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Bin spatial data and determine statistics per bin
gmt binstats [ table ] -Goutgrid -Iincrement -Ca|d|g|i|l|L|m|n|o|p|q[quant]|r|s|u|U|z -Rregion -Sradius [ -Eempty ] [ -N ] [ -T[h|r] ] [ -V[level] ] [ -W[+s] ] [ -aflags ] [ -bibinary ] [ -dinodata[
+ccol] ] [ -eregexp ] [ -fflags ] [ -ggaps ] [ -hheaders ] [ -iflags ] [ -qiflags ] [ -rreg ] [ -wflags ] [ -:[i|o] ] [ --PAR=value ]
Note: No space is allowed between the option flag and the associated arguments.
binstats reads arbitrarily located (x, y[, z][, w]) points (2-4 columns) from standard input [or table] and for each node in the specified grid layout determines which points are within the given
radius. These points are then used in the calculation of the specified statistic. The results may be presented as is or may be normalized by the circle area to perhaps give density estimates.
Alternatively, select hexagonal tiling instead or a rectangular grid layout.
Required Arguments
A 2-4 column ASCII file(s) [or binary, see -bi] holding (x, y[, z][, w]) data values. You must use -W to indicate that you have weights. Only -Cn will accept 2 columns only. If no file is
specified, binstats will read from standard input.
Choose the statistic that will be computed per node based on the points that are within radius distance of the node. Append one directive among these candidates:
□ a: Mean (i.e., average).
□ d: Median absolute deviation (MAD).
□ g: The full (max-min) range.
□ i: The 25-75% interquartile range.
□ l: Minimum (lowest value).
□ L: Minimum of positive values only.
□ m: Median value.
□ n: The number of values per bin.
□ o: Least median square (LMS) scale.
□ p: Mode (maximum likelihood estimate).
□ q: Selected quantile (append desired quantile in 0-100% range [50]).
□ r: Root mean square (RMS).
□ s: Standard deviation.
□ u: Maximum (highest value).
□ U: Maximum of negative values only.
□ z: The sum of the values.
Optionally, append =ID for writing a specific file format. The following modifiers are supported:
☆ +d - Divide data values by given divisor [Default is 1].
☆ +n - Replace data values matching invalid with a NaN.
☆ +o - Offset data values by the given offset, or append a for automatic range offset to preserve precision for integer grids [Default is 0].
☆ +s - Scale data values by the given scale, or append a for automatic scaling to preserve precision for integer grids [Default is 1].
Note: Any offset is added before any scaling. +sa also sets +oa (unless overridden). To write specific formats via GDAL, use =gd and supply driver (and optionally dataType) and/or one or more
concatenated GDAL -co options using +c. See the “Writing grids and images” cookbook section for more details.
Set the grid spacing as x_inc [and optionally y_inc].
Geographical (degrees) coordinates: Optionally, append an increment unit. Choose among:
☆ d - Indicate arc degrees
☆ m - Indicate arc minutes
☆ s - Indicate arc seconds
If one of e (meter), f (foot), k (km), M (mile), n (nautical mile) or u (US survey foot), the the increment will be converted to the equivalent degrees longitude at the middle latitude of the
region (the conversion depends on PROJ_ELLIPSOID). If y_inc is not given or given but set to 0 it will be reset equal to x_inc; otherwise it will be converted to degrees latitude.
All coordinates: The following modifiers are supported:
☆ +e - Slightly adjust the max x (east) or y (north) to fit exactly the given increment if needed [Default is to slightly adjust the increment to fit the given domain].
☆ +n - Define the number of nodes rather than the increment, in which case the increment is recalculated from the number of nodes, the registration (see GMT File Formats), and the domain.
Note: If -Rgrdfile is used then the grid spacing and the registration have already been initialized; use -I and -R to override these values.
Specify the region of interest. (See full description) (See technical reference).
Optional Arguments
Set the value assigned to empty nodes [NaN].
Normalize the resulting grid values by the area represented by the search radius [no normalization].
Sets the search radius that determines which data points are considered close to a node. Append the distance unit (see Units). Not compatible with -T.
Instead of circular, possibly overlapping areas, select non-overlapping tiling. Choose between rectangular and hexagonal binning. For -Tr, set bin sizes via -I and we write the computed
statistics to the grid file named in -G. For -Th, we write a table with the centers of the hexagons and the computed statistics to standard output (or to the file named in -G). Here, the -I
setting is expected to set the y increment only and we compute the x-increment given the geometry. Because the horizontal spacing between hexagon centers in x and y have a ratio of \(\sqrt{3}\),
we will automatically adjust xmax in -R to fit a whole number of hexagons. Note: Hexagonal tiling requires Cartesian data.
Select verbosity level [w]. (See full description) (See technical reference).
Input data have an extra column containing observation point weight. If weights are given then weighted statistical quantities will be computed while the count will be the sum of the weights
instead of number of points. If your weights are actually uncertainties (\(1\sigma\)) then append +s and we compute weight = \(\frac{1}{\sigma}\).
-a[[col=]name[,…]] (more …)
Set aspatial column associations col=name.
-birecord[+b|l] (more …)
Select native binary format for primary table input. [Default is 3 (or 4 if -W is set) columns].
-dinodata[+ccol] (more …)
Replace input columns that equal nodata with NaN.
-e[~]“pattern” | -e[~]/regexp/[i] (more …)
Only accept data records that match the given pattern.
-f[i|o]colinfo (more …)
Specify data types of input and/or output columns.
-gx|y|z|d|X|Y|Dgap[u][+a][+ccol][+n|p] (more …)
Determine data gaps and line breaks.
-h[i|o][n][+c][+d][+msegheader][+rremark][+ttitle] (more …)
Skip or produce header record(s).
-icols[+l][+ddivisor][+sscale|d|k][+ooffset][,…][,t[word]] (more …)
Select input columns and transformations (0 is first column, t is trailing text, append word to read one word only).
-qi[~]rows|limits[+ccol][+a|t|s] (more …)
Select input rows or data limit(s) [default is all rows].
-r[g|p] (more …)
Set node registration [gridline].
-wy|a|w|d|h|m|s|cperiod[/phase][+ccol] (more …)
Convert an input coordinate to a cyclical coordinate.
-:[i|o] (more …)
Swap 1st and 2nd column on input and/or output.
-^ or just -
Print a short message about the syntax of the command, then exit (Note: on Windows just use -).
-+ or just +
Print an extensive usage (help) message, including the explanation of any module-specific option (but not the GMT common options), then exit.
-? or no arguments
Print a complete usage (help) message, including the explanation of all options, then exit.
Temporarily override a GMT default setting; repeatable. See gmt.conf for parameters.
For map distance unit, append unit d for arc degree, m for arc minute, and s for arc second, or e for meter [Default unless stated otherwise], f for foot, k for km, M for statute mile, n for nautical
mile, and u for US survey foot. By default we compute such distances using a spherical approximation with great circles (-jg) using the authalic radius (see PROJ_MEAN_RADIUS). You can use -jf to
perform “Flat Earth” calculations (quicker but less accurate) or -je to perform exact geodesic calculations (slower but more accurate; see PROJ_GEODESIC for method used).
Grid Values Precision
Regardless of the precision of the input data, GMT programs that create grid files will internally hold the grids in 4-byte floating point arrays. This is done to conserve memory and furthermore most
if not all real data can be stored using 4-byte floating point values. Data with higher precision (i.e., double precision values) will lose that precision once GMT operates on the grid or writes out
new grids. To limit loss of precision when processing data you should always consider normalizing the data prior to processing.
Note: Below are some examples of valid syntax for this module. The examples that use remote files (file names starting with @) can be cut and pasted into your terminal for testing. Other commands
requiring input files are just dummy examples of the types of uses that are common but cannot be run verbatim as written.
To examine the population inside a circle of 1000 km radius for all nodes in a 5x5 arc degree grid, using the remote file @capitals.gmt, and plot the resulting grid using default projection and
colors, try:
gmt begin map
gmt binstats @capitals.gmt -a2=population -Rg -I5 -Cz -Gpop.nc -S1000k
gmt grdimage pop.nc -B
gmt end show
To do hexagonal binning of the data in the file mydata.txt and counting the number of points inside each hexagon, try:
gmt binstats mydata.txt -R0/5/0/3 -I1 -Th -Cn > counts.txt | {"url":"https://docs.generic-mapping-tools.org/dev/gmtbinstats.html","timestamp":"2024-11-03T04:23:14Z","content_type":"text/html","content_length":"34756","record_id":"<urn:uuid:c1a260e4-67d3-4c77-94c5-5d855d9bf19a>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00087.warc.gz"} |
achieving ~e-10 KKT feasibility for well-scaled problem
Would you like to run ALGLIB with logging enabled, i.e. call alglib::trace_file("sqp","trace.log") prior to running the optimizer? I would like to take a closer look at the optimization process. The
fact that the objective value shows good agreement, but KKT error is not so good, suggests that there is some kind of degeneracy here.
Apologies for the lengthy context before asking my question.
I have a simple/small subproblem that I will solve thousands of times within a framework to solve a larger problem; the subproblem maximizes the distance (for this purpose, Euclidean only) between
two convex neighborhoods. For simplicity's sake, let them be two ellipsoids with explicitly fairly different radii with no additional linear constraints imposed.
I have multiple means of solving this problem to global optimality including, now (thank you!!!!), alglib's options (filter SQP, AUL, etc) that I just implemented/tested; I also have tested SNOPT via
dll, SNOPT, CONOPT, MINOS and COUENNE via GAMS, and a geometric parameter search method I wrote. I say 'globally optimal' on purpose because I have a geometric partitioning scheme that mimics what a
branch/cut implementation would do here when used in conjunction with a local optimizer. I provide more context in this recent unanswered post: https://math.stackexchange.com/questions/4917464/
analytical-solution-to-a-toy-sized-but-interesting-optimization-problem. Note that I've correctly modified the object function and gradient to solve as a minimization within ALGLIB and match the
expected solution.
I solve the problem in what I call unit-canonical space (the ellipsoids are transformed into axis-aligned unit spheres centered at the Cartesian origin) because this provides numerical stability over
many orders of magnitude difference in radii sizes (so my ALGLIB scaling is simply a unit-vector). The KKT conditions are quite 'simple' in that I have only 6 variables (the xyz-position for each
neighborhood) and 2 quadratic constraints, so I have 6 equations to evaluate: df/dx -lambda1*dg1/dx -lambda2*dg2/dx. A further simplification is that since the dg1 vector is non-zero for only
ellipsoid 1 and the dg2 vector is non-zero for only ellipsoid 2, I end up with 3 equations for df-lambda1*dg1 and 3 for df-lambda2*dg2 (because the other lambda is irrelevant in each case).
Here are two ALGLIB questions that are probably simple to you but not to me:
1. I have the analytical gradient (it's linear) and the analytical Hessian (it's a constant matrix) for f and g but there seem to be no algorithms making use of the exact Hessian - e.g. filter SQP
algorithms like SNOPT talk about an approximate Hessian and a modified Lagrangian to deal with large and/or highly non-linear problems; I see nothing in minnlcsetalgosqp or the other algorithms where
I can use that. Is there a means of using this within ALGLIB's optimization schemes? Are there algorithms for simple problems like mine that utilize the analytical Hessian?
2. ALGLIB and SNOPT's optimal (termination status 1) objective function value matches mine but the KKT conditions that I calculate are infeasible at the 1e-3 to 1e-5 level whereas in my (slow)
geometric algorithm, I can sometimes get to 1e-11 to 1e-13 even for highly ellipsoidal cases. Even when I hack the ALGLIB sqp code to force things like egap and edual to be < 1e-14 instead of
relative, the solution still is KKt feasible only at ~1e-4.
Where can I extract the options for KKT multipliers that ALGLIB computes so I can see what ALGLIB says df/dx-lambda1*dg1/dx-lambda2*dg2/dx are? All I can find are the box constraint and linear
constraint KKT multipliers which are not part of my current problem? What can I do in ALGLIB to find a solution feasible to the KKT conditions at the ~e-10 level?
Thank you,
Thank you for reading my post and or the suggestion. I've attached the trace file as well as a more detailed one.
FWIW, to show how similar the solution values are, here is the alglib solution. Please excuse the many decimal places. I generally ignore Visual Studio C++ numerical places beyond 1e-13 but I value
extreme precision in my problem.
Alglib filter sqp:
Unit-Canonical Ellipsoid 1: (-0.329176838420615, 0.94401746027854, 0.0217633576606146)
Unit-Canonical Ellipsoid 2: (0.930895021061298, 0.329232641233283, -0.158241359036905)
In general space, this corresponds to
General Ellipsoid 1: (-10.5967582185939, 206.347913283321, 6.77869831250489)
General Ellipsoid 2: (18.8893523034911, -49.9697297257856, 570.805604711329)
The Euclidean norm is: 620.237467405261
In code, of the 6 KKT condition equations, the worst feasibility is 9.7502106655156240e-06 and in my Excel "implementation" of my geometric convergence algorithm, the KKT feasibility evaluates as
0.0001294260728173 or roughly 1e-4.
In contrast, the point I find in my Excel "algorithm" differs from the ALGLIB solution by at most e-9 in unit-canonical space and e-8 in general space!
My solution:
Unit-Canonical Ellipsoid 1: (-0.32917684136241, 0.944017459191415, 0.0217633603207913)
Unit-Canonical Ellipsoid 2: (0.930895019622935, 0.329232643072075, -0.158241363672689)
In general space, this corresponds to
General Ellipsoid 1: (-10.5967582026675, 206.347913213863, 6.77869828010731)
General Ellipsoid 2: (18.8893522318179, -49.9697297427802, 570.805604707352)
The Euclidean norm is: 620.237467405261
In code, of the 6 KKT condition equations, the worst feasibility is 9.0949470177292824e-12 and in my Excel "implementation" of my geometric convergence algorithm, the KKT feasibility evaluates as
0.00000000003547029 or roughly 3.5e-11.
Does this give you any hints? I realize it may seem like polishing an already very nice apple, but I'd love to squeeze out the extra accuracy from alglib if at all possible...
File comment: trace_file("SQP.DETAILED,SQP.PROBING,PREC.F15",
logDirectory + "trace.log");
trace_15.log [168.53 KiB]
Downloaded 1365 times
File comment: trace_file("SQP",
logDirectory + "trace.log");
trace.log [47.92 KiB]
Downloaded 1437 times
I forgot to give you an idea of the original problem.
Ellipsoid 1: radii: (16.765, 55.8273, 1.3), origin: (-20.0, 155.9, 19.99), rotations: (1.4, 236.83383, 12.999) degrees
Ellipsoid 2: radii: (8.7, 10.1, 15.1617), origin: (20.09, -45.6, 562.94), rotations: (1.23456, 98.764523, 186.34) degrees
So a 'large' difference in radii which I like because it shows robustness of the approach. More spherical ellipsoids solve to e-13 KKT feasibility by any method I mentioned above.
I just tested in GAMS (CONOPT and SNOPT) a "post-processing" NLP that makes the KKT multiplier calculations actual variables and constraints and minimizes the infeasibility as the objective function
while requiring my previous objective function value be maintained...A poor person's dual formulation in a way. I have to check for divide by zero bu
It seems to only take a couple internal iterations (I guess that's the QP solver) to achieve what I want. However, it seems I should be able to get this from a primal solution. Recognizing ALGLIB
advertises the "no tuning needed" for its excellent implementations, any thoughts on internal settings that might help here? Or parts of the code I could look into different from what I reference
Thank you!!
Hi! Actually, I thought for several days about using arbitrary precision to further polish solution obtained in a double precision. ALGLIB has legacy version (2.x branch) supporting MPFR, probably if
you use run a 128-bit Newton iteration straight from the solution returned by ALGLIB it will converge in several iterations even without merit functions, line search and other kinds of safeguards
that make optimizer so difficult to implement. That legacy version does not support advanced optimization algos, but you can use its linear algebra to implement Newton method yourself.
Your approach is also a good option, though.
Thank you! Yours is also an excellent idea. I will try to implement that if it doesn't take me too long to get it running.
In the guts of the current alglib release, when I comment out the termination conditions that specify if the change in objective function is too small and if the trust region is smaller than the sqrt
of machine epsilon (optimization.cpp ~line 63417 and line 63438), the solution will be perturbed until the trust region is smaller than the actual epsilon I input (rather than the sqrt of epsilon).
Of course, since it's not optimizing anything really at that point, it's "random"....but perhaps you could add a "solution polishing" step under these conditions that used extra numerical precision,
per your idea, or explicitly minimizes the infeasibility per an extended formulation.
I'll let you know how your idea goes once I get to it - but may be a few weeks.
Thank you for thinking about it!
Just an update - I think your Newton-Raphson suggestion is superior. Although my idea is certainly robust, it's far more complicated whereas this is, so far, 1-2 iterations of Newton-Raphson to get
to 1e-13 roughly...and that's simply prototyping it in Excel.
I was unable to get the ALGLIB 2.6.0 mpr fully imported (I create dlls out of all 3rd party software I use and export methods I want but I can't figure out some of the unresolved externals I'm
getting)...but given the success using regular 64-bit Excel accuracy, I expect to be just fine without it. Hopefully, if I ever have to use it, I'll be able to just pull in some of the specific code
that had no issues linking.
So for now, I think this should be good enough.
Now to figure out how to calculate KKT multipliers for additional linear constraints.
Many thanks!! | {"url":"http://forum.alglib.net/viewtopic.php?f=2&t=4606","timestamp":"2024-11-03T00:59:51Z","content_type":"application/xhtml+xml","content_length":"43820","record_id":"<urn:uuid:2d59472a-b91a-4236-9eb4-cfe3104754ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00168.warc.gz"} |
A thread for little questions
How so? I admitted it's a troll ship, instead of saying that all is good and balanced about it... But some players get to a point where they crave greater challenge, that is what troll ships are for.
And the prize for defeating a troll ship is real, which IMO makes up for the effort. | {"url":"https://openxcom.org/forum/index.php?topic=5345.60","timestamp":"2024-11-02T15:41:33Z","content_type":"application/xhtml+xml","content_length":"66061","record_id":"<urn:uuid:7e606dd1-ac23-4602-9b61-68bfa2213856>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00424.warc.gz"} |
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Quantitative Aptitude Quiz For ESIC- UDC, Steno, MTS Prelims 2022-18th January
Directions (1-5): Study the pie-chart carefully and answer the questions.
Pie-chart given below shows the percentage distribution of students in different class of school ‘RPM’
Note → Ratio of total student of school RPM to school SVM is 2 : 3 and both school has 5 class only i.e. class 6 to class 10
Number of students in 10th class of RPM school is 360
Q1. If total student in 6th class in school SVM is 20% of total student then find difference of student in 6th class in both school?
(a) 120
(b) 310
(c) 230
(d) 210
Q3. If ratio of students in class 10th of school SVM to that of school RPM is 1 : 2 then students in class 10th in SVM is what percent total student in school?
(a) 9%
(b) 6%
(c) 8%
(d) 4%
(e) 12%
Q4. Find the average no. of students in class 8th and 9th of school SVM if percentage distribution is same as school RPM?
(a) None of these
(b) 425
(c) 400
(d) 250
(e) 450
Q5. If total girls in school RPM is 600 then total boys in school RPM is what percent of total student in SVM?
(a) 40%
(b) 30%
(c) 35%
(d) None of these
(e) 50%
Directions (6-10): Study the passage carefully and answer the questions.
Given below is the data for four countries i.e. India, Pakistan, Iran, France
Price of diesel in Pakistan is Rs50/L which is 37 ½ % less than price of India. Ratio of price of diesel in Pak to Iran is 5 : 2 and average of diesel price including all countries is Rs65/L. Ratio
of price of petrol to price of diesel in India is 9 : 8. Price of petrol in India is 260% more them price of petrol in Iran. Average price of petrol in all countries is thrice of price of petrol in
Q6. What is the difference in sum of price of petrol from all countries & sum of price of diesel from all countries?
(a) Rs 40
(b) Rs 35
(c) None of these
(d) Rs 30
(e) Rs 45
Q7. If price of petrol in India is 50% more than price of petrol in Pak then find difference in price of petrol in Pak and France?
(a) None of these
(b) Rs 45
(c) Rs 65
(d) Rs 60
(e) Rs 75
Q8. What is ratio of price of petrol in Pak and France together to price of diesel in both countries together?
(a) 32 : 27
(b) 37 : 31
(c) 3 : 2
(d) 37 : 32
(e) None of these
Q9. Price of petrol is what percent of price of diesel in Iran?
(a) 25%
(b) 150%
(c) None of these
(d) 75%
(e) 125%
Q10. Difference of price of petrol & diesel in India is what percent of difference in price of petrol & diesel in Pak?
(a) 100%
(b) 50%
(c) None of these
(d) Cannot be determined
(e) 125%
Directions (11–15): Given bar graph shows the number of student passed in Xth class from 6 different school.
Q11. Pass percentage of school S is equal to that of school Q. Find total strength of school Q is what % more than that of school S.
(a) 20%
(b) 40%
(c) 50%
(d) 25%
(e) 60%
Q12. If fail percentage of school P is 65% then, find number of student failed from school P is what percentage of number of student passed from school T.
(a) 30%
(b) 100%
(c) 120%
(d) 130%
(e) 70%
Q13. If ratio between total student who passed to who failed from all school is 7 : 3, then find the total number of failed student from all schools together.
(a) 225
(b) 125
(c) 250
(d) 275
(e) None of these
Q14. Student passed from school P, Q, U and T together is how much more than that of school R and S together.
(a) 220
(b) 250
(c) 190
(d) 220
(e) None of these
Q15. Failed student of school U is 15 more than that of school R. If ratio between total strength of school U to school R is 3 : 2, then find the total number of failed student from both schools
(a) 57
(b) 23
(c) 45
(d) 63
(e) 35
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3D Parallelism
In practice
• Letβ s say you have 128 GPUs with 8 GPUs per node. Letβ s say 64 GPUs are required for one batch (70B model). Then you have
□ Data parallelism , =2.
□ We want to tensor parallelism because we have 8 GPUs per node
□ Thus, we have pipeline parallelism where .
□ This means that the layers are distributed among 8 nodes, and each node distributes the tensor computation between 8 GPUs.
• How to compute the needed number of GPUs for a batch ?
□ If we do mixed-precision training with a 70B model, as explained in Memory usage (VRAM) we need num_GPUS = model_size_in_B * 18 * 1.25 / gpu_size_in_GB
□ For a 70B model and A100s with 80GB RAM, this gives us 19.68 GPUs
☆ Could be 8.75 GPUs if using only bf16/fp16
□ 1.25 is very rough estimate for the activation overhead, actual overhead is dependent linearly on batch and quadratically on sequence length.
□ We can reduce this by using Activation checkpointing in a smart manner + doing tensor and sequence parallelism.
How to choose your 3D Parallelism (ZeRO-3 or ZeRO-1+PP+TP)
• Increasing TP and PP implicitly β shardsβ the model across GPUs, thus, it is quite memory efficient
□ Main constraint is that TP is fairly communication intensive, and thus should usually stay within the boundary of a single node, to only use inter-node communication
☆ *This might become irrelevant as intra-node networking performance approaches inter-node *
□ Thus, the maximum granularity at which we can shard a model is the number of GPUS in a node.
☆ Depending on the size, a single transformer layer may not fit within a node
• In the case of extreme size, we may have to use ZeRO-3, as it allows for arbitrary size model.
□ For a model of size and devices, we just need
# ZeRO Data Parallelism + Pipeline Parallelism + Tensor Parallelism
• When ZeRO-DP is combined with PP (and optionally TP) it typically enables only ZeRO stage 1 (optimizer sharding).
□ At the end of the training iteration,
☆ Each process sends parameters of volume to all the processes β volume
• In DeepSpeed, Pipeline Parallelism can work with ZeRO stage 1 but not stage 2
□ This is due to the gradient accumulation in PP that requires that all gradients be present across multiple forward/backward passes.
□ Since zero stage 2 partitions the gradients, they are simply incompatible unfortunately.
□ Indeed, in PP, each device accumulates the gradients correspond to its layers across the microbatches.
□ When replicating the pipeline across multiple clusters of nodes to do DP, each pipeline needs to hold on to its gradients throughout the training iteration to be able to do the backward
passes appropriately (communicating the gradients in between the boundaries)
PTD-P (Megatron-LM)
• β Efficient Large-Scale Language Model Training on GPU Clusters Using Megatron-LMβ
• Data parallelism + Tensor Parallelism + Interleaved 1F1B Pipeline Parallelism
β ’ (π , π ‘, π ): Parallelization dimensions. π for the pipeline-model- parallel size, π ‘ for the tensor-model-parallel size, and π for the data-parallel size. β ’ π : Number of GPUs.
We require π Β· π ‘ Β· π = π . β ’ π ΅: Global batch size (provided as input). β ’ π : Microbatch size. β ’ π = 1/π Β· π ΅/π : Number of microbatches in a batch per pipeline
Tensor and Pipeline Model Parallelism interactions
Bubble size
• As stated in Pipeline Parallelism, using pipeline parallelism with periodic flushes results in a pipeline bubble of size
• Letβ s assume (data-parallel), consequently
• The pipeline bubble size in terms of is:
• As increases, the pipeline bubble thus decreases for fixed , and
□ Indeed, the pipeline depth decreases if we have more tensor-parallelism.
• The amount of communication performed between different GPUs is also affected by the values of and .
• Pipeline model parallelism features cheaper point-to-point communication.
□ With pipeline parallelism, the total amount of communication that needs to be performed between every pair of consecutive devices (for either the forward or backward pass) for each microbatch
is , where π is the sequence length and β is the hidden size
• Tensor model parallelism, on the other hand, uses all-reduce communication (two all-reduce operations each in the forward and backward pass, two all-gather)
□ With tensor model parallelism, tensors of total size π π β need to be all-reduced among π ‘ model replicas twice each in the forward and backward pass for each layer (MLP= 2 forward),
leading to a total communication of per layer per device for each micro-batch.
□ Each device typically has multiple layers; the total amount of tensor-parallel-communication per device for each microbatch is then , where is the number of layers in a pipeline stage
• We see that tensor model parallelism increases the amount of communication between devices.
Takeaway #1
• When considering different forms of model parallelism, tensor model parallelism should generally be used up to degree π when using π -GPU servers, and then pipeline model parallelism can be
used to scale up to larger models across servers. | {"url":"https://notes.haroldbenoit.com/ml/engineering/training/parallelism/3d-parallelism","timestamp":"2024-11-09T20:11:00Z","content_type":"text/html","content_length":"104997","record_id":"<urn:uuid:731cee38-8a48-416d-a52f-71a7692b7191>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00378.warc.gz"} |
Network Dynamics and Learning
A.A. 2024/25
Course Language
Degree programme(s)
Master of science-level of the Bologna process in Ingegneria Informatica (Computer Engineering) - Torino
Course structure
Teaching Hours
Lezioni 40
Esercitazioni in aula 20
Esercitazioni in laboratorio 20
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
Fagnani Fabio Professore Ordinario MATH-03/A 40 0 0 0 5
SSD CFU Activities Area context
MAT/05 3 D - A scelta dello studente A scelta dello studente
MAT/05 3 F - Altre attività (art. 10) Altre conoscenze utili per l'inserimento nel mondo del lavoro
This course aims at presenting the mathematical foundations of networks with special emphasis on dynamical models both of deterministic and stochastic nature, as well to illustrate their main
engineering applications. The course covers the basics of: graph theory, non-negative matrices, Markov probabilistic models, game theory, distributed optimization. It introduces the student to
several applications: distributed optimization and inference algorithms, opinion dynamics, epidemics models, learning in games.
This course aims at presenting the mathematical foundations of networks with special emphasis on dynamical models both of deterministic and stochastic nature, as well to illustrate their main
engineering applications. The course covers the basics of: graph theory, non-negative matrices, Markov probabilistic models, game theory, distributed optimization. It introduces the student to
several applications: distributed optimization and inference algorithms, opinion dynamics, epidemics models, learning in games.
Knowledge of: - Elements of algebraic and topological graph theory with special emphasis on connectivity, centrality, and network flows. - Elements of non-negative matrix theory (stochastic and
substochastic matrices), their spectral properties, and their use in the mathematical modelling of deterministic and stochastic network dynamics. - Elements of Markov processes over networks (random
walks, epidemic spreading, randomized algorithms) - Convex optimization problems over large scale network (flow optimization, inference, learning). Distributed algorithms. - Elements of game theory
and learning dynamics. Mechanism design. - Basics of probabilistic graphical models: Bayesan networks and random Markov fields. - Basics of models of random graphs and their fundamental properties
Ability to: - Construct and compare mathematical models for interconnected systems arising in information, social, economic, biological, and infrastructure networks. - Apply basics notions of
algebraic and topological graph theory, Markov processes, game theory, and dynamical systems in order to analyze networks and interconnected dynamical systems. - Critically identify and evaluate
network properties such as centrality, connectivity, emerging behaviors, and scalability. - Design and propose cooperative distributed algorithms on networks, to evaluate their asymptotics, speed of
convergence, and complexity, and to implement them on modern platforms for numerical simulation.
Knowledge of: - Elements of algebraic and topological graph theory with special emphasis on connectivity, centrality, and network flows. - Elements of non-negative matrix theory (stochastic and
substochastic matrices), their spectral properties, and their use in the mathematical modelling of deterministic and stochastic network dynamics. - Elements of Markov processes over networks (random
walks, epidemic spreading, randomized algorithms) - Convex optimization problems over large scale network (flow optimization, inference, learning). Distributed algorithms. - Elements of game theory
and learning dynamics. Mechanism design. - Basics of probabilistic graphical models: Bayesan networks and random Markov fields. - Basics of models of random graphs and their fundamental properties
Ability to: - Construct and compare mathematical models for interconnected systems arising in information, social, economic, biological, and infrastructure networks. - Apply basics notions of
algebraic and topological graph theory, Markov processes, game theory, and dynamical systems in order to analyze networks and interconnected dynamical systems. - Critically identify and evaluate
network properties such as centrality, connectivity, emerging behaviors, and scalability. - Design and propose cooperative distributed algorithms on networks, to evaluate their asymptotics, speed of
convergence, and complexity, and to implement them on modern platforms for numerical simulation.
Basic knowledge of linear algebra, probability theory and calculus is a prerequisite.
Basic knowledge of linear algebra, probability theory and calculus is a prerequisite.
1. Topological and algebraic graph theory - Basics of graph theory. Directed acyclic graphs, undirected graphs: properties and examples. Combinatorial properties of trees and bipartite graphs.
Eulerian circuits. Connected components and condensation graph. Cliques, independent sets, coverings, matchings, and colorings. - Matrices associated to a graph: weight, adjacency, stochastic, and
Laplacian matrices. Perron-Frobenius theorem and spectral properties of nonnegative matrices. - Network centrality (degree, eigenvector, Bonacich, Page-rank, Katz). - Network connectivity and flows:
Menger’s theorem, max-flow/min-cut theorem. - Applications: constraints satisfaction problems (e.g., scheduling, allocation), social and economic networks, resilience of infrastructure networks. 2.
Distributed averaging and linear flow dynamics - Distributed averaging and compartmental systems for closed networks. Convergence theorems, role of centrality. - Dynamics with exogeneous inputs.
Properties of sub-stochastic matrices, convergence results. - Applications: distributed estimation and computation on a network, opinion dynamics, wisdom of crowds, social influence, flow diffusion.
3. Distributed optimization - Optimization of convex separable functions with linear coupling constraints. - Network flow optimization: duality, Lagrange multipliers as marginal costs. - Electrical
networks: effective resistance, Rayleigh’s variational principle. - Network flow optimization: system optimum traffic assignment, Wardrop equilibrium, price of anarchy, marginal cost mechanisms.
-Applications: energy and transportation systems. 4. Markov chains - Discrete- and continuous-time Markov chains: stationary distributions, convergence, ergodic theorem, reversibility, hitting and
return times, absorbing probabilities. - Notable examples: birth-and-death chains, random walks on graphs. - Mixing time, conductance, rapid mixing, coupling, - Applications: Markov Chain Monte
Carlo, Metropolis-Hastings algorithm, Gibbs sampling. 5. Contagion and diffusion over networks - Pairwise interacting models: general definition and properties. - Epidemic models (SI, SIS, SIR),
voter model, evolutionary dynamics. - Computation of the absorbing times and absorbing probabilities for simple topologies (complete, line, star). - Detection and control policies. - Applications:
diffusion of innovations, evolutionary competition, spread of rumors and fake news, network security. 6. Game theory and learning - Non-cooperative games in strategic form, best response, dominant
strategies, Nash equilibria. mixed strategies, correlated strategies. - Classical examples. Symmetric games. Potential and congestion games. - Network games. Examples: coordination and
anti-coordination, graph coloring, quadratic games. - Learning: best response dynamics, log-linear learning, fictitious play. - Mechanism design. - Applications: behavior of social and economic
networks, pricing mechanisms in markets, algorithms for combinatorial optimization. 7. Probabilistic graphical models - Bayesian networks - Markov random fields. Ising model and Glauber dynamics.
Boltzmann machines. - Distributed inference: computation of marginal and posterior distributions, likelihood maximization, 8. Random graphs - Branching processes. - Erdos-Renyi random graph: phase
transitions for connectivity and for the existence of a giant component. - Random graphs with preassigned degree distribution (configuration model). - Preferential attachment and small world models.
1. Topological and algebraic graph theory - Basics of graph theory. Directed acyclic graphs, undirected graphs: properties and examples. Combinatorial properties of trees and bipartite graphs.
Eulerian circuits. Connected components and condensation graph. Cliques, independent sets, coverings, matchings, and colorings. - Matrices associated to a graph: weight, adjacency, stochastic, and
Laplacian matrices. Perron-Frobenius theorem and spectral properties of nonnegative matrices. - Network centrality (degree, eigenvector, Bonacich, Page-rank, Katz). - Network connectivity and flows:
Menger’s theorem, max-flow/min-cut theorem. - Applications: constraints satisfaction problems (e.g., scheduling, allocation), social and economic networks, resilience of infrastructure networks. 2.
Distributed averaging and linear flow dynamics - Distributed averaging and compartmental systems for closed networks. Convergence theorems, role of centrality. - Dynamics with exogeneous inputs.
Properties of sub-stochastic matrices, convergence results. - Applications: distributed estimation and computation on a network, opinion dynamics, wisdom of crowds, social influence, flow diffusion.
3. Distributed optimization - Optimization of convex separable functions with linear coupling constraints. - Network flow optimization: duality, Lagrange multipliers as marginal costs. - Electrical
networks: effective resistance, Rayleigh’s variational principle. - Network flow optimization: system optimum traffic assignment, Wardrop equilibrium, price of anarchy, marginal cost mechanisms.
-Applications: energy and transportation systems. 4. Markov chains - Discrete- and continuous-time Markov chains: stationary distributions, convergence, ergodic theorem, reversibility, hitting and
return times, absorbing probabilities. - Notable examples: birth-and-death chains, random walks on graphs. - Mixing time, conductance, rapid mixing, coupling, - Applications: Markov Chain Monte
Carlo, Metropolis-Hastings algorithm, Gibbs sampling. 5. Contagion and diffusion over networks - Pairwise interacting models: general definition and properties. - Epidemic models (SI, SIS, SIR),
voter model, evolutionary dynamics. - Computation of the absorbing times and absorbing probabilities for simple topologies (complete, line, star). - Detection and control policies. - Applications:
diffusion of innovations, evolutionary competition, spread of rumors and fake news, network security. 6. Game theory and learning - Non-cooperative games in strategic form, best response, dominant
strategies, Nash equilibria. mixed strategies, correlated strategies. - Classical examples. Symmetric games. Potential and congestion games. - Network games. Examples: coordination and
anti-coordination, graph coloring, quadratic games. - Learning: best response dynamics, log-linear learning, fictitious play. - Mechanism design. - Applications: behavior of social and economic
networks, pricing mechanisms in markets, algorithms for combinatorial optimization. 7. Probabilistic graphical models - Bayesian networks - Markov random fields. Ising model and Glauber dynamics.
Boltzmann machines. - Distributed inference: computation of marginal and posterior distributions, likelihood maximization, 8. Random graphs - Branching processes. - Erdos-Renyi random graph: phase
transitions for connectivity and for the existence of a giant component. - Random graphs with preassigned degree distribution (configuration model). - Preferential attachment and small world models.
Theoretical lectures and practice classes. Theoretical lectures are devoted to the presentation of the topics, with definitions, properties, introductory examples, as well as a number of selected
proofs which are believed to facilitate the learning process. The practice classes are devoted to train the students’ abilities to solve problems and exercises and to implement computations and
Theoretical lectures and practice classes. Theoretical lectures are devoted to the presentation of the topics, with definitions, properties, introductory examples, as well as a number of selected
proofs which are believed to facilitate the learning process. The practice classes are devoted to train the students’ abilities to solve problems and exercises and to implement computations and
Part of the course topics are covered by lecture notes that will be made available to the students through the Portale della Didattica. Other material will be suggested in class and made avalaible
thorugh the Portale della Didattica.
Part of the course topics are covered by lecture notes that will be made available to the students through the Portale della Didattica. Other material will be suggested in class and made avalaible
thorugh the Portale della Didattica.
Slides; Dispense; Esercizi; Esercitazioni di laboratorio; Video lezioni tratte da anni precedenti;
Lecture slides; Lecture notes; Exercises; Lab exercises; Video lectures (previous years);
Modalità di esame: Prova orale obbligatoria; Elaborato progettuale individuale;
Exam: Compulsory oral exam; Individual project;
... Exam: compulsory oral exam; individual essay; The assessment consists of two parts: 1. Three homework assignments (HW1, HW2, HW3) that will be assigned to the students during the course HW1 and
HW2 consist of exercises that are aimed at assessing the students’ achieved learning level of the theoretical aspects and their ability to select relevant principles to solve the proposed problems.
In addition to an exercise part of the same nature as in HW1 and HW2, the HW3 assignment includes a small project that involves numerical simulations as well as deductive and analytical reasonings
and is aimed at testing the mathematical modeling abilities of the students and their ability to present their analyses and results in a proper format of a written report such that a technically
qualified person can follow and obtain similar findings. Students will have two weeks each to submit their HW1 and HW2 reports, and three weeks to submit their HW3 report. 2. An oral test consisting
of two parts: (2a) a discussion of the submitted HW1, HW2, and HW3 reports, aimed at testing the depth of the students’ understanding of the subjects and their ability to explain, defend, reflect,
critically evaluate, and possibly improve their work; (2b) a presentation of a topic studied in the course covering both theoretical aspects and possibly their applications. The topic is chosen by
the examination committee and communicated to the student, who has 30 minutes to prepare her/his presentation: during this time the student has access to books, notes, and other learning material.
This part of the oral test is aimed at evaluating the breadth and depth of the knowledge acquired by the student, and her/his ability to effectively present and communicate it to a technically
qualified audience. Grading: the maximum grade for HW1, HW2, and HW3, upon the discussion detailed at point (2a) above, is of 4,4, and 6 points, respectively. If HW1, HW2, and HW3 are not submitted
by their deadline, the maximum grades above are lowered to 3, 3, and 5 points, respectively. The maximum grade for part (2b) of the oral test is 18 points. The final course grade is then obtained by
summing up the final grades in HW1, HW2, HW3, and of part (2b) of the oral test: if such sum is less than or equal to 30, the final grade coincides with it, if the sum is strictly larger than 30 the
final grade is 30 cum laude.
Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente
al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine
di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.
Exam: Compulsory oral exam; Individual project;
Exam: compulsory oral exam; individual essay; The assessment consists of two parts: 1. Three homework assignments (HW1, HW2, HW3) that will be assigned to the students during the course HW1 and HW2
consist of exercises that are aimed at assessing the students’ achieved learning level of the theoretical aspects and their ability to select relevant principles to solve the proposed problems. In
addition to an exercise part of the same nature as in HW1 and HW2, the HW3 assignment includes a small project that involves numerical simulations as well as deductive and analytical reasonings and
is aimed at testing the mathematical modeling abilities of the students and their ability to present their analyses and results in a proper format of a written report such that a technically
qualified person can follow and obtain similar findings. Students will have two weeks each to submit their HW1 and HW2 reports, and three weeks to submit their HW3 report. 2. An oral test consisting
of two parts: (2a) a discussion of the submitted HW1, HW2, and HW3 reports, aimed at testing the depth of the students’ understanding of the subjects and their ability to explain, defend, reflect,
critically evaluate, and possibly improve their work; (2b) a presentation of a topic studied in the course covering both theoretical aspects and possibly their applications. The topic is chosen by
the examination committee and communicated to the student, who has 30 minutes to prepare her/his presentation: during this time the student has access to books, notes, and other learning material.
This part of the oral test is aimed at evaluating the breadth and depth of the knowledge acquired by the student, and her/his ability to effectively present and communicate it to a technically
qualified audience. Grading: the maximum grade for HW1, HW2, and HW3, upon the discussion detailed at point (2a) above, is of 4,4, and 6 points, respectively. If HW1, HW2, and HW3 are not submitted
by their deadline, the maximum grades above are lowered to 3, 3, and 5 points, respectively. The maximum grade for part (2b) of the oral test is 18 points. The final course grade is then obtained by
summing up the final grades in HW1, HW2, HW3, and of part (2b) of the oral test: if such sum is less than or equal to 30, the final grade coincides with it, if the sum is strictly larger than 30 the
final grade is 30 cum laude.
In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the
special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to
provide students with the most suitable arrangements for each specific type of exam.
Esporta Word | {"url":"https://didattica.polito.it/pls/portal30/gap.pkg_guide.viewGap?p_cod_ins=02TXLOV&p_a_acc=2025&p_header=S&p_lang=IT&multi=N","timestamp":"2024-11-07T19:11:44Z","content_type":"text/html","content_length":"65101","record_id":"<urn:uuid:535b5efd-2bce-4b4c-a54d-34a100aa8764>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00090.warc.gz"} |
For input level 0 V:
As a bias of approximately 0V is sufficient to cut off a silicon emitter junction, it follows that transistor is cut off when vi=0
When vi=0,the output voltage is vo=VCC=10 V
This indicates that the output is in state 1 when the input is in state 0
For input level 10 V:
Since iB>iB(min),it is verified that the transistor is in saturation
When vi=10,the output voltage is vo=VCE(sat)=0.2 V
This indicates that the output is in state 0 when the input is in state 1
Overall it has been thus verified that the circuit has performed the NOT operation | {"url":"https://tbc-python.fossee.in/convert-notebook/Electronics_Fundamentals_and_Applications_by_D._Chattopadhyay_and_P._C._Rakshit/ch17.ipynb","timestamp":"2024-11-05T15:12:02Z","content_type":"text/html","content_length":"228829","record_id":"<urn:uuid:457a7836-7b6a-4d0f-b574-b5f3ede8e73b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00018.warc.gz"} |
Witold Hurewicz - Biography
Quick Info
29 June 1904
Łódź, Russian Empire (now Poland)
6 September 1956
Mérida, Mexico
Witold Hurewicz made important contributions to algebraic topology including discovering higher homotopy groups.
Witold Hurewicz's father, Mieczyslaw Hurewicz, was an industrialist. Mieczyslaw was born in Wilno, Poland on 4 April 1872 to Serge Hurewicz and Fannie Eisenstat. He married Katarzyna Finkelsztain
(born Bila Tserkva, Russian Empire, 26 April 1877) on 4 September 1900 at Warsaw, Poland. Mieczyslaw and Katarzyna Hurewicz had two children, Stefan (born 3 October 1901 at Łódź, Poland) and
Witold (the subject of this biography). The family was Jewish.
Witold attended elementary school in Łódź in a Russian controlled Poland but with World War I beginning before he had begun secondary school, major changes occurred in Poland. At the outbreak of
war, the Hurewicz family left Łódź and travelled to Moscow where Witold attended secondary school from 1914 to February 1919. In August 1915 the Russian forces which had held Poland for many
years withdrew. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly a strong school
of mathematics grew up in the University of Warsaw, with topology being one of the main topics. The Hurewicz family returned to Łódź in February 1919 and Witold completed his secondary school
studies at the Oswiata Gymnasium in that city. He passed the matriculation examination in May 1921 and graduated from the Oswiata Gymnasium. Although Hurewicz knew by this time that he wanted to
specialise in mathematics and fully understood that there was a vigorous school of mathematics in Poland, nevertheless he chose to go to Vienna to continue his studies. He left Łódź to travel to
Vienna on 16 July 1921.
In Vienna, he studied under Hans Hahn, receiving a Ph.D. in 1926 for his thesis Über eine Verallgemeinerung des Borelschen Theorems Ⓣ. Karol Borsuk explains how the Vienna school of mathematics
determined the direction of Hurewicz's research [5]:-
At that time Vienna was a very prosperous place for mathematics, and besides Hahn (who authored the excellent exposition 'Theorie der reellen Funktionen' Ⓣ), there were also many other
outstanding mathematicians, including N Hofreiter and W Wirtinger. Under Hahn's powerful influence, an active mathematics research centre of set theory formed in Vienna. Among the many
eminent set theorists there were Karl Menger, one of the creators of dimension theory, and his colleagues Kurt Gödel (who consequently became famous as one of the most thorough investigators
of the principles of set theory), Georg Nöbeling, and Abraham Wald. In this environment Hurewicz turned to set theory.
Although Borsuk gives a good indication of Hahn's Vienna School in this quote, it is a little confusing since not all those mentioned were teaching there when Hurewicz was a student. Wilhelm
Wirtinger was appointed to a chair at the University of Vienna in 1905 and Hahn was appointed in 1921. Karl Menger entered the University of Vienna in 1920 to study physics but changed to write a
doctoral thesis on dimension advised by Hahn which he completed in 1924. Nikolaus Hofreiter (1904-1990) studied from 1923 in Vienna with Hans Hahn, Wilhelm Wirtinger, Emil Müller and Philipp
Furtwängler. He was a student at the same time as Hurewicz and was awarded his doctorate in 1927. Abraham Wald entered the University of Vienna in 1927 to study with Karl Menger and was awarded
his doctorate in 1929.
Hurewicz began to work on extending theorems of Menger and Urysohn on dimension, which they had proved for Euclidean spaces, to separable metric spaces. To do this he had to produce new
techniques and he began to publish his results in a series of papers. The first few are: Über schnitte von Punktmengen Ⓣ (1926), Stetige bilder von Punktmengen Ⓣ (1926), Grundiss der Mengerschen
Dimensionstheorie Ⓣ (1927), Normalbereiche und Dimensionstheorie Ⓣ (1927), Stetige bilder von Punktmengen. II Ⓣ (1927), Verhalten separabler Räume zu kompakten Räumen Ⓣ (1927), and Über Folgen
stetiger Funktion Ⓣ (1927). As Borsuk writes that, through these papers [4]:-
... Hurewicz became known as one of the creators of dimension theory, next to Menger and Urysohn.
He was awarded a Rockefeller scholarship which allowed him to spend the year 1927-28 in Amsterdam. He remained in Amsterdam being appointed as a docent and an assistant to L E J Brouwer from 1928
to 1936. Samuel Eilenberg writes about this period in [6]:-
I first met Hurewicz when I was a student at the University of Warsaw. It was around 1932-1933. To me he was an idol, a Jew from Poland who became a prominent world mathematician in a field I
was in love with: an ideal to admire and to follow. Hurewicz was then in Holland and came to Warsaw almost once a year. We talked about mathematics, and I discussed what I was doing. He was
supportive and helpful. Once when I proved something good, I wrote to him and received a very congratulatory reply. I still have that letter. At the same time I met Lefschetz who visited
Warsaw on several occasions. We all three met in Oslo in 1936 on the occasion of the International Mathematical Congress. At that time my future was discussed, and it was agreed that I should
visit Western Europe first (Paris, Zürich, Oxford, and Cambridge) before moving to America. In the fall of 1936 I started implementing this plan and went to Paris for a six-month stay. I was
helped in various ways by Professor Wacław Sierpiński. At the time Hurewicz was already in America.
He was given study leave for a year which he decided to spend in the United States. In September 1936 he sailed from Rotterdam to New York on the Statendam. He visited the Institute for Advanced
Study in Princeton and spent the year 1936-37 as a fellow there. He decided to remain in the United States and not return to his position in Amsterdam but he came back to Europe in the summer of
1937, returning from Le Havre to New York on the SS President Roosevelt in October 1937. Given the impending war in Europe this was clearly a wise decision. He returned to the Institute for
Advanced Study in Princeton and, in January 1938, he applied for citizenship of the United States. At this time he was living at 1 Evelyn Place, Princeton. He describes himself as having brown
hair, brown eyes, height 5 ft 6 in, weight 145 lbs, of Hebrew race and Polish nationality. He remained at the Institute until 1939 although he again visited Europe in the summer of 1938,
returning to New York from Le Havre in September of that year on the Champlain. In April 1939 he went with his colleague Henry Wallman to meet his friend Samuel Eilenberg when he arrived in New
York on the SS Manhattan. Eilenberg writes [6]:-
I arrived to New York on April 27, 1939, and there at the pier were Hurewicz and Wallman to take me by car for about ten days to Princeton, which was then the undisputed mathematical mecca of
the world. On the way we stopped for a snack, and I was introduced to cinnamon toast, which just became a big fad. I was also introduced to car trouble, as the lights refused to work when we
were ready to continue.
After his years at Princeton, Hurewicz was appointed first to the University of North Carolina being an assistant professor there from 1939 to 1942. His parents, Mieczyslaw and Katarzyna
Hurewicz, sailed to New York from Rotterdam on the Niew-Amsterdam in April 1939. They returned to Europe but sailed from Genoa, Italy to New York in June 1940 on the SS Manhattan. His brother
Stefan also emigrated to the United States via the Philippines in January 1941. This route via the Philippines was a common one for Jews fleeing Nazi persecution. Hurewicz was registered for the
draft on 16 February 1942. During World War II he contributed to the war effort with research on applied mathematics, in particular the work he did on servomechanisms at that time was classified
because of its military importance. He was still officially on the staff of the University of North Carolina from 1942 to 1945 although he was given leave of absence for government service. In
fact he was promoted to Associate Professor in 1942. He did have time to accept the position of Visiting Professor at Brown University, Providence, Rhode Island from January 1943 to June 1944
living at 1 Megee, Providence. He gave a series of lectures at Brown University in 1943 and these were published in mimeographed form by Brown University as Ordinary differential equations in the
real domain with emphasis on geometric method. The notes covered existence theorems, linear systems, and geometrical aspects of non-linear systems in the plane. During 1944-45 he worked at the
Radiation Laboratory at the Massachusetts Institute of Technology in Cambridge, Massachusetts.
From 1945 until his death he worked at the Massachusetts Institute of Technology. He was an Associate Professor of Mathematics there from 1945 to 1948 when he was promoted to full professor. He
lived at 993 Memorial Drive, Cambridge, Massachusetts with his mother who was now calling herself Catherine. In September 1950 he attended the International Congress of Mathematicians at
Cambridge, Massachusetts and delivered the invited plenary lecture Homology and Homotopy in the Topology Section of the Congress. During the summer of 1953 he was in Paris, lecturing on
'Homotopy' at the Collège de France, then flying back from Paris to Boston on 7 August on Air France. During the autumn of 1953 he again visited the Institute for Advanced Study in Princeton. He
also spent the summer of 1954 visiting Europe, attending the International Congress of Mathematicians in Amsterdam from 2 September to 9 September, then visiting Paris from where he flew back to
Boston on 16 September on Air France. Hurewicz died falling off a ziggurat (a Mexican pyramid) while on a conference outing at the 'International Symposium on Algebraic Topology' in Mexico. His
fall was on the 4 September 1956 and he was taken to the medical centre in Mérida, Mexico with serious injuries, dying two days later. The death certificate, signed by Dr Fernando
Guzmán-Espinoso, gives the cause of death as "traumatic and hemorrhagic shock resulting from severe fractures." In [1] it is suggested that his legendary absentmindedness was a factor:-
Hurewicz, who never married, was a highly cultured and charming man, and a paragon of absentmindedness, a failing that probably led to his death.
Eilenberg writes [6]:-
When Hurewicz died in Mérida after the Mexico City conference of 1956, I and several other participants were still in Mexico City. I remember sitting with a group of friends in the gardens of
a hotel (a converted cloisterlike seventeenth-century hospital) when the news arrived. It was a black day.
At his brother Stefan's request, Hurewicz was cremated and his ashes were shipped back to Mount Auburn Cemetery, Cambridge, Massachusetts. Rather strangely, it is noted on his death certificate
that, "Neither a United States passport nor any other evidence of citizenship of the deceased was found among his personal effects."
As we have already noted, Hurewicz's early work was on set theory and topology and [1]:-
... a remarkable result of this first period [1930] is his topological embedding of separable metric spaces into compact spaces of the same (finite) dimension.
In the field of general topology his contributions are centred around dimension theory. He wrote, in collaboration with Henry Wallman, an important text Dimension theory published in 1941. The
authors write in the Preface:-
In this book it has been the aim of the authors to give a connected and simple account of the most essential parts of dimension theory. Only those topics were chosen which are of interest to
the general worker in mathematics as well as the specialist in topology. Since the appearance of Karl Menger's well-known 'Dimensions theorie' in 1928, there have occurred important advances
in the theory, both in content and in method. These advances justify a new treatment, and in the present book great emphasis has been laid on the modern techniques of function spaces and
mappings on spheres. The algebraically minded reader will find in Chapter VIII a concise exposition of modern homology theory, with applications to dimension. Historical references are made
solely for the guidance of the beginning student, and no attempt has been made to attain completeness in this respect.
A reviewer writes that the book:-
... is truly a classic. It presents the theory of dimension for separable metric spaces with what seems to be an impossible mixture of depth, clarity, precision, succinctness, and
Karol Borsuk writes [5]:-
... for separable metric spaces, the book by Hurewicz and Wallman remains a model of clarity and strict parallels between the theory and the geometric insight. Their book also contains
algebraic topology notions and methods introduced to dimension theory by P S Aleksandrov.
In addition to this book, Hurewicz is best remembered for two remarkable contributions to mathematics, his discovery of the higher homotopy groups in 1935-36, and his discovery of exact sequences
in 1941. His work led to homological algebra. It was during Hurewicz's time as Brouwer's assistant in Amsterdam that he did the work on the higher homotopy groups [1]:-
... the idea was not new, but until Hurewicz nobody had pursued it as it should have been. Investigators did not expect much new information from groups, which were obviously commutative ...
Hurewicz had a second textbook published, but this was not until 1958 after his death. Lectures on ordinary differential equations was a reprinting, with minor revisions, of the mimeographed
notes of his Brown University lectures. Perhaps it is worth noting that the mimeographed notes had been reissued by the Mathematics department of the Massachusetts Institute of Technology in 1956
. This textbook is a beautiful introduction to ordinary differential equations which again reflects the clarity of his thinking and the quality of his writing.
Let us end our biography of Hurewicz by quoting from the Preface to [2] written by Krystyna Kuperberg:-
Witold Hurewicz is known for his contributions to dimension theory, algebraic topology (mainly, the 1930s papers on homotopy theory and the work on fibrations), and applied mathematics. Among
his published works are two excellent books ... Both books, beautifully written in a clear and concise manner so characteristic of Hurewicz, are continuously popular, and have been reprinted
more than once. ...
1. H Freudenthal, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. W Hurewicz, Collected works of Witold Hurewicz (Providence, RI, 1995).
3. Biography of Witold Hurewicz, in Collected works of Witold Hurewicz (Providence, R.I., 1995), xix-xx.
4. K Borsuk, Witold Hurewicz - life and activity (Polish), Wiadomosci matematyczne (2) 23 (1980), 69-74.
5. K Borsuk, Witold Hurewicz - life and work, in Handbook of the history of general topology 1 (Dordrecht, 1997), 79-84.
6. S Eilenberg, Witold Hurewicz-personal reminiscences, in Collected works of Witold Hurewicz (Providence, R.I., 1995), xlv-xlvi.
7. R Engelking and R Pol, Hurewicz's contributions to dimension theory, in Collected works of Witold Hurewicz (Providence, R.I., 1995), xxi-xxvii.
8. E Fadell, The contributions of Witold Hurewicz to algebraic topology, in Collected works of Witold Hurewicz (Providence, R.I., 1995), xxxiii-xl.
9. S Lefschetz, Witold Hurewicz In Memoriam, Bull. Amer. Math. Soc. 63 (1957), 77-82.
10. S Lefschetz, Witold Hurewicz, in memoriam, in Collected works of Witold Hurewicz (Providence, R.I., 1995), xli-xliv.
11. R Pol, Hurewicz's papers on descriptive set theory, in Collected works of Witold Hurewicz (Providence, R.I., 1995), xxix-xxxi.
12. Publications of Witold Hurewicz, in Collected works of Witold Hurewicz (Providence, R.I., 1995), xlvii-lii.
Additional Resources (show)
Other websites about Witold Hurewicz:
Honours awarded to Witold Hurewicz
Written by J J O'Connor and E F Robertson
Last Update January 2014 | {"url":"https://mathshistory.st-andrews.ac.uk/Biographies/Hurewicz/","timestamp":"2024-11-11T08:29:39Z","content_type":"text/html","content_length":"45433","record_id":"<urn:uuid:e252bd5b-30da-4d3b-8931-1c313889cbcb>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00740.warc.gz"} |
Math Expressions
iViewer embeds a powerful and versatile math expression engine. Math expressions can be inserted in commands and feedback items. The Math expression parser features:
Conditional logic using the if() function (like if / then /else statements)
Basic math expressions can be written in a standard way using numbers and operators. You can use tokens which are dynamically replaced by their actual value before the math evaluation begins.
Multiple expressions can be separated by a comma. You will use this feature to declare local variables and reuse them in the other expressions. When multiple expressions are present, the final result
is the value of the last expression. Within commands, math expressions must be enclosed in double curly braces, the result of the math evaluation is converted to a string (and can be optionally
formatted the way you need it). Math expression can also be used in the transform field of 'capture group' feedback items (where the incoming data type is set to Analog), in which case you won't use
the double curly braces to enclose the expression.
Do NOT use the double curly braces in the capture group > transform property in your feedback parsing. They are not needed in the transform property because this field can only contain math
expressions. Read the above paragraph for more details.
Simple addition: {{ 1 + [currentValue] }} Math functions: {{ round(1.5 * [currentValue]) }} Multiple expressions: {{ a=100, b=2*[inputValue], max(a, b) }} Conditions: {{ a=[currentValue]+[delta], if
(a<5, a=0, a) }} Boolean logic: {{ above=([currentValue] > 100), below=([currentValue] < 0), if (above or below, 0, [currentValue]) }}
When using a math expression, you can specify an output format prefixed by a colon before the closing accolades. If you omit the format, the result will be formatted as an integer.
{{math expression}} -> will output an integer without any decimal points {{math expression:}} -> will output a double with full decimal points {{math expression:N}} -> will output double with N
decimals (can be 0, in this case this amounts to outputing an int) {{math expression::printf format}} -> will output a string using a full printf format string, with one argument (the math expression
double result).
For more info on the printf format, see here: Printf Format on Wikipedia
Math expressions can also be used in feedback 'capture group' items, within the transform field. This works same as above, but you don't have to wrap the expression in curly braces. With transforms,
one predefined constant is set (“value”) which is the value extracted from the capture group, and can be used in the math computation.
Note that the 'value' constant does not support decimal values when using an Analog data type. The constant will always be rounded to a whole number before any transform can take place. So you need
to capture it into a serial join first, then use capture group name referencing to perform any math on this value if you need the initial captured value to support decimals.
In addition to the various number formatting options shown above, math expression can generate a hexadecimal ASCII string, or even raw bytes that you can use to send to a remote system. The options
should be appended to the math expression, separated by a colon, just before the closing curly braces.
:h Convert the result to an unsigned long value (4 bytes) then output as few raw bytes as possible to represent this number. Bytes are output in network order (big endian). Examples: {{12+1:h}} ->
result=0x0000000d, output = "\x0d" (1 byte) {{1005*2:h}} -> result=0x000007da, output="\x07\xda" (2 bytes) :h<count> Convert the result to an unsigned long value (4 bytes) the output the requested
number of bytes, regardless of the result value. count should be 1 to 4 inclusive. Examples: {{12+1:h3}} -> result=0x0000000d, output = "\x00\x00\x0d" (3 bytes)
:hs Convert the result to an unsigned long value (4 bytes) then output an ASCII representation, using as few characters as possible to represent the value, using lowercase letters when needed.
{{12+1:hs}} -> result=0x0000000d, output = "0d" (2 characters) {{1005*2:hs}} -> result=0x000007da, output="07da" (4 characters) :hs<count> Convert the result to an unsigned long value (4 bytes) then
output an ASCII representation, showing the <count> least significant bytes of the result, using lowercase letters when needed. {{12+1:hs3}} -> result=0x0000000d, output = "00000d" (6 characters)
{{1005*2:hs4}} -> result=0x000007da, output="000007da" (8 characters) :Hs Same as :hs but text is output using uppercase characters: {{12+1:Hs}} -> result=0x0000000d, output = "0D" (2 characters)
{{1005*2:Hs}} -> result=0x000007da, output="07DA" (4 characters) :Hs<count> Same as :hs<count> but text is output using uppercase characters. {{12+1:Hs3}} -> result=0x0000000d, output = "00000D" (6
characters) {{1005*2:Hs4}} -> result=0x000007da, output="000007DA" (8 characters)
Finally, an additional 's' specifier asks the engine to output as few characters as possible by stripping the leading nibble (half-byte) if its value is 0:
:hss Same as :hs, stripping the leading nibble is possible: {{12+1:hss}} -> result=0x0000000d, output = "d" (2 characters) {{1005*2:hss}} -> result=0x000007da, output="7da" (4 characters) :Hss Same
as :Hs, stripping the leading nibble is possible: {{12+1:hss}} -> result=0x0000000d, output = "D" (2 characters) {{1005*2:hss}} -> result=0x000007da, output="7DA" (4 characters)
The math expression parser recognizes the following operators. They are listed in inverse priority order (lowest priority ones come first).
= assignment (creates or updates a local variable) and logical AND or logical OR xor logical XOR <= less than or equal >= greater than or equal != not equal == equal > greater than < less than +
addition - subtraction * multiplication / division % modulo ^ power (raise x to the power of y)
The math parser provides several built-in functions. Let us know if the function you need is missing from this list! Trigonometric functions take an angle expressed in radians, use the dtor() and
rtod() functions to convert between degrees and radians.
dtor(a) convert angle from degrees to radians rtod(a) convert angle from radians to degrees sin(a) sine function cos(a) cosine function tan(a) tangent function asin(a) arc sine function acos(a) arc
cosine function atan(a) arc tangent function sinh(a) hyperbolic sine function cosh(a) hyperbolic cosine function tanh(a) hyperbolic tangent function asinh(a) hyperbolic arc sine function acosh(a)
hyperbolic arc cosine function atanh(a) hyperbolic arc tangent function
log(n) natural logarithm log2(n) logarithm to base 2 log10(n) logarithm to base 10 ln(n) logarithm to base e (2.71828...) exp(n) e raised to the power of x sqrt(n) square root abs(n) absolute value
trunc(n) truncate to integral value rint(n) round to integral value near(n) same as rint(n) round(n) round to integral value, regardless of rounding direction ceil(n) round to smallest integral value
not less than n floor(n) round to largest integral value not greater than n sign(n) sign function: -1 if n<0, 0 if n=0, 1 if n>0
min(n1,n2,...) smallest of all arguments. Use as many arguments as needed. max(n1,n2,...) largest of all arguments. Use as many arguments as needed. sum(n1,n2,...) sum of all arguments. Use as many
arguments as needed. avg(n1,n2,...) mean value of all arguments. Use as many arguments as needed.
if(t,a,b) if t is > 0 then result is a, otherwise result is b.
Assuming the captured value of the feedback group = 125.266335454
"value / 100:" -> "1.25266335454" "value / 100" -> "1" "value / 100:0" -> "1" "value / 100:3" -> "1.252" "value / 100::%03.1f" -> "001.2" "value / 100::your value is %.2f. Isn't it cool?" -> "your
value is 1.25. Isn't it cool?"
Token names can be used anywhere in the math expression, and will be replaced with their value before the math expression is evaluated. For example, if you had a global token named [level] defined
with a value of 125.266335454, you could create the following examples:
"[level] / 100:" -> "1.25266335454" "[level] / 100" -> "1" "[level] / 100:0" -> "1" "[level] / 100:3" -> "1.252" "[level] / 100::%03.1f" -> "001.2" "[level] / 100::your value is %.2f. Isn't it cool?"
-> "your value is 1.25. Isn't it cool?"
Within feedback group transform expressions, you can also reference the value of any other capture group defined in the same feedback item. To reference a group, it must first be given a name, and
must also be listed above the current group in the feedback processing order. Reference the group by surrounding the name of the group in dollar signs. So if we had another capture group named
[temp], which captured the value 1.25266335454 we could reference it like so:
"$[temp]$ / 100:3" -> "1.252" "$[temp]$ / 100::%03.1f" -> "001.2" "$[temp]$ / 100::your temp value before transform is $[temp]$ and after transform is %.2f. Isn't it cool?" -> "your value before
transform is 1.25266335454 and after transform is 1.25. Isn't it
Commands (and commands within macros), string join assignments and token assignments are being run through the math parser which looks for any math expression(s) enclosed in double curly braces
within the text. Text can include multiple math blocks, each enclosed in double curly braces. When processing the text, all math local variables are reset at beginning, then kept accross multiple
math blocks. This means that you can perfectly send text with intertwined math result values, and that each calculation can reuse local variables from the previous step. Here is an example:
"Going from [currentValue] to {{a=min(1,max(100, [currentValue]+[delta]))}}. Final delta is {{abs(a - [currentValue])}} (was [delta])."
In the case above, we compute a new value from a current value and a delta, cap it to a minimum of 1 and a maximum of 100, and output text showing the new value, as well as the delta between the
capped new value and the current value. Assuming that [currentValue] is 17 and [delta] is 100, the output text will be:
"Going from 17 to 100. Final delta is 83 (was 100)." | {"url":"https://commandfusion.com/wiki2/software/gui-designer/math-expressions","timestamp":"2024-11-06T00:45:41Z","content_type":"text/html","content_length":"79051","record_id":"<urn:uuid:f82f8c00-ea09-4d99-bc52-44e4431c6817>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00054.warc.gz"} |
Algorithms Homework #5 solved
1. In the flow network shown below, the number beside an edge denotes its corresponding capacity. Apply
the Edmonds-Karp algorithm to find a maximum flow from s to t in the network. Show every
augmentation path (but you do NOT need to show the whole network to save time) and
explain why the flow you found is maximum.
d 3
2. Problem 26-1 (pages 760–761).
3. Problem 26-3 (pages 761–762).
4. A realtor would like to maximize the number of apartments sold. She has p apartments to sell and q
potential customers for these apartments. She has m salesmen working for her. Each salesman is assigned
a list of apartments and clients interested in these apartments. A salesman can sell an apartment to
any of his customers. Salesman i can sell at most bi apartments. Also, any apartment cannot be owned
by more than one person. For m = 2, p = 4, q = 5, b1 = 3, b2 = 1, and the following assignments of
customers and apartments to the salesmen, construct the flow network for the underlying problem. How
to find the maximum number of apartments that can be sold? (Hint: How can you constrain that
salesman i can sell at most bi apartments in this flow network?)
Salesman Customers Apartments
1 1, 2, 3, 4 1, 2, 3
2 3, 4, 5 3, 4
5. The figure below shows a segmented routing structure in a row-based field-programmable gate array
(FPGA). There are five connections, c1, c2, . . . , c5, to be routed on three segmented tracks, t1, t2, and
t3, with eight segments s11, s12, . . . , s32 in the row-based FPGA. A track can be partitioned into a set
of segments by using switches. If a switch incident on two adjacent segments is “ON”, then the two
segments are electrically connected; otherwise, the two segments can be used independently. You are
asked to route (place) the five connections on the three segmented tracks. Suppose each connection can
use at most one segment for routing, i.e., 1-segment routing. In other words, a connection ck of the
column span [lk, rk] is said to be routed on a segment sij of track ti
if ck = [lk, rk] is placed within the
column span of sij . For example, c3 = [2, 5] can be routed on segment s31 of track t3 (which consumes
only one segment) while it cannot route on track t1 or t2 (which would have consumed two segments,
thus violating the constraint of 1-segment routing). Give an efficient algorithm to solve the 1-segment
routing problem. What is the time complexity of your algorithm?
Problem 7. (10 pts total) The figure below shows a segmented routing structure in a row-based
field-programmable gate array (FPGA). There are five connections, c1, c2, . . . , c5, to be routed on
three segmented tracks, t1, t2, and t3, with eight segments s11, s12, . . . , s32 in the row-based FPGA. A
track can be partitioned into a set of segments by using switches. If a switch incident on two adjacent
segments is “ON”, then the two segments are electrically connected; otherwise, the two segments can
be used independently. You are asked to route (place) the five connections on the three segmented
tracks. Suppose each connection can use at most one segment for routing, i.e., 1-segment routing.
In other words, a connection ck of the column span [lk, rk] is said to be routed on a segment sij of
track ti if ck = [lk, rk] is placed within the column span of sij . For example, c3 = [2, 5] can be routed
on segment s31 of track t3 (which consumes only one segment) while it cannot route on track t1 or t2
(which would have consumed two segments, thus violating the constraint of 1-segment routing). Give
an efficient algorithm to solve the 1-segment routing problem. (Hint: Think about the resource
assignment between key components.)
Problem 8. (bonus) (This problem can be answered by email by 1pm January 14 after
the final exam.) Please list the corrections to the class notes and lectures you made in this semester,
if any. Please give specific information on the corrections, e.g., page numbers of the class notes, if
6. Concepts on polynomial-time complexity. (a) Exercise 34.1-4 (page 1060). (b) Professor Right finds a
fast algorithm for the maximum flow problem on the network G = (V, E) with the capacity c(u, v) for
the edge (u, v), which runs in O(V E(lg C)
) time, where C = max(u,v)∈E c(u, v). Is it a polynomial-time
algorithm? Justify your claim.
7. Exercise 34.4-7 (page 1086).
8. Problem 34-1 (pages 1101–1102).
9. Problem 34-3 (pages 1103–1104).
10. (a) Exercise 17.1-3 (page 456). (b) Exercise 17.2-2 (page 459). (c) Exercise 17.3-2 (page 462).
11. Exercise 17.4-3 (page 471).
12. Problem 17-3 (pages 473–474).
13. (DIY Problem) For this problem, you are asked to design a problem set related to Chapter(s) 17, 26,
and/or 34, and give a sample solution to your problem set. Grading on this problem will be based upon
the quality of the designed problem as well as the correctness of your sample solution. | {"url":"https://codeshive.com/questions-and-answers/algorithms-homework-5-solved/","timestamp":"2024-11-04T01:38:08Z","content_type":"text/html","content_length":"104396","record_id":"<urn:uuid:695a51bd-ce47-4da0-a370-0adf5343bcb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00260.warc.gz"} |
You are likely limited by
05-20-2019 01:24 PM
I have a problem where I need to compute many (1e4 - 1e6) small matrix-matrix and matrix-vector products (matrix dimensions around ~15 - 35). This problem seems "embarrassingly parallel" to me, and
so I am confused as to why I am seeing the following performance issue: on a Google Cloud compute server with 48 physical cores (96 logical cores), performance plateaus at 10-16 threads. Adding
additional threads does not reduce computation time. I have tried several different approaches: (1) cblas_dgemm_batch; (2) calling cblas_dgemm within a tbb::parallel_for, with both sequential and
TBB-threaded MKL; (3) JIT-compiled problem-specific dgemm kernel (created with mkl_jit_create_dgemm) within a parallel_for; (4) mkl_dgemm_compact (along with mkl_dgepack and mkl_dgeunpack).
All of these yield roughly comparable performance (except for the compact functions--there, packing and unpacking time completely dominates computation time), but none of them seems to yield
performance that scales linearly with the number of threads I specify, as I would expect. The maximum performance I see is around 50 GFLOPS on a system capable of around 1-2 TFLOPS. (Indeed,
multiplying two large matrices achieves performance in the teraflop range.) Is this the best I can expect? Why do I not see performance scaling linearly with thread count on this embarrassingly
parallel problem?
05-23-2019 11:29 PM
05-24-2019 10:13 PM
05-26-2019 10:43 AM | {"url":"https://community.intel.com/t5/Intel-oneAPI-Math-Kernel-Library/Batched-dgemm-performance-plateaus/m-p/1150161","timestamp":"2024-11-13T18:42:31Z","content_type":"text/html","content_length":"215022","record_id":"<urn:uuid:b266ce14-98d1-4008-9f4f-df3ac4705d43>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00818.warc.gz"} |
A triangle has sides A, B, and C. Sides A and B have lengths of 10 and 8, respectively. The angle between A and C is (13pi)/24 and the angle between B and C is (pi)24. What is the area of the triangle? | Socratic
A triangle has sides A, B, and C. Sides A and B have lengths of 10 and 8, respectively. The angle between A and C is #(13pi)/24# and the angle between B and C is # (pi)24#. What is the area of the
1 Answer
Since triangle angles add to $\pi$ we can figure out the angle between the given sides and the area formula gives
$A = \setminus \frac{1}{2} a b \sin C = 10 \left(\sqrt{2} + \sqrt{6}\right)$.
It helps if we all stick to the convention of small letter sides $a , b , c$ and capital letter opposing vertices $A , B , C$. Let's do that here.
The area of a triangle is $A = \frac{1}{2} a b \sin C$ where $C$ is the angle between $a$ and $b$.
We have $B = \setminus \frac{13 \setminus \pi}{24}$ and (guessing it's a typo in the question) $A = \setminus \frac{\pi}{24}$.
Since triangle angles add up to ${180}^{\setminus} \circ$ aka $\setminus \pi$ we get
$C = \setminus \pi - \setminus \frac{\pi}{24} - \frac{13 \pi}{24} = \setminus \frac{10 \pi}{24} = \setminus \frac{5 \pi}{12}$
$\setminus \frac{5 \pi}{12}$ is ${75}^{\setminus} \circ .$ We get its sine with the sum angle formula:
$\sin {75}^{\circ} = \sin \left(30 + 45\right) = \sin 30 \cos 45 + \cos 30 \sin 45$
$= \left(\setminus \frac{1}{2} + \frac{\sqrt{3}}{2}\right) \setminus \frac{\sqrt{2}}{2}$
$= \setminus \frac{1}{4} \left(\sqrt{2} + \sqrt{6}\right)$
So our area is
$A = \setminus \frac{1}{2} a b \sin C = \setminus \frac{1}{2} \left(10\right) \left(8\right) \setminus \frac{1}{4} \left(\sqrt{2} + \sqrt{6}\right)$
$A = 10 \left(\sqrt{2} + \sqrt{6}\right)$
Take the exact answer with a grain of salt because it's not clear we guessed correctly what the asker meant by the angle between $B$ and $C$.
Impact of this question
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C : Minimum number of jumps to reach the end of the array
C Exercises: Return the minimum number of jumps to reach the end of the array
C Array: Exercise-56 with Solution
Write a program in C to return the minimum number of jumps to reach the end of the array.
Expected Output :
The given array is : 1 3 5 8 9 2 6 7 6 8 9 1 1 1
The minimum of number of jumps is required to reach the end is: 3
The task is to write a C program that calculates the minimum number of jumps required to reach the end of an array. Each element in the array represents the maximum number of steps that can be taken
forward from that element. The program should determine the smallest number of jumps needed to traverse the array from the first to the last element.
Sample Solution:
C Code:
#include <stdio.h>
#include <limits.h>
// Function to calculate the minimum number of jumps needed to reach the end
int noOfJumps(int arr1[], int low, int high) {
// If the start and end point are the same, no jump is needed
if (high == low)
return 0;
// If the current position is 0, it's impossible to move forward
if (arr1[low] == 0)
return INT_MAX;
int min = INT_MAX; // Initialize the minimum jumps to a maximum value
// Iterate through all possible steps from the current position
for (int i = low + 1; i <= high && i <= low + arr1[low]; i++) {
// Recursively find the minimum jumps needed from the next position
int jumps = noOfJumps(arr1, i, high);
// If it's possible to jump from the next position and it's minimum, update min
if (jumps != INT_MAX && jumps + 1 < min)
min = jumps + 1; // Update the minimum jumps
return min; // Return the minimum number of jumps
int main() {
int arr1[] = {1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9, 1, 1, 1};
int n = sizeof(arr1) / sizeof(arr1[0]);
int i;
//------------- print original array ------------------
printf("The given array is: ");
for (i = 0; i < n; i++) {
printf("%d ", arr1[i]);
// Calculate and display the minimum number of jumps needed to reach the end
printf("The minimum number of jumps required to reach the end is: %d\n", noOfJumps(arr1, 0, n - 1));
return 0;
The given array is : 1 3 5 8 9 2 6 7 6 8 9 1 1 1
The minimum of number of jumps is required to reach the end is: 3
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Errors and Confidence Levels: Step-by-Step - CIAO 4.16 Sherpa
Step-by-Step Guide to Estimating Errors and Confidence Levels
Sherpa Threads (CIAO 4.16 Sherpa)
This thread uses the "native" Sherpa interface to estimate errors or confidence levels for parameters in a fit (to data of any dimensionality).
Related Links:
Last Update: 5 Dec 2022 - reviewed for CIAO 4.15, updated screen output.
Getting Started
To obtain the sample data files used in this thread, download the sherpa.tar.gz file as described in the "Sherpa Getting Started" thread.
Finding the best fit
First, we check the current Sherpa settings. By default, Sherpa uses the Levenberg-Marquardt method to optimize the fit of a model to data with χ^2 statistics, using the Gehrels variance function. In
this example, we choose to fit with the default optimization method, and change the fit statistic from default to the Chi2Xspecvar statistic.
sherpa> clean()
sherpa> show_method()
Optimization Method: LevMar
name = levmar
ftol = 1.1920928955078125e-07
xtol = 1.1920928955078125e-07
gtol = 1.1920928955078125e-07
maxfev = None
epsfcn = 1.1920928955078125e-07
factor = 100.0
numcores = 1
verbose = 0
sherpa> set_stat("chi2datavar")
sherpa> show_stat()
Statistic: Chi2DataVar
Chi Squared with data variance.
The variance in each bin is estimated from the data value in that
If the number of counts in each bin is large, then the shape of
the Poisson distribution from which the counts are sampled tends
asymptotically towards that of a Gaussian distribution, with
sigma(i)^2 = N(i,S) + [A(S)/A(B)]^2 N(i,B)
where N is the number of on-source (and off-source) bins included
in the fit. The background term appears only if an estimate of the
background has been subtracted from the data.
A(B) is the off-source "area", which could be
the size of the region from which the background is extracted, or
the length of a background time segment, or a product of the two,
etc.; and A(S) is the on-source "area". These terms may be defined
for a particular type of data: for example, PHA data sets A(B) to
`BACKSCAL * EXPOSURE` from the background data set and A(S) to
`BACKSCAL * EXPOSURE` from the source data set.
See Also
Chi2Gehrels, Chi2ModVar, Chi2XspecVar
sherpa> load_pha("source_grouped_pi.fits")
WARNING: systematic errors were not found in file 'source_grouped_pi.fits'
statistical errors were found in file 'source_grouped_pi.fits'
but not used; to use them, re-read with use_errors=True
read ARF file arf.fits
read RMF file rmf.fits
sherpa> notice(0.5, 8.0)
dataset 1: 0.00146:14.9504 -> 0.00146:8.76 Energy (keV)
sherpa> plot_data()
The data we will be fitting is read into the session with the load_data command, which automatically loads the instrument response associated with a source data set when the ARF and RMF filenames are
recorded in the header of the source data file, as shown above; this also applies to background data associated with the source (not considered in this example). The plot in Figure 1 results from the
plot_data command above, showing the source data that is to be fit, between 0.5-8.0 keV.
Figure 1: Source spectrum
The source model is set to an absorbed power-law with the set_source command, and fit with the fit command, producing the results shown below.
sherpa> set_source(xsphabs.abs1 * powlaw1d.p1)
sherpa> fit()
Dataset = 1
Method = levmar
Statistic = chi2datavar
Initial fit statistic = 6.65323e+09
Final fit statistic = 123.621 at function evaluation 29
Data points = 133
Degrees of freedom = 130
Probability [Q-value] = 0.640847
Reduced statistic = 0.950933
Change in statistic = 6.65323e+09
abs1.nH 2.25583 +/- 0.117429
p1.gamma 1.47466 +/- 0.0818852
p1.ampl 0.00185329 +/- 0.000224598
The fit and δχ residuals (\(\frac{data-model}{errors}\), also refered to as σ residuals which is not the same σ as standard deviation) may be plotted with the plot or plot_fit_delchi command. Here we
use Matplotlib commands to adjust the axes after the plot has been created, resulting in Figure 2:
sherpa> plot_fit_delchi(xlog=True, ylog=True)
sherpa> plt.xlim(0.3, 10)
(0.3, 10)
Figure 2: Best fit model with residuals
The show_fit and get_fit_results commands are also available for checking the quality of the fit:
sherpa> print(get_fit_results())
datasets = (1,)
itermethodname = none
methodname = levmar
statname = chi2datavar
succeeded = True
parnames = ('abs1.nH', 'pl.gamma', 'pl.ampl')
parvals = (2.2558840821480937, 1.4746943632186176, 0.0018533814206418062)
statval = 123.62123838387117
istatval = 6653225321.935967
dstatval = 6653225198.314729
numpoints = 133
dof = 130
qval = 0.6408472990001428
rstat = 0.9509326029528552
message = successful termination
nfev = 29
Confidence limits for individual parameters
After finding the best-fit model parameter values, we calculate the confidence limits (parameter bounds) for these parameters using either the confidence, projection or covariance methods. The
confidence limits are defined by a required confidence level in our analysis, typically the 68.3% or 90% level, corresponding to 1σ and 1.6σ for the normal distribution. The table below displays the
relationship between the standard deviation, σ, and confidence level for one significant parameter. The relationship between the confidence level, Δχ^2, and the general log-likelihood, \(\Delta \log
Confidence intervals for a normal distribution
Confidence \(\sigma\) \(\Delta \chi^{2}\) \(\Delta \log{\mathcal{L}}\)
68.3% 1.0 1.00 0.50
90.0% 1.6 2.71 1.36
95.5% 2.0 4.00 2.00
99.0% 2.6 6.63 3.32
99.7% 3.0 9.00 4.50
We will use the confidence method to estimate 1σ errors on the gamma parameter of the power-law model component:
sherpa> print(get_conf())
name = confidence
sigma = 1
eps = 0.01
maxiters = 200
soft_limits = False
remin = 0.01
fast = False
parallel = True
numcores = 4
maxfits = 5
max_rstat = 3
tol = 0.2
verbose = False
openinterval = False
sherpa> conf(p1.gamma)
pl.gamma lower bound: -0.0830705
pl.gamma upper bound: 0.0849456
Dataset = 1
Confidence Method = confidence
Iterative Fit Method = None
Fitting Method = levmar
Statistic = chi2datavar
confidence 1-sigma (68.2689%) bounds:
Param Best-Fit Lower Bound Upper Bound
----- -------- ----------- -----------
pl.gamma 1.47469 -0.0830705 0.0849456
To access the 90% confidence limits on a parameter, the sigma field of the conf_opt variable should be changed to 1.6.
sherpa> set_conf_opt("sigma", 1.6)
sherpa> conf(p1.gamma, abs1.nH)
abs1.nH lower bound: -0.183538
abs1.nH upper bound: 0.197266
pl.gamma lower bound: -0.13186
pl.gamma upper bound: 0.136806
Dataset = 1
Confidence Method = confidence
Iterative Fit Method = None
Fitting Method = levmar
Statistic = chi2datavar
confidence 1.6-sigma (89.0401%) bounds:
Param Best-Fit Lower Bound Upper Bound
----- -------- ----------- -----------
pl.gamma 1.47469 -0.13186 0.136806
abs1.nH 2.25588 -0.183538 0.197266
We have also used conf to calculate the uncertainty on the optimized hydrogen column density parameter of the absorption model, abs1.nH.
To estimate errors on all the thawed parameters, conf should be called with no parameter names. Since sigma is still set to 1.6 for the confidence method, the following calculates the 90% confidence
limits for all the thawed parameters:
sherpa> conf()
abs1.nH lower bound: -0.183538
abs1.nH upper bound: 0.197266
pl.ampl lower bound: -0.000329436
pl.gamma lower bound: -0.13186
pl.gamma upper bound: 0.136806
pl.ampl upper bound: 0.000413355
Dataset = 1
Confidence Method = confidence
Iterative Fit Method = None
Fitting Method = levmar
Statistic = chi2datavar
confidence 1.6-sigma (89.0401%) bounds:
Param Best-Fit Lower Bound Upper Bound
----- -------- ----------- -----------
abs1.nH 2.25588 -0.183538 0.197266
pl.gamma 1.47469 -0.13186 0.136806
pl.ampl 0.00185338 -0.000329436 0.000413355
The covariance command behaves similarly to conf, although the fields in the state object are different. While it is quicker than the confidence method, it is less accurate; i.e., it is always
symmetric because it uses the diagonal elements of the covariance matrix and ignores correlations between the parameters.
Note that the computationally intensive confidence function has been parallelized in Sherpa, to make use of multi-core systems (i.e., laptops or desktops with 2 or more cores).
Common WARNING messages returned by confidence methods
WARNING: hard minimum hit for parameter <parameter name>
When the confidence, projection, and covariance methods are used to estimate confidence intervals for thawed model parameters after a fit, sometimes a hard upper or lower limit will be reached for
one or more parameter. This produces the message "WARNING: hard minimum hit for parameter <parameter name>", along with a row of dashes in the appropriate place in the function output.
The covariance method can also return a null value for an upper/lower limit when the parameter-space at the minimum is non-quadratic for a given parameter. The covariance matrix calculations assume
that the parameters follow the normal distribution. If the parameter-space is non-smooth, then the covariance calculations fail and Sherpa returns "-----".
Example confidence output:
sherpa> conf()
WARNING: hard minimum hit for parameter bpow1.gamma2
WARNING: hard maximum hit for parameter bpow1.gamma2
WARNING: hard minimum hit for parameter bpow1.eb
WARNING: hard maximum hit for parameter bpow1.eb
Dataset = 1
Confidence Method = confidence
Iterative Fit Method = None
Fitting Method = neldermead
Statistic = cstat
confidence 1-sigma (68.2689%) bounds:
Param Best-Fit Lower Bound Upper Bound
----- -------- ----------- -----------
bpow1.gamma1 1.54147 -0.0292891 0.0292709
bpow1.gamma2 8.10056 ----- -----
bpow1.eb 9.49083 ----- -----
bpow1.ampl 0.022806 -0.000378395 0.000383854
This occurs when the parameter bound found by one of the confidence methods lies outside the hard limit boundary for a model parameter—this could result from an issue with the signal-to-noise of the
data, the applicability of the model to the data, systematic errors in the data, among others things.
A parameter hard limit represents either a hard physical limit (e.g., temperature is not allowed to go below zero), a mathematical limit (e.g., prevent a number from going to zero or below, when the
logarithm of that number will be taken), or the limit of what a float or double can hold (the fit should not be driven above or below the maximum or minimum values a variable can hold). For this
reason, model parameter hard limits should not be changed by the user.
WARNING: The confidence level lies within <interval>
Another warning message which may be returned by confidence is that a model parameter lies within the stated range:
sherpa> conf(g15.Sigma)
g15.Sigma -: WARNING: The confidence level lies within (8.706380e-05,9.252185e-05)
Datasets = 1, 2
Confidence Method = confidence
Iterative Fit Method = None
Fitting Method = levmar
Statistic = chi2datavar
confidence 1.64-sigma (89.8995%) bounds:
Param Best-Fit Lower Bound Upper Bound
----- -------- ----------- -----------
g15.Sigma 0.000997626 -0.000907834 0.000597058
This occurs where confidence cannot locate the root (minimum value of the fit statistic function) even though the root is bracketed within an interval (perhaps due to poor resolution of the data or a
discontinuity). In such cases, when the openinterval option of confidence is set to False (default), the confidence function will not be able to find the root within the set tolerance and the
function will return the average of the open interval which brackets the root. If the option openinterval is set to True, then confidence will print the minimal open interval which brackets the root
(not to be confused with the lower and upper bound of the confidence interval). The most accurate thing to do is to return an open interval where the root is localized/bracketed rather than the
average of the open interval (since the average of the interval is not a root within the specified tolerance).
The output from confidence may be checked by setting 'set_conf_opt('verbose',1)', and then re-running confidence for the relevant parameter(s).
sherpa> set_conf_opt('verbose',1)
sherpa> conf(g15.Sigma)
# f[ 2.931742e+00 2.957942e-02 1.471941e+00 9.976265e-04
1.840837e+00 3.667986e-03 7.820442e-05 1.864564e+00 2.562143e-03
1.415699e-04 2.004898e+00 6.288115e-04 1.259512e-04] =
# sigma = 1.640000e+00
# target_stat = 8.596825e+02
# tol = 1.000000e-02
# smin = [-2. 0. 1.45 0. 0. 1.82 0. 0. 1.85 0.
0. 1.96
0. 0. ]
# smax = [ 9.000000e+00 1.000000e+24 1.500000e+00 1.000000e-02
1.850000e+00 1.000000e-02 1.000000e-03 1.900000e+00 1.000000e-02
1.000000e-03 2.040000e+00 1.000000e-03 1.000000e-03]
# hmin = [ -3.402823e+38 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00]
# hmax = [ 3.402823e+38 3.402823e+38 3.402823e+38 3.402823e+38
3.402823e+38 3.402823e+38 3.402823e+38 3.402823e+38 3.402823e+38
3.402823e+38 3.402823e+38 3.402823e+38 3.402823e+38]
# Note: for the intermediate steps, the notation:
par.name -/+: f( x ) = stat
# ==> `stat` is the statistic when parameter `par.name` is frozen at `x`
# while searching for the `lower/upper` confidence level, repectively.
g15.Sigma -: f( 4.812517e-04 ) = -2.565633e+00
g15.Sigma -: f( 0.000000e+00 ) = 3.505966e+01
g15.Sigma -: f( 2.406259e-04 ) = -2.565434e+00
g15.Sigma -: f( 2.113694e-04 ) = -2.565393e+00
g15.Sigma -: f( 1.653199e-04 ) = -2.565381e+00
g15.Sigma -: f( 1.343754e-04 ) = -2.564794e+00
g15.Sigma -: f( 1.079722e-04 ) = -2.565100e+00
g15.Sigma -: f( 8.706380e-05 ) = 3.778055e+01
g15.Sigma -: f( 1.068516e-04 ) = -2.563338e+00
g15.Sigma -: f( 1.022595e-04 ) = -2.562500e+00
g15.Sigma -: f( 9.961908e-05 ) = -2.563027e+00
g15.Sigma -: f( 9.721570e-05 ) = -2.564524e+00
g15.Sigma -: f( 9.532733e-05 ) = -2.562276e+00
g15.Sigma -: f( 9.377811e-05 ) = -2.562519e+00
g15.Sigma -: f( 9.252185e-05 ) = -2.563134e+00
g15.Sigma -: f( 9.149979e-05 ) = -2.563034e+00
g15.Sigma -: f( 9.066944e-05 ) = -2.563196e+00
g15.Sigma -: WARNING: The confidence level lies within (8.706380e-05, 9.252185e-05)
g15.Sigma lower bound: -0.000907834
g15.Sigma +: f( 1.514001e-03 ) = -8.109195e-01
g15.Sigma +: f( 2.546751e-03 ) = 2.553184e+01
g15.Sigma +: f( 2.030376e-03 ) = 7.845440e+00
g15.Sigma +: f( 2.030376e-03 ) = 7.845440e+00
g15.Sigma +: f( 2.030376e-03 ) = 7.845440e+00
g15.Sigma +: f( 1.772189e-03 ) = 2.467092e+00
g15.Sigma +: f( 1.772189e-03 ) = 2.467092e+00
g15.Sigma +: f( 1.772189e-03 ) = 2.467092e+00
g15.Sigma +: f( 1.643095e-03 ) = 5.832027e-01
g15.Sigma +: f( 1.643095e-03 ) = 5.832027e-01
g15.Sigma +: f( 1.643095e-03 ) = 5.832027e-01
g15.Sigma +: f( 1.578548e-03 ) = -1.721177e-01
g15.Sigma +: f( 1.578548e-03 ) = -1.721177e-01
g15.Sigma +: f( 1.578548e-03 ) = -1.721177e-01
g15.Sigma +: f( 1.610822e-03 ) = 1.906909e-01
g15.Sigma +: f( 1.610822e-03 ) = 1.906909e-01
g15.Sigma +: f( 1.610822e-03 ) = 1.906909e-01
g15.Sigma +: f( 1.594685e-03 ) = 5.746982e-03
g15.Sigma upper bound: 0.000597058
Datasets = 1, 2
Confidence Method = confidence
Iterative Fit Method = None
Fitting Method = levmar
Statistic = chi2datavar
confidence 1.64-sigma (89.8995%) bounds:
Param Best-Fit Lower Bound Upper Bound
----- -------- ----------- -----------
g15.Sigma 0.000997626 -0.000907834 0.000597058
method = levmar, stat = chi2datavar, in 17.921696 secs
How does the fit surface vary for a parameter (interval-projection)?
In order to visually inspect the model parameter-space we can "project" the statistics into the 1-D or 2-D plane with int_proj and reg_proj, respectively. This allows for checking the shape of the
parameter-space around the best fit parameter values, as well as evaluate correlations between parameters.
Here we use the int_proj (interval-projecton) method to see how the fit statistic varies with the gamma parameter of the power-law component. Since we already know that the 90% errors of p1.gamma are
approximately \(\pm 0.13\), we choose to set the axis range manually:
sherpa> print(get_int_proj())
x = None
y = None
min = None
max = None
nloop = 20
delv = None
fac = 1
log = False
sherpa> int_proj(p1.gamma, min=1, max=2)
sherpa> print(get_int_proj())
x = [1. ,1.0526,1.1053,1.1579,1.2105,1.2632,1.3158,1.3684,1.4211,1.4737,
1.5263,1.5789,1.6316,1.6842,1.7368,1.7895,1.8421,1.8947,1.9474,2. ]
y = [159.736 ,151.8122,144.9428,139.0959,134.2386,130.3377,127.3595,125.27 ,
124.0353,123.6214,123.9943,125.1203,126.9661,129.4985,132.685 ,136.4935,
min = 1
max = 2
nloop = 20
delv = None
fac = 1
log = False
The resulting plot is shown in Figure 3.
Figure 3: Plot of interval-projection results
The "confidence intervals" table above lists a range of common confidence levels and the corresponding change in χ^2 values (i.e., the statistic value on the y-axis in this plot). The parameters
displayed by print(get_int_proj()) show how the fit statistic versus single model parameter plot is calculated. The parameters min and max are the minimum and maximum grid boundary values; if set to
the default values of None, the grid boundaries are calculated automatically from the covariance. nloop is the bin size, which by default is used with the min and max grid boundaries in order to
determine the step size, delv (default is delv=None).
The int_unc (interval-uncertainty) command behaves similarly to int_proj , although the fields in the state object for the two methods are different.
How are two parameters correlated (region-projection)?
In this section we use the reg_proj (region-projection) method of Sherpa to see whether the p1.gamma and abs1.nh parameters are correlated.
From our earlier run we know that the 90% errors on the two parameters—when evaluated independently—are approximately 0.14 (gamma) and 0.2 (nH). However we decide to let the routine calculate plot
limits automatically, and choose to display contours at the 1 and 1.6 σ level (68.3% and 90% confidence levels).
sherpa> print(get_reg_proj())
x0 = None
x1 = None
y = None
min = None
max = None
nloop = (10, 10)
fac = 4
delv = None
log = (False, False)
sigma = (1, 2, 3)
parval0 = None
parval1 = None
levels = None
sherpa> reg_proj(p1.gamma, abs1.nH, sigma=[1, 1.6])
The resulting plot is shown in Figure 4.
Figure 4: Plot of region-projection results
sherpa> print(get_reg_proj())
x0 = [1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127,
1.1367,1.2118,1.2869,1.362 ,1.4371,1.5122,1.5874,1.6625,1.7376,1.8127]
x1 = [1.7772,1.7772,1.7772,1.7772,1.7772,1.7772,1.7772,1.7772,1.7772,1.7772,
2.309 ,2.309 ,2.309 ,2.309 ,2.309 ,2.309 ,2.309 ,2.309 ,2.309 ,2.309 ,
y = [143.7825,145.3247,155.4125,174.083 ,201.2284,236.5969,279.7989,330.3201,
224.294 ,265.4248,313.5139,367.9944,147.9872,135.0252,129.4146,131.3832,
178.5517,155.3186,138.257 ,127.6825,123.8383,126.8861,136.8982,153.8533,
177.6349,208.033 ,200.0785,172.9645,151.4788,135.9554,126.6704,123.8318,
133.0762,139.2095,307.2241,270.4337,237.5076,208.7693,184.521 ,165.0362,
min = [1.1366511 1.77720269]
max = [1.81270991 2.73452195]
nloop = (10, 10)
fac = 4
delv = None
log = [False False]
sigma = [1, 1.6]
parval0 = 1.4746805027612861
parval1 = 2.2558623214598748
levels = [125.91698772 128.04310045]
The automatically-chosen limits have resulted in a poor-quality plot: there are not enough data points close to the best-fit location. The easiest way to improve on this is to change and re-run the
function, increasing the number of points. We also elect to use a smaller parameter range along both axes to reduce the amount of wasted computation. In a complex case with a larger grid, it may be
worthwhile to manually set the limits before running reg_proj, since it may take longer to create a plot.
sherpa> reg_proj(p1.gamma, abs1.nH, min=[1.2, 1.9], max=[1.8, 2.6], nloop=[51, 51], sigma=[1, 1.6])
The resulting plot is shown in Figure 5, which is a smooth contour plot.
Figure 5: Improved region-projection results (nloop=[51, 51])
When we utilize print(get_reg_proj()), we are given information on the most recent confidence contour plot produced by reg_proj. The min and max lists of grid boundaries are calculated automatically
from the covariance when they are set to the default "None". However, they may be set manually as min=[x[min],y[min]] and max=[x[max],y[max]]. nloop is a list of bin sizes of the x- and y-axes, which
by default are used with the min and max grid boundaries to determine the x- and y-axes step sizes, given as the list, delv. The parameter of most interest for reg_proj is sigma, which is a list of
the number of σ; at which to plot contours. The levels parameter is subsequently determined after executing reg_proj, which are the confidence level z-values for each σ.
Log-space for int_proj and reg_proj
In the current version of Sherpa, the log parameter should be left at its default value of False in int_proj and reg_proj, as the tools do not properly scale plots with logarithmic spacing.
The reg_unc (region-uncertainty) command behaves similarly, although the fields in the state object for the two methods are different. The two commands differ in that reg_unc fixes all other thawed
parameters to their best-fit values, rather than being allowed to float to new best-fit values as in reg_proj. This makes reg_unc contours less accurate, but quicker to create.
Scripting It
The file fit.py is a Python script which performs the primary commands used above; it can be executed by typing %run -i fit.py on the Sherpa command line.
The Sherpa script command may be used to save everything typed on the command line in a Sherpa session:
sherpa> script(filename="sherpa.log", clobber=False)
(Note that restoring a Sherpa session from such a file could be problematic since it may include syntax errors, unwanted fitting trials, et cetera.)
14 Jan 2005 reviewed for CIAO 3.2: no changes
21 Dec 2005 reviewed for CIAO 3.3: no changes
01 Dec 2006 reviewed for CIAO 3.4: no changes
02 Dec 2008 reviewed for CIAO 4.1: updated syntax for CIAO4.1
29 Apr 2009 new script command is available with CIAO 4.1.2
21 Jan 2010 updated for CIAO 4.2: the conf command is available
13 Jul 2010 updated for CIAO 4.2 Sherpa v2: removal of S-Lang version of thread.
15 Jul 2010 updated to include information about warning messages returned by the confidence method.
03 Sep 2010 figures moved inline with text
30 Jan 2012 reviewed for CIAO 4.4 (no changes)
13 Dec 2012 reviewed for CIAO 4.5 (no changes)
11 Dec 2013 reviewed for CIAO 4.6: updated formatting, content unchanged
18 Mar 2015 reviewed for CIAO 4.7; updated plots and fixed typos, no content change.
10 Dec 2015 reviewed for CIAO 4.8; no content change.
03 Nov 2016 reviewed for CIAO 4.9; updated outputs and fixed typos.
01 Jun 2018 reviewed for CIAO 4.10; no content change.
01 Jun 2018 reviewed for CIAO 4.11; updated screen output and added information about the confidence interval table.
12 Dec 2019 Updated for CIAO 4.12: switched from ChIPS to Matplotlib for plotting.
22 Dec 2020 Updated for CIAO 4.13: use new plot style for PHA data
31 Mar 2022 reviewed for CIAO 4.14, add clarification on delchi.
05 Dec 2022 reviewed for CIAO 4.15, updated screen output. | {"url":"https://asc.harvard.edu/sherpa/threads/confidence_manual/","timestamp":"2024-11-03T00:04:32Z","content_type":"text/html","content_length":"58467","record_id":"<urn:uuid:d6f8ed1d-1cab-44bb-a5e8-2b903cd6d6ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00172.warc.gz"} |
Frontiers | Machine learning-based personalized composite score dissects risk and protective factors for cognitive and motor function in older participants
• ^1Department of Psychological Medicine and Clinical Neuroscience, UK Dementia Research Institute, Cardiff University, Cardiff, United Kingdom
• ^2Department of Neurodegeneration and Hertie-Institute for Clinical Brain Research, Center of Neurology, University of Tübingen, Tübingen, Germany
• ^3German Center for Neurodegenerative Diseases (DZNE), University of Tübingen, Tübingen, Germany
• ^4Department of Neurology, Kiel University, Kiel, Germany
• ^5Department of Psychiatry, University of Tübingen, Tübingen, Germany
• ^6Geriatric Centre at the University Hospital Tübingen, Tübingen, Germany
• ^7Institute for Computer Science, University Göttingen, Göttingen, Germany
• ^8Institute Bioinformatics and Medical Informatics (IBMI), University of Tübingen, Tübingen, Germany
Introduction: With age, sensory, cognitive, and motor abilities decline, and the risk for neurodegenerative disorders increases. These impairments influence the quality of life and increase the need
for care, thus putting a high burden on society, the economy, and the healthcare system. Therefore, it is important to identify factors that influence healthy aging, particularly ones that are
potentially modifiable through lifestyle choices. However, large-scale studies investigating the influence of multi-modal factors on a global description of healthy aging measured by multiple
clinical assessments are sparse.
Methods: We propose a machine learning model that simultaneously predicts multiple cognitive and motor outcome measurements on a personalized level recorded from one learned composite score. This
personalized composite score is derived from a large set of multi-modal components from the TREND cohort, including genetic, biofluid, clinical, demographic, and lifestyle factors.
Results: We found that a model based on a single composite score was able to predict cognitive and motor abilities almost as well as a classical flexible regression model specifically trained for
each single clinical score. In contrast to the flexible regression model, our composite score model is able to identify factors that globally influence cognitive and motoric abilities as measured by
multiple clinical scores. The model identified several risk and protective factors for healthy aging and recovered physical exercise as a major, modifiable, protective factor.
Discussion: We conclude that our low parametric modeling approach successfully recovered known risk and protective factors of healthy aging on a personalized level while providing an interpretable
composite score. We suggest validating this modeling approach in other cohorts.
Neuropsychiatric diseases are currently the leading cause of disability and dependency worldwide. Among them, the neurodegenerative diseases Parkinson’s disease (PD) and Alzheimer’s dementia (AD) are
rising the fastest (Dorsey et al., 2018). Aging represents one of the strongest risk factors for both diseases. Predictions indicate that the prevalence will double worldwide in the next 20years (
Ferri et al., 2005). Since there is considerable diversity in the rate at which we age, the identification, and effect size of risk and protective factors that indicate the dynamic processes from
aging to neurodegeneration is of high interest. While some factors, such as sex and genetic status, are immutable, a large proportion can be influenced by lifestyle. These factors include
cardiometabolic, physical, and educational profiles (Mukadam et al., 2024; Cova et al., 2017; Livingston et al., 2020). This offers the opportunity to focus on these factors for preventive healthcare
strategies. However, many of these factors are interdependent. Moreover, the heterogeneity of human subjects and intervals of data collection in longitudinal studies make it difficult to extract
suitable data for robust statistical predictions. Conventional statistical methods cannot accommodate these complex relationships. Therefore, we used an unbiased machine learning approach by
developing a Bayesian model to simultaneously predict aging-related key functions such as motor and cognitive function from a single composite score that reflects a large set of multi-modal factors,
including genetic, biofluid, clinical, demographic, and lifestyle factors.
Similar models have been used in high-dimensional medical settings using imaging or genetic data (Goh et al., 2017) and also to investigate dietary patterns by fat types (Brayner et al., 2021).
However, such methods have not been used in multi-modal settings assessing aging- and neurodegeneration-related profiles. Importantly, we primarily focused on factors that were already identified in
epidemiological and genetic studies by standard statistical approaches in order to facilitate a proof-of-concept for the Bayesian model.
Study population
We used the data from the TREND study (Gaenslen et al., 2014) which is a prospective longitudinal study initiated in 2009 with biennial assessments of older participants aged between 50 and 80years
without neurodegenerative diseases at study recruitment. Newspaper announcements and public events were used to recruit participants from Tübingen and the surrounding area. Between 2009 and 2012,
1,201 participants underwent baseline assessments. For study inclusion, participants had to be free of a diagnosis of a neurodegenerative disorder, history of stroke, inflammatory disorders affecting
the central nervous system (such as multiple sclerosis, encephalitis, meningitis, vasculitis), and inability to walk without aids. The study has been performed at the Department of Neurology and the
Department of Psychiatry of the University Hospital Tuebingen, Germany. A large assessment battery with quantitative, unobtrusive measurements for repeated objective application was designed. To
avoid bias in data acquisition, all investigators were blinded to the results of all other examinations. For more details about the TREND study please visit https://www.trend-studie.de/.
Supplementary Figure S1 summarizes the exclusion criteria and selection of participants for the current analysis.
Clinical investigations
Motor function: gait
For the assessment of motor function, we decided to focus on gait as representative of axial motor performance which is key for maintaining independence in older participants. Gait assessments were
performed in an at least 1.5 meters wide corridor allowing obstacle-free 20-meter walking. All subjects performed four single-task conditions: 1. walking with habitual speed, 2. walking with maximum
speed, 3. checking boxes with maximum speed while standing, and 4. subtracting serial 7s with maximum speed while standing. Additionally, two dual-task conditions were performed: 1. walking with
maximum speed and checking boxes with maximum speed and 2. walking with maximum speed and subtracting serial 7s with maximum speed (Hobert et al., 2011).
Based on the two dual-task conditions, we extracted the respective four dual-task speeds: 1. checking boxes when walking (number of boxes per second), 2. walking when checking boxes (meters per
second), 3. subtracting when walking (number of serial 7s subtractions per second), 4. walking when subtracting (meters per second). The single and dual-task speed parameters were then used to
calculate dual-task costs and overall speed according to the following formulae:
• Overall speed: dual-task speed + single-task speed
• Dual-task cost: dual-task speed – single-task speed
A detailed assessment of cognitive function was implemented using the standardized German version of the extended Consortium to Establish a Registry for Alzheimer’s Disease (CERAD)-Plus
neuropsychological battery (Morris et al., 1989; Rossetti et al., 2010). This comprehensive battery includes the following cognitive subtests: semantic and phonematic verbal fluency tasks, the Boston
Naming Test, Mini-Mental Status Examination, word list learning, word list recall, word list recognition, figure drawing, figure recall, and the Trail Making Test (TMT) A and B (Welsh-Bohmer and
Mohs, 1997; Ehrensperger et al., 2010). The TMT consists of two parts and evaluates executive function, cognitive flexibility, and working memory (Bowie and Harvey, 2006). In part A, participants
connect randomly spread numbers from 1 to 25 in ascending order. In part B, participants are asked to connect randomly spread numbers () and letters (A to L) in alternating numeric and alphabetical
order (1-A-2-B-3-C-…-13-L). In case of an error, the examiner draws the attention of the participant to the error, to allow completion of the task without errors at the expense of additional time.
The maximum time allowed is 180s for part A and 300s for part B. After this time, the investigator discontinues the experiment. Two parameters were calculated from the TMT A and TMT B tests:
• Overall speed: TMT A+TMT B
• Cognitive flexibility: TMT B – TMT A
Next, to the CERAD total score, subscores of the different CERAD domains were included in the analysis. Ordinal variables were measured on a Likert scale and indicated the number of items completed
Medical condition and lifestyle
Lifetime diagnosis of hypertension (medical history) and/or intake of anti-hypertensive medication was defined as the presence of hypertension.
Body mass index (BMI) was calculated by: mass [kg]/(height [m])^2.
Body composition (fat/skeleton muscle mass)
Body composition was assessed by bioelectrical impedance analysis using a body impedance analyzer (BIA 101, Akern, Germany) for two out of four visits. Therefore, ohmic resistance was measured
between the dominant hand wrist and dorsum and the dominant foot angle and dorsum in the supine position. Muscle mass in kg was then calculated according to Janssen et al. (1985) and subsequently
normalized to subjects’ body height squared (skeletal muscle index: SMIBIA): with body height in centimeters, resistance in Ω, for gender: male=1 and female=0, and age in years.
Assessment of physical activity
Physical activity was assessed by a self-administered questionnaire. This questionnaire is part of the Bundes-Gesundheits Survey (national health survey) and allows to rate physical activity between
0 and 4 (0=no activity, 1=0.5–1h per week, 2=1–2h per week, 3=2–4h per week, 4=more than 4h per week) (Mensink, 1999).
Smoking and drinking
Personal history of smoking and alcohol-drinking behavior was assessed by a self-administered questionnaire. Pack-years were calculated by quantifying the packs (20 cigarettes/pack) smoked per day
multiplied by years as a smoker. The frequency of drinking alcohol was assessed on a scale from 0 to 4, which indicates the number of drinks per month.
Genetic risk factors for Parkinson’s disease and Alzheimer’s disease
Pathogenic variants in LRRK2 and GBA are the most common PD-associated genes. DNA was isolated from EDTA blood by salting out and stored at 4°C. All participants were analyzed by NeuroChip.
Pathogenic variants in LRRK2 and GBA were confirmed by Sanger sequencing. None of the participants carried a LRRK2 mutation. Fifty-seven participants carried a GBA variant. We further grouped those
according to known PD-specific mutation severity: wild type (0), low risk (Dorsey et al., 2018), and mild/severe (Ferri et al., 2005). Moreover, the most relevant single-nucleotide polymorphisms in
genes for PD (SNCA rs356220 or proxy rs356219) and AD (ApoE, MAPT) were investigated to explore the effect on motor and cognitive function. We grouped the number of risk alleles according to an
additive model: SNCA rs356220 (or proxy rs356219) minor allele C (0, 1, 2), ApoE4 allele (0, 1, 2), and MAPT haplotype (H1/H1, H1/H2, H2/H2).
Measurement of neurofilament light chain in blood
Neurofilament light (NFL) chain protein is an unspecific biofluid marker that reflects the extent of neuronal/axonal damage. Blood samples were collected on the day of the study visit, cooled,
centrifuged (4°C, 10min, 2000g), aliquoted, and stored at −80°C within 4h after collection. They were analyzed without any previous thaw–freeze cycle. Serum levels of NFL as a marker for
neuronal-axonal damage were measured in duplicates using the SIMOA NF-light KIT (Quanterix, Product number: 103186) on the SIMOA HD-1 Analyzer (Quanterix, Lexington, MA) as established previously (
Kuhle et al., 2016). Technicians were blinded to all other tests of the participants.
Definition of age-related key functions and model overview
The aim was to simultaneously predict aging-related key functions of motor and cognitive performance from a large set of multiple multi-modal factors including genetic, biofluid, clinical,
demographic, and lifestyle factors (Supplementary Table S1).
As outcome measures for motor function, we defined the different gait conditions:
• Overall speed: dual-task speed + single-task speed
o Walk while subtract serial 7s dual + single walk
o Subtract serial 7s while walk dual + single subtract serial 7s
o Walk while cross boxes dual + single walk
o Cross boxes while walk dual + single cross boxes
• Dual-task cost: dual-task speed – single-task speed
o Walk while subtract serial 7s dual—single walk
o Subtract serial 7s while walk dual—single subtract serial 7s
o Walk while cross boxes dual—single walk
o Cross boxes while walk dual—single cross boxes
As outcome measures for cognitive function, we used the CERAD and defined the different CERAD subdomains:
• Overall cognitive function: Total score
• Memory function: Word list learning and word list recall
• Executive function: TMT A+B and TMT B—A
All scores were transformed such that higher values reflect worse performance by flipping the scale.
Our goal was to develop a Bayesian RRR model that simultaneously predicts all motor and cognitive outcome measures from a single composite score extracted as a linear combination of lifestyle and
genetic factors and compare this model to conventional statistical approaches (Supplementary Table S1). This restricts the flexibility of the model but increases its ability to identify a key feature
extractor by using several prediction targets. Figure 1 illustrates the rationale for Bayesian RRR compared to classical multivariate linear regressions.
Figure 1
Figure 1. Schematic comparison of multivariate regression and reduced rank regression. The number of coefficients to learn (number of arrows) is illustrated for (a) multivariate regression and (b)
reduced rank regression. The latent variable $θ$ , the composite risk is a linear combination of the predictors and is projected via linear multiplications to the targets.
We used python 3 (3.8.2) in combination with the probabilistic modeling library PyMC3 (3.9.3) (Salvatier and Fonnesbeck, 2016) to implement the Bayesian RRR model. The ordinary least squares (OLS)
models were implemented with statsmodels (0.11.1) (Seabold, 2010). Model evaluation was performed using scikit-learn (0.23.1) (Pedregosa et al., 2011). We used datajoint (0.12.6) (Yatsenko et al.,
2015) to build our data processing pipeline.
Handling missing data
Missing data of predictor variables were handled in the same way for both models. After subject and visit exclusion as detailed in Supplementary Figure S1, we assessed the missingness of the
predictors across all remaining visits. The percentage of missingness for time-varying predictor variables can be found in Supplementary Table S1. To increase the amount of available data points, we
performed imputation. We did so only for subjects with at least one value available for each predictor. Based on the assumption that predictors only change when a new value is given, we first applied
forward filling and then backward filling.
Reduced rank regression model and Bayesian reduced rank regression
Our reduced rank regression model is based on the observation that the outcome measures/clinical tests are correlated and thus can be represented through a smaller set of latent variables. Therefore,
we used an RRR model that allows us to predict multiple response variables from the same set of predictor variables while reducing the amount of model parameters (Figure 1). RRR can be seen as a
multivariate regression model with a coefficient matrix of reduced rank (Velu, 2013). RRR is a computationally efficient method that increases statistical power in settings where the number of
dimensions is large compared to the number of examples. In such m≫j settings, RRR is nowadays a state-of-the-art method in fields with high-dimensional data, such as genetics and imaging (Zhu et
al., 2019; Kobak et al., 2021). Given j observations of m predictors and n outcome measures, standard multivariate regression requires fitting m n coefficients Y=XC+E, with Y being the response
matrix of size j×n, X being the j×m predictor matrix, C the m x n coefficient matrix and E being the error term matrix of size j×n. The RRR is obtained by adding a rank constraint rank(C)=k,
k≤min (n,m). The rank constraint decreases the dimensionality of the model and improves the statistical power. Using the rank constraint, C can be rewritten as C=AB^T, with A of size m×k and B
having size n×k. Hence, the model can be expressed as Y=(XA) B^T +E.
This decomposition allows for interpretations of A and B. A is a mapping from the predictor matrix X to a latent representation of dimension k. B is a mapping from the latent scores to the responses
Y. The latent scores XA display the low-dimensional predictor variability that is predictive of the response variability.
We used Bayesian inference for our RRR model to obtain parameter uncertainty and handle missing data. This means that given observations X and Y, we sampled model parameters Ψ from the posterior
distribution p(Ψ|X,Y) ∝ p(Y|Ψ,X)·p(Ψ). The variable Ψ generically denotes all parameters of the model. The Bayesian framework requires a prior distribution p(Ψ) that embodies our prior knowledge
about these parameters and the behavior that we want the model to exhibit. We are specifying these choices in the following paragraphs.
Least squares regression models can easily be transformed into Bayesian models by rewriting the model as Y∼N(XAB^T,σ^2), where N(μ,Σ) denotes the normal distribution with mean μ and covariance Σ.
Given a high-dimensional data setting, it is likely that some of the predictors are non-informative for some of the outcome measures. A Laplace prior to A can realize this desired sparsity as it
promotes element-wise sparsity: Certain elements in A are set to 0, resulting in a latent composite score depending on certain predictors but not on others.
Suppose we have a predictor matrix X which holds information about $m$ predictors (time-varying and static) for j visits of subjects. Through A those are mapped to the latent space of size k, such
that we obtain k composite scores for each visit, θ. For visit i, we thus get for composite score f: θ[if] =∑[l=1]^m X[il] A[lf]. A priori, each element of A is sampled from a Laplace distribution
Laplace(x,μ,b)= $1 2 b exp − x − μ b$ with b=1 to enforce element-wise sparsity. The matrix B maps back from the latent space to the response space. A lognormal distribution ln(e^μ+σZ) with σ=
0.25 is used prior to enforcing the positivity of the coefficients. By centering the real responses prior to learning, an offset can be omitted. For visit i we get a prediction for the response o
via Y[io] = θ’[io]B[o] for which we assume a Gaussian observation noise with σ=0.908, which is informed by the MSE on the training data of the multiple OLS models. We trained a model with k=1,
for which we present the results in the main text, but we also trained a model with k=2 to check how this increase in complexity improves the performance. We further trained a model with k=1 and
a deterministic B=1 to check whether our Bayesian model with similar complexity to the OLS models performs as well as those. The results can be found in Supplementary Table S3.
Ordinal predictors
We further improved our Bayesian RRR model through the way it handles ordinal predictors. Ordinal variables are commonly used in clinical settings. However, in most modeling approaches, they are
encoded as either nominal or interval variables. The former disregards the ordering information, and the latter assumes regular spacing, which may not be given. To correctly use ordinal predictors,
one can use monotonic effects (Burkner and Charpentier, 2020) (Supplementary Figure S2). This transformation ensures a monotonic increase or decrease, while adjacent categories can be arbitrarily
spaced. For an ordinal predictor x taking values x[n]∈{0,…,D} a monotonic transformation is defined as $m o : 0 … D → 0 D , x n → m o x n ζ = D ∑ i = 1 x n ζ i$ where ζ is the element of a simplex,
meaning it satisfies $∑ i = 1 l ζ i = 1$ and ζ_i∈[0,1]. It can be interpreted as the normalized distances between adjacent categories. As D can be absorbed into the regression coefficients A and lead
to redundancies, we instead encoded ordinal variables in our model with: $c m o x n ζ = ∑ i = 1 x n ζ i .$
This still ensures a monotonic transformation with arbitrary spacing; however, the effect and sign will be inferred through the regression coefficient. In our Bayesian RRR model, we chose a Dirichlet
prior for the ζ[i] as it is the natural choice for a prior on simplex parameters. By choosing a constant α=1, we effectively used a uniform (equal probability) prior to the probability simplex,
i.e., all vectors ζ that sum to one are equally likely. The a priori expectation of ζ is given by $w i = E ζ i = α i ∑ i = 1 D α i$ . With α=1, we have $w i = 1 D$ . This prior centers the category
distances, ζ around a linear trend but allows for high variations around this. This transformation was applied to the ordinal predictors in X prior to the RRR.
We decided to model the genetic data as ordinal predictors as well. The monotonic transformation allows us to consider dominant (0 vs. 1), additive (0 vs. 1 vs. 2), as well as recessive (0 vs. 2)
effects simultaneously.
Model comparison
To compare the predictive performance of the Bayesian RRR model against a more flexible and traditional approach, we trained 13 OLS models that each predict a single outcome measure from the set of
predictors. To handle the longitudinal data, we decided to include all available visits of each subject, thereby having subjects unequally represented in the dataset. The models thus treat each visit
as an independent data point, disregarding the correlation arising from repeated measures of the same subject. As the outcome measures have variable availability (Supplementary Table S2), for each
model, the valid visits to include for training were selected separately in order to maximize the number of overall data points. For each outcome measure, we kept all data points where the outcome
measure itself was available. We thus trained the OLS models on differently sized datasets. In contrast, the Bayesian RRR model was trained on all targets simultaneously. Due to the nature of the
Bayesian framework, we can include data points where parts of the outcome measures are missing. Thereby the entire Bayesian RRR is trained on the union of the datasets for the OLS models but for each
outcome measure, only the same visits as for the corresponding OLS model are used for training. For the OLS models, we decided to use dummy encoding for nominal and ordinal predictors and include an
intercept term. For a predictor with $n$ categories, we thus included $n − 1$ coefficient in the model. Real-valued predictors were standardized. All models were fit using mean-squared error loss. We
performed 5-fold cross-validation using 20% as test and 80% as training data, ensuring that visits of the same subject are grouped into either one. For each fold, the outcome measures and real-valued
predictors were standardized on the training set. As no hyperparameters (values that we set to control the learning process) were learned, cross-validation yielded a measure of uncertainty for the
prediction performance from the 5-folds. All performance evaluations and comparisons were conducted through this cross-validation. We retrained all 13 OLS models on their respective complete datasets
(training and test) to obtain the final coefficients for the predictors.
Sampling from the posterior
We sampled from the posterior p(Ψ|X,Y) through NUTS sampling (Hoffman and Gelman, 2014) with two chains, each with a burn-in of 2000 samples and 500 retained samples. We thus obtained 1,000 samples
from the posterior distributions of each parameter. Subsequently, we obtained the posterior predictive distribution by feeding the samples through the generative model: p(Y|X)=∫ p(Y|Ψ)p(Ψ|X)dΨ.
These predictions were used for performance evaluation. To assess the generalization of our model, we performed a 5-fold cross-validation where for each split the data were randomly split into a test
(20%) and train (80%) set, ensuring that each subject is only in either one. We retrained the model using the whole dataset to obtain the final posterior distributions for the coefficients.
Performance evaluation
We used the coefficient of determination R^2 to compare our model performances. Suppose we have n data points with y[i] being the true value for visit i, y ̂[i] being our predicted value, and y^−
being the mean of the true values. R^2 is defined as $R 2 = 1 − ∑ i = 1 n y i − y ̂i 2 ∑ i = 1 n y i − y − 2 .$
It is 1 for perfect prediction and 0 when the mean is predicted. Note that, R2 can be negative if the prediction is worse than the mean, i.e., the constant predictor. We calculated R^2 as the
standardized mean-squared error (MSE): $R 2 = 1 − M S E v a r y$ where y are the true values. To make the measure more robust, we decided to normalize the MSE by the variance of the whole dataset,
i.e., train and test set. This better captures the true variation regardless of the applied train/test split. We compared the performance over the 5-folds of the OLS models and the Bayesian RRR model
for each clinical test with a t-test.
We evaluated the significance of predictors of the multiple linear regression models with t-tests and a type-I error threshold for a p-value of 0.05 that is corrected by the number of tests performed
using Bonferroni correction. For our Bayesian RRR model, where we obtained posterior samples for our coefficients, we calculated the highest posterior density (Turkkan, 1993) (95%) and defined
significance as this interval not crossing 0.t.
Model performance
The Bayesian RRR composite score model achieved comparable performance to classical linear regressions (OLS) per clinical outcome measure (Figure 2). It showed similar levels of explained variance
for all cognitive outcome measures (Supplementary Table S3). Separate linear regressions significantly outperformed the composite score model in the four gait-related outcome measures (walk while
cross dual – single, subtract while walk dual – single, walk while subtract dual + single, walk while cross dual + single), Figure 2. Our composite score model performed on par with linear
regressions in predicting the gait-related outcome measure “Subtract while Walk Dual + Single” which is the gait measure that has the highest cognitive load as it measures the speed of mathematical
calculations while walking. Overall, the composite score model performed well on cognitive outcome measures, on par with the OLS models, and worse on gait-related measures. This indicates that a
composite score RRR model with a single explaining factor performs almost equally well as several individual OLS models.
Figure 2
Figure 2. Performance comparison of Bayesian RRR and multiple OLS. The mean 5-fold CV R^2 on the test sets is shown for each of the outcome measures. Error bars denote the 95% confidence interval
across 5-folds. For the Bayesian RRR, we show the mean across 5-folds of the mean R^2 over 1,000 samples and the corresponding 95% confidence interval across 5-folds.
To assess whether the worse performance in the four gait-related measures is due to the reduced complexity of our model or due to other model specifications, we trained a separate Bayesian RRR for
each outcome measure. These performed as well as the OLS models on all tasks (see A6).
All models recover known protective and risk factors
Multiple linear regressions
The multiple linear regressions showed an overall agreement for the effect direction, i.e., whether a factor is a risk or a protective factor (Figure 3). Factors identified as protective are female
sex, longer time in education, and a higher level of physical fitness (hours of exercise per week, higher skeleton muscle mass). Only for some movement speed-related outcome measures (walk while
subtract/cross dual—single) female sex negatively impact the outcome measure (i.e., decreased performance). Risk factors that were in agreement between the majority of the OLS models are older age, a
higher number of cigarette pack-years, and a higher BMI.
Figure 3
Figure 3. Regression coefficients of multiple linear models. The influence of the predictive factors (x-axis) on the outcome measures (y-axis) is shown. The color indicates the size and direction of
the effect (protective=blue, risk=red), with the size showing the importance (abs(coefficient)/standard error) and a black outline indicating significance (Bonferroni-corrected p-threshold 0.05).
Bayesian reduced rank regression composite score model
The composite score Bayesian RRR model merged this overall agreement of the OLS models into one composite score (Figures 4, 5). In addition to the identified factors from the OLS models, the Bayesian
model identified hypertension, ApoE4 genotype, and higher NFL values as significant risk factors. The number of relatives with PD or dementia was not significant in any OLS model but was identified
as a significant protective factor in our composite risk model. The Bayesian RRR further identified genetic variants in GBA and MAPT (H2 haplotype) as protective factors.
Figure 4
Figure 4. Composite score model The composite score model recovers the overall agreement of the coefficients across the multiple OLS models. The color indicates the direction and size of the effect
of a predictor (x-axis) on a target (y-axis). The size of the square indicates its importance as the absolute ratio of mean and standard deviation (the larger the further away from 0).
Figure 5
Figure 5. Regression coefficients for the composite score model. The estimated effect sizes of the predictors on the composite score are displayed. The highest posterior density is plotted. The
coloring indicates significance (95% highest posterior density contains 0, not significant=gray) and direction of the effect (blue=protective, red=risk).
Monotonic transformation reveals a proportional effect on exercise
The encoding of ordinal factors in our model allowed for fine-grained information on their effect on the composite score not addressed by a nominal encoding as used in the OLS models. By learning the
distance between the categories of ordinal predictors through monotonic transformation, we obtained flexible spacing of the different levels with additional meaning (Figure 6). We saw a steep
reduction of risk for people who drink at least two drinks per month but increasing the number of drinks did not further reduce the risk substantially. In contrast, for physical exercise, we observed
no such saturation and can conclude that more exercise is more protective. We also note a steep increase in risk for carriers of 2 ApoE4 alleles compared to carriers of one allele. Contradictorily,
heterozygous carriers of mild and severe GBA variants seem to be more protected than those with GBA wildtype.
Figure 6
Figure 6. Flexible spacing of ordinal predictors. For each ordinal predictor, we show the distance between the categories modeled through a monotonic transformation in the composite score model. We
plot the learned distances multiplied by the predictor’s effect size (A). The mean of the samples alongside the highest posterior density interval (95%) is shown.
We analyzed the combined influence of a large set of multi-modal factors, including environmental, lifestyle, biofluid, and genetic data, on aging-related key functions, cognitive, and gait
performance, measured by multiple clinical tests. To this end, we compared two approaches: independent prediction of each outcome measure with a linear regression model (OLS) and joint prediction of
all outcome measures from one composite score learned by a Bayesian RRR model. We could show that the predictive performance of the Bayesian RRR model with one single composite score was comparable
to classical multiple OLS models. The most relevant factors that showed a protective effect on complex gait and cognitive abilities in older participants included female sex, a higher degree of
physical activity, more skeletal muscle mass, and more years of education. Contrary, higher age, body mass index and more smoking pack-years, the presence of hypertension, having two ApoE4 alleles,
and higher serum levels of NFL were predictors for impaired gait and reduced cognitive performance.
We primarily included well-known risk and protective factors to check the validity of the composite modeling. However, some factors showed an influence in an unexpected direction. For example, the
number of relatives with dementia or PD should serve as a proxy of the genetic risk and thus be a risk factor. Our model as well as the OLS models instead revealed a protective effect. This could be
due to the high motivation of individuals with a family member affected by a neurodegenerative disease, as they have an increased personal interest in performing well and taking care of one’s health.
However, we did not directly measure motivation. Such motivational influences were not measured directly within the TREND study, but the literature supports this assumption (Soule et al., 2016).
Carriers of mild and severe variants in the GBA gene had a reduced risk according to our model, which could be a reflection of these carriers being significantly younger than the other groups (mild
and severe GBA variant carriers (N=30) vs. wildtype (N=4,294): t-statistic=−3.12, p-value=1.8e-3; mild and severe GBA variant carriers (N=30) vs. low-risk GBA variant carriers (N=166): t
-statistic=−2.54, p-value=1.2e-2).
Our modeling of the ordinal predictors allowed for interpretations of the effect sizes of each category. For example, ApoE4 is a well-known genetic risk factor for cognitive decline, with carriers of
one allele having an odds ratio of approximately 3 for developing AD and carriers of two alleles having an odds ratio of approximately 15 (Farrer et al., 1997). This steep increase in risk for
carriers of two alleles was replicated in our model despite the small sample size (11 persons with two ApoE4 alleles and 193 with one ApoE4 allele). We investigated how much complexity is needed to
achieve similar performance to the OLS models in all outcome measures by training a model with two composite scores and by training separate Bayesian models for each outcome measure. Increasing the
latent space and allowing for two composite scores slightly improved the performance for a subset of the clinical tests, albeit not significantly. This mainly affected gait outcome measures and
revealed distinct effects of factors for different clinical tests (Supplementary Table S3). For example, female sex was identified as a risk factor for walking speed in general but a protective
factor across all cognitive tests (Figure 3). This effect was found in the OLS models and the two composite scores models (Supplementary Figures S3, S4) alike and could reflect the height difference
and thus step size differences between males and females. This was not measured within the TREND study and can thus not be corrected for. The single composite score model thus prioritized the
cognitive measures over the gait-related measures, leading to a composite score that performs well on cognitive measures and worse than the OLS models on the gait measures. The good performance of
the single composite score model to predict the gait measure “Subtract while Walk Dual + Single” might be explained by the high cognitive load of this task. It might thus be better represented by a
cognitive composite score.
Our approach of a joint prediction of all outcome measures from one composite score learned by a Bayesian RRR model performed comparably well to the more classical flexible individually fitted
regression models. This suggests that already one composite score can capture a substantial part of the complex effects on cognitive and motor function in an aging cohort. This finding is in line
with recent studies. Data-driven techniques applied in archival clinical datasets may outperform classical models and could enhance diagnostic procedures in regions with limited resources (Maito et
al., 2023; Javeed et al., 2023). Our model unbiasedly identified known risk and protective factors of aging.
The fact that our less flexible model performed comparably well to individual OLS models indicates that the explored factors either share similar mechanistic pathways and/or are interrelated to each
other. This further highlights a global underlying risk for aging processes where motor and cognitive abilities are affected alike.
Our Bayesian RRR has several strengths compared to traditional approaches, such as its handling of missing data, its reduced complexity and thus its interpretability, and its handling of ordinal
predictors. In Bayesian models, missing values in the outcome measures can be imputed through the model’s parameter estimates. As we can include incomplete data in the model, we increased the total
amount of data the model uses but did not alter the data distribution artificially by learning from imputed outcome measures (covariates were imputed). Through our assumption of a composite risk, we
decreased the complexity of the model and thus made it more scalable and better suited for medical data, which are scarce and high-dimensional. This assumption of a low rank further increased the
model’s interpretability, as the single composite score can be interpreted as an estimate of the true underlying risk. The modeling of the ordinal predictors better captured the true scale level of
the data, indicating that physical exercise has an additive effect.
We acknowledge the following limitations: (1) Our Bayesian RRR model does not test for causality but merely identifies associations between the included predictors and outcome measures. (2) Our
Bayesian RRR model currently assumes a linear relation between the predictors and outcomes, although this is not necessarily true. For example, it would be reasonable to assume that drinking a small
amount of alcohol could have a protective effect, but excessive drinking could be a risk factor for cognitive performance. Such reverse effects cannot be captured through our linear models. Using a
quadratic link function could better account for such scenarios, other non-linearities could be further explored through neural networks. Drinking two drinks might be a confounding fitness factor, as
many seniors avoid drinking if they are multimorbid or take multiple medications. (3) Another limitation is the handling of longitudinal data, where we treat visits from the same individual as
independent. This disregards the correlation within a subject. A potential improvement of our model could be the adaptation of a mixed model where visits are grouped by individuals and identified
through a subject-specific identifier, i.e., random effects. This would further allow us to make statements about a subject’s temporal slope. A different approach to modeling the longitudinal data
would be a stacked model where a linear mixed model first learns the trajectory over time for each subject, and this estimated change over time is then used as the outcome measure in our Bayesian
RRR. (4) While our Bayesian RRR model reduces complexity, it may not capture all nuances and interactions between predictors as effectively as more flexible models. Additionally, there is a trade-off
between interpretability and performance, which might lead to overlooking some significant interactions and non-linear relationships between variables. (5) While the model is designed to be
generalizable, its performance and findings are based on this single cohort. Therefore, a validation in further worldwide cohorts is necessary (Santamaria-Garcia et al., 2023; Ibanez et al., 2024).
(6) While the overall number of genetic risk carriers (GBA, SNCA, and MAPT) was in the expectant range of the known prevalence, this sample size is too small to robustly recover their effect on
age-related functions.
Currently, our model achieved similar performance to multiple OLS models, however, several adaptations could be explored to improve our model’s performance. As our model requires fewer parameters, we
could increase the number of predictors and targets without the need to increase the sample size. Especially exploring the effect of various aging-related genetic markers and their interaction could
be a promising future project. A similar model has been used before for modeling genotype–phenotype associations (Goh et al., 2017); however, they did not use monotonic transformations but instead
assumed additive effects for the SNPs.
We conclude that our low parametric modeling approach successfully recovered known risk and protective factors of healthy aging on a personalized level while providing an interpretable composite
score. An extension of this model using more predictors and clinical tests could further identify unknown factors and distinct aging-related processes. To this end, more sensitive tests are needed to
better capture the variation with a healthy cohort. Digital sensors such as wrist-worn acceleration devices could provide such sensitive data. The modeling approach is generalizable and could also be
applied to other cohorts to investigate the complex interplay of risk and protective factors along with effect sizes from different dimensions such as lifestyle, medical, genetic, and biochemical
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Ethics statement
The study was approved by the Ethics Committee of the Faculty of Medicine at the University of Tübingen (TREND: 90/2009BO2). The studies were conducted in accordance with the local legislation and
institutional requirements. The participants provided their written informed consent to participate in this study.
Author contributions
A-KS: Formal analysis, Software, Writing – original draft, Writing – review & editing. SL: Data curation, Investigation, Writing – original draft, Writing – review & editing. IW: Investigation,
Writing – review & editing. BR: Investigation, Writing – review & editing. MZ: Investigation, Writing – review & editing. FF: Investigation, Writing – review & editing. A-KT: Data curation,
Investigation, Writing – review & editing. GE: Conceptualization, Supervision, Writing – review & editing. WM: Conceptualization, Investigation, Supervision, Writing – review & editing. DB:
Conceptualization, Supervision, Writing – review & editing. FS: Methodology, Software, Writing – original draft, Writing – review & editing. KB: Conceptualization, Investigation, Supervision, Writing
– original draft, Writing – review & editing.
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work was supported by the BMBF-funded de.NBI Cloud within the German
Network for Bioinformatics Infrastructure (de.NBI) (031A532B, 031A533A, 031A533B, 031A534A, 031A535A, 031A537A, 031A537B, 031A537C, 031A537D, 031A538A). This work was further partially supported from
the Else Kröner-Fresenius-Stiftung within the Project “ClinBrAIn: Künstliche Intelligenz für Klinische Hirnforschung” (KB and FS). FS was supported by the Carl-Zeiss-Stiftung and acknowledges the
support of the DFG Cluster of Excellence “Machine Learning – New Perspectives for Science”, EXC 2064/1, project number 390727645. KB received support by the DFG for "Psychosoziale und
gesundheitsbezogene Auswirkungen der SARS-CoV-2 Pandemie, Antikörper und Impfung bei älteren Menschen (CORO-TREND)”. We acknowledge support from the Open Access Publishing Fund of the University of
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnagi.2024.1447944/full#supplementary-material
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Keywords: healthy, aging, machine learning, physical activity, cognition
Citation: Schalkamp A-K, Lerche S, Wurster I, Roeben B, Zimmermann M, Fries F, von Thaler A-K, Eschweiler G, Maetzler W, Berg D, Sinz FH and Brockmann K (2024) Machine learning-based personalized
composite score dissects risk and protective factors for cognitive and motor function in older participants. Front. Aging Neurosci. 16:1447944. doi: 10.3389/fnagi.2024.1447944
Copyright © 2024 Schalkamp, Lerche, Wurster, Roeben, Zimmermann, Fries, von Thaler, Eschweiler, Maetzler, Berg, Sinz and Brockmann. This is an open-access article distributed under the terms of the
Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the
original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Fabian H. Sinz, sinz@cs.uni-goettingen.de; Kathrin Brockmann, kathrin.brockmann@uni-tuebingen.de
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Understanding Mathematical Functions: How To Find The Function Value
Mathematical functions are essential concepts in the field of mathematics, and they play a crucial role in various scientific and engineering applications. A function is a relation between a set of
inputs and a set of permissible outputs, with the property that each input is related to exactly one output. Understanding how to find the value of a mathematical function is important for solving
equations, analyzing data, and making predictions in various real-life scenarios.
Key Takeaways
• Mathematical functions are vital in various scientific and engineering applications.
• A function is a relation between inputs and permissible outputs, with each input related to exactly one output.
• Understanding how to find the value of a mathematical function is crucial for solving equations, analyzing data, and making predictions in real-life scenarios.
• Common mathematical functions include linear, quadratic, exponential, and trigonometric functions.
• Practical applications of understanding function value include engineering, economics, and science.
Understanding Mathematical Functions
Mathematical functions are essential in the field of mathematics and are used to describe relationships between variables. Understanding how to evaluate a function and find its value is crucial for
various mathematical calculations and problem-solving.
A. Definition of a mathematical function
A mathematical function is a rule or correspondence that assigns to every element in a set A exactly one element in a set B. In other words, for every input, there is only one output. The input is
typically represented by the variable x, and the output is represented by the function notation f(x).
1. Examples of mathematical functions
• Linear functions: f(x) = mx + b
• Quadratic functions: f(x) = ax^2 + bx + c
• Exponential functions: f(x) = a^x
• Trigonometric functions: sin(x), cos(x), tan(x)
C. Notation for functions
Functions are commonly denoted by the letter f, followed by the input variable in parentheses. For example, f(x) represents a function of x. The input variable x can be any real number within the
domain of the function, and the output is the value of the function evaluated at x.
Understanding Mathematical Functions: How to Find the Function Value
Mathematical functions are fundamental in various fields, and understanding how to find the function value is essential for solving problems and making informed decisions. In this chapter, we will
explore the concept of function value, the steps to find it, and its importance.
A. Understanding the concept of function value
The function value, also known as the output or dependent variable, represents the result of applying the function to a specific input. In simple terms, it is the y-value that corresponds to a given
x-value on the graph of the function. Understanding the concept of function value is crucial for interpreting the behavior of a function and analyzing its relationships with other variables.
B. Steps to find the function value
1. Identify the input value
Before finding the function value, you need to determine the input or independent variable for which you want to calculate the output. This could be a specific number, variable, or expression.
2. Substitute the input value into the function
Once you have the input value, substitute it into the function to find the corresponding output. This involves replacing the independent variable with the given value and simplifying the expression
to obtain the function value.
3. Evaluate the function
After substituting the input value, perform the necessary arithmetic operations to evaluate the function and determine the function value. This may involve using mathematical operations such as
addition, subtraction, multiplication, division, exponentiation, and more.
C. Importance of finding the function value
Finding the function value is essential for various reasons:
• 1. Problem-solving: It allows you to solve equations, analyze patterns, and make predictions based on the behavior of the function.
• 2. Understanding relationships: It helps in understanding the relationship between the input and output variables and how they change in response to each other.
• 3. Real-world applications: It has practical applications in fields such as science, engineering, economics, and finance, where functions model real-world phenomena.
In conclusion, understanding how to find the function value is a fundamental skill that enhances your ability to analyze and interpret mathematical functions, making it an invaluable tool in various
academic and professional contexts.
Common Mathematical Functions
When dealing with mathematical functions, there are several types that are commonly encountered. Understanding how to find the function value for each type is essential in mathematics. The following
are the most common mathematical functions:
A. Linear functions
Linear functions are the simplest type of mathematical functions and are represented in the form y = mx + b. In a linear function, the rate of change is constant. To find the function value for a
linear function, simply substitute the input value into the function and solve for the output value.
B. Quadratic functions
Quadratic functions are represented in the form y = ax^2 + bx + c. These functions form a parabola and have a constant rate of change. To find the function value for a quadratic function, substitute
the input value into the function and solve for the output value using the quadratic formula if necessary.
C. Exponential functions
Exponential functions are represented in the form y = a * b^x, where a and b are constants. These functions grow at an increasing rate as x increases. To find the function value for an exponential
function, substitute the input value into the function and use exponent rules to simplify the expression.
D. Trigonometric functions
Trigonometric functions, such as sine, cosine, and tangent, are used to model periodic phenomena. To find the function value for a trigonometric function, substitute the input value into the function
and use the unit circle or trigonometric identities to evaluate the expression.
Using Input to Find Output
Understanding how input and output work in mathematical functions is essential for solving problems and making predictions. In this chapter, we will explore the concept of finding the function value
through input and output.
A. Defining the input and output in a function
In a mathematical function, the input is the value that is plugged into the function, and the output is the resulting value that the function produces. The input is typically represented by the
variable x, and the output is represented by the variable f(x) or y.
B. Importance of input-output relationship
The input-output relationship is crucial in understanding how a function behaves and how different inputs produce different outputs. By analyzing this relationship, we can make predictions, solve
equations, and understand the behavior of a function.
C. Examples of using input to find output
Let's consider an example where the function f(x) = 2x + 3. If we plug in an input value of x = 4, we can find the output by substituting the value into the function: f(4) = 2(4) + 3 = 11. Therefore,
the output for the input x = 4 is 11.
• Another example is the function g(x) = x^2. If we input x = 3, we can find the output by substituting the value into the function: g(3) = 3^2 = 9. In this case, the output for the input x = 3 is
• One more example is the function h(x) = √x. If we input x = 16, we can find the output by substituting the value into the function: h(16) = √16 = 4. Here, the output for the input x = 16 is 4.
Practical Applications of Understanding Function Value
Understanding how to find the value of a mathematical function has numerous practical applications across various fields. Whether it's engineering, economics, or science, the ability to calculate
function values is essential for making informed decisions and solving real-world problems.
A. Engineering
• Design and Analysis
In engineering, mathematical functions are used to model and analyze systems, structures, and processes. Understanding how to find function values allows engineers to make accurate predictions
about the behavior of these systems and evaluate their performance.
• Optimization
Engineers use mathematical functions to optimize designs and processes to achieve the best possible outcomes. Calculating function values helps in determining the optimal settings and parameters
for various engineering applications.
B. Economics
• Market Analysis
Economists use mathematical functions to model demand, supply, and market behavior. By calculating function values, they can forecast trends, analyze economic data, and make predictions about
market conditions.
• Cost-Benefit Analysis
Understanding function values is crucial for evaluating the costs and benefits of different economic decisions and policies. Whether it's investment analysis or budget planning, economists rely
on mathematical functions to make informed choices.
C. Science
• Physical Modeling
In the field of science, mathematical functions are used to model physical phenomena and natural processes. Calculating function values helps scientists describe and predict the behaviors of
these systems.
• Data Analysis
Scientists use mathematical functions to analyze experimental data and extract meaningful insights. Understanding how to find function values is crucial for interpreting and drawing conclusions
from scientific observations.
In conclusion, understanding function value is crucial in solving real-world problems and making informed decisions. By knowing how to find the function value, you can calculate the relationship
between different variables and predict future outcomes. It is important to grasp the concept of mathematical functions to excel in various fields such as engineering, economics, and science.
I encourage you to explore and practice various mathematical functions to improve your problem-solving skills and gain a deeper understanding of their applications. The more you engage with
mathematical functions, the better you will become at using them to analyze and interpret data.
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Mutual Inductance and Self Inductance | Formula & Example | Electrical Academia
Mutual Inductance and Self Inductance | Formula & Example
Electromagnetic induction occurs when a magnetic flux in motion with respect to a single conductor or a coil induces an emf in the conductor or coil. Because the growth or decline of current through
a coil generates a changing flux, an emf is induced in the coil by its own current change. The same effect can induce an emf in an adjacent coil. The level of emf induced in each case depends on the
self inductance of the coil, or on the mutual inductance between the two coils. In all cases, the polarity of the induced emf is such that it opposes the original change that induced the emf.
Components called inductors or chokes are constructed to have specified values of inductance. Inductors can be operated in series or in parallel. Even the shortest of conductors has an inductance.
This is usually an unwanted quantity and is termed stray inductance.
What is Self Inductance?
Coil and Conductor Inductance
It has been shown that an emf is induced in a conductor moving through a magnetic field and that the growth of current in a coil can induce an emf in another magnetically coupled coil. It is also
possible for a coil to induce a voltage in itself as its current level changes. This phenomenon is known as self inductance, and the principle is illustrated in Figure 1.
Figure 1: Current Carrying Coil and its Cross-Sectional Area
Magnetic flux growing outwards around the turns of a coil cuts (or brushes over) the other coil turns and induces emf in the coil
A coil and its cross-sectional area are shown in Figure 1, with arrow tails and points indicating the current directions in each turn. Every turn of the coil has a flux around it produced by the
current flowing through the coil. However, for convenience, the illustration shows the growth of flux around only one turn on the coil. It is seen that as the current grows, the flux expands outward
and cuts (or brushes over) the other turns. This causes currents to be induced in the other turns, and the direction of the induced currents is such that they set up a flux that opposes the flux
inducing them.
Remembering that the current through the coil causes the flux to grow around all turns at once, it is seen that the flux from every turn induces a current that opposes it in every other turn.
To set up opposing fluxes, the induced current in a coil must be in opposition to the current flowing through the coil from the external source of supply. The induced current is, of course, the result
of an induced emf. Thus, it is seen that the self inductance of a coil sets up an induced emf that opposes the external emf that is driving current through the coil. Because this induced emf is in
opposition to the supply voltage, it is usually termed the counter-emf or back-emf. The counter-emf occurs only when the coil current is growing or declining. When the current has reached a constant
level, the flux is no longer changing and no counter-emf is generated.
Even a single conductor has self inductance. Figure 2 shows that when current is growing in a conductor, flux may grow outward from the center of the conductor. This flux cuts other portions of the
conductor and induces a counter-emf.
Figure 2: Conductor Cross Section
The growth of current within a conductor induces emfs in other portions of the conductor.
In Figure 3, the polarity of the counter-emf induced in a coil is illustrated for a given supply voltage polarity. In Figure 3(a), the switch is closed and current I commences to grow from zero. The
polarity of the counter-emf (e[L]) is such that it opposes the growth of I, thus it is series-opposing with the supply voltage. When the switch is opened (figure 3(b)), the current tends to fall to
zero. But now the polarity of e[L] is such that it opposes the decline of I. it is series-aiding with the supply voltage. In fact, e[L] may cause arcing at the switch terminals as it depends on the
coil’s inductance.
Figure 3: Induced EMF Polarity
The counter-emf induced in a coil always opposes the growth or decline of the current.
The SI unit of inductance is the Henry (H).
The inductance of a circuit is one Henry when an emf of 1 V is induced by the current changing at the rate of 1 A/s.
Thus the relationship among inductance, induced voltage, and rate of change of current is:
\[\begin{matrix} L=\frac{{{e}_{L}}}{{\Delta i}/{\Delta t}\;} & {} & \left( 1 \right) \\\end{matrix}\]
Where L is the inductance in Henry, e[L] is the induced counter-emf in volts and is the rate of change of current in A/s. a negative sign is sometimes included in front of e[L] to show that the
induced emf is in opposition to the applied emf. When e[L]=1V, and =1A/s, L=1H. If the rate of change of current is 2 A/s and e[L]=1V, the inductance is 0.5 H.
A coil constructed to have a certain inductance is usually referred to as an inductor or choke. Note the graphic symbols for an inductor shown in figure 3.
Self Inductance Formula
A formula for self inductance can be derived involving the coil dimensions and the number of turns [see figure 4].
Figure 4: Number of turns in a coil
The self inductance of a coil depends on the number of turns and on the flux and current changes.
From equation (2):
\[\begin{matrix} {{e}_{L}}=N\frac{\Delta \phi }{\Delta t} & {} & \left( 2 \right) \\\end{matrix}\]
Substituting for e[L] into equation (1) gives
\[L=N\frac{{\Delta \phi }/{\Delta t}\;}{{\Delta i}/{\Delta t}\;}\]
\[\begin{matrix} L=N\frac{\Delta \phi }{\Delta i} & {} & \left( 3 \right) \\\end{matrix}\]
\[\phi =B\times A\]
$B={{\mu }_{o}}\times {{\mu }_{r}}\times H={{\mu }_{o}}\times {{\mu }_{r}}\times \frac{IN}{l}$
$\phi ={{\mu }_{o}}\times {{\mu }_{r}}\times IN\times \frac{A}{l}$
Since I is a maximum current level, it also represents the change in current (∆i) from zero to the maximum level. Therefore, change in flux is
$\Delta \phi ={{\mu }_{o}}\times {{\mu }_{r}}\times \Delta i\times N\times \frac{A}{l}$
Substituting for ∆ϕ in equation (3) gives
\[L=\frac{\left( {{\mu }_{o}}\times {{\mu }_{r}}\times \Delta i\times N\times {}^{A}/{}_{l} \right)\times N}{\Delta i}\]
\[\begin{matrix} L={{\mu }_{o}}\times {{\mu }_{r}}\times {{N}^{2}}\times {}^{A}/{}_{l} & {} & \left( 4 \right) \\\end{matrix}\]
Note that, as illustrated in Figure 5, the self inductance is proportional to the cross-sectional area of a coil and to the square of the number of turns. It is also inversely proportional to the
coil length. Therefore, maximum inductance is obtained with a short coil that has a large cross-sectional area and a large number of turns.
Figure 5: Coil Dimensions
Coil inductance can be calculated from its dimensions and its core permeability.
Equation (4) now affords a means of calculating the self inductance of a coil of known dimensions. Alternatively, it can be used to determine the required dimensions for a coil to have a given
inductance. However, it is not so easily applied to iron-cored coils, because the permeability of ferromagnetic material changes when the flux density changes. Consequently, the inductance of an
iron-cored coil is constantly changing as the coil current increases and decreases.
Non-inductive Coil
In many cases it is desired to have a non-inductive coil; for example, precision resistors are usually non-inductive. To construct such a coil, the winding is made of two side-by-side conductors, as
illustrated in Figure 6. Every coil turn has an adjacent turn carrying current in the Opposite direction. The magnetic fields generated by adjacent turns cancel each other out. Therefore, no
counter-emf is generated, and the coil is non-inductive.
Figure 6: Non-Inductive Coil
Self-Inductance Example
A solenoid with 900 turns has a total flux of 1.33 X 10^-7 Wb through its air core when the coil current is 100 mA. If the flux takes 75 ms to grow from zero to its maximum level, calculate the self
inductance of the coil. Also, determine the counter-emf induced in the coil during the flux growth.
$ & \Delta \phi =1.33\times {{10}^{-7}}Wb \\ & \Delta i=100mA \\ & \Delta t=75ms \\$
Equation (3):
\[L=N\frac{\Delta \phi }{\Delta i}=900\frac{1.33\times {{10}^{-7}}}{100\times {{10}^{-3}}}=1.2mH\]
From equation (2)
\[{{e}_{L}}=N\frac{\Delta \phi }{\Delta t}=900\frac{1.33\times {{10}^{-7}}}{75\times {{10}^{-3}}}=1.6mV\]
What is Mutual Inductance?
When the flux from one coil cuts another adjacent (or magnetically coupled) coil, an emf is induced in the second coil. Following Lenz’s law, the emf induced in the second coil sets up a flux that
opposes the original flux from the first coil. Thus, the induced emf is again a counter-emf, and in this case the inductive effect is referred to as mutual inductance. Figure 7 shows the graphic
symbols used for coils with mutual inductance, also termed coupled coils.
Figure 7: Graphic symbols for Air and Iron Cored Coils
Like self-inductance, mutual inductance is measured in Henry (H).
Mutual Inductance Formula
Two coils have a mutual inductance of 1H when an emf of 1V is induced in one coil by current changing at the rate of 1 A/s in the other coil.
This definition gives rise to the equation relating mutual inductance to induced voltage and rate of change of current:
\[\begin{matrix} M=\frac{{{e}_{L}}}{{\Delta i}/{\Delta t}\;} & {} & \left( 5 \right) \\\end{matrix}\]
Where M is the mutual inductance in Henry, e[L] is the emf in volts induced in the secondary coil and is the rate of change of current in the primary coil in A/s.
The coil through which a current is passed from an external source is termed the primary, and the coil that has an emf induced in it is referred to as the secondary.
An equation for the emf induced in the secondary coil can be written as:
\[\begin{matrix} {{e}_{L}}={{N}_{s}}\frac{\Delta \phi }{\Delta t} & {} & \left( 6 \right) \\\end{matrix}\]
Here ∆ϕ is the total change in flux linking with the secondary winding, N[s] is the number of turns in the secondary winding, and ∆t is the time required for the flux change.
Substituting for e[L] from equation (6) into equation (5) gives
\[M={{N}_{s}}\frac{{\Delta \phi }/{\Delta t}\;}{{\Delta i}/{\Delta t}\;}\]
\[\begin{matrix} M={{N}_{s}}\frac{\Delta \phi }{\Delta i} & {} & \left( 7 \right) \\\end{matrix}\]
Figure 8(a) illustrates the fact that when the two coils are wound on a single ferromagnetic core, effectively all of the flux generated by the primary coil links with the secondary coil. However,
when the coils are air-cored, only a portion of the flux from the primary may link with the secondary [see figure 8 (b)]. Depending on how much of the primary flux cuts the secondary, the coils may be
classified as loosely coupled or tightly coupled. One way to ensure tight coupling is shown in Figure 8(c), where each turn of the secondary winding is side by side with one turn of the primary
winding. Coils wound in this fashion are said to bifilar.
Figure 8: Flux linkages in primary and secondary coils
The amount of flux from a primary winding that links with a secondary depends on how closely the coils are coupled. The coefficient of coupling defines the linkage.
The amount of flux linking from primary to secondary is also defined in terms of a coefficient of coupling, k. If all the primary flux links with the secondary, the coefficient of coupling is 1. When
only 50% of the primary flux links with the secondary coil, the coefficient of coupling is 0.5. Thus,
\[k=\frac{flux\text{ }linkages\text{ }between\text{ }primary\text{ }and\text{ }\sec ondary}{total\text{ }fluxproduced\text{ }by\text{ }primary}\]
Returning to equation (7). When ∆ϕ is the total flux change in the primary coil, the flux linking with the secondary is k∆ϕ. Therefore, the equation for M
\[\begin{matrix} M=k{{N}_{s}}\frac{\Delta \phi }{\Delta i} & {} & \left( 8 \right) \\\end{matrix}\]
Also, substituting for $\Delta \phi ={{\mu }_{o}}\times {{\mu }_{r}}\times \Delta i\times N\times \frac{A}{l}$ into equation (8) gives
\[M=\frac{k{{N}_{s}}}{\Delta i}\times{{\mu }_{o}}\times {{\mu }_{r}}\times \Delta i\times {{N}_{p}}\times \frac{A}{l}\]
\[\begin{matrix} M=k\times {{N}_{p}}\times {{N}_{s}}\times {{\mu }_{o}}\times {{\mu }_{r}}\times \frac{A}{l} & {} & \left( 9 \right) \\\end{matrix}\]
Each winding considered alone has a self-inductance that can be calculated from equation (4). Thus, for the primary coil,
${{L}_{1}}=N_{p}^{2}\times {{\mu }_{o}}\times {{\mu }_{r}}\times \frac{A}{l}$
And for the secondary
${{L}_{2}}=N_{s}^{2}\times {{\mu }_{o}}\times {{\mu }_{r}}\times \frac{A}{l}$
Assuming that the two windings share the common core (magnetic or non-magnetic as in figure 9), the only difference in the expression for L[1] and L[2] is the number of turns.
Figure 9: Two windings on the same core
${{L}_{1}}\times {{L}_{2}}=N_{p}^{2}\times N_{p}^{2}\times {{\left( {{\mu }_{o}}\times {{\mu }_{r}}\times \frac{A}{l} \right)}^{2}}$
\[\begin{matrix} \sqrt{{{L}_{1}}\times {{L}_{2}}}={{N}_{p}}\times {{N}_{s}}\times {{\mu }_{o}}\times {{\mu }_{r}}\times \frac{A}{l} & {} & \left( 10 \right) \\\end{matrix}\]
Comparing equations 9 and 10, it is seen that,
\[\begin{matrix} M=k\sqrt{{{L}_{1}}\times {{L}_{2}}} & {} & \left( 11 \right) \\\end{matrix}\]
Mutual Inductance Example
Two identical coils are wound on a ring-shaped iron core that has a relative permeability of 500. Each coil has 100 turns, and the core dimensions are: cross-sectional area A=3 cm^2 and magnetic path
length l=20cm. Calculate the inductance of each coil and the mutual inductance between the coils.
From equation (4):
\[ & {{L}_{1}}={{L}_{2}}={{N}^{2}}\times {{\mu }_{o}}\times {{\mu }_{r}}\times \frac{A}{l} \\ & ={{100}^{2}}\times 500\times 4\pi \times {{10}^{-7}}\times \frac{3\times {{10}^{-4}}}{20\times {{10}^
{-2}}}\cong 9.42mH \\\]
As the coils are wound on the same iron core, k=1. Equation (11):
$M=k\sqrt{{{L}_{1}}\times {{L}_{2}}}=\sqrt{9.42\times 9.42}=9.42mH$
Key Takeaways of Mutual and Self Inductance
• Mutual inductance refers to the phenomenon where the change in current flow in one coil induces a voltage in an adjacent coil. It occurs when two coils are placed close to each other and have a
magnetic field that interacts with each other.
• Self inductance is the property of a coil to induce a voltage in itself when the current through it changes. It is a measure of the coil’s ability to oppose changes in current flow.
• Mutual inductance is responsible for the operation of transformers, where voltage is stepped up or down by changing the number of turns in the coils. It enables efficient energy transfer between
• Self inductance is present in all coils and is the basis for the operation of inductors in electronic circuits. It stores energy in its magnetic field and resists changes in current flow.
• The unit of inductance is the Henry (H). Mutual and self inductance can be quantified using mathematical equations and are influenced by factors such as the number of turns, the geometry of the
coils, and the permeability of the core material.
• Mutual and self inductance play crucial roles in various applications, including power transmission, wireless communication, electric motors, generators, and inductive sensors.
• Proper understanding and control of mutual and self inductance are essential in designing and optimizing electrical and electronic systems for efficient and reliable operation.
• Remember, mutual and self inductance are fundamental concepts in electromagnetism that have significant implications in various fields of electrical engineering.
Mutual and Self Inductance FAQs
What is the difference between mutual inductance and self inductance?
Mutual inductance refers to the interaction between two coils where the change in current flow in one coil induces a voltage in the other. Self inductance, on the other hand, is the ability of a
single coil to induce a voltage in itself when the current through it changes.
How is mutual inductance useful in practical applications?
Mutual inductance is crucial in the operation of transformers, which are used for voltage transformation in power distribution systems. It allows efficient energy transfer between coils and enables
stepping up or stepping down of voltages.
In what devices is self inductance commonly found?
Self inductance is present in all coils and is the fundamental principle behind the operation of inductors. Inductors are used in various electronic circuits for energy storage, filtering, and
current regulation.
Can you provide an example of mutual inductance?
An example of mutual inductance is the operation of a transformer. When alternating current flows through the primary coil, it creates a changing magnetic field that induces a voltage in the
secondary coil, resulting in energy transfer.
How does self inductance affect the behavior of an inductor?
Self inductance resists changes in current flow through an inductor. When the current changes, the inductor generates an opposing voltage, causing a delay in the current response. This property is
utilized in circuits to control the rate of change of current.
Are mutual and self inductance dependent on the physical properties of the coils?
Yes, both mutual and self inductance are influenced by factors such as the number of turns in the coil, the geometry of the coil, and the permeability of the core material. These factors determine
the magnitude of inductance in a given configuration.
What are some practical applications of mutual and self inductance?
Mutual and self inductance find applications in various fields, including power transmission, wireless communication, electric motors, generators, and inductive sensing technologies.
How is inductance measured?
The unit of inductance is the Henry (H). Inductance can be measured using specialized instruments such as an LCR meter or calculated using mathematical formulas based on the coil’s physical
These frequently asked questions provide a clearer understanding of mutual and self inductance and their relevance in different aspects of electrical engineering and technology.
2 thoughts on “Mutual Inductance and Self Inductance | Formula & Example”
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Equalize the Array | HackerRank
Given an array of integers, determine the minimum number of elements to delete to leave only elements of equal value.
Delete the elements and leaving . If both twos plus either the or the are deleted, it takes deletions to leave either or . The minimum number of deletions is .
Function Description
Complete the equalizeArray function in the editor below.
equalizeArray has the following parameter(s):
• int arr[n]: an array of integers
• int: the minimum number of deletions required
The first line contains an integer , the number of elements in .
The next line contains space-separated integers .
STDIN Function
----- --------
5 arr[] size n = 5
3 3 2 1 3 arr = [3, 3, 2, 1, 3]
Delete and to leave . This is minimal. The only other options are to delete elements to get an array of either or . | {"url":"https://www.hackerrank.com/challenges/equality-in-a-array/problem","timestamp":"2024-11-12T12:24:31Z","content_type":"text/html","content_length":"879478","record_id":"<urn:uuid:0ed9d31d-b246-4944-b8b5-f3b07aeefb9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00848.warc.gz"} |
STEP Support Foundation modules
These Foundation modules are designed to develop your problem solving skills and provide an introduction to solving STEP (and STEP-like) problems. Most of the questions are taken from old STEP 1
papers, with some STEP 2 questions appearing in later modules. The assignments also introduce mathematical ideas beyond the syllabus with lots of opportunities for extra reading.
Please note that STEP 1 has now been discontinued, and the last STEP 1 paper was in 2019.
A lot of these assignments only require GCSE or AS knowledge (though they will ask you to use it in unusual ways!), and the first 10 to 15 Assignments are aimed at year 12 students. If you think
there is a possibility that you will be sitting STEP 2 or STEP 3 in the summer of year 13 then we strongly advise that you start working on these assignments in year 12, or in the summer before you
start year 13.
There are 25 Foundation modules, and the intention is that you work through them in order. There are also three collections of STEP questions (one each for pure, mechanics and statistics) which can
be found after Assignment 25. Once you feel ready, you can move onto the 21 STEP 2 and STEP 3 modules.
If you have any questions about the assignments, or STEP in general, or feedback about our resources you can email us at step@maths.cam.ac.uk, or contact us via twitter @stepsupportcam.
There is more information about the different module types here. | {"url":"https://maths.org/step/assignments","timestamp":"2024-11-12T12:18:39Z","content_type":"application/xhtml+xml","content_length":"46708","record_id":"<urn:uuid:e3707a40-745f-4dfd-b623-622b5e2b12b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00457.warc.gz"} |
Excel Formula for Unique Date Ranges
Formula for Assigning Unique Values to Date Ranges in Excel
This article explains how to write an Excel formula in Python that assigns a unique value to each of five date ranges. The formula uses nested IF and AND functions to check if a date falls within a
specific range and returns the corresponding unique value. The step-by-step explanation provides a detailed breakdown of how the formula works, and examples demonstrate its usage.
To use this formula in Python, you can adapt it to your specific requirements and integrate it into your code. By understanding the logic behind the formula, you can customize it to handle different
date ranges and assign different unique values.
Let's dive into the details of the formula and explore its functionality.
Step-by-Step Explanation
1. The formula starts with the outermost IF function. It checks if the date falls within the first range (January 1, 2022, to March 31, 2022). If it does, it returns the unique value for Range 1.
2. If the date does not fall within the first range, the formula moves to the next nested IF function. It checks if the date falls within the second range (April 1, 2022, to June 30, 2022). If it
does, it returns the unique value for Range 2.
3. If the date does not fall within the second range, the formula moves to the next nested IF function. It checks if the date falls within the third range (July 1, 2022, to September 30, 2022). If
it does, it returns the unique value for Range 3.
4. If the date does not fall within the third range, the formula moves to the next nested IF function. It checks if the date falls within the fourth range (October 1, 2022, to December 31, 2022). If
it does, it returns the unique value for Range 4.
5. If the date does not fall within the fourth range, the formula moves to the last nested IF function. It checks if the date falls within the fifth range (January 1, 2023, to March 31, 2023). If it
does, it returns the unique value for Range 5.
6. If the date does not fall within any of the five ranges, the formula returns a message indicating that the date is not in any range.
To illustrate the functionality of the formula, consider the following dates:
Applying the formula to these dates would yield the following results:
Date Result
1/15/2022 Range 1
4/5/2022 Range 2
7/20/2022 Range 3
10/10/2022 Range 4
2/1/2023 Range 5
The formula correctly identifies the range for each date and returns the corresponding unique value.
By understanding the logic and structure of this formula, you can adapt it to handle different date ranges and assign different unique values based on your specific requirements. This flexibility
allows you to efficiently process and categorize dates in Excel using Python.
An Excel formula
=IF(AND(A1>=DATE(2022,1,1), A1<=DATE(2022,3,31)), "Range 1",
IF(AND(A1>=DATE(2022,4,1), A1<=DATE(2022,6,30)), "Range 2",
IF(AND(A1>=DATE(2022,7,1), A1<=DATE(2022,9,30)), "Range 3",
IF(AND(A1>=DATE(2022,10,1), A1<=DATE(2022,12,31)), "Range 4",
IF(AND(A1>=DATE(2023,1,1), A1<=DATE(2023,3,31)), "Range 5", "Not in any range")))))
Formula Explanation
This formula uses nested IF and AND functions to check if a date falls within one of the five unique date ranges. If the date falls within a range, it returns a unique value for that range. If the
date does not fall within any of the ranges, it returns "Not in any range".
Step-by-step explanation
1. The formula starts with the outermost IF function. It checks if the date in cell A1 falls within the first range (January 1, 2022 to March 31, 2022). If it does, it returns "Range 1".
2. If the date does not fall within the first range, the formula moves to the next nested IF function. It checks if the date falls within the second range (April 1, 2022 to June 30, 2022). If it
does, it returns "Range 2".
3. If the date does not fall within the second range, the formula moves to the next nested IF function. It checks if the date falls within the third range (July 1, 2022 to September 30, 2022). If it
does, it returns "Range 3".
4. If the date does not fall within the third range, the formula moves to the next nested IF function. It checks if the date falls within the fourth range (October 1, 2022 to December 31, 2022). If
it does, it returns "Range 4".
5. If the date does not fall within the fourth range, the formula moves to the last nested IF function. It checks if the date falls within the fifth range (January 1, 2023 to March 31, 2023). If it
does, it returns "Range 5".
6. If the date does not fall within any of the five ranges, the formula returns "Not in any range".
For example, if we have the following dates in column A:
| A |
| |
| 1/15/2022 |
| 4/5/2022 |
| 7/20/2022 |
| 10/10/2022 |
| 2/1/2023 |
The formula =IF(AND(A1>=DATE(2022,1,1), A1<=DATE(2022,3,31)), "Range 1", IF(AND(A1>=DATE(2022,4,1), A1<=DATE(2022,6,30)), "Range 2", IF(AND(A1>=DATE(2022,7,1), A1<=DATE(2022,9,30)), "Range 3", IF(AND
(A1>=DATE(2022,10,1), A1<=DATE(2022,12,31)), "Range 4", IF(AND(A1>=DATE(2023,1,1), A1<=DATE(2023,3,31)), "Range 5", "Not in any range"))))) would return the following results:
| A | B |
| | |
| 1/15/2022 | Range 1 |
| 4/5/2022 | Range 2 |
| 7/20/2022 | Range 3 |
| 10/10/2022 | Range 4 |
| 2/1/2023 | Range 5 |
The formula correctly identifies the range for each date and returns the corresponding unique value. | {"url":"https://codepal.ai/excel-formula-generator/query/2UNFGY1e/excel-formula-unique-date-ranges","timestamp":"2024-11-04T13:27:10Z","content_type":"text/html","content_length":"111270","record_id":"<urn:uuid:191ac73d-e326-4306-8e73-6368eae2feab>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00569.warc.gz"} |
Optimizing Membership Functions using Learning Automata for Fuzzy Association Rule Mining
Document Type : Original/Review Paper
^1 Department of Computer Engineering and Information Technology, Payame Noor University (PNU), P. OBox,19395-4697 Tehran, Iran
^2 Department of Computer Engineering, Khoy Branch, Islamic Azad University, Khoy, Iran.
^3 Department of Computer Engineering, Shabestar Branch, Islamic Azad University, Shabestar, Iran.
^4 Department of Computer Engineering, Urmia Branch, Islamic Azad University, Urmia, Iran
The Transactions in web data often consist of quantitative data, suggesting that fuzzy set theory can be used to represent such data. The time spent by users on each web page is one type of web data,
was regarded as a trapezoidal membership function (TMF) and can be used to evaluate user browsing behavior. The quality of mining fuzzy association rules depends on membership functions and since the
membership functions of each web page are different from those of other web pages, so automatic finding the number and position of TMF is significant. In this paper, a different reinforcement-based
optimization approach called LA-OMF was proposed to find both the number and positions of TMFs for fuzzy association rules. In the proposed algorithm, the centers and spreads of TMFs were considered
as parameters of the search space, and a new representation using learning automata (LA) was proposed to optimize these parameters. The performance of the proposed approach was evaluated and the
results were compared with the results of other algorithms on a real dataset. Experiments on datasets with different sizes confirmed that the proposed LA-OMF improved the efficiency of mining fuzzy
association rules by extracting optimized membership functions.
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I still don’t understand exactly how I can implement this. From what I can understand you choose the energy not the eigenstate for the density function.
I am unclear what you mean here. In your code you do the following:
ham_mat = sys.hamiltonian_submatrix(sparse=True, params = params)
evals, evecs = sorted_eigs(sla.eigsh(ham_mat.tocsc(), k=20, sigma=0))
kwant.plotter.map(sys, np.abs(evecs[:, 9])**2
You take the hamiltonian of the scattering region, diagonalize it, and then plot *a single eigenvector*. Your problem was that because you have >1 degree of freedom per site you cannot simply plot
the eigenvector as-is. You need to first compute the density per site for this eigenvector, which you can do using the operator module:
rho = kwant.operator.Density(np.kron(-s_z, s_0)) # holes count +1 to the density, and electrons -1
kwant.plotter.map(sys, rho(evecs[:, 9))
Note that you call the density operator with a single eigenvector.
Independently of the above it's not clear to me that you should be diagonalizing just the scattering region Hamiltonian, given that your system has leads. It seems to me that you should be
calculating the scattering states and the density of those.
Happy Kwanting, | {"url":"https://mail.python.org/archives/list/kwant-discuss@python.org/message/S46A2CUTVXIDPFEFTVCLICNDXJOX53TX/attachment/4/attachment.htm","timestamp":"2024-11-05T20:01:32Z","content_type":"text/html","content_length":"3103","record_id":"<urn:uuid:7ca2854b-66d2-44b4-a072-7cef39b59c47>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00784.warc.gz"} |
Alexandre Vinogradov
Contribution to mathematics and to the mathematical community
Alexandre Vinogradov, a remarkable mathematician and extraordinary man, was born on February 18, 1938 in Novorossiysk, but almost all his life lived in Moscow. In 1955 he became a student of Mekhmat
of Moscow State University and, in 1960, a graduate (PhD track) student there. After obtaining his PhD in 1964, he was soon invited to take a teaching position at the Chair of Higher Geometry and
Topology, which he held until he left the Soviet Union for Italy in 1990. He obtained the habilitation degree in 1984. From 1993 to 2010, he held the position of professor at the University of
Salerno in Italy.
Being a second year undergraduate student, Vinogradov published two works (with B.N. Delaunay and D.B. Fuchs) in number theory, but by the end of undergraduate years his research interests changed:
he began working in algebraic topology. His PhD was devoted to the homotopic properties of the embedding spaces of circles into the 2-sphere or the 3-disk. One of Vinogradov's first works was devoted
to the Adams spectral sequence. In 1960, Vinogradov announced the solution of J.F. Adams' problem concerning the relationship between the higher cohomological operations and the Adams filtration in
the stable homotopy groups of spheres. Adams wrote a favorable review of that note.
Vinogradov radically changed the direction of his research between the sixties and the seventies. Inspired by the ideas of Sophus Lie, he began to think about the foundations of the geometric theory
of PDEs; having become familiar with the work of Spencer, Goldschmidt, and Quillen on formal solvability, he turned his attention to the algebraic (in particular, cohomological) component of that
theory. In 1972, the short note "The logic algebra of the theory of linear differential operators" introduced what Vinogradov himself called the main functors of the differential calculus in
commutative algebras. On four pages, it was elegantly shown that for the definition and the study of such fundamental notions as vector field, differential form, jet, linear differential operator,
etc., the category of modules over a commutative algebra with unit provides an appropriate setting, while the geometric prototypes of these notions occur when, for the algebra, one chooses the
algebra of smooth functions on a manifold, and for the modules, the spaces of sections of vector bundles over the manifold.
Vinogradov's approach to nonlinear differential equations as geometric objects, with general theory and applications, is developed in several monographs and articles. He combined infinitely prolonged
differential equations into a category. Its objects, diffieties (differential varieties), are studied in the framework of what he called the secondary calculus. One of the central parts of this
theory is based on the ${\displaystyle {\mathcal {C}}}$-spectral sequence (Vinogradov spectral sequence). The term ${\displaystyle E_{1}}$ of this sequence gives a unified cohomological approach to
many scattered concepts and statements, including the Lagrangian formalism with constraints, conservation laws, cosymmetries, the Noether theorems, and the Helmholtz criterion in the inverse problem
of the calculus of variations (for arbitrary nonlinear differential operators. The ideas underlying the construction of the ${\displaystyle {\mathcal {C}}}$-spectral sequence and the results
following from these ideas were the first decisive steps in the direction of what is now called "cohomological physics".
Vinogradov introduced the construction of a new bracket on the graded algebra of linear transformations of a cochain complex. The construction preceded the general concept of derived bracket on a
differential Loday algebra. The Vinogradov bracket is a skew-symmetric version of the derived bracket generated by the coboundary operator. Derived brackets and their generalizations play an
exceptionally important role in modern applications of homotopy Lie algebras, Lie algebroids, etc., and Vinogradov's results are pioneering in this direction. In particular, Vinogradov showed that
the classical Schouten bracket (on multivector fields) and the Nijenhuis bracket (on vector fields with coefficients in differential forms) are restrictions of his bracket onto the corresponding
subalgebras of superdifferential operators on the algebra of differential forms.
In two last papers he developed a theory of compatibility of Lie algebra structures and proved that any finite-dimensional Lie algebra over an algebraically closed field or over ${\displaystyle \
mathbb {R} }$ can be assembled in a few steps from two elementary constituents, that he called dyons and triadons. Furthermore, Vinogradov speculated that this particle-like structures could be
related to the ultimate structure of elementary particles.
Generally speaking, a significant part of Vinogradov's work was highly motivated by the complex and important problems of modern physics. In particular, much attention was paid to the mathematical
understanding of the fundamental physical concept of the observable in the book "Smooth manifolds and observables", written by A.M. Vinogradov in co-authorship with the participants of his seminar
and published under the pseudonym Jet Nestruev.
Vinogradov's published heritage consists of over a hundred articles and ten monographs. Whatever he worked on, be it the geometry of differential equations, the Schouten and Nijenhuis brackets,
mathematical questions of gravitation theory, ${\displaystyle n}$-ary generalizations of Lie algebras or the structural analysis of the latter, he produced work characterized by a very unorthodox
approach, depth, and nontriviality of the obtained results.
The scientific activity of Vinogradov was not limited to the writing of books and articles. For many years he headed a research seminar at Mekhmat at Moscow State University; the seminar was in two
parts - mathematical and physical - and became a notable phenomenon in Moscow's mathematical life between 1960 and 1980. He had numerous students (in Russia, Italy, Switzerland, and Poland), nineteen
of whom obtained their PhD's under his guidance, six obtained the higher habilitation degree, and one became a corresponding member of the Russian Academy of Sciences. Vinogradov organized and headed
Diffiety Schools in Italy, Russia, and Poland. He was the soul of a series of small "Current Geometry" conferences that took place in Italy from 2000 to 2010, as well as of the large Moscow
conference "Secondary Calculus and Cohomological Physics".
A.M. Vinogradov was one of the initial organizers of the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna, as well as of the Journal of Differential Geometry and
Applications, remaining one of the editors to his last days. In 1985 he created a department that studied various aspects of the geometry of differential equations at the Institute of Programming
Systems in Pereslavl-Zalessky and was its scientific supervisor until he left for Italy. He was one of the organizers and first lecturers in the unofficial school for students who were not accepted
to Mekhmat because they were ethnically Jewish.
Alexandre Vinogradov was a versatile person — he played the violin, wrote poetry in Russian and Italian, played for the Mekhnat water-polo team, was an enthusiastic football player. But the most
important thing for him was, undoubtedly, mathematics. He was full of bright and fruitful ideas and actively worked until his death on September 20, 2019.
List of publications
• A. De Paris, A. M. Vinogradov, Fat Manifolds and Linear Connections, World Scientific, 2008, xii+297 pp., DOI: 10.1142/6904.
• Jet Nestruev, Smooth manifolds and observable, Grad. Texts in Math., 220, New York: Springer-Verlag, pp. XIV+222, 2003, DOI: 10.1007/b98871.
Russian original: Moscow, MCCMO Publ., 317 pp., 2000.
Second extended and revised English edition: Grad. Texts in Math., 220, New York: Springer-Verlag, pp. XVIII+433, 2020, DOI: https://doi.org/10.1007/978-3-030-45650-4.
• A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, AMS, series: Translations of Mathematical Monograph, 204, 2001, AMS bookstore.
• I. S. Krasil'shchik , A. M. Vinogradov (eds.), Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, AMS, Translations of Mathematical Monograph series, 182,
xiv+333 pp., 1999, AMS bookstore, zbl 0911.00032.
Parallel Russian edition: Moscow, Factorial Publ. House, 461 pp., 1997.
Second extended and revised Russian edition: Moscow, Factorial Publ. House, 380 pp., 2005.
• D. V. Alekseevski, V. V. Lychagin, A. M. Vinogradov, Basic ideas and concepts of differential geometry, Geometry I. Encycl. Math. Sci. 28, 255 pp., 1991, Mi intf108, MR 1315081, Zbl 0675.53001.
Russian original: «Modern problems of mathematics: fundamental directions», Vol. 28, 1988, 298 pp., Moscow, VINITI.
• I. S. Krasil'shchik, V. V. Lychagin, A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics, 1, Gordon and Breach, New York,
London. xx+441 pp, 1986.
• A. M. Vinogradov, I. S. Krasil'shchik, V. V. Lychagin, Introduction to geometry of nonlinear differential equations (Russian), «Nauka», Moscow, 336 pp, 1986.
• A. M. Vinogradov, I. S. Krasil'shchik, V. V. Lychagin, Geometry of nonlinear differential equations (Russian), Moscow Institute of Electronic Engineering, 86 pp, 1982.
• A. M. Vinogradov, I. S. Krasil'shchik, V. V. Lychagin, Application of nonlinear differential equations in civil aviation (Russian), Moscow Insitute of Civil Aviation Engineering, 123 pp., 1977.
• A. M. Vinogradov, Algebraic Topology (Russian), Moscow Institute of Electronic Engineering, 232 pp., 1970.
2015 — 2019
• A. M. Vinogradov, Logic of differential calculus and the zoo of geometric structures, «Geometry of Jets and Fields», Banach Center Publications, 2016, 110, 257-285, 2015, arXiv:1511.06861.
2010 — 2014
• A. M. Vinogradov, Some remarks on contact manifolds, Monge-Ampáre equations and solution singularities, International Journal of Geometric Methods in Modern Physics, 14 pp., 2014, arXiv:1403.1742
• A. M. Vinogradov, What are symmetries of nonlinear PDEs and what are they themselves? In «Lie and Klein: The Erlangen program and its impact on mathematics and physics» (ed: A. Papadopoulos and
L. Ji), European Mathematical Society Publishing House, 45 pp., 2014, arXiv:1308.5861.
• A. M. Vinogradov, Assembling Lie algebras from lieons, arXiv:1205.6096v1 [math.DG], 99 pp., 2012.
• D. Catalano Ferraioli, A. M. Vinogradov, Differential invariants of generic parabolic Monge-Ampere equations, J. Phys. A: Math. Theor., 45, 265204, 24 pp., 2012, arXiv:0811.3947.
• A. De Paris, A. M. Vinogradov, Scalar differential invariants of symplectic Monge-Ampere equations, Cent. Eur. J. Math., 9, no.4, 731-751, 2011, arXiv:1102.0426.
2005 — 2009
• A. M. Vinogradov, On geometry of second order parabolic differential equations in two independent variables, Doklady Akademii Nauk, 2008, 423:5, 588-591, Mi dan189, MR 2498570 (Russian).
English trans. Doklady Mathematics, 2008, 78, no. 3, 887-890, DOI: 10.1134/S1064562408060227, DIPS-01/08.
• C. Di Pietro, A. M. Vinogradov, A spectral sequence associated with a symplectic manifold, Dokl. Acad. Nauk, 2007, 413:5, 591-593 (Russian), Mi dan662, MR 2458550.
English translation: Doklady Mathematics, 2007, 75:2, 287-289, arXiv:math/0611138, DOI: 10.1134/S1064562407020287.
• G. Moreno, A. M. Vinogradov, Domains in infinite jet spaces: ${\displaystyle {\cal {C}}}$-spectral sequence, Dokl. Acad. Nauk, 2007, 4132, 154-157, (Russian), Mi dan689, MR 2456137.
English translation: Doklady Mathematics, 2007, 75:2, 204-207,arXiv:math/0609079, DOI: 10.1134/S1064562407020081.
• M. Marvan, A. M. Vinogradov and V. A. Yumaguzhin, Differential invariants of generic hyperbolic Monge-Ampere equations, Cent. Eur. J. Math., 2007, 5, no. 1, 105-133, arXiv:nlin/0604038, DOI:
• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: Λ[k-1]${\displaystyle {\cal {C}}}$-spectral sequence on infinitely prolonged equations, Dokl. Acad. Nauk, 2007, 416:3, 298-301, Mi
dan547, 2458866 MR 2458866, (Russian).
English translation: Doklady Mathematics, 2007, 76:, 692-695, arXiv:math/0703761, DOI: 10.1134/S1064562407050146.
• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: Λ[k-1]${\displaystyle {\cal {C}}}$-spectral sequence on infinite jets, Dokl. Acad. Nauk, 2007, 416:2, 161-165 (Russian), Mi dan556,
MR 2450915.
English translation: Doklady Mathematics, 2007, 76:2, 673-677, arXiv:math/0703661, DOI: 10.1134/S1064562407050092.
• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: the ${\displaystyle {\cal {C}}}$-spectral sequence, Dokl. Acad. Nauk, 2007, 414:1, 447-450 (Russian), Mi dan629, MR 2451933.
English translation: Doklady Mathematics, 2007, 75:3, 403-406, arXiv:math/0610917, DOI: 10.1134/S1064562407030192.
• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: integral calculus, Dokl. Acad. Nauk, 2007, 413:1, 7-10 (Russian), Mi dan697, MR 2447059.
English translation: Doklady Mathematics, 2007, 75:2, 177-180, arXiv:math/0610914, DOI: 10.1134/S1064562407020019.
• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: Riemannian geometry revisited, Dokl. Acad. Nauk, 2006, 407:2, 151-153 (Russian), Mi dan988, MR 2348307.
English translation: Doklady Mathematics, 2006, 73:2, 182-184, arXiv:math/0609287, DOI: 10.1134/S1064562406020074.
• A. M. Vinogradov, L. Vitagliano, Iterated differential forms: tensors, Dokl. Acad. Nauk, 2006, 407:1, 16-18 (Russian), Mi dan998, MR 2347355.
English translation: Doklady Mathematics, 2006, vol. 73, no. 2, pp.169-171, arXiv:math/0605113, DOI: 10.1134/S1064562406020037.
• D. Catalano Ferraioli, A. M. Vinogradov, Ricci flat 4-metrics with bidimensional null orbits. Part II: the Abelian case, Acta Applicandae Mathematicae, 2006, 92:3, 223-239, DIPS 8/2004.
• D. Catalano Ferraioli, A. M. Vinogradov, Ricci flat 4-metrics with bidimensional null orbits. Part I: General aspects and nonabelian case, Acta Applicandae Mathematicae, 2006, 92:3, 209-223, DIPS
• A. M. Vinogradov, M. Marvan, V. A. Yumagughin, Differential invariants of generic hyperbolic Monge-Ampere equations, Dokl. Acad. Nauk, 2005, 405:3, 299-301, (Russian), Mi dan1076, MR 2264293.
English translation: Doklady Mathematics, 2005, 72:3, 883-885, arXiv:nlin/0604038, DOI: 10.2478/s11533-006-0043-4.
2000 — 2004
• F. Pugliese, A. M. Vinogradov, Discontinuous trajectories of Lagrangian systems with singular hypersurface, J. Math.Phys., 2001, 42(1), 309-329, DOI: 10.1063/1.1324653.
• G.Sparano, G.Vilasi, A. M. Vinogradov, Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries, Phys. Lett., Sec. B. 2001, 513, 142-146, arXiv:gr-qc/0102112, DOI: 10.1016/
1995 — 1999
• F. Pugliese, A. M. Vinogradov, Jumping oscilator, arXiv:math/9902115 [math.DG], 27 pp., 1999.
• A. M. Vinogradov, Introduction to Secondary Calculus, Contemporary Mathematics, 1998, 219, 241-272, Amer. Math. Soc., Providence, Rhode Island, DIPS-05/98, DOI: 10.1090/conm/219/03079, MR 1640456
• A. M. Vinogradov, M. M. Vinogradov, 'On multiple generalizations of Lie algebras and Piosson manifolds, Contemporary Mathematics,1998, 219, 273-287, Amer.Math.Soc., Providence, Rhode Island,,
DIPS-06/98 DOI: 10.1090/conm/219/03080, MR 1640457.
• G. Vezzosi, A. M. Vinogradov, Infinitesimal Stokes' formula for higher order de Rham complexes, Acta Applicandae Mathematicae, 1997, 49(3), 311-329, DOI: 10.1023/A:1005811010161.
1990 — 1994
• F. Lizzi, G. Marmo, G. Sparano, A. M. Vinogradov, Eikonal type equations for geometrical singularities of solutions in field theory, J. Geom. and Phys., 1994, 14, 211-235, preprint ESI 46(1993),
DOI: 10.1016/0393-0440(94)90008-6.
• A. M. Vinogradov, From symmetries of partial differential equations towards secondary («quantized») calculus, J. Geom. and Phys., 1994, 14, 146-194, DOI: 10.1016/0393-0440(94)90005-1.
• A. Cabras, A. M. Vinogradov, Extension of the Poisson bracket to differential forms and multi-vector fields, J. Geom. and Phys., 1992, 9(1), 75-100, DOI: 10.1016/0393-0440(92)90026-W.
• A.M. Verbovetsky, A. M. Vinogradov and D. M. Gessler, Scalar differential invariants and characteristic classes of homogeneous geometric structures, Mat. Zametki, 1992, 51:6, 15-26, Mi mz4625, MR
1187472, Zbl 0814.57019 (Russian).
English translation in Math. Notes (1992), 51(5-6), 543-549, DOI: 10.1007/BF01263295.
• A. M. Vinogradov, Scalar differential invariants, diffeties, and characteristic classes, In: Francaviglia M. (Ed.), Mechanics, Analysis and Geometry: 200 Years after Lagrange, Elsevier,
Amsterdam, 1991, 379-416, DOI: 10.1016/B978-0-444-88958-4.50020-3.
• A. M. Vinogradov, V. A. Yumaguzhin, Differential invariants of webs on two-dimensional manifolds, Mat. Zametki, 1990, 48:1, 26-37, Mi mz3280, MR 1081890, Zbl 0714.53019 (Russian).
English translation in Math. Notes, 1991, 48(1), 639-647, DOI: 10.1007/BF01164260.
• A. M. Vinogradov, A common generalization of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators, Mat. Zametki, 1990, 47:6, 138-140, Mi mz3270, MR 1074539, Zbl
0712.58059, (Russian).
1985 — 1989
• I. S. Krasil'shchik, A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math., 1989, 15:1,
161-209, DOI: 10.1007/BF00131935.
Also in: «Symmetries of Partial Differential Equations», ed. by A. M. Vinogradov, Kluwer Acad. Publ., Dordrecht, Boston, London, 1989, 161-209.
• V. N. Gusyatnikova, A. V. Samokhin, V. S. Titov, A. M. Vinogradov, V. A. Yumaguzhin, Symmetries and conservation laws of Kadomtsev-Pogutse equations (their computation and first applications),
Acta Appl. Math. 1989, 15(1), 23-64, DOI: 10.1007/BF00131929.
• A. M. Vinogradov, Symmetries and conservation laws of partial differential equations: basic notions and results, Acta Appl. Math., 1989, 15(1), 3-21, DOI: 10.1007/BF00131928.
• A. M. Vinogradov, An informal introduction to the geometry of jet spaces, Conference on Differential Geometry and Topology (Sardinia, 1988). Rend. Sem. Fac. Sci. Univ. Cagliari, 1988, 15, suppl.,
• A. M. Vinogradov, Integrability and symmetries, in «Nonlinear waves. Structures and bifurcations», Moscow, «Nauka», 1987, 279-290 (Russian).
• A. M. Astashov, A. M. Vinogradov, On the structure of Hamiltonian operator in field theory, J. Geom. and Phys., (1986), 3:2, 263-287, DOI: 10.1016/0393-0440(86)90022-7.
• A. M. Vinogradov, Geometric singularities of solutions of nonlinear partial differential equations, Proc. Conf. «Differential geometry and its applications», Brno, 1986, Math. Appl. (East
European Ser.), Reidel, Dordrecht-Boston, MA, 1987, 27, 359-379,.
• A. M. Vinogradov, A. V. Samokhin, The Cartan-Kähler theorem, Transactions of Seminar on Algebra and Geometry of Differential Equations, Moscow, VINITI, 1986,858-B, 112-132 (Russian).
• A. M. Vinogradov, A. V. Samokhin, On quotiening of partial differential equations, Transactions of the Seminar of Algebra and Geometry of Differential Equations, Moscow, VINITI, 1986,858-B,
133-146 (Russian).
• A. M. Vinogradov, Why is the space 3-dimensional and how may groups be seen?, Acta Appl. Math., 1986, 5(2), 169-180, DOI: 10.1007/BF00046586.
• A. M. Vinogradov, Geometry of differential equations, secondary differential calculus and quantum field theory, Soviet Mathematics (Izvestiya VUZ. Matematika), 1986, 1, 13-21, Mi ivm7465, MR
838427, Zbl 0616.58009 (Russian).
English translation Soviet Math. (Iz. VUZ), 1986, 30:1, 14–25.
• A. M. Vinogradov, V. N. Gusyatnikova, V. A. Yumaguzhin, Secondary differential operators, Dokl. Akad. Nauk SSSR, 1985, 283:4, 801-805, Mi dan9040, MR 802682, Zbl 0598.58009 (Russian).
English transl. in Soviet Math.Dokl., 1985, 32:1, 198-202.
1980 — 1984
• A. M. Vinogradov, Category of partial differential equations, «Global Analysis - Studies and Applications I», Lecture Notes in Math., 1984, 1108, 77-102, DOI: 10.1007/BFb0099553.
• A. M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., 1984, 3, 21-78, DOI: 10.1007/BF01405491.
• I. S. Krasil'shchik, A. M. Vinogradov, Nonlocal symmetries and the theory of coverings: addendum to to A. M. Vinogradov's «Local symmetries and conservation laws», Acta Appl. Math., 1984, 2(1),
79-96, DOI: 10.1007/BF01405492.
• I. S. Krasil'shchik, A. M. Vinogradov, On the theory of nonlocal symmetries of nonlinear partial differential equations, Dokl. Akad. Nauk SSSR, 1984, 275:5, 1044-1049, Mi dan9705, MR 745842, Zbl
0604.58053 (Russian).
English transl. in Sov. Math. Dokl., 1984, 20:2, 337-341.
• A. M. Vinogradov, Category of differential equations and its significance for physics, In: Krupka D. (Ed.), Proc. Conf. Diff. Geom. Appl. (Brno, 1984), J.E. Purkynue Univ., Brno, Czechoslovakia,
• A. M. Vinogradov, Category of nonlinear differential equations (Russian), addendum to the Russian translation of: J.-F. Pommaret, «Systems of partial differential equations and Lie pseudogroups»
(translated by A. V. Bocharov, M. M. Vinogradov and I. S. Krasil'shchik), Moscow, Mir, 400 pp., 1983.
• A. M. Vinogradov, Higher symmetries and conservation laws, in «Group-theoretic methods in physics», 1983, 2, 414-420, Moscow, Nauka, (Russian).
• A. M. Vinogradov, Category of nonlinear differential equations, Equations on manifolds, Novoe v Global. Anal., Voronezh. Gos. Univ., Voronezh, 1982, 26-51 (Russian).
• A. M. Vinogradov, Conservation laws, the Spencer cohomology and the ${\displaystyle {\cal {C}}}$-spectral sequence, in «Leningrad international topology conference», Leningrad, Nauka, 1982, p.
166 (Russian).
• A. M. Vinogradov, Geometry of nonlinear differential equations, Itogi Nauki i Tekhniki. Ser. Probl. Geom., 11, Moscow, VINITI, 1980, 89–134, Mi intg121, MR 579929, Zbl 0475.58025|0461.58012.
English translation: Journal of Soviet Mathematics, 17(1), 1624-1649, 1981, DOI: 10.1007/BF01084594
• A. M. Vinogradov, Category of nonlinear differential equations, «XV Voronezh winter mathematical school», Voronezh Gos. Univ. Publ., Moscow, VINITI, 5691, 1981, 9-10 (Russian).
• A. M. Vinogradov, I. S. Krasil'shchik, A method of computing higher symmetries of nonlinear evolution equations and nonlocal symmetries, Dokl.Akad.Nauk SSSR (1980) 253, 1089-1093, Mi dan43819, MR
0583788, Zbl 0498.35076 (Russian).
English transl. in Soviet Math. Dokl., 1980, 22, 235-239.
• A. M. Vinogradov, Geometry of nonlinear differential equations, Problems in geometry, 1980, 11, 89-134, Moscow, VINITI, Mi intg121, MR 579929, Zbl 0475.58025|0461.58012 (Russian).
English transl. in J.Sov.Math., 1981, 17, 1624-1649, DOI: 10.1007/BF01084594.
1975 — 1979
• A. M. Vinogradov, Some new homological systems associated with differential calculus over commutative algebras, Uspechi Mat. Nauk, 1979, 34:6, 145-150, Mi umn4163, MR 562827, Zbl 0475.58024|
0476.58028, (Russian).
English transl. in Russian Math. Surveys, 1979, 34:6, 250-255, DOI: 10.1070/RM1979v034n06ABEH003355.
• A. M. Vinogradov, Theory of higher infinitesimal symmetries of nonlinear partial differential equations, Dokl.Akad.Nauk SSSR, 1979, 248:2, 274-278, Mi dan42982, MR 0553187, Zbl 0445.58030
English transl. in Soviet Math. Dokl., 1979, 20, 985-989.
• A. M. Vinogradov, A spectral system associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Dokl.Akad.Nauk SSSR,
1978, 238:5, 1028-1031, Mi dan41521, MR 0483733, Zbl 0406.58015 (Russian).
English translation in Soviet Math. Dokl., 1978, 19, 144-148.
• A. M. Vinogradov, Hamiltonian structures in field theory, Dokl.Akad.Nauk SSSR, 1978, 241:1, 18-21, Mi dan41816, MR 0510883, Zbl 0421.70026 (Russian).
English transl. in Soviet Math. Dokl., 1978, 19:4, 790-794.
• A. M. Vinogradov, On the algebro-geometric foundations of Lagrangian field theory, Dokl.Akad.Nauk SSSR,1977, 236:2, 284-287, Mi dan41214, MR 0501142, Zbl 0403.58005 (Russian).
English transl. in Soviet Math. Dokl., 1977, 18:5 , pp. 1200-1204.
• A. M. Vinogradov, B. A. Kupershmidt, The structure of Hamiltonian mechanics, Uspechi Mat.Nauk, 1977, 32:4, 175-228, Mi umn3221, MR 501143, Zbl 0365.70016|0383.70020 (Russian).
English transl. in Russian Math. Surveys, 1977, 32:4, 177--232; also in London Math. Soc. Lect. Notes, 1981, 60, 173-228, DOI: 10.1070/RM1977v032n04ABEH001642.
• A. V. Bocharov, A. M. Vinogradov, The Hamiltonian form of mechanics with friction, non-holonomic mechanics, invariant mechanics, the theory of refraction and impact, addendum II in A. M.
Vinogradov, B. A. Kupershmidt, The structure of Hamiltonian mechanics, Uspechi Mat.Nauk, 1977, 32:4, 228-236, Mi umn3221 (Russian).
English transl. in Russian Math. Surveys, 1977, 32:4, 232-243; also in London Math. Soc. Lect. Notes, 1981, 60, 229-239, DOI: 10.1070/RM1977v032n04ABEH001642.
• A. M. Vinogradov, Theory of symmetries of non-linear differential equations, DEP 2855-74, Moscow, VINITI, 1974, 16 pp. (Russian).
• A. M. Vinogradov, I. S. Krasil'shchik, What is the Hamiltonian formalism?, Uspechi Mat.Nauk, 1975, 30:1, 173-198, Mi umn4140, MR 650307, Zbl 0327.70006 (Russian).
English transl. in Russian Math. Surveys,1975, 30, 177-202, also in London Math. Soc. Lect. Notes,1981, 60, 241-266, DOI: 10.1070/RM1975v030n01ABEH001403.
1970 — 1974
• A. M. Vinogradov, Multivalued solutions and a principle of classification of nonlinear differential equation, Dokl.Akad.Nauk, 1973, 210:1, 11-14, Mi dan37624, MR 0348799, Zbl 0306.35003,
English transl. in Soviet Math. Dokl., 1973, 14:3, 661-665.
• A. M. Vinogradov, The logic algebra for the theory of linear differential operators, Dokl.Akad.Nauk, 1972, 205:5, 1025-1028, Mi dan37058, MR 0304363, Zbl 0267.58013 (Russian).
English transl. in Soviet Math. Dokl., 1972, 13:4, 1058-1062.
1965 — 1969
• A. M. Vinogradov, S. P. Novikov, Geometric and differential topology, in «History of Soviet Mathematics», Naukova Dumka, Kiev, 1968, 3, 511-529, (Russian).
• A. M. Vinogradov, Some properties of knots, addendum to Russian transl. of «Introduction to knot theory» by R.Crowell and R.Fox, Moscow, Mir, 1967, 284-309 (Russian).
1960 — 1964
1958 — 1959
• B. N. Delaunay, A. M. Vinogradov, Uber den Zussamenhang zwishen den Lagrangeschen Klassen der Irrationalitaten mit begretzen Leilennern und Markoffschen Klassen der extremen Formen, in «Ehren 250
Geburtstages L. Eulers», Akad. Verlag, Berlin, 1959, 101-106.
Edited collections and proceedings
• M. Henneaux, I. S. Krasil'shchik, A. M. Vinogradov (Eds.), Secondary calculus and cohomological physics, Proc. conf. «Secondary calculus and cohomological physics», August 24-31, 1997, Moscow;
Contemporary Mathematics, 1998, vol. 219.
• I. S. Krasil'shchik, A. M. Vinogradov (Eds.), Algebraic aspects of differential calculus, special issue of Acta Applicandae Mathematicae, 1997, 49:3. Also in The Diffety Inst. Preprint Series,
DIPS 1/96 -DIPS 8/96.
• A. M. Vinogradov (Ed.), Symmetries of partial differential equations: conservation laws, applications, algorithms, Kluwer Acad. Publ., Dordrecht, Boston, London, 1989, vi+456 pp.
• A. M. Vinogradov (Ed.), Transactions of the seminar «Algebra and geometry of differential equations», VINITI, 1986, Dep. 858-B, Moscow.
Addendum: doctoral dissertation | {"url":"https://gdeq.org/Alexandre_Vinogradov","timestamp":"2024-11-13T22:09:45Z","content_type":"text/html","content_length":"94110","record_id":"<urn:uuid:9a609a33-5d45-4519-978e-de0d7d4676ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00272.warc.gz"} |
Section 6.2 in Matter and Interactions (4th edition)
The Simplest System: A Single Particle
The energy principle is widely applicable and helps to explain or to predict the motion of systems by considering how the system exchanges energy with its surroundings. For now, you will read about
the simplest of systems, that of a single particle. In these notes, you will read about the total energy of a particle, the energy due to its motion, and how those energies are connected in
situations where we can neglect the heat exchanges.
Lecture Video
The Total Energy of a Single Particle
The systems that you will consider will be approximated by a single object, the point particle. The point particle is an object that has no size of its own, but carries the mass of the object it is
meant to represent. This point particle experiences the same force that the real object experiences, and thus models the motion of that real physical system to the extent that you only care about how
the object translates (moves without rotation). Point particles do not spin or change their shape. Later, we will relax these conditions.
Thanks to Einstein, we know the total energy of a single particle system is given by,
$$E_{tot} = \gamma m c^2$$
where $m$ is the mass of the particle, $c$ is the speed of light in vacuum (3$\times10^8$ m/s), and $\gamma$ is the correction due to relativity when the particle is moving near the speed of light.
If a system of one particle is at rest ($v=0$) then,
$$E_{tot} = \gamma m c^2 = \dfrac{1}{\sqrt{1-(v^2/c^2)}}mc^2 = \dfrac{1}{\sqrt{1-(0^2/c^2)}} mc^2 = mc^2$$
Evidently, a particle at rest has a total energy that is simply associated with its mass. This is called the rest mass energy of that particle and really matters when particles change their identity
(e.g., in chemical or nuclear reactions).
$$E_{rest} = mc^2$$
It appears that the rest of the energy is associated with the motion of the particle. As such, it is refereed to as the kinetic energy (J) of the particle.
$$K = E_{tot} - E_{rest} = \gamma m c^2 - mc^2 = (\gamma - 1)mc^2$$
This is probably not the form of the kinetic energy that you are used to seeing. This is because for most purposes, objects are moving slowly enough where the relativistic correction doesn't matter.
At low speeds,
$$K = (\gamma - 1)mc^2 = \left(\dfrac{1}{\sqrt{1-v^2/c^2}}-1\right) mc^2 \approx \left(\left(1+\dfrac{1}{2}\dfrac{v^2}{c^2}\right)-1\right)mc^2 = \dfrac{1}{2}\dfrac{v^2}{c^2} mc^2 = \dfrac{1}{2}mv^
This definition of kinetic energy is due to Newton, but was confirmed by Coriolis and others. The total energy of a particle is thus the sum of its rest mass energy and its kinetic energy, which at
low speeds is given by,
$$E_{tot} = E_{rest} + K = mc^2 + \dfrac{1}{2}mv^2$$
For the time being you will neglect heat exchanges (although you will later relax that assumption), so that the system of a single particle system changes its total energy as a result of work by the
$$\Delta E_{tot} = \Delta E_{rest} + \Delta K = W_{surr}$$
If the particle does not change its identity, then there is no change in rest mass energy and you are left with,
$$\Delta K = K_f - K_i = W_{surr}$$
This is often called the “Work-Kinetic Energy Theorem”, but it's just a restricted version of the energy principle.
Work: Mechanical Energy Transfer
Let's consider the case where a particle doesn't change its identity, so the system simply changes its kinetic energy,
$$\Delta K = W_{surr} = W$$
where we can drop the subscript for “surroundings” knowing full well that the work is done by the interactions the system has with its surroundings.
Consider this analogy. From the momentum principle, the net force acting over some time results in a change in momentum,
$$\Delta \vec{p}_{sys} = \vec{F}_{net}\Delta t$$
This expression relates the net force and the time over which it acts to change in momentum, and thus, a change in velocity. Is there a quantity that is related to distance over which the net force
$$\Delta \mathrm{??} = (\mathrm{net\:force})*(\mathrm{distance})$$
As it turns out, this thing is the energy of the system, or in this restricted case, the kinetic energy of the particle.
$$\Delta K = W = Fd$$
Update form of the Energy Principle
You can rewrite the energy principle to predict the final kinetic energy if you know the initial kinetic energy and the work done by the surroundings.
$$\Delta K = K_f - K_i = W$$
$$K_f = K_i + W$$
This is the update form of the energy principle for a single particle that doesn't change its identity. | {"url":"https://msuperl.org/wikis/pcubed/doku.php?id=183_notes:point_particle","timestamp":"2024-11-05T19:52:16Z","content_type":"application/xhtml+xml","content_length":"39203","record_id":"<urn:uuid:d2d0fc8a-e3f4-4660-b2c2-673adac578b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00228.warc.gz"} |
body surface area (Costeff)
`BSA_"Costeff" = (4 * "w" + 7)/(90 + "w" )`
Enter a value for all fields
The Costeff Body Surface Area calculator computes the body surface area of the human body based on the person's weight (W) using the Costeff formula (below).
INSTRUCTIONS: Choose your preferred weight units and enter the following:
• (W) Enter the weight of the person.
BSA:The weight is converted to kilograms (kg), and the BSA is expressed in units of square meters (m²). However, the BSA can be automatically converted into different area via the pull-down menus.
General Information
The Costeff formula is as follows:
BSA = (4W +7)/(90 +W)
BSA is used in physiology and medicine. The default input units are kilograms (kg) for weight, and meters squared (m^2). However, the calculator provides automatic conversions for both the inputs
and outputs to other units via the pull-down menus.
Body Surface Area (BSA) is the measured or calculated surface area of a human body frequently used in physiology and medicine. For many clinical purposes BSA is a better indicator of metabolic mass
than body weight because it is less affected by abnormal adipose mass. Estimation of BSA is simpler than many measures of volume.
Demographic BSA Means
Mean Body Surface Area (male and
│ Age Range │ Male │ Female │
│Neonate │0.243 m²│0.234 m²│
│2 years old │0.563 m²│0.54 m² │
│5 years old │0.787 m²│0.771 m²│
│10 years old │1.236 m²│1.245 m²│
│13 years old │1.603 m²│1.55 m² │
│18 years old │1.98 m² │1.726 m²│
│20 to 79 years old │2.06 m² │1.83 m² │
│80 and above │1.92 m² │1.638 m²│
Other BSA Applications
• The BSA Compare function lets you enter a body surface area and choose a demographic (above) to compute the percent compared to the mean.
• Wallace Rule of Nines: This computes a percentage of the human body for burn victims based on percentages allocated to different body parts.
• Rule of Fives: This computes a percentage of the human body for obese burn victims.
• Parkland Replacement Fluid: This compute the volume of replacement fluids needed in the first 24 hours based on the patients weight (mass) and the percent of their body that has been burned.
• BSA Percent: This computes a body surface area based on the total body surface area and a percent..
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Natural Roasts
People typically talk about the art of coffee roasting. We're interested in the science. Therefore, we keep adding features to Artisan that allow to analyze and compare roasts. Here we want to
introduce our latest observation and the tools now available in Artisan v0.9.8 to play with the underlying idea. On September 2015 Rafael Cobo added a post on HomeBarista titled Natural logarithm
curve for roasting triggered by a discussion about Scott Rao on The Flick.
"A decreasing deltaBT would theoretically approximate a natural logarithmic curve since the ln() function has a positive slope that decreases steadily as time increases."
This is an interesting obvervation, because the natural logarithm
is based on Euler's number
, which is understood as one of the fundamental mathematical constants, similar to π. Further, there is an interesting
work on chemical kinetics
that explains the relation of logarithms to the development of chemical reaction.
Natural Roast Model
Rafael proposed the following general formula to compute the temperature T in degrees (in C or F) to describe any profile that is steadily decreasing.
T = A*log(B*t-C,e)
T: the temperature in degrees (in C or F)
t: the time in seconds since CHARGE
e: Euler's constant as base of the natural logarithm (e=2.71828)
A: a multiplication factor
B: rate of change in combination with A
C: time shift to the right
The example Rafael gave for such a "natural" ln()-roast approximation
T = 65*log(1.618*t-350,e)
is plotted as thin black curve in the following profile using the
Artisan plotter
. Note that the above formula (without the "
" can be directly pasted into one of the plotter fields in Artisan).
Note that this ln() curve fits nicely the bean temperature (BT) from about 300F (DRY) on. Note also that this roast mostly fullfils Rao's criteria of an always declining BT rate-of-rise (RoR) curve
drawn in light blue as computed by Artisan from the BT curve displayed in dark blue. Only towards the end of the roast, at the point the BT deviates from the ln()-curve, its RoR rises slightly.
The following shows the shape the RoR (in light blue) corresponding to this logarithmic curve (here in dark blue) as computed by Artisan.
To summarize, the observations that drove Rafael to its natural roasts model were:
1. The ln() curve has an always positive slope that decreases steadily (the RoR incrementally decreases over time). This models the current trend in roasting in terms of RoR for best results based
on empirical results.
2. The concentration of chemical products in a first order reaction follows a logarithmic curve. Most reactions speed up as temperature increases and the concentration of reactants decreases. There
could be some parallels between RoR and speed of the chemical reaction.
Natural Roasts in Artisan
Looking at my recent roast profiles I could see that most of my successful ones seem to follow that natural roast model. However, I had a hard time to find the right constants
A, B
to draw the corresponding natural logarithm curve on my profiles using the plotter.
Therefore, I added a function in v0.9.8 that calculates the best approximation of the current profile and thus automatically determining those three constants. The function takes the following three
time/temperature points as input from the current profile:
1. CHARGE (the moment the beans are filled into the machine): assuming the BT of the green beens to be at room temperature standardized to 22C/70F (ignoring the BT reading of the profile here)
2. DRY (the moment the beans turn yellow)): an early point at which the measured BT can be assumed to be close enough to the real temperature of the beans
3. FCs (the moment the beans start to make a cracking noise for the first time): a second point that can be determined relatively accurate
All one has to do now, is to load a profile and open the math menu (menu Tools >> Extras; Math tab) and click the Show flag in the ln() part.
Note that if the points DRY and FCs are not defined in the current profile (or erased by putting 0:00 in the corresponding time edit element of the Roast Properties dialog), those points are taken
from the BT at the intersection with the corresponding phases limits. As those limits can be freely defined in the Phases Dialog (menu Config >> Phases) this allows to define those points as needed.
The resulting ln() approximation is shown as formula in the math tab (see the screenshot above), including the computed values for the constants
A, B
. The curve itself is drawn on top of the current profile as black dotted line with the three anchor points marked by red dots.
The calculated ln() formula displayed in the math tab can be copied directly into one of the plotter fields. If we choose the field P1 or P2, we can finally establish the curve as background profile
by pressing the "Background" button, visualize the corresponding ln() RoR curve, and use it as template for further roasts.
The only thing that still remains to be decide upon is the temperature at which to DROP the roast best. Here, Rao's 20-25% commandment could be applied as supported in Artisan by the live display of
the phases ratios in the Phases LCDs displayed on the top of the main window on them. Remember that one needs to right-click on those Phases LCDs to get the ratios displayed.
Looking at the last profile above one can see how close this ln() approximation fits to my roast P164. That observation holds for a lot of roasts done on my 80 years old cast-iron Probat drum roasts,
but also on those I did on the latest Probatone.
It has been observed that this natural roast style better fits the slow-start and fast-finish roasts (SSFF) than to the fast-start slow-finish roasts (FSSF) as advocated by Rao and some others. While
faster and slower roasts are possible along those ln() approximations as shown below, a real FSSF roast would suggest a lower RoR then given by the ln() formula at least from FCs on. But lowering the
RoR during the FCs too much has also been found to result in weaker roasting results (cf. Rob Hoos' book on Modulating the Flavor Profile of Coffee).
Here is a profile roasted on a small dynamic air roaster. A typical FSSF profile. It remains to been seen if a roast along this ln() approximation leads to better results than the original profile or
if a compromise, roasting along the ln() approximation until FCs and then following a slightly lower RoR towards the end of the roast, would win.
An approximation using a quadratic function of the form
T = A*t^2 + B*t + C
was also discussed on HomeBarista and has been added to Artisan. While closer to the
FSSF roasting style, this
one seems to have the problem that it drops
at the end of the roast
even if the DROP point is considered in the regression as shown in the following.
The quadratic approximation seems to work better for that air roaster profile from above. Just the first phase seem to deviate significantly as can see below. | {"url":"https://artisan-roasterscope.blogspot.com/2015/10/natural-roasts.html","timestamp":"2024-11-05T09:24:29Z","content_type":"application/xhtml+xml","content_length":"85195","record_id":"<urn:uuid:cf88750e-e265-454b-aaee-cc0bb81a0be8>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00213.warc.gz"} |
Conference Proceedings
Conference Proceedings–Raghu Raghavan
• [59] J. H. Sampson, D. A. Reardon, A.H.Friedman, H.S. Friedman, J.M.Provenzale, D. D. Bigner, M. Brady, R. Raghavan, C. Pedain, G. Archer, D. Lally-Batts,A. Grahn, T. Cohen, J. L. Dul, D.
Croteau, and R. K. Puri. Convection-enhanced delivery of IL13-PE38QQR in malignant glioma: effect of catheter placement on drug distribution. Proceedings of the American Association of
Neurosurgeons, 2004.
• [58] F. Weber, E. Bauer, M. Brady, R. Raghavan, A. Hartlep, and C. Pedain. Assessing drug disposition in convection enhanced drug delivery using Gadolinium DTPA and computerized modeling.
Proceedings of the Society of Neurooncology, 2003.
• [57] Z. Ram, Y. Mardor, T. Jonas, M. Brady, R. Raghavan, C. Pedain, and P. Tanner. Preliminary assessment of a model for prediction of convection-enhanced delivery. Proceedings of the American
Association of Neurosurgeons, 2003.
• [56] R. Raghavan, N. Konyer, G. Stanisz, M. Brady, and M. Bronskill. Magnetic resonance veri.cation of intra-parenchymal infusions. Proceedings of the International Society of magnetic resonance
in medicine, 2003.
• [55] P. Tanner, F. W. Kreth, R. Goldbrunner, M. Holtmannspotter, R. Raghavan, M. Brady, C. Pedain, and J. C. Tonn. MR basierter Simulationsalgoithmus zur konvektions-gestutzten intratumoralen
Pharmakotherapie. DGNC Neuroonkologie, 2002.
• [54] Zvi Ram, Zvi Lidar, Raphael Pfeffer, Yiftach Roth, Tali Jonas, Dvora Nass, Martin Brady, Raghu Raghavan, Christoph Pedain, Philipp Tanner, and Yael Mardor. Convection-enhanced taxol delivery
for treatment of recurrent glioblastoma; radiological/clinical experience and applications for treatment optimization. Proceedings of the annual meeting of the Israel society of oncology and
radiotherapy, to be published, 2003.
• [53] N. B. Conyer, G. Stanisz, M. Brady, M. J. Bronskill, and R. Raghavan. Magnetic resonance tracking of intra-parenchymal infusions. magnetic resonance in medicine, 2003.
• [52] Zvi Ram, Zvi Lidar, Raphael Pfeffer, Yiftach Roth, Tali Jonas, Dvora Nass, Martin Brady, Raghu Raghavan, Christoph Pedain, Philipp Tanner, and Yael Mardor. Convection-enhanced taxol delivery
for treatment of recurrent glioblastoma; radiological/clinical experience and applications for treatment optimization. Proceedings of the annual meeting of the Israel society of oncology and
radiotherapy, to be published, 2003.
• [51] Zvi Ram, Yael Mardor, Tali Jonas, Martin Brady, Raghu Raghavan, Christoph Pedain, and Philipp Tanner. Preliminary assessment of a model for predicting convection-enhanced delivery.
Proceedings of the American Association of Neurosurgery, to be published, 2003.
• [50] N. B. Conyer, G. Stanisz, M. Brady, M. J. Bronskill, and R. Raghavan. Magnetic resonance tracking of intra-parenchymal infusions. Biomedical Research Opportunities Workshop, to be published,
• [49] N. B. Conyer, R. Viswanathan, R. Raghavan, G. Mills, M. Brady, and M. J. Bronskill. Improved RF coils for internal imaging. Proceedings of the First IEEE International Symposium on
Biomedical Imaging, pages WP–125, 2002.
• [48] N. B. Conyer, R. Viswanathan, R. Raghavan, G. Mills, M. Brady, and M. J. Bronskill. Improved Interstitial and Intravascular RF coils. Proceedings of the Tenth ISMRM meeting, page 2262, 2002.
• [47] N. Conyer, N. J. Lobaugh, G. Sela, R. Raghavan, G. Mills, M. Brady, R. Viswanathan, W. K. Sootsman, and M. J. Bronskill. Interstitial RF coil incorporating multiple microcatheters.
Proceedings of the ISMRM-ESMRMB joint annual meeting, page 2170, 2001.
• [46] M. Brady, R. Raghavan, S. R. Ranjan, and R. Viswanathan. Multiscale mechanics in structural biology (Abstract only). Fourth Annual Hilton Head Workshop on Computational Modeling in
Biological Systems, page 62, 2000.
• [45] M. Brady, K. Jung, W. Lawton, R. Mullick, H.T. Nguyen, T. Poston, R. Raghavan, S. R. Ranjan, K. Schulz, S. Venkataraman, R. Viswanathan, Y. Yu, and G. Zhu. Interactive haptic modeling of
tensegrities and network structures . Siggraph ’99 Technical Sketch and Creative Applications, page 252, 1999.
• [44] S. Meiyappan, R. Raghavan, R. Viswanathan, and Y. Yu. Proteinmorphosis: a mechanical model for protein conformational changes. Proceedings of the Paci.c Symposium on Biocomputing, ed. R.
Altman et. al., 2:341–353, 1999.
• [43] James C. Anderson and Raghu Raghavan. A Vascular Catheterization Simulator for Training and Treatment Planning. Proceedings of the Society of Computer Assisted Radiology, 11, Supp.
1:120–123, 1998.
• [42] Raghu Raghavan, Timothy Poston, and Rakesh Mullick. Biomedical image computing, medical simulators, and Image-guided therapies. Proceedings of the 9’th International Conference on Biomedical
Engineerin, pages 68–71, 1997.
• [41] R. Viswanathan, T. Poston, R. Raghavan, and Y. Yu. Robot Tentacles: dynamics and control of robot arms. Proceedings of the IEEE International Symposium on Control Theory and Applications,
pages 480–484, 1997.
• [40] Raghu Raghavan. Old Mathematics and New Appications: System Modeling and Control in Biomedicine and Pattern Analysis. Proceedings of the IEEE International Symposium on Control Theory and
Applications, pages 137–145, 1997.
• [39] Y. Y. Cai, Y. P. Wang, C. K. Chui, R. Viswanathan, and R. Raghavan. Integration of Geometric and Physical Modeling in realtime Medical Simulation. Proceedings of the 4th International
Conference on Computer Integrated Manufacturing, Singapore, 1997.
• [38] Y. Y. Cai, R. Viswanathan, Y. P. Wang, C. K. Chui, and R. Raghavan. Simulation of catheter-guidewire interaction for catheterization by arc parametrization. Fourth International Conference
on Control, Automation, Robotics, and Vision, pages 2466– 2470, 1996.
• [37] J. A. Anderson, R. Raghavan, W. R. Brody, C. J. Kriz, Y. P. Wang, Y. Y. Cai, R. Viswanathan, and C. K. Chui. daVinci: A vascular catheterization and interventional radiology-based training
and patient pretreatment planning simulator. Journal of Vascular and Interventional Radiology JVIR, 7, Part 2:373, 1996.
• [36] R. Mullick, H. T. Nguyen, Y. P. Wang, J. K. Raphel, and R. Raghavan. Overview of Visible HumanTM based applications at CIeMed. Proceedings of the First Visible Human Project Conference,
Bethesda, Maryland, pages 119–120, 1996.
• [35] C. K. Chui, H. T. Nguyen, Y. P. Wang, R. Mullick, and R. Raghavan. Potential .eld of Vascular anatomy for real-time computation of catheter navigation. Proceedings of the First Visible Human
Project Conference, Bethesda, Maryland, pages 113–114, 1996.
• [34] Y. P. Wang, C. K. Chui, Y. Y. Cai, R. Raghavan, and R. Viswanathan. Potential .eld supported contact calculations in FEA of catheters. Proceedings of the International Congress of Applied
and Theoretical Mechanics, Kyoto, KS4-04:596, 1996.
• [33] S. Fang, R. Raghavan, and J. T. Richtsmeier. Volume Morphing methods for landmark based 3D image deformation. Proceedings of SPIE, Medical Imaging, 2710:404– 415, 1996.
• [32] J. T. Richtsmeier, J. Ohman, C. vanderKolk, B. Carson, P. Pang, S. Fang, R. Raghavan, D. Hauser, and S. Lele. 3D growth analysis and visualization in craniosynostosis. Proceedings of the
American Cleft Palate Craniofacial Association meeting, San Diego, April, 1996.
• [31] Craig R. Dufresne, Raghu Raghavan, Shiaofen Fang, Pingli Pang, and Joan T. Richtsmeier. Computerized dynamical skeletal modeling for craniofacial surgical planning: new tools to predict
growth following surgery. International Congress of Craniofacial Surgery, 1995.
• [30] R. N. Bryan, C. Davatzikos, M. Vaillant, J. L. Prince, S. Letovsky, R. Raghavan, W. L. Nowinski, G. Salamon, N. Murayama, and O. Levrier. Creation of population-based atlases with a brain
image database (BRAID). Proceedings First International Conference on functional Mapping of the Human Brain, 1:72, 1995.
• [29] W. L. Nowinski, A. Fang, B. T. Nguyen, and R. Raghavan. Three-dimensional electronic brain atlas of human cerebral deep structures. Proceedings XIVth International Conference on Information
Processing in Medicine IPMI ’95, pages 51–62, 1995.
• [28] W. L. Nowinski, A. Fang, B. T. Nguyen, R. Raghavan, R. N. Bryan, and J. Miller. A multiple-atlas neuroimaging system. 7th Asian and Oceanic Cogress of Radiology AOCR ’95, 1995.
• [27] R. N. Bryan, W. L. Nowinski, A. Fang, B. T. Nguyen, R. Raghavan, J. Miller, H. L. Lim, and J. Raphel. Talairach-Tournoux/Schaltenbrand-Wahren based electronic brain atlas system. Lecture
Notes in Computer Science (Proceedings of the First International Conference on Computer Vision, Virtual Reality and Robotics in Medicine, CVRMed ’95), 905:257 – 261, 1995.
• [26] R. Raghavan. Bottom-up Design: from Cellular Automata to Pattern Recognition. Workshop on Parallel computing by Cellular Automata Arrays PARCELLA, pages 51–62, 1994.
• [25] C. K. Chui, S. Jain, and R. Raghavan. Topology Independent Models for Parallel Computation. Workshop on Parallel computing by Cellular Automata Arrays PARCELLA, pages 119–129, 1994.
• [24] W. L. Nowinski, R.Raghavan,C.P.Yu, W.S. Fok, P. Pillay,and S. K. Tan. Computer Systems for Neuro-Cranio-and Orthopedic Surgeries. Hand-book of the 18th International Congress of Radiology
ICR’94, 1994.
• [23] C.K. Chui, S. Jain, H.T. Nguyen, R. Raghavan, M.K. Sridhar, U. Sridhar, and R. Srinivasan. High Performance ComputingResearchinthe InstituteofSystems Science in Singapore. TCPP Newsletter,
2, no. 2:3–4, 1994.
• [22] C. K. Chui, S. Jain, H.T. Nguyen, and R. Raghavan. Research Activities in Accelerated Computing and theory Program (ACT Program). Journal of High-Performance computing, 1:50–57, 1994.
• [21] P. A. Heng, L. H. Ngoh, B. Nguyen, W. Nowinski, R. Raghavan, and C. P. Yu. An ATM-based Multimedia Medical Imaging System. Workshop on Multimedia in Medical Education, Anaheim, California,
• [20] R. Raghavan. Computational Learning Theory. SPIE Conference on the Science of Artificial Neural Networks, 1710:2–17, 1992.
• [19] R. Raghavan. Dynamics, Learning and Control: A Case Study, Intelligent Engineering Systems through Artificial Neural Networks. ASME,edited byC.H. Dagli and S. R. T. Kumara and Y.C.Shin,
pages 119–125, 1991.
• [18] R. Raghavan. Dynamics, Learning and Control in a Cellular Automata Network. SPIE Conference on Applications of Artificial Neural Networks, 1469:89–101, 1991.
• [17] R. Raghavan. Linear Programming for Learning in Neural Networks. SPIE Conference on Image Understanding and the Man-Machine Interface, 1472:139–148, 1991.
• [16] R. Raghavan and W. R. Lawton. The Algebra of Image Transformations. Proceedings of the SPIE, Image Algebra and Morphological Image Processing, 1350:455–466, 1990.
• [15] R. Raghavan, F. W. Adams Jr., and H. T. Nguyen. Target Recognition in Parallel Networks. Proceedings of SPIE , Applications of Artificial Neural Networks, 1294:94– 109, 1990.
• [14] O. Farotimi and R. Raghavan. Learning in a Recognition Network. Proceedings of the IEEE/INNS International Joint Conference on Neural Networks, 3:217–224, 1990.
• [13] R. Raghavan, K. K. Jung, and H. T. Nguyen. Fine Grain Parallel Processors in Real-Time Applications. Proceedings of IEEE Tenth International Conference on Pattern Recognition, 2:324–331,
• [12] K.K. Jung, H.T. Nguyen, and R. Raghavan. Massively Parallel Processors in Real Time Applications. Proceedings of SPIE, Parallel Architectures for Image Processing, 1246:107–119, 1990.
• [11] R. Raghavan, Jr. F. W. Adams, H. T. Nguyen, and J. Slawny. Image Recognition and Learning in Parallel Networks. Proceedings of SPIE, Nonlinear Image Processing, 1247:258–273, 1990.
• [10] H. T. Nguyen, K. K. Jung, and R. Raghavan. Fast Parallel Algorithms: From Images to Level Sets and Labels. Proceedings of SPIE, Parallel Architectures for Image Processing, 1246:107–119,
• [9] M. Brady, R. Raghavan, and J. Slawny. Probabilistic Cellular Automata in Pattern Recognition. Proceedings of the International Joint Conference on Neural Networks, 1:177–182, 1989.
• [8] M. Brady, R. Raghavan, and J. Slawny. Statistical Mechanics and Pattern Recognition: A New Feedback Technique. Proceedings of the SPIE, 1099:89–99, 1989.
• [7] M. Brady, R. Raghavan, and J. Slawny. Gradient Descent Fails to Separate. Proceedings of the IEEE Second International Conference on Neural Networks, 1:649–656, 1988.
• [6] W. Holsztynski and R. Raghavan. The Distributed Macro Controller for GSIMD Arrays. Concurrent Computations : Algorithms, Architecture and Technology, pages 689–696, 1988.
• [5] H. T. Nguyen, R. Raghavan, C. H. Ting, and H. S. Truong. High Density Parallel Processing. Proceedings of 7th Rochester Conference on Advanced Architectures, 1987.
• [4] R. Raghavan, W. Holsztynski, P. Mancuso, C.H. Ting, and P. Wong. Parallelizations: Old Transformations and a New Parallel Processing System. VLSI Signal Processing II, IEEE Press, S.Y. Kung,
R.E. Owen, and J.G. Nash, eds, pages 165–176, 1986.
• [3] R. Raghavan. Processing on Geometric Single Instruction Multiple Data Machines. Proceedings of the IEE Colloquium on Parallel Processors, London, 1986.
• [2] K. S. Miller, R. Raghavan, and M. M. Rochwarger. Invariance Methods in Signal Processing. Proceedings of the 9th Strategic Space Symposium (Advanced Research Project Agency, page 272, 1984.
• [1] R. Raghavan and D.L. Huber. Spin Dynamics in Uniaxial Ferromagnets. Proceedings of the American Institute of Physics (AIP) Conference on Magnetism, Philadelphia, 1975. | {"url":"https://therataxis.com/publications/conference_raghu.html","timestamp":"2024-11-13T12:13:54Z","content_type":"application/xhtml+xml","content_length":"23641","record_id":"<urn:uuid:0e580f80-ce11-4b56-afdc-35cb3eab70a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00858.warc.gz"} |
Coverage Rates | PaintScout Help Center
What is a coverage rate?
A coverage rate refers to the amount of surface area that a product can effectively cover. You should be able to find the coverage rate on the paint can or on the manufacturer's website. The coverage
rate is sqft/gallon or lnft/gallon.
This will allow PaintScout to automatically calculate the number of gallons needed depending on the room's dimensions and the surface's number of coats.
Walls, Trim, and Ceilings
When considering coverage rates for these surfaces, it's like asking, "How many square feet or linear feet can I cover with one can of paint using just one coat?" That's the figure you should enter
as your coverage rate.
A room that is 12 x 15 x 8 feet. How many gallons of paint are needed for 1 coat?
Surface Dimensions Coverage Rate Gallons
Walls 432 sqft 400 sqft/gal 1.08
Baseboards 54 lnft 650 lnft/gal 0.17
Ceilings 180 sqft 400 sqft/gal 0.45
In PaintScout, the coverage rate for item-based rates (such as doors, windows, etc.) is measured in gallons per item. This calculation is based on estimating how much product from a gallon of paint
would be used for a single item.
Let's break it down using a standard window frame as an example. Ask yourself, "How many window frames can I paint with one can of paint, applying just one coat?"
Next, you'll calculate this by dividing 1 by the number you came up with. For instance, if you can paint 20 window frames with one coat using one gallon of paint, your coverage rate would be:
The calculation to figure out the total number of gallons is:
# items x Coverage rate x # coats
You are painting 18 window frames in the living room, 2 coats. How many gallons of paint will you need?
Using the coverage rate calculated above, you can easily find your number of gallons!
18 window frames x 0.05 coverage x 2 coats = 1.8 gallons of paint
Things to keep in mind
• When entering your coverage rates, input them for one coat of paint. PaintScout automatically considers the number of coats when calculating the total gallons needed.
• The materials calculated on the work order may slightly exceed your actual requirements since it doesn't initially factor in spaces like doors and windows. This aligns with the PCA rules of
• A helpful guideline is to order slightly less product than calculated on the work order (around 60-70%). It's always more convenient to buy additional paint if needed rather than ending up with
For any questions, get in touch with our Support Team π ©β π » | {"url":"https://help.paintscout.com/en/articles/6063072-coverage-rates","timestamp":"2024-11-11T06:59:19Z","content_type":"text/html","content_length":"79214","record_id":"<urn:uuid:14bbe427-94ca-4ae2-9cb4-a293d906bdca>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00838.warc.gz"} |
Demystifying ARIMA Model Parameters: A Step-by-Step Guide - Data Magic AI Blog
Data ScienceMachine LearningPython
Demystifying ARIMA Model Parameters: A Step-by-Step Guide
ARIMA, which stands for AutoRegressive Integrated Moving Average, is a widely-used statistical method for time series forecasting. It combines autoregressive, differencing, and moving average
components to model data patterns.
What is ARIMA?
ARIMA is a mathematical model that describes a time series as a combination of autoregressive (AR), differencing (I), and moving average (MA) components. These components are denoted by the
parameters p, d, and q, respectively.
Components of ARIMA
1. Autoregressive Component (AR): This component models the relationship between an observation and several lagged observations.
2. Integrated Component (I): This component represents the number of differences needed to make the time series data stationary.
3. Moving Average Component (MA): This component accounts for the error term as a linear combination of previous error terms.
By understanding and appropriately selecting these components, one can create an effective ARIMA model tailored to their specific dataset.
Autoregressive (AR) Component
The autoregressive component of an ARIMA model focuses on modeling the relationship between an observation and a lagged version of itself. This is expressed mathematically as:
[X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + … + \phi_p X_{t-p} + \epsilon_t]
Here, (\phi_1, \phi_2, …, \phi_p) are the autoregressive parameters, (X_t) represents the current observation, and (\epsilon_t) is the error term.
Explanation of Autoregressive Component
The autoregressive component essentially looks at how the previous observations contribute to the current value. For instance, in a stock market context, it would analyze how past prices affect the
present price.
Selecting the Order of AR (p)
Determining the order of the autoregressive component ((p)) involves identifying how many lagged observations are significant. This can be accomplished using techniques like the partial
autocorrelation function (PACF) plot.
By choosing the right order of AR, you refine the model’s ability to capture dependencies from previous time points.
Integrated (I) Component
The integrated component of an ARIMA model focuses on differencing the time series data to make it stationary. Stationarity is crucial for accurate modeling as it ensures that the statistical
properties of the data remain constant over time.
Understanding Integrated Component
Differencing involves subtracting the current observation from the previous observation. The number of differences required to achieve stationarity is denoted by the parameter ‘d’. Mathematically, it
can be represented as:
[Y_t = (1 – B)^d X_t]
Where (Y_t) represents the differenced series, (X_t) is the original series, and ‘B’ is the backshift operator.
Selecting the Order of Integration (d)
Choosing the appropriate order of integration ((d)) is a crucial step in ARIMA modeling. It determines how many times differencing needs to be applied to achieve stationarity. This can be determined
through visual inspection of the data and using statistical tests like the Augmented Dickey-Fuller test.
If the data appears to have a trend or seasonality, a higher order of integration may be required. It’s important to strike a balance – too much differencing can lead to information loss, while too
little may result in a non-stationary model.
Moving Average (MA) Component
The moving average component of an ARIMA model focuses on modeling the error term as a linear combination of previous error terms. This component helps capture short-term fluctuations in the data.
Demystifying Moving Average Component
Mathematically, the moving average component is represented as:
[X_t = \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2} + … + \theta_q \epsilon_{t-q}]
Where (\epsilon_t) represents the current error term, and (\theta_1, \theta_2, …, \theta_q) are the moving average parameters.
The moving average component helps in filtering out short-term noise and isolating the underlying patterns in the data.
Selecting the Order of MA (q)
Determining the order of the moving average component ((q)) is a crucial step in building an effective ARIMA model. It signifies how many lagged error terms to include in the model.
This can be done through methods like the autocorrelation function (ACF) and partial autocorrelation function (PACF) plots. These plots provide insights into the lag values that significantly
influence the current observation.
By carefully selecting the order of the moving average component, you can enhance the model’s ability to capture short-term fluctuations.
Stationarity and Differencing
Achieving stationarity is a crucial step in time series analysis. A stationary time series has constant statistical properties over time, which simplifies modeling.
The Concept of Stationarity
A time series is considered stationary if its mean, variance, and autocovariance remain constant over time. Stationarity ensures that the underlying patterns in the data are not changing.
Applying Differencing for Stationarity
Differencing is a technique used to remove trends or seasonality from a time series. By subtracting the previous observation from the current one, you can eliminate any linear trends.
Identifying Seasonality
Seasonality refers to recurring patterns that occur at regular intervals within a time series. Recognizing these patterns is crucial for accurate forecasting.
Recognizing Seasonal Patterns
Seasonal patterns can be observed in various domains, such as retail sales (spikes during holidays) or weather data (temperature fluctuations across seasons).
Seasonal Differencing in ARIMA
In cases where seasonality is present, seasonal differencing can be applied in addition to regular differencing. This involves subtracting the observation from the same season in the previous year.
Choosing the Right Order
Selecting the right order of the ARIMA model is a critical step in building an accurate forecasting model. The AIC and BIC criteria are commonly used methods for model selection.
AIC and BIC Criteria
The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are statistical measures used to compare the goodness of fit of different models. Lower values indicate better-fitting
Grid Search Method
Grid search involves systematically testing a range of hyperparameters to identify the combination that produces the best model performance. This method is particularly useful for automating model
Fitting the ARIMA Model
Once the components and their respective orders are determined, the ARIMA model can be fitted to the data using various software packages.
Implementing the ARIMA Model in Python
Python offers libraries like statsmodels that provide functionalities for ARIMA modeling. This includes functions for model fitting, forecasting, and model evaluation.
Evaluating Model Fit
Model fit can be assessed using statistical measures like Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). Additionally, visual inspection of the residuals can
provide insights into model performance.
Forecasting with ARIMA
After fitting the ARIMA model, it can be used to make future predictions based on the historical data.
Making Future Predictions
Using the model, you can forecast future data points. This is valuable for planning and decision-making in various domains.
Visualizing Forecasted Data
Visualizing the forecasted data alongside the actual data allows for a clear understanding of the model’s predictive capabilities. This can help identify any areas where the model may need further
Model Validation
Validating the ARIMA model is crucial to ensure its accuracy and reliability in making forecasts.
Out-of-Sample Testing
Out-of-sample testing involves evaluating the model’s performance on data that it hasn’t seen before. This provides a realistic assessment of how the model will perform in real-world scenarios.
Measuring Forecast Accuracy
Forecast accuracy can be assessed using metrics like Mean Absolute Percentage Error (MAPE) and Forecast Bias. These metrics quantify the level of accuracy achieved by the model.
Handling Anomalies and Outliers
Anomalies and outliers in the data can significantly impact the performance of an ARIMA model.
Impact on ARIMA Model
Outliers can introduce noise and lead to inaccurate predictions. It’s essential to identify and handle them appropriately.
Strategies for Outlier Handling
Techniques like winsorization, data transformation, or using robust models can be employed to mitigate the effects of outliers on the model.
Fine-Tuning ARIMA Models
Fine-tuning the ARIMA model involves making adjustments to improve its performance and accuracy.
Model Refinement Techniques
Techniques like seasonal decomposition, parameter optimization, and incorporating exogenous variables can enhance the model’s forecasting capabilities.
Adjusting Parameters for Improved Performance
Iteratively adjusting the AR, I, and MA orders, as well as considering seasonal components, can lead to a more accurate model.
Common Pitfalls and Challenges
While ARIMA is a powerful tool, there are common pitfalls that practitioners should be aware of.
Overfitting and Underfitting
Overfitting occurs when the model is too complex and captures noise in the data. Underfitting, on the other hand, happens when the model is too simple to capture the underlying patterns.
Dealing with Noisy Data
Noisy data can obscure meaningful patterns. Data cleaning and preprocessing techniques are crucial for effective modeling.
In conclusion, understanding the various parameters of an ARIMA model is essential for accurate time series forecasting. By breaking down the components and following a systematic approach, you can
effectively apply ARIMA to your own datasets.
Can ARIMA handle seasonal data patterns?
• Yes, ARIMA can be extended to incorporate seasonal components through seasonal differencing.
How do I know if my time series data needs differencing?
• You can perform a visual inspection of the data plot to check for any obvious trends or seasonality. Additionally, statistical tests like the Augmented Dickey-Fuller test can be used.
What is the significance of the AIC and BIC criteria in ARIMA model selection?
• The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are used to compare the goodness of fit of different models. Lower values indicate better-fitting models.
How can outliers impact the performance of an ARIMA model?
• Outliers can distort the modeling process, leading to inaccurate predictions. It’s important to identify and handle outliers appropriately.
Are there automated tools available for ARIMA modeling?
• Yes, there are various libraries and software packages, such as Python’s statsmodels and R’s forecast package, that provide functionalities for ARIMA modeling.
Machine Learning books from this Author: | {"url":"https://datamagiclab.com/demystifying-arima-model-parameters-a-step-by-step-guide/","timestamp":"2024-11-07T05:35:19Z","content_type":"text/html","content_length":"150173","record_id":"<urn:uuid:01df797a-b05d-46e8-adff-41f17bd2a2cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00284.warc.gz"} |
Notes on the Hermite-based poly-Euler polynomials with a
Notes on the Hermite-based poly-Euler polynomials with a q-parameter
Burak Kurt
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 74–82
DOI: 10.7546/nntdm.2020.26.3.74-82
Full paper (PDF, 206 Kb)
Authors and affiliations
Burak Kurt
Mathematics of Department, Akdeniz University
Antalya TR-07058, Turkey
We introduce and investigate the Hermite-based poly-Euler polynomials with a q-parameter. We give some basic properties and identities for these polynomials. Furthermore, we prove two explicit
• Bernoulli polynomials and numbers
• Euler polynomials and numbers
• 2-variable Hermite–Kampé de Feriét polynomials
• Polylogarithm function
• Poly-Euler polynomials
• Stirling numbers of the second kind
2010 Mathematics Subject Classification
1. Bayad, A., & Hamahata, Y. (2011). Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math., 65, 15–34.
2. Bayad, A., & Kim, T. (2012). Higher Recurrences for Apostol–Bernoulli–Euler polynomials, Russ. J. of Math. Phys., 19 (1), 1–10.
3. Cenkci, M., & Komatsu, T. (2015). Poly-Bernoulli numbers and polynomials with a q-parameter, J. Number Theory, 152, 38–54.
4. Duran, U., & Acikgoz, M. (2018). On (p, q)-Euler numbers and polynomials associated with (p, q)-Volkenborn integral, Int. J. of Number Theory, 14 (1), 241–253.
5. Duran, U., Acikgoz, M., & Araci, S. (2018). Hermite based poly-Bernoulli polynomials with q-parameter, Advanced Stud. in Contemp. Math., 28 (2), 285–296.
6. Duran, U., Acikgoz, M., Esi, A., & Araci, S. (2018). A note on the (p, q)-Hermite
polynomials, App. Math. and Information Sciences, 12, 227–231.
7. Hamahata, Y. (2014). Poly-Euler polynomials and Arakawa–Kaneko type zeta functions, Functione et. App. Commentarii Mathematica, 51 (1), 7–27.
8. Jolany, H., Corcino, R. B., & Komatsu, T. (2015). More properties on multi-Poly-Euler polynomials, Bull. Soc. Math. Mex., 21, 149–162.
9. Kim, D. S., & Kim, T. (2012). Some identities of Frobenius–Euler polynomials arising from umbral calculus, Advances in Diff. Equa., 2012, Article No. 196.
10. Kim, D. S., & Kim, T. (2013). Higher-order Frobenius–Euler and Poly-Bernoulli mixed type polynomials, Advances in Diff. Equa., 2013, Article No. 251.
11. Kim, D. S., & Kim, T.(2015). A note on poly-Bernoulli and higher order poly-Bernoulli polynomials, Russ. J. Math., 22 (1), 26–33.
12. Kim, D. S., & Kim, T. (2015). Higher order Bernoulli and poly-Bernoulli mixed type polynomials, Georgian Math. J., 22, 265–272.
13. Kim, D. S., Kim, T., Dolgy, D. V., & Rim, S. H. (2013). Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus, Advances in Diff. Equa., 2013, Article No.
14. Kim, D. S., & Kim, T. (2013). Hermite and Poly-Bernoulli mixed type polynomials, Advances in Diff. Equa., 2013, Article No. 343.
15. Kim T., Kim D. S., Kim H. Y., & Jang L.-C. (2020). Degenerate poly-Bernoulli numbers and polynomials, Informatica, 31 (3), 2–8.
16. Kurt, B. (2018). Identities and relation on the Poly-Genocchi polynomials with a
q-parameter, J. Inequa. Special Func., 9 (1), 1–8.
17. Kurt, B. & Simsek, Y. (2013). On the generalized Apostol-type Frobenius–Euler
polynomials, Advances in Diff. Equa., 2013, Article No. 1.
18. Ozarslan, M. A. (2013). Hermite-based unified Apostol–Bernoulli, Euler and Genocchi polynomials, Advances in Diff. Equa., 2013, Article No. 116.
19. Sanchez-Peregrino, R. (2002). Closed formula for Poly-Bernoulli numbers, Fibonacci Quart., 40, 362–364.
20. Srivastava, H. M. (2011). Some generalization and basic (or q−) extension of the
Bernoulli, Euler and Genocchi polynomials, App. Math. Inform. Sci., 5 (3), 390–444.
21. Srivastava, H. M. & Choi, J. (2001). Series Associated with the Zeta and Related Functions, Kluwer Academic Pub., Dordrecht, Boston and London.
22. Srivastava, H. M. & Manocho, H. L. (1984) A Treatise on Generating Functions, Halsted Press, Chichester, West Sussex, England.
Related papers
Cite this paper
Kurt, B. (2020). Notes on the Hermite-based poly-Euler polynomials with a q-parameter. Notes on Number Theory and Discrete Mathematics, 26(3), 74-82, DOI: 10.7546/nntdm.2020.26.3.74-82. | {"url":"https://nntdm.net/volume-26-2020/number-3/74-82/","timestamp":"2024-11-08T04:44:19Z","content_type":"text/html","content_length":"33776","record_id":"<urn:uuid:8d3f58be-aa1e-4459-8fcd-e4ed729a2aee>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00371.warc.gz"} |
All quantum states are equally defined
All quantum states are equally defined
TL;DR – No quantum state is more uncertain than the other. All states can be identified by a set of perfectly prepared quantities.
When studying quantum mechanics, you may get the (wrong) impression that some states are better defined than others. Some are eigenstates while others are just a superposition. Or that the gaussian
packets, are more determined than the other states, since some satisfy the uncertainty principle with an equal. Well, that’s just not the case.
The confusion stems from the idea that quantum states are like classical statistical distribution, in which you have some well defined elements (which are the objects that are well defined) upon
which you assign probabilities. Quantum states are not like this at all. We are going to see that any state is always the eigenstate of some Hermitian operator, and therefore they always have a
quantity that is perfectly prepared. And they also have a symmetry under the transformation generated by that operator, so there is always another quantity for which no value is more likely than the
1. Prepared quantities and unprepared conjugates
Suppose that you have a quantum state for which the position is perfectly prepared. This corresponds to the eigenstate $|x\rangle$ of the operator $X$. The transformation generated by $X$ is $1+\frac
{Xdp}{\imath\hbar}$ which corresponds to increasing momentum by $dp$. Now, since $|x\rangle$ is an eigenstate of $X$, it is also an eigenstate of the transformation generated by $X$. If we imagine
the distribution of $|x\rangle$ over momentum, then, this has to be a distribution that does not change if we increase momentum. But the only distribution that is symmetric under that change is the
one that has the same value for all possible values of momentum: the eigen state $|x\rangle$ is uniformly distributed in momentum. This can be generalized to any quantity. For example, an eigenstate
of spin in the $z$ direction will be symmetric along the angle on the $(x,y)$ plane. This will also hold for any function of position and momentum and indeed for any Hermitian operator.
So, for a particular space of quantum states, we can imagine that, instead of specifying a wavefunction, we can give a set of operators and eigenvalues. Given that information, we can identify the
corresponding eigenstate. This would not give the value for all quantities since the conjugate quantities will be left completely unspecified. The question is: can all states can be specified in this
way? Which is equivalent to asking: are all states eigenstates of some Hermitian operator?
Suppose we have a state $|\psi\rangle$. We can construct the operator $O=|\psi\rangle a \langle \psi |$ where $a$ is a real number. $O$ is Hermitian and $|\psi\rangle$ is the eigenstate corresponding
to the eigenvalue $a$. Yes, the operator is trivial, but we can indeed construct it. Which shows that for any state there exists Hermitian operators that allow that state as an eigenstate. In fact,
there are infinitely many.
Overall, a state always has some well specified quantities and it also has some unspecified ones. For example, suppose we have an eigenstate for the operator $X + a P$, a linear combination of
position and momentum. This means that the quantity $x + ap$ is perfectly prepared while the conjugate quantity, $\frac{1}{2a}(x-ap)$, is a uniform distribution. We can verify the two quantities are
conjugates by calculating the commutator.
\frac{\left[\frac{1}{2a}(X-aP), X+aP \right]}{\imath \hbar} &= \frac{1}{2a \imath \hbar} \left[X-aP, X+aP\right] \\
&= \frac{1}{2a \imath \hbar} \left( \left[X, X\right] + \left[-aP, X\right] + \left[X, aP\right] + \left[-aP, aP\right] \right) \\
&= \frac{1}{2a \imath \hbar} \left( 0 + a \imath \hbar + a \imath \hbar + 0 \right) \\
&= 1
Now consider the two distributions for that state over position and momentum. Since all the values of $\frac{1}{2a}(x-ap)$ are equally likely, then all values of $x$ and $p$ are also equally likely.
That is: the distribution is uniform for both quantities. The variance for both $x$ and $p$ is infinite. If we only know that, we would think that state to be infinitely less defined than an
eigetnstate of position. What happens is that for that state the uniform distribution in $x$ and $p$ are strongly correlated, so much so that the distribution over $x+ap$ admits a single value.
Therefore this state is no less defined of an eigenestate of $x$: what changes is what quantities are well defined and which aren’t.
What is significantly different in quantum mechanics is that all states are distributions but no distribution is more defined than the other. Even if a state has an extremely large spread in a pair
of conjugate variables, it will have a quantity somewhere that will be perfectly defined. There is no base that is mathematically better then the other: all states can be part of a basis and all
states are superpositions in some other basis. Whenever we are superposing two states, we are always adding some other correlation such that the new states is as well defined as the other two.
This actually makes a lot of sense conceptually. It’s telling us that each quantum state is a distribution that cannot be decomposed into smaller independent ones. Each quantum system is an
irreducible unit. This is the main difference from classical mechanics, where each distribution over phase space can be divided into smaller pieces that can be studied independently. | {"url":"https://sufficientlywise.org/2018/02/25/all-quantum-states-are-equally-defined/","timestamp":"2024-11-09T02:47:38Z","content_type":"application/xhtml+xml","content_length":"37918","record_id":"<urn:uuid:c11fa520-1ad9-474c-b6e2-fdbc6876e37d>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00197.warc.gz"} |
mixed-effects model
Class: GeneralizedLinearMixedModel
Generate random responses from fitted generalized linear mixed-effects model
ysim = random(glme) returns simulated responses, ysim, from the fitted generalized linear mixed-effects model glme, at the original design points.
ysim = random(glme,tblnew) returns simulated responses using new input values specified in the table or dataset array, tblnew.
ysim = random(___,Name,Value) returns simulated responses using additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes. For example, you can specify
observation weights, binomial sizes, or offsets for the model.
Input Arguments
tblnew — New input data
table | dataset array
New input data, which includes the response variable, predictor variables, and grouping variables, specified as a table or dataset array. The predictor variables can be continuous or grouping
variables. tblnew must contain the same variables as the original table or dataset array, tbl, used to fit the generalized linear mixed-effects model glme.
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
BinomialSize — Number of trials for binomial distribution
ones(m,1) (default) | m-by-1 vector of positive integer values
Number of trials for binomial distribution, specified as the comma-separated pair consisting of 'BinomialSize' and an m-by-1 vector of positive integer values, where m is the number of rows in tblnew
. The 'BinomialSize' name-value pair applies only to the binomial distribution. The value specifies the number of binomial trials when generating the random response values.
Data Types: single | double
Offset — Model offset
zeros(m,1) (default) | vector of scalar values
Model offset, specified as a vector of scalar values of length m, where m is the number of rows in tblnew. The offset is used as an additional predictor and has a coefficient value fixed at 1.
Weights — Observation weights
m-by-1 vector of nonnegative scalar values
Observation weights, specified as the comma-separated pair consisting of 'Weights' and an m-by-1 vector of nonnegative scalar values, where m is the number of rows in tblnew. If the response
distribution is binomial or Poisson, then 'Weights' must be a vector of positive integers.
Data Types: single | double
Output Arguments
ysim — Simulated response values
m-by-1 vector
Simulated response values, returned as an m-by-1 vector, where m is the number of rows in tblnew. random creates ysim by first generating the random-effects vector based on its fitted prior
distribution. random then generates ysim from its fitted conditional distribution given the random effects. random takes into account the effect of observation weights specified when fitting the
model using fitglme, if any.
Simulate Random Responses from a GLME Model
Load the sample data.
This simulated data is from a manufacturing company that operates 50 factories across the world, with each factory running a batch process to create a finished product. The company wants to decrease
the number of defects in each batch, so it developed a new manufacturing process. To test the effectiveness of the new process, the company selected 20 of its factories at random to participate in an
experiment: Ten factories implemented the new process, while the other ten continued to run the old process. In each of the 20 factories, the company ran five batches (for a total of 100 batches) and
recorded the following data:
• Flag to indicate whether the batch used the new process (newprocess)
• Processing time for each batch, in hours (time)
• Temperature of the batch, in degrees Celsius (temp)
• Categorical variable indicating the supplier (A, B, or C) of the chemical used in the batch (supplier)
• Number of defects in the batch (defects)
The data also includes time_dev and temp_dev, which represent the absolute deviation of time and temperature, respectively, from the process standard of 3 hours at 20 degrees Celsius.
Fit a generalized linear mixed-effects model using newprocess, time_dev, temp_dev, and supplier as fixed-effects predictors. Include a random-effects term for intercept grouped by factory, to account
for quality differences that might exist due to factory-specific variations. The response variable defects has a Poisson distribution, and the appropriate link function for this model is log. Use the
Laplace fit method to estimate the coefficients. Specify the dummy variable encoding as 'effects', so the dummy variable coefficients sum to 0.
The number of defects can be modeled using a Poisson distribution
${\text{defects}}_{ij}\sim \text{Poisson}\left({\mu }_{ij}\right)$
This corresponds to the generalized linear mixed-effects model
$\mathrm{log}\left({\mu }_{ij}\right)={\beta }_{0}+{\beta }_{1}{\text{newprocess}}_{ij}+{\beta }_{2}{\text{time}\text{_}\text{dev}}_{ij}+{\beta }_{3}{\text{temp}\text{_}\text{dev}}_{ij}+{\beta }_{4}
{\text{supplier}\text{_}\text{C}}_{ij}+{\beta }_{5}{\text{supplier}\text{_}\text{B}}_{ij}+{b}_{i},$
• ${\text{defects}}_{ij}$ is the number of defects observed in the batch produced by factory $i$ during batch $j$.
• ${\mu }_{ij}$ is the mean number of defects corresponding to factory $i$ (where $i=1,2,...,20$) during batch $j$ (where $j=1,2,...,5$).
• ${\text{newprocess}}_{ij}$, ${\text{time}\text{_}\text{dev}}_{ij}$, and ${\text{temp}\text{_}\text{dev}}_{ij}$ are the measurements for each variable that correspond to factory $i$ during batch
$j$. For example, ${\text{newprocess}}_{ij}$ indicates whether the batch produced by factory $i$ during batch $j$ used the new process.
• ${\text{supplier}\text{_}\text{C}}_{ij}$ and ${\text{supplier}\text{_}\text{B}}_{ij}$ are dummy variables that use effects (sum-to-zero) coding to indicate whether company C or B, respectively,
supplied the process chemicals for the batch produced by factory $i$ during batch $j$.
• ${b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)$ is a random-effects intercept for each factory $i$ that accounts for factory-specific variation in quality.
glme = fitglme(mfr,'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1|factory)','Distribution','Poisson','Link','log','FitMethod','Laplace','DummyVarCoding','effects');
Use random to simulate a new response vector from the fitted model.
rng(0,'twister'); % For reproducibility
ynew = random(glme);
Display the first 10 rows of the simulated response vector.
ans = 10×1
Simulate a new response vector using new input values. Create a new table by copying the first 10 rows of mfr into tblnew.
The first 10 rows of mfr include data collected from trials 1 through 5 for factories 1 and 2. Both factories used the old process for all of their trials during the experiment, so newprocess = 0 for
all 10 observations.
Change the value of newprocess to 1 for the observations in tblnew.
tblnew.newprocess = ones(height(tblnew),1);
Simulate new responses using the new input values in tblnew.
ynew2 = random(glme,tblnew)
ynew2 = 10×1
More About
Conditional Distribution Method
random generates random data from the fitted generalized linear mixed-effects model as follows:
• Sample ${b}_{sim}\sim P\left(b|\stackrel{^}{\theta },{\stackrel{^}{\sigma }}^{2}\right)$, where $P\left(b|\stackrel{^}{\theta },{\stackrel{^}{\sigma }}^{2}\right)$ is the estimated prior
distribution of random effects, and $\stackrel{^}{\theta }$ is a vector of estimated covariance parameters, and ${\stackrel{^}{\sigma }}^{2}$ is the estimated dispersion parameter.
• Given b[sim], for i = 1 to m, sample ${y}_{sim_i}\sim P\left({y}_{new_i}|{b}_{sim},\stackrel{^}{\beta },\stackrel{^}{\theta },{\stackrel{^}{\sigma }}^{2}\right)$, where $P\left({y}_{new_i}|{b}_
{sim},\stackrel{^}{\beta },\stackrel{^}{\theta },{\stackrel{^}{\sigma }}^{2}\right)$ is the conditional distribution of the ith new response y[new_i] given b[sim] and the model parameters. | {"url":"https://au.mathworks.com/help/stats/generalizedlinearmixedmodel.random.html","timestamp":"2024-11-13T09:41:25Z","content_type":"text/html","content_length":"104720","record_id":"<urn:uuid:addfde92-3f0b-4867-8143-3a591212c4bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00105.warc.gz"} |
Python: Find the maximum and minimum product from the pairs of tuple within a given list - w3resource
Python: Find the maximum and minimum product from the pairs of tuple within a given list
Python List: Exercise - 124 with Solution
Write a Python program to find the maximum and minimum product of pairs of tuples within a given list.
Sample Solution:
Python Code:
# Define a function 'tuple_max_val' that finds the maximum and minimum product of pairs in a list of tuples
def tuple_max_val(nums):
# Calculate the maximum product using list comprehension with abs
result_max = max([abs(x * y) for x, y in nums])
# Calculate the minimum product using list comprehension with abs
result_min = min([abs(x * y) for x, y in nums])
# Return the maximum and minimum product as a tuple
return result_max, result_min
# Create a list 'nums' containing tuples of integers
nums = [(2, 7), (2, 6), (1, 8), (4, 9)]
# Print a message indicating the original list of tuples
print("The original list, tuple:")
# Print the contents of 'nums'
# Print a message indicating the operation to find the maximum and minimum product
print("\nMaximum and minimum product from the pairs of the said tuple of list:")
# Call the 'tuple_max_val' function with 'nums' and print the result
Sample Output:
The original list, tuple :
[(2, 7), (2, 6), (1, 8), (4, 9)]
Maximum and minimum product from the pairs of the said tuple of list:
(36, 8)
Python Code Editor:
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Next: Write a Python program to calculate the product of the unique numbers of a given list.
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Is stratified randomisation in trials (at least large ones) pointless?
I recently wrote a blog post giving some comments about the revised FDA guidance on covariate adjustment in randomised trials. One of the things I commented on was recent work by different authors on
the impacts of stratified randomisation on inference. It is well known (or should be!) that if stratified randomisation is used, the statistical analysis should adjust for the variables used in the
stratified randomisation in order for the efficiency gain to be realised.
A fact that was new to me is that if in the analysis one adjusts (in linear models for continuous outcomes) for dummy variables corresponding to each of the randomisation strata, the true variance of
the resulting estimator of treatment effect is the same whether simple (non stratified) or stratified randomisation is used, in the most common situation that randomisation to the two arms is 1:1.
This was shown by one of the results of Bugni et al 2018. Wang et al 2019 subsequently showed it also holds if additional baseline covariates (not used in the stratified randomisation) are also
adjusted for, and that it also holds for the standardisation type estimator of the marginal risk difference based on a logistic regression working model.
These results mean that, at least for large sample sizes, if one considers strata defined by baseline covariates, provided you adjust for these strata as covariates in the analysis, performing the
randomisation stratified on these gains you no additional efficiency compared to using simple randomisation. These theoretical results are in agreement with simulation results mentioned on Twitter
today by Jack Wilkinson that prompted this post:
Following up on this… yeah, I can’t get stratification to reduce the SE for an adjusted mean difference by more than a negligible amount. So it’s just a fun thing we do.
— Jack Wilkinson (@jd_wilko) August 4, 2021
The theoretical results in the above papers are asymptotic results. Thus I can imagine it could well be the case that with small sample sizes stratified randomisation does buy you some additional
Moreover, in practice I believe it is more common in the analysis model to adjust for only main effects of the variables used to define the randomisation strata, rather than dummy variables for each
of their combinations. My guess is that if an assumption that the outcome only depends on these variables via their main effects is correct, the theoretical results mentioned above that imply no
(asymptotic) benefit to using stratified randomisation would also hold true for this type of analysis model.
I don’t have time to actively investigate this myself right now, but if anyone is interested in pursuing it and leading on the work for this, please get in touch.
Postscript – a small simulation illustration
The following small simple simulation study in R follows. The setup is very simple – one binary baseline covariate (X) which influences the outcome and either is ignored in the randomisation (simple
randomisation) or randomisation is performed stratified on it to ensure balance. In both cases, the analysis is a linear regression adjusting for treatment (Z) and this baseline covariate (X).
n <- 250
nSim <- 10000
est <- array(0, dim=nSim)
#simple randomisation
for (sim in 1:nSim) {
#simulate binary baseline covariate
x <- 1*(runif(n)<0.5)
#simulate treatment assignment, using simple randomisation
z <- 1*(runif(n)<0.5)
y <- x+z+rnorm(n)
mod <- lm(y~x+z)
est[sim] <- coef(mod)[3]
#look at empirical SE of estimates
#stratified block randomisation
for (sim in 1:nSim) {
x <- 1*(runif(n)<0.5)
#stratified block randomisation
z <- rep(0,n)
n0 <- sum(x==0)
z[x==0] <- as.numeric(blockrand(n=n0, num.levels=2)$treatment)[1:n0]-1
n1 <- sum(x==1)
z[x==1] <- as.numeric(blockrand(n=n1, num.levels=2)$treatment)[1:n1]-1
y <- x+z+rnorm(n)
mod <- lm(y~x+z)
est[sim] <- coef(mod)[3]
The empirical SE from simple randomisation (based on 10,000 simulations) was 0.1259364 and for stratified randomisation was 0.1254624. This shows that, at least in this setup, the stratified
randomisation, does not materially reduce the (true) variability of the treatment effect estimates.
These results are in accordance with a 1982 paper I just came across: ‘A note on stratifying versus complete random assignment in clinical trials’ by Grizzle.
12 thoughts on “Is stratified randomisation in trials (at least large ones) pointless?”
1. Jonathan, I’ve just skimmed this and haven’t digested yet but at a glance it doesn’t make sense to me, so I’ll write down my confusion…
You note above that ‘if stratified randomisation is used, the statistical analysis should adjust for the variables used in the stratified randomisation in order for the efficiency gain to be
realised’. Exactly! We expect e.g. overcoverage due to a mismatch between the empirical SE and model SE. This doesn’t (I think) go away with large sample sizes*, which implies that stratified
randomisation still reduced the empirical SE compared with simple randomisation. And if there were a ‘catching up’ of simple randomisation as n increases then that would surely also lead to
overcoverage – which we know doesn’t happen.
*Figures 1 and 2 of Anthony Atkinson’s paper ‘Optimum biased-coin designs for sequential treatment allocation with covariate information’ spring to mind, where the loss of simple randomisation
vs. other methods actually gets worse as n increases.
□ Thanks Tim. What analysis model did you have in mind? I probably wasn’t very clear in the post about what I meant. Imagine you have one binary baseline covariate, and you are going to adjust
for it (and treatment group) in the statistical analysis (whichever randomisation scheme you use). Now compare the the true/empirical SE of the estimator under simple randomisation vs.
stratified randomisation (stratified on the binary baseline variable). This theory says (I think) asymptotically the empirical SE of this estimator will be the same (and in particular no
lower) whether you use stratified randomisation or not. I.e. once you commit to adjusting for it in the analysis model, there is no (asymptotically at least) benefit to using the more complex
stratified randomisation.
2. Thanks Jonathan. I can’t reply to your comment above so this is in response to ‘Thanks Tim. What analysis model did you have in mind?’ Yes, I had in mind what you wrote (i.e. both adjusted).
After a bit of thought, I have some intuition. In Atkinson’s paper above, he quantified loss as ‘number of participants on whom information is lost due to imbalance’. In his figure 1, loss
(eventually) reaches ~5 for simple randomisation. But this is an absolute number and the results of Bugni and of Wang are not in the same terms. If they took Atkinson’s loss of 5, the difference
in variance with n–5 vs. n participants becomes vanishingly small as n goes to infinity. (I realise this is very hand-wavy!)
Here’s code showing how this works out for loss = 5:
# Stata
twoway function (1/(x-5)) / (1/x), range(10 100)
# R
eq = function(x){(1/(x-5))/(1/x)}
plot(eq(10:100), type=’l’)
□ Thanks Tim! I need to look more carefully at Atkinson’s paper, but your suggested explanation makes sense to me.
3. This is really interesting! I’ve always had the mantra ‘analyse as you randomise’ so if you stratify by a discrete covariate you should include it in the analysis, but because of randomisation
within strata there will be little effect of including the covariate in the model on the treatment estimate. (Interestingly with this mantra really one should also include blocks in block
randomisation in the analysis, which I must confess I don’t do).
One assumes the ‘true’ mode includes the covariate but that is an assumption one makes when doing stratified randomisation. Omitting the covariate is a ‘collapsed’ model and not all estimates for
treatment effects are ‘collapsible’.
. What I wondered was about multi-centre trials. I generally suggest stratifying by centre but then wondered about including centre effect in the analysis. If centre effect is included should it
be as a fixed effect or a random effect?
4. The theory was covered by Emmanuel Lesaffre and me in
Lesaffre E, Senn S. A note on non-parametric ANCOVA for covariate adjustment in randomized clinical trials. Statistics in Medicine. 2003;22(23):3583-3596.
As you note, it’s not quite true that there is no difference. It is asymptotically true. The lower bound for the variance is proprtional to n/(n_1 x n_2) x lambda where n_1 and n_2 are the sample
sizes in the two arms and n = n_1+n_2 and lambda is the penalty for loss of orthogonality. For a formula for lambda see
Senn SJ. Modelling in drug development. In: Christie M, Cliffe A, Dawid AP, Senn SJ, eds. Simplicity Complexity and Modelling. Chichester: Wiley; 2011:35-49.
However, the really important work on this Carl-Fredrik Burman’s PhD thesis
Burman C-F. On Sequential Treatment Allocations in Clinical Trials [PhD]. Gothenburg: Department of Mathematics, Chalmers University of Technology; 1996.
The expected loss compared to perfect balance is about one patient per covariate so not really important for moderate;y sized trials. this is one of the reasons why I hate minimisation: a
pointless and overhyped procedure.
I like to think that I am responible for Anthony Atkinson investigating the loss. He remarked that he thought it was a scandal that medical statisticians didn’t use his biased coin approach. I
said to him it’s optimal but how much better is it than randomisation. Not much, it turns out.
5. Here’s my understanding of the excellent discussion. I hope someone will correct me, then I have a question.
* adjustment for blocking/stratification variables is essential to have in the model
* blocking during randomization has a low probability of helping
* the way in which it would help is that it makes (in a linear model) the covariance matrix off diagonal terms that involve treatment be close to zero, which lowers the variance of the treatment
effect estimate; i.e. it makes the stratification variable not collinear with treatment
Now my question: suppose that there is only a 0.02 chance in an unstratified randomization for a noticeable imbalance in an important baseline covariate. Then you would be right, on the average,
to ignore this problem. But in the 0.02 of trials in which it does occur, stratified randomization would have prevented damage to the treatment effect variance term. So why not stratify as an
insurance policy?
□ Thanks Frank, and all for all the great contributions.
Frank wrote that blocking has a low probability of helping, but as far as I understand now this statement applies for large trials, but not small ones. As the trial gets smaller, the
potential for stratified randomisation to be helpful is greater.
As to the pertinent and good question of ‘why not use stratified randomisation?’: the only thing that comes to mind is that from a practical/logistical perspective it might be easier to use
simple randomisation. Someone with actual experience of organising the randomisation (i.e. not me!) may be able to usefully comment on this point.
6. It might also be worth looking at Senn SJ, Anisimov VV, Fedorov VV. Comparisons of minimization and Atkinson’s algorithm. Statistics in Medicine. 2010;29(7-8):721-730. https://
onlinelibrary.wiley.com/doi/abs/10.1002/sim.3763 This discusses the issue of efficiency in section 2. This shows that where more than one stratifying factor is involved, it is actually not
necessary for the strata defined by the combinations to be balanced, provided that only main effects (and not interactions) of the covariates will be adjusted for. An example is the following
Treatment Sex Steroids used
Experimental Female No
Experimental Male Yes
Control Female Yes
Control Male No
with n patients allocated to each of the four combination shown.This is balanced by sex and by steroid use but not by their combination.
This paper also shows that the loss involved in randomisation compared to perfect balance (which is usually unatainable) is typically less than one patient per covariate. The comparsion is made
assuming that whether or not stratisfication has taken place, the covariate is adjusted for. Unfortunately many of the comparisons of minimisation and randomisation have failed to use the same
model for analysis, thus failing to separate design and analysis issues. this paper also shows that Atkinson’s algorithm is superior to minimisation.
7. Thanks very much Stephen and Jonathan. Stephen I think that’s the paper I need.
8. This is really interesting thanks, surprising and thought provoking. I’ve started to do some regressions and I’m not sure stratified randomisation has any benefits on power, bias or type 1 error
rates (maybe things depend on sample size, effect of strata covariates on outcome, their intercorrelation). Are power, bias or type 1 error rates the metrics we should base a design on ? So why
do we continue to do stratified randomisation given it must have a cost (eg logistical). If there is no reason to do it will guidance from eg ICH or FDA/EMA or (UK) https://www.ct-toolkit.ac.uk/
routemap/trial-planning-and-design/ be likely to change – thanks again for this interesting article and discussion
9. A very interesting discussion indeed. All this pertains, presumably, to a linear regression modelling framework. I have been investigating whether the same applies to Poisson regression and
Negative Binomial regression. It turns out, based on simulation, that it does. If anyone is interested, try running the code below (3 covariates, one with six levels and the others dichotomous,
sample size of 150 and 10k iterations). It would be an interesting exercise to establish this analytically.
mean.sd <- function(vec, digits = 4, digits.sd = digits – 1, cnt = FALSE) {
mn % signif(digits = digits)
std.dev % signif(digits = digits.sd);
if (!cnt) str <- paste(mn, '(SD =', std.dev, ')')
if (cnt) str <- paste(mn, '(SD =', std.dev, 'n =', length(vec), ')')
## Set the sample size, and the covariate effect size and treatment effect sizes ####
nn <- 150;
n.sites <- 6;
eff.sz <- 1.5;
tx.names <- c('SIRO', 'PLAC')
prior.SCC.cnt =10′, ‘<10')
Use.5FU <- c('No', 'Yes')
Site.name <- LETTERS[1:n.sites]
risk.eff <- 0.5*c(-1,1)
Use.5FU.eff <- 0.5*c(-1,1);
Site.eff <- 2*(1:n.sites – n.sites/2 – 0.5)/2
tx.eff <- eff.sz*c(-1,1);
## List of models to be simulated ####
mdl.types <- c('gaussian', 'poisson', 'quasipoisson');
disper <- 2;
## Perform simulation ####
iters <- 10000;
se.0.strat <- se.1.strat <- est.0.strat <- est.1.strat <- disp.0.strat <- disp.1.strat <- numeric(iters)
se.0.simp <- se.1.simp <- est.0.simp <- est.1.simp <- disp.0.simp <- disp.1.simp <- numeric(iters)
st <- proc.time()['elapsed'];
for (mdl.type in mdl.types) {
et <- proc.time()['elapsed'];
cat('\nModelling', mdl.type, ', time elapsed', round(et-st), 'secs. \n')
for (ii in 1:iters) {
dta <- data.frame(ix = 1:nn);
dta$risk <- prior.SCC.cnt[1+rbinom(nn,1, 0.5)];
dta$Use.5FU <- Use.5FU[1+rbinom(nn,1, 0.5)];
dta$Site <- Site.name[ceiling(runif(n = nn)*n.sites)];
dta$stratum <- paste(dta$Site, dta$Use.5FU, dta$risk)
## Assign treatments using simple randomisation, and then by stratified randomisation
dta$tx.simp <- rep(tx.names, length.out=nn);
dta % group_by(stratum) %>% mutate(tx.strat = rep(sample(tx.names), length.out=length(stratum)));
log.lamb.simp <- tx.eff[match(dta$tx.simp, tx.names)] +
risk.eff[match(dta$risk, prior.SCC.cnt)] +
Use.5FU.eff[match(dta$Use.5FU, Use.5FU)] +
Site.eff[match(dta$Site, Site.name)];
dta$lambd.simp <- exp(log.lamb.simp)
log.lamb.strat <- tx.eff[match(dta$tx.strat, tx.names)] +
risk.eff[match(dta$risk, prior.SCC.cnt)] +
Use.5FU.eff[match(dta$Use.5FU, Use.5FU)] +
Site.eff[match(dta$Site, Site.name)];
dta$lambd.strat <- exp(log.lamb.strat)
if (mdl.type=='gaussian') {
dta$SSCs.simp <- rnorm(n = nn, mean = log(dta$lambd.simp), sd = 2)
dta$SSCs.strat <- rnorm(n = nn, mean = log(dta$lambd.strat), sd = 2)
if (mdl.type=='poisson') {
dta$SSCs.simp <- rpois(n = nn, lambda = dta$lambd.simp)
dta$SSCs.strat <- rpois(n = nn, lambda = dta$lambd.strat)
if (mdl.type=='quasipoisson') {
dta$SSCs.simp <- rnbinom(n = nn, mu = dta$lambd.simp, size = dta$lambd.simp/(disper-1))
dta$SSCs.strat <- rnbinom(n = nn, mu = dta$lambd.strat, size = dta$lambd.strat/(disper-1))
m.pois.0.simp <- glm(formula = SSCs.simp ~ tx.simp, family = mdl.type, data = dta)
m.pois.1.simp <- glm(formula = SSCs.simp ~ tx.simp + risk + Use.5FU + Site, family = mdl.type, data = dta)
m.pois.0.strat <- glm(formula = SSCs.strat ~ tx.strat, family = mdl.type, data = dta)
m.pois.1.strat <- glm(formula = SSCs.strat ~ tx.strat + risk + Use.5FU + Site, family = mdl.type, data = dta)
est.0.simp[ii] <- coef(m.pois.0.simp)['tx.simpSIRO'];
est.1.simp[ii] <- coef(m.pois.1.simp)['tx.simpSIRO'];
est.0.strat[ii] <- coef(m.pois.0.strat)['tx.stratSIRO'];
est.1.strat[ii] <- coef(m.pois.1.strat)['tx.stratSIRO'];
## Extract the standard errors for the treatment effect
se.0.simp[ii] <- summary(m.pois.0.simp)$coef['tx.simpSIRO', 'Std. Error']
se.1.simp[ii] <- summary(m.pois.1.simp)$coef['tx.simpSIRO', 'Std. Error']
se.0.strat[ii] <- summary(m.pois.0.strat)$coef['tx.stratSIRO', 'Std. Error']
se.1.strat[ii] <- summary(m.pois.1.strat)$coef['tx.stratSIRO', 'Std. Error']
if (mdl.type=='quasipoisson') {
disp.0.simp[ii] <- summary(m.pois.0.simp)$dispersion;
disp.1.simp[ii] <- summary(m.pois.1.simp)$dispersion;
disp.0.strat[ii] <- summary(m.pois.0.strat)$dispersion;
disp.1.strat[ii] <- summary(m.pois.1.strat)$dispersion;
if (ii%%5000==0) {
et <- proc.time()['elapsed'];
cat(ii, round(et-st), 'secs. \n')
## Use summary statistics to check that the stratification is working
## Stratum specific means and SD's should be approximately the same under both simple and stratified randomisation
## But stratum specific counts should be balanced by arm under stratified randomisation only
smry.strat % group_by(stratum, tx.strat) %>% summarise(mn.strat = mean.sd(SSCs.strat, cnt = T))
smry.simp % group_by(stratum, tx.simp) %>% summarise(mn.simp = mean.sd(SSCs.simp, cnt = T))
smry <- merge(smry.simp, smry.strat, by.x = c('stratum', 'tx.simp'), by.y = c('stratum', 'tx.strat'))
assign(paste0('smry.', mdl.type), smry)
## Report results ####
multi.strat.sd <- mean.sd(est.1.strat, digits.sd = 4)
uni.strat.sd <- mean.sd(est.0.strat, digits.sd = 4)
multi.simp.sd <- mean.sd(est.1.simp, digits.sd = 4)
uni.simp.sd <- mean.sd(est.0.simp, digits.sd = 4)
cat('\n\n**', paste(mdl.type, 'model treatment effect'), '**')
cat('\nSimple randomisation\n')
cat('Univariate model point estimate:', uni.simp.sd, '\n')
cat('Multivariate model point estimate:', multi.simp.sd, '\n')
cat('\nStratified randomisation\n')
cat('Univariate model point estimate:', uni.strat.sd, '\n')
cat('Multivariate model point estimate:', multi.strat.sd, '\n')
uni.simp.mdl.sd % mean
multi.simp.mdl.sd % mean
uni.strat.mdl.sd % mean
multi.strat.mdl.sd % mean
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ML: Naive Bayes classification
ML: Naive Bayes classification¶
Classification is one form of supervised learning. The aim is to annotate all data points with a label. Those points that have the same label belong to the same class. There can be two or more
labels. For example, a lifeform can be classified (coarsely) with labels animal, plant, fungi, archaea, bacteria, protozoa, and chromista. The data points are observed to have certain features that
can be used to predict their labels. For example, if it is has feathers, then it is most likely an animal.
In supervised learning an algorithm is first given a training set of data points with their features and labels. Then the algorithm learns from these features and labels a (probabilistic) model,
which can afterwards be used to predict the labels of previously unseen data.
Naive Bayes classification is a fast and simple to understand classification method. Its speed is due to some simplifications we make about the underlying probability distributions, namely, the
assumption about the independence of features. Yet, it can be quite powerful, especially when there are enough features in the data.
Suppose we have for each label L a probability distribution. This distribution gives probability for each possible combination of features (a feature vector):
\[P(features | L).\]
The main idea in Bayesian classification is to reverse the direction of dependence: we want to predict the label based on the features:
\[P(L | features)\]
This is possible by the Bayes theorem:
\[P(L | features) = \frac{P(features | L)P(L)}{P(features)}.\]
Let’s assume we have to labels L1 and L2, and their associated distributions: \(P(features | L1)\) and \(P(features | L2)\). If we have a data point with “features”, whose label we don’t know, we can
try to predict it using the ratio of posterior probabilities:
\[\frac{P(L1 | features)}{P(L2 | features)} = \frac{P(features | L1)P(L1)}{P(features | L2)P(L2)}.\]
If the ratio is greater than one, we label our data point with label L1, and if not, we give it label L2. The prior probabilities P(L1) and P(L2) of labels can be easily found out from the input
data, as for each data point we also have its label. Same goes for the probabilities of features conditioned on the label.
We first demonstrate naive Bayes classification using Gaussian distributions.
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.datasets import make_blobs
X,y = make_blobs(100, 2, centers=2, random_state=2, cluster_std=1.5)
colors=np.array(["red", "blue"])
plt.scatter(X[:, 0], X[:, 1], c=colors[y], s=50)
for label, c in enumerate(colors):
plt.scatter([], [], c=c, label=str(label))
from sklearn.naive_bayes import GaussianNB
from sklearn.naive_bayes import MultinomialNB
model = GaussianNB()
#model = MultinomialNB()
model.fit(X, y);
Naive Bayes algorithm fitted two 2-dimensional Gaussian distribution to the data. The means and the variances define these distributions completely.
print("Means:", model.theta_)
print("Standard deviations:", model.sigma_)
Means: [[-1.64939095 -9.36891451]
[ 1.29327924 -1.24101221]]
Standard deviations: [[ 2.06097005 2.47716872]
[ 3.33164807 2.22401384]]
Let’s plot these distributions. First we define a helper function to draw an ellipse that gives the standard deviation in each direction from the origo.
def plot_ellipse(ax, mu, sigma, color="k", label=None):
Based on
from matplotlib.patches import Ellipse
# Compute eigenvalues and associated eigenvectors
vals, vecs = np.linalg.eigh(sigma)
# Compute "tilt" of ellipse using first eigenvector
x, y = vecs[:, 0]
theta = np.degrees(np.arctan2(y, x))
# Eigenvalues give length of ellipse along each eigenvector
w, h = 2 * np.sqrt(vals)
ax.tick_params(axis='both', which='major', labelsize=20)
ellipse = Ellipse(mu, w, h, theta, color=color, label=label) # color="k")
return ellipse
Then we do the actual plotting:
plt.xlim(-5, 5)
plt.ylim(-15, 5)
plot_ellipse(plt.gca(), model.theta_[0], np.identity(2)*model.sigma_[0], color="red")
plot_ellipse(plt.gca(), model.theta_[1], np.identity(2)*model.sigma_[1], color="blue");
Accuracy score gives a measure about how well we managed to predict the labels. The maximum value is 1.0.
from sklearn.metrics import accuracy_score
y_fitted = model.predict(X)
print("Accuracy score is", acc)
The score was the best possible, which is not a surprise, since we tried to predict the data we had already seen! Later we will split our data into two parts: one for learning the model and the other
for testing its predictive skills.
Another example¶
Let’s generate some more data using multivariate normal distributions.
cov=np.array([[ 4.68, -4.32],
[-4.32, 4.68]])
mean1 = [0,0]
mean2 = [0,4]
x1 = np.random.multivariate_normal(mean1, cov, n).T
x2 = np.random.multivariate_normal(mean2, cov, n).T
y=np.hstack([[0]*n, [1]*n]).T
plt.scatter(X[:n,0], X[:n,1], color="red", label=0)
plt.scatter(X[n:,0], X[n:,1], color="blue", label=1)
The two clusters seem to be quite separate. Let’s try naive Bayesian classification on this data.
model = GaussianNB()
#model = MultinomialNB()
model.fit(X, y);
print("Means:", model.theta_)
print("Standard deviations:", model.sigma_)
Means: [[ 0.03197033 -0.06105273]
[-0.06051563 4.06189544]]
Standard deviations: [[ 4.72397818 4.72222037]
[ 4.55421873 4.7211965 ]]
y_fitted = model.predict(X)
colors=np.array(["red", "blue"])
plt.scatter(X[:,0], X[:,1], color=colors[y_fitted])
plt.scatter([], [], color="red", label="0")
plt.scatter([], [], color="blue", label="1")
from sklearn.metrics import accuracy_score
print("Accuracy score is", acc)
Even thought the score is quite good, we can see from the plot that the algorithm didn’t have good models for the data. We can plot the models the algorithm used:
plt.xlim(-10, 10)
plt.ylim(-15, 10)
e1=plot_ellipse(plt.gca(), model.theta_[0], np.identity(2)*model.sigma_[0], color="red", label="0")
e2=plot_ellipse(plt.gca(), model.theta_[1], np.identity(2)*model.sigma_[1], color="blue", label="1")
plt.legend([e1, e2], ["0", "1"]);
The problem with naive Bayesian classification is that it tries to model the data using Gaussian distributions, which are aligned along the x and y axes. With this example data we would have needed
Gaussian distributions which are “tilted”.
Text classification¶
We next try to classify a set of messages that were posted on a public forum. The messages were divided into groups by the topics. So, we have a data set ready for classification testing. Let’s first
load this data using scikit-learn and print the message categories.
from sklearn.datasets import fetch_20newsgroups
data = fetch_20newsgroups()
Downloading 20news dataset. This may take a few minutes.
Downloading dataset from https://ndownloader.figshare.com/files/5975967 (14 MB)
We concentrate on four message categories only. The tool fetch_20newsgroups allows us to easily split the data into training and testing data.
categories = ['comp.graphics', 'rec.autos', 'sci.electronics', 'sci.crypt']
train = fetch_20newsgroups(subset='train', categories=categories)
test = fetch_20newsgroups(subset='test', categories=categories)
Let’s see what we got:
print("Training data:", "Data:", str(type(train.data)), len(train.data), "Target:", str(type(train.target)), len(train.target))
print("Test data:", "Data:", str(type(test.data)), len(test.data), "Target", str(type(test.data)), len(test.target))
Training data: Data: <class 'list'> 2364 Target: <class 'numpy.ndarray'> 2364
Test data: Data: <class 'list'> 1574 Target <class 'list'> 1574
We use as features the frequencies of each word in the dataset. That is, there are as many features as there are distinct words in the dataset. We denote the number of features by \(f\). As the
features are now counts, it is sensible to use multinomial distribution instead of Gaussian.
Let’s try to model these messages using multinomial distributions. Each message category has its own distribution. A multinomial distribution has \(f\) non-negative parameters \(\theta_1,\ldots , \
theta_f\), which sum up to one. For example, the parameter \(\theta_3\) might tell the the probability of the word “board” appearing in a message of the category this distribution is describing.
In scikit-learn there is a class CountVectorizer that converts messages in form of text strings to feature vectors. We can integrate this conversion with the model we are using (multinomial naive
Bayes), so that the conversion happens automatically as part of the fit method. We achive this integration using the make_pipeline tool.
#from sklearn.feature_extraction.text import TfidfVectorizer # an alternative feature extractor
from sklearn.feature_extraction.text import CountVectorizer
from sklearn.naive_bayes import MultinomialNB
from sklearn.pipeline import make_pipeline
#model = make_pipeline(TfidfVectorizer(), MultinomialNB())
model = make_pipeline(CountVectorizer(), MultinomialNB())
model.fit(train.data, train.target)
labels_fitted = model.predict(test.data)
print("Accuracy score is", accuracy_score(labels_fitted, test.target))
Accuracy score is 0.920584498094
The classifier seem to work quite well! Notice that now we used separate data for testing the model.
Let’s have a closer look at the resulting feature vectors.
print("Type of feature matrix:", type(features))
print(features[0,:]) # print the features of the first sample point
Type of feature matrix: <class 'scipy.sparse.csr.csr_matrix'>
(0, 20579) 1
(0, 19220) 1
(0, 29697) 1
(0, 6320) 1
(0, 25926) 1
(0, 34222) 1
(0, 31398) 1
(0, 17883) 1
(0, 16809) 1
(0, 34425) 1
(0, 23460) 1
(0, 21787) 1
(0, 11068) 1
(0, 29494) 1
(0, 29505) 1
(0, 18436) 1
(0, 24025) 1
(0, 25336) 1
(0, 12577) 1
(0, 27517) 1
(0, 30641) 1
(0, 5980) 1
(0, 29104) 1
(0, 27521) 1
(0, 11100) 1
: :
(0, 17310) 1
(0, 25400) 1
(0, 23118) 1
(0, 31686) 6
(0, 27158) 1
(0, 18085) 1
(0, 12580) 1
(0, 2100) 1
(0, 20381) 1
(0, 32729) 1
(0, 23854) 2
(0, 11079) 1
(0, 15109) 2
(0, 20509) 1
(0, 23858) 1
(0, 26624) 1
(0, 30377) 1
(0, 16034) 1
(0, 19099) 1
(0, 13317) 6
(0, 34790) 6
(0, 9553) 4
(0, 21852) 5
(0, 18962) 3
(0, 15373) 1
The feature matrix is stored in sparse format, that is, only the nonzero counts are stored. How many words were in the first message?
print("Number of words:", features[0,:].sum())
col = vec.vocabulary_["it"] # Get the column of 'it' word in the feature matrix
print(f"Word 'it' appears in the first message {features[0, col]} times.")
print(train.data[0]) # Let's print the corresponding message as well
Number of words: 177
Word 'it' appears in the first message 2 times.
From: jgfoot@minerva.cis.yale.edu (Josh A. Goldfoot)
Subject: Re: Organized Lobbying for Cryptography
Organization: Yale University
Lines: 21
Distribution: inet
Reply-To: jgfoot@minerva.cis.yale.edu
NNTP-Posting-Host: minerva.cis.yale.edu
X-Newsreader: TIN [version 1.1 Minerva PL9]
Shaun P. Hughes (sphughes@sfsuvax1.sfsu.edu) wrote:
: In article <1r3jgbINN35i@eli.CS.YALE.EDU> jgfoot@minerva.cis.yale.edu writes:
: >Perhaps these encryption-only types would defend the digitized porn if it
: >was posted encrypted?
: >
: >These issues are not as seperable as you maintain.
: >
: Now why would anyone "post" anything encrypted? Encryption is only of
: use between persons who know how to decrypt the data.
: And why should I care what other people look at?
I was responding to another person (Tarl Neustaedter) who held that the
EFF wasn't the best organization to fight for crytography rights since the
EFF also supports the right to distribute pornography over the internet,
something some Crypto people might object to. In other words, he's
implying that there are people who will protect any speech, just as long
as it is encrypted.
Write function blob_classification that gets feature matrix X and label vector y as parameters. It should then return the accuracy score of the prediction. Do the prediction using GaussianNB, and use
train_test_split function from sklearn to split the dataset in to two parts: one for training and one for testing. Give parameter random_state=0 to the splitting function so that the result is
deterministic. Use training set size of 75% of the whole data.
Write function plant_classification that does the following:
• loads the iris dataset using sklearn (sklearn.datasets.load_iris)
• splits the data into training and testing part using the train_test_split function so that the training set size is 80% of the whole data (give the call also the random_state=0 argument to make
the result deterministic)
• use Gaussian naive Bayes to fit the training data
• predict labels of the test data
• the function should return the accuracy score of the prediction performance (sklearn.metrics.accuracy_score)
This exercise can give four points at maximum!
In this exercise we create a model that tries to label previously unseen words to be either Finnish or English.
Part 1.
Write function get_features that gets a one dimensional np.array, containing words, as parameter. It should return a feature matrix of shape (n, 29), where n is the number of elements of the input
array. There should be one feature for each of the letters in the following alphabet: “abcdefghijklmnopqrstuvwxyzäö-“. The values should be the number of times the corresponding character appears in
the word.
Part 2.
Write function contains_valid_chars that takes a string as a parameter and returns the truth value of whether all the characters in the string belong to the alphabet or not.
Part 3.
Write function get_features_and_labels that returns the tuple (X, y) of the feature matrix and the target vector. Use the labels 0 and 1 for Finnish and English, respectively. Use the supplied
functions load_finnish() and load_english() to get the lists of words. Filter the lists in the following ways:
• Convert the Finnish words to lowercase, and then filter out those words that contain characters that don’t belong to the alphabet.
• For the English words first filter out those words that begin with an uppercase letter to get rid of proper nouns. Then proceed as with the Finnish words.
Use get_features function you made earlier to form the feature matrix.
Part 4.
We have earlier seen examples where we split the data into learning part and testing part. This way we can test whether the model can really be used to predict unseen data. However, it can be that we
had bad luck and the split produced very biased learning and test datas. To counter this, we can perform the split several times and take as the final result the average from the different splits.
This is called cross validation.
Create word_classification function that does the following:
Use the function get_features_and_labels you made earlier to get the feature matrix and the labels. Use multinomial naive Bayes to do the classification. Get the accuracy scores using the
sklearn.model_selection.cross_val_score function; use 5-fold cross validation. The function should return a list of five accuracy scores.
The cv parameter of cross_val_score can be either an integer, which specifies the number of folds, or it can be a cross-validation generator that generates the (train set,test set) pairs. What
happens if you pass the following cross-validation generator to cross_val_score as a parameter: sklearn.model_selection.KFold(n_splits=5, shuffle=True, random_state=0).
Why the difference?
This exercise gives two points if solved correctly!
In the src folder there are two files: ham.txt.gz and spam.txt.gz. The files are preprocessed versions of the files from https://spamassassin.apache.org/old/publiccorpus/. There is one email per
line. The file ham.txt.gz contains emails that are non-spam, and, conversely, emails in file spam.txt are spam. The email headers have been removed, except for the subject line, and non-ascii
characters have been deleted.
Write function spam_detection that does the following:
• Read the lines from these files into arrays. Use function open from gzip module, since the files are compressed. From each file take only fraction of lines from the start of the file, where
fraction is a parameter to spam_detection, and should be in the range [0.0, 1.0].
• forms the combined feature matrix using CountVectorizer class’ fit_transform method. The feature matrix should first have the rows for the ham dataset and then the rows for the spam dataset. One
row in the feature matrix corresponds to one email.
• use labels 0 for ham and 1 for spam
• divide that feature matrix and the target label into training and test sets, using train_test_split. Use 75% of the data for training. Pass the random_state parameter from spam_detection to
• train a MultinomialNB model, and use it to predict the labels for the test set
The function should return a triple consisting of
• accuracy score of the prediction
• size of test sample
• number of misclassified sample points
Note. The tests use the fraction parameter with value 0.1 to ease to load on the TMC server. If full data were used and the solution did something non-optimal, it could use huge amounts of memory,
causing the solution to fail. | {"url":"https://csmastersuh.github.io/data_analysis_with_python_spring_2020/bayes.html","timestamp":"2024-11-08T23:54:31Z","content_type":"text/html","content_length":"70394","record_id":"<urn:uuid:88625c34-ee7f-44e1-ae49-b2954ac2edc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00678.warc.gz"} |
Ontology in a modern website
Anywhere there are products, ontology provides a meaningful way to understand and organize them. No search or browse experience on any marketplace is possible without ontology giving people and
computes shared context. One of the greatest temptations you have to fight when designing an ontology is to create a fixed structure embodying everything you know about your business. And why not?
After all, zoologists have their zoological classification, botanists have their classification of plants with their kingdoms,geologists classify their rocks. Every time a new plant is discovered it
fits neatly into the classification. Number of known plants increases but the classification itself almost never changes. All classifications represent languages — vocabularies people use to discuss
classified things. That’s the key difference between biology and a website selling products consumed by regular people: all biologists use the same vocabulary while a layperson might have no idea how
to name or even describe a product they are actually looking for. I most often run into this problem at home improvements stores where the only way for me to find something is to describe it to an
employee by a function it performs and have them walk me to the proper shelf.
A website’s internal ontology must connect products and the vocabulary of its users. As such, it changes as often as the language itself. For instance, in the case of job search, changes in some
occupations are as frequent as (for example) publishing a new web development framework for web developers. Then it’s the task of our ontology to put it in the right place (somewhere under “web
development”), connect it to similar frameworks, to programming language it uses and so on. Not only the skills change, but large parts of ontology need to be updated relatively often. Just think if
“development” in “cloud” is the same as “on premise”, as “hybrid” and as “private cloud” and if it isn’t then where the difference will show up. The whole area might spring practically overnight and
rise from obscurity to mainstream, like for example “reinforcement learning”. Clearly, dramatic changes are not specific to job search and home improvement but exist in any area where a website needs
to understand the vocabulary of regular people.
Another temptation is to plug special people (curators) in the loop for every change. There are many reasons it sounds tempting: people will help avoid SEO (search engine optimization) problems,
potential embarrassment when terms show up in wrong places (as constantly happens on Facebook) and so on. The problem, of course, is that people become a bottleneck. Not just everyone can curate an
ontology: such a person requires understanding of ontology’s structure as a whole and expertise in the area of curation. Now your site has to wait until a person looks at every change. Meanwhile your
site has no idea what its users are looking for. All these risks can be mitigated without human involvement. We can set up multiple sources of automated updates: parsing various pages on the
Internet, analysing user queries, taking user’s suggestions and so on. Every change runs through ML models that establish its location in ontology. The change is then automatically plugged into the
right place.. Every so often we want to rearrange things to fight entropy and make sure we are in the best possible place and that’s where the curated release comes in.
Speaking of the ontology’s structure, the idea that comes to everyone is to model it based on other known taxonomies: plants, rocks, and so on. Let’s consider the difference a true ontology brings to
the site’s functionality. For example, let’s take a website matching job offers to candidate profiles. One of the main friction points is getting candidates to specify skills they have and employers
— skills they need for a particular job. One of the approaches is to build a taxonomy solving that particular problem and guiding users from general to specific in such a way that at the end all of
them speak the same language and the website can match offers to profiles. An example of such a taxonomy is below. While it certainly works there are a few obvious issues: if something (like
“pandas”, 1) fits into several places you need to build several hierarchies (2,3) leading to it. After you found it in one place finding all other places where it is requires a separate query. You
can’t build any kind of hypothesis based on your taxonomy as you don’t know how close “Data Mining” is to “Machine Learning” so you can’t suggest anything to your users based on where “pandas” is in
the hierarchy the user actually followed to find it. Similarly, your matching algorithm has to rely on the precise skills match. For example, if a job requires “pandas” and “python” while a profile
specifies “pandas” only, matching algorithm has no basis for assuming that anybody having experience with “pandas” necessarily has to know “python”. Suppose there is another tool called “SciPi”. In a
single taxonomy model we don’t know if “SciPi” is really similar to “pandas” or performs a different function entirely. Consecutively we don’t know if it needs to be present in all the places
“pandas” is present or if people knowing “pandas” can master “SciPi” easily. We never know how complete our taxonomy is as it’s likely built by curators and can only be updated from similar
taxonomies (unlikely as this is purpose-built) or manually.
All of these questions and problems can be resolved if we re-plan the taxonomy into more connected ontology. In fact, some of these issues can be resolved even if we add different types of links into
the existing structure. For example, adding “depends on” type of link (dotted green arrow) helps us figure out if people who know “pandas” and “SciFi” know “python” as well. Perhaps they can not only
do Data Science jobs but other types of jobs that require “python” as well!
We can go one step further and rearrange our taxonomy into several connected ones and create a real ontology as in the picture below.
While the ontology here is far from ideal it solves practically all issues we noticed in the taxonomy above. Taxonomies presented here can be measured in completeness. They already allow independent
automated updates. For example, we can add new programming languages as they appear. Adding a new occupation can be done by parsing and processing texts related to that occupation, such as textbooks,
specialized websites, other job search sites to set up links between the occupation and its tools and other aspects. Continuing a job search example, a well-developed ontology makes it possible to
answer a query “develop eCommerce website” with several proposals for a team composition and individuals ready to be hired for that team. Links need to have attributes. The most widely used attribute
is the version numbers expressing ontology’s versions. With such links you can easily find the differences between ontology versions (as one query to the graph), recover state of the graph at any
date or version in the past (very useful for various ML purposes). Having different link types and interconnected nature allows you to use the graph for different purposes: helping users browse your
product, search for best fit, suggest search criteria or keywords, convert text terms to semantics.
In conclusion, it’s hard to imagine a modern website with search and browse capabilities without using ontology for semantic search and match, as well as for directing a user in their discovery of
your products. | {"url":"https://a-v-goldberg.medium.com/ontology-in-a-modern-website-6edc840aa6ad","timestamp":"2024-11-09T03:53:57Z","content_type":"text/html","content_length":"106155","record_id":"<urn:uuid:fc808a66-eef1-4dc0-b1bc-adc42a5112ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00669.warc.gz"} |
How do you calculate grade slope?
How do you calculate the percentage grade of a slope? Slope can be calculated as a percentage which is calculated in much the same way as the gradient. Convert the rise and run to the same units and
then divide the rise by the run. Multiply this number by 100 and you have the percentage slope.
How do you calculate average grade elevation?
How to Find Grade of an Elevation. Grade can be found by measuring the horizontal length of an elevation, the run, and the vertical height of the elevation, the rise. Grade is expressed as rise/run,
so if the rise is 25 and the run is 80 the grade is 25/80.
What is the gradient calculator?
Gradient calculator lets you measure the steepness of a line going through two points.
How is stream gradient measured?
Gradient is the slope of the stream and is measured by the difference in elevation between two points on a stream divided by the distance between the two points that the water actually flows.
Gradient is usually expressed in feet per mile of meters per kilometer.
Can a gradient be a decimal?
Gradient is usually expressed as a simplified fraction. It can also be expressed as a decimal fraction or as a percentage.
What is the formula for gradient calculation?
The gradient can be found by finding how much the line goes up – the rise, and dividing it by how much the line goes down – the run. Hence, the equation for the gradient = rise / run, or gradient =
change in y / change in x.
How to measure gradient?
Gradients can be calculated by dividing the vertical height by the horizontal distance . Gradient is a measure of how steep a slope is. The greater the gradient the steeper a slope is. The smaller
the gradient the shallower a slope is.
How do you calculate gradient of function?
To find the gradient, take the derivative of the function with respect to x, then substitute the x-coordinate of the point of interest in for the x values in the derivative. For example, if you want
to know the gradient of the function y = 4×3−2×2+7 at the point (1,9) we would do the following:
How do you calculate road gradient?
Calculating road Gradient. The Gradient of a road, or Slope angle, can be calculated simply by taking the ratio of the horizontal speed against the vertical velocity. | {"url":"https://yycnewcentrallibrary.com/how-do-you-calculate-grade-slope/","timestamp":"2024-11-03T10:25:16Z","content_type":"text/html","content_length":"41047","record_id":"<urn:uuid:292619c3-3d7e-4e26-a36c-6d72a3824b56>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00103.warc.gz"} |
zla_herpvgrw - Linux Manuals (3)
zla_herpvgrw (3) - Linux Manuals
zla_herpvgrw.f -
DOUBLE PRECISION function zla_herpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
Function/Subroutine Documentation
DOUBLE PRECISION function zla_herpvgrw (character*1UPLO, integerN, integerINFO, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )
IPIV, double precision, dimension( * )WORK)
ZLA_HERPVGRW computes the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute element" norm is used. If this is
much less than 1, the stability of the LU factorization of the
(equilibrated) matrix A could be poor. This also means that the
solution X, estimated condition numbers, and error bounds could be
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
INFO is INTEGER
The value of INFO returned from ZHETRF, .i.e., the pivot in
column INFO is exactly 0.
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF is COMPLEX*16 array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by ZHETRF.
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by ZHETRF.
WORK is COMPLEX*16 array, dimension (2*N)
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 123 of file zla_herpvgrw.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-zla_herpvgrw/","timestamp":"2024-11-10T21:38:00Z","content_type":"text/html","content_length":"9163","record_id":"<urn:uuid:b8aee9b0-395c-46ee-b607-214fc727cf13>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00337.warc.gz"} |
Back to School - Basic Pool Math – Aquatic Council, LLC – Pool Operator Certification
Basic pool math isn’t rocket science… it’s not brain surgery… but it can be complicated. Even seasoned operators find themselves stumped by seemingly basic pool math problems. Troubleshooting pool
issues often starts with a bit of math – and getting that math right may make all the difference. To help, we’ve dug deep and compiled a summary of the basic pool math you need to know, with simple
explanations for how the numbers impact your job.
Surface Area and Volume – Let’s start with the basics. Before we dose chemicals or buy pump room equipment, we’re going to need to know how big your pool is. We’ll find that both surface area and
volume are tremendously useful numbers to know. Much of the later math we’ll discuss use these values.
To figure out surface area, we’ll use a couple basic formulas:
Surface Area of a Rectangular Pool = Length x Width
Surface Area of a Circular Pool = Radius x Radius x 3.14
Tip – Make sure all of your measurements are starting in linear feet, not meters or yards. Not sure how to convert? See the conversion section below.
Tip – A radius is the distance from the center of a circle to its outside edge (halfway across the circle). When sizing circles, 3.14 is a constant number in the formula, representing the number Pi.
Our volume formulas are very similar:
Volume of a Rectangular Pool = Length x Width x Average Depth x 7.5
Volume of a Circular Pool = Radius x Radius x 3.14 x Average Depth x 7.5
Tip – The number 7.5 at the end of both equations is a constant representing the amount of water in a cubic foot. If this value was not included in the formula, your answer would be given in cubic
feet, as opposed to gallons.
Tip – Average depth is an easier value to determine than you may think. Simply add the deep end depth, to the shallow end depth and divide by two.
Average Depth = (Shallow End Depth + Deep End Depth) / 2
Tip – Multiple sections or depths in your pool? Break it up into simple areas and repeat the math.
Tip – Have an odd shaped pool? The math for figuring out your surface area or volume may be very complex. Check your pool permit, engineering specs or blueprints. Many of these documents show your
surface area and volume.
Conversions – Pool operators are frequently tasked with performing a mathematical conversion. Make sure you know the most common poolside conversions:
Dry Ounces to Pounds Ounces / 16 = Pounds
Fluid Ounces to Gallons Fluid Ounces / 128 = Gallons
Yards to Feet Yards x 3 = Feet
Meters to Feet Meters x 3.28 = Feet
Chemical Adjustment – Ready to add a chemical to your pool? You’ll need to consider three variables:
1. The volume of your pool
2. The desired change in chemical value, measured in parts per million (ppm)
3. Chemical dosing recommendations (from the product label or an adjustment guide)
Example: You are planning to add calcium hypochlorite (dry chlorine) to your 35,000 gallon pool. You would like to raise your ppm from 2 to 5. The product label instructions note that you would add 2
ounces per 10,000 gallons of water for a 1 ppm change.
Here’s how the variables would produce factors and a final dosage:
Your Pool Recommendations Factor
Volume 35,000 Gallons / 10,000 Gallons = 3.5
Desired Change 3 ppm / 1 ppm = 3
Dosage Recommendation………………………………………………………………………… = 2 oz.
Simply multiply your three factors for your final dosage amount:
3.5 x 3 x 2 ounces = 21 ounces
Breakpoint Chlorination – You can smell a poorly treated pool from a mile away. The smell associated some pools is often blamed on chlorine, but that’s not what your patrons are smelling. The smell
is actually attributed to chloramines, the gaseous bi-product of combined chlorine. Although there are a few methods for chloramine reduction, one of the most effective is a chemical process known as
breakpoint chlorination. Raising your chlorine levels rapidly can separate combined chlorines and dramatically improve air quality in your facility. Here’s the basic formula that helps us figure out
how much chlorine we need to add to achieve breakpoint.
Total Chlorine
– Free Chlorine
= Combined Chlorine
X 10__________
= Breakpoint
– Free Chlorine
= PPM Change
Tip – This formula only needs two numbers to work, your total chlorine and your free chlorine (which is used twice in the equation). Both can be determined with a common pool test kit. Your final
answer is your desired ppm adjustment and can be used in the standard chemical adjustment formula described above.
Heating – In the market for a new heater? They’re sized in BTU’s or British Thermal Units. Figuring out how many BTU’s your system needs uses a simple formula:
BTU’s = Gallons x 8.33 x Change in Temperature
Tip – Like many of our other formulas, this equation features a constant (8.33) that is used in determining BTU’s when considering a temperature change in Fahrenheit.
Turnover, Volume and Flow Rate – Pools are designed to set specifications with regards to turnover, volume and flow. Because of this, there are consistent relationships between these numbers that
become useful in pool operations. These relationships can be represented in the following set of formulas.
Turnover Rate = Volume / Flow Rate / 60
Flow Rate = Volume / Turnover Rate / 60
Tip – Because we’re moving between minutes and hours in these equations, the number 60 is included as a constant in both formulas.
Tip – Turnover rate is measured in hours, flow rate is measured in gallons per minute and volume is measured in gallons in these equations.
Filtration – Like turnover, volume and flow rate, our filters also follow a set of design standards that can be useful in day to day operations. These relationships yield yet another set of formulas.
Filter Area = Flow Rate / Filter Media Rate
Filter Media Rate = Flow Rate / Filter Area
Flow Rate = Filter Area x Filter Media Rate
Tip – Although written three different ways, these are actually the same formula. When tasked with using one of these equations to determine a value, use the one that starts with the value that you
are looking for.
Formula Summary List
Area and Volume
Area of a Rectangle = Length x Width
Area of a Circle = Radius x Radius x 3.14
Volume of a Rectangle = Length x Width x Average Depth x 7.5
Volume of a Circle = Radius x Radius x 3.14 x Average Depth x 7.5
Breakpoint Chlorination Formul
Total Chlorine
– Free Chlorine
= Combined Chlorine
X 10__________
= Breakpoint
– Free Chlorine
= PPM Change
Turnover, Flow and Volume Relationships
Turnover Rate = Volume / Flow Rate / 60
Flow Rate = Volume / Turnover Rate / 60
Turnover Rate is measured in Hours
Flow Rate is measured in Gallons Per Minute (gpm)
Volume is measured in Gallons
Filter Sizing
Filter Area = Flow Rate / Filter Media Rate
Filter Media Rate = Flow Rate / Filter Area
Flow Rate = Filter Area x Filter Media Rate
Gas Heater Sizing
BTU’s = Volume x 8.33 x Change in Temperature
© 2020 Aquatic Council, LLC. All rights reserved. This proprietary information is not to be duplicated for commercial use. | {"url":"https://www.aquaticcouncil.com/news/back-school-pool-math-101/","timestamp":"2024-11-11T01:22:28Z","content_type":"text/html","content_length":"99950","record_id":"<urn:uuid:90c15534-680c-4afb-b1dc-42af5a15b0b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00194.warc.gz"} |
Excel relative reference | MBT
Cell references are an integral part of excel use. In financial modeling, there is usually a need to refer to cells for various things like completing a function, data validation list, conditional
formatting, named ranges and so on, and without knowing which type of reference you require, you may end up spending a lot of time making corrections to references initially selected. In this post,
we would walk you through what relative references are and how to go about it. Before we begin however, Let’s touch on the three main types of cell references:
Types of Cell references in Excel
There are three main types of cell references in excel namely:
• Relative references: This is the default referencing style of excel when you select cells within the same workbook. In this, excel automatically calculates the movement from left or right and top
or bottom from the originating cell (where you copy the cell containing the reference) to the destination cell (final destination you paste the formula copied). Confused? Don’t worry, we would
treat in detail in the next section.
• Absolute references: They are references to a particular cell or range of cells that do not change regardless of where you copy from and paste to. They remain fixed and are denoted by a dollar
($) sign appearing before and after the column letter. Please see our post on Absolute references for more details.
• Mxed references: These are a combination of both relative and absolute references. In this case, the columnn could be fixed and row relative or the opposite could be the case. This could be
suitable when you have all inputs on a particular column and you need to copy and paste formulas on adjacent columns. Please see our post on mixed references for more details
Relative reference example
In our example for this section, we have information on data offers of a Telecommunications company. The unit price and sales count have been provided and our task is to calculate revenue for each
plan using a simple multiplication formula. We are however not going to manually type in the formula on each row, as we would be relying on excels relative referencing to copy and paste the formula
to other cells.
Let’s see how we can create a formula in cell D3 to multiply the price and sales count, and then copy it across from cells D4:D10 using relative referencing.
Step 1: Type in the multiplication formula B3 X C3
Step 2: Because the input cells are not anchored (locked in), we can copy and paste with CTRL + C and CTRL + V respectively.
Result: The other plans automatically pick the price and sales count of their respective rows.
We have come to the end of this tutorial, remember that there is no need to do anything when usng relative references. It is the default of excel, so you can just copy and paste. If however you
desire to keep a cell, column or row fixed, then you would need to use absolute references for the first and mixed references for the latter two (column or row fixed).
Leave a Comment | {"url":"https://www.modelsbytalias.com/excel-relative-reference/","timestamp":"2024-11-13T02:06:58Z","content_type":"text/html","content_length":"83335","record_id":"<urn:uuid:b4544c39-22fe-4482-bdd3-2de9ade09fcb>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00831.warc.gz"} |
Refactor rule for arithmetic operators. (!27) · Merge requests · Iris / RefinedC · GitLab
Refactor rule for arithmetic operators.
This PR does a bit of refactoring of the rules for binary (arithmetic) operators on integers. This will probably be useful to @paulzhu, and I needed this first step for the handling of right shifts.
The changes are the following:
• Define a function arith_op_result that, given the two operands and the op computes the result.
• Define a predicate arith_op_sidecond that gives the side conditions for a given op.
• Use the above functions in the rule type_arith_op_int_int, and factor in the rules for / and %.
• Give more precise side conditions for the right and left shift operators (this is the goal of this PR). Note that the standard leaves it as implementation-define what is the result of
right-shifting a negative number. I thing we should consider that UB.
Some further factoring of code is possible with the operational semantics, but we can do that as a second step.
Merge request reports | {"url":"https://gitlab.rts.mpi-sws.org/iris/refinedc/-/merge_requests/27","timestamp":"2024-11-08T10:31:46Z","content_type":"text/html","content_length":"64082","record_id":"<urn:uuid:8a9d5b48-8414-4d99-b8ec-fe7f3729ff05>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00791.warc.gz"} |
Infinite Monkeys and Alphabet Permutations
Given enough time, a chimp punching keys on a typewriter at random will almost surely type out all of Shakespeare's plays.
Of course, the modern chimpanzee would choose a laptop over a typewriter for this task.
This metaphor describes what's known as the "
infinite monkey theorem
", which basically states that any finite string of characters must be contained within an infinite string of random characters. Here's the proof:
Let's start with looking at the probability of typing a particular finite string of letters on the first try. Let's also ignore all the other keys on a typewriter (or keyboard for those of you who've
never seen a
) and consider just the 26 letters of the alphabet. There is a 1 in 26 chance of any particular letter being typed. We are assuming that the letters are selected randomly and independently, so the
chance of typing any two particular letters is (1/26) * (1/26) = 1 in 676. Any three particular letters: (1/26)
= 1 in 17,576. For some number "
" particular letters: 1 in 26 to the
th power. Now the probability of the inverse situation - not typing a particular letter (or block of letters) - is simply 100% minus the probability of successfully typing a particular letter (or
block of letters). I've summarized in the tables below and included notes to help you understand the magnitudes of some of the numbers.
It's clear that randomly typing a short word on the first attempt is extremely unlikely. A complete sentence is practically impossible from our perspective. The chances of typing complete book at
random are so exceedingly small that a physical analogy doesn't exist. Let's go off on a bit of tangent to look at some really big and really small numbers in the physical universe. The
observable universe
is a sphere with a diameter of approximately 92 billion light years, or (to employ some of Mr. Spock's absurd precision) approximately 870,387,203,477,433,600,000,000,000 metres. The
Planck length
represents the shortest length that, theoretically, could ever possibly be measured. It's approximately equal to 0.000 000 000 000 000 000 000 000 000 000 000 016 161 99 metre. However difficult to
fathom the magnitudes of these numbers might be, just keep in mind that I can still fit them on one line in this blog post without resorting to scientific notation. The number representing the chance
of successfully typing Hamlet at random on the first attempt contains 55,059 more digits than the play Hamlet does letters. Similarly, the number representing the chance of successfully typing the
Bible at random on the first attempt contains 1,467,549 more digits than the Bible does letters. Here are some other really big and really small numbers to show just how small the observable universe
is compared to the number of ways you can fail to type a complete work of fiction at random:
Ok, back to proving the infinite monkey theorem. We've calculated the chances of typing a string of
letters on the first attempt. What if we had more monkeys? Let's say there are
monkeys, each with a MacBook Air to type their random strings of letters. The probability of at least one of
monkeys successfully typing a particular string of
letters at random on the first attempt is:
The limit of
approaches infinity is 100%:
This means that, given enough monkeys typing randomly, the probability that at least one will successfully type a particular string of
letters on the first attempt approaches 100%. We can also rearrange the equation above to solve for the number of monkeys necessary to ensure a given likelihood that at least one monkey will be
successful on the first attempt:
I've calculated how large
has to be to give a certain probability of success by at least one of the monkeys and summarized below.
Number of monkeys required to ensure a given probability of success on the first attempt.
The numbers of monkeys needed to achieve a reasonable probability of success are mindbogglingly large, but they are still finite and calculable.
Ok, we've seen what happens with many monkeys, but we can look at this in a different way. What if instead of many monkeys, we have a single monkey with infinite lifespan, typing randomly and
continuously. This problem is a little more interesting and the exact probability is a function of the particular pattern we're looking for. First I'll demonstrate how the probability depends on the
particular pattern. Suppose you're playing "
Penney's Game
" with a fair coin and want to know the chances of getting the sequence HHH or THH in a continuous sequence of tosses. The chance of getting either in the first 3 tosses is equal to 1/2 * 1/2 * 1/2 =
1/8. But as you keep going, the likelihood of THH increases because you have more potential starting points. Let's look at HHH. If you get one H, there's a 50% chance that the next toss is an H, and
an equal chance it's a T. Now if you toss a second H, you have a 50% chance of completing the sequence, but if you toss a T, you have to wait at least one more toss to start over with another H. Put
another way, if you're trying to get HHH but toss H and then T, you have to toss at least three more times to succeed in getting HHH. When you're after THH, if you toss a T and then another T, you're
still potentially only two tosses away from success. In either case, your chance of success approaches 100% as your total number of tosses increases, but THH approaches 100% faster than HHH.
The same situation occurs with our random letters of the alphabet. The probability of finding a certain sequence of letters in a continuous random sequence depends on the sequence that you're looking
for. However, the effect is less significant here because there are 26 possible outcomes per keystroke instead of just 2 and the finite string we're really after is thousands of characters long.
Consider the pairs of letters AA and AS. They have equal chance of appearing in the first two letters (0.148%) of a random sequence. However, in a random three-letter sequence, AS has a 0.290% chance
of appearance, compared to 0.284% for AA.
Despite the complication, there is still hope for our analysis of the monkey that ceaselessly types random letters. We can estimate a very conservative lower bound on the probability by dividing the
sequence of
letters into
non-overlapping blocks. This basically assumes that the string we're searching for must start at some multiple of
-letters into the full sequence.
Conservative lower bound probability
Now it'd be nice if we had an upper bound on the probability. I can't prove that this is an upper bound, and it might not necessarily always be an upper bound, but I think that it is probably likely
to be an upper bound. Instead of assuming there are n/k independent trial starting points, let's assume that every letter is an independent trial starting point. Then subtract (k -1) so that we
eliminate the final few letters as possible starting points (because if you start fewer than k letters from the end, you can't possibly complete the string). To give an example, if the string is PAS,
you can't possibly get PAS at the end of a random letter sequence if the third-to-last letter is not 'P'. So that gives us an estimated upper limit of n - k + 1 independent trials.
Upper bound (?) probability
The limit of both of these equations as n approaches infinity is 100%. This confirms that after typing a sufficient number of letters at random, the probability that you happened to type some finite
string of letters approaches 100%.
We can also take the upper and lower bound probability and estimate the number of letters the monkey would have to type to achieve a given probability of success.
The high estimate of n, based on the low estimate of p.
The low estimate of n, based on the high estimate of p.
Total number of letters required in the sequence to ensure a given probability of matching a string of k letters.
So is there any conceivable way we could actually type something like Hamlet at random? Let's forget about our metaphorical monkeys now and discuss this in terms of computing power. CPU speeds today
are commonly on the order of 3 GHz. A computer with a 3 GHz CPU would not actually be able to generate random letters at a rate of 3 billion per second, but I'll be very conservative anyway to
demonstrate how unlikely it is that the randomly duplicated work of fiction will ever exist. Let's assume that our computers will be able to generate random letters indefinitely at a rate of
3,000,000,000,000,000,000 (3 billion billion) letters per second. According to
this 2008 article
, there were over 1 billion PCs in use at the time and there would be an estimated 2 billion in use by 2014. So let's be really conservative and assume that we employ 4,000,000,000,000,000,000 (4
billion billion) computers with the task of generating random works of literature. I'll even use the upper bound probability estimate here. How long would it take before we had a reasonable
probability that at least one computer matched a particular string of
letters at least once? Well putting it all together gives us this lovely looking equation to estimate the probability of success at any time
(in seconds) after embarking on this endeavor:
Approximate probability of success after t seconds in our hypothetical scenario.
Solving for
from the approximate equation above gives us:
Which gives us an estimated lower bound time limit on matching a particular string of
letters. The number of years it would take before we could reasonably expect a duplication of Hamlet is still mindbogglingly large (it contains 187,694 digits!). The estimated
age of the universe
is only an 11 digit number of years (about 13.8 billion years). Even matching a complete sentence would take thousands of years.
Okay, let's give it one more chance. Surely the universe could duplicate Hamlet if we could enlist alien races to help out. The number of stars in the universe is estimated to be between 10 to the
22nd and 10 to the 24th powers. Let's take 100 times the high estimate and assume 10 to the 26th power. Now let's assume that 10 intelligent races exist around each star and match the computing power
from our previous hypothetical scenario, and that we all coordinate to devote our total computing power to duplicating Hamlet by random letter selection. So in a grand universal waste of time,
effort, and resources, we've employed {4 followed by 45 zeroes} computers spitting out random letters at a rate of 3 billion billion letters per second each. Now
And we still can't duplicate Hamlet within 100 billion years with only a 1 in 1,000,000 probability. In fact, we probably can't even duplicate a short paragraph.
Conservative estimate of the minimum time required to match a particular string of k letters
with given probability using the universe's combined computing power.
So there you have it. By the Infinite Monkey Theorem, duplication of Shakespeare's work is possible with enough computing power. However, actual duplication is practically impossible from a physical
In 2003 the
University of Plymouth actually spent grant money
(about 2,000 British pounds, equal to about $3,270 USD or $4,580 CAD at the time) to give a computer to six macaques for a month and study their "literary output". The monkeys produced only five
pages of text, apparently were fond of the letter 'S', and preferred using the computer as a lavatory to doing any actual typing. Even when they were typing, they made lousy random letter generators.
Some confused creationists,
such as this one
, have used the results of this "study" as evidence against evolution. First, they fail to recognize that the monkeys in the "infinite monkey theorem" metaphor are meant to represent unthinking
generators of random events, not actual monkeys. Actual monkeys do not act randomly. Their past experiences and their environmental conditions will influence their actions. Second, the study involved
only six monkeys sharing one computer for only one month. That's hardly enough time or "monkey power" to generate a random string of letters long enough to expect anything resembling a word, let
alone an entire work of fiction. Anyone who thinks this study tells us anything useful about the infinite monkey theorem is making the absurd assumption that either six monkeys is approximately equal
to infinite monkeys or one month is approximately equal to infinite time. | {"url":"http://alohonyai.blogspot.com/2014/06/infinite-monkeys-and-alphabet.html","timestamp":"2024-11-05T04:35:05Z","content_type":"text/html","content_length":"77348","record_id":"<urn:uuid:6a7c6d80-1305-4e50-9864-a5effbf77f4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00164.warc.gz"} |
High-Speed Digital PCB Project
Last Updated
för 10 år sedan
Creative Commons CC BY 4.0
In this experiment I looked into a way fo measuring the ground bounce generated by capacitively loading 5 outputs of a Texas Instrument small outline package Hex-Flip Flop chip and discharging them
all simultaneously. | {"url":"https://sv.overleaf.com/articles/high-speed-digital-pcb-project/vvgwmpnfsvzm","timestamp":"2024-11-06T20:37:55Z","content_type":"text/html","content_length":"55944","record_id":"<urn:uuid:72dc1bb5-c153-4fe0-a08d-f6202dab3ac7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00088.warc.gz"} |
Frequency Response
What Is a Frequency Response?
A frequency response describes the steady-state response of a system to sinusoidal inputs of varying frequencies and lets control engineers analyze and design control systems in the frequency domain.
To understand why the frequency domain is important consider an acoustic guitar. If we place a microphone close to its soundboard and pluck one of the strings (Fig. 1. left), the vibrating string
will resonate in the guitar cavity and produce a sound wave that is captured by the microphone. Looking at the time trace of the captured signal (Fig. 1, right) it is difficult to quickly extract
information about what is going on.
When we look at that same signal in the frequency domain on a spectrum analyzer or by taking a fast Fourier transform (FFT) of the time domain signal, we see an amplitude peak at some frequency (Fig.
2, left). This peak frequency is the underlying tone that forms the note we just played. When we adjust the tuner knob or press the string to the neck of the guitar, we change the preload or the
effective length of that string. This will shift the frequency at which the string resonates up or down, and we produce a different note (Fig. 2, right). With this simple analysis in the frequency
domain, we can see how the guitar (the system) responds to plucking (the system input).
This analogy can be carried over to other systems where we are interested in the system’s response to inputs or stimuli from the environment. We can get insights into the system dynamics such as
frequency of a resonant peak, DC gain, bandwidth, phase delay, and phase and gain margins for a closed-loop system.
Getting the Frequency Response of a System
The chart below helps identify an approach (shown in gray) to obtain the frequency response of a system using MATLAB^® and Simulink^®.
1. If you have a linear representation of the system in the form of a transfer function or state-space model, you can plot the frequency response using one of the three plots: a Bode plot, Nyquist
plot, or a Nichols chart. The Bode plot displays magnitude and phase as functions of the frequency of the excitation signal (Fig. 4).
For example, given the transfer function representation of a system \((H)\),
$$H(s) = {s^2+ 0.1s + 7.5\over s^4+0.12s^3+9s^2}.$$
you can plot its frequency response in MATLAB using the following commands:
\(H = {tf([1 \quad 0.1\quad 7.5], [1 \quad 0.12 \quad 9 \quad 0 \quad 0]});\)
In some situations, a linear representation of the system might not be available.
2. In that case, if you have access to input-output test data from the physical system, you can use data-driven modeling approaches with System Identification Toolbox™ to identify transfer function,
state-space representations, and frequency response models of the system.
3. If you use Simulink to model the system dynamics, you can use the Model Linearizer app in Simulink Control Design™ to linearize your model to create a linear state-space approximation of your
Simulink model and plot the frequency response.
4. In case the Simulink models cannot linearize due to discontinuities, you can use frequency response estimation to directly estimate a frequency response model.
Simulink Control Design provides two approaches for estimating a frequency response model of your system.
Offline frequency response estimation
The Model Linearizer app excites the system with an input perturbation signal at specified frequencies and logs the response at the model output during simulation (Fig. 5). Post simulation, the
logged input and output signals are processed to compute a frequency response of the model.
Online frequency response estimation
The frequency response of a physical plant is estimated during real-time operation with the Frequency Response Estimator block. This blocks injects sinusoidal test signals into the plant at the
nominal operating point and the frequency response is continually refined as the output signal data is collected
The following table shows the perturbation signals that you can inject based on your estimation needs of frequency range, accuracy, and speed of estimation.
Input signal Availability of offline/online Frequency range (narrowband/ Accuracy Speed of Useful when…
type estimation wideband) estimation
Scale of 1 (low) to 5
Sinestream Offline, Online Narrowband ★★★★★ ★ System contains strong nonlinearities or you require highly accurate frequency response models.
Chirp Offline Wideband ★★ ★★★ System is nearly linear in the frequency range. Also useful when you want to obtain a response
quickly for a lot of frequency points.
PRBS Offline Wideband ★★ ★★★ System contains high frequency switching components such as with communications and power
electronics systems.
Step Offline Wideband ★ ★★★ Exciting the system uniformly at all frequencies up to Nyquist frequency
Random Offline Wideband ★★ ★★★ You don’t have much knowledge about the system you are estimating.
In summary, computing a frequency response of a system is important for control analysis and design. MATLAB and Simulink provide different approaches you can use to get the frequency response of your
system. To learn more about these approaches, see the examples and the references below.
Examples and How To
Simulink Model Linearization
Offline Frequency Response Estimation
Online Frequency Response Estimation | {"url":"https://nl.mathworks.com/discovery/frequency-response.html","timestamp":"2024-11-04T02:59:04Z","content_type":"text/html","content_length":"87497","record_id":"<urn:uuid:001e9575-64d8-401f-9b3b-fd1a61e03efe>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00406.warc.gz"} |
Find the slope of a… - QuestionCove
Ask your own question, for FREE!
Mathematics 60 Online
OpenStudy (anonymous):
Find the slope of a line that passes through the points (4, 5) and (4, –6).
Still Need Help?
Join the QuestionCove community and study together with friends!
hartnn (hartnn):
same formula : The slope of the line through points (x1,y1) and (x2,y2) is given by : \(\huge m=\frac{y_1-y_2}{x_1-x_2}\) now,just put the values and find m.
OpenStudy (anonymous):
the m cannot define x = 4
OpenStudy (anonymous):
Is the answer no slope?
hartnn (hartnn):
yeah, because the denominator comes out to be 0 the slope is infinite or undefined or no slope.
OpenStudy (anonymous):
Thank you!
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hartnn (hartnn):
welcome ^_^
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Automatic (Differentiation) for the COS Method - Matthias Thul's Homepage
Automatic (Differentiation) for the COS Method
In the last post, I provided a brief introduction to forward mode automatic differentiation with CppAD. In this post, I propose to use automatic differentiation for the computation of cumulants of
option pricing models based on characteristic functions. This is useful, for example, when pricing European vanilla options using the Fang and Oosterlee (2008) COS method. Here, the first four
cumulants are used to determine the integration range.
While analytical expressions for the cumulants can usually be obtained with the help of some computer algebra system, they can easily become very long. Consider for example the characteristic
function for the Heston (1993) stochastic volatility model. The cumulant generating function of the martingale component is
see e.g. Albrecher et al. (2007). The second cumulant is
compare to for example Le Floc’h (2014). The fourth order cumulant already spans over half a page. Coding these formulas is tedious, error prone and not efficient.
We instead suggest to use the general differentiation function in the last post to compute the cumulants directly from the cumulant generating function. The advantage of this approach is that it can
be applied to almost any model. It works even when the characteristic function itself is not available analytically but needs to be computed numerically through e.g. quadrature.
Code Example
Consider for example the following incomplete interfaces.
template<typename Implementation>
class FourierBaseModel
template<size_t order>
array<double, order + 1> cumulants(double maturity) const;
class FourierHestonModel
: public FourierBaseModel<FourierHestonModel>
// implementation of the cumulant generating function
template<typename Type>
Type cumulantFunction(double maturity,
Type omega) const;
We implement static polymorphism using the curiously recurring template pattern. This way, the function cumulants(...) has to be implemented only once in the base class. It can invoke the function
cumulantFunction(...) in the implementation without the overhead of dynamic dispatch. The implementation of cumulants(...) is relatively short.
template<typename Implementation>
template<size_t order>
array<double, order + 1> FourierBaseModel<Implementation>::cumulants(double maturity) const {
static complex<double> _oneOverI = 1.0 / complex<double>(0.0, 1.0);
auto function = [=](auto omega) {
return static_cast<Implementation*>(this)->cumulantFunction(maturity, omega);
auto const derivatives = differentiate<order>(move(function), complex<double>(0.0, 0.0));
array<double, order + 1> cumulants;
cumulants[0] = 0.0;
complex<double> scalingFactor(1.0, 0.0);
for (size_t i = 1; i < order + 1; i++) {
scalingFactor *= _oneOverI;
cumulants[i] = (scalingFactor * derivatives[i]).real();
return cumulants;
Here, we used that the
That is all! We just implemented a general framework for the computation of arbitrary order cumulants for any Fourier model.
Albrecher, Hansjörg, Philipp Mayer, Wim Schoutens and Jurgen Tistaert (2007) “The Little Heston Trap,” Wilmott Magazine, pp. 83-92
Fang, Fang and Cornelis W. Oosterlee (2008) “A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions,” SIAM Journal on Scientific Computing, Vol. 31, No. 2, pp. 826-848
Heston, Steven L. (1993) “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, pp. 327-343
Le Floc’h, Fabien (2014) “Fourier Integration and Stochastic Volatility Calibration,” Working Paper, available at SSRN http://ssrn.com/abstract=2362968 | {"url":"http://www.matthiasthul.com/wordpress/2016/05/17/automatic-cos-method/","timestamp":"2024-11-06T18:42:05Z","content_type":"text/html","content_length":"37231","record_id":"<urn:uuid:cc5c4135-1d78-4434-bad1-ed912ed147d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00563.warc.gz"} |
seminars - An ordering problem on triangulated surfaces from quantum Teichmüller theory
I will introduce an ordering problem about a simple loop on a triangulated surface. Then I will explain Allegretti-Kim's construction of regular functions on quantum Teichmüller space of a punctured
surface, using Bonahon-Wong's quantum trace map. Finally, I will briefly explain how the ordering problem yields the positivity of the coefficients of these regular functions, when written as Laurent
polynomials in quantum cluster X variables. This talk is based on the joint paper with S. Cho, H. Kim, and D. Oh (arXiv:1710.06217). | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=date&order_type=desc&page=46&l=ko&document_srl=803220","timestamp":"2024-11-06T07:34:20Z","content_type":"text/html","content_length":"45562","record_id":"<urn:uuid:070b24dd-b816-4ddb-a931-ac934fbe1795>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00286.warc.gz"} |
If an electron were increased to the size of an apple, how big, proportionately, would a human being be?
Asked by: Lou Spadaccini
Well, the classical radius of an electron (this is the 'electo-magnetic field' type of radius - nobody has actually measured the exact radius of an electron.) is about 2.82 x 10
m. An average size of an apple is about 4 cm or 0.04 m in radius (at least the apples I just got today from the supermarket:-).
So the scalling factor is just:
radius of the apple
radius of the electron
which is: 4 x 10
m / 2.82 x 10
m = 1.42 x 10
This means that in the universe where the electron is as big as an apple in ours everything will be bigger by a factor of 1.42 x 10
or 14,200,000,000,000 (fourteen trillion and two hundred billion times bigger.)
So now you can calculate how big would the human be: for example I am 6 ft (1.83 m) tall so in your apple-sized-electron universe I would be: 1.83 m x 1.42 x 10
= 2.6 x 10
m tall! Just to give you an idea how tall I would be: it would take light a full day to travel from my toes to my nose! (and it only takes about 8 minutes for light to travel from the Sun to the
Earth.) Also, I would be about 3.5 times taller than the diameter of our Solar System (farthest reaches of the Pluto orbit are at about 7.37 x 10
Answered by: Anton Skorucak, M.S. Physics, PhysLink.com Creator
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Mathematician Hurls Structure and Disorder Into Century-Old Problem | Quanta Magazine
Rose Wong for Quanta Magazine
The mathematician Ben Green of the University of Oxford has made a major stride toward understanding a nearly 100-year-old combinatorics problem, showing that a well-known recent conjecture is “not
only wrong but spectacularly wrong,” as Andrew Granville of the University of Montreal put it. The new paper shows how to create much longer disordered strings of colored beads than mathematicians
had thought possible, extending a line of work from the 1940s that has found applications in many areas of computer science.
The conjecture, formulated about 17 years ago by Ron Graham, one of the leading discrete mathematicians of the past half-century, concerns how many red and blue beads you can string together without
creating any long sequences of evenly spaced beads of a single color. (You get to decide what “long” means for each color.)
This problem is one of the oldest in Ramsey theory, which asks how large various mathematical objects can grow before pockets of order must emerge. The bead-stringing question is easy to state but
deceptively difficult: For long strings there are just too many bead arrangements to try one by one.
“Sometimes there’s these very basic-looking questions where we really don’t understand almost anything,” said Jacob Fox of Stanford University. “I think this was one of those questions that really
surprised a lot of people, how little we understood.”
Mathematicians have known for nearly a century that you can’t keep stringing beads indefinitely. Once you’ve chosen your parameters for each color, you can string only so many beads before being
forced to create an evenly spaced sequence that is longer than you are willing to tolerate. As you increase the red and blue parameters, the overall number of beads you can string increases — but how
In a version of the problem in which you forbid even the shortest evenly spaced blue sequences, Graham speculated that a simple relationship holds: The length of the longest possible bead string is
roughly the square of the red-bead parameter. All the numerical data mathematicians have accumulated supports Graham’s conjecture.
But now Green has proved the conjecture wrong. In a 68-page paper, he has shown how to create much longer bead strings than Graham predicted.
“I was shocked when Ben sent me a draft,” said Sarah Peluse of the Institute for Advanced Studies in Princeton, New Jersey. “I think it’s amazing.”
Green’s construction, which blends geometry and dynamical systems to fashion the disordered bead strings, builds on an earlier bead-stringing construction that has found applications in subjects from
matrix multiplication to cryptography. This kind of construction is “very important for questions in computer science,” Fox said.
An Implausible Pattern
If you have a strong taste for disorder, you might prohibit any evenly spaced sequences at all in your string. It doesn’t make sense to talk about evenly spaced sequences of only two beads, so you’re
trying to avoid sequences of three or more beads.
You can string a few beads easily, but you’ll soon run into difficulties. For example, if your first six beads are RBBRBR, there’s no way to continue. Stringing a blue bead puts evenly spaced beads
in spots 3, 5 and 7; stringing a red bead puts evenly spaced beads in spots 1, 4 and 7. A simple computer search shows that no matter how you start your bead string, you’ll be stuck by bead 9.
If you want to string more than eight beads, you have to give a little. Perhaps you’ll decide that you’re OK with evenly spaced blue sequences of fewer than five beads and red sequences of fewer than
12 beads. In 1927, Bartel Leendert van der Waerden proved that for any pair of parameters you pick, there is some finite length by which you’ll get stuck. These lengths are now called van der Waerden
numbers. (Like the mathematicians who came after him, van der Waerden phrased this problem in terms of coloring numbers rather than stringing beads, but the two formulations are equivalent.)
It’s hard for mathematicians to figure out precisely how the van der Waerden numbers change as you change the parameters. But if you decide not to tolerate any evenly spaced blue sequences — in other
words, if you fix the blue parameter at 3 — then a pattern seems to emerge. We saw that if the red parameter is also 3, you’ll get stuck by bead 9. Mathematicians have calculated that if the red
parameter is 10, you’ll get stuck by bead 97; if it’s 15, you’re stuck by bead 218; and if it’s 20, you’re stuck by bead 389. In each case, the number of beads you can string is remarkably close to
the square of the red parameter. All the data collected so far fits this pattern.
Sometime around 2004, Graham conjectured that the pattern continues for all values of the red parameter (let’s call it r). Within a few years, mathematicians did find ways to make bead strings whose
length was close to r^2. But that’s not enough to prove the conjecture. It shows that you can string approximately r^2 beads without getting stuck, but it leaves open the possibility that you could
continue stringing beads for much longer.
When Graham told Green about the conjecture, Green’s gut instinct was that it must be wrong. “It just didn’t seem at all plausible,” he said.
Green thought it should be possible to disprove the conjecture quickly using constructions developed for a related problem — one where you’re trying to avoid evenly spaced blue sequences, but you
don’t care what the red beads do. For that problem, researchers had found ways to pack in many blue beads without creating blue sequences. Green suspected that these abundant blue beads would disrupt
any potential long red sequences, even though the bead strings hadn’t been specifically designed for that purpose.
But when he looked closely at these bead strings, he found that the blue beads were distributed in ways that left wide swaths of red territory in which patterns could form. These examples, he
realized, would not lead to an easy answer for the van der Waerden problem.
Green periodically returned to the problem, and he spent a while trying to prove Graham’s conjecture, since he couldn’t disprove it. He included it on a list of 100 unsolved problems in mathematics,
writing, “I now believe that the answer to this question may be affirmative.”
Even so, he said, “I don’t think I ever felt it with any great conviction, that it was true.” Repeating the conjecture was less a statement of his beliefs than “a challenge to other people,” he said.
Structure and Randomness
Green wasn’t content to leave the problem to other mathematicians. When he strongly believes a conjecture is false but all the data says it’s true, “I find that a very attractive situation to try and
work on,” he said.
His intuition said there should be much longer disordered bead strings than Graham had predicted. If the earlier constructions couldn’t disprove the conjecture, he still felt that some modification
of them might work.
These earlier bead strings started with a 1946 construction by Felix Behrend that relies on a basic geometric fact. Imagine a blue circle on a red sheet of paper. If you connect two points on the
circle with a line segment, the midpoint of the segment lies inside the circle, so it is red. These three points are evenly spaced, and they’re not all blue. It’s an elementary observation, but by
translating points on the plane (or in higher dimensions) into bead locations, Behrend used this geometry as the basis for constructing bead strings with no evenly spaced blue sequences.
Over the years, this construction has found a wide range of applications. Late last year, for example, it formed one of the key components in a record-breaking algorithm for multiplying matrices.
“Behrend’s construction comes up in a surprising number of places,” said David Conlon of the California Institute of Technology.
To tackle the van der Waerden problem, Green zeroed in on an extension of Behrend’s work developed in 2008 by Michael Elkin of Ben-Gurion University of the Negev in Israel, and then adapted later
that year by Green himself and Julia Wolf of the University of Cambridge. In this adaptation, we again envision a blue circle (slightly thickened) on a red background. But this time, we picture this
design as a square tile, and use identical copies of the tile to fill the entire plane, creating a repeating pattern of blue circles on a red background.
Then we envision knotting the starting end of our bead string at some point in the plane and pulling the string tight in a randomly chosen direction, so it lies flat on the plane, crossing the red
and blue terrain unpredictably. We slide beads onto the string, choosing each bead’s color to match the color at the point where the bead’s center will land. The dynamics of the string’s path across
the tiles, Green and Wolf showed, will often generate bead strings that have many blue beads but no evenly spaced blue sequences.
The problem with this construction, from the point of view of the van der Waerden question, is that to prevent blue sequences from forming, the blue circles must be kept fairly small. That leaves
huge expanses of red, making it impossible to string many beads without creating a long red sequence.
But in the final days of 2020, while out for a leisurely walk with his wife and children, Green suddenly had an insight: What if instead of one smallish blue circle per tile, you used many minuscule
circles, scattered randomly? Over the following month, Green figured out that if you choose just the right size and quantity of circles and manage a few additional wrinkles (for instance, the circles
get slightly squashed), then the scattered blue circles will thoroughly disrupt the red territory without creating significant opportunities for blue sequences to form. This makes it possible to
string many beads without creating any long red sequences, or any blue sequences at all.
Green was able to show that as the red parameter (r) increases, the longest bead strings will eventually grow bigger than r^2. Then, as r continues to increase, the bead strings will eventually grow
bigger than r^3 — and then r^4, r^5, and every higher power of r. In other words, for large values of r, the bead strings are vastly longer than Graham predicted.
These jumps to higher powers of r occur only after r gets very large. This may explain why Graham was fooled in the first place: The data mathematicians have collected only goes through the first few
dozen values of r, which are way too small to undergo the jumps to higher exponents.
“It’s a really great result,” Conlon said. When Green posted his paper in February, Conlon emailed him: “I’m rarely surprised by results anymore, but that surprised me.”
The construction lives halfway between structure and randomness. There’s the carefully chosen geometry of the circles, plus an assortment of random choices: the direction of the string, the size of
the beads, how the circles are squashed and where they are scattered. It’s “a random union of structured objects,” Green said. “I think this kind of intuition is something that probably comes up in
other problems.”
Most coloring constructions in Ramsey theory lean more exclusively on randomness, Peluse said. “It’s really hard to come up with colorings that aren’t random,” she said. “You have to have a really,
really insightful idea, like Ben.”
Green’s construction is not the final word on the van der Waerden problem. Just as with earlier constructions, he can’t prove that there aren’t significantly longer bead strings out there. Already,
last month, Zach Hunter, an Oxford undergraduate, managed to nudge the length of the bead strings upward by modifying Green’s construction so the circle diameters are varied randomly. But Fox is
inclined to think that Green’s result may be in the same ballpark as the true van der Waerden numbers. “I find it a very satisfying answer,” he said.
Graham died last year at age 84, seven months before Green posted his paper. “Ron would have been very excited,” Fox said. | {"url":"https://www.quantamagazine.org/oxford-mathematician-advances-century-old-combinatorics-problem-20211215/","timestamp":"2024-11-07T22:36:27Z","content_type":"text/html","content_length":"220986","record_id":"<urn:uuid:52030150-7406-43dc-bdd3-9d036e762434>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00413.warc.gz"} |
1990 Technical Reports
YALEU/DCS/TR756 * Projecting Plans for Uncertain Worlds
Steve Hanks
January 1990
YALEU/DCS/TR757 * CS661 Lecture Notes
Prabhakar Raghavan
January 1990
YALEU/DCS/TR758 * Non-Strict Monoloitic Arrays in a Strict Context
Steve Anderson
January 1990
YALEU/DCS/TR759 [.pdf] Memo-Functions in Alfl
Pradeep Varma
Paul Hudak
January 1990
YALEU/DCS/TR760 * Theory and Pragmatics of Compiling Efficient Parallel Code
Marina Chen
Young-il Choo
Jingke Li
January 1990
YALEU/DCS/TR761 [.pdf] Static and Dynamic Semantics Processing
Charles Consel
Oliver Danvy
February 1990
YALEU/DCS/TR762 [.pdf] Some Remarks on the Generalised Bareiss and Levinson Algorithms
Ilse Ipsen
February 1990
YALEU/DCS/TR763 [.pdf] Linear Algorithms for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs
Heather Booth
Jeffery Westbrook
February 1990
YALEU/DCS/TR764 [.pdf] Boolean Cube Emulation of Butterfly Networks Encoded by Gray Code
Lennart Johnsson
Ching-Tien Ho
February 1990
YALEU/DCS/TR765 [.pdf] Mathematical Foundations for Fast Algorithms for the Biharmonic Equation
Peter Farkas
February 1990
YALEU/DCS/TR766 * Melinda: Linda with Multiple Tuple Spaces
Susanne C. Hupfer
February 1990
YALEU/DCS/TR767 [.pdf] Improving the Accuracy of Inverse Iteration
Elizabeth R. Jessup
Ilse C.F. Ipsen
February 1990
YALEU/DCS/TR768 [.pdf] Routing Multiple Paths in Hypercubes
David Greenberg
Sandeep N. Bhatt
March 1990
YALEU/DCS/TR769 * Evaluating Explanations
David Leake
March 1990
YALEU/DCS/TR770 [.pdf] Incremental Computation via Partial Evaluation
R.S. Sundaresh
Paul Hudak
March 1990
YALEU/DCS/TR771 [.pdf] The Wakeup Problem
Michael Fischer
Shlomo Moran
Steven Rudich
Gadi Taubenfeld
March 1990
YALEU/DCS/TR772 [.pdf] Sub-domain Dependency Test and Scheduling Algorithms for Massively Parallel Computing
Lee-Chung Lu
Marina Chen
March 1990
YALEU/DCS/TR773 [.pdf] Parallel Performance of Domain-Decomposed Preconditioned Krylov Methods for PDEs with Adaptive Refinement
William Gropp
David Keyes
March 1990
YALEU/DCS/TR774 [.pdf] From Interpreting to Compiling Binding Times
Charles Consel
Olivier Danvy
March 1990
YALEU/DCS/TR775 [.pdf] Data Parallel Algorithms for Finite Element Method
Kapil K. Mathur
S. Lennart Johnsson
March 1990
YALEU/DCS/TR776 [.pdf] Domain Decomposition Algorithms for Elliptic Partial Differential Equations (Thesis)
Diana C. Resasco
March 1990
YALEU/DCS/TR777 * Report on the Programming Language Haskell
Paul Hudak
et al
April 1990
YALEU/DCS/TR778 [.pdf] Supercomputers: Past and Future
S. Lennart Johnsson
April 1990
YALEU/DCS/TR779 * Optimal Communication Primitives and Graph Embeddings on Hypercubes (Thesis)
Ching-Tien Ho
April 1990
YALEU/DCS/TR780 [.pdf] Binding time Analysis for Higher Order Untyped Functional Languages
Charles Consel
April 1990
YALEU/DCS/TR781 [.pdf] Semantics-Directed Generation of a Prolog Compiler
Charles Consel
Siau Cheng Khoo
April 1990
YALEU/DCS/TR782 [.pdf] TupleScope: A Graphical Monitor and Debugger for Linda- Based Parallel Programs
Paul Bercovitz
Nicholas Carriero
April 1990
YALEU/DCS/TR783 * Semantics and Analyst of First-Class Tuple-Spaces
Suresh Jagannathan
April 1990
YALEU/DCS/TR784 * Data Dependencies and Space-Time Algebras in Parallel Programming
Magne Haveraaen
April 1990
YALEU/DCS/TR785 * Automated Reasoning About Machines
Andrew Gelsey
April 1990
YALEU/DCS/TR786 [.pdf] Meta-Crystal – A Metalanguange for Parallel-Program Optimazation
J. Allan Yang
Young-il Choo
April 1990
YALEU/DCS/TR787 [.pdf] The Complexity of Reshaping Arrays on Boolean Cubes
Lennart Johnsson
Ching-Tien Ho
April 1990
YALEU/DCS/TR788 * The Semantics of Tuple Space and Correctness of an Implementation
Keld Kondrup Jensen
April 1990
YALEU/DCS/TR789 * Constraints for the Early Detection of Discontinuity from Motion
Michael J. Black
P. Anandan
April 1990
YALEU/DCS/TR790 * Neutral Networks for Model-Based Recognition
Gene Gindi
Eric Mjolsness
P. Anandan
May 1990
YALEU/DCS/TR791 [.pdf] Embedding Meshes into Small Boolean Cubes
Ching-Tien Ho
Lennart Johnsson
May 1990
YALEU/DCS/TR792 [.pdf] Secret Bit Transmission Using a Random Deal of Cards
Michael J. Fischer
Michael S. Paterson
Charles Rackoff
May 1990
YALEU/DCS/TR793 [.pdf] Embedding Three-Demensional Meshes in Boolean Cubes by Graph Decomposition
Ching-Tien Ho
Lennart Johnson
May 1990
YALEU/DCS/TR794 * Linda Coordination Language; Subsystem Kernal Architecture (on transputers)
Steven Ericsson Zenith
May 1990
YALEU/DCS/TR795 [.pdf] A Formal Model for Divide-and-Conquer and its Parallel Realization
Z.G. Mou
May 1990
YALEU/DCS/TR796 * A Connectionist Model of Morphogenesis
Eric Mjolsness
David H. Sharp
John Reinitz
May 1990
YALEU/DCS/TR797 [.pdf] Multiscale Optimization in Neural Nets
Eric Mjolsness
Charles Garrett
Willard L. Miranker
May 1990
YALEU/DCS/TR798 * Modularity & Concurrency through Environment- based reflection
Suresh Jagannathan
May 1990
YALEU/DCS/TR799 [.pdf] How to Create a Failure Tolerant Distributed System
Jonathan Hochman
June 1990
YALEU/DCS/TR800 * Knowledge in Distributed Byzantine Environments
Ruben Michel
June 1990
YALEU/DCS/TR801 [.pdf] The Trade off Between Processor Speed and Paralellism for Supercomputers
Min-You Wu
June 1990
YALEU/DCS/TR802 [.tex] [.pdf] On the Numerical Solution of Two-Point Boundary Value Problems II
V. Rokhlin
P. Starr
June 1990
YALEU/DCS/TR803 [.tex] [.pdf] PP is Closed Under Intersection. Suppt. by: NSF CCR-8601920
Richard Beigel
Nick Reingold
Daniel Spielman
June 1990
YALEU/DCS/TR804 [.tex] [.pdf] Randomized Algorithms for The List Update Problem
Nick Reingold
Jeffery Westbrook
June 1990
YALEU/DCS/TR805 [.tex] [.pdf] Optimal Off-line Algorithms for List Update Problems
Nick Reingold
Jeffery Westbrook
June 1990
YALEU/DCS/TR806 * A Scheme for Supporting Automatic Data Migration on Multicomputers
S. Mirchandaney
J. Saltz
P. Mehrotra
S. Berryman
June 1990
YALEU/DCS/TR807 * Execution Time Support for Adaptive Scientific Algorithms on Distributive Memory Machines
S. Berryman
J. Saltz
J. Scroggs
June 1990
YALEU/DCS/TR808 * Real-Time Performance, Parallelism and Program Visualization in Medical Monitoring
M. Factor
D. Gelernter
C. Kolb
P. Miller
D. Sittig
July 1990
YALEU/DCS/TR809 * Programming with Ease: Semiotic Definition of the Language
S. Ericsson Zenith
July 1990
YALEU/DCS/TR810 [.pdf] A Comparison of Three Column-based Distributed Sparse Ractorization Schemes
Cleve Ashcraft
Stanley Eisenstat
Joseph H. Liu
Andrew H. Sherman
July 1990
YALEU/DCS/TR811 [.pdf] Numerical Techniques for the Solution of the Time- dependent Schrodinger Equation and their Parallel Implementation
Faisal Saied
July 1990
YALEU/DCS/TR812 [.pdf] A Linear Time Algorithm for DNA Sequencing
David E. Foulser
July 1990
YALEU/DCS/TR813 [.tex] [.pdf] The Perceptron Strikes Back
Richard Beigel
Nick Reingold
Daniel Spielman
July 1990
YALEU/DCS/TR814 [.pdf] Sparse Representation of Smooth Linear Operators
Bradley Keith Alpert
August 1990
YALEU/DCS/TR815 * Parallel Computation and FASTA: Confronting the Problems of Parallel Data-base Search for a Fast-Sequence Comparison Algorithm
Perry Miller
Prakash Nadkarni
Nicholas Carriero
August 1990
YALEU/DCS/TR816 * Systematic Applications of Loop Transactions
Marina Chen
Lee-Chung Lu
August 1990
YALEU/DCS/TR817 * Domain Morphisms: A New Construct for Parallel Programming and Formalizing Program Optimization
Marina Chen
Young-il Choo
August 1990
YALEU/DCS/TR818 * Experience with the Process Trellis Software Architecture
Michael Factor
David Gelernter
August 1990
YALEU/DCS/TR819 [.tex] [.pdf] Improved Bounds on Coherence and Checkability
Richard Beigel
Joan Feigenbaum
September 1990
YALEU/DCS/TR820 [.pdf] Partial Evaluation in Parallel
Charles Consel
Olivier Danvy
September 1990
YALEU/DCS/TR821 * True Bigness
Michael Factor
David Gelernter
September 1990
YALEU/DCS/TR822 * A Model for Incremental Motion Estimation
Michael Black
P. Anandan
September 1990
YALEU/DCS/TR823 [.tex] On ACC
Richard Beigel
September 1990
YALEU/DCS/TR824 [.pdf] Parellel Programming Transformation Using a Metalanguage
J. Allen Yang
Young-il Choo
September 1990
YALEU/DCS/TR825 * Parellel Programming with Theory Morphisms
J. Allen Yang
Young-il Choo
September 1990
YALEU/DCS/TR826 * The Hypercomputer: A Network Process Management System
David Kaminsky
September 1990
YALEU/DCS/TR827 [.pdf] Convergence Rate Estimate for A Domain Decomposition Method
Xiao-Chuan Cai
William D. Gropp
David E. Keyes
October 1990
YALEU/DCS/TR828 [.pdf] Building Incremental Programs Using Partial Evaluation
Raman Sundaresh
October 1990
YALEU/DCS/TR829 * Automating the Coordination of Interprocessor Communication
Jinge Li
Marina Chen
October 1990
YALEU/DCS/TR830 * YALE - The Yale Automated Linda Editor
Shakil Ahmed
October 1990
YALEU/DCS/TR831 * Inside Linda
Paolo Ciancarini Consiglio
October 1990
YALEU/DCS/TR832 * The Process Trellis Software Architecture for Parallel, Real-Time Monitors
Michael E. Factor
October 1990
YALEU/DCS/TR833 * New Loop Transformation Techniques for Massive Parallelism
Lee-Chung Lu
Marina Chen
October 1990
YALEU/DCS/TR834 * Global Optimization of Interprocedural Data Movement
Marina Chen
Jan-Jan Wu
October 1990
YALEU/DCS/TR835 [.pdf] Robust Dynamic Motion Estimation Over Time
Michael J. Black
P. Anandan
November 1990
YALEU/DCS/TR836 * Compiling Inheritance using Partial Evaluation
Siau Cheng
R.S. Sundaresh
December 1990
YALEU/DCS/TR837 [.pdf] Wavelets for the Fast Solution of Second Kind Integral Equations
B. Alpert
C. Beylkin
R. Coifman
V. Rokhlin
December 1990
YALEU/DCS/TR838 [.pdf] On the Inverse Scattering Problem for the Helmholtz Equation in One Demension
Y. Chen
V. Rokhlin
December 1990
YALEU/DCS/TR839 [.tex] An Informal Operational Semantics of C-Linda V2.3.5
J. Narem Jr.
December 1990 | {"url":"https://cpsc.yale.edu/research/technical-reports/1990-technical-reports","timestamp":"2024-11-11T02:03:38Z","content_type":"text/html","content_length":"44226","record_id":"<urn:uuid:516d4c8b-2516-4948-a956-c44275b1d2a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00092.warc.gz"} |
Tensor networks get entangled with quantum gravity | Science News
Tensor networks get entangled with quantum gravity
First of two parts
One of the first steps toward becoming a scientist is discovering the difference between speed and velocity.
To nonscientists, it’s usually a meaningless distinction. Fast is fast, slow is slow. But speed, technically, refers only to rate of motion. Velocity encompasses both speed and direction. In science,
you usually want to know more than just how fast something is going; you also want to know where it is going. Hence the need to know direction, and to analyze velocity, not just speed. Numbers like
velocity that express both a magnitude and a direction are known as vectors.
Vectors are great for describing the motion of a particle. But now suppose you need to analyze something more complicated, where multiple magnitudes and directions are involved. Perhaps you’re an
engineer calculating stresses and strains in an elastic material. Or a neuroscientist tracing the changing forces on water flow near nerve cells. Or a physicist attempting to describe gravity in the
cosmos. For all that, you need tensors. And they might even help you unify gravitational theory with quantum physics.
Tensors accommodate multiple numerical values (a vector is actually a simple special case of a tensor). While the ideas behind tensors stretch back to Gauss, they were first fully described in the
1890s by the Italian mathematician Gregorio Ricci-Curbastro, with the help of his student Tullio Levi-Civita. (Tensors were given their name in 1898 by Woldemar Voigt, a German crystallographer, who
was studying stresses and strains in nonrigid bodies.)
Ricci (as he is commonly known) was influenced by the German mathematician Bernhard Riemann in developing advanced calculus with applications to complicated geometrical problems. In particular, this
approach proved valuable in studying coordinate systems. Tensors help make sense of the relationships in the system that stay the same when you change the coordinates. That turned out to be just the
thing Einstein needed in his theory of gravity, general relativity. His friend Marcel Grossmann explained tensors to him and they became the essential feature of general relativity’s mathematics.
And now, in a recent development, some physicists think tensors of a sort could help solve the longstanding problem of unifying general relativity with quantum mechanics. It’s part of a popular new
line of research using tensors to quantify quantum entanglement, which some physicists believe has something to do with gravity.
Entanglement is that spooky connection between separated particles that disturbed Einstein so much. Somehow a measurement of one of a pair of particles affects what you’ll find when you measure its
distant partner, or so it seems. But this “entanglement” is a clear-cut consequence of quantum physics for particles that share a common origin or interaction. It leads to some weird phenomena, but
it’s all very sensible mathematically, as described by the “quantum state.” Entangled particles belong to a single quantum state.
A quantum state determines the mathematical expression (called the wave function) that can be used to predict the outcome of measurements of a particle — whether the direction that it spins is
pointing up or down, for instance. When describing multiple particles — such as those in materials exhibiting quantum properties such as superconductivity — quantum states can get very complicated.
Coping with them is made easier by analyzing the network of entanglement among those many particles. And patterns of such network connections can be described using tensors.
“Tensor networks are representations of quantum many-body states of matter based on their local entanglement structure,” physicist Román Orús writes in a recent paper posted at arXiv.org. “In a way,
we could say that one uses entanglement to build up the many-body wave function.”
Put another way, Orús says, the entire wave function can be thought of as built from smaller tensor subnetworks, kind of like Legos. Entanglement is the glue holding the Legos together.
“Tensor network methods represent quantum states in terms of networks of interconnected tensors, which in turn capture the relevant entanglement properties of a system,” Orús writes in another recent
paper, to be published in Annals of Physics.
While the basic idea of tensor networks goes back decades, they became more widely used to study certain quantum systems in the 1990s. In the last few years, ideas from quantum information theory
have spawned an explosion of new methods using tensor networks to aid various calculations. Instead of struggling with complicated equations, physicists can analyze systems using tensor network
diagrams, similar to the way Feynman diagrams are used in other aspects of quantum physics.
“This is a new language for condensed matter physics (and in fact, for all quantum physics) that makes everything much more visual and which brings new intuitions, ideas and results,” Orús writes.
Most recently, tensor networks have illuminated the notion that quantum entanglement is related to gravity. In Einstein’s general relativity, gravity is the effect of the geometry of spacetime.
Analyses suggest that the geometry in which a quantum state exists is determined by the entanglement tensor network.
“By pushing this idea to the limit,” Orús notes, “a number of works have proposed that geometry and curvature (and hence gravity) could emerge naturally from the pattern of entanglement present in
quantum states.”
If so, tensor networks could be the key to unlocking the mystery of quantum gravity. And in fact, another clue to quantum gravity, known as the holographic principle, seems naturally linked to a
particular type of tensor network. That’s a connection worth exploring further.
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College of Science and Mathematics
Department of Mathematics
The Putnam Competition
What is It?
The following information is taken from the Mathematical Association of America website (https://www.maa.org/math-competitions/putnam-competition), where you can also see detailed overall results
from previous years.
The William Lowell Putnam Mathematical Competition is the preeminent mathematics competition for undergraduate students in the United States and Canada. Each year the Putnam Exam is held on the first
Saturday in December. The competition consists of two 3-hour sessions, one in the morning and one in the afternoon. During each session, participants work individually on 6 challenging mathematical
The Putnam began in 1938 as a competition between mathematics departments at colleges and universities. Now the competition has grown to be the leading university-level mathematics examination in the
world. Although participants work independently on the problems, there is a team aspect to the competition as well. Each institution with at least three participants identifies three participants who
comprise its team. Prizes are awarded to the participants with the highest scores and to the departments of mathematics of the five institutions whose teams obtain the highest rankings.
Fresno State and the Putnam
The Fresno State Mathematics Department offers $200 and $100 prizes to the Fresno State students who place first and second (respectively) among Fresno State students. (To be eligible, the students
must receive a score of at least five points. In case of a tie for top score, the tied students will share a $300 award and there will be no second place prize. In case of a tie for second place, the
tied students will share the $100 award.)
One of the recent editions of the Putnam Exam was held on Saturday, December 3, 2016. At Fresno State, ten undergraduate students and one high school student participated in the exam. Juhoon Chung
received $200 as first price with a score of 20 points. Andres Zumba scored 3, John Fausone scored 2, and Olivia Krohn scored 1. These scores are quite impressive since 4164 students participated in
the Putnam Exam in the United States and Canada and the overall median score was 1. Juhoon Chung ranked 733rd.
The Department of Mathematics wants to congratulate all participants of this prestigious contest.
For more information contact:
Dr. Stefaan Delcroix
PB 354
The Putnam website is http://math.scu.edu/putnam/ | {"url":"https://csm.fresnostate.edu/math/activities-and-events/news-and-events/putnam/index.html","timestamp":"2024-11-10T13:00:48Z","content_type":"text/html","content_length":"30591","record_id":"<urn:uuid:7caed5e2-2dff-4672-871f-c048f69a290b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00060.warc.gz"} |
"Codeforces 624D" Array GCD
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Coordinate vector
Jump to navigation Jump to search
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified
relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices,
hence are useful in calculations.
The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below.
Let V be a vector space of dimension n over a field F and let
${\displaystyle B=\{b_{1},b_{2},\ldots ,b_{n}\}}$
be an ordered basis for V. Then for every ${\displaystyle v\in V}$ there is a unique linear combination of the basis vectors that equals v:
${\displaystyle v=\alpha _{1}b_{1}+\alpha _{2}b_{2}+\cdots +\alpha _{n}b_{n}.}$
The coordinate vector of v relative to B is the sequence of coordinates
${\displaystyle [v]_{B}=(\alpha _{1},\alpha _{2},\cdots ,\alpha _{n}).}$
This is also called the representation of v with respect of B, or the B representation of v. The α-s are called the coordinates of v. The order of the basis becomes important here, since it
determines the order in which the coefficients are listed in the coordinate vector.
Coordinate vectors of finite-dimensional vector spaces can be represented by matrices as column or row vectors. In the above notation, one can write
${\displaystyle [v]_{B}={\begin{bmatrix}\alpha _{1}\\\vdots \\\alpha _{n}\end{bmatrix}}}$
${\displaystyle [v]_{B}={\begin{bmatrix}\alpha _{1}&\alpha _{2}&\dots &\alpha _{n}\end{bmatrix}}.}$
The standard representation[edit]
We can mechanize the above transformation by defining a function ${\displaystyle \phi _{B}}$, called the standard representation of V with respect to B, that takes every vector to its coordinate
representation: ${\displaystyle \phi _{B}(v)=[v]_{B}}$. Then ${\displaystyle \phi _{B}}$ is a linear transformation from V to F^n. In fact, it is an isomorphism, and its inverse ${\displaystyle \phi
_{B}^{-1}:F^{n}\to V}$ is simply
${\displaystyle \phi _{B}^{-1}(\alpha _{1},\ldots ,\alpha _{n})=\alpha _{1}b_{1}+\cdots +\alpha _{n}b_{n}.}$
Alternatively, we could have defined ${\displaystyle \phi _{B}^{-1}}$ to be the above function from the beginning, realized that ${\displaystyle \phi _{B}^{-1}}$ is an isomorphism, and defined ${\
displaystyle \phi _{B}}$ to be its inverse.
Example 1[edit]
Let P3 be the space of all the algebraic polynomials in degree at most 3 (i.e. the highest exponent of x can be 3). This space is linear and spanned by the following polynomials:
${\displaystyle B_{P}=\left\{1,x,x^{2},x^{3}\right\}}$
${\displaystyle 1:={\begin{bmatrix}1\\0\\0\\0\end{bmatrix}};\quad x:={\begin{bmatrix}0\\1\\0\\0\end{bmatrix}};\quad x^{2}:={\begin{bmatrix}0\\0\\1\\0\end{bmatrix}};\quad x^{3}:={\begin{bmatrix}0\
then the coordinate vector corresponding to the polynomial
${\displaystyle p\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}}$
${\displaystyle {\begin{bmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\end{bmatrix}}.}$
According to that representation, the differentiation operator d/dx which we shall mark D will be represented by the following matrix:
${\displaystyle Dp(x)=P'(x);\quad [D]={\begin{bmatrix}0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\\\end{bmatrix}}}$
Using that method it is easy to explore the properties of the operator: such as invertibility, hermitian or anti-hermitian or none, spectrum and eigenvalues and more.
Example 2[edit]
The Pauli matrices which represent the spin operator when transforming the spin eigenstates into vector coordinates.
Basis transformation matrix[edit]
Let B and C be two different bases of a vector space V, and let us mark with ${\displaystyle \lbrack M\rbrack _{C}^{B}}$ the matrix which has columns consisting of the C representation of basis
vectors b[1], b[2], …, b[n]:
${\displaystyle \lbrack M\rbrack _{C}^{B}={\begin{bmatrix}\lbrack b_{1}\rbrack _{C}&\cdots &\lbrack b_{n}\rbrack _{C}\end{bmatrix}}}$
This matrix is referred to as the basis transformation matrix from B to C. It can be regarded as an automorphism over V. Any vector v represented in B can be transformed to a representation in C as
${\displaystyle \lbrack v\rbrack _{C}=\lbrack M\rbrack _{C}^{B}\lbrack v\rbrack _{B}.}$
If E is the standard basis, the notation can be simplified by omitting it, with the transformation from B to E being represented:
${\displaystyle v=\lbrack M\rbrack ^{B}\lbrack v\rbrack _{B}.\,}$
{\displaystyle {\begin{aligned}v&=\lbrack v\rbrack _{E},\\\lbrack M\rbrack ^{B}&=\lbrack M\rbrack _{E}^{B}.\end{aligned}}}
Under the transformation of basis, notice that the superscript on the transformation matrix, M, and the subscript on the coordinate vector, v, are the same, and seemingly cancel, leaving the
remaining subscript. While this may serve as a memory aid, it is important to note that no such cancellation, or similar mathematical operation, is taking place.
The matrix M is an invertible matrix and M^−1 is the basis transformation matrix from C to B. In other words,
{\displaystyle {\begin{aligned}&\operatorname {Id} \\[3pt]={}&\lbrack M\rbrack _{C}^{B}\lbrack M\rbrack _{B}^{C}=\lbrack M\rbrack _{C}^{C}\\[3pt]={}&\lbrack M\rbrack _{B}^{C}\lbrack M\rbrack _{C}
^{B}=\lbrack M\rbrack _{B}^{B}\end{aligned}}}
Infinite-dimensional vector spaces[edit]
Suppose V is an infinite-dimensional vector space over a field F. If the dimension is κ, then there is some basis of κ elements for V. After an order is chosen, the basis can be considered an ordered
basis. The elements of V are finite linear combinations of elements in the basis, which give rise to unique coordinate representations exactly as described before. The only change is that the
indexing set for the coordinates is not finite. Since a given vector v is a finite linear combination of basis elements, the only nonzero entries of the coordinate vector for v will be the nonzero
coefficients of the linear combination representing v. Thus the coordinate vector for v is zero except in finitely many entries.
The linear transformations between (possibly) infinite-dimensional vector spaces can be modeled, analogously to the finite-dimensional case, with infinite matrices. The special case of the
transformations from V into V is described in the full linear ring article.
See also[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/tenzorszorzatok/en.wikipedia.org/wiki/Coordinate_vector.html","timestamp":"2024-11-03T22:23:52Z","content_type":"text/html","content_length":"87709","record_id":"<urn:uuid:f60c4ff3-a5af-4074-800b-cbcc336e325c>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00048.warc.gz"} |
Linear probing: searching for a key
□ When searching for a key K in a table of size N, with hash function H(K) :
1. Set indx = H(K)
2. If table location indx contains the key, return FOUND.
3. Else if table location indx is empty, return NOT FOUND.
4. Else set indx = (indx + 1) mod M.
5. If indx == H(K), return NOT FOUND. Else go to 2. | {"url":"https://cseweb.ucsd.edu/~kube/cls/100/Lectures/lec16/lec16-22.html","timestamp":"2024-11-07T03:01:05Z","content_type":"text/html","content_length":"1854","record_id":"<urn:uuid:fc8cb982-1c8e-4fcc-a078-d99d07368413>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00001.warc.gz"} |
Update a Bank Statement | Veryfi Docs: API Reference, Tutorials, and Integration Guides
Update a Bank Statement
PUT /api/v8/partner/bank-statements/:document_id
Veryfi's Update a Document by ID endpoint allows you to change a Document that Veryfi's Machine Learning models have already processed. This feature will enable users to update Documents previously
processed by Veryfi to ensure data extraction accuracy. Updating a Document is especially useful for correcting mistakes and updating information over time. By changing a processed Document, Veryfi's
Machine Learning models can re-learn the updated information, allowing them to stay accurate and up-to-date.
Path Parameters
document_id int64required
The unique identifier of the document.
Query Parameters
• 200
• 400
• 404
• 429
• 499
• 503
• 504
• default
A processed Bank Statement response.
• Schema
• Example (from schema)
pdf_url uri
Possible values: non-empty and <= 2083 characters
A signed URL to access the auto-generated PDF created from the submitted document. This URL expires 15 minutes after the response object is returned and is resigned during every GET request.
id integerrequired
The unique number created to identify the document.
external_id string
Possible values: non-empty
A custom identification value. Use this if you would like to assign your own ID to documents. This parameter is useful when mapping this document to a service or resource outside Veryfi.
created_date date-timerequired
updated_date date-timerequired
img_thumbnail_url uri
Possible values: non-empty and <= 2083 characters
A signed URL to access the auto-generated thumbnail created for the submitted document. This URL expires 15 minutes after the response object is returned and is resigned during every GET request.
account_holder_address object
The address of the account holder.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
account_holder_name object
The name of the account holder.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
account_number object
The account number associated with the bank statement.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
account_numbers object[]required
• Array [
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
• ]
account_type object
The type of account associated with the bank statement.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
account_vat_number object
The unique identifier for businesses used for tax purposes
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
bank_address object
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
bank_name object
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
bank_vat_number object
The unique identifier assigned to a bank for tax purposes
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
bank_website object
The URL for the website of the bank.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
beginning_balance object
The balance at the beginning of the statement period.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
currency_code object
The currency code associated with the bank statement.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
due_date object
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
ending_balance object
The balance at the end of the statement period.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
iban_number object
The International Bank Account Number
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
minimum_due object
The minimum amount due for the statement period.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
period_end_date object
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
period_start_date object
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
routing_number object
The routing number associated with the bank statement.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
routing_numbers object[]required
• Array [
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
• ]
statement_date object
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
statement_number object
The unique identifier associated with the bank statement.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
swift object
The unique identifier for a bank used in international transactions
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
transactions object[]required
A list of transactions associated with the bank statement.
• Array [
order integerrequired
The value indicating the position of where the transaction appears on the bank statement.
account_number object
The account number associated with the bank statement.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
balance object
The balance after any credit or debits have been applied from this transaction.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
card_number object
A credit card number associated with this transaction.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
credit_amount object
The amount credited from this transaction.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
date object
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
debit_amount object
The amount debited from this transaction.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
description object
The description of the transaction.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
transaction_id object
The unique identifier of the transaction.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
text object
The OCR text extracted from the transaction.
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
• ]
summaries object[]required
• Array [
name object
The title or label that captures the main information
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
value stringrequired
Possible values: non-empty
total object
The overall amount calculated from all transactions
bounding_region number[]
Possible values: >= 8, <= 8
An array containing (x,y) coordinates in the format [x1,y1,x2,y2,x3,y3,x4,y4]` for skewed images and handwritten fields. The bounding region is more precise than bounding box, otherwise it's the
bounding_box object[]
Possible values: >= 5, <= 5
An array containing relative coordinates in the format [page_number,x1,y1,x2,y2] for the extracted field from img_url before any rotation.
• Array [
• ]
score number
The score shows how confident the model is that the predicted value belongs to the field. See confidence scores explained for more information.
rotation integer
Possible values: [0, 90, 180, 270]
The angle of rotation of the document in degrees.
• ]
• Schema
• Example (from schema)
error string
Default value: Malformed parameters
details undefined[]
Default value: [object Object]
• Schema
• Example (from schema)
□ NOT_FOUND
□ DOCUMENT_NOT_FOUND
error string
Default value: Not found.
error string
Default value: Document Not Found
• Schema
• Example (from schema)
□ YOU_HAVE_BEEN_RATE_LIMITED
error string
Default value: You have been rate limited
details undefined[]
Default value: [object Object]
• Schema
• Example (from schema)
□ CLIENT_CLOSED_REQUEST_OR_LOST_CONNECTION
error string
Default value: Client closed request or lost connection
Service is temporaly unavailable
• Schema
• Example (from schema)
□ SERVICE_IS_TEMPORALY_UNAVAILABLE_PLEASE_TRY_AGAIN_LATER
error string
Default value: Service is temporaly unavailable. Please try again later
Gateway timeout. Returned if request takes more than 150 seconds. The request might finish successfully later.
• Schema
• Example (from schema)
error string
Default value: Gateway timeout
• Schema
• Example (from schema)
"status": "fail",
"error": "string",
"details": [ | {"url":"https://docs.veryfi.com/api/update-a-bank-statement/","timestamp":"2024-11-03T21:58:34Z","content_type":"text/html","content_length":"217942","record_id":"<urn:uuid:38bcf4be-aa01-4bb5-add0-c817c099c95d>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00710.warc.gz"} |
Fahrenheit to Celsius conversion (39 °F to °C) - StudySaga
Fahrenheit to Celsius conversion (39 °F to °C)
Fahrenheit to Celsius conversion
39 °F is equal to 3.88 °C
Convert here from other Fahrenheit to Celcius
Temperature Converter
Type a value in the Fahrenheit field to convert the value to Celsius:
Fahrenheit is a scale for measuring temperature, in which water freezes at 32 degrees and boils at 212 degrees. It is represented by the symbol °F.
Fahrenheit temperature scale is a scale based on 32 for the freezing point of water and 212 for the boiling point of water, the interval between the two being divided into 180 parts. The 18th-century
German physicist Daniel Gabriel Fahrenheit originally took as the zero of his scale the temperature of an equal ice-salt mixture and selected the values of 30 and 90 for the freezing point of water
and normal body temperature, respectively; these later were revised to 32 and 96, but the final scale required an adjustment to 98.6 for the latter value.
Until the 1970s the Fahrenheit temperature scale was in general common use in English-speaking countries; the Celsius, or centigrade, scale was employed in most other countries and for scientific
purposes worldwide. Since that time, however, most English-speaking countries have officially adopted the Celsius scale. The conversion formula for a temperature that is expressed on the Celsius (C)
scale to its Fahrenheit (F) representation is: F = 9/5C + 32.
Celsius is a scale for measuring temperature, in which water freezes at 0 degrees and boils at 100 degrees. It is represented by the symbol °C.
Celsius temperature scale also called centigrade temperature scale, is the scale based on 0 for the freezing point of water and 100 for the boiling point of water. Invented in 1742 by the Swedish
astronomer Anders Celsius, it is sometimes called the centigrade scale because of the 100-degree interval between the defined points. The following formula can be used to convert a temperature from
its representation on the Fahrenheit ( F) scale to the Celsius (C) value: C = 5/9(F – 32). The Celsius scale is in general use wherever metric units have become accepted, and it is used in scientific
work everywhere.
Kelvin temperature scale is the base unit of thermodynamic temperature measurement in the International System (SI) of measurement. It is defined as 1/ 273.16 of the triple point (equilibrium among
the solid, liquid, and gaseous phases) of pure water. The kelvin (symbol K without the degree sign []) is also the fundamental unit of the Kelvin scale, an absolute temperature scale named for the
British physicist William Thomson, Baron Kelvin. Such a scale has as its zero point absolute zero, the theoretical temperature at which the molecules of a substance have the lowest energy. Many
physical laws and formulas can be expressed more simply when an absolute temperature scale is used; accordingly, the Kelvin scale has been adopted as the international standard for scientific
temperature measurement. The Kelvin scale is related to the Celsius scale. The difference between the freezing and boiling points of water is 100 degrees in each, so that the kelvin has the same
magnitude as the degree Celsius.
How to Convert Fahrenheit to Celcius ?
The procedure to use the Fahrenheit to Celcius Converter is as follows:
Step 1: Enter the Fahrenheit value the input field
Step 2: Now there is automatically converted to Celcius value
Step 3: Finally, the conversion from Fahrenheit to Celcius value will be displayed in the output field.
Relation between Fahrenheit and Celcius ?
The conversion formula for a temperature that is expressed on the Celsius (C) scale to its Fahrenheit (F) representation is: F = 9/5C + 32.
Kelvin to Fahrenheit Conversion Formula F= 9/5 (°K-273)+32
Fahrenheit to Kelvin Conversion Formula K= 5/9 (°F-32)+273
Celsius to Kelvin Conversion Formula K= °C+273
Fahrenheit to Celcius Conversion Table
• 1 F = -17.222222 C
• 10F = -12.22222 C
• 20F = -6.666666 C
• 30F = -1.111111111 C
• 40F = 4.44444 C
• 50F = 10 C
• 60F = 15.55555 C
• 70F = 21.1111111 C
• 80F = 26.666666 C
• 90F = 32.2222 C
• 100F = 37.7777 C
• 110F = 43.3333333 C
• 120F = 48.8888888 C
• 130F = 54.4444444 C
• 140F = 60 C
• 150F = 65.555555 C | {"url":"https://studysaga.in/fahrenheit-to-celsius-conversion-39-f-to-c/","timestamp":"2024-11-06T17:31:04Z","content_type":"text/html","content_length":"125746","record_id":"<urn:uuid:b8fe00de-c175-47c5-9ea8-ba08b2432047>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00792.warc.gz"} |
Micro Pillar
Micro Pillar¶
The example is inspired by Gregersen et al. [1] where a quantum dot is placed in a micro pillar to produce a single photon source. However, we have simplified the problem so that the 3D computations
run smoothly on a laptop computer:
The following figure shows the field intensities for
The far field data is the electromagnetic field on a infinitely far distant hemisphere above or below the micro pillar. As normalization, the far field data as returned by the FarField post process,
refers to a hemisphere with distance FarField yields these fields in 2D polar coordinates. JCMsuite visualizes the far field on a polar disk:
Parameter Scan
[1] 14. Gregersen, T. R. Nielsen, et al., Quality factors of nonideal micro pillars, APPLIED PHYSICS LETTERS 91, 011116 (2007) | {"url":"https://docs.jcmwave.com/JCMsuite/html/EMTutorial/067d8839b1eadfbb7068eedf58469057.html?version=6.4.1","timestamp":"2024-11-09T19:34:20Z","content_type":"text/html","content_length":"15755","record_id":"<urn:uuid:8b2cdcaf-21c4-4104-a87d-af7bb4537027>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00017.warc.gz"} |
Memory graphs (4D)
SuperMemo 17 uses a new spaced repetition algorithm denoted Algorithm SM-17. Unlike all prior algorithms that were either theoretical or "inspired by data", this algorithm has been developed entirely
on the basis of prior records of repetitions collected by users of SuperMemo. This data-driven effort required untold hours of analysis while processing millions of repetition samples. Tools : Memory
: 4D Graphs was instrumental in that analysis and debugging process. If you want to understand the algorithm and help improve it further, please study those tools and keep analyzing your own data and
your own memory. In a stochastic system of memory, perfection is impossible, but we should always try to come closer to the optimum.
1. At the moment of writing (April 2016), SuperMemo 17 does not use incremental adjustments to optimization matrices in Algorithm SM-17. This is why you should execute Tools : Memory : 4D Graphs :
Stability : Compute from time to time to adjust the algorithm to newly available data. In the future, the adjustments will be made at each repetition.
2. To see nice graphs as shown in the pictures below, you also need to use a collection with a mature learning process. New collections have no memory data to show.
Available memory graphs
All memory graphs provide a 3-dimensional view with rotation along all 3 axes (X, Y and Z), and a slider for animation in the 4th dimension along item difficulty.
The following memory graphs are available with Tools : Memory : 4D Graphs on its individual tabs:
Graph analysis controls
• X, Y, Z axis rotation (top 3 sliders)
• Difficulty slider (for animation in the 4th dimension)
• Repetition cases in consideration (bottom slider)
• Cases: the label showing the total number of repetition cases in consideration
• Compute: recompute the graph using the data in the collection
• Reset: reset the memory matrices
• Smoothing: average neighboring entries in matrices
• Subset: select a subset of elements for which matrices should be computed
• Reset Cases: reset the count of repetition cases without changing the data (i.e. values of entries in matrices)
• Export: export data for analysis in Excel
• Average checkbox: the "golden mean" average of the data with:
1. the best-fit approximation, and
2. data-rich neighboring entries in proportion to available information
Stability increase function
Figure: 3D graph of SInc[] matrix based on 60,167 repetition cases for items with difficulty=0.5. The increase is dramatic at low stabilities (17-fold increase is unheard of in earlier
SuperMemos), and peaks at retrievability of 0.85. In some cases, the SInc drops below 1.0, which corresponds with a drop in stability (i.e. memory lability). Those low values of SInc do not
depend on retrievability. Those huge variations in SInc[] are the main reason why SuperMemo 17 beats SuperMemo 16 in learning metrics by a wide margin..
Stability increase function contour map
Figure: A "from above" view at the SInc[] matrix providing a contour map. Red zones indicate high stability increase at review. The picture shows that the greatest stability increase occurs for
lower stability levels and retrievabilities around 70-90%.
Stability increase approximation
Figure: Approximating the SInc[] matrix with the best-fit function used by default in SuperMemo to compute the increase in stability (e.g. in cases of lack of data). The approximation procedure
uses a hill-climbing algorithm with parameters A, B, C, D displayed in the picture. Least squares deviation is obtained to asses the progress. Green circles represent the Sinc[] matrix at a
chosen difficulty level. Their size corresponds with repetition cases investigated. The blue surface is the best fit of the studied function to the SInc[] data.
Figure: The Recall[] matrix graph shows that the actual recall differs from predicted retrievability. For higher stabilities and difficulties, it is harder to reach the desired recall level.
Recall approximation
Figure: Approximating the Recall[] matrix with the best-fit function to compute default recall in conditions of data scarcity. The approximation procedure uses a hill-climbing algorithm with
parameters A, B, C, D displayed in the picture. Least squares deviation is obtained to asses the progress. The circles represent the Recall[] matrix at a chosen difficulty level. Their size
corresponds with repetition cases investigated. The red surface is the best fit of the studied function to the Recall[] data.
Recall approximation curve
Figure: Approximating the Recall[] matrix with the best-fit function to compute default recall in conditions of data scarcity. By choosing the right viewing angle, the curve that reflects the
changes to recall with retrievability can be seen in abstraction of stability. In this case the relationship is almost linear (the logarithmic bend is a result of the log scale used for
First interval
First interval approximation | {"url":"https://super-memory.com/help/graphs4D.htm","timestamp":"2024-11-14T15:03:02Z","content_type":"text/html","content_length":"20512","record_id":"<urn:uuid:64d51e8e-b499-461b-b2d4-c5342e53d571>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00877.warc.gz"} |
g22yaf (lm_formula)
NAG FL Interface
g22yaf (lm_formula)
FL Name Style:
FL Specification Language:
1 Purpose
parses a text string containing a formula specifying a linear model and outputs a G22 handle to an internal data structure. This G22 handle can then be passed to various routines in
Chapter G22
. In particular, the G22 handle can be passed to
to produce a design matrix or
to produce a vector of column inclusion flags suitable for use with routines in
Chapter G02
2 Specification
Fortran Interface
Subroutine g22yaf ( hform, formula, ifail)
Integer, Intent (Inout) :: ifail
Character (*), Intent (In) :: formula
Type (c_ptr), Intent (Inout) :: hform
C Header Interface
#include <nag.h>
void g22yaf_ (void **hform, const char *formula, Integer *ifail, const Charlen length_formula)
The routine may be called by the names g22yaf or nagf_blgm_lm_formula.
3 Description
3.1 Background
Let $D$ denote a data matrix with $n$ observations on $md$ independent variables, denoted $V1, V2, …, Vmd$. Let $y$ denote a vector of $n$ observations on a dependent variable.
A linear model, $M$, as the term is used in this routine, expresses a relationship between the independent variables, $Vj$, and the dependent variable. This relationship can be expressed as a series
of additive terms $T1+ T2+ ⋯$, with each term, $Tt$, representing either a single independent variable $Vj$, called the main effect of $Vj$, or the interaction between two or more independent
variables. An interaction term, denoted here using the $.$ operator, allows the effect of an independent variable on the dependent variable to depend on the value of one or more other independent
variables. As an example, the three-way interaction between $V1, V2$ and $V3$ is denoted $V1. V2. V3$ and describes a situation where the effect of one of these three variables is influenced by the
value of the other two.
This routine takes a description of
, supplied as a text string containing a formula, and outputs a G22 handle to an internal data structure. This G22 handle can then be passed to
to produce a design matrix for use in analysis routines from other chapters, for example the regression routines of
Chapter G02
A more detailed description of what is meant by a G22 handle can be found in
Section 2.1
in the
Chapter Introduction.
3.2 Syntax
In its most verbose form $M$ can be described by one or more variable names, $Vj$, and the two operators, $+$ and $.$. In order to allow a wide variety of models to be specified compactly this syntax
is extended to six operators ($+$, $.$, $*$, $-$, $:$, $^$) and parentheses.
A formula describing the model is supplied to g22yaf via a character string which must obey the following rules:
1. 1.Variables can be denoted by arbitrary names, as long as
1. (i)The names used are a subset of those supplied to g22ybf when describing $D$.
2. (ii)The names do not contain any of the characters in $+.*-:^()@$.
2. 2.The $.$ operator denotes an interaction between two or more variables or terms, with $V1. V2. V3$ denoting the three-way interaction between $V1$, $V2$ and $V3$.
3. 3.A term in $M$ can contain one or more variable names, separated using the $.$ operator, i.e., a term can be either a main effect or an interaction term between two or more variables.
1. (i)If a variable appears in an interaction term more than once, all subsequent appearances, after the first, are ignored, therefore, $V1. V2. V1$ is the same as $V1. V2$.
2. (ii)The ordering of the variables in an interaction term is ignored when comparing terms, therefore, $V1. V2$ is the same as $V2. V1$. This ordering may have an effect when the resulting G22
handle is passed to another routine, for example g22ycf.
3. (iii)Applying the $.$ operator to two terms appends one to the other, for example, if $T1= V1. V2$ and $T2= V3. V4$, $T1. T2= V1. V2. V3. V4$.
4. 4.The $+$ operator allows additional terms to be included in $M$, therefore, $T1+T2$ is a model that includes terms $T1$ and $T2$.
1. (i)If a term is added to $M$ more than once, all subsequent appearances, after the first, are ignored, therefore, $T1+ T2+ T1$ is the same as $T1+ T2$.
2. (ii)The ordering of the terms is ignored whilst parsing the formula, therefore, $T1+ T2$ is the same as $T2+ T1$. This ordering may have an effect when the resulting G22 handle is passed to
another routine, for example g22ycf.
3. (iii)Internally, the terms are reordered so that all main effects come first, followed by two-way interactions, then three-way interactions, etc. The ordering within each of these categories
is preserved.
5. 5.The $*$ operator can be used as a shorthand notation denoting the main effects and all interactions between the variables involved. Therefore, $T1* T2$ is equivalent to $T1+ T2+ T1. T2$ and
$T1* T2* T3$ is equivalent to $T1+ T2+ T3+ T1. T2+ T1. T3+ T2. T3+ T1. T2. T3$.
6. 6.The $-$ operator removes a term from $M$, therefore, $T1* T2* T3- T1. T2. T3$ is equivalent to $T1+ T2+ T3+ T1. T2+ T1. T3+ T2. T3$ as the three-way interaction, $T1. T2. T3$, usually present
due to $T1* T2* T3$ has been removed.
7. 7.The $:$ operator is a shorthand way of specifying a series of variables, with $V1: Vj$ being equivalent to $V1+ V2+ ⋯+ Vj$.
1. (i)This operator can only be used if the variable names end in a numeric, therefore, $VAR2:VAR4$ would be valid, but $FVAR:LVAR$ would not.
2. (ii)The root part of both variable names (i.e., the part before the trailing numeric, so $VAR$ in the valid example above) must be the same.
3. (iii)The trailing numeric parts of the two variable names must be in ascending order.
8. 8.The $^$ operator is a shorthand notation for a series of $*$ operators. $(T1+T2+T3) ^2$ is equivalent to $(T1+T2+T3) * (T1+T2+T3)$ which in turn is equivalent to $T1+ T2+ T3+ T1. T2+ T1. T3+
T2. T3$.
1. (i)This notation is present primarily for use with the $:$ operator in examples of the form, $(V1:V5) ^3$ which specifies a model containing the main effects for variables $V1$ to $V5$ as
well as all two- and three-way interactions.
2. (ii)Using the $^$ operator on a single term has no effect, therefore, $T2^2$ is the same as $T2$.
3.2.1 Precedence
Each operator has an associated default precedence, but this can be overridden through the use of parentheses. The default precedence is:
1. 1.The $:$ operator, with the resulting expression is treated as if it was surrounded by parentheses. Therefore, $V1+ V3: V6* V7$ is equivalent to $V1+ (V3+V4+V5+V6) * V7$.
2. 2.The $^$ operator, with the resulting expression is treated as if it was surrounded by parentheses. Therefore, $(T1+T2+T3) ^2. T4$ is equivalent to $((T1+T2+T3)^2) . T4$, which is the equivalent
to $T1. T4+ T2. T4+ T3. T4+ T1. T2. T4+ T1. T3. T4+ T2. T3. T4$.
3. 3.The $.$ operator, so $T1* T2. T3$ is equivalent to $T1* (T2.T3)$.
4. 4.The $*$ operator.
1. (i)When using parentheses with the $*$ or $.$ operators the usual rules of multiplication apply, therefore, $(T1+T3.T4) . (T5+T7)$ is equivalent to $T1. T5+ T1. T7+ T3. T4. T5+ T3. T4. T7$
and $(T1+T3.T4) * (T5+T7)$ is equivalent to $T1+ T5+ T7+ T3. T4+ T1. T5+ T1. T7+ T3. T4. T5+ T3. T4. T7$.
2. (ii)Syntax of the following form is invalid: $T1 o (T2) o T3$, where $o$ indicates an operator, unless one or more of those operators are $+$ and/or $-$. Therefore, $T1. (T2+T3) * T4$ is
invalid, whilst $T1. (T2+T3)+ T4$ is valid.
5. 5.The $+$ and $-$ operators have equal precedence.
1. (i)If the terms associated with a $-$ operator do not occur in the current expression they are ignored, therefore, $T1+ (T2-T1)$ is the equivalent to $T1+ T2$; the $(T2-T1)$ part of the
expression is calculated first and results in $T2$ as the $T1$ term does not exist in this particular sub-expression so cannot be removed.
3.2.2 Mean Effect / Intercept Term
A mean effect (or intercept term) can be explicitly added to a formula by specifying $1$ and can be explicitly excluded from the formula by specifying $-1$. For example, $1+V1+V2$ indicates a model
with the main effects of two variables and a mean effect, whereas $V1+V2-1$ denotes the same model, but without the mean effect. The mean indicator can appear anywhere in the formula string as long
as it is not contained within parentheses.
If the mean effect is not explicitly mentioned in the model formula, the model is assumed to include a mean effect.
3.3 Optional Parameters
accepts a number of optional parameters described in
Section 11
. Usually these parameters are set via call to
, however when specifying a subject term in a mixed effects linear regression model it is often more convenient to supply the information along with the rest of the formula. Therefore, writeable
optional parameters can be set via the
argument. The delimiter
must be used between the main formula and the optional parameter. For example, supplying a formula of the form
, would specify a model formula of
and set the optional parameter
4 References
5 Arguments
1: $hform$ – Type (c_ptr) Input/Output
On entry
: must be set to
, alternatively an existing G22 handle may be supplied in which case this routine will destroy the supplied G22 handle as if
had been called.
On exit
: holds a G22 handle to the internal data structure containing a description of the model
as specified in
. You
must not
change the G22 handle other than through routines in
Chapter G22
2: $formula$ – Character(*) Input
On entry
: a string containing the formula specifying
. See
Section 3
for details on the allowed model syntax.
3: $ifail$ – Integer Input/Output
On entry
must be set to
to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a
value of $1$ means that it is not.
If halting is not appropriate, the value
is recommended. If message printing is undesirable, then the value
is recommended. Otherwise, the value
is recommended.
When the value $-1$ or $1$ is used it is essential to test the value of ifail on exit.
On exit
unless the routine detects an error or a warning has been flagged (see
Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by
Errors or warnings detected by the routine:
On entry,
is not
or a recognised G22 handle.
The formula contained a mismatched parenthesis.
The position in the formula string of the error is $⟨value⟩$.
An operator was missing.
The position in the formula string of the error is $⟨value⟩$.
Invalid use of an operator.
The position in the formula string of the error is $⟨value⟩$.
Invalid specification for the power operator.
The position in the formula string of the error is $⟨value⟩$.
Invalid specification for the colon operator.
The position in the formula string of the error is $⟨value⟩$.
Invalid specification for the mean.
The position in the formula string of the error is $⟨value⟩$.
Invalid variable name.
The position in the formula string of the error is $⟨value⟩$.
Missing variable name.
The position in the formula string of the error is $⟨value⟩$.
After processing, the model contains no terms.
An invalid contrast specifier has been supplied.
The position in the formula string of the error is $⟨value⟩$.
A term contained a repeated variable with a different contrast specifier.
On entry, an invalid
was supplied in
On entry, an
was supplied in
, but the expected delimiter ‘
’ was not found.
On entry, an
was supplied in
, but the supplied
was invalid.
An unexpected error has been triggered by this routine. Please contact
Section 7
in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
Section 8
in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
Section 9
in the Introduction to the NAG Library FL Interface for further information.
7 Accuracy
Not applicable.
8 Parallelism and Performance
Background information to multithreading can be found in the
g22yaf is not threaded in any implementation.
9 Further Comments
10 Example
This example reads in and parses a formula specifying a model,
, and displays the processed formula. A data matrix,
, is then read in and a design matrix constructed from
The design matrix includes an explicit term for the mean effect.
See also the examples for
10.1 Program Text
10.2 Program Data
10.3 Program Results
11 Optional Parameters
As well as the optional parameters common to all G22 handles described in
, a number of additional optional parameters can be specified for a G22 handle holding the description of a model, as returned by
Each writeable optional parameter has an associated default value; to set any of them to a non-default value, use
. The value of any optional parameter can be queried using
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in
Section 11.1
All routines that make use of the G22 handle returned by g22yaf combine it with a description of a data matrix, $D$, to construct a design matrix, $X$.
11.1 Description of the Optional Parameters
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
• a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
• the default value.
Keywords and character values are case and white space insensitive.
Contrast $a$ Default $=FIRST$
This parameter controls the default contrasts used for the categorical independent variables appearing in the model. Six types of contrasts and dummy variables are available:
Treatment contrasts relative to the first level of the variable will be used.
Treatment contrasts relative to the last level of the variable will be used.
$SUM FIRST$
Sum contrasts relative to the first level of the variable will be used.
$SUM LAST$
Sum contrasts relative to the last level of the variable will be used.
Helmert contrasts will be used.
Polynomial contrasts will be used.
Dummy variables will be used rather than a contrast.
for more information on contrasts, their effect on the design matrix and how they are constructed.
This parameter may have an
instance identifier
associated with it (see
). The
instance identifier
must be the name of one of the variables appearing in the model supplied in
when the G22 handle was created. For example,
CONTRAST : VAR1 = HELMERT
would set Helmert contrasts for the variable named
If no instance identifier is specified, the default contrast for all categorical variables in the model is changed, otherwise only the default contrast for the named variable is changed.
In some situations it might be necessary for a variable to use a different contrast, depending on where it appears in the model formula. In order to allow contrasts to be specified on a term by term
basis the $@$ operator can be used in the model formula. The syntax for this operator is $Vj@c$, where $c$ is one of: F, L, SF, SL, H, P or D, corresponding to treatment contrasts relative to the
first and last levels, sum contrasts relative to the first and last levels, Helmert contrasts, polynomial contrasts or dummy variables respectively.
If the contrast has not been explicitly specified via the
operator, the value obtained from the optional parameter
is used.
For example, setting
VAR1 + VAR1@H.VAR2@P + VAR2@H.VAR3
, specifies that the variable named
should use the default contrasts in the first term and Helmert contrasts in the second term. The variable named
should use polynomial contrasts in the second term and Helmert contrasts in the third term. The variable named
should use the default contrasts in the third term.
Constraint: $Contrast=FIRST$, $LAST$, $SUM FIRST$, $SUM LAST$, $HELMERT$, $POLYNOMIAL$ or $DUMMY$.
Explicit Mean $a$ Default $=NO$
If $Explicit Mean=YES$, any mean effect included in the model will be explicitly added to the design matrix, $X$, as a column of $1$s.
$Explicit Mean=NO$
, it is assumed that the routine to which
will be passed treats the mean effect as a special case, see
for example.
Constraint: $Explicit Mean=YES$ or $NO$.
This parameter returns a verbose version of the model formula specified in
, expanded and simplified to only contain variable names, the operators
and any contrast identifiers present.
Storage Order $a$ Default $=OBSVAR$
This optional parameter controls how the design matrix,
, should be stored in its output array and only has an effect if the design matrix is being constructed using
If $Storage Order=OBSVAR$, $Xij$, the value for the $j$th variable of the $i$th observation of the design matrix is stored in $x(i,j)$.
If $Storage Order=VAROBS$, $Xij$, the value for the $j$th variable of the $i$th observation of the design matrix is stored in $x(j,i)$.
is the output parameter of the same name in
Constraint: $Storage Order=OBSVAR$ or $VAROBS$.
This parameter gives the subject terms associated with the
in a linear mixed effects model.
The supplied value must consist of a single term, representing either a single independent variable, or a single interaction term between two or more independent variables. All variables in the
subject term must not also appear in the model formula. | {"url":"https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g22/g22yaf.html","timestamp":"2024-11-06T15:34:35Z","content_type":"text/html","content_length":"75836","record_id":"<urn:uuid:3217c8d1-6454-4546-a84a-55b6a37dc899>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00164.warc.gz"} |
Printing Fractals on MiniFactory
Last week, we had our 8th International IT Week for Students here at HAAGA-HELIA where I work. We had teams from Spain, Denmark, and Finland, and we looked into issues like mobile games development,
robot building on the Arduino set, and on the Danish day, fractals. My good friend and colleague from Coopenhagen North, Anders Kalhauge, presented a lecture, and the students then led a workshop
into fractals.
Fractals are odd creatures. Wikipedia says that
“A fractal is a mathematical set that typically displays self-similar patterns, which means it is “the same from near as from far”. Fractals may be exactly the same at every scale, or, […] they may
be nearly the same at different scales. The concept of fractal extends beyond self-similarity and includes the idea of a detailed pattern repeating itself.”
To show you a fractal, I will grab a copy of one from Wikimedia Commons. This is the Mandelbrot set, the most famous of fractals.
Mandelbrot set, the most famous of all fractals
During his lecture, Anders showed us how a simple recursive formula will eventually yield this image, and how you can edit the parameters in the formula to change the resulting image. This is
fascinating stuff, even if the mathematics are beyond many people, and most definitely beyond my feeble grasp of math. But anyhow, Anders got to thinking about Blender.
Since Blender is built on the Python language, and Anders knows Python well, he wanted to see if he could run the Mandelbrot math and come up with a Blender mesh, ie. a solid virtual object, where
the colors of the fractal would be represented with different elevations. The red outer area would be the zero elevation, and it would build up gradually to the black plateau in the middle. If that
were possible, he thought, maybe we could then export the mesh from Blender and print a solid object. He worked on this idea overnight and got it working, and on the next day, he presented me with
this Blender file:
Anders’ script and the mesh
I was amazed. This little script on the right creates this mesh you see on the left, and you can edit the parameters to come up with a different fractal form if you wish. The current one runs on a
twodimensional array of 1000 x 1000 vertices, ie. one million points. But when Anders showed me this, I knew it could be turned into a printable piece. We did some experimenting, and after fifteen
minutes, we had the print-ready STL file.
Now, most of the time, meshes need to be manifold to print. Manifold means there are no loose edges, no holes, and no vertices that are unattached. The Blender 3D Print Tools add-on reported that
there were about 400 loose edges, which usually would be a showstopper for printing. However, this time, the resolution of the printer is not even close to the resolution of the mesh. Therefore the
extrusion of material would fill in any of these gaps anyway, and even if the Slic3r program alerted us to such problems, we forged ahead.
The STL file looked okay when I placed it on the table and scaled it by 25. I could have scaled it in Blender just as well, but I decided to do it here and see how it sat on the table.
The STL file imported to Repetier Host and scaled
A moment later Slic3r finished, and we had the G-code that would turn the virtual object into a very real one fith 53,000 moves of the table and the printer nozzle. I hit Print.
Finished print of Mandelbrot set
So, after 25 minutes, we could see the finished object on the table. Given its small scale, and the infill set at 60%, the job finished very fast, but the outcome was better than I had hoped for. It
shows all the classic features of the Mandelbrot set, just as it should, since it was created using the very formula. When you hold this against the window, you can see how its inner forms conform to
the image on the top even better:
Printed fractal on window
So, our little test turned out to work straight off the box. My hat’s off to Anders, whose knowledge of math and fractals, and Python of course, enabled him to pull this off at one go. For those who
are interested in the code, it is inside this Blender file, and if you just press the Run Script button, you can see it create another mesh in seconds (delete the previous one before you do).
And this is the code:
import bpy
import math
vr, vi = 200, 200
limit, iterations, max_height = 10000000.0, 1024, 0.25
p = -2.25 - 1.5j
d = 3.0 + 3.0j
def index(x, y):
return y*(vr + 1) + x
def mandelpoint(c):
z = c
for h in range(iterations):
if z.real*z.real + z.imag*z.imag > limit: break
z = z*z + c
return (c.real, c.imag, max_height*math.log(h)/math.log(iterations))
vertices = [ mandelpoint(x*d.real/vr + (y*d.imag/vi)*1j + p)
for y in range(vi + 1) for x in range(vr + 1) ]
faces = [
(index(x, y), index(x + 1, y), index(x + 1, y + 1), index(x, y + 1))
for y in range(vi) for x in range(vr)]
# Code to create a solid base
base = 0.0
pos = len(vertices)
pos_base = pos
vertices.append((p.real, p.imag, -base))
for x in range(1, vr + 1):
vertices.append((p.real + x*d.real/vr, p.imag, -base))
faces.append((index(x - 1, 0), index(x, 0), pos + x, pos + x - 1))
pos = len(vertices) - 1
for y in range(1, vi + 1):
vertices.append((p.real + d.real, p.imag + y*d.imag/vi, -base))
faces.append((index(vr, y - 1), index(vr, y), pos + y, pos + y - 1))
pos = len(vertices) - 1
for x in range(1, vr + 1):
vertices.append((p.real + (vr - x)*d.real/vr, p.imag + d.imag, -base))
faces.append((index(vr - x + 1, vi), index(vr - x, vi), pos + x, pos + x - 1))
pos = len(vertices) - 1
for y in range(1, vi):
vertices.append((p.real, p.imag + (vi - y)*d.imag/vi, -base))
faces.append((index(0, vi - y + 1), index(0, vi - y), pos + y, pos + y - 1))
faces.append((index(0, 1), index(0, 0), pos_base, len(vertices) - 1))
faces.append( tuple([i for i in range(pos_base, len(vertices))]) )
#code to create mesh and object and place the object in the scene
brot = bpy.data.meshes.new("Brot")
mandel = bpy.data.objects.new("Mandelbrot", brot)
mandel.location = (0.0, 0.0, 0.0)
#induce vertices, edges (empty list), and faces in the mesh
brot.from_pydata(vertices, [], faces)
brot.update(calc_edges = True)
0 thoughts on “Printing Fractals on MiniFactory”
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A ``Concrete'' Open problem
(Guest Post by Ken Regan) pdf file available
Computational complexity theory is the study of information flow and the
required for it to reach desired conclusions. Computational models like cellular automata, Boolean or algebraic circuits, and other kinds of fixed networks exemplify this well, since they do not have
"moving parts" like Turing machine tape heads, so the flow's locations are fixed. Measures of effort include the time for the flow, the amount of space or hardware needed, and subtler considerations
such as time/space to prepare the network, or energy to overcome possible dissipation during its operation. These models and measures have fairly tight relations to Turing machines and their familiar
complexity measures. For an example and open problem, consider the general task of moving all "garbage bits" to the end of a string, leaving the "good bits" in their original sequence. We can model
this as computing the function f: {0,1,2}
® {0,1,2}
exemplified by f(1020212) = 1001222, f(2200) = 0022, f(101) = 101, etc., with 0,1 as "good bits" and 2 as "garbage." A rigorous inductive definition, using e for the empty string, is f(e) = e, f(0x)
= 0f(x), f(1x) = 1f(x), and f(2x) = f(x)2. This is the "topological sort" of the partial order B = {0 < 2, 1 < 2} that is
, meaning that subsequences of incomparable elements are preserved. The problem is, can we design circuits C
, each computing f(x) on strings x of length n, that have size O(n)? The circuits C
input gates
labeled x
which receive the corresponding "trits" (0, 1, or 2) of the input string x, and
output gates
giving y = f(x). The first question is, what interior computational gates can C
have? A
comparator gate
g for a partial order (P, < ) has two input and two output wires, maps (a,b) either to (a,b) or (b,a), and never maps to (d,c) when c < d. The unique
stable comparator
maps (a,b) to (a,b) unless b < a. The following slightly extends the famous 0-1 law for comparator networks:
Theorem 1.
If a circuit C
of comparator gates computes f(x) correctly for all x Î {0,2}
(not even including any 1s), then for every partial order (P, < ), the circuit C
with each comparator replaced by g
computes the stable topological sort of P.
First suppose C
errs for a total order (P, < ). Then there are x,y Î P
such that C
(x) = y, but for some j, y
+1 < y
. Take the permutation p such that x
= y
for all indices i. Define a binary string y¢ Î {0,2}
by y¢
= 0 if y
< y
, y¢
= 2 otherwise, and x¢ by x¢
= y¢
for all i. Then C
(x¢) = y¢ (exercise: prove this by induction taking gates one at a time), contradicting that the original C
was correct on {0,2}*.
For (P, < ) not a total order, an error C
(x) = y (which might violate only stability) is also an error in the total order (P
, < ¢) with P
= {(a,i): x
= a} and (a,i) < ¢(b,j) if a < b or a is not comparable to b and i < j. [
Corollary 2.
Circuits C
of comparator gates computing f require size n*log
(n) - O(n). [
This follows by applying the standard sorting lower bound to C
. It's interesting that we did not need 1s in x to argue stability, and the lower bound allows gates g in C
to be arbitrary when either input is 1. For general circuits, however, the argument doesn't hold, and all bets are off! To see why, consider sorting the total order {0 < 1 < 2}. Clever O(n)-size
circuits can
the numbers a,b,c of 0s, 1s, and 2s in the input string x, respectively, and then assemble the correct output y = 0
. For the basic idea see
Muller-Preparata, 1975
, and various sources on the "Dutch National Flag Problem." Applying this counting idea to our poset B reduces our task to "nice" strings z of length N = 2
with exactly N/2 2s.
Theorem 3.
If s(N)-size circuits D
can compute f(z) for "nice" z, then f has circuits of size at most s(4n) + O(n).
We can build O(n)-size circuits E
that on inputs x of length n count b,c as above and find k such that m = 2
is the least power of 2 above n. Make E
(x) output z = x1
, which gives |z| = N < 4n. Then compute y¢ = D
(z) and re-use the computed b,c,m to pluck off the n bits of f(x). [
This reduction to nice z enhances the "flow" metaphor. The m-many 2s in z can be advance-routed to the last m places of y¢, so the whole issue is how the m-many 0s and 1s in z flow together into the
first m places of y¢.
this flow progress (without loss of circuit-size generality) by "squeezing out 2s" in an intuitively plane-filling fashion, allowing "mileposts" whose forced spacing might mandate having n*log
(n) - O(n) gates? Or can linear-size networks rise above the planar view? No one I've asked has known, and lack of them frustrates a desired general linear-size circuit simulation of my
"Block Move" model
. Issues
may be involved. Nor do I know nicer descriptions of O(nlogn)-sized circuits than "use ancillas to tag bits of x and work in P
as in the proof of Theorem 1, employing ideas of Theorem 3 and/or mapping into the O(nlogn)-sized
Ajtai-Komlos-Szemeredi networks
." Those seeking an o(nlogn) upper bound may be my guest, but those believing a super-linear circuit lower bound must reflect that no such bounds are known for string functions whose graphs belong to
NP or to E. The above inductive definition of f yields a linear-time algorithm on any model that simulates each operation of a double-ended queue in O(1) time. But is booting a 2 to the rear in f(2x)
= f(x)2 really in constant time, even amortized? True, our technical issues shrink away on passing from linear to polynomial time, so all this may seem to have nothing to do with P versus NP. But
the Baker-Gill-Solovay "oracle" obstacle may mean nothing more than that standard "diag-sim" and timing techniques are insensitive to internal information flow. The "Natural Proofs" obstacle
ultimately say only that network-preparation/"nonuniformity" is a subtly powerful consideration. Honing tools for information-flow analysis on incrementally more-general cases that yield super-linear
lower bounds may be the walk to walk before trying to run.
File translated from T
X by
, version 3.77.
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Savant Syndrome of Neural NetworksData Science News
Savant Syndrome of Neural Networks
For this, we calculate the correlation coefficient R between two weight vectors given byDefinition of correlation between weight vectors of two pixelsThe correlation coefficient values between -1 and
If two pixels trigger similar node areas in the hidden layer the correlation of the associated weight tends to be high and targets 1.
However, if two pixels address different zones in the hidden layer R tends to zero or even -1.
Now we are on the last mile.
We assume the grid structure to be reflected in the hidden layer.
Accordingly, the correlation of adjacent pixels should be high if pixel pairs are located within the same grid block and low if pixels are located cross-border in different grid blocks.
So we calculate for each pixel the average of the correlation with its nearest neighbors to the right and downwards.
Definition of weight vector correlation between adjacent pixel pairsIf we visualize this value for both NNs we obtain the following heatmaps:Weight vector correlation between adjacent pixel pairs for
original NN and shuffled NNThis shows that the shuffled NN evolves a separated hidden layer with areas dedicated for each grid segment whereas the original NN considers the image en bloc.
Final RemarksPlease notice that we have used a simple 3-layer dense NN with purpose of classification.
In particular, we have not used a convolutional network.
Nevertheless, our network is able to recognize puzzled images.
One could argue that puzzling is just an additional degree of freedom for image representation, i.
a shoe can be shown original but also in several puzzled variants, however, all these pictures are still just a shoe (although the puzzled versions are very uncommon in real life).
But if so the NN would treat the input pictures en bloc and not show the separated weight structure.
Furthermore, I ask myself if the applied puzzling algorithm is also a form of Data Augmentation.
At least it’s not a classical transformation like rotation or scaling to enlarge the amount of training data.
Finally, feel free to browse my code.
So you can reproduce all of my findings presented above.
It’s in Python using TensorFlow Keras.
Thanks for reading this article!.If you like this neural session I would be pleased to hear you clap.
But maybe my findings are an old hat?.If you can provide further insights, alternative approaches or know related topics feel free to add your comments!.
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RTD Calculator: Converting Measured Resistance to Process Temperature
Instrumentation Calculators
RTD Calculator: Converting Measured Resistance to Process Temperature
• A Resistance Temperature Detector (RTD) is a sensor used to measure temperature by correlating the resistance of the RTD element with temperature.
• Resistance Temperature Detectors (RTDs) are widely used in industrial and scientific applications to measure temperature precisely.
Formula for calculation
One common formula for estimating temperature based on RTD resistance is
T=Rref + {(RT- Rref)/α Rref}
In this formula,
• RT represents the resistance of the RTD at temperature T,
• T is the temperature at which you want to calculate with measured resistance.
• Rref is the reference resistance at the reference temperature Tref,
• α denotes the temperature coefficient factor of the RTD,
• Tref is the reference temperature.
This formula assumes a linear relationship between resistance and temperature, making it a simple and useful tool for quick temperature estimation.
Example Calculation
The most common type of RTD is the PT100, which has a resistance of 100 ohms at 0 degrees Celsius.
Let’s consider a practical example using a PT100 RTD with specific values for the formula components. Assume that:
• Rref = 100 ohms (at Tref = 0°C)
• α = 0.00385 (a common value for PT100 RTDs)
• We have a given that measured resistance RT = 119.25 ohms.
• Tref = 0°C (reference temperature of the RTD )
Follow these steps to calculate the process temperature:
Understanding the Formula
The formula relates the process temperature (T) to the measured resistance
• T=Rref + {(RT- Rref)/α Rref}
Substituting Known Values and Calculating Temperature
• Insert the provided values into the formula:
T = 0+(119.25−100)/(0.00385X100)
T = 19.25/0.385 ≈50°C
With the measured resistance of 119.25 ohms, the calculated process temperature is approximately 50°C. This result is consistent with what we would expect from a PT100 RTD.
While the formula provides a practical way to estimate process temperature, keep in mind a few important considerations:
• Lead wire resistance and other factors may affect the accuracy of the measurement.
• The formula assumes a linear relationship between resistance and temperature, which might not hold true over a wide temperature range.
• The T=Rref + {(RT- Rref)/α Rref} formula provides a way to estimate process temperature based on RTD resistance.
• The example demonstrates its application with a PT100 RTD. However, for accurate results, factors like lead wire resistance and non-linearity should be considered, and alternative equations like
the Callendar-Van Dusen equation might be more suitable.
RTD Calculator for converting Measured Resistance to Process Temperature
When an RTD is put in a process, the measured resistance to process temperature can be calculated with the help of the calculator that can be found below.
Ref the below link Converting Process Temperature to Measured output Resistance RTD
RTD Calculator: Converting Process Temperature to Measured Output Resistance
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Student Estimates of Probability and Uncertainty in Advanced Laboratory and Statistical Physics Courses
Conference Proceedings Detail Page
Student Estimates of Probability and Uncertainty in Advanced Laboratory and Statistical Physics Courses
written by Donald B. Mountcastle, Brandon Bucy, and John R. Thompson
Equilibrium properties of macroscopic systems are highly predictable as n, the number of particles approaches and exceeds Avogadro's number; theories of statistical physics depend on these
results. Typical pedagogical devices used in statistical physics textbooks to introduce entropy (S) and multiplicity () (where S = k ln()) include flipping coins and/or other equivalent
binary events, repeated n times. Prior to instruction, our statistical mechanics students usually gave reasonable answers about the probabilities, but not the relative uncertainties, of
the predicted outcomes of such events. However, they reliably predicted that the uncertainty in a measured continuous quantity (e.g., the amount of rainfall) does decrease as the number of
measurements increases. Typical textbook presentations assume that students understand that the relative uncertainty of binary outcomes will similarly decrease as the number of events
increases. This is at odds with our findings, even though most of our students had previously completed mathematics courses in statistics, as well as an advanced electronics laboratory
course that included statistical analysis of distributions of dart scores as n increased.
Physics Education Research Conference 2007
Part of the PER Conference series
Greensboro, NC: August 1-2, 2007
Volume 951, Pages 152-155
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@inproceedings{ Author = "Donald Mountcastle and Brandon Bucy and John Thompson", Title = {Student Estimates of Probability and Uncertainty in Advanced Laboratory and Statistical Physics
Courses}, BookTitle = {Physics Education Research Conference 2007}, Pages = {152-155}, Address = {Greensboro, NC}, Series = {PER Conference}, Volume = {951}, Month = {August 1-2}, Year =
{2007} }
%A Donald Mountcastle %A Brandon Bucy %A John Thompson %T Student Estimates of Probability and Uncertainty in Advanced Laboratory and Statistical Physics Courses %S PER Conference %V 951
%D August 1-2 2007 %P 152-155 %C Greensboro, NC %U https://www.compadre.org/Repository/document/ServeFile.cfm?ID=9092&DocID=2009 %O Physics Education Research Conference 2007 %O August 1-2
%O application/pdf
%0 Conference Proceedings %A Mountcastle, Donald %A Bucy, Brandon %A Thompson, John %D August 1-2 2007 %T Student Estimates of Probability and Uncertainty in Advanced Laboratory and
Statistical Physics Courses %B Physics Education Research Conference 2007 %C Greensboro, NC %V 951 %P 152-155 %S PER Conference %8 August 1-2 %U https://www.compadre.org/Repository/
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How to Convert Tensorflow Models to TensorRT - reason.townHow to Convert Tensorflow Models to TensorRT
How to Convert Tensorflow Models to TensorRT
TensorRT is a high performance neural network inference optimizer and runtime engine. In this blog, we’ll show you how to convert your Tensorflow models to TensorRT for faster inference.
Checkout this video:
This tutorial is an introduction to converting a Tensorflow* model to TensorRT*. You will learn how to take a pre-trained Tensorflow model and convert it into a format that is optimized for
Inferencing on Jetson devices.
What is TensorRT?
TensorRT™ is an SDK that optimizes deep learning models for inference and creates a runtime for deploying them on GPUs. It includes a deep learning inference optimizer and runtime that delivers low
latency and high-throughput for deep learning inference applications.
Why Convert Tensorflow Models to TensorRT?
There are several reasons why you might want to convert a Tensorflow model to TensorRT. First, TensorRT can provide significant performance improvements over Tensorflow for inference, especially on
GPUs. Second, TensorRT can also help to save memory by optimizing the model for inference. Finally, converting a Tensorflow model to TensorRT can help to improve the portability of the model, as
TensorRT is supported on a wide variety of platforms.
How to Convert Tensorflow Models to TensorRT?
Tensorflow is a powerful tool for machine learning, but it can be challenging to get the most out of it on mobile or embedded devices. TensorRT is a toolkit that helps optimize machine learning
models for these devices, providing up to 40x faster performance.
Here’s how you can convert your Tensorflow models to TensorRT:
first, install the TensorRT Python package:
$ pip install tensorrt
next, import the package and create a converter object:
import tensorrt as trt
# create converter object
converter = trt.Converter()
# specify model file and input/output nodes
model_file = ‘model.pb’
input_nodes = [‘input’]
output_nodes = [‘output’]
# parse model file
converter.parse(model_file, input_nodes, output_nodes)
# convert model to TensorRT format
tensorrt_model = converter.convert()[‘engine’]
# save converted model to file
save_path = ‘tensorrt_model.bin’
with open(save_path, ‘wb’) as f:
TensorRT Inference
Graphical processing units (GPUs) are ideal for accelerating inferencing of deep learning (DL) models because they can parallelize many computations. TensorRT™ is an SDK from NVIDIA® that optimizes
DL models and minimizes gaps in processing capabilities between training and inferencing. TensorRT takes a trained network, which consists of a set of layers, and creates a new graph with some nodes
that are optimized for inferencing. This process can significantly speed up inferencing time while reducing computational resources usage on the edge devices like Jetson TX2.
TensorRT supports both C++ and Python and developers using either can easily convert their models to TensorRT using one of the provided parsers. TensorRT inference can be performed on CPUs, GPUs, and
other supported hardware devices. The CPU implementation is not as efficient as the GPU implementation though because it lacks the ability to perform optimizations specifically for CPUs.
TensorRT Optimizations
NVIDIA’s TensorRT is a platform for high-performance deep learning inference. It includes a deep learning inference optimizer and runtime that delivers low latency and high-throughput for deep
learning inference applications.
TensorRT can be used to accelerate diverse applications including recommendation systems, image classification, object detection, machine translation, and question answering. With TensorRT, you can
get up to 40x faster performance over CPU-only platforms for inferencing of popular models like ResNet50, SSD-Mobilenet, and U-Net.
This guide provides instructions on how to convert existing Tensorflow models to TensorRT format and run the inference using TensorRT.
TensorRT Engines
TensorRT is a high performance neural network inference accelerator. It is designed to work with deep learning frameworks such as TensorFlow, Caffe, and PyTorch. TensorRT accelerates deep learning
inference by performing linear algebra operations on hardware that is specifically designed for deep learning. This results in faster and more efficient inference than would be possible using a
general purpose CPU or GPU.
TensorRT engines can be created from TensorFlow models using the convert_to_tensorrt() function in the tensorrt package. This function takes as input a TensorFlow model, a set of input tensors, and a
list of output tensors. It produces as output a TensorRT engine which can be used for inference.
TensorRT Integration
TensorRT is a library created by NVIDIA that optimizes neural network models for inference. It can be used to improve performance on GPUs for both training and deployment. TensorRT integration is
available as a feature in TensorFlow starting from version 1.8.
TensorRT can be used to improve the performance of both training and deployment for machine learning models. In TensorFlow, there are two ways to use TensorRT: through the Grappler plugin or through
the C++ API. The Grappler plugin is the more recommended approach as it is easier to use and integrates better with TensorFlow’s graph optimization system. The C++ API can be used if more control
over model conversion is needed.
To use the Grappler plugin, you will need to set the following environment variables:
– GRAPPLER_TRT_ENABLE=1
– GRAPPLER_TRT_MAX_BATCH=1
– GRAPPLER_TRT_PRECISION=32
– LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/usr/local/cuda/lib64:/usr/lib/nvidia-cuda-toolkit:/usr/local/lib64
TensorRT Benchmarks
If you are looking for ways to speed up your neural network inference, you may be interested in using TensorRT. TensorRT is a toolkit developed by NVIDIA that allows you to optimize neural network
models for faster inference. In this article, we will show you how to convert your Tensorflow models to TensorRT and run benchmark tests to see the performance gains.
1. Install the TensorRT package. You can find the latest version here.
2. Convert your Tensorflow model to a UFF file. You can do this using the convert-to-uff script that comes with the TensorRT package.
3. Create a TensorRT engine using the UFF file. You can do this using the create-engine command-line tool that comes with the TensorRT package.
4. Optimize your neural network for inference using the engine you created in step 3. You can do this using the optimize-network command-line tool that comes with the TensorRT package
TensorRT Use Cases
There are a few use cases for running TensorRT models:
-Optimizing existing TensorFlow models: If you have a model that is already trained and you want to deploy it on a platform that supports TensorRT (like the Jetson TX2), you can convert the model to
TensorRT format and run it using the TRT engine.
-Training new models: You can use TensorRT to accelerate training of new models by converting the models to TensorRT format and running them on supported hardware.
-Inference only: You can use TensorRT for inference only, without retraining or fine-tuning your model. | {"url":"https://reason.town/tensorflow-to-tensorrt/","timestamp":"2024-11-13T03:04:55Z","content_type":"text/html","content_length":"97844","record_id":"<urn:uuid:e4204670-cb5c-48e4-bf43-989e8d2c7eb6>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00041.warc.gz"} |
What Is the Long-Term Fish Population in This Infinite Series Problem?
• Thread starter Burjam
• Start date
So you can choose a specific value for one Fn and use it to find the other unknown. For example, if you choose Fn = 4500, you will get two equations in two unknowns (a and c). Solve those equations.
Then you can use the values you get for a and c to find the long-term population of the fish.In summary, two individuals are discussing a question about a fishery manager using an infinite series to
find the long-term population of fish. The question is unclear about whether the population is measured after a harvest or before, but ultimately it is determined that the population will either
continue to increase indefinitely or eventually die out. The individuals also discuss a method for solving the equation and finding
Homework Statement
A fishery manager knows that her fish population naturally increases at a rate of 1.4% per month, while 119
fish are harvested each month. Let F[n] be the fish population after the nth month, where F[0] = 4500 fish. Assume that that process continues indefinitely. Use the infinite series to find the
long-term (steady-state) population of the fish exactly.
Homework Equations
The Attempt at a Solution
My issue is that I can't seem to set up an expression to evaluate the series. I know that the expression will involve subtracting 119 and use 0.014 to represent the percent increase. If it were only
the percent increase, I would be able to set up an expression. But the -119 is really throwing me off.
Burjam said:
Homework Statement
A fishery manager knows that her fish population naturally increases at a rate of 1.4% per month, while 119
fish are harvested each month. Let F[n] be the fish population after the nth month, where F[0] = 4500 fish. Assume that that process continues indefinitely. Use the infinite series to find the
long-term (steady-state) population of the fish exactly.
Homework Equations
The Attempt at a Solution
My issue is that I can't seem to set up an expression to evaluate the series. I know that the expression will involve subtracting 119 and use 0.014 to represent the percent increase. If it were
only the percent increase, I would be able to set up an expression. But the -119 is really throwing me off.
The question does not make clear whether the F
represent the population just after a harvest or just before. I would take it as just after.
If the population is F
after the n
month what will it be after one more month?
F[n+1] = F[n](1 + 0.014) - 119?
Burjam said:
F[n+1] = F[n](1 + 0.014) - 119?
Right. Do you know a way to solve such equations? If not, an easy thing to try is to see if you can add a constant to each F
so that it reduces to a simple geometric progression.
haruspex said:
Right. Do you know a way to solve such equations? If not, an easy thing to try is to see if you can add a constant to each F[n] so that it reduces to a simple geometric progression.
I don't know how to write this equation without F
being in terms of F
or F
Burjam said:
I don't know how to write this equation without F[n] being in terms of F[n+1] or F[n-1].
It will be the same equation, but written in the form (F
+c) for some pair of constants a and c.
haruspex said:
It will be the same equation, but written in the form (F[n+1]+c)=a(F[n]+c) for some pair of constants a and c.
How will adding the c to both sides eliminate the F
? None of the problems I have done or have examples of with infinite series so far have anything like this, so I don't really have anything to go by.
Burjam said:
How will adding the c to both sides eliminate the F[n+1]?
I did not suggest it would.
You have this equation: F
= F
(1 + 0.014) - 119
and I am suggesting this form of it: (F
What do you get if you combine them?
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
Burjam said:
F[n+1] = F[n](1 + 0.014) - 119?
You have ##F_{n+1} = 1.014 F_n - 119## with ##F_0 = 4500##. Try calculating ##F_1, F_2, F_3## (keeping ##F_0## symbolic instead of 4500). In fact, it might make everything much clearer if you keep
all parameters symbolic, so that ##F_{n+1} = r F_n - k##. Using symbols like that instead of numbers helps keep separate the different effects.
However, I think there is something very wrong with the original problem statement: for ##r > 1## (for example, for ##r = 1.014##) you must have a very special relationship between ##F_0,r,k## in
order to obtain a finite limit; otherwise you will either have ##F_n \to +\infty## as ##n \to \infty## (for some combinations of ##F_0##, ##r##, and ##k##) or else ##F_n \to -\infty## for for other
combinations. Of course, the latter case really means that ##F_n## hits zero at some finite ##n## and so the fish population dies out completely and the problem ends; ##F_n## does not actually go to
haruspex said:
I did not suggest it would.
You have this equation: F[n+1] = F[n](1 + 0.014) - 119
and I am suggesting this form of it: (F[n+1]+c)=a(F[n]+c)
What do you get if you combine them?
By combine them, do you mean take a F
in the second equation as F
= F
(1+0.014) - 119 and then try to solve for a and C?
Ray Vickson said:
I think there is something very wrong with the original problem statement
I assumed it was intended that:
Ray Vickson said:
you will either have ##F_n \to +\infty## as n→∞... [or] ... the fish population dies out completely
Burjam said:
By combine them, do you mean take a F[n+1] in the second equation as F[n+1] = F[n](1+0.014) - 119 and then try to solve for a and C?
Yes. You will have one equation with two unknowns, but remember that the equation has to be true for all F
FAQ: What Is the Long-Term Fish Population in This Infinite Series Problem?
What is an infinite series word problem?
An infinite series word problem is a mathematical problem that involves an infinite sequence of numbers or terms. The goal is to find the sum of the infinite series, which may or may not converge to
a finite value.
What is the difference between a convergent and divergent infinite series?
A convergent infinite series is one in which the sum of its terms approaches a finite value as the number of terms increases. A divergent infinite series is one in which the sum of its terms does not
approach a finite value, but instead either tends to infinity or oscillates between positive and negative values.
How do you test for convergence or divergence of an infinite series?
The most commonly used tests for convergence or divergence of an infinite series are the comparison test, the ratio test, and the integral test. These tests compare the given series to a known
convergent or divergent series and use mathematical techniques to determine the behavior of the given series.
Can an infinite series word problem have multiple solutions?
Yes, an infinite series word problem can have multiple solutions. This is because there are often multiple ways to manipulate and rearrange the given series, which can lead to different sums.
Additionally, some series may have multiple convergent values depending on the starting point or the number of terms used in the calculation.
What real-world applications use infinite series word problems?
Infinite series word problems have various real-world applications, such as in physics, engineering, and economics. For example, they can be used to calculate the trajectory of a projectile, the
electrical resistance of a circuit, or the value of an investment over time. Infinite series also play a crucial role in the development of calculus and other mathematical concepts. | {"url":"https://www.physicsforums.com/threads/what-is-the-long-term-fish-population-in-this-infinite-series-problem.897693/","timestamp":"2024-11-12T04:35:41Z","content_type":"text/html","content_length":"123207","record_id":"<urn:uuid:209dda94-6f04-46eb-abb8-983e8133c594>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00777.warc.gz"} |
Vishwesh Nath^1, Kurt G Schilling^2, Prasanna Parvathaneni^3, Allison E Hainline^4, Colin B Hansen^1, Camilo Bermudez^2, Andrew J Plassard^1, Justin A Blaber^1, Vaibhav Janve^2, Yurui Gao^2, Iwona
Stepniewska^5, Adam W Anderson^2, and Bennett A Landman^3
^1Computer Science, Vanderbilt University, Nashville, TN, United States, ^2Biomedical Engineering, Vanderbilt University, Nashville, TN, United States, ^3Electrical Engineering, Vanderbilt
University, Nashville, TN, United States, ^4Biostatistics, Vanderbilt University, Nashville, TN, United States, ^5Psychology, Vanderbilt University, Nashville, TN, United States
Confocal histology provides an opportunity to establish intra-voxel fiber orientation distributions that can be used to quantitatively assess the biological relevance of diffusion-weighted MRI
models, e.g., constrained spherical deconvolution (CSD). Here, we apply deep learning to investigate the potential of single shell diffusion-weighted MRI to explain histologically observed fiber
orientation distributions (FOD) and compare the derived deep learning model with a leading CSD approach. This study (1) demonstrates that there exists additional information in the diffusion signal
that is not currently exploited by CSD, and (2) provides an illustrative data-driven model that makes use of this information.
Understanding the relationship between observed diffusion-weighted MRI signals and true tissue microarchitecture is of fundamental concern for biophysical modeling, detecting microstructural
differences, and brain tractography. Substantial efforts have been invested in interpreting the diffusion signal from both model-based (e.g., constrained spherical deconvolution - CSD [1,2], Q-ball
[3] , persistent angular structure - PAS [4]) and data-driven [5] perspectives. Recently, multi-layer neural networks (or informally, deep learning or deep neural networks - DNN) have emerged as a
leading class of machine learning approaches. Moreover, advances combining MRI and whole brain histology have enabled volumetric registration between MRI and histological processes, while
co-registered confocal microscopy allows direct 3-D observation of intra-voxel tissue orientation. Here, we apply deep learning to investigate the potential information content in single shell
diffusion-weighted MRI to explain histologically observed fiber orientation distribution (FOD) functions.
Three ex vivo squirrel monkey brains were imaged on a Varian 9.4T scanner. Briefly, data were acquired with a 3D diffusion-weighted EPI sequence (b-value=6,000 s/mm2, 100 directions) at 300um
isotropic resolution. After scanning, the tissue was sectioned, stained with the fluorescent DiI, and imaged on an LSM710 Confocal microscope following the procedures outlined in [6]. The
histological FOD was extracted using structure tensor analysis. Finally, a multi-step registration procedure [6] was used to determine the corresponding diffusion MRI signal. A total of 567
histological voxels were processed, and a hundred random rotations were applied to each one of them for both the MR signal and the histology FOD to augment the data bringing the total to 57267 voxels
For qualitative validation, a single healthy human volunteer was scanned for a single session using a 3T (Achieva, Philips Medical Systems, Best, The Netherlands) with a 32-channel head coil. Four
scans acquired were at a b-value of 2000 s/mm2 (which approximates the diffusion contrast of a fixed ex vivo scan at a b-value of 6000 s/mm2) with 96 gradient directions and an additional b0 per scan
(2.5mm isotropic resolution, matrix of 96x96, 38 slices, Multi-Band=2; SENSE=2.2;TR= 2650 ms; TE=94 ms; partial Fourier=0.7). Standard pre-processing with FSL (topup, eddy correction, registration,
averaging across scans) was performed before analysis.
Both ex vivo and in vivo HARDI acquisitions were fit with 8th order real spherical harmonics. Outliers were manually reviewed for imaging artifacts, and 54 voxels were removed. FOD’s from the
histology were fitted with a 10th order real spherical harmonics. Histology data was divided into training/validation (44,541 voxels) and testing sets (7,272 voxels) without mixing augmented data
(rotations). For training/validation, a 20% percent split was used with 5 folds. Mean squared error was used to assess model accuracy [8].
The median angular correlation coefficient (ACC) for CSD (0.7965) was significantly (p<0.05, non-parametric signed rank test) lower than for the deep approach (0.8165) (Fig 2), which corresponded to
a lower root mean squared error for the deep approach (0.539 versus 0.561). Qualitatively, the predicted FOD’s on the human in vivo data demonstrate anatomical consistency (Fig 3), indicating that
the deep learning approach is predicting structure in line with prior observations.
By demonstrating superiority of a deep learning approach over a leading model-based approach, CSD, we show that (1) there exists additional information in the diffusion signal that is not currently
exploited by CSD, and (2) provide an illustrative data-driven model that makes use of this information. In a preliminary analysis, we applied the same network to ex vivo imaging at a b-value of 9000
s/mm2 and found a significantly higher ACC (0.850, p<0.05, non-parametric signed rank test) for deep learning which is 6.7% higher than CSD. Hence, generalizing the deep learning to use multiple
shells and adapt to high b-values is a promising area of exploration. To enable others to investigate our results, the derived TensorFlow models that describe the identified MRI:histology
relationships are available on the NITRC project “masimatlab” . Perhaps most importantly, this deep learning analysis demonstrates that current models for identifying fiber orientation distributions
do not make all possible use of existing information, and additional innovation is possible. The deep learning models presented herein are preliminary and have not guaranteed optimality properties,
and further exploration of the space of multi-layer neural networks is warranted. Additionally, continued refinement of deep learning approaches could make use of both traditional data augmentation
of ground truth (e.g., rotations as used herein), but also physics/diffusion simulations of modeled geometry along with image acquisition models.
Research reported in this publication was supported in part by the National Institutes of Health R01EB017230 (Landman) and R01NS058639 (Anderson). The content is solely the responsibility of the
authors and does not necessarily represent the official views of the National Institutes of Health. Confocal imaging was performed in the Digital Histology Shared Resource at Vanderbilt University
Medical Center (www.mc.vanderbilt.edu/dhsr)
1.) Tournier, J-Donald, et al. "Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data." Neuroimage 42.2 (2008): 617-625.
2.) Anderson, Adam W. "Measurement of fiber orientation distributions using high angular resolution diffusion imaging." Magnetic Resonance in Medicine 54.5 (2005): 1194-1206.
3.)Descoteaux, Maxime, et al. "Regularized, fast, and robust analytical Q‐ball imaging." Magnetic resonance in medicine 58.3 (2007): 497-510.
4.) Jansons, Kalvis M., and Daniel C. Alexander. "Persistent angular structure: new insights from diffusion magnetic resonance imaging data." Inverse problems 19.5 (2003): 1031.
5.) Koppers, Simon, Christoph Haarburger, and Dorit Merhof. "Diffusion MRI Signal Augmentation: From Single Shell to Multi Shell with Deep Learning." International Conference on Medical Image
Computing and Computer-Assisted Intervention. Springer, Cham, 2016.
6.) Schilling, Kurt, et al. "Comparison of 3D orientation distribution functions measured with confocal microscopy and diffusion MRI." Neuroimage 129 (2016): 185-197.
7.) Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton. "Imagenet classification with deep convolutional neural networks." Advances in neural information processing systems. 2012.
8.) Tieleman, Tijmen, and Geoffrey Hinton. "Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude." COURSERA: Neural networks for machine learning 4.2 (2012): 26-31. | {"url":"https://cds.ismrm.org/protected/18MProceedings/PDFfiles/5225.html","timestamp":"2024-11-11T20:32:22Z","content_type":"application/xhtml+xml","content_length":"15889","record_id":"<urn:uuid:8158ca22-753f-4305-91d4-511fc6dab40d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00877.warc.gz"} |
Questions about moisture budget analysis using WRF outputs
Dear all,
I am doing moisture budget analysis with WRF outputs. I have a few questions:
The moisture budget equation I used is
Seager and Henderson 2013
Q1: When calculating the vertically integrated moisture flux, I modified the code of vertical integration from (
NCL example
Can I use the method to do vertical integration on (ua * Qvapor) / dx and (va * Qvapor) / dy, and then add them up?
Q2: In WRF output, how can I calculate the last term in the Eq (9)?
Should I use the wa (z-wind component") at the lowest level or should I use ua * gradient of terrain height?
Q3: Dose the variable P in the Eq (9) mean the RAINNC(t) - RAINNC(t-1) in WRF output? Cumulus parameterization is not used.
And how about E? What is the corresponding variable in WRF output?
Any information would be greatly appreciated.
Please see my answers below:
(1) Q1: When calculating the vertically integrated moisture flux, I modified the code of vertical integration from (
NCL example
Can I use the method to do vertical integration on (ua * Qvapor) / dx and (va * Qvapor) / dy, and then add them up?
Personally I think this is a reasonable approach. So the answer is yes.
(2) Q3: Dose the variable P in the Eq (9) mean the RAINNC(t) - RAINNC(t-1) in WRF output? Cumulus parameterization is not used.
And how about E? What is the corresponding variable in WRF output?
P is precipitation. It should be RAINNC(t2) - RAINNC(t1), t2 and t1 are the beginning and end time of the period you calculate the moisture budget.
Note that RAINNCV is accumulative precipitation from the initial time of model run.
Hi Ming,
Thank you so much for your information. I may have some follow up questions:
Related to previous Q2, I was wondering how to calculate omega (vertical velocity in pressure coordinate).
It seems like I need to use omega to compute the last term in Eq (9).
In your reply, you mentioned RAINNCV. Could you go into a little more detail about RAINNCV?
I do not see it in my WRF output.
Thank you!
I am really sorry for the typo. RAINNCV should be RAINNC, which is the accumulative resolved-scale precipitation.
For Q2, please see the formula below, which is what I used to calculate omega based on other wrfout variables:
in which:
grav=9.81 ! m/s**2
prs is pressure in [mb]
rgas=287.04 !J/K/kg
tmk isTemperature in [K].
qvp is water vapor mixing ratio in [kg/kg]
w is vertical velocity in [m/s]
Hope this is helpful for you.
I have a following question: what is the wrfout variable for surface evaporation?
SFCEVP(t) - SFCEVP(t-1) equals to the "E" in the equation, right?
Thanks in advance.
Thank you for confirming that! | {"url":"https://forum.mmm.ucar.edu/threads/questions-about-moisture-budget-analysis-using-wrf-outputs.12224/","timestamp":"2024-11-04T08:39:07Z","content_type":"text/html","content_length":"66505","record_id":"<urn:uuid:cc494973-fd6d-4197-8bdd-37d6f4f7473a>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00509.warc.gz"} |
A spring with Spring Constant k=340N/m is used to Weigh a 6.7-kg fish - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day
A spring with Spring Constant k=340N/m is used to Weigh a 6.7-kg fish
This question aims to find the change in the length of spring (used to weigh 6.7-kg fish), which is displaced from its mean position. The value of the spring constant is given as k=340N/m.
Hooke’s law states that the force exerted by the spring when stretched or compressed from its mean position is directly proportional to the distance it covers from its mean position.
Spring is called ideal if it has an equilibrium length. The spring in compression is directed towards its mean position, and its length changes from its equilibrium length. This change in length
shows a decrease in the equilibrium length.
On the other hand, the spring in a stretched state exerts a force away from its mean position, and the change in length is always greater than the equilibrium length.
The spring in a stretched or compressed state exerts a force to restore the equilibrium length of the spring and to make it come back to its mean position is called the restoring force.
F = -k{x}
Where k is called the spring constant, x represents the change in length from its equilibrium length, and F is the force exerted on the spring. The spring constant measures the stiffness of the
spring. At the mean position, the spring has no displacement i.e, x=0, and it changes when the spring is at extreme positions.
The elastic limit is reached when the displacement becomes very large. Stiff objects show very small displacement before the elastic limit is reached. Pulling or pushing an object beyond its elastic
limit causes a permanent change in the shape of the spring.
Expert Answer
The force exerted by the spring on the object is equal to the mass of the object attached to that spring. Since the mass is pulled by gravitational force, we will use:
\[F = K x\] , \[F= m g\]
\[k x = m g\]
\[x = \frac{m \times g}{k}\]
Value of spring constant $k$ = $340 N/m$
Mass of the fish $m$ = $6.7 kg$
The change in length $x$.
Numerical Solution
By putting the given values of $k$ and $m$ and $g$ = $9.8ms^{-1}$ in the formula, we will get:
\[x = \frac{ 6.7 \times 9.8}{340}\]
\[x = 0.193 m\]
The change in length of the spring stretched by the fish will be $x$ = $0.193$.
A spring having force $100N$ is stretched and displaced by $0.8m$. Find the spring constant.
The given values are:
\[Force( F) = 100N\]
\[Displacement (x) = 0.8m\]
To find the spring constant,
\[F = -kx\]
\[k = \frac{-F}{x}\]
\[k = \frac{-100}{0.8}\]
\[k = -125 N/m\]
The value of spring constant is $k$ = $-125 N/m$.
Image/Mathematical drawings are created in Geogebra. | {"url":"https://www.storyofmathematics.com/a-spring-with-spring-constant-k-340n-m-is-used-to-weigh-a-6-7-kg-fish/","timestamp":"2024-11-08T11:24:05Z","content_type":"text/html","content_length":"136285","record_id":"<urn:uuid:71f568b6-1f1e-42ec-8aee-0d546833f630>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00308.warc.gz"} |
FIR Filter Design in Simulink
In this tutorial, we will discuss filters, their uses, and their benefits. At the start, we provide a brief and general introduction to filters. Then we will discuss Finite Impulse Response (FIR)
filters. After that, we provide an explanation of the different orders of FIR filters. Using the information provided in the introduction, a simple and comprehensive FIR filter of second order is
designed in the subsection “Explanation with Example”. In this subsection, we provide a step-by-step explanation along with the results of the filter. At the end, a simple and easy-to-perform
exercise is provided for the reader to do on their own related to the concept provided in the tutorial.
Introduction to Filters
Filters are a very basic component used by almost every single electrical engineer. As the name suggests, a filter is used to filter out unwanted or noisy components and features from the input. In
general, the filters can take any input, but when we talk about signal processing specifically, the input must be an electrical signal. Now, redefining the filter, it is a process of removing
unwanted components or noise from an input signal. There are various types of signals, but we will only discuss a few of them here.
Types of Filters
Filters can be classified into the following types:
FIR Filters (Finite Impulse Response) IIR Filters (infinite impulse response filter)
High-pass and Low-pass Filters Band-pass Filters
Stop-band Filters Notch Filters
Comb Filters All-pass Filters
To name a few but we will only discuss FIR filters here.
FIR Filters
A finite impulse response filter (FIR filter) can easily be understood by its name. It provides a finite-length output response to an input impulse. In simple words, FIR filters give a
finite-duration output in response to an impulse, as we will see shortly in the example below. Coming over to the order of FIR filters, it is defined as the order of their transfer function. For an
Nth-order FIR filter, the output is only dependent on the first N input samples. We will design a second-order FIR filter in this tutorial. A general design of a FIR filter is shown in the figure
Designing FIR Filters in Simulink
Now let’s design a second-order FIR filter. A comparison of an equation with its general form is done below. So, to find the coefficients b[0], b[1], and b[2], we will write them as follows: the
general form on the left side and its corresponding coefficient on the right side.
b[0 ]= 2, b[1] = 1, b[2 ]= 2
Placing Components
Now, let’s design this filter in MATLAB’s Simulink. First of all, open MATLAB and then Simulink, as we have been doing in previous tutorials. Create a blank model to design a simple FIR filter. Open
the library browser of Simulink, and from the commonly used blocks, select the constant block as shown in the figure below. This block will serve as the coefficient of the equation.
Constant block
Delay Block
Next, we will need a delay block, which will serve as the x[n-1] and x[n-2] delayed samples. The order of the filter will determine the number of delay blocks to be used in the filter design. In our
example, the order of filters is 2, hence the number of delay blocks. From the commonly used blocks section, select the delay block and place it on the model as shown in the figure below.
Delay block
Sum Block
The number of stages of an FIR filter to be designed will also depend on the order of the filter. If the order of the filter is N, then the number of stages to be used in the filter design will be
N+1. In our case, the order of the filter is 2, and hence the stages will be 3. At the end of the filter stages, we will have to sum up the output of all the stages, as is obvious from the equation
of our system. For this summation process, we have to use a sum block. From the Math Operation section in the Simulink library browser, select the sum block and add it to the model as shown in the
figure below.
Sum block
Product Block
Also, in order to multiply the coefficient of each stage with the delayed input, we have to use some product blocks. In the library browser, from the section of Math Operations, select the Product
block and place it in the model as shown in the figure below.
Product block
Input Source
The next step is to add some input sources in order to see the correct response of the system. The input source we will use here is Repeating Sequence Stairs. The purpose of using this block is to
generate an impulse, as we will see shortly. From the sources section of the library browser of Simulink, select the Repeating Sequence Stairs block and add it to the model as shown in the figure
In order to display the output of the filter, we also need some kind of oscilloscope to display input along with output. From the sinks section of the library browser, select scope and add it to the
model as shown in the figure below.
Now let’s move to the model we created at the start to jump towards the designing part of the filter. As we have discussed previously, our filter will have three stages. We will design each stage
step by step. First of all, place the input source on the left and double-click on it to change its input to an impulse. In the parameter block, add the sequence of inputs as shown below.
Stages of FIR Filter
Now, using the equation given for the filter, we will start designing from left to right. The first part of the equation will be the first stage, as shown in the figure below.
The same input will then be used with a single delay to make the second stage and multiplied by the constant value of 1, as shown in the figure below.
The same will happen in the case of the third stage with one more delay block at the output of the first delay block to provide the twice delayed input, i.e., x[n-2], as shown in the figure below.
Complete Block Diagram
Now we will add the output of all three stages using the sum block. But first, change the number of inputs in the sum block, as we have done previously. And at the output of the sum block, connect a
scope with two inputs (one for the input and one for the output), as shown in the figure below.
Now change the simulation stop time to 1, as we have done before, and run the Simulink model. After the completion of the run, double-click on the scope, and the output of the filter will look like
the one shown in the figure below.
The output of the filter is in accordance with the explanation in the introduction.
• Design a third order FIR filter of the system given in the equation below.
(Hint: Number of delay blocks to be used will be 3)
In conclusion, this tutorial provides an in-depth overview of designing and simulating FIR filter in Simulink. It covers step-by-step procedures along with explanation of an example to helps us
better understand the concept. You can utilize this concept to design various other types of filters in Simulink. At the end, we have provided an exercise to reinforce the concept of this tutorial.
Hopefully, this was helpful in expanding your knowledge of Simulink.
You may also like to read:
This concludes today’s article. If you face any issues or difficulties, let us know in the comment section below.
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A Treatise on the Circle and the Spheresearch
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A Treatise on the Circle and the Sphere
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Hardcover ISBN: 978-0-8218-3488-6
Product Code: CHEL/236.H
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
Click above image for expanded view
A Treatise on the Circle and the Sphere
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Hardcover ISBN: 978-0-8218-3488-6
Product Code: CHEL/236.H
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
• AMS Chelsea Publishing
Volume: 236; 1916; 602 pp
MSC: Primary 51
Circles and spheres are central objects in geometry. Mappings that take circles to circles or spheres to spheres have special roles in metric and conformal geometry. An example of this is Lie's
sphere geometry, whose group of transformations is precisely the conformal group.
Coolidge's treatise looks at systems of circles and spheres and the geometry and groups associated to them. It was written (1916) at a time when Lie's enormous influence on the field was still
widely felt. Today, there is a renewed interest in the geometry of special geometric configurations. Coolidge has examined many of the most intuitive: linear systems of circles, circles
orthogonal to a given sphere, and so on. He also examines the differential and projective geometry of the space of all spheres in a given space.
Through the simple vehicles of circles and spheres, Coolidge makes contact with diverse areas of mathematics: conformal transformations and analytic functions, projective and contact geometry,
and Lie's theory of continuous groups, to name a few. The interested reader will be well rewarded by a study of this remarkable book.
Graduate students and research mathematicians.
□ The author has fully carried out the high aim he has set before himself: “The present work is an attempt, perhaps the first, to present a consistent and systematic account of the various
theories [those of Steiner, Feuerbach, Chasles, Lemoine, Casey, ... Reye, Fiedler, Loria, Mobius, Lie, Stephanos, Castelnuovo, Cosserat, Ribaucour, Darboux, Guichard ...].”
The Mathematical Gazette
□ Not a list of results, but a well digested account of theories and methods ... is what he has given us for leisurely study and enjoyment.
Bulletin of the AMS
□ The book provides a wealth of information from both a historical and mathematical perspective including many early ideas from the theory of algebraic curves and surfaces.
Zentralblatt MATH
• Permission – for use of book, eBook, or Journal content
• Book Details
• Reviews
• Requests
Volume: 236; 1916; 602 pp
MSC: Primary 51
Circles and spheres are central objects in geometry. Mappings that take circles to circles or spheres to spheres have special roles in metric and conformal geometry. An example of this is Lie's
sphere geometry, whose group of transformations is precisely the conformal group.
Coolidge's treatise looks at systems of circles and spheres and the geometry and groups associated to them. It was written (1916) at a time when Lie's enormous influence on the field was still widely
felt. Today, there is a renewed interest in the geometry of special geometric configurations. Coolidge has examined many of the most intuitive: linear systems of circles, circles orthogonal to a
given sphere, and so on. He also examines the differential and projective geometry of the space of all spheres in a given space.
Through the simple vehicles of circles and spheres, Coolidge makes contact with diverse areas of mathematics: conformal transformations and analytic functions, projective and contact geometry, and
Lie's theory of continuous groups, to name a few. The interested reader will be well rewarded by a study of this remarkable book.
Graduate students and research mathematicians.
• The author has fully carried out the high aim he has set before himself: “The present work is an attempt, perhaps the first, to present a consistent and systematic account of the various theories
[those of Steiner, Feuerbach, Chasles, Lemoine, Casey, ... Reye, Fiedler, Loria, Mobius, Lie, Stephanos, Castelnuovo, Cosserat, Ribaucour, Darboux, Guichard ...].”
The Mathematical Gazette
• Not a list of results, but a well digested account of theories and methods ... is what he has given us for leisurely study and enjoyment.
Bulletin of the AMS
• The book provides a wealth of information from both a historical and mathematical perspective including many early ideas from the theory of algebraic curves and surfaces.
Zentralblatt MATH
Permission – for use of book, eBook, or Journal content
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This package is a simple and efficient implementation of the Coordinate Descent Full Configuration Interaction (CDFCI) algorithm using modern C++ (C++14). CDFCI is an efficient algorithm for the
electronic structure ground-state calculation in the configuration interaction framework. CDFCI solves an unconstrained nonconvex optimization problem, which is a reformulation of the full
configuration interaction eigenvalue problem, via an adaptive coordinate descent method with a deterministic compression strategy. CDFCI captures and updates appreciative determinants with different
frequencies proportional to their importance. This is joint work with Jianfeng Lu and Zhe Wang.
KSSOLV is a MATLAB toolbox for solving Kohn-Sham density functional theory based electronic structure eigenvalue problems. It uses an object oriented features of MATLAB to represent atom, molecules,
wavefunctions and Hamiltonians and their operations. It is designed to make it easier for users to prototype and test new algorithms for solving the Kohn-Sham problem. KSSOLV2.0 contains significant
improvement over the original KSSOLV. In addition to performing ground state calculation for small molecules, it can also perform geometry optimization for both molecules and solids. It uses standard
pseudopotentials and implements local density approximation, generalized gradient approximation and hybrid functionals. Future releases will also include time-dependent DFT and post DFT calculations
such as the GW quasi-particle energy calculation and Bethe-Salpeter equation solver for optical absorption. This is joint work with KSSOLV Team.
ELSI-RCI provides and enhances open-source software packages which iteratively solve or circumvent eigenvalue problems in self-consistent field calculations based on the Kohn-Sham density-functional
theory. This is joint work with Jianfeng Lu and ELSI Team.
ELSI provides and enhances scalable, open-source software library solutions for electronic structure calculations in materials science, condensed matter physics, chemistry, molecular biochemistry,
and many other fields. ELSI focuses on methods that solve or circumvent the Kohn-Sham eigenvalue problem in density-functional theory. The ELSI infrastructure should also be useful for other
challenging eigenvalue problems. This is joint work with ELSI Team.
ButterflyNet [github] and ButterflyNet2 [github]
Butterfly-Net and Butterfly-Net2 are convolutional neural network structures with sparse channel connection. The architectures are inspired by Butterfly algorithms. Both Butterfly-Nets are especially
useful for signal processing and image processing tasks, where CNNs are widely used. The overall parameter complexity is $\mathcal{O}(K \log N)$ for $N$ and $K$ being the input and output sizes. Both
codes are implemented using Tensorflow 2 and accept json files for input configuration. This is joint work with Xiuyuan Cheng, Jianfeng Lu, and Zhongshu Xu.
Fast Butterfly Factorization, which is also known as the interpolative butterfly factorization (IBF), gives a data-sparse representation of matrices that satisfy the complementary low-rank property.
Given the explicit expression of the kernel, IBF factorizes the kernel matrix in $\mathcal{O}(N\log N)$ operations. And the final factorization admits $\mathcal{O}(N\log N)$ operations and memory
complexity with nearly optimal prefactor. This code supports the interpolative butterfly factorization of any dimensional problems with and without singularity. Several examples are provided in the
test folders. This is joint work with Haizhao Yang.
Butterfly Factorization (BF) gives a data-sparse representation of matrices that satisfy the complementary low-rank property. The factorization approximates such a kernel matrix of size $N\times N$
with a product of $\mathcal{O}(\log N)$ sparse matrices, each of which contains $\mathcal{O}(N)$ nonzero entries. Hence the application only requires $\mathcal{O}(N\log N)$ operations and memory.
This code supports the butterfly factorization of $d$ dimensional matrices for $d\leq 2$. Several examples are provided in the test folders. This is joint work with Haizhao Yang, Eileen Martin(one
dimensional code), Kenneth L. Ho(one dimensional code), and Lexing Ying.
Multiscale Butterfly Algorithm (MBA) is a code for the fast evaluation of Fourier Integral Operators (FIOs). Both 2D and 3D FIOs can be fast evaluated via this code. Several examples are provided in
the test folders. This is joint work with Haizhao Yang and Lexing Ying.
Distributed-Memory Hierarchical Matrices (DMHM) is a code for mpi based hierarchical matrix algebra. This code support both 2D and 3D $\mathcal{H}$-matrices. $\mathcal{H}$-matrix application,
composition, addition and inversion are completed. This is joint work with Jack Poulson and Lexing Ying.
这部分的程序和我科研有关, 但是和文章无关.
MuFiM is code for the MultiFrontal method for general sparse matrices in Matlab. This code currently supports the fast factorization of symmetric matrices, Hermitian matrices, and pattern symmetric
matrices. The complexity analysis of the algorithm is available if the sparse matrix is discretized from PDEs with a local numerical scheme. For two-dimensional problems, the factorization and
solving/application are of complexities $\mathcal{O}(N^{3/2})$ and $\mathcal{O}(N\log N)$ respectively. For three-dimensional problems, the factorization and solving/application are of complexities $
\mathcal{O}(N^2)$ and $\mathcal{O}(N^{4/3})$ respectively. The factorization phase of MuFiM is about 4 times slower than Matlab default "/" or "\" operation. While once the factors are available, the
solving/application is of lower complexity than Matlab default solving. In practice, the running time is also much faster.
Meshpart is a Matlab toolbox for several graph and mesh partitioning methods, including geometric, spectral, geometric spectral, and coordinate bisections. It also has routines to generate recursive
multiway partitions, vertex separators, and nested dissection orderings. It functions similar to MetisMex, but purely in Matlab. This is joint work with John R. Gilbert and Shang-Hua Teng.
Metis Mex is mex files for METIS. This code currently supports 64-bit Linux, Mac, and Windows. If any support for 32-bit computer is needed, please email me.
FastSpecialMat [github]
Fast Special Matrices is code for fast application of special matrices. This code currently supports the fast application of circulant matrices, Hankel matrices, Hankel circulant matrices, Toeplitz
matrices and Toeplitz symmetric matrices. The complexities of all these applications are $\mathcal{O}(N \log N)$ for $N$ by $N$ matrices. | {"url":"https://yingzhouli.com/cn/software.html","timestamp":"2024-11-12T02:19:46Z","content_type":"text/html","content_length":"14387","record_id":"<urn:uuid:3282a002-a43a-44b9-98a3-2bf1ab49e8bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00572.warc.gz"} |
Increasing ADC resolution using oversampling.
After a few post dedicated exclusively to the FPGA, on this post I going to talk about a technique that can help us to improve the ADC resolution. To study this technique, we will use as a example
the ADC which is based the Digilent’s ZMOD ADC, the AD9648.
First of all, we need to know what is exactly the resolution of the ADC. In general, if we want to know the resolution, we only have to check the number of bits of the ADC, and the divide the full
scale and the total of binary combinations. We can normalize the resolution by changing the full scale by 1. Being the resolution only a function of the number of bits.
This resolution, is the maximal theoretical resolution of the ADC converter itself, that is, the resolution that ADC can achieve with a clean signal. In the real world, clean signals are like a
unicorns, so we never can talk about clean signals in a real development. There are waste that will be added to the signal, and the ability of the ADC to discard that waste in favor of the real
signal we want to acquire, is what make an ADC have a better or worst resolution. This is easy to understand if we think in a 4 bit ADC, with a full scale of 1V. This ADC has 16 binary combinations
to translate a signal, that is, the ADC will increment one binary step every 0.0625 voltage steps at the input. If the ADC has an input white noise of 0.07 volts, we never could know the real value
of the LSB, because the level of noise is greater than the theoretical resolution, so the real resolution will be decreased in 1 bit. Notice that this is a simple explanation of the problem, but in
essence this is very close to the reality.
For this case, we will assume that the input noise on the ADC is less than LSB. Even in this case, we will have an error due to changes less than step value won’t be acquired, This error is named
quantification noise, and the ability of the ADC to discard this noise is the Signal-to-noise Ratio. The temporal shape of that error is a sawtooth, and it is explained here, and the equation that
describes this error is the next.
It is important to notice that this equation describes the signal-to-noise ratio between 0 and the Nyquist frequency, and here comes the magic. If we have an acquisition system with an interest
bandwidth that corresponds exactly to the Nyquist frequency, the shape of the noise in the frecuency domain looks like:
Our signal-to-noise ratio will be the worst as possible, because all the noise is located on our interest band. Even if we apply a digital filtering, we only will discard the half of the noise. To
improve that, we need to apply a technique named oversampling, that is increase the acquisition frequency, to reduce the noise on the interest bandwidth. The effect is shown on the next image.
As we can see, the noise is distributed along the new acquisition bandwidth, making the interest bandwidth cleaner. This is translated in a SNR increase. The new increased SNR looks like the next.
In this new equation, the sampling frequency is added in a new term named the process gain. Studying the equation, we can notice that an increase in the SNR of 6.02dB, is equivalent to increase the
number of bits in 1, due to the first term. So, if we make the process gain equal to 6.02 will be equivalent to increase 1 bit resolution. According the equation, if BW is equal to fs/2, the process
gain is 0, but if we multiply by 4 the fs, that is an oversampling ratio of 4, the process gain is equal to 6.02, and if we multiply the fs by 16, the resulting value is 12.04, so we can say that the
over sampling ratio (OSR) we have to apply is 4 raised to the number of resolution bits we want to increase.
\[OSR=2^{\delta nBits}\]
To test, I have used a Digilent Eclypse Z7, and the ZMOD ADC, that has a sampling frequency of 100MHZ, with 14 bits of theoretical resolution. I will create an square signal and test the algorithm on
the steady part of the signal. The module will perform a decimation, and every 4 samples at the input of 14 bits, module will compute 1 sample of 15 bits.
Module name: decimate_x4_v1_0
Author: P Trujillo ([email protected])
Date: Nov 2020
Description: Module to decimate by 4 increasing 1 bit resolution.
Revision: 1.0 Module created.
module decimate_x4_v1_0 #(
parameter pw_input_width = 14
input clk,
input rstn,
input signed [pw_input_width-1:0] i_data,
input i_data_valid,
output o_data_valid,
output reg signed [pw_input_width:0] or_data
reg signed [pw_input_width:0] rp_data_0;
reg signed [pw_input_width:0] rp_data_1;
reg signed [pw_input_width:0] rp_data_2;
reg signed [pw_input_width:0] rp_data_3;
reg [1:0] r2_data_counter;
always @(posedge clk)
if (!rstn) begin
rp_data_1 <= 0;
rp_data_2 <= 0;
rp_data_3 <= 0;
r2_data_counter <= 2'd0;
if (i_data_valid) begin
rp_data_0 <= {i_data[13], i_data};
rp_data_1 <= rp_data_0;
rp_data_2 <= rp_data_1;
rp_data_3 <= rp_data_2;
r2_data_counter <= r2_data_counter + 2'b1;
always @(posedge clk)
if (!rstn)
or_data <= 0;
if (&r2_data_counter)
or_data <= rp_data_0[14:1] + rp_data_1[14:1] + rp_data_2[14:1] + rp_data_3[14:1];
The block design looks like the next:
The result of this test can be checked on the next diagram, where the red dot represents the acquired signal, and the blue dots are corresponding with the resampled signal. We can see if, we scale
both with the same factor, the blue signal can has a resolution of 0.5, however, the red signal is moving over the integer values.
Now, the question is, may I acquire a dynamic signal with this method to improve my ADC resolution? The answer is it depends. This method is based on take 4 samples where before only 1 samples was
taken, and this has a limitation. The extra bit comes from the discarding the noise of the signal, but if is the interest signal the reason of the signal variation, this method is not valid, because
we would be discarding the signal of interest. Despite this, if the sampling frequency is higher than the interest signal variation, in other words, the variation if the signal in the sampling period
is low, we can use this method without problem.
Related posts | {"url":"https://www.controlpaths.com/2020/11/23/increasing-adc-resolution-using-oversampling/","timestamp":"2024-11-04T21:33:51Z","content_type":"text/html","content_length":"27876","record_id":"<urn:uuid:67d1b7d1-a289-416c-9ec8-f5b780d96af5>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00583.warc.gz"} |
KKT Conditions - (Linear Algebra for Data Science) - Vocab, Definition, Explanations | Fiveable
KKT Conditions
from class:
Linear Algebra for Data Science
KKT conditions, short for Karush-Kuhn-Tucker conditions, are a set of mathematical conditions that provide necessary and sufficient criteria for optimality in constrained optimization problems. These
conditions are crucial in identifying the points at which an objective function achieves maximum or minimum values while adhering to specific constraints. In data science, they help optimize models
and algorithms that rely on constraints, ensuring that solutions not only fit the data but also comply with real-world limitations.
congrats on reading the definition of KKT Conditions. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. KKT conditions encompass primal feasibility, dual feasibility, complementary slackness, and stationarity conditions, which together form a comprehensive framework for solving constrained
optimization problems.
2. These conditions apply to both equality and inequality constraints, making them versatile tools in various fields, including economics and engineering.
3. When KKT conditions are satisfied at a feasible point, it indicates that the point is a local optimal solution under the given constraints.
4. In convex optimization problems, if the KKT conditions hold, they guarantee global optimality of the solution.
5. Understanding KKT conditions is essential for implementing machine learning algorithms that require optimization under constraints, such as support vector machines and regression models.
Review Questions
• How do KKT conditions contribute to solving constrained optimization problems?
□ KKT conditions provide a structured approach to identifying optimal solutions in constrained optimization by ensuring that solutions meet both the objective function's requirements and any
imposed constraints. These conditions evaluate primal feasibility by checking if potential solutions satisfy constraint equations, dual feasibility by assessing Lagrange multipliers, and
complementary slackness to examine the relationship between active constraints and variable values. Understanding how these aspects interact helps in determining whether a solution is optimal
or not.
• Discuss the significance of KKT conditions in the context of convex optimization and their implications for global optimality.
□ In convex optimization problems, KKT conditions are particularly significant because they not only indicate local optimality but also ensure global optimality when satisfied. This means that
if the KKT conditions hold at a feasible point within a convex set, it guarantees that this point is indeed the best possible solution across all feasible points. This property simplifies
many optimization tasks in data science, as it allows practitioners to confidently rely on KKT conditions when designing models that optimize under constraints.
• Evaluate how knowledge of KKT conditions can enhance model building in data science, particularly in algorithms like support vector machines.
□ Knowledge of KKT conditions is crucial for enhancing model building in data science because it provides a foundation for understanding how to optimize complex algorithms like support vector
machines (SVMs). In SVMs, KKT conditions help determine the optimal separating hyperplane by identifying support vectors while considering margin constraints. By applying these conditions
during the training phase, practitioners can ensure that their models effectively balance accuracy and compliance with imposed limitations, ultimately leading to better-performing predictive
"KKT Conditions" also found in:
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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/linear-algebra-for-data-science/kkt-conditions","timestamp":"2024-11-12T03:55:34Z","content_type":"text/html","content_length":"151851","record_id":"<urn:uuid:21055941-ff45-4d4c-9a7e-831e18556c97>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00188.warc.gz"} |
Avient Corporation, with 2020 pro forma revenues of $3.8 billion, provides specialized and sustainable material solutions that transform customer challenges into opportunities, bringing new products
to life for a better ...
All financial data is based on trailing twelve months (TTM) periods - updated quarterly, unless otherwise specified. Data from | {"url":"https://fullratio.com/stocks/nyse-avnt/avient","timestamp":"2024-11-09T23:42:32Z","content_type":"text/html","content_length":"57164","record_id":"<urn:uuid:d93f9acc-913a-4a0c-90da-3e1ead918f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00057.warc.gz"} |
How to Play Poker | Poker Hand Probability Calculation
Have you every felt overwhelmed by the numbers in odds charts? When seeing the winning percentages, probabilities, and various other mathematical probabilities in poker games, have you ever wondered
where they come from?
In this article, we will discuss some common probability principles in Texas Hold'em poker games and how they are used.
Don't worry about the poker math. Everything we cover in this article can be easily understood, allowing you to better understand how to calculate a hand’s probability before the flop.
Starting from some basics
The two numbers we use most often in calculating hand probability are how many cards are in a deck and how many cards we want to be dealt in the game. For example, when we want to be dealt one
specific card:
Remember there are 52 cards in total in a deck:
• If we want to be dealt a 9♦，the probability would be 1/52, because there is one 9♦ in the whole deck.
• On the other hand, if we want to be dealt an Ace, regardless of its suit, the probability would be 4/52, because there are four Aces among the 52 cards.
• The same logic also applies to the situation where we only need one card from a certain playing card suit, diamond♦ for example. Then the probability of dealing any ♦ would be 13/52, because
there are 13 cards of the same suit.
To sum up, the probability of dealing any random card is 1/52, the probability of hitting any specific card such as Ace or King, Queen is 4/52, and the probability of dealing any suit is 13/52.
Now, let’s level up a bit
Suppose you are holding a 9♦, and you want to be dealt a pair. Now you are wondering what is the probability that you are dealt a 9 again.
• We’ve already known that the probability of being dealt a 9 is 4/52, which is the one you are holding now.
• And the probability that you are dealt another 9, regardless of its suit, would be 3/51.
Have you noticed that the numerator and the denominator of hitting the second card changed? When we got the first card, there were only 51 of 52 cards left. And when we hit the first 9, there are
only 3 other 9s left in the whole deck.
These numbers will keep changing as we move along in the game. Always pay attention to adjust the numbers based on the information you learn.
Tips for calculating probability
• When we use the word "and", we will use multiplication for the calculation.For example, we want to be dealt a King and a Queen.
• When we use the word "or", we would use addition for calculation.For example, we want to be dealt a 9♦ or a 9♣.
Moving on, we will talk about how to work them out in detail.
Probability of being dealt two specific cards
Suppose we want to know the probability of being dealt a King♦ and a Queen♣, (pay attention to the “and” we use here).
1. Probability of being dealt a King♦ first would be 1/52.
2. Probability of being dealt a Queen♣ would then be 1/51.
So the probability of hitting a specific poker hand, one king and one queen, is 1/2652.
P = (1/52)*(1/51)
P = 1/2652
So, the probability of being dealt any combination of hands in poker is 1/2652. However, this number represents that you get a King♦ first and then a Queen♣. But do you really care about their order?
Not really.
In other words, there are actually two ways to hit this hand ( get a King♦ first or get a Queen♣ first). Therefore, the probability that we are dealt this hand should be multiplied by 2.
P = 1/2652*2
P = 1/1326
To sum up, there are 1,326 possibilities for any starting hand we hit in the Texas Hold'em poker game.
Poker probability for a certain hand
What if you only need a specific hand and don’t care about their suits?
Again, we use multiplication.
Taking KQ as an example:
• The probability of hitting any King, regardless of its suit, would be 4/52.
• Probability of hitting any Queen, regardless of its suit, would be 4/51 (Because one King has already been dealt and there are only 51 cards remaining).
P = (4/52)*(4/51)
P = 16/2652 = 1/166
Same as before, what is calculated here is actually the probability of getting a King first and then a Queen. Of course, the probability is the same if you do it vice versa, so just multiply it by 2.
P = 16/2652 * 2
P = 32/2652
P = 1/83
As a result, the probability of being dealt any hand type in poker is 1/83, regardless of the poker hand ranking.
Poker probability for a certain range
What if you only need a KQ or a QJ? Pay attention to the”or” in the sentence.
What you need to do now is to calculate all the probabilities of hands you want and add them together
Since we have discussed before, when seeing “or”, we will use addition for calculation, therefore:
• The probability of being dealt KQ is 4/52 * 4/51 = 4/663
• The probability of being dealt QJ is 4/52 * 4/51 = 4/663
P = 4/663 + 4/663
P = 8/663
If you want to add more hand types into your range, simply add up their probabilities like we have done here. Pay attention when you want to calculate the probability of a pair, for example AA. The
probability of being dealt a pair is slightly different from KQ.
• The probability of being dealt AA is 4/52 * 3/51 = 1/221
It should be noted that these arithmetic problems are not for you to sort out when you sit at the poker table. They are the basis for you to review and organize your Texas Hold'em poker strategy in
your spare time. You need to know these principles in order to learn and practice more advanced strategies. Keep learning and gradually you will become more and more experienced and confident.
Poker Hand Probability Overview
What we covered today are some of the most common concepts and principles that will help you work out the probabilities of being dealt various hands before the flop. What’s more important is that you
try and use this knowledge in your games.
You may feel that this article does not actually affect your skills or style of play, but the underlying logic of skills and style of play is supported by these basic mathematical concepts.
I hope this article can serve as a stepping stone to more advanced study, helping you to more efficiently absorb and learn more advanced and complex poker theories.
Keep learning, keep practicing and start building up your own strategy. Otherwise these are merely abstract concepts which cannot help you win. | {"url":"https://www.slowplay.store/en-eu/blogs/news/poker-hand-probability","timestamp":"2024-11-05T07:48:01Z","content_type":"text/html","content_length":"180390","record_id":"<urn:uuid:d39e198b-55f9-4334-9dea-d21df75ef8f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00166.warc.gz"} |
Random neural networks – The Dan MacKinlay stable of variably-well-consider’d enterprises
Random neural networks
February 17, 2017 — October 12, 2021
dynamical systems
feature construction
machine learning
neural nets
probabilistic algorithms
stochastic processes
If you do not bother to train your neural net, what happens? In the infinite-width limit you get a Gaussian process. There are a number of net architectures that do not make use of that argument and
are still random.
1 Recurrent: Echo State Machines / Random reservoir networks
This sounds deliciously lazy; At a glance, it sounds like the process is to construct a random recurrent network, i.e. a network of random saturating IIR filters. Let the network converge to a steady
state for a given stimulus. These are the features to which you fit your classifier/regressor/etc.
Easy to implement, that. I wonder when it actually works, constraints on topology etc.
Some of the literature claims these are based on spiking (i.e. event-driven) models, but AFAICT this is not necessary, although it might be convenient for convergence.
Various claims are made about how hard they avoid the training difficulty of similarly basic RNNs by being essentially untrained; you use them as a feature factory for another supervised output
Suggestive parallel with random projections. Not strictly recurrent, but same general idea: He, Wang, and Hopcroft (2016).
Lukoševičius and Jaeger (2009) mapped out various types as at 2009:
From a dynamical systems perspective, there are two main classes of RNNs. Models from the first class are characterised by an energy-minimising stochastic dynamics and symmetric connections. The
best known instantiations are Hopfield networks, Boltzmann machines, and the recently emerging Deep Belief Networks. These networks are mostly trained in some unsupervised learning scheme.
Typical targeted network functionalities in this field are associative memories, data compression, the unsupervised modelling of data distributions, and static pattern classification, where the
model is run for multiple time steps per single input instance to reach some type of convergence or equilibrium (but see e.g., Taylor, Hinton, and Roweis (2006) for extension to temporal data).
The mathematical background is rooted in statistical physics. In contrast, the second big class of RNN models typically features a deterministic update dynamics and directed connections. Systems
from this class implement nonlinear filters, which transform an input time series into an output time series. The mathematical background is nonlinear dynamical systems. The standard training
mode is supervised. | {"url":"https://danmackinlay.name/notebook/nn_random.html","timestamp":"2024-11-10T22:15:37Z","content_type":"application/xhtml+xml","content_length":"41829","record_id":"<urn:uuid:a9cd6adb-234c-47b2-b6fc-2bb666877b4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00199.warc.gz"} |
75,513 research outputs found
We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact $p$-adic Lie group is finitely generated. In particular, the category of coadmissible
modules does not have enough projectives. In the Appendix a "generalized Robba ring" for uniform pro-$p$ groups is constructed which naturally contains the locally analytic distribution algebra as a
subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a self-dual locally
convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.Comment: with an appendix by Peter Schneider; revised; new titl
By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible
filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a
type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and
only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic
representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm
implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified
Langlands functoriality and therefore generalizes to arbitrary split reductive groups
We present mass reconstructions from weak lensing for the galaxy clusters A1835 and A2204 over 34'x34' fields using data from the ESO/MPG Wide Field Imager. Using a background galaxy population of 22
<R<25.5 we detect the gravitational shear of A1835 at 8.8 sigma significance, and obtain best-fit mass profiles of sigma_v=1233^{+66}_{-70} km/s for a singular isothermal sphere model and r_{200}=
1550 h^{-1} kpc, c=2.96 for a `universal' CDM profile. Using a color-selected background galaxy population of 22<R<25.8 we detect the gravitational shear of A2204 at 7.2 sigma significance, and
obtain best-fit mass profiles of sigma_v=1035^{+65}_{-71} km/s for a SIS model and r_{200}=1310 h^{-1} km/s, c=6.3 for a `universal' CDM profile. The gravitational shear at distances greater than 10'
is significantly detected for both clusters. The best fit weak lensing cluster masses agree well with both X-ray and dynamical mass measurements, although the central concentration of A1835 is much
lower in the weak lensing mass profile than that measured by recent Chandra results. We suggest that this lower concentration is most likely a combination of contamination of the 'background' galaxy
population with cluster dwarf galaxies and the effect of a prolate or tri-axial cluster core with the major axis lying near the plane of the sky. We also detect a number of additional structures at
moderate significance, some of which appear to be sub-haloes associated with the clusters.Comment: accepted to A&A, 14 pages, 13 figures, version with higher quality images can be found at http://
We analyze the data for the pressure and boron isotope effect on the temperature dependence of the magnetization near $T_{c}$. Invoking the universal scaling relation for the magnetization at fixed
magnetic field it is shown that the relative shift of $T_{c}$, induced by pressure or boron isotope exchange, mirrors essentially that of the anisotropy. This uncovers a novel generic property of
anisotropic type II superconductors, inexistent in the isotropic case. For MgB$_{2}$ it implies that the renormalization of the Fermi surface topology due to pressure or isotope exchange is dominated
by a mechanism controlling the anisotropy.Comment: 7 pages, 3 figure
We analyze the magnetization, magnetic torque and susceptibility data of La2-xSrxCu(16,18)O4 and YBa2(63,65)CuO7-x near Tc in terms of the universal 3D-XY scaling relations. It is shown that the
isotope effect on Tc mirrors that on the anisotropy. Invoking the generic behavior of the anisotropy the doping dependence of the isotope effects on the critical properties, including Tc, correlation
lengths and magnetic penetration depths are traced back to a change of the mobile carrier concentration.Comment: 5 pages, 3 figure
Since Ludwieg tubes have been around for many years, and NASA has already established the feasibility of creating quiet-flow wind tunnels, the major question addressed was the cost of the proposed
facility. Cost estimates were obtained for major system components, and new designs which allowed fabrication at lower cost were developed. A large fraction of the facility cost comes from the
fabrication of the highly polished quiet-flow supersonic nozzle. Methods for the design of this nozzle were studied at length in an attempt to find an effective but less expensive design. Progress
was sufficient to show that a quality facility can be fabricated at a reasonable cost | {"url":"https://core.ac.uk/search/?q=authors%3A(Schneider%20P.)","timestamp":"2024-11-02T03:24:44Z","content_type":"text/html","content_length":"123586","record_id":"<urn:uuid:29f34143-5019-49d0-9c02-3a7188cda39b>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00330.warc.gz"} |
Standard Equation Of A Circle Formula at Sarah Cordero blog
Standard Equation Of A Circle Formula. This equation can be used for a circle. The standard equation for a circle contains pertinent information about the circle's center and radius. standard
equation of a circle. standard equation of a circle. the standard equation of a circle is given by: How to express the standard form equation of a circle of a given radius. standard equation of a
circle. Where (h,k) is the coordinates of center of the circle and r is. equation of a cirle. how do you write the standard form of equation of a circle? Practice problems with worked out solutions,
pictures and. learn how to write and graph the standard equation of a circle using the center point coordinates and the radius.
from www.tessshebaylo.com
standard equation of a circle. Practice problems with worked out solutions, pictures and. the standard equation of a circle is given by: This equation can be used for a circle. The standard equation
for a circle contains pertinent information about the circle's center and radius. How to express the standard form equation of a circle of a given radius. learn how to write and graph the standard
equation of a circle using the center point coordinates and the radius. standard equation of a circle. how do you write the standard form of equation of a circle? equation of a cirle.
Standard Equation Of A Circle Tessshebaylo
Standard Equation Of A Circle Formula equation of a cirle. How to express the standard form equation of a circle of a given radius. standard equation of a circle. learn how to write and graph the
standard equation of a circle using the center point coordinates and the radius. standard equation of a circle. the standard equation of a circle is given by: how do you write the standard form of
equation of a circle? standard equation of a circle. This equation can be used for a circle. Practice problems with worked out solutions, pictures and. equation of a cirle. Where (h,k) is the
coordinates of center of the circle and r is. The standard equation for a circle contains pertinent information about the circle's center and radius. | {"url":"https://ceogtxbl.blob.core.windows.net/standard-equation-of-a-circle-formula.html","timestamp":"2024-11-14T05:15:50Z","content_type":"text/html","content_length":"35661","record_id":"<urn:uuid:a0de6c80-fde8-41c5-98c5-ee2a24dca54e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00556.warc.gz"} |
Rising Factorial
#include <boost/math/special_functions/factorials.hpp>
namespace boost{ namespace math{
template <class T>
calculated-result-type rising_factorial(T x, int i);
template <class T, class Policy>
calculated-result-type rising_factorial(T x, int i, const Policy&);
}} // namespaces
Returns the rising factorial of x and i:
rising_factorial(x, i) = Γ(x + i) / Γ(x)
rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i-1)
Note that both x and i can be negative as well as positive.
The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more
May return the result of overflow_error if the result is too large to represent in type T.
The return type of these functions is computed using the result type calculation rules: the type of the result is double if T is an integer type, otherwise the type of the result is T.
The accuracy will be the same as the tgamma_delta_ratio function.
The spot tests for the rising factorials use data generated by functions.wolfram.com.
Rising and factorials are implemented as ratios of gamma functions using tgamma_delta_ratio. Optimisations for small integer arguments are handled internally by that function. | {"url":"https://live.boost.org/doc/libs/1_83_0/libs/math/doc/html/math_toolkit/factorials/sf_rising_factorial.html","timestamp":"2024-11-11T01:45:14Z","content_type":"text/html","content_length":"10894","record_id":"<urn:uuid:5f05df48-6d34-4e1d-8666-4d9004a02d1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00246.warc.gz"} |
Introduction to Population Growth - Fluctuation, Growth Models, and Pyramids
Population growth is one of the main concerns of this world because the human population isn't a static factor. Rather, it's growing at a really alarming rate. In spite of the increasing world
population, the resources of the world remain constant. Thus, the power to take care of sustainable development is becoming a serious challenge to mankind today.
Human increase is the increase in the number of people in a particular area. There has been a decrease in the death rate over the past 200 years due to the changes in public health and sanitation.
The advent of antibiotics and vaccines has led to a decrease in the chances of infections in humans. Urbanization and advancements in agriculture have also led to a rise in population.
Factors that Influence Population Fluctuation
The fluctuations within the population during a given area are influenced by four major factors, which include the following:
Natality – it's the number of births during a given period of your time during a population
Mortality – It is defined as the number of deaths that takes place in a population at a given period of time.
Immigration – it's defined to be the number of people who come from another population and increase the population in consideration during a period of time.
Emigration – it's defined as the number of people from a population who leave the habitat and attend a special habitat at a given period of time.
Thus, it's clearly visible that the two factors Natality (N) and Immigration (I) increase a population, thus increasing population whereas, Mortality (M) and Emigration (E) decrease the population.
The population density (Pt) at a given point of time is often given as:
Pt =P0 + (N + I) – (M + E)
Where P0 is the initial population density.
We Have Two Growth Models Which Describe the Essential Growth Trend During a Population. These Are
Exponential Growth
In a perfect condition where there's a vast supply of food and resources, the increase will follow an exponential order. Consider a population of size N and birth rate represented as b, death rate as
d, Rate of change of N are often given by the equation.
dN/dt = (b-d) x N
If, (b–d) = r,
dN/dt = rN
Where r = intrinsic rate of natural increase
This equation is often represented with a graph that features a J shaped curve. According to calculus
Where, Nt = Population density at time t
N0= Population density at time zero
r = intrinsic rate of natural increase
e = base of natural logarithms
Logistic Growth
This model defines the concept of ‘survival of the fittest’. Thus, it considers the very fact that resources in nature are exhaustible.
The limit of resources beyond which it cannot support any number of organisms can be defined as the carrying capacity. The carrying capacity can be represented as K.
The availability of limited resources cannot show exponential growth. As a result to which the graph will have a lag phase, followed by an exponential phase, then a declining phase and ultimately an
asymptote. This is referred to as Verhulst-Pearl Logistic Growth and is represented using the equation:
dN/dt = rN((K-N) /K) | {"url":"https://www.vedantu.com/biology/introduction-to-population-growth?utm_source=Topicpages","timestamp":"2024-11-10T04:21:34Z","content_type":"text/html","content_length":"241570","record_id":"<urn:uuid:f0a85029-bc37-4b0a-badc-ec1aebdd9566>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00469.warc.gz"} |
Program for Monday, April 4th
PROGRAM FOR MONDAY, APRIL 4TH
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09:00-11:05 Session 3A: Krylov Methods
GPMR: An Iterative Method for Unsymmetric Partitioned Linear Systems
09:00 ABSTRACT. We introduce an iterative method named GPMR for solving 2x2 block unsymmetric linear systems. GPMR is based on a new process that simultaneously reduces two rectangular matrices to
upper Hessenberg form and is closely related to the block-Arnoldi process. We compare the performance of GPMR with GMRES on linear systems from the SuiteSparse Matrix Collection. In our
experiments, GPMR terminates significantly earlier than GMRES on a residual-based stopping condition with an improvement ranging from around 10% up to 50% in terms of number of iterations.
Convergence of GMRES with respect to the right-hand-side vector
Shikhar Shah
ABSTRACT. The generalized minimum residual method (GMRES) has a rich body of convergence theory related to the spectrum and normality of the coefficient matrix. In preconditioning, GMRES can be
used to perform linear solves with the preconditioning matrices (e.g., the incomplete LU factors). Here, the GMRES polynomial must be fixed between preconditioning iterations. In this case, the
convergence of GMRES can have strong dependence on the right-hand-side vector. Specifically, the locations of the roots of the GMRES polynomial with respect to the pseudospectrum of the
preconditioning matrix affects the quality of the linear solve. We characterize this dependence and employ it to choose a robust preconditioner.
Linear Asymptotic Convergence Analysis of Anderson Acceleration, with Krylov Formulation in the Linear Case
ABSTRACT. We consider Anderson acceleration (AA) with a moving window of size m, and investigate the linear asymptotic convergence behaviour of AA(m) applied to linear and nonlinear fixed-point
methods. Anderson acceleration has been shown empirically to be very effective for accelerating nonlinear solvers and optimization methods, with broad applications in computational science,
data analysis and machine learning, but there is no theory to explain and quantify the asymptotic convergence improvement that is observed in practice.
We first observe numerically that the root-linear convergence factor of sequences generated by AA(m) strongly depends on the initial condition, and that the acceleration coefficients oscillate
while the approximation converges to the fixed point. To shed light on this behaviour, we write AA(m) itself as an augmented fixed-point method and establish that the iteration function of the
AA(m) fixed-point iteration is not differentiable at the fixed point (but the directional derivatives exist). This allows the root-linear convergence factor to be strongly dependent on the
initial guess. We also find that the acceleration coefficient function is not continuous, thus allowing the coefficients to oscillate while the approximation converges.
09:50 To further investigate AA(m) convergence, we consider the case of accelerating linear fixed-point methods and write AA(m) with the usual recursive initial guess process as a Krylov space
method. We obtain polynomial residual update formulas for AA(m) and derive an (m+2)-term recurrence relation for the AA(m) polynomials. A direct consequence is that k steps of AA(m) cannot
produce a residual that is smaller in the 2-norm than the residual obtained by GMRES(k). AA(m) with m arbitrary initial guesses is a multi-Krylov space method. The recurrence relations also
reveal that AA(m) possesses a memory effect: as a result of the windowing in AA(m), the damping effect provided by fixed-point iterations accumulates multiplicatively (contrary to the case of
restarted GMRES(m)), and is combined with polynomial acceleration to lead to effective convergence. We further derive orthogonality relations for the AA(m) residuals, and for AA(1) we find a
lower bound for the acceleration coefficient. Results are also obtained for the influence of the initial guess on the convergence speed of AA(1).
While these results reveal many interesting findings about asymptotic convergence of AA(m), the main question to quantify the asymptotic convergence acceleration provided by AA(m) remains an
open problem. Specifically, we now know numerically that the root-linear convergence factor of sequences generated by AA(m) strongly depends on the initial condition and for many problems the
worst-case convergence factor over all initial guesses is much smaller than the convergence factor of the initial fixed-point method, but we don't know how to compute this worst-case
convergence factor.
GCR equivalent multiple right-hand side GMRES
ABSTRACT. In this paper we present a GMRES-like iterative Krylov subspace method for solving a system of linear equations with a non-symmetric matrix and multiple right-hand sides. It minimizes
10:15 residual norms and in exact arithmetic is equivalent to a GCR method. Unlike GCR, the new method requires only one basis storage. Also, as it does not use a blockwise matrix-vector product, it
can be applied to sequentially accessed right-hand sides. We provide a detailed algorithm description and its template implementation in C++ with MPI and OpenMP parallelism. Several numerical
experiments and a comparison with a GCR implementation are presented as well.
Post-Modern PM-GMRES
ABSTRACT. The GMRES algorithm of Saad and Schultz (1986) for nonsymmetric linear systems relies on the Arnoldi expansion for the Krylov basis. The algorithm computes the $QR$ factorization of
the matrix $B = [\: \vect{r}_0, AV_m\:]$. Despite an ${\cal O}(\eps)\kappa(B)$ loss of orthogonality, the modified Gram-Schmidt (MGS) formulation was shown to be backward stable in the seminal
papers by Paige, et al. (2006) and Paige and Strako\v{s} (2002). Classical Gram-Schmidt (CGS) exhibits an ${\cal O}(\eps)\kappa^2(B)$ loss of orthogonality, whereas DCGS-2 (CGS with delayed
10:40 reorthogonalization) reduces this to ${\cal O}(\eps)$ in practice (without a formal proof). We present a post-modern (viz not classical) GMRES algorithm based on Ruhe (1983) and the low-synch
algorithms of Swirydowicz et al (2020) that achieves ${\cal O}(\eps) \|Av_k \|_2 / h_{k+1,k}$ loss of orthogonality. By projecting the vector $Av_m$ with Gauss-Seidel onto the orthogonal
complement of the space spanned by the computed Krylov vectors $\bv_m$ where $\bv_m^T\bv_m = I + L_m + L_m^T$, we can further demonstrate that the loss of orthogonality closely follows ${\cal
O}(\eps)$. For a broad class of matrices, unlike MGS-GMRES, significant loss of orthogonality does not occur and the relative residual no longer stagnates for highly non-normal systems. The
Krylov vectors remain linearly independent and the smallest singular value of $\tv_m$ is close to one. We also demonstrate that Henrici's departure from normality of the lower triangular matrix
$T_m \approx (\:V_m^TV_m\:)^{-1}$ in the Gram-Schmidt projector $P = I - V_mT_mV_m^T$ is an appropriate quantity for detecting the loss of orthogonality.
09:00-11:05 Session 3B: Scalable Solvers for Coupled Multiphysics 1
An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD
Qi Tang
ABSTRACT. The magnetohydrodynamics (MHD) equations are continuum models used in the study of a wide range of plasma physics systems, including the evolution of complex plasma dynamics in
tokamak disruptions. However, efficient numerical solution methods for MHD are extremely challenging due to disparate time and length scales, strong hyperbolic phenomena, and nonlinearity.
09:00 Therefore the development of scalable, implicit MHD algorithms and high-resolution adaptive mesh refinement strategies is of considerable importance. In this work, we develop a high-order
stabilized finite-element algorithm for the reduced visco-resistive MHD equations based on the MFEM finite element library (mfem.org). The scheme is fully implicit, solved with the
Jacobian-free Newton-Krylov (JFNK) method with a physics-based preconditioning strategy. Our preconditioning strategy is a generalization of the physics-based preconditioning methods in [Chac ́
on, et al, JCP 2002] to adaptive, stabilized finite elements. Algebraic multigrid methods are used to invert sub-block operators to achieve scalability. A parallel adaptive mesh refinement
scheme with dynamic load-balancing is implemented to efficiently resolve the multi-scale spatial features of the system. Our implementation uses the MFEM framework, which provides
arbitrary-order polynomials and flexible adaptive conforming and non- conforming meshes capabilities. Results demonstrate the accuracy, efficiency, and scalability of the implicit scheme in the
presence of large scale disparity. The potential of the AMR approach is demonstrated on an island coalescence problem in the high Lundquist-number regime (>=1e7) with the successful resolution
of plasmoid instabilities and thin current sheets.
Augmented Lagrangian preconditioners for incompressible resistive magnetohydrodynamics
ABSTRACT. The equations of magnetohydrodynamics are generally known to be difficult to solve numerically. They are highly nonlinear and exhibit strong coupling between the electromagnetic and
hydrodynamic variables, especially for high Reynolds and coupling numbers.
In this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretization of the single-fluid incompressible viscoresistive MHD equations. For stationary
09:25 problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. We extend our method to fully implicit
methods for time-dependent problems which we solve robustly in both two and three dimensions. Our approach relies on specialized parameter-robust multigrid methods for the hydrodynamic and
electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances.
We confirm the robustness of our solver by numerical experiments in which we consider fluid and magnetic Reynolds numbers and coupling numbers up to 10,000 for stationary problems and up to
100,000 for transient problems in two and three dimensions.
Block Preconditioning and a Monolithic AMG Method for Magnetic Confinement Fusion Relevant Resistive MHD Simulations
Peter Ohm
ABSTRACT. The mathematical basis for the continuum fluid modeling of resistive magnetohydrodynamic of multifluid plasma physics systems is the solution of the governing partial differential
equations (PDEs) describing conservation of mass, momentum, and thermal energy, along with various reduced forms of Maxwell’s equations for the electromagnetic fields. The resulting systems are
characterized by strong nonlinear and nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that these interactions produce.
These characteristics make scalable and efficient iterative solution, of the resulting poorly-conditioned discrete systems, extremely difficult. In this talk we consider the use of both block
09:50 preconditioners and an algebraic monolithic multigrid approach for solving the coupled physics block systems.
Monolithic multigrid methods can also benefit from this block linear structure. In this context we present a framework for the construction of an algebraic monolithic multigrid utilizing the
natural block linear structure arising from coupled multiphysics problems. Multigrid components are constructed first on matrix subblocks corresponding to individual physics. The resulting AMG
sub-components are then composed together to define a monolithic AMG preconditioner. We demonstrate this approach through an implementation in MueLu for various Resistive MHD problems that that
are relevant to magnetic confinement fusion applications and compare the performance with alternative preconditioning methods.
Time permitting, we will discuss a block preconditioner based on an approximate operator splitting that factors the system into block sub-systems, with approximate Schur complements that
explicitly encode critical coupling into the sub-systems. In the case of MHD this is the Alfven wave physics. Early results indicate a better scaling of iterations for the linear solves with
longer time steps compared to the Alfven time-scale.
Beyond radiation-hydrodynamics: Coupling kinetic plasma and radiation transport
Hans Hammer
ABSTRACT. Radiation-hydrodynamics (rad-hydro) is insufficient to describe accurately weakly collisional environments in high-energy-density physics (HEDP) experiments. These environments are
common, for instance, in the hohlraum in inertial confinement fusion (ICF) indirectly-driven experiments, which is a key component of the energy delivery system to the capsule that controls,
for instance, implosion symmetry. For high-fidelity, a kinetic description of the plasma is necessary, which additionally must be coupled with a (in principle, also kinetic) radiation model.
We consider a hybrid plasma description [1], comprised of kinetic ions and fluid electrons, to be coupled with thermal radiative transfer fully implicitly and nonlinearly. Tight coupling is
orchestrated via a low-order (LO) moment description for all involved physics: ions, electrons, and radiation. This LO system (which is largely equivalent to a multifluid rad-hydro system) is
informed by the high-order (HO) kinetic descriptions via so-called consistency terms, which close the moment equations and enforce consistency with the fully kinetic solution. This HOLO
approach is algorithmically efficient [2] because the stiff nonlinear coupling is addressed in the (computationally cheaper) LO solver, while allowing the computationally expensive HO solvers
10:15 to decouple from one another.
We present here results from the coupling of the hybrid-kinetic plasma model with a gray radiation-diffusion model (i.e., neglecting kinetic radiation physics) [3]. Our implementation features
a novel mesh-motion strategy for the coupled plasma-radiation system [4] that is able to effectively adapt to solution features dynamically in time. We demonstrate that our implementation is
able to deal with strong radiative shocks (Mach 45), delivering excellent agreement with a self-similar rad-hydro shock test problem. We will characterize the efficiency of the solver, as well
as convergence of the solution with timestep and mesh refinement.
[1] W. T. Taitano et al., Comput. Phys. Comm., 263, 107861 (2021)
[2] L. Chacón et al. J. Comput. Phys., 330, 21-25 (2017)
[3] Hans Hammer et al., Proceedings of the ANS Mathematics & Computation (M&C) 2021, Raleigh, North Carolina, October 3–7, 2021, pp. 1153-1162
[4] Hans Hammer et al., Transactions of the American Nuclear Society. ANS Winter Meeting. Washington, D.C.: ANS, Nov. 17–21, 2019.
Monolithic Multigrid Methods for Higher-Order Discretizations of Time-Dependent Maxwell’s Equations
ABSTRACT. Maxwell’s equations arise in many challenging applications in computational science. Several paths exist to achieve high-order discretizations, including the use of specialized
10:40 finite-element basis (such as Raviart-Thomas, Brezzi-Douglas-Marini, and both first- and second-kind Nédélec elements) and high-order implicit temporal integration schemes. While significant
effort has been invested in the past 25 years in developing efficient multigrid methods for various spatial discretizations, much of the work in the time-dependent case has been focused on
multi-step (or diagonally implicit) temporal discretizations. In this talk, we present recent work on extending monolithic multigrid methods to fully implicit Runge-Kutta temporal
discretizations. Particular focus is paid to extending the common overlapping Schwarz relaxation strategy to these discretizations, as well as their use on non-nested multigrid hierarchies, as
needed to accurately model complex geometries.
09:00-11:05 Session 3C: Topics in Optimization, Inversion
From challenges to edge-preserving methods for large-scale dynamic inverse problems
Mirjeta Pasha
09:00 ABSTRACT. Inverse problems are ubiquitous in many scientific fields such as engineering, biology, medical imaging, atmospheric science, and geophysics. Three emerging challenges on obtaining
meaningful solutions to large-scale and data-intensive inverse problems are ill-posedness of the problem, large dimensionality of the parameters, and the complexity of the model constraints. In
this talk we discuss efficient methods for computing solutions to dynamic inverse problems, where both the quantities of interest and the forward operator (measurement process) may change at
different time instances. We consider large-scale ill-posed problems that are made more challenging by their dynamic nature and, possibly, by the limited amount of available data per
measurement step. To remedy these difficulties, we apply efficient regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each time
instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy. Numerical examples from a wide range of applications, such as
limited-angle computerized tomography (CT), space-time image deblurring, and photoacoustic tomography (PAT), will be used to illustrate the effectiveness of the described approaches.
An ℓp Variable Projection Method for Large-Scale Separable Nonlinear Inverse Problems
ABSTRACT. Variable projection methods are among the classical and efficient methods to solve separable nonlinear least squares problems such as blind deconvolution, system identification, and
machine learning. In this talk, we present a modified variable projection method for large-scale separable nonlinear inverse problems, that promotes edge-preserving and sparsity properties on
09:25 the desired solution, and enhances the convergence of the parameters that define the forward problem. Specifically, we adopt a majorization minimization method that relies on constructing
quadratic tangent majorants to approximate an ℓp regularization term, by a sequence of ℓ2 problems that can be solved by the aid of generalized Krylov subspace methods at a relatively low cost
compared to the original unprojected problem. In addition, more potential generalized regularizers including total variation (TV), framelet, and wavelet operators can be used, and the
regularization parameter can be defined automatically at each iteration with the aid of generalized cross validation. Numerical examples on large-scale two-dimensional imaging problems arising
from blind deconvolution are used to highlight the performance of the proposed method in both quality of the reconstructed image as well as the reconstructed forward operator.
Bayesian Level Set Approach for Inverse Problems with Piecewise Constant Reconstructions
William Reese
09:50 ABSTRACT. There are several challenges associated with inverse problems in which we seek to reconstruct a piecewise constant field, and which we model using multiple level sets. Adopting a
Bayesian viewpoint, we impose prior distributions on both the level set functions that determine the piecewise constant regions as well as the parameters that determine their magnitudes. We
develop a Gauss-Newton approach with a backtracking line search to efficiently compute the maximum a priori (MAP) estimate as a solution to the inverse problem. We use the Gauss-Newton Laplace
approximation to construct a Gaussian approximation of the posterior distribution and use preconditioned Krylov subspace methods to sample from the resulting approximation. To visualize the
uncertainty associated with the parameter reconstructions we compute the approximate posterior variance using a matrix-free Monte Carlo diagonal estimator, which we develop in this paper. We
will demonstrate the benefits of our approach and solvers on synthetic test problems (photoacoustic and hydraulic tomography, respectively a linear and nonlinear inverse problem) as well as an
application to X-ray imaging with real data.
Hybrid Projection Methods for Solution Decomposition in Large-scale Bayesian Inverse Problems
Jiahua Jiang
ABSTRACT. In this work, we develop hybrid projection methods for computing solutions to large-scale inverse problems, where the solution represents a sum of different stochastic components.
10:15 Such scenarios arise in many imaging applications (e.g., anomaly detection in atmospheric emissions tomography) where the reconstructed image can be represented as a combination of two or more
images and each image contains different smoothness or stochastic properties. In an inversion or inverse modeling framework, these assumptions correspond to different regularization terms for
each image in the sum. Although various prior assumptions can be included in our framework, we focus on the scenario where the solution is a sum of a sparse image and a smooth image; thus, we
require $\ell_1$ and $\ell_2$ regularization for separate components, respectively. For computing solution estimates, we develop hybrid projection methods for solution decomposition that are
based on a combined flexible and generalized Golub-Kahan process. This approach integrates techniques from the generalized Golub-Kahan bidiagonalization and the flexible Krylov methods. The
benefits of the proposed methods are that the decomposition of the solution can be done iteratively, and the regularization terms and regularization parameters are adaptively chosen at each
iteration. Numerical examples from image processing demonstrate the potential for these methods to be used for anomaly detection.
Minimisation of || b-Ax||_max ||b- Ax||_1 with Krylov-Simplex and column generation
Wim Vanroose
ABSTRACT. We minimise the residual, r=b−Ax, over a subspace in ||r||_\max, i.e. maximum of the absolute residuals, and the ||r||_1-norm, i.e least absolute residuals. Optimised over a Krylov
subspace this leads to a small linear programming problem that can be solved by a specialised simplex algorithm that finds the optimal linear combination of Krylov basis vectors to approximate
the solution. The resulting simplex algorithm requires the solution of a series of small dense linear systems that only differ by rank-one updates. We compare the method with a column
generation approach where the basis vectors of the subspace are selected based on the steepest descent direction that is orthogonal to the current subpace. We illustrate the methods with
applications from inverse problems.
11:25-13:30 Session 4A: Asynchronous Methods
On Iterative Sparse Triangular Solves
ABSTRACT. Sparse triangular solves are a key kernel in numerical linear algebra, especially for preconditioning with incomplete factorizations. The traditional sequential algorithm is not
11:25 suitable for parallel computing. Recently, several authors have suggested iterative solution methods, which are more suitable for GPUs and similar architectures. Unfortunately, these methods
are not very robust and may not converge. We discuss blocking and scaling strategies that improve robustness. We further discuss reordering schemes that are useful when solving a sequence of
systems. Our work applies to both the synchronous and the asynchronous case.
Asynchronous Jacobi Methods with Resilience to Data Corruption and Robustness to Data Delay
ABSTRACT. Over the past decade, the proliferation of smart devices ranging from residential smart thermostats to industrial smart power grid meters has motivated research into migrating
scientific computing from high-performance computing (HPC) and cloud computing (CC) to these devices. Computing on these smart devices, also referred to as edge devices, allows for computation
on data in-situ, avoiding the need to aggregate that data to HPC and/or CC facilities for processing. One bottleneck of this migration is that the same fault tolerance approaches that provide
reliable results on HPC and CC systems do not necessarily apply to edge computing environments. As an example, consider checkpoint and restart functionality is a standard approach for
mitigating the disruption of an HPC or CC node failing. Such an approach is not feasible for many edge computing environments where data storage is more limited and synchronization more costly
than with the massive file systems and high-speed interconnects in HPC and CC systems. Thus, new fault tolerance approaches are needed to reliably perform standard scientific computing tasks on
the edge.
This work leverages recently developed approaches in the literature to formulate asynchronous Jacobi (ASJ) methods that can solve linear systems in the presence of data corruption or
communication delay. Motivated by the ideas of the alternating direction method of multipliers to enable rejection of corrupt information, this work derives a rejection criterion for the ASJ
methods from convergence theory. The resulting ASJ method is shown to restore convergence that is lost in the original ASJ method when data corruption is introduced. Motivated by work on
average consensus reformulated to use evolution information, this work derives a reformulated ASJ method to incorporate the push-sum approach. The resulting ASJ method is shown to restore the
convergence rate that is degraded in the original ASJ method when data delay is introduced. The results of this work serve as a proof of concept for similar resilience and robustness
improvements in more powerful solvers, such as the conjugate gradient method.
Funded by LLNL LDRD projects 21-FS-007 and 22-ERD-045. Prepared by LLNL under Contract DE-AC52-07NA27344. LLNL-ABS-831426.
Asynchronous Chebyshev Methods
ABSTRACT. Iterative methods typically contain many synchronization points which can scale poorly on massively parallel computers. Additionally, these synchronization costs can be further
amplified when some cores are delayed, e.g., when faults occur or load imbalances are present. While asynchronous methods have recently gained interest, asynchronous versions of certain
12:15 state-of-the-art iterative solvers have yet to be developed. We present the first asynchronous Chebyshev method which uses Chebyshev to accelerate an asynchronous version of the BPX multigrid
method. Our initial experiments show that, as the problem scales up, using the Jacobi preconditioner within asynchronous Chebyshev can result in divergence. This indicates the need for a
preconditioner that provides grid-size independent convergence which is why we use a multigrid preconditioner. We present experimental results from an OpenMP implementation of asynchronous
The Algorithmic Development of a Fully Asynchronous Conjugate Gradient Method
ABSTRACT. Decentralized computing environments (DCE), or environments, such as edge computing, which utilize spatially distributed, non-hierarchical, and potentially heterogeneous computational
units, present additional challenges such as communication delays and data locality which require asynchronous and resilient iterative methods to be practical. Such algorithms must be capable
of progressing toward a solution even if one or more computational units is experiencing slowdowns, either due to poor communication connections or under-performing hardware. Previous work has
shown that these challenges can be overcome for stationary iterative methods, such as asynchronous Jacobi (ASJ). In high-performance computing (HPC) settings, Krylov subspace iterative solvers,
such as the conjugate gradient method (CG), are preferred for solving linear systems with symmetric positive-definite matrices due to their strong convergence guarantees. Despite requiring
12:40 synchronous communication across all processors at every iteration, CG performs exceptionally well in shared memory and HPC environments. In DCE, the communication costs of synchronization may
be considerably higher, as delays from even one computational unit prevent any other units from continuing. Further, the connection speeds between devices in edge computing will be considerable
slower than the high-speed interconnects found in the HPC setting. We have developed an asynchronous CG (ACG) algorithm which removes these synchronization points by utilizing the partial
direction vectors received at each iteration to form the next, mutually orthogonal search direction. Initial numerical results demonstrate the number of iterations required for convergence
scales with the square root of the condition number of the matrix, as in the traditional CG algorithm. Further, we will present numerical results which compare ACG to CG and ASJ in both
reliable computing environments and with injected delays and corruption to demonstrate the algorithmic performance of each for different computing environments.
Prepared by LLNL under Contract DE-AC52-07NA27344. Funded by LLNL LDRD project 22-ERD-045. LLNL-ABS-831528
Dynamic Non-Uniform Randomization in Asynchronous Linear Solvers
Evan Coleman
ABSTRACT. One approach to improving the performance of stationary linear solvers is to utilize asynchronous communication between the processors. In order to establish bounds on convergence
rate, randomized various of asynchronous linear solvers have been studied. This approach has been examined previously for the case where the random selection is done uniformly [1, 5]. A
non-uniform random selection has been used for the case of synchronous algorithms [2, 3], however in both studies the distribution remains fixed and does not dynamically change. The idea behind
the solvers considered here is for each processor to select the next component to update randomly, using a distribution that more heavily weights selection of components that are somehow more
important to the solution. The main contribution is to analyze the potential performance benefit for using a non-uniform distribution, to evaluate the viability of changing the update order
13:05 dynamically, and to investigate methods for ranking the contribution of each component.
This updating procedure is motivated in part by the Southwell iteration, which selects the component with the largest contribution to the residual at each iteration and which can converge in
fewer iterations than traditional relaxation schemes; previous work has also shown that Southwell type iterations can converge faster than uniform random selection [4]. There is a balance
between the extra computational time required to intelligently select components, and the savings in total iterations. Updating all the residuals every iteration likely introduces too much
computational overhead to be of practical use, and techniques are explored for lessening this computational burden. The thrust of the new algorithms is to focus on techniques that achieve this
dynamic focus on components that contribute more to the residual but do so in an efficient manner.
[1] Haim Avron, Alex Druinsky, and Anshul Gupta. Revisiting asynchronous linear solvers: Provable convergence rate through randomization. Journal of the ACM (JACM), 62(6):51, 2015. [2] Michael
Griebel and Peter Oswald. Greedy and randomized versions of the multiplicative schwarz method. Linear Algebra and its Applications, 437(7):1596–1610, 2012. [3] Dennis Leventhal and Adrian S
Lewis. Randomized methods for linear constraints: convergence rates and conditioning. Mathematics of Operations Research, 35(3):641–654, 2010. [4] Julie Nutini, Mark Schmidt, Issam Laradji,
Michael Friedlander, and Hoyt Koepke. Coordinate descent converges faster with the gauss-southwell rule than random selection. In International Conference on Machine Learning, pages 1632–1641,
2015. [5] John C Strikwerda. A probabilistic analysis of asynchronous iteration. Linear algebra and its applications, 349(1-3):125–154, 2002.
11:25-13:30 Session 4B: Scalable Solvers for Coupled Multiphysics 2
An implicit, conservative, asymptotic-preserving electrostatic particle-in-cell algorithm for arbitrarily magnetized plasmas
Guangye Chen
ABSTRACT. We introduce a new electrostatic particle-in-cell algorithm capable of using large timesteps compared to particle gyro-period under a (to begin, uniform) large external magnetic
field. The algorithm extends earlier electrostatic fully implicit PIC implementations [1] with a new asymptotic-preserving (AP) particle-push scheme [2] that allows timesteps much larger than
particle gyro-periods. In the large-timestep limit, the AP integrator preserves all the averaged particle drifts, while recovering the standard CN scheme for small timesteps, while recovering
particle full orbits with small timesteps. The scheme allows for a seamless, efficient treatment of particles in coexisting magnetized and unmagnetized regions, conserves energy and charge
11:25 exactly, and does not spoil implicit solver performance. Key to the approach is the generalization of the particle sub-stepping approach introduced in Ref. [1] to allow for orbit segments much
larger than cell sizes without spoiling conservation properties. The uniform-magnetic-field assumption allows us to use the standard CN update in Ref. [1] without modification, which is a
necessary preliminary step to demonstrate the viability of the approach for more general magnetic field topologies (which will require the implementation in Ref. [2], currently underway). We
demonstrate by numerical experiment with several strongly magnetized problems (e.g., diocotron instability, modified two-stream instability, drift instability, etc.) that two orders of
magnitude wall-clock-time speedups are possible vs. the standard fully implicit electrostatic PIC algorithm [1] without sacrificing solution quality and while preserving strict charge and
energy conservation. We will also discuss possible extensions to the electromagnetic context.
[1] Chen, Guangye, Luis Chacón, and Daniel C. Barnes. "An energy-and charge-conserving, implicit, electrostatic particle-in-cell algorithm." Journal of Computational Physics 230.18 (2011):
7018-7036. [2] Ricketson, Lee F., and Luis Chacón. "An energy-conserving and asymptotic-preserving charged-particle orbit implicit time integrator for arbitrary electromagnetic fields." Journal
of Computational Physics 418 (2020): 109639.
Fluid Preconditioning for a Fully Implicit Electromagnetic Gyrokinetic Particle-in-Cell Method
ABSTRACT. A fully implicit particle-in-cell (PIC) method, based on the work in [1], has been implemented in the full-volume fusion plasma code XGC [2] and was recently demonstrated to provide
numerically stable and accurate solutions to the electromagnetic gyrokinetic equations in tokamak geometry [3]. In this talk, we will present a preconditioned Picard iteration scheme,
accelerated with Anderson mixing, for handling the resulting system of nonlinear equations at each timestep. The preconditioner is designed from a simplified electron fluid model, which
11:50 captures the stiff modes of the kinetic system, and accounts for additional numerical effects originating from the PIC method. In addition, we will present our recent work in designing and
implementing a discrete formulation for eliminating finite-grid instabilities, which can be problematic in certain physical parameter regimes [4]. The new discrete formulation has promise for
enabling the use of a Schur complement form of the fluid preconditioner equations, which can be solved using a semi-coarsening multigrid approach.
[1] G. Chen and L. Chacón, Comput. Phys. Comm., 197, 73-87, 2015. [2] S. Ku, C.S. Chang, and P.H. Diamond, Nucl. Fusion, 49, 115021, 2009. [3] B. Sturdevant, S. Ku, L. Chacón, et al. Phys.
Plasmas, 28, 072505, 2021. [4] B. Sturdevant and L. Chacón. submitted to J. Comput. Phys., 2021.
Block Preconditioning of a Semi-Implicit Gyrokinetic Model of Fusion Plasmas
Lee Ricketson
ABSTRACT. We describe the block preconditioning approach used in our COGENT code, whose primary application is the simulation of the edge plasma region of tokamak fusion reactors. The
underlying model is a continuum gyrokinetic system describing the evolution of plasma species distribution functions in 4D axisymmetric or 5D phase space, where the configuration space geometry
includes open and closed field line regions spanning the magnetic separatrix and X-point. COGENT combines a high-order, finite-volume, mapped-multiblock spatial discretization with an additive
12:15 Runge-Kutta (ARK) time integrator to advance the distribution functions, fluid and neutral species moments, and an electrostatic potential, all with a variety of collision operators. Central to
the ARK approach is the identification of fast time-scales to be included in the implicit component of the fully coupled system, which is advanced consistently with the slower explicit terms to
a specified temporal accuracy. Updates of the implicit ARK stages require the solution of nonlinear systems, for which COGENT employs a Newton-Krylov algorithm. Block preconditioners are
employed to accelerate the Krylov iteration, where the blocks approximate the operators responsible for the various fast time scales being treated implicitly. In addition to disjoint operators
resulting in diagonal preconditioner blocks, the framework anticipates the possible overlap of implicit terms, coordinating the construction of linear systems and their solution using multigrid
solvers provided by the Hypre library. We present several examples of the application of this approach and the preconditioning strategies required for each, including the simulation of kinetic
microturbulence in diverted tokamak geometries.
*This material is based on work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program under Contract
DE-AC52-07NA27344 at Lawrence Livermore National Laboratory.
Enabling optimizations for reduced kinetic spectral models
ABSTRACT. Efficient coupling of the microscopic physics into the macroscopic system-scale dynamics (called "fluid-kinetic" coupling) is probably the most important and unresolved problem of
computational plasma physics. It impacts most of plasma physics areas including space physics and fusion systems. Majority of conventional simulation tools capable of describing large scale
dynamics are usually limited to simplified fluid/magnetohydrodynamics description of the plasma, because of the large spatial and temporal scale separation typical of plasmas. Yet, fluid models
lack the microscopic physics, which is known to be important in many applications (e.g., reconnection, shock physics, etc.) A way forward is to build models that combine kinetic and fluid
description in one consistent framework. The development of such methods can bridge the scale gap to successfully handle coupling of large-scale dynamics and microscopic processes.
In this presentation, we describe a novel simulation method, where the kinetic equation is solved using a spectral expansion of the plasma distribution function. The low-order terms in the
expansion capture the large-scale dynamics of the system, while higher-order terms add microscopic physics incrementally, similar to classical fluid-moment expansion. Such a method is ideally
suited for problems involving fluid-kinetic coupling, since the number of expansion terms could be adapted in space and time. Furthermore, the spectral basis itself adaptively changes in space
and time, adjusting to plasma mean flow and temperature, thus making the representation of the particle distribution function very efficient. We show that our reduced kinetic model with just a
~(4-6)^3 velocity-space moments agrees well with results from fully-kinetic simulations on some examples. In addition to describing the method, we will present several examples illustrating its
application to various problems, including solar wind-magnetosphere interaction.
Newton-Krylov-Multigrid approaches for mixed formulations of smectic A liquid crystals
ABSTRACT. In recent years, energy-minimization finite-element methods have been proposed for the computational modelling of equilibrium states of several types of liquid crystals (LCs). Here,
we consider a four-field formulation for models of smectic A liquid crystals, based on the free-energy functionals proposed by Pevnyi, Selinger, and Sluckin, and by Xia et al. The
13:05 Euler-Lagrange equations for these models include fourth-order terms acting on the smectic order parameter (or density variation of the LC) and second-order terms acting on the Q-tensor or
director field. While $H^2$ conforming or $C^0$ interior penalty methods can be used to discretize the fourth-order terms, we investigate introducing the gradient of the smectic order parameter
as an explicit variable and constraining its value using a Lagrange multiplier. In this talk, we focus on the construction of solvers for the nonlinear systems that result from the
discretization of these models. We consider a Newton-Krylov-Multigrid approach, using Newton's method to linearize the systems, and developing monolithic geometric multigrid preconditioners for
the resulting saddle-point systems. We demonstrate this to be an effective solver strategy when using a “star” relaxation scheme for the coupled system.
11:25-13:30 Session 4C: Efficient Optimization Algorithms
Golub-Kahan bidiagonalization with hybrid projection for streaming problems
11:25 ABSTRACT. We use the Golub-Kahan bidiagonalization algorithm with hybrid projection to solve inverse problems with regularization with only a limited part of the matrix available at a time and
using only limited memory. The algorithm allows accessing the matrix one block at a time, this can be a block of rows or a block of columns. The possibility of solving a linear system with only
a subset of columns of the matrix available at a time also allows a range of interesting solution strategies for general matrices.
Fast algorithms for initial value control problems in diffeomorphic registration
Andreas Mang
ABSTRACT. We propose fast numerical methods for optimal control problems with initial value control. Our contributions are the design of efficient second-order optimization methods and their
11:50 analysis. The inverse problem we consider is referred to as diffeomorphic image registration. Here, we seek a diffeomorphism $y$ that establishes a meaningful spatial correspondence between two
views (images) of the same scene. In our formulation the diffeomorphism $y$ is parametrized by its velocity $v$. The control is the initial momentum $u_0$ of an integro-differential
equation---the Euler--Poincar\'e equation associated with the diffeomorphism group. This equation together with a transport equation for the image intensities represent the state equations of
our problem. We present a Newton--Krylov method for numerical optimization. The bottleneck of our method is the computation of the search direction, i.e., the solution of the reduced space
Hessian system. We present unconditionally stable numerical methods for evaluating the forward and adjoint operators. We study the spectral properties of the Hessian operator and introduce
different strategies for preconditioning the reduced space Hessian. More precisely, we explore the performance of low-rank approximations that exploit randomized linear algebra ideas,
multi-level/multi-grid approaches, as well as their combination. We showcase results for applications in medical imaging sciences.
A new robust collective multigrid SQP-type approach for risk-adverse optimal control problems
ABSTRACT. In this contribution, a class of risk-adverse Optimal Control Problems (OCPs) under uncertainty is considered, involving the conditional value at risk measure of level $\beta$, that
is the expectation of a quantity of interest conditioned above the $\beta$ quantile. Such problem can be approximated by a nonlinear optimization problem over the classical state, adjoint and
control variables plus a further scalar unknown.
12:15 Classical full-space Sequential Quadratic Programming approaches (SQP) are not robust since they may lead to singular Hessian matrices along the iterations. To overcome this difficulty, a new
SQP procedure is proposed, where a first SQP is performed on the state, control and adjoint variables, and the scalar quantity is sequentially updated. The SQP step requires the solution of a
large saddle-point systems for which few preconditioners have been studied recently. In this talk, the Collective MultiGrid (CMG) algorithm is extended to OCP under uncertainty. Each step of
CMG involves the solution a reduced system of dimension N times N, where N is the number of collocation points in the discretization of the continuous expectation. It is shown that this reduced
system can be solved with linear complexity in $N$.
Numerical experiments support the proposed methodology, showing robustness of the global nonlinear procedure and interior linear solver with respect to the several parameters of the problem.
History Matching Relative Permeability Curves with Bayesian Optimization
Steven Samoil
ABSTRACT. In the field of reservoir engineering one of the primary time-consuming steps is process of history matching to update the mathematical models describing fluid flow in the subsurface
reservoir to match the existing historical production data more closely. This task usually involves either data assimilation or the repeated tuning of parameters through numerous simulations
(Oliver 2011). Bayesian Optimization has not yet seen much use in the petroleum industry and is a useful approach to optimization that allows finding an optimal set of parameters for cost
functions that are especially expensive (Brochu 2010). This ability to optimize cost functions that may take extensive run time may be very well suited to the task of history matching reservoir
models. In this study a framework has been developed to assist with the evaluation of Bayesian Optimization for the task of history matching relative permeability curves. This framework was
designed to enable testing on both local and remote machines with a range of capabilities from desktop workstations to large scale clusters. Simulations are conducted utilizing the Reservoir
Simulation Group parallel black oil simulator (Wang 2015). This simulator is designed to run on distributed memory computers and utilizes the finite difference method for discretization of the
black oil model. At each timestep in the simulation, the simulator utilizes the inexact Newton method to approximately solve the linearized form of the nonlinear systems in the black oil model
(Wang 2015). The coupling between the unknowns in the system (pressure and saturation/bubble point pressure) needs to first be weakened through the application of a multi-stage preconditioner
(Wang 2015). The preconditioner chosen for this study is the constrained pressure residual (CPR) method that separates the pressure block from the full system and utilizes the algebraic
multigrid solver (AMG) to solve the pressure block before utilizing a global smoother applied to the full system (Wang 2015, Wallis 1985, Cao 2005). The restarted general minimal residual
method (GMRES) is then used to find the solution to the linear system (Wang 2015, Saad 2003). This whole process is repeated for the next timestep in the simulation until a stopping condition
12:40 is met. When conducting history matching of a reservoir model, a search of possible parameters is conducted to find the best possible fit. This task typically involves either the iterative
assimilation of data during a simulation or the repeated tuning and re-running of simulations until an appropriate match is found (Oliver 2011). Bayesian Optimization has been chosen for this
study due to its suitability for optimization of the extremely long running simulations required in reservoir simulation. Bayesian Optimization utilizes Bayes theorem to determine the
likelihood of an output based on prior knowledge of the cost function. Several sets of initial parameters are chosen and then the costs are found by running a simulation with each set of
parameters. Based on this newly found knowledge of the cost function the Bayesian Optimization algorithm will use an acquisition function to choose a new set of parameters that are expected to
provide an improvement. At this stage the optimization process can be configured to be more exploratory in selecting parameters to avoid local minimums or more exploitative to conduct a more
directed search of the parameter space. Multiple iterations of this process are conducted until a stopping condition is met (Brochu 2010). For the cost function the fluid production and
injection rate curves for the water, oil, gas, and injected fluids are compared against the production data curves using FastDTW (Salvador 2007). The distances between the test and production
curves are combined using the Euclidean norm to find the cost of the current set of parameters. At the end of the optimization process, the parameters with the minimum cost are chosen as the
optimal parameter set. Due to the randomness in selecting new parameters the average number of required epochs for the optimization is found through repeated runs of the entire process. The SPE
9 black oil model has been utilized for the initial development in this study as a test and benchmark model due to its reasonable complexity but still manageable simulation time (Killough
1995). To simulate the real-world task of history matching, the standard SPE 9 model is set as the “real” production data. For the remainder of the study the relative permeability curves are
assumed to be unknown and history matching will be conducted to determine these curves. To aid the process of history matching the relative permeability curves are reduced to a smaller set of
10 parameters with the application of Stone’s Model II (Reynolds 2004). In this model the curves are parameterized to an approximate analytical form that closely matches the values of the
relative permeability curves (Reynolds 2004). Current early results are showing promising behavior to find the optimal parameters, however further work is in progress to finalize the
experiments. We have begun to evaluate the performance of this Bayesian Optimization framework on larger scale models that require a significant amount of simulation time (on the order of days/
weeks on standard workstations or hours on large-scale computation resources). Due to the expensive time cost of these simulations, it is expected the Bayesian Optimization framework will show
benefits over more traditional approaches such as the Ensemble Kalman Filter (ENKF) method (Lorentzen 2001, Aanonsen 2009). A full comparison between the Bayesian Optimization history matching
framework and the ENKF method will need to be conducted.
Multigrid preconditioning for distributed optimal control of fractional parabolic equations
ABSTRACT. Optimal control problems constrained by fractional parabolic operators have attracted a lot of attention over the last couple of years. Such problems arise naturally in optimal
control of anomalous diffusion and in machine learning. While solving fractional parabolic equations poses significant challenges on its own, having them appear as constraints in an
optimization problem increases the complexity dramatically, further limiting the size of the problems that can be solved in practice.
In general there are two approaches for solving optimal control problems constrained by PDEs, namely the all-at-once pathway, and the reduced-type methods. In the all-at-once approach one has
to solve the KKT system representing the first order optimality conditions. The main advantage lies in the fact that for PDE constraints this system is sparse, and no PDE solves are necessary
during the process, as the PDE and its adjoint equation are solved as a coupled system. Several works favor this approach for classical parabolic control. However, for fractional parabolic
control, the KKT system is no longer sparse, and we prefer the reduced approach using the control-to-state map. Building on our earlier work on multigrid preconditioning for classical parabolic
control and on fractional elliptic control, we show how multigrid can be used to solve fractional parabolic control problems very efficiently. A partial analysis and numerical results support
our claim.
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2 Digit By 2 Digit Division Worksheets - Divisonworksheets.com
2 Digit By 2 Digit Division Worksheets
2 Digit By 2 Digit Division Worksheets – With the help of worksheets for division, you can help your child learn and master their division abilities. There are numerous types of worksheets to choose
from as well as the option to make your own. You can download the worksheets for free and customize them as you like. These worksheets are ideal for students in kindergarten and first grade.
Two people can produce massive numbers
A child should practice on worksheets and divide huge numbers. The worksheets often only support three, two or four divisors. The child will not have to worry about forgetting how to divide the large
number or making mistakes in their tables of times because of this method. It is possible to find worksheets on the internet or download them onto your personal computer to assist your child in
developing this mathematical skill.
Multi-digit division worksheets can be used by children to practice and increase their knowledge. This is an essential mathematical skill that is needed for maths that are complex and everyday
calculations. These worksheets offer interactive questions and activities to strengthen the understanding.
Students find it difficult to divide large numbers. The worksheets usually are constructed using a similar algorithm and provide step-by–step instructions. They may not get the intellectual
understanding they require from these. Long division can be taught by using base ten blocks. Understanding the steps will simplify long division for students.
Large numbers that are divided could be taught to pupils through various worksheets and exercises. Additionally, fractional findings stated in decimals can be found on the worksheets. There are
worksheets which can be used to calculate hundredths. This is especially useful when you need to divide large sums of money.
Divide the numbers into smaller ones.
Putting a number into small groups might be challenging. While it sounds good on paper, the small facilitators of groups hate it. It truly reflects how our bodies develop, and the procedure could aid
in the Kingdom’s limitless expansion. It inspires others and motivates people to reach out to the forgotten.
It can also be useful to brainstorm ideas. It’s possible to make groups of people who have the same traits and experience. This could lead to some extremely innovative ideas. After you’ve created
your groups, present yourself to each participant. It’s a good way to inspire creativity and encourage creative thinking.
Divide big numbers into smaller numbers is the basic principle of division. It is useful when you want to make the same amount of items for multiple groups. For instance, the large class which could
be divided into smaller groups of five students. The initial 30 pupils are created by adding the groups.
It is crucial to keep in mind that there are two types of numbers that you can divide: the divisor, and the quotient. The result of dividing one number by another will be “ten/five,” while dividing
two by two yields the exact same result.
It is an excellent idea to utilize the power of ten for big numbers.
The splitting of huge numbers into powers can make it easier to compare them. Decimals are an integral part of the shopping process. They can be located on receipts as well as price tags. They are
used by petrol pumps to indicate the cost per gallon as well as the amount of fuel being delivered via a pipe.
It is possible to split big numbers into powers of ten using two different methods: shift the decimal point to the left or multiply by 10-1. The other method utilizes the associative component of
powers of 10. Once you’ve learned to use the powers of ten’s associative function, you can split enormous numbers into smaller power.
The first method uses mental computation. Divide 2.5 by the power of 10 to find the pattern. The decimal points will shift left when the power of ten grows. You can apply this concept to solve any
The other method involves mentally dividing extremely large numbers into powers of ten. The other method involves rapidly writing large numbers using scientific notation. If you are using scientific
notation, huge numbers must be written using positive exponents. For example, if you move the decimal points five spaces to your left, you can turn 450,000 into 4.5. To split large numbers into
smaller powers, use the exponent 5.
Gallery of 2 Digit By 2 Digit Division Worksheets
Long Division Worksheet With Double Digit Divisors Set 2 Homeschool
Division With Two Digit Divisors Teacher Printables
Division By Two Digits Worksheets
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Does base n have any Rotate-Left-Double numbers? ~ Code Golf ~ TransWikia.com
Python 3, ^[S:19:S] 18 bytes
Uses Kevin Cruijssen's formula.
Returns True/False.
Saved a byte thanks to dingledooper!!!
lambda n:n-2&n-3>0
Answered by Noodle9 on December 17, 2020
C (gcc), 19 bytes
Uses Kevin Cruijssen's formula.
Returns $$1$$ for falsy and $$0$$ for truthy.
Answered by Noodle9 on December 17, 2020
Bash, ^[S:22:S] 20 bytes
Uses Kevin Cruijssen's formula.
Returns $$1$$ for falsy and $$0$$ for truthy.
Saved 2 bytes thanks to dingledooper!!!
echo $[!($1-2&$1-3)]
Answered by Noodle9 on December 17, 2020
Ruby, 14 bytes
Port of Kevin Cruijssen's answer, remember to upvote them!
Answered by user92069 on December 17, 2020
JavaScript (Node.js), 10 bytes
In JS 0 is falsy and everything else is truthy. Again, another port of Kevin Cruijssen's answer!
Answered by user92069 on December 17, 2020
GolfScript, 5 bytes
In GolfScript 0 is falsy while any other value is truthy.
Answered by user92069 on December 17, 2020
Retina 0.8.2, 21 bytes
Try it online! Link includes test cases. Explanation: The first stage converts to unary, while the last stage uses @KevinCruijssen's observation that a solution exists if n-2 has a nontrivial odd
Answered by Neil on December 17, 2020
Charcoal, 8 bytes
Try it online! Link is to verbose version of code. Outputs a Charcoal boolean, i.e. - if RLD numbers exist otherwise no output. Explanation:
Ｎ Input as a number
⊖⊖ Decremented twice
⍘ ² Converted to base 2
Σ Digital sum
‹¹ Is greater than 1
The only binary numbers with a digital sum of 1 or less are 0 and powers of 2, so by @KevinCruijssen's proof a solution exists for all other values of n.
Answered by Neil on December 17, 2020
APL (Dyalog Extended), 9 bytes
⊃∧/⊤⎕-2 3
A full program that takes a single number $$n$$ from stdin, and prints 1 for true, 0 otherwise. APL doesn't have bitwise functions, so we need to explicitly convert to binary and apply boolean
functions on each bit.
How it works
⊃∧/⊤⎕-2 3 ⍝ Input: n (from stdin)
⎕-2 3 ⍝ [n-2, n-3]
⊤ ⍝ Convert to binary
⍝ (each number becomes a column in a matrix, aligned to bottom)
⊃∧/ ⍝ Check if the MSB of both numbers are 1,
⍝ i.e. the bit lengths of the two are the same
Answered by Bubbler on December 17, 2020
Husk, 6 bytes
This one got simplified really quickly.
Answered by Razetime on December 17, 2020
Jelly, 5 bytes
Uses the fact that a given $$n$$ returns $$0$$ iff $$n-2$$ is a power of $$2$$, as pointed out by Kevin Cruijssen and the n-2 & n-3 trick
How they work
_2&’$ - Main link. Takes n on the left
_2 - n-2
$ - To n-2:
’ - Decrement; n-3
& - n-2 & n-3
Answered by caird coinheringaahing on December 17, 2020
-1 byte thanks to @xnor and @Noodle9.
Try it online or verify the first $$[2,100]$$ test cases.
Í # Decrease the (implicit) input-integer by 2
# Check that this input-2 is a power of 2 by:
D # Duplicating it
< # Decrease the copy by 1 (so integer-3)
& # Take the bitwise-AND of input-2 and input-3
Ā # Check that this is NOT 0
# (after which the result is output implicitly)
But wait, I don't see any use of bases nor rotation!
When I saw the challenge in the Sandbox and was working on a solution, I noticed that the only falsey values in the first $$n=[2,500]$$ bases formed the sequence A056469: number of elements in the
continued fraction for $$sum_{k=0}^n (frac{1}{2})^{2^k}$$, which could be simplified to $$a(n)=leftlfloor2^{n-1}+2rightrfloor$$. Here a copy of the first 25 numbers in that sequence as reference:
2, 3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610
It can also be note that all the numbers in this sequence are of the form $$a(n)=2^n+2$$, so checking whether $$n-2$$ is a power of $$2$$ will verify whether it's in this sequence. Since we want to
do the invert here, and having a falsey result if it's in this sequence (or truthy if it's NOT in this sequence), we'll do just that, resulting in the code above.
Mathematical proof that all falsey cases of the Left-Rotate-Double numbers are of the form $$2^n+2$$:
Quote from @saulspatz at the Math SE, who provided me with this Mathematical proof to back up my theory I based on the first $$n=[2,500]$$ test cases. So all credit for this proof goes to him/her.
If $$m$$ is a $$(d+1)$$-digit Rotate-Left-Double number in base $$n$$, then $$m=xn^d+ytag1$$ where $$dgeq1, 0<x<n, 0leq y<n^d$$. (Includes the rule that the number can't start with $$0$$.) Rotating
$$m$$ gives $$ny+x$$, so we have $$2xn^d+2y=ny+x$$ or $$(n-2)y=(2n^d-1)xtag2$$ If $$n=2^k+2$$ then $$(2)$$ gives $$(n-2)|x$$ (which means $$x$$ is divisible by $$(n-2)$$), since $$2n^s-1$$ is odd.
But then $$ygeq 2n^d-1$$ which contradicts $$y<n^d$$.
To show that these are the only falsey numbers, let $$p$$ be an odd prime dividing $$n-2$$. (Such a $$p$$ exists because $$n-2$$ is not a power of $$2$$.) In $$(2)$$ we can take $$x=frac{n-2}p<n$$
and we have to show that there exist an exponent $$d>0$$ and $$0leq y<n^d$$ such that $$py = 2n^d-1$$ If we can find a $$d$$ such that $$p|(2n^d-1)$$, we are done, for we can take $$y = frac{2n^d-1}p
By assumption, $$n-2equiv0pmod{p}$$ so $$nequiv 2pmod p$$. Therefore, $$2n^dequiv1iff 2cdot2^dequiv1 iff 2^{d+1}equiv 1pmod p,$$ and by Fermat's little theorem, which states that $$a^{p-1}equiv 1pmod
p$$, we can take $$d=p-2$$, because $$2^{p-2+1}equiv 1 iff 2^{p-1}equiv 1 pmod p$$
This completes the proof.
Answered by Kevin Cruijssen on December 17, 2020 | {"url":"https://transwikia.com/code-golf/does-base-n-have-any-rotate-left-double-numbers/","timestamp":"2024-11-07T02:43:48Z","content_type":"text/html","content_length":"69083","record_id":"<urn:uuid:dad05d41-41df-4bbb-9ba0-bc95da337980>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00674.warc.gz"} |
David Barton : Numerical continuation for investigating nonlinear systems: from model to experiment
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David Barton : Numerical continuation for investigating nonlinear systems: from model to experiment
Numerical continuation is a tool for investigating the bifurcation structure of a nonlinear dynamical system with respect to the system parameters. It is most often used to "carve up" parameter space
into regions of qualitatively different behaviour by finding and tracking bifurcations (e.g., Hopf bifurcations) as the system parameters change. This talk will give an introduction to the theory
behind numerical continuation and go on to discuss recent developments in the field.
Particular attention will be paid to numerical continuation of systems with non-smoothness, motivated by the example of intermittent contacts in a model of orthogonal cutting (turning). Rich
dynamical behaviour is present in this model due to the presence of a grazing bifurcation which denotes the transition point from constant contact of the cutting tool with the workpiece to
intermittent contact. Using numerical continuation it is possible to elucidate the full bifurcation structure of the system, something that would be extremely difficult with other methods.
Finally, numerical continuation will be demonstrated as applied to a physical experiment (so-called control-based continuation): a nonlinear energy harvesting device. Numerical continuation in this
context allows the investigation of a physical device without prior knowledge of a model. Both stable and unstable motions can be investigated and bifurcations found directly. As such these
investigations may aid in establishing what an appropriate mathematical model could be.
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Headphone Test Lab test details harmans ppr
Harman's Predicted Preference Rating
As part of Harman International’s research into defining new target frequency responses for both insert and circumaural/supra-aural (over-ear/on-ear) headphones, led by Sean Olive, it has developed
equations which can be used to calculate a Predicted Preference Rating (PPR), based on the headphone’s frequency response as measured at the DRP (drum reference point) of an artificial ear. For the
measurements used in developing this metric, Harman used similar GRAS artificial ear hardware to that which HTL employs (but unique pinnae).
Harman’s two equations, for insert and circumaural/supra-aural headphones respectively, are applied to the error response (measured frequency response minus Harman target response) and have similar
but not identical forms. That for insert headphones is:
A – B.s – C.g – D.m
where A, B, C and D are constants, 's' is the standard deviation of the error response, 'g' is the absolute value of the gradient (slope) of the logarithmic regression line fitted to the error
response, and 'm' is the mean error of the error response. (Harman uses different abbreviations for these quantities.) For circumaural/supra-aural (over-ear/on-ear) headphones the equation is a
little simpler:
A – B.s – C.g
as including mean error did not improve the fit of the model.
The values of A, B, C and D for insert headphones are 68.685, 3.238, 4.473 and 2.658, and 's', 'g' and 'm' are defined over the frequency ranges 20Hz to 10kHz, 20Hz to 10kHz, and 40Hz to 10kHz
respectively. For circumaural/supra-aural headphones the values of A, B and C are 114.49, 12.62 and 15.52, and 's' and 'g' are both calculated over the frequency range 50Hz-10kHz. (The higher LF
limit in this case is because variations in earpad sealing can make the measured frequency response at lower frequencies too variable.)
Note that if a headphone is ‘perfect’ – ie, if its response exactly matches the Harman target – then 's', 'g', and 'm' are all zero and the maximum PPR values are therefore 69 and 114 respectively
(to the nearest integer, as usually stated). So it is incorrect to refer to PPR values as percentages, and values for insert and circumaural/supra-aural headphones are not directly comparable. To
circumvent this, HTL's PPR scores are quoted as both 'raw' figures (direct from Harman's equations), and as percentage figures to allow comparison across all headphone types. The results are
presented in this form:
84/82 ≡ 73%/72% (L/R)
where the mathematical symbol ≡ means 'is equivalent to'.
The equations assume logarithmically spaced data points along the frequency axis, Harman having used 1/48th-octave data for its measurements and development of the PPR equations. So the first step in
calculating PPR values from FFT response data, where the frequency spacing is linear, is interpolation to frequencies matching those of Harman’s target. The two responses can then be subtracted to
create the error response, and truncated to the appropriate frequency range.
From this point the calculation of PPR is straightforward using standard equations for 's', 'g', and 'm':
But there is a wrinkle, in that the equation for 'g' stated in Harman’s papers (references below) is instead:
I have queried this with Harman but not received a reply. The PPR values quoted in HTL’s test results are calculated using the standard equation for 'g'.
Because of how the PPR calculations are formulated, it is possible to generate a negative PPR if the values of 's' and/or 'g' (and/or 'm') are sufficiently high. In this case, the outcome is stated
in HTL's test results as '<0' (ie, less than zero).
S Olive, T Welti and O Khonsaripour, "A Statistical Model That Predicts Listeners’ Preference Ratings of In-Ear Headphones: Part 1 – Listening Test Results and Acoustic Measurements", Audio
Engineering Society 143rd Convention, October 2017 (available here)
S Olive, T Welti and O Khonsaripour, "A Statistical Model That Predicts Listeners’ Preference Ratings of In-Ear Headphones: Part 2 – Development and Validation of the Model", Audio Engineering
Society 143rd Convention, October 2017 (available here)
S Olive, T Welti and O Khonsaripour, "A Statistical Model that Predicts Listeners’ Preference Ratings of Around-Ear and On-Ear Headphones", Audio Engineering Society 144th Convention, May 2018
(available here) | {"url":"https://headphonetestlab.co.uk/test-details-harmans-ppr","timestamp":"2024-11-10T17:43:36Z","content_type":"text/html","content_length":"40072","record_id":"<urn:uuid:6dbd0393-423b-4541-a25a-c6ce775fa29c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00321.warc.gz"} |
As above, so below
David's comment "as above, so below" made me think of above as Monad and below as monad, and so I think of "m/Monad" as a representation of a reality that incorporates all possible variables in
terms of interpretations, dimensions, karma, kama, maya, manvataras, etc.
Comment by Mark Kusek on February 20, 2015 at 4:15pm
Hi Mauri, Unlike our rather sustained dialog on this topic over the years on Theos-l, these views have not yet been presented or exchanged very much here. It's a new thread.
Comment by Mauri on February 19, 2015 at 10:59pm
I'm tending to think in terms of a link, or a sort of quantum continuum, between the monad/s (individual/s) and Monad (Commonality) in terms of interpretation and experience or, in other words,
in terms of "reality as we know it." Hence "as above, so below." In other words I tend to think of that link as if it's like the dot in the circumpunct, or the laya centre in Leon's theory, or
the dot in Mark's ( + . - ) symbol, or the forward slash in my x/X symbol. But of course I'm just "exoterically speaking," lol, so ....
Comment by David Allen on February 19, 2015 at 9:46am
I have to apologize for being unfamiliar with your terminology. If I understand what you said correctly, you are suggesting a repeating pattern? | {"url":"https://theosophy.net/profiles/blogs/as-above-so-below-1","timestamp":"2024-11-05T00:58:40Z","content_type":"text/html","content_length":"50100","record_id":"<urn:uuid:872524f8-c66b-4f4f-94a5-7cd7f8dcb88f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00077.warc.gz"} |
ThmDex – An index of mathematical definitions, results, and conjectures.
Let $M = (X, \mathcal{F}, \mu)$ be a
D1158: Measure space
such that
(i) $E, F, G \in \mathcal{F}$ are each a D1109: Measurable set in $M$
(ii) $F, G$ is a D5143: Set partition of $X$
Then $$\mu(E) = \mu(E \cap F) + \mu(E \cap G)$$ | {"url":"https://thmdex.com/r/4927","timestamp":"2024-11-13T21:34:16Z","content_type":"text/html","content_length":"6261","record_id":"<urn:uuid:0f533c53-0d63-41f8-9f5a-6a0c95301e54>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00873.warc.gz"} |
Functionality to clean up expressions so that they satisfy the requirements of a consistent expression tree.
◆ cleanup_dispatch()
void cadabra::cleanup_dispatch ( const Kernel & k,
Ex & ,
Ex::iterator & it
Central cleanup dispatch routine, which calls the other cleanup functions defined later.
These cleanup routines do NOT use normal cadabra algorithms; they are completely independent of them to prevent circular dependence or infinite recursion.
These algorithms clean up the tree at the current node and the first layer of child nodes, but do NOT descend deeper down the tree, UNLESS that would leave the tree in an inconsistent state. An
example is acting at the top node of \prod{4}{\sum{a}{b}}, which would push the 4 to the multiplier of the sum, but that is not allowed, so it needs to go further down. Sibling nodes of 'it' remain
untouched as well.
◆ cleanup_dispatch_deep()
void cadabra::cleanup_dispatch_deep ( const Kernel & k,
Ex & ,
dispatcher_t disp = &cleanup_dispatch
More general cleanup of an entire tree.
Walks depth-first along the entire tree and call cleanup_dispatch at every node. | {"url":"https://cadabra.science/doxygen/html/group__cleanup.html","timestamp":"2024-11-07T19:28:01Z","content_type":"application/xhtml+xml","content_length":"9532","record_id":"<urn:uuid:460b3723-3b47-47b5-b6d1-9de4ca83c36a>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00342.warc.gz"} |
Jacobians, Hessians, hvp, vhp, and more: composing functorch transforms
Jacobians, Hessians, hvp, vhp, and more: composing functorch transforms¶
Computing jacobians or hessians are useful in a number of non-traditional deep learning models. It is difficult (or annoying) to compute these quantities efficiently using a standard autodiff system
like PyTorch Autograd; functorch provides ways of computing various higher-order autodiff quantities efficiently.
Computing the Jacobian¶
import torch
import torch.nn as nn
import torch.nn.functional as F
from functools import partial
_ = torch.manual_seed(0)
Let’s start with a function that we’d like to compute the jacobian of. This is a simple linear function with non-linear activation.
def predict(weight, bias, x):
return F.linear(x, weight, bias).tanh()
Let’s add some dummy data: a weight, a bias, and a feature vector x.
D = 16
weight = torch.randn(D, D)
bias = torch.randn(D)
x = torch.randn(D) # feature vector
Let’s think of predict as a function that maps the input x from \(R^D -> R^D\). PyTorch Autograd computes vector-Jacobian products. In order to compute the full Jacobian of this \(R^D -> R^D\)
function, we would have to compute it row-by-row by using a different unit vector each time.
def compute_jac(xp):
jacobian_rows = [torch.autograd.grad(predict(weight, bias, xp), xp, vec)[0]
for vec in unit_vectors]
return torch.stack(jacobian_rows)
xp = x.clone().requires_grad_()
unit_vectors = torch.eye(D)
jacobian = compute_jac(xp)
print(jacobian[0]) # show first row
torch.Size([16, 16])
tensor([-0.5956, -0.6096, -0.1326, -0.2295, 0.4490, 0.3661, -0.1672, -1.1190,
0.1705, -0.6683, 0.1851, 0.1630, 0.0634, 0.6547, 0.5908, -0.1308])
Instead of computing the jacobian row-by-row, we can use vmap to get rid of the for-loop and vectorize the computation. We can’t directly apply vmap to PyTorch Autograd; instead, functorch provides a
vjp transform:
from functorch import vmap, vjp
_, vjp_fn = vjp(partial(predict, weight, bias), x)
ft_jacobian, = vmap(vjp_fn)(unit_vectors)
# lets confirm both methods compute the same result
assert torch.allclose(ft_jacobian, jacobian)
In future tutorial a composition of reverse-mode AD and vmap will give us per-sample-gradients. In this tutorial, composing reverse-mode AD and vmap gives us Jacobian computation! Various
compositions of vmap and autodiff transforms can give us different interesting quantities.
functorch provides jacrev as a convenience function that performs the vmap-vjp composition to compute jacobians. jacrev accepts an argnums argument that says which argument we would like to compute
Jacobians with respect to.
from functorch import jacrev
ft_jacobian = jacrev(predict, argnums=2)(weight, bias, x)
# confirm
assert torch.allclose(ft_jacobian, jacobian)
Let’s compare the performance of the two ways to compute the jacobian. The functorch version is much faster (and becomes even faster the more outputs there are).
In general, we expect that vectorization via vmap can help eliminate overhead and give better utilization of your hardware.
Vmap does this magic by pushing the outer loop down into the functions primitive operations in order to obtain better performance.
Let’s make a quick function to evaluate performance and deal with microseconds and milliseconds measurements:
def get_perf(first, first_descriptor, second, second_descriptor):
""" takes torch.benchmark objects and compares delta of second vs first. """
faster = second.times[0]
slower = first.times[0]
gain = (slower-faster)/slower
if gain < 0: gain *=-1
final_gain = gain*100
print(f" Performance delta: {final_gain:.4f} percent improvement with {second_descriptor} ")
And then run the performance comparison:
from torch.utils.benchmark import Timer
without_vmap = Timer(stmt="compute_jac(xp)", globals=globals())
with_vmap = Timer(stmt="jacrev(predict, argnums=2)(weight, bias, x)", globals=globals())
no_vmap_timer = without_vmap.timeit(500)
with_vmap_timer = with_vmap.timeit(500)
<torch.utils.benchmark.utils.common.Measurement object at 0x7fa9a911b350>
2.25 ms
1 measurement, 500 runs , 1 thread
<torch.utils.benchmark.utils.common.Measurement object at 0x7fa9a6a99d50>
jacrev(predict, argnums=2)(weight, bias, x)
884.34 us
1 measurement, 500 runs , 1 thread
Lets do a relative performance comparison of the above with our get_perf function:
get_perf(no_vmap_timer, "without vmap", with_vmap_timer, "vmap");
Performance delta: 60.7170 percent improvement with vmap
Furthemore, it’s pretty easy to flip the problem around and say we want to compute Jacobians of the parameters to our model (weight, bias) instead of the input.
# note the change in input via argnums params of 0,1 to map to weight and bias
ft_jac_weight, ft_jac_bias = jacrev(predict, argnums=(0, 1))(weight, bias, x)
reverse-mode Jacobian (jacrev) vs forward-mode Jacobian (jacfwd)¶
We offer two APIs to compute jacobians: jacrev and jacfwd:
• jacrev uses reverse-mode AD. As you saw above it is a composition of our vjp and vmap transforms.
• jacfwd uses forward-mode AD. It is implemented as a composition of our jvp and vmap transforms.
jacfwd and jacrev can be substituted for each other but they have different performance characteristics.
As a general rule of thumb, if you’re computing the jacobian of an \(𝑅^N \to R^M\) function, and there are many more outputs than inputs (i.e. \(M > N\)) then jacfwd is preferred, otherwise use
jacrev. There are exceptions to this rule, but a non-rigorous argument for this follows:
In reverse-mode AD, we are computing the jacobian row-by-row, while in forward-mode AD (which computes Jacobian-vector products), we are computing it column-by-column. The Jacobian matrix has M rows
and N columns, so if it is taller or wider one way we may prefer the method that deals with fewer rows or columns.
from functorch import jacrev, jacfwd
First, let’s benchmark with more inputs than outputs:
Din = 32
Dout = 2048
weight = torch.randn(Dout, Din)
bias = torch.randn(Dout)
x = torch.randn(Din)
# remember the general rule about taller vs wider...here we have a taller matrix:
using_fwd = Timer(stmt="jacfwd(predict, argnums=2)(weight, bias, x)", globals=globals())
using_bwd = Timer(stmt="jacrev(predict, argnums=2)(weight, bias, x)", globals=globals())
jacfwd_timing = using_fwd.timeit(500)
jacrev_timing = using_bwd.timeit(500)
print(f'jacfwd time: {jacfwd_timing}')
print(f'jacrev time: {jacrev_timing}')
torch.Size([2048, 32])
jacfwd time: <torch.utils.benchmark.utils.common.Measurement object at 0x7fa9a5d792d0>
jacfwd(predict, argnums=2)(weight, bias, x)
1.32 ms
1 measurement, 500 runs , 1 thread
jacrev time: <torch.utils.benchmark.utils.common.Measurement object at 0x7fa9a4dee450>
jacrev(predict, argnums=2)(weight, bias, x)
12.46 ms
1 measurement, 500 runs , 1 thread
and then do a relative benchmark:
get_perf(jacfwd_timing, "jacfwd", jacrev_timing, "jacrev", );
Performance delta: 842.8274 percent improvement with jacrev
and now the reverse - more outputs (M) than inputs (N):
Din = 2048
Dout = 32
weight = torch.randn(Dout, Din)
bias = torch.randn(Dout)
x = torch.randn(Din)
using_fwd = Timer(stmt="jacfwd(predict, argnums=2)(weight, bias, x)", globals=globals())
using_bwd = Timer(stmt="jacrev(predict, argnums=2)(weight, bias, x)", globals=globals())
jacfwd_timing = using_fwd.timeit(500)
jacrev_timing = using_bwd.timeit(500)
print(f'jacfwd time: {jacfwd_timing}')
print(f'jacrev time: {jacrev_timing}')
jacfwd time: <torch.utils.benchmark.utils.common.Measurement object at 0x7fa9a5d64790>
jacfwd(predict, argnums=2)(weight, bias, x)
7.99 ms
1 measurement, 500 runs , 1 thread
jacrev time: <torch.utils.benchmark.utils.common.Measurement object at 0x7fa9a5d67b50>
jacrev(predict, argnums=2)(weight, bias, x)
1.09 ms
1 measurement, 500 runs , 1 thread
and a relative perf comparison:
get_perf(jacrev_timing, "jacrev", jacfwd_timing, "jacfwd")
Performance delta: 635.2095 percent improvement with jacfwd
Hessian computation with functorch.hessian¶
We offer a convenience API to compute hessians: functorch.hessian. Hessians are the jacobian of the jacobian (or the partial derivative of the partial derivative, aka second order).
This suggests that one can just compose functorch’s jacobian transforms to compute the Hessian. Indeed, under the hood, hessian(f) is simply jacfwd(jacrev(f)).
Note: to boost performance: depending on your model, you may also want to use jacfwd(jacfwd(f)) or jacrev(jacrev(f)) instead to compute hessians leveraging the rule of thumb above regarding wider vs
taller matrices.
from functorch import hessian
# lets reduce the size in order not to blow out colab. Hessians require significant memory:
Din = 512
Dout = 32
weight = torch.randn(Dout, Din)
bias = torch.randn(Dout)
x = torch.randn(Din)
hess_api = hessian(predict, argnums=2)(weight, bias, x)
hess_fwdfwd = jacfwd(jacfwd(predict, argnums=2), argnums=2)(weight, bias, x)
#hess_revrev = jacrev(jacrev(predict, argnums=2), argnums=2)(weight, bias, x)
Let’s verify we have the same result regardless of using hessian api or using jacfwd(jacfwd())
torch.allclose(hess_api, hess_fwdfwd)
Batch Jacobian and Batch Hessian¶
In the above examples we’ve been operating with a single feature vector. In some cases you might want to take the Jacobian of a batch of outputs with respect to a batch of inputs. That is, given a
batch of inputs of shape (B, N) and a function that goes from \(R^N \to R^M\), we would like a Jacobian of shape (B, M, N).
The easiest way to do this is to use vmap:
batch_size = 64
Din = 31
Dout = 33
weight = torch.randn(Dout, Din)
print(f"weight shape = {weight.shape}")
bias = torch.randn(Dout)
x = torch.randn(batch_size, Din)
weight shape = torch.Size([33, 31])
compute_batch_jacobian = vmap(jacrev(predict, argnums=2), in_dims=(None, None, 0))
batch_jacobian0 = compute_batch_jacobian(weight, bias, x)
If you have a function that goes from (B, N) -> (B, M) instead and are certain that each input produces an independent output, then it’s also sometimes possible to do this without using vmap by
summing the outputs and then computing the Jacobian of that function:
def predict_with_output_summed(weight, bias, x):
return predict(weight, bias, x).sum(0)
batch_jacobian1 = jacrev(predict_with_output_summed, argnums=2)(weight, bias, x).movedim(1, 0)
assert torch.allclose(batch_jacobian0, batch_jacobian1)
If you instead have a function that goes from \(𝑅^𝑁 \to 𝑅^𝑀\) but inputs that are batched, you compose vmap with jacrev to compute batched jacobians:
Finally, batch hessians can be computed similarly. It’s easiest to think about them by using vmap to batch over hessian computation, but in some cases the sum trick also works.
compute_batch_hessian = vmap(hessian(predict, argnums=2), in_dims=(None, None, 0))
batch_hess = compute_batch_hessian(weight, bias, x)
torch.Size([64, 33, 31, 31])
Computing Hessian-vector products¶
The naive way to compute a Hessian-vector product (hvp) is to materialize the full Hessian and perform a dot-product with a vector. We can do better: it turns out we don’t need to materialize the
full Hessian to do this. We’ll go through two (of many) different strategies to compute Hessian-vector products:
• composing reverse-mode AD with reverse-mode AD
• composing reverse-mode AD with forward-mode AD
Composing reverse-mode AD with forward-mode AD (as opposed to reverse-mode with reverse-mode) is generally the more memory efficient way to compute a hvp because forward-mode AD doesn’t need to
construct an Autograd graph and save intermediates for backward:
from functorch import jvp, grad, vjp
def hvp(f, primals, tangents):
return jvp(grad(f), primals, tangents)[1]
Here’s some sample usage.
def f(x):
return x.sin().sum()
x = torch.randn(2048)
tangent = torch.randn(2048)
result = hvp(f, (x,), (tangent,))
If PyTorch forward-AD does not have coverage for your operations, then we can instead compose reverse-mode AD with reverse-mode AD:
def hvp_revrev(f, primals, tangents):
_, vjp_fn = vjp(grad(f), *primals)
return vjp_fn(*tangents)
result_hvp_revrev = hvp_revrev(f, (x,), (tangent,))
assert torch.allclose(result, result_hvp_revrev[0]) | {"url":"https://pytorch.org/functorch/0.2.0/notebooks/jacobians_hessians.html","timestamp":"2024-11-02T11:28:12Z","content_type":"text/html","content_length":"59500","record_id":"<urn:uuid:6d16551f-4c3b-4f37-a0d7-fab96db7576f>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00715.warc.gz"} |
Identity matrix - (College Algebra) - Vocab, Definition, Explanations | Fiveable
Identity matrix
from class:
College Algebra
An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning any matrix multiplied by an identity
matrix remains unchanged.
congrats on reading the definition of identity matrix. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The identity matrix is always square, meaning it has the same number of rows and columns.
2. In notation, the identity matrix of size $n \times n$ is often denoted as $I_n$ or simply $I$ if the size is clear from context.
3. For any matrix $A$ of compatible dimensions, multiplying by an identity matrix satisfies $AI = IA = A$.
4. The inverse of a matrix $A$, if it exists, can be found such that $AA^{-1} = A^{-1}A = I$.
5. Identity matrices play a crucial role in solving systems of linear equations using inverse matrices.
Review Questions
• What properties make the identity matrix unique in terms of its elements?
• How does multiplying a matrix by an identity matrix affect the original matrix?
• Why is the concept of an identity matrix important when discussing inverses?
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/college-algebra/identity-matrix","timestamp":"2024-11-13T12:14:39Z","content_type":"text/html","content_length":"160966","record_id":"<urn:uuid:d2d241ed-9158-4543-980c-b4ecb305c821>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00381.warc.gz"} |
What is a dice building game? - Explained
The Lord of the Rings Dice Building Game is a semi-cooperative game—players will work together to defeat the game (or to put it another way—if Sauron wins, everybody loses!). If players defeat the
game, the player who has earned the most Glory will be declared the winner.
How do you play the dice game?
What games can be played with just dice?
Best Dice Games (Table of Contents)
• Bar Dice (aka Ship, Captain and Crew)
• Bunco.
• Balut.
• Yahtzee.
• Liar’s Dice.
• Shut the Box.
• DAGZ.
How do you play dice Forge?
What is a dice building game? – Related Questions
What are the rules for dice craft?
How to Play: Drag and drop the dominoes onto the grid. Place three dice together, regardless of the horizontal or vertical lines or triangles, can be combined to create a higher number of dice. You
can rotate dices before placing them.
How do you play building blocks with dice?
The main goal is for each player to roll the dice then take out the block with the number that indicates the multiplication of both dice . For example, if the person rolls a 6 and a 3, the block they
must take out is the one numbered 18 because 6 x 3 = 18.
How do you play the dice game in AC Valhalla?
Play starts by one player rolling their six dice, selecting which ones they want to play, then letting the other player roll. This happens three times in total, at which point the option to use a God
Favor is presented. After that, resolutions take place, with each dice effect and God Favor happening in order.
How do you set up dice forge dice?
Is Dice forge a good game?
Dice Forge is an enjoyable game to play. Changing out the die faces puts players in control of their luck. While you may not roll your upgraded die faces every turn, you can further forge your dice
to increase the likelihood of rolling better die faces.
How does the dice game work in Kingdom Come Deliverance?
Both players take it in turns to roll six dice, and when it’s your turn, you accumulate points towards that 2000 goal. To score points when it’s your turn, you need to roll either a 1, which equates
to one hundred points, or a 5, which equals fifty points scored.
What is the point of a dice tower?
A dice tower is a tool used by gamers to roll dice fairly. Dice are dropped into the top of the tower, and bounce off of various hidden platforms inside it before emerging from the front. Dice towers
eliminate some methods of cheating which may be performed when rolling dice by hand.
What is snake eye in dice?
A roll of two 1s (the lowest roll possible) on a pair of six-sided dice. The probability of rolling snake eyes is 1/36, or 2.777 %.
How does 7/11 dice work?
The first player throws the dice. If they roll a 7, an 11 or a double, the roller chooses a player to drink. If the roll is none of those, then the roller passes the dice to the left. Once a player
rolls a 7, 11 or a double, they choose a player to drink.
Why is 1 and 4 red on a dice?
Chinese and Korean dice will have a red 4-spot as well as the 1. The Chinese custom of painting the 4-spot red is said to have originated when an Emperor playing sugoruku with his queen was about to
lose and desperately needed fours to win the game. He cried out, threw the dice and they came up accordingly.
What does 11 mean in craps?
The Yo 11 bet is one of the most popular bets on a craps game. As its name implies, an 11 must roll in order for this wager to be a winner. This bet is good for only one roll. If it wins, dealers
will pay you, and themselves, at 15:1 odds and leave the original bet(s) on the table.
Is 7/11 the same as craps?
You win if a 7 or 11 roll, or lose if 2, 3, or 12 roll (known as “craps”). Any other number that rolls becomes the “point” and the point must roll again before a 7 to win.
What is the safest bet in craps?
The simplest, most fundamental bet in the game of craps, the pass bet, is also one of the very safest, with a low house edge of 1.41%. Pass bets pay even money – in other words, if you bet $10, you
win $10. With a pass bet, if the come out roll is 7 or 11, you win, while if the come out roll is 2, 3, or 12, you lose.
What numbers hit the most in craps?
Placing 6 & 8
The reason for this is that other than 7, the 6 and 8 are the most frequently rolled numbers. The house edge is higher on placing 6 and 8 at 1.52 per cent, which is still lower than most bets you can
make in a casino.
What does 12 mean in craps?
Two Craps Or Aces: If two aces or 2 is rolled, you win and are paid 30 to 1. Twelve Craps: If two sixes or 12 is rolled, you win and are paid 30 to 1. Three Craps Or Ace-Deuce: If ace-deuce or 3 is
thrown, you win and are paid 15 to 1. Eleven: If 11 is rolled, you win and are paid 15 to 1.
What is the best bet in craps?
What are the best craps bets in terms of odds? The best craps bet in terms of odds is the don’t pass/don’t come bet, which gives the house an edge of just 1.36%. To make things even better, you can
use the “free odds” option once the point is set and reduce the house edge to as low as 0.01%.
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Published on Apr 02, 2024
The Objective : The goal of this project was to see what effects rotational inertia would have on the velocity of a softball pitched fastball. Another goal of the project was that by finding the
results of the experiment to use the data to help younger pitchers understand more about pitching. I myself also learned a lot which was very helpful in the end. By slowly spreading the mass of the
pitcher farther and farther away from their axis of rotation, the results of this project were found.
The method is this project was that pitchers from within my county would pitch five fastballs.
The five pitches were a regular fastball, fastball finishing with their left arm out, fastball finishing with their right arm out, fastball finishing with their hip back, and fastball finishing doing
all three of the above (left, right, hip.)
A radar gun was placed behind them and at the end of each pitch the velocity was recorded.
Materials used for this project were quite simple, a softball, a radar gun, a pitching area, and a pitcher.
The results showed that the farther away mass was from the pitcher's axis of rotation the slower the resulting velocity would be.
The left arm, and doing all three of the finishes proved to result in the slowest velocity.
This was because in these two pitches the mass was farthest from the axis of rotation.
The hypothesis was proven correct through this experiment. It was proven that the farther away a pitcher's mass is from their axis of rotation their resulting velocity will be much slower than if
they finish tightly with all their mass closer to their axis of rotation. There were not many surprises that were encountered in the conducting of this experiment, and the results turned out very
good and helpful.
This project was to see what effects rotational inertia would have on the velocity of a softball pitched fastball.
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+y3+yz2−xy2−y2z−xyz+x2z+y2z+z3−xyz−yz2−xz2=x3+y3+z3−3xyz (On si... | Filo
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So, we obtain the following identity: Identity VIII: Example 25 : Factorise : Solution : Here, we have EXERCISE
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Question Text So, we obtain the following identity: Identity VIII: Example 25 : Factorise : Solution : Here, we have EXERCISE
Updated On Feb 27, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 135
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At St Mary’s and St Peter’s school we believe that Mathematics equips children with a uniquely powerful set of tools to understand and change the world. A high-quality mathematics education provides
a foundation for understanding the world, the ability to reason mathematically (building endurance), an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity
about the subject.
SMSP follows the National Curriculum, primarily through the use of White Rose and Abacus Evolve, which provides detailed guidance, planning and resources for the implementation of mathematics. This
ensures consistency, continuity and progression in the teaching of mathematics.
Additional materials and guidance come from Maths Mastery, NCETM and the DfE. In Early Years, the curriculum is guided by the Early Years Foundation Stage Framework.
Mathematics is taught as a discrete lesson (with cross curricular links where appropriate, e.g. Science).
• Knowledge in mathematics is broken down into three types:
• Declarative knowledge: facts, concepts and formulae, e.g. number bonds, times-tables.
• Procedural knowledge: methods, procedures, algorithms.
• Conditional knowledge: strategies formed from the combination of facts and methods to reason and problem solve.
Mathematical concepts are explored in a variety of contexts to give children a richer and deeper learning experience. The concrete, pictorial, abstract approach is employed across the school.
Children use objects and pictures to demonstrate and visualise abstract ideas, alongside numbers and symbols. Children are involved in a broad range of activities in order to learn mathematical
concepts and develop numeracy fluency, with the intention that pupils ‘keep up, not catch up’. They are given practical experience, investigative tasks, regular practice and memorisation of
mathematical facts and procedures to build mathematical
In order for children to make sense of a new idea or relationship, they need to incorporate it into their current understanding and see how it connects with ideas and relationships they have
encountered previously. The greater their understanding of what has been taught previously, the more sense they will be able to make of increasingly complex mathematics in the future. Therefore, we
believe that the key to knowing more mathematics lies in understanding.
We also believe that children who make sense of the mathematics they are learning have more memorable and enjoyable experiences that are more likely to be remembered in the long term. They will also
be able to do more as they understand how to push the boundaries of what they know and apply it to solve problems.
The curriculum is organised to be cumulative. This means that mathematical concepts that are taught earlier in the curriculum are revisited in the context of a new area of mathematics. This helps the
children to make connections between different mathematical concepts. Retrieving, using and applying concepts regularly, transferring to new concepts helps develop fluency as well as conceptual
We recognise that not all children come to each lesson at the same starting point and for this reason we adapt and scaffold learning according to the needs of the learners. Children benefit from
explicit, systematic instruction and from practice in using declarative and procedural knowledge. Pre-teaching and same-day interventions are effective examples of additional help. Children are
supported by teaching assistants/learning support staff where appropriate.
Pupils are taught new vocabulary and are provided with opportunities for meaningful dialogue to take place in lessons. It is by giving children opportunity to talk and by listening carefully to what
they have to say, that we can gather some of the richest data on their understanding and can plan the next steps.
Due to the impact of Covid-19 on children’s learning, the Recovery Curriculum was implemented at SMSP to ensure that the children were secure in the key areas of mathematics.
Curriculum Content
• Awareness of time
• Counting/Concept of number/Recognising/Sequencing of numbers
• 2D and 3D shape/Symmetry/Repeating patterns
• Comparison of quantities
• Weighing/Capacity/Distance
• Number Lines
• One more/one less
• Estimating
• Addition/Subtraction
• Halving/Sharing/Doubling
• Counting in 2s and 10s/Odd and even numbers
• Money
Year 1
• Revisiting and consolidating all topics covered in EYFS
• Counting up to and operating up to 100
• Read, write, compare and know number names to 20 and beyond
• Times tables as repeated addition for 2s 5s 10s
• Number bonds to 10, 20 and beyond
• Sorting using Venn/Carroll diagrams
• Position and direction
• Measuring using standard/non-standard units
• 1 and 2 more/less than numbers up to 100/counting on and back in 10s from any number
• Finding halves/quarters/3 quarters of shape
• Describe properties of 2D and 3D shape
• Read and write time to the hour/half past/quarter hour; analogue/digital
• Name, know value and solve problems using coins/give change up to 20p/make amounts using coins
• Read, interpret and correct pictograms and block graphs
Year 2
• Revisiting and consolidating all topics covered in EYFS/Year One
• Use < > = to compare numbers
• Locate numbers on landmarked line and grid
• Rounding numbers
• Count in 3s 4s to record multiplication problems/write x to go with arrays/groupings to show division and use the division sign
• Make amounts using coins and notes/money problems/ find change up to £20
• Know seconds/tell the time in multiples of 5 mins past and to the hour
• Use tally charts
• Weighing/Capacity/Distance/Volume using standard measurements
• Place halves on a number line/count in halves and quarters/understand and write mixed numbers/find a quarter/half and three quarters of a number
• Partition to add two 2 digit numbers/find the difference between the 2 digit numbers
• Use thermometers
• Subtraction using addition facts/add three or more small numbers
Year 3
• Revisiting and consolidating all topics covered in EYFS/Year One/Year Two
• The four operations using 2/3 digit numbers
• Counting up method for subtraction/subtract three digits using counting up
• Division facts from 2s 3s 4s 5s 8s 10s times tables including remainders/Use chunking to divide
• Partition 3 digit numbers
• Word problems involving all areas of maths
• Add 2 digit numbers using expanded column addition/add 3 digit numbers using column addition
• Grid method for multiplying 1 digit numbers
• Identify, place on a number line, find amounts for 1/3, 1/6, 1/8/compare, order and add fractions with the same denominator
• Tell the time to the nearest minute/recognise am/pm and 24hr clock times
• Introduce perimeter of shapes/understand horizontal/vertical/perpendicular/parallel/diagonal lines
• Recognise degrees and understand right-angles
Year 4
• Revisiting and consolidating all topics covered in EYFS/Year One/Year Two/Year Three
• Four operations using 4 digit numbers/read, write, compare 4 digit numbers/the four operations using 4 digit numbers
• Round 4 digit numbers to nearest 10, 100, 1000/Number bonds to 100
• Count on and back in 10, 100, 1000, 25 and 50/ find complements to multiples of 1000
• Derive factors of 2 digit numbers
• Find change from £10, £20 and £50
• Multiply and divide (2/3 digit numbers) ; 6, 7 and 9, 11 and 12, with remainders/understand that division is the inverse of multiplication
• Partition 4 digit numbers
• Add and subtract using column method (borrowing/carrying/decomposition)
• Reduce fractions to their simplest forms/understand compare and order decimal numbers/multiply and divide decimals by 10 and 100/read decimals to 2d.p/recognise decimal and fraction equivalents
• Calculate time intervals/convert pm times to 24 hour clock
• Converts between units of length/convert g/kg/read scales to the nearest 100ml
• Recognise and compare; acute, right and obtuse angles, perpendicular and parallel lines
• Draw lines of symmetry/identify and use coordinates to draw regular and irregular polygons/find coordinates of a shape after translation
• Draw and interpret line graphs
Year 5
• Revisiting and consolidating all topics covered in EYFS/Year One/Year Two/Year Three/Year Four
• Four operations using 5, 6 digits/read, write, compare and order 5/6 digit numbers/the four operations using 4 digit numbers/read, write, order and compare 3 place decimal numbers/negative
numbers in the context of temperature/write dates using Roman numerals
• Identify prime numbers
• Use counting up strategy to perform mental addition of 2 place decimals to next whole number/2-step word problems choosing appropriate methods
• Multiply and divide by 4 by doubling and halving twice/use mental multiplication strategies to multiply by 20, 25 and 9/use grid method to multiply pairs of 2 digit numbers/use short and long
multiplication and division/express remainders as fractions
• Place mixed numbers on a number line/convert improper fractions to mixed numbers and vice versa/multiply proper fractions by whole numbers/compare and order fractions with related denominators
• Add and subtract 0.1 and 0.01/know and recognise equivalent fractions and decimals to half, tenths and fifths.
• Understand what percentages are/find percentages of amounts of money/find equivalent fractions/solve problems involving fraction and percentage equivalents
• Find perimeters and convert cm to mm/use scales to weigh amounts to nearest half interval/conversion of measurements, distance and weight/understand imperial units and relate to daily life/solve
scaling problems involving measure and fractions
• Name parts of a circle including diameter, radius and circumference/draw circles using radius and a pair of compass
• Use a protractor to measure and draw angles in degrees/use terms and classify angles as obtuse, acute and reflex
• Draw polygons using co-ordinates/ read and mark co-ordinates in the first two quadrants/recognise and classify quadrilaterals/perpendicular and parallel sides/reflect simple shapes in the y axis
or in a line/note what happens to co-ordinates when translated
• Draw and interpret line conversion graphs to show change in temperature over time
Year 6
• Revisiting and consolidating all topics covered in EYFS/Year One/Year Two/Year Three/Year Four/Year Five
• Add and subtract negative numbers
• Multiply and divide by 10, 100 and 1000; use mental multiplication strategies to multiply by numbers such as 4, 8, 5, 25, 19, 29 and 99; use mental strategies to divide by 2, 4, 8, 5, 20 and 25.
• use long multiplication to multiply 3-digit and 4-digit numbers by teens numbers; use short division to divide 3- and 4-digit numbers by 1-digit numbers, including those which leave a remainder;
express a remainder as a fraction, simplifying where possible.
• Compare fractions with unlike, denominators; correctly use the terms fraction, denominator and numerator; understand improper fractions and mixed numbers and add fractions with the same
denominator, writing the answer as a mixed number; find non-unit fractions of amounts; add and subtract unit fractions with different denominators including mixed numbers; multiply fractions less
than 1 by whole numbers, converting improper fractions to whole numbers; use commutativity to efficiently multiply fractions by whole numbers; divide unit and non-unit fractions by whole numbers;
solve word problems involving fractions
• Round decimals to nearest tenth and whole number and place on a number line; convert decimals (up to 3 places) to fractions and vice-versa; use mental strategies to find simple percentages of
amounts, including money.
• Convert between grams and kilograms, millilitres and litres, millimetres and centimetres, centimetres and metres, metres and kilometres, and miles and kilometres; revise reading the 24-hour clock
and convert 12-hour times to 24-hour; read and write Roman numerals; find time intervals using the 24-hour clock;
• Calculate the perimeter, area and volume of shapes, and know their units of measurement; calculate the area of a triangle using the formula A = 1/2 b × h ; find the area of parallelograms using
the formula A = b × h ; name and describe properties of 3D shapes; systematically find and compare nets for different 3D shapes.
• Solve problems involving number up to 3 decimal places; express missing number problems algebraically and find pairs of numbers that satisfy equations involving two unknowns; find missing lengths
and angles; understand how brackets can be used in calculation problems; use knowledge of the order of operations to carry out calculations involving the four operations; use mathematical
reasoning to investigate; solve word problems involving multiplication including two-step problems and finding change
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