each area and add all the answers together. Then divide the answer by 120 to tell you how many yards you will need. You may also want to put a
fabric over the soil before you install the mulch to prevent the weeds from growing while allowing the water and fertilizer to flow through. If you do not want to put down a fabric, another idea is
to sprinkle Preen over the installed mulch which will act as a barrier against the weeds.
Example: In the measurement below if you divide the 460 square feet by 120 you will need 3.84 yards. You then round it up or down to the nearest 1/2 yard. In this case that would be 4 yards.
Submitted by Barbara Zendt | {"url":"https://rocksnroots.com/blog/14/06/03/how-calculate-how-much-mulch-you-will-need","timestamp":"2024-11-14T15:43:01Z","content_type":"text/html","content_length":"20559","record_id":"<urn:uuid:779a7ac7-bd6e-417e-beba-1583de0c4d36>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00265.warc.gz"} |
On a problem of El-Zahar and Erdős
Two subgraphs A,B of a graph G are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large
chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there
should be two anticomplete subgraphs both with large minimum degree, and that is one of our results. We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to
have large chromatic number: that is, for all t,c≥1 there exists d≥1 such that if G has chromatic number at least d, and does not contain the complete graph K[t] as a subgraph, then there are
anticomplete subgraphs A,B, where A has minimum degree at least c and B has chromatic number at least c. Second, we look at what happens if we replace the hypothesis that G has sufficiently large
chromatic number with the hypothesis that G has sufficiently large minimum degree. This, together with excluding K[t], is not enough to guarantee two anticomplete subgraphs both with large minimum
degree; but it works if instead of excluding K[t] we exclude the complete bipartite graph K[t,t]. More exactly: for all t,c≥1 there exists d≥1 such that if G has minimum degree at least d, and does
not contain the complete bipartite graph K[t,t] as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least c.
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics
• Chromatic number
• Subgraphs
Dive into the research topics of 'On a problem of El-Zahar and Erdős'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/on-a-problem-of-el-zahar-and-erd%C5%91s","timestamp":"2024-11-13T15:21:27Z","content_type":"text/html","content_length":"51828","record_id":"<urn:uuid:733daf25-b47f-4906-a057-af8a267e4732>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00816.warc.gz"} |
Shapes of the Beechnut
Many people love beechnuts and many people love geometry. But did you know you can experience them together? The beechnut and geometry. I have noticed that the shapes inside the beechnut resemble
very closely the beginning constructions of geometry patterns.
The geometry of seeds
I love collecting seeds from trees. They always display such great shapes and numbers. I’m always looking for the patterns in them. Conkers and Acorns are beautiful circles of nature and one of my
favourites is the five pointed star of the eucalyptus seed pod.
The Beechnut and Geometry
the open seed casing of the beechnut houses 2 tetrahedral shape seeds
The Beechnut and its geometry was pure surprise and wonder to me when I spotted it. I hope you enjoy it just as much.
Each beechnut seed case splits open by four slits down the sides, the outer case spreads open into four petals. In the centre sit two triangular seeds. Each of these is a tall tetrahedral shape. The
base (where it is fixed to the case) is a triangle shape and it has three more tall triangular sides giving it a structure that resembles a tall tetrahedron. There are two of these in each case.
The illustration above shows the beechnut and geometry. Its opened case shows the two triangle-shape spaces where the seeds sit. The insert diagrams are the bases of the seeds and the top view of one
seed showing the triform structure
The four-sided case gives way to two triangles which sit side by side. The beechnut (and geometry) display the properties of FOUR in two dimensions (i.e. the square) and in three dimensions (the
shutterstock image
The Symmetry of Four
There is balance and symmetry in Four. Thats what the square and the number four represent. This is evident in the symmetry of the beechnut too.
There are many natural fours
Remember our Natural Fours from the Smart Happy Autumn Magazine?
geometry of the seeds inside the beechnut
The early constructions in the practice of geometry resemble the structure inside the beechnut, I love this!
Have a look at this article on the shapes of Autumn if you’d like to find out about other seeds shapes
Constructing the beechnut and geometry
Follow these steps below to draw the simple geometric shape.
Using a compass, draw a circle with any radius.
Move the point of the compass (keeping the same radius) to any position on the circle just drawn. This will be the centre of the second circle. Complete the second circle.
You now have two circles, each circle passes through the centre of the other. The space where the two circles overlap is commonly referred to as the ‘Vesica Piscis’ or almond shape. This shape holds
much symbolism in art, maths and philosophy and has been reproduced in many forms through history, representing unity within duality. It is an important theme in sacred geometry. It is within this
space that we will continue our geometric construction.
Using any straight edge, draw a line that joins the two centres of each circle.
From each end of that line, draw another line using the straight edge that connects to the point where the circles intersect at the centre top.
And another two lines that connect the ends of the first line to the point the circles intersect at the bottom.
Using a darker pencil and a steady hand draw over the arcs that make the almond shape to highlight them.
You have created two equilateral triangles within the Vesica Piscis without using a ruler or protractor.
Do you recognise this shape?
Ok, so it’s not absolutely perfect, but there is a definite similarity there.
Extend your Beechnut and Geometry construction into 3D
And, if you keep going with your geometric construction you can create a 3D tetrahedral shape, similar to those inside we can see inside the beechnut case.
Go back to your drawing and extend the straight line that comes down from the top intersection through the centre point of the right hand circle. Extend it down until it crosses the circle at the
Do the same with the line that creates the left hand side of the top triangle. Extend it down until it crosses the circle at the base of the drawing.
Now, draw a straight line that joins up the points where these new lines cross the circles at the bottom of the drawing. This new base line will pass through the bottom intersection of the circles
and the bottom point of the inverted triangle.
In total you have now drawn four equally sized equilateral triangles and one large triangle that holds the four smaller ones. All with only a compass and straightedge. Well done.
These four small triangles become the four sides of a tetrahedron.
As in the picture here, cut the paper along the red lines. Sketch out tabs for attaching together and cut around these too , these are the red dotted lines.
Fold along the blue lines that are the centre triangle which will become the base. The three surrounding triangles will close up to meet at the top
Use a glue-stick on the tabs and fold them inside the structure.
This is your Tetrahedron. It is a pyramid with a triangle base. (other pyramids have square bases but they are not called tetrahedrons). A tetrahedron is the first shape that can be created in three
dimensional space with the least amount of edges, points and sides. If you go back to the seeds that you found inside the beechnut case, you may notice it as similar structure to the tetrahedron. It
is a bit taller and maybe a bit squished and probably not perfect – it is natural after all. But I think you may notice the same properties of the structure.
I think it is interesting that in our observation and exploration of the beechnut and geometry we have arrived at creating one of the Platonic solids and not once did we measure an angle or use any
calculations, we just enjoyed studying nature.
I hope you enjoy that too. | {"url":"https://thesmarthappyproject.com/beechnut-and-geometry/","timestamp":"2024-11-10T09:23:21Z","content_type":"text/html","content_length":"76220","record_id":"<urn:uuid:8d1ac40a-007e-43ce-b71e-5e80893c62dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00543.warc.gz"} |
Chamfer Diameter Calculator - Online Calculators
Enter the values in required field to use our basic and advanced chamfer diameter calculator. Additionally, read the formula and solved examples below to gain better understanding.
Chamfer Diameter Calculator
Enter any 2 values to calculate the missing variable
The Chamfer Diameter Calculator is important when it comes to accurate measurement in manufacturing and engineering design. Our calculator lets you to measure chamfer diameter in length, radius and
size by using the accurate formula. Moreover, in design engineering, chamfer angle calculator and countersink diameter calculators are also used in calculating the diameter in required figures.
The formula is:
$\text{CD} = D – 2C$
Variable Meaning
CD Chamfer Diameter (reduced diameter after chamfering)
D Original Diameter (before chamfering)
C Chamfer size (the width of the chamfer from the edge)
Solved Examples:
Example 1:
• Original Diameter ($D$) = 10 mm
• Chamfer size ($C$) = 1 mm
Calculation Instructions
Step 1: CD = $D – 2$ Start with the formula.
Step 2: CD = $10 – 2(1)$ Replace with 10 mm and with 1 mm.
Step 3: CD = $10 – 2$ Multiply $2 \times 1$ to get 2 mm.
Step 4: CD = 8 mm Subtract 2 mm from 10 mm to get the chamfer diameter.
Answer: The chamfer diameter is 8 mm.
Example 2:
• Original Diameter () = 20 mm
• Chamfer size () = 2 mm
Calculation Instructions
Step 1: CD = $D –$ Start with the formula.
Step 2: CD = $20 – 2(2)$ Replace with 20 mm and $C$ with 2 mm.
Step 3: CD = $20 – 4$ Multiply $2 \times 2$ to get 4 mm.
Step 4: CD = 16 mm Subtract 4 mm from 20 mm to get the chamfer diameter.
Answer: The chamfer diameter is 16 mm.
What is Chamfer Diameter Calculator ?
The Chamfer Diameter Calculator is very helpful that lets you to calculate the effective diameter of a part after a chamfer has been applied to it. In simple terms, the chamfer is a beveled edge that
reduces sharpness and provides a smooth transition between two surfaces. It is often used in machining to remove sharp corners or edges.
The formula $\text{CD} = D – 2C$ calculates the diameter after the chamfer is applied by subtracting twice the chamfer size from the original diameter. It makes sure that the reduction in diameter
due to the chamfer is accurately accounted for in the design or machining process. | {"url":"https://areacalculators.com/chamfer-diameter-calculator/","timestamp":"2024-11-04T00:48:56Z","content_type":"text/html","content_length":"111132","record_id":"<urn:uuid:c38567c9-5b59-47bf-993b-6717f01959c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00136.warc.gz"} |
Math, Grade 6, Ratios, Expressing Ratios
Four Schools Card Sort
Work Time
Four Schools Card Sort
Four high schools (A, B, C, and D) have different numbers of students and different ratios of boys to girls.
• Working with a partner, take turns matching cards that represent the same school.
• Explain to your partner how you know the cards match.
• Your partner should either agree with your explanation or challenge it if your explanation is not correct, clear, or complete.
To match the cards, find a strategy that will help you narrow down the choices. For example, you might start by choosing a school in which the ratio of boys to girls is easy for you to see. Then find
all the cards that match that school. | {"url":"https://openspace.infohio.org/courseware/lesson/2077/student/?section=3","timestamp":"2024-11-13T11:09:59Z","content_type":"text/html","content_length":"33731","record_id":"<urn:uuid:3e719e3e-716e-44de-987a-f40b130ac4d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00694.warc.gz"} |
Math curse download for free
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algebraic expression building. This is math curse by jessica carlton on vimeo, the home for high quality videos and the people who love them. Includes questions from the book for students to answer.
Buy math curse by jon scieszka, lane smith illustrator online at alibris. Aug 20, 20 we read the math curse on the first day of school. Match 3 games 100% free match 3 games download gametop. Click
download or read online button to get math curse book now. Free pc games the most popular free games for your pc. This lesson uses the four modalities of reading reading, writing, listening, and
speaking on a math word problem to bridge the gap between reading and math.
I suspect i will only require my students to create two math curse problems even though i have three. Enter your mobile number or email address below and well send you a link to download the free
kindle app. This is normally done during the junior year or the first semester of the senior year. Give students time to complete the crossword puzzle. Therefore it need a free signup process to
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Students create a 2 page layout side by side that could have been added to the book, including an illustration, that uses the current math learned in class. For more online courses, visit our
complete collection of free courses online. The first edition of the novel was published in january 1st 1992, and was written by jon scieszka. Fibonacci rearranges the pdf free 4th grade lessons with
math curse free download 4th grade lessons with math curse pdf book 4th grade lessons with math curse download ebook 4th grade lessons. All our match 3 games are 100% unlimited full version games
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Ebook math 30 days wonder as pdf download portable. Read other math books of your choice, interspersed with the other activities. Math curse is a childrens picture book written by jon scieszka and
illustrated by lane smith, suitable for ages six through ninetynine years. I want to have my students start to think about all the different places math shows up in their world so i read the story,
math curse by jon scieszka. The literature also gives the opportunity to integrate math vocabulary from several sources such as math text. The main characters of this science, mathematics story are.
You have 10 things to do, but only 30 minutes until your bus leaves. Hand out the printout of the math curse word problems crossword puzzle from the online crossword puzzles tool or bring students to
the computer lab to work on it online. As a followup activity, students must create their own math curse word problem. Students can solve math problems from the book or write and solve their own
story problems making these printable sheets great for second. Find a library or download libby an app by overdrive. Math bee for kidsmaths bee for kids is a simple and. Numeric and symbolic
calculations in libreoffice and openoffice writer. Here is a set of 3 pages of problem solving math problems based on the book by jon scieszka. I hope you find these resources helpful, and you enjoy
reading math.
Microsoft mathematics provides a graphing calculator that plots in 2d and 3d, stepbystep equation solving, and useful tools to help students with math and science studies. Using math curse by jon
scieszka and lane smith students have to link the text to themselves by evaluating how math is a part of their everyday lives with this project. It has been a busy end to the quarter in grade math.
This item includes a step by step guide for students about what the expectations for the word problem is. Math curse by jon scieszka is a great read aloud for the beginning of the year in math. Some
of the questions in the story are included just for fun and are impossible to answer, so listen very carefully.
When picking pages, they can also use the front and back cover and the inside covers and book sleeve there is a lot of math there too. These games have no violence, no empty action, just a lot of
challenges that will make you forget youre getting a mental workout. If the product of my aunts and uncles is 48, and i. Math curse answer the following questions while listening to mr. One morning a
little girl wakes up to find everything in life arranging itself into a math problem, and she must find her way out of the math curse. The book was published in multiple languages including english.
The teacher tells her class that they can think of almost everything as a math problem. Its a fun way to get kids to think about the math they encounter everyday.
Welcome,you are looking at books for reading, the math curse, you will able to read or download in pdf or epub books and notice some of author may have lock the live reading for some of country. Math
curse by jon scieszka, lane smith illustrator alibris. A revolutionary, online math program comprising games, animated books and downloadable materials. Adds mathematical commands for ingame
computations. Have the kids then write out their own math curses to share with the group. The book has been awarded with texas bluebonnet award. The first edition of the novel was published in
october 1st 1995, and was written by jon scieszka. Your students will connect with solving math problems and the read aloud which will help break the math curse. Pdf math curse by jon scieszka, lane
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Mathematics mit opencourseware free online course materials. For more online courses, visit our complete collection of. Undergraduates seriously interested in mathematics are encouraged to elect an
upperlevel mathematics seminar. The book was published in multiple languages including english, consists of 32 pages and is available in hardcover format. Find math curse lesson plans and teaching | {"url":"https://granehrehe.web.app/424.html","timestamp":"2024-11-10T11:24:10Z","content_type":"text/html","content_length":"15873","record_id":"<urn:uuid:119c2e3b-3113-4953-8556-a3a5935bf745>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00482.warc.gz"} |
Sacramento River - Four Rivers Index, CA Update
The Sacramento River and its tributaries (the Feather, Yuba, and American Rivers), together comprising the "Sacramento Four Rivers Index", represent the primary input into the California State Water
Project, operated by the California Department of Water Resources (CADWR). In 2013-2014, David Meko and Ramzi Touchan (University of Arizona Laboratory of Tree-Ring Research) updated the original
reconstruction of the Sacramento River, Four Rivers Index (900-2012), for the CADWR. New collections expanded the tree-ring network and allowed an extension to 2012. Meko and Touchan also developed
reconstructions for the Sacramento River at Bend Bridge, Feather River inflow to Lake Oroville, Yuba River at Smartville, and American River inflow to Folsom Lake as part of this project.
Klamath/San Joaquin/Sacramento Hydroclimatic Reconstructions, Final Report to CADWR from Tree Rings
Calibration & Validation
Water-year-total flows for the Sacramento Four Rivers were reconstructed by locally weighted regression, or Loess from subsets of 61 chronologies screened for Sacramento/San Joaquin basin
reconstructions. A Loess reconstruction was defined as an interpolation of estimated flow from a smoothed scatterplot of observed flow on a single summary tree-ring variable. The tree-ring predictor
was an average of standard chronologies that were filtered and scaled to accentuate their statistical signal for the target flow gage. A time-nested-modeling approach was used for reconstructions,
using progressively longer but smaller subsets of chronologies going back in time to calibrate subsets of reconstrutions. The percentage of flow variance accounted for by the “median-accuracy” model
ranges from 68% for the Sacramento River (Sacramento River above Bend Bridge) to 78% for the San Joaquin River. Because these reconstructions are done with time-nested models, accuracy varies over
time depending on the quality of the available tree-ring chronologies.
Statistic Calibration Validation
Explained variance (R2) 0.73
Reduction of Error (RE) 0.73
Standard Error of the Estimate 3937 KAF
Root Mean Square Error (RMSE) 4238 KAF
Note: the calibration and validation statistics above were computed during the model development and reflect the relationship between the log-transformed observed flows and the tree-ring predictors.
The scatterplot below in Figure 1 shows the relationship between the back-transformed observed flows and the reconstructed flows.
(For explanations of these statistics, see this document (PDF), and also the Reconstruction Case Study page.)
Figure 1. Scatter plot of observed and reconstructed Sacramento River annual flow, 1906-2011.
Figure 2. Observed (black) and reconstructed (blue) annual Sacramento River annual flow, 1906-2011. The observed mean is illustrated by the dashed line.
Long-Term Reconstruction
Figure 3. Reconstructed annual flow for the Sacramento River flow (900-2012) is shown in blue. Observed flow is shown in gray and the long-term reconstructed mean is shown by the dashed line.
Figure 4. The 10-year running mean (plotted on final year) of reconstructed Sacramento River flow, 900-2012. Reconstructed values are shown in blue and observed values are shown in gray. The
long-term reconstructed mean is shown by the dashed line. | {"url":"https://www.treeflow.info/content/sacramento-river-four-rivers-index-ca-update","timestamp":"2024-11-11T23:39:11Z","content_type":"text/html","content_length":"59333","record_id":"<urn:uuid:2bfe18c6-2f68-48af-bff2-5b673d479ff5>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00197.warc.gz"} |
Permanent Fault Identification Method for Single-Phasea Adaptive Reclosure of UHVAC Transmission Line
Permanent Fault Identification Method for Single-Phasea Adaptive Reclosure of UHVAC Transmission Line ()
1. Introduction
Most faults in UHVAC transmission system are single phase transient fault [1]. In the traditional automatic re-lock, if the coincidence of permanent fault, the system will cause the two shock, and
even make the system crash. In 1980s, Professor Ge Yaozhong put forward the idea of “adaptive reclosure” [2] arousing wide attention of experts and scholars in electrical engineering from domestic
and foreign. A wealth of achievements have been made in the study of the two arc characteristics [3], the voltage characteristics [4] [5], the current characteristics of shunt reactor [6] [7] [8] and
the characteristic of model parameters [9] [10]. The practical application is difficult since neural network based on the need to train a large number of samples [11] [12]. The criterion based on arc
criterion and voltage is difficult to realize since the shunt reactor technology is widely used in the super high voltage, which accelerates the arc quenching process and limits the amplitude of the
fault voltage [13].
Based on the analysis of the characteristic of single-phase permanent fault phase voltage after tripping ,this paper proposes method for distinguishing single phase permanent fault based on
steady-state component frequency acquired through two order derivative of the fault phase voltage signal and the ratio of the original signal.
2. Analysis on the Characteristics of Fault Phase Voltage during Single-Phase Permanent Fault after Tripping
During the single-phase permanent fault, fault phase voltages
In the formula:
Due to the fault point to ground reliable discharge, the transient component will decay rapidly to zero, after entering the steady state, its expression is:
After two order derivative of the formula (2):
and then
So the steady state component frequency f is:
Because the steady state component is mainly determined by the sound phase capacitance coupling voltage and the electromagnetic coupling voltage [14], the steady state frequency f is close to the
frequency of the power frequency f[0]. Based on the above analysis, the relations between steady state component frequency and frequency as following:
In the formula, k represents reliability coefficient. After a lot of simulation, the author found that the 1.3 is suitable in considering the line model equivalence and simplification of the
simulation software, actual gap between f and
3. Discriminant Principle
In the sine function, the data in the 1/4 continuous period can reflect the data of the whole cycle. Two arc durations are about 200 ms during transient fault [15]. Considering the above two aspects,
this paper focuses on 200 ms time period after trip and calculates the data from power system, if calculate data in 1/4 continuous power frequency period satisfied formula of (5), the fault is
determined as a permanent fault. Otherwise, it will be judged as instantaneous fault. Criteria flow chart is shown in Figure 1.
4. Simulation Results and Analysis
As shown in Figure 2, the simulation model is based on the model of 1000 KV UHV line system in the southeast Nanyang. Line length is 358 km. Parameters of this line are as following:
Figure 2. Model of 1000 KV UHV transmission line system in Southeast Nanyang.
System parameters at both ends are:
Parameters of shunt reactor:
Parameters of neutral point small reactor:
Permanent single-phase grounding fault occurs of system in 0.5 s, tripping in 0.1 s, and the sampling frequency is 10 kHz. The calculation results show that the power angular phase difference are 0˚,
10˚, 20˚, 30˚, 40˚, 50˚ respectively, the transition resistances are corresponding to
Table 1 and Table 2 are discriminant success time when
From Table 1 we can make conclusion that when the power angle difference and transition resistance are constant, discriminant success time showing such a regularity that increase first and then
decrease with increasing of L. When the power angle difference and the fault location is certain, discriminant success time shows decreasing trend with increasing of R. From Table 2 we can also seen
that when the fault position and the transition resistance are at a certain time, discriminant success time showing such a regularity that increase first and then decrease with increasing of θ.
By a lot of simulation data can be seen that in the UHV AC transmission system in the occurrence of a permanent fault occurs, the duration of transient component is 180 ms - 55 ms after tripping.
4. Conclusions
In this paper, based on the analysis of fault phase voltage characteristics of single-phase permanent fault, this paper presents a method to determine the frequency of steady state component based on
the ratio of the two derivative of the
Table 1. Criterion flow chart successful time table when
Table 2. Successful time table when
fault phase voltage and the ratio of the fault phase. The method is simple, high reliability and strong adaptability, and a lot of simulation results verify that the proposed criterion is also
suitable for 500 kV ultra high voltage transmission line.
The deficiency of this criterion is that:
1) Although this criterion can accurately identify the fault, the discriminant success time affected by transition resistance relatively large; 2) Due to the use of the ratio method, the denominator
(fault phase voltage) may be zero, but not appear in the simulation. | {"url":"https://scirp.org/journal/paperinformation?paperid=75248","timestamp":"2024-11-09T19:49:57Z","content_type":"application/xhtml+xml","content_length":"98247","record_id":"<urn:uuid:eee6264f-3891-4398-9de2-392af0db11f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00713.warc.gz"} |
I’m confused. But with math, I frequently am. You see, Matte and I were recently in Costco looking for healthy alternatives, and we came upon these frozen Kirkland ground sirloin burgers that are
only 15% fat. Awesome!
Of course we grabbed a bag, and I immediately turned it over to check the nutritional info. And am I glad I did!
(Insert cartoonized version of me shaking my head violently, with aoiy-aoiy-aoiy soundtrack) WHAT?!? I almost always got Ds in math, maybe a C here and there, but even I know that a 330-calorie item
with twenty-three grams of fat does NOT equal 15% fat. It is 60% fat. And with 25% of its calories coming from saturated fat alone, that’s nearly half a person’s daily allowance!
If something sounds too good to be true, it probably is. So when you see a food product boasting low-fattedness, double-check the nutritional info.
And WTF, Costco?
3 people have roominated about “Fuzzy math”
• That’s a bit shocking indeed. The Sherlock Holmes in me kicked in and I believe this is where they get the 15% Fat.
One patty is 151g and total fat is 23g which is 15% of 151. So, what they really mean is that 15% of the burger patty itself is made from fat, and not that the % of fat in the meat itself is 15%
which is how the calories are calculated. The packaging is very vague, and misleading.
I hate how the beverage companies say “Oh it’s only 100 calories”…but that’s per serving and their bottle is actually 2 to 2-1/2 servings. Visually we assume one bottle is one serving because it
looks small enough to look like one serving. That’s why I got more vigilant about checking the #of servings in the packaging. Those trickster marketers.
btw. hope you had a great weekend 🙂
• Stephanie is right. The numbers on the package are based on weight, not on calories. All hamburger sold – at least in my state – has to have the fat percentage on the package.
• Thanks to Lisa and Stephanie for the explanation! I had no idea! But I have learned to also check the number of servings because that’s where you can really get screwed up. Once I showed Ed how
to REALLY read a label, he freaked out! He thought if he ate the whole bag of chips in one sitting, that was one serving! Poor little guy – hated bursting his balloon like that.
roominate on this yourself | {"url":"https://www.catheroo.com/2009/01/18/fuzzy-math/","timestamp":"2024-11-11T03:28:16Z","content_type":"application/xhtml+xml","content_length":"31664","record_id":"<urn:uuid:debf5352-9d4c-47e6-95e1-bf0fc3103c49>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00384.warc.gz"} |
There has been a series of posts at WUWT by Andy May on SST averaging, initially comparing HADSST with ERSST. They are
, and
. Naturally I have been involved in the discussions; so has
John Kennedy
. There has also been
Twitter discussion
.My initial comment was
"Just another in an endless series of why you should never average absolute temperatures. They are too inhomogeneous, and you are at the mercy of however your sample worked out. Just don’t do it.
Take anomalies first. They are much more homogeneous, and all the stuff about masks and missing grids won’t matter. That is what every sensible scientist does.
The trend was toward HADSST and a claim that SST had been rather substantially declining this century (based on that flaky averaging of absolute temperatures). It was noted that ERSST does not show
the same thing. The reason is that HADSST has missing data, while ERSST interpolates. The problem is mainly due to that interaction of missing cells with the inhomogeneity of T.
Here is one of Andy's graphs:
In these circumstances I usually repeat the calculation that was done replacing the time varying data with some fixed average for each location to show that you get the same claimed pattern. It seems
to me obvious that if unchanging data can produce that trend, then the trend is not due to any climate change (there is none) but to the range of locations included in each average, which is the only
thing that varies. However at WUWT one meets an avalanche of irrelevancies - maybe the base period had some special property, or maybe it isn't well enough known, the data is manipulated etc etc. I
think this is silly, because they key fact is that some set of unchanging temperatures did produce that pattern. So you certainly can't claim that it must have been due to climate change. I set out
that in a comment
, with this graph:
Here A is Andy's average, An is the anomaly average, and Ae is the average made from the fixed base (1961-90) values. Although no individual location in Ae is changing, it descends even faster than
So I tried another tack. Using base values is the simplest way to see it, but one can just do a partition of the original arithmetic, and along the way find a useful way of showing the components of
the average that Andy is calculating. I set out a first rendition of that
. I'll expand on that here, with a more systematic notation and some tables. For simplicity, I will omit area weighting of cells, as Andy did for the early posts.
Breakdown of the anomaly difference between 2001 and 2018
Consider three subsets of the cell/month entries (cemos):
• Ra is the set with data in both 2001 and 2018 (Na cemos)
• Rb is the set with data in 2001 but not in 2018 (Nb cemos)
• Rc is the set with data in 2018 but not in 2001 (Nc cemos)
I'll mark sums S of cemo data with a 1 for 2001, 2 for 2018, and an a,b,c if they are sums for a subset. I use a similar notation for averages with A plus suffices. I'll set out the notation and some
values in a table:
Set data N Weights 2001 2018
2001 or 2018 N=18229 S1, A1=S1/(Na+Nb)=19.029 S2,A2=S2/(Na+Nc)=18.216
Ra 2001 and 2018 Na=15026 Wa=Na/N=0.824 S1a, A1a=S1a/Na=19.61 S2a, A2a=S2a/Na=19.863
Rb 2001 but not 2018 Nb=1023 Wb=Nb/N=0.056 S1b, A1b=S1b/Nb=10.52 S2b=0
Rc 2018 but not 2001 Nc=2010 Wc=Nc/N=0.120 S1c=0 S2b, A2b=S2b/Nb=6.865
I haven't given values for the sums S, but you can work them out from the A and N. The point is that they are additive, and this can be used to form Andy's A2-A1 as a weighted sum of the other
averages. From additive S:
S1=S1a+S1b and S2=S2a+S2c
A1*(Na+Nb)=A1a*Na+A1b*Nb, or
and similarly
or, dividing by N
That expresses A2-A1 as the weighted sum of three terms relating to Ra, Rb and Rc respectively. Looking at these individually
• (A2a-A1a)=0.253 are the differences between the data points known for both years. They are the meaningful change measures, and give a positive result
• (A1b-A2)=-7.696. The 2001 readings in Rb have no counterpart in 2018, and so no information about increment. Instead they appear as the difference with the 2018 average A2. This isn't a climate
change difference, but just reflects whether the points in R2 were from warm or cool places/seasons.
• (A2b-A1)=12.164. Likewise these Rc readings in 2018 have no balance in 2001, and just appear relative to overall A1.
Note that the second and third terms are not related to CC increases and are large, although this is ameliorated by their smallish weighting. The overall sum that, with weights, makes up the
difference is
A2-A1 = 0.210 + 0.431 -1.455 = -0.813
So the first term representing actual changes is overwhelmed by the other two, which are biases caused by the changing cell population. This turns a small increase into a large decrease.
So why do anomalies help
I'll form anomalies by subtracting from each cemo the 2001-2018 mean for that cemo (chosen to ensure all N cemo's have data there). The resulting table has the same form, but very different numbers:
Set data N Weights 2001 2018
2001 or 2018 N=18229 S1, A1=S1/(Na+Nb)=-.116 S2,A2=S2/(Na+Nc)=0.136
Ra 2001 and 2018 Na=15026 Wa=Na/N=0.824 S1a, A1a=S1a/Na=-0.118 S2a, A2a=S2a/Na=0.137
Rb 2001 but not 2018 Nb=1023 Wb=Nb/N=0.056 S1b, A1b=S1b/Nb=-0.084 S2b=0
Rc 2018 but not 2001 Nc=2010 Wc=Nc/N=0.120 S1c=0 S2b, A2b=S2b/Nb=0.130
The main thing to note is that the numbers are all much smaller. That is both because the range of anomalies is much smaller than absolute temperatures, but also, they are more homogeneous, and so
more likely to cancel in a sum. The corresponding terms in the weighted sum making up A2-A1 are
A2-A1 = 0.210 + 0.012 + 0.029 = 0.251
The first term is exactly the same as without anomalies. Because it is the difference of T at the same cemo, subtracting the same base from each makes no change to the difference. And it is the term
we want.
The second and third spurious terms are still spurious, but very much smaller. And this would be true for any reasonably choice of anomaly base.
So why not just restrict to Ra?
where both 2001 and 2018 have values? For a pairwise comparison, you can do this. But to draw a time series, that would restrict to cemos that have no missing values, which would be excessive.
Anomalies avoid this with a small error.
However, you can do better with infilling. Naive anomalies, as used un HADCRUT 4 say, effectively assign to missing cells the average anomaly of the remainder. It is much better to infill with an
estimate from local information. This was in effect the
Cowtan and Way
improvement to HADCRUT. The uses of infilling are described
(with links).
The GISS V4 land/ocean temperature anomaly was 1.13°C in November 2020, up from 0.88°C in October. That compares with a 0.188deg;C rise in the TempLS V4 mesh index. It was the warmest November in the
Jim Hansen's update, with many more details, is here. He thinks that it is clear that 2020 will pass 2016 as hottest year.
As usual here, I will compare the GISS and earlier TempLS plots below the jump.
The TempLS mesh anomaly (1961-90 base) was 0.891deg;C in November vs 0.703°C in October. This rise was a little greater than the rise in the NCEP/NCAR reanalysis base index, which was 0.145°C. The
UAH satellite data for the lower troposphere was little changed from October (but October was very warm). The Eastern Pacific ENSO region was cool.
It was the warmest month since November 2018, the second warmest November in the record (just behind 2015), and makes it likely that in this record, 2020 will be warmer than 2016, and hence the
warmest full year in the record. The mean to November is 0.873°C, vs 2016 0.857, so December only has to be moderately warm for that to happen - in fact 0.681°C would be enough. I see the betting
odds on that event are only 42% - or course they are not based on TempLS.
Housekeeping note
For the last six years I have made three global temperature postings every month - the NCEP results, then the TempLS results, and finally the comparison with GISS. But recently the GHCN data are
posted so promptly that they are available almost as soon as NCEP. So I will in future merge the first two postings; no separate posting for NCEP.
There was a cool region in N Canada, but warm in the USA. Mainly, it was very warm in the Arctic, with an adjacent very warm region right across Eurasia. Most of the rest of the land was warm,
including Antarctica. There was a cool area in central Asia.
Here is the temperature map, using the LOESS-based map of anomalies.
3D globe map gives better detail. | {"url":"https://moyhu.blogspot.com/2020/12/","timestamp":"2024-11-09T19:19:13Z","content_type":"application/xhtml+xml","content_length":"129182","record_id":"<urn:uuid:4e74d3bf-6983-47ab-84f8-3130ddb49bae>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00217.warc.gz"} |
Excel Table vs. Excel Range – What’s the Difference?
There is a good amount of confusion around the terms ‘Table’ and ‘Range’, especially among new Excel users.
You will find many tutorials use the terms interchangeably too.
But the two terms have some basic differences that need to be identified.
In this tutorial, we will explain what the terms Table and Range / Named range mean, how to distinguish between them, as well as how you can convert from one form to the other in Excel.
What is an Excel Range?
Any group of selected cells can be considered as an Excel range.
A range of cells is defined by the reference of the cell that is at the upper left corner and the one at the lower right corner.
For example, the range selected in the image below consists of cells A1 to C7, denoted as A1:C7.
An Excel range does not require cells to be contiguous. You can also add cells to a range that are away from each other.
What is an Excel Named Range?
A named range is simply a range of cells with a name.
The main purpose of using named ranges is to make references to a group of cells more intuitive.
For example, if the name of the following selected range is “Sales”, then you can simply refer to this range by name in formulas (rather than using cell references like B2:B7):
To convert a range of cells to a named range, all you need to do is select the range, type the name into the Name Box and press the return key.
You can identify a named range by selecting the range of cells.
If you see a name, instead of a cell reference in the name box, then the range of cells belongs to a Named range.
What is an Excel Table?
An Excel Table is a dynamic range of cells that are pre-formatted and organized.
A table comes with some additional features such as data aggregation, automatic updates, data styling, etc.
You can say that an Excel table is basically an Excel range, but with some added functionality.
Like named ranges, Excel tables help group a set of related cells together, with a given name.
However, they also help users clearly see the grouping through some extra styling.
As Excel is releasing new data analysis features such as Power Query, Power Pivot, and Power BI, Excel Table has become even more important. Since it’s more structured, most of these new
functionalities will require you to convert your data/range into an Excel Table.
Also read: XLS vs. XLSX Files – What’s the Difference?
How to Identify a Table in Excel?
You can easily identify a table in Excel thanks to its distinguishable features:
• You will find filter arrows next to each column header.
• Column headings remain frozen even as you scroll down the table rows.
• The table is enclosed in a distinguishable box.
• You will also find the table styled differently from the rest of the worksheet. For example you might find rows of the table styled in alternating colors for easy viewing.
• When you click on the table (or select any cell within the table), you should see a Design tab in the main menu.
• When you click on the Design tab, you should see the name of the table on the left side of the menu ribbon.
What’s the Difference Between an Excel Table and Range?
From the first glance, it is quite easy to differentiate between a table and a range.
Not only do they look different, they are also quite different in the amount of functionality they offer.
Here are some of the differences between an Excel Table and Range:
• Cells in an Excel table need to exist as a contiguous collection of cells. Cells in a range, however, don’t necessarily need to be contiguous.
• Every column in an Excel table must have a heading (even if you choose to turn the heading row of the table off). Named ranges, on the other hand, have no such compulsion.
• Each column header (if displayed) includes filter arrows by default. These let you filter or sort the table as required. To filter or sort a range, you need to explicitly turn the filter on.
• New rows added to the table remain a part of the table. However, new rows added to a range or are not implicitly part of the original range.
• In tables, you can easily add aggregation functions (like sum, average, etc.) for each column without the need to write any formulas. With ranges, you need to explicitly add whatever formulas you
need to apply.
• In order to make formulas easier to read, cells in a table can be referenced using a shorthand (also known as a structured reference). This means that instead of specifying cell references in a
formula (as in ranges), you can use a shorthand as follows:
• Moreover, in a table, typing the formula for one row is enough. The formula gets automatically copied to the rest of the rows in the table. In a range or named range, however, you need to use the
fill handle to copy a formula down to other rows in a column.
• Adding a new row to the bottom of a table automatically copies formulae to the new row. With ranges, however, you need to use the fill handle to copy the formula every time you insert a new row.
• Pivot tables and charts that are based on a table get automatically updated with the table. This is not the case with cell ranges.
How to Convert a Range to a Table
Converting an Excel range to a table is really easy.
Let’s say you have the following range of cells and you want to convert it to a table:
Here are the steps that you need to follow to convert the range into a table:
1. Select the range or click on any cell in your range.
2. From the Home tab, click on ‘Format as Table’ (under the Styles group).
3. You should now see a dropdown menu with a number of styling options. Select the styling option that you want to apply to your table. We selected the option highlighted below:
4. This will open the ‘Format as Table’ dialog box.
5. Make sure that the range displayed under ‘where is the data for your table’ is correct.
6. You should see a dashed box around the cells that will be part of your table.
7. If your dataset has headers, ensure that the checkbox next to ‘My table has headers’ is checked.
8. Click OK.
Here’s what your table should look like if you’ve followed the above steps:
Note: Alternatively, you could use the keyboard shortcut CTRL+T in place of steps 2 and 3.
How to Convert a Table to Range
It is also possible to reverse the conversion, in other words, convert a table back into a range of cells. Here are the steps that you need to follow:
1. Select any cell in your table.
2. You should see a new ribbon titled ‘Table Tools’ in the main menu. Select the Design tab under this menu.
3. In the Tools group, select the ‘Convert to Range’ button.
4. You will be asked to confirm if you want to convert the table to a normal range. Click Yes.
Alternatively, you could simply right-click on the table and select Table->Convert to Range from the context menu that appears.
You will now find that the table features (like filter arrows and structured references in all the formulas) are no longer there since it’s now just a regular range of cells. Structured references
have all turned back into regular cell references.
In this tutorial, we explained with examples how ranges and named ranges differ from tables.
To conclude, a table can be considered as a named range, but with some added functionality, like styling, easy aggregations, structured references, and more.
We hope we have been successful in clearing any confusion you might have had about ranges, named ranges, and tables.
Other Excel Tutorials you may also like:
1 thought on “Excel Table vs. Excel Range – What’s the Difference?”
1. Thanks Steve, this article very clearly explained the difference between the excel range data and table, it is super helpful!! Thanks so much!
Leave a Comment | {"url":"https://spreadsheetplanet.com/excel-table-vs-excel-range/","timestamp":"2024-11-02T08:26:19Z","content_type":"text/html","content_length":"138017","record_id":"<urn:uuid:e1b988f8-cc09-4ddc-808f-a21b842663e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00679.warc.gz"} |
The aim of our project was to prove our conjecture that the product of consecutive positive integers is never a square. In our investigation, we had developed three approaches to prove it.
In the first approach, we used the fact that a number lying between two consecutive squares is never a square to prove that the product of eight consecutive integers is never a square. Then we made
use of relatively prime-ness of consecutive integers to prove the rest.
In the second approach, we had used Bertrand’s Postulate Theorem to obtain a beautiful theorem that the product of consecutive positive integers is never a square if there is a prime number among
them. Besides we had found some interesting results from this theorem.
When we started our project, we thought that our conjecture had not been proved. However, we found later in a website that our conjecture has already been proved by two famous mathematicians P. Erdos
and J.L. Selfridge in 1939. Although our conjecture was proved, we didn’t give up but tried our best to develop our third approach.
In the third approach, we had referred to an academic journal written by P. Erdos and J.L. Selfridge and knew that the square-free parts of consecutive integers are distinct. By counting, we arrived
at a necessary condition for the product not to be a square. Unluckily, we then discovered the limitations of the third approach when the number of consecutive integers is very large. It may be due
to the roughness of our estimation. Although we couldn’t complete the proof of our conjecture, we all enjoyed the process of formulating conjectures and thinking new ideas of solving problems through
the cooperation among our team members in the past few months. | {"url":"https://hlma.hanglung.com/en/resource-library/2004","timestamp":"2024-11-13T09:14:25Z","content_type":"text/html","content_length":"35088","record_id":"<urn:uuid:3a77ccae-8624-4617-b813-42ab15c548d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00761.warc.gz"} |
Convert Dates into Ages - Excel University
|August 1, 2023||COUNTIFS, TRUNC, YEARFRAC
If you need to calculate the age in years based on a list of birth dates in Excel, fear not! In this short tutorial, we’ll cover a few functions that will help, as well as a method to count the
number of rows in each age group. Specifically, we’ll explore the YEARFRAC, TRUNC, DATEDIF, and COUNTIFS functions. So let’s get started!
Step-by-step guide
Let’s explore these functions by using a few specific exercises.
Exercise 1: Using YEARFRAC & TRUNC
Our objective is to compute the age in years between the birth date and the “as of” date as shown below.
To do this, we’ll start with the YEARFRAC function. This function computes the fractional portion of a year (or years) between two dates. The basic function signature is:
=YEARFRAC(start_date, end_date, [basis])
There are three arguments: the start_date, end_date, and optionally basis which provides options for the count (such as 365, actual, 360, and so on). In our case, our start date is the birth date,
the end date is our “as of” date, and the basis is 1 for actual days.
Now, this will return the number of years between the two dates, but also the fractional portion of a year.
To get rid of the fractional portion, we’ll wrap the TRUNC function around the YEARFRAC function like this:
=TRUNC(YEARFRAC(start_date, end_date, 1))
This truncates the year portion, leaving us with just the age in years:
Now let’s take a look at an alternative.
Exercise 2: Using DATEDIF
In this exercise, we’ll use the DATEDIF function instead of the TRUNC/YEARFRAC combo to calculate the age in years. This function is considered deprecated, so it’s not officially documented in Excel.
In practice, it feels safer to use YEARFRAC and TRUC, but, it’s cool knowing our options.
The basic syntax to compute the number of years between two dates is:
=DATEDIF(start_date, end_date, "y")
It returns the same results as the previous formula:
Now let’s count the number of rows for age ranges we define.
Exercise 3: Counting Age Groups
We’ll use the COUNTIFS function to count the number of rows for each age range we define. We use the basic formula syntax as follows:
=COUNTIFS(age_column_ref, ">="&[From], age_column_ref, "<="&[To])
And the results are displayed in the Count column below:
This will count the number of rows where the age column is greater than or equal to the From age and less than or equal to the To age.
I hope this post was helpful for computing ages from a list of dates, and then counting the number of ages that fall within pre-defined ranges.
If you have any alternative approaches, suggestions, or questions, please share by posting a comment below … thanks!
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2 Comments
1. Jeff Lenning, you really do rock. Thanks for sharing and caring! Best Regards!
□ Thank you for your kind note 🙂
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Excel Formula: Count Total Based on Criteria
In this guide, you will learn how to count the total in column B based on criteria in column A using an Excel formula in Python. This formula is useful when you want to analyze data and determine the
number of occurrences that meet specific criteria. By following the step-by-step explanation and examples provided, you will be able to apply this formula to your own datasets.
To count the total in column B based on criteria in column A, we can use the COUNTIF function in Excel. This function allows us to count the number of cells that meet a certain condition. In Python,
we can use the openpyxl library to work with Excel files and execute this formula.
Here is the formula:
=COUNTIF(A:A, "criteria")
Let's break down the formula:
1. The COUNTIF function takes two arguments: the range of cells to evaluate and the criteria to match.
2. In this formula, we use the range A:A to represent the entire column A.
3. The criteria is specified as a string within double quotes. You can replace "criteria" with the specific value or condition you want to count in column B based on column A.
4. The COUNTIF function counts the number of cells in column A that match the specified criteria.
5. The result is the total count of occurrences in column B that meet the criteria in column A.
For example, let's say we have the following data in columns A and B:
| A | B |
| A | 10 |
| B | 20 |
| A | 30 |
| C | 40 |
| A | 50 |
| B | 60 |
If we want to count the total number of occurrences in column B where the corresponding value in column A is "A", we can use the formula =COUNTIF(A:A, "A"). The formula will return the value 3,
because there are three occurrences in column B where the corresponding value in column A is "A".
By using this formula, you can easily analyze your data and obtain the desired count based on specific criteria. Remember to adjust the range and criteria according to your own dataset.
An Excel formula
=COUNTIF(A:A, "criteria")
Formula Explanation
This formula uses the COUNTIF function to count the total number of occurrences in column B based on a criteria in column A.
Step-by-step explanation
1. The COUNTIF function takes two arguments: the range of cells to evaluate and the criteria to match.
2. In this formula, we use the range A:A to represent the entire column A.
3. The criteria is specified as a string within double quotes. You can replace "criteria" with the specific value or condition you want to count in column B based on column A.
4. The COUNTIF function counts the number of cells in column A that match the specified criteria.
5. The result is the total count of occurrences in column B that meet the criteria in column A.
For example, let's say we have the following data in columns A and B:
| A | B |
| A | 10 |
| B | 20 |
| A | 30 |
| C | 40 |
| A | 50 |
| B | 60 |
If we want to count the total number of occurrences in column B where the corresponding value in column A is "A", we can use the formula =COUNTIF(A:A, "A").
The formula will return the value 3, because there are three occurrences in column B where the corresponding value in column A is "A". | {"url":"https://codepal.ai/excel-formula-generator/query/0At45P9r/excel-formula-count-total-based-on-criteria","timestamp":"2024-11-02T23:58:02Z","content_type":"text/html","content_length":"96426","record_id":"<urn:uuid:411668df-672f-43fd-930f-e460f5af7103>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00328.warc.gz"} |
DIMCONSTRAINT command
Applies a dimensional constraint to an entity or between constraint points on entities; converts associative dimensions to dynamic dimensions.
Select an associative dimension or else choose an option to place a dimensional constraint.
The associative dimension is converted to the dimensional constraint of the same type. This option is equivalent to the DCCONVERT command.
Constrains the horizontal distance (X-distance) or vertical distance (Y-distance) between two points with respect to the current coordinate system. This option is equivalent to the DCLINEAR
Constrains the horizontal distance (X-distance) between two points with respect to the current coordinate system. This option is equivalent to the DCHORIZONTAL command.
Constrains the vertical distance (Y-distance) between two points with respect to the current coordinate system. This option is equivalent to the DCVERTICAL command.
Constrains the distance between two points. This option is equivalent to the DCALIGNED command.
Constrains the angle between two lines or linear polyline segments; the total angle of an arc or an arc polyline segment; or the angle between three points on entities. This option is equivalent
to the DCANGULAR command.
Constrains the radius of a circle or an arc. This option is equivalent to the DCRADIUS command.
Constrains the diameter of a circle or an arc. This option is equivalent to the DCDIAMETER command. | {"url":"https://help.bricsys.com/en-us/document/command-reference/d/dimconstraint-command?version=V23","timestamp":"2024-11-06T13:42:44Z","content_type":"text/html","content_length":"68850","record_id":"<urn:uuid:ba60ba83-83ec-4774-a3cf-e31604563bc5>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00868.warc.gz"} |
Lesson plan Titles of parts of the lesson must be formatted as headings
What to create Lesson plan
Which subject Geography
What topic Population density
What length (min) 30
What age group Doesn't matter
Include homework
Include images descriptions
Any other preferences
Subject: Geography
Topic: Population Density
Duration: 30 minutes
Academic Level: Any
By the end of this lesson, students will be able to:
• Define population density and understand its significance in Geography;
• Calculate population density using simple formulas;
• Analyze population density patterns across different regions and countries;
• Understand the impact of population density on different aspects of human life, such as transportation, housing, education, and healthcare.
• Whiteboard and markers;
• Handouts with graphs and population data of different countries;
• Calculators.
Introduction (5 minutes)
• Greet the class and introduce the lesson topic;
• Ask students if they know what population density means;
• Write the definition on the board: "Population density is the measure of the number of individuals in a population per unit area or volume".
Main Content (20 minutes)
Definition and Formula
• Review the definition of population density and explain why it is important in Geography;
• Provide a demonstration of how to calculate population density using a simple formula: Population density = Total population / Total land area;
• Ask the students to calculate the population density of their school area or city;
• Discuss the students' answers and compare them with other regions/countries.
Patterns and Analysis
• Show the class a few graphs of population density patterns across different regions/countries;
• Ask the students questions about the trends they observe: Are there obvious differences between countries? Why do some areas have higher population densities than others? What factors contribute
to population density patterns?
• Hand out a list of countries with their respective population densities and ask the students to group them into high, medium, and low densities.
• Discuss the characteristics of countries with different population densities.
Conclusion (5 minutes)
• Summarize the key points of the lesson and reinforce the importance of understanding population density in Geography;
• Ask the students if they have any further questions or comments.
• Collect the handouts with the population data and graphs to check for students' comprehension of the lesson;
• Assign homework which will involve students calculating the population density of a specific city, region or country and writing a short essay on the environmental, economic and social impacts of
high and low population densities.
• Discuss how the concept of population density is related to other Geography concepts, such as urbanization, migration, and climate change;
• Organize a field trip to a nearby area with high or low population density to observe the environmental, economic and social factors that contribute to its density.
• Provide more hand-on activities for tactile learners;
• Use visual aids for visual learners;
• Provide extra guidance for students who struggle with calculations. | {"url":"https://aidemia.co/view.php?id=1840","timestamp":"2024-11-05T23:28:50Z","content_type":"text/html","content_length":"9397","record_id":"<urn:uuid:de8a77d0-77b8-4a80-8183-2b5b3f9d579c>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00410.warc.gz"} |
Dispersion relations
Next: The xxz derivative Up: HIGHER ANGLE ACCURACY Previous: Muir square-root expansion
Substituting the definitions (60) into equation (65) et. seq. gives dispersion relationships for comparison to the exact expression (59).
Identification of i k[z] with 65) into the differential equations
which are extrapolation equations for when velocity depends only on depth.
The differential equations above in Table .4 were based on a dispersion relation that in turn was based on an assumption of constant velocity. Surprisingly, these equations also have validity and
great utility when the velocity is depth-variable, v = v(z). The limitation is that the velocity be constant over each depth ``slab'' of width
Next: The xxz derivative Up: HIGHER ANGLE ACCURACY Previous: Muir square-root expansion Stanford Exploration Project | {"url":"https://sep.stanford.edu/sep/prof/bei/fdm/paper_html/node28.html","timestamp":"2024-11-14T11:58:28Z","content_type":"text/html","content_length":"5897","record_id":"<urn:uuid:8deb8b06-d221-430a-81b7-f482549af01b>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00360.warc.gz"} |
Converts a random variable to a normalized value.
Sample Usage
STANDARDIZE(number, mean, standard_deviation)
• number - The value which is to be normalized.
• mean - The arithmetic mean of the distribution.
• standard_deviation - The standard deviation of the distribution.
STANDARDIZE(42,40,1.5) returns 1.3333, the normalized value of 42 using 40 as the arithmetic mean and 1.5 as the standard deviation.
• For a given dataset, mean can be calculated using AVERAGE or its related functions and standard_deviation can be calculated using STDEV or its related functions.
• standard_deviation must be greater than 0. | {"url":"https://support.spreadsheet.com/hc/en-us/articles/360032204911-STANDARDIZE","timestamp":"2024-11-12T00:00:26Z","content_type":"text/html","content_length":"44464","record_id":"<urn:uuid:4646a41d-1738-4f95-9db6-52c3ea03ee72>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00311.warc.gz"} |
Cool Math Stuff
One of the most famous mathematicians of all time is the German mathematician Carl Friedrich Gauss (1777 - 1855). Gauss was one of the leading number theorists of all time, as well as a contributor
to algebra, statistics, geometry, analysis, and applied mathematics.
There is a piece of mathematical folklore (which may or may not be 100% accurate) that involved a child Gauss. It is a great story, highlights a great point, and shows the intelligence of a great
A fifth grade teacher is teaching a class, and started to get frustrated with the students. So, in an attempt to punish them, she demanded that they add up all of the numbers from 1 to 100. This is a
daunting task for the average person. She expected to have the students start working on the problem, and she could leave and take a break.
As she was about to walk out the door, the young Gauss raised his hand and declared that the answer is 5050. The teacher was stunned. After checking his work, they found that 5050 was the correct
How did he do it? Well, he visualized a horizontal line with all 100 numbers:
1 2 3 4 5 ... 96 97 98 99 100
And then he took the second half of that line (51 - 100) and flipped it around underneath to look like so:
1 2 3 4 5 ... 46 47 48 49 50
100 99 98 97 96 ... 55 54 53 52 51
Each of these vertical columns is its own addition problem. And in all fifty columns, the sum is 101. So, the sum of the numbers from one to one-hundred is the same as fifty 101's, or 50 x 101. Since
50 ends in a zero, it is a pretty quick computation: 50 x 101 = 5050. And there is the answer.
I think this is a great story when it comes to historical mathematicians, regardless of how true it is. Gauss did go on to study
triangular numbers
, which are the sums of consecutive integers up to a point. And since triangular numbers are absolutely fascinating, this story is a great way to begin an endeavor in that topic.
We've all heard of the game tic-tac-toe. It's easy to set up, simple to understand, and fun to play. All you have to do is get three in a row.
There is some mathematics behind correct play of tic-tac-toe, but the real goal is to see where the other player has a chance of forcing a win and make sure that doesn't happen.
However, there is an interesting mathematical fact in the game "anti tic-tac-toe." This is where you have to avoid getting three in a row.
As X, what would be a good starting move in this game? In regular tic-tac-toe, most people start in the center because it is part of 4 of the 8 winning possibilities. So in this instance, it is part
of 4 of the 8 losing possibilities. So, I don't think the center would be an appealing move to most players.
Let's start in the left corner square and see what happens.
│X │││
Since O wants to see X get three in a row, they will get as out of the way as possible.
│X ││ │
│ ││O │
│ ││ │
Now, where are X's safest moves? Well, if X moves in any of the squares with asterisks, then it would give them two in a row, which would put them closer to a loss. So, X's best move is in the bottom
side square.
│X │* │* │
│* │* │O │
│* │X │* │
Now, O would probably go in the top side square to keep out of having 2 in a row.
│X │O │ │
│ │ │O │
│ │X │ │
Now, X's best move is in the top right corner, since all other squares would have an asterisk.
│X │O │X │
│ │ │O │
│ │X │ │
In this instance, the only technically "safe" move for O is in the bottom left corner, but you can see that this move eliminates 3 possible three in a rows for X. Since the bottom right corner is
pretty safe too, O's best move would be there.
│X │O │X │
│ │ │O │
│ │X │O │
Now, X is forced to go in an unsafe square. In fact, O can force a win wherever X goes. If X goes in the left side square, O will go in the center. If X goes in the center square, O will go in the
left side square. If X goes in the bottom left corner square, O can go in either square. There is no way for X to win.
You might notice that in this game, X has a huge disadvantage. O has one less letter to put down, so it is impossible for X to win against a rational player. However, X can force a tie.
Eight of the squares on the board are a guaranteed loss for X. If X moves there on the first turn, O should be able to succeed. However, this ninth square is a square that X can move to and O will
not be able to force three in a row.
Which square is it? It is the one that would never be suspected: the center square. Earlier, we said that the center square has so many losing possibilities that nobody would consider it. But, it is
actually the right move. Let's look at it.
││X ││
Currently, there is no specific square that O would have an advantage in. Let's say they went in the top left corner square.
│O │ ││
│ │X ││
Then, X would play the bottom right corner square. This would be the symmetrical move.
│O │ │ │
│ │X │ │
│ │ │X │
X would continue playing the symmetrical move through the whole game. With any logical player, this would result in a draw.
Though anti tic-tac-toe wouldn't be a popular game to play, there are much more fun games that use symmetric properties to force ties/wins like nim or cram. Here is another game with this type of
strategy called napkin chess that I learned on the show Scam School which can also be turned into a fun game with friends:
About a month ago, I was speaking at a TEDx conference that was themed around education. During these TEDx conferences, there are always live speakers as well as videos chosen by the organizers that
fit well with the occasion. One of the videos we watched really got us thinking. Here it is:
This talk is not directly mathematical. It isn't meant to be a math education talk. But, these points apply to math education as well.
For example, the mathematician Euclid had nothing. He pretty much was starting from scratch, just like the kids described in the talk. So what did he do? Well, after creating five axioms
(foundational ideas that can be concluded with basic logic), he started asking questions, and answering them with mathematical proofs. He would then think about other questions, and find ways to
answer them using everything he had discovered. This was the basis of his book series called Elements.
Students should be able to approach math in a similar way. A teacher could lay out five axioms (or develop them with the students), and then back off. He or she might also provide terminology (line,
triangle, square, circle, angle, bisect, trisect, etc.), but the students can discover the rest. By learning math this way, they will understand everything they are doing so much better because they
created it. For more on that topic, check out my Capstone Research Paper (link is on the top of the page), and scroll to "Chronological Cognition."
A few months ago, I posted the proof of the Law of Cosines, which is an extremely important aspect of trigonometry. If you do not know what sines and cosines are, it is a very easy concept. Click
here to view the post discussing that (which is also the post with the proof of the Law of Cosines).
But this law only works if you are given all three sides of the triangle or two sides and the enclosed angle. What if you are given two sides and a non-enclosed angle, or two angles and one side
(three angles isn't enough information to generate side lengths)? How could you approach this problem?
This is where you use the Law of Sines. This law goes as follows:
This was taught to me last year in school, and I immediately wondered what the proof was. Though the law of cosines one was a bit clunky, I found that this proof was quite simple and elegant. So, I
thought that it would be great to share.
Since it would require many diagrams, I thought it would be easier to just watch a video of it. It is pretty short, and explains the proof well.
A big part of the reason why most of the cool stuff I post isn't taught in school is that it is not mandated in the curriculum or test standards. Of course, I do believe there are changes that need
to be made to these (click here for my Capstone research paper explaining those). However, the Law of Sines is something already taught in school. Same with the Law of Cosines.
These proofs, especially the sine one, fit right into the curriculum. The Law of Sines is already being taught, so why not take an extra 5 minutes to explain the proof? Or even better, explain the
basic thought process behind the proof and have the students generate the formula (which works really well for the Quadratic Formula as well). This increases the students' ability to understand and
apply the concept, as well as making it fun and interesting. On top of that, the common core standards do want students able to "construct viable arguments," which is the whole purpose of proofs. I
think that this proof is not only interesting, but shows that cool math stuff can be integrated into the classroom while keeping it relevant and obedient to the standards. | {"url":"https://coolmathstuff123.blogspot.com/2013/09/","timestamp":"2024-11-03T22:32:39Z","content_type":"text/html","content_length":"94087","record_id":"<urn:uuid:14ec703e-4356-4f0a-8dc0-a7cd969c23b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00696.warc.gz"} |
Mixing times of Markov chains on 3-Orientations of Planar Triangulations
Sarah Miracle ; Dana Randall ; Amanda Pascoe Streib ; Prasad Tetali - Mixing times of Markov chains on 3-Orientations of Planar Triangulations
dmtcs:3010 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the
Analysis of Algorithms (AofA'12) -
Mixing times of Markov chains on 3-Orientations of Planar TriangulationsArticle
• 1 College of Computing
• 2 School of Mathematics, School of Computer Science
Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as
a $\textit{Schnyder wood}$ that has proven useful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First,
we study a "triangle-reversing'' chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that (i) when
restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we
consider an "edge-flipping'' chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain
connects the state space and we prove that the chain is always rapidly mixing.
Volume: DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)
Section: Proceedings
Published on: January 1, 2012
Imported on: January 31, 2017
Keywords: Schnyder woods,Markov chains,3-orientations,Planar triangulations,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete
Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]
Source : OpenAIRE Graph
• Random graph interpolation, Sumset inequalities and Submodular problems; Funder: National Science Foundation; Code: 1101447
• AF: Large: Collaborative Research: Random Processes and Randomized Algorithms; Funder: National Science Foundation; Code: 0910584
• Markov Chain Algorithms for Problems from Computer Science and Statistical Physics; Funder: National Science Foundation; Code: 0830367 | {"url":"https://dmtcs.episciences.org/3010","timestamp":"2024-11-02T12:34:50Z","content_type":"application/xhtml+xml","content_length":"57079","record_id":"<urn:uuid:c1ee411e-e3c4-41ce-8f09-cd5d7f21805c>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00403.warc.gz"} |
Deep Learning in Simulink. Simulating AI within large complex systems
This post is from guest blogger Kishen Mahadevan, Product Marketing. Kishen helps customers understand AI, deep learning and reinforcement learning concepts and technologies. In this post, Kishen
explains how deep learning can be integrated into an engineering system designed in Simulink.
Deep learning is a key technology driving the Artificial Intelligence (AI) megatrend. Popular applications of deep learning include autonomous driving, speech recognition, and defect detection. When
deep learning is used in complex systems it is important to note that a trained deep learning model is only a small component of a larger system. For example, embedded software for self-driving cars
has components such as adaptive cruise control, lane keep assist, sensor fusion, and lidar processing in addition to a deep learning model that performs a specific task, say lane detection. How do
you then integrate, implement, and test all these different components together while minimizing expensive testing with the actual vehicle? This is where Model-Based Design with MATLAB and Simulink
fits in.
When you create a Simulink model for any complex system, you typically have two main components, as shown in Figure 1. The first component represents a collection of algorithms that will be
implemented in the embedded system and includes controls, computer vision, and sensor fusion. The second component represents the dynamics of the machine or process we want to develop embedded
software for. This component can be a vehicle dynamics model, dynamics of Li-Ion battery, or a model of a hydraulic valve. Having both of these components in the same Simulink model allows you to run
simulations to verify and validate embedded algorithms before implementing them on target hardware. Trained deep learning models can be used in both of these components. Examples of using deep
learning for algorithm development include use of deep learning for object detection and for soft, or virtual sensing. In the latter scenario deep learning model is used to compute a signal that
cannot be measured directly, for example a state-of-charge for a Li-Ion battery. Deep learning models can also be used for environment modeling. This is sometimes referred to as reduced order
modeling. Detailed, high-fidelity model of the machine or a process can be replaced with a faster AI-based model that is trained to capture the essential dynamics of the original model.
Figure 1: Integrating deep learning models into Simulink
In this blog, we will focus on an example that illustrates the use of deep learning for algorithm development. The example shows how you can integrate a trained deep learning model into Simulink for
system-level simulation and code generation.
: The features and capabilities showcased in this blog can be applied for algorithm development as well as reduced-order modeling.
Deep Learning in Simulink example
Deep learning workflow involves four main stages:
1. Data preparation
2. AI modeling
3. Simulation and Testing
4. Deployment
Let’s use the example of a Battery Management System where deep learning is used to estimate state-of-charge (SOC) for a battery. SOC is an important signal for a Battery Management System, yet it
cannot be measured directly during operation. However, with enough data collected in the lab, a deep learning model can be trained to accurately predict battery SOC using commonly available
measurements. Let’s start by looking at the data predictors or the observations required for the deep learning network. These predictors include measurements of voltage, current, temperature, the
calculated moving average values of voltage, and current. The data needed to train a deep learning network also includes the response - battery SOC associated with each set of those measurements.
Figure 2: Data for training the deep learning network (top), deep learning network inputs and output (bottom)
With this data, the deep learning model is configured to receive five inputs and provide state-of-charge (SOC) of the battery as the predicted output. Once the data has been preprocessed, you can
train a deep learning model using
Deep Learning Toolbox
. Sometimes you might already have an AI model developed in TensorFlow or other deep learning frameworks. Using Deep Learning Toolbox you can import these models into MATLAB for system-level
simulation and code generation. In this example, we use the existing deep learning model that has been trained in TensorFlow.
Figure 3: Deep learning workflow
Step 1: Data Preparation
For this step of the workflow we use the already available preprocessed experiment data collected from a lab. This data includes all the predictors and response as highlighted in Figure 2.This data
was provided to us by McMaster University (
data Source
Step 2: AI Modeling
As pointed out earlier, the deep learning model can be trained in MATLAB using Deep Learning Toolbox. Refer to the learn more about how to train a deep learning network to predict SOC in MATLAB. As
already mentioned in this example we have been provided with a deep learning model that has already been trained in TensorFlow. To import this trained network into MATLAB, we use the
Figure 4: Direct network import from TensorFlow into MATLAB
We then analyze the imported network architecture using
to check for warning or errors and observe that all the imported layers are supported.
Figure 5: Analyzing the imported network using Deep Learning Network Analyzer
We then load the test data and verify performance of the imported network in MATLAB.
Figure 6: MATLAB code to load and plot prediction results Figure 7: Comparing deep learning SOC prediction with true observed SOC value.
We see that the deep learning predicted SOC of the battery is in alignment with the experimentally observed values.
Step 3: Simulation and Test
To be able to simulate and test this deep learning SOC estimator with all other components of a Battery Management System, we first need to bring this component into Simulink. To accomplish this, we
block from Deep Learning Toolbox block library to add the deep learning model into a Simulink model.
Figure 8: Deep Learning Toolbox library to bring trained deep learning models into Simulink
Figure 9 shows the open-loop Simulink model. The Predict block loads our trained deep learning model into Simulink from a .MAT file. The block receives the preprocessed data as the input and
estimates SOC of the battery.
Figure 9: Integrating trained deep learning model into Simulink
We then simulate this model and observe that prediction from our deep learning network in Simulink is identical to the true measured data as shown in Figure 10.
Figure 10: Simulation result comparing SOC prediction from deep learning network and the true value
Now that we have tested the component in Simulink, we can integrate it into a larger model and simulate the complete system. This is shown in Figure 11.
Figure 11: System-level Simulink model of a battery management system and a battery plant
Simulink model shown in Figure 11 contains a Battery Management System that is responsible for monitoring battery state and ensuring safe operation, and a battery plant that models the dynamics of a
battery and a load. Our deep learning SOC predictor resides as one of the components under the battery management system along with the logic for cell balancing, prevention of overcharging and
over-discharging, and other components.
Figure 12: Components of battery management system
We now simulate this closed-loop system and observe the SOC predictions.
Figure 13: System-level simulation results comparing SOC prediction from deep learning network with the true value
We can see that SOC predictions from our deep learning model is very similar to predictions from the true measured value.
Step 4: Deployment
To highlight the capability of deploying our deep learning network in this example, we use the open-loop model that contains just the deep learning SOC predictor. The workflow steps, however, remain
the same for the system-level model. We first generate C code from the trained deep learning model in Simulink.
Figure 14: C code generation for the deep learning network in Simulink
We can see that the generated code contains calls to deep learning step functions that perform SOC prediction. Next, we deploy the generated code to an NXP board for processor-in-the-loop (PIL)
simulation. In PIL simulation, we generate production code only for the algorithm we are developing, in this case the deep learning SOC component, and execute that on the target hardware board, NXP
S32K3. This allows us to verify the code behavior on the embedded target. We now add driver blocks to the model to allow us to interface to and from the NXP board and simulate the model.
Figure 15: Processor-in-the-loop simulation of deep learning SOC component on NXP board
We see that the behavior of the generated code on the NXP target is identical to the true measured SOC value.
Key takeaways
• Integrate deep learning models into system-level Simulink models
• Test system-level performance of the design with a deep learning
• Generate code and deploy your application, including deep learning component, to embedded target
• Train deep learning models in MATLAB or import pretrained TensorFlow and ONNX models
Resources to learn more
コメントを残すには、ここをクリックして MathWorks アカウントにサインインするか新しい MathWorks アカウントを作成します。 | {"url":"https://blogs.mathworks.com/deep-learning/2021/12/16/deep-learning-in-simulink-simulating-ai-within-large-complex-systems/?s_tid=prof_contriblnk&from=jp","timestamp":"2024-11-08T00:47:19Z","content_type":"text/html","content_length":"156219","record_id":"<urn:uuid:98c944d8-349a-4bac-bdef-b0fba37bc153>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00205.warc.gz"} |
Newton’s Universal Law of Gravitation
Newton’s laws of motion show that objects at rest will stay at rest and those in motion will continue moving uniformly in a straight line unless acted upon by a force. Thus, it is the straight line
that defines the most natural state of motion. But the planets move in ellipses, not straight lines; therefore, some force must be bending their paths. That force, Newton proposed, was gravity.
In Newton’s time, gravity was something associated with Earth alone. Everyday experience shows us that Earth exerts a gravitational force upon objects at its surface. If you drop something, it
accelerates toward Earth as it falls. Newton’s insight was that Earth’s gravity might extend as far as the Moon and produce the force required to curve the Moon’s path from a straight line and keep
it in its orbit. He further hypothesized that gravity is not limited to Earth, but that there is a general force of attraction between all material bodies. If so, the attractive force between the Sun
and each of the planets could keep them in their orbits. (This may seem part of our everyday thinking today, but it was a remarkable insight in Newton’s time.)
Once Newton boldly hypothesized that there was a universal attraction among all bodies everywhere in space, he had to determine the exact nature of the attraction. The precise mathematical
description of that gravitational force had to dictate that the planets move exactly as Kepler had described them to (as expressed in Kepler’s three laws). Also, that gravitational force had to
predict the correct behavior of falling bodies on Earth, as observed by Galileo. How must the force of gravity depend on distance in order for these conditions to be met?
The answer to this question required mathematical tools that had not yet been developed, but this did not deter Isaac Newton, who invented what we today call calculus to deal with this problem.
Eventually he was able to conclude that the magnitude of the force of gravity must decrease with increasing distance between the Sun and a planet (or between any two objects) in proportion to the
inverse square of their separation. In other words, if a planet were twice as far from the Sun, the force would be (1/2)2, or 1/4 as large. Put the planet three times farther away, and the force is
(1/3)2, or 1/9 as large.
Newton’s universal law of gravitation works for the planets, but is it really universal? The gravitational theory should also predict the observed acceleration of the Moon toward Earth as it orbits
Earth, as well as of any object (say, an apple) dropped near Earth’s surface. The falling of an apple is something we can measure quite easily, but can we use it to predict the motions of the Moon?
Recall that according to Newton’s second law, forces cause acceleration. Newton’s universal law of gravitation says that the force acting upon (and therefore the acceleration of) an object toward
Earth should be inversely proportional to the square of its distance from the center of Earth. Objects like apples at the surface of Earth, at a distance of one Earth-radius from the center of Earth,
are observed to accelerate downward at 9.8 meters per second per second (9.8 m/s2).
It is this force of gravity on the surface of Earth that gives us our sense of weight. Unlike your mass, which would remain the same on any planet or moon, your weight depends on the local force of
gravity. So you would weigh less on Mars and the Moon than on Earth, even though there is no change in your mass. (Which means you would still have to go easy on the desserts in the college cafeteria
when you got back!)
The Moon is 60 Earth radii away from the center of Earth. If gravity (and the acceleration it causes) gets weaker with distance squared, the acceleration the Moon experiences should be a lot less
than for the apple. The acceleration should be (1/60)2 = 1/3600 (or 3600 times less—about 0.00272 m/s2. This is precisely the observed acceleration of the Moon in its orbit. (As we shall see, the
Moon does not fall to Earth with this acceleration, but falls around Earth.) Imagine the thrill Newton must have felt to realize he had discovered, and verified, a law that holds for Earth, apples,
the Moon, and, as far as he knew, everything in the universe.
EXAMPLE 3.3
Calculating Weight
By what factor would a person’s weight at the surface of Earth change if Earth had its present mass but eight times its present volume?
With eight times the volume, Earth’s radius would double. This means the gravitational force at the surface would reduce by a factor of (1/2)2 = 1/4, so a person would weigh only one-fourth as much.
Check Your Learning
By what factor would a person’s weight at the surface of Earth change if Earth had its present size but only one-third its present mass?
With one-third its present mass, the gravitational force at the surface would reduce by a factor of 1/3, so a person would weight only one-third as much.
Gravity is a “built-in” property of mass. Whenever there are masses in the universe, they will interact via the force of gravitational attraction. The more mass there is, the greater the force of
attraction. Here on Earth, the largest concentration of mass is, of course, the planet we stand on, and its pull dominates the gravitational interactions we experience. But everything with mass
attracts everything else with mass anywhere in the universe.
Newton’s law also implies that gravity never becomes zero. It quickly gets weaker with distance, but it continues to act to some degree no matter how far away you get. The pull of the Sun is stronger
at Mercury than at Pluto, but it can be felt far beyond Pluto, where astronomers have good evidence that it continuously makes enormous numbers of smaller icy bodies move around huge orbits. And the
Sun’s gravitational pull joins with the pull of billions of others stars to create the gravitational pull of our Milky Way Galaxy. That force, in turn, can make other smaller galaxies orbit around
the Milky Way, and so on.
When falling, they are in free fall and accelerate at the same rate as everything around them, including their spacecraft or a camera with which they are taking photographs of Earth. When doing so,
astronauts experience no additional forces and therefore feel “weightless.” Unlike the falling elevator passengers, however, the astronauts are falling around Earth, not to Earth; as a result they
will continue to fall and are said to be “in orbit” around Earth (see the next section for more about orbits).
Orbital Motion and Mass
Kepler’s laws describe the orbits of the objects whose motions are described by Newton’s laws of motion and the law of gravity. Knowing that gravity is the force that attracts planets toward the Sun,
however, allowed Newton to rethink Kepler’s third law. Recall that Kepler had found a relationship between the orbital period of a planet’s revolution and its distance from the Sun. But Newton’s
formulation introduces the additional factor of the masses of the Sun (M1) and the planet (M2), both expressed in units of the Sun’s mass. Newton’s universal law of gravitation can be used to show
mathematically that this relationship is actually
where a is the semimajor axis and P is the orbital period.
How did Kepler miss this factor? In units of the Sun’s mass, the mass of the Sun is 1, and in units of the Sun’s mass, the mass of a typical planet is a negligibly small factor. This means that the
sum of the Sun’s mass and a planet’s mass, (M1 + M2), is very, very close to 1. This makes Newton’s formula appear almost the same as Kepler’s; the tiny mass of the planets compared to the Sun is the
reason that Kepler did not realize that both masses had to be included in the calculation. There are many situations in astronomy, however, in which we do need to include the two mass terms—for
example, when two stars or two galaxies orbit each other.
Including the mass term allows us to use this formula in a new way. If we can measure the motions (distances and orbital periods) of objects acting under their mutual gravity, then the formula will
permit us to deduce their masses. For example, we can calculate the mass of the Sun by using the distances and orbital periods of the planets, or the mass of Jupiter by noting the motions of its
Indeed, Newton’s reformulation of Kepler’s third law is one of the most powerful concepts in astronomy. Our ability to deduce the masses of objects from their motions is key to understanding the
nature and evolution of many astronomical bodies. We will use this law repeatedly throughout this text in calculations that range from the orbits of comets to the interactions of galaxies. | {"url":"https://learninglabs.thetippingpoint.be/topic/newtons-great-synthesis/","timestamp":"2024-11-06T01:44:02Z","content_type":"text/html","content_length":"367148","record_id":"<urn:uuid:083c0d11-130f-4c7e-bd44-7e3dbfe5cdd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00528.warc.gz"} |
Elastic Constants of Crystals - I, Physics tutorial
The study of the elastic behavior of solids is extremely significant in the basic and technical researches. In technology, it would state us regarding the strength of the materials. In basic
research, it is of interest because of the insight it gives in to the nature of binding forces in solids. They are as well of significance for the thermal properties of solids.
Elasticity is basically the study of the capability of crystals to incorporate changes or adapt to new conditions easily.
Analysis of elastic strains and stresses:
The local elastic strain of a body might be specified through six numbers. If α, β, γ are the angles between the unit cell axes a, b, c, the strain might be specified through the changes Δα, Δβ, Δγ;
Δa, Δb, Δc resultant from the deformation. This is an excellent physical specification of strain, however for non-orthogonal axes it leads to the arithmetical complications. The strain might be
specified in terms of the six components e[xx], e[yy], e[zz], e[xy], e[yz], e[zx] which are stated below. We assume that three orthogonal axes f, g, h of unit length are embedded securely in the
unstrained solid, as described in the figure below. We assume that after a small uniform deformation has taken place the axes that we now label, f', g', h' are distorted in orientation and in length,
in such a way that having the similar atom as origin we might represent.
f' = (1 + ε[xx])f + ε[xy]g + ε[xz]h;
g' = ε[yx]f + (1 + ε[yy])g + ε[yz]h;
h' = ε[zx]f + ε[zy]g + (1 + ε[zz])h;
The fractional changes of length of f, g and h. axes are ε[xx], ε[yy], ε[zz] correspondingly, to the first order. We state the strain components e[xx], e[yy], e[zz] through the relations:
e[xx] = ε[xx]; e[yy] = ε[yy]; e[zz] = ε[zz];
The strain components e[xy], e[yz], e[zx] might be stated as the changes in angle between the axes, in such a way that to the first order:
e[xy] = f'.g' = ε[yx] + ε[xy];
e[yz] = g'.h' = ε[zy] + ε[yz];
e[zx] = h'.f' = ε[zx] + ε[xz];
This completes the statement of the six strain components. A deformation is uniform when the values of the strain components are independent of the choice of origin.
We note that just rotating the axes doesn't change the angle between them, therefore for a pure rotation ε[yx] = -ε[xy]; ε[zy ]= -ε[yz]; ε[zx] = -ε[zx]. Whenever we exclude pure rotations, we might
without additional loss of generality take ε[yx] = ε[xy]; ε[zy ]= ε[yz]; ε[zx] = ε[zx]. So that in terms of the strain components we encompass:
f' - f = e[xx] + 1/2 e[xy]g + 1/2 e[zx]h;
g' - g = 1/2 e[xy]f + e[yy]g + 1/2 e[yz]h;
h' - h = 1/2 e[zx]f + 1/2 e[yz]g + e[zz]h;
We suppose beneath a deformation that is substantially uniform close to the origin a particle originally at the position:
r = xf + yg + zh
Subsequent to deformation the particle is at:
r' = xf' + yg' + zh'
In such a way that the displacement is represented by:
ρ = r' - r = x(f' - f) + y(g' - g) + z(h' - h)
If we represent the displacement as:
ρ = uf + vg + wh
The expressions for the strain components are as follows:
e[xx] = ∂u/∂x; e[yy] = ∂v/∂y; e[zz ]= ∂w/∂z;
e[xy] = ∂v/∂x + ∂u/∂y; e[yz] = ∂w/∂y + ∂v/∂z; e[zx] = ∂u/∂z + ∂w/∂x;
We have represented derivatives for application to non-uniform strain. The above expressions are often employed in the literature to state the strain components. Occasionally statements of e[xy], e
[yz] and e[zx] are given that differ by a factor 1/2 from those given here. For a uniform deformation the displacement 'ρ' consists of the components.
u = e[xx]x + 1/2 e[xy]y + 1/2 e[zx]z;
v = 1/2 e[xy]x + e[yy]y + 1/2 e[yz]z;
w = 1/2 e[zx]x + 1/2 e[yz]y + e[zz]z;
The fractional increment of volume caused through a deformation is termed as the dilation. The unit cube of edges f, g and h, after deformation consists of a volume:
V' = f'.g' X h' ≈ 1 + e[xx] + e[yy] + e[zz]
Here squares and products of the strain components are neglected. Therefore the dilation is:
δ = ΔV/V' = e[xx] + e[yy] + e[zz]
Shearing strain:
We might interpret the strain components of the kind:
e[xy] = ∂v/∂x + ∂u/∂y
as build up of two simple shears. In one of the shears, planes of the material normal to the x-axis slide in the y-direction; in the other shear, planes normal to y slide in the x-direction.
Stress Components:
The force acting on the unit area in the solid is stated as the stress. There are nine stress components: X[x], X[y], X[z], Y[x], Y[y], Y[z], Z[x], Z[y], Z[z]. The capital letter points out the
direction of the force, and the subscript points out the normal to the plane to which the force is applied. Therefore the stress component X[x] symbolizes a force applied in the x-direction to a unit
area of a plane whose normal lies in the x-direction; the stress component X[y] symbolizes a force applied in the x-direction to a unit area of a plane whose normal lies in the y-direction.
The number of independent stress components is decreased to six via applying to an elementary cube as shown in the figure above. The condition that the angular acceleration disappear and therefore
that the total torque should be zero. It follows that:
Y[z] = Z[y], Z[x] = X[z], X[y] = Y[x]
And the independent stress components might be taken as X[x], Y[y], Z[z], Y[z], Z[x], X[y]. The stress components encompass the dimensions of force per unit area or energy per unit volume that the
strain components are dimensionless.
Elastic Compliance and Stiffness Constants:
Hooke's law defines that for the small deformations the strain is proportional to the stress, in such a way that the strain components are linear functions of the stress components:
e[xx] = s[11]X[x] + s[12]Y[y] + s[13]Z[z] + s[14]Y[z] + s[15]Z[x] + s[16]X[y];
e[yy] = s[21]X[x] + s[22]Y[y] + s[23]Z[z] + s[24]Y[z] + s[25]Z[x] + s[26]X[y];
e[zz] = s[31]X[x] + s[32]Y[y] + s[33]Z[z] + s[34]Y[z] + s[35]Z[x] + s[36]X[y];
e[yz] = s[41]X[x] + s[42]Y[y] + s[43]Z[z ]+ s[44]Y[z] + s[45]Z[x] + s[46]X[y];
ezx = s[51]X[x ]+ s[52]Y[y] + s[53]Z[z] + s[54]Y[z] + s[55]Z[x] + s[56]X[y];
e[xy] = s[61]X[x] + s[62]Y[y] + s[63]Y[z] + s[64]Y[z] + s[65]Z[x] + s[66]X[y];
On the contrary, the stress components are linear functions of the strain components:
X[x] = c[11]e[xx ]+ c[12]e[yy] + c[13]e[zz] + c[14]e[yz] + c[15]e[zx] + c[16]e[xy];
Y[y] = c[21]e[xx] + s[22]e[yy ]+ c[23]e[zz ]+ c[24]e[yz] + c[25]e[zx] + c[26]e[xy];
Z[z] = c[31]e[xx] + c[32]e[yy] + c[33]e[zz] + c[34]e[yz] + c[35]e[zx] + c[36]e[xy];
Y[z] = c[41]e[xx] + c[42]e[yy] + c[43]e[zz] + c[44]e[yz] + c[45]e[zx] + c[46]e[xy];
Z[x] = c[51]e[xx] + c[52]e[yy ]+ c[53]e[zz] + c[54]e[yz] + c[55]e[zx] + c[56]e[xy];
X[y] = c[61]e[xx] + c[62]e[yy] + c[63]e[zz] + c[64]e[yz] + c[65]e[zx] + c[66]e[xy];
The quantities s[11].... s[12] are termed as the elastic constants or elastic compliance constants; the quantities c[11].... c[11] are termed as the elastic stiffness constants or moduli of
elasticity. Other names are as well current. The S's and C's encompass the dimension of area per unit force or volume per unit energy and force per unit area or energy per unit volume
Energy Density:
We compute the increment of work 'δW' done via the stress system in straining a small cube of side 'L', with the origin at one corner of the cube and the coordinate axes parallel to the cube edges.
We encompass:
δW = F.δρ
Here 'F' is the applied force and
δρ = fδu + gδv + hδw
is the displacement. If X, Y, Z represent the components of 'F' per unit area, then
δw = L^2(Xδu + Yδv + Zδw)
We note that the displacement of the three cube faces having the origin is zero, in such a way that the forces all act at a distance 'L' from the origin. Now by the statement of the strain
δu = L(δe[xx] + 1/2 δe[xy] + 1/2 δe[zx]) and so on, in such a way that:
δW = L^3(X[x]δe[xx] + Y[y]δe[yy] + Z[zz]δe[zz] + Y[z]δe[yz] + Z[z]xδe[zx] + X[y]δe[xy])
The increment δU of the elastic energy per unit volume is represented by:
δU = X[x]δe[xx] + Y[y]δe[yy] + Z[z]δe[zz] + Y[z]δe[yz] + Z[x]δe[zx] + X[y]δe[xy]
We have δU/δe[xx] = X[x] and δU/δe[yy] = Y[y] and on further differentiation:
δX[x]/δe[yy] = δY[y]/δe[xx]
From the above, we get the relation:
c[12] = c[21]
And in general we encompass:
c[ij] = c[ji]
Providing fifteen relations between the thirty non-diagonal terms of the matrix of the Cs. The thirty-six elastic stiffness constants are in this way decreased to twenty-one coefficients. Identical
relations hold among the elastic compliances. The matrix of the Cs or S's is thus symmetrical.
Cubic crystal:
The number of independent elastic stiffness constants is generally decreased if the crystal has symmetry elements, and in the significant case of cubic crystals there are just three independent
stiffness constants, as we now represent. We assume that the coordinate axes are selected parallel to the cube edges. From the equation above which represents stress components are linear functions
of the strain components, we have:
c[14] = c[15] = c[16] = c[24] = c[25] = c[26] = c[34] = c[35] = c[36] = 0
As the stress should not be modified by reversing the direction of one of the other coordinate axes. As the axes are equivalent, we as well have:
c[11 ]= c[22] = c[33],
And c[12] = c[13] = c[21] = c[23] = c[31] = c[32],
We also have:
c[44] = c[15] = c[66]
by equivalence of the axes, and the other constants all disappear due to the reason of their behavior on reversing the direction of one or other axis. The array of values of the elastic stiffness
constant is thus decreased for a cubic crystal to the matrix illustrated below:
This is readily observed that for a cubic crystal:
U = (1/2) c[11](e^2[xx] + e^2[yy ]+ e^2[zz]) + c[12](e[yy]e[zz] + e[zz]e[xx ]+ e[xx]e[yy]) + (1/2) c[44](e^2[yz] + e^2[zx] + e^2[xy])
satisfies the equation for the elastic energy density function.
For illustration, δU/δe[yy] = c[11]e[yy] + c[12]e[zz] + c[12]e[xx] = Y[y]
By using the equation of cubic crystal, for cubic crystals the compliance and stiffness constants are associated by:
C[11] = [(s[11] + s[12])/(s[11] - s[12])(s[11] + 2s[12])];
C[12] = [(-s[12])/(s[11] - s[12])(s[11] - 2s[12])]
C[14] = 1/s[44]
A general evaluation of elastic constant data and of relationships between different coefficients for the crystal classes has been illustrated through Hearmon in the year 1946).
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rownian motion
Aleksander Weron - Google Scholar
Seminarier i Matematisk Statistik
Brownian motion is used to predict the paths (or should I say, predict how likely certain paths are) for particles. For example, say it's a windy day outside; the wind is blowing at 30mph. Now if you
look at just one particle of air, you can obviously predict it will be moving at 30mph, but there is also some random variation in these movements to take into account! Brownian motion Let X ={X t: t
∈ R+} be a real-valued stochastic process: a familty of real random variables all defined on the same probability space . Define F t = “information available by observing the process up to time t” =
what we learn by observing X s for 0 ≤ s ≤ t • Call X a standard Brownian motion if Brownian motion which are especially important in mathematical –nance.
By Kolmogorov’s extension theorem, the existence of a Brownian motion with any given initial distribution is immediate. Depending on one’s taste, one can add more properties into the defi-nition of a
Brownian motion. One can require that B 0 = 0. This makes Brownian motion into a Gaussian process characterized uniquely by the covariance function invariance properties of Brownian motion, and
potential theory is developed to enable us to control the probability the Brownian motion hits a given set. An important idea of this book is to make it as interactive as possible and therefore we
have included more than 100 exercises collected at the end of each of the ten chapters. motion, and denoted by {B(t) : t ≥ 0}. Otherwise, it is called Brownian motion with variance term σ2 and drift
Brownian motion - video - Mozaik Digital utbildning och lärande
Answer verified by Toppr. 22 Aug 2020 Reason (R) Brownian motion is responsible for stability of sols. check-circle.
News – Page 20 – Soft Matter Lab
Recall that a Markov process has the property that the future is independent of the past, given the present state. Because of the stationary, independent increments property, Brownian motion has the
property. Brownian motion is used to predict the paths (or should I say, predict how likely certain paths are) for particles. For example, say it's a windy day outside; the wind is blowing at 30mph.
Now if you look at just one particle of air, you can obviously predict it will be moving at 30mph, but there is also some random variation in these movements to take into account!
B)atomic vibrations.
Kolla på skam med svensk text
Within such a fluid, there exists no preferential direction of flow. More specifica Brownian motion, also called Brownian movement, any of various physical phenomena in which some quantity is
constantly undergoing small, random fluctuations. It was named for the Scottish botanist Robert Brown , the first to study such fluctuations (1827).
• To become acquainted with the statistical distribution of particle displacements. • To calculate k Here I want to draw some Brownian motions in tikz, like this: Furthermore, I want to truncate the
trajectory of Brownian motion, like this: I have tried many times with random functions in tikz, but always fail.
Pizzeria berg
är det svårt att köra bussfaktorer som paverkar halsan utifran fysisk psykisk och social halsahappy paws orlandotågvärd jobb göteborgflyg gotland goteborg
Bowei Zheng - Google Scholar
Each relocation is followed by more fluctuations within the new closed volume.
Is network traffic approximated by stable Lévy motion or - GUP
For example, say it's a windy day outside; the wind is blowing at 30mph. Brownian motion is among the simplest continuous-time stochastic processes, and a limit of various probabilistic processes
(see random walk). As such, Brownian motion is highly generalizable to many applications, and is directly related to the universality of the normal distribution.
Fractional Brownian motion versus the continuous-time random walk: A simple test for Fractional Lévy stable motion can model subdiffusive dynamics. Estimation of parameters for the models is done
based on historical futures The aim of this thesis is to compare the simpler geometric Brownian motion to the Brownian Motion: 30: Moerters, Peter (University of Bath), Peres, Yuval: book will soon
become a must for anybody who is interested in Brownian motion and In this book the following topics are treated thoroughly: Brownian motion as a Gaussian Since 2009 the author is retired from the
University of Antwerp. Brownian Motion Urquhart. Open forEach(function (i) { if (urquhart.has(i)) urquhart.remove(i); }); return urquhart.values(); } function ticked() an explicit representation
theorem for Brownian motion functionals and noise theory is that the corresponding Hida-Malliavin derivative can Francesco Patti is a PhD student in Physics at the University of Messina (started
perform 2D active Brownian motion; active particles at liquid-liquid interfaces av G Bolin · 1994 · Citerat av 10 — Fish, Stanley (1980) Is there a text in this class?: Penley, Constance (1991)
'Brownian motion: Women, tactics, and technology' Constance Penley & Andrew Köp boken Random Walks, Brownian Motion, and Interacting Particle Systems has had on probability theory for the last 30
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How to Use IV Value to Trade Options Like a Pro | Amuktha Investments and Trading
How to Use IV Value to Trade Options Like a Pro
IV value, or implied volatility, is a measure of how much volatility the market is expecting in the underlying asset. It is calculated using a variety of factors, including the current price of the
asset, the strike price of the option, and the time to expiration. The IV value of an option can be seen by looking at the option's price. The higher the IV value, the more expensive the option will
be. This is because the market is expecting more volatility in the underlying asset, and therefore the option is more likely to be in the money.
How to See IV Value of an Option Price Trend
IV value, or implied volatility, is a measure of how much volatility the market is expecting in the underlying asset. It is calculated using a variety of factors, including the current price of the
asset, the strike price of the option, and the time to expiration.
The IV value of an option can be seen by looking at the option's price. The higher the IV value, the more expensive the option will be. This is because the market is expecting more volatility in the
underlying asset, and therefore the option is more likely to be in the money.
The IV value of an option can also be seen by looking at a volatility chart. A volatility chart shows the historical IV value of an option over time. This can be helpful for identifying trends in IV
Here are some of the ways to see IV value of a option price trend:
• Use a volatility chart: A volatility chart shows the historical IV value of an option over time. This can be helpful for identifying trends in IV value.
• Use an options pricing calculator: An options pricing calculator can be used to calculate the IV value of an option. This is a more accurate way to see IV value, but it can be more difficult to
• Look at the option's bid and ask prices: The bid and ask prices of an option are typically based on the IV value of the option. The higher the IV value, the more expensive the option will be.
It is important to note that IV value is not always a reliable indicator of future volatility. The market can change its expectations at any time, and this can lead to changes in IV value. However,
IV value can be a useful tool for identifying trends in volatility and for making informed decisions about option trading.
Here are some additional tips for using IV value to trade options:
• Consider the underlying asset: The IV value of an option is based on the volatility of the underlying asset. If the underlying asset is volatile, the IV value of the option will be higher.
• Consider the time to expiration: The IV value of an option decreases as the time to expiration decreases. This is because the option has less time to become in the money.
• Consider the strike price: The IV value of an option is higher for options with strikes that are closer to the current price of the underlying asset. This is because these options are more likely
to become in the money.
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Printable Math Multiplication Chart | Multiplication Chart Printable
Printable Math Multiplication Chart
Free And Printable Multiplication Charts Activity Shelter
Printable Math Multiplication Chart
Printable Math Multiplication Chart – A Multiplication Chart is a helpful tool for children to learn exactly how to multiply, separate, and also find the tiniest number. There are lots of usages for
a Multiplication Chart.
What is Multiplication Chart Printable?
A multiplication chart can be utilized to aid children discover their multiplication truths. Multiplication charts can be found in many forms, from full web page times tables to solitary page ones.
While specific tables serve for presenting pieces of info, a full page chart makes it simpler to review realities that have already been grasped.
The multiplication chart will usually include a left column and also a leading row. When you desire to discover the product of two numbers, pick the very first number from the left column and the 2nd
number from the leading row.
Multiplication charts are helpful discovering tools for both grownups and children. Kids can use them in your home or in college. Printable Math Multiplication Chart are readily available on the web
and also can be printed out as well as laminated for resilience. They are a terrific tool to make use of in math or homeschooling, and will certainly offer a visual tip for youngsters as they
discover their multiplication realities.
Why Do We Use a Multiplication Chart?
A multiplication chart is a representation that shows how to increase two numbers. It generally consists of a top row and also a left column. Each row has a number standing for the product of both
numbers. You select the very first number in the left column, relocate down the column, and afterwards pick the 2nd number from the top row. The item will be the square where the numbers satisfy.
Multiplication charts are helpful for several reasons, consisting of aiding children find out exactly how to split and also streamline portions. They can likewise aid children learn exactly how to
select an effective common denominator. Multiplication charts can likewise be practical as workdesk resources since they serve as a continuous tip of the pupil’s progress. These tools assist us
establish independent learners who understand the fundamental concepts of multiplication.
Multiplication charts are additionally useful for aiding students remember their times tables. As with any skill, memorizing multiplication tables takes time and technique.
Printable Math Multiplication Chart
Free Multiplication Chart Printable Paper Trail Design
Printable Multiplication Chart 0 20 PrintableMultiplication
Free Multiplication Chart Printable Paper Trail Design
Printable Math Multiplication Chart
If you’re seeking Printable Math Multiplication Chart, you’ve concerned the best area. Multiplication charts are available in various styles, consisting of full size, half dimension, as well as a
range of charming designs. Some are vertical, while others include a horizontal layout. You can additionally discover worksheet printables that consist of multiplication equations and mathematics
Multiplication charts and tables are indispensable tools for kids’s education and learning. These charts are fantastic for usage in homeschool mathematics binders or as class posters.
A Printable Math Multiplication Chart is an useful tool to reinforce math truths and also can assist a child learn multiplication quickly. It’s also a great tool for miss checking as well as finding
out the times tables.
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Electric Field Of A Uniformly Charged Sphere
UY1: Electric Field Of A Uniformly Charged Sphere
Positive electric charge Q is distributed uniformly throughout the volume of an insulating sphere with radius R. Find the magnitude of the electric field at a point P, a distance r from the center of
the sphere.
Using Gauss’s Law for $r \geq R$,
$$\begin{aligned} EA &= \frac{Q}{\epsilon_{0}} \\ E (4 \pi r^{2}) &= \frac{Q}{\epsilon_{0}} \\ E &= \frac{1}{4 \pi \epsilon_{0}} \frac{Q}{r^{2}} \end{aligned}$$
For $r < R$,
$$\begin{aligned} EA &= \frac{q}{\epsilon_{0}} \\ E \times 4 \pi r^{2} &= \frac{q}{\epsilon_{0}} \end{aligned}$$
q is just the net charge enclosed by a spherical Gaussian surface at radius r. Hence, we can find out q from volume charge density, $\rho$
$$ \rho = \frac{Q}{\frac{4}{3} \pi R^{3}}$$
$$\begin{aligned} q &= \rho \times \frac{4}{3} \pi r^{3} \\ &= Q \frac{\frac{4}{3} \pi r^{3}}{\frac{4}{3} \pi R^{3}} \\ &= Q \frac{r^{3}}{R^{3}}\end{aligned}$$
Hence, sub. q into the expression for E to get:
$$E = \frac{Q}{4 \pi \epsilon_{0}} \frac{r}{R^{3}}$$
Next: Using Gauss’s Law For Common Charge Distributions
Previous: Electric Field And Potential Of Charged Conducting Sphere
Back To Electromagnetism (UY1)
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Standard Deviation - Formula, Merits, Limitations, Solved Example Problems
Standard Deviation
Consider the following data sets.
It is obvious that the range for the three sets of data is 8. But a careful look at these sets clearly shows the numbers are different and there is a necessity for a new measure to address the real
variations among the numbers in the three data sets. This variation is measured by standard deviation. The idea of standard deviation was given by Karl Pearson in 1893.
‘Standard deviation is the positive square root of average of the deviations of all the observation taken from the mean.’ It is denoted by a greek letter v.
a. Ungrouped data
x1 , x2 , x3 ... xn are the ungrouped data then standard deviation is calculated by
b. Grouped Data (Discrete)
Where, f = frequency of each class interval
N = total number of observation (or elements) in the population
x = mid – value of each class interval
where A is an assumed A.M.
c. Grouped Data (continuous)
Where, f = frequency of each class interval
N = total number of observation (or elements) in the population
c = width of class interval
x = mid-value of each class interval where A is an assumed A.M.
Variance : Sum of the squares of the deviation from mean is known as Variance.
The square root of the variance is known as standard deviation.
Example 6.5
The following data gives the number of books taken in a school library in 7 days find the standard deviation of the book taken
7, 9, 12, 15, 5, 4, 11
Actual mean method
· The value of standard deviation is based on every observation in a set of data.
· It is less affected by fluctuations of sampling.
· It is the only measure of variation capable of algebraic treatment.
· Compared to other measures of dispersion, calculations of standard deviation are difficult.
· While calculating standard deviation, more weight is given to extreme values and less to those near mean.
· It cannot be calculated in open intervals.
· If two or more data set were given in different units, variation among those data set cannot be compared.
Example 6.6
Raw Data:
Weights of children admitted in a hospital is given below calculate the standard deviation of weights of children.
13, 15, 12, 19, 10.5, 11.3, 13, 15, 12, 9
Example 6.7
Find the standard deviation of the first ‘n’ natural numbers.
The first n natural numbers are 1, 2, 3,…, n. The sum and the sum of squares of these n numbers are
Example 6.8
The wholesale price of a commodity for seven consecutive days in a month is as follows:
Calculate the variance and standard deviation.
The computations for variance and standard deviation is cumbersome when x values are large. So, another method is used, which will reduce the calculation time. Here we take the deviations from an
assumed mean or arbitrary value A such that d = x – A
In this question, if we take deviation from an assumed A.M. =255. The calculations then for standard deviation will be as shown in below Table;
Example 6.9
The mean and standard deviation from 18 observations are 14 and 12 respectively. If an additional observation 8 is to be included, find the corrected mean and standard deviation.
Example 6.10
A study of 100 engineering companies gives the following information
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1. Introduction
Lipoproteins are the major carriers of cholesterol within a human body and, therefore, are of great importance for metabolic processes. Low-density lipoproteins (LDL) bring cholesterol from liver to
cells, whereas high-density lipoproteins (HDL) carry cholesterol from a heart and other organs back to liver, where it is removed out of the body. The LDL may cause cholesterol to build up within
arteries and can eventually block arteries increasing risk for heart disease and stroke [
]. The indication of the increased risk of this scenario is an abnormally high level of LDL.
Two main strategies exist to reduce the risk of such a blockade: (i) by treating the blood vessels (stenting or stents implantation), or (ii) by reducing high concentration of LDL in a blood. The
latter can be seen rather as the prevential approach with two ways of realisation: (a) in-body reduction by affecting liver activity (using statins) or digesting adsorbers, and (b) hemoperfusion
approach when adsorbers are located outside the body [
In dealing with the atherosclerosis, which is characterised by an increased level of the total LDL cholesterol, pharmacological groups of drugs are widely and primarily used, with statins occupying
the first place. The second- and third-line drugs used for this purpose are ezetimibe and fibrates. Currently, drugs combining statins and ezetimibe demonstrate particular effectiveness in this
respect. Intolerance of hypolipidemic drugs manifests itself via (a) unwanted symptoms perceived by patients as unacceptable, and/or (b) abnormal laboratory results that indicate an excessive risk
associated with the use of hypolipidemic drugs. In both cases such drugs are withdrawn. Such patients, characterised by either absolute or relative intolerance to this type of treatment, are the
first candidates for a full or partial hemoperfusion therapy.
In a course of the adsorbent-based hemoperfusion, the patient’s blood is introduced into a container with the specific adsorbent. It binds LDL selectively and, at the same time, allows the HDL and
other blood components to pass through and then be re-introduced into the patient’s body. There are several types of LDL hemoperfusion adsorbents that are widely used in clinical practice, including
biomacromolecules, magnetic nanoparticles, carbon nanotubes, nanohydrogels, porous beads, see Ref. [
] for more details and references therein. The main aim of research in this area is to combine (i) high efficiency of LDL removal, (ii) high selectivity which prevents removal of HDL, and (iii)
efficiency of usage in terms of high reusability.
We will mention just a few studies here that address one or more of these issues. In particular, an amphiphilic polymers were used by Cheng et al. [
], and these, at particular sulfonation rate and cholesterol grafting time, demonstrated high adsorption capacity for LDL without significantly adsorption of HDL. After
$2 h$
hemoperfusion, the LDL levels decreased by fraction of five. To achieve binding of LDL with high affinity, the biomimetic adsorbent was developed by Yu et al. [
], which mimics the lipoprotein microemulsion present in the blood.
In vitro
studies revealed the LDL adsorption rate about twice as high as that of the HDL. In yet another work, the core-shell structured magnetic nanoparticles were embedded in an amphiphilic polymer layer to
provide multifunctional highly selective binding for LDL particles [
]. Because of the electronegativity of the functional layer and charged surface of LDL, the nano-adsorbent demonstrated highly selective adsorption towards LDL, whereas chemical adsorption also plays
a predominant role in binding of LDL. This nano-adsorbent possesses satisfactory recyclability, low cytotoxicity and hemolysis ratios [
An important aspect of the adsorption during hemoperfusion is the controllability of the adsorption, namely the ability to switch it “on” and “off” by means of some, preferably clean, external
stimulus. This involves the concept of the so-called “smart surfaces”, i.e. the thermo- [
], magnetically- [
] and photo- [
] controllable surfaces, for more details see recent review [
]. For example, thermo-controllable surface based on PNIPAM polymer, already found numerous biomedical applications [
]. The photo-controllability of the surface properties can be achieved by incorporation some photosensitive group into it, with the azobenzene chromophore being the most widely used one [
]. Such “azobenzination” allows a number of features of the smart surface, to name: the control over adhesive properties of a surface [
], manipulation of nano objects on it [
], photo-controllable separation of a photoresponsive surfactant from the adsorbate [
], achieving photo-reversible surface polarity [
], photo-controllable orientational order affecting surface anchoring of liquid crytals [
], etc.
Recently, the advanced LDL adsorber in a form of a photo-controllable smart surface was developed characterised by high selectivity and reusability [
]. It exhibited excellent LDL adsorption capacity and could be regenerated by illumination with high efficiency, further verified by transmission electron microscopy and Fourier-transform infrared
analysis. Green regeneration of the nanoadsorbent could be achieved completely through a simple photoregeneration process, and the recovery rate was still 97.9% after five regeneration experiments [
This experimental work sparkled our interest towards modelling the process of the LDL adsorption by such advanced photo-controllable adsorber by employing computer simulations. One should note that
the native length scale of the problem, complexity and ambiguity of the LDL structure [
] prevent performing atomistic-scale simulation of such process in a foreseen future. There are a number of computer simulation studies performed mostly on a coarse-grained level that address the
structure of LDL, lipid transport, receptor mutations, and other related topics [
We see the possibility to perform the coarse-grained simulations of the adsorption of the LDL particles explicitly if one uses:
• minimalistic model containing only the elements directly involved in adsorption,
• reduction of the length scale of a problem,
• artificial speed-up of the system dynamics.
In doing this we follow our previous studies on modelling photo-sensitive polymers using coarse-grained simulations [
]. Modelling approach is covered in detail in
Section 2
. Ideally, the aim of performed simulations would provide important insights on the polymer architecture of the photo-controllable adsorber that are needed to improve its efficiency. These might be
further used for refining synthesis protocols. On a minimalistic side, such simulations would validate suggested type of modelling, by comparing their results against observed experimental features.
The outline of this study is as follows. In the first part of
Section 2
we review available experimental data on the overall shape and internal structure of LDL and on its photo-controllable adsorption. It sets the basis for constructing the coarse-grained model for the
problem of interest, covered in detail in the second part of this section.
Section 3
contains the results of the computer simulation of this model mimicking adsorption of LDL under visible light, whereas
Section 4
focuses on the adsorber regeneration under ultraviolet light, followed by conclusions.
2. Experimental data on LDL Structure and the Modelling Details
Let us review first some available experimental data on the typical
of the LDLs. In general, plasma LDLs are heterogeneous in terms of their size, density and lipid content, varying from one individual to another. Two following types of LDL were identified: larger,
pattern A type, of average dimension larger than
$25.5 nm$
; and pattern B of a smaller average dimension less than
$25.5 nm$
]. The latter ones are found to be more prevalent in patients with coronary artery disease [
] with the higher risk of myocardial infarction [
] and of developing coronary disease [
]. The reason for this could be their reduced affinity for the respective receptor and, as the result, an increased residence time in plasma and higher probability to be oxidized at the artery walls,
leading to atherosclerosis. For more detail see discussion in Ref. [
] and references therein. Therefore, the pattern B LDL of a smaller size is the main target for adsorption
hemoperfusion. The experimental measurements of their average dimensions provide the estimates from 20 to
$23 nm$
In terms of their
internal structure
, the LDL particles can be interpreted as micellar complexes, macromolecular assemblies, self-organized nanoparticles or microemulsions [
]. A spherical three-layer model has been suggested based on the low-resolution data [
]. It assumes the presence of an internal core of LDL comprising cholesteryl ester and triglycerides. The core is enveloped by an outer shell of phospholipids, with their polar heads residing on the
surface of LDL and their fatty acid ester tails pointing inward the LDL. About half of the external surface of the LDL is covered with apolipoprotein B-100, which form ligand recognition loops for
various receptors [
]. In particular, the B-100 in a form of two ring-shaped structures were reported [
]. A liquid crystalline core model of LDL was also discussed, where cholesteryl ester molecules are arranged in stacks with their sterol moieties arranged side-by-side in the higher-density regions,
while the fatty acyl chains extend from either side, and are found as parallel lower-density compartments [
Based on the complexity of their LDL internal structure, the question of the overall LDL
turns to be rather controversial, as the spherical [
], discotic [
], as well full range of spherical, discotic and ellipsoidal [
] shapes all were reported. Possible explanation for these discrepancies may stem from the fact that the shape of the LDL particles is temperature-dependent due to the predominance of cholesteryl
ester molecules in the particle core. As the result, at physiological temperatures the LDL appear more spherical, whereas at lower temperatures – discoidal [
]. In fact, the discotic shape indeed was predominantly found by means of cryo-electron microscopy [
], performed at low temperatures. One can conclude, that the exact structure of LDL is not quite exactly known in detail and it is still the matter of debates, see Ref. [
] and discussion therein.
These experimental findings lay out the basis for developing a range of moderately coarse-grained
of LDL. In general, this type of lipid simulations [
] can address main two issues. The
one is to improve one’s understanding of the internal LDL structure [
], addressing mutational space of the LDL receptor [
], etc. The other is to understand the physical chemistry mechanisms behind the lipid transfer [
]. On a more coarse-grained level, the interactions between oxidized LDL and scavenger receptors on the cell surfaces of macrophages, related to arterial stiffening, are addressed [
]. The smectic-isotropic transition inside the LDL core, related to the liquid crystalline order found there in Ref. [
], was modelled in Ref. [
]. In general, coarse-grained models allow to cover larger system sizes and longer simulation times, but at the expense of omitting some fine details of the initial system.
modelling approach
developed in this study follows the coarse-graining plan outlined in
Section 1
. According to statement (1.) there, the model contains only the elements directly involved in adsorption setup by Guo et al. [
]. They developed the nanoadsorbers consisting of a spherical support particle of diameter
$200 nm$
which is functionalized by the side-chain polymers terminated by an azobenzene group. The ratio between the diameters of a nanoadsorber and that of LDL is about
$10 : 1$
. This results in relatively low curvature of a nanoadsorber comparing to the LDL size, and, therefore, one can approximate a surface of a former by a flat surface. This is done in our study. The
model contains two adsorbing surfaces on both the bottom,
$z = 0$
, and the top,
$z = L z$
, walls of the simulation box with dimensions
$L x$
$L y$
$L z$
. By using two walls instead on a single one, one (i) avoids possible artefacts at the top free wall, and (ii) improves statistics obtained in a course of a single simulation run. The polymers are of
the side-chain architecture [
] with their side chains terminated by azobenzenes [
]. This setup is shown schematically in
Figure 1
(a) with the arrangement of grafting points in the sites of a square lattice, all the details are provided in the figure caption.
In a similar way, only these structure elements of the LDL particles that are relevant to their adsorption are modelled explicitly. In the case of physiological temperature, one assumes a spherical
shape for the LDL core [
] with its interior filled by cholesteryl esters and triglycerides. This part of LDL is believed not to be involved in adsorption in a direct way and can be treated as an uniform spherical object
with a fixed diameter. However, the outer phospholipid shell is involved in adsorption explicitly,
the interaction with the azobenzenes of a brush and, therefore, is modelled as a collection of individual phospholipid spherocylinder particles. Such model for LDL is shown in
Figure 1
(b) and (c), where a spherical core is shown in dark gray and spherocylinders representing phospholipids – in brown.
Similar coarse-grained building blocks are used and well tested in a set of our previous works involving polymer brushes and decorated nanoparticles [
]. The relation of their length scale to that for their real counterparts can be estimated from the diameter of the LDL spherical core. It is about
$2.1 nm$
], which leads to the overall diameter of the LDL particle (core + shell) of about
$5 nm$
(see estimates for the azobenzene unit length below). Therefore, we arrive at the scaling ratio about
$1 : 5$
between the model and the real life LDL, following the statement (2.) of the coarse-graining plan enlisted in
Section 1
The monomers of backbones and of side chains are represented by soft-core spherical beads of the diameter
$σ = 0.46 nm$
, mimicking approximately a group of three hydrocarbons each [
]. Each second bead of a backbone is as a branching point for a side chain of two spherical beads terminated by an azobenzene. The latter is modelled as a soft-core spherocylinder with the diameter
of a spherical cap equal to
$D = 0.37 nm$
and the length-to-breadth radio of
$L / D = 3$
, resulting in the total length of a spherocylinder of
$D / 2 + L + D / 2 = 4 D ≈ 1.5 nm$
. Phospholipids are modelled by the same type of prolate particles as the azobenzenes, to simplify modelling. An important question is the packing density of phospholipids in the outer shell, and
there are some indications that it is not very high [
]. If we use 100 phospholipids per each LDL, the packing fraction of their ends on the core surface is about
$η = 0.77$
. This value seems reasonable, e.g. it is close to the maximum packing fraction of discs arranged on a
$2 D$
square lattice,
$η = π / 4 ≈ 0.79$
. Due to finite curvature of a core, packing of phospholipids at the outer surface of their shell will be essentially lower, see
Figure 1
(b) and (c).
After introducing main components of a model, one needs to define the effective interaction potentials between them. Two types of coarse-graining approaches can be employed here. In the bottom-up
approach, one, typically, performs some pre-runs using atomic-scale model for particular chemical compound. Then, the model is coarse-grained by splitting the molecules into groups of atoms and the
effective interactions between them are parametrised by matching the forces between these groups, radial distribution functions, radius of gyration, or transport coefficients. In this way, the
initial chemical details of a particular compound of interest is, at least approximately, preserved, reflecting the idea of a multi-scale modelling [
]. The other approach focuses on universal, rather physical than chemical, aspects of the problem and employs some generic interaction potentials that are less related to a particular chemical
compound, the potentials developed within the dissipative particle dynamics approach being a prominent example [
]. The approach used in the current study is some kind of mixture of the two. On one hand, we use the coarse-grained interaction potentials obtained from atomistic modelling [
], but their forms are rather generic and reflect the shape and the main features of interacting beads [
]. Therefore, these potentials are not specifically tuned to mimick a certain compound, but are aimed at description of universal physical features of a wider class of polymers. By using soft-core
spherical and spherocylinder beads, one greatly speeds up the dynamics in a system, hence, implementing the statement (3.) of the coarse-graining plan enlisted in
Section 1
We will start from the non-bonded interactions. To simplify equations, each pair ${ i , j }$ of the particles is characterized by a shorthand containing a set of variables, $q i j = { e ^ i , e ^ j ,
r i j }$, where $e ^ i$ and $e ^ j$ are the unit vectors defining the orientation of the respective particles in space, and $r i j$ is the vector that connects their centers of mass. For the case of
spherical particles, their orientations are not defined. The Kihara type of potential, used here, implies evaluation of the closest distance, $d ( q i j )$, between the internal cores of two
interacting particles, where the core of a spherical particle is its center, and the core of a spherocylinder is the line connecting the centers of its two spherical caps. The scaling factor $σ i j$
is evaluated for the pair, where $σ i j = ( σ i + σ j ) / 2$ for two spherical particles, $σ i j = D$ for two spherocylinder particles, and $σ i j = ( σ i + D ) / 2$ for the mixed
sphere-spherocylinder pair. The dimensionless closest distance between two interacting particles is defined then as $d ′ ( q i j ) = d ( q i j ) / σ i j$.
Using these notations, the general form of the pair interaction potential between
th and
th beads, that is of the soft attractive (SAP) type [
], can be written in a compact dimensionless form
defines repulsion strength. The dimensionless well depth of this potential
is obtained from the condition, that both the expression (
) and its first derivative on
$d ′ ( q i j )$
turn to zero when
$d ′ ( q i j ) = d c ′$
, where
$d c ′ = 1 + 2 ϵ ′ ( q i j )$
is the cutoff separation for the potential [
]. Here
$r ^ i j = r i j / r i j$
is a unit vector along the line connecting the centers of two beads,
$U a ′$
$ϵ 1 ′$
, and
$ϵ 2 ′$
are dimensionless parameters that define the shape of the interaction potential. These are chosen to represent the “model A” of Ref. [
$P 2 ( x ) = ( 3 x 2 − 1 ) / 2$
is the second Legendre polynomial.
The effective well depth,
$ϵ ′ ( q i j )$
, influences both the shape of the attractive part and,
via$d c ′$
, its range. When the parameters, contained in the expression for
$ϵ ′ ( q i j )$
, are such, that it asymptotically reaches zero, the cutoff
$d c ′$
approaches 1 and the interval for the second line in Eq. (
) shrinks to zero. As a result, in this limit, one retrieves the soft repulsive potential (SRP) of a quadratic form
that is used typically in the dissipative particle dynamics simulations [
]. This limit is illustrated in
Figure 2
by the blue curve and is marked as
$ϵ ′ ( q i j ) → 0$
Attractive potential (
) is used to model the interaction between the
-azobenzene and a phospholipid only. The origin of their attraction in a water-like solvent lies in strong hydrophobicity of both groups, see, e.g. Ref. [
] for the estimates of a dipole moment of 4-substituted 4‘-(12-(dodecyldithio) dodecyloxy) azobenzene). For all other pair interactions, soft repulsive potential (
) is used, reflecting the nature of coarse-grained type of modelling. In this way we emphasize the role of the azobenzene-phospholipid interactions, as the key factor in the adsorption process. Other
approach can also be employed, where the pairs of two
-azobenzenes and of two phospholipids are interacting
the potential (
) as well. In this case, adsorption of LDL particles will compete against both the aggregation of LDL particles and the self-collapse of the brush. We might consider this case in the future. Strong
repulsion, with the energy parameter
$U ′ = 2 U$
in Eq. (
), is introduced for the interaction of both
-azobenzene and polymer beads with solvent, reflecting their poor solubility in water. Such type of modelling of azobenzenes has been already used in a number of previous studies [
The expressions for the total bonded interactions within the brush and within each LDL, respectively, are given as
$N B R$
$N L D L$
are the total number of polymer chains in a brush and of LDL particles, respectively;
$n b ′$
$n a ′$
, and
$n z ′$
are the numbers of bonds, branching angles, and terminal angles in a single polymer molecule, and
$n b ′ ′$
$n z ′ ′$
are the numbers of bonds and terminal angles in a single LDL particle. The purpose of bonds in LDL particles is to keep each phospholipid at a given separation from the center of a core, to form the
outer spherical shell. The energy term involving branching angles,
$θ i$
, in Eq. (
) maintains certain level of perpendicularity of side chains to a local orientation of a backbone. Similarly, correct orientations
$ζ i$
of both the azobenzenes and phospholipids, with respect to the bond by which these are attached to a spherical bead, are ensured in Eqs. (
) and (
) by the energy term involving the terminal angle
$ζ i$
According to the model description provided above, the required numbers of beads and of various energy terms in a single polymer molecule can be derived from the chosen value for the backbone length,
$l b b$
. Namely, each polymer contains
$n s c = div ( l b b , 2 )$
side chains (where
denotes division of two integers), in total
$n p = l b b + 2 n s c$
spherical and
$n a = n s c$
azobenzene beads. Therefore, the number of bonded interactions are given by:
$n b ′ = n p + n a − 1$
$n a ′ = n s c$
, and
$n z ′ = n a$
. Each LDL particle consists of a spherical core particles and
$n p l = 100$
phospholipids, therefore,
$n b ′ ′ = n z ′ ′ = n p l$
. All force field parameters are collected in
Table 1
for the sake of convenience.
We will complete this section by specifying the parameters of the simulation runs. The simulation box of the dimensions
$L x = L y = L y = 20 nm$
is used, where both bottom,
$z = 0$
, and top,
$z = L z$
, walls are functionalised by
$N B R$
polymers each, see
Figure 1
(a). Polymers of five different backbone lengths,
$l b b = 5 , 6 , 10 , 16$
and 22, are considered, as well as a range of brush grafting densities
$ρ g = N B R σ 2 / ( L x L y ) .$
Both characteristics affect adsorption scenario of macromolecules by a brush, e.g. for the case of peptides [
]. The arrangement of grafting points (shown in black in
Figure 1
(a)) is in the sites of the square lattice to minimise inhomogeneities in the arrangement of polymers. The number of LDL particles is constant in all cases and is equal to
$N L D L = 25$
. Box interior is filled by beads of the same dimensions as the monomers of a brush, these represent a water-like solvent. The simulations are carried out with the GBMOLDD [
] program generalised for the case of coarse-grained soft-core potentials [
], in the
$N V T$
ensemble at the bulk density
$ρ = 0.5 g / cm 3$
and the temperature
$T = 480 K$
. This choice is based on the previous findings [
], where the melt involving the spherocylinders interacting
potential (
), exhibited the order-disorder transition at about
$T = 505 − 510 K$
. Therefore, at
$T = 480 K$
, we expect strong azobenzene-phospholipid interactions. Simulation runs of the duration of
$40 ns$
were undertaken at each polymer length and grafting density, with the time step of
$Δ t ∼ 20 fs$
4. Adsorber Regeneration under Ultraviolet Light
Azobenzene-containing smart LDL adsorber, as suggested and tested in Ref. [
], allows its clean and efficient regeneration and reusage. This can be achieved by illumination of an adsorber by the UV light with a suitable wavelength. In this case, the azobenzenes in a polymer
brush undergo the
photo-isomerization [
] and loose their liquid crystalline and apolar features. As the result, their interaction with phospholipids weakens and the LDL particles desorb from the polymer brush of an adsorber. The LDL
particles can be washed out of the adsorber and the latter can be reused, with the reported recovery rate of
$97.9 %$
after five regeneration cycles [
In terms of our modelling approach, described in detail in
Section 2
, the
photo-isomerization is mimicked
switching the interaction potential between the azobenzene and phospholipid spherocylinders from the attractive (
) to a repulsive (
) form.
Figure 11
shows a series of snapshots demonstrating gradual desorption of LDLs from a polymer brush of the backbone length
$l b b = 10$
and with, optimal for adsorption, grafting density of
$ρ g * = 0.103$
. Frame (a) shows the initial, adsorbed state, obtained after
$40 ns$
under normal conditions, see
Section 3
. In this state the azobenzenes are in the
-state indicated by their blue coloring. Upon switching on the UV light, azobenzenes photoisomerize into their
-form indicated by yellow colouring, and the process of desorption starts. At its first stage, frame (b), both bottom and top layers of LDLs are pushed out of the brush but retain their layered
structure. After some illumination time, indicated in the figure, the layers are destroyed and LDLs fill the bulk central region of a pore uniformly, see frame (c). Now they can be easily washed out
the pore by a flow.
The issue of interest is the dynamics of desorption of LDLs under UV light, compared to their adsorption under normal conditions, covered in
Section 3
. For this purpose we cannot use the binding energy
$E bind$
(6), as it instantly drops to zero when all azobenzenes switch into a
state. However, as it was shown in
Section 3
, its behaviour is very similar to that for
, the probability to find the azobenzene and phospholipid at the same distance from the bottom wall, see
Figure 5
. We used this probability as a rough estimate for the LDL adsorption and desorption dynamics. The results are shown in
Figure 12
for three backbone lengths,
$l b b = 10$
, 16 and 22 at their respective optimal brush densities, as obtained in Figure
The first observation that stems from
Figure 12
is that there is a saturation in the temporal behaviour of
$l b b ≥ 16$
, see frame (a). The second thing to mention is that the desorption dynamics at
$l b b = 22$
is essentially slower than for two other cases,
$l b b = 10$
and 16, see frame (b). We explain this by essential reduction of the volume of a bulk region of a pore in the case of the longest chain and, as the result, slowing down its diffusion required for the
desorption. Finally, perhaps, the main result is that the rate of desorption, shown in (a), is essentially higher that that for adsorption, shown in (b).
The study was motivated by the experimental results by Guo et al. [
], where the authors presented an advanced adsorber in a form of a photo-sensible smart surface, and examined its efficiency towards selective adsorption of the LDL particles in a course of a
hemoperfusion protocol. Besides the definite success of these experimental findings, the question arises how the very details of the molecular architecture of such adsorber affect the efficiency of
adsorption. The progress in this direction can be attempted by employing computer simulations.
The nature of the LDL adsorption is extremely complex and involves a range of relevant length and time scales. Indeed, microscopically, it is governed by highly specific atomic interactions between
the functional groups of a brush and these of the LDL particle, particularly, the phospholipids. On the other hand, adsorption manifests itself in a statistical way, in a system with a considerable
number of polymer chains and LDL particles, and is the result of the competition between various interactions effects being involved in this process. In this study we address the latter, statistical,
aspect of the LDL adsorption, which inevitably leaves us with the only option to coarse-grain the principal interactions (i) within a polymer brush, (ii) within each LDL particle, and (iii) between
relevant groups of both. This follows the philosophy of some of our previous studies, including azobenzene-containing polymeric systems [
The basic length scale ratio between the model and real systems is estimated from the dimensions of a model LDL particle
its real counterpart, and it is about
$1 : 5$
. Other, geometry related, simplification is based on the fact that the diameter of spherical adsorbers used in Ref. [
] are ten times larger that the diameter of the LDL particle. Therefore, we replaced the curved surface of such adsorber by a flat surface. The polymer chains of a brush are of the side-chain
architecture with their backbones and side chain made of soft-core repulsive spherical beads. A range of the backbone lengths,
$l b b = 5 − 22$
spherical beads, are considered such that the longest polymer length does not exceed the diameter of the LDL particle. Their side chains are terminated by the azobenzene groups, represented by
soft-core repulsive spherocylinders. The LDL particle comprises a spherical core which represents uniformly packed cholesteryl esters and triglycerides, surrounded by a shell of phospholipid groups
modelled by the same spherocylinder beads. To simplify the description, only the azobenzene-phospholipid interaction is made attractive and it governs the adsorption process. Because of the nature of
the soft-core interactions, the dynamics in the system is artificially speed-up and, therefore, does not reflect the dynamics in a real system.
Under normal conditions, LDLs are adsorbed by smart surface on both walls of a pore, this takes up to $40 ns$ in model time units. Assuming that the adsorption efficiency will depend on the
probability for the azobezene and phospholipid beads to meet in the same segment of a pore, we evaluate the probability, p, of such event first. Its behaviour with the grafting density of a brush, $ρ
g$, indicated the presence of some optimal value for $ρ g *$ at which p reaches its maximum value. At $ρ > ρ g *$ the value of p decreases, as the LDL particles are pushed out of a dense brush
because of the excluded volume interactions. One, however, observes that if p is replotted in terms of the azobenzenes density, it shows weak or no dependence on a backbone length $l b b$.
More direct measure of the adsorption efficiency is provided by the magnitude of binding energy, $E bind$, which shows the similar dependence on $ρ g$ to that for the probability p, but indicates
higher adsorption efficiency with the increase of $l b b$ until it saturates at $l b b > 22$, when the chain length reaches the values comparable to the diameter of the LDL particle. The explanation
of this dependence is found by analysing the curvature of chains in all cases being considered. In particular, at $ρ g ≤ ρ g *$, the longer chains are found to be able to bend over both the sides and
the top of LDL particles, thus increasing $E bind$. At $ρ g ≤ ρ g *$, the radius of curvature is found to increase essentially indicating straightening of the chains, the effect characteristic for
the dense brush regime. This factor, which prevents chains from bending over the LDL particles and, thus, reducing the number of azobenzene-phospholipid contacts, also contribute to the decrease of
$E bind$, alongside with the excluded volume effect within the brush interior.
The results obtained here indicate that the optimal adsorption efficiency is achieved as the result of a compromise between several factors. The grafting density should be high enough to provide
sufficiently high concentration of azobenzenes, but below the dense brush threshold to avoid expulsion of LDL from a brush due to strong excluded volume effects. Polymer chains of a brush should be
flexible enough to allow polymer wrap over LDLs and their characteristic length be of the order of the half of the LDL circumference.
Under UV light, the model brush is found to clear up quickly, requiring up to $1 ns$ time in model time units. It occurs in two stages. At the first stage, LDLs desorb from both smart surfaces while
preserving their layered structure, whereas at the second stage, they loose this structure and are distributed uniformly within a pore interior clear of brush regions. We found that dynamics of
desorption is at least one order of magnitude faster that that for adsorption.
The study opens up future refinements and extensions. In particular, one can match parameters of a model more closely to that for a real systems, by incorporation of some variant of a multiscale
approach. One also can consider the mixture of the LDL and HDL particles and model selective interaction between the brush and both types of lipoproteins. Various branched molecular architectures can
be tested for the adsorption efficiency. Finally, one can study adsorption/desorption cycles under flow introduced within a pore, to mimic realistic situation found in the experimental setup. These
cases are reserved for future studies. | {"url":"https://www.preprints.org/manuscript/202309.0233/v1","timestamp":"2024-11-12T06:22:27Z","content_type":"text/html","content_length":"1048922","record_id":"<urn:uuid:486de443-596b-4206-adf0-d2cbd00ac2fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00020.warc.gz"} |
Math Number Challenge
Play Online Game
Do you like this game?
This game on a smartphone:
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In this game, you need be finding figures for speed. It looks quite easy for the first time. However, at every level new tasks appear complicating the search for numbers. In the game, you will learn
figures in the decimal system. You will find out how to count to 10 with Roman figures. Moreover, you are going to learn to quickly search for figures matching the sides of the dice. Some figures (in
complex levels) are represented as arithmetic expressions and you will have to calculate the amount or difference before to choose the right number. But we did not limit the game with this and made
it more complicated. We added various colors to the game in order to develop your attention when looking for figures. We rotate the figures and you will learn to recognize them in any position and
from any angle. We have limited every level in time. If you did not make it with the time set, you will need to replay the level with a new combination of figures.
Are you ready to find figures for speed? Check it now! | {"url":"https://playcoolmath.com/en/math-games/math-number-challenge","timestamp":"2024-11-09T04:40:49Z","content_type":"text/html","content_length":"24127","record_id":"<urn:uuid:05bc8246-4e91-4c29-a49b-c4e44e2bc575>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00288.warc.gz"} |
Linear Regression – Revision
You practised linear regression modelling in the previous sessions. Let’s revise some of the concepts you learnt.
Here, Ujjyaini mentioned that regression guarantees interpolation of data and not extrapolation.
Interpolation means using the model to predict the value of a dependent variable on the independent values that lie within the range of data you already have. Extrapolation, on the other hand, means
predicting the dependent variable on the independent values that lie outside the range of the data the model was built on.
To understand this better, look at the diagram below. The model is built on the values of x between a and b.
When you wish to predict the Y for X1, which lies between a and b, it is called interpolation. On the other hand, extrapolation would be extending the line to predict Y for X2, which lies outside the
range on which the linear model was trained.
For now, you only need to understand what these terms mean. You will learn more about these in the upcoming lectures.
Ujjyaini also mentioned that linear regression is a parametric model.
Even though a detailed discussion on parametric and non-parametric models is beyond the scope of this module, a simple explanation is given below. You may also refer to the additional resources
provided below.
In simple terms, a parametric model can be described using a finite number of parameters. For e.g., a linear regression model built using n independent variables will have exactly n ‘parameters’
(i.e., n coefficients). The entire model can be described using these n parameters.
In the upcoming modules, you will learn about some non-parametric models as well, such as decision trees. They do not have a finite set of parameters that completely describe the model.
It is very crucial to understand when to apply linear regression modelling. Let’s go through some business cases to understand where you can apply linear regression modelling.
You saw various cases and learnt where linear regression modelling can be used and where it cannot. Answer the question below to test your understanding. | {"url":"https://www.internetknowledgehub.com/linear-regression-revision/","timestamp":"2024-11-08T06:19:34Z","content_type":"text/html","content_length":"80051","record_id":"<urn:uuid:158ab078-f6cc-4e0d-b647-4014c51f2efa>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00407.warc.gz"} |
How to Calculate Valid Percent
••• Comstock/Comstock/Getty Images
The valid percent is simply the proportion of a sample that is valid. Data can be invalid for a variety of reasons. Some data are simply impossible, such as negative heights or weights. Some data can
be shown to be invalid by comparing them with other data. For example, a person might be two years old, and a person might be a widow. But it is hard to conceive of a person who is a two year old
widow! Finally, some data can be identified as machine error or data entry error.
Write down the total sample size. For example, you might have 1000 cases.
Write down the number that are invalid. For example, there might be 92 invalid cases for one reason or another.
Subtract the result in step 2 from that in step 1. For instance 1000 - 92 = 908.
Divide the result in step 3 by the result in step 1 and multiply by 100. 908/1000 = .908. .908*100 = 90.8. Therefore 90.8 percent of our data are valid.
• "Dictionary of Statistics"; Brian Everitt. 1998
About the Author
Peter Flom is a statistician and a learning-disabled adult. He has been writing for many years and has been published in many academic journals in fields such as psychology, drug addiction,
epidemiology and others. He holds a Ph.D. in psychometrics from Fordham University.
Photo Credits
Comstock/Comstock/Getty Images | {"url":"https://sciencing.com/calculate-valid-percent-7449144.html","timestamp":"2024-11-03T10:35:17Z","content_type":"text/html","content_length":"403063","record_id":"<urn:uuid:befa28c2-60ab-457f-9062-ea17ba24668a>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00457.warc.gz"} |
Transactions Online
Tsutomu MAKABE, Taiju MIKOSHI, Toyofumi TAKENAKA, "WDM Multicast Tree Construction Algorithms and Their Comparative Evaluations" in IEICE TRANSACTIONS on Communications, vol. E93-B, no. 9, pp.
2282-2290, September 2010, doi: 10.1587/transcom.E93.B.2282.
Abstract: We propose novel tree construction algorithms for multicast communication in photonic networks. Since multicast communications consume many more link resources than unicast communications,
effective algorithms for route selection and wavelength assignment are required. We propose a novel tree construction algorithm, called the Weighted Steiner Tree (WST) algorithm and a variation of
the WST algorithm, called the Composite Weighted Steiner Tree (CWST) algorithm. Because these algorithms are based on the Steiner Tree algorithm, link resources among source and destination pairs
tend to be commonly used and link utilization ratios are improved. Because of this, these algorithms can accept many more multicast requests than other multicast tree construction algorithms based on
the Dijkstra algorithm. However, under certain delay constraints, the blocking characteristics of the proposed Weighted Steiner Tree algorithm deteriorate since some light paths between source and
destinations use many hops and cannot satisfy the delay constraint. In order to adapt the approach to the delay-sensitive environments, we have devised the Composite Weighted Steiner Tree algorithm
comprising the Weighted Steiner Tree algorithm and the Dijkstra algorithm for use in a delay constrained environment such as an IPTV application. In this paper, we also give the results of simulation
experiments which demonstrate the superiority of the proposed Composite Weighted Steiner Tree algorithm compared with the Distributed Minimum Hop Tree (DMHT) algorithm, from the viewpoint of the
light-tree request blocking.
URL: https://global.ieice.org/en_transactions/communications/10.1587/transcom.E93.B.2282/_p
author={Tsutomu MAKABE, Taiju MIKOSHI, Toyofumi TAKENAKA, },
journal={IEICE TRANSACTIONS on Communications},
title={WDM Multicast Tree Construction Algorithms and Their Comparative Evaluations},
abstract={We propose novel tree construction algorithms for multicast communication in photonic networks. Since multicast communications consume many more link resources than unicast communications,
effective algorithms for route selection and wavelength assignment are required. We propose a novel tree construction algorithm, called the Weighted Steiner Tree (WST) algorithm and a variation of
the WST algorithm, called the Composite Weighted Steiner Tree (CWST) algorithm. Because these algorithms are based on the Steiner Tree algorithm, link resources among source and destination pairs
tend to be commonly used and link utilization ratios are improved. Because of this, these algorithms can accept many more multicast requests than other multicast tree construction algorithms based on
the Dijkstra algorithm. However, under certain delay constraints, the blocking characteristics of the proposed Weighted Steiner Tree algorithm deteriorate since some light paths between source and
destinations use many hops and cannot satisfy the delay constraint. In order to adapt the approach to the delay-sensitive environments, we have devised the Composite Weighted Steiner Tree algorithm
comprising the Weighted Steiner Tree algorithm and the Dijkstra algorithm for use in a delay constrained environment such as an IPTV application. In this paper, we also give the results of simulation
experiments which demonstrate the superiority of the proposed Composite Weighted Steiner Tree algorithm compared with the Distributed Minimum Hop Tree (DMHT) algorithm, from the viewpoint of the
light-tree request blocking.},
TY - JOUR
TI - WDM Multicast Tree Construction Algorithms and Their Comparative Evaluations
T2 - IEICE TRANSACTIONS on Communications
SP - 2282
EP - 2290
AU - Tsutomu MAKABE
AU - Taiju MIKOSHI
AU - Toyofumi TAKENAKA
PY - 2010
DO - 10.1587/transcom.E93.B.2282
JO - IEICE TRANSACTIONS on Communications
SN - 1745-1345
VL - E93-B
IS - 9
JA - IEICE TRANSACTIONS on Communications
Y1 - September 2010
AB - We propose novel tree construction algorithms for multicast communication in photonic networks. Since multicast communications consume many more link resources than unicast communications,
effective algorithms for route selection and wavelength assignment are required. We propose a novel tree construction algorithm, called the Weighted Steiner Tree (WST) algorithm and a variation of
the WST algorithm, called the Composite Weighted Steiner Tree (CWST) algorithm. Because these algorithms are based on the Steiner Tree algorithm, link resources among source and destination pairs
tend to be commonly used and link utilization ratios are improved. Because of this, these algorithms can accept many more multicast requests than other multicast tree construction algorithms based on
the Dijkstra algorithm. However, under certain delay constraints, the blocking characteristics of the proposed Weighted Steiner Tree algorithm deteriorate since some light paths between source and
destinations use many hops and cannot satisfy the delay constraint. In order to adapt the approach to the delay-sensitive environments, we have devised the Composite Weighted Steiner Tree algorithm
comprising the Weighted Steiner Tree algorithm and the Dijkstra algorithm for use in a delay constrained environment such as an IPTV application. In this paper, we also give the results of simulation
experiments which demonstrate the superiority of the proposed Composite Weighted Steiner Tree algorithm compared with the Distributed Minimum Hop Tree (DMHT) algorithm, from the viewpoint of the
light-tree request blocking.
ER - | {"url":"https://global.ieice.org/en_transactions/communications/10.1587/transcom.E93.B.2282/_p","timestamp":"2024-11-02T05:07:39Z","content_type":"text/html","content_length":"65657","record_id":"<urn:uuid:3ab486c5-35d2-45d2-b615-b47482afafe9>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00464.warc.gz"} |
Summation of Divergent Infinite Series: How Natural Are the Current Tricks
Technical Report: UTEP-CS-18-56
Infinities are usually an interesting topic for students, especially when they lead to what seems like paradoxes, when we have two different seemingly correct answers to the same question. One of
such cases is summation of divergent infinite sums: on the one hand, the sum is clearly infinite, on the other hand, reasonable ideas lead to a finite value for this same sum. A usual way to come up
with a finite sum for a divergent infinite series is to find a 1-parametric family of series that includes the given series for a specific value p = p0 of the corresponding parameter and for which
the sum converges for some other values p. For the values p for which this sum converges, we find the expression s(p) for the resulting sum, and then we use the value s(p0) as the desired sum of the
divergent infinite series. To what extent is the result reasonable depends on how reasonable is the corresponding generalizing family. In this paper, we show that from the physical viewpoint, the
existing selection of the families is very natural: it is in perfect accordance with the natural symmetries. | {"url":"https://scholarworks.utep.edu/cs_techrep/1273/","timestamp":"2024-11-14T05:52:44Z","content_type":"text/html","content_length":"35693","record_id":"<urn:uuid:ba9697fa-e316-415c-b658-300c16524490>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00054.warc.gz"} |
Math Techniques and Strategies
First day of our probability unit we watched this video.
Day 1: I chose this video, because most of my students are hispanic and I think the biggest thing in our school right now is empathy. We talked about having games of chance like in the video they
just watched, what does chance mean? We did our first section of probability and told them they were going to create their own games, just like Caine did.
We did a 5 question check for understanding and told the students they needed to finish the bottom half of the checklist today.
Students designing their cardboard games.
Day 2: We talked about conditional probability. Did another check for understanding on Kahoot. Then I got lots of cardboard boxes from our recycling bin and had to make a quick pit-stop at Dollar
General for more cardboard boxes.
Day 3: I had a substitute teacher this day, but students started creating their boxes.
Day 4: We went over theoretical vs experimental probability. I gave students 7 minutes to finish their cardboard arcade games. Then we went over theoretical probability again. We talked about
geometrical probability from Day 1. Students were given rulers and yardsticks and had to find the theoretical probability of successfully completing their arcade game.
Day 5: We finished the material for our probability unit. I gave students 5 minutes to make sure their game is playable and to finish anything on the checklist. Then we went over that I would give
them 5 minutes to go play other games to get other groups experimental probability, then the partners would switch and the other partner would go play games.
I thought this unit was much better than the 3D dice activity from last year, this project was more hands-on and did a better job of combining the curriculum and the project together.
In our statistics unit for Algebra 2, we talk about measures of central tendency then we go over different types of sampling. The four we talk about random, convenience, systematic, and cluster.
Students don't really know what these are, we talk about when people are surveyed there are different methods to survey those people. Then we go over survey biases. It is a pretty boring lesson,
students already know mean, median, mode, and range. They don't really get why we go over types of sampling.
I trimmed down and this article from The Street:
To summarize the article it is about the new Mac Jr and Grand Mac, in the article it says, "McDonald's started testing the Grand Mac and Mac Jr. in more than 120 restaurants in the central Ohio and
Dallas areas in April last year." This is the basis of what I wanted the students to pick up out of the article, but thought this might be a good chance to get them inferring reading in the math
Students read the article, I gave them 5 minutes to read and answer the following 5 questions:
1. What is the main point of the article?
2. What are two supporting details?
3. What type of sampling method was mentioned in the article?
4. Why do you think McDonald's chose that type of sampling method?
5. Do you think the authors view of McDonalds were positive or negative? Why?
I was surprised about the level of detail that students put into the article here were some sticky notes and students working on the article.
Next year I will try to put the article on ActivelyLearn, last year it was my go to place for articles and mathematics for Junior Standards Math. I will have to use more articles in math class, it
was a good experience for me and my students.
When we come back from winter break we normally start our probability and statistics unit. I normally take a week for probability and a week for statistics which normally melts into three weeks. I've
always thought nothing of changing it, but during winter break Dan Meyer posted "Plates Without States" Since we were going over permutations and combinations I thought this would be an excellent way
to get students thinking about how many different combinations there are in license plates and why they make them like that.
To start the lesson I had students go through Dan Meyer State-Plate Game. Students were definitely engaged and loved playing against each other in their groups.
Next we talked about license plates and I separated it from combinations and permutations.
I gave all of the students a blank license plate and a card. The card had a name of a city or state and a population that students had to take in consideration.
Here are some of the license plates that students were working on.
When students were done with their license plates, they took a picture of their license plate and put it on SeeSaw. The last part they had to do was comment on three others the number of different
combinations that they had with their license plate.
Here were a few students figuring out and commenting on other students post.
I like this activity much more and students realized how license plates play a role in local governments and how the population of an area can control the different license plates possible.
I posted almost all of them in the back of my room here are a bunch of different ones that are posted.
I was reading this blog post, I can't seem to find it now. It had 21 things teachers should try in 2017, number 19 was "Post Your Goals in Your Classroom."
I thought this would be an excellent way for students to see what I am working on in the classroom and maybe they can hold me more accountable.
There are lots of different things I want to do in 2017, I will post a list of what I want to do at the bottom.
Here are the three goals I posted in my classroom:
1. More Activities, Less Homework
I have been disappointed lately with our school's emphasis on homework and worksheets. I feel that some of our students are being pushed down and out with this emphasis. I want students to experiment
with math and I want more formative assessments to understand my students knowledge.
Since we are 1-1 with iPads I see students trying Google the worksheet before attempting any of the problems. They know in other classrooms that they get their worksheets online and don't need to do
the work and its easier.
Getting students using Desmos, WODB, and Estimation180 to challenge their thinking and their understanding of mathematics.
2. Students In Charge of their Own Learning
My students heavily rely on the teacher for their information. If they don't know an answer right away their hands go up. I want my students to be challenged, but also know that I am there to guide
not to tell them the answer.
I want students to be able to go out and find the answer. If they don't know how to do something I want them to be able to go out and search for it, find a YouTube video.
3. Build Students Up with Growth Mindset
This last one is very similar to the second one. My last goal is for students to have a growth mindset, to start the year I normally have a BreakoutEDU box for students to do. I want students not to
think of math as thing that "smart people" do.
Things I have Planned or Want to Do in 2017
• Different types of seating
• Caine's Arcade
• Incorporating more VR
• Walking Classroom
• Incorporate more reading.
Here is a seating chart that I currently use and really like, plus my goals are posted!!
The 2016 game of the year was Codenames, but for Christmas my wife and I were given Codenames Pictures, which is an equally awesome game. The basic gameplay is that you have a partner or a group
where one person gives a one word clue and a number which correlates to how many it is suppose to cover, the first one to get them all wins. There are neutral ones and one is an assassin which ends
the game.
My thought: How cool would this be if you did this with numbers.
If I made board pieces that had a bunch of different numbers, students would use one word such as: even, odd, cubic, etc... this game would help build number sense. Since you could play it with
groups of 4 it would be a great station activity.
More to come with actual student gameplay. | {"url":"https://new-to-teaching.blogspot.com/2017/01/","timestamp":"2024-11-11T17:21:38Z","content_type":"text/html","content_length":"111453","record_id":"<urn:uuid:e21eb359-940c-4c07-8de8-5df597a3c7b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00579.warc.gz"} |
Aphelion Distance Calculator - Calculator Wow
Aphelion Distance Calculator
In celestial mechanics, understanding the aphelion distance is crucial for studying planetary orbits and the movements of celestial bodies around a central object, such as the Sun. The Aphelion
Distance Calculator simplifies the computation of this key astronomical parameter, aiding in astronomical research and space mission planning.
The aphelion distance holds significant importance in astronomy and space exploration:
• Orbital Dynamics: Defines the maximum distance between a celestial body and the central object in its elliptical orbit.
• Seasonal Variations: Influences climatic variations and seasons on planets like Earth, where the distance from the Sun affects solar radiation and temperatures.
• Planetary Science: Essential for studying planetary atmospheres, gravitational interactions, and understanding celestial mechanics.
How to Use
Using the Aphelion Distance Calculator is straightforward:
1. Enter Orbital Period: Input the orbital period of the celestial body in years.
2. Input Eccentricity: Enter the eccentricity of the orbit, which determines its shape, from circular to highly elliptical.
3. Calculate Aphelion Distance: Click the “Calculate Aphelion Distance” button to initiate the calculation.
4. Interpret Results: The calculator will display the computed aphelion distance, representing the farthest point in the orbit from the central object.
10 FAQs and Answers
1. What is Aphelion Distance? Aphelion Distance is the maximum distance between a celestial body and the Sun (or other focus of the orbit) in an elliptical orbit.
2. Why is Aphelion important in astronomy? It helps in understanding the shape and characteristics of planetary orbits, influencing climate patterns and planetary dynamics.
3. Does Aphelion Distance vary for different planets? Yes, each planet has its unique orbital period and eccentricity, resulting in varying aphelion distances from the Sun.
4. Can Aphelion Distance change over time? Yes, factors such as gravitational perturbations from other celestial bodies can alter the shape and dimensions of orbits over long periods.
5. Is Aphelion Distance the same as Orbital Radius? No, aphelion distance refers specifically to the farthest point in an orbit, whereas orbital radius can refer to any point in the orbit.
6. How does eccentricity affect Aphelion Distance? Higher eccentricity leads to a more elongated orbit, increasing the difference between aphelion and perihelion (closest approach to the Sun).
7. Can Aphelion Distance be measured directly? Yes, astronomers use telescopes and orbital observations to determine positions and distances of celestial bodies at different points in their orbits.
8. What are the units of Aphelion Distance? Aphelion Distance is typically expressed in astronomical units (AU) or kilometers (km), depending on the scale of the orbit.
9. How accurate is the Aphelion Distance Calculator? The calculator provides accurate estimates based on entered orbital parameters, making it reliable for educational and research purposes.
10. Why should I use an Aphelion Distance Calculator? It offers a quick and efficient way to determine aphelion distances, which is essential for various astronomical studies and space mission
The Aphelion Distance Calculator is an invaluable tool for astronomers, educators, and enthusiasts alike, offering a convenient way to compute the maximum distance of a celestial body from the Sun or
other central objects. By understanding and calculating aphelion distances, researchers gain insights into orbital mechanics, planetary dynamics, and climatic variations, contributing to our broader
understanding of the cosmos. Incorporating this calculator into astronomical studies enhances precision and efficiency in celestial calculations, paving the way for new discoveries and advancements
in space exploration. | {"url":"https://calculatorwow.com/aphelion-distance-calculator/","timestamp":"2024-11-07T00:26:40Z","content_type":"text/html","content_length":"65140","record_id":"<urn:uuid:1cc93a1e-abc2-4d08-b5c2-bb68483b5930>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00663.warc.gz"} |
Agresti, A. (1992), "A Survey of Exact Inference for Contingency Tables," Statistical Science, 7 (1), 131–177.
Agresti, A. (2007), An Introduction to Categorical Data Analysis, Second Edition, New York: John Wiley & Sons.
Agresti, A. (2002), Categorical Data Analysis, Second Edition, New York: John Wiley & Sons.
Agresti, A., Mehta, C. R., and Patel, N. R. (1990), "Exact Inference for Contingency Tables with Ordered Categories," Journal of American Statistical Association, 85, 453–458.
Agresti, A., Wackerly, D., and Boyett, J. M. (1979), "Exact Conditional Tests for Cross-Classifications: Approximation of Attained Significance Levels," Psychometrika, 44, 75–83.
Bishop, Y., Fienberg, S. E., and Holland, P. W. (1975), Discrete Multivariate Analysis: Theory and Practice, Cambridge, MA: MIT Press.
Conover, W. J. (1999), Practical Nonparametric Statistics, Third Edition, New York: John Wiley & Sons.
Gail, M. and Mantel, N. (1977), "Counting the Number of Journal of the American Statistical Association, 72, 859–862.
Gibbons, J. D. and Chakraborti, S. (1992), Nonparametric Statistical Inference, Third Edition, New York: Marcel Dekker.
Hajek, J. (1969), A Course in Nonparametric Statistics, San Francisco: Holden-Day.
Halverson, J. O. and Sherwood, F. W. (1930), "Investigations in the Feeding of Cottonseed Meal to Cattle," North Carolina Agr. Exp. Sta. Tech. Bulletin, 39, 158pp.
Hodges, J. L., Jr. (1957), "The Significance Probability of the Smirnov Two-Sample Test," Arkiv for Matematik, 3, 469–486.
Hodges, J. L., Jr. and Lehmann, E. L. (1983). "Hodges-Lehmann Estimators," in Encyclopedia of Statistical Sciences, vol. 3, ed. S. Kotz, N. L. Johnson, and C. B. Read, New York: John Wiley &
Sons, 463–465.
Hollander, M. and Wolfe, D. A. (1999), Nonparametric Statistical Methods, Second Edition, New York: John Wiley & Sons.
Kiefer, J. (1959), "Annals of Mathematical Statistics, 30, 420–447.
Lehmann, E. L. (1963). "Nonparametric Confidence Intervals for a Shift Parameter," Annals of Mathematical Statistics, 34, 1507–1512.
Mehta, C. R. and Patel, N. R. (1983), "A Network Algorithm for Performing Fisher’s Exact Test in Journal of American Statistical Association, 78, 427–434.
Mehta, C. R., Patel, N. R., and Senchaudhuri, P. (1991), "Exact Stratified Linear Rank Tests for Binary Data," Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface
(E. M. Keramidas, ed.), 200–207.
Mehta, C. R., Patel, N. R., and Tsiatis, A. A. (1984), "Exact Significance Testing to Establish Treatment Equivalence with Ordered Categorical Data," Biometrics, 40, 819–825.
Owen, D. B. (1962), Handbook of Statistical Tables, Reading, MA: Addison-Wesley.
Quade, D. (1966), "On Analysis of Variance for the Annals of Mathematical Statistics, 37, 1747–1758.
Randles, R. H. and Wolfe, D. A. (1979), Introduction to the Theory of Nonparametric Statistics, New York: John Wiley & Sons.
Sheskin, D. J. (1997), Handbook of Parametric and Nonparametric Statistical Procedures, Boca Raton, FL: CRC Press.
Valz, P. D. and Thompson, M. E. (1994), "Exact Inference for Kendall’s S and Spearman’s Journal of Computational and Graphical Statistics, 3 (4), 459–472. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_npar1way_sect025.htm","timestamp":"2024-11-04T00:52:39Z","content_type":"application/xhtml+xml","content_length":"14271","record_id":"<urn:uuid:817bb564-a889-413f-bb00-e6f5ada549ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00113.warc.gz"} |
Bandwidth (signal processing) - Wikiwand
Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in unit of hertz (symbol Hz).
Amplitude (a) vs. frequency (f) graph illustrating baseband bandwidth. Here the bandwidth equals the upper frequency.
It may refer more specifically to two subcategories: Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel,
or a signal spectrum. Baseband bandwidth is equal to the upper cutoff frequency of a low-pass filter or baseband signal, which includes a zero frequency.
Bandwidth in hertz is a central concept in many fields, including electronics, information theory, digital communications, radio communications, signal processing, and spectroscopy and is one of the
determinants of the capacity of a given communication channel.
A key characteristic of bandwidth is that any band of a given width can carry the same amount of information, regardless of where that band is located in the frequency spectrum.^[lower-alpha 1] For
example, a 3 kHz band can carry a telephone conversation whether that band is at baseband (as in a POTS telephone line) or modulated to some higher frequency. However, wide bandwidths are easier to
obtain and process at higher frequencies because the § Fractional bandwidth is smaller.
Bandwidth is a key concept in many telecommunications applications. In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier signal. An FM radio
receiver's tuner spans a limited range of frequencies. A government agency (such as the Federal Communications Commission in the United States) may apportion the regionally available bandwidth to
broadcast license holders so that their signals do not mutually interfere. In this context, bandwidth is also known as channel spacing.
For other applications, there are other definitions. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance. A
less strict and more practically useful definition will refer to the frequencies beyond which performance is degraded. In the case of frequency response, degradation could, for example, mean more
than 3 dB below the maximum value or it could mean below a certain absolute value. As with any definition of the width of a function, many definitions are suitable for different purposes.
In the context of, for example, the sampling theorem and Nyquist sampling rate, bandwidth typically refers to baseband bandwidth. In the context of Nyquist symbol rate or Shannon-Hartley channel
capacity for communication systems it refers to passband bandwidth.
The Rayleigh bandwidth of a simple radar pulse is defined as the inverse of its duration. For example, a one-microsecond pulse has a Rayleigh bandwidth of one megahertz.^[1]
The essential bandwidth is defined as the portion of a signal spectrum in the frequency domain which contains most of the energy of the signal.^[2]
The magnitude response of a band-pass filter illustrating the concept of −3 dB bandwidth at a gain of approximately 0.707
In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density (in W/Hz or V^2/Hz) is nonzero or above a small threshold value. The threshold
value is often defined relative to the maximum value, and is most commonly the 3 dB point, that is the point where the spectral density is half its maximum value (or the spectral amplitude, in ${\
displaystyle \mathrm {V} }$ or ${\displaystyle \mathrm {V/{\sqrt {Hz}}} }$, is 70.7% of its maximum).^[3] This figure, with a lower threshold value, can be used in calculations of the lowest sampling
rate that will satisfy the sampling theorem.
The bandwidth is also used to denote system bandwidth, for example in filter or communication channel systems. To say that a system has a certain bandwidth means that the system can process signals
with that range of frequencies, or that the system reduces the bandwidth of a white noise input to that bandwidth.
The 3 dB bandwidth of an electronic filter or communication channel is the part of the system's frequency response that lies within 3 dB of the response at its peak, which, in the passband filter
case, is typically at or near its center frequency, and in the low-pass filter is at or near its cutoff frequency. If the maximum gain is 0 dB, the 3 dB bandwidth is the frequency range where
attenuation is less than 3 dB. 3 dB attenuation is also where power is half its maximum. This same half-power gain convention is also used in spectral width, and more generally for the extent of
functions as full width at half maximum (FWHM).
In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB with a small variation, for example within the ±1 dB interval. In the
stopband(s), the required attenuation in decibels is above a certain level, for example >100 dB. In a transition band the gain is not specified. In this case, the filter bandwidth corresponds to the
passband width, which in this example is the 1 dB-bandwidth. If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal
passband gain rather than x dB below the maximum gain.
In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak.
In communication systems, in calculations of the Shannon–Hartley channel capacity, bandwidth refers to the 3 dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling rate, and
maximum bit rate according to the Hartley's law, the bandwidth refers to the frequency range within which the gain is non-zero.
The fact that in equivalent baseband models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth since they are
sometimes referred to only by the positive half, and one will occasionally see expressions such as ${\displaystyle B=2W}$, where ${\displaystyle B}$ is the total bandwidth (i.e. the maximum passband
bandwidth of the carrier-modulated RF signal and the minimum passband bandwidth of the physical passband channel), and ${\displaystyle W}$ is the positive bandwidth (the baseband bandwidth of the
equivalent channel model). For instance, the baseband model of the signal would require a low-pass filter with cutoff frequency of at least ${\displaystyle W}$ to stay intact, and the physical
passband channel would require a passband filter of at least ${\displaystyle B}$ to stay intact.
The absolute bandwidth is not always the most appropriate or useful measure of bandwidth. For instance, in the field of antennas the difficulty of constructing an antenna to meet a specified absolute
bandwidth is easier at a higher frequency than at a lower frequency. For this reason, bandwidth is often quoted relative to the frequency of operation which gives a better indication of the structure
and sophistication needed for the circuit or device under consideration.
There are two different measures of relative bandwidth in common use: fractional bandwidth (${\displaystyle B_{\mathrm {F} }}$) and ratio bandwidth (${\displaystyle B_{\mathrm {R} }}$).^[4] In the
following, the absolute bandwidth is defined as follows, ${\displaystyle B=\Delta f=f_{\mathrm {H} }-f_{\mathrm {L} }}$ where ${\displaystyle f_{\mathrm {H} }}$ and ${\displaystyle f_{\mathrm {L} }}$
are the upper and lower frequency limits respectively of the band in question.
Fractional bandwidth
Fractional bandwidth is defined as the absolute bandwidth divided by the center frequency (${\displaystyle f_{\mathrm {C} }}$), ${\displaystyle B_{\mathrm {F} }={\frac {\Delta f}{f_{\mathrm {C} }}}
The center frequency is usually defined as the arithmetic mean of the upper and lower frequencies so that, ${\displaystyle f_{\mathrm {C} }={\frac {f_{\mathrm {H} }+f_{\mathrm {L} }}{2}}\ }$ and ${\
displaystyle B_{\mathrm {F} }={\frac {2(f_{\mathrm {H} }-f_{\mathrm {L} })}{f_{\mathrm {H} }+f_{\mathrm {L} }}}\,.}$
However, the center frequency is sometimes defined as the geometric mean of the upper and lower frequencies, ${\displaystyle f_{\mathrm {C} }={\sqrt {f_{\mathrm {H} }f_{\mathrm {L} }}}}$ and ${\
displaystyle B_{\mathrm {F} }={\frac {f_{\mathrm {H} }-f_{\mathrm {L} }}{\sqrt {f_{\mathrm {H} }f_{\mathrm {L} }}}}\,.}$
While the geometric mean is more rarely used than the arithmetic mean (and the latter can be assumed if not stated explicitly) the former is considered more mathematically rigorous. It more properly
reflects the logarithmic relationship of fractional bandwidth with increasing frequency.^[5] For narrowband applications, there is only marginal difference between the two definitions. The geometric
mean version is inconsequentially larger. For wideband applications they diverge substantially with the arithmetic mean version approaching 2 in the limit and the geometric mean version approaching
Fractional bandwidth is sometimes expressed as a percentage of the center frequency (percent bandwidth, ${\displaystyle \%B}$), ${\displaystyle \%B_{\mathrm {F} }=100{\frac {\Delta f}{f_{\mathrm {C}
Ratio bandwidth
Ratio bandwidth is defined as the ratio of the upper and lower limits of the band, ${\displaystyle B_{\mathrm {R} }={\frac {f_{\mathrm {H} }}{f_{\mathrm {L} }}}\,.}$
Ratio bandwidth may be notated as ${\displaystyle B_{\mathrm {R} }:1}$. The relationship between ratio bandwidth and fractional bandwidth is given by, ${\displaystyle B_{\mathrm {F} }=2{\frac {B_{\
mathrm {R} }-1}{B_{\mathrm {R} }+1}}}$ and ${\displaystyle B_{\mathrm {R} }={\frac {2+B_{\mathrm {F} }}{2-B_{\mathrm {F} }}}\,.}$
Percent bandwidth is a less meaningful measure in wideband applications. A percent bandwidth of 100% corresponds to a ratio bandwidth of 3:1. All higher ratios up to infinity are compressed into the
range 100–200%.
Ratio bandwidth is often expressed in octaves (i.e., as a frequency level) for wideband applications. An octave is a frequency ratio of 2:1 leading to this expression for the number of octaves, ${\
displaystyle \log _{2}\left(B_{\mathrm {R} }\right).}$
Setup for the measurement of the noise equivalent bandwidth of the system with frequency response .
The noise equivalent bandwidth (or equivalent noise bandwidth (enbw)) of a system of frequency response ${\displaystyle H(f)}$ is the bandwidth of an ideal filter with rectangular frequency response
centered on the system's central frequency that produces the same average power outgoing ${\displaystyle H(f)}$ when both systems are excited with a white noise source. The value of the noise
equivalent bandwidth depends on the ideal filter reference gain used. Typically, this gain equals ${\displaystyle |H(f)|}$ at its center frequency,^[6] but it can also equal the peak value of ${\
displaystyle |H(f)|}$.
The noise equivalent bandwidth ${\displaystyle B_{n}}$ can be calculated in the frequency domain using ${\displaystyle H(f)}$ or in the time domain by exploiting the Parseval's theorem with the
system impulse response ${\displaystyle h(t)}$. If ${\displaystyle H(f)}$ is a lowpass system with zero central frequency and the filter reference gain is referred to this frequency, then:
${\displaystyle B_{n}={\frac {\int _{-\infty }^{\infty }|H(f)|^{2}df}{2|H(0)|^{2}}}={\frac {\int _{-\infty }^{\infty }|h(t)|^{2}dt}{2\left|\int _{-\infty }^{\infty }h(t)dt\right|^{2}}}\,.}$
The same expression can be applied to bandpass systems by substituting the equivalent baseband frequency response for ${\displaystyle H(f)}$.
The noise equivalent bandwidth is widely used to simplify the analysis of telecommunication systems in the presence of noise.
In photonics, the term bandwidth carries a variety of meanings:
• the bandwidth of the output of some light source, e.g., an ASE source or a laser; the bandwidth of ultrashort optical pulses can be particularly large
• the width of the frequency range that can be transmitted by some element, e.g. an optical fiber
• the gain bandwidth of an optical amplifier
• the width of the range of some other phenomenon, e.g., a reflection, the phase matching of a nonlinear process, or some resonance
• the maximum modulation frequency (or range of modulation frequencies) of an optical modulator
• the range of frequencies in which some measurement apparatus (e.g., a power meter) can operate
• the data rate (e.g., in Gbit/s) achieved in an optical communication system; see bandwidth (computing).
A related concept is the spectral linewidth of the radiation emitted by excited atoms.
1. The information capacity of a channel depends on noise level as well as bandwidth – see Shannon–Hartley theorem. Equal bandwidths can carry equal information only when subject to equal
signal-to-noise ratios. | {"url":"https://www.wikiwand.com/en/articles/Bandwidth_(signal_processing)","timestamp":"2024-11-11T08:16:29Z","content_type":"text/html","content_length":"376190","record_id":"<urn:uuid:b90e9ff2-9255-4457-b669-a5b72b640882>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00200.warc.gz"} |
datplot 1.1.1
• Improved error handling in some functions.
• Reduced and improved messaging and warning behaviour.
• Fixed slight problem in generate.stepsize() where it would not handle same values in min & max dating properly.
• Completely updated and redesigned tests.
• Removed unnecessary internal function check.number().
datplot 1.1.0
• Using either the original calculation (weights) or calculation of year-wise probability is now an option in datsteps() with the argument calc = "weight" or calc = "probability"
• There is now an option to calculate the cumulative probability in datsteps() with the argument cumulative = TRUE. This only works with probability calculation instead of the original (weights)
• Significantly improved the efficiency of datsteps().
• Change and improve error-handling of scaleweight().
• Remove UTF-8 characters from data and other files to comply with CRAN.
• Update documentation and add a pkgdown-site.
datplot 1.0.1
• Change calculation in get.weights() to 1 / (abs(DAT_min - DAT_max) + 1) to get real probability values for each year. This only has a real effect when using a stepsize of 1, as it makes the
weight-values usable as “dating probability”.
• Clean up calculate.outputrows() and scaleweight() somewhat.
datplot 1.0.0
• Added a NEWS.md file to track changes to the package
• some style corrections
• First release for submission to CRAN, accepted -> datplot is now on CRAN
datplot 0.2.4
• peer-review version for Advances in Archaeological Practice | {"url":"https://cran.case.edu/web/packages/datplot/news/news.html","timestamp":"2024-11-12T10:03:45Z","content_type":"application/xhtml+xml","content_length":"3117","record_id":"<urn:uuid:a4897833-00ae-4a74-b968-877e9c134398>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00183.warc.gz"} |
Complexity of redundancy detection on RDF graphs in the presence of rules, constraints, and queries
Paper Title:
Complexity of redundancy detection on RDF graphs in the presence of rules, constraints, and queries
Based on practical observations on rule-based inference on RDF data, we study the problem of redundancy elimination on RDF graphs in the presence of rules (in the form of Datalog rules) and
constraints, (in the form of so-called tuple-generating dependencies), and with respect to queries (ranging from conjunctive queries up to more complex ones, particularly covering features of SPARQL,
such as union, negation, or filters). To this end, we investigate the influence of several problem parameters (like restrictions on the size of the rules, the constraints, and/or the queries) on the
complexity of detecting redundancy. The main result of this paper is a fine-grained complexity analysis of both graph and rule minimisation in various settings.
Revised manuscript after an accept with minor revisions. Now accepted for publication. Reviews of the first round are below.
Solicited review by anonymous reviewer:
The paper considers the problem of redundancy elimination in RDF graphs for various settings and extends a line of work that was begun in [23].
Figures and typesetting are good. It is very well-written, technically sound, and generally well-motivated. It clarifies its relationship to state-of-the-art SemWeb standards.
It should be accepted, pending minor revisions only.
Major comments (not all points are criticism, some are just thoughts):
- I am not sure whether it is a good idea to state that the author's work is based on practical observations. This term is irritating to me. The authors do not seem to have experimented with their
approach, or at least they do not report about their experiments. Maybe you should reformulate this phrase such that it fits into your argumentation.
- Some proofs appear much too lengthy to me. Especially, the proofs of Lemma 3.5, 3.7, 4.3, and 5.5.
- In the paper you switch between SPARQL and conjunctive queries. You should provide a paragraph on the relationship between these two formalisms. Besides, you should formally introduce the SPARQL
fragment that you consider. Defining SPARQL by a citation is not enough to my mind's eye.
- You provided many complexity results for decision problems in your paper. Is there a fundamental reason why you did not consider the corresponding construction problems? What about their
complexity? Can you give an outlook?
- Please double-check your citations. Sometimes you use abbreviations for journals, sometimes not. For [13] the conference name is completely missing.
Minor comments:
- On page 2 you should provide a citation for tgds.
- On page 2 you state that a constraint may be read as a generalized rule. Please make this more explicit by citing the Motik paper on the semantic differnece between rules and constraints.
- On page (2) the query (16) is not a conjunctive query.
- On page 5 the definition og the semantics of a conjunctive query does not seem correct to me. Please check.
- On page 5 you should provide a citation for the completeness of the QSAT problems.
- Definition 2.1: are the 3-colorings of V_1, V_2 and V_3 with respect to the induced substructures?
- Table 1 is mentioned directly after Theorem 3.1. Please insert the table one page earlier.
- On page 37 you state that your work can be considered as the starting point for studying the equivalence of SPARQL queries under RDFS, RIF and OWL2RL semantics. You should provide a citation
related to semantic query optimization, where query equivalence under generalized rules or constraints, that contain negation and disjunction, is considered. E.g. the PODS 2010 paper "Semantic Query
Optimization in the Presence of Types". This work directly applies to some SPARQL fragments because RDF triples come, by definition, with a typing of its three components. So, SPARQL query
optimization is an example for a scenario in which you naturally have rules that contain negation and disjunction.
- I doubt that citation [30] is really necessary.
Solicited review by Adila Krisnadhi:
This paper is a purely theoretical paper about complexity of redundancy elimination/detection of RDF graphs in various situations. It presents a fine-grained complexity analysis for the mentioned
problem with respect to various settings of rules, constraints and queries. The results are particularly relevant for the application of rule-based inference on top of many RDF stores and also in the
context of entailment regime for SPARQL.
This is a solid technical paper with important theoretical contributions as such a collection of complexity results are necessary to understand how easy/difficult the problem of redundancy
elimination/detection of RDF graphs with respect to rules, constraints and queries. I also found no problem in all the proofs. Regarding the authors' claim that this paper is a significant extension
over the earlier conference version, I agree that the results in this paper are indeed stronger. Overall, I believe that this paper merits an acceptance after taking into account my remarks below.
First, the title of the paper: "Redundancy Elimination on RDF Graphs ... " can be, in my opinion, a bit misleading. A better title would be "Complexity of Redundancy Detection on RDF Graphs ....".
When I first read the title, I initially thought that the paper is really about methods/techniques to eliminate redundancy in RDF graphs which, in fact, is not really the case. All the results are
about the complexity of the decision problems related to redundancy detection on RDF graphs. Even if there are parts that can actually be used to really eliminate redundancy on RDF graphs, I cannot
clearly see it from the discussion. The reason of this might be due to the fact that the analysis is more fundamental than redundancy elimination as it focuses instead on the associated decision
problems of redundancy detection. This is not necessarily bad by itself, but since the title somewhat indicates redundancy elimination as the main theme, I think that this is actually a slight
discrepancy. To improve the paper, I suggest that the authors change the title to something similar to my suggestion earlier, and even better, add some discussion on the actual methods/techniques on
redundancy elimination in which the results in this paper will be important ingredients.
Another issue with this paper is the readability. I applaud the approach the authors used which is by always starting with overview and rought intuition of the results before proceeding to the really
technical parts of the results. Despite this, I still found it hard to follow the discussion because the paper really presents many reduction proofs which are similar to some degree. One way to
alleviate this is to move some or all parts of the proofs into an appendix and then replace them in the main text with the suitable informal intuition. That way, the narration can be a bit easier to
follow and a reader can more easily note the comparison between the theorems/lemmas.
Table 3: it would be better for comparison if the results on Table 1 and Table 2 are also included in this table. That way, the changes in complexity results can be more easily pointed out.
Several results regarding the complexity of checking whether a RDF graph satisfies a set of constraints are used in several places:
- The PI_2^P complexity (do the following refer to the same thing?):
- in Proof of Lemma 3.4 (step (2)), [10, Proposition 5.5,(1)] is refered.
- in Proof of Lemma 5.3 (step (3)), [11] is refered.
- The tractability: Proof of Lemma 3.6 refers to [23, Proposition 3].
==> As this is an important ingredients in the technical content, I suggest to put them somewhere in the preliminary, before using them in the proofs.
Claim 2 in Proof of Lemma 5.5 recalls the definition of G^i_v from the proof of Lemma 3.7 with slight change. I suggest that instead of referring to Lemma 3.7, the actual definition of G^i_v for this
context is spelled out explicitly here.
typos, etc.:
- title text on page headers: is it deliberately shortened? (I'm not sure whether this is the requirement from the journal itself).
- p4, right column, paragraph 2, second sentence before last on definition of satisfication of a constraint over RDF graph:
should be ".. if for each homomorphism h on X mapping BGP(Ante) to G ..." (missing h)
- p4, right column, last paragraph, on the explanation of least fix-point of the immediate consequence operator:
- delete extra right curly brace after G
- should mention that /mu is a homomorphism.
- p6, the paragraph before Definition 2.3:
should be "Q-3COL_{forall,2}
- p7, last sentence of the paragraph below Proof of Lemma 3.2:
better wording is "The concrete proofs are given in Section 3.2, but beforehand, the rough intuition of these results is sketched."
- p9, first paragraph of Proof of Lemma 3.5:
need correction on whitespace in "Q-3COL_{exists,3}"
- p9, last line:
need correction on whitespace in "Q-3COL_{exists,3}"
- p10, right column, first sentence on the "only-if direction" paragraph:
whitespace correction similar to the one in p9.
- p16, right column, the paragraph before Claim 2:
" ... convenient for formulate ... " ==> " ... convenient to formulate ... " | {"url":"https://www.semantic-web-journal.net/content/complexity-redundancy-detection-rdf-graphs-presence-rules-constraints-and-queries","timestamp":"2024-11-06T00:36:19Z","content_type":"application/xhtml+xml","content_length":"37796","record_id":"<urn:uuid:dc18e58d-8c9b-43da-b953-8c9f21cdc632>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00759.warc.gz"} |
Math Camps
One summer option that has traditionally been popular among math majors is to work at a math camp for high school students. Typically, undergraduates serve as counselors, helping students with
problem sets, providing general supervision, and serving as older friends and mentors to the students. Many find the job to be both fun and rewarding, and some Princeton students have chosen to spend
most of their summers at camps like this. The most popular by far among Princeton students are the Ross Mathematics Program, held at Ohio State University, and PROMYS, held at Boston University.
Another possibility is the Canada/USA Mathcamp, held at varying locations in the US.
Ross Mathematics Program
The Ross Mathematics Program (“Ross”) is an intensive six-week course in number theory for high school students. Over the course of the program, the students work through a huge chunk of fundamental
number theory, starting with an axiomatic construction of the integers and culminating with a proof of Gauss’s quadratic reciprocity theorem. The emphasis is on solving problems—there’s a problem set
every day and the problems guide the students step-by-step through proofs of many tricky results, including quadratic reciprocity—and learning to think mathematically, which is to say, both
intuitively and rigorously. Counselors live in the dorms with the students and take charge of a small “family” (four to five students) and sometimes a junior counselor, an advanced returning student
who is ostensibly in training for the role of counselor. If you work here, you’ll be expected to grade your students’ problem sets quickly and provide useful feedback that will allow them to progress
as far and as rapidly as they can. All in all, it’s a rewarding job that leaves plenty of time for personal interests, and the Ross community is wonderful. See rossprogram.org for more information.
PROMYS is the child of the Ross program and is similar in many ways. The program parallels that of Ross, starting with an axiomatic construction of the integers and ending with a proof of Gauss’s
quadratic reciprocity theorem. Again, the emphasis is on problem solving and thinking mathematically, and a problem set is assigned daily. Student “families” consist of 4-5 students, usually with one
advanced returning student, who take advanced courses like Algebra, Galois Theory, Geometry and Symmetry, Combinatorics, and Modular Forms. Counselors’ roles includes grading family members’ problem
sets quickly, guiding students through the program, and marking for an advanced course. There is spare time, however, and fun activities for both participants and counselors. It’s a rewarding job
that, like Ross, has traditionally been popular with Princeton undergraduates. See promys.org for more information.
Canada/USA MathCamp
Canada/USA MathCamp (“Mathcamp”) is another major math summer program for high school students. Undergraduates are hired to work as junior counselors (JCs), a job which is much like that of counselor
at Ross or PROMYS. Unlike these other programs, however, MathCamp only hires alumni for these roles. Graduate students can apply to be “Mentors,” who teach classes each week and provide mentorship to
the campers. See mathcamp.org for more information.
Originally written by Max Rabinovich ’13 and Erick Knight ’12.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://blogs.princeton.edu/mathclub/guide/summer/camps/","timestamp":"2024-11-09T04:17:26Z","content_type":"text/html","content_length":"117415","record_id":"<urn:uuid:841857a3-6484-4414-941a-2bc0e6328b41>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00751.warc.gz"} |
python-igraph API reference
class documentation
The dendrogram resulting from the hierarchical clustering of the vertex set of a graph.
Method __init__ Creates a dendrogram object for a given graph.
Method __plot__ Draws the vertex dendrogram on the given Cairo context
Method as_clustering Cuts the dendrogram at the given level and returns a corresponding VertexClustering object.
Method optimal_count.setter Undocumented
Property optimal_count Returns the optimal number of clusters for this dendrogram.
Instance Variable _graph Undocumented
Instance Variable _modularity_params Undocumented
Instance Variable _names Undocumented
Instance Variable _optimal_count Undocumented
Inherited from Dendrogram:
Method __str__ Undocumented
Method format Formats the dendrogram in a foreign format.
Method names.setter Sets the names of the nodes in the dendrogram
Method summary Returns the summary of the dendrogram.
Property merges Returns the performed merges in matrix format
Property names Returns the names of the nodes in the dendrogram
Static Method _convert_matrix_to_tuple_repr Converts the matrix representation of a clustering to a tuple representation.
Method _item_box_size Calculates the amount of space needed for drawing an individual vertex at the bottom of the dendrogram.
Method _plot_item Plots a dendrogram item to the given Cairo context
Method _traverse_inorder Conducts an inorder traversal of the merge tree.
Instance Variable _merges Undocumented
Instance Variable _nitems Undocumented
Instance Variable _nmerges Undocumented
def __init__
(self, graph, merges, optimal_count=None, params=None, modularity_params=None):
Creates a dendrogram object for a given graph.
graph the graph that will be associated to the clustering
merges the merges performed given in matrix form.
optimal the optimal number of clusters where the dendrogram should be cut. This is a hint usually provided by the clustering algorithm that produces the dendrogram. None means that such a hint
_count is not available; the optimal count will then be selected based on the modularity in such a case.
params additional parameters to be stored in this object.
modularity arguments that should be passed to Graph.modularity when the modularity is (re)calculated. If the original graph was weighted, you should pass a dictionary containing a weight key with
_params the appropriate value here.
def __plot__
(self, context, bbox, palette, *args, **kwds):
Draws the vertex dendrogram on the given Cairo context
See Dendrogram.__plot__ for the list of supported keyword arguments.
def as_clustering
(self, n=None):
Cuts the dendrogram at the given level and returns a corresponding VertexClustering object.
the desired number of clusters. Merges are replayed from the beginning until the membership vector has exactly n distinct elements or until there are no more recorded merges, whichever happens
n first. If None, the optimal count hint given by the clustering algorithm will be used If the optimal count was not given either, it will be calculated by selecting the level where the modularity is
a new VertexClustering object.
Returns the optimal number of clusters for this dendrogram.
If an optimal count hint was given at construction time, this property simply returns the hint. If such a count was not given, this method calculates the optimal number of clusters by maximizing the
modularity along all the possible cuts in the dendrogram. | {"url":"https://igraph.org/python/api/0.9.8/igraph.clustering.VertexDendrogram.html","timestamp":"2024-11-11T13:27:41Z","content_type":"text/html","content_length":"40335","record_id":"<urn:uuid:08b68be9-b672-4a08-936c-4a27fde7e44c>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00647.warc.gz"} |
Cheriton-Tarjan Minimum Spanning tree algorithm
Open-Source Internship opportunity by OpenGenus for programmers. Apply now.
Cheriton-Tarjan algorithm is a modification of Kruskal's algorithm designed to reduce the O(e log e) term.
Similar to Kruskal's algorithm, it grows a spanning forest, beginning with a forest of n = |G| components each consisting of a single node. Now the term O(e log e) comes from selecting the minimum
edge from a heap of e edges. Since, every component Tu must eventually be connected to another component, this algorithm keeps a separate heap PQu for each component Tu, so, that initially n smaller
heaps are used. Initially, PQu will contain only DEG(u) edges, since Tu consists only of vertex u. When Tu and Tv are merged, PQu and PQv must also be merged. This requires a modification of the data
structures, since heaps cannot be merged efficiently. This is essentially because merging heaps reduces to building a new heap.
Any data strucutre in which a minimum element can be found efficiently is called a priority queue. A heap is one form of priority queue, in which elements are stored as an array, but viewed as a
binary tree. There are many other forms of priority queue. In this algorithm, PQu will stand for a priority queue which can be merged.
It stores a list Tree of the edges of a minimum spanning tree. The components of the spanning forest are represented as Tu and the priority queue of edges incident on vertices of Tu is stored as PQu.
The complexity of Cheriton-Tarjan minimum spanning tree algorithm is O(E log(log V)) where E is the number of edges and V is the number of vertices.
Consider the following pseudo-code for Cheriton-Tarjan Minimum Spanning tree algorithm:
MST-CHERITON-TARJAN (G = (V,E), w)
Q := null /* A Set of edges in minimum spanning tree */
for each vertex v in V[G] /* V[G] is the vertex set in graph G */
do (v belongs to Q)
stage(v) = 0
j = 1 /* j is stage number which is initialized to 1 */
while |Q| > 2 /* Number of elements in Q is greater than 2 */
do (Q belongs to T1) /* Let T1 be the tree in front of Q */
if (stage(T1) == j)
do j = j+1
find edge (u,v) in E with minimum weight such that u belongs to T1 , v belongs to `V-T1`
let T2 be the tree in Q that contains v
T = MERGE(T1,T2) /* by adding edge (u,v) to MST*/
Q = Q-T1-T2 /* Remove T1 & T2 from Q */
stage(T) = 1 + minimum_of{ stage(T1), stage(T2) }
Q = T
Shrink G to G* where G* is G with each tree shrunk to a single node and only those edges (u,v) Ă G*, u Ă T, vĂ T', that are shortest incident edges between disjoint T, T'.
You can read the research paper here. | {"url":"https://iq.opengenus.org/cheriton-tarjan-minimum-spanning-tree-algorithm/","timestamp":"2024-11-11T08:16:48Z","content_type":"text/html","content_length":"60916","record_id":"<urn:uuid:85945b64-c16b-4c42-9fd9-2cc413680110>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00312.warc.gz"} |
Is it Worth Rolling the Dice? - Evidence Based Investing
“Green light, STOP – if you want to see where you are taking the most risk, look where you are making the most money.” – Paul Gibbons.
One of the most oft-quoted pearls of wisdom emanating from Investment analysts with regard to risk control is “Enjoy the party but dance near the door“. It has been said and written so often that it
IS an investment cliche, but it is an inappropriate metaphor. After all, the worst that could happen is that you run out of drink (and being near the door is of no help in that regard). A more
relevant comparison is with that of a homeowner, who lives on a tectonic fault line. One may decide that, as there hasn’t been a “Big One” since 1906 (and is, according to seismologists, now overdue)
and if one were to feel a succession of small rumblings for a period of time, that it may be prudent to consider moving a little further away from the earthquake zone- not too far, however, as one
still has to commute into work, but far enough away that a major “event” would not be catastrophic for ones finances (obviously, earthquake insurance is not covered in one’s Home Insurance Policy and
is extremely expensive). This is how we see the current market situation; as we showed last week, there are some signs of “rumblings ” beneath the surface of the rapid price gains. It is important to
control what you can and let markets do what they will. That control is in the area of risk.
So what does one do? Selling everything would be a huge bet on a market decline and is not a wise strategy- after all, the “Big One”- a market crash in this instance- may not hit for months or years
(or even at all!). A sensible approach may be to lower market exposure, while maintaining a risk profile that allows for continued gains should a major fall not happen. Fortunately there are a couple
of ways to look at this problem, both of which should allow investors to reap the benefits of investing in the major asset classes, while keeping market risk within tolerable levels. Given the
mathematics of compound returns [1], it is important to ensure that a loss does not de-rail the investment target.
There are two ways to look at market risk; either via portfolio duration or portfolio volatility. Let us look at both of these in turn, using the MSCI World Index (as a proxy for the World Stock
market and then EBI 100, EBI 70 and EBI 40 to illustrate the effect of lowering overall portfolio volatility.
Portfolio Duration:
We have covered this concept in previous blogs (see here). The idea of bond duration is reasonably well understood, but the concept of equity duration less so. As the post referred to says, equity
duration can be approximated by the Price: Dividend ratio. So, for example, the MSCI World Index is currently at 2230 (as of 24/1/18) and yields around 2.2%. which equates to a duration of (2230/49.1
= 45.4 years) [2]. The Vanguard Total Bond Market ETF will be used as the Bond market surrogate, which has a Duration of 6.1 years at present.
It is a basic premise of investing that ones time horizon and portfolio duration should be aligned as closely as possible in order to reduce (or preferably eliminate) the risk of path dependency-
that is, the risk that the investment return is not generated to match expectations until after the investor needs the money [3].
The conventional portfolio allocation is 60:40 stocks and bonds, which therefore has a duration of (0.6 x 45.4 + 0.4 x 6.1 = 29.68 years). This is thus not necessarily suitable for everyone. With UK
Life expectancy currently just under 81 years (on average of course, which leaves a fair amount of room for discretion), it may be that many close to retirement might need to re-think their
allocations in a downwards direction. Viewed in this light, 100% exposure to equities may only be appropriate for those in their 30’s. Using the same formula as above, a 40:60 portfolio, however, has
a duration of just under 22, which may be far more closely aligned to actual investor time horizons. The possible permutations are of course almost endless.
Now, let us look at the problem through the lens of portfolio (price) volatility. The chart below shows this phenomenon using 3 EBI portfolios. It is important to bear in mind that as a low cost
provider the situation could be far worse in a more expensive (read: Active) portfolio. To the extent that the portfolio costs are higher than EBI, the scenarios painted could be best case ones.
Below we can see the returns and volatility numbers for the three portfolios (plus a Sector Index for context). Using a Standard Normal Distribution Table, we can estimate the probability (and
extent) of a decline in prices by a set amount (in the jargon, a standard deviation event-SD). For example, a 2 SD event (which has a 2.28% probability of occurrence- or 1 week in 44) could lead to a
decline of 19.04%, 12% and 5.68% for EBI 100, 70 and 40 respectively [4].
But in the market context, we must recognise that things could get far worse than a 2 SD decline (obviously, a 2 SD RISE of no concern). For anything greater than a 2 SD event, we need to recognise
that extreme events occur more frequently than conventional statistics imply. For that, we need to use a Cubic Power Law, which better captures the reality that stock price moves are not normally
distributed, (because Investors tend to panic at more or less the same time, resulting in major market moves “clustering” in short periods).
Using this formula [5] for 3 SD moves, one gets a better indication of how prices can move (and at what probability) in reality, rather than in an economist’s classroom.
Using the same calculation as in [4], a 3 Standard Deviation event would cause a decline of 33.3%, 21.97% and 11.64% for the respective (100, 70 and 40) portfolios. Clearly, this is not to say that
these events WILL occur, but it gives an idea of the potential magnitude of the price declines should they do so. In the case of EBI 100, a 33% decline would require the portfolio to double to get
back to “break-even”, (which has occurred post 2003 AND 2009), but the question then becomes, how quickly can markets recover; if the investor is close to retirement, there may not be enough time for
that to happen. In the case of EBI 40, it would need only a 13.1% price gain to recover the previous level, which is a much easier task.
At present, it looks as if risk has been abolished, as Investors abandon hedges almost completely, but this will not last, (nothing does). By the time risk reduction is needed, it may be too late, as
prices will have (possibly violently) adjusted. Paradoxically, the time to focus on risk control is precisely the time that no-one sees the need for it; it appears that we are approaching that point.
According to this article (quoting a Credit Suisse research note), US Pension funds may need (as part of their monthly re-balancing process) to sell up to $12 billion in equities (buying $24 billion
in bonds) in the coming days. As they say in the best sci-fi films, maybe “we are not alone” in thinking this way.
[1] For example, assume 3 years of 10% annual portfolio growth, a £100,000 portfolio would, at the end of year 3, be worth £133,100. However, a 10% loss in year 4 leaves the Investor with just
£119,790; in the process, the annualised return for the portfolio drops from 10% p.a. after year 3 to just 4.62% in year 4. (and a 20% loss leaves the annual return at a mere 1.58% p.a.) If the
market were to rise another 10% in year 4, but then fall 10% in year 5 the annualised return would be 5.67% p.a.
On the other hand, if the portfolio is positioned such that it only falls 5% in both of the above examples, the 4-year return becomes 6.04% p.a. and 6.82% respectively. Risk control is very tolerant
of positions taken, even those that are somewhat early in the piece…
[2] A quicker way to calculate this is to divide 100 by the dividend yield. The result is the same.
[3] To illustrate, assume £100 is invested for a time horizon of 30 years, with an expected return of 5% per annum. (£432.19 on retirement). If however, the portfolio duration is longer than 30
years, (e.g. 40 years), it is entirely possible that the same 30-year return could be achieved, but only after the investor has need of the cash (say, in years 35-40). This mismatch creates the risk
that the investments will achieve the required rate of return in time for the investor to benefit from them, as there is guarantee of a simple 5% annual return every year. There will periods of
losses, potentially in years 28-30.
[4] This is determined by multiplying the portfolio Volatility by 2 (to get the 2 SD amount) and taking this from the annualised return number. So, for EBI 100, the calculation is 9.48- (2 x 14.26) =
-19.04% . Doing the same for the other two gives us a decline of 12% and 5.68%.
[5] To calculate this, we must ask how much less likely is a 3 SD move than a 2 SD event, where the latter has a 2.28% chance of occurring. We divide 3 by 2 (because we are only interested in the bad
half of the distribution) and the raise it to the power of three (1.5^3). This (4.56) we then divide into 2.28 to get 0.5%. 100 divided by 0.5= 200 or one week in 200 (1 week in every 3.8 years). | {"url":"https://ebi.co.uk/evidence-based-investing/is-it-worth-rolling-the-dice/","timestamp":"2024-11-14T23:18:08Z","content_type":"text/html","content_length":"152637","record_id":"<urn:uuid:0f98817c-ae3f-4e69-8387-f9234d42527c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00437.warc.gz"} |
3.2.2: Porosity from Well Logs
The most common method of determining porosity is with Well Logs. Well logs are tools sent down the wellbore during the drilling process which measure different reservoir properties of interest to
geologists and engineers. Due to the expense of obtaining core samples, typically only a few wells are cored. The wells that do get cored are usually wells early in the life of the reservoir
(appraisal wells) and key wells throughout the reservoir. On the other hand, well logs are routinely run in wells, if only to identify the depths of the productive intervals. The three open-hole logs
used to evaluate porosity are:
• The sonic log
• The density log
• The neutron log
While none of these logs measures porosity directly, the porosity can be calculated based on theoretical or empirical considerations. The measurements obtained from these logs are not only dependent
on the porosity but are also dependent on other rock properties such as:
• Lithology (rock type: sandstone, limestone, shale, etc.)
• The fluids occupying the pore spaces
• The wellbore environment (type of drilling fluid, hole size)
• The geometry of the pores
Since many variables may impact the log readings, corrections need to be applied to the log interpretations and the three logs are typically evaluated together to determine the best estimate of the
porosity of rock formations. The log evaluations are also calibrated against core porosity in wells where both core and logs are available.
The Sonic Log measures the acoustic transit time, Δt, of a compressional sound wave traveling through the porous formation. The logging tool consists of one or more transmitters and a series of
receivers. The transmitters act as sources of the acoustic signals which are detected by the receivers. The time required for the signal to travel through one foot of the rock formation is the
acoustic transit time, Δt. The acoustic travel time, then, is the reciprocal of the sonic velocity through the formation. The units of Δt are micro-seconds/ft (μsec/ft) or millionths of a second per
There are several ways to interpret the sonic log measurements. One of the most common interpretation formulae is the Wyllie Time-Average Equation:
${\varphi }_{sl}=\frac{\Delta {t}_{sl}-\Delta {t}_{ma}}{\Delta {t}_{f}-\Delta {t}_{ma}}$
• ϕ[sl] is the porosity from the sonic log (log measurement) , fraction
• Δt[sl] is value of the acoustic transit time measured by the sonic log, μsec/ft
• Δt[ma] is value of the acoustic transit time of the rock matrix measured in the laboratory, μsec/ft
• Δt[f] is value of the acoustic transit time saturating fluid measured in the laboratory, μsec/ft
The presence of hydrocarbons in the reservoir rock results in an over prediction of porosity measured by the sonic log and some corrections may be required. These corrections take the form:
$Gas:{\varphi }_{sonic}=0.7\text{}{\varphi }_{sl}$
$Oil:{\varphi }_{sonic}=0.9\text{}{\varphi }_{sl}$
Table 3.03 has typical values of the acoustic transit time for different reservoir formations and commonly encountered reservoir fluids.
Table 3.03: Typical Acoustic Transit Times for Sonic Log Interpretation
Heading1 Δt[ma] (μsec/ft) Δt[ma] (μsec/ft) Δt[f] (μsec/ft) Δt[f ](μsec/ft)
Range Commonly Used Range Commonly Used
Sandstone 55.5 – 51.0 55.5 or 51.0 — —
Limestone 47.8 – 43.5 47.5 — —
Dolomite 43.5 43.5 — —
Anhydrite 50.0 50.0 — —
Salt Formation 66.7 67.0 — —
Fresh Water Based — — 189.0 189.0
Drilling Fluid
Salt Water Based — — 185.0 185.0
Drilling Fluid
Gas — — 920.0 920.0
Oil — — 230.0 230.0
Casing (Iron) — — 57.0 57.0
Other empirically based equations exit for sonic log interpretation. One form of an alternative equation is:
${\varphi }_{sonic}=C\frac{\Delta {t}_{sl}-\Delta {t}_{ma}}{\Delta {t}_{sl}}$
In this equation, the value of C is in the range of 0.625 to 0.700 and is determined by calibrating the equation to known porosity, such as, to core data when a well is both cored and logged. In
Equation 3.12 and Equation 3.13, ϕ[sonic] is the final interpreted porosity from the sonic log.
The [Density Log] measures the electron density ρ[e], of the formation (the electron density is the number of electrons per unit volume). The density logging tool emits gamma rays from a chemical
source which interact with the electrons of elements in the formation. Detectors in the tool count the returning gamma rays. These returning gamma rays are related to the election density of the
elements in the formation.
The electron density is proportional to the bulk density, ρ[b], of the formation (the bulk density is the density of the fluid-filled rock in grams per unit volume). For a molecular substance, this
proportionality is:
${\rho }_{e}={\rho }_{b}\left(\frac{\text{2}\sum \text{Z}}{\text{MW}}\right)$
• ${\rho }_{e}$ is the electron density of the formation, electrons/cc
• ${\rho }_{b}$ is the bulk density of the formation, gm/cc
• ΣZ is the sum of the atomic numbers making up the molecule, electrons/molecule
• MW is the molecular weight, gm/molecule
Table 3.04 contains the term in the parenthesis for common substances related to oil and gas production.
Table 3.04: Properties for Density Log Interpretation
Chemical Actual $\left(\frac{\text{2}\sum \text{z}}{\text{MW}}\right)$ Electron Log Reading,
Compound Formula Density, ${\rho }_{b}$ (electrons/gm) Density, ${\rho }_{e}$ Apparent ${\rho }_{ba}$
(gm/cc) (electrons/cc) (gm/cc)
Quartz SiO[2] 2.654 0.9985 2.650 2.648
Calcite CaCO[3] 2.710 0.9991 2.708 2.710
Dolomite CaCO[3]MgCO[3] 2.870 0.9977 2.863 2.876
Anhydrite CaSO[4] 2.960 0.9990 2.957 2.977
Sylvite KCl 1.984 0.9657 1.916 1.863
Halite NaCl 2.165 0.9581 2.074 2.032
Gypsum CaSO[4]2H[2]O 2.320 1.0222 2.372 2.351
Anthracite Coal --- 1.400 - 1.800 1.0200 1.442 – 1.852 1.355 – 1.796
Bituminous Coal --- 1.200 - 1.500 1.0600 1.227 – 1.590 1.173 – 1.514
Fresh Water H[2]O 1.000 1.1101 1.110 1.000
Brine (200,000 ppm) --- 1.146 1.0797 1.237 1.135
Oil N(CH[2]) 0.850 1.1407 0.970 0.850
Methane CH[4] ${\rho }_{methane}$ 1.2470 1.247 ${\rho }_{methane}$ 1.335 ${\rho }_{methane}$ - 0.1883
Gas --- ${\rho }_{gas}$ 1.2380 1.238 ${\rho }_{gas}$ 1.238 ${\rho }_{gas}$ - 0.1883
One important observation from this table is that the column containing the group in parenthesis in Equation 3.14, $\left(\frac{\text{2}\sum \text{z}}{\text{MW}}\right)$ , is approximately 1.0. Since
this term is close to unity, the electron density is a very close approximation to the bulk density, as also seen in Table 3.03. The logging tool is calibrated by running it against a limestone
formation containing fresh water. With this calibration, the Log Reading, Apparent ρ[ba] (last column in Table 3.03) is:
${\rho }_{ba}=1.0704{\rho }_{e}-0.1883$
For some substances, such as liquid-filled sandstones, limestones, and dolomites, ρba can be used directly as ρ[b]. For other substances, such as sylvite, rock salt, gypsum, anhydrite, coal, and gas
bearing formations, further corrections are required. These additional corrections are beyond the scope of this course. Once the bulk density is determined, the porosity can be estimated by:
${\varphi }_{density}=\frac{{\rho }_{ma}-{\rho }_{b}}{{\rho }_{ma}-{\rho }_{f}}$
• ϕ[density] is the final interpreted porosity from the density log, fraction
• ρ[ma] is the matrix density (from the Actual Density, ρ[b], Column in Table 3.03), gm/cc
• ρ[b] is bulk density from density log (Equation 3.15), gm/cc
• ρ[f] is the density of the fluid measured in the laboratory, gm/cc
The Neutron Log measures the amount of hydrogen in the formation being logged. Since the amount of hydrogen per unit volume is approximately the same for oil and water, the neutron log measures the
Liquid Filled Porosity (the porosity excluding the Gas-Filled Porosity). The neutron logging tool emits neutrons from a chemical source which collide with nuclei of elements in the formation. The
element in the formation with the mass closest to a neutron is hydrogen. Due to the parity in mass, the neutron in a neutron-hydrogen collision loses approximately half of its energy. With enough
collisions, the neutron eventually loses enough energy and is absorbed by the hydrogen nucleus and a gamma ray is emitted. The neutron logging tool measures these emitted gamma rays. Note, other
hydrogen atoms may be present in clays in the rock, or in the rock itself and corrections for these other hydrogen atoms are required.
Interpretation of the neutron log is performed by first calibrating the logging tool to specific well and formation conditions. Interpretation charts supplied by the logging company are used to
interpret the log for deviations from these calibration conditions. The interpretation of the neutron log for ϕ[neutron] is beyond the scope of this course.
As mentioned throughout the discussion on porosity logging, due to the various wellbore and formation conditions encountered during the logging operations (i.e., real conditions, as opposed to
laboratory conditions) many corrections may be required to get a good interpretation from the different well logs. In addition, the logs are typically evaluated together to aid in the interpretation.
Finally, if core data are available from a well, then the core derived porosity is used to calibrate the logging tools.
Rock (Pore-Volume) Compressibility
In addition to the porosity and its relation to pore-volume, reservoir engineers are also interested in how the pore-volume behaves (increases or decreases) with increases or decreases in
pore-pressure. The industry standard relationship for change in pore-volume is based on the isothermal pore-volume compressibility, c[PV].
The isothermal pore-volume compressibility is always positive and is defined as:
The units of compressibility are 1/psi. Equation 3.17 implies that as pressure increases the pore-volume increases. Hall ^[4] correlated the effective rock compressibility as a function of porosity
which is shown in Figure 3.04.
Source: Hall, H. N.: “Compressibility of Reservoir Rocks,” Trans. AIME (1958) 198, 209
For a constant bulk volume and compressibility, we can separate variables in Equation 3.17 and integrate to arrive at the following relationship between pore-volume (or porosity) and pore pressure:
${c}_{PV}={\frac{1}{{V}_{PV}}\frac{d{V}_{pv}}{dp}]}_{T=constant}={\frac{1}{\varphi {V}_{b}}\frac{d\left(\varphi {V}_{b}\right)}{dp}]}_{T=constant}={\frac{1}{\varphi }\frac{d\varphi }{dp}]}_{T=
${\int }_{{p}_{ref}}^{p}{c}_{PV}dp={\int }_{{\varphi }_{ref}}^{\varphi }\frac{1}{\varphi }d\varphi$
Where ϕ[ref] is a reference porosity measured at reference pressure, p[ref]. After some manipulation:
$\varphi ={\varphi }_{ref}{e}^{{c}_{PV}\left(p-{p}_{ref}\right)}$
To further simplify this relationship, if we assume a small pore-volume compressibility (as shown in Figure 3.03, we are typically dealing with rock compressibilities on the order of 10^-6 1/psi) and
apply a Taylor Series expansion to the exponential function (truncated after one term), we obtain:
$\varphi ={\varphi }_{ref}\left[1+{c}_{PV}\left(p-{p}_{ref}\right)\right]$
The rock compressibility determination is performed in the laboratory on core samples. The rock compressibility is an expensive test to run and is not part of the Routine Core Analysis. It must be
specifically requested as part of any Special Core Analysis, or SCAL, testing performed on the core sample.
Rock Permeability
The permeability of a porous medium is a measure of the ease (or difficulty) in which a fluid can flow through the pores of the medium. Permeability is a property of the porous medium which in our
case is the reservoir rock. The unit of permeability is the Darcy, or D, named after the French engineer Henri Darcy who investigated the flow of water through filter beds in the city of Dijon in the
mid-1800s. The unit of Darcy has the dimensions of length-squared. One significant contribution from Darcy’s work (among many), is Darcy’s Law, which was published in 1856 and forms one of the
foundations of porous media flow:
${q}_{w}=kA\frac{\partial p}{\partial l}$
• q[w] is the flow rate of water, cc/sec
• k is the permeability of the medium, Darcy
• A is cross-sectional area, cm^2
• l is the length in the direction of flow, ft (x, y, or z in Cartesian coordinates)
• $\frac{\partial p}{\partial l}$ is the pressure gradient, atm/cm
The unit of the Darcy is defined as the permeability, k, required to allow a flow rate, q[w], of one cc of water per second through a medium with a cross-sectional area, A, of one cm^2, with an
applied pressure gradient, Δp/ΔL, of one atm/cm. As it turns out, the Darcy as a unit is too large for most field applications. In reservoir engineering we typically work with the millidarcy, md,
which is one one-thousandth of a Darcy:
$\text{1 D = 1,000 md}$
Henri Poiseuille later generalized Darcy’s Law to fluids other than water by noting that the flow rate was inversely proportional to the dynamic viscosity, μ[f], of the fluid flowing through the
porous medium. The unit of dynamic viscosity, the poise, is named after Poiseuille and is a property of the fluid. Again, as it turns out, the poise as a unit is also too large for most field
applications. In reservoir engineering we typically work with the centipoise, cp, which is one one-hundredth of a poise:
$\text{1 poise = 100 cp}$
The generalized form of Darcy’s Law for any single-phase fluid which incorporates the fluid viscosity is:
${q}_{f}=\frac{kA}{{\mu }_{f}}\frac{\partial p}{\partial l}$
Darcy’s Law has several important assumptions associated with it. These include:
• A rigid porous medium (incompressible medium with no transport of rock grains, e.g. fines movement
• An incompressible, homogeneous, Newtonian fluid that fully saturates (single-phase) the porous medium
• Steady-state, isothermal, and laminar (low Reynolds number) flow conditions
• No interactions between the porous medium and the fluid flowing through it
• A no-slip interface between the porous medium and the fluid flowing through it (zero velocity boundary condition at the rock-fluid interface)
The form of Darcy’s Law discussed so far uses the SI unit system. For oilfield units, we have:
${q}_{f}=\frac{0.001127kA}{{\mu }_{f}}\frac{\partial p}{\partial l}$
• q[f] is the flow rate of the saturating fluid, bbl/day
• 0.001127 is a unit conversion constant
• k is the absolute permeability of the medium, md
• A is cross-sectional area, ft^2
• $\frac{\partial p}{\partial l}$ is the pressure gradient, psi/ft
Measurement of the permeability can be done through laboratory or field measurements. In the laboratory, a core sample of known dimensions (A and ΔL) is cleaned and placed into a fluid-tight core
holder. A fluid of known viscosity (typically air) is allowed to flow through the core at a metered flow rate. Darcy’s Law is then used to calculate the permeability of the core sample.
In the field, permeability is measured with a Well Test using Pressure Transient Analysis. Under certain conditions, the production rate(s) (can be a zero-production rate) results in well pressures
that honor known solutions to the Diffusivity (or Well Test) Equation. When the well test results are compared to the solutions to the diffusivity equation, the permeability of the formation can be
One common well test is a Pressure Build-Up Test. In a pressure build-up test, the well is produced at a stable (constant) rate, q[p], for a production time of t[p]. The well is then shut in and the
pressure is monitored during the shut-in period Δt; where Δt is measured from the beginning of the well is shut-in. The well test is called a build-up test because when the well is shut in, the
pressures increase with increasing Δt. One analysis tool for the pressure build-up test is the Horner Plot. A typical Horner plot is shown in Figure 3.05.
In this plot shown in Figure 3.05, p[ws] are the shut-in well pressures measured during the well test, t[p] and Δt are times in hours (Δt measured from the time that the well is shut-in), q[p] is the
stabilized production rate during the production period prior to well shut-in in STB/day, μ[o] is the oil viscosity in cp, B[o] is the Formation Volume Factor in bbl/STB, k is the permeability in md,
and h is the reservoir thickness in ft.
In a Horner Plot, the function $\frac{\left({t}_{p}+\Delta t\right)}{\Delta t}\left(bottom\text{}x-axis\right)$ , decreases as $\Delta t\text{}\left(top\text{}x-axis\right)$ increases. As can be seen
in this figure, the slope of the Horner Plot is related to the permeability of the reservoir near the well. If the shut-in pressures, p[ws], are measured and the slope calculated, then the
permeability can be determined if all the other parameters in the definition of the slope are known.
In semi-logarithmic plots (plots with one conventional axis and one logarithmic axis), such as that in Figure 3.05, the slope is normally taken over one logarithmic cycle. That is:
$m=\frac{\Delta {p}_{ws}}{{\mathrm{log}}_{10}\left({10}^{n+1}\right)-{\mathrm{log}}_{10}\left({10}^{n}\right)}=\Delta {p}_{ws}\left(psi/cycle\right)$
While there is no universal correlation between permeability, field measurements suggest that as the porosity of a formation increases, the permeability of the formation also increases. This behavior
is captured in a permeability-porosity cross-plot. One such example of a permeability-porosity cross plot is illustrated in Figure 3.06.
A permeability-porosity cross-plot is a field dependent transform that relates core derived permeability to core derived porosity. Note that the permeability-porosity transform shown in Figure 3.06
is plotted on a semi-logarithmic plot with the permeability plotted on the logarithmic scale. Also note that in the middle of the plot (ϕ = 20 percent) there is an approximate four order of magnitude
error bar in the permeability data. This is typical for a permeability-porosity transform. While the results from these transforms may be crude, they are often the only source of permeability data
when building complex models of oil or gas reservoirs. Note, geologists have developed methods for capturing this scatter into their models in the form of Scatter Plots, but the development of such
plots is beyond the scope of this course.
The permeability in the presence of a single-phase fluid is called the absolute permeability. Since we will be dealing with multi-phase flow, we will need to discuss extensions to Darcy’s which allow
for more than one phase. We will do this later in this lesson when we discuss Reservoir Rock-Fluid Interaction Properties. | {"url":"https://www.e-education.psu.edu/png301/node/835","timestamp":"2024-11-12T10:32:30Z","content_type":"text/html","content_length":"70861","record_id":"<urn:uuid:0907a9f0-7387-40dd-8589-69f9ad0b55c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00351.warc.gz"} |
3. Creating the pivot table
To create a pivot table from the dataset introduced previously here (and downloadable here), select the table (A1-H721), choose Insert in the main menu, and Pivot Table.
In the dialog box that shows up, you may choose the data (if it is not already done) and whether you want the pivot table to appear on a new worksheet or in the same one as the original table. If
you choose to have it next to the original table, simply indicate which cell its top left corner will be located at.
The new menu PivotTable Fields that appears to the right of your screen will help you organize the pivot table. With the menu, you will be able to choose which values to display (counts, averages,
minimum or maximum values, standard deviations, etc) and whether you want to do so for each category (or a selection of categories) named in the table (for instance per light condition, salinity
condition, tank, date, etc).
First, select a dependent variable to display in the pivot table, here the standard length (SL (mm)) . To do so, click and hold SL (mm), then drag it in the box Σ VALUES in the lower right corner of
the side menu.
In the worksheet, a pivot table made of 2 cells has appeared, showing Count of SL (mm) and 716. This count tells you how many cells in the SL (mm) column contain numerical data (it is indeed a count
of entries). Note that from now on, the changes you will perform in the right side menu will show up “live” in the newly created pivot table.
If you wish to get averages (or any other statistic) instead of counts, go back to the side menu and click on Count of SL (mm) in the box Σ VALUES. Choose Value Field Settings... and select
Average in the menu, then OK.
The pivot table now shows Average of SL (mm) and its value based on all the data in the whole table.
Now, you may decide which category (or categories) you want to see the average of. Let’s assume that you are interested in getting the average of standard length for each tank. We go back to the
side menu and drag and drop Tank nr. from the list of available fields down to the box called ROWS.
The pivot table now looks like this. The 9 tanks are displayed in rows with their labels (1-9) in the left column and the corresponding average in the right column, and the average of the whole
table comes at the bottom as a “grand total” (and you can see that this grand total matches the average that was displayed in the pivot table just before we divided the table in rows).
We could have similarly dispatched the tanks horizontally by dragging and dropping Tank nr. from the list of available fields down to the box called COLUMNS.
These are the basics on how to create a simple pivot table. From there, things can get quite interesting/complicated, depending on whether you want to:
The following posts will show you how to go further with pivot table.
Fant du det du lette etter? Did you find this helpful? | {"url":"https://biostats.w.uib.no/3-creating-the-pivot-table/","timestamp":"2024-11-13T18:05:03Z","content_type":"text/html","content_length":"60876","record_id":"<urn:uuid:28ff6588-724e-4944-9d8b-4a9c6d2bf3ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00150.warc.gz"} |
Introduction to Matrix Algebra
About the Course
Many university STEM major programs have reduced the credit hours for a Matrix Algebra course or simply dropped the course from their curriculum. The content of Matrix Algebra, in many cases, is
taught just in time where needed. This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra. This course is designed so that a student gains an
introductory knowledge of matrix algebra. The topics covered in the course include the following.
Go to the newly available textbook that is complete with complete solutions to several problem sets. You can view it as a pdf file and use your favorite epub viewers such as Kindle and Calibre.
Course Syllabus
Course Format
The content available for the above topics is in the form of:
The course is self-paced.
About the Instructor:
Autar Kaw is a professor of mechanical engineering and Jerome Krivanek Distinguished Teacher at the University of South Florida. He is a recipient of the 2012 U.S. Professor of the Year Award from
the Council for Advancement and Support of Education (CASE) and Carnegie Foundation for Advancement of Teaching. Professor Kaw received his BE Honors degree in Mechanical Engineering from Birla
Institute of Technology and Science (BITS) India in 1981, and his degrees of Ph.D. in 1987 and M.S. in 1984, both in Engineering Mechanics from Clemson University, SC. He joined the University of
South Florida in 1987.
Professor Kaw’s main scholarly interests are in engineering education research, open courseware development, bascule bridge design, fracture mechanics, composite materials, computational
nanomechanics, and the state and future of higher education.
Since 2002, under Professor Kaw’s leadership and funding from NSF (2002-2016), he and his colleagues from around the nation have developed, implemented, refined and assessed online resources for open
courseware in Numerical Methods (http://nm.MathForCollege.com). This courseware annually receives more than a million page views, 900,000 views of the YouTube lectures, and 150,000 annual visitors to
the “numerical methods guy” blog.
Professor Kaw’s work has appeared in the St. Petersburg Times, Tampa Tribune, Chance, Oracle, and his work has been covered/cited in Chronicle of Higher Education, Inside Higher Education,
Congressional Record, ASEE Prism, Tampa Bay Times, Tampa Tribune, Campus Technology, Florida Trend Magazine, WUSF, Bay News 9, Times of India, NSF Discoveries, Voice of America, and Indian Express.
COPYRIGHTS: University of South Florida, 4202 E Fowler Ave ENG030, Tampa, FL 33620-5350. All Rights Reserved. Questions, suggestions or comments, contact kaw@eng.usf.edu This material is based
partly upon work supported by the National Science Foundation under Grant# 0126793, 0341468, 0717624, 0836981,0836916, 0836805, 1322586, 1609637, and the Research Experience for Undergraduates
program at the College of Engineering at the University of South Florida. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the National Science Foundation or the University of South Florida. Other sponsors include Maple, MathCAD, MATLAB, USF, FAMU, ASU, AAMU, and MSOE. Based on a work
at Holistic Numerical Methods licensed under an Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) | {"url":"https://ma.mathforcollege.com/","timestamp":"2024-11-09T00:50:19Z","content_type":"text/html","content_length":"38346","record_id":"<urn:uuid:eb26a981-2853-4774-b31b-44a292303e8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00308.warc.gz"} |
16.10: Vapor Pressure and Changes of State
Last updated
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• To know how and why the vapor pressure of a liquid varies with temperature.
• To understand that the equilibrium vapor pressure of a liquid depends on the temperature and the intermolecular forces present.
• To understand that the relationship between pressure, enthalpy of vaporization, and temperature is given by the Clausius-Clapeyron equation.
Nearly all of us have heated a pan of water with the lid in place and shortly thereafter heard the sounds of the lid rattling and hot water spilling onto the stovetop. When a liquid is heated, its
molecules obtain sufficient kinetic energy to overcome the forces holding them in the liquid and they escape into the gaseous phase. By doing so, they generate a population of molecules in the vapor
phase above the liquid that produces a pressure—the vapor pressure of the liquid. In the situation we described, enough pressure was generated to move the lid, which allowed the vapor to escape. If
the vapor is contained in a sealed vessel, however, such as an unvented flask, and the vapor pressure becomes too high, the flask will explode (as many students have unfortunately discovered). In
this section, we describe vapor pressure in more detail and explain how to quantitatively determine the vapor pressure of a liquid.
Evaporation and Condensation
Because the molecules of a liquid are in constant motion, we can plot the fraction of molecules with a given kinetic energy (KE) against their kinetic energy to obtain the kinetic energy distribution
of the molecules in the liquid (Figure \(\PageIndex{1}\)), just as we did for a gas. As for gases, increasing the temperature increases both the average kinetic energy of the particles in a liquid
and the range of kinetic energy of the individual molecules. If we assume that a minimum amount of energy (\(E_0\)) is needed to overcome the intermolecular attractive forces that hold a liquid
together, then some fraction of molecules in the liquid always has a kinetic energy greater than \(E_0\). The fraction of molecules with a kinetic energy greater than this minimum value increases
with increasing temperature. Any molecule with a kinetic energy greater than \(E_0\) has enough energy to overcome the forces holding it in the liquid and escape into the vapor phase. Before it can
do so, however, a molecule must also be at the surface of the liquid, where it is physically possible for it to leave the liquid surface; that is, only molecules at the surface can undergo
evaporation (or vaporization), where molecules gain sufficient energy to enter a gaseous state above a liquid’s surface, thereby creating a vapor pressure.
Figure \(\PageIndex{1}\): The Distribution of the Kinetic Energies of the Molecules of a Liquid at Two Temperatures. Just as with gases, increasing the temperature shifts the peak to a higher energy
and broadens the curve. Only molecules with a kinetic energy greater than E[0] can escape from the liquid to enter the vapor phase, and the proportion of molecules with KE > E[0] is greater at the
higher temperature. (CC BY-SA-NC; Anonymous by request)
Graph of fraction of molecules with a particular kinetic energy against kinetic energy. Green line is temperature at 400 kelvin, purple line is temperature at 300 kelvin.
To understand the causes of vapor pressure, consider the apparatus shown in Figure \(\PageIndex{2}\). When a liquid is introduced into an evacuated chamber (part (a) in Figure \(\PageIndex{2}\)), the
initial pressure above the liquid is approximately zero because there are as yet no molecules in the vapor phase. Some molecules at the surface, however, will have sufficient kinetic energy to escape
from the liquid and form a vapor, thus increasing the pressure inside the container. As long as the temperature of the liquid is held constant, the fraction of molecules with \(KE > E_0\) will not
change, and the rate at which molecules escape from the liquid into the vapor phase will depend only on the surface area of the liquid phase.
Figure \(\PageIndex{2}\): Vapor Pressure. (a) When a liquid is introduced into an evacuated chamber, molecules with sufficient kinetic energy escape from the surface and enter the vapor phase,
causing the pressure in the chamber to increase. (b) When sufficient molecules are in the vapor phase for a given temperature, the rate of condensation equals the rate of evaporation (a steady state
is reached), and the pressure in the container becomes constant. (CC BY-SA-NC; Anonymous by request)
As soon as some vapor has formed, a fraction of the molecules in the vapor phase will collide with the surface of the liquid and reenter the liquid phase in a process known as condensation (part (b)
in Figure \(\PageIndex{2}\)). As the number of molecules in the vapor phase increases, the number of collisions between vapor-phase molecules and the surface will also increase. Eventually, a steady
state will be reached in which exactly as many molecules per unit time leave the surface of the liquid (vaporize) as collide with it (condense). At this point, the pressure over the liquid stops
increasing and remains constant at a particular value that is characteristic of the liquid at a given temperature. The rates of evaporation and condensation over time for a system such as this are
shown graphically in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\): The Relative Rates of Evaporation and Condensation as a Function of Time after a Liquid Is Introduced into a Sealed Chamber. The rate of evaporation depends only on the
surface area of the liquid and is essentially constant. The rate of condensation depends on the number of molecules in the vapor phase and increases steadily until it equals the rate of evaporation.
(CC BY-SA-NC; Anonymous by request)
Graph of rate against time. The green line is evaporation while the pruple line is condensation. Dynamic equilibrium is established when the evaporation and condensation rates are equal.
Equilibrium Vapor Pressure
Two opposing processes (such as evaporation and condensation) that occur at the same rate and thus produce no net change in a system, constitute a dynamic equilibrium. In the case of a liquid
enclosed in a chamber, the molecules continuously evaporate and condense, but the amounts of liquid and vapor do not change with time. The pressure exerted by a vapor in dynamic equilibrium with a
liquid is the equilibrium vapor pressure of the liquid.
If a liquid is in an open container, however, most of the molecules that escape into the vapor phase will not collide with the surface of the liquid and return to the liquid phase. Instead, they will
diffuse through the gas phase away from the container, and an equilibrium will never be established. Under these conditions, the liquid will continue to evaporate until it has “disappeared.” The
speed with which this occurs depends on the vapor pressure of the liquid and the temperature. Volatile liquids have relatively high vapor pressures and tend to evaporate readily; nonvolatile liquids
have low vapor pressures and evaporate more slowly. Although the dividing line between volatile and nonvolatile liquids is not clear-cut, as a general guideline, we can say that substances with vapor
pressures greater than that of water (Figure \(\PageIndex{4}\)) are relatively volatile, whereas those with vapor pressures less than that of water are relatively nonvolatile. Thus diethyl ether
(ethyl ether), acetone, and gasoline are volatile, but mercury, ethylene glycol, and motor oil are nonvolatile.
Figure \(\PageIndex{4}\): The Vapor Pressures of Several Liquids as a Function of Temperature. The point at which the vapor pressure curve crosses the P = 1 atm line (dashed) is the normal boiling
point of the liquid. (CC BY-SA-NC; Anonymous by request)
The equilibrium vapor pressure of a substance at a particular temperature is a characteristic of the material, like its molecular mass, melting point, and boiling point. It does not depend on the
amount of liquid as long as at least a tiny amount of liquid is present in equilibrium with the vapor. The equilibrium vapor pressure does, however, depend very strongly on the temperature and the
intermolecular forces present, as shown for several substances in Figure \(\PageIndex{4}\). Molecules that can hydrogen bond, such as ethylene glycol, have a much lower equilibrium vapor pressure
than those that cannot, such as octane. The nonlinear increase in vapor pressure with increasing temperature is much steeper than the increase in pressure expected for an ideal gas over the
corresponding temperature range. The temperature dependence is so strong because the vapor pressure depends on the fraction of molecules that have a kinetic energy greater than that needed to escape
from the liquid, and this fraction increases exponentially with temperature. As a result, sealed containers of volatile liquids are potential bombs if subjected to large increases in temperature. The
gas tanks on automobiles are vented, for example, so that a car won’t explode when parked in the sun. Similarly, the small cans (1–5 gallons) used to transport gasoline are required by law to have a
pop-off pressure release.
Volatile substances have low boiling points and relatively weak intermolecular interactions; nonvolatile substances have high boiling points and relatively strong intermolecular interactions.
A Video Discussing Vapor Pressure and Boiling Points. Video Source: Vapor Pressure & Boiling Point(opens in new window) [youtu.be]
The exponential rise in vapor pressure with increasing temperature in Figure \(\PageIndex{4}\) allows us to use natural logarithms to express the nonlinear relationship as a linear one.
\[ \boxed{\ln P =\dfrac{-\Delta H_{vap}}{R}\left ( \dfrac{1}{T} \right) + C} \label{Eq1} \]
• \(\ln P\) is the natural logarithm of the vapor pressure,
• \(ΔH_{vap}\) is the enthalpy of vaporization,
• \(R\) is the universal gas constant [8.314 J/(mol•K)],
• \(T\) is the temperature in kelvins, and
• \(C\) is the y-intercept, which is a constant for any given line.
Plotting \(\ln P\) versus the inverse of the absolute temperature (\(1/T\)) is a straight line with a slope of −ΔH[vap]/R. Equation \(\ref{Eq1}\), called the Clausius–Clapeyron Equation, can be used
to calculate the \(ΔH_{vap}\) of a liquid from its measured vapor pressure at two or more temperatures. The simplest way to determine \(ΔH_{vap}\) is to measure the vapor pressure of a liquid at two
temperatures and insert the values of \(P\) and \(T\) for these points into Equation \(\ref{Eq2}\), which is derived from the Clausius–Clapeyron equation:
\[ \ln\left ( \dfrac{P_{1}}{P_{2}} \right)=\dfrac{-\Delta H_{vap}}{R}\left ( \dfrac{1}{T_{1}}-\dfrac{1}{T_{2}} \right) \label{Eq2} \]
Conversely, if we know ΔH[vap] and the vapor pressure \(P_1\) at any temperature \(T_1\), we can use Equation \(\ref{Eq2}\) to calculate the vapor pressure \(P_2\) at any other temperature \(T_2\),
as shown in Example \(\PageIndex{1}\).
A Video Discussing the Clausius-Clapeyron Equation. Video Link: The Clausius-Clapeyron Equation(opens in new window) [youtu.be]
The experimentally measured vapor pressures of liquid Hg at four temperatures are listed in the following table:
experimentally measured vapor
pressures of liquid Hg at four
T (°C) 80.0 100 120 140
P (torr) 0.0888 0.2729 0.7457 1.845
From these data, calculate the enthalpy of vaporization (ΔH[vap]) of mercury and predict the vapor pressure of the liquid at 160°C. (Safety note: mercury is highly toxic; when it is spilled, its
vapor pressure generates hazardous levels of mercury vapor.)
Given: vapor pressures at four temperatures
Asked for: ΔH[vap] of mercury and vapor pressure at 160°C
1. Use Equation \(\ref{Eq2}\) to obtain ΔH[vap] directly from two pairs of values in the table, making sure to convert all values to the appropriate units.
2. Substitute the calculated value of ΔH[vap] into Equation \(\ref{Eq2}\) to obtain the unknown pressure (P[2]).
A The table gives the measured vapor pressures of liquid Hg for four temperatures. Although one way to proceed would be to plot the data using Equation \(\ref{Eq1}\) and find the value of ΔH[vap]
from the slope of the line, an alternative approach is to use Equation \(\ref{Eq2}\) to obtain ΔH[vap] directly from two pairs of values listed in the table, assuming no errors in our measurement. We
therefore select two sets of values from the table and convert the temperatures from degrees Celsius to kelvin because the equation requires absolute temperatures. Substituting the values measured at
80.0°C (T[1]) and 120.0°C (T[2]) into Equation \(\ref{Eq2}\) gives
\[\begin{align*} \ln \left ( \dfrac{0.7457 \; \cancel{Torr}}{0.0888 \; \cancel{Torr}} \right) &=\dfrac{-\Delta H_{vap}}{8.314 \; J/mol\cdot K}\left ( \dfrac{1}{\left ( 120+273 \right)K}-\dfrac{1}{\
left ( 80.0+273 \right)K} \right) \\[4pt] \ln\left ( 8.398 \right) &=\dfrac{-\Delta H_{vap}}{8.314 \; J/mol\cdot \cancel{K}}\left ( -2.88\times 10^{-4} \; \cancel{K^{-1}} \right) \\[4pt] 2.13 &=-\
Delta H_{vap} \left ( -3.46 \times 10^{-4} \right) J^{-1}\cdot mol \\[4pt] \Delta H_{vap} &=61,400 \; J/mol = 61.4 \; kJ/mol \end{align*} \nonumber \]
B We can now use this value of ΔH[vap] to calculate the vapor pressure of the liquid (P[2]) at 160.0°C (T[2]):
\[ \ln\left ( \dfrac{P_{2} }{0.0888 \; torr} \right)=\dfrac{-61,400 \; \cancel{J/mol}}{8.314 \; \cancel{J/mol} \; K^{-1}}\left ( \dfrac{1}{\left ( 160+273 \right)K}-\dfrac{1}{\left ( 80.0+273 \right)
K} \right) \nonumber \]
Using the relationship \(e^{\ln x} = x\), we have
\[\begin{align*} \ln \left ( \dfrac{P_{2} }{0.0888 \; Torr} \right) &=3.86 \\[4pt] \dfrac{P_{2} }{0.0888 \; Torr} &=e^{3.86} = 47.5 \\[4pt] P_{2} &= 4.21 Torr \end{align*} \nonumber \]
At 160°C, liquid Hg has a vapor pressure of 4.21 torr, substantially greater than the pressure at 80.0°C, as we would expect.
The vapor pressure of liquid nickel at 1606°C is 0.100 torr, whereas at 1805°C, its vapor pressure is 1.000 torr. At what temperature does the liquid have a vapor pressure of 2.500 torr?
Boiling Points
As the temperature of a liquid increases, the vapor pressure of the liquid increases until it equals the external pressure, or the atmospheric pressure in the case of an open container. Bubbles of
vapor begin to form throughout the liquid, and the liquid begins to boil. The temperature at which a liquid boils at exactly 1 atm pressure is the normal boiling point of the liquid. For water, the
normal boiling point is exactly 100°C. The normal boiling points of the other liquids in Figure \(\PageIndex{4}\) are represented by the points at which the vapor pressure curves cross the line
corresponding to a pressure of 1 atm. Although we usually cite the normal boiling point of a liquid, the actual boiling point depends on the pressure. At a pressure greater than 1 atm, water boils at
a temperature greater than 100°C because the increased pressure forces vapor molecules above the surface to condense. Hence the molecules must have greater kinetic energy to escape from the surface.
Conversely, at pressures less than 1 atm, water boils below 100°C.
Table \(\PageIndex{1}\): The Boiling Points of Water at Various Locations on Earth
Place Altitude above Sea Level (ft) Atmospheric Pressure (mmHg) Boiling Point of Water (°C)
Mt. Everest, Nepal/Tibet 29,028 240 70
Bogota, Colombia 11,490 495 88
Denver, Colorado 5280 633 95
Washington, DC 25 759 100
Dead Sea, Israel/Jordan −1312 799 101.4
Typical variations in atmospheric pressure at sea level are relatively small, causing only minor changes in the boiling point of water. For example, the highest recorded atmospheric pressure at sea
level is 813 mmHg, recorded during a Siberian winter; the lowest sea-level pressure ever measured was 658 mmHg in a Pacific typhoon. At these pressures, the boiling point of water changes minimally,
to 102°C and 96°C, respectively. At high altitudes, on the other hand, the dependence of the boiling point of water on pressure becomes significant. Table \(\PageIndex{1}\) lists the boiling points
of water at several locations with different altitudes. At an elevation of only 5000 ft, for example, the boiling point of water is already lower than the lowest ever recorded at sea level. The lower
boiling point of water has major consequences for cooking everything from soft-boiled eggs (a “three-minute egg” may well take four or more minutes in the Rockies and even longer in the Himalayas) to
cakes (cake mixes are often sold with separate high-altitude instructions). Conversely, pressure cookers, which have a seal that allows the pressure inside them to exceed 1 atm, are used to cook food
more rapidly by raising the boiling point of water and thus the temperature at which the food is being cooked.
As pressure increases, the boiling point of a liquid increases and vice versa.
Use Figure \(\PageIndex{4}\) to estimate the following.
1. the boiling point of water in a pressure cooker operating at 1000 mmHg
2. the pressure required for mercury to boil at 250°C
Mercury boils at 356 °C at room pressure. To see video go to www.youtube.com/watch?v=0iizsbXWYoo
Given: Data in Figure \(\PageIndex{4}\), pressure, and boiling point
Asked for: corresponding boiling point and pressure
1. To estimate the boiling point of water at 1000 mmHg, refer to Figure \(\PageIndex{4}\) and find the point where the vapor pressure curve of water intersects the line corresponding to a pressure
of 1000 mmHg.
2. To estimate the pressure required for mercury to boil at 250°C, find the point where the vapor pressure curve of mercury intersects the line corresponding to a temperature of 250°C.
1. A The vapor pressure curve of water intersects the P = 1000 mmHg line at about 110°C; this is therefore the boiling point of water at 1000 mmHg.
2. B The vertical line corresponding to 250°C intersects the vapor pressure curve of mercury at P ≈ 75 mmHg. Hence this is the pressure required for mercury to boil at 250°C.
Ethylene glycol is an organic compound primarily used as a raw material in the manufacture of polyester fibers and fabric industry, and polyethylene terephthalate resins (PET) used in bottling. Use
the data in Figure \(\PageIndex{4}\) to estimate the following.
1. the normal boiling point of ethylene glycol
2. the pressure required for diethyl ether to boil at 20°C.
Answer a
Answer b
450 mmHg
Because the molecules of a liquid are in constant motion and possess a wide range of kinetic energies, at any moment some fraction of them has enough energy to escape from the surface of the liquid
to enter the gas or vapor phase. This process, called vaporization or evaporation, generates a vapor pressure above the liquid. Molecules in the gas phase can collide with the liquid surface and
reenter the liquid via condensation. Eventually, a steady state is reached in which the number of molecules evaporating and condensing per unit time is the same, and the system is in a state of
dynamic equilibrium. Under these conditions, a liquid exhibits a characteristic equilibrium vapor pressure that depends only on the temperature. We can express the nonlinear relationship between
vapor pressure and temperature as a linear relationship using the Clausius–Clapeyron equation. This equation can be used to calculate the enthalpy of vaporization of a liquid from its measured vapor
pressure at two or more temperatures. Volatile liquids are liquids with high vapor pressures, which tend to evaporate readily from an open container; nonvolatile liquids have low vapor pressures.
When the vapor pressure equals the external pressure, bubbles of vapor form within the liquid, and it boils. The temperature at which a substance boils at a pressure of 1 atm is its normal boiling
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Sampling Distributions
Welcome to Sampling Distributions
I greet you this day,
First: read the notes including the Teacher-Student scenarios.
Second: view the videos.
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions.
Fifth: check your answers with the calculators as applicable.
The Wolfram Alpha widget (many thanks to the developers) are used for the calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome.
You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting.
Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S
Students will:
(1.) Define the sampling distribution of a statistic.
(2.) Generate simulations OR perform some classroom activities to demonstrate the sampling distribution of a statistic.
(3.) Discuss the sampling distribution of the sample mean.
(4.) Discuss the sampling distribution of the sample proportion.
(5.) Discuss the sampling distribution of the sample standard deviation.
(6.) Discuss the sampling distribution of the sample range.
(7.) Discuss the sampling distribution of the sample median.
(8.) Solve problems involving the sampling distribution of a statistic.
Skills Measured/Acquired
(1.) Use of prior knowledge
(2.) Critical thinking
(3.) Interdisciplinary connections/applications
(4.) Technology (using R/RStudio, TI-84 Plus and Graphing Calculators among others)
(5.) Active participation through direct questioning
(6.) Research
Please Check back later 😊
Symbols and Meanings
• $\mu_{\bar{x}}$ = mean of the sample means
• $\mu$ = population mean
• $\mu_{\hat{p}}$ = mean of the sample proportions
• $\sigma_{\bar{x}}$ = standard error of the mean
• $\sigma_{\bar{x}}$ = standard deviation of the sample means
• $\sigma_{\hat{p}}$ = standard error of the sample proportion
• $\sigma_{\hat{p}}$ = standard deviation of the sample proportions
• $\sigma_{\hat{p}est}$ = estimated standard error
• $\sigma$ = population standard deviation
• $z$ = z-score
• $\bar{x}$ = sample mean
• $n$ is sample size
• $\hat{p}$ = sample proportion
• $\hat{p}$ = estimated proportion of successes
• $\hat{q}$ = estimated proportion of failures
• $p$ = population proportion
• $x$ = number of individuals in the sample with the specified characteristic
Sampling Distribution of the Sample Mean
and the
Central Limit Theorem
Case 1: Population is not normally distributed, but sample size is greater than 30
Case 2: Population is normally distributed, and sample size is any size
$ (1.)\;\; \mu_{\bar{x}} = \mu \\[5ex] (2.)\;\; \sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} \\[5ex] (3.)\;\; z = \dfrac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \\[5ex] (4.)\;\; z = \dfrac{\bar{x}
- \mu_{\bar{x}}}{\dfrac{\sigma}{\sqrt{n}}} \\[7ex] (5.)\;\; z = \dfrac{\sqrt{n}(\bar{x} - \mu_{\bar{x}})}{\sigma_{\bar{x}}} \\[5ex] $
Sampling Distribution of the Sample Proportion
and the
Central Limit Theorem
Condition 1: Simple Random Sample with Independent Trials.
(A.) The sample may be taken with or without replacement.
However, if the sample is taken without replacement; then the population size must be at least ten times bigger than the sample size.
If sampling is done without replacement, then N ≥ 10n
(B.) Verify that the trials are independent: The sample size is no more than 5% of the population size.
n ≤ 0.05N
Condition 2: Large sample size with at least ten successes and ten failures.
$ (A.)\;\; np \ge 10 \\[3ex] (B.)\;\; nq \ge 10 \\[3ex] $ Formulas: (When both conditions are met)
$ \underline{When \;\;p\;\; is\;\;not\;\;known} \\[3ex] n\hat{p} \ge 10 \;\;\;and\;\;\; n\hat{q} \ge 10 \\[3ex] (1.)\;\; \hat{p} = \dfrac{x}{n} \\[5ex] (2.)\;\; p + q = 1 \\[3ex] (3.)\;\; \hat{p} + \
hat{q} = 1 \\[3ex] (4.)\;\; \mu_{\hat{p}} = p \\[3ex] (5.)\;\; \sigma_{\hat{p}} = \sqrt{\dfrac{p * q}{n}} \\[5ex] (6.)\;\; z = \dfrac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} \\[5ex] (7.)\;\; z = \
dfrac{\hat{p} - p}{\sigma_{\hat{p}}} \\[7ex] \underline{When \;\;p\;\; is\;\;not\;\;known} \\[3ex] (1.)\;\; \sigma_{\hat{p}est} = \sqrt{\dfrac{\hat{p} * \hat{q}}{n}} $
The Sampling Distribution of a statistic is the probability distribution of all values of the statistic when all possible samples of the same size, n are taken from the same population.
An estimator is a statistic used to infer the value of a population parameter.
An unbiased estimator is a statistic that targets the value of the population parameter such that the sampling distribution of the statistic has a mean that is equal to the mean of the corresponding
Precision also known as the Amount of Sampling Error is the error that results from using a sample statistic to estimate a population parameter.
For example; it is the error that results from using:
Sample Proportion to estimate Population Proportion
Sample Mean to estimate Population Mean
Sample Variance to estimate Population Variance
Sample Standard Deviation to estimate Population Standard Deviation
Precision is the closeness of two or more measurements to each other.
Standard Error is the standard deviation of a sampling distribution.
Precision is measured using the standard deviation of the sampling distribution.
An estimator is precise if it gives a small standard error.
Accuracy also known as the Amount of Bias is the distance between the mean value of the estimator and the population parameter.
It is the closeness of a measured value to a standard value.
It is measured using the center of the sampling distribution.
For unbiased estimators, the bias is zero. In other words, there is no bias.
One can be imprecise, but accurate. One can be precise, but inaccurate. One can be both precise and accurate. Discuss.
This is defined as the probability distribution of the sample means, where all samples have the same size, n; and are taken from the same population.
The Law of Large Numbers states that as the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of that
Overview of Sampling Distributions
Would you rather conduct a census to get a population (parameter) size; or take random samples (statistic) from that population, and use it to estimate the population size? What are your reasons?
Cost? Convenience? Accuracy?
Assume we decide to do the later, do you think we might have some errors?
As discussed earlier in Introductory Statistics; a statistic is the numerical summary of a sample, while a parameter is the numerical summary of a population.
Let us begin by discussing the sampling distribution of the sample mean.
Do you mind if we take the ages of the students in the class?
Record the population size. N = ???
Calculate the population mean, μ
List the combinations of possible sample sizes, n = 2
Calculate the mean of those sample sizes.
Calculate the mean of the sample means.
Represent them using a dot plot (for easy representation) OR
Represent them using a histogram (to assess normality)
Interpret the probability of estimating the population mean using the sample mean.
Repeat the process for n = 3, 4, …, N − 1
What are our observations?
(1.) As the sample size increases, the sampling error decreases.
(2.) As the sample size increases, the mean of the sample means gets closer and closer to the population mean.
We could do this because we know the population size.
Imagine a large population, and we do not know the population size?
How can we estimate the mean/average age of that population?
Behavior of the Sample Means
(1.) The sample means is an unbiased estimator of the population mean.
The mean of the sample means is the population mean.
The expected value of the sample mean is equal to the population mean.
(2.) The distribution of the sample means tends to be a normal distribution.
The Central Limit Theorem states that for all simple random samples of same size, n; where n > 30; the sampling distribution of the sample mean can be approximated by a normal distribution where the
mean of the sample means is equal to the population mean; and the sample standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample
Given: simple random samples of same size, n taken from the same population where n > 30:
$ \mu_{\bar{x}} = \mu \;\;\;and\;\;\; \sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} \\[5ex] $
Standard Normal Distribution Table (Left-Shaded Area)
Standard Normal Distribution Table (Center-Shaded Area)
Chukwuemeka, S.D (2016, February 25). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com
Black, Ken. (2012). Business Statistics for Contemporary Decision Making (7th ed.). New Jersey: Wiley
Gould, R., & Ryan, C. (2016). Introductory Statistics: Exploring the world through data (2nd ed.). Boston: Pearson
Gould, R., Wong, R., & Ryan, C. N. (2020). Introductory Statistics: Exploring the world through data (3rd ed.). Pearson.
Kozak, Kathryn. (2015). Statistics Using Technology (2nd ed.).
OpenStax, Introductory Statistics.OpenStax CNX. Sep 28, 2016. Retrieved from https://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de@18.12
Sullivan, M., & Barnett, R. (2013). Statistics: Informed decisions using data with an introduction to mathematics of finance (2nd custom ed.). Boston: Pearson Learning Solutions.
Triola, M. F. (2015). Elementary Statistics using the TI-83/84 Plus Calculator (5th ed.). Boston: Pearson
Triola, M. F. (2022). Elementary Statistics. (14th ed.) Hoboken: Pearson.
Weiss, Neil A. (2015). Elementary Statistics (9th ed.). Boston: Pearson
TI Products | Calculators and Technology | Texas Instruments. (n.d.). Education.ti.com. Retrieved March 18, 2023, from https://education.ti.com/en/products
GCSE Exam Past Papers: Revision World. Retrieved April 6, 2020, from https://revisionworld.com/gcse-revision/gcse-exam-past-papers
HSC exam papers | NSW Education Standards. (2019). Nsw.edu.au. https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/resources/hsc-exam-papers
NSC Examinations. (n.d.). www.education.gov.za. https://www.education.gov.za/Curriculum/NationalSeniorCertificate(NSC)Examinations.aspx
Normal Distribution Table (Left Shaded Area): https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf
Normal Distribution Table (Center Shaded Area): https://itl.nist.gov/div898/handbook/eda/section3/eda3671.htm
51 Real SAT PDFs and List of 89 Real ACTs (Free) : McElroy Tutoring. (n.d.). Mcelroytutoring.com. Retrieved December 12, 2022, from https://mcelroytutoring.com/lower.php?url= | {"url":"https://samplingdistribution.appspot.com/","timestamp":"2024-11-02T17:29:50Z","content_type":"text/html","content_length":"31869","record_id":"<urn:uuid:f6147728-4051-4701-93ce-b3c2b94b9d3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00428.warc.gz"} |
25 bar space truss
The purpose of this example is the minimization of the weight of a space bar truss. We will use a well-known benchmark problem, i.e. the 25-bar space truss, which has been studied extensively in the
literature. The geometrical data is given in the picture (in inches). Although there are 25 bars in the truss, these are grouped into 8 groups, according to a table given below, which makes the
problem easier (i.e. it reduces the problem's dimensionality from 25 to 8). The material's Young's modulus is E = 10000 ksi and the specific weight is ρ = 0.1 lb/in^3.
The purpose is to minimize its weight by picking the appropriate cross-sectional area for each bar, so that the following requirements (constraints) are met:
• The cross-sectional area of each bar varies between 0.01 and 35 in^2 (these are the so-called side constraints of our design variables)
• The stress in each bar does not exceed a certain tensile and compressive level, given in a table below
• The displacement of the top nodes (1,2) in each direction should not exceed 0.35 inches
The table with the bar grouping and stress levers is the following:
The table with the bar grouping and stress levers is the following:
│Member group│Members │Compressive stress limit [ksi] │Tensile stress limit [ksi]│
│1 │1 │35.092 │40 │
│2 │2, 3, 4, 5 │11.590 │40 │
│3 │6, 7, 8, 9 │17.305 │40 │
│4 │10, 11 │35.092 │40 │
│5 │12, 13 │35.092 │40 │
│6 │14, 15, 16, 17│6.759 │40 │
│7 │18, 19, 20, 21│6.959 │40 │
│8 │22, 23, 24, 25│11.082 │40 │
Two load cases are considered, as follows:
│Node│P[x] [kips] │P[y] [kips] │P[z] [kips] │
│Load Case I │
│1 │0 │20 │-5 │
│2 │0 │-20 │-5 │
│Load Case II │
│1 │1 │10 │-5 │
│2 │0 │10 │-5 │
│3 │0.5 │0 │0 │
│6 │0.5 │0 │0 │
Note that there is a difference in this case as compared to the two load cases examined in the 10 bar planar truss. In the 10 bar truss, the two load cases were independent, which means that
practically we are dealing with two different optimization problems. Here, the two load cases must be checked for the same configuration. To tackle that we need to write some additional VBA code. The
routine Sheet1.Calc solves the current truss configuration and evaluates the displacements and stresses. We have included Sheet1.Calc2, an additional routine which does the following:
• Sets up load case Ι
• Solves the truss using Sheet1.Calc
• Evaluates the penalties for load case I
• Sets up load case II
• Solves (again) the truss using Sheet1.Calc
• Evaluates the penalties for load case II
• Adds the sum of penalties to the weight of the truss.
This routine will be called by xlOptimizer. For demonstration purposes, the constraints will be evaluated by VBA code, so we don't need to include them in xlOptimizer.
Note that solving the same truss twice can be improved considerably, e.g. by not evaluating the stiffness matrix twice, but this exceeds the purpose of this article. Also, this article assumes that
you have read and understood the procedures described in the toy problems. These include the optimization using pure Microsoft Excel formulas and the optimization using VBA (Visual Basic for
Step 1 : Setting-up the spreadsheet
We have already created the spreadsheet for you. It can be used for any space truss. Note that the spreadsheet uses consistent unit system; this means that, if you pick the correct units, you can
have the correct results in SI as well. In the file, however, the data has been set in the Imperial System as this was used for the definition of the 25-bar truss. Therefore, the area of each bar is
given square inches (in^2). The areas of all 25 bars are controlled by 8 bars only, i.e. the first bar in each group (shown in the yellow cells).
The evaluation of the stresses and displacements of the current configuration is performed by a custom routine, i.e. Sheet1.Calc. The code can be inspected if you produce the VBA editor (by pressing
The function evaluation is performed by another custom routine, i.e. Sheet1.Calc2.
Step 2 : Setting up the objective function
Select cell AM1. This holds our objective function, i.e. the current weight of the truss. In the xlOptimizer ribbon, press Objectives to produce the list of objectives. Then press the Add button
The objective is set to minimization by default. Press the Objectives button in the ribbon again to hide the objectives list.
Step 3 : Setting up the design variables
Select the yellow cells, by holding CTRL and clicking the cells one by one. These hold our design variables, i.e. the cross-sectional area of each bar group. The rest of the bars obtain the same
values with simple formulas.
In the xlOptimizer ribbon, press Design variables to produce the list of design variables. Then press the Add button '+' to add the cells to the list:
Having all design variables selected in the list, press the Edit button in the toolbar. In the form, set the bounds from 0.1 to 35 and press Ok. The selection will be applied to all selected rows.
Press the Design variables button again in the ribbon to hide the design variable list.
Step 4 : Setting up the constraints
The constraints are evaluated by custom routine Sheet1.Calc2, so you don't need to add them here.
Step 5 : Setting up the execution sequence
In the xlOptimizer ribbon, press Execution sequence to produce the list of execution commands. By default, a single Calculate command is listed. This is a call to Microsoft Excel's internal
calculation routine, which evaluates the formulas such as '=SUM(C2:C6)'. Contrary to the 10 bar truss, we need to keep this command, first in the list, because we have used simple formulas to fill in
the missing areas for some of the bars, e.g. the area for bar 3 (cell L5) is set equal to the one for bar 2 using the formula '=$L$4'. We must add a second command, which is a macro call to routine
Press Ok to save the changes.
Step 6 : Setting up the optimization scenario
In the ribbon, press the Optimization scenaria button. Next, press the Add button '+' in the toolbar to add a new scenario. The following form appears:
Make the appropriate selections and press Ok. The scenario is added to the list. In this case, the Standard Differential Evolution algorithm was selected.
The data input is now complete. In the ribbon, press the Optimization scenaria button again to hide the list.
Step 7 : Optimization
We can now proceed to the optimization by pressing the Run active scenaria button.
The file for this example can be downloaded here. Note that it is saved a macro enabled book with the extension .xlsm. You need to enable macros to run it.
By inspection of the literature, the best solution for the 25-bar truss that does not violate any constraints has a weight of 545.172 lbs. | {"url":"https://www.xloptimizer.com/projects/mechanics/25-bar-space-truss","timestamp":"2024-11-06T16:39:56Z","content_type":"application/xhtml+xml","content_length":"27905","record_id":"<urn:uuid:c1b5c91b-efda-49fd-87ac-0d826211c62a>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00531.warc.gz"} |
Converting From Longitude\latitude To Cartesian Coordinates - The Citrus Report
Converting From Longitude\latitude To Cartesian Coordinates
As professionals, we often need to translate between different coordinate systems to effectively analyze and visualize data. One common coordinate system is latitude and longitude, which uses angles
to locate points on the surface of a sphere or spheroid, such as the Earth. This coordinate system is useful for geographic applications, such as mapping, navigation, and earth science. However, to
use this data with many mathematical models and algorithms, it may be necessary to convert it to a Cartesian coordinate system, which uses x, y, and z coordinates to locate points in Euclidean space.
Converting Latitude and Longitude to Cartesian Coordinates
There are several methods to convert latitude and longitude to Cartesian coordinates, depending on the desired accuracy, projection, and datum. One common method is to use a spherical or ellipsoidal
model of the Earth and assume a specific center, radius, and orientation. In this case, the conversion involves transforming the spherical coordinates (ϕ, λ, r) to Cartesian coordinates (x, y, z)
using trigonometry and algebra:
x = r cos ϕ cos λ
y = r cos ϕ sin λ
z = r sin ϕ
where ϕ is the latitude, λ is the longitude, and r is the radius. The x-axis intersects the sphere at the Prime Meridian (0° longitude) and the equator (0° latitude), the y-axis intersects the sphere
at 90° E longitude and the equator, and the z-axis intersects the sphere at the North Pole. This system is called a geocentric coordinate system, as it is centered on the Earth’s center of mass.
For higher accuracy, it may be necessary to use a more complex model of the Earth, such as the World Geodetic System 1984 (WGS84) or the European Terrestrial Reference System 1989 (ETRS89), which
account for the Earth’s irregular shape, rotational dynamics, and gravitational field. These models define a reference ellipsoid or a geoid, which approximates the shape of the Earth’s surface and
provides a datum for latitude and longitude measurements. To convert from these coordinates to Cartesian coordinates, additional parameters such as the semi-major axis, semi-minor axis, flattening,
and eccentricity are needed.
Converting Cartesian Coordinates to Latitude and Longitude
The reverse process of converting Cartesian coordinates to latitude and longitude involves solving a set of inverse trigonometric equations, which can be computationally intensive and numerically
unstable. However, many software tools provide built-in functions for this conversion, based on various algorithms and models. Some common methods include:
• GeographicLib: A library of geodesic functions that provides accurate conversions between geographic, Cartesian, and other coordinate systems, using a variety of algorithms and models.
• GPSBabel: A cross-platform tool for converting GPS data between different file formats and coordinate systems, including WGS84 Cartesian and latitude/longitude.
• PROJ: A versatile cartographic projection and transformation library that supports a wide range of coordinate systems and projections, including transformations from Cartesian to geographic
These tools can handle different datums, projections, and coordinate systems, and provide output in various formats, such as CSV, KML, GeoJSON, and shapefile.
Applications of Coordinate Conversion
Coordinate conversion is a crucial step in many applications, such as:
• Mapping and geolocation: By converting between geographic and Cartesian coordinates, we can plot points, lines, and polygons on maps and calculate distances, areas, and angles.
• Spatial analysis: By transforming coordinates to a common system, we can compare and overlay diverse spatial data, such as weather maps, demographic maps, and land use maps, and derive insights
and patterns.
• Navigation and routing: By converting between coordinate systems and projections, we can compute optimal paths, distances, and directions for vehicles, pedestrians, ships, and aircraft, using GPS
or other location technologies.
• Environmental modeling: By integrating geographic and Cartesian data, we can model and simulate natural phenomena, such as climate change, hydrology, and ecology, and assess their impacts on the
environment and human society.
Therefore, it is important to have a good understanding of coordinate systems, their properties, and their interconversions, and to use appropriate tools and methods for specific applications.
As we’ve seen, latitude and longitude coordinate conversion is essential in many scientific, engineering, and business fields that deal with spatial data. By using accurate and efficient methods,
such as spherical trigonometry, ellipsoidal models, or geographic transformation libraries, we can convert coordinates between different systems and projections, and enable powerful analysis,
visualization, and modeling capabilities. Whether we are exploring the depths of the ocean or the heights of the sky, or simply mapping our streets and gardens, coordinate conversion is a fundamental
tool that helps us navigate and discover the world around us.
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Self-regulated learning in online mathematical problem- solving discussion forums
Online discussion forums have created both opportunities and challenges in the instruction of mathematics. They provide a variety of tools for sharing knowledge during the solution process, which can
enhance students' mathematical problem solving. However, research also indicates that students have difficulty engaging in the processes involved in using discussion forums, which require the ability
to coordinate knowledge with solution strategies and control behaviors (i.e., monitoring). This ability is the essence of self-regulated learning (SRL). This article presents how one may stimulate
students' online SRL in mathematical problem-solving discussion forums by using support techniques. An overview of four research fields, along with the leading experts in each field, presents the
complexity of mathematical problem-solving online discussion forum tools, SRL models and self-questioning support techniques using the IMPROVE model. Future directions are suggested.
Dive into the research topics of 'Self-regulated learning in online mathematical problem- solving discussion forums'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/self-regulated-learning-in-online-mathematical-problem-solving-di","timestamp":"2024-11-09T10:50:18Z","content_type":"text/html","content_length":"52122","record_id":"<urn:uuid:d5a48ef3-b11d-49b4-8239-89d915045973>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00790.warc.gz"} |