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Explaining the Chi-Square Test: What it is and How it Works
We know there are a number of statistical tests for establishing the relationship between continuous data variables. But what if you have categorical variables? The Chi-Square Test allows you to
explore the relationship or association between categorical variables. This article will explain what the Chi-Square Test is, how it works, and how you can apply it in your organization.
Overview: What is the Chi-Square Test?
When you want to see whether there is an association between two categorical variables, you can use a Chi-Square test for association. This tests if the probabilities of items being classified for
one variable depends on the classification of the other variable.
The Chi-Square Test is a form of a hypothesis test. As in all hypothesis testing, there is a null hypothesis and an alternate or alternative hypothesis. It is written this way:
• Ho: There is no association between two variables
• Ha: There is association between two variables
The data is captured and formatted into a table. The general design is as follows:
As an example, let’s see what it would look like if you wanted to test whether there was any association between types of promotional materials and action by the customer.
Notice that we have two categorical variables: promotional item and customer action. The values in the table cells are the number of times a specific customer action occurred for a specific type of
promotional item.
You would usually run this type of analysis with statistical software. The output would be presented in a tabular format showing:
1. Observed values
2. Calculated expected values
3. Calculated contribution to the overall Chi-Square value
4. Calculated p-value used to determine whether to reject or not reject the Ho
Here is the output for our example:
Note that the p=value is zero. Since the null hypothesis stated that there was no association, the p-value tells us to reject the null hypothesis and conclude that the alternate hypothesis is true.
There is an association between promotional items and customer action.
3 benefits of the Chi-Square Test
You want to replace your intuition with solid statistical analysis. The Chi-Square test gives you the following benefits over using your intuition.
1. Lets you explore the relationships between categorical variables
You are familiar and probably comfortable looking at relationships between continuous variables. The Chi-Square Test gives you a solid option for looking at relationships and associations between
categorical or discrete variables.
2. Using statistical software makes the calculations and conclusions easy to understand
By using statistical software to explore the relationship between categorical variables, you will get an output that allows you to get insight and take action if there is a relationship and
3. Allows a versatile use of data
Since the data is formatted in a table, you can explore any number of categorical variables and are not restricted to just using a 2×2 table.
Why is the Chi-Square Test important to understand?
The output of the Chi-Square Test can help you better understand what your data is telling you.
Primary method for establishing relationship and association between categorical variables
If you are using categorical variables, you don’t really have a choice of using another statistical tool. While you might not need to understand all the underlying statistical calculations, you
should understand when it is appropriate to use the Chi-Square Test.
Useful tool for dealing with survey data
Many survey results are in a categorical or attribute format (Gender, Income Range, Age Range, Ethnicity, Location, etc.). The Chi-Square test allows you to analyze this information and do the
necessary cross tabulations to determine whether there is a statistical difference between the segments/categories in how the respondent answered the question.
Points out the variables that have the strongest association
One of the outputs of the Chi-Square Test is the percent contribution to the Total Chi-Square value. The variable with the highest contribution can be considered to have the strongest association
although it should be interpreted as a relative association rather than an absolute value.
An industry example of using the Chi-Square Test
A large consumer products company was interested in whether there was a relationship or association between their portfolio of products and when they are used by the customer. They were planning to
use this insight to guide their advertising and promotion budgets and content. A national consulting firm completed extensive surveys, and the company captured data about when and under what
circumstances or occasions customers used their various products.
Although the organization ultimately used more sophisticated statistical tools, the Chi-Square Test was an easy and quick first step to understand the data and relationships. The outcome and insights
were very interesting. Here are a few examples of what they found out:
1. One product which they thought was applicable for any time of day, was associated and used by the customer primarily as a breakfast item.
2. Rather than being associated with group occasions, one product turned out to be viewed as a personal reward after a hard day at work.
3. A popular product was used more often as a mixer rather than being consumed on its own.
These and many other insights allowed the marketing department to shift its advertising focus and content to be consistent with how and when their key products were being consumed. The result was a
nice increase in product sales.
3 best practices when thinking about the Chi-Square Test
As in all statistical testing, there are some assumptions and watch-outs when doing a Chi-Square Test.
1. Agreement with the operational definitions of the categorical variables
Since the variables used in a Chi-Square Test are categorical in nature, it is important that there is a common and agreed upon operational definition of these variables. For example, if one of the
variables is “Customer Satisfaction” be sure that everyone agrees with the definition of the phrase so when you collect the data, there is consistency.
2. Test assumptions
Because of the underlying statistical assumptions of the Chi-Square Test, verify that all the assumptions are satisfied. For example, the number of observations in a cell must be five or greater in
number to satisfy the underlying assumptions of the distribution.
3. Keep your variables independent
Your categories need to be independent and the data randomly selected for the Chi-Square test to be valid.
Frequently Asked Questions (FAQ) about the Chi-Square Test
What type of data do I use for a Chi-Square Test?
The data variable must be categorical. For example; Male/Female, North/South/East/West, Red/Green/Blue, Crispy/Soggy, etc.
Am I restricted to only using a 2×2 table for the Chi-Square Test?
No. You can use any number of levels within your category type.
What happens if I have small sample sizes?
Unfortunately, the Chi-Square Test is sensitive to small sample sizes. You must have at least 5 observations in any cell. Variables must also be independent. In the event that you do have small
sample sizes, you will need to use the Fisher’s Exact Test to explore the association between your categorical variables.
Let’s summarize the Chi-Square Test
The Chi-Square Test is a handy tool for establishing the relationship or association between categorical variables. If you have a larger enough sample size and your variables are independent then
statistical software can be used to do the calculations.
You can use the p-value to determine whether there is a statistically significant association or not. In addition to determining an overall association, you can also gain insight into what
characteristics of your categories might be the biggest contributors to your association. | {"url":"https://www.isixsigma.com/dictionary/chi-square-test/","timestamp":"2024-11-08T04:51:42Z","content_type":"text/html","content_length":"218108","record_id":"<urn:uuid:56e1d0e1-fee0-40ce-b596-aa051d1da5c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00880.warc.gz"} |
Beyond Worst-Case Analysis
Data and Information Review articles
Beyond Worst-Case Analysis
The need for deeply understanding when algorithms work (or not) has never been greater.
Comparing different algorithms is hard. For almost any pair of algorithms and measure of algorithm performance like running time or solution quality, each algorithm will perform better than the other
on some inputs.^a For example, the insertion sort algorithm is faster than merge sort on already-sorted arrays but slower on many other inputs. When two algorithms have incomparable performance, how
can we deem one of them “better than” the other?
Key Insights
• Worse-case analysis takes a “Murphy’s Law” approach to algorithm analysis, which is too crude to give meaningful algorithmic guidance for many important problems, including linear programming,
clustering, caching, and neural network training.
• Research going “beyond worst-case analysis” articulates properties of realistic inputs, and proves rigorous and meaningful algorithmic guarantees for inputs with these properties.
• Much of the present and future research in the area is motivated by the unreasonable effectiveness of machine learning algorithms.
Worst-case analysis is a specific modeling choice in the analysis of algorithms, where the overall performance of an algorithm is summarized by its worst performance on any input of a given size. The
“better” algorithm is then the one with superior worst-case performance. Merge sort, with its worst-case asymptotic running time of Θ(n log n) for arrays of length n, is better in this sense than
insertion sort, which has a worst-case running time of Θ(n^2).
While crude, worst-case analysis can be tremendously useful, and it is the dominant paradigm for algorithm analysis in theoretical computer science. A good worst-case guarantee is the best-case
scenario for an algorithm, certifying its general-purpose utility and absolving its users from understanding which inputs are relevant to their applications. Remarkably, for many fundamental
computational problems, there are algorithms with excellent worst-case performance guarantees. The lion’s share of an undergraduate algorithms course comprises algorithms that run in linear or
near-linear time in the worst case.
For many problems a bit beyond the scope of an undergraduate course, however, the downside of worst-case analysis rears its ugly head. Here, I review three classical examples where worst-case
analysis gives misleading or useless advice about how to solve a problem; further examples in modern machine learning are described later. These examples motivate the alternatives to worst-case
analysis described in the article.^b
The simplex method for linear programming. Perhaps the most famous failure of worst-case analysis concerns linear programming, the problem of optimizing a linear function subject to linear
constraints (Figure 1). Dantzig’s simplex method is an algorithm from the 1940s that solves linear programs using greedy local search on the vertices on the solution set boundary, and variants of it
remain in wide use to this day. The enduring appeal of the simplex method stems from its consistently superb performance in practice. Its running time typically scales modestly with the input size,
and it routinely solves linear programs with millions of decision variables and constraints. This robust empirical performance suggested the simplex method might well solve every linear program in a
polynomial amount of time.
Figure 1. A two-dimensional linear programming problem.
In 1972, Klee and Minty showed by example that there are contrived linear programs that force the simplex method to run in time exponential in the number of decision variables (for all of the common
“pivot rules” for choosing the next vertex). This illustrates the first potential pitfall of worst-case analysis: overly pessimistic performance predictions that cannot be taken at face value. The
running time of the simplex method is polynomial for all practical purposes, despite the exponential prediction of worst-case analysis.
To add insult to injury, the first worst-case polynomial-time algorithm for linear programming, the ellipsoid method, is not competitive with the simplex method in practice.^c Taken at face value,
worst-case analysis recommends the ellipsoid method over the empirically superior simplex method. One framework for narrowing the gap between these theoretical predictions and empirical observations
is smoothed analysis, discussed later in this article.
Clustering and NO-hard optimization problems. Clustering is a form of unsupervised learning (finding patterns in unlabeled data), where the informal goal is to partition a set of points into
“coherent groups” (Figure 2). One popular way to coax this goal into a well-defined computational problem is to posit a numerical objective function over clusterings of the point set, and then seek
the clustering with the best objective function value. For example, the goal could be to choose k cluster centers to minimize the sum of the distances between points and their nearest centers (the k
-median objective) or the sum of the squared such distances (the k-means objective). Almost all natural optimization problems that are defined over clusterings are NP-hard.
Figure 2. One possible way to group data points into three clusters.
In practice, clustering is not viewed as a particularly difficult problem. Lightweight clustering algorithms, like Lloyd’s algorithm for k-means and its variants, regularly return the intuitively
“correct” clusterings of real-world point sets. How can we reconcile the worst-case intractability of clustering problems with the empirical success of relatively simple algorithms?”^d
One possible explanation is that clustering is hard only when it doesn’t matter.^18 For example, if the difficult instances of an NP-hard clustering problem look like a bunch of random unstructured
points, who cares? The common use case for a clustering algorithm is for points that represent images, or documents, or proteins, or some other objects where a “meaningful clustering” is likely to
exist. Could instances with a meaningful clustering be easier than worst-case instances? This article surveys recent theoretical developments that support an affirmative answer.
Cache replacement policies. Consider a system with a small fast memory (the cache) and a big slow memory. Data is organized into blocks called pages, with up to k different pages fitting in the cache
at once. A page request results in either a cache hit (if the page is already in the cache) or a cache miss (if not). On a cache miss, the requested page must be brought into the cache. If the cache
is already full, then some page in it must be evicted. A cache policy is an algorithm for making these eviction decisions. Any systems textbook will recommend aspiring to the least recently used
(LRU) policy, which evicts the page whose most recent reference is furthest in the past. The same textbook will explain why: real-world page request sequences tend to exhibit locality of reference,
meaning that recently requested pages are likely to be requested again soon. The LRU policy uses the recent past as a prediction for the near future. Empirically, it typically suffers fewer cache
misses than competing policies like first-in first-out (FIFO).
Sleator and Tarjan^37 founded the area of online algorithms, which are algorithms that must process their input as it arrives over time (like cache policies). One of their first observations was that
worst-case analysis, straightforwardly applied, provides no useful insights about the performance of different cache replacement policies. For every deterministic policy and cache size k, there is a
pathological page request sequence that triggers a page fault rate of 100%, even though the optimal clairvoyant replacement policy (known as Bélády’s algorithm) would have a page fault rate of at
most (1/k)%. This observation is troublesome both for its absurdly pessimistic performance prediction and for its failure to differentiate between competing replacement policies (like LRU vs. FIFO).
One solution, discussed next, is to choose an appropriately fine-grained parameterization of the input space and to assess and compare algorithms using parameterized guarantees.
Models of Typical Instances
Maybe we shouldn’t be surprised that worst-case analysis fails to advocate LRU over FIFO. The empirical superiority of LRU is due to the special structure in real-world page request
sequences—locality of reference—and traditional worst-case analysis provides no vocabulary to speak about this structure.^e This is what work on “beyond worst-case analysis” is all about:
articulating properties of “real-world” inputs, and proving rigorous and meaningful algorithmic guarantees for inputs with these properties.
Research in the area has both a scientific dimension, where the goal is to develop transparent mathematical models that explain empirically observed phenomena about algorithm performance, and an
engineering dimension, where the goals are to provide accurate guidance about which algorithm to use for a problem and to design new algorithms that perform particularly well on the relevant inputs.
One exemplary result in beyond worst-case analysis is due to Albers et al.,^2 for the online paging problem described in the introduction. The key idea is to parameterize page request sequences
according to how much locality of reference they exhibit, and then prove parameterized worst-case guarantees. Refining worst-case analysis in this way leads to dramatically more informative results.^
Locality of reference is quantified via the size of the working set of a page request sequence. Formally, for a function f : N → N, we say that a request sequence conforms to f if, in every window of
w consecutive page requests, at most f(w) distinct pages are requested. For example, the identity function f(w)= w imposes no restrictions on the page request sequence. A sequence can only conform to
a sublinear function like or f(w) = [1 + log[2] w] if it exhibits locality of reference.^g
The following worst-case guarantee is parameterized by a number α[f](k), between 0 and 1, that we discuss shortly; recall that k denotes the cache size. It assumes that the function f is “concave” in
the sense that the number of inputs with value x under f (that is, |f^-1(x)|) is nondecreasing in x.
Theorem 1 (Albers et al.^2)
(a) For every f and k and every deterministic cache replacement policy, the worst-case page fault rate (over sequences that conform to f) is at least α[f](k).
A. For every f and k and every sequence that conforms to f, the page fault rate of the LRU policy is at most α[f](k).
B. There exists a choice of f and k, and a page request sequence that conforms to f, such that the page fault rate of the FIFO policy is strictly larger than α[f](k).
Parts (a) and (b) prove the worst-case optimality of the LRU policy in a strong sense, f-by-f and k-by-k. Part (c) differentiates LRU from FIFO, as the latter is suboptimal for some (in fact, many)
choices of f and k.
The guarantees in Theorem 1 are so good that they are meaningful even when taken at face value—for sublinear f‘s, α[f](k) goes to 0 reasonably quickly with k. For example, if , then α[f] (k) scales
with Thus, with a cache size of 10,000, the page fault rate is always at most 1%. If f(w) = [1 + log[2] w], then α[f](k) goes to 0 even faster with k, roughly as k/2^k.^h
Stable Instances
Are point sets with meaningful clusterings easier to cluster than worst-case point sets? Here, we describe one way to define a “meaningful clustering,” due to Bilu and Linial;^12 for others, see
Ackerman and Ben-David,^1 Balcan et al.,^9 Daniely et al.,^18 Kumar and Kannan,^29 and Ostrovsky et al.^34
The maximum cut problem. Suppose you have a bunch of data points representing images of cats and images of dogs, and you would like to automatically discover these two groups. One approach is to
reduce this task to the maximum cut problem, where the goal is to partition the vertices V of a graph G with edges E and nonnegative edge weights into two groups, while maximizing the total weight of
the edges that have one endpoint in each group. The reduction forms a complete graph G, with vertices corresponding to the data points, and assigns a weight w[e] to each edge e indicating how
dissimilar its endpoints are. The maximum cut of G is a 2-clustering that tends to put dissimilar pairs of points in different clusters.
There are many ways to quantify “dissimilarity” between images, and different definitions might give different optimal 2-clusterings of the data points. One would hope that, for a range of reasonable
measures of dissimilarity, the maximum cut in the example above would have all cats on one side and all dogs on the other. In other words, the maximum cut should be invariant under minor changes to
the specification of the edge weights (Figure 3).
Figure 3. In a perturbation-stable maximum cut instance, the optimal solution is invariant under small perturbations to the edges’ weights.
Definition 2 (Bilu and Linial^12). An instance G = (V, E, w) of the maximum cut problem is γ-perturbation stable if, for all ways of multiplying the weight w[e] of each edge e by a factor a[e] ϵ [1,
γ], the optimal solution remains the same.
A perturbation-stable instance has a “clearly optimal” solution—a uniqueness assumption on steroids—thus formalizing the idea of a “meaningful clustering.” In machine learning parlance, perturbation
stability can be viewed as a type of “large margin” assumption.
The maximum cut problem is NP-hard in general. But what about the special case of γ-perturbation-stable instances? As γ increases, fewer and fewer instances qualify as γ-perturbation stable. Is there
a sharp stability threshold—a value of γ where the maximum cut problem switches from NP-hard to polynomial-time solvable?
Makarychev et al.^30 largely resolved this question. On the positive side, they showed that if γ is at least a slowly growing function of the number of vertices n, then the maximum cut problem can be
solved in polynomial time for all γ-perturbation stable instances.^i Makarychev et al. use techniques from the field of metric embeddings to show that, in such instances, the unique optimal solution
of a certain semidefinite programming relaxation corresponds precisely to the maximum cut.^j Semi-definite programs are convex programs, and can be solved to arbitrary precision in polynomial time.
There is also evidence that the maximum cut cannot be recovered in polynomial time in γ-perturbation-stable instances for much smaller values of γ.^30
Other clustering problems. Bilu and Linial^12 defined γ-perturbation-stable instances specifically for the maximum cut problem, but the definition makes sense more generally for any optimization
problem with a linear objective function. The study of γ-perturbation-stable instances has been particularly fruitful for NP-hard clustering problems in metric spaces, where interpoint distances are
required to satisfy the triangle inequality. Many such problems, including the k-means, k-median, and k-center problems, are polynomial-time solvable already in 2-perturbation-stable instances.^5,10
The algorithm in Angelidakis et al.,^5 like its precursor in Awasthi et al.,^8 is inspired by the well known single-linkage clustering algorithm. It computes a minimum spanning tree (where edge
weights are the interpoint distances) and uses dynamic programming to optimally remove k – 1 edges to define k clusters. To the extent that we are comfortable identifying “instances with a meaningful
clustering” with 2-perturbation-stable instances, these results give a precise sense in which clustering is hard only when it doesn’t matter.^k
The unreasonable effectiveness of modern machine learning algorithms has thrown down the gauntlet to algorithms researchers, and there is perhaps no other problem domain with a more urgent need
for the beyond worst-case approach.
Overcoming NP-hardness. Polynomial-time algorithms for γ-perturbation-stable instances continue the age-old tradition of identifying “islands of tractability,” meaning polynomial-time solvable
special cases of NP-hard problems. Two aspects of these results diverge from a majority of 20^th century research on tractable special cases. First, perturbation-stability is not an easy condition to
check, in contrast to a restriction like graph planarity or Horn-satisfiability. Instead, the assumption is justified with a plausible narrative about why “real-world instances” might satisfy it, at
least approximately. Second, in most work going beyond worst-case analysis, the goal is to study general-purpose algorithms, which are well defined on all inputs, and use the assumed instance
structure only in the algorithm analysis (and not explicitly in its design). The hope is the algorithm continues to perform well on many instances not covered by its formal guarantee. The results
here for mathematical programming relaxations and single-linkage-based algorithms are good examples of this paradigm.
Analogy with sparse recovery. There are compelling parallels between the recent research on clustering in stable instances and slightly older results in a field of applied mathematics known as sparse
recovery, where the goal is to reverse engineer a “sparse” object from a small number of clues about it. A common theme in both areas is identifying relatively weak conditions under which a tractable
mathematical programming relaxation of an NP-hard problem is guaranteed to be exact, meaning the original problem and its relaxation have the same optimal solution.
For example, a canonical problem in sparse recovery is compressive sensing, where the goal is to recover an unknown sparse signal (a vector of length n) from a small number m of linear measurements
of it. Equivalently, given an m x n measurement matrix A with m << n and the measurement results b = Az, the problem is to figure out the signal z. This problem has several important applications,
for example in medical imaging. If z can be arbitrary, then the problem is hopeless: since m < n, the linear system Ax = b is underdetermined and has an infinite number of solutions (of which z is
only one). But many real-world signals are (approximately) k-sparse in a suitable basis for small k, meaning that (almost) all of the mass is concentrated on k coordinates.^l The main results in
compressive sensing show that, under appropriate assumptions on A, the problem can be solved efficiently even when m is only modestly bigger than k (and much smaller than n).^15,20 One way to prove
these results is to formulate a linear programming relaxation of the (NP-hard) problem of computing the sparsest solution to Ax = b, and then show this relaxation is exact.
Planted and Semi-Random Models
Our next genre of models is also inspired by the idea that interesting instances of a problem should have “clearly optimal” solutions, but differs from the stability conditions in assuming a
generative model—a specific distribution over inputs. The goal is to design an algorithm that, with high probability over the assumed input distribution, computes an optimal solution in polynomial
The planted clique problem. In the maximum clique problem, the input is an undirected graph G = (V, E), and the goal is to identify the largest subset of vertices that are mutually adjacent. This
problem is NP-hard, even to approximate by any reasonable factor. Is it easy when there is a particularly prominent clique to be found?
Jerrum^27 suggested the following generative model: There is a fixed set V of n vertices. First, each possible edge (u, v) is included independently with 50% probability. This is also known as an
Erdös-Renyi random graph with edge density 1/2. Second, for a parameter k ∈ {1, 2, …, n}, a subset Q ⊆ V of k vertices is chosen uniformly at random, and all remaining edges with both endpoints in Q
are added to the graph (thus making Q a k-clique).
How big does k need to be before Q becomes visible to a polynomial-time algorithm? The state of the art is a spectral algorithm of Alon et al.,^3 which recovers the planted clique Q with high
probability provided k is at least a constant times Recent work suggests that efficient algorithms cannot recover Q for significantly smaller values of k.^11
An unsatisfying algorithm. The algorithm of Alon et al.^3 is theoretically interesting and plausibly useful. But if we take k to be just a bit bigger, at least a constant times then there is an
uninteresting and useless algorithm that recovers the planted clique with high probability: return the k vertices with the largest degrees. To see why this algorithm works, think first about the
sampled Erdös-Renyi random graph, before the clique Q is planted. The expected degree of each vertex is ≈ n/2, with standard deviation Textbook large deviation inequalities show that, with high
probability, the degree of every vertex is within standard deviations of its expectation (Figure 4). Planting a clique Q of size for a sufficiently large constant a, then boosts the degrees of all
of the clique vertices enough that they catapult past the degrees of all of the non-clique vertices.
Figure 4. Degree distribution of an Erdős-Rényi graph with edge density 1/2, before planting the k-clique Q. If then the planted clique will consist of the k vertices with the highest degrees.
What went wrong? The same thing that often goes wrong with pure average-case analysis—the solution is brittle and overly tailored to a specific distributional assumption. How can we change the input
model to encourage the design of algorithms with more robust guarantees? Can we find a sweet spot between average-case and worst-case analysis?
Semi-random models. Blum and Spencer^13 proposed studying semi-random models, where nature and an adversary collaborate to produce an input. In many such models, nature first samples an input from a
specific distribution (like the probabilistic planted clique model noted here), which is then modified by the adversary before being presented as an input to an algorithm. It is important to restrict
the adversary’s power, so that it cannot simply throw out nature’s starting point and replace it with a worst-case instance. Feige and Killian^24 suggested studying monotone adversaries, which can
only modify the input by making the optimal solution “more obviously optimal.” For example, in the semi-random version of the planted clique problem, a monotone adversary is only allowed to remove
edges that are not in the planted clique Q—it cannot remove edges from Q or add edges outside Q.
Semi-random models with a monotone adversary may initially seem no harder than the planted models that they generalize. But let’s return to the planted clique model with where the “top-k degrees”
algorithm succeeds with high probability when there is no adversary. A monotone adversary can easily foil this algorithm in the semi-random planted clique model, by removing edges between clique and
non-clique vertices to decrease the degrees of the former back down to ≈ n/2. Thus the semi-random model forces us to develop smarter, more robust algorithms.^m
For the semi-random planted clique model, Feige and Krauthgamer^24 gave a polynomial-time algorithm that recovers the clique with high probability provided The spectral algorithm by Alon et al.^3
achieved this guarantee only in the standard planted clique model, and it does not provide any strong guarantees for the semi-random model. The algorithm of Feige and Krauthgamer^24 instead uses a
semidefinite programming relaxation of the problem. Their analysis shows that this relaxation is exact with high probability in the standard planted clique model (provided and uses the monotonicity
properties of optimal mathematical programming solutions to argue this exactness cannot be sabotaged by any monotone adversary.
Smoothed Analysis
Smoothed analysis is another example of a semi-random model, now with the order of operations reversed: an adversary goes first and chooses an arbitrary input, which is then perturbed slightly by
nature. Smoothed analysis can be applied to any problem where “small perturbations” make sense, including most problems with real-valued inputs. It can be applied to any measure of algorithm
performance, but has proven most effective for running time analyses.
Like other semi-random models, smoothed analysis has the benefit of potentially escaping worst-case inputs (especially if they are “isolated”), while avoiding overfitting a solution to a specific
distributional assumption. There is also a plausible narrative about why real-world inputs are captured by this framework: whatever problem you would like to solve, there are inevitable inaccuracies
in its formulation (from measurement error, uncertainty, and so on).
The simplex method. Spielman and Teng^38 developed the smoothed analysis framework with the specific goal of proving that bad inputs for the simplex method are exceedingly rare. Average case analyses
of the simplex method from the 1980s (for example, Borgwardt^14) provide evidence for this thesis, but smoothed analysis provides more robust support for it.
The perturbation model in Spielman and Teng^38 is: independently for each entry of the constraint matrix and right-hand side of the linear program, add a Gaussian (that is, normal) random variable
with mean 0 and standard deviation σ.^n The parameter σ interpolates between worst-case analysis (when σ = 0) and pure average-case analysis (as σ → ∞, the perturbation drowns out the original linear
program). The main result states that the expected running time of the simplex method is polynomial as long as typical perturbations have magnitude at least an inverse polynomial function of the
input size (which is small!).
Theorem 3 (Spielman and Teng^38)
For every initial linear program, in expectation over the perturbation to the program, the running time of the simplex method is polynomial in the input size and in ^1/σ.
The running time blow-up as σ → 0 is necessary because the worst-case running time of the simplex method is exponential. Several researchers have devised simpler analyses and better polynomial
running times, most recently Dadush and Huiberts.^17 All of these analyses are for a specific pivot rule, the “shadow pivot rule.” The idea is to project the high-dimensional feasible region of a
linear program onto a plane (the “shadow”) and run the simplex method there. The hard part of proving Theorem 3 is showing that, with high probability over nature’s perturbations, the perturbed
instance is well-conditioned in the sense that each step of the simplex method makes significant progress traversing the boundary of the shadow.
Local search. A local search algorithm for an optimization problem maintains a feasible solution, and iteratively improves that solution via “local moves” for as long as possible, terminating with a
locally optimal solution. Local search heuristics are ubiquitous in practice, in many different application domains. Many such heuristics have an exponential worst-case running time, despite always
terminating quickly in practice (typically within a sub-quadratic number of iterations). Resolving this disparity is right in the wheelhouse of smoothed analysis. For example, Lloyd’s algorithm for
the k-means problem can require an exponential number of iterations to converge in the worst case, but needs only an expected polynomial number of iterations in the smoothed case (see Arthur et al.^7
and the references therein).^o
Much remains to be done, however. For a concrete challenge problem, let’s revisit the maximum cut problem. The input is an undirected graph G = (V, E) with edge weights, and the goal is to partition
V into two groups to maximize the total weight of the edges with one endpoint in each group. Consider a local search algorithm that modifies the current solution by moving a single vertex from one
side to the other (known as the “flip neighborhood”), and performs such moves as long as they increase the sum of the weights of the edges crossing the cut. In the worst case, this local search
algorithm can require an exponential number of iterations to converge. What about in the smoothed analysis model, where a small random perturbation is added to each edge’s weight? The natural
conjecture is that local search should terminate in a polynomial number of iterations, with high probability over the perturbation. This conjecture has been proved for graphs with maximum degree O
(log n)^21 and for the complete graph;^4 for general graphs, the state-of-the-art is a quasi-polynomial-time guarantee (meaning n^O(log n) iterations).^22
More ambitiously, it is tempting to speculate that for every natural local search problem, local search terminates in a polynomial number of iterations in the smoothed analysis model (with high
probability). Such a result would be a huge success story for smoothed analysis and beyond worst-case analysis more generally.
On Machine Learning
Much of the present and future of research going beyond worst-case analysis is motivated by advances in machine learning.^p The unreasonable effectiveness of modern machine learning algorithms has
thrown down the gauntlet to algorithms researchers, and there is perhaps no other problem domain with a more urgent need for the beyond worst-case approach.
To illustrate some of the challenges, consider a canonical supervised learning problem, where a learning algorithm is given a dataset of object-label pairs and the goal is to produce a classifier
that accurately predicts the label of as-yet-unseen objects (for example, whether or not an image contains a cat). Over the past decade, aided by massive datasets and computational power, deep neural
networks have achieved impressive levels of performance across a range of prediction tasks.^25 Their empirical success flies in the face of conventional wisdom in multiple ways. First, most neural
network training algorithms use first-order methods (that is, variants of gradient descent) to solve nonconvex optimization problems that had been written off as computationally intractable. Why do
these algorithms so often converge quickly to a local optimum, or even to a global optimum?^q Second, modern neural networks are typically over-parameterized, meaning that the number of free
parameters (weights and biases) is considerably larger than the size of the training dataset. Over-parameterized models are vulnerable to large generalization error (that is, overfitting), but
state-of-the-art neural networks generalize shockingly well.^40 How can we explain this? The answer likely hinges on special properties of both real-world datasets and the optimization algorithms
used for neural network training (principally stochastic gradient descent).^r
There are compelling parallels between the recent research on clustering in stable instances and slightly older results in a field of applied mathematics known as sparse recovery, where the goal
is to reverse engineer a “sparse” object from a small number of clues around it.
Another interesting case study, this time in unsupervised learning, concerns topic modeling. The goal here is to process a large unlabeled corpus of documents and produce a list of meaningful topics
and an assignment of each document to a mixture of topics. One computationally efficient approach to the problem is to use a singular value decomposition subroutine to factor the term-document matrix
into two matrices, one that describes which words belong to which topics, and one indicating the topic mixture of each document.^35 This approach can lead to negative entries in the matrix factors,
which hinders interpretability. Restricting the matrix factors to be nonnegative yields a problem that is NP-hard in the worst case, but Arora et al.^6 gave a practical factorization algorithm for
topic modeling that runs in polynomial time under a reasonable assumption about the data. Their assumption states that each topic has at least one “anchor word,” the presence of which strongly
indicates that the document is at least partly about that topic (such as the word “Durant” for the topic “basketball”). Formally articulating this property of data was an essential step in the
development of their algorithm.
The beyond worst-case viewpoint can also contribute to machine learning by “stress-testing” the existing theory and providing a road map for more robust guarantees. While work in beyond worst-case
analysis makes strong assumptions relative to the norm in theoretical computer science, these assumptions are usually weaker than the norm in statistical machine learning. Research in the latter
field often resembles average-case analysis, for example when data points are modeled as independent and identically distributed samples from some (possibly parametric) distribution. The semi-random
models described earlier in this article are role models in blending adversarial and average-case modeling to encourage the design of algorithms with robustly good performance. Recent progress in
computationally efficient robust statistics shares much of the same spirit.^19
With algorithms, silver bullets are few and far between. No one design technique leads to good algorithms for all computational problems. Nor is any single analysis framework—worstcase analysis or
otherwise—suitable for all occasions. A typical algorithms course teaches several paradigms for algorithm design, along with guidance about when to use each of them; the field of beyond worst-case
analysis holds the promise of a comparably diverse toolbox for algorithm analysis.
Even at the level of a specific problem, there is generally no magical, always-optimal algorithm—the best algorithm for the job depends on the instances of the problem most relevant to the specific
application. Research in beyond worst-case analysis acknowledges this fact while retaining the emphasis on robust guarantees that is central to worst-case analysis. The goal of work in this area is
to develop novel methods for articulating the relevant instances of a problem, thereby enabling rigorous explanations of the empirical performance of known algorithms, and also guiding the design of
new algorithms optimized for the instances that matter.
With algorithms increasingly dominating our world, the need to understand when and why they work has never been greater. The field of beyond worst-case analysis has already produced several striking
results, but there remain many unexplained gaps between the theoretical and empirical performance of widely used algorithms. With so many opportunities for consequential research, I suspect the best
work in the area is yet to come.
Acknowledgments. I thank Sanjeev Arora, Ankur Moitra, Aravindan Vijayaraghavan, and four anonymous reviewers for several helpful suggestions. This work was supported in part by NSF award CCF-1524062,
a Google Faculty Research Award, and a Guggenheim Fellowship. This article was written while the author was at Stanford University.
Figure. Watch the author discuss this work in the exclusive Communications video. https://cacm.acm.org/videos/beyond-worst-case-analysis
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1,285 research outputs found
The spectra of a particular class of PT symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper we present an explanation for these
spectral properties in terms of quantisation conditions obtained from the complex WKB method. In particular, we consider the relation of the quantisation conditions to the reality and positivity
properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al. in [1,2].Comment: 22 pages, 13 figures. Added references,
minor revision
In this paper we consider excited state g-functions, that is, overlaps between boundary states and excited states in boundary conformal field theory. We find a new method to calculate these overlaps
numerically using a variation of the truncated conformal space approach. We apply this method to the Lee-Yang model for which the unique boundary perturbation is integrable and for which the TBA
system describing the boundary overlaps is known. Using the truncated conformal space approach we obtain numerical results for the ground state and the first three excited states which are in
excellent agreement with the TBA results. As a special case we can calculate the standard g-function which is the overlap with the ground state and find that our new method is considerably more
accurate than the original method employed by Dorey et al.Comment: 21 pages, 6 figure
The classical trajectories of a particle governed by the PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) have been studied in depth. It is known that almost all trajectories that
begin at a classical turning point oscillate periodically between this turning point and the corresponding PT-symmetric turning point. It is also known that there are regions in $\epsilon$ for which
the periods of these orbits vary rapidly as functions of $\epsilon$ and that in these regions there are isolated values of $\epsilon$ for which the classical trajectories exhibit spontaneously broken
PT symmetry. The current paper examines the corresponding quantum-mechanical systems. The eigenvalues of these quantum systems exhibit characteristic behaviors that are correlated with those of the
associated classical system.Comment: 11 pages, 7 figure
For the pig respiratory tract pathogens, Actinobacillus pleuropneumoniae and Pasteurella multocida, Minimum Inhibitory Concentration (MIC) of marbofloxacin was determined in recommended broths and
pig serum at three inoculum strengths. MICs in both growth matrices increased progressively from low, through medium to high starting inoculum counts, 104, 106 and 108 CFU/mL, respectively. P.
multocida MIC ratios for high:low inocula were 14:4:1 for broth and 28.2:1 for serum. Corresponding MIC ratios for A. pleuropneumoniae were lower, 4.1:1 (broth) and 9.2:1 (serum). MIC high:low ratios
were therefore both growth matrix and bacterial species dependent. The effect of alterations to the chemical composition of broths and serum on MIC were also investigated. Neither adjusting broth or
serum pH in six increments over the range 7.0 to 8.0 nor increasing calcium and magnesium concentrations of broth in seven incremental steps significantly affected MICs for either organism. In
time-kill studies, the killing action of marbofloxacin had the characteristics of concentration dependency against both organisms in both growth matrices. It is concluded that MIC and time-kill data
for marbofloxacin, generated in serum, might be preferable to broth data, for predicting dosages of marbofloxacin for clinical use
It is shown that if a Hamiltonian $H$ is Hermitian, then there always exists an operator P having the following properties: (i) P is linear and Hermitian; (ii) P commutes with H; (iii) P^2=1; (iv)
the nth eigenstate of H is also an eigenstate of P with eigenvalue (-1)^n. Given these properties, it is appropriate to refer to P as the parity operator and to say that H has parity symmetry, even
though P may not refer to spatial reflection. Thus, if the Hamiltonian has the form H=p^2+V(x), where V(x) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has
PT symmetry. This shows that PT symmetry is a generalization of Hermiticity: All Hermitian Hamiltonians of the form H=p^2+V(x) have PT symmetry, but not all PT-symmetric Hamiltonians of this form are
Pharmacodynamic properties of marbofloxacin were established for six isolates each of the pig respiratory tract pathogens, Actinobacillus pleuropneumoniae and Pasteurella multocida. Three in vitro
indices of potency were determined; Minimum Inhibitory Concentration (MIC), Minimum Bactericidal Concentration (MBC) and Mutant Prevention Concentration (MPC). For MIC determination Clinical
Laboratory Standards Institute guidelines were modified in three respects: (1) comparison was made between two growth media, an artificial broth and pig serum; (2) a high inoculum count was used to
simulate heavy clinical bacteriological loads; and (3) five overlapping sets of two-fold dilutions were used to improve accuracy of determinations. Similar methods were used for MBC and MPC
estimations. MIC and MPC serum:broth ratios for A. pleuropneumoniae were 0.79:1 and 0.99:1, respectively, and corresponding values for P. multocida were 1.12:1 and 1.32:1. Serum protein binding of
marbofloxacin was 49%, so that fraction unbound (fu) serum MIC values were significantly lower than those predicted by correction for protein binding; fu serum:broth MIC ratios were 0.40:1 (A.
pleuropneumoniae) and 0.50:1 (P. multocida). For broth, MPC:MIC ratios were 13.7:1 (A. pleuropneumoniae) and 14.2:1 (P. multocida). Corresponding ratios for serum were similar, 17.2:1 and 18.8:1,
respectively. It is suggested that, for dose prediction purposes, serum data might be preferable to potency indices measured in broths
We investigate the sub-leading contributions to the free energy of Bethe Ansatz solvable (continuum) models with different boundary conditions. We show that the Thermodynamic Bethe Ansatz approach is
capable of providing the O(1) pieces if both the density of states in rapidity space and the quadratic fluctuations around the saddle point solution to the TBA are properly taken into account. In
relativistic boundary QFT the O(1) contributions are directly related to the exact g-function. In this paper we provide an all-orders proof of the previous results of P. Dorey et al. on the
g-function in both massive and massless models. In addition, we derive a new result for the g-function which applies to massless theories with arbitrary diagonal scattering in the bulk.Comment: 28
pages, 2 figures, v2: minor corrections, v3: minor corrections and references adde
In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from
H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum.
However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original
PT-invariant Hamiltonian, with potential V=igx^3. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent
potential. We again explore the relationship between the canonical variables x and p and the observables X and P.Comment: 9 pages, no figure | {"url":"https://core.ac.uk/search/?q=authors%3A(P.%20Dorey)","timestamp":"2024-11-03T04:15:04Z","content_type":"text/html","content_length":"148307","record_id":"<urn:uuid:0ad81fe4-7e94-4998-8016-06251a05836e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00570.warc.gz"} |
How to find lim
How to find lim (e^t-1)/t as t->0 using l'Hospital's Rule?
Talon Cannon
Answered question
How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?
Answer & Explanation
We have
$L=\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}$
To apply L'Hôpital's rule, we must have a $0\text{/}0$ or $\infty \text{/}\infty$ situation. If we plug in $t=0$ we find that:
So, we can apply the L'Hôpital's rule, which says:
$L=\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}=\frac{\frac{\text{d}}{\text{d}t}\left({e}^{t}-1\right)}{\frac{\text{d}}{\text{d}t}t}$
We know that ${e}^{x}$ is one of the functions with the property that $f\prime \left(x\right)=f\left(x\right)$, and as $-1$ is just a constant, it will vanish when we take the derivative.
$\therefore L=\underset{t\to 0}{lim}\frac{{e}^{t}}{1}=\underset{t\to 0}{lim}{e}^{t}$
$\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}{=}_{DLH}^{\left(\frac{0}{0}\right)}\underset{t\to 0}{lim}{e}^{t}={e}^{0}=1$ | {"url":"https://plainmath.org/calculus-1/104204-how-to-find-lim-e-t-1-t-as-t","timestamp":"2024-11-09T19:23:28Z","content_type":"text/html","content_length":"166072","record_id":"<urn:uuid:5c5c6358-cff5-437e-9061-a80eb76ac8a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00337.warc.gz"} |
Year 2 & 3 Maths Parents Workshop Canford Heath First School April 2012 I think of a number and add 6. My answer is 45, what number did I start with? - ppt download
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5 km to mile
To convert kilometers (km) to miles, you can use the following step-by-step instructions:
Step 1: Understand the conversion factor
1 kilometer is equal to 0.621371 miles. This conversion factor will be used to convert kilometers to miles.
Step 2: Set up the conversion equation
To convert 5 kilometers to miles, you can set up the equation as follows:
5 km * (0.621371 miles / 1 km)
Step 3: Cancel out the units
In the equation, the “km” unit cancels out, leaving only “miles” as the desired unit:
5 * 0.621371 miles
Step 4: Perform the calculation
Multiply 5 by 0.621371 to get the equivalent value in miles:
5 * 0.621371 = 3.106855 miles
Step 5: Round the answer (if necessary)
In this case, the answer is already rounded to six decimal places. However, if you need to round it further, you can do so according to the desired level of precision.
Therefore, 5 kilometers is equal to approximately 3.106855 miles.
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Word Ladder (C# and Python3) » ayoubb
Word Ladder (C# and Python3)
A transformation sequence from word beginWord to word endWord using a dictionary wordList is a sequence of words beginWord -> s[1] -> s[2] -> ... -> s[k] such that:
• Every adjacent pair of words differs by a single letter.
• Every s[i] for 1 <= i <= k is in wordList. Note that beginWord does not need to be in wordList.
• s[k] == endWord
Given two words, beginWord and endWord, and a dictionary wordList, return the number of words in the shortest transformation sequence from beginWord to endWord, or 0 if no such sequence exists.
Example 1:
Input: beginWord = "hit", endWord = "cog", wordList = ["hot","dot","dog","lot","log","cog"]
Output: 5
Explanation: One shortest transformation sequence is "hit" -> "hot" -> "dot" -> "dog" -> cog", which is 5 words long.
• C# solution is commented and explained
• Python solution is similar to the c# solution (not explained)
Graph problem + BFS to find the shortest path | {"url":"https://ayoubb.com/algorithms-and-data-structures/word-ladder-c-and-python3/","timestamp":"2024-11-09T11:05:35Z","content_type":"text/html","content_length":"91075","record_id":"<urn:uuid:11c79538-7597-48a2-8bb4-9570420e7ee8>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00749.warc.gz"} |
(7.) Find the number of common terms to the two seque 17 21. 25... | Filo
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Question Text (7.) Find the number of common terms to the two seque 17 21. and .
Updated On Aug 6, 2024
Topic Sequence Series and Quadratic
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 149
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How Many Amps Does a 1000 Watt Inverter Draw? - RVing Beginner
Many inverters in a solar power system perform the same task, which is to change direct current (DC) into alternating current (AC) for use by AC appliances and gadgets.
Inverters come in all shapes and sizes.
A 1000 watt inverter uses how many amps, though? Is your inverter big enough to handle the system amp demands? Or is a bigger system required?
Depending on the inverter efficiency, a 1000 watt load on a 1000 watt 12V inverter consumes 100 to 110 amps.
The same 1000 watt load will use 40 to 60 amps on a 24V system.
How to Calculate the Amp Draw of a 1000W Inverter
Until a load is connected to an inverter, it does not use amps.
Utilize the following formula to determine the amps:
Watt load / input voltage / inverter efficiency rating = amps drawn
It would look like this if you had a 400W blender at 12V and a 1000W inverter with an efficiency rating of 85%:
400W / 12V / .85 = 39.2 amps
The blender will use 40 amps per hour, or 39.2 amps, rounded up.
Since running the blender for an hour is obviously implausible, the real amp demand will be smaller.
But it’s simple to estimate the amp needs by beginning with 40 amps per hour (20 amps for 30 minutes, 10 amps for 15 minutes etc.).
The following conversion is often used when estimating the needs for solar panels on appliances:
Watts / volts = amps
And it’s effective.
That does not, however, take into consideration inverter inefficiency, which results in energy loss and increased amp draws.
Let’s stick with the previous illustration.
400W x 12V x.85 equals 39.2 amps
If we ignore the efficiency rating of 85%, the outcome will be:
400W at 12V equals 33 amps.
That is nearly 7 amps of difference.
Is that relevant? Every amp counts if you use your 1000W inverter to its maximum capacity.
But it won’t matter if you don’t if you don’t.
Do You Need a 1000W Inverter?
Consider upgrading to a 1500W inverter, such as the Energizer 1500W, if you often load the system with 1000 watts.
Instead of pushing the load to its limit, it is preferable to have some extra power on hand in case you need to operate appliances.
The inverter should never be overworked, just like batteries.
When the load is close to 1000 watts, the overload indicator may blink even if it is not technically overloaded.
A 1000 watt inverter can theoretically power a 1000 watt load.
However, doing so often may harm the system.
Never, unless necessary, should batteries, solar panels, or charge controllers be overtaxed.
A load on your inverter of no more than 50% or 70% is acceptable with an efficiency of 80%.
The inverter can easily manage the load of a 400W blender, a 100W laptop, and a few light bulbs.
While a laptop may be used for hours at a time, most kitchen gadgets are only utilized for brief periods of time.
Keep in mind that appliance power recommendation tables are for watts per hour when estimating the number of amps that will be required.
A coffee maker uses 900 watts an hour, but after a few minutes, it won’t come close to that amount.
However, 900W of overall load is pushing it to its maximum.
And do not be shocked if a 1000W inverter cannot power your 1000 watt load.
Why? due to inefficiency and energy loss, which we shall discuss in more detail below.
Rating for Inverter Efficiency
Knowing an appliance’s efficiency rating is essential if you wish to power it – be it a refrigerator or another applianceโ with an inverter.
80 percent is the minimum allowable inverter efficiency rating, however 85 percent is obviously preferable.
Although they are more expensive, some inverters have an efficiency of 90% to 95%.
Does the difference in efficiency of 5% to 10% matter? It does in the long term.
Although a 1000W inverter may theoretically load 1000 watts, in practice the load limit may only be 900W or such.
Inverter inefficiency has an impact on both amp draws and watts load.
The difference is less the higher the efficiency rating.
However, while calculating system losses, you must take the whole system into consideration.
Even if your inverter is 95 percent efficient, 5 percent of solar energy will still be wasted.
During transmission, solar cables and wires lose energy.
Additionally, the efficiency of solar panels varies, and output is weather-dependent.
You shouldn’t let this stop you from utilizing inverters, however.
It only implies that you should allow for some flexibility when estimating how many amps a gadget may use.
You can determine the size of the inverter you require by adding the total watt load and the efficiency rating.
For A 1000W Inverter, How Many Batteries Will I Need?
Appliances can only operate as long as there is electricity in the batteries since inverters use power from them to do so.
A 100ah battery with a 50% depth of discharge can power a 1000W inverter for 45 to 55 minutes while driving a 700W load.
The running duration will be greater if your battery permits a higher discharge rate of 30%.
Once again, the effectiveness of the inverter and the solar power system overall will affect the run time.
The amount of hours you need to operate the load constantly determines how many batteries you need.
Runtime x watts equals watts/volts equals required battery amps
You are using a 700 watt load with a 1000 watt 12 volt inverter.
700 watts divided by 12 volts equals 58.3 amps/hour.
To calculate run time, divide the amps per hour by the battery.
A 100ah battery will last for 1.71 hours, or 1 and 45 minutes more or less, when divided by 58.3 amps.
If the battery is entirely discharged, which you shouldn’t do, it will only last 1.7 hours.
The depth of discharge is 50% for AGM and other FLA batteries.
Thus, multiply 1.7 hours by. 50.
1.7 / .50 =3.4
That is around 45 to 55 minutes, to be exact.
Modified Wave Inverter VS. 1000W Pure Sine Inverter
If you wish to utilize an inverter, this is one of the first decisions you will need to make.
Do you choose the more expensive but better pure sine wave inverter or the less expensive but less efficient modified sine wave inverter?
Bottom line: An inverter that produces a pure sine wave, such the Renogy 1000W 12V Inverter, is a preferable option.
Although it costs more, you can be confident that it is compatible with contemporary equipment.
Modified sine wave inverters may potentially produce losses of up to 30%, which is considered undesirable by many.
For rudimentary electronics, outdated equipment, and appliances, a modified sine wave inverter is suitable.
You may utilize a modified sine inverter if the system has no sensitive components.
However, modern appliances operate more effectively on pure sine waves.
Being energy efficient is key when using solar electricity, thus pure sine makes sense.
The following appliances and pieces of equipment cannot be powered by modified sine wave inverters.
For these, pure sine is required.
• Medical supplies
• Home auto systems X 10
• Any device that has a microprocessor
• Any device having speed controls
• Digital timepieces
• Electronic furnaces
• Chargers for cordless tool batteries
• Certain fluorescent light kinds
• Several laptops and desktops
• Almost anything SCR (silicone controlled rectifier)
• Anything that makes use of the electrical component thyristor
These are just a few of the gadgets and appliances that won’t work with an inverter that produces modified sine waves.
To make sure they are compatible, you need first verify your appliance.
New gadgets and appliances work with pure sine inverters, of that you may be sure.
Cost is the main barrier for most individuals.
But you should think of this as a solar panel system investment.
Additionally, replacing your equipment due to their incompatibility with modified sine wave is less expensive than purchasing a pure sine inverter.
Advice on Purchasing a 1000W Inverter
• Get the appropriate inverter type. A 1000 watt inverter is not created equal. For the same wattage, a 95 percent efficient inverter will perform better than an 80 percent efficient unit. The
performance of a pure sine wave is superior to that of a modified sine wave, of course.
• Warranty. Along with the charge controller, inverters are the solar system’s most delicate component. For this reason, you should purchase one from a reputable manufacturer who also offers a
thorough warranty coverage.
• Usage. Determine the amps and watts you will be loading into the inverter. Use the above table as a reference to determine if 1000 watts will be enough. Consider your long-term goals and if you
may need a larger inverter.
• Size of the battery. An inverter can only function as long as the battery bank has electricity, as was previously mentioned. Take note of the depth discharge and decide if lead acid or lithium
ion batteries are required. The longer your inverter can operate, the better the DOD.
Getting a precise number is challenging due to the system’s intrinsic energy losses.
However, you may get a very good idea of how many amps a 1000 watt inverter will use using the method given here. | {"url":"https://rvingbeginner.com/how-many-amps-does-a-1000-watt-inverter-draw","timestamp":"2024-11-02T09:34:05Z","content_type":"text/html","content_length":"228780","record_id":"<urn:uuid:062e3158-c7d6-4eef-a066-17ea9ef211cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00126.warc.gz"} |
Rapid SCADA
Viewing 15 posts - 1 through 15 (of 25 total)
• Author
• July 28, 2024 at 3:04 am #15036
I have connected a device to Rapid Scada using Modbus.
There are 8 Statuses stored in the same address 1080.
Status A is in BIT0-1
Status B is in BIT2-3
Status C is in BIT4-6
Status D is in BIT7-9
Status E is in BIT10
Status F is in BIT11-12
Status G is in BIT13
Status H is in BIT14-15
Using Bitmask, individual channels have been automatically created by Rapid Scada for each BIT with GetBit formula.
Please suggest formula to get 2 and 3 BITS in one Channel as required above
Many Thanks
July 28, 2024 at 6:34 am #15037
If you always need to pull out 2 bits each, then you can make one formula, just like getBit. If your pairs may differ, then you will need to make a formula and install it with your hands. Modify
getBit into a different formula
public double GetTwoBit(double val, int n)
ulong ulVal = (ulong)val;
return (ulVal >> n) & 3ul;
Here 3ul is the number 3 (00000011) and by shifting the bits by the right amount you make a logical one with two bits at once, and not one as in getBit. You can look at how much you need to shift
in a calculator or even on a piece of paper.
□ This reply was modified 3 months, 2 weeks ago by manjey73.
July 28, 2024 at 6:38 am #15039
public double GetThreeBit(double val, int n)
ulong ulVal = (ulong)val;
return (ulVal >> n) & 7ul;
Here the number 7 (00000111) is used as a mask
July 28, 2024 at 6:43 am #15040
public double GetAnyBits(double val, int n, int mask = 1)
ulong ulVal = (ulong)val;
return (ulVal >> n) & (ulong)mask;
I haven’t checked, but the mask should work by default. The mask is 1. Which fully corresponds to the getBit formula
You can set 3, 7, and so on.
n is the offset
July 28, 2024 at 7:04 am #15041
Thanks Manjey
Since I am new, I have few questions:
1. I will add the GetAnyBits .. to the scripts by editing the Scripts source code
2. For usage, lets say I want the 3rd, 4th and 5th bit, what would the syntax be?
Thanks again
July 28, 2024 at 7:25 am #15042
The 0th bit is on the right. You need to shift by 3
GetAnyBits(Val(XXX), 3, 7) like this
□ This reply was modified 3 months, 2 weeks ago by manjey73.
July 28, 2024 at 7:44 am #15044
Hi Manjey
As per manual of the device, Register 1080 size is 2 therefore has 16 bits and I need to read the (bit 3, bit 4 and bit 5), (bit 6 and bit 7) and so on.
I am however not clear of the (XXX) and 7 in the syntax you provided.
I also request you for Syntax for the examples above
Please help
Thanks in advance
July 28, 2024 at 8:01 am #15045
GetAnyBits(Val(1080), 3, 7)
Val(XXX) – Where is the XXX number of your channel
00000111 is the number 7 in the bit representation
When you need to select 3 bits in a value, you need to make a logical AND with the bits you need, and depending on which bits from the beginning you want to compare, you need to shift the bits
you need so that they turn out to be starting from the zero bit
00001111 is the number 15, which will correspond to four consecutive bits
00000011 is the number 3, which will correspond to two consecutive bits
01010110 – Your certain number for checking 3 bits. You need to shift this representation to position 0 (n = 2) 00010101 and make AND with the number 7 (00000111)
July 28, 2024 at 10:57 am #15046
Hi Manjey
Many thanks. I have understood.
Now there is only a small issue and request for your input.
The value which i am getting after the formula is in decimal eg 01 is shown as 1; 10 is coming as 2. Please suggest weak in the script so that I get the output as 01 or 10.
July 28, 2024 at 11:08 am #15047
00000010 in the bit representation is the number 2 in the decimal representation. Do you need to see the number in some other format?
July 28, 2024 at 11:32 am #15048
yes please. I need to see it is 10 i.e. in bit representation
July 28, 2024 at 2:31 pm #15049
Set the HEX format in the channel, perhaps this will be enough for you.
There is a choice there
Hexadecimal 2 digits
Hexadecimal 4 digits
Hexadecimal 8 digits
□ This reply was modified 3 months, 2 weeks ago by manjey73.
July 28, 2024 at 3:12 pm #15051
00000010 is a binary display of bits of a number, I don’t even know if there is support for such a display in the system. I wrote some kind of formula for this and displayed it as a string
July 28, 2024 at 3:45 pm #15052
Format Channel – String
Data Type = ASCII
public string BitsView(double number)
byte[] data = BitConverter.GetBytes(Convert.ToUInt16(number));
uint N = BitConverter.ToUInt16(data, 0);
string bits = Convert.ToString(N, 2);
return bits;
Make a calculation channel with the specified parameters. Use the specified formula. Displaying a string in the channel, but only for a byte. In order for the whole number to be displayed
correctly, the formula needs to be refined.
July 29, 2024 at 3:47 am #15053
Many Thanks for all your help
• Author
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Understanding Mathematical Functions: Which Of The Following Is Not A
Introduction to Mathematical Functions
Mathematical functions are fundamental concepts in mathematics that relate input values to output values. Understanding functions is crucial in various areas of mathematics and has practical
applications in real-world scenarios.
A Definition and Importance of Understanding Functions in Mathematics
A mathematical function is a relation between a set of input values (domain) and a set of output values (range), where each input value corresponds to exactly one output value. Functions are
represented using mathematical notation such as f(x) = x^2, where f is the function, x is the input, and x^2 is the output.
Understanding functions in mathematics is essential for solving equations, analyzing data, and modeling real-world phenomena. Functions help in describing relationships and patterns, making
predictions, and solving problems systematically.
Overview of Group Functions and Their Role in Various Mathematical and Real-World Applications
Group functions, also known as group homomorphisms, are functions between two groups that preserve the group structure. In group theory, a branch of abstract algebra, group functions play a
significant role in studying symmetries, transformations, and properties of groups.
Group functions have diverse applications in mathematics, including cryptography, coding theory, and quantum mechanics. They are also used in computer science, physics, and chemistry for solving
problems related to symmetries and transformations.
Setting the Stage for Identifying Functions That Do Not Qualify as Group Functions
While group functions have specific properties that make them unique in group theory, not all functions meet the criteria to be classified as group functions. Some functions may not preserve the
group structure or operations, making them ineligible for the title of group functions.
In the following sections, we will explore functions that do not qualify as group functions and examine the reasons behind their exclusion from this special class of functions.
Key Takeaways
• Key Takeaways:
• Functions map input to output
• Group functions have specific properties
• Not all functions are group functions
• Understanding functions is essential in mathematics
• Identifying group functions requires specific criteria
Understanding the Concept of Group Functions
Group functions are an essential concept in mathematics that play a significant role in various mathematical operations. In this chapter, we will delve into the definition of a group in mathematical
terms, explore the characteristics of group functions, and provide examples of typical group functions.
A Definition of a group in mathematical terms
In mathematics, a group is defined as a set equipped with a binary operation that satisfies four fundamental properties. These properties include closure, associativity, the existence of an identity
element, and the availability of an inverse element for each element in the set.
Characteristics of group functions
Closure: One of the key characteristics of group functions is closure. This property states that when two elements from the set are combined using the binary operation, the result is also an element
of the set.
Associativity: Group functions exhibit associativity, meaning that the way in which elements are grouped does not affect the outcome of the operation. In other words, for any elements a, b, and c in
the set, (a * b) * c = a * (b * c).
Existence of an identity element: Every group function must have an identity element, denoted as e, such that for any element a in the set, a * e = e * a = a.
Inverse element availability: Lastly, group functions require the availability of an inverse element for each element in the set. For every element a, there exists an element b such that a * b = b *
a = e, where e is the identity element.
Examples of typical group functions
Two common examples of group functions are addition and multiplication operations for numbers. In the case of addition, the set of integers forms a group under addition, as it satisfies all four
properties of a group. Similarly, the set of non-zero rational numbers forms a group under multiplication, meeting the criteria of closure, associativity, identity element, and inverse element
Identifying Non-Group Functions
When it comes to mathematical functions, not all operations qualify as group functions. Group functions have specific characteristics that set them apart from other mathematical operations. In this
chapter, we will explore key features that disqualify certain operations from being group functions, common misconceptions about group functions in mathematics, and practical examples highlighting
operations that are not considered group functions.
A. Key features that disqualify certain operations from being group functions
Group functions in mathematics must satisfy four fundamental properties: closure, associativity, identity element, and inverse element. If an operation fails to meet any of these criteria, it cannot
be classified as a group function. For example, if an operation does not have an identity element or if it is not associative, it cannot be considered a group function.
B. Common misconceptions about group functions in mathematics
One common misconception about group functions is that all mathematical operations are group functions. However, this is not true. While many operations in mathematics do form groups, there are also
operations that do not meet the criteria to be classified as group functions. It is important to understand the specific properties that define a group function in order to accurately identify them.
C. Practical examples highlighting operations that are not considered group functions
One practical example of an operation that is not considered a group function is division in certain contexts. Division does not always have an inverse element, as division by zero is undefined.
Therefore, division does not satisfy the criteria for being a group function in all cases.
Another example is subtraction. While subtraction may seem like a simple operation, it does not always have an identity element. For example, subtracting a number from itself does not result in a
unique identity element. Therefore, subtraction does not meet the requirements to be classified as a group function.
The Importance of Distinguishing Group Functions
Understanding mathematical functions is essential in solving complex problems and developing new theories in mathematics. One crucial aspect of functions is distinguishing between group functions and
non-group functions. Misidentifying a function as a group function can lead to errors in problem-solving and analysis, impacting the accuracy of mathematical research.
A. The role of group functions in solving mathematical problems and theories
Group functions play a significant role in various mathematical disciplines, including algebra, number theory, and geometry. These functions exhibit specific properties that make them essential for
solving mathematical problems efficiently. In group theory, for example, group functions help mathematicians analyze the symmetries and structures of mathematical objects.
By understanding and correctly identifying group functions, mathematicians can apply group theory to solve complex mathematical problems, such as finding solutions to equations, proving theorems, and
studying abstract algebraic structures. Group functions provide a framework for organizing mathematical concepts and relationships, making it easier to analyze and interpret mathematical phenomena.
B. How misidentifying a function as a group function can lead to errors in problem-solving and analysis
One common mistake in mathematical analysis is misidentifying a function as a group function when it does not satisfy the necessary properties. Group functions must adhere to specific criteria, such
as closure, associativity, identity element, and inverse element. Failing to recognize these properties in a function can lead to errors in problem-solving and analysis.
For instance, assuming a non-group function as a group function may result in incorrect conclusions, faulty proofs, and inaccurate mathematical models. Misidentifying functions can hinder progress in
mathematical research and lead to misleading results. It is crucial for mathematicians to accurately distinguish between group functions and non-group functions to ensure the validity and reliability
of their mathematical analyses.
C. The impact of correctly identifying non-group functions on the understanding and advancement of mathematical research
Correctly identifying non-group functions is essential for advancing mathematical research and developing new theories. Non-group functions may exhibit different properties and behaviors that require
unique mathematical approaches for analysis. By accurately recognizing non-group functions, mathematicians can explore new avenues of research, discover novel mathematical concepts, and make
significant contributions to the field.
Furthermore, understanding the limitations of non-group functions can lead to the development of new mathematical frameworks and theories. By acknowledging the diverse range of functions in
mathematics, researchers can broaden their perspectives, foster innovation, and push the boundaries of mathematical knowledge. Correctly identifying non-group functions is crucial for the growth and
advancement of mathematical research.
Practical Applications and Implications
A Real-world scenarios where the distinction between group and non-group functions plays a critical role
Understanding the difference between group and non-group functions is essential in various real-world scenarios. For instance, in finance, group functions are used to analyze market trends and make
predictions based on historical data. On the other hand, non-group functions may be used in areas such as social media algorithms to personalize content for users based on their preferences.
B Case studies highlighting the application of group functions in fields such as cryptography, physics, and computer science
In the field of cryptography, group functions are utilized to encrypt and decrypt sensitive information securely. For example, the Diffie-Hellman key exchange algorithm relies on group functions to
establish a shared secret key between two parties without the need to transmit the key over the communication channel.
In physics, group functions play a crucial role in understanding the symmetries and conservation laws of physical systems. For instance, the concept of rotational symmetry in three-dimensional space
can be described using group theory, which helps physicists analyze the behavior of particles and forces.
In computer science, group functions are used in various applications such as data compression, error correction codes, and network protocols. For example, the RSA encryption algorithm relies on the
mathematical properties of group functions to ensure secure communication over the internet.
C The influence of understanding group functions on technological advancements and innovations
The understanding of group functions has significantly impacted technological advancements and innovations in various fields. For instance, in the development of artificial intelligence algorithms,
group functions are used to optimize neural networks and improve machine learning models.
In the field of robotics, group functions are utilized to design efficient motion planning algorithms that enable robots to navigate complex environments and perform tasks autonomously. By
understanding the principles of group theory, engineers can develop more sophisticated and reliable robotic systems.
Overall, the knowledge of group functions has paved the way for groundbreaking advancements in technology, leading to the creation of innovative solutions that have revolutionized industries and
improved the quality of life for people around the world.
Troubleshooting Common Misunderstandings
When it comes to understanding mathematical functions, distinguishing between group functions and non-group functions can be a challenging task. Here are some tips and strategies to help identify
common pitfalls and effectively teach the differences:
A Tips for identifying common pitfalls in distinguishing group functions
• Understand the definition: Make sure to have a clear understanding of what constitutes a group function. A group function is a function that satisfies the properties of closure, associativity,
identity element, and inverse element.
• Check for closure: One common pitfall is failing to check if the function is closed under the operation. If the result of applying the function to two elements is not within the same set, then it
is not a group function.
• Verify associativity: Another common mistake is assuming associativity without verifying it. Make sure to check if the function satisfies the associativity property.
B Strategies for effectively teaching and communicating the differences between group and non-group functions
• Use examples: Provide concrete examples of group functions and non-group functions to illustrate the differences. This can help students visualize and understand the concepts better.
• Engage in hands-on activities: Encourage students to participate in activities that involve group operations. This hands-on approach can help solidify their understanding of group functions.
• Encourage critical thinking: Ask thought-provoking questions that require students to analyze and differentiate between group and non-group functions. This can help them develop a deeper
understanding of the concepts.
C Resources for further study and clarification on complex mathematical functions
• Textbooks: Utilize textbooks that cover group theory and mathematical functions in depth. These resources can provide additional explanations and examples to enhance understanding.
• Online courses: Enroll in online courses or tutorials that focus on group theory and mathematical functions. These courses often offer interactive lessons and quizzes to reinforce learning.
• Consult with experts: Reach out to mathematics professors or experts in the field for clarification on complex mathematical functions. They can provide valuable insights and guidance on
challenging concepts.
Conclusion & Best Practices
A. Summarizing the significance of accurately identifying non-group functions in mathematics
Understanding the concept of group functions and accurately identifying non-group functions in mathematics is crucial for various reasons. By recognizing non-group functions, mathematicians can avoid
errors in calculations and ensure the validity of their mathematical operations. This knowledge also helps in distinguishing between functions that follow specific mathematical properties and those
that do not, leading to a deeper understanding of mathematical structures and relationships.
B. Best practices for studying and applying the concept of group functions in various mathematical and practical contexts
• Study Different Examples: To enhance your understanding of group functions, explore various examples and practice identifying non-group functions. This hands-on approach will help solidify your
knowledge and improve your ability to recognize patterns and properties.
• Utilize Mathematical Software: Take advantage of mathematical software tools to analyze functions and determine their properties. These tools can assist in verifying whether a function satisfies
the criteria of a group function or not, making your learning process more efficient.
• Engage in Collaborative Learning: Discussing concepts related to group functions with peers or instructors can provide different perspectives and insights. Collaborative learning environments can
help clarify doubts, deepen understanding, and foster a supportive learning community.
C. Encouragement for ongoing learning and exploration in the vast field of mathematical functions to enhance problem-solving skills and theoretical knowledge
Mathematics is a dynamic and ever-evolving field that offers endless opportunities for exploration and discovery. By continuously engaging with mathematical functions and expanding your knowledge,
you can enhance your problem-solving skills and develop a deeper understanding of theoretical concepts. Embrace the challenges that come with exploring new mathematical territories, as they can lead
to personal growth and intellectual fulfillment. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-which-is-not-a-group-function","timestamp":"2024-11-11T17:57:54Z","content_type":"text/html","content_length":"227342","record_id":"<urn:uuid:3b66de3a-8c32-40a4-aa53-f4886201168f>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00792.warc.gz"} |
Summer 2009
Hi, This doesnt quite fit in with the topic but does anyone have any predictions for summer 2009? I am trying to set a date to get married, when do you predict the nicest weather - either late May,
start of August or end of August? Thanks
Useful links: Will it thunder? -- Weather Radar -- UKV model -- Local hour-by-hour forecast
way too far off for any idea...
Hi, This doesnt quite fit in with the topic but does anyone have any predictions for summer 2009? I am trying to set a date to get married, when do you predict the nicest weather - either late
May, start of August or end of August? Thanks
I'd always noticed that when I was at school, there was always a scorching half term followed by decent weather when taking GCSEs and A levels - come the 15th June when exams were over, the weather
always turned! Therefore, I picked June 3rd 2006 to get married to Mrs Plum and the weather gods smiled at me then!
I am so Looking forward to summer 2009
I love it when its warm and sunny and the long days.
Roll on next month when the clocks go forward a hour.
Far to early to be talking about summer, still got the rest of winter and spring to go yet.
lots more snow,new ice age
Ages away yet so no idea what it'll be like.
Understand the longing for it at this time of year though. Snow or no snow - by this point in winter I'm definitely beginning to pine for some light & warmth again (although wintry weather is
certainly still welcome in spring - it's no sudden switch!).
We've had two fairly naff (average) summer the last couple of years, I for one am hoping for something drier and hotter. Doesn't need to be 35c+ everyday but a high frequency of 25c+ days would be
I have found that June is often quite a cool/chilly month. What is more, I have seen indications that the relatively cool/cold start to 2009 is set to continue up to and including July. Haven't seen
anything for August yet, mind.
Yep as others said its too early, though I'd actually like to see something of a warmer summer this time round after two poor summers in a row, not sure I want anything exceptional but as Bottesford
said a higher number of warm days would be nice.
Far to early to yet, but hopefully it's far better than the last two. Though with continuing low solar activity I wouldn't like to bet on it!
September...always seems to be nice now days....+ that's when I picked my wedding date 2 years ago to Prescila
The Met office have said Summer 2009 will be a scorcher. They have been wrong before.
The latest CFS charts are suggesting:
June: wet in the south, average rainfall in north. Temperatures close to or slightly below average.
July: below average in the east, warmer further west. Wet in the north, a little drier than average in the south.
August: wet in the north-east, progressively drier as you go south-east. Temperatures average or slightly below.
So much like last summer if we're honest, if a little drier. May is looking very wet aswell.
no one can help you with your question as you omit to say WHERE?
and as has already been said its far too early to give even a rough idea of temperature levels, wind, sun etc etc
welcome to the site, enjoy but drop your nearest town into your avatar please.
Hi, This doesnt quite fit in with the topic but does anyone have any predictions for summer 2009? I am trying to set a date to get married, when do you predict the nicest weather - either late
May, start of August or end of August? Thanks
What ever the weather on your special day... you'll have a great time. Some of the best weddings I've photographed have been in the rain, and some of the worst were where everyone can't wait to take
off suit jackets and hefty dresses to cool down. Tis difficult to predict the weather so far ahead... could be a heatwave or freak hailstorm.... Pick a day and enjoy what ever nature throws at you.
by this point in winter I'm definitely beginning to pine for some light & warmth again
Same here my friend, same here!!!
So far this season we have had snow in October, November, December, January and February - perhaps it will continue through March and April, which are always good bets and into May thus ending up
with snow falling in 8 months of the year.
As for the summer, can't really say - it depends where the jet streams get stuck - let hope they go a bit further north this year.
Give me a generally cool summer with the occasional hot blast to bring some nice storms. Hate heat.
But way too far out to make any predictions!
I cant wait for Summer now...the amount of Snow dissapointments here is pretty much unacceptable...not a single forecast barely got last night right in terms of snow depths etc. All of last week
wasnt correct in snowfall depths either.
Summer will probably be cool and wet tbh, we are currentey in a cooling trend that will last for about 10 years (thats what the meto hope), which will increase the percip amonts, i would expect a
reapet of the summer floods, maybe not to such an extreme, but never the less another summer to be remebered for all the wrong reasons !!!!!!
Cool, average, some warmer spells, up to 30 degrees in London area in July. Rain most of the time but drier than last year.. Thats about it.
I chose first weekend of July last year for my wedding - BIG mistake!! Writing was on the wall when a week before, a deep low showed up on the models, centred right over the south of the UK
Hopefully, August 2009 will be a lot sunnier than the August of 2008!
The last two were poor for this area, 2007 for the rainfall and last summer for the lack of sunshine.
So long as we get 2 lots of H.P. dominated fortnights and temps/sunshine reminiscent of summer I'd be happy, esp.. after the last two washouts!!!
WAYYY tooo far out to tell, but it should really be a mix of cool, warm, hot, wet, windy and dry calm.....its just pinning down dates which will be nearer the time.
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clang/lib/Headers/mmintrin.h - llvm-project - Git at Google
/*===---- mmintrin.h - MMX intrinsics --------------------------------------===
* Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
* See https://llvm.org/LICENSE.txt for license information.
* SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
#ifndef __MMINTRIN_H
#define __MMINTRIN_H
#if !defined(__i386__) && !defined(__x86_64__)
#error "This header is only meant to be used on x86 and x64 architecture"
typedef long long __m64 __attribute__((__vector_size__(8), __aligned__(8)));
typedef long long __v1di __attribute__((__vector_size__(8)));
typedef int __v2si __attribute__((__vector_size__(8)));
typedef short __v4hi __attribute__((__vector_size__(8)));
typedef char __v8qi __attribute__((__vector_size__(8)));
/* Define the default attributes for the functions in this file. */
#define __DEFAULT_FN_ATTRS __attribute__((__always_inline__, __nodebug__, __target__("mmx"), __min_vector_width__(64)))
/// Clears the MMX state by setting the state of the x87 stack registers
/// to empty.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> EMMS </c> instruction.
static __inline__ void __attribute__((__always_inline__, __nodebug__, __target__("mmx")))
/// Constructs a 64-bit integer vector, setting the lower 32 bits to the
/// value of the 32-bit integer parameter and setting the upper 32 bits to 0.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> MOVD </c> instruction.
/// \param __i
/// A 32-bit integer value.
/// \returns A 64-bit integer vector. The lower 32 bits contain the value of the
/// parameter. The upper 32 bits are set to 0.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cvtsi32_si64(int __i)
return (__m64)__builtin_ia32_vec_init_v2si(__i, 0);
/// Returns the lower 32 bits of a 64-bit integer vector as a 32-bit
/// signed integer.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> MOVD </c> instruction.
/// \param __m
/// A 64-bit integer vector.
/// \returns A 32-bit signed integer value containing the lower 32 bits of the
/// parameter.
static __inline__ int __DEFAULT_FN_ATTRS
_mm_cvtsi64_si32(__m64 __m)
return __builtin_ia32_vec_ext_v2si((__v2si)__m, 0);
/// Casts a 64-bit signed integer value into a 64-bit integer vector.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> MOVQ </c> instruction.
/// \param __i
/// A 64-bit signed integer.
/// \returns A 64-bit integer vector containing the same bitwise pattern as the
/// parameter.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cvtsi64_m64(long long __i)
return (__m64)__i;
/// Casts a 64-bit integer vector into a 64-bit signed integer value.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> MOVQ </c> instruction.
/// \param __m
/// A 64-bit integer vector.
/// \returns A 64-bit signed integer containing the same bitwise pattern as the
/// parameter.
static __inline__ long long __DEFAULT_FN_ATTRS
_mm_cvtm64_si64(__m64 __m)
return (long long)__m;
/// Converts 16-bit signed integers from both 64-bit integer vector
/// parameters of [4 x i16] into 8-bit signed integer values, and constructs
/// a 64-bit integer vector of [8 x i8] as the result. Positive values
/// greater than 0x7F are saturated to 0x7F. Negative values less than 0x80
/// are saturated to 0x80.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PACKSSWB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16]. Each 16-bit element is treated as a
/// 16-bit signed integer and is converted to an 8-bit signed integer with
/// saturation. Positive values greater than 0x7F are saturated to 0x7F.
/// Negative values less than 0x80 are saturated to 0x80. The converted
/// [4 x i8] values are written to the lower 32 bits of the result.
/// \param __m2
/// A 64-bit integer vector of [4 x i16]. Each 16-bit element is treated as a
/// 16-bit signed integer and is converted to an 8-bit signed integer with
/// saturation. Positive values greater than 0x7F are saturated to 0x7F.
/// Negative values less than 0x80 are saturated to 0x80. The converted
/// [4 x i8] values are written to the upper 32 bits of the result.
/// \returns A 64-bit integer vector of [8 x i8] containing the converted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_packs_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_packsswb((__v4hi)__m1, (__v4hi)__m2);
/// Converts 32-bit signed integers from both 64-bit integer vector
/// parameters of [2 x i32] into 16-bit signed integer values, and constructs
/// a 64-bit integer vector of [4 x i16] as the result. Positive values
/// greater than 0x7FFF are saturated to 0x7FFF. Negative values less than
/// 0x8000 are saturated to 0x8000.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PACKSSDW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [2 x i32]. Each 32-bit element is treated as a
/// 32-bit signed integer and is converted to a 16-bit signed integer with
/// saturation. Positive values greater than 0x7FFF are saturated to 0x7FFF.
/// Negative values less than 0x8000 are saturated to 0x8000. The converted
/// [2 x i16] values are written to the lower 32 bits of the result.
/// \param __m2
/// A 64-bit integer vector of [2 x i32]. Each 32-bit element is treated as a
/// 32-bit signed integer and is converted to a 16-bit signed integer with
/// saturation. Positive values greater than 0x7FFF are saturated to 0x7FFF.
/// Negative values less than 0x8000 are saturated to 0x8000. The converted
/// [2 x i16] values are written to the upper 32 bits of the result.
/// \returns A 64-bit integer vector of [4 x i16] containing the converted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_packs_pi32(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_packssdw((__v2si)__m1, (__v2si)__m2);
/// Converts 16-bit signed integers from both 64-bit integer vector
/// parameters of [4 x i16] into 8-bit unsigned integer values, and
/// constructs a 64-bit integer vector of [8 x i8] as the result. Values
/// greater than 0xFF are saturated to 0xFF. Values less than 0 are saturated
/// to 0.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PACKUSWB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16]. Each 16-bit element is treated as a
/// 16-bit signed integer and is converted to an 8-bit unsigned integer with
/// saturation. Values greater than 0xFF are saturated to 0xFF. Values less
/// than 0 are saturated to 0. The converted [4 x i8] values are written to
/// the lower 32 bits of the result.
/// \param __m2
/// A 64-bit integer vector of [4 x i16]. Each 16-bit element is treated as a
/// 16-bit signed integer and is converted to an 8-bit unsigned integer with
/// saturation. Values greater than 0xFF are saturated to 0xFF. Values less
/// than 0 are saturated to 0. The converted [4 x i8] values are written to
/// the upper 32 bits of the result.
/// \returns A 64-bit integer vector of [8 x i8] containing the converted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_packs_pu16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_packuswb((__v4hi)__m1, (__v4hi)__m2);
/// Unpacks the upper 32 bits from two 64-bit integer vectors of [8 x i8]
/// and interleaves them into a 64-bit integer vector of [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PUNPCKHBW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8]. \n
/// Bits [39:32] are written to bits [7:0] of the result. \n
/// Bits [47:40] are written to bits [23:16] of the result. \n
/// Bits [55:48] are written to bits [39:32] of the result. \n
/// Bits [63:56] are written to bits [55:48] of the result.
/// \param __m2
/// A 64-bit integer vector of [8 x i8].
/// Bits [39:32] are written to bits [15:8] of the result. \n
/// Bits [47:40] are written to bits [31:24] of the result. \n
/// Bits [55:48] are written to bits [47:40] of the result. \n
/// Bits [63:56] are written to bits [63:56] of the result.
/// \returns A 64-bit integer vector of [8 x i8] containing the interleaved
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_unpackhi_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_punpckhbw((__v8qi)__m1, (__v8qi)__m2);
/// Unpacks the upper 32 bits from two 64-bit integer vectors of
/// [4 x i16] and interleaves them into a 64-bit integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PUNPCKHWD </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// Bits [47:32] are written to bits [15:0] of the result. \n
/// Bits [63:48] are written to bits [47:32] of the result.
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// Bits [47:32] are written to bits [31:16] of the result. \n
/// Bits [63:48] are written to bits [63:48] of the result.
/// \returns A 64-bit integer vector of [4 x i16] containing the interleaved
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_unpackhi_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_punpckhwd((__v4hi)__m1, (__v4hi)__m2);
/// Unpacks the upper 32 bits from two 64-bit integer vectors of
/// [2 x i32] and interleaves them into a 64-bit integer vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PUNPCKHDQ </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [2 x i32]. The upper 32 bits are written to
/// the lower 32 bits of the result.
/// \param __m2
/// A 64-bit integer vector of [2 x i32]. The upper 32 bits are written to
/// the upper 32 bits of the result.
/// \returns A 64-bit integer vector of [2 x i32] containing the interleaved
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_unpackhi_pi32(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_punpckhdq((__v2si)__m1, (__v2si)__m2);
/// Unpacks the lower 32 bits from two 64-bit integer vectors of [8 x i8]
/// and interleaves them into a 64-bit integer vector of [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PUNPCKLBW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8].
/// Bits [7:0] are written to bits [7:0] of the result. \n
/// Bits [15:8] are written to bits [23:16] of the result. \n
/// Bits [23:16] are written to bits [39:32] of the result. \n
/// Bits [31:24] are written to bits [55:48] of the result.
/// \param __m2
/// A 64-bit integer vector of [8 x i8].
/// Bits [7:0] are written to bits [15:8] of the result. \n
/// Bits [15:8] are written to bits [31:24] of the result. \n
/// Bits [23:16] are written to bits [47:40] of the result. \n
/// Bits [31:24] are written to bits [63:56] of the result.
/// \returns A 64-bit integer vector of [8 x i8] containing the interleaved
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_unpacklo_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_punpcklbw((__v8qi)__m1, (__v8qi)__m2);
/// Unpacks the lower 32 bits from two 64-bit integer vectors of
/// [4 x i16] and interleaves them into a 64-bit integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PUNPCKLWD </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// Bits [15:0] are written to bits [15:0] of the result. \n
/// Bits [31:16] are written to bits [47:32] of the result.
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// Bits [15:0] are written to bits [31:16] of the result. \n
/// Bits [31:16] are written to bits [63:48] of the result.
/// \returns A 64-bit integer vector of [4 x i16] containing the interleaved
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_unpacklo_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_punpcklwd((__v4hi)__m1, (__v4hi)__m2);
/// Unpacks the lower 32 bits from two 64-bit integer vectors of
/// [2 x i32] and interleaves them into a 64-bit integer vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PUNPCKLDQ </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [2 x i32]. The lower 32 bits are written to
/// the lower 32 bits of the result.
/// \param __m2
/// A 64-bit integer vector of [2 x i32]. The lower 32 bits are written to
/// the upper 32 bits of the result.
/// \returns A 64-bit integer vector of [2 x i32] containing the interleaved
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_unpacklo_pi32(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_punpckldq((__v2si)__m1, (__v2si)__m2);
/// Adds each 8-bit integer element of the first 64-bit integer vector
/// of [8 x i8] to the corresponding 8-bit integer element of the second
/// 64-bit integer vector of [8 x i8]. The lower 8 bits of the results are
/// packed into a 64-bit integer vector of [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PADDB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8].
/// \param __m2
/// A 64-bit integer vector of [8 x i8].
/// \returns A 64-bit integer vector of [8 x i8] containing the sums of both
/// parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_add_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_paddb((__v8qi)__m1, (__v8qi)__m2);
/// Adds each 16-bit integer element of the first 64-bit integer vector
/// of [4 x i16] to the corresponding 16-bit integer element of the second
/// 64-bit integer vector of [4 x i16]. The lower 16 bits of the results are
/// packed into a 64-bit integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PADDW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [4 x i16] containing the sums of both
/// parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_add_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_paddw((__v4hi)__m1, (__v4hi)__m2);
/// Adds each 32-bit integer element of the first 64-bit integer vector
/// of [2 x i32] to the corresponding 32-bit integer element of the second
/// 64-bit integer vector of [2 x i32]. The lower 32 bits of the results are
/// packed into a 64-bit integer vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PADDD </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [2 x i32].
/// \param __m2
/// A 64-bit integer vector of [2 x i32].
/// \returns A 64-bit integer vector of [2 x i32] containing the sums of both
/// parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_add_pi32(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_paddd((__v2si)__m1, (__v2si)__m2);
/// Adds each 8-bit signed integer element of the first 64-bit integer
/// vector of [8 x i8] to the corresponding 8-bit signed integer element of
/// the second 64-bit integer vector of [8 x i8]. Positive sums greater than
/// 0x7F are saturated to 0x7F. Negative sums less than 0x80 are saturated to
/// 0x80. The results are packed into a 64-bit integer vector of [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PADDSB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8].
/// \param __m2
/// A 64-bit integer vector of [8 x i8].
/// \returns A 64-bit integer vector of [8 x i8] containing the saturated sums
/// of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_adds_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_paddsb((__v8qi)__m1, (__v8qi)__m2);
/// Adds each 16-bit signed integer element of the first 64-bit integer
/// vector of [4 x i16] to the corresponding 16-bit signed integer element of
/// the second 64-bit integer vector of [4 x i16]. Positive sums greater than
/// 0x7FFF are saturated to 0x7FFF. Negative sums less than 0x8000 are
/// saturated to 0x8000. The results are packed into a 64-bit integer vector
/// of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PADDSW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [4 x i16] containing the saturated sums
/// of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_adds_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_paddsw((__v4hi)__m1, (__v4hi)__m2);
/// Adds each 8-bit unsigned integer element of the first 64-bit integer
/// vector of [8 x i8] to the corresponding 8-bit unsigned integer element of
/// the second 64-bit integer vector of [8 x i8]. Sums greater than 0xFF are
/// saturated to 0xFF. The results are packed into a 64-bit integer vector of
/// [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PADDUSB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8].
/// \param __m2
/// A 64-bit integer vector of [8 x i8].
/// \returns A 64-bit integer vector of [8 x i8] containing the saturated
/// unsigned sums of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_adds_pu8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_paddusb((__v8qi)__m1, (__v8qi)__m2);
/// Adds each 16-bit unsigned integer element of the first 64-bit integer
/// vector of [4 x i16] to the corresponding 16-bit unsigned integer element
/// of the second 64-bit integer vector of [4 x i16]. Sums greater than
/// 0xFFFF are saturated to 0xFFFF. The results are packed into a 64-bit
/// integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PADDUSW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [4 x i16] containing the saturated
/// unsigned sums of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_adds_pu16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_paddusw((__v4hi)__m1, (__v4hi)__m2);
/// Subtracts each 8-bit integer element of the second 64-bit integer
/// vector of [8 x i8] from the corresponding 8-bit integer element of the
/// first 64-bit integer vector of [8 x i8]. The lower 8 bits of the results
/// are packed into a 64-bit integer vector of [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSUBB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8] containing the minuends.
/// \param __m2
/// A 64-bit integer vector of [8 x i8] containing the subtrahends.
/// \returns A 64-bit integer vector of [8 x i8] containing the differences of
/// both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sub_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_psubb((__v8qi)__m1, (__v8qi)__m2);
/// Subtracts each 16-bit integer element of the second 64-bit integer
/// vector of [4 x i16] from the corresponding 16-bit integer element of the
/// first 64-bit integer vector of [4 x i16]. The lower 16 bits of the
/// results are packed into a 64-bit integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSUBW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16] containing the minuends.
/// \param __m2
/// A 64-bit integer vector of [4 x i16] containing the subtrahends.
/// \returns A 64-bit integer vector of [4 x i16] containing the differences of
/// both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sub_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_psubw((__v4hi)__m1, (__v4hi)__m2);
/// Subtracts each 32-bit integer element of the second 64-bit integer
/// vector of [2 x i32] from the corresponding 32-bit integer element of the
/// first 64-bit integer vector of [2 x i32]. The lower 32 bits of the
/// results are packed into a 64-bit integer vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSUBD </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [2 x i32] containing the minuends.
/// \param __m2
/// A 64-bit integer vector of [2 x i32] containing the subtrahends.
/// \returns A 64-bit integer vector of [2 x i32] containing the differences of
/// both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sub_pi32(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_psubd((__v2si)__m1, (__v2si)__m2);
/// Subtracts each 8-bit signed integer element of the second 64-bit
/// integer vector of [8 x i8] from the corresponding 8-bit signed integer
/// element of the first 64-bit integer vector of [8 x i8]. Positive results
/// greater than 0x7F are saturated to 0x7F. Negative results less than 0x80
/// are saturated to 0x80. The results are packed into a 64-bit integer
/// vector of [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSUBSB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8] containing the minuends.
/// \param __m2
/// A 64-bit integer vector of [8 x i8] containing the subtrahends.
/// \returns A 64-bit integer vector of [8 x i8] containing the saturated
/// differences of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_subs_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_psubsb((__v8qi)__m1, (__v8qi)__m2);
/// Subtracts each 16-bit signed integer element of the second 64-bit
/// integer vector of [4 x i16] from the corresponding 16-bit signed integer
/// element of the first 64-bit integer vector of [4 x i16]. Positive results
/// greater than 0x7FFF are saturated to 0x7FFF. Negative results less than
/// 0x8000 are saturated to 0x8000. The results are packed into a 64-bit
/// integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSUBSW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16] containing the minuends.
/// \param __m2
/// A 64-bit integer vector of [4 x i16] containing the subtrahends.
/// \returns A 64-bit integer vector of [4 x i16] containing the saturated
/// differences of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_subs_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_psubsw((__v4hi)__m1, (__v4hi)__m2);
/// Subtracts each 8-bit unsigned integer element of the second 64-bit
/// integer vector of [8 x i8] from the corresponding 8-bit unsigned integer
/// element of the first 64-bit integer vector of [8 x i8].
/// If an element of the first vector is less than the corresponding element
/// of the second vector, the result is saturated to 0. The results are
/// packed into a 64-bit integer vector of [8 x i8].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSUBUSB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8] containing the minuends.
/// \param __m2
/// A 64-bit integer vector of [8 x i8] containing the subtrahends.
/// \returns A 64-bit integer vector of [8 x i8] containing the saturated
/// differences of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_subs_pu8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_psubusb((__v8qi)__m1, (__v8qi)__m2);
/// Subtracts each 16-bit unsigned integer element of the second 64-bit
/// integer vector of [4 x i16] from the corresponding 16-bit unsigned
/// integer element of the first 64-bit integer vector of [4 x i16].
/// If an element of the first vector is less than the corresponding element
/// of the second vector, the result is saturated to 0. The results are
/// packed into a 64-bit integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSUBUSW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16] containing the minuends.
/// \param __m2
/// A 64-bit integer vector of [4 x i16] containing the subtrahends.
/// \returns A 64-bit integer vector of [4 x i16] containing the saturated
/// differences of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_subs_pu16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_psubusw((__v4hi)__m1, (__v4hi)__m2);
/// Multiplies each 16-bit signed integer element of the first 64-bit
/// integer vector of [4 x i16] by the corresponding 16-bit signed integer
/// element of the second 64-bit integer vector of [4 x i16] and get four
/// 32-bit products. Adds adjacent pairs of products to get two 32-bit sums.
/// The lower 32 bits of these two sums are packed into a 64-bit integer
/// vector of [2 x i32].
/// For example, bits [15:0] of both parameters are multiplied, bits [31:16]
/// of both parameters are multiplied, and the sum of both results is written
/// to bits [31:0] of the result.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PMADDWD </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [2 x i32] containing the sums of
/// products of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_madd_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pmaddwd((__v4hi)__m1, (__v4hi)__m2);
/// Multiplies each 16-bit signed integer element of the first 64-bit
/// integer vector of [4 x i16] by the corresponding 16-bit signed integer
/// element of the second 64-bit integer vector of [4 x i16]. Packs the upper
/// 16 bits of the 32-bit products into a 64-bit integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PMULHW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [4 x i16] containing the upper 16 bits
/// of the products of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_mulhi_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pmulhw((__v4hi)__m1, (__v4hi)__m2);
/// Multiplies each 16-bit signed integer element of the first 64-bit
/// integer vector of [4 x i16] by the corresponding 16-bit signed integer
/// element of the second 64-bit integer vector of [4 x i16]. Packs the lower
/// 16 bits of the 32-bit products into a 64-bit integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PMULLW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [4 x i16] containing the lower 16 bits
/// of the products of both parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_mullo_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pmullw((__v4hi)__m1, (__v4hi)__m2);
/// Left-shifts each 16-bit signed integer element of the first
/// parameter, which is a 64-bit integer vector of [4 x i16], by the number
/// of bits specified by the second parameter, which is a 64-bit integer. The
/// lower 16 bits of the results are packed into a 64-bit integer vector of
/// [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSLLW </c> instruction.
/// \param __m
/// A 64-bit integer vector of [4 x i16].
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector of [4 x i16] containing the left-shifted
/// values. If \a __count is greater or equal to 16, the result is set to all
/// 0.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sll_pi16(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_psllw((__v4hi)__m, __count);
/// Left-shifts each 16-bit signed integer element of a 64-bit integer
/// vector of [4 x i16] by the number of bits specified by a 32-bit integer.
/// The lower 16 bits of the results are packed into a 64-bit integer vector
/// of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSLLW </c> instruction.
/// \param __m
/// A 64-bit integer vector of [4 x i16].
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector of [4 x i16] containing the left-shifted
/// values. If \a __count is greater or equal to 16, the result is set to all
/// 0.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_slli_pi16(__m64 __m, int __count)
return (__m64)__builtin_ia32_psllwi((__v4hi)__m, __count);
/// Left-shifts each 32-bit signed integer element of the first
/// parameter, which is a 64-bit integer vector of [2 x i32], by the number
/// of bits specified by the second parameter, which is a 64-bit integer. The
/// lower 32 bits of the results are packed into a 64-bit integer vector of
/// [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSLLD </c> instruction.
/// \param __m
/// A 64-bit integer vector of [2 x i32].
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector of [2 x i32] containing the left-shifted
/// values. If \a __count is greater or equal to 32, the result is set to all
/// 0.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sll_pi32(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_pslld((__v2si)__m, __count);
/// Left-shifts each 32-bit signed integer element of a 64-bit integer
/// vector of [2 x i32] by the number of bits specified by a 32-bit integer.
/// The lower 32 bits of the results are packed into a 64-bit integer vector
/// of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSLLD </c> instruction.
/// \param __m
/// A 64-bit integer vector of [2 x i32].
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector of [2 x i32] containing the left-shifted
/// values. If \a __count is greater or equal to 32, the result is set to all
/// 0.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_slli_pi32(__m64 __m, int __count)
return (__m64)__builtin_ia32_pslldi((__v2si)__m, __count);
/// Left-shifts the first 64-bit integer parameter by the number of bits
/// specified by the second 64-bit integer parameter. The lower 64 bits of
/// result are returned.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSLLQ </c> instruction.
/// \param __m
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector containing the left-shifted value. If
/// \a __count is greater or equal to 64, the result is set to 0.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sll_si64(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_psllq((__v1di)__m, __count);
/// Left-shifts the first parameter, which is a 64-bit integer, by the
/// number of bits specified by the second parameter, which is a 32-bit
/// integer. The lower 64 bits of result are returned.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSLLQ </c> instruction.
/// \param __m
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector containing the left-shifted value. If
/// \a __count is greater or equal to 64, the result is set to 0.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_slli_si64(__m64 __m, int __count)
return (__m64)__builtin_ia32_psllqi((__v1di)__m, __count);
/// Right-shifts each 16-bit integer element of the first parameter,
/// which is a 64-bit integer vector of [4 x i16], by the number of bits
/// specified by the second parameter, which is a 64-bit integer.
/// High-order bits are filled with the sign bit of the initial value of each
/// 16-bit element. The 16-bit results are packed into a 64-bit integer
/// vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRAW </c> instruction.
/// \param __m
/// A 64-bit integer vector of [4 x i16].
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector of [4 x i16] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sra_pi16(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_psraw((__v4hi)__m, __count);
/// Right-shifts each 16-bit integer element of a 64-bit integer vector
/// of [4 x i16] by the number of bits specified by a 32-bit integer.
/// High-order bits are filled with the sign bit of the initial value of each
/// 16-bit element. The 16-bit results are packed into a 64-bit integer
/// vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRAW </c> instruction.
/// \param __m
/// A 64-bit integer vector of [4 x i16].
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector of [4 x i16] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srai_pi16(__m64 __m, int __count)
return (__m64)__builtin_ia32_psrawi((__v4hi)__m, __count);
/// Right-shifts each 32-bit integer element of the first parameter,
/// which is a 64-bit integer vector of [2 x i32], by the number of bits
/// specified by the second parameter, which is a 64-bit integer.
/// High-order bits are filled with the sign bit of the initial value of each
/// 32-bit element. The 32-bit results are packed into a 64-bit integer
/// vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRAD </c> instruction.
/// \param __m
/// A 64-bit integer vector of [2 x i32].
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector of [2 x i32] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_sra_pi32(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_psrad((__v2si)__m, __count);
/// Right-shifts each 32-bit integer element of a 64-bit integer vector
/// of [2 x i32] by the number of bits specified by a 32-bit integer.
/// High-order bits are filled with the sign bit of the initial value of each
/// 32-bit element. The 32-bit results are packed into a 64-bit integer
/// vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRAD </c> instruction.
/// \param __m
/// A 64-bit integer vector of [2 x i32].
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector of [2 x i32] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srai_pi32(__m64 __m, int __count)
return (__m64)__builtin_ia32_psradi((__v2si)__m, __count);
/// Right-shifts each 16-bit integer element of the first parameter,
/// which is a 64-bit integer vector of [4 x i16], by the number of bits
/// specified by the second parameter, which is a 64-bit integer.
/// High-order bits are cleared. The 16-bit results are packed into a 64-bit
/// integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRLW </c> instruction.
/// \param __m
/// A 64-bit integer vector of [4 x i16].
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector of [4 x i16] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srl_pi16(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_psrlw((__v4hi)__m, __count);
/// Right-shifts each 16-bit integer element of a 64-bit integer vector
/// of [4 x i16] by the number of bits specified by a 32-bit integer.
/// High-order bits are cleared. The 16-bit results are packed into a 64-bit
/// integer vector of [4 x i16].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRLW </c> instruction.
/// \param __m
/// A 64-bit integer vector of [4 x i16].
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector of [4 x i16] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srli_pi16(__m64 __m, int __count)
return (__m64)__builtin_ia32_psrlwi((__v4hi)__m, __count);
/// Right-shifts each 32-bit integer element of the first parameter,
/// which is a 64-bit integer vector of [2 x i32], by the number of bits
/// specified by the second parameter, which is a 64-bit integer.
/// High-order bits are cleared. The 32-bit results are packed into a 64-bit
/// integer vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRLD </c> instruction.
/// \param __m
/// A 64-bit integer vector of [2 x i32].
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector of [2 x i32] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srl_pi32(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_psrld((__v2si)__m, __count);
/// Right-shifts each 32-bit integer element of a 64-bit integer vector
/// of [2 x i32] by the number of bits specified by a 32-bit integer.
/// High-order bits are cleared. The 32-bit results are packed into a 64-bit
/// integer vector of [2 x i32].
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRLD </c> instruction.
/// \param __m
/// A 64-bit integer vector of [2 x i32].
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector of [2 x i32] containing the right-shifted
/// values.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srli_pi32(__m64 __m, int __count)
return (__m64)__builtin_ia32_psrldi((__v2si)__m, __count);
/// Right-shifts the first 64-bit integer parameter by the number of bits
/// specified by the second 64-bit integer parameter.
/// High-order bits are cleared.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRLQ </c> instruction.
/// \param __m
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \param __count
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \returns A 64-bit integer vector containing the right-shifted value.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srl_si64(__m64 __m, __m64 __count)
return (__m64)__builtin_ia32_psrlq((__v1di)__m, __count);
/// Right-shifts the first parameter, which is a 64-bit integer, by the
/// number of bits specified by the second parameter, which is a 32-bit
/// integer.
/// High-order bits are cleared.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PSRLQ </c> instruction.
/// \param __m
/// A 64-bit integer vector interpreted as a single 64-bit integer.
/// \param __count
/// A 32-bit integer value.
/// \returns A 64-bit integer vector containing the right-shifted value.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_srli_si64(__m64 __m, int __count)
return (__m64)__builtin_ia32_psrlqi((__v1di)__m, __count);
/// Performs a bitwise AND of two 64-bit integer vectors.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PAND </c> instruction.
/// \param __m1
/// A 64-bit integer vector.
/// \param __m2
/// A 64-bit integer vector.
/// \returns A 64-bit integer vector containing the bitwise AND of both
/// parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_and_si64(__m64 __m1, __m64 __m2)
return __builtin_ia32_pand((__v1di)__m1, (__v1di)__m2);
/// Performs a bitwise NOT of the first 64-bit integer vector, and then
/// performs a bitwise AND of the intermediate result and the second 64-bit
/// integer vector.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PANDN </c> instruction.
/// \param __m1
/// A 64-bit integer vector. The one's complement of this parameter is used
/// in the bitwise AND.
/// \param __m2
/// A 64-bit integer vector.
/// \returns A 64-bit integer vector containing the bitwise AND of the second
/// parameter and the one's complement of the first parameter.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_andnot_si64(__m64 __m1, __m64 __m2)
return __builtin_ia32_pandn((__v1di)__m1, (__v1di)__m2);
/// Performs a bitwise OR of two 64-bit integer vectors.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> POR </c> instruction.
/// \param __m1
/// A 64-bit integer vector.
/// \param __m2
/// A 64-bit integer vector.
/// \returns A 64-bit integer vector containing the bitwise OR of both
/// parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_or_si64(__m64 __m1, __m64 __m2)
return __builtin_ia32_por((__v1di)__m1, (__v1di)__m2);
/// Performs a bitwise exclusive OR of two 64-bit integer vectors.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PXOR </c> instruction.
/// \param __m1
/// A 64-bit integer vector.
/// \param __m2
/// A 64-bit integer vector.
/// \returns A 64-bit integer vector containing the bitwise exclusive OR of both
/// parameters.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_xor_si64(__m64 __m1, __m64 __m2)
return __builtin_ia32_pxor((__v1di)__m1, (__v1di)__m2);
/// Compares the 8-bit integer elements of two 64-bit integer vectors of
/// [8 x i8] to determine if the element of the first vector is equal to the
/// corresponding element of the second vector.
/// The comparison yields 0 for false, 0xFF for true.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PCMPEQB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8].
/// \param __m2
/// A 64-bit integer vector of [8 x i8].
/// \returns A 64-bit integer vector of [8 x i8] containing the comparison
/// results.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cmpeq_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pcmpeqb((__v8qi)__m1, (__v8qi)__m2);
/// Compares the 16-bit integer elements of two 64-bit integer vectors of
/// [4 x i16] to determine if the element of the first vector is equal to the
/// corresponding element of the second vector.
/// The comparison yields 0 for false, 0xFFFF for true.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PCMPEQW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [4 x i16] containing the comparison
/// results.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cmpeq_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pcmpeqw((__v4hi)__m1, (__v4hi)__m2);
/// Compares the 32-bit integer elements of two 64-bit integer vectors of
/// [2 x i32] to determine if the element of the first vector is equal to the
/// corresponding element of the second vector.
/// The comparison yields 0 for false, 0xFFFFFFFF for true.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PCMPEQD </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [2 x i32].
/// \param __m2
/// A 64-bit integer vector of [2 x i32].
/// \returns A 64-bit integer vector of [2 x i32] containing the comparison
/// results.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cmpeq_pi32(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pcmpeqd((__v2si)__m1, (__v2si)__m2);
/// Compares the 8-bit integer elements of two 64-bit integer vectors of
/// [8 x i8] to determine if the element of the first vector is greater than
/// the corresponding element of the second vector.
/// The comparison yields 0 for false, 0xFF for true.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PCMPGTB </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [8 x i8].
/// \param __m2
/// A 64-bit integer vector of [8 x i8].
/// \returns A 64-bit integer vector of [8 x i8] containing the comparison
/// results.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cmpgt_pi8(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pcmpgtb((__v8qi)__m1, (__v8qi)__m2);
/// Compares the 16-bit integer elements of two 64-bit integer vectors of
/// [4 x i16] to determine if the element of the first vector is greater than
/// the corresponding element of the second vector.
/// The comparison yields 0 for false, 0xFFFF for true.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PCMPGTW </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [4 x i16].
/// \param __m2
/// A 64-bit integer vector of [4 x i16].
/// \returns A 64-bit integer vector of [4 x i16] containing the comparison
/// results.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cmpgt_pi16(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pcmpgtw((__v4hi)__m1, (__v4hi)__m2);
/// Compares the 32-bit integer elements of two 64-bit integer vectors of
/// [2 x i32] to determine if the element of the first vector is greater than
/// the corresponding element of the second vector.
/// The comparison yields 0 for false, 0xFFFFFFFF for true.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PCMPGTD </c> instruction.
/// \param __m1
/// A 64-bit integer vector of [2 x i32].
/// \param __m2
/// A 64-bit integer vector of [2 x i32].
/// \returns A 64-bit integer vector of [2 x i32] containing the comparison
/// results.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_cmpgt_pi32(__m64 __m1, __m64 __m2)
return (__m64)__builtin_ia32_pcmpgtd((__v2si)__m1, (__v2si)__m2);
/// Constructs a 64-bit integer vector initialized to zero.
/// \headerfile <x86intrin.h>
/// This intrinsic corresponds to the <c> PXOR </c> instruction.
/// \returns An initialized 64-bit integer vector with all elements set to zero.
static __inline__ __m64 __DEFAULT_FN_ATTRS
return __extension__ (__m64){ 0LL };
/// Constructs a 64-bit integer vector initialized with the specified
/// 32-bit integer values.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __i1
/// A 32-bit integer value used to initialize the upper 32 bits of the
/// result.
/// \param __i0
/// A 32-bit integer value used to initialize the lower 32 bits of the
/// result.
/// \returns An initialized 64-bit integer vector.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_set_pi32(int __i1, int __i0)
return (__m64)__builtin_ia32_vec_init_v2si(__i0, __i1);
/// Constructs a 64-bit integer vector initialized with the specified
/// 16-bit integer values.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __s3
/// A 16-bit integer value used to initialize bits [63:48] of the result.
/// \param __s2
/// A 16-bit integer value used to initialize bits [47:32] of the result.
/// \param __s1
/// A 16-bit integer value used to initialize bits [31:16] of the result.
/// \param __s0
/// A 16-bit integer value used to initialize bits [15:0] of the result.
/// \returns An initialized 64-bit integer vector.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_set_pi16(short __s3, short __s2, short __s1, short __s0)
return (__m64)__builtin_ia32_vec_init_v4hi(__s0, __s1, __s2, __s3);
/// Constructs a 64-bit integer vector initialized with the specified
/// 8-bit integer values.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __b7
/// An 8-bit integer value used to initialize bits [63:56] of the result.
/// \param __b6
/// An 8-bit integer value used to initialize bits [55:48] of the result.
/// \param __b5
/// An 8-bit integer value used to initialize bits [47:40] of the result.
/// \param __b4
/// An 8-bit integer value used to initialize bits [39:32] of the result.
/// \param __b3
/// An 8-bit integer value used to initialize bits [31:24] of the result.
/// \param __b2
/// An 8-bit integer value used to initialize bits [23:16] of the result.
/// \param __b1
/// An 8-bit integer value used to initialize bits [15:8] of the result.
/// \param __b0
/// An 8-bit integer value used to initialize bits [7:0] of the result.
/// \returns An initialized 64-bit integer vector.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_set_pi8(char __b7, char __b6, char __b5, char __b4, char __b3, char __b2,
char __b1, char __b0)
return (__m64)__builtin_ia32_vec_init_v8qi(__b0, __b1, __b2, __b3,
__b4, __b5, __b6, __b7);
/// Constructs a 64-bit integer vector of [2 x i32], with each of the
/// 32-bit integer vector elements set to the specified 32-bit integer
/// value.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __i
/// A 32-bit integer value used to initialize each vector element of the
/// result.
/// \returns An initialized 64-bit integer vector of [2 x i32].
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_set1_pi32(int __i)
return _mm_set_pi32(__i, __i);
/// Constructs a 64-bit integer vector of [4 x i16], with each of the
/// 16-bit integer vector elements set to the specified 16-bit integer
/// value.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __w
/// A 16-bit integer value used to initialize each vector element of the
/// result.
/// \returns An initialized 64-bit integer vector of [4 x i16].
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_set1_pi16(short __w)
return _mm_set_pi16(__w, __w, __w, __w);
/// Constructs a 64-bit integer vector of [8 x i8], with each of the
/// 8-bit integer vector elements set to the specified 8-bit integer value.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __b
/// An 8-bit integer value used to initialize each vector element of the
/// result.
/// \returns An initialized 64-bit integer vector of [8 x i8].
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_set1_pi8(char __b)
return _mm_set_pi8(__b, __b, __b, __b, __b, __b, __b, __b);
/// Constructs a 64-bit integer vector, initialized in reverse order with
/// the specified 32-bit integer values.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __i0
/// A 32-bit integer value used to initialize the lower 32 bits of the
/// result.
/// \param __i1
/// A 32-bit integer value used to initialize the upper 32 bits of the
/// result.
/// \returns An initialized 64-bit integer vector.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_setr_pi32(int __i0, int __i1)
return _mm_set_pi32(__i1, __i0);
/// Constructs a 64-bit integer vector, initialized in reverse order with
/// the specified 16-bit integer values.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __w0
/// A 16-bit integer value used to initialize bits [15:0] of the result.
/// \param __w1
/// A 16-bit integer value used to initialize bits [31:16] of the result.
/// \param __w2
/// A 16-bit integer value used to initialize bits [47:32] of the result.
/// \param __w3
/// A 16-bit integer value used to initialize bits [63:48] of the result.
/// \returns An initialized 64-bit integer vector.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_setr_pi16(short __w0, short __w1, short __w2, short __w3)
return _mm_set_pi16(__w3, __w2, __w1, __w0);
/// Constructs a 64-bit integer vector, initialized in reverse order with
/// the specified 8-bit integer values.
/// \headerfile <x86intrin.h>
/// This intrinsic is a utility function and does not correspond to a specific
/// instruction.
/// \param __b0
/// An 8-bit integer value used to initialize bits [7:0] of the result.
/// \param __b1
/// An 8-bit integer value used to initialize bits [15:8] of the result.
/// \param __b2
/// An 8-bit integer value used to initialize bits [23:16] of the result.
/// \param __b3
/// An 8-bit integer value used to initialize bits [31:24] of the result.
/// \param __b4
/// An 8-bit integer value used to initialize bits [39:32] of the result.
/// \param __b5
/// An 8-bit integer value used to initialize bits [47:40] of the result.
/// \param __b6
/// An 8-bit integer value used to initialize bits [55:48] of the result.
/// \param __b7
/// An 8-bit integer value used to initialize bits [63:56] of the result.
/// \returns An initialized 64-bit integer vector.
static __inline__ __m64 __DEFAULT_FN_ATTRS
_mm_setr_pi8(char __b0, char __b1, char __b2, char __b3, char __b4, char __b5,
char __b6, char __b7)
return _mm_set_pi8(__b7, __b6, __b5, __b4, __b3, __b2, __b1, __b0);
#undef __DEFAULT_FN_ATTRS
/* Aliases for compatibility. */
#define _m_empty _mm_empty
#define _m_from_int _mm_cvtsi32_si64
#define _m_from_int64 _mm_cvtsi64_m64
#define _m_to_int _mm_cvtsi64_si32
#define _m_to_int64 _mm_cvtm64_si64
#define _m_packsswb _mm_packs_pi16
#define _m_packssdw _mm_packs_pi32
#define _m_packuswb _mm_packs_pu16
#define _m_punpckhbw _mm_unpackhi_pi8
#define _m_punpckhwd _mm_unpackhi_pi16
#define _m_punpckhdq _mm_unpackhi_pi32
#define _m_punpcklbw _mm_unpacklo_pi8
#define _m_punpcklwd _mm_unpacklo_pi16
#define _m_punpckldq _mm_unpacklo_pi32
#define _m_paddb _mm_add_pi8
#define _m_paddw _mm_add_pi16
#define _m_paddd _mm_add_pi32
#define _m_paddsb _mm_adds_pi8
#define _m_paddsw _mm_adds_pi16
#define _m_paddusb _mm_adds_pu8
#define _m_paddusw _mm_adds_pu16
#define _m_psubb _mm_sub_pi8
#define _m_psubw _mm_sub_pi16
#define _m_psubd _mm_sub_pi32
#define _m_psubsb _mm_subs_pi8
#define _m_psubsw _mm_subs_pi16
#define _m_psubusb _mm_subs_pu8
#define _m_psubusw _mm_subs_pu16
#define _m_pmaddwd _mm_madd_pi16
#define _m_pmulhw _mm_mulhi_pi16
#define _m_pmullw _mm_mullo_pi16
#define _m_psllw _mm_sll_pi16
#define _m_psllwi _mm_slli_pi16
#define _m_pslld _mm_sll_pi32
#define _m_pslldi _mm_slli_pi32
#define _m_psllq _mm_sll_si64
#define _m_psllqi _mm_slli_si64
#define _m_psraw _mm_sra_pi16
#define _m_psrawi _mm_srai_pi16
#define _m_psrad _mm_sra_pi32
#define _m_psradi _mm_srai_pi32
#define _m_psrlw _mm_srl_pi16
#define _m_psrlwi _mm_srli_pi16
#define _m_psrld _mm_srl_pi32
#define _m_psrldi _mm_srli_pi32
#define _m_psrlq _mm_srl_si64
#define _m_psrlqi _mm_srli_si64
#define _m_pand _mm_and_si64
#define _m_pandn _mm_andnot_si64
#define _m_por _mm_or_si64
#define _m_pxor _mm_xor_si64
#define _m_pcmpeqb _mm_cmpeq_pi8
#define _m_pcmpeqw _mm_cmpeq_pi16
#define _m_pcmpeqd _mm_cmpeq_pi32
#define _m_pcmpgtb _mm_cmpgt_pi8
#define _m_pcmpgtw _mm_cmpgt_pi16
#define _m_pcmpgtd _mm_cmpgt_pi32
#endif /* __MMINTRIN_H */ | {"url":"https://llvm.googlesource.com/llvm-project/+/77ff6f7df8691a735a2dc979cdb44835dd2d41af/clang/lib/Headers/mmintrin.h","timestamp":"2024-11-14T18:35:54Z","content_type":"text/html","content_length":"430865","record_id":"<urn:uuid:778630f5-27a9-4d4e-b22f-f218ca619dbb>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00882.warc.gz"} |
Fuel Cell Handbook by U.S. Department of Energy
Fuel Cell Handbook
by U.S. Department of Energy
Publisher: University Press of the Pacific 2004
ISBN/ASIN: 1410219607
ISBN-13: 9781410219602
Number of pages: 427
Fuel cells are an important technology for a potentially wide variety of applications including micropower, auxiliary power, transportation power, stationary power for buildings and other distributed
generation applications, and central power. These applications will be in a large number of industries worldwide. This edition includes calculation examples for fuel cells for the wide variety of
possible applications. The handbook includes a separate section on alkaline fuel cells.
Download or read it online for free here:
Download link
(5MB, PDF)
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Leetcode 441: Arranging Coins - Leetcode Detailed Solutions - Cse Nerd
Leetcode 441: Arranging Coins
Category: Easy
You have a total of n coins that you want to form in a staircase shape, where every k-th row must have exactly k coins.
Given n, find the total number of full staircase rows that can be formed.
n is a non-negative integer and fits within the range of a 32-bit signed integer.
Example 1
2 n = 5
3 The coins can form the following rows:
4 ¤
5 ¤ ¤
6 ¤ ¤
7 Because the 3rd row is incomplete, we return 2.
Example 2
2 n = 8
3 The coins can form the following rows:
4 ¤
5 ¤ ¤
6 ¤ ¤ ¤
7 ¤ ¤
8 Because the 4th row is incomplete, we return 3.
Solution Approach
This question looks pretty easy in the first glance. Although it is easy I will show fast mathematical techniques to solve questions like these.
Arithmetic Progression
If you look at the question and the examples carefully you will detect a pattern in the number of steps. It always starts with 1 and 1 is incremented every time until it reaches a point when there
are no coins for further steps or there are not enough coins for the next step. This forms an AP or Arithmetic Progression with initial value as 1 and step as 1. If you are not familiar with AP you
can check here. Now to solve the problem using AP just keep on subtracting step value from N and incrementing step by 1 until the step value is greater than the value of N.
Completing the square
We can also use completing the square technique which is used in quadratic equations to solve the problem. Learn more about it here.
To solve the problem an equation is given in the question which we will use and complete the square for k.
(K(K+1) / 2) <= N
Using completing the square technique on it we get
(k + 0.5)^2 – 0.25 <= 2*n
After solving this equation we will get
K = Sqrt(2*N + 0.25) – 0.5
Now using this equation we can get the answer directly.
Solution Code
Arithmetic Progression
Arranging coins using AP
1 class Solution {
2 public int arrangeCoins(int n) {
3 int step = 1;
4 while(n>=step){
5 n-=step; // n = n-step;
6 step++;
7 }
8 return step-1;
9 }
10 }
Completing the square
Arranging Coins using Completing the square
1 class Solution {
2 public int arrangeCoins(int n) {
3 return (int)(Math.sqrt(2 * (long)n + 0.25) - 0.5);
4 }
5 }
For more Leetcode explained solutions visit Leetcode Solutions. If you like capture the flag challenges visit here.
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Practice Question
• Subject 11. Tests Concerning the Equality of Two Variances (F-Test)
CFA Practice Question
There are 676 practice questions for this topic.
CFA Practice Question
A large value for F will result in ______.
A. a rejection of H[0]
B. an acceptance of H[0]
C. the test statistic falling in the acceptance region
Explanation: A large value for F will result in a rejection of H[0].
User Contributed Comments 3
User Comment
mchu A F-test checks whether or not the ratio of the test statistic, (s1)2/(s2)2, is close to 1. If it is, you will obtain a non-rejection of H0; if not, a rejection of H0.
teje but is also depends on the significance level that you choose...in fact if you get a F statistic below 1, you should invert it, i.e. 1/f stat; be sure to also invert the degrees of freedom!
How is a big value of F always gonna cause a rejection of Ho. It depends on the critical value and the significance level you choose.
Unless they mean a HUGE number. I also didnt know if they meant the F value calculated as the test statistic or the F critical value. Without being specific you cant answer the question.
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Resubstitution classification edge
e = resubEdge(Mdl) returns the weighted resubstitution Classification Edge (e) for the trained classification model Mdl using the predictor data stored in Mdl.X, the corresponding true class labels
stored in Mdl.Y, and the observation weights stored in Mdl.W.
e = resubEdge(Mdl,'IncludeInteractions',includeInteractions) specifies whether to include interaction terms in computations. This syntax applies only to generalized additive models.
Estimate Resubstitution Edge of SVM Classifiers
Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').
Train a support vector machine (SVM) classifier. Standardize the data and specify that 'g' is the positive class.
SVMModel = fitcsvm(X,Y,'Standardize',true,'ClassNames',{'b','g'});
SVMModel is a trained ClassificationSVM classifier.
Estimate the resubstitution edge, which is the mean of the training sample margins.
Select Naive Bayes Classifier Features by Comparing In-Sample Edges
The classifier edge measures the average of the classifier margins. One way to perform feature selection is to compare training sample edges from multiple models. Based solely on this criterion, the
classifier with the highest edge is the best classifier.
Load the ionosphere data set. Remove the first two predictors for stability.
load ionosphere
X = X(:,3:end);
Define these two data sets:
• fullX contains all predictors.
• partX contains the 10 most important predictors.
fullX = X;
idx = fscmrmr(X,Y);
partX = X(:,idx(1:10));
Train a naive Bayes classifier for each predictor set.
FullMdl = fitcnb(fullX,Y);
PartMdl = fitcnb(partX,Y);
FullMdl and PartMdl are trained ClassificationNaiveBayes classifiers.
Estimate the training sample edge for each classifier.
fullEdge = resubEdge(FullMdl)
partEdge = resubEdge(PartMdl)
The edge of the classifier trained on the 10 most important predictors is larger. This result suggests that the classifier trained using only those predictors has a better in-sample fit.
Compare GAMs by Examining Training Sample Margins and Edge
Compare a generalized additive model (GAM) with linear terms to a GAM with both linear and interaction terms by examining the training sample margins and edge. Based solely on this comparison, the
classifier with the highest margins and edge is the best model.
Load the 1994 census data stored in census1994.mat. The data set consists of demographic data from the US Census Bureau to predict whether an individual makes over $50,000 per year. The
classification task is to fit a model that predicts the salary category of people given their age, working class, education level, marital status, race, and so on.
census1994 contains the training data set adultdata and the test data set adulttest. To reduce the running time for this example, subsample 500 training observations from adultdata by using the
datasample function.
rng('default') % For reproducibility
NumSamples = 5e2;
adultdata = datasample(adultdata,NumSamples,'Replace',false);
Train a GAM that contains both linear and interaction terms for predictors. Specify to include all available interaction terms whose p-values are not greater than 0.05.
Mdl = fitcgam(adultdata,'salary','Interactions','all','MaxPValue',0.05)
Mdl =
PredictorNames: {'age' 'workClass' 'fnlwgt' 'education' 'education_num' 'marital_status' 'occupation' 'relationship' 'race' 'sex' 'capital_gain' 'capital_loss' 'hours_per_week' 'native_country'}
ResponseName: 'salary'
CategoricalPredictors: [2 4 6 7 8 9 10 14]
ClassNames: [<=50K >50K]
ScoreTransform: 'logit'
Intercept: -28.5594
Interactions: [82x2 double]
NumObservations: 500
Mdl is a ClassificationGAM model object. Mdl includes 82 interaction terms.
Estimate the training sample margins and edge for Mdl.
M = resubMargin(Mdl);
E = resubEdge(Mdl)
Estimate the training sample margins and edge for Mdl without including interaction terms.
M_nointeractions = resubMargin(Mdl,'IncludeInteractions',false);
E_nointeractions = resubEdge(Mdl,'IncludeInteractions',false)
E_nointeractions =
Display the distributions of the margins using box plots.
boxplot([M M_nointeractions],'Labels',{'Linear and Interaction Terms','Linear Terms Only'})
title('Box Plots of Training Sample Margins')
When you include the interaction terms in the computation, all the resubstitution margin values for Mdl are 1, and the resubstitution edge value (average of the margins) is 1. The margins and edge
decrease when you do not include the interaction terms in Mdl.
Input Arguments
includeInteractions — Flag to include interaction terms
true | false
Flag to include interaction terms of the model, specified as true or false. This argument is valid only for a generalized additive model (GAM). That is, you can specify this argument only when Mdl is
The default value is true if Mdl contains interaction terms. The value must be false if the model does not contain interaction terms.
Data Types: logical
More About
Classification Edge
The classification edge is the weighted mean of the classification margins.
One way to choose among multiple classifiers, for example to perform feature selection, is to choose the classifier that yields the greatest edge.
Classification Margin
The classification margin for binary classification is, for each observation, the difference between the classification score for the true class and the classification score for the false class. The
classification margin for multiclass classification is the difference between the classification score for the true class and the maximal classification score for the false classes.
If the margins are on the same scale (that is, the score values are based on the same score transformation), then they serve as a classification confidence measure. Among multiple classifiers, those
that yield greater margins are better.
resubEdge computes the classification edge according to the corresponding edge function of the object (Mdl). For a model-specific description, see the edge function reference pages in the following
Extended Capabilities
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2012a
R2024b: Specify GPU arrays for neural network models (requires Parallel Computing Toolbox)
resubEdge fully supports GPU arrays for ClassificationNeuralNetwork.
R2023b: Observations with missing predictor values are used in resubstitution and cross-validation computations
Starting in R2023b, the following classification model object functions use observations with missing predictor values as part of resubstitution ("resub") and cross-validation ("kfold") computations
for classification edges, losses, margins, and predictions.
Model Type Model Objects Object Functions
Discriminant analysis classification model ClassificationDiscriminant resubEdge, resubLoss, resubMargin, resubPredict
ClassificationPartitionedModel kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Ensemble of discriminant analysis learners for classification ClassificationEnsemble resubEdge, resubLoss, resubMargin, resubPredict
ClassificationPartitionedEnsemble kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Gaussian kernel classification model ClassificationPartitionedKernel kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
ClassificationPartitionedKernelECOC kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Linear classification model ClassificationPartitionedLinear kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
ClassificationPartitionedLinearECOC kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Neural network classification model ClassificationNeuralNetwork resubEdge, resubLoss, resubMargin, resubPredict
ClassificationPartitionedModel kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
Support vector machine (SVM) classification model ClassificationSVM resubEdge, resubLoss, resubMargin, resubPredict
ClassificationPartitionedModel kfoldEdge, kfoldLoss, kfoldMargin, kfoldPredict
In previous releases, the software omitted observations with missing predictor values from the resubstitution and cross-validation computations.
R2022a: resubEdge returns a different value for a ClassificationSVM model with a nondefault cost matrix | {"url":"https://in.mathworks.com/help/stats/classificationsvm.resubedge.html","timestamp":"2024-11-11T19:35:54Z","content_type":"text/html","content_length":"121284","record_id":"<urn:uuid:828897ae-2245-488b-996f-dcc2679ce2be>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00340.warc.gz"} |
Exploring Young's Modulus in Engineering
Formula:E = stress / strain
Understanding Young’s Modulus
Young's Modulus, also known as the modulus of elasticity, is a fundamental property of materials that measures their stiffness and elastic behavior. This critical concept in engineering helps us
understand how materials deform under mechanical stress and return to their original shape when the stress is removed. Let’s break down its significance, formula, and real life applications.
What is Young’s Modulus?
Young’s Modulus (E) is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. For engineers and scientists, it’s an indispensable tool for
predicting how materials will behave in different situations.
In more approachable terms, imagine you have a rubber band and a metal wire. If you apply the same stretching force to both, the rubber band will stretch much more than the metal wire. This
difference in their stretching behavior is captured by Young’s Modulus; the metal wire has a higher Young’s Modulus than the rubber band, indicating it is stiffer and less elastic.
The Formula
The formula for Young’s Modulus is:
E = stress / strain
• stress is defined as the force applied per unit area, measured in Pascals (Pa) or Newtons per square meter (N/m²).
• strain is the deformation or change in length divided by the original length, a dimensionless quantity.
Inputs and Outputs
• stress (Input): The force (in Newtons, N) applied to the material, divided by the cross sectional area (in square meters, m²) on which the force is acting. Stress can be thought of as the
intensity of the internal forces within the material when it is loaded.
• strain (Input): The relative deformation or change in length (dimensionless). It is calculated by dividing the change in length (in meters, m) by the original length (in meters, m).
• Young’s Modulus (E) (Output): This is the ratio of stress to strain and gives an indication of the stiffness of the material. It is measured in Pascals (Pa) or Newtons per square meter (N/m²).
Real Life Examples
Let’s put this into perspective with some real life examples:
• Steel: Steel has a very high Young’s Modulus, around 200 GPa (Gigapascals). This means it takes a lot of stress (force per unit area) to produce even a small amount of strain (deformation) in
steel, indicating it’s a very stiff material.
• Rubber: Rubber, on the other hand, has a much lower Young’s Modulus, around 0.01 GPa. It deforms easily under low stress, showing it’s very elastic.
How to Use the Formula: A Step by Step Example
Here’s a step by step process for using the Young’s Modulus formula:
1. Identify the force applied and the cross sectional area: For example, a force of 1000 Newtons is applied to a rod with a cross sectional area of 0.01 square meters.
2. Calculate the stress: Stress = Force / Area = 1000 N / 0.01 m² = 100,000 N/m² (Pascal).
3. Measure the original length and the change in length: Suppose the rod was originally 2 meters long and it elongated by 0.001 meters under the load.
4. Calculate the strain: Strain = Change in Length / Original Length = 0.001 m / 2 m = 0.0005.
5. Compute Young’s Modulus: E = Stress / Strain = 100,000 N/m² / 0.0005 = 200,000,000 N/m² or 200 MPa (Megapascals).
Data Validation
It's vital to ensure the values used are physically plausible:
• Stress and strain should be numerical and positive, as negative values would indicate incorrect application of force and deformation measures.
• The original length should be a positive number; zero or negative lengths are not realistic.
Q: Why is Young’s Modulus important in engineering?
A: Young's Modulus helps engineers choose the right material for construction projects and other applications by predicting how much a material will deform under a given load.
Q: What units are used for Young’s Modulus?
A: It is typically measured in Pascals (Pa), Megapascals (MPa), or Gigapascals (GPa) depending on the material in question.
Q: Can Young’s Modulus be zero?
A: In practical terms, no real material has a Young’s Modulus of zero; that would mean the material offers no resistance to deformation.
Young's Modulus provides critical insights into material stiffness and elasticity, forming the backbone of many engineering applications. Whether you're designing skyscrapers, crafting medical
devices, or working in any field that requires knowledge of material properties, understanding Young's Modulus is essential. Armed with this knowledge and the practical examples provided, you are
well equipped to apply this concept to real world challenges.
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Class Groups: A Comprehensive Primer
Table of Contents
In 2018, Benedikt Bunz and Ben Fisch, both PhD students at Stanford University, released a paper along with Dan Boneh titled “Batching Techniques for Accumulators with Applications to IOPs and
Stateless Blockchains”. In it they use some basic group theory to build a dynamic accumulator, which allows for storing and deleting elements in addition to the use of both membership and
non-membership proofs. It can be used to create a vector commitment data structure analogous to Merkle trees, with the main difference being that it allows for constant-sized inclusion proofs, where
a Merkle tree has \(O(\log n)\) sized inclusion proofs where \(n\) is the number of elements being stored. If the stored set is big enough, this can be a pretty big deal.
One of the main security assumptions their construction uses is that it relies on a group of unknown order. In particular the strong RSA assumption must hold, meaning it is hard to compute a chosen
root of a random group element. There are two good candidates for such a group. The simpler of the two is known as an RSA group, or a multiplicative group of order \(N\) where \(N = pq\) for some
unknown factorization. This however requires a trusted setup, as whoever generates the modulus N must be trusted to discard \(p\) and \(q\). The alternative is known as a class group of imaginary
quadratic order, which eliminates the need for a trusted setup and which we will be exploring in this post.
Class groups come from a long line of mathematical research, and were originally discovered by Gauss in his Disquisitiones Arithmeticae in 1801. The math that’s been developed on top of his work in
the past two centuries involves some decently complex algebra. I’ll explain most of the concepts used but will expect that you know what groups, rings, fields, homomorphisms, and isomorphisms are in
an algebraic context. Feel free to look them up if not.
This post is meant to be an introduction to what a class group is and to summarize the most important results to consider when implementing them as a group of unknown order; detailed proofs can be
found in the list of further readings below.
What is a class group?
There are two equivalent ways to understand the class group which are isomorphic to one another. One coming from a subfield of mathematics known as algebraic number theory is known as the ideal class
group, which is the quotient of fractional ideals by fractional principal ideals \(J_K / P_K\) of a ring of integers \(O_K\) of a quadratic extension \(K = \mathbb{Q}(\sqrt{d})\) of the rational
numbers. Later we’ll walk through what this means step-by-step. The other way to look at the class group is known as the form class group and comes from the study of binary quadratic forms, or
equations of the form
\[ ax^2 + bxy + cy^2 = n \]
by working with an equivalence relation over forms which all have the same discriminant \(b^2 - 4ac\).
These two views are actually pretty different; it’s not at all obvious that they represent the same object! Either is paramaterized by its integer discriminant \(\Delta\), as there is a one-to-one
correspondence between class groups and valid discriminants. When using the group for cryptographic purposes as a group of unknown order there are three main choices to be made:
1. What discriminant should be chosen
2. How to represent the group in a numerical setting
3. What algorithms should be used when performing the group operation
The ideal class group makes it easier to understand how and why a particular form of discriminant should be chosen. As we will see, we want the discriminant to be the negation of a prime congruent to
3 mod 4, or \(-p\) where \(p \equiv 3\) (mod 4). On the other hand, the form class group is easier to represent and perform operations on numerically. With this in mind, we will start with the ideal
class group and move on to the form class group from there.
Ideal Class Group
In basic field theory there is the concept of a field extension, or a field containing another field as a subfield. The degree of an extension is the size of the larger field’s basis over the smaller
base field. An example of this is the complex numbers \(\mathbb{C}\), which extend the real numbers \(\mathbb{R}\) by adding in \(\sqrt{-1} = i\), so that \(\mathbb{C} = \mathbb{R}(i)\). This is
known as a degree 2 or quadratic extension, because the basis for \(\mathbb{C}\) as a vector space over \(\mathbb{R}\) is of size 2, and is given by \((1, i)\).
Similarly, we can construct generalizations of the rational numbers by adding in \(\sqrt{d}\) to \(\mathbb{Q}\), for some square-free number \(d\). In this case a basis for \(\mathbb{Q}(\sqrt{d})\)
over \(\mathbb{Q}\) is also size 2 and is given by \((1, \sqrt{d})\), making \(\mathbb{Q}(\sqrt{d})\) a quadratic extension of \(\mathbb{Q}\). More formally, \(\mathbb{Q}(\sqrt{d})\) is the smallest
field containing both \(\mathbb{Q}\) and \(\sqrt{d}\).
Once we have our quadratic extension \(K = \mathbb{Q}(\sqrt{d})\), we can also get a corresponding generalization of the integers known as the ring of integers of \(K\), denoted \(O_K\). To obtain \
(O_K\), we look at \(K\) and take all of its elements which are the roots of some monic polynomial with integer coefficients, also known as an integral polynomial, i.e. the set of all \(\alpha \in K
\) such that there exists some polynomial
\[ p(x) = x^n + b_{n-1}x^{n-1} + ... b_1x + b_0 \]
where \(b_{n-1}, ..., b_0 \in \mathbb{Z}\) and \(p(\alpha) = 0\). It turns out that \(O_\mathbb{Q} = \mathbb{Z}\), so that the ring of integers of the rational numbers is simply the integers. In
fact, for any finite-degree extension \(K\) of \(\mathbb{Q}\), \(O_K\) contains \(\mathbb{Z}\). As its name suggests, \(O_K\) forms a ring under addition and multiplication.
Something that would help us to understand \(O_K\) would be to find some analogue of a basis over a vector space for it. We are helped here by the concept of a module, which is a generalization of
vector spaces for rings. As an example, a \(\mathbb{Z}\) -module \(M\) consists of the integers \(\mathbb{Z}\), which form a ring, acting on some abelian group \((M,+)\). In other words, the ring of
integers acts like a set of scalars and the abelian group like a set of vectors. Modules do not have as many guarantees as vector spaces; for example, a module may not have a basis.
Because every ring is an abelian group under addition, every ring is a \(\mathbb{Z}\) -module. This implies we can act on any ring of integers \(O_K\) by the integers \(\mathbb{Z}\) in a way
analogous to acting on vectors by scalars. While not every module has a basis as a \(\mathbb{Z}\) -module (a “\(\mathbb{Z}\) -basis”), it is true that every ring of integers \(O_K\) has a \(\mathbb
{Z}\) -basis. More specifically, if \(K\) is a degree \(n\) extension of \(\mathbb{Q}\), then there is some \(b_1, ..., b_n \in O_K\) such that any \(x \in O_K\) can be uniquely written as
\[ x = \sum_{i=1}^n a_i b_i \]
where \(a_1, ..., a_n \in \mathbb{Z}\).
What are the possible \(\mathbb{Z}\) -bases for a ring of integers of a quadratic extension? This turns out to be pretty simple. We know \(\sqrt{d} \in O_K\) since \(\sqrt{d} \in K = \mathbb{Q}(\sqrt
{d})\) and it is the root of the integral polynomial \(x^2 - d\). It also happens to be the case that if \(d \equiv 1\) (mod 4) then for any \(x + y \sqrt{d} \in O_K\) where \(x,y \in \mathbb{Z}\) we
can write
\[ x + y \sqrt{d} = a + b \frac{1 + \sqrt{d}}{2} \]
for some \(a,b \in \mathbb{Z}\). It can be shown that no other elements are in \(O_K\), giving us two possible \(\mathbb{Z}\) -basis depending on \(d\). If \(d \equiv 2,3\) (mod 4) then we have a \(\
mathbb{Z}\) -basis of \((1, \sqrt{d})\), and if \(d \equiv 1\) (mod 4) we have a \(\mathbb{Z}\) -basis of \((1, \frac{1+\sqrt{d}}{2})\). Another way of phrasing this is to say that if \(d \equiv 1\)
(mod 4), then \(O_K = \mathbb{Z}[\frac{1+\sqrt{d}}{2}]\) and \(O_K = \mathbb{Z}[\sqrt{d}]\) otherwise.
To define the discriminant of a finite extension \(K\) of \(\mathbb{Q}\), we first need the concept of an embedding of \(K\) into some field \(\mathbb{F}\), which is simply an injective ring
homomorphism from \(K\) into \(\mathbb{F}\). If we have \(n\) embeddings \(\sigma_1, ..., \sigma_n\) from \(K\) into the complex numbers \(\mathbb{C}\) and a basis \(b_1, ..., b_n\) of \(O_K\) as a \
(\mathbb{Z}\) -module, then the discriminant of \(K\) is given by
\[ \Delta_K = \begin{vmatrix} \sigma_1(b_1) & \sigma_1(b_2) & \dots & \sigma_1(b_n) \\ \sigma_2(b_1) & \ddots & & \sigma_2(b_n) \\ \vdots & & \ddots & \vdots \\ \sigma_n(b_1) & \dots & \dots & \
sigma_n(b_n) \end{vmatrix}^2 \]
where the notation above means we take the square of the determinant of the given matrix.
For our case where \(K = \mathbb{Q}(\sqrt{d})\), we have two embeddings of \(K\) into \(\mathbb{C}\). Letting \(a + b\sqrt{d} \in \mathbb{Q}(\sqrt{d})\), one is given by \(\sigma_1(a + b\sqrt{d}) = a
+ b\sqrt{d}\) and the other by \(\sigma_2(a + b\sqrt{d}) = a - b\sqrt{d}\). If \(d > 0\) then we can consider these to be embeddings of \(K\) into \(\mathbb{R}\). In practice we want \(d < 0\). As
we’ll see later this gives us a nicer structure by letting us take a unique “reduced” form for any element when using the form class group as a numerical representation. It also makes it less likely
that the class group will be trivial and contain only one element.
Given these embeddings and the \(\mathbb{Z}\) -bases above it is easy to check that if \(d \equiv 1\) (mod 4) then \(\Delta_K = d\) and if \(d \equiv 2,3\) (mod 4) then \(\Delta_K = 4d\). When
choosing a discriminant, this makes it more convenient to pick some square-free value \(\Delta \equiv 1\) (mod 4), as otherwise we need to ensure that \(\Delta / 4\) is square-free.
We’re almost ready to construct the class group itself! To do so we’ll need the concept of an ideal, which is an additive subgroup of a ring with an additional multiplicative “absorption” property.
Specifically if we have an ideal \(I \subset O_K\) of a ring of integers then for any \(r \in R\) and \(x \in I\), \(rx \in I\).
For any \(n \in \mathbb{Z}\), \(n\mathbb{Z}\) is an ideal.
We can generalize the concept of an ideal of a ring to get the idea of a fractional ideal. Intuitively a fractional ideal \(J\) of a ring \(O_K\) is a subset of the ring’s enclosing field \(K\) such
that we can “clear the denominators” of \(J\). More formally \(J\) is a nonzero subset of \(K\) such that for some \(r \not= 0 \in O_K\), \(rJ \subset O_K\) is an ideal of \(O_K\).
As an example \(\frac{5}{4} \mathbb{Z}\) is a fractional ideal of \(\mathbb{Z}\), since it is a subset of the smallest field \(\mathbb{Q}\) containing \(\mathbb{Z}\) and has the property that \(4(\
frac{5}{4}\mathbb{Z}) = 5\mathbb{Z}\) is an ideal of \(\mathbb{Z}\).
For any two ideals or fractional ideals \(I,J \subset O_K\) we can define a form of multiplication on them:
\[ IJ = \{\sum a_i b_i | a_i \in I, b_i \in J\} \]
Since ideals are closed under addition, this gives the smallest ideal containing every element of both \(I\) and \(J\).
A fractional ideal \(I\) of \(O_K\) is called principal if there is some \(r \in O_K\) such that \(rO_K = I\). In other words, \(I\) is generated by a single element of \(O_K\).
Finally we need one more concept from group theory known as a quotient group. While I won’t define it formally here, a basic example is given by \(\mathbb{Z} / 3 \mathbb{Z}\) which has the structure
of arithmetic mod 3. It can be formed by taking every element \(n \in \mathbb{Z}\), computing \(n3\mathbb{Z}\), then forming a group operation having the same structure as \((\mathbb{Z},+)\) over the
3 resulting distinct sets of integers.
Let \(J_K\) denote the set of fractional ideals of \(O_K\) and \(P_K\) the set of principal fractional ideals of \(O_K\). Both form abelian groups under ideal multiplication defined above. This means
it makes sense to take the quotient group \(J_K / P_K\). This quotient group is the ideal class group.
There are two extreme cases we can consider here. If \(J_K = P_K\), then \(J_K / P_K\) is the trivial group with one element. On the other hand, if no fractional ideals are principal then \(J_K / P_K
= J_K\). The order of \(J_K / P_K\) can be interpreted as a measurement of the extent to which \(O_K\) fails to be a principal ideal domain, or to have all of its ideals be principal. We can even
take this a bit further, since for any ring of integers \(O_K\) being a principal ideal domain is equivalent to every element of \(O_K\) having a unique factorization. In other words, the order of a
class group of \(K\) is a measurement of how much its ring of integers \(O_K\) fails to give each element a unique factorization! Given that \(O_K\) is effectively a generalization of the integers,
it’s pretty neat that we can define this rigorously.
The order or number of elements of \(J_K / P_K\) is known as its class number, and is known to be hard to compute for large discriminants. As with all cryptographic assumptions, this has not been
proven but is assumed to be true because no one has broken it efficiently. Currently, a discriminant of at least 687 bits provides as much security as a 1024-bit RSA key, and a discriminant of at
least 1208 bits about as much security as a 2048-bit RSA key. These numbers may change with improved attacks on class group constructions, and should not be taken as advice.
Form Class Group
Next we will discuss the form class group, and see how it is related to the ideal class group. The form class group was originally discovered in the study of binary quadratic forms, or functions of
the form
\[ f(x,y) = ax^2 + bxy + cy^2 \]
where \(a,b,c \in \mathbb{R}\). For convenience we will write \(f = (a,b,c)\) and call \(f\) a form. In practice we want to restrict ourselves to the case where \(a,b,c \in \mathbb{Z}\), as our end
goal is to use forms to represent the class group in a computer and being able to store forms as integer triples rather than triples of floating point values simplifies things.
All binary quadratic forms in a given form class group have the same discriminant, given by \(\Delta_f = b^2 - 4ac\). This is identical to the discriminant of the corresponding ideal class group.
A form represents an integer \(n\) if \(f(x,y) = n\) for some \(x,y \in \mathbb{Z}\).
The form class group is made up of equivalence classes of forms, or sets of forms considered to be equivalent. This is similar to how, when doing arithmetic mod 3, the symbol 1 represents the set of
all integers congruent to 1 (mod 3). We say that two forms \(f_1, f_2\) are equivalent if they represent the same set of integers.
In order to be a valid group, we need a group operation \(*\) that will give us some representative \(f_3\) of an appropriate equivalence class given forms \(f_1, f_2\) so that \(f_1 * f_2 = f_3\).
We also need an identity form \(g\) so that \(f * g = f\) for all forms \(f\), and for any form \(f\) we need an inverse \(f^{-1}\) so that \(f * f^{-1} = g\).
We mentioned above the the ideal class group with discriminant \(\Delta \in \mathbb{Z}\) is isomorphic to the form class group with the same discriminant \(\Delta\). In fact, this is only true when \
(\Delta < 0\), and all forms \(f = (a,b,c)\) in the group are positive definite, meaning \(f(x,y) > 0\) for all possible \(x,y \in \mathbb{Z}\). This is equivalent to having both \(\Delta_f < 0\) and
\(a > 0\). From now on, we will assume that any form is positive definite.
As the form class group is composed of equivalence classes of forms, it would also be good if we could reduce any form of an equivalence class down to one unique element. Since we also want to
represent elements \(f = (a,b,c)\) in terms of bits, it would also be good if this reduced element was reasonably small. It turns out that we can define a reduction operation that gives us both of
these desirable properties.
We can break down the reduction operation into two pieces; a “normalization” operation and a “reduction” operation, with the requirement that a form must be normalized before it is reduced.
A form \(f = (a,b,c)\) is normal if \(-a < b \leq a\). We can define a normalization operation by
\[ \eta(a,b,c) := (a, b+2ra, ar^2 + br + c) \]
where \(r = {\lfloor \frac{a-b}{2a} \rfloor}\).
When we normalize a form \(f\), we want the resulting normalized form \(f'\) to be equivalent to \(f\). How do we know this is actually the case? It turns out we can understand equivalence of forms
using actions by a certain class of matrices known as \(SL_2\), or the “special linear group” of invertible matrices with determinant equal to 1.
Two forms \(f_1, f_2\) are in fact equivalent if there exists \(\alpha, \beta, \gamma, \delta \in \mathbb{Z}\) such that
\[ f_1(\alpha x + \beta y, \gamma x + \delta y) = f_2(x,y) \] \[ \alpha \delta - \beta \gamma = 1 \]
This is actually only partially true, as we can relax the second requirement to be \(\alpha \delta - \beta \gamma = \pm 1\). However, this won’t actually give us a valid equivalence relation, i.e. if
we let \(\equiv\) denote equivalence then \(f_1 \equiv f_2\) and \(f_2 \equiv f_3\) wouldn’t necessarily imply \(f_1 \equiv f_2 \equiv f_3\).
The “correct” form of equivalence mentioned above gives us an action of \(SL_2\) on forms, so that if \(f_1,f_2\) are equivalent then there is some invertible
\[ A= \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix} \]
with det\((A) = 1\) such that \(f_1 = Af_2\).
With this is mind, we can show that \(f\) and its normalized form \(f' = \eta(f)\) are equivalent by the matrix
\[ A= \begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix} \]
Once we’ve normalized a form, we can then reduce it to obtain its unique reduced equivalent form. A form \(f = (a,b,c)\) is reduced if it is normal and \(a < c\) or \(a = c\) and \(b \geq 0\).
We can define a reduction operation as
\[ \rho(a,b,c) := (c, -b + 2sc, cs^2-bs + a) \]
where \(s = {\lfloor \frac{c+b}{2c} \rfloor}\). Unlike the normalization operation \(\eta\), it may need multiple iterations before our form is reduced. To reduce a form \(f\), we can compute \(f \
leftarrow \eta(f)\) then repeatedly compute \(f \leftarrow \rho(f)\) until \(f\) is reduced.
Similar to normalized forms, we can see that a reduced form \(f'\) of \(f\) is equivalent to \(f\) using the matrix
\[ A= \begin{bmatrix} 0 & -1 \\ 1 & r \end{bmatrix} \]
How do we know reduced elements are relatively small? If \(\Delta_f < 0\) then for a reduced form \(f = (a,b,c)\) we have that
\[ a \leq \sqrt{\frac{|\Delta_f|}{3}} \]
Because in a reduced form \(|b| \leq a\) and \(c = \frac{b^2 - \Delta_f}{4a}\), the above upper bound on the size of \(a\) implies that reduce elements as a whole tend to be small, having a bit
representation at least as small as that of \(\Delta_f\). This makes the group operation on reduced elements relatively efficient.
There is also a reasonable upper bound on the number of reduction steps required, given by \(\log_2 (\frac{a}{\sqrt{|\Delta|}}) + 2\).
The group operation itself, known as “form composition”, is a bit complicated. The basic idea is that, given two forms \(f_1, f_2\) with the same discirminant \(\Delta\), we can multiply them
together and use a change of variables to obtain a third form \(f_3\) such that \(f_1f_2 = f_3\). More exactly, the game is to find integers \(p,q,r,s,p',q',r',s',\alpha,\beta,\gamma\) so that
\begin{eqnarray*} X &=& px_1x_2 + qx_1y_2 + ry_1x_2 + sy_1y_2\\ Y &=& p'x_1x_2 + q'x_1y_2 + r'y_1x_2 + s'y_1y_2\\ f_3(x,y) &=& \alpha x^2 + \beta xy + \gamma y^2\\ f_3(X,Y) &=& f_1(x,y)f_2(x,y)\\ \
Let LinCong\((a,b,m)\) be an algorithm which solves a linear congruence of the form \(ax \equiv b\) (mod \(m\)) by finding some \(x = \mu + \nu n\) where \(n \in \mathbb{Z}\) and outputs \((\mu, \nu)
\). Given two forms \(f_1 = (a,b,c)\) and \(f_2 = (\alpha, \beta, \gamma)\), we can define a group operation on them as follows:
1. Set \(g \leftarrow \frac{1}{2}(b + \beta)\), \(h \leftarrow -\frac{1}{2}(b - \beta)\), $w ← \(gcd\)(a, α, g)$
2. Set \(j \leftarrow w\), \(s \leftarrow \frac{a}{w}\), \(t \leftarrow \frac{\alpha}{w}\), \(u \leftarrow \frac{g}{w}\)
3. Set $(μ, ν) ← \(LinCong\)(tu, hu + sc, st)$
4. Set $(λ, ρ) ← \(LinCong\)(tv, h - tμ, s)$
5. Set \(k \leftarrow \mu + \nu \lambda\), \(l \leftarrow \frac{kt - h}{s}\), \(m \leftarrow \frac{tuk - hu - cs}{st}\).
6. Return \(f_3 = (st, ju - (kt + ls), kl - jm)\).
In practice it’s best to always reduce the result \(f_3\) after performing composition. This way we are guaranteed that the multiplication of two forms takes \(O(\log^2 |\Delta|)\) bit operations
where \(\Delta\) is the discriminant of the group being used. This is not guaranteed if the two inputs forms are not reduced.
In order to be a group operation forms under composition must have an identity element. If \(\Delta < 0\) this turns out to be \(f = (1, k, \frac{k^2-\Delta}{4})\) where \(k = \Delta\) (mod 2).
For any form \(f = (a,b,c)\), its inverse under form composition is given by \(f^{-1} = (a, -b, c)\).
We’re now done constructing the form class group! We have a group operation, a way to get a unique representative element from each equivalence class of forms, an identity, and inverses.
There is one very important optimization we can do. Above we mentioned that we want to choose the negation of a prime \(p\) as our discriminant. It turns out that this lets us simplify our
composition algorithm when \(f_1 = f_2\), since using \(\Delta = -p\) implies that gcd\((a,b) = 1\) for any form \(f = (a,b,c)\). Unlike much of the math discussed in this post, this is pretty easy
to see. If gcd\((a,b) = n \not= 1\) then for some \(a', b' \in \mathbb{Z}\) we have that \(a = na'\) and \(b = nb'\), implying \(\Delta = b^2 - 4ac = n(b'b - 4a'c)\) which is impossible because \(-p
\) can’t be divisible by \(n\).
This, in addition to the fact that \(a = \alpha\), \(b = \beta\), \(c = \gamma\) gives us the simplified squaring algorithm below:
1. Set \((\mu, \nu) \leftarrow\) LinCong\((b,c,a)\)
2. Return \(f^2 = (a^2, b-2\alpha \mu, \mu^2 - \frac{b \mu - c}{a})\).
This is a big deal for efficiency, since we can compute exponentiation using repeated squaring.
Going back to the ideal class group, there is another construction equivalent to the quotient group \(J_K / P_K\). Similar to equivalence of forms, we can define an equivalence of ideals in \(J_K\)
by saying that two fractional ideals \(I,J\) of \(J_K\) are equivalent if there is some \(\alpha \not= 0 \in K\) such that \(\alpha I = J\). The equivalence classes formed by this relation are
exactly the elements of \(J_K / P_K\). We can similarly represent fractional ideals by their at most 2 generating elements, so that if \(I\) is generated by \(\alpha, \beta\) we can represent it by \
((\alpha, \beta)\). We can also get a unique “reduced” ideal from each equivalence class.
Ideal and form class groups are isomorphic when the discriminant \(\Delta \in \mathbb{Z}\) being used is negative. In some sense this means multiplication of fractional ideals in the ideal setting
and form composition in the form class group are really the same operation, since we can move back and forth between corresponding equivalence classes of fractional ideals and of forms.
We need just a few more tools before defining the isomorphism itself. If \(K\) is a finite field extension of \(\mathbb{F}\), so that \(K\) is a finite-dimensional vector space over \(\mathbb{F}\),
then for any \(\alpha \in K\) the map \(m_\alpha (x) = \alpha x\) is an \(\mathbb{F}\) -linear transformation from \(K\) into itself. The field norm \(N(\alpha)\) is the determinant of the matrix of
this linear transformation. The trace \(Tr\) of \(\alpha\) is the trace of this matrix.
In our case where \(K = \mathbb{Q}(\sqrt{d})\), for any \(x = a + b\sqrt{d} \in K\) we have \(N(x) = a^2 - db^2\), implying \(N(x)\) is positive when \(d < 0\).
Next, if \(I\) is a non-zero fractional ideal of \(O_K\), the absolute norm of \(I\) is given by the mapping \(N(I) = |O_K / I|\), or the order of the quotient of \(O_K\) by its ideal \(I\).
As their names and notation suggests, the field and absolute norms are related. If an ideal \(I\) of ring of integers \(O_K\) is principal so that there is some \(\alpha \in O_K\) such that \(\alpha
O_K = I\), then \(N(I) = |N(\alpha)|\).
The isomorphism between ideal and form class groups is as follows. If \(f = (a,b,c)\) where \(a,b,c \in \mathbb{Z}\) and \(\Delta_f < 0\), then we can map \(f\) to a fractional ideal of the ideal
class group with same discriminant by
\[ \Phi(a,b,c) := (a \mathbb{Z} + \frac{-b + \sqrt{\Delta_f}}{2}\mathbb{Z}) \]
with inverse
\[ \Phi^{-1}(I) := \frac{N(\alpha x - \beta y)}{N(I)} \]
where \((\alpha, \beta)\) is some \(\mathbb{Z}\) -basis of the fractional ideal \(I\). We can see that the inverse maps a fractional ideal to a binary quadratic form using the following identity:
\[ N(\alpha x + \beta y) = N(\alpha)x^2 + Tr(\alpha \beta')xy + N(\beta)y^2 \]
where for some element of a ring of integers \(x = a + b\sqrt{d}\) we denote its conjugate by \(x' = a - b\sqrt{d}\).
If \(\Delta < 0\) then the coefficient of \(x^2\) given by \(N(\alpha)/N(I)\) will be positive, meaning the resulting form \(f = (a,b,c)\) will be positive definite as \(\Delta_f < 0\) and \(a > 0\).
It can be shown that when \(\Delta < 0\), \(\Phi\) is a bijection which preserves the group structure of binary quadratic forms under form composition in mapping to fractional ideals of a ring of
integers under ideal multiplication. In other words, \(\Phi\) is an isomorphism.
Solved and Open Conjectures
We’ll conclude with a handful of conjectures made in some form by Gauss in 1801 which show further why we want to use a negative discriminant:
1. The class number of the ideal class group of \(\mathbb{Q}(\sqrt{d})\) converges to infinity as \(d \rightarrow -\infty\).
2. There are exactly 13 negative discriminants having a class number of 1, in particular -3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163.
3. There are infinitely many positive discriminants associated with class groups having a class number of 1.
The first was proven in 1934 by Hans Heilbronn, the second in 1967 by Heegner, Stark and Baker. The third remains open. | {"url":"https://www.michaelstraka.com/classgroups","timestamp":"2024-11-02T06:07:23Z","content_type":"application/xhtml+xml","content_length":"34246","record_id":"<urn:uuid:5db1512e-23d1-47f8-b32a-1002d22b7360>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00822.warc.gz"} |
Empirical Bounds on Linear Regions of Deep Rectifier Networks
One form of characterizing the expressiveness of a piecewise linear neural network is by the number of linear regions, or pieces, of the function modeled. We have observed substantial progress in
this topic through lower and upper bounds on the maximum number of linear regions and a counting procedure. However, these bounds only account for the dimensions of the network and the exact counting
may take a prohibitive amount of time, therefore making it infeasible to benchmark the expressiveness of networks. In this work, we approximate the number of linear regions of specific rectifier
networks with an algorithm for probabilistic lower bounds of mixed-integer linear sets. In addition, we present a tighter upper bound that leverages network coefficients. We test both on trained
networks. The algorithm for probabilistic lower bounds is several orders of magnitude faster than exact counting and the values reach similar orders of magnitude, hence making our approach a viable
method to compare the expressiveness of such networks. The refined upper bound is particularly stronger on networks with narrow layers.
AAAI 2020
View Empirical Bounds on Linear Regions of Deep Rectifier Networks | {"url":"https://optimization-online.org/2018/10/6852/","timestamp":"2024-11-03T15:31:06Z","content_type":"text/html","content_length":"84226","record_id":"<urn:uuid:abab7bcf-866a-460b-b6cd-64b49bb7959b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00868.warc.gz"} |
Using SSI Work Incentives
Nuts & Bolts
This tool describes work incentives used by the Supplemental Security Income (SSI). These incentives create an economic safety net for SSI beneficiaries on a path to work and financial independence.
Reviewing How Income Affects SSI Payments
Tool 3 describes how SSI payment rates are calculated by subtracting a person’s Countable Income from their state’s SSI Base Rate. As a review, consider the example of Carolina. Carolina is 23 years
old and receives $440 in Social Security disability benefits. She has no other income. Here is how Carolina’s monthly SSI payment is calculated:
Explanation Calculation
Unearned Income (Social Security) $440
Subtract the $20 General Income Exclusion $440 – $20 =$420
Carolina’s Countable Unearned Income $420
Subtract Carolina’s Countable Unearned Income from the SSI Federal Benefit Rate (2024) (no State Supplement) $943 – $420 = $523
Carolina’s SSI payment $523
Generally, the income received in a month is used to determine the SSI payment 2 months later (i.e., if Carolina received this income in June, it would affect her SSI payment in August). Since
Carolina receives at least $1 of SSI, she will qualify for automatic Medicaid in most states.
Note: For information about how earned income can affect Medicaid, see Tool 6.
Earned Income Disregards: $20, $65, and 50 Percent
disregards (also called exclusions). These disregards are used to reduce the amount of Countable Earned Income:
1. If the recipient has no unearned income, the $20 General Income Exclusion is applied to earned income.
2. The Earned Income Exclusion calls for $65 of earned income to be subtracted.
3. And, 50 percent of the remaining earned income is excluded.
The result is Countable Earned Income. Countable Earned Income, along with any Countable Unearned Income, is subtracted from the SSI Base Rate to determine the SSI payment.
Example of Earned Income Disregards: Lisa
Consider Lisa, who was previously unemployed and receiving monthly SSI payments. Lisa begins a part-time job earning $885 gross per month. This is how Lisa's new SSI payment will be calculated:
Explanation Calculation
Lisa’s Gross Earned Income $885
Subtract the $20 General Income Exclusion $885 – $20 = $865
Subtract the $65 Earned Income Exclusion $865 – $65 =$800
Divide by 2 to find the Countable Earned Income $800/2 = $400
Subtract Lisa’s Countable Earned Income from the SSI Federal Benefit Rate (2024) (no State Supplement) $943 – $400 = $543
Lisa’s SSI payment $543
Lisa’s net monthly gain from working is $485 (the amount of the earned income exclusions) less any tax withholding. In the 41 states where Medicaid is automatic for SSI beneficiaries, Lisa’s Medicaid
will continue. In the remaining nine states, her Medicaid eligibility will be determined separately.
Note: When a person who gets SSI begins earning wages, their net gain will be more than half their gross wages.
Calculating SSI Payments on the Web
SSI Calculation Worksheet can help you calculate Countable Unearned Income and Countable Earned Income.
It can also calculate an expected monthly SSI payment.
About the Student Earned Income Exclusion (SEIE)
The powerful Student Earned Income Exclusion (SEIE) is for students who are under age 22 and “regularly attending school”—see Tool 5 for a definition of regularly attending school.
The SEIE rewards a student who stays in school and starts working. In most cases, they keep the full SSI payment and full paycheck. It applies to the first $2,290 of gross earnings per month, up to
$9,230 per year in 2024.
Let’s go back to Lisa, from the previous example. Assume she is age 17 and attending high school (or college). She makes $585 gross monthly by working after school and on weekends. The SSI program
will exclude the entire $585 each month for a full 12 months from her Countable Income, as her total annual earnings of $7,020 remain below the annual $9,230 limit. In this example, Lisa’s SSI
payment will remain at $943 each month. Even though Lisa’s paycheck is $300 less in this example, the SEIE causes her net gain to be more.
Note: For details about the SEIE, see Tool 5.
About Impairment-Related Work Expenses (IRWEs)
To be deductible as an IRWE, the expense must meet a three-part test:
• The expense must be paid by the worker and not paid or reimbursed by another source.
• The expense must relate to the individual’s disability or a condition for which they are receiving treatment.
• It must be true that without the item or service, the person would be unable to work.
Example of IRWEs: Martin
Martin earns $885 gross per month at a part-time job. He has cerebral palsy, uses a cane for mobility, and cannot walk well enough to reach a nearby bus stop to take public transportation. He pays a
local agency $135 per month to take him to work and back several times per month. Without this transportation he could not work.
This will meet SSI’s criteria as an IRWE, and his SSI payment is calculated as follows:
Explanation Calculation
Martin’s Gross Earned Income $885
Subtract the $20 General Income Exclusion $885 – $20 = $865
Subtract the $65 Earned Income Exclusion $865 – $65 =$800
Subtract the IRWE deduction from the $800 running total $800 – $135 = $665
Divide by 2 to compute the Countable Earned Income $665/2 = $332.50
Subtract Martin’s Countable Earned Income from the SSI Federal Benefit Rate (2024) (no State Supplement) $943 – $332.50 = $610.50
Martin’s SSI payment $610.50
Note that Martin is earning the same amount per month as Lisa in the first example above. With the $135 IRWE deduction, Martin’s SSI payment is $67.50 higher than Lisa’s—half the value of the IRWE.
About Blind Work Expenses (BWEs)
not have to be related to blindness.
Common BWEs include:
• Federal, state, and local income taxes
• Social Security and Medicare taxes (FICA)
• Meals consumed during work hours
• Guide dog expenses (including food, licenses, and veterinary services)
• Medical devices, medical supplies, and therapy
• Training to use a disability-related item (e.g., cane travel) or an item attributable to work (e.g., computer training)
Example of Blind Work Expenses: Veda
Veda, who is statutorily blind, works and earns $18,420 per year ($1,535 per month). She has the following expenses that meet SSI criteria as BWEs:
Item or Service Cost
Income taxes (federal, state, local) $57
Social Security and Medicare taxes (FICA) $117
Transportation $95
Guide dog $45
Lunches $132
Readers $105*
Total of Veda’s BWEs $551
*If Veda’s employer pays for readers as a reasonable accommodation under the Americans with Disabilities Act, this cost will not qualify as a BWE.
Here is how Veda’s monthly SSI payment is calculated:
Explanation Calculation
Veda’s Gross Earned Income $1,535
Subtract the $20 General Income Exclusion $1,535 – $20 = $1,515
Subtract the $65 Earned Income Exclusion $1,515 – $65 =$1,450
Divide by 2 to compute the additional 50% exclusion $1,450/2 = $725
Subtract the BWEs from the running total. The Countable Earned Income is $174. $725 – $551 = $174
Subtract the Countable Earned Income from the SSI Federal Benefit Rate (2024) (no State Supplement) $943 – $174 = $769
Veda’s SSI payment $769
By using BWEs, Veda’s SSI payment increases by $551 (i.e., by the full amount of the BWEs). | {"url":"https://ssi.disabilitybenefitsatwork.org/tool/4/nuts-and-bolts","timestamp":"2024-11-11T23:46:57Z","content_type":"text/html","content_length":"28155","record_id":"<urn:uuid:268095ee-f1fb-4073-bf62-11efb223b909>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00875.warc.gz"} |
Why Base Current is Weak Then Collector Current? [Answered] - electronicstalk.org
Why Base Current is Weak Then Collector Current? [Answered]
Transistors, the backbone of electronic circuits, operate on principles governed by intricate equations. If you’ve ever wondered why the base current is weaker than the collector current, you’re
about to embark on a journey into the world of transistor equations and dynamics.
Concept of Base Current in Electronics
Base current (IB) is the current flowing into the base terminal of a transistor, initiating the flow of collector current (IC). Mathematically, this relationship is expressed by the equation:
β represents the transistor’s current gain.
Figure: Base current in a transistor.
Relationship with Ic and IB
The equation I[E]=I[C]+I[B] illustrates the interdependence of base, collector, and emitter currents. It emphasizes the weaker nature of I[B] compared to the more substantial I[c].
Collector Current: The Powerhouse
Collector current (I[c]) is the driving force behind the amplification capabilities of a transistor. Its influence on the output voltage is determined by the equation:
This equation highlights the role of I[c] in signal amplification.
Expressed by the equation
where V[CC] is the collector supply voltage, V[CE] is the collector-emitter voltage, and R[C] is the collector resistor, I[c] holds significant sway in transistor operation.
Weakness of Base Current
The equation I[c]=β*I[B] inherently signifies the limitations of I[B]. Due to its dependence on β, the current gain of the transistor, I[B] is weaker and imposes constraints on the overall efficiency
of the transistor.
Impact on Efficiency:
The efficiency (η) of a transistor is defined by the equation:
This equation underscores the critical role of balancing I[B] and I[C] for optimal transistor efficiency.
Amplification Process of Base Current and Collector Current
Equations governing the amplification process shed light on the relationship between I[B] and I[C]. The gain (β) of a transistor, represented by the equation A[V]=V[out]/V[in], is directly
influenced by I[B].
Factors Influencing Current Strength
The equation I[B]=I[CBO]/β encapsulates the influence of semiconductor characteristics on I[B]. I[CBO] represents the reverse collector current, and β is the transistor gain.
Balance Between I[B] and I[C]:
Achieving balance is crucial, as highlighted by the equation I[C]=β*I[B]. Deviations from this balance can lead to inefficiencies and signal distortion.
Analogies and Metaphors
Analogies and metaphors enriched with equations provide a deeper understanding. Imagine I[B] as the conductor orchestrating a symphony of electrons represented by I[C]. This analogy aligns with the
equation I[C]=β*I[B], showcasing the conductor’s influence.
Comparing Base and Collector Currents
Quantitative analysis using equations reinforces the disparity between I[B] and I[C]. The graphical representation of I[B] and I[C] magnitudes visually demonstrates their relationship.
The intricate relationship between I[B] and I[C] in transistors unravels when viewed through equations. The mathematical underpinnings provide a nuanced understanding, emphasizing the importance of
balancing these currents for efficient and reliable transistor operation.
Additional Questions May Ask
How is base current defined in a transistor?
Base current (I[B]) is the current flowing into the base terminal of a transistor, initiating the flow of collector current (I[C]). This relationship is expressed by the equation I[C]=β*I[B].
What is the role of collector current in transistor amplification?
The collector current is the driving force behind the amplification capabilities of a transistor. Its influence on the output voltage is determined by the equation V[out]=-β*V[in].
How does the efficiency of a transistor relate to base and collector currents?
The efficiency of a transistor is defined by the equation η=P[out]/P[in]. This equation underscores the critical role of balancing I[B] and I[C] for optimal transistor efficiency.
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What are Lines and Planes? (Video & Practice Questions)
Lines and Planes
Hi, and welcome to this video on Lines and Planes! The study of geometry is very much language-based, meaning that there are countless terms, relationships, and figures with meanings that are
dependent on an understanding of other concepts. It can get pretty confusing if the foundational terms are not understood. In this video, we’re going to start with the most basic figures: a point, a
line, and a plane. These “undefined” terms are described, rather than being defined, and they support the definitions of all other geometric terms.
To start off, what is a point?
A point is described as a very specific location, or position, in a plane. The notation for a point is a dot, but that dot does not have any dimension (length, width, circumference). A point is named
with a capital letter, as in “point A”
A line is described as a “path,” as if a point was dragged or is moving. A straight line extends infinitely in opposite directions. A line is typically named with a lowercase letter, or by
referencing two points on the line, with a line symbol above. The line notation has arrows on either end to indicate that they extend forever. Points that lie on a line are referred to as collinear.
A plane surface, has length and width, and extends infinitely in all directions. A flat surface, like a wall, floor, or ceiling, can be imagined as finite planes where geometric figures, like points
and lines, can be drawn. A plane is typically named with a letter in script or italics (plane m) or by naming three points that lie on the plane, (plane ABC). Using three points in the naming of a
plane lends to the perception of a two-dimensional surface. Points that lie in the same plane are said to be coplanar.
Planes that intersect do so at a line, and it is possible for three planes to intersect at exactly one point.
Now that we know these basic components, we can build our knowledge with terms that incorporate them in their definitions. For example:
A line segment is the portion of a line that lies between two points on the line. The two points are called endpoints, and are included in the line segment, as are all the points that are between
them. A line segment with endpoints A and B would be referenced as \(\overline{AB}\).
A ray starts at one point and extends infinitely in one direction on a plane. The ray symbol has one arrow indicating the starting point and the direction of the ray.
When two lines on a plane cross each other, they are referred to as intersecting lines. Intersecting lines on a plane cross at exactly one point.
Because a line segment has length that can be measured between the endpoints, the exact midpoint of the segment can be determined. A point, line, or ray, or plane that crosses a line segment at the
midpoint is called a bisector.
Intersecting lines on a plane that cross at 90° angles, or “right angles,” are perpendicular to each other. Examples of perpendicular lines can be found on window panes, or on door frames.
Lines on a plane that never cross are called parallel. These lines are exactly the same distance apart at all points, like the double yellow lines on a road, or tire tracks of a car.
A line that crosses two lines in a plane at two distinct points is called a transversal line. Transversal lines in combination with special angle relationships are used to determine whether lines in
a plane are parallel.
As you can see, it is essential to understand the relationships between the “undefined” terms of a point, a line and a plane in order to strengthen and expand your understanding of other geometry
concepts. It’s important to review these frequently from the ground up to keep pace and to retain your knowledge.
Thanks for watching, and happy studying!
Frequently Asked Questions
What are lines?
In the context of mathematics, a line is an infinitely long collection of points. A line has no width or depth*, and it will continue to run in opposite directions forever. We designate that
something is a line by marking arrows on both (visible) ends of a line segment.
*Because a line only has length as a dimension, it is a 1-dimensional object.
would not necessarily be considered a line in a math course because we don’t know if this object has specific endpoints or if it runs on forever.
However, we can turn this into a line by strategically placing 2 arrows:
As you might guess, a line never has a visible ‘ending.’ In fact, a line doesn’t have any ending because it is infinitely long!
How do you define a point?
A point is a set position (or “coordinate”) within a space. Traditionally, a point is designated by a simple ‘dot’ on some surface; and it’s either named by a single letter and/or described in \
((x,y)\)– or \((x,y,z)\)-coordinate form.
Remember that a point is a dimensionless object because it doesn’t have any width, length, or depth.
What is a collinear point?
Collinear points are points that sit on the same line, line segment, or ray.
The points A, B, and C are collinear.
The points X and Y are not collinear because Y isn’t on the same line as X.
What is a plane in math?
Think of a plane as the surface of an ever-lasting piece of paper: a flat surface that you can only move up and down or right and left on. You couldn’t move “in” or “out.”
A plane is the collection of an infinite amount of points and lines, and it has both length and width (but no depth). Hence, a plane has only 2 dimensions.
We usually call it ‘the \(xy\)-plane,’ because we describe points and lines in relation to the horizontal axis, \(x\), and the vertical axis, \(y\).
How do you identify a plane?
There are a few different ways to identify/construct a plane:
• Using 3 non-collinear points: When looking at a collection of 3 points that don’t sit on a line together, one might notice that the only way to “connect” the points to one another is to have them
sitting on the same plane together.
• Using a line and a separate point: This method is very similar to the one outlined above. Imagine that we drew a line connecting 2 out of the 3 points from above; the only way to connect the two
new objects would, again, be to draw a plane.
• Using a pair of intersecting lines: Imagine that you have two lines that cross over one another at some shared point. There is only one way to sit these lines on the same flat surface, and so we
construct a plane that gives the two lines ‘common ground.’
• Using a pair of parallel lines: Once again, this is similar to the intersecting lines method we just discussed. There is only one way to set up a plane for these parallel lines to sit on
Does a plane always have 3 points?
Technically, yes- a plane always has at least 3 points; because a plane is a collection of infinitely-many points. However, we can’t identify or construct a plane given less than 3 points.
Let’s say that we’ve been given the point A,
Now let’s say that we’ve been given the points E and F
and told to find the plane (like above). While there’s only 1 unique line that connects this pair, again: we run into the problem that there are infinitely-many possible planes that the two points
could be sitting on together.
When you are given 3 points, you have made certain that the space you are looking at is a plane because there is only one unique plane that all 3 points can lie on
Does one line define a plane?
No, a single line cannot be used to define a unique plane. As mentioned above, 1 line can sit on a countless amount of possible planes.
What are line segments?
It might seem silly at first, but a line segment is actually quite different from a line in math. This distinction is important: while a line continues infinitely in both directions, a line segment
has a finite length. More specifically, line segments run from one “endpoint” to another, and these endpoints are the points that sit on both ends of the line segment.
What does a ray look like in math?
A ray is kind of like the combination of a line and a line segment; it has 1 endpoint, but on the opposite end it continues on forever.
What is the difference between intersecting lines and perpendicular lines?
While 2 lines are considered intersecting lines if they cross over one another at a particular point, they are only considered perpendicular to one another if all 4 angles formed at the intersection
point are right angles (each measures 90°).
Keep in mind: all perpendicular lines are intersecting lines, but not all intersecting lines are perpendicular.
The lines A and B are simply intersecting. However, the lines C and D are perpendicular.
What are parallel and intersecting lines?
There are only 2 possible relationships that a pair of lines can have between themselves: they are either parallel or they are intersecting (or will eventually intersect) one another.
If it’s impossible for 2 lines to ever cross over one another, they are considered parallel. If the lines cross over one another at some point (we call this point the “intersection point”), we call
them “intersecting lines.”
The lines K and L are parallel to one another; and while K’ and L’ are not yet intersecting, they will eventually meet at the intersection point to the right.
How do you identify a transversal line?
A transversal line is one which intersects at least 2 other lines. In the following figures, the dashed line is the transversal:
Lines and Planes Practice Questions
Question #1:
A ________ is a part of a line that has one fixed starting point, and extends infinitely in one direction.
A ray is a part of a line that has one fixed starting point, and extends infinitely in one direction.
Question #2:
Name the plane in the image below.
A plane can be named by an italicized letter such as T, or by three non-collinear points that lie on the plane, such as EFG.
Question #3:
Which capital letters in the alphabet have parallel lines.
E, F, H, M, N, X, Y, and Z
Parallel lines will never cross each other even when extended infinitely. The letters E, F, H, M, N, W, and Z consist of parallel lines.
Question #4:
Which geometric term best describes how the city of Austin, Texas would be represented on a globe?
A point describes a location, such as Austin, Texas on a globe. In geometry, a point does not take up space, but in pictures or diagrams they are drawn as dots.
Question #5:
Andrew wants to build a model of a skyscraper using paper. He decides to design the building as a triangular prism. How many planes will be used to create this model?
The model will be built from five planes: top, bottom, and three sides. | {"url":"https://www.mometrix.com/academy/lines-and-planes/","timestamp":"2024-11-12T05:54:04Z","content_type":"text/html","content_length":"104089","record_id":"<urn:uuid:1f351e91-4856-4756-a477-8023c0a9e9bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00807.warc.gz"} |
Overview of Binomial Distribution in Excel 2010 and Excel 2013
This is one of the following four articles on the Binomial Distribution in Excel
Overview of the Binomial Distribution in Excel 2010 and Excel 2013
Solving Problems With the Binomial Distribution in Excel 2010 and Excel 2013
Normal Approximation of the Binomial Distribution in Excel 2010 and Excel 2013
Distributions Related to the Binomial Distribution
Overview of the Binomial
The binomial distribution is the discrete probability distribution of the number of successes in n successive independent binary trials. Each trial has the same probability p of a successful outcome
of the binary trial’s two possible outcomes.
p = probability of success in each and every binary trial
n = the total number of binary trials
k = a specific number of successful outcomes of n binary trials
X = the actual number of successful outcomes of n binary trials
The probability that X (the actual number of successful outcomes of n successive, independent binary trials each having probability p of a successful outcome) equals a specific k is given as follows:
(Click On Image To See a Larger Version)
for k = 0, 1, 2, …, n,
(Click On Image To See a Larger Version)
is sometimes referred to as “n choose k.” This is called the binomial coefficient and the source of the name of the binomial distribution. The binomial coefficient is equal to the number ways that k
items can be combined to create n elements. This is the number combinations that k items can be arranged to create n total elements.
A different ordering of the same k items does not create a new combination. For example, the three elements A, B, and C can only be arranged into one combination of three total elements. ABC is not a
different combination than CAB is. A different ordering of the same three elements does, however, create a new permutation.
The Excel formula for the total possible number of combinations of k elements into n total elements is given by the following:
(Click On Image To See a Larger Version)
COMBIN(n,k) is the Excel combination formula.
A combination should be differentiated from a permutation. The number of possible combinations is the total number of ways that k elements can be arranged into n total elements when order does not
matter. The number of possible permutations is the total number of ways that k elements can be arranged into n total elements when order does matter.
There are always more permutations of k objects than combinations because, for example, ABC and CAB are two different permutations of the letters A, B, and C but not two different combinations of
those three letters. Re-arrangement of the same elements within a set creates a new permutation but does not create a new combination.
The Excel formula for the total possible number of permutations of k elements into n total elements is given by the following:
(Click On Image To See a Larger Version)
PERMUT(n,k) is the Excel permutation formula.
Note that the following is true:
(Click On Image To See a Larger Version)
Each Trial is a Bernoulli Trial
Each individual trial is called a Bernoulli trial. The outcome of a Bernoulli trial can be described by the binomial distribution with n = 1. A Bernoulli trial follows the Bernoulli distribution,
which is a special case of the binomial distribution where n = 1.
Maximum Sample Size Rule
The binomial distribution should not be applied to a single random sample taken from a population unless the population is at least 10 times larger than the sample size.
Binomial Distribution Has Two Parameters n and p
A binomial distribution is fully described by two parameters n (the total number of trials) and p (the constant and unchanging probability of success on each trial). The binomial distribution is
denoted as follows:
The binomial distribution is a discrete distribution because its probability equation Pr(X = k) is calculated only for values of k, which can only assume integer values.
The binomial distribution describes the distribution of the number of successes, X, if the following four conditions exists:
1) Each trial is a single Bernoulli trial, which is a binary event having only one of two outcomes: success or failure.
2) The total number of trials = n.
3) Each trial is independent of all other trials.
4) The probability of success in each trial is p, which is constant in all trials. The probability of failure = q = 1 – p.
The binomial distribution requires that each sample taken is returned to the population before the next sample is taken. Samples are always taken from the same population. This is called sampling
with replacement.
If samples taken are not returned to the population, the hypergeometric distribution is used in place of the binomial distribution. This is called sampling without replacement. If sample size is much
smaller than the population from the sample was drawn, the binomial distribution provides a good approximation of the hypergeometric distribution when sampling without replacement is performed. The
population size should be at least ten times as large of the sample size for this substitution to be valid.
Population Parameters of the Binomial Distribution
n = number of trial
p = probability of success on each trial
q = 1 – p = probability of failure on each trial
If X ~ B(n,p) (X is a binomially-distributed variable having n trials and the probability p of success on each trial. X is a variable representing the number of successes given n and p), then the
following is true:
Expected Value(X) = Mean(X) = μ[X] = np
Variance(X) = σ^2[X] = npq
Applying these formulas to a basic example illustrates the intuitive nature of these formulas. If a coin were flipped 12 times (n = 12) and the coin was fair (p = 0.5 and q = 1 – p = 0.5), then the
following are true regarding X, the number of successes (heads) in n trials (coin flips) given the probability p of success (heads) on each trial (coin flip):
Expected Value(X) = Mean(X) = μ[X] = 12 * 0.5 = 6
Variance(X) = σ^2[X] = npq = 12 * 0.5 * 0.5 = 3
This makes sense because 6 heads is the number of heads that one would expect to occur with 12 flips of a fair coin.
Binomial Distribution’s PDF and CDF
PDF = Probability Density Function
CDF = Cumulative Distribution Function
Binomial Distribution’s PDF - Probability Density Function
The binomial distribution’s PDF is given by the following:
(Click On Image To See a Larger Version)
for k = 0, 1, 2, …, n, where
(Click On Image To See a Larger Version)
f(k;n,p) = Pr(X = k) is the probability that X, the number of successes, equals k for n independent, successive Bernoulli trials each having the probability p of success.
For example, for the following parameters:
k = 4
n = 10
p = 0.5
Each unique binomial distribution is fully described by two parameters n and p. This binomial distribution is the distribution X successes in n = 10 trials with p = 0.5 probability of a successful
outcome in each and every one of the 10 trials.
This binomial distribution’s PDF calculates that the probability that X (the actual number of successful outcomes) equal k which is 4.
The Excel formula to calculate the binomial distribution’s PDF is the following:
f(k;n,p) = Pr(X = k) = BINOM.DIST(k, n, p , FALSE)
FALSE indicates that this Excel formula is calculating the binomial distribution’s PDF and not the CDF for this k, n, and p. “False” answers the question of the calculation is cumulative (which is
the case if calculating the CDF – Cumulative Distribution Function) on not cumulative (which is the case if calculating the PDF – Probability Density Function).
Excel 2010 and later also use the formula BINOM.DIST() which is equivalent to BINOMDIST() that is used by earlier versions of Excel. It should be noted that many of the equivalent but upgraded
formulas in Excel 2010 are more accurate than the original versions and should be used when possible. Microsoft recommends using the latest possible versions of any statistical formulas.
BINOM.DIST(4,10,0.5,FALSE) = 0.2051
This Excel binomial PDF calculation indicates that there is a 20.51 percent chance of exactly 4 successes in 10 independent, successive Bernoulli trials with 50 percent probability of success on each
A graph of the binomial distribution’s PDF for this unique binomial distribution (n= 10 and p = 0.5) shows the probability that X = 4 is 0.2051.
(Click On Image To See a Larger Version)
The binomial distribution’s curve shifts from left to right as p increases. This is shown is the following graphs of the binomial’s PDF for n = 10 when p = 0.2 and then as p = 0.8. Note that the
graphs of p = 1/5 (0.2) and p = 4/5 (0.8) are mirror images of each other about the mean of 5.
(Click On Image To See a Larger Version)
(Click On Image To See a Larger Version)
Binomial Distribution’s CDF - Cumulative Distribution Function
The binomial distribution’s CDF is as follows:
F(k;n,p) = Pr(X ≤ k). This is the probability that X, the number of successes in n Bernoulli trials each having the probability p of a successful outcome, equals up to k.
The binomial distribution’s CDF is given by the following:
(Click On Image To See a Larger Version)
is the “floor” under k, i.e., the greatest integer less than or equal to k.
The Excel formula to calculate the binomial distribution’s CDF is the following:
F(k;n,p) = Pr(X ≤ k) = BINOM.DIST(k, n, p , TRUE)
TRUE indicates that this Excel formula is calculating the binomial distribution’s CDF and not the PDF for this k, n, and p.
BINOM.DIST(4,10,0.5, TRUE) = 0.3769
There is a 37.69 percent chance of up to 4 successes in 10 independent, successive Bernoulli trials with 50 percent probability of success on each trial. A graph of the binomial distribution’s CDF
for this unique binomial distribution (n= 10 and p = 0.5) shows the probability that X = 4 is 0.3769.
(Click On Image To See a Larger Version)
The CDF for any distribution always varies from a minimum value of 0 on the left to a maximum value of 1 on the right. Just as with the PDF, the binomial distribution’s CDF shifts from left to right
as p increases. This is shown is the following graphs of the binomial’s CDF for n = 10 when p = 0.2 and then as p = 0.8.
(Click On Image To See a Larger Version)
(Click On Image To See a Larger Version)
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beta coefficient regression | Excelchat
i need data analysis help with linear regression, 6. Using Data > Data Analytics in Excel, do a linear regression for each type of car. Put this output on the same page as your scatter plot. NOTE: If
your regression coefficients do not match the trendline equation for each car type, double check your work! 7.Use Excel to calculate the correlation coefficient using the =CORREL command for each car
type. Additionally,calculate the coefficient of determination for each car type as well.Put these values near your regression output. Do these values match what you found in your regression output?
Solved by M. Q. in 30 mins | {"url":"https://www.got-it.ai/solutions/excel-chat/excel-help/how-to/beta/beta-coefficient-regression","timestamp":"2024-11-14T03:55:06Z","content_type":"text/html","content_length":"339416","record_id":"<urn:uuid:322aa6a0-1803-4841-8571-e26afd83f604>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00330.warc.gz"} |
Derivative Graph Calculator
Introduction to Derivative Graph Calculator
It is an online calculator that calculates the graph of a function with its derivative. It plots the graph of a function by finding its derivative at every point. It provides you the information on
the rate of change at every point. It uses the derivative formula to plot the graph.
In calculus, the derivative is an important concept that calculates the rate of change of a function. The plot of the derivative of a function is also an important factor in derivatives. We introduce
a tool that can plot the derivative graph quickly and accurately.
Formula used by Derivative Graph Calculator
The derivative graph is a graph of a function that is drawn by finding the derivative of that function and substituting the values in it. It helps to optimize a function with the derivative at every
function. The function calculator uses the following derivative formula to plot a graph between the values of its derivative and the y-axis.
$ f'(x) \;=\; \frac{f(x+δx)-f(x)}{δy} $
It plots the curve line by using the values of the function and its derivative. Then it compares both curve lines. If the line is normal in a graph you can also use a normal line calculator to find
the value of f(x).
How to find the Derivative Graph Calculator with steps?
You can find this calculator online easily by using the following steps.
1. Open your preferred browser and navigate to the search engine google.
2. Now, on the google search bar, write the Derivative Graph Calculator and press enter.
3. Google will provide you a list of websites offering derivative tools to draw a derivative plot.
How does the Derivative Graph Calculator work?
The working of the derivative calculator to plot graphs depends on the input function. It uses the fundamental derivative formula in the backend to calculate rate of change. It plots a graph at each
point according to the change in the given function.
When you input a function in the calculator, it calculates the derivative at every point in the function's domain. After calculating the derivatives, the derivative graph calculator plots a graph
that represents the variation of rate of change in the given function. Hence it provides you a quick way to visualize any rate of change.
Why use Derivative Plot Calculator?
A function's derivative is used to find the rate of change. But when we want to find how the derivative and the function are related, we usually use the graphical method. But it is a lengthy and
time-consuming method. Many students skip this method because of the lengthy procedure. So they always feel the need for external help. It would be helpful for them to use this tool because it can
handle long-term calculations easily. You can also use a slope graph calculator to calculate the curve line quickly.
Benefits of using Implicit Derivative Graph Calculator
Using an online tool in mathematics, especially in plotting graphs, is a smart way. It is always a helpful way to solve mathematical problems. It has many other useful benefits for you that are:
1. Curve derivative calculator is easy to use; you don’t have to perform long calculations to plot a graph.
2. You can get the derivative graph within a minute by clicking the calculate button.
3. This online derivative calculator computes the derivative graph quickly and 100% accurately.
4. It helps you to improve your graph understanding skills.
How to use Curve Derivative Calculator?
It is an easy method to plot a derivative graph with manual calculation. All you need is to follow the given steps.
• In the first step, enter the value of the function.
• Or use the load examples options.
• Review the function that appears below the input box.
• Click on the calculate button.
You can obtain the derivative graph after clicking the calculate button.
Frequently asked questions
What is the Derivative Graph?
The derivative graph is a graphical representation of a function with its derivative. It helps to compute the derivative at any point of the function’s graph. It describes the relationship between
the derivative and the function and tells how the derivative varies with the function values.
What is the rule for derivative graphs?
There are following rules to draw a derivative graph. These are:
• If the slope of the function f(x) is positive then the derivative graph of f'(x) will lie above the x-axis.
• If the slope of the function f(x) is negative then the derivative graph of f'(x) will lie below the x-axis.
What is the derivative graph of a horizontal line?
The derivative graph of a horizontal line is a straight line parallel to the x-axis. It is because the slope of a horizontal line is a constant zero and there is no rate of change in the direction of
the y-axis. Therefore, the derivative graph of a horizontal line is always a straight line parallel to x-axis.
0 Comment | {"url":"https://calculator-derivative.com/derivative-graph-calculator","timestamp":"2024-11-03T23:20:53Z","content_type":"text/html","content_length":"81322","record_id":"<urn:uuid:4d62485e-f46f-4e3b-bea2-88d4235d08e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00517.warc.gz"} |
Round a number up
The ROUNDUP function rounds a number up to a given number of places. The number of places is controlled by the number of digits provided in the second argument (num_digits). For example, these
formulas round the number 5.13 up to 1 and zero places:
=ROUNDUP(5.13,1) // returns 5.2
=ROUNDUP(5.13,0) // returns 6
In the example, the formula in cell D7 is
This tells Excel to take the value in B7 (PI) and round it to the number of digits in cell C7 (3) with a result of 3.142 Notice that even though the number in the 4th position to the right of the
decimal is 1, it is still rounded up to 2.
In the table, the ROUNDUP function is used to round the same number (pi, created with the PI function) to a decreasing number of digits, starting at 4 and moving down past zero to -1. Note that
positive numbers round to the right of the decimal point, while digits less than or equal to zero round to the left.
You can see that ROUNDUP is a rather heavy-handed function, so use with care. You can use the CEILING function to round a number up to a given multiple. If you want to discard the decimal portion of
a number, you can use the TRUNC function. | {"url":"https://exceljet.net/formulas/round-a-number-up","timestamp":"2024-11-09T10:44:53Z","content_type":"text/html","content_length":"44556","record_id":"<urn:uuid:99587535-b3c8-476b-ad69-2d41ac601028>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00856.warc.gz"} |
Linear or nonlinear?
07/06/19 20:36
This question might surprise you. But there are good arguments to discuss this mathematical concept in the photographic environment. Assume you have a 24 Megapixel sensor, like the one in the current
M series. The image quality (a very elusive concept) should be replaced by the information capacity, but for now it are the Image Engineering calculations that dominate the discussion. The Image
Engineering analysis of the sensor performance (always including a lens) has a good correlation with perceived image quality. The German magazine Color Foto is a true believer of this software. The
recent issue has a report of the Leica M10-D. The results are, for ISO100, 1931 line pairs per image height (LP/IH). One would assume that when doubling the number of pixels on the sensor, the lp/im
would also double. This is the linear aproach: when x=y the 2x would be 2y. As it happens, there is also a report of the Nikon Z7 with 45.7 Megapixels. Almost twice the amount of the pixels on the
Leica sensor. The result? At ISO 100 it is 2822 lp/ih. The Z6 (with 24.5 Mp) has 1988 lp/ih. The lenses used are different of course. This might have some influence, as is the selection of the JPG
format. The results of the Z7 are intriguing. There is a 90% difference between the pixel amount of the Leica M10 compared to the pixel amount of the Z7, but only a 46% increase in resolution. A
non-linear result! So a roughly 2 times the amount of pixels results in only 1.5 times resolution.
Another comparison: the Leica has pixel pitch of 6 micron, the Z6 of 5.9 and the Z7 of 4.3 micron. The APS-C sensor of the Ricoh GR III has a pixel pitch of 3.9 micron with an APS-C sensor of 24 Mp
and a resolution of 2075 lp/ih. Presumably the pixel size is more important than the sensor size. The Leica M8 is a living proof for this argument!
If and when Leica will decide on an increase of the amount of pixel for the next generation of the M camera it will be somewhere between the 24 Mp of the current model and the ±65 Mp of the Leica S
models. Not being in the position to being allowed to compete with the SL in future versions (let us assume 45 Mp) the final amount would be somewhere between 24 and 45: 34.5, which happens to be
neatly between both extremes. Then the increase in amount of pixels will be ±40%. The predicted increase in resolution will be around 0.5*40% and 0.75 * 40% = between ± 20 and ±30% or 1900 *1.25 =
2375 lp/ih, not the result to be really happy with. Assuming the usual tolerance of 5% for the bandwidth of the measured results, these figures only give the direction of thinking. The exact values
are less important.
The same argument can be found in the discussion about film emulsions that can record 200 lp/mm and film emulsions that can ‘only’ record 80 lp/mm. With 80 lp/mm almost every detail, that is visually
relevant in a scene, can be captured. But the price for the higher resolution is slow speed, careful focusing and the use of a tripod. In handheld shooting, the increase in resolution can not be
exploited. Again, assuming that the M camera will be the champion of handheld snapshot style of photography, the current level of resolution that is supported by the sensor is more than adequate for
the task. Leica could improve the imaging chain and especially the demosaicing section for enhanced clarity and best results.
UPDATE June 10: There is some confusion here:
Let us first get the basic figures about the measurements, based on the IE software, related to the fixed ISO 100
Ricoh GR-III: 24 Mp and 2075 lp/ih (pixel pitch 3.9 micron)
Leica M10-D: 24 MP and 1911 lp/ih (pixel pitch 6 micron)
Nikon Z6: 24 Mp and 1988 lp/ih (pixel pitch 5.9 micron)
Nikon Z7: 46 MP and 2822 lp/ih (pixel pitch 4.3 micron)
It is universally assumed that in order to double the resolution, one needs a four times increase in area: to double the resolution of the Leica sensor (24 Mp) one needs a sensor size of 4 * 24 = 96
Mp. This increase in size would (theoretically!) elevate the resolution from 1900 to 3800 lp/ih.
The Nikon Z7, which has only twice the area of the Leica sensor and therefore its resolution would be less: it is in fact 2800 lp/ih. The sensor of the Ricoh with 24 Mp has 2075 lp/ih with a
comparable pixel pitch. This is important to note, because the Nyquist limit is related to the pixel pitch. For a pixel pitch of 4 micron, the Nyquist frequency is 0.008 mm per mm (or cycle) = 125 lp
per mm. (application of the Kell factor of 0.7 gives 87.5 lp per mm). 2000 lp/ih is 64 lp/mm for a 15.6 mm image height. So there is some room for improvement, at least theoretically. The pixel size
of 6 micron for the Leica would produce 0.012 lp per mm or 83,3 lp per mm. The 1900 lp are for the image height of 24 mm, which is 79 lp/mm. Including the Kell factor which says that you can only
reliably resolve 70% of the Nyquist frequency, the practical resolution limit of the Leica sensor would be .7* 83 = 58 lp for every mm. The Leica imaging chain is better than that of the Ricoh! Or
one could also claim that the JPG demosaicing of the Leica is more aggressive and that spurious resolution is spoiling the results.
The Nikon Z7 with its 4 micron pixel pitch would be able to resolve 0.7*125 lp/mm = 87.5 lp per mm. The image height is 24 mm and the resolution is 2800 lp/ih. This would result in 117 lp per mm
compared to a Nyquist frequency of 125 lp per mm. Compare the measured resolution with the calculated Nyquist limit and the Kell factor:
Leica M10-D: 79 lp/mm; 83.3 lp/mm; 58 lp/mm
Nikon Z7: 117 lp/mm; 125 lp/mm; 87.5 lp/mm
The measured resolution is quite close to the Nyquist number. This is not surprising because the IE software uses the Nyquist calculation as the limiting factor in their calculations. This limiting
value would result at the point where the contrast is almost zero. Not very useful! The Kell factor is used because there is a contrast level below which there is no visual difference between two
adjacent lines. A contrast difference of 15% is the minimum and the Kell factor is in many cases too conservative.
Now the calculation. Doubling the size (from 24 Mp to 46 Mp) produces an increase in resolution of 1900 lp/ih to 2800 lp/ih. That is an increase of 47% or a factor of about 1.5. This is indeed a
one-dimensional relation. It compares only one direction and not the area. But here is the confusion. The resolution is measured one dimensional in line pairs per mm. This resolution is identical in
the horizontal and the vertical direction.The pixel pitch is a square measure (the 6 micron length of the Leica are the same for both directions. The pixel has a square area!) Now an example: assume
that we would like to have the resolution of the Z7 for a new Leica sensor. Going from 1900 to 2800 lp per mm and increasing the resolution in both directions would require that the amount of pixels
for the same size of the sensor grows2800/1900*24 = 35.4 Mp to get the same resolution of 2800 lp/ih. This value is less than expected. But the Leica processing chain might be more effective. The 35
Mp number would require a pixel pitch of 4.1 micron. This would result in a Nyquist value of 122 lp/mm or 2920 lp/ih. If we require to double the resolution of 1900 lp per mm, we would need a
decrease in pixel pitch from 6 to 3 micron to get a resolution of 166.7 lp per mm. (Nyquist limit). This would imply an increase in amount of pixels to 96 Mp or 4 times the actual Mp or 24 Mp.
Mixing the concept of the number of pixels in a certain area and the concept of the resolution of the pixel itself (in a one dimensional line) may be the reason for much confusion. The Nyquist
frequency is a one dimensional measure, assuming a square sized pixel, and will calculate the resolution of the system. The resulting pixel pitch will define the number of pixels per sensor area. | {"url":"https://photo.imx.nl/blog/files/5881e09e81ef66eec972c389656deb7f-140.html","timestamp":"2024-11-13T18:13:10Z","content_type":"text/html","content_length":"30982","record_id":"<urn:uuid:a6e58ef3-8ab5-49d4-96db-2892a2d2796f>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00239.warc.gz"} |
CAT Permutation and Combination Formulas PDF, Check Now
CAT Permutations & Combinations Formula is an important part of the cat exam. In the current 22-question format of the CAT Quant section, one can expect around 2-3 P&C; questions. One can download
the CAT P&C; Formula PDF free to practice these questions. Candidates often fear permutation and combination formulas, but they are not very difficult if you understand the basics well.
Practicing free CAT mocks will give you a fair idea of the type of permutation and combination questions asked in the CAT Exam. Permutation-based questions appear in the CAT test almost every year.
Many candidates avoid this topic but remember that one can solve the easy probability questions if familiar with the basics.
This blog explores the importance of a CAT Permutations & Combinations Formula PDF, provides essential tips, and discusses the significance of CAT permutation and combination Formulas.
Download CAT Permutations and Combination Formula Pdf
Importance of CAT Permutations & Combinations Formula PDF
The CAT exam is notorious for its ability to test candidates on their conceptual understanding rather than mere formulaic application. However, P&C; formulas act as the backbone for solving problems
efficiently. They provide a structured approach to tackling complex problems, saving valuable time during the exam.
• Quick Reference: CAT Permutations & Combinations Formulas are the core tools for solving P&C; problems. A PDF provides a handy, organized list for easy access during study and practice.
• Efficient Revision: Before exams, formulas are what you need to jog your memory. Cracku's CAT Permutations & Combinations PDF allows for focused revision without needing to search through
• Clarity and Consistency: Cracku's Well-structured PDFs often present formulas with clear explanations and examples, aiding understanding.
• Saves Time: Referring to a concise PDF is faster than flipping through pages in a book, maximizing your study efficiency.
Permutations & Combinations Formula PDF not only helps in solving QA questions but also help aspirants in LRDI sets. A strong grasp of these formulas will not only help you solve questions directly
but also develop the intuition required to approach unconventional problems.
Tips to Ace CAT P&C; Formula Problems
• Conceptual Clarity: While formulas are essential, it's equally important to understand the underlying concepts. Know the difference between permutations (order matters) and combinations (order
doesn't matter).
• Formula Practice: Practice using the formulas regularly. Solve a variety of problems to gain proficiency in applying the correct formula to different scenarios.
• Visualization: Many P&C; problems can be solved by visualizing the problem. Draw diagrams or use objects to represent the elements involved. This can help you break down complex problems into
simpler steps.
• Shortcut Development: Look for patterns and shortcuts while solving problems. These shortcuts can save you time during the exam.
• Time Management: Practice solving P&C; problems within a time limit. This will help you develop a sense of urgency and improve your speed.
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Complete CAT Quant Formula List
Preparing for the CAT exam means you need to understand different math topics, and we’re here to help. We’ve gathered all the important CAT quant formulas in one place to make your study easier.
Below is a table with links to download PDFs for topics like Progressions, Interest, Geometry, and more.
CAT Permutations & Combinations Formula Shortcuts
While specific formulas cannot be shared in this blog, here are some general tips for developing shortcuts:
• Learn the factorial properties: Understanding the properties of factorials can help simplify calculations.
• Master the concept of division principle: This principle can be applied in many P&C; problems to reduce calculations.
• Utilize symmetry: If a problem exhibits symmetry, exploit it to reduce the number of calculations.
• Practice mental calculations: Improving your mental math skills can significantly enhance your speed.
Weightage of P&C; in CAT
The weightage of P&C; in CAT fluctuates from year to year. However, it is consistently a part of the Quantitative Aptitude section. You can expect 2-3 questions based on this topic. While it might
not seem like a large portion, mastering P&C; can make a significant difference in your overall score.
Year Number of Questions
Mastering permutations and combinations formula is essential for acing the CAT quantitative section. A dedicated CAT formula PDF can be your secret weapon for quick reference and efficient revision.
Cracku's free permutations and combinations formula PDF for CAT offers a comprehensive collection of formulas, making it an invaluable resource for your preparation. Remember, understanding the
concepts is equally important as memorizing formulas. Consistent practice will solidify your grasp on the topic.
Start your CAT preparation on the right foot by downloading Cracku's free permutations and combinations formula PDF and dedicating time to mastering permutations and combinations. | {"url":"https://cracku.in/cat-permutation-and-combination-formula-pdf/","timestamp":"2024-11-02T07:34:17Z","content_type":"text/html","content_length":"126202","record_id":"<urn:uuid:13db155d-cfbc-43f9-98fd-94bdd32d4c74>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00021.warc.gz"} |
Tangram Template
Tangram template - Use all of the tangram pieces to make a square. Make a trapezoid with the seven tangram pieces. Use the seven tangram pieces to form a parallelogram. A comprehensive and coherent
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pieces to make a triangle. Do not look at the existing pattern. Use two tangram pieces to make a triangle. Common themes are animals, numbers, and letters. The printable below provides four printable
outlines that can be solved using the tangram template above.
No two lines on a tile have the same colour. Use four tangram pieces to make a triangle.
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Use three tangram pieces to make a triangle. Common themes are animals, numbers, and letters. Use all of the tangram pieces to make a square.
Do not look at the existing pattern. How to solve a tangram puzzle… the traditional way to play with tangram puzzles is to attempt to use all seven pieces, with no overlapping, to fill a given
outline. Use two tangram pieces to make a triangle. A comprehensive and coherent set of mathematics standards for each and every student from prekindergarten through grade 12, principles and
standards is the first set of rigorous, college and career readiness standards for the 21st century. Common themes are animals, numbers, and letters. Use all of the tangram pieces to make a square.
Make a trapezoid with the seven tangram pieces. Use the seven tangram pieces to form a parallelogram. Use three tangram pieces to make a triangle. No two lines on a tile have the same colour.
Use four tangram pieces to make a triangle. The printable below provides four printable outlines that can be solved using the tangram template above. | {"url":"https://templates.esad.edu.br/en/tangram-template.html","timestamp":"2024-11-04T00:53:29Z","content_type":"text/html","content_length":"112796","record_id":"<urn:uuid:42d7c939-55f1-40f7-baa0-3249e99625b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00191.warc.gz"} |
[Confidence interval calculation for small numbers of observations or no observations at all].
Confidence interval calculation is a common statistics measure, which is frequently used in the statistical analysis of studies in medicine and life sciences. A confidence interval specifies a range
of values within which the unknown population parameter may lie. In most situations, especially those involving normally-distributed data or large samples of data from other distributions, the normal
approximation may be used to calculate the confidence interval. But, if the number of observed cases is small or zero, we recommend that the confidence interval be calculated in more appropriate
ways. In such cases, for example, in clinical trials where the number of observed adverse events is small, the criterion for approximate normality is calculated. Confidence intervals are calculated
with the use of the approximated normal distribution if this criterion is met, and with the use of the exact binomial distribution if not. This article, accompanied by examples, describes the
criteria in which the common and known method cannot be used as well as the stages and methods required to calculate confidence intervals in studies with a small number of observations.
Original language English
Pages (from-to) 289-291, 304, 303
Journal Harefuah
Volume 153
Issue number 5
State Published - May 2014
ASJC Scopus subject areas
Dive into the research topics of '[Confidence interval calculation for small numbers of observations or no observations at all].'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/confidence-interval-calculation-for-small-numbers-of-observations","timestamp":"2024-11-09T10:41:36Z","content_type":"text/html","content_length":"51728","record_id":"<urn:uuid:0c14854c-9dfb-4e23-8d0d-c7f9c0f04bc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00853.warc.gz"} |
2024-11-12T03:20:36Z https://u-ryukyu.repo.nii.ac.jp/oai oai:u-ryukyu.repo.nii.ac.jp:02001473 2022-10-31T00:48:55Z 1642837622505:1642837745608:1642837760806 1642838403551:1642838405037 即時応答型音程
トレーナーの為の周波数解析手法について Real-Time Response Pitch Discrimination Training System : A Pitch Computing Method 仲間, 正浩 Nakama, Masahiro It is meaningful to develop a Real-Time Response
Pitch Descrimination System by utilizing a computing sound pitch. There are many approaches in computing sound pitch. But, no approach was seen in developing enough response-time and accuracy of
frequency discrimination for pitch discrimination training. Therefore new methodology is considered to improve response-time and accuracy of frequency discrimination. In FFT (Fast Fourier
Transformation), there exist mathematical relations such that\n(1) time_range_of_analysis^*frequency_discrimination = 1 and\n(2) response_time = time_range_of_analysis/2^*computing_time,\nwhich shows
no improvement in both response-time and accuracy of frequency discrimination. Fortunately, there exist the fact such that many peaks of power spectrum : f[Hz], f^*2[Hz], f^*3[Hz], ･･････ as the
results of FFT from a single f[Hz] pitch sound.\nTo improve both response-time and accuracy of frequency discrimination, the following new method is considered in this paper. Step1 : Compute pitch
roughly from f[Hz] power spectrum peak. Step2 : Compute pitch accurately from f^*n[Hz]. 紀要論文 http://purl.org/coar/resource_type/c_6501 琉球大学 1994-10-31 NA http://hdl.handle.net/20.500.12000/
1374 0386-5738 AN10144831 琉球大学教育学部紀要第一部・第二部 45 418 411 http://hdl.handle.net/20.500.12000/1374 https://u-ryukyu.repo.nii.ac.jp/records/2001473 jpn open access | {"url":"https://u-ryukyu.repo.nii.ac.jp/oai?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai:u-ryukyu.repo.nii.ac.jp:02001473","timestamp":"2024-11-12T03:20:37Z","content_type":"application/xml","content_length":"4895","record_id":"<urn:uuid:6b0db34f-f7db-48e0-9629-fc4ee5ce0602>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00215.warc.gz"} |
Tree Data Structure and its Applications
What is Tree Data Structure?
A tree is a non-linear data structure as it stores data in a hierarchical manner that consists of nodes connected through edges.
It is the best alternative for the linear data structure like arrays, stacks, queues, etc as these linear data structures have high time and space complexity.
The data stored in nodes of the tree are easier to access thus reducing the time complexity.
Terms related to tree data structure
1. Node
A node is an entity that stores a data element and links to its child and parent nodes.
2. Edge
The link between any two nodes is called the edge.
3. Root
The root is the transparent node in a tree that is a root node doesn't have any parent node. Here no. 1 is the Root node.
4. Parent and Child Node
The node which contains sub-nodes is called the parent node. Here 1 is the parent node for 2 and 3 and 2 is the parent node for 4 and 5.
The node which is a descendant of any node is called a child node. Here 2 and 3 are child nodes and 1 and 4 and 5 are child nodes of 2.
5. Internal node
A node that has at least one child node is called an Internal node. Here 1 is an internal node whereas 3 is not an internal node.
6. Leaf node (external node)
A node that doesn't have a child node is called a leaf node. It is the bottom-most node of a tree. Leaf nodes are commonly referred to as external nodes.
7. Height of a node
The number of edges from the node to the deepest leaf (leaf node) is called the height of a node. Here eight are node 1 and node 2. The height of node 2 is 1.
8. Depth of a node
The no. of edges from the root to that node is called the depth of the node. Here the depth of node 2 is 1. The depth of node 5 is 2.
9. Height of a tree
The height of the tree is the height of the root node to the deepest leaf node. Here the height of the tree is 2.
10. Degree of node
The degree of a node is the total number of branches of that node. Here node 1 has 2 degrees. Node 3 has 0 degrees as no branches are there.
Application of a Tree
1. Trees are used to quickly check whether a data element is present in the set or not.
2. Heap tree is used to perform heap sort of elements.
3. A modified version of a tree, called tries, is utilized in modern routers for storing routing information.
4. Compilers use a syntax tree to validate the syntax of every program. | {"url":"https://www.sciencedoze.com/2023/12/tree-data-structure-and-its-applications.html","timestamp":"2024-11-11T09:48:08Z","content_type":"application/xhtml+xml","content_length":"133626","record_id":"<urn:uuid:571bc002-c99a-4769-95b4-f75854cbc311>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00330.warc.gz"} |
Comparison of GMP, Apfloat and Threeway for the multiplication of large integers
Comparison of three public-domain multiprecision libraries: BigNum, Gmp and Pari We describe the use of three multiprecision libraries on two sample problems: compute digits of Pi and compute a
recursive sequence of integers. Here is a revised version of this comparison.
Numerical evaluation of special functions
Multiplication rapide en LeLisp
An implementation of Schönhage's multiplication algorithm [patch for gmp-4.1.3] Note: a faster implementation was made with Pierrick Gaudry and Alexander Kruppa and is available here.
mpn_mul_fftw.c, a fast integer multiplication for GMP using the FFTW package. Guillermo Ballester Valor made a similar package named YEAFFT.
Implantation de l'algorithme de Schönhage en C avec la bibliotheque BigNum
Implantation de l'algorithme de Schönhage en Maple
FFT Patch for gmp-5.0.1 [README]
Other interesting links
The GMP home page.
The multi-precision number library CLN from Bruno Haible.
The MPI library from Michael Fromberger (no longer maintained, but IMath is) and the derived LibTomMath library from Tom St Denis.
The Crack program from Joris van der Hoeven.
Multidigit multiplication for mathematicians by Dan Bernstein, and a comparison of integer multiplication with various libraries
MPSolve 2.0, Multiprecision Polynomial Solver developed by Bini and Fiorentino in the Frisco project. | {"url":"https://members.loria.fr/PZimmermann/bignum/","timestamp":"2024-11-10T21:17:00Z","content_type":"text/html","content_length":"2881","record_id":"<urn:uuid:5c8b3082-ca3b-4bc0-a553-10751d6c4c3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00503.warc.gz"} |
Cirruculum Vitae
Academic Qualifications
Ph. D. (Mathematics). Department of Mathematics, University Of Delhi, India. Title of Ph. D. Thesis: Behavior of Functions and Their Fourier Transform (Harmonic Analysis). Academic Supervisor: Prof.
Ajay Kumar, Department of Mathematics, University of Delhi, India.
M.Sc. (Mathematics); First Division. Central Department of Mathematics, Tribhuvan University, Kirtipur, Kathmandu, Nepal.
B.Sc. (Physical Group); First Division with Distinction. H.P.T Arts and R.Y.K. Science College Nasik, Maharashtra, University of Poona, India.
I. Sc. (Physical Group); First Division. H.P.T. Arts and R.Y.K science College, Nasik, Higher Secondary Board, Pune (1986), India.
S.L.C.; First Division with District Top. His Majesty Government of Nepal SLC Board, Sahastraling Madhamik Vidhayala, Chamada Dadeldhura, Nepal.
Working Experiences
• Head of the Department of Master Degree in Mathematics, Siddhanath Science Campus, Mahendranagar, Kanchanpur from 1999-2000.
• Teaching various Mathematics courses such as Functional Analysis, Operator Theory, and Abstract Harmonic Analysis in M. Phil. and Ph.D. level.
• Teaching various Mathematics courses such as Functional Analysis, Real Analysis, Algebra, Mathematical Analysis, Integral Transforms, Complex variables and Differential Equations, Harmonic
Analysis in M.Sc. level
• Teaching various mathematics courses such as Algebra, Calculus, Mathematical Analysis, and Vector Calculus in graduate level and under graduate level.
• Supervision of M.Phil./Ph.D. Dissertation/Thesis of Students of Kathmandu University and Tribhuvan University and M.Sc. Dissertation of Students at Central Department of Mathematics, Tribhuvan
Computer Skill:
Microsoft word, Power Point, Latex, Email and internet
Country Visited:
Spain, Switzerland, South Korea, China and India | {"url":"http://chetraj.cdmathtu.edu.np/cirruculum-vitae.html","timestamp":"2024-11-14T23:04:55Z","content_type":"text/html","content_length":"8192","record_id":"<urn:uuid:239b5ee5-3829-4eae-be91-024c107a91ba>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00845.warc.gz"} |
How do I bulk apply multiple Cost Codes to my timesheets?
This article will detail how to bulk-apply cost codes across multiple workers for completed shifts
Scenario: A foreman is managing timesheets for two workers and needs to apply productivity for their shifts. The foreman needs to allocate time across two cost codes with a fixed hr. amount assigned
to Cost Code 1 for safety-related tasks. The remainder of the hours will be allocated to Cost Code 2, which will be input as a percentage, indicating all remaining hours.
• Worker 1: Worked 10 hours total
• Worker 2: Worked 11 hours total
Step 1: The foreman selects the workers on the productivity tab that they will be assigning productivity to and clicks "Bulk Productivity."
Step 2: The foreman assigns 0.5 hours to Cost Code 1, which will apply to each worker, and adds Cost Code 2 by clicking the blue + on the top right of the screen. For this Cost Code, you will select
the % allocation type and input 100% for the remaining time for each worker.
• Worker 1 (11 total hours):
□ Cost Code 1 (Safety): 0.5 hours
□ Cost Code 2: 100% of remaining hours (10.5 hours)
• Worker 2 (10 total hours):
□ Cost Code 1 (Safety): 0.5 hours
□ Cost Code 2: 100% of remaining hours (9.5 hours)
On the bottom left of the screen, the "Max hours to Allocate" will reflect the specific hours applied in Cost Code 1, and the "Allocated Percent" should now be 100% on the bottom left of the screen.
Step 3: Click Apply
On the timesheet, the cost codes will now be updated based on the total amounts for the shift
For Worker 1:
• Cost Code 1: 0.5 hours
• Cost Code 2: 100% of remaining hours (which is 9.5 hours)
For Worker 2:
• Cost Code 1: 0.5 hours
• Cost Code 2: 100% of remaining hours (which is 10.5 hours)
Key Point: The final cost code (or the only Cost Code) will always be a percentage. In this case, Cost Code 2 will always represent 100% of the remaining hours after the allocation to Cost Code 1.
By using percentages for the final cost code, the system ensures that any remaining hours are properly allocated without needing to manually calculate exact numbers, reducing the risk of errors and
keeping the process streamlined. | {"url":"https://ask.smartbarrel.io/en/how-do-i-apply-cost-codes-in-bulk","timestamp":"2024-11-05T07:11:29Z","content_type":"text/html","content_length":"54188","record_id":"<urn:uuid:5bb9e68f-89a5-47f3-a7c6-b46d635fc515>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00086.warc.gz"} |
EViews Help: @mstdev
Trailing moving sample standard deviation (d.f. adjusted; ignore NAs).
n-period trailing moving square roots of Pearson product moment sample variances, with d.f. correction, ignoring NAs.
Syntax: @mstdev(x, n)
x: series
n integer, series
Return: series
and ignoring missing values (NAs).
If n is not an integer, the integer floor
show @mstdev(x, 12)
produces a linked series of the moving sample standard deviation of the series x where NAs are ignored.
For the NA-propagating variant of this function, see | {"url":"https://help.eviews.com/content/functionref_m-@mstdev.html","timestamp":"2024-11-05T05:43:37Z","content_type":"application/xhtml+xml","content_length":"10778","record_id":"<urn:uuid:2f1f6c83-aa78-49c7-8f48-d457c74afb21>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00638.warc.gz"} |
Graphical Summarization of Continuous Variables Against a
summaryRc {Hmisc} R Documentation
Graphical Summarization of Continuous Variables Against a Response
summaryRc is a continuous version of summary.formula with method='response'. It uses the plsmo function to compute the possibly stratified lowess nonparametric regression estimates, and plots them
along with the data density, with selected quantiles of the overall distribution (over strata) of each x shown as arrows on top of the graph. All the x variables must be numeric and continuous or
nearly continuous.
summaryRc(formula, data=NULL, subset=NULL,
na.action=NULL, fun = function(x) x,
na.rm = TRUE, ylab=NULL, ylim=NULL, xlim=NULL,
nloc=NULL, datadensity=NULL,
quant = c(0.05, 0.1, 0.25, 0.5, 0.75,
0.90, 0.95), quantloc=c('top','bottom'),
cex.quant=.6, srt.quant=0,
bpplot = c('none', 'top', 'top outside', 'top inside', 'bottom'),
trim=NULL, test = FALSE, vnames = c('labels', 'names'), ...)
An R formula with additive effects. The formula may contain one or more invocations of the stratify function whose arguments are defined below. This causes the entire analysis to be
formula stratified by cross-classifications of the combined list of stratification factors. This stratification will be reflected as separate lowess curves.
data name or number of a data frame. Default is the current frame.
subset a logical vector or integer vector of subscripts used to specify the subset of data to use in the analysis. The default is to use all observations in the data frame.
function for handling missing data in the input data. The default is a function defined here called na.retain, which keeps all observations for processing, with missing variables or
na.action not.
fun function for transforming lowess estimates. Default is the identity function.
na.rm TRUE (the default) to exclude NAs before passing data to fun to compute statistics, FALSE otherwise.
ylab y-axis label. Default is label attribute of y variable, or its name.
ylim y-axis limits. By default each graph is scaled on its own.
a list with elements named as the variable names appearing on the x-axis, with each element being a 2-vector specifying lower and upper limits. Any variable not appearing in the list
xlim will have its limits computed and possibly trimmed.
nloc location for sample size. Specify nloc=FALSE to suppress, or nloc=list(x=,y=) where x,y are relative coordinates in the data window. Default position is in the largest empty space.
datadensity see plsmo. Defaults to TRUE if there is a stratify variable, FALSE otherwise.
quant vector of quantiles to use for summarizing the marginal distribution of each x. This must be numbers between 0 and 1 inclusive. Use NULL to omit quantiles.
quantloc specify quantloc='bottom' to place at the bottom of each plot rather than the default
cex.quant character size for writing which quantiles are represented. Set to 0 to suppress quantile labels.
srt.quant angle for text for quantile labels
if not 'none' will draw extended box plot at location given by bpplot, and quantiles discussed above will be suppressed. Specifying bpplot='top' is the same as specifying bpplot='top
bpplot inside'.
height.bpplot height in inches of the horizontal extended box plot
The default is to plot from the 10th smallest to the 10th largest x if the number of non-NAs exceeds 200, otherwise to use the entire range of x. Specify another quantile to use other
trim limits, e.g., trim=0.01 will use the first and last percentiles
test Set to TRUE to plot test statistics (not yet implemented).
vnames By default, plots are usually labeled with variable labels (see the label and sas.get functions). To use the shorter variable names, specify vnames="names".
... arguments passed to plsmo
no value is returned
Frank Harrell
Department of Biostatistics
Vanderbilt University
See Also
plsmo, stratify, label, formula, panel.bpplot
sex <- factor(sample(c("m","f"), 500, rep=TRUE))
age <- rnorm(500, 50, 5)
bp <- rnorm(500, 120, 7)
units(age) <- 'Years'; units(bp) <- 'mmHg'
label(bp) <- 'Systolic Blood Pressure'
L <- .5*(sex == 'm') + 0.1 * (age - 50)
y <- rbinom(500, 1, plogis(L))
summaryRc(y ~ age + bp)
# For x limits use 1st and 99th percentiles to frame extended box plots
summaryRc(y ~ age + bp, bpplot='top', datadensity=FALSE, trim=.01)
summaryRc(y ~ age + bp + stratify(sex),
label.curves=list(keys='lines'), nloc=list(x=.1, y=.05))
y2 <- rbinom(500, 1, plogis(L + .5))
Y <- cbind(y, y2)
summaryRc(Y ~ age + bp + stratify(sex),
label.curves=list(keys='lines'), nloc=list(x=.1, y=.05))
version 5.1-3 | {"url":"https://search.r-project.org/CRAN/refmans/Hmisc/html/summaryRc.html","timestamp":"2024-11-13T15:22:13Z","content_type":"text/html","content_length":"8998","record_id":"<urn:uuid:dfc30442-963b-4304-badd-5563621527ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00286.warc.gz"} |
Essentially unique
In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the
property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which is often formalized using an equivalence relation. A related notion is a universal
property, where an object is not only essentially unique, but unique up to a unique isomorphism^[1] (meaning that it has trivial automorphism group). In general there can be more than one isomorphism
between examples of an essentially unique object.
Set theory
At the most basic level, there is an essentially unique set of any given cardinality, whether one labels the elements [math]\displaystyle{ \{1,2,3\} }[/math] or [math]\displaystyle{ \{a,b,c\} }[/
math]. In this case, the non-uniqueness of the isomorphism (e.g., match 1 to [math]\displaystyle{ a }[/math] or 1 to [math]\displaystyle{ c }[/math]) is reflected in the symmetric group.
On the other hand, there is an essentially unique totally ordered set of any given finite cardinality that is unique up to unique isomorphism: if one writes [math]\displaystyle{ \{1 \lt 2 \lt 3\} }[/
math] and [math]\displaystyle{ \{a\lt b\lt c\} }[/math], then the only order-preserving isomorphism is the one which maps 1 to [math]\displaystyle{ a }[/math], 2 to [math]\displaystyle{ b }[/math],
and 3 to [math]\displaystyle{ c }[/math].
Number theory
The fundamental theorem of arithmetic establishes that the factorization of any positive integer into prime numbers is essentially unique, i.e., unique up to the ordering of the prime factors.^[2]^
Group theory
In the context of classification of groups, there is an essentially unique group containing exactly 2 elements.^[3] Similarly, there is also an essentially unique group containing exactly 3 elements:
the cyclic group of order three. In fact, regardless of how one chooses to write the three elements and denote the group operation, all such groups can be shown to be isomorphic to each other, and
hence are "the same".
On the other hand, there does not exist an essentially unique group with exactly 4 elements, as there are in this case two non-isomorphic groups in total: the cyclic group of order 4 and the Klein
Measure theory
There is an essentially unique measure that is translation-invariant, strictly positive and locally finite on the real line. In fact, any such measure must be a constant multiple of Lebesgue measure,
specifying that the measure of the unit interval should be 1—before determining the solution uniquely.
There is an essentially unique two-dimensional, compact, simply connected manifold: the 2-sphere. In this case, it is unique up to homeomorphism.
In the area of topology known as knot theory, there is an analogue of the fundamental theorem of arithmetic: the decomposition of a knot into a sum of prime knots is essentially unique.^[5]
Lie theory
A maximal compact subgroup of a semisimple Lie group may not be unique, but is unique up to conjugation.
Category theory
An object that is the limit or colimit over a given diagram is essentially unique, as there is a unique isomorphism to any other limiting/colimiting object.^[6]
Coding theory
Given the task of using 24-bit words to store 12 bits of information in such a way that 7-bit errors can be detected and 3-bit errors can be corrected, the solution is essentially unique: the
extended binary Golay code.^[7]
See also
Original source: https://en.wikipedia.org/wiki/Essentially unique. Read more | {"url":"https://handwiki.org/wiki/Essentially_unique","timestamp":"2024-11-05T18:20:53Z","content_type":"text/html","content_length":"43025","record_id":"<urn:uuid:79849b1f-891b-48d9-8734-5a098ea1a969>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00494.warc.gz"} |
320 Linear Feet to Square Feet - GEGCalculators
320 Linear Feet to Square Feet
Linear feet are a measure of length, while square feet are a measure of area. To convert linear feet to square feet, you need to know the width (in feet) of the area you’re measuring.
Let’s say the width is 4 feet. To convert 320 linear feet to square feet:
320 linear feet * 4 feet (width) = 1,280 square feet.
320 linear feet to square feet
Result: square feet
How many linear feet is in a square foot?
There is no direct conversion between linear feet and square feet. Linear feet measures length, while square feet measures area.
Can you convert linear feet to square feet?
No, linear feet and square feet are different units of measurement, and they cannot be directly converted without additional information about the width or depth of the object.
How do you convert linear square feet to square feet?
“Linear square feet” is not a standard unit of measurement. It’s possible that there might be a misunderstanding in the terminology.
What is 500 square feet in linear feet?
It is not possible to convert square feet directly to linear feet without additional information about the width or depth of the object being measured.
How many feet is 1 linear feet?
One linear foot is equal to one foot in length. There is no conversion needed between linear feet and feet as they are the same measurement.
How many linear feet is a 10×10 deck?
To determine the linear feet of a 10×10 deck, you would need to know the width or depth of the boards used for the decking. With that information, you can calculate the linear feet by multiplying the
number of boards needed by their length.
How much is 100 linear feet?
The value of 100 linear feet depends on what is being measured. Linear feet is a measure of length, so the value will depend on the specific object being referenced (e.g., 100 linear feet of a board,
100 linear feet of fencing, etc.).
What is a linear square foot?
“Linear square foot” is not a standard unit of measurement. It’s possible that there might be a misunderstanding in the terminology.
How many linear feet is a 2x4x8?
A 2x4x8 is a piece of lumber that is 2 inches thick, 4 inches wide, and 8 feet long. Therefore, it is already measured in linear feet, and its length is 8 feet.
Is linear square feet the same as square feet?
There is no standard unit of measurement called “linear square feet.” It is possible that this term is being used incorrectly.
How do you convert linear to board feet?
Linear feet and board feet are different units of measurement and cannot be directly converted without additional information about the width and thickness of the board.
How do I estimate square feet?
To estimate square feet, measure the length and width of the area in feet and then multiply those two measurements together. This will give you the square footage of the area.
What size is a 500 sq ft room?
A 500 square feet room could have various dimensions depending on its shape. For example, a room that is 25 feet by 20 feet would have an area of 500 square feet.
What size is 500 sq feet?
500 square feet is a measure of area, so it does not have a specific size or dimensions. It could be in the shape of a square, rectangle, or any other irregular shape as long as the total area is 500
square feet.
How to design a 500 square feet apartment?
Designing a 500 square feet apartment involves making efficient use of space and choosing furnishings that are appropriate for smaller living areas. Consider using multifunctional furniture and
maximizing storage options to make the most of the available space.
What is the linear foot rule?
The linear foot rule is a method used to calculate the number of board feet in a piece of lumber based on its dimensions. It takes into account the width, thickness, and length of the lumber to
determine its board footage.
What is 62 linear inches in feet?
62 linear inches is equal to 5 feet and 2 inches. To convert inches to feet, divide the number of inches by 12 (since there are 12 inches in a foot). In this case, 62 divided by 12 equals 5 with a
remainder of 2 inches.
Is a 400 square foot deck big?
A 400 square foot deck is a moderate size and provides ample space for outdoor activities and entertaining.
How much is a 1,000 square foot deck?
The cost of a 1,000 square foot deck can vary widely depending on factors such as the materials used, design complexity, and location. It is best to consult with a contractor or supplier to get an
accurate estimate.
How big is a 200 square foot deck?
A 200 square foot deck is a small to medium-sized deck and can comfortably accommodate a small seating area or outdoor dining space.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
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How to Use the AVERAGE.WEIGHTED Function: A Comprehensive Guide
Last Modified: October 14, 2024 - 5 min read
Julian Alvarado
Are you struggling to calculate weighted averages in your spreadsheets? The AVERAGE.WEIGHTED function is a powerful tool that can simplify this process. In this guide, we’ll walk you through
everything you need to know about using AVERAGE.WEIGHTED in Excel and Google Sheets.
Step-by-Step Guide to Using AVERAGE.WEIGHTED
Let’s walk through the process of using the AVERAGE.WEIGHTED function in your spreadsheet.
1. Prepare your data
First, ensure your data is organized properly:
• Place your values in one column or row
• Place the corresponding weights in an adjacent column or row
• Make sure you have an equal number of values and weights
2. Enter the AVERAGE.WEIGHTED function
In the cell where you want your result to appear, type:
3. Select your value range
Click and drag to select the range containing your values, or type the range manually (e.g., A1:A5).
4. Select your weight range
After the values range, type a comma, then select or type the range containing your weights.
5. Verify and calculate
Your formula should now look something like this:
=AVERAGE.WEIGHTED(A1:A5, B1:B5)
Press Enter to calculate the weighted average.
Practical Examples of AVERAGE.WEIGHTED
To help you understand how to apply AVERAGE.WEIGHTED in real-world scenarios, let’s explore some practical examples.
Calculating student grades
Suppose you’re a teacher calculating final grades for your students. Each assignment has a different weight:
• Homework: 20%
• Midterm Exam: 30%
• Final Project: 20%
• Final Exam: 30%
Here’s how you might set up your spreadsheet:
Assignment Grade Weight
Homework 85 0.20
Midterm Exam 78 0.30
Final Project 92 0.20
Final Exam 88 0.30
To calculate the weighted average, you would use:
=AVERAGE.WEIGHTED(B2:B5, C2:C5)
This would give you the student’s final grade, taking into account the different weights of each assignment.
Analyzing stock portfolio performance
Let’s say you’re evaluating the performance of your stock portfolio. You have different amounts invested in various stocks, and you want to calculate the weighted average return:
Stock Return Investment
AAPL 12% $5000
GOOGLE 8% $3000
MSFT 15% $4000
AMZN 10% $2000
To calculate the weighted average return, you would use:
=AVERAGE.WEIGHTED(B2:B5, C2:C5)
This gives you the overall portfolio return, weighted by the amount invested in each stock.
Evaluating employee performance
Imagine you’re a manager conducting performance reviews. Different aspects of job performance have varying importance:
Criteria Score Weight
Quality of Work 4.5 0.30
Productivity 4.0 0.25
Communication 3.8 0.20
Initiative 4.2 0.15
Teamwork 4.7 0.10
To calculate the overall performance score:
=AVERAGE.WEIGHTED(B2:B6, C2:C6)
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This provides a comprehensive score that reflects the relative importance of each performance criterion.
Understanding Weighted Averages
Before diving into the AVERAGE.WEIGHTED function, it’s essential to understand the concept of weighted averages and why they’re important.
What is a weighted average?
A weighted average is a calculation that takes into account the relative importance or significance of each value in a dataset. Unlike a simple average, which treats all values equally, a weighted
average assigns different weights or levels of importance to each value.
For example, imagine you’re calculating your final grade for a course. If your midterm exam is worth 30% of your grade and your final exam is worth 70%, you’d use a weighted average to determine your
overall score. This approach ensures that the final exam, which carries more weight, has a greater impact on your final grade.
Why use weighted averages?
Weighted averages are crucial in many real-world scenarios where not all data points are equally significant. They provide a more accurate representation of data by accounting for the varying
importance of different factors. Some common applications include:
1. Academic grading systems
2. Financial portfolio analysis
3. Performance evaluations in the workplace
4. Market research and customer satisfaction surveys
5. Quality control in manufacturing
Differences between simple and weighted averages
The key difference between simple and weighted averages lies in how they treat each value in a dataset:
1. Simple average: All values are considered equally important. It’s calculated by summing all values and dividing by the number of values.
2. Weighted average: Each value is assigned a weight that reflects its importance. The calculation involves multiplying each value by its weight, summing these products, and then dividing by the sum
of the weights.
Here’s a quick example to illustrate the difference:
Suppose you have three test scores: 80, 90, and 70.
Simple average: (80 + 90 + 70) / 3 = 80
Now, let’s say the tests have different weights: 20%, 50%, and 30% respectively.
Weighted average: (80 * 0.2) + (90 * 0.5) + (70 * 0.3) = 82
As you can see, the weighted average provides a different result that takes into account the varying importance of each test.
Master AVERAGE.WEIGHTED
In conclusion, the AVERAGE.WEIGHTED function is a powerful tool for calculating weighted averages in Excel and Google Sheets. By understanding its syntax, applications, and limitations, you can make
more accurate and meaningful calculations in various fields, from finance to education and beyond.
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Introduction to JustifyAlpha
Daniel Lakens & Maximilian Maier
Vignette Accompanying “Justify Your Alpha: A Primer on Two Practical Approaches”
The goal of JustifyAlpha is to provide ways for researchers to justify their alpha level when designing studies. Two approaches are currently implemented. The first function optimal_alpha allows
users to computed balanced or minimized Type 1 and Type 2 error rates. The second approach uses the function ttestEvidence or ftestEvidence to lower the alpha level as a function of the sample size
to prevent Lindley’s paradox.
You can install the released version of JustifyAlpha from GitHub with:
Minimizing Error Rates
Assume we plan to perform an independent t-test, where our smallest effect size of interest is d = 0.5, and we are planning to collect 64 participants in each condition. We would normally calculate
power as follows:
pwr.t.test(d = 0.5, n = 64, sig.level = 0.05, type = 'two.sample', alternative = 'two.sided')$power
This analysis tells us that we have 80% power with a 5% alpha level for our smallest effect size of interest, d = 0.5, when we collect 64 participants in each condition.
If we design 2000 studies like this, the number of Type 1 and Type 2 errors we make depend on how often the null hypothesis is true, and how often the alternative hypothesis is true. Let’s assume
both are equally likely. This means that in 0.5 × 2000 = 1000 studies the null hypothesis is true, and we will make 0.05 × 1000 = 50 Type 1 errors, so in 50 studies we will find a significant result,
even though there is no true effect. In 0.5 × 2000 = 1000 studies the alternative hypothesis is true, and with 80% power we will make 0.2 × 1000 = 200 Type 2 errors, so in 200 studies we will not
observe a significant result even if there is a true effect. Combining Type 1 and Type 2 errors, in the long run, we should expect 50 + 200 = 250 of our 2000 studies to yield an error. The combined
error rate is therefore 250/2000 = 0.125.
The goal in Neyman-Pearson hypothesis testing is to control the number of errors we make, as we perform hypothesis tests. Researchers often rely on convention when setting error rates, and there is
no special reason to set the Type 1 error rate at 5% and the Type 2 error rate at 20%. Indeed, there might be better choices when designing studies. For example, when collecting 64 participants per
condition, we can set the Type 1 and Type 2 error rates in a way that reduced the number of errors we make, such that from the 2000 studies less than 250 studies yield misleading conclusions.
We can use the optimal_alpha function to compute the minimized error rates. The optimal_alpha function takes as input a power function, the relative cost of Type 1 and Type 2 errors (the default is
1, meaning both errors are equally costly), The prior odds of H1 versus H0 (the default is 1, meaning H1 and H0 are believed to be equally likely). We can convert odds to probabilities by calculating
\(odds/(1+odds)\). For example, if the prior odds are 1, the prior probability is 1/(1+1) = 0.5. If the prior odds of H1 compared to H0 are 2, the prior probability of H1 is 2/(2+1) = 0.66.
Analogous, we can convert from probability to odds by dividing the probability by 1 minus the probability. For example, if the prior probability of H1 is 0.66. The prior odds of H1 compared to H0 are
0.66/(1-0.66) = 2. Apart from the prior odds, we also need to specify whether to compute the minimized combined error rate (“minimize”) or balanced error rates (“balanced”), and whether to provide
output for each iteration of the optimization function or not, and a plot or not. An example of the us of the function is provided below:
res1 <- optimal_alpha(power_function = "pwr.t.test(d=0.5, n=64, sig.level = x, type='two.sample', alternative='two.sided')$power",
error = "minimize",
costT1T2 = 1,
priorH1H0 = 1,
verbose = FALSE,
printplot = TRUE)
## [1] 0.09978841
## [1] 0.1215426
## [1] 0.1106655
The output indicates that given the specific settings (e.g., weighing Type 1 and Type 2 errors equally, expecting H1 and H0 to be equally likely to be true) the combined error rate is minimized when
(rounding to 3 decimal places) the alpha level is set to 0.10 (indicated by res$alpha), and the Type 2 error rate is set to 0.122 (indicated by res$beta). The expected error rate is then 0.111. In
other words, if a researcher is interested in effects of d = 0.5, and plans to collect 64 participants in each condition, setting the Type 1 error rate to 10% will increase the power to 87.8%. If we
would perform 2000 studies designed with these error rates, we would observe 0.5 (the prior probability that H0 is true) × 0.100 (the alpha level) × 2000 = 100 Type 1 errors, and 0.5 (the prior
probability that H1 is true) × 0.122 (the Type 2 error rate) × 2000 = 122 Type 2 errors, for a total of 222 errors instead of 250. The combined error rate is therefore 222/2000 = 0.111. Indeed ,this
is the value provided as the errorrate. In other words, by choosing a more optimal alpha level, we can design lines of research more efficiently, because we are less likely to make errors in our
statistical inferences. This did increase the probability of making a Type 1 error (because we increased the alpha level), but this is compensated by reducing the probability of a Type 2 error even
The figure below recreates Figure 2 in Mudge et al. (2012).
# Note that printing plots is suppressed with rmarkdown here and it is simply used to generate the data.
resplot1 <- optimal_alpha(power_function = "pwr::pwr.t.test(d = 1, n = 3, sig.level = x, type = 'two.sample', alternative = 'two.sided')$power", error = "minimize", costT1T2 = 1, printplot = TRUE)
resplot2 <- optimal_alpha(power_function = "pwr::pwr.t.test(d = 1, n = 10, sig.level = x, type = 'two.sample', alternative = 'two.sided')$power", error = "minimize", costT1T2 = 1, printplot = TRUE)
resplot3 <- optimal_alpha(power_function = "pwr::pwr.t.test(d = 1, n = 30, sig.level = x, type = 'two.sample', alternative = 'two.sided')$power", error = "minimize", costT1T2 = 1, printplot = TRUE)
plot_data <- rbind(resplot1$plot_data, resplot2$plot_data, resplot3$plot_data)
plot_data$n <- as.factor(rep(c(3, 10, 30), each = 9999))
w_c_alpha_plot <- ggplot(data=plot_data, aes(x=alpha_list, y=w_c_list)) +
geom_line(size = 1.3, aes(linetype = n)) +
geom_point(aes(x = resplot1$alpha, y = (1 * resplot1$alpha + 1 * (resplot1$beta)) / (1 + 1)), color="red", size = 3) +
geom_point(aes(x = resplot2$alpha, y = (1 * resplot2$alpha + 1 * (resplot2$beta)) / (1 + 1)), color="red", size = 3) +
geom_point(aes(x = resplot3$alpha, y = (1 * resplot3$alpha + 1 * (resplot3$beta)) / (1 + 1)), color="red", size = 3) +
theme_minimal(base_size = 16) +
scale_x_continuous("alpha", seq(0,1,0.1)) +
scale_y_continuous("weighted combined error rate", seq(0,1,0.1), limits = c(0,1))
Balancing Error Rates
You can choose to minimize the combined error rates, but you can also decide that it makes most sense to balance the error rates. For example, you might think a Type 1 error is just as problematic as
a Type 2 error, and therefore, you want to design a study that has balanced error rates for a smallest effect size of interest (e.g., a 5% Type 1 error rate and a 95% Type 2 error rate). The
optimal_alpha function can compute the alpha level that would lead to a balanced Type 2 error rate.
res2 <- optimal_alpha(power_function = "pwr.t.test(d=0.5, n=64, sig.level = x, type='two.sample', alternative='two.sided')$power",
error = "balance",
costT1T2 = 1,
priorH1H0 = 1,
verbose = FALSE,
printplot = TRUE)
## [1] 0.1111217
## [1] 0.1110457
## [1] 0.1110837
This balances the Type 1 and Type 2 error rates (with a maximum difference between the two of 0.0001). The alpha level is 11.11%, and the power is 88.9% (or the Type 2 error rate is 11.1%). Choosing
to balance error rates is only slightly less efficient than minimizing the combined error rate in this example, with a combined error rate when balancing Type 1 and 2 errors of 22.22% compared to a
minimized error rate of 22.13%.
Relative costs of Type 1 and Type 2 errors
So far we have assumed a Type 1 error and Type 2 error are equally costly. This means that we consider the consequences of a false positive just as bad as the consequences of a false negative. This
is not the default in psychology, where researchers typically treat Type 1 errors as 4 times as bad as Type 2 errors. This is based on Cohen (1988), who proposed to aim for 80% power, because we use
an alpha of 5%. The optimal_alpha function users to set the relative cost of Type 1 and Type 2 errors, costT1T2. By default this parameter is set to 1, meaning both types of errors are weighed
equally. Changing the value to 4 means that Type 1 errors are weighed 4 times as much as Type 2 errors, or 4:1. This will change the weight of Type 1 errors compared to Type 2 errors, and thus also
the alpha level at which combined costs of Type 1 and Type 2 errors are minimized.
res3 <- optimal_alpha(power_function = "pwr.t.test(d=0.5, n=64, sig.level = x, type='two.sample', alternative='two.sided')$power",
error = "minimize",
costT1T2 = 4,
priorH1H0 = 1,
verbose = FALSE,
printplot = TRUE)
## [1] 0.03268853
## [1] 0.2524248
## [1] 0.07663579
Now, the alpha level that minimizes the combined Type 1 and Type 2 error rates is 3.27%. With 64 participants in each condition of an independent t-test the Type 2 error is 25.24%, and the expected
combined error rate is 7.66%. If we would perform 2000 studies designed with these error rates, we would observe 0.5 (the prior probability that H0 is true) × 0.033 (the alpha level) × 2000 = 33 Type
1 errors, and 0.5 (the prior probability that H1 is true) × 0.252 (the Type 2 error rate) × 2000 = 252 Type 2 errors. Since we weigh Type 1 errors 4 times as much as Type 2 errors, we multiple the
cost of the 33 Type 1 errors by 4, which makes 4×33 = 132, and to keep the weighted error rate between 0 and 1, we also multiple the 1000 studies where we expect H0 to be true by 4, such that the
weighted combined error rate is (132+252)/(4000 + 1000) = 0.0768.
If we choose to compute balanced error rates, we not surprisingly get, with n = 64 per conditions, which gives us 80% power with an alpha of 5%, exactly these error rates, as this scenario actually
put the cost of a Type 1 error are 4 times the cost of a Type 2 error.
res4 <- optimal_alpha(power_function = "pwr.t.test(d=0.5, n=64, sig.level = x, type='two.sample', alternative='two.sided')$power",
error = "balance",
costT1T2 = 4,
priorH1H0 = 1,
verbose = FALSE,
printplot = TRUE)
## [1] 0.04974484
## [1] 0.1991642
## [1] 0.07962871
If we would perform 2000 studies designed with these error rates, we would observe 0.5 (the prior probability that H0 is true) × 0.05 (the alpha level) × 2000 = 50 Type 1 errors, and 0.5 (the prior
probability that H1 is true) × 0.200 (the Type 2 error rate) × 2000 = 200 Type 2 errors, for a total of 250 errors. However, we weigh Type 1 errors 4 times as much as Type 2 errors (or 4:1). So the
cost is or the 50 Type 1 errors is 4 × 50 = 200 (hence the balanced error rates, as the costs of Type 1 errors is now balanced with the cost for Type 2 errors). Because Type 1 errors are weighed four
times as much as Type 2 errors, 0.8 of the weight is determined by Type 1 errors, and 0.2 of the weight is determined by Type 2 errors, and the weighted combined error rate is (0.8 × 0.05 + 0.2 ×
0.20) = 0.08.
Prior probabilities of H0 and H1
So far, we have assumed that H0 and H1 are equally likely. We can change the prior probabilities that H0 is true (and you will observe a Type 1 error), or that the alternative hypothesis (H1) is true
(and you will observe a Type 2 error). By incorporating these expectations, you can minimize or balance error rates in the long run (assuming your priors are correct). Priors can be specified using
the priorH1H0 argument, which by default is 1 (H1 and H0 are equally likely). Setting it to 4 means you think the alternative hypothesis (and hence, Type 2 errors) are 4 times more likely than that
the null hypothesis is true (and hence, Type 1 errors).
res5 <- optimal_alpha(power_function = "pwr.t.test(d=0.5, n=64, sig.level = x, type='two.sample', alternative='two.sided')$power",
error = "minimize",
costT1T2 = 1,
priorH1H0 = 4,
verbose = FALSE,
printplot = TRUE)
## [1] 0.2461469
## [1] 0.04831345
## [1] 0.08788013
If you think H1 is four times more likely to be true than H0, you need to worry less about Type 1 errors, and now the alpha that minimizes the weighted error rates is 24.6%, with a power of 4.8%. If
we would perform 2000 studies designed with these error rates, and we expect H1 is true 4 times as often as H0, then we expect H0 to be true in 20% (or 400) of the studies, and H1 to be true in 80%
(or 1600) of the studies (as 1:4 = 20:80 = 400:1600). So, we should expect to observe 0.2 (the prior probability that H0 is true) × 0.246 (the alpha level) × 2000 = 98.4 Type 1 errors, and 0.8 (the
prior probability that H1 is true) × 0.0483 (the Type 2 error rate) × 2000 = 77.3 Type 2 errors, for a total of 175.7 errors. Because we expect H1 to be four times as likely as H0, we weigh Type 2
errors for 0.8, and Type 1 errors for 0.2. The weighted combined error rate is (0.2 × 0.246 + 0.8 × 0.0483) = 0.08784.
The decision about priors is always subjective, as is the case in any decision under uncertainty. The more certain you are about auxiliary hypotheses and the stronger the theory, the higher your
prior might be that you are right. Also, when you conduct a replication study, the prior probability of the hypothesis should usually be higher than when attempting to find a novel effect. The more a
prediction goes against things we know or expect to be true, the more likely H0 should be. But when designing any study, we need to consider prior probabilities. You always make a choice - even if
you choose to assume H0 is equally likely as H1.
Sample Size Justification
So far we have only discussed how to justify the alpha level given a fixed sample size. However, in practice researchers usually want to conduct power analysis. This can be incorporated smoothly when
minimizing or balancing error rates. To do so, you simply need to specify the weighted combined error rate you are aiming for and the function optimal_sample will return the sample size as well as
the alpha and beta required to achieve the desired weighted combined error rate.
res6 <- optimal_sample(power_function = "pwr.t.test(d=0.5, n = sample_n, sig.level = x, type='two.sample', alternative='two.sided')$power",
error = "minimize",
errorgoal = 0.05,
costT1T2 = 1,
priorH1H0 = 1)
## $alpha
## [1] 0.046575
## $beta
## [1] 0.05311184
## $errorrate
## [1] 0.04984342
## $objective
## [1] 0.04984342
## $samplesize
## [1] 105
Using the code above, we see that if we aim to achieve a combined error rate of 0.05, we need 105 participants in each of two conditions for an independent t-test.
Specifying power functions
So far we have used only one function from the pwr package to specify the power function. However, any function can be entered. Below we illustrate some additional approaches. The trickiest part is
entering the correct power function. You can provide an analytic power function, either programmed yourself, or from an existing package loading on the server. Then, make sure the alpha value is not
set, but specified as x, and that the function itself returns a single value, the power of the test. Finally, if you use existing power functions the shiny app needs to know which package this
function is from, and thus the call to the function needs to be precended by the package and ‘::’, so ‘pwr::’ or ‘TOSTER::’. Some examples that work are provided below.
res <- optimal_alpha(power_function = "TOSTER::powerTOSTtwo(alpha=x, N=200, low_eqbound_d=-0.4, high_eqbound_d=0.4) ",
error = "minimize",
costT1T2 = 1,
priorH1H0 = 1)
For a more challenging power function, we can use the Superpower package by Daniel Lakens and Aaron Caldwell. The power function in the ANOVAexact function is based on a simulation, which takes a
while to perform. The optimization function used in this Shiny app needs to perform the power calculation multiple times. Thus, the result takes a minutes to calculate. Press calculate, and check the
results 5 to 10 minutes later. Furthermore, the output of the ANOVA_exact function prints power as 80%, not 0.8, and thus we actually have to divide the power value by 100 for the Shiny app to return
the correct results. Nevertheless, it works if you are very patient (this code is not run to prevent .
res <- optimal_alpha(power_function = "Superpower::ANOVA_exact( (Superpower::ANOVA_design(design = '2b', n = 64, mu = c(0, 0.5), sd = 1, plot = FALSE)), alpha_level = x, verbose = FALSE)$main_results$power/100",
error = "minimize",
costT1T2 = 1,
priorH1H0 = 1)
Avoiding the Lindley Paradox
Sometimes we don’t know the prior odds or the effect size of interest. In this case we can justify the alpha level by aiming to avoid the Lindley paradox. The Lindley paradox arises from the
difference between error rate control and likelihood ratios in statistics. When the power is high, it is possible that a significant p-value is actually more likely to occur under the alternative
than under the null hypothesis. This situation, where we reject the null hypothesis because p < alpha, but the evidence based on the likelihood suggests the observed data is more likely under the
null hypothesis than the alternative hypothesis, is considered the Lindley paradox. The figure below shows that the steeper the p-value distribution (which occurs as the sample size increases) the
more to the left the point where the expected p-value distribution under the alternative will cross the uniform p-value distribution under H0. The solution to prevent Lindley’s paradox is to lower
the alpha level as a function of the sample size.
A Bayes factor and a p-value are directly related, given a prior, and a sample size. They can both be computed directly from the t-value. In the plot below we see the Bayes factor (plotted on the
vertical axis) and the p-value (plotted on the horizontal axis) for an independent t-test with 100 participants in each condition.
n1 <- 100
n2 <- 100
loops <- seq(from = 0, to = 3, by = 0.001)
p <- numeric(length(loops))
bf <- numeric(length(loops))
d <- numeric(length(loops))
tval <- numeric(length(loops))
i <- 0
for(t in loops){
i <- i+1
bf[i] <- exp(BayesFactor::ttest.tstat(t, n1, n2)$bf)
p[i] <- 2*pt(t, ((n1 + n2) - 2), lower=FALSE)
tval[i] <- t
d[i] <- t * sqrt((1/n1)+(1/n2))
plot(p, bf, type="l", lty=1, lwd=2, log = "y")
abline(v = seq(0,1,0.1), h = c(0, 1/10, 1/3, 1, 3, 10), col = "gray", lty = 1)
We can easily see which Bayes factor threshold corresponds to a specific p-value. For a Bayes factor of 1, we get a p-value of:
## [1] 0.0462172
So, as long as the observed p-value is smaller than 0.046 the data will provide stronger support for H1 than for H0.
JustifyAlpha allows users to directly compute the alpha level that enables them to avoid the Lindley paradox through the functions ftestEvidence and ttestEvidence, which calculate the alpha level
that is needed so that a significance p-value is always more likely under the alternative hypothesis than under the null hypothesis, based on a specification of a prior for the alternative model. For
a two sample t-test with 100 participants per group, we can do this using the following function.
## [1] 0.04625576
This shows that an alpha level of at least 0.046 would be required to avoid the Lindley paradox. Of course, researchers can also use an informed prior, or use the the likelihood of the data under the
effect size of the alternative hypothesis. There are many approaches to prevent Lindley’s paradox. All boil down to lowering the alpha level as a function of the sample size, but they differ slightly
the relation between the sample size and the alpha level. | {"url":"https://cran.fhcrc.org/web/packages/JustifyAlpha/vignettes/Introduction_to_JustifyAlpha.html","timestamp":"2024-11-08T02:23:26Z","content_type":"text/html","content_length":"123125","record_id":"<urn:uuid:383fd775-c98b-4e29-a651-06f5ad4761cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00411.warc.gz"} |
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If you’ve found your way to these pages, you are interested in one of the courses MTH102: Elementary Functions, MTH111: Calculus 1, and MTH112: Calculus 2. On this site, you’ll find a variety of
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Volume of a Box
Volume of a box
Given the length, the height, and the width, the volume of a box also called rectangular prism can be found by using the following formula in the figure below.
It is not always straightforward to label the height, the width, and the length. It is just a matter of perspective!
Looking at the box below, what I labeled as length could also be called width and vice versa.
And it you rotate the box by 90 degrees, what looks like the length right now will look like the height.
Finally, the volume is expressed in cubit unit.
Therefore, if the unit you are using is meter, the volume is expressed in cubic meter or meter
If the unit you are using is ft, the volume is expressed ft
Enough talking! Let us compute some volume
Some examples showing how to find the volume of a box or rectangular prism.
Example #1:
Find the volume with a length of 5 inches, a height of 2 inches, and a width of 4 inches
= l × w × h
= 5 inches × 4 inches × 2 inches
= 20 inches
× 2 inches
= 40 inches
^3 Example #2:
An LCD tv was put in a box with a length of 2 feet, a height of 3 feet, and a width of 0.5 foot. What is the volume of the box?
= l × w × h
= 2 × 0.5 × 3
= 1 ft
× 3 ft
= 3 ft
^3 Example #3:
A swimming pool is shaped like a big box with a length of 10 feet, a height of 8 feet, and a width of 20 foot. What is the volume of the swimming pool?
= l × w × h
= 10 × 20 × 8
= 200 ft
× 8 ft
= 1600 ft
^3 Example #4:
A packaging box has a length of 9 inches, a height of 5 inches, and a width of 8 inches. What is the volume of the packaging box?
= l × w × h
= 9 × 8 × 5
= 72 in
× 5 in
= 360 in
Take the quiz below to see how well you can find the volume of a box or rectangular prism.
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Measuring Inequality at the Sub-National Level in Bolivia - SDSN Bolivia
Measuring Inequality at the Sub-National Level in Bolivia
March 20th, 2020
The scale of inequality in the world is staggering. In 2018, the average inhabitant of Denmark, Norway, Sweden, Iceland, and Ireland earned more in two days than what the average inhabitant of Malawi
and Burundi earned during an entire year [1].
However, the inequality within some countries is even bigger than the inequality between countries. As we show in the upcoming Municipal Atlas of the SDGs in Bolivia, there are larger differences in
the Sustainable Development Index between municipalities within Bolivia than there are between all the countries in the world. This means that within Bolivia we find municipalities with levels of
development similar to the most advanced countries in the World, but also municipalities similar to some of the least developed countries in the World.
In addition to this astounding inequality between municipalities within Bolivia, there are also huge inequalities within each municipality, and that is what we focus on in this blog.
Measuring inequality at the sub-national level is difficult, because it requires income/consumption data that is representative at the municipal level, and this is neither available from household
surveys nor population censuses. We get around this problem by using data on electricity consumption by each household in Bolivia, under the assumption that electricity consumption in each household
is a reliable predictor of general consumption/income/welfare in the household. The analysis was made possible due to a research project sponsored by the Centre for Social Research (CIS) of the
Bolivian Vice-presidency [2].
The CIS project calculated Gini coefficients of electricity consumption to estimate the level of inequality within each municipality. Figure 1 shows two examples. To the left is the Lorenz curve and
the Gini coefficient for a big urban municipality, Santa Cruz de la Sierra. To the right is the Lorenz curve and Gini coefficient for a poor rural municipality, Tinguipaya, with low electricity
coverage and low electricity consumption for the ones that actually do have access to the electricity grid.
Figure 1: Examples of electricity consumption Lorenz curves, 2016
Source: Andersen, Branisa y Guzman (2019) [2].
According to these Lorenz curves, the big, urban municipality of Santa Cruz de la Sierra is clearly more equal in electricity consumption (and thus likely consumption in general) than the small,
rural municipality of Tinguipaya, as the Lorenz curve of the former is much closer to the diagonal line of perfect equality.
Indeed, as can be seen in Figure 2, this is the general tendency across all municipalities in Bolivia. Urban municipalities tend to have lower levels of poverty (poverty being measured as extremely
low electricity consumption) and lower levels of inequality (as measured by the Gini coefficient of electricity consumption).
Figure 2: Energy inequalty versus energy poverty in Bolivian municipalities, 2016
Authors’ elaboration based on data from Andersen, Branisa y Guzman (2019).
This strong positive correlation between energy poverty and energy inequality seems suspicious, and perhaps even counter-intuitive. Consider this extreme hypothetical example: If all households in a
municipality had zero electricity consumption, except the mayor, who had a moderate electricity consumption covering a few light bulbs and a refrigerator, then the Gini coefficient would be 0.99,
suggesting extremely high inequality in the municipality, whereas common sense would suggest that the population is very equal in their extremely high level of poverty.
The Gini coefficient is by far the most widely used measure of inequality, and easy to interpret, so it was the logical metric to use. But these results made us wonder if the Gini coefficient tends
to confound extreme poverty with extreme inequality.
Thus, in this blog we set out to explore whether there are alternative measures of inequality that might correspond better to intuition. There are surprisingly many different measures of inequality,
so it became another long post.
There are four basic principles that one would expect from an inequality measure [3]:
• Symmetry (or anonymity): If two people switch incomes, the index level should not change.
• Population invariance (or replication invariance): If the population is replicated or “cloned” one or more times, the index level should not change.
• Scale invariance (or mean independence): If all incomes are scaled up or down by a common factor (for example, doubled), the index level should not change.
• The Pigou-Dalton Transfer Principle: If income is transferred from one person to another who is richer, the index level should increase. In other words, in the face of a regressive transfer, the
index level must rise.
It can be shown that the Gini coefficient satisfies these four basic principles, as does several other inequality measures, such as the Atkinson Index, the Theil entropy measure, and the Theil mean
log deviation measure. These inequality measures are all called strongly Lorenz-consistent [4].
Some frequently used inequality measures are only weakly Lorenz-consistent, however, as they do not fully comply with the fourth principle. When a poorer person makes a transfer to a richer person,
the inequality indicator need not rise, but at least it should not fall [4]. The Palma ratio (the income of the 10% richest divided by the income of the 40% poorest) and other Kuznets ratios (X%
richest/Y% poorest) are only weakly Lorenz-consistent.
Other inequality measures are plainly Lorenz-inconsistent. This is the case for quantile ratios (p90/p10, p75/p25, etc.) and the Variance of Logarithms. For both of these measures it is possible that
a regressive transfer from a poorer person to a richer person would cause a fall in the inequality measure, which is clearly counter intuitive [4]. The Absolute Gini proposed by Jason Hickel [5] is
even worse, as it also violates the third principle of scale invariance.
From the theoretical arguments mentioned above, we would expect all the Lorenz-consistent measures to yield pretty much the same results as the Gini coefficient, so we decided to also include weakly
Lorenz-consistent and Lorenz-inconsistent measures in our comparisons.
However, we quickly ran into an important problem: Many inequality measures cannot handle zero values (e.g. Theil, Atkinson, percentile ratios). Some algorithms circumvent this problem simply by
ignoring zero values. But we do not consider this a reasonable strategy, since the lack of access to electricity is one of the fundamental problems we want to highlight rather than ignore. An
alternative way around this problem is simply to add a small value, for example 1 kWh per year, to all observations, which does not change overall patterns, but will make all computational algorithms
work [6].
During the following sections, we will compare many different inequality measures with the Gini coefficient. All inequality measures are calculated on household level data on annual electricity
consumption + 1 kWh for a selection of 25 municipalities that span the whole range shown in Figure 2. For each of the alternative inequality measures we discuss the intuition behind the measure, its
advantages and disadvantages, and show its relationship to the Gini coefficient.
Atkinson Indices
The Atkinson Index, A(e), is actually a whole class of inequality measures, differentiated by a parameter, e, that measures the degree of inequality aversion. When e = 0, there is no aversion to
inequality, and A(0) = 0. When e = ∞, there is infinite aversion to inequality and A(∞) = 1. The Atkinson Index thus varies between 0 and 1, like the Gini coefficient, which facilitates
interpretation. The Atkinson Index has the advantage of being sub-group decomposable, which the Gini coefficient is not.
The Stata package ineqdeco (created by Stephen P. Jenkins at London School of Economics) calculates the Atkinson Index for three different parameters: 0.5, 1 and 2. Figure 3 shows how the Gini
coefficient (of annual electricity consumption + 1 kWh) compares to these three Atkinson Indices.
Figure 3: Comparing the Gini coefficient with three Atkinson Indices for a sample of 25 municipalities
Authors’ elaboration.
We see that for moderate inequality aversion (e = 0.5), the Atkinson Index behaves very similarly to the Gini coefficient. For stronger inequality aversion (e = 1), the Atkinson index increases at
all levels, but especially for the ones that had high Gini coefficients, thus exaggerating the counter-intuitive results of the Gini coefficient. For very high inequality aversion (e = 2), the
Atkinson Index is very close to maximum for all our municipalities, and thus do not provide any useful information about differences in inequality.
In conclusion, the A(0.5) index seems the most useful of the three, but it behaves very much like the Gini coefficient and thus does not solve our initial problem of counter-intuitive results.
Generalized Entropy Indices (including mean log deviation and Theil index)
The Generalized Entropy Index, GE(α), is another class of inequality measures. As with the Atkinson Indices, the GE Indices involve a parameter, α, that can shift the sensitivity to different parts
of the distribution. For lower values of α, GE is more sensitive to differences in the lower tail of the distribution, and for higher values GE is more sensitive to differences that affect the upper
tail. The most common values of α used are 0, 1 and 2 [7].
The GE Index has several other inequality metrics as special cases. For example, GE(0) is the mean log deviation (also sometimes called Theil’s L), GE(1) is the Theil index, or Theil’s T, and GE(2)
is half the squared Coefficient of Variation.
The values of GE indices can vary between 0 and ∞, with zero representing an equal distribution and higher value representing a higher level of inequality.
Like the Atkinson Index, the GE Index is decomposable, which is an advantage over the Gini coefficient. However, it is not bounded above, and the interpretation is not at all intuitive.
The Stata package ineqdeco calculates several different GE Indices. Figure 4 shows how the Gini coefficient (of annual electricity consumption + 1 kWh) compares to the three most common ones.
Figure 4: Comparing the Gini coefficient with the three most common
Generalized Entropy Indices for a sample of 25 municipalities
Source: Authors’ elaboration.
In all three cases the GE indices agree with the Gini coefficient that the poorest municipalities with the highest Gini coefficients have the highest levels of inequality. The GE(0) index moderates
the relationship slightly, whereas the GE(1) exaggerates it, and the GE(2) exaggerates it even more. Thus, none of the three most commonly used GE indices suggests that poorer municipalities would be
more equal.
However, α can also take on negative values, making it more sensitive to differences in the lower end of the distribution. Figure 5 shows that the GE(-1) index is completely different from the Gini
coefficient. One municipality that stands out with extremely high inequality (GE(-1) = 250) is Santa Cruz de la Sierra (Bolivia’s most populous municipality, home to both extremely rich and extremely
poor households), whereas the poorer municipalities in our sample have the lowest levels of inequality according to this measure (still high, though, as the scale is completely different for this GE
Figure 5: Comparing the Gini coefficient with the GE(-1) index for a sample of 25 municipalities
Source: Authors’ elaboration.
This rarely used inequality measure, GE(-1), potentially fits better with common intuition about which municipalities are more unequal. We will explore it in more detail further below.
Income shares, Kuznets ratios and Palma ratio
Thomas Piketty’s best-selling book “Capital in the Twenty-First Century” made the use of percentile shares popular for analyzing inequality. Piketty and collaborators focused on top-percentage
shares, using varying percentages as thresholds (top 10%, top 1%, top 0.1%, etc.).
A related concept is the Kuznets ratio, which compares the income of the top X% with the bottom Y% of the population. A special case of this is the Palma ratio (the income of the 10% richest divided
by the income of the 40% poorest) (8). The top-percentage shares are a special case of Kuznets ratio, as it is the top X% divided by the bottom 100%. So all these measures can be called Kuznets
Kuznets ratios are weakly Lorenz-consistent, since progressive or regressive transfers within each group analyzed will not affect the various indicators.
Ben Jann of the University of Bern created a convenient Stata command, pshare, to calculate the share of income received by any group along the income distribution [9]. The default is the income
shares of the five quintiles (20% poorest to 20% richest). However, due to the high level of inequality in electricity consumption indicated by the Gini coefficient, we find it important to
disaggregate the top quintile, and see the share of electricity consumed by the top 10%, top 5% and top 1% of households in each municipality.
In addition, in order to calculate the Palma ratio, we use the 40% and 90% cut-offs.
Figure 6 shows how the Gini coefficient (of annual electricity consumption + 1 kWh) compares to the following three Kuznets ratios: Top 5% of electricity consumption share; top 1% share; and the
Palma ratio.
Figure 6: Comparing the Gini coefficient with three Kuznets ratios
Source: Authors’ elaboration.
The first two measures, which focus on the top end of the distribution, confirms -and even exaggerates- the finding that the poorest municipalities are the most unequal.
However, the Palma ratio shows a completely different pattern. The extreme outlier in our small sample is Uyuni, with a Palma ratio of 777, followed by Ixiamas, Machacamarca and Patacamaya. In
contrast, all the mayor cities of Bolivia have very low levels of inequality by this measure. There is nothing about these results that seems even remotely related to intuition.
In the range of Gini coefficients between 0.4 and 0.63, there seem to be a positive relationship with the Palma ratio, but for higher Gini’s the relationship becomes completely random and unrelated
to common sense.
Other inequality measures
There are several standard measures of dispersion reported by statistical packages, but which do not have fancy names nor particularly intuitive interpretations, and thus are not widely used in the
inequality literature.
The first three indicators reported by the Stata command inequal (developed by Edward Whitehouse of OECD in Paris), are the Relative Mean Deviation (RMD), the Coefficient of Variation (CV) and the
Standard Deviation of Logs (SDL). Figure 7 plots these inequality measures against the Gini for our sample of 25 municipalities.
Figure 7: Comparing RMD, CV, and SDL to the Gini coefficient
Source: Authors’ elaboration.
The first of these measures, RMD, are closely related to the Gini coefficient; the second one, CV, exaggerates the relationship between inequality and poverty, but the third paints a different
picture that might possibly correspond better to intuition. According to the bland indicator called “Standard Deviation of logs”, Santa Cruz de la Sierra has high inequality, while Poroma (one of the
poorest municipalities in Bolivia) has low inequality, which seem to correspond to intuition.
The same inequal Stata command reports the Mehran, Piesch and Kakwani measures of inequality. They are not well known, nor widely used, and since they all correlate strongly with the Gini coefficient
(see Figure 8), we don´t think they provide much additional insights into inequality within Bolivian municipalities.
Figure 8: Comparing Mehran, Piesch and Kakwani to the Gini coefficient
Authors’ elaboration.
In this blog we have investigated the properties of 16 alternative measures to the Gini coefficient of inequality. Most of them agree with the Gini coefficient that poor, rural municipalities are
more unequal than rich, urban municipalities.
However, we found two inequality measure that provide different, yet plausible patterns of inequality. Those are the Standard Deviation of logs and the Generalized Entropy Index with a parameter of
-1 (strong emphasis on the lower end of the distribution).
Since the results seemed intuitive for our sample of 25 municipalities, we decided to crunch the numbers for all 339 municipalities. Figure 9 shows how Figure 2 would look if we use GE(-1) instead of
the Gini coefficient. And Figure 10 shows the same, but using the Standard Deviation of logs measure instead.
Figure 9: Comparing poverty and inequality using GE(-1) for all municipalities in Bolivia
Source: Authors’ elaboration.
Figure 10: Comparing poverty and inequality using Standard Deviation of logs for all municipalities in Bolivia
Source: Authors’ elaboration.
It is interesting to note that the two measures GE(-1) and Standard Deviation of logs agree on many of the most unequal municipalities (e.g. Rurrenabaque, Reyes, Cobija, Santa Cruz de la Sierra,
Puerto Suarez), while they completely disagree with the Gini coefficient about the high level of inequality in small, poor, completely rural municipalities. Both are in reasonable agreement
concerning the relative rankings of the large urban municipalities, with all agreeing that Cochabamba is the most equal of the 10 capital cities (+ El Alto) and Cobija the most unequal. The
correlation between the two is 0.83, which is high, but not so high as to be redundant.
Of the 17 different ways to measure inequality that we have tried out in this blog, GE(-1) and Standard Deviation of logs seem to correspond best to intuition about inequality within Bolivian
municipalities. The Gini coefficient, and all the other measures that correlate strongly with the Gini coefficient, do not seem to correspond to an intuitive perception of inequality, at least not
for very poor municipalities.
It is clear that the optimal choice of inequality measures depends greatly on the distribution of the data, and that the Gini coefficient is not automatically the best choice.
After having processed millions of data points for 300+ municipalities in 17 different ways, we conclude that the best indicators to include in our Municipal Atlas of the SDGs in Bolivia under SDG 10
are: 1) Inequality Index 1: Standard Deviation of log household electricity consumption, and 2) Inequality Index 2: Generalized Entropy (α = -1) of household electricity consumption.
We do recognize, however, that we have engaged in what could be called “method-mining” in order to obtain results that correspond better to our intuition and expectations. Thus, in order not to give
too much weight to these electricity-consumption-based inequality measures, we recommend to include other measures of inequality under SDG 10. Next week we will explore within-municipality inequality
measures based on education levels.
[1] World Development Indicators, Gross National Income per capita, Atlas method, 2018.
[2] Andersen, L. E., B. Branisa & F. Calderón (2019) “Estimaciones del PIB per cápita y de la actividad económica a nivel municipal en Bolivia en base a datos de consumo de electricidad.”
Investigación ganadora presentada al Centro de Investigaciones Sociales (CIS) de la Vicepresidencia del Estado Plurinacional de Bolivia. Mayo. [3] In the latest
Human Development Report 2019
, which analyzes inequality, James Foster and Nora Lustig provide a useful overview to help decide between alternative inequality measures. See Spotlight 3.2, pp. 136-138. [4] See Francisco
Ferreira’s blog “In defense of the Gini coefficient”:
. [5] See
[6] We also considered the possibility of adding more than 1 kWh per year to each observation, because all households receive a transfers from the sun amounting to at least a couple of lightbulbs for
12 hours per day, and perhaps a whole lot more. In cold regions, the sun also provides free heat, but in hot regions, that may not be perceived as a benefit, but rather as a disservice. Thus, it is
virtually impossible to take into account these varying contributions from the sun, and any values we might chose would be arbitrary. Researchers face the same dilemma when calculating income
inequality, or wealth inequality, as there are so many public services or environmental services that contribute to the wellbeing of families, but are extremely difficult to quantify and include in
each family’s income/wealth. It is major research problem that we do not pretend to be able to solve here, so in this blog we will just add 1 kWh to each observation in order to overcome the
technical problem of zeros and be able to actually compare all the different inequality statistics.
[7] See
. [8] See
. [9] See
* SDSN Bolivia.
The viewpoints expressed in the blog are the responsibility of the authors and do not necessarily reflect the position of their institutions. These posts are part of the project “Atlas of the SDGs in
Bolivia at the municipal level” that is currently carried out by the Sustainable Development Solutions Network (SDSN) in Bolivia. | {"url":"https://sdsnbolivia.org/en/measuring-inequality-at-the-sub-national-level-in-bolivia/","timestamp":"2024-11-11T20:49:30Z","content_type":"text/html","content_length":"105987","record_id":"<urn:uuid:cb6ea2e2-3628-466d-8ca7-96cdf12f9286>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00867.warc.gz"} |
1st PUC Maths Question Bank with Answers Karnataka
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Content area to select questions for PART D and PART E
(a) In PART D
1. Relations and functions: Problems on drawing graph of a function and writing its domain and range.
2. Trigonometric functions: Problems on Transformation formulae.
3. Principle of Mathematical Induction: Problems.
4. Permutation and Combination: Problems on combinations only.
5. Binomial theorem: Derivation/problems on Binomial theorem.
6. Straight lines: Derivations.
7. Introduction to 3D geometry: Derivations.
8. Limits and Derivatives: Derivation / problems.
9. Statistics: Problems on finding mean deviation about mean or median.
10. Linear inequalities: Problems on solution of system of linear inequalities in two variables.
(b) In PARTE
6 mark questions must be taken from the following content areas only.
1. Derivations on trigonometric functions.
2. Definitions and derivations on conic sections.
4 mark questions must be taken from the following content areas only.
1. Problems on algebra of derivatives.
2. Problems on summation of finite series.
Unit-I: Sets and Functions
Chapter 1 Sets:
Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set of real numbers especially intervals (with notations). Power set. Universal set. Venn
diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement Sets. Practical Problems based on sets. (8 Hours)
Chapter 2 Relations & Functions:
Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the sets of real (upto R x R). Definition of relation, pictorial
diagrams, domain, co-domain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain and range of a function.
Real valued functions, domain and range of these functions: constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum,
difference, product and quotients of functions. (10 Hours)
Chapter 3 Trigonometric Functions
Positive and negative angles. Measuring angles in radians and in degrees and conversion of one into other. Definition of trigonometric functions with the help of unit circle. Truth of the sin2x+cos2x
=1, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sin x, sin y, cos x & cos y and their
simple application. Deducing identities like the following:
Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x. General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a and problems. Proofs
and simple applications of sine and cosine rule. (18 Hours)
Unit-II: Algebra
Chapter 4 Principle of Mathematical Induction:
Process of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and
simple applications. (4 Hours)
Chapter 5 Complex Numbers and Quadratic Equations:
Need for complex numbers, especially √1, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers. Argand plane and polar representation of
complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system. Square root of a complex number. (8 Hours)
Chapter 6 Linear Inequalities:
Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Graphical
solution of system of linear inequalities in two variables. (8 Hours)
Chapter 7 Permutations and Combinations:
Fundamental principle of counting. Factorial n. Permutations and combinations, derivation of formulae and their connections, simple applications. (9 Hours)
Chapter 8 Binomial Theorem:
History, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle, General and middle term in binomial expansion, simple applications. (7 Hours)
Chapter 9 Sequence and Series:
Sequence and Series. Arithmetic Progression (A.P.). Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of n terms of a G.P., Arithmetic and Geometric series infinite
G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Formula for the following special sum: (9 Hours)
Unit-III: Coordinate Geometry
Chapter 10 Straight Lines:
Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope
form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Equation of family of lines passing through the point of intersection of two lines. Distance of
a point from a line. (10 Hours)
Chapter 11 Conic Sections:
Sections of a cone: circles, ellipse, parabola, hyperbola; a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties
of parabola, ellipse and hyperbola. Standard equation of a circle. (8 Hours)
Chapter 12 Introduction to Three–dimensional Geometry:
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula. (5 Hours)
Unit-IV: Calculus
Chapter 13 Limits and Derivatives:
Derivative introduced as rate of change both as that of distance function and geometrically.
Intutive idea of limit. Limits of polynomials and rational functions, trignometric, exponential and logarithmic functions. Definition of derivative, relate it to slope of tangent of a curve,
derivative of sum, difference, product and quotient of functions. The derivative of polynomial and trignometric functions. (14 Hours)
Unit-V: Mathematical Reasoning
Chapter 14 Mathematical Reasoning:
Mathematically acceptable statements. Connecting words/ phrases – consolidating the understanding of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”,
“or”, “there exists” and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words difference between contradiction,
converse and contrapositive. (6 Hours)
Unit-VI: Statistics and Probability
Chapter 15 Statistics:
Measures of dispersion; Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances. (7 Hours)
Chapter 16 Probability:
Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic)
probability, connections with the theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events. (8 Hours)
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Magnetic Circuit Calculators | List of Magnetic Circuit Calculators
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Magnetic Circuit calculators give you a list of online Magnetic Circuit calculators. A tool perform calculations on the concepts and applications for Magnetic Circuit calculations.
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Synchronous characteristics of a vibration piling system with electromechanical coupling
From the point of view of frequency capture, the nonlinear dynamic models of the self-synchronous vibrating pile system are presented for the analysis of the nonlinear stiffness of the soil, which is
induced by the relationship between the nonlinear stress and the strain in the soil. And the nonlinear dynamic models of the self-synchronous vibrating pile system with electromechanical coupling
also are presented for the analysis of the pile- soil- electric coupling. The nonlinear characteristics of the vibrating pile in the self-synchronous vibrating pile system with frequency capture are
analyzed, and the periodic solutions for the self-synchronous system with frequency capture are investigated using the nonlinear models. The synchronization condition for the self-synchronous
vibrating pile system with frequency capture is theoretical analyzed using the rotor-rotation equations of the two-excited motors, and the synchronization stability condition also is theoretical
analyzed using Jacobi matrix of the phase difference equation of the two-excited motors. Using Matlab/Simlink, the reverse rotation synchronization of the two-excited motors and the stability of
synchronization of the self-synchronous vibrating pile system with electromechanical coupling are analyzed through the selected parameters. The nonlinear phenomena in the self-synchronous vibrating
pile system with electromechanical coupling, such as frequency capture and the limit cycles, are reproduced. Various synchronous phenomena are obtained through the difference rates of the two-excited
motors, which are induced by the relationship between the phases of the two-excited motors and the rotation speeds of the two-excited motors. It has been shown that the research results can provide
theoretical basis for the design and research of the self-synchronous vibratory system.
1. Introduction
Frequency capture has been an important factor in the vibrating system, and frequency capture is usually explained that the vibration frequency is captured by the first natural frequency when the
excitation frequency is close to the definite range of the first natural frequency in the vibrating system [1, 2]. The large amplitude of the vibrating system is very favorable in many engineering
field, especially in the self-synchronous vibrating pile system, and the large amplitude can be obtained in the self-synchronous vibrating pile system with frequency capture phenomenon. So frequency
capture obtained by the self-synchronous vibrating pile system is very important for performances of the sinking pile. In addition, the exciting force of the vibrating pile system should be vertical
sinking into the soil, and a better effect of the sinking pile can be obtained in the vibrating pile system. When the eccentric-blocks on the two excited-motors are the reverse and synchronous
rotation in the self-synchronous vibrating pile system, the centrifugal force generated by the rotating eccentric-block can be cancelled by each other on the horizontal direction. And the friction
resistance of the pile side can be reduced and the resistance can also be overcome, so the exciting force in the vertical direction is obtained in the vibrating pile system. This vibrating pile
system is called as the self-synchronous vibrating pile system, which is including two excited-motor that the eccentric blocks on two excited-motors are the reverse and synchronous rotation. The
synchronous rotation can be explained that the phase difference of the two eccentric-blocks is 0 or constant. So investigation on the phase difference of the two eccentric-blocks has become one of
the key issues in the self-synchronous vibrating pile system with frequency capture.
Many models of the self-synchronous vibrating pile system, such as the linear model with the linear stiffness of the soil and the simplified ideal model, have been investigated and can be found in
many references [3-5]. For the previous research, it had been assumed that the interaction between the pile and the soil was coupled together by linear springs and dampers, and the acting force of
the soil in the self-synchronous vibrating pile system was expressed as a linear function [6, 7]. It is no doubt that linear model is a very good model for the analysis of the vibrating pile system,
but it is of very limited to describe the acting force of the soil in the vibrating pile system. Firstly, the interaction between the piles and the soil is very complex, and the stress in the soil
has a non-linear relation with the strain in the soil, therefore the linear model is not perfect enough. Secondly, the vibrating pile in the soil is self-synchronous vibration, and with the nonlinear
nature. If the linear model was used to describe the sinking pile, the self-synchronous vibrating pile system would lose the most essential dynamic phenomena. This linear model cannot describe the
nonlinear characteristics, such as the jump phenomenon, frequency synchronization and so on. In addition, the system with the phase synchronization (or speed synchronization) is named as a
self-synchronous vibration system, so the investigations on the relationship between the phase difference and the amplitude are essential in the self-synchronous vibrating pile system with frequency
capture [8-11], but these investigations can be rarely found in many references.
In this paper, from the point of view of frequency capture, the nonlinear dynamic models of the self-synchronous vibrating pile system with electromechanical coupling also are presented for the
analysis of the pile- soil- electric coupling. The synchronization condition and the synchronization stability condition for the self-synchronous vibrating pile system with frequency capture is
theoretical analyzed using the rotor-rotation equations of the two-excited motors. Using Matlab/Simlink, the reverse rotation synchronization of the two-excited motors and the stability of
synchronization of the self-synchronous vibrating pile system with electromechanical coupling are analyzed through the selected parameters.
2. Mathematical model
The interaction between the piles and the soil has been a nonlinear issue in the self-synchronous vibrating pile system with frequency capture, and which cannot be analyzed using the traditional
linear model. From the point of view of the nonlinear relation between the stress and the strain in the soil, the nonlinear characteristics of the interaction between the pile and the soil are
expressed using the flexible nonlinear springs. The elastic force of the flexible nonlinear spring can be defined as $k\left(y\right)=ky-\epsilon {k}^{"}{y}^{3}$, where, $k$ is the linear elastic
stiffness of the soil, $y$ is the displacement of the piles, $ky$ is the linear elastic force, $\epsilon$ is the nonlinear coefficient of the soil (which is a small integer), $\epsilon {k}^{"}{y}^{3}
$ is the nonlinear elastic force, and it is usually less than $ky$. The vibrating force in the self-synchronous vibrating pile system will be generated by the eccentric blocks with the reverse
rotation on the double excited-motors, and this vibrating force is in the vertical direction. The dynamic model of the self-synchronous vibrating pile system is shown in Fig. 1, as shown in Fig. 1,
$oxy$ is the coordinate system of the nonlinear vibration system, $O$ is the center of the system (it is also the midpoint of the line of the rotating shaft for the two exciter), ${O}_{1}$, ${O}_{2}$
is the center of the rotating shaft for the two exciter. Using Lagrange equation, the differential equations of the self-synchronous vibrating pile system under the action of the nonlinear elastic
force of the soil are defined as:
$m\mathrm{}\stackrel{¨}{y}+c\stackrel{˙}{y}+ky-\mathrm{\epsilon }{k}^{\mathrm{"}{y}^{3}}={m}_{1}{r}_{1}\left(-{\stackrel{¨}{\phi }}_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}+{\stackrel{˙}{\phi }}_
{1}^{2}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\right)-{m}_{2}{r}_{2}\left({\stackrel{¨}{\phi }}_{2}\mathrm{c}\mathrm{o}\mathrm{s}{\mathrm{\phi }}_{2}-{\stackrel{˙}{\phi }}_{2}^{2}\mathrm{s}\mathrm
{i}\mathrm{n}{\mathrm{\phi }}_{2}\right),$${{J}_{0}}_{1}{\stackrel{¨}{\phi }}_{1}={T}_{m1}\left({\stackrel{˙}{\phi }}_{1}\right)-{T}_{f1}\left({\stackrel{˙}{\phi }}_{1}\right)-{c}_{1}{\stackrel{˙}{\
phi }}_{1}-{m}_{1}{r}_{1}\stackrel{¨}{y}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}-{m}_{1}{r}_{1}g\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1},$${{J}_{0}}_{2}{\stackrel{¨}{\phi }}_{2}={T}_{m2}\left({\
stackrel{˙}{\phi }}_{2}\right)-{T}_{f2}\left({\stackrel{˙}{\phi }}_{2}\right)-{c}_{2}{\stackrel{˙}{\phi }}_{2}-{m}_{2}{r}_{2}\stackrel{¨}{y}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}-{m}_{2}{r}_{2}g\
mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}.$
Fig. 1The dynamic model of the vibratory pile system
In this equation, $y$, $\stackrel{˙}{y}$ and $\stackrel{¨}{y}$ are the vibration displacement, the velocity and the acceleration of the pile of the vibrating pile system in the vertical direction,
respectively, and $m$ is the total mass of the vibrating pile system and the total mass $m$ is composed of two components, the mass of the vibratory pile hammer $M$ and the mass of two eccentric
blocks on two excited-motors ${m}_{1}$ and ${m}_{2}$, so $m$ can be written as $m=M+{m}_{1}+{m}_{2}$, ${r}_{i}$ ($i=$ 1, 2) is the radiuses of the eccentric block around ${O}_{i}$ ($i=$ 1, 2), ${\phi
}_{i}$ ($i=$ 1, 2) is the angular phase of the eccentric block $i$, ${\stackrel{˙}{\phi }}_{i}$ ($i=$ 1, 2) is the angular velocity of the eccentric block $i$, respectively, ${\stackrel{¨}{\phi }}_
{i}$ ($i=$ 1, 2) are the angular acceleration of the eccentric block $i$, $\mathrm{\Delta }\alpha$ is the phase difference angle of the eccentric blocks on the two excited motors, $c$ is the damping
of the soil on the vibrating pile hammer, ${c}_{i}$ ($i$ = 1, 2) is the rotating damping of the excited motor $i$, ${J}_{0i}$ ($i=$ 1, 2) is the moment of the inertia of eccentric block $i$, ${T}_
{mi}\left({\stackrel{˙}{\phi }}_{i}\right)$ ($i=$ 1, 2) is the electromagnetic torque on the excited-motor $i$, ${T}_{fi}\left({\stackrel{˙}{\mathrm{\phi }}}_{i}\right)$ ($i=$ 1, 2) is the load
torque on the excited-motor $i$.
In the paper, three-phase asynchronous motors are chosen as the excited motors in the self-synchronous vibrating pile system with frequency capture, the state equation of the motor is expressed as:
$\left[\begin{array}{c}{u}_{M1}\\ {u}_{T1}\\ 0\\ 0\end{array}\right]=\left[\begin{array}{cccc}{R}_{1}+p{L}_{s}& -{\omega }_{s}{L}_{s}& p{L}_{m}& -{\omega }_{s}{L}_{m}\\ {\omega }_{s}{L}_{s}& {R}_{1k}
+p{L}_{sk}& {\omega }_{s}{L}_{m}& p{L}_{m}\\ p{L}_{m}& -\left({\omega }_{s}-{n}_{p}{\stackrel{˙}{\phi }}_{i}\right){L}_{m}& {R}_{2k}^{"}+p{L}_{r}& -\left({\omega }_{s}-{n}_{p}{\stackrel{˙}{\phi }}_
{i}\right){L}_{r}\\ \left({\omega }_{s}-{n}_{p}{\stackrel{˙}{\phi }}_{i}\right){L}_{m}& p{L}_{m}& \left({\omega }_{s}-{n}_{p}{\stackrel{˙}{\phi }}_{i}\right){L}_{r}& {R}_{2}^{"}+p{L}_{r}\end{array}\
right]\left[\begin{array}{c}{i}_{M1}\\ {i}_{T1}\\ {i}_{M2}\\ {i}_{T2}\end{array}\right],$
where, ${u}_{M1}$, ${u}_{T1}$ can be expressed as the voltage of the motor in the $MT$ coordinate system, ${i}_{M1}$, ${i}_{M2}\text{,}$${i}_{T1}\text{,}$${i}_{T2}$ are described as the electric
current in the rotating $MT$ coordinate system, respectively, and the subscript 1 and 2 of the electric current are expressed as the stator and rotor respectively. ${L}_{s}$, ${L}_{r}$ and ${L}_{m}$
are described as the stator winding of the electric machine, the rotor winding of the electric machine and the mutual inductance between the stator winding and the rotor winding. ${n}_{p}$ can be
described as the pole number of the motor, ${\omega }_{s}$ is the electric angular velocity of the motor stator, ${R}_{1}$and ${R}_{2}^{\mathrm{"}}$ are described as the stator resistance and the
rotor conversion resistor of the motor, respectively. $p$ can be expressed as the differential operator, namely, $p=d/dt$. The electromagnetic torque equation of the motor is expressed as the
${T}_{mi}\left({\stackrel{˙}{\phi }}_{i}\right)=1.5\cdot {n}_{p}{L}_{m}\left[{i}_{T1}\left({\stackrel{˙}{\phi }}_{i}\right){i}_{M2}\left({\stackrel{˙}{\phi }}_{i}\right)-{i}_{M1}\left({\stackrel{˙}{\
phi }}_{i}\right){i}_{T2}\left({\stackrel{˙}{\phi }}_{i}\right)\right].$
The rotor motion equation of the motor is expressed as the following:
$\frac{{J}_{0}}{{n}_{p}}{\stackrel{˙}{\omega }}_{i}={T}_{mi}\left({\stackrel{˙}{\phi }}_{i}\right)-{T}_{fi}\left({\stackrel{˙}{\phi }}_{i}\right).$
where, ${J}_{0}$ is described as the moment of inertia of the rotor system.
Eqs. (1)-(4) constitute a nonlinear dynamic model of the self-synchronous vibrating pile system with the pile- soil-electric coupling, which is named as the model of the self-synchronous vibrating
pile system with the electromechanical coupling.
The angular velocity ${\stackrel{˙}{\phi }}_{i}$ is generated through the rotation of the eccentric block on the exciter-motor $i$ and can be expressed as ${\stackrel{˙}{\phi }}_{i}={\omega }_{i}\
text{,}$ and two angular velocities in the self-synchronous vibration system should be synchronized, which can be named as ${\omega }_{1}={\omega }_{2}$. The average angularphase of the angular
phases may be expressed as $\phi$, Namely, ${\phi }_{1}=\phi +\mathrm{\Delta }\alpha /2$, ${\phi }_{2}=\phi -\mathrm{\Delta }\alpha /2$. $\stackrel{˙}{\phi }$ is named as the average speed of two
eccentric blocks and can be expressed as $\stackrel{˙}{\phi }=\stackrel{-}{\omega }$, and can been defined as $\stackrel{-}{\omega }=\left({\omega }_{1}+{\omega }_{2}\right)/2$. Where, ${\omega }_{n}
=\sqrt{k/m}$, ${\omega }_{0}=\sqrt{k\mathrm{"}/m}$, $2\xi {\omega }_{n}=c/m$. When ${m}_{1}={m}_{2}={m}_{0}$, ${r}_{1}={r}_{2}={r}_{0}$, the eccentricity of the eccentric blocks can be defined as $F=
\left(\left({\sum }_{i=1}^{2}{m}_{i}{r}_{i}{\stackrel{˙}{\phi }}_{i}^{2}\right)\mathrm{c}\mathrm{o}\mathrm{s}\frac{1}{2}\mathrm{\Delta }\alpha \right)/m=\left(2{m}_{0}{r}_{0}{\stackrel{-}{\omega }}^
{2}\mathrm{c}\mathrm{o}\mathrm{s}\frac{1}{2}\mathrm{\Delta }\alpha \right)/m$. The first equation in Eq. (1) can be transformed into:
$\stackrel{¨}{y}+{\omega }_{n}^{2}y=-2\xi {\omega }_{n}\stackrel{˙}{y}+\epsilon {\omega }_{0}{y}^{3}+F\mathrm{c}\mathrm{o}\mathrm{s}\left(\phi -{90}^{\mathrm{o}}\right).$
An approximate solution of the self-synchronous vibrating pile system with frequency capture is solved using multi-scale method, and which can be calculated to solve the following equation:
$y=\alpha \mathrm{s}\mathrm{i}\mathrm{n}\left(\stackrel{-}{\omega }t-\gamma \right),$
where, $\alpha ={m}_{0}{r}_{0}{\stackrel{-}{\omega }}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\frac{1}{2}\mathrm{\Delta }\alpha /m\sqrt{\left(\xi {\omega }_{n}^{2}{\right)}^{2}+\left(\stackrel{-}{\omega }{\
omega }_{n}-{\omega }_{e}^{2}{\right)}^{2}}$, $\gamma =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(-\xi {\omega }_{n}^{2}/\stackrel{-}{\omega }{\omega }_{n}-{\omega }_{e}^{2}\
In addition, ${\omega }_{e}=\sqrt{{k}_{e}/m}$ and is named as the first order equivalent natural frequency of the self-synchronous vibrating pile system with frequency capture. Where, ${k}_{e}=k-\
left(2\epsilon {k}^{"}{\alpha }^{2}\right)/8$ and ${k}_{e}$ is named as the equivalent stiffness of the self-synchronous vibrating pile system with frequency capture. When the excitation frequency is
close to the definite range of the first order equivalent natural frequency, namely, $\stackrel{-}{\omega }={\omega }_{e}$. At this time, the excitation frequency is captured by the natural frequency
in the self-synchronous vibrating pile system, and frequency capture occurs in the self-synchronous vibrating pile system.
3. Theoretical analysis
3.1. Theoretical analysis about the synchronization condition
When the phase difference of the two-excited motors is in a certain range, the reverse synchronous operation of the dual excited motor can be realized to run safely and stably in the self-synchronous
vibrating pile system. The rotor rotational motion equations (the last two formulas in Eq. (1)) is transformed to obtain the synchronization condition through the theoretical analysis. If frequency
capture occurred in the self-synchronous vibrating pile system, the angle frequency of the two excited motor would be equal. $\stackrel{-}{\omega }$ is named as average angular frequency of the two
excited motor, so the angle frequency of the two-excited motor can be replaced by $\stackrel{-}{\omega }$. It has been assumed that the relevant parameters of the two-excited motor is equal, namely $
{{J}_{0}}_{1}={{J}_{0}}_{2}$, ${c}_{1}={c}_{2}$, ${m}_{1}={m}_{2}={m}_{0}$, ${r}_{1}={r}_{2}={r}_{0}$. Then, the last two formulas in Eq. (1) are subtracted from each other, and the rotary motion
equation in one $2\pi$ period is averaged, and small variables is neglected. Finally, using the last two formulas in Eq. (1), it has been assumed that the relevant parameters of the two-excited motor
is equal, the rotary motion equation about the phase difference can be defined as:
${{J}_{0}}_{1}\mathrm{\Delta }\stackrel{¨}{\alpha }=\left[{T}_{m1}\left({\stackrel{˙}{\phi }}_{1}\right)-{T}_{m2}\left({\stackrel{˙}{\phi }}_{2}\right)\right]-\left[{T}_{f1}\left({\stackrel{˙}{\phi
}}_{1}\right)-{T}_{f2}\left({\stackrel{˙}{\phi }}_{2}\right)\right]-{c}_{1}\mathrm{\Delta }\stackrel{˙}{\alpha }$$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}-\frac{\left({m}_{0}
{r}_{0}{\stackrel{-}{\omega }}^{2}{\right)}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\gamma \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Delta }\alpha }{m\sqrt{\left(\xi {\omega }_{n}^{2}{\right)}^{2}+\left(\
stackrel{-}{\omega }{\omega }_{n}-{\omega }_{e}^{2}{\right)}^{2}}}.$
If ${e}_{1}=\mathrm{\Delta }\alpha$, ${e}_{2}=\mathrm{\Delta }\stackrel{˙}{\alpha }$, Eq. (7) can be transformed into and the state equation of Eq. (7) can be expressed as:
${\stackrel{˙}{e}}_{1}={e}_{2},{\stackrel{˙}{e}}_{2}=\frac{1}{{{J}_{0}}_{1}}\left(\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}\right)-\frac{{c}_{1}}{{{J}_{0}}_{1}}{e}_{2}-\frac{\left({m}_{0}{r}_
{0}{\stackrel{-}{\omega }}^{2}{\right)}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\gamma \mathrm{s}\mathrm{i}\mathrm{n}{e}_{1}}{{{J}_{0}}_{1}m\sqrt{\left(\xi {\omega }_{n}^{2}{\right)}^{2}+\left(\stackrel{-}
{\omega }{\omega }_{n}-{\omega }_{e}^{2}{\right)}^{2}}},$
where, $\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}$ is expressed as the difference between the electromagnetic torque and the equivalent load torque. If the state of the stable equilibrium is
achieved in the self-synchronous vibrating pile system, ${\stackrel{˙}{e}}_{1}={\stackrel{˙}{e}}_{2}=0$, which must be satisfied, namely:
$\mathrm{s}\mathrm{i}\mathrm{n}{e}_{1}=\frac{\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}}{{\left({m}_{0}{r}_{0}{\stackrel{-}{\omega }}^{2}\right)}^{2}W}=\frac{1}{D},$
$W=\frac{\left({\omega }_{e}^{2}-\stackrel{-}{\omega }{\omega }_{n}\right)}{m\left[\left(\xi {\omega }_{n}^{2}{\right)}^{2}+\left(\stackrel{-}{\omega }{\omega }_{n}-{\omega }_{e}^{2}{\right)}^{2}\
The necessary condition for synchronous operation is that the absolute value of $D$ is greater than or equal to 1. So the synchronization condition in the self-synchronous vibrating pile system with
frequency capture can be expressed as:
$\left|D\right|=\left|\frac{{\left({m}_{0}{r}_{0}{\stackrel{-}{\omega }}^{2}\right)}^{2}W}{\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}}\right|\ge 1.$
As shown in Eq. (10), if the synchronous and stable operation is achieved in the self-synchronous vibrating pile system with frequency capture, which can be achieved by reducing $\mathrm{\Delta }{T}_
{m}-\mathrm{\Delta }{T}_{f}$ or increasing $W$. In addition, the difference between the electromagnetic torque and the equivalent load torque is zero, namely, $\mathrm{\Delta }{T}_{m}-\mathrm{\Delta
}{T}_{f}=0$, and it is the ideal condition, but the most unsatisfactory condition is that $W=0$, namely, ${\omega }_{e}^{2}-\stackrel{-}{\omega }{\omega }_{n}=0$ in Eq. (10).
The excitation frequency is close to the definite range of the first order equivalent natural frequency in the self-synchronous vibrating pile system with frequency capture, namely, $\stackrel{-}{\
omega }={\omega }_{e}$. The self-synchronous vibrating pile system with frequency capture is a flexible nonlinear system, so the first order equivalent natural frequency cannot be identical to the
natural frequency in a flexible nonlinear system. Namely, ${\omega }_{n}e {\omega }_{e}e \stackrel{-}{\omega }$. Therefore $W$ is unequal to 0, namely, $We 0$, and the most unsatisfactory condition
(namely, $W=$0) cannot be presented in the self-synchronous vibrating pile system with frequency capture. Eq. (10) can be transformed into and be rewritten as:
$\frac{\left({m}_{0}{r}_{0}\stackrel{-}{\omega }{\right)}^{2}\left|\left(1-\frac{{\omega }_{n}}{\stackrel{-}{\omega }}\right)\right|}{\left|\left(\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}\
right)\right|m\left\{{\left[\xi {\left(\frac{{\omega }_{n}}{\stackrel{-}{\omega }}\right)}^{2}\right]}^{2}+{\left(\frac{{\omega }_{n}}{\stackrel{-}{\omega }}-1\right)}^{2}\right\}}\ge 1.$
If ${\omega }_{n}/\stackrel{-}{\omega }=\varpi$, Eq. (11) can be rewritten as:
$\frac{\left({m}_{0}{r}_{0}\stackrel{-}{\omega }{\right)}^{2}\left|\left(1-\varpi \right)\right|}{\left|\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}\right|m\left[\left(\xi {\varpi }^{2}{\right)}^
{2}+\left(\varpi -1{\right)}^{2}\right]}\ge 1.$
The self-synchronous vibrating pile system with frequency capture is a flexible nonlinear system, namely, ${\omega }_{n}>{\omega }_{e}$, and the excitation frequency is close to the definite range of
the first order equivalent natural frequency in the self-synchronous vibrating pile system with frequency capture, namely, $\stackrel{-}{\omega }={\omega }_{e}$, so ${\omega }_{n}>\stackrel{-}{\omega
}$, namely, $\varpi ={\omega }_{n}/\stackrel{-}{\omega }>1$. Therefore, that ($\varpi >1$) can only be achieved in the self-synchronous vibrating pile system with frequency capture. Here, Eq. (12)
can be rewritten as:
$\left|\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}\right|\le \frac{\left({m}_{0}{r}_{0}\stackrel{-}{\omega }{\right)}^{2}\left(\varpi -1\right)}{m\left[\left(\xi {\varpi }^{2}{\right)}^{2}+\left
(\varpi -1{\right)}^{2}\right]}.$
When Eq. (13) is satisfied in the self-synchronous vibrating pile system with frequency capture, the reverse synchronous rotation of the two-excited motor can be achieved.
In addition, as shown in Eqs. (10)-(13), if $D$ is positive, namely, $\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}$ is negative, the phase different $\mathrm{\Delta }\alpha =\mathrm{}$[0°, 90°] or
$\mathrm{\Delta }\alpha =$[90°, 180°]. If $D$ is negative, namely, $\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}$is positive, the phase different $\mathrm{\Delta }\alpha =\mathrm{}$[–90°, 0°] or $
\mathrm{\Delta }\alpha =\mathrm{}$[180°, 270°]. Namely, there are two phase difference at each $D$ value, and only one solution is stable, and the other one is unstable. Therefore, the synchronous
stability condition is analyzed in the self-synchronous vibrating pile system with frequency capture.
3.2. Synchronous stability condition
Using Jacobian matrix in Eq. (8), the synchronous stability condition is deduced in the self-synchronous vibrating pile system with frequency capture. Namely, the synchronous stability condition for
the phase different of the two-excited motor are analyzed. Jacobian matrix of Eq. (8) can be expressed as:
$\left|\begin{array}{l}01\\ -\frac{\left({m}_{0}{r}_{0}{\stackrel{-}{\omega }}^{2}{\right)}^{2}\left({\omega }_{e}^{2}-\stackrel{-}{\omega }{\omega }_{n}\right)cos{e}_{1}}{{{J}_{0}}_{1}m\left[\left(\
xi {\omega }_{n}^{2}{\right)}^{2}+\left(\stackrel{-}{\omega }{\omega }_{n}-{\omega }_{e}^{2}{\right)}^{2}\right]}-\frac{{c}_{1}}{{{J}_{0}}_{1}}\end{array}\right|.$
The characteristic equation of Jacobi matrix can be written as:
${\lambda }^{2}+\frac{{c}_{1}}{{{J}_{0}}_{1}}\lambda -\frac{\left({m}_{0}{r}_{0}{\stackrel{-}{\omega }}^{2}{\right)}^{2}\left({\omega }_{e}^{2}-\stackrel{-}{\omega }{\omega }_{n}\right)\mathrm{c}\
mathrm{o}\mathrm{s}{e}_{1}}{{{J}_{0}}_{1}m\left[\left(\xi {\omega }_{n}^{2}{\right)}^{2}+\left(\stackrel{-}{\omega }{\omega }_{n}-{\omega }_{e}^{2}{\right)}^{2}\right]}=0.$
When the real part of the characteristic root is negative in Eq. (15), which can be solved using Hurwits theorem, the phase difference equation is asymptotically stable. The real part of the
characteristic root is negative in Eq. (15), which can be satisfied using Hurwits theorem. And the synchronous stability condition must be satisfied and can be expressed as the following:
$-\frac{\left({m}_{0}{r}_{0}\stackrel{-}{\omega }{\right)}^{2}\left(1-\varpi \right)\mathrm{c}\mathrm{o}\mathrm{s}{e}_{1}}{{{J}_{0}}_{1}m\left[\left(\xi {\varpi }^{2}{\right)}^{2}+\left(\varpi -1{\
The self-synchronous vibrating pile system with frequency capture is a flexible nonlinear system, and $\varpi >1$ for the self-synchronous vibrating pile system. When $\varpi >1$, $\mathrm{c}\mathrm
{o}\mathrm{s}{e}_{1}>0$ in Eq. (16). Namely, the phase different $\mathrm{\Delta }\alpha$ is at [–90°, 90°] (namely, $\mathrm{\Delta }\alpha =\mathrm{}$[0°, 90°] and $\mathrm{\Delta }\alpha =\mathrm
{}$[–90°, 0°]). The mean value of the phase difference is 0 at [–90°, 90°], and The self-synchronous vibrating pile system with frequency capture is stable when the phase difference $\mathrm{\Delta }
\alpha$ is at [–90°, 90°], but the self-synchronous vibrating pile system is unstable when the phase difference $\mathrm{\Delta }\alpha$ is at $\mathrm{\Delta }\alpha =\mathrm{}$[90°, 270°]. So if $\
mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}$ is negative and the phase difference $\mathrm{\Delta }\alpha$ is at [0°, 90°], the synchronization condition and the synchronization stability
condition can be satisfied in the self-synchronous vibrating pile system with frequency capture. If $\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}$ is positive and the phase difference $\mathrm{\
Delta }\alpha$ is at [–90°, 0°], the synchronization condition and the synchronization stability condition can be satisfied in the self-synchronous vibrating pile system with frequency capture. In
addition, when $\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}$ is 0 and the phase difference is stable at 0, which is the ideal state to meet the reverse synchronous operation of the two-excited
motors in the self-synchronous vibrating pile system with frequency capture.
4. Simulation analysis
Using the model of the self-synchronous vibrating pile system with electromechanical coupling (Eqs. (1)-(4)), Some parameters for the self-synchronous vibrating pile system with electromechanical
coupling is select as follows: $m=$86 kg, ${m}_{1}={m}_{2}=$ 3.5 kg, ${r}_{1}={r}_{2}=$0.08 m, $k=$1552000 N/m, ${k}_{1}=\text{0.8}k\text{,}$$\epsilon =$0.5, $c=$ 100N∙s/m, ${c}_{1}=$0.01N∙s/m, ${c}_
{2}=$0.01N∙s/m, ${J}_{01}={J}_{02}=$0.01kg∙m^2. And the magnetic pole number ${n}_{p}$ of the selected motor is 4 and the excitation frequency of the two-excited motors is 25 Hz (about 157 rad/s).
The simulation is performed using Matlab/Simlink, and the responses of the parameters in the model of the self-synchronous vibrating pile system with electromechanical coupling (Eqs. (1)-(4)) has
been obtained.
Fig. 2Parameters simulation of the system without frequency capture
Fig. 3Parameters simulation of the system without frequency capture under different initial conditions
a) Initial velocity (1.5 m/s) and initial displacement (0.03 m)
b) Initial velocity (1.5 m/s) and initial displacement (–0.03 m)
From the point of view of simulation analysis, when the damping of the soil is increased (namely, $c=$10000N∙s/m), the responses of the parameters and the phase plane of the vibration displacement in
the self-synchronous vibrating pile system with electromechanical coupling (Eqs. (1)-(4)) has been obtained and shown in Figs. 2-3. As shown in Fig. 2, the displacement of the periodic motion is
eventually stabilized at about 8mm. But the excitation frequency in the self-synchronous vibrating pile system with electromechanical coupling is about 25 Hz, namely, the speeds of the two-excited
motors is eventually stabilized at about 157 rad/s, and the excitation frequency is not captured by the natural frequency, and frequency capture does not occur in the self-synchronous vibrating pile
system with electromechanical coupling. In addition, when the initial displacement and the initial velocity is changed in the self-synchronous vibrating pile system with electromechanical coupling,
the amplitude of the displacement is not changed, as shown in Figs. 2-3.
When the damping of the soil is 1000 N∙s/m, the responses of the parameters and the phase plane of the vibration displacement in the self-synchronous vibrating pile system with electromechanical
coupling (Eqs. (1)-(4)) has been obtained and shown in Fig. 4. As shown in Fig. 4, after starting the system slowly, the large oscillations of the amplitude in the self-synchronous vibrating pile
system with electromechanical coupling can be obtained at the resonance point (about 22.35 Hz), thus the angular frequency of the two-excited motors is captured by the natural frequency, and
frequency capture occurs in the self-synchronous vibrating pile system with electromechanical coupling. Then the oscillation of the amplitude is reduced to be stabilized at about 25 Hz (namely, the
angular frequency of the two-excited motors), and the speeds of the two-excited motors is eventually stabilized at about 157 rad/s. Finally, the self-synchronous vibrating pile system with
electromechanical coupling is still performed under the condition of the angular frequency of the two-excited motors (157 rad/s), the angular frequency of the two-excited motors is not also captured
by the natural frequency. The self-synchronous vibrating pile system with electromechanical coupling has occurred frequency capture phenomenon, which is named as critical frequency capture.
Fig. 4Parameter simulation of the system with critical frequency capture
Fig. 5Parameter simulation of the system with frequency capture
When the damping of the soil is 100 N∙s/m, the responses of the parameters and the phase plane of the vibration displacement in the self-synchronous vibrating pile system with electromechanical
coupling (Eqs. (1)-(4)) has been obtained and shown in Fig. 5. When certain parameters of the self-synchronous vibrating pile system with electromechanical coupling are changed appropriately, such as
the damping of the soil, the excitation frequency is close to the definite range of the first natural frequency, and the excitation frequency is captured by the first order equivalent natural
frequency in the self-synchronous vibrating pile system with electromechanical coupling. As shown in Fig. 5, when the system is slowly started, and the working frequency is not changed after the
resonance point (about 21.25 Hz), the response tends to be stable and does periodic motion in the self-synchronous vibrating pile system with electromechanical coupling, and the displacement of the
periodic motion is eventually stabilized at about 0.28 m. As shown in Fig. 5 (motor speed diagram), the speeds of the two-excited motors is eventually stabilized at about 132 rad/s and also slightly
less than the first order natural frequency (about 134 rad/s), it also has been confirmed that the speeds of the two-excited motors are not the speed (about 157 rad/s) under the condition of their
own vibration frequency, and the angular frequency of the two-excited motors is captured by first order equivalent natural frequency (about 21.25 Hz). The self-synchronous vibrating pile system with
electromechanical coupling occur frequency capture phenomenon, which is named as frequency capture. As shown in Figs. 2-5, when frequency capture occurs in the self-synchronous vibrating pile system
with electromechanical coupling, the amplitude of the system is much greater than the amplitude of the system without the frequency capture. The large amplitude is beneficial to the speed of the pile
and the efficiency of the pile, so the self-synchronous vibrating pile system with frequency capture should be rational use. But the synchronous rotation of the two excited motor is very sensitive to
the resonance of the system, and it is easy to cause the instability when frequency capture occurs the self-synchronous vibrating pile system with electromechanical coupling. So the reverse
synchronous operation of the two-excited motors and the synchronization stability of the self-synchronous vibrating pile system should be analyzed in detail.
The simulation diagrams of the parameters in the self-synchronous vibrating pile system with frequency capture are shown in Fig. 6. The first diagram in Fig. 6 is obtained by Eq. (13), the difference
between the electromagnetic torque and the equivalent load torque $\left|\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}\right|$ must be in the shadow range, and the reverse synchronous rotation of
the two-excited motor can be achieved and frequency capture can occur in the self-synchronous vibrating pile system. When frequency capture occurs in the self-synchronous vibrating pile system with
electromechanical coupling, the excited frequency (namely, the angular frequency of the two-excited motors) is captured by the first order equivalent natural frequency (about 132 rad/s), namely, the
excited frequency can be replaced by the first order equivalent natural frequency. Using the selected parameters, when the ratio between the first natural frequency (134 rad/s) and the excitation
frequency (132 rad/s) is 1.0176, the difference between the electromagnetic torque and the equivalent load torque $\left|\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}\right|$ must be less than
0.845 and $D$ in Eq. (10) can be greater than 1, and the reverse synchronous rotation of the two-excited motor can be achieved in the self-synchronous vibrating pile system. As shown in the second
diagram of Fig. 6, when the initial phase difference and the initial rotational speed difference also are 0, the phase difference (or the rotational speed difference) of the two-excited motors are
always 0. As shown in the last diagram of Fig. 6, the relationship between the phase difference and the amplitude of the periodic solution for the self-synchronous vibrating pile system with
electromechanical coupling has been obtained by using implicit equations between the phase difference and the amplitude. When the phase difference is at 0 or 2$\pi$, the maximum amplitude can be
obtained at 0 or 2$\pi$. But the phase difference is at$\mathrm{\pi }$, the amplitude is 0, namely, if the two-excited motors are the reverse synchronous operation, the phase difference must be in
the range of 0 (or 2$\pi$), and the maximum amplitude can be obtained. Thus the phase difference is at 0 or 2$\pi$, It is very ideal to meet the reverse synchronous operation of the two-excited
motors in the self-synchronous vibrating pile system with frequency capture, which is consistent with theoretical analysis.
In theoretical analysis, it has been discussed that the self-synchronous vibrating pile system with frequency capture tend to be stable when the phase difference $\mathrm{\Delta }\alpha$ is at [–90°,
90°]. And when $\left|\mathrm{\Delta }{T}_{m}-\mathrm{\Delta }{T}_{f}\right|$ is 0, it is the ideal state for the self-synchronous vibrating pile system with frequency capture. Namely, the difference
rate of the two-excited motors is 0 and the phase difference of the two-excited motors is 0, and the ideal state for the self-synchronous vibrating pile system with frequency capture can be obtained.
But the two excited motors cannot be completely consistent in actual engineering, the difference rate of the two-excited motors cannot be avoided. So the synchronous operation of the two-excited
motors and the stability of synchronization for the self-synchronous vibrating pile system with electromechanical coupling must be quantitative analyzed.
Fig. 6The simulation diagrams of the parameters
When the initial phase (namely, the initial phase difference is 1rad) of the two-excited motors and the initial rotational speed (namely, the initial rotational speed difference is 0.5 rad) of the
two-excited motors are different, the parameters responses of the self-synchronous vibrating pile system with electromechanical coupling are shown in Fig. 7. As shown in Fig. 7, after a big shock has
been presented for the rotational speeds of the two-excited motors, finally, the rotational speeds of the two-excited motors also are stable at around 132 rad/s and do periodic motion. Hence, when
the difference rate of the two-excited motors is considered, the excited frequency can still be captured by the first order equivalent natural frequency, namely, frequency capture can still occur in
the self-synchronous vibrating pile system with electromechanical coupling, and the reverse synchronous operation of the two-excited motors and the stability of synchronization of the
self-synchronous vibrating pile system with electromechanical coupling can be achieved.
Fig. 7The system parameter response in different initial conditions
When the difference rates of the two-excited motors are shown in the self-synchronous vibrating pile system with electromechanical coupling, the stabilities problems of the phase difference and the
rotational speed difference for the two-excited motors are analyzed in the following. The simulation of the system parameters is performed using Matlab/Simlink, and the responses and the phase plane
of the phase difference and the rotational speed difference are shown in Figs. 8-11, when the difference rates of the two-excited motors are selected, such as the different initial phase, the
different initial rotational speed, the parameter difference of the two-excited motors and the different rotational damping of the excited motor rotor. As shown in Figs. 8-11, as a result of some
differences between the two-excited motors, wherever the initial point of the phase difference and the rotational speed difference are, the phase difference and the rotational speed difference of the
two-excited motor are all experienced a gentle transition process, then a high fluctuations of the phase difference and the rotational speed difference are obtained in the self-synchronous vibrating
pile system with electromechanical coupling, finally, the phase difference is stable at 0 or 2$\pi$ and the rotational speed difference is stable at 0, which is consistent with theoretical analysis.
Fig. 8Simulation of the system with frequency capture in different initial phase conditions
a) The initial phase difference (5 rad)
b) The initial phase difference (1.57 rad)
Fig. 9Simulation of the system with frequency capture in different initial rotational speed conditions
a) The initial phase difference (–0.5 rad) and the initial rotational speed difference (0.5 rad)
b) The initial phase difference (0.5 rad) and the initial rotational speed difference (1 rad)
Fig. 10Simulation of the system with frequency capture in the difference of the motors parameters
So when the difference rates of the two-excited motors is in a certain range, the two-excited motor can restore the synchronization by itself, and the phase difference is stable at 0 or 2$\pi$ and
the rotational speed difference is stable at 0, namely, when frequency capture occurs, the reverse synchronous operation of the two-excited motors and the synchronous stability operation of the
self-synchronous vibrating pile system with electromechanical coupling can be obtained to achieve the large amplitude of the pile and the pile speed, and to improve the efficiency of the pile.
Fig. 11Simulation of the system with frequency capture in the difference of the rotating damping
5. Conclusions
1. From the point of view of frequency capture (the excited frequency of the excited motor is captured by the first order equivalent natural frequency, namely, $\stackrel{-}{\omega }={\omega }_{e}$),
the synchronization condition of the two-excited motors and the synchronous stability condition is theoretical discussed, and using the relationship between the phase difference and the amplitude of
an approximate solution, the synchronization and the synchronous stability are analyzed. It has been shown that the phase difference is stable at 0 or 2$\pi$, the synchronous stability condition can
be achieved in the self-synchronous vibrating pile system.
2. Based on the model of the self-synchronous vibrating pile system with the electromechanical coupling, the reverse synchronous operation of the two-excited motors and the synchronous stability of
the self-synchronous vibrating pile system have been quantitative analyzed in the self-synchronous vibrating pile system with electromechanical coupling. It has shown that the amplitude of the system
is much greater than the amplitude of the system without the frequency capture when frequency capture occurs in the self-synchronous vibrating pile system with electromechanical coupling. The large
amplitude is beneficial to the speed of the pile and the efficiency of the pile, so the self-synchronous vibrating pile system with frequency capture should be rational use. when the difference rate
of the two-excited motors is considered, the excited frequency can still be captured by the first order equivalent natural frequency, namely, frequency capture can still occur in the self-synchronous
vibrating pile system with electromechanical coupling, and the reverse synchronous operation of the two-excited motors and the stability of synchronization of the self-synchronous vibrating pile
system with electromechanical coupling can be achieved.
3. Frequency capture phenomenon in the self-synchronous vibrating pile system with electromechanical coupling are presented for the changes of the system parameters (including the damping of the
soil), and various synchronous phenomena of the self-synchronous vibrating pile system with electromechanical coupling are obtained through the difference rates of the two-excited motors (including
the initial phase difference, the initial rotational speed difference, the difference of the motors parameters and the difference of the rotating damping). When the difference rates of the
two-excited motors is in a certain range, the two-excited motor can restore the synchronization by itself, and the phase difference is stable at 0 or 2$\pi$ and the rotational speed difference is
stable at 0, namely, when frequency capture occur, the reverse synchronous operation of the two-excited motors and the synchronous stability operation of the self-synchronous vibrating pile system
with electromechanical coupling can be obtained to achieve the large amplitude of the pile and the pile speed, and to improve the efficiency of the pile.
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About this article
Chaos, nonlinear dynamics and applications
vibration synchronization
frequency capture
electromechanical coupling
the stability of synchronization
The author gratefully acknowledges that the work was supported by the science research foundation of Beijing University of Civil Engineering and Architecture under the Project No. 00331616043.
Copyright © 2016 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/16737","timestamp":"2024-11-03T16:40:29Z","content_type":"text/html","content_length":"208626","record_id":"<urn:uuid:45a253af-ff9c-4f63-bd0d-549c1bc08d6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00184.warc.gz"} |
How to integrate (x^2+1)/(x^2-1)^2? | TutorChase
How to integrate (x^2+1)/(x^2-1)^2?
To integrate (x^2+1)/(x^2-1)^2, use partial fractions and substitution.
First, factor the denominator: (x^2-1)^2 = (x-1)^2(x+1)^2
Then, use partial fractions to write the integrand as:
(x^2+1)/(x^2-1)^2 = A/(x-1) + B/(x-1)^2 + C/(x+1) + D/(x+1)^2
To find A, multiply both sides by (x-1) and let x=1:
A = lim(x->1) [(x^2+1)/(x+1)^2] = 1/4
Similarly, find B, C, and D:
B = lim(x->1) [(x^2+1)/(x+1)^3] = -1/8
C = lim(x->-1) [(x^2+1)/(x-1)^3] = 1/8
D = lim(x->-1) [(x^2+1)/(x-1)^2] = -1/4
Now, substitute the partial fractions into the integral:
∫(x^2+1)/(x^2-1)^2 dx = ∫[1/4(x-1) - 1/8(x-1)^2 + 1/8(x+1) - 1/4(x+1)^2] dx
Simplify and integrate each term:
= 1/4 ln|x-1| - 1/24(x-1)^3 + 1/8 ln|x+1| - 1/12(x+1)^3 + C
Therefore, the final answer is:
∫(x^2+1)/(x^2-1)^2 dx = 1/4 ln|x-1| - 1/24(x-1)^3 + 1/8 ln|x+1| - 1/12(x+1)^3 + C
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Parent Hamiltonian Reconstruction of Ja
SciPost Submission Page
Parent Hamiltonian Reconstruction of Jastrow-Gutzwiller Wavefunctions
by Xhek Turkeshi, Marcello Dalmonte
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Marcello Dalmonte · Xhek Turkeshi
Submission information
Preprint Link: https://arxiv.org/abs/1909.11327v3 (pdf)
Date accepted: 2020-02-28
Date submitted: 2020-02-05 01:00
Submitted by: Turkeshi, Xhek
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Quantum Physics
Approaches: Theoretical, Computational
Variational wave functions have been a successful tool to investigate the properties of quantum spin liquids. Finding their parent Hamiltonians is of primary interest for the experimental simulation
of these strongly correlated phases, and for gathering additional insights on their stability. In this work, we systematically reconstruct approximate spin-chain parent Hamiltonians for
Jastrow-Gutzwiller wave functions, which share several features with quantum spin liquid wave-functions in two dimensions. Firstly, we determine the different phases encoded in the parameter space
through their correlation functions and entanglement content. Secondly, we apply a recently proposed entanglement-guided method to reconstruct parent Hamiltonians to these states, which constrains
the search to operators describing relativistic low-energy field theories - as expected for deconfined phases of gauge theories relevant to quantum spin liquids. The quality of the results is
discussed using different quantities and comparing to exactly known parent Hamiltonians at specific points in parameter space. Our findings provide guiding principles for experimental Hamiltonian
engineering of this class of states.
Author comments upon resubmission
Dear Editor,
thank you for handling our submission.
We are pleased to thank the referees, whose insightful comments and observations helped us further improve and clarify our work. In this resubmission, we believe to have addressed all the point
raised by the Referees. We append below a detailed response to the reports and the list of changes.
Yours sincerely,
Xhek Turkeshi and Marcello Dalmonte
Reply to referee report I:
We thank the referee for their thoughtful reading on the manuscript. Here we reply to the corresponding points of their report, specifying the changes included in the resubmitted manuscript:
1) We updated the bibliography with other relevant works we have missed, including the ones suggested by the referee.
2) We included additional system sizes in Fig.2 (Fig.1 of previous submission) of the paper. We also performed a better system size analysis, included in Sec. (2.3) of the new manuscript. Since the
scaling is very slow, we cannot give a thermodynamic answer on the intermediate regime, which seems representative of a Luttinger liquid phase.
3) In our work we use the definition of the correlation length in terms of the real space correlation functions (eq.(21-22) of the new manuscript). These formulae are meaningful only when the cluster
decomposition principle holds. However, due to the functional form of the JG wave functions, in the symmetry broken phases this state would result in a coherent superposition of states in the
different symmetry broken sectors. In fact these are the GHZ states, which are a remarkable exception of the cluster decomposition. All the data presented are in the critical regime, where this
principle holds (and the correlation length is well defined). To confirm this fact, we added a new subplot of the correlation functions in LogLog scale. The seemingly algebraic decay guarantees the
principle is holding. We added a discussion to further clarify this point in Sec.(2.4). Finally, for presentation convenience we decided to present in the new manuscript only L multiples of 4, with
system sizes up to L=36.
4) As the referee observe, the optimization with only nearest-neighbors indeed requires only one free parameter. Nevertheless, to benchmark the algorithm, we decided to present the results of an
unconstrained optimization. From the table one can see that the symmetries are always respected and the relative ratios of the converged couplings are consistent with those expected in the exact
cases. In the table caption we further comment on this point. We apologize for not stating this before, as this was clearly confusing.
5) Motivated by the insightful comment of the Referee, we computed the relative error between the exact ground state energy of the reconstructed Hamiltonian and its variational energy on the
Jastrow-Gutzwiller state. The results are presented in a new paragraph of section 4.3. Below we summarize the included discussion, and refer to the new version of the paper for further details.
All the cases considered in our studies lie within 1\% of relative error in the energy landscape. In addition, we also present data of the relative error as a function on nearest-neighbors considered
in the reconstruction. Interestingly, this quantity seems to increase with larger basis considered, although the two states are closer and closer (see Fig.5 and Fig.6). This is reminiscent of what
happens for the correlation functions (Section 4.3, paragraph: Correlation functions). In the same perspective, at present we cannot fully understand and characterize such counterintuitive behavior.
As we mention, this may be due to the algorithm forcing the optimization on a finite size landscape and creating frustration effects. For example, this is probably what happens in the case $\alpha=2$
, which should converge to the Haldane-Shastry prefactors. Another possibility is that new operator content is needed, and the chosen basis cannot grasp the thermodynamic properties of the systems.
Further investigations on this problem are left for future studies.
Reply to referee report II:
We thank the referee for their interesting comments. We address the requested changes below.
1) We added a reliable system size scaling in Sec.(2.3). We construct a mesh of the critical exponent and the critical value of alpha e consider the optimal fit over different polynomial degree and
system sizes. The optimal fit is chosen using the least-square test. Values and error of the different estimates are the average and standard deviation over the set of optimal fits obtained with
different L-ranges and deg-ranges. In particular, the new estimated critical alpha is around 4.3(1).
Nonetheless, we also argued that this scaling is very slow, since is an algebraic function of Log
(L). As such, further system sizes are needed to fully characterize the transition point.
3) We added a new subsection about the participation spectrum in Sec.(2.2). In particular we discuss the relation to the logarithmic potential observed in the domain walls sector of the XXZ model. We
also discuss their relationship to the recently proposed conjecture that JG states are representatives of a Luttinger liquid phase. At the time of writing this paper we were not aware of the work of
Ref.[64-65]. In particular in the latter the authors use the Resta polarization to numerically estimate a critical point between a Luttinger phase to a Neel ordered phase at alpha_c=4. Our
simulations for similar system sizes in the entanglement entropy seems to give another value of critical transition point. We believe that further studies are needed to resolve the nature of the
variational JG wave functions.
3) We have split Fig.5 ( Fig.3 of the previous version) in two LogLog subplots presenting respectively the connected correlation functions and the inverse correlation length. For notational
convenience we present only L multiples of 4, with system sizes up to L=36.
4) We added a comment in Sec.(3.3) on other choices of estimators (including the KL divergence). In particular we remark this is not unique and we have chosen the relative entropy for implementation
convenience. In fact the latter is both convex and has a simple derivative (used in the gradient descent). Furthermore, we mention that at present the relationship between the KL divergence and the
quantum relative entropy has not been studied yet and deserve independently to be investigated.
5) As discussed in the work by Chertkov and Clark, methods based on the quantum covariance matrix, do not return the parent Hamiltonian, but rather, an Hamiltonian whose input vector is a generic
eigenstate, not necessarily the ground state. In order to verify that the given state is the ground state, a separate procedure is required. As such, we feel that a direct comparison is not possible,
as the methods are addressing quite different questions (out method targets the ground state by construction).
Following the Referee’s remarks, we have expanded our discussion on this issue in the
beginning of Sec.(3).
List of changes
List of changes:
-Corrected various typos
-Added Sec(2.2) discussing the participation spectrum (new figure Fig.1) (comments requested by Referee II)
-Added a discussion on the finite size scaling in sec 2.3 (new figure Fig.4) (to respond to both referees)
-Removed the appendix on finite size scaling
-Added new subfigure in fig.5 (old fig.3), both in Loglog scale (to respond to both the referees)
-Added a discussion in sec (2.4) about the correlation length (requested by Referee I)
-Added a discussion in the beginning of sec (3) to comment on the quantum covariance matrix approach in relationship to our work (requested by Referee II)
-Added a comment on other estimators in sec (3.3), especially the KL divergence (requested by Referee II)
-Added a clarification in the label of Table 1 (requested by Referee I)
-Added a new paragraph in sec (4.3) discussing the relative error between the variational energy of the JG wave functions and the ground state energy of the parent Hamiltonian (requested by Referee
-Updated bibliography
Published as SciPost Phys. 8, 042 (2020)
Reports on this Submission
Pedagogical and thorough study of an interesting family of 1D wave functions
In this new version, Turkeshi and Dalmonte have addressed several new aspects of this interesting problem. They certainly raised many new interesting effects, which deserve further studies.
I believe this manuscript is now ready for publication. | {"url":"https://scipost.org/submissions/1909.11327v3/","timestamp":"2024-11-13T15:49:59Z","content_type":"text/html","content_length":"42753","record_id":"<urn:uuid:3c2e086f-9d4d-4e6c-a973-96b5046809b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00807.warc.gz"} |
Haskell Fast & Hard (Part 3)
The hard part can now begin.
In this section, I will give a short example of the impressive refactoring ability provided by Haskell. We will select a problem and solve it using a standard imperative way. Then I will make the
code evolve. The end result will be both more elegant and easier to adapt.
Let's solve the following problem:
Given a list of integers, return the sum of the even numbers in the list.
example: [1,2,3,4,5] ⇒ 2 + 4 ⇒ 6
To show differences between the functional and imperative approach, I'll start by providing an imperative solution (in Javascript):
function evenSum(list) {
var result = 0;
for (var i=0; i< list.length ; i++) {
if (list[i] % 2 ==0) {
result += list[i];
return result;
But, in Haskell we don't have mutable variables, nor for loop. One solution to achieve the same result without loops is to use recursion.
Remark: Recursion is generally perceived as slow in imperative languages. But it is generally not the case in functional programming. Most of the time Haskell will handle recursive functions
Here is a C version of the recursive function. Note that for simplicity, I assume the int list ends with the first 0 value.
int evenSum(int *list) {
return accumSum(0,list);
int accumSum(int n, int *list) {
int x;
int *xs;
if (*list == 0) { // if the list is empty
return n;
} else {
x = list[0]; // let x be the first element of the list
xs = list+1; // let xs be the list without x
if ( 0 == (x%2) ) { // if x is even
return accumSum(n+x, xs);
} else {
return accumSum(n, xs);
Keep this code in mind. We will translate it into Haskell. But before, I need to introduce three simple but useful functions we will use:
even :: Integral a => a -> Bool
head :: [a] -> a
tail :: [a] -> [a]
even verifies if a number is even.
even :: Integral a => a -> Bool
even 3 ⇒ False
even 2 ⇒ True
head returns the first element of a list:
head :: [a] -> a
head [1,2,3] ⇒ 1
head [] ⇒ ERROR
tail returns all elements of a list, except the first:
tail :: [a] -> [a]
tail [1,2,3] ⇒ [2,3]
tail [3] ⇒ []
tail [] ⇒ ERROR
Note that for any non empty list l, l ⇔ (head l):(tail l)
The first Haskell solution. The function evenSum returns the sum of all even numbers in a list:
-- Version 1
evenSum :: [Integer] -> Integer
evenSum l = accumSum 0 l
accumSum n l = if l == []
then n
else let x = head l
xs = tail l
in if even x
then accumSum (n+x) xs
else accumSum n xs
main = print $ evenSum [1..10]
Here is an example of execution ; (I know I'm cheating. But I will talk about non-strict later) :
*Main> evenSum [1..5]
accumSum 0 [1,2,3,4,5]
1 is odd
accumSum 0 [2,3,4,5]
2 is even
accumSum (0+2) [3,4,5]
3 is odd
accumSum (0+2) [4,5]
4 is even
accumSum (0+2+4) [5]
5 is odd
accumSum (0+2+4) []
l == []
Coming from an imperative language all should seem right. In reality many things can be improved. First, we can generalize the type.
-- show
evenSum :: Integral a => [a] -> a
-- /show
main = do print $ evenSum [1..10]
Next, we can use sub functions using where or let. This way our accumSum function won't pollute the global namespace.
-- show
-- Version 2
evenSum :: Integral a => [a] -> a
evenSum l = accumSum 0 l
{-hi-} where {-/hi-} accumSum n l =
if l == []
then n
else let x = head l
xs = tail l
in if even x
then accumSum (n+x) xs
else accumSum n xs
-- /show
main = print $ evenSum [1..10]
Next, we can use pattern matching.
-- show
-- Version 3
evenSum l = accumSum 0 l
accumSum {-hi-}n []{-/hi-} = n
accumSum {-hi-}n (x:xs){-/hi-} =
if even x
then accumSum (n+x) xs
else accumSum n xs
-- /show
main = print $ evenSum [1..10]
What is pattern matching? Use values instead of general parameter names (For the brave, a more complete explanation of pattern matching can be found here).
Instead of saying: foo l = if l == [] then <x> else <y> You simply state:
foo [] = <x>
foo l = <y>
But pattern matching goes even further. It is also able to inspect the inner data of a complex value. We can replace
foo l = let x = head l
xs = tail l
in if even x
then foo (n+x) xs
else foo n xs
foo (x:xs) = if even x
then foo (n+x) xs
else foo n xs
This is a very useful feature. It makes our code both terser and easier to read.
In Haskell you can simplify function definition by η-reducing them. For example, instead of writing:
f x = (some expresion) x
you can simply write
f = some expression
Simplify the function evenSum by η-reducing it.
-- show
-- Version 3
evenSum {-hi-}l{-/hi-} = accumSum 0 {-hi-}l{-/hi-}
accumSum n [] = n
accumSum n (x:xs) =
if even x
then accumSum (n+x) xs
else accumSum n xs
-- /show
main = print $ evenSum [1..10]
We use this method to remove the l:
-- show
-- Version 4
evenSum :: Integral a => [a] -> a
evenSum = accumSum 0
accumSum n [] = n
accumSum n (x:xs) =
if even x
then accumSum (n+x) xs
else accumSum n xs
-- /show
main = print $ evenSum [1..10]
To make things even better we should use higher order functions. What are these beasts? Higher order functions are functions taking functions as parameter.
Here are some examples:
filter :: (a -> Bool) -> [a] -> [a]
map :: (a -> b) -> [a] -> [b]
foldl :: (a -> b -> a) -> a -> [b] -> a
Let's proceed by small steps.
-- show
-- Version 5
evenSum l = mysum 0 (filter even l)
mysum n [] = n
mysum n (x:xs) = mysum (n+x) xs
-- /show
main = print $ evenSum [1..10]
filter even [1..10] ⇔ [2,4,6,8,10]
The function filter takes a function of type (a -> Bool) and a list of type [a]. It returns a list containing only elements for which the function returned true.
Our next step is to use another way to simulate a loop. We will use the foldl function to accumulate a value. The function foldl captures a general coding pattern:
myfunc list = foo initialValue list
foo accumulated [] = accumulated
foo tmpValue (x:xs) = foo (binop tmpValue x) xs
Which can be replaced by:
myfunc list = foldl binop initialValue list
If you really want to know how the magic works. Here is the definition of foldl.
foldl f z [] = z
foldl f z (x:xs) = foldl f (f z x) xs
foldl f z [x1,...,xn]
⇔ f (... (f (f z x1) x2) ...) xn
But as Haskell is lazy, it doesn't evaluate (f z x) and pushes it to the stack. This is why we generally use foldl' instead of foldl; foldl' is a strict version of foldl. If you don't understand what
lazy and strict means, don't worry, just follow the code as if foldl and foldl' where identical.
Now our new version of evenSum becomes:
-- show
-- Version 6
-- foldl' isn't accessible by default
-- we need to import it from the module Data.List
import Data.List
evenSum l = foldl' mysum 0 (filter even l)
where mysum acc value = acc + value
-- /show
main = print $ evenSum [1..10]
We can simplify by using directly a lambda notation. This way we don't have to create the temporary name mysum.
-- show
-- Version 7
-- Generally it is considered a good practice
-- to import only the necessary function(s)
import Data.List (foldl')
evenSum l = foldl' (\x y -> x+y) 0 (filter even l)
-- /show
main = print $ evenSum [1..10]
And of course, we note that
(\x y -> x+y) ⇔ (+)
-- show
-- Version 8
import Data.List (foldl')
evenSum :: Integral a => [a] -> a
evenSum l = foldl' (+) 0 (filter even l)
-- /show
main = print $ evenSum [1..10]
foldl' isn't the easiest function to intuit. If you are not used to it, you should study it a bit.
To help you understand what's going on here, a step by step evaluation:
evenSum [1,2,3,4]
⇒ foldl' (+) 0 (filter even [1,2,3,4])
⇒ foldl' (+) 0 [2,4]
⇒ foldl' (+) (0+2) [4]
⇒ foldl' (+) 2 [4]
⇒ foldl' (+) (2+4) []
⇒ foldl' (+) 6 []
⇒ 6
Rewrite the following program using foldl'
import Data.List (foldl')
-- show prod [3,4,5] will return 3*4*5=60
prod :: [Integer] -> Integer
prod [] = 1
prod (x:xs) = x*prod xs
main = print $ prod [3,4,5]
import Data.List (foldl')
-- show
prod = foldl' (*) 1
-- /show
main = print $ prod [3,4,5]
Another useful higher order function is (.). The (.) function corresponds to the mathematical composition.
(f . g . h) x ⇔ f ( g (h x))
We can take advantage of this operator to η-reduce our function:
-- show
-- Version 9
import Data.List (foldl')
evenSum :: Integral a => [a] -> a
evenSum = (foldl' (+) 0) . (filter even)
-- /show
main = do print $ evenSum [1..10]
Also, we could rename some parts to make it clearer:
-- show
-- Version 10
import Data.List (foldl')
sum' :: (Num a) => [a] -> a
sum' = foldl' (+) 0
evenSum :: Integral a => [a] -> a
evenSum = sum' . (filter even)
-- /show
main = do print $ evenSum [1..10]
It is time to discuss a bit. What did we gain by using higher order functions?
At first, you can say it is terseness. But in fact, it has more to do with better thinking. Suppose we want to modify slightly our function. We want to get the sum of all even square of element of
the list.
[1,2,3,4] ▷ [1,4,9,16] ▷ [4,16] ▷ 20
Update the version 10 is extremely easy:
squareEvenSum = sum' . (filter even) . (map (^2))
squareEvenSum' = evenSum . (map (^2))
We just had to add another "transformation function"[^0216].
map (^2) [1,2,3,4] ⇔ [1,4,9,16]
The map function simply apply a function to all element of a list.
We didn't had to modify anything inside the function definition. It feels more modular. But in addition you can think more mathematically about your function. You can then use your function as any
other one. You can compose, map, fold, filter using your new function.
To modify version 1 is left as an exercise to the reader ☺.
If you believe we reached the end of generalization, then know you are very wrong. For example, there is a way to not only use this function on lists but on any recursive type. If you want to know
how, I suggest you to read this quite fun article: Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire by Meijer, Fokkinga and Paterson. You could also just get a bit of the idea
by viewing my presentation about Category Theory.
This example should show you how great pure functional programming is. Unfortunately, using pure functional programming isn't well suited to all usages. Or at least such a language hasn't been found
One of the great powers of Haskell is the ability to create DSLs (Domain Specific Language) making it easy to change the programming paradigm.
In fact, Haskell is also great when you want to write imperative style programming. Understanding this was really hard for me when learning Haskell. A lot of effort has been done to explain to you
how much functional approach is superior. Then when you start the imperative style of Haskell, it is hard to understand why and how.
But before talking about this Haskell super-power, we must talk about another essential aspect of Haskell: Types. | {"url":"https://www.schoolofhaskell.com/school/to-infinity-and-beyond/pick-of-the-week/haskell-fast-hard/haskell-fast-hard-part-3","timestamp":"2024-11-03T00:38:31Z","content_type":"text/html","content_length":"39412","record_id":"<urn:uuid:5872fefa-459e-429a-9cbe-db91d4b1c171>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00030.warc.gz"} |
CN118074195B - Distributed energy storage converter integrated system and power distribution method thereof
- Google Patents
CN118074195B - Distributed energy storage converter integrated system and power distribution method thereof - Google Patents
Distributed energy storage converter integrated system and power distribution method thereof Download PDF
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CN118074195B CN118074195B CN202410459406.XA CN202410459406A CN118074195B CN 118074195 B CN118074195 B CN 118074195B CN 202410459406 A CN202410459406 A CN 202410459406A CN 118074195 B CN118074195
B CN 118074195B
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energy storage
storage converter
phase angle
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□ 238000004146 energy storage Methods 0.000 title claims abstract description 337
□ 238000000034 method Methods 0.000 title claims abstract description 22
□ 238000006243 chemical reaction Methods 0.000 claims abstract description 11
□ 230000007935 neutral effect Effects 0.000 claims abstract description 7
□ 230000009466 transformation Effects 0.000 claims abstract description 5
□ 230000002457 bidirectional effect Effects 0.000 claims abstract description 4
□ 238000004364 calculation method Methods 0.000 claims description 35
□ 230000014509 gene expression Effects 0.000 claims description 35
□ 238000007665 sagging Methods 0.000 claims description 24
□ 230000010354 integration Effects 0.000 claims description 8
□ 230000000694 effects Effects 0.000 abstract description 5
□ 230000009286 beneficial effect Effects 0.000 description 9
□ 230000001012 protector Effects 0.000 description 5
□ 238000010586 diagram Methods 0.000 description 3
□ 238000010891 electric arc Methods 0.000 description 2
□ 238000002955 isolation Methods 0.000 description 2
□ 238000012423 maintenance Methods 0.000 description 2
□ 230000007547 defect Effects 0.000 description 1
□ 238000006467 substitution reaction Methods 0.000 description 1
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
☆ H02J3/00—Circuit arrangements for ac mains or ac distribution networks
☆ H02J3/28—Arrangements for balancing of the load in a network by storage of energy
☆ H02J3/32—Arrangements for balancing of the load in a network by storage of energy using batteries with converting means
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
☆ H02J3/00—Circuit arrangements for ac mains or ac distribution networks
☆ H02J3/36—Arrangements for transfer of electric power between ac networks via a high-tension dc link
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
☆ H02J3/00—Circuit arrangements for ac mains or ac distribution networks
☆ H02J3/38—Arrangements for parallely feeding a single network by two or more generators, converters or transformers
☆ H02J3/381—Dispersed generators
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO
SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
☆ H02M7/00—Conversion of ac power input into dc power output; Conversion of dc power input into ac power output
☆ H02M7/66—Conversion of ac power input into dc power output; Conversion of dc power input into ac power output with possibility of reversal
☆ H—ELECTRICITY
☆ H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
☆ H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
☆ H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
☆ H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]
□ Engineering & Computer Science (AREA)
□ Power Engineering (AREA)
□ Supply And Distribution Of Alternating Current (AREA)
The invention discloses a distributed energy storage converter integrated system and a power distribution method thereof, belonging to the technical field of integrated energy storage converters,
wherein the distributed energy storage converter integrated system comprises a plurality of energy storage converter modules which are mutually connected in parallel, an alternating current
circuit breaking module and a metering device; each energy storage conversion module is respectively connected with the alternating current circuit breaking module and the metering device; each
energy storage current transformation module comprises a battery cluster, a direct current breaker and an energy storage current transformer which are sequentially connected; the positive
electrode and the negative electrode of the battery cluster are correspondingly connected with the positive electrode and the negative electrode of the direct current side of the energy storage
converter through a direct current breaker; the alternating current side of the energy storage converter is connected in parallel by adopting a three-phase four-wire system and is respectively
connected with the alternating current circuit breaking module and the input side of the metering device; the output side of the metering device and the neutral line are externally connected with
a power distribution system, so that grid connection is realized; the energy storage converter is used for carrying out bidirectional conversion on direct current and alternating current. The
system solves the problem of insufficient equalization effect of mixed use of new and old battery clusters by combining a power distribution method.
Distributed energy storage converter integrated system and power distribution method thereof
The invention belongs to the technical field of integrated energy storage converters, and particularly relates to a distributed energy storage converter integrated system and a power distribution
method thereof.
The existing energy storage converter generally adopts a centralized integration mode, and power devices therein are assembled in a centralized way, so that only one path of interface is arranged
on the direct current side. When the existing energy storage converter is used, the battery clusters are connected in parallel and converged, and then connected into a direct current interface of
the centralized energy storage converter.
The existing energy storage converter is mainly balanced and controlled by the BMS due to the fact that the existing energy storage converter is required to be connected in parallel at the direct
current side, and the existing energy storage converter has a certain balancing effect on a new battery system no matter in active balancing or passive balancing, but the BMS is difficult to
achieve an ideal balancing effect after the batteries are aged, and particularly the balancing of deviation among battery clusters is very difficult. When the system capacity reaches a certain
degree, the problems of direct current arc discharge, parallel capacity loss at the direct current side, parallel circulation among battery clusters and the like can occur, the safety and the
efficiency of the energy storage power station are affected, and the whole system is down if the whole system fails due to the fact that the power devices of the existing energy storage current
transformer are piled up in a concentrated mode. The distributed energy storage is low in general capacity and dispersed in regions, and once the energy storage converter fails, the problems of
long after-sale period and difficult maintenance are met.
Disclosure of Invention
In order to overcome the defects in the prior art, the distributed energy storage converter integrated system and the power distribution method thereof provided by the invention have the
advantages that by configuring an independent energy storage converter module for each battery cluster and connecting the alternating current sides of the energy storage converters in parallel,
the consistency requirements among the battery clusters are greatly reduced, even the mixed use of new and old battery clusters can be realized based on the power distribution method, and the
problem of insufficient mixed use balance effect of the new and old battery clusters is solved.
In order to achieve the aim of the invention, the invention adopts the following technical scheme:
on one hand, the distributed energy storage converter integrated system provided by the invention comprises a plurality of energy storage converter modules which are mutually connected in
parallel, an alternating current circuit breaking module and a metering device;
each energy storage conversion module is respectively connected with the alternating current circuit breaking module and the metering device;
Each energy storage current transformation module comprises a battery cluster, a direct current breaker and an energy storage current transformer which are sequentially connected; the positive
electrode and the negative electrode of the battery cluster are correspondingly connected with the positive electrode and the negative electrode of the direct current side of the energy storage
converter through a direct current breaker; the alternating current side of the energy storage converter is connected in parallel by adopting a three-phase four-wire system and is respectively
connected with the alternating current circuit breaking module and the input side of the metering device; the output side of the metering device and the neutral line are externally connected with
a power distribution system, so that grid connection is realized; the energy storage converter is used for carrying out bidirectional conversion on direct current and alternating current.
The beneficial effects of the invention are as follows: according to the distributed energy storage converter integrated system, when one or a part of energy storage converter modules fail, the
rest energy storage converter modules can still continue to operate, the system is maintained to be stable, the size of a single energy storage converter module is small, the weight is light,
factory returning maintenance is facilitated, and the reliability and maintainability of the distributed energy storage converter integrated system are ensured; by matching a single energy
storage converter for each battery cluster and connecting the alternating current sides of all the energy storage converters in parallel, the direct current sides of the energy storage converters
are not required to be connected in parallel, the consistency requirement among the battery clusters is greatly reduced, the flexibility of power configuration of the energy storage converters is
greatly enhanced, and even the mixed use of a new battery cabinet and an old battery cabinet can be realized; according to the invention, the current of the direct current side of the energy
storage converter is reduced, the arc discharge and the capacity loss of the direct current side are effectively improved, and the safety performance and the efficiency of the system are
Further, the power of each energy storage current transformation module is not identical.
The beneficial effects of adopting the further scheme are as follows: the distributed energy storage converter integrated system provided by the invention does not require the power to be
completely consistent for each energy storage converter module connected in parallel, and can be combined with a power distribution method to realize the mixed use of new and old battery
On the other hand, the invention also provides a power distribution method based on the distributed energy storage converter integrated system, which comprises the following steps:
S1, acquiring active power and reactive power output to grid-connected points by an alternating current side of each energy storage converter, and constructing a sagging control model;
S2, constructing a phase angle difference model of the parallel energy storage converters based on active power and sagging control models output to grid-connected points at alternating current
sides of the energy storage converters;
S3, neglecting system loss, and obtaining a phase angle power correlation model of the parallel energy storage converter based on a phase angle difference model of the parallel energy storage
S4, obtaining a fuzzy active power ratio model of the parallel energy storage converter based on a phase angle power correlation model of the parallel energy storage converter according to
high-resistance low-inductance characteristics of the power distribution network line;
S5, obtaining an output power distribution model and a target power model of the parallel energy storage converter based on the fuzzy active power ratio model of the parallel energy storage
and S6, performing power distribution of the energy storage converters with different rated powers based on an output power distribution model and a target power model of the parallel energy
storage converters so as to mix and connect the energy storage converters with different powers in parallel.
The beneficial effects of the invention are as follows: according to the power distribution method based on the distributed energy storage converter integrated system, the balanced coordination
and parallel mixed use of the battery clusters with different powers can be realized through the analysis of the power and the phase angle based on the distributed energy storage converter
integrated system, so that the balanced performance between the old battery and the new battery which are used for a period of time is effectively improved, and the stable operation of the
distributed energy storage converter integrated system is effectively ensured.
Further, the step S1 includes the following steps:
S11, acquiring active power and reactive power from an alternating current output side of each energy storage converter to a grid-connected point;
The calculation expressions of the active power and the reactive power from the alternating current output side of the energy storage converter to the grid-connected point are as follows:
Wherein P represents the active power output to the grid-connected point by the alternating current side of the energy storage converter, U represents the voltage amplitude of the alternating
current side of the energy storage converter, U [W] represents the voltage of the grid-connected point, delta represents the phase angle of the alternating current side of the energy storage
converter, delta [W] represents the phase angle of the grid-connected point, and X [W] represents the reactance of a line between the energy storage converter and the grid-connected point;
S12, enabling the difference value between the phase angle of the grid-connected point and the phase angle of the alternating current side of the energy storage converters to be smaller than a
preset phase angle difference threshold value, and obtaining an active power phase angle correlation model and a reactive power amplitude correlation model based on active power and reactive
power output to the grid-connected point by the alternating current side of each energy storage converter;
the calculation expressions of the active power phasor correlation model and the reactive power voltage amplitude correlation model are respectively as follows:
s13, constructing a sagging control model based on the active power phase angle correlation model and the reactive power amplitude correlation model;
the computational expression of the droop control model is as follows:
Wherein m represents a phase angle control parameter, δ [e] represents a rated phase angle, P [e] represents a rated active power, n represents an amplitude control parameter, U [e] represents a
rated voltage amplitude, and Q [e] represents a rated reactive power.
The beneficial effects of adopting the further scheme are as follows: according to the invention, according to the reactance relation from the energy storage converter to the grid-connected point
in each energy storage converter module, the active power and reactive power from the alternating-current output side of each energy storage converter to the grid-connected point are obtained,
and the active power phase angle correlation model, the reactive power amplitude correlation model and the sagging control model are constructed based on the condition that the phase angle
difference between the phase angle of the grid point and the alternating-current side of the energy storage converter is smaller by setting the phase angle difference threshold value, so that a
foundation is provided for analyzing the relation between the phase angle and the power of each energy storage converter module in a sagging control mode and realizing the parallel mixed use of
new and old battery clusters.
Further, the step S2 includes the following steps:
s21, constructing a grid-connected point phase angle difference model based on active power output to the grid-connected point by the output side of each energy storage converter;
The calculation expression of the grid-connected point phase angle difference model of the parallel energy storage converter is as follows:
Wherein δ [i] represents a phase angle of the ac side of the ith parallel energy storage converter, δ [j] represents a phase angle of the ac side of the jth parallel energy storage converter, P
[i] represents an active power output from the ac side of the ith parallel energy storage converter to the grid-connected point, P [j] represents an active power output from the ac side of the
jth parallel energy storage converter to the grid-connected point, ω represents an angular frequency, L [i] represents an output inductance corresponding to the ac side of the ith parallel energy
storage converter, U [i] represents a voltage amplitude of the ac side of the ith energy storage converter, L [Di] represents a line inductance corresponding to the jth parallel energy storage
converter, L [j] represents a voltage amplitude of the ac side of the jth parallel energy storage converter, L [Dj] represents a line inductance in the jth energy storage converter module,
wherein i, j=1, 2, …, N, and i+noteq, N are the total number of energy storage converter modules;
S22, constructing a sagging model of the parallel energy storage converter based on the sagging control model;
The calculation expression of the sagging model of the parallel energy storage converter is as follows:
Wherein δ [ei] represents a rated droop phase angle corresponding to the ith parallel energy storage converter, m [i] represents a phase angle control parameter corresponding to the ith parallel
energy storage converter, P [ei] represents a rated active power corresponding to the ith parallel energy storage converter, δ [ej] represents a rated droop phase angle corresponding to the jth
parallel energy storage converter, m [j] represents a phase angle control parameter corresponding to the jth parallel energy storage converter, and P [ej] represents a rated active power
corresponding to the jth parallel energy storage converter;
s23, constructing a phase angle difference model of the parallel energy storage converter based on a grid-connected point phase angle difference model and a sagging model of the parallel energy
storage converter;
the calculation expression of the phase angle difference model of the parallel energy storage converter is as follows:
The beneficial effects of adopting the further scheme are as follows: the invention provides a specific method for constructing a phase angle difference model of parallel energy storage
converters based on active power and sagging control models output to grid-connected points at alternating sides of the energy storage converters, and provides a basis for analyzing the phase
angle power relation of the parallel energy storage converters on the basis of the phase angle difference model of the parallel energy storage converters under the condition of neglecting system
Further, the calculation expression of the phase angle power correlation model of the parallel energy storage converter is as follows:
Wherein P [L] represents the total power of the parallel energy storage converters.
The beneficial effects of adopting the further scheme are as follows: the invention provides a calculation method of a phase angle power model of a parallel energy storage converter, which
provides a basis for phase angle power analysis among parallel energy storage converter modules.
Further, the step S4 includes the following steps:
s41, obtaining an active power ratio model of the parallel energy storage converter based on a phase angle power correlation model of the parallel energy storage converter;
the calculation expression of the active power ratio model of the parallel energy storage converter is as follows:
s42, based on an active power ratio model of the parallel energy storage converter, combining high-resistance low-inductance characteristics of a power distribution network line to obtain a fuzzy
active power ratio model of the parallel energy storage converter;
The calculation expression of the fuzzy active power ratio model of the parallel energy storage converter is as follows:
The beneficial effects of adopting the further scheme are as follows: the invention provides a method for obtaining a fuzzy active power ratio model of a parallel energy storage converter based
on a phase angle power correlation model of the parallel energy storage converter according to high resistance and low inductance characteristics of a power distribution network line.
Further, the calculation expression of the output power distribution model of the parallel energy storage converter is as follows:
Wherein P [1] represents the active power output from the ac side of the 1 st parallel energy storage converter to the grid-connected point, m [1] represents the phase angle control parameter
corresponding to the 1 st parallel energy storage converter, P [2] represents the active power output from the ac side of the 2 nd parallel energy storage converter to the grid-connected point, m
[2] represents the phase angle control parameter corresponding to the 2 nd parallel energy storage converter, P [k] represents the active power output from the ac side of the k parallel energy
storage converter to the grid-connected point, m [k] represents the phase angle control parameter corresponding to the k parallel energy storage converter, P [N] represents the active power
output from the ac side of the N parallel energy storage converter to the grid-connected point, m [N] represents the phase angle control parameter corresponding to the N parallel energy storage
converter, P [e1] represents the rated active power corresponding to the 1 st parallel energy storage converter, P [e2] represents the rated active power corresponding to the 2 nd parallel energy
storage converter, P [ek] represents the active power corresponding to the k parallel energy storage converter, P [eN] represents the rated active power corresponding to the N energy storage
converter, wherein n=1, n=98.
The beneficial effects of adopting the further scheme are as follows: compared with the prior art that parallel operation is required to be performed by the same rated power, the calculation
method of the output power distribution model of the parallel energy storage converter provided by the invention can be used for distributing the output power among energy storage conversion
modules with different rated powers, namely, equalization among battery clusters is realized.
Further, the calculation expression of the target power model of the parallel energy storage converter is as follows:
Wherein P [target] represents the converter integration target power.
The beneficial effects of adopting the further scheme are as follows: the invention provides a calculation method of a target power model of a parallel energy storage converter, wherein the
target power of the parallel energy storage converter is the sum of the power of all the parallel energy storage converters, and the equalization effect of a battery cluster can be improved by
combining an output power distribution model of the parallel energy storage converter.
Other advantages that are also present with respect to the present invention will be more detailed in the following examples.
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings that are needed in the embodiments will be briefly described below, it being
understood that the following drawings only illustrate some embodiments of the present invention and therefore should not be considered as limiting the scope, and other related drawings may be
obtained according to these drawings without inventive effort for a person skilled in the art.
Fig. 1 is a schematic diagram of a distributed energy storage converter integrated system according to embodiment 1 of the present invention.
Fig. 2 is a flow chart of steps of a power distribution method based on a distributed energy storage converter integrated system in embodiment 2 of the present invention.
Fig. 3 is a schematic diagram of a line power reactance between a single energy storage converter module and a grid-connected point in embodiment 2 of the present invention.
Fig. 4 is a schematic diagram of a line power reactance between a parallel energy storage converter module and a grid-connected point in embodiment 2 of the present invention.
Wherein, CT, current transformer; a PA ammeter; QF1, a first alternating current breaker; QF2, a second ac circuit breaker; SPD, surge protector; PEN, neutral.
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments
described are only some embodiments of the present invention, but not all embodiments. The components of the embodiments of the present invention generally described and illustrated in the
figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the invention, as presented in the
figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be made by a
person skilled in the art without making any inventive effort, are intended to be within the scope of the present invention.
BMS (Battery MANAGEMENT SYSTEM), battery management system.
Example 1:
As shown in fig. 1, in one aspect, in an embodiment of the present invention, the present invention provides a distributed energy storage converter integrated system, including a plurality of
energy storage converter modules connected in parallel with each other, and an ac disconnection module and a metering device;
each energy storage conversion module is respectively connected with the alternating current circuit breaking module and the metering device;
Each energy storage current transformation module comprises a battery cluster, a direct current breaker and an energy storage current transformer which are sequentially connected; the positive
electrode and the negative electrode of the battery cluster are correspondingly connected with the positive electrode and the negative electrode of the direct current side of the energy storage
converter through a direct current breaker; the alternating current side of the energy storage converter is connected in parallel by adopting a three-phase four-wire system and is respectively
connected with the alternating current circuit breaking module and the input side of the metering device; the output side of the metering device and the neutral line are externally connected with
a power distribution system, so that grid connection is realized; the energy storage converter is used for carrying out bidirectional conversion on direct current and alternating current.
According to the invention, the alternating current sides of the energy storage converters are connected in parallel by adopting a three-phase four-wire system, so that an isolation transformer
is omitted, the cost of the system is effectively reduced, and the potential safety hazard of the fault of the isolation transformer to the system is eliminated.
In this embodiment, the ac short circuit module includes a second ac breaker QF2 and a surge protector SPD, one end of the second ac breaker QF2 is connected to the grid-connected point of each
energy storage converter, the other end of the second ac breaker QF2 is connected to the surge protector SPD, in this embodiment, the stationary end of the second ac breaker QF2 is connected to
the grid-connected point of each energy storage converter, the moving end of the second ac breaker QF2 is connected to the surge protector, and the surge protector is grounded; the metering
device comprises a current transformer CT, an ammeter PA, a first alternating current breaker QF1 and a neutral line PEN; the current transformers CT are arranged on A-phase lines, B-phase lines
and C-phase lines between grid connection points of alternating current side output ends of the energy storage converters and the power distribution system, the ammeter PA is arranged on neutral
lines PEN between grid connection points of alternating current side output ends of the energy storage converters and the power distribution system, and the first alternating current circuit
breaker QF1 is arranged between the current transformers CT on the A-phase lines, the B-phase lines and the C-phase lines and the power distribution system.
In this embodiment, the power of each energy storage converter module is not exactly the same. The distributed energy storage converter integrated system provided by the invention is also
suitable for the situations that the power of each energy storage converter module is completely different, and the power of each energy storage converter module is partially the same or
completely the same.
Example 2:
As shown in fig. 2, in another aspect, the present invention further provides a power distribution method based on a distributed energy storage converter integrated system, including the
following steps:
S1, acquiring active power and reactive power output to grid-connected points by an alternating current side of each energy storage converter, and constructing a sagging control model;
The step S1 comprises the following steps:
S11, acquiring active power and reactive power from an alternating current output side of each energy storage converter to a grid-connected point;
As shown in fig. 3, reactance exists in a line between a single energy storage converter module and a grid-connected point, and active power and reactive power from the alternating current output
end of each energy storage converter to the grid-connected point can be obtained from parameters such as voltage peak value and phase angle at the grid-connected point.
The calculation expressions of the active power and the reactive power from the alternating current output side of the energy storage converter to the grid-connected point are as follows:
Wherein P represents the active power output to the grid-connected point by the alternating current side of the energy storage converter, U represents the voltage amplitude of the alternating
current side of the energy storage converter, U [W] represents the voltage of the grid-connected point, delta represents the phase angle of the alternating current side of the energy storage
converter, delta [W] represents the phase angle of the grid-connected point, and X [W] represents the reactance of a line between the energy storage converter and the grid-connected point;
S12, enabling the difference value between the phase angle of the grid-connected point and the phase angle of the alternating current side of the energy storage converters to be smaller than a
preset phase angle difference threshold value, and obtaining an active power phase angle correlation model and a reactive power amplitude correlation model based on active power and reactive
power output to the grid-connected point by the alternating current side of each energy storage converter;
the calculation expressions of the active power phasor correlation model and the reactive power voltage amplitude correlation model are respectively as follows:
s13, constructing a sagging control model based on the active power phase angle correlation model and the reactive power amplitude correlation model;
the computational expression of the droop control model is as follows:
Wherein m represents a phase angle control parameter, δ [e] represents a rated phase angle, P [e] represents a rated active power, n represents an amplitude control parameter, U [e] represents a
rated voltage amplitude, and Q [e] represents a rated reactive power.
S2, constructing a phase angle difference model of the parallel energy storage converters based on active power and sagging control models output to grid-connected points at alternating current
sides of the energy storage converters;
The step S2 comprises the following steps:
s21, constructing a grid-connected point phase angle difference model based on active power output to the grid-connected point by the output side of each energy storage converter;
The calculation expression of the grid-connected point phase angle difference model of the parallel energy storage converter is as follows:
Wherein δ [i] represents a phase angle of the ac side of the ith parallel energy storage converter, δ [j] represents a phase angle of the ac side of the jth parallel energy storage converter, P
[i] represents an active power output from the ac side of the ith parallel energy storage converter to the grid-connected point, P [j] represents an active power output from the ac side of the
jth parallel energy storage converter to the grid-connected point, ω represents an angular frequency, L [i] represents an output inductance corresponding to the ac side of the ith parallel energy
storage converter, U [i] represents a voltage amplitude of the ac side of the ith energy storage converter, L [Di] represents a line inductance corresponding to the jth parallel energy storage
converter, L [j] represents a voltage amplitude of the ac side of the jth parallel energy storage converter, L [Dj] represents a line inductance in the jth energy storage converter module,
wherein i, j=1, 2, …, N, and i+noteq, N are the total number of energy storage converter modules;
S22, constructing a sagging model of the parallel energy storage converter based on the sagging control model;
The calculation expression of the sagging model of the parallel energy storage converter is as follows:
Wherein δ [ei] represents a rated droop phase angle corresponding to the ith parallel energy storage converter, m [i] represents a phase angle control parameter corresponding to the ith parallel
energy storage converter, P [ei] represents a rated active power corresponding to the ith parallel energy storage converter, δ [ej] represents a rated droop phase angle corresponding to the jth
parallel energy storage converter, m [j] represents a phase angle control parameter corresponding to the jth parallel energy storage converter, and P [ej] represents a rated active power
corresponding to the jth parallel energy storage converter;
In the embodiment, when the active power output and the phase angle of the parallel energy storage converter are both 0, the rated output power of the parallel energy storage converter can be
the calculation expression of the rated output power of the parallel energy storage converter is as follows:
s23, constructing a phase angle difference model of the parallel energy storage converter based on a grid-connected point phase angle difference model and a sagging model of the parallel energy
storage converter;
the calculation expression of the phase angle difference model of the parallel energy storage converter is as follows:
S3, neglecting system loss, and obtaining a phase angle power correlation model of the parallel energy storage converter based on a phase angle difference model of the parallel energy storage
the calculation expression of the phase angle power correlation model of the parallel energy storage converter is as follows:
Wherein P [L] represents the total power of the parallel energy storage converters.
As shown in fig. 4, in this embodiment, the case where there are only two parallel energy storage converters is exemplified, the voltage amplitude of the ac side of the 1 st energy storage
converter is U [1], the phase angle of the ac side of the 1 st parallel energy storage converter is δ [1], and the output inductance corresponding to the 1 st parallel energy storage converter is
L [1], The active power output to the grid-connected point by the alternating-current side of the 1 st parallel energy storage converter is P [1], the reactive power output to the grid-connected
point by the alternating-current side of the 1 st parallel energy storage converter is Q [1], the voltage amplitude of the alternating current side of the 2 nd energy storage converter is U [2],
the phase angle of the alternating current side of the 2 nd parallel energy storage converter is delta [2], the output inductance corresponding to the 2 nd parallel energy storage converter is L
[2], the line inductance L [D2] in the 2 nd energy storage converter module and the line resistance R [D2] in the 2 nd energy storage converter module, The active power output from the
alternating-current side of the 2 nd parallel energy storage converters to the grid-connected point is P [2], the reactive power output from the alternating-current side of the 2 nd parallel
energy storage converters to the grid-connected point is Q [2], the total power of the corresponding parallel energy storage converters at the load of the final parallel point is P [L], The
reactive power of the parallel energy storage converters corresponding to the load of the parallel point is Q [L], if P [L]=P[1]+P[2] exists, the calculation expression of the phase angle power
correlation model when two parallel energy storage converters exist is as follows:
Wherein, P [1] represents the active power output from the ac side of the 1 st parallel energy storage converter to the grid-connected point, P [2] represents the active power output from the ac
side of the 2 nd parallel energy storage converter to the grid-connected point, L [1] represents the output inductance corresponding to the 1 st parallel energy storage converter, U [1]
represents the voltage amplitude of the ac side of the 1 st parallel energy storage converter, L [D1] represents the line inductance in the 1 st energy storage converter module, L [2] represents
the output inductance corresponding to the 2 nd parallel energy storage converter, U [2] represents the voltage amplitude of the ac side of the 2 nd energy storage converter, L [D2] represents
the line inductance in the 2 nd energy storage converter module, m [1] represents the phase angle control parameter corresponding to the 1 st parallel energy storage converter, m [2] represents
the phase angle control parameter corresponding to the 2 nd parallel energy storage converter; the active power output to the grid-connected point from the alternating current side of the 1 st
energy storage converter and the 2 nd energy storage converter which are connected in parallel can be obtained respectively on the basis of the second time, and the active power is as follows:
S4, obtaining a fuzzy active power ratio model of the parallel energy storage converter based on a phase angle power correlation model of the parallel energy storage converter according to
high-resistance low-inductance characteristics of the power distribution network line;
the step S4 comprises the following steps:
s41, obtaining an active power ratio model of the parallel energy storage converter based on a phase angle power correlation model of the parallel energy storage converter;
the calculation expression of the active power ratio model of the parallel energy storage converter is as follows:
s42, based on an active power ratio model of the parallel energy storage converter, combining high-resistance low-inductance characteristics of a power distribution network line to obtain a fuzzy
active power ratio model of the parallel energy storage converter;
The calculation expression of the fuzzy active power ratio model of the parallel energy storage converter is as follows:
S5, obtaining an output power distribution model and a target power model of the parallel energy storage converter based on the fuzzy active power ratio model of the parallel energy storage
The calculation expression of the output power distribution model of the parallel energy storage converter is as follows:
Wherein P [1] represents the active power output from the ac side of the 1 st parallel energy storage converter to the grid-connected point, m [1] represents the phase angle control parameter
corresponding to the 1 st parallel energy storage converter, P [2] represents the active power output from the ac side of the 2 nd parallel energy storage converter to the grid-connected point, m
[2] represents the phase angle control parameter corresponding to the 2 nd parallel energy storage converter, P [k] represents the active power output from the ac side of the k parallel energy
storage converter to the grid-connected point, m [k] represents the phase angle control parameter corresponding to the k parallel energy storage converter, P [N] represents the active power
output from the ac side of the N parallel energy storage converter to the grid-connected point, m [N] represents the phase angle control parameter corresponding to the N parallel energy storage
converter, P [e1] represents the rated active power corresponding to the 1 st parallel energy storage converter, P [e2] represents the rated active power corresponding to the 2 nd parallel energy
storage converter, P [ek] represents the active power corresponding to the k parallel energy storage converter, P [eN] represents the rated active power corresponding to the N energy storage
converter, wherein n=1, n=98.
The calculation expression of the target power model of the parallel energy storage converter is as follows:
Wherein P [target] represents the converter integration target power.
And S6, performing power distribution of the energy storage converters with different rated powers based on an output power distribution model and a target power model of the parallel energy
storage converters so as to mix and connect the energy storage converters with different powers in parallel.
The foregoing is merely illustrative of the present invention, and the present invention is not limited thereto, and any person skilled in the art will readily recognize that variations or
substitutions are within the scope of the present invention.
Claims (7)
1. The power distribution method based on the distributed energy storage converter integrated system is characterized in that the distributed energy storage converter integrated system comprises
a plurality of energy storage conversion modules which are mutually connected in parallel, an alternating current circuit breaking module and a metering device;
each energy storage conversion module is respectively connected with the alternating current circuit breaking module and the metering device;
each energy storage current transformation module comprises a battery cluster, a direct current breaker and an energy storage current transformer which are sequentially connected; the positive
electrode and the negative electrode of the battery cluster are correspondingly connected with the positive electrode and the negative electrode of the direct current side of the energy storage
converter through a direct current breaker; the alternating current side of the energy storage converter is connected in parallel by adopting a three-phase four-wire system and is respectively
connected with the alternating current circuit breaking module and the input side of the metering device; the output side of the metering device and the neutral line are externally connected with
a power distribution system, so that grid connection is realized; the energy storage converter is used for carrying out bidirectional conversion on direct current and alternating current; the
power of each energy storage current conversion module is not completely the same;
the power distribution method comprises the following steps:
S1, acquiring active power and reactive power output to grid-connected points by an alternating current side of each energy storage converter, and constructing a sagging control model;
S2, constructing a phase angle difference model of the parallel energy storage converters based on active power and sagging control models output to grid-connected points at alternating current
sides of the energy storage converters;
S3, neglecting system loss, and obtaining a phase angle power correlation model of the parallel energy storage converter based on a phase angle difference model of the parallel energy storage
S4, obtaining a fuzzy active power ratio model of the parallel energy storage converter based on a phase angle power correlation model of the parallel energy storage converter according to
high-resistance low-inductance characteristics of the power distribution network line;
S5, obtaining an output power distribution model and a target power model of the parallel energy storage converter based on the fuzzy active power ratio model of the parallel energy storage
and S6, performing power distribution of the energy storage converters with different rated powers based on an output power distribution model and a target power model of the parallel energy
storage converters so as to mix and connect the energy storage converters with different powers in parallel.
2. The power distribution method based on the distributed energy storage converter integrated system according to claim 1, wherein S1 comprises the steps of:
S11, acquiring active power and reactive power from an alternating current output side of each energy storage converter to a grid-connected point;
The calculation expressions of the active power and the reactive power from the alternating current output side of the energy storage converter to the grid-connected point are as follows:
Wherein P represents the active power output to the grid-connected point by the alternating current side of the energy storage converter, U represents the voltage amplitude of the alternating
current side of the energy storage converter, U [W] represents the voltage of the grid-connected point, delta represents the phase angle of the alternating current side of the energy storage
converter, delta [W] represents the phase angle of the grid-connected point, and X [W] represents the reactance of a line between the energy storage converter and the grid-connected point;
S12, enabling the difference value between the phase angle of the grid-connected point and the phase angle of the alternating current side of the energy storage converters to be smaller than a
preset phase angle difference threshold value, and obtaining an active power phase angle correlation model and a reactive power amplitude correlation model based on active power and reactive
power output to the grid-connected point by the alternating current side of each energy storage converter;
the calculation expressions of the active power phasor correlation model and the reactive power voltage amplitude correlation model are respectively as follows:
s13, constructing a sagging control model based on the active power phase angle correlation model and the reactive power amplitude correlation model;
the computational expression of the droop control model is as follows:
Wherein m represents a phase angle control parameter, δ [e] represents a rated phase angle, P [e] represents a rated active power, n represents an amplitude control parameter, U [e] represents a
rated voltage amplitude, and Q [e] represents a rated reactive power.
3. The power distribution method based on the distributed energy storage converter integrated system according to claim 2, wherein the step S2 includes the steps of:
s21, constructing a grid-connected point phase angle difference model based on active power output to the grid-connected point by the output side of each energy storage converter;
The calculation expression of the grid-connected point phase angle difference model of the parallel energy storage converter is as follows:
Wherein δ [i] represents a phase angle of the ac side of the ith parallel energy storage converter, δ [j] represents a phase angle of the ac side of the jth parallel energy storage converter, P
[i] represents an active power output from the ac side of the ith parallel energy storage converter to the grid-connected point, P [j] represents an active power output from the ac side of the
jth parallel energy storage converter to the grid-connected point, ω represents an angular frequency, L [i] represents an output inductance corresponding to the ac side of the ith parallel energy
storage converter, U [i] represents a voltage amplitude of the ac side of the ith energy storage converter, L [Di] represents a line inductance corresponding to the jth parallel energy storage
converter, L [j] represents a voltage amplitude of the ac side of the jth parallel energy storage converter, L [Dj] represents a line inductance in the jth energy storage converter module,
wherein i, j=1, 2, …, N, and i+noteq, N are the total number of energy storage converter modules;
S22, constructing a sagging model of the parallel energy storage converter based on the sagging control model;
The calculation expression of the sagging model of the parallel energy storage converter is as follows:
Wherein δ [ei] represents a rated droop phase angle corresponding to the ith parallel energy storage converter, m [i] represents a phase angle control parameter corresponding to the ith parallel
energy storage converter, P [ei] represents a rated active power corresponding to the ith parallel energy storage converter, δ [ej] represents a rated droop phase angle corresponding to the jth
parallel energy storage converter, m [j] represents a phase angle control parameter corresponding to the jth parallel energy storage converter, and P [ej] represents a rated active power
corresponding to the jth parallel energy storage converter;
s23, constructing a phase angle difference model of the parallel energy storage converter based on a grid-connected point phase angle difference model and a sagging model of the parallel energy
storage converter;
the calculation expression of the phase angle difference model of the parallel energy storage converter is as follows:
4. The distributed energy storage converter integration system-based power distribution method according to claim 3, wherein the calculation expression of the phase angle power correlation model
of the parallel energy storage converter is as follows:
wherein, & represents and P [L] represents the total power of the energy storage converters connected in parallel.
5. The method for power distribution based on a distributed energy storage converter integration system according to claim 4, wherein S4 comprises the steps of:
s41, obtaining an active power ratio model of the parallel energy storage converter based on a phase angle power correlation model of the parallel energy storage converter;
the calculation expression of the active power ratio model of the parallel energy storage converter is as follows:
s42, based on an active power ratio model of the parallel energy storage converter, combining high-resistance low-inductance characteristics of a power distribution network line to obtain a fuzzy
active power ratio model of the parallel energy storage converter;
The calculation expression of the fuzzy active power ratio model of the parallel energy storage converter is as follows:
6. The distributed energy storage converter integration system-based power distribution method according to claim 5, wherein the calculation expression of the output power distribution model of
the parallel energy storage converters is as follows:
Wherein P [1] represents the active power output from the ac side of the 1 st parallel energy storage converter to the grid-connected point, m [1] represents the phase angle control parameter
corresponding to the 1 st parallel energy storage converter, P [2] represents the active power output from the ac side of the 2 nd parallel energy storage converter to the grid-connected point, m
[2] represents the phase angle control parameter corresponding to the 2 nd parallel energy storage converter, P [k] represents the active power output from the ac side of the k parallel energy
storage converter to the grid-connected point, m [k] represents the phase angle control parameter corresponding to the k parallel energy storage converter, P [N] represents the active power
output from the ac side of the N parallel energy storage converter to the grid-connected point, m [N] represents the phase angle control parameter corresponding to the N parallel energy storage
converter, P [e1] represents the rated active power corresponding to the 1 st parallel energy storage converter, P [e2] represents the rated active power corresponding to the 2 nd parallel energy
storage converter, P [ek] represents the active power corresponding to the k parallel energy storage converter, P [eN] represents the rated active power corresponding to the N energy storage
converter, wherein n=1, n=98.
7. The distributed energy storage converter integration system-based power distribution method according to claim 6, wherein the calculation expression of the target power model of the parallel
energy storage converter is as follows:
Wherein P [target] represents the converter integration target power.
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ArkStream Capital: A milestone in zero-knowledge proof technology development over the past 40 years | Bee.com
ArkStream Capital: A milestone in zero-knowledge proof technology development over the past 40 years
Анализ3 месяца назадreleased 6086см... 46 0 0
Original author: @renkingeth
Краткое содержание
Zero-knowledge proof (ZKP) is widely regarded as one of the most important technological innovations in the blockchain field since distributed ledger technology, and it is also a key area of venture
capital. This article systematically reviews the historical literature and latest research on zero-knowledge proof technology over the past four decades.
First, the basic concepts and historical background of zero-knowledge proofs are introduced. Then, the circuit-based zero-knowledge proof technology is analyzed in detail, including the design,
application, and optimization methods of models such as zkSNARK, Ben-Sasson, Pinocchio, Bulletproofs, and Ligero. In the field of computing environment, this article introduces ZKVM and ZKEVM, and
discusses how they can improve transaction processing capabilities, protect privacy, and improve verification efficiency. The article also introduces the working mechanism and optimization methods of
zero-knowledge Rollup (ZK Rollup) as a Layer 2 expansion solution, as well as the latest progress in hardware acceleration, hybrid solutions, and dedicated ZK EVM.
Finally, this paper looks ahead to emerging concepts such as ZKCoprocessor, ZKML, ZKThreads, ZK Sharding, and ZK StateChannels, and explores their potential in blockchain scalability,
interoperability, and privacy protection.
By analyzing these latest technologies and разработка trends, this article provides a comprehensive perspective for understanding and applying zero-knowledge proof technology, demonstrates its great
potential in improving the efficiency and security of blockchain systems, and provides an important reference for future investment decisions.
Today, as the Internet is entering the Web3 era, blockchain applications (DApps) are developing rapidly, with new applications emerging almost every day. In recent years, blockchain platforms have
hosted millions of user activities and processed billions of transactions every day. The large amount of data generated by these transactions usually includes sensitive personal information such as
user identity, transaction amount, account address, and account balance. Given the openness and transparency of blockchain, these stored data are open to everyone, which has caused a variety of
security and privacy issues.
Currently, there are several cryptographic techniques that can address these challenges, including homomorphic encryption, ring signatures, secure multi-party computation, and zero-knowledge proofs.
Homomorphic encryption allows operations to be performed without decrypting ciphertext, which helps to protect the security of account balances and transaction amounts, but cannot protect the
security of account addresses. Ring signatures provide a special form of digital signature that can hide the identity of the signer, thereby protecting the security of account addresses, but cannot
protect account balances and transaction amounts. Secure multi-party computation allows computing tasks to be distributed among multiple participants without any participant knowing the data of other
participants, effectively protecting the security of account balances and transaction amounts, but also cannot protect the security of account addresses. In addition, homomorphic encryption, ring
signatures, and secure multi-party computation cannot be used to verify whether the prover has sufficient transaction amounts in a blockchain environment without revealing transaction amounts,
account addresses, and account balances (Sun et al., 2021).
Zero-knowledge proof is a more comprehensive solution. This verification protocol allows the correctness of certain propositions to be verified without revealing any intermediary data (Goldwasser,
Micali Rackoff, 1985). The protocol does not require complex public key facilities, and its repeated implementation does not provide malicious users with the opportunity to obtain additional useful
information (Goldreich, 2004). Through ZKP, the verifier is able to verify whether the prover has sufficient transaction amount without revealing any private transaction data. The verification
process involves generating a proof containing the transaction amount claimed by the prover, and then passing the proof to the verifier, who performs a predefined calculation on the proof and outputs
the final calculation result to conclude whether to accept the provers statement. If the provers statement is accepted, it means that they have sufficient transaction amount. The above verification
process can be recorded on the blockchain without any falsification (Feige, Fiat Shamir, 1986).
This feature of ZKP makes it play a core role in blockchain transactions and cryptocurrency applications, especially in terms of privacy protection and network expansion, making it not only the focus
of academic research, but also widely regarded as one of the most important technological innovations since the successful implementation of distributed ledger technology, especially Bitcoin. It is
also a key track for industry applications and venture capital (Konstantopoulos, 2022).
As a result, many ZKP-based network projects have emerged, such as ZkSync, StarkNet, Mina, Filecoin, and Aleo. As these projects develop, algorithmic innovations about ZKP are endless, and new
algorithms are reportedly released almost every week (Lavery, 2024; AdaPulse, 2024). In addition, hardware development related to ZKP technology is also progressing rapidly, including chips optimized
for ZKP. For example, projects such as Ingonyama, Irreducible, and Cysic have completed large-scale fundraising. These developments not only demonstrate the rapid progress of ZKP technology, but also
reflect the shift from general-purpose hardware to specialized hardware such as GPUs, FPGAs, and ASICs (Ingonyama, 2023; Burger, 2022).
These advances demonstrate that zero-knowledge proof technology is not only an important breakthrough in the field of cryptography, but also a key driving force for realizing broader applications of
blockchain technology, especially in improving privacy protection and processing capabilities (Zhou et al., 2022).
Therefore, we decided to systematically organize the relevant knowledge of zero-knowledge proof (ZKP) to better assist us in making future investment decisions. To this end, we comprehensively
reviewed the core academic papers related to ZKP (sorted by relevance and number of citations); at the same time, we also analyzed in detail the materials and white papers of leading projects in this
field (sorted by their financing scale). This comprehensive data collection and analysis provides a solid foundation for the writing of this article.
1. Zero-knowledge proof basics
1 Overview
In 1985, scholars Goldwasser, Micali and Rackoff first proposed zero-knowledge proof (Zero-Knowledge Proof, ZKP) and interactive knowledge proof (InteractiveZero-Knowledge, IZK) in the paper The
Knowledge Complexity of Interactive Proof-Systems. This paper is the foundation of zero-knowledge proof and defines many concepts that have influenced subsequent academic research. For example, the
definition of knowledge is the output of unfeasible computation, that is, knowledge must be an output and an unfeasible computation, which means that it cannot be a simple function but a complex
function. Infeasible computation can usually be understood as an NP problem, that is, a problem whose solution can be verified in polynomial time. Polynomial time means that the running time of the
algorithm can be expressed as a polynomial function of the input size. This is an important criterion for measuring the efficiency and feasibility of algorithms in computer science. Since the
solution process of NP problems is complex, they are considered to be computationally infeasible; however, their verification process is relatively simple, so they are very suitable for
zero-knowledge proof verification (Goldwasser, Micali Rackoff, 1985).
A classic example of an NP problem is the traveling salesman problem, where the shortest path to visit a series of cities and return to the starting point is found. While finding the shortest path
can be difficult, given a path, verifying that the path is the shortest is relatively easy. This is because verifying the total distance of a specific path can be done in polynomial time.
Goldwasser et al. introduced the concept of knowledge complexity in their paper to quantify the amount of knowledge leaked by the prover to the verifier in an interactive proof system. They also
proposed an interactive proof system (IPS), in which the prover and the verifier prove the truth of a statement through multiple rounds of interaction (Goldwasser, Micali Rackoff, 1985).
In summary, the definition of zero-knowledge proof summarized by Goldwasser et al. is a special interactive proof in which the verifier does not obtain any additional information except the truth
value of the statement during the verification process; and three basic characteristics are proposed, including:
• Completeness: the fact that an honest prover can convince an honest verifier if the argument is true;
• Soundness: If the prover does not know the content of the statement, he can deceive the verifier only with negligible probability;
• Zero-knowledge: After the proof process is completed, the verifier only obtains the information that the prover has this knowledge and cannot obtain any additional content (Goldwasser, Micali
Rackoff, 1985).
2. Zero-knowledge proof example
To better understand zero-knowledge proofs and their properties, here is an example of verifying that a prover has some private information in three stages: setup, challenge, and response.
Step 1: Setup
In this step, the provers goal is to create a proof that he knows a secret number s without directly revealing s. Let the secret number be;
Choose two large prime numbers p and q, and calculate their product . Let the prime numbers and , calculated;
Compute,Here,v is sent to the verifier as part of the proof, but it is not sufficient for the verifier or any bystander to infer s. ;
Randomly select an integer r, calculate and send it to the verifier. This value x is used in the subsequent verification process, but s is also not exposed. Let the random integer be calculated.
Step 2: Challenge
The verifier randomly selects a bit a (which can be 0 or 1) and sends it to the prover. This challenge determines the next steps the prover needs to take.
Step 3: Response
Based on the value a sent by the verifier, the prover responds:
If , the prover sends (here r is a number he chose randomly before).
If, the prover calculates and sends. Let the random bit sent by the verifier, according to the value of a, the prover calculates;
Finally, the verifier verifies whether it is equal to g according to the received g. If the equality is established, the verifier accepts this proof. When , the verifier calculates the verifier
calculation, and verifies the right side; when , the verifier calculates the verifier calculation, and verifies the right side.
Here, we see that the verifier calculated that the prover successfully passed the verification process without revealing his secret number s. Here, since a can only be 0 or 1, there are only two
possibilities, the probability of the prover passing the verification by luck (when a is 0). But the verifier then challenges the prover again, and the prover keeps changing the relevant numbers and
submitting them to the verifier, and always successfully passes the verification process. In this way, the probability of the prover passing the verification by luck (infinitely close to 0), and the
conclusion that the prover does know a secret number s is proved. This example proves the integrity, reliability and zero-knowledge of the zero-knowledge proof system (Fiat Shamir, 1986).
2. Non-interactive Zero-Knowledge Proof
1. Background
Zero-knowledge proof (ZKP) is usually an interactive and online protocol in the traditional concept; for example, the Sigma protocol usually requires three to five rounds of interaction to complete
the authentication (Fiat Shamir, 1986). However, in scenarios such as instant transactions or voting, there is often no opportunity for multiple rounds of interaction, especially in blockchain
technology applications, offline verification functions are particularly important (Sun et al., 2021).
2. Proposal of NIZK
In 1988, Blum, Feldman, and Micali first proposed the concept of non-interactive zero-knowledge (NIZK) proof, proving the possibility that the prover and the verifier can complete the authentication
process without multiple rounds of interaction. This breakthrough makes instant transactions, voting, and blockchain applications feasible (Blum, Feldman Micali, 1988).
They proposed that non-interactive zero-knowledge proof (NIZK) can be divided into three stages:
1. set up
2. calculate
3. verify
The setup phase uses a computation function to convert the security parameters into public knowledge (available to both the prover and the verifier), usually encoded in a common reference string
(CRS). This is how the proof is computed and verified using the correct parameters and algorithm.
The calculation phase uses the calculation function, input and proof key, and outputs the calculation result and proof.
In the verification phase, the validity of the proof is verified by verifying the key.
The common reference string (CRS) model they proposed is a non-interactive zero-knowledge proof that implements NP problems based on a string shared by all participants. The operation of this model
relies on the trusted generation of CRS, and all participants must have access to the same string. The scheme implemented according to this model can only ensure security if the CRS is generated
correctly and securely. For a large number of participants, the generation process of CRS can be complex and time-consuming, so although such schemes are generally easy to operate and have a small
proof size, their setup process is quite challenging (Blum, Feldman Micali, 1988).
Subsequently, NIZK technology has experienced rapid development, and a variety of methods have emerged to transform interactive zero-knowledge proofs into non-interactive proofs. These methods differ
in the construction of the system or the assumptions of the underlying encryption model.
3. Fiat-Shamir Transform
Fiat-Shamir Transformation, also known as Fiat-Shamir Heurisitc (heuristic), or Fiat-Shamir Paradigm (paradigm); proposed by Fiat and Shamir in 1986, is a method that can convert interactive
zero-knowledge proofs into non-interactive ones. This method reduces the number of interactions by introducing hash functions, and relies on security assumptions to ensure the authenticity of the
proof and its difficult-to-forge properties. Fiat-Shamir Transformation uses public cryptographic hash functions to replace some randomness and interactivity, and its output can be regarded as CRS to
some extent. Although this protocol is considered secure in the random oracle model, it relies on the assumption that the hash function output is uniformly random and independent of different inputs
(Fiat Shamir, 1986). Canetti, Goldreich, and Halevis research in 2003 showed that although this assumption is valid in theoretical models, it may encounter challenges in practical applications, so
there is a risk of failure when used (Canetti, Goldreich Halevi, 2003). Micali later improved this method by compressing multiple rounds of interaction into a single round, further simplifying the
interaction process (Micali, 1994).
4. Jens Groth and his research
Jens Groths subsequent research has greatly promoted the application of zero-knowledge proofs in cryptography and blockchain technology. In 2005, he, Ostrovsky and Sahai jointly proposed the first
perfect non-interactive zero-knowledge proof system applicable to any NP language, which can guarantee universal combinatorial security (UC) even in the face of dynamic/adaptive adversaries. In
addition, they used the number theory complexity assumption to design a concise and efficient non-interactive zero-knowledge proof system, which significantly reduced the size of CRS and proofs
(Groth Sahai, 2005).
In 2007, Groth, Cramer and Damgård began to commercialize these technologies. Through experimental verification, their public key encryption and signature schemes have significantly improved
efficiency and security, although these schemes are based on the assumption of bilinear groups (Groth Sahai, 2007). In 2011, Groth further explored how to combine fully homomorphic encryption with
non-interactive zero-knowledge proofs and proposed a scheme to reduce communication overhead, making the size of NIZK consistent with the size of the proof witness (Groth, 2011). In the following
years, he and other researchers have further studied pairing-based techniques, providing compact and efficient non-interactive proofs for large-scale statements, although these proofs still do not
leave the bilinear group framework (Bayer Groth, 2012; Groth, Kohlweiss Pintore, 2016; Bootle, Cerulli, Chaidos, Groth Petit, 2015; Groth, Ostrovsky Sahai, 2012; Groth Maller, 2017).
5. Other studies
In specific application scenarios, non-interactive zero-knowledge proofs for specific verifiers have shown their unique practical value. For example, Cramer and Shoup used a public key encryption
scheme based on a universal hash function to effectively resist selective ciphertext attacks in 1998 and 2002. In addition, in the key registration model, a new non-interactive zero-knowledge proof
method was successfully developed, which is suitable for solving all NP-class problems. The key is that participants need to register their own keys for subsequent verification (Cramer Shou, 1998,
In addition, Damgård, Fazio, and Nicolosi proposed a new method to improve the existing Fiat-Shamir transformation in 2006, allowing non-interactive zero-knowledge proofs without direct interaction.
In their method, the verifier first needs to register a public key to prepare for subsequent encryption operations. The prover uses additive homomorphic encryption technology to operate on the data
without knowing it and generate encrypted information containing the answer as a response to the challenge. The security of this method is based on the complexity leverage hypothesis, which believes
that for opponents with extraordinary computing resources, some computational problems that are considered difficult to solve may be solved (Damgård, Fazio Nicolosi, 2006).
The concept of weakly attributable reliability proposed by Ventre and Visconti in 2009 is an alternative to this assumption. It requires that when an adversary presents a false proof, he must not
only be aware of its falsity, but also be clear about how he successfully fabricated the false proof. This requirement significantly increases the difficulty of deception because the adversary must
be clear about his means of deception. In practлед, an adversary using this concept needs to provide a specific proof that contains ciphertext information for a specified verifier. It is difficult to
complete the proof without the private key of the verifier, so that when the adversary attempts to forge a proof, his behavior is exposed through detection (Ventre and Visconti, 2009).
The Unruh transform is an alternative to the Fiat-Shamir transform proposed in 2015. The Fiat-Shamir method is generally not safe against quantum computation and can produce insecure schemes for some
protocols (Unruh, 2015). In contrast, the Unruh transform provides non-interactive zero-knowledge proofs (NIZKs) that are provably secure against quantum adversaries for any interactive protocol in
the random oracle model (ROM). Similar to the Fiat-Shamir method, the Unruh transform does not require additional setup steps (Ambainis, Rosmanis Unruh, 2014).
In addition, Kalai et al. proposed an argumentation system for arbitrary decision problems based on private information retrieval technology. This method adopts the multi-prover interactive proof
system (MIP) model and converts MIP into an argumentation system through the method of Aiello et al. This construction runs in the standard model and does not need to rely on the random oracle
assumption. This method has been applied to some zero-knowledge arguments based on Proofs-for-Muggles (Kalai, Raz Rothblum, 2014).
Based on these technologies, non-interactive zero-knowledge proofs (NIZK) have been widely used in various fields that require high security and privacy protection, such as financial transactions,
electronic voting, and blockchain technology. By reducing the number of interactions and optimizing the proof generation and verification process, NIZK not only improves the efficiency of the system,
but also enhances security and privacy protection capabilities. In the future, with the further development and improvement of these technologies, we can expect NIZK to play an important role in more
fields and provide a solid technical foundation for more secure and efficient information processing and transmission (Partala, Nguyen Pirttikangas, 2020).
3. Circuit-based zero-knowledge proof
1. Background
In the field of cryptography, the traditional Turing machine model shows certain limitations, especially when dealing with tasks that require high parallelization and specific types of computing
(such as large-scale matrix operations). The Turing machine model needs to simulate infinitely long paper tapes through complex memory management mechanisms, and is not suitable for directly
expressing parallel computing and pipeline operations. In contrast, the circuit model, with its unique computing structure advantages, is more suitable for certain specific cryptographic processing
tasks (Chaidos, 2017). This article will discuss in detail the circuit-based zero-knowledge proof system (Zero-Knowledge Proof Systems Based on Circuit Models), which places special emphasis on the
use of circuits (usually arithmetic circuits or Boolean circuits) to express and verify the computing process.
2. Basic concepts and characteristics of circuit models
In the circuit-based computing model, a circuit is defined as a special computing model that can convert any computing process into a series of gates and wires that perform specific logical or
arithmetic operations. Specifically, circuit models are mainly divided into two categories:
• Arithmetic circuits: They are mainly composed of addition and multiplication gates and are used to process elements over finite fields. Arithmetic circuits are suitable for performing complex
numerical operations and are widely used in encryption algorithms and numerical analysis.
• Logic circuit: It is composed of basic logic gates such as AND gate, OR gate, NOT gate, etc., and is used to process Boolean operations. Logic circuits are suitable for performing simple judgment
logic and binary calculations, and are often used to implement various control systems and simple data processing tasks (Chaidos, 2017).
3. Circuit design and application in zero-knowledge proof
In a zero-knowledge proof system, the process of circuit design involves expressing the problem to be proved as a circuit. This process requires a lot of reverse thinking to design zk circuits: If
the claimed output of a computation is true, the output must satisfy certain requirements. If these requirements are difficult to model with just addition or multiplication, we ask the prover to do
extra work so that we can more easily model these requirements. The design process usually follows these steps (Chaidos, 2017):
• Problem representation: First, convert the problem to be proved, such as the calculation process of cryptographic hash functions, into the form of a circuit. This includes decomposing the
calculation steps into basic units in the circuit, such as gates and wires.
• Circuit optimization: Through technical means such as gate merging and constant folding, the circuit design is optimized to reduce the number of gates and calculation steps required, thereby
improving the operating efficiency and response speed of the system.
• Convert to polynomial representation: To adapt to zero-knowledge proof technology, the optimized circuit is further converted to polynomial form. Each circuit element and connection corresponds
to a specific polynomial constraint.
• Generate a Common Reference String (CRS): During the system initialization phase, a common reference string including a proof key and a verification key is generated for use in the subsequent
proof generation and verification process.
• Proof generation and verification: The prover performs computation on the circuit based on its private input and CRS to generate a zero-knowledge proof. The verifier can verify the correctness of
the proof based on the public circuit description and CRS without knowing the provers private information (Chaidos, 2017).
Zero-knowledge proof circuit design involves converting a specific computational process into a circuit representation and ensuring the accuracy of the computational results by constructing
polynomial constraints while avoiding the disclosure of any additional personal information. In circuit design, the key task is to optimize the structure of the circuit and generate an effective
polynomial representation in order to improve the efficiency of proof generation and verification. Through these steps, zero-knowledge proof technology can verify the correctness of the calculation
without leaking additional information, ensuring that the dual needs of privacy protection and data security are met (Chaidos, 2017).
4. Potential pitfalls and challenges
Disadvantages include:
• Circuit complexity and scale: Complex computations require large circuits, which significantly increases the computational cost of proof generation and verification, especially when dealing with
large-scale data;
• Difficulty of optimization: Although technical means (such as gate merging, constant folding, etc.) can optimize circuits, designing and optimizing efficient circuits still requires deep
• Adaptability to specific computing tasks: Different computing tasks require different circuit designs. Designing efficient circuits for each specific task can be time-consuming and difficult to
• Difficulty in implementing cryptographic algorithms: Implementing complex cryptographic algorithms (such as hash functions or public key encryption) may require a large number of logic gates,
making circuit design and implementation difficult;
• Resource consumption: Large-scale circuits require a lot of hardware resources and may encounter bottlenecks in actual hardware implementation in terms of power consumption, heat, and physical
space (Goldreich, 2004; Chaidos, 2017; Partala, Nguyen, Pirttikangas, 2020; Sun et al., 2021).
Solutions and improvement directions:
• Circuit compression technology: Reduce the number of logic gates and computing resources required by studying and applying efficient circuit compression technology;
• Modular design: By designing circuits in a modular way, the reusability and scalability of circuit design can be improved, and the workload of redesigning circuits for different tasks can be
• Hardware acceleration: Using specialized hardware (such as FPGA or ASIC) to accelerate circuit computation and improve the overall performance of zero-knowledge proofs (Goldreich, 2004; Chaidos,
2017; Partala, Nguyen Pirttikangas, 2020; Sun et al., 2021).
4. Zero-knowledge proof model
1. Background
Circuit-based zero-knowledge proofs have poor versatility and require the development of new models and algorithms for specific problems. There are a variety of high-level language compilers and
low-level circuit combination tools to generate circuits and design algorithms. The conversion of related calculations can be completed by manual circuit construction tools or automatic compilers.
Manual conversion usually produces more optimized circuits, while automatic conversion is more convenient for developers. Performance-critical applications usually require manual conversion tools
(Chaidos, 2017; Partala, Nguyen Pirttikangas, 2020; Sun et al., 2021).
This article will discuss the most notable ones. In general, these models are extensions or variations of zkSNARKs technology, each trying to provide optimizations in specific application
requirements (such as proof size, computational complexity, setup requirements, etc.).
Each protocol has its specific applications, advantages, and limitations, especially in terms of setup requirements, proof size, verification speed, and computational overhead. They are used in
various fields, ranging from cryptocurrency privacy and secure voting systems to general computation verified in a zero-knowledge manner (Čapko, Vukmirović Nedić, 2019).
2. Common algorithm models
1. zkSNARK model: In 2011, cryptographer Bitansky et al. proposed zkSNARK as an abbreviation of Zero-Knowledge Succinct Non-Interactive Argument of Knowledge. It is an improved zero-knowledge proof
mechanism. If there is an extractable collision-resistant hash (ECRH) function, it is possible to implement SNARK for NP problems. It also demonstrates the applicability of SNARK in various scenarios
such as computational delegation, succinct non-interactive zero-knowledge proof, and succinct two-party secure computation. This study also shows that the existence of SNARK implies the necessity of
ECRH, establishing the fundamental connection between these cryptographic primitives (Bitansky et al., 2011).
The zkSNARK system consists of three parts: setup, prover, and verifier. The setup process generates a proving key (PK) and a verification key (VK) using predefined security parameters l and an
F-arithmetic circuit C. All inputs and outputs of this circuit are elements in the field F. PK is used to generate verifiable proofs, while VK is used to verify the generated proofs. Based on the
generated PK, the prover generates a proof p using input x ∈ Fn and witness W ∈ Fh, where C(x, W) = 0 l. Here, C(x, W) = 0 l means that the output of circuit C is 0 l, and x and W are the input
parameters of circuit C. n, h, and l represent the dimensions of the output of x, W, and C, respectively. Finally, the verifier uses VK, x, and p to verify p, and decides to accept or reject the
proof based on the verification result (Bitanskyet al., 2011).
In addition, zkSNARKs have some additional features. First, the verification process can be completed in a short time, and the size of the proof is usually only a few bytes. Second, there is no need
for synchronous communication between the prover and the verifier, and any verifier can verify the proof offline. Finally, the prover algorithm can only be implemented in polynomial time. Since then,
a variety of improved zkSNARK models have emerged, further optimizing its performance and application scope (Bitanskyet al., 2011).
2. Ben-Sassons model: Ben-Sasson et al. proposed a new zkSNARK model for program execution on von Neumann RISC architecture in 2013 and 2014. Then, based on the proposed universal circuit generator,
Ben-Sasson et al. built a system and demonstrated its application in verifying program execution. The system consists of two components: a cryptographic proof system for verifying the satisfiability
of arithmetic circuits, and a circuit generator that converts program execution into arithmetic circuits. The design is superior to previous work in terms of functionality and efficiency, especially
the universality of the circuit generator and the additive dependence of the output circuit size. Experimental evaluation shows that the system can process programs of up to 10,000 instructions and
generate concise proofs at a high security level with a verification time of only 5 milliseconds. Its value lies in providing an efficient, universal and secure zk-SNARKs solution for practical
applications such as blockchain and privacy-preserving smart contracts (Ben-Sasson et al., 2013, 2014).
3. Pinocchio model: Parno et al. (2013) proposed a complete non-interactive zero-knowledge argument generation suite (Parno etal., 2013). It includes a high-level compiler that provides developers
with an easy way to convert computations into circuits. These compilers accept code written in high-level languages, so both new and old algorithms can be easily converted. However, there may be some
restrictions on the code structure to generate circuits of appropriate size.
Another feature of Pinocchio is the use of a mathematical structure called Quadratic Arithmetic Programs (QAPs), which can efficiently convert computational tasks into verification tasks. QAPs can
encode arbitrary arithmetic circuits as sets of polynomials, and only linear time and space complexity are required to generate these polynomials. The proof size generated by Pinocchio is 288 bytes,
which does not change with the complexity of the computational task and the input and output size. This greatly reduces the overhead of data transmission and storage. Pinocchios verification time is
typically 10 milliseconds, which is 5-7 orders of magnitude less than previous work. For some applications, Pinocchio can even achieve faster verification speeds than local execution. Reduce worker
proof overhead: Pinocchio also reduces the workers overhead of generating proofs, which is 19-60 times less than previous work (Parno et al., 2013).
4. Bulletproofs model: In 2017, Benedikt Bünz et al. (2018) designed a new non-interactive ZKP model. No trusted setup is required, and the proof size grows logarithmically with the size of the
witness value. Bulletproofs is particularly suitable for interval proofs in confidential transactions, and can prove that a value is within a certain range by using the minimum number of group and
field elements. In addition, Bulletproofs also supports the aggregation of interval proofs, so that a single proof can be generated through a concise multi-party computing protocol, greatly reducing
communication and verification time. The design of Bulletproofs makes it highly efficient and practical in distributed and trustless environments such as cryptocurrencies. Bulletproofs are not
strictly traditional circuit-based protocols. They are not as concise as SNARKs, and it takes longer to verify Bulletproofs than to verify SNARK proofs. But it is more efficient in scenarios where a
trusted setup is not required.
5. Ligero model: A lightweight zero-knowledge proof model proposed by Ames et al. (2017). The communication complexity of Ligero is proportional to the square root of the size of the verification
circuit. In addition, Ligero can rely on any collision-resistant hash function. In addition, Ligero can be a zkSNARK scheme in the random oracle model. This model does not require a trusted setup or
public key cryptosystem. Ligero can be used for very large verification circuits. At the same time, it is suitable for moderately large circuits in applications.
3. Solutions based on linear PCP and discrete logarithm problems
Ishai and Paskin (2007) proposed using additive homomorphic public key encryption to reduce the communication complexity of interactive linear PCP. Subsequently, Groth et al. published several
studies from 2006 to 2008 and proposed the NIZK scheme based on the discrete logarithm problem and bilinear pairing, which achieved perfect completeness, computational correctness and perfect zero
knowledge. The scheme represents the statement as an algebraic constraint satisfaction problem, and uses a cryptographic commitment scheme similar to the Pedersen commitment to achieve sublinear
proof length and non-interactivity without the need for the Fiat-Shamir heuristic. Although a large CRS and strong cryptographic assumptions of exponential knowledge are required, a sufficiently long
CRS can achieve a constant proof length. The verification and proof costs are high, and it is recommended to adopt the simulated extractability security model. This type of scheme is based on linear
PCP and/or discrete logarithm problems, but neither has quantum security (Groth, 2006, 2006, 2008; Groth Sahai, 2007).
6. Groth 16 model: It is an efficient non-interactive zero-knowledge proof system proposed by Jens Groth in 2016. The protocol is based on elliptic curve pairing and quadratic arithmetic program
(QAP), aiming to provide concise, fast and secure zero-knowledge proof.
7. Sonic model: M. Maller et al. (2019) proposed a Groth-based updatable CRS model using a polynomial commitment scheme, pairing, and arithmetic circuits. A trusted setup is required, which can be
implemented through secure multi-party computation. Once the CRS is generated, it supports circuits of arbitrary size.
8. PLONK model: A general zk-SNARK proposed in 2019, which uses permutation polynomials to simplify arithmetic circuit representation, making proofs simpler and more efficient; it is versatile and
supports recursive proof combination (Gabizon, Williamson Ciobotaru, 2019). The PLONK model claims to reduce the proof length of Sonic and improve the proof efficiency, but has not yet passed peer
9. Marlin model: An improved zk-SNARK protocol that combines the efficiency of algebraic proof systems with the universal and updatable setting properties of Sonic and PLONK, providing improvements
in proof size and verification time (Chiesa et al., 2019).
10. SLONK model: A new protocol introduced by Zac and Ariel in a paper on ethresear, an extension of PLONK that aims to solve specific computational efficiency problems and enhance the functionality
of the original PLONK system, usually involving changes in underlying cryptographic assumptions or implementations (Ethereum Research, 2019).
11. SuperSonic model: A novel polynomial commitment scheme is used to transform Sonic into a zero-knowledge scheme that does not require a trusted setup. It is not quantum-safe (Bünz, Fisch
Szepieniec, 2019).
4. Solutions based on ordinary people’s proof
Proofs-for-Muggles is a new zero-knowledge proof method proposed by Goldwasser, Kalai, and Rothblum in 2008. This method constructs interactive proofs for polynomial-time provers in the original
interactive proof model and is applicable to a wide range of problems. Through the transformation of Kalai et al., these proofs can be turned into non-interactive zero-knowledge proofs (Kalai, Raz
Rothblum, 2014).
12. Hyrax model: Based on ordinary peoples proof, Wahby et al. (2018) first designed a low-communication, low-cost zero-knowledge proof scheme Hyrax, which is low cost for provers and verifiers. In
this scheme, there is no trusted setup in this proof. If applied to batch statements, the verification time has a sublinear relationship with the arithmetic circuit size, and the constant is very
good. The running time of the prover is linear with the arithmetic circuit size, and the constant is also very good. Non-interactivity is achieved using the Fiat-Shamir heuristic, based on the
discrete logarithm problem, and quantum security is not achieved.
13. Libra model: The first ZKP model with linear prover time, concise proof size, and verification time. In Libra, in order to reduce the verification overhead, the zero-knowledge mechanism is
implemented through a method that can mask the provers response with a slightly random polynomial. In addition, Libra requires a one-time trusted setup, which only depends on the input size of the
circuit. Libra has excellent asymptotic performance and excellent efficiency of the prover. Its performance in proof size and verification time is also very efficient (Xie et al., 2019).
In terms of the computational complexity of the prover algorithm, Libra outperforms Ben-Sassons model, Ligero, Hyrax, and Aurora. In addition, the computational complexity of Libras prover algorithm
is independent of the circuit type (Partala, Nguyen Pirttikangas, 2020).
14. Spartan model: A zero-knowledge proof system proposed by Srinath Setty (2019) that aims to provide efficient proofs without the need for a trusted setup; it uses the Fiat-Shamir transformation to
achieve non-interactivity. It is known for its lightweight design and ability to efficiently handle large circuits.
5. Zero-knowledge based on probabilistically verifiable proofs (PCP)
Kilian (1992) constructed the first interactive zero-knowledge argument scheme for NP, implementing polylogarithmic communication. The scheme used collision-resistant hash functions, interactive
proof systems (IP), and probabilistically checkable proofs (PCP). The prover and the verifier (as a randomized algorithm) communicate through multiple rounds, and the verifier tests the provers
knowledge of the statement. Usually only one-sided faults are considered: the prover can always defend a true statement, but the verifier may accept a false statement with low probability. In 2000,
Micali used the Fiat-Shamir transformation to transform the scheme into a single-message non-interactive scheme. The following implementation can be considered to adopt this approach:
15. STARK model: In 2018, ZK-STARKs (Scalable Transparent ARgument of Knowledge) technology was proposed by Ben-Sasson et al. to solve the inefficiency of zk-SNARKs in processing complex proofs. At
the same time, it solves the problem of verifying the integrity of computations on private data, and can provide transparent and post-quantum secure proofs without relying on any trusted party.
In the same year, Ben-Sasson and others founded StarkWareIndustries and developed the first scalability solution StarkEx based on ZK-STARKs. According to Ethereums official documentation, it can
achieve non-interactivity in the random oracle model through the Fiat-Shamir paradigm. This construction is quantum-resistant, but its security relies on non-standard cryptographic assumptions about
Reed-Solomon codes. ZK-STARKs has the same characteristics as ZK-SNARKs, but includes the following advantages: a) Scalability: The verification process is faster. Transparency: The verification
process is public. Larger proof size: requires higher transaction fees (StarkWare Industries, 2018, 2018)
16. Aurora model: Ben-Sasson et al. (2019) proposed a succinct non-interactive argument based on STARK (SNARG). The non-interactivity is based on the Fiat-Shamir construction. It applies to the
satisfiability of arithmetic circuits. The argument size of Aurora is polylogarithmically related to the circuit size. In addition, Aurora has several attractive features. In Aurora, there is a
transparent setting. There is no effective quantum computing attack that can crack Aurora. In addition, fast symmetric encryption is used as a black box. Aurora optimizes the proof size. For example,
if the security parameter is 128 bits, the proof size of Aurora is at most 250 kilobytes. Aurora and Ligero optimize the proof size and computational overhead, making them very suitable for
zero-knowledge proofs on resource-limited devices. These optimizations not only improve efficiency, but also expand the scope of application of zero-knowledge proof technology, enabling it to be
applied in more practical scenarios.
17. Succinct Aurora Model: Ben-Sasson et al. (2019) proposed in the same paper: An extension of the Aurora protocol that provides a more optimized proof size and verification process. It maintains
Aurora’s transparent setup and security features while enhancing efficiency.
18. Fractal Model: Chiesa et al. (2020) proposed a preprocessing SNARK that uses recursive composition to improve efficiency and scalability. It takes advantage of logarithmic proof size and
verification time, and is particularly suitable for complex computations.
6. Classification based on the CPC (Common Proof Construction) setup phase
• Generation 1 (G 1) – Each circuit requires a separate trusted setup. zkSNARK, Pinocchio and Groth 16
• Generation 2 (G 2) – initially set once for all circuits. PlonK, Sonic, Marlin, Slonk and Libra
• Third generation (G 3) – proof systems that do not require a trusted setup. Bulletproofs, STARKs, Spartan, Fractal, Supersonic, Ligero, Aurora and SuccinctAurora (Čapko, Vukmirović Nedić, 2019;
Partala, Nguyen Pirttikangas, 2020).
5. Overview and Development of Zero-Knowledge Virtual Machines
1. Background
The previous part is more about the development of zero-knowledge proof ZKP in cryptography. Next, we will briefly introduce its development in the computer field.
In 2019, Andreev et al. first proposed the concept of ZK-VM at the ZkVM: Fast, Private, Flexible Blockchain Contracts conference as a way to implement a zero-knowledge proof system. The goal of ZK-VM
is to generate zero-knowledge proofs by running virtual machine programs to verify the correctness of program execution without leaking input data.
VM (Virtual Machine) is a software-simulated computer system that can execute programs, similar to a physical computer. VMs are often used to create independent operating system environments, perform
software testing and development, etc. VM or VM abstraction can be equivalent to CPU abstraction in most cases. It refers to the abstraction of the complex operations and architecture of the
computers processing unit (CPU) into a set of simple, operational instruction set architectures (ISA) to simplify the design and execution of computer programs. In this abstraction, computer programs
can be run through virtual machines (VMs) that simulate the operating behavior of real CPUs (Henderson, 2007).
Zero-knowledge proofs (ZKPs) often require execution via CPU abstraction. The setting is that the prover runs a public program on private inputs and wants to prove to the verifier that the program
executed correctly and produced the asserted output, without revealing the inputs or intermediate states of the computation. CPU abstraction is very useful in this context because it allows the
program to be run in a controlled virtual environment while generating proofs (Arun, Setty Thaler, 2024).
Example: The prover wishes to prove that he possesses a hashed password without revealing the password:
Password → Hash function → Hash value
Private → Public
In general, the prover should be able to run code that performs the hashing operation and produce a proof that allows anyone to verify the correctness of the proof, i.e., that the prover does have a
valid preimage for a given hash value.
Systems that generate these VM abstract proofs are often called zkVMs. This name is actually misleading because ZKVM does not necessarily provide zero knowledge. In short, ZKVM is a virtual machine
focused on zero-knowledge proofs, which extends the functionality of traditional VMs, can generally lower the threshold for the development of zero-knowledge circuits, and can instantly generate
proofs for any application or calculation (Zhang et al., 2023).
2. Classification of existing ZKVMs
According to the design goals, it is mainly divided into three categories:
1. Mainstream ZKVM
These ZKVMs leverage existing standard instruction set architectures (ISAs) and compiler toolchains, making them suitable for a wide range of applications and development environments.
• RISCZero (2021): uses the RISC-V instruction set and has a rich compiler ecosystem (Bögli, 2024).
• PolygonMiden (2021): Based on standard ISA, it enables simple and efficient development (Chawla, 2021).
• zkWASM (2022): zkWASM implements zero-knowledge proofs for the WebAssembly (WASM) instruction set, a widely adopted standard instruction set (DelphinusLab, 2022).
2. EVM-equivalent ZKVM
These ZKVMs are specifically designed to be compatible with the Ethereum Virtual Machine (EVM) and are able to run Ethereum’s bytecode directly.
• zkEVM projects: Several projects are working on achieving bytecode-level compatibility with the EVM, such as zkSync (MatterLabs, 2020) and Polygon Hermez (Polygon Labs, 2021).
3. Zero-knowledge-optimized (zero-knowledge-friendly) ZKVM
These ZKVMs optimize the efficiency and performance of zero-knowledge proofs and are designed for specific application scenarios.
• Cairo-VM (2018): Simple and compatible with SNARK proofs, its instruction set is specially designed to be arithmetic-friendly, making it easy to implement basic arithmetic operations such as
addition, multiplication, etc. in zero-knowledge circuits (StarkWare, 2018).
• Valida (2023): Optimized for specific applications, such as reducing the computing resources and time required to generate proofs by optimizing algorithms; its lightweight design makes it
suitable for a variety of hardware and software environments (LitaFoundation, 2023).
• TinyRAM (2013): Not dependent on standard toolchains: Due to its simplified and optimized design, it is generally not supported by LLVM or GCC toolchains and can only be used for small-scale
custom software components (Ben-Sasson et al., 2013).
The prevailing view is that simpler VMs can be transformed into circuits with fewer gates per step. This is most evident in the design of particularly simple and apparently SNARK-friendly VMs such as
TinyRAM and Cairo-VM. However, this comes with additional overhead, as implementing the primitive operations of a real-world CPU on a simple VM requires many primitive instructions (Arun, Setty,
Thaler, 2024).
3. Front-end and back-end paradigms
From the perspective of programming, ZKP systems can generally be divided into two parts: the frontend and the backend. The frontend part of the ZKP system mainly uses low-level languages to
represent high-level languages. For example, a general computational problem can be represented using a lower-level circuit language, such as R 1 CS circuit constraints to construct computations (for
example, circom uses R 1 CS to describe its frontend circuit). The backend part of the ZKP system is the cryptographic proof system, which mainly converts the circuit described by the low-level
language constructed by the frontend into generating proofs and verifying correctness. For example, commonly used backend system protocols include Groth 16 and Plonk (Arun, Setty Thaler, 2024; Zhang
et al., 2023).
Typically, the circuit will incrementally “execute” each step of a computational program (with the help of untrusted “suggested input”). Executing a CPU step conceptually involves two tasks: (1)
identifying the basic instructions that should be executed for that step, and (2) executing the instructions and updating the CPU state appropriately. Existing front ends implement these tasks via
carefully designed gates or constraints. This is time-consuming and error-prone, and also results in circuits that are much larger than they actually need to be (Arun, Setty, Thaler, 2024; Zhang et
al., 2023).
4. Advantages and Disadvantages of the ZKVM Paradigm
• Leverage existing ISAs: For example, RISC-V and EVM instruction sets can leverage existing compiler infrastructure and toolchains, without having to build infrastructure from scratch. Existing
compilers can be directly called to convert witness check programs written in high-level languages into assembly code for the ISA and benefit from previous audits or other verification work.
• Single circuit supports multiple programs: zkVM allows one circuit to run all programs until a certain time limit is reached, while other approaches may need to re-run the front end for each
• Circuits with repetitive structures: The front-end outputs circuits with repetitive structures, which the back-end can process faster (Arun, Setty, Thaler, 2024; Zhang et al., 2023).
• Cost of universality: In order to support all possible CPU instruction sequences, zkVM circuits need to pay the price for their universality, resulting in an increase in circuit size and proof
• Expensive operations: Some important operations, such as cryptographic operations, are very expensive to implement in zkVM. For example, ECDSA signature verification takes 100 microseconds on a
real CPU and millions of instructions on RISC-V. Therefore, the zkVM project contains hand-optimized circuits and lookup tables for computing specific functions.
• High proof cost: Even for very simple ISAs, the prover cost of existing zkVMs is still very high. For example, the prover of Cairo-VM needs to encrypt and submit 51 domain elements per step,
which means that executing one original instruction may require millions of instructions on a real CPU, limiting its applicability in complex applications (Arun, Setty, Thaler, 2024; Zhang et
al., 2023).
6. Overview and Development of Zero-Knowledge Ethereum Virtual Machine
1. Background
ZKEVM (Zero-Knowledge Ethereum Virtual Machine) and ZKVM (Zero-Knowledge Virtual Machine) are both virtual machines that apply zero-knowledge proof (ZKP) technology. The Ethereum Virtual Machine
(EVM) is part of the Ethereum blockchain system and is responsible for handling the deployment and execution of smart contracts. EVM has a stack-based architecture and is a computational engine that
provides computation and storage of a specific set of instructions (such as log operations, execution, memory and storage access, control flow, logging, calling, etc.). The role of EVM is to update
the state of Ethereum after applying the operations of smart contracts. ZKEVM is designed specifically for Ethereum and is mainly used to verify the correctness of smart contract execution while
protecting transaction privacy. ZKEVM converts the EVM instruction set into the ZK system for execution, and each instruction requires proof, including state proof and execution correctness proof
(Čapko, Vukmirović Nedić, 2019).
The current mainstream solutions for ZKEVM include STARKWARE, ZkSync, Polygen-Hermez, Scroll, etc. The following is a brief introduction to these projects (Čapko, Vukmirović Nedić, 2019):
• STARKWARE: Founded by Ben-Sasson et al. (2018), dedicated to using STARK zero-knowledge proof technology to improve the privacy and scalability of blockchain
• zkSync: Founded by Alex Gluchowski (2020) and others, Matter Labs proposed an Ethereum Layer 2 scaling solution based on zk-rollups.
• Polygon-Hermez: Hermez was originally an independent project and was released in 2020. After being acquired by Polygon in August 2021, it became PolygonHermez, focusing on high-throughput
zk-rollups solutions.
• Scroll: Founded by Zhang and Peng (2021), it achieves higher transaction throughput and lower gas fees, thereby improving the overall performance and user experience of Ethereum.
Generally, they can be divided into the following categories according to the level of compatibility with EVM (Čapko, Vukmirović Nedić, 2019):
• EVM-EVM-compatibility Smart contract function level compatibility, such as STARKWARE, zkSync
• EVM-equivalence, EVM instruction level compatibility (equivalence), such as polygen-Hrmez, scroll
See Figure 1 for the improved solution of the Ethereum system based on zero knowledge
Figure 1 Ethereum system improvement solution based on zero knowledge
2. How ZKEVM works
• Node program processing: The node program processes and verifies execution logs, block headers, transactions, contract bytecodes, Merkle proofs, etc., and sends this data to zkEVM for processing.
• Generate ZK proofs: zkEVM uses circuits to generate ZK proofs of execution results (state and execution correctness proofs). These circuit functions are mainly implemented using tables and
special circuits.
• Aggregate proofs: Use aggregate circuits to generate smaller proofs from large proofs, such as using recursive proofs.
• Send to L1 contract: The aggregated proof is sent to the L1 contract in the form of a transaction for execution (Čapko, Vukmirović Nedić, 2019).
3. ZKEVM Implementation Process
• Get data: Get data from the Ethereum blockchain system, including transactions, block headers, contracts, etc.
• Processing data: Processing and verifying execution logs, block headers, transactions, contract bytecode, Merkle proofs, etc.
• Generate proof: Use circuits to generate ZK proofs to ensure the state update and execution correctness of each instruction.
• Recursive proofs: Compress the generated large proof into smaller aggregate proofs.
• Submit proof: Submit the aggregate proof to the L1 contract to complete the transaction verification (Čapko, Vukmirović Nedić, 2019).
4. Features of ZKEVM
• Improve transaction processing capabilities: Execute transactions through ZKEVM on L2, reducing the load on L1.
• Privacy protection: Protect transaction privacy while verifying smart contract execution.
• Efficient Verification: Use zero-knowledge proof techniques to achieve efficient state and execution correctness verification (Čapko, Vukmirović Nedić, 2019).
7. Overview and Development of Zero-Knowledge Layer 2 Network Solutions
1. Background
The Ethereum blockchain is one of the most widely adopted blockchain ecosystems. However, Ethereum faces serious scalability issues, which makes it expensive to use. ZK Rollup is based on
zero-knowledge proof (ZKP) and is a Layer 2 solution for Ethereum expansion. It overcomes the defect of OptimisticRollups transaction final confirmation time being too long (Ganguly, 2023).
2. How ZK Rollup works
ZK Rollup allows scalability within a single transaction. The smart contract on L1 is responsible for processing and verifying all transfers, ideally generating only one transaction. This is done by
executing transactions off-chain to reduce the use of computing resources on Ethereum and putting the final signed transaction back on-chain. This step is called Validity Proof. In some cases,
verification may not be completed within a single proof, and additional transactions are required to publish the data on the rollup to the Ethereum main chain to ensure the availability of the data
(Ganguly, 2023).
In terms of space, using ZK Rollup improves efficiency since there is no need to store data like normal smart contracts. Each transaction only requires verification of the proof, which further
confirms the minimization of data, making them cheaper and faster (Ganguly, 2023).
Although ZK Rollup contains the term ZK (zero-knowledge) in its name, they mainly utilize the simplicity of zero-knowledge proofs to improve the processing efficiency of blockchain transactions,
rather than focusing primarily on privacy protection (Ganguly, 2023).
3. Disadvantages and optimizations of ZKRollup
ZK Rollup (Zero Knowledge Rollup) is a Layer 2 solution for Ethereum scalability. Although it excels in improving transaction processing efficiency, its main problem is that the computational cost is
very high. However, through some optimization solutions, the performance and feasibility of ZK Rollup can be significantly improved (Čapko, Vukmirović Nedić, 2019).
1. Optimize the calculation of cryptographic algorithms
Optimizing the computational process of cryptographic algorithms can improve the efficiency of ZK Rollup and reduce computing time and resource consumption. For example, Plonky 2, proposed by
PolygonZero (formerly MIR), is a decentralized ZK Rollup solution. Plonky 2 is a recursive SNARK that is 100 times faster than other Ethereum-compatible alternatives and combines the best features of
STARKs and SNARKs:
• Plonk and FRI: Providing fast proofs without trustless setup.
• Support recursion: Improve efficiency through recursive proof.
• Low verification cost: Efficient proof is achieved by combining 64-bit recursive FRI with Plonk.
2. Hybrid Optimistic and ZK Rollup
For example, PolygonNightfall is a hybrid Rollup that combines features of Optimistic and ZK Rollups, aiming to increase transaction privacy and reduce transfer fees (up to 86%).
3. Develop a dedicated ZK EVM
The dedicated ZK EVM is designed to improve the ZK Rollup algorithm and optimize the zero-knowledge proof process. Here are a few specific solutions:
• AppliedZKP: An open source project funded by the Ethereum Foundation that implements ZK for Ethereum EVM native opcodes, using cryptographic algorithms such as Halo 2, KZG, and Barreto-Naehrig
(BN-254) elliptic curve pairing.
• zkSync: zkEVM, developed by Matter Labs, is a custom EVM that implements the compilation of contract code into YUL (the intermediate language of the Solidity compiler) and then into supported
custom bytecode, using ultraPlonk, an extended version of Plonk.
• Polygon Hermez: Custom EVM-compatible decentralized Rollup that compiles contract code into supported microinstruction sets, using Plonk, KZG and Groth 16 proof systems.
• Sin 7 Y zkEVM: Implements ZK of EVM native opcodes and optimizes specialized opcodes, using halo 2, KZG, and RecursivePlonk.
• Polygon Miden: A universal zero-knowledge virtual machine based on STARK.
4. Hardware Optimization
Hardware optimization can significantly improve the performance of ZK Rollup. Here are several hardware optimization solutions:
• DIZK (DIstributedZero Knowledge): Optimizes zkSNARK proofs by distributing them on a computing cluster. The hardware architecture includes two subsystems, one for polynomial computation (POLY)
with large-scale number theoretic transforms (NTTs), and the other for performing multi-scalar multiplication (MSM) on elliptic curves (ECs). PipeMSM is a pipelined MSM algorithm for
implementation on FPGAs.
• FPGA-based ZKP hardware accelerator design: including multiple FFT (Fast Fourier Transform) units and decomposition of FFT operations, multiple MAC (Multiply-Add Circuit) units, and multiple ECP
(Elliptic Curve Processing) units to reduce computational overhead. The FPGA-based zk-SNARK design reduces the proof time by about 10 times.
• Hardware acceleration of the Bulletproof protocol: via a CPU-GPU collaboration framework and parallel Bulletproofs on GPU (Čapko, Vukmirović Nedić, 2019).
8. Future Development Direction of Zero-Knowledge Proof
1. Accelerate the development of computing environment
Zero-knowledge proof protocols (such as ZKSNARKs and ZKSTARKs) usually involve a large number of complex mathematical operations during execution, which need to be completed in a very short time,
placing extremely high demands on computing resources (such as CPU and GPU), resulting in high computational complexity and long computation time. In addition, generating and verifying zero-knowledge
proofs requires frequent access to large amounts of data, which places high demands on memory bandwidth. The limited memory bandwidth of modern computer systems cannot efficiently support such
high-frequency data access requirements, resulting in performance bottlenecks. Ultimately, high computational loads lead to high energy consumption, especially in blockchains and decentralized
applications, when a large number of proof calculations need to be performed continuously. Therefore, although software optimization solutions can partially alleviate these problems, it is difficult
to achieve the high efficiency and low energy consumption levels of hardware acceleration due to the physical limitations of general-purpose computing hardware. Hybrid solutions can achieve higher
performance improvements while maintaining flexibility (Zhang et al., 2021).
ZK-ASIC (Application Specific Integrated Circuit)
During 2020, several projects emerged, aiming to improve efficiency by accelerating the generation and verification process of zero-knowledge proofs (ZKP) through hardware such as GPUs or FPGAs
(Filecoin, 2024; Coda, 2024; GPU groth 16 prover, 2024; Roy et al., 2019; Devlin, 2024; Javeed Wang, 2017).
2021: Zhang et al. proposed a zero-knowledge proof acceleration scheme based on a pipeline architecture, using the Pippenger algorithm to optimize multi-scalar multiplication (MSM) and reduce data
transmission delay by unrolling the fast Fourier transform (FFT) (Zhang et al., 2021).
Axiom (2022) proposed the concept of ZKCoprocessor, or ZK coprocessor. A coprocessor is a separate chip that enhances the CPU and provides specialized operations such as floating-point operations,
cryptographic operations, or graphics processing. Although the term is no longer commonly used as CPUs become more powerful, GPUs can still be considered a coprocessor for the CPU, especially in the
context of machine learning.
The term ZK coprocessor extends the analogy of physical coprocessor chips to blockchain computation, allowing smart contract developers to statelessly prove off-chain computations on existing
on-chain data. One of the biggest bottlenecks facing smart contract developers remains the high cost of on-chain computation. Since gas is calculated for each operation, the cost of complex
application logic can quickly become prohibitive. ZK coprocessors introduce a new design pattern for on-chain applications, removing the limitation that computations must be done in the blockchain
virtual machine. This enables applications to access more data and operate at a larger scale than before (Axiom, 2022).
2. The proposal and development of ZKML
Concepts of ZKML
Zero-Knowledge Machine Learning (ZKML) is an emerging field that applies zero-knowledge proof (ZKP) technology to machine learning. The core idea of ZKML is to allow machine learning calculation
results to be verified without revealing data or model details. This not only protects data privacy, but also ensures the credibility and correctness of calculation results (Zhang et al., 2020).
The development of ZKML
In 2020, Zhang et al. systematically proposed the concept of ZKML for the first time at the 2020 CCS conference, demonstrating how to perform zero-knowledge proof of decision tree predictions without
revealing data or model details. This laid the theoretical foundation for ZKML.
In 2022, Wang and Hoang further studied and implemented ZKML and proposed an efficient zero-knowledge machine learning reasoning pipeline, showing how to implement ZKML in real-world applications.
The study showed that although ZKP technology is complex, through reasonable optimization, acceptable computing performance can be achieved while ensuring data privacy and computational correctness.
3. ZKP Scaling Technology Development
The concept of ZKThreads
In 2021, StarkWare proposed the concept of ZKThreads, which aims to combine zero-knowledge proof (ZKP) and sharding technology to provide scalability and customization for decentralized applications
(DApps) without fragmentation problems. ZKThreads improves security and composability by directly falling back on the base layer to ensure real-time performance at every step.
ZKThreads has been optimized mainly in three aspects: single-chain structure, rollup liquidity issues, and Proto-Danksharding.
• Single-chain solution: In the traditional single-chain architecture, all transactions are processed on one chain, resulting in excessive system load and poor scalability. ZKThreads significantly
improves processing efficiency by distributing data and computing tasks to multiple shards.
• ZK-rollups solution: Although ZK-rollups have significantly increased transaction processing speed and reduced costs, they are usually run independently, resulting in liquidity fragmentation and
interoperability issues. ZKThreads provides a standardized development environment that supports interoperability between different shards, solving the problem of liquidity fragmentation.
• Proto-Danksharding technology: This is an internal improvement plan of Ethereum that reduces the transaction cost of zk-rollups by temporarily storing data blocks. ZKThreads further improves on
this basis, reducing the reliance on temporary data storage through a more efficient sharding architecture, and improving the overall efficiency and security of the system (StarkWare, 2021).
The concept of ZK Sharding
Later, in 2022, NilFoundation proposed the concept of ZK Sharding, which aims to achieve Ethereums scalability and faster transaction speed by combining zero-knowledge proof (ZKP) and sharding
technology. This technology aims to divide the Ethereum network into multiple parts to process transactions in a cheaper and more efficient way. The technology includes zkSharding, which uses
zero-knowledge technology to generate proofs to ensure that transactions across different shards are valid before being submitted to the main chain. This approach not only improves transaction speed,
but also reduces the fragmentation of on-chain data, ensuring economic security and liquidity.
4. Development of ZKP interoperability
ZK State Channels
In 2021, the concept of ZK StateChannels was proposed by Virtual Labs, which combines zero-knowledge proof (ZKP) and state channel technology. It aims to achieve efficient off-chain transactions
through state channels while using zero-knowledge proof to ensure the privacy and security of transactions.
ZK State Channels replace the original solution
1. Traditional State Channels:
• Original solution: Traditional state channels allow two users to conduct peer-to-peer (P2P) transactions in a smart contract by locking funds. Since the funds are locked, signature exchanges
between users can be carried out directly without any gas fees and delays. However, this method requires predefined addresses, and the opening and closing of channels requires on-chain
operations, which limits its flexibility.
• Alternative: ZK StateChannels provides support for unlimited participants, allowing dynamic entry and exit without pre-defined user addresses. In addition, through zero-knowledge proof, ZK
StateChannels provides instant cross-chain access and self-verified proof, solving the flexibility and scalability problems of traditional state channels.
2. Multi-chain support:
• Original solution: Traditional state channels usually only support transactions on a single chain and cannot implement cross-chain operations, limiting the users operating scope.
• Alternative: ZK StateChannels uses zero-knowledge proof technology to achieve instant cross-chain transactions and asset flows without the need for intermediate bridges, greatly improving
multi-chain interoperability.
3. Predefined address restrictions:
• Original solution: In traditional state channels, the addresses of transaction participants must be predefined when the channel is created. If new participants join or leave, the channel must be
closed and reopened, which increases operational complexity and costs.
• Alternative: ZK StateChannels allows dynamic joining and exiting. New participants can join existing channels at any time without affecting the operations of current users, greatly improving the
flexibility of the system and user experience.
4.ZK Omnichain InteroperabilityProtocol
In 2022, ZKOmnichain Interoperability Protocol was proposed by Way Network to achieve cross-chain asset and data interoperability based on zero-knowledge proof. The protocol achieves full-chain
communication and data transmission by using zkRelayer, ZK Verifier, IPFS, Sender and Receiver.
The Omnichain project focuses on cross-chain interoperability and aims to provide a low-latency, secure network that connects different blockchains. It introduces a standardized cross-chain
transaction protocol that allows assets and data to be transferred seamlessly between blockchains. This approach not only improves the efficiency of transactions, but also ensures the security of
cross-chain operations.
Way Network can be seen as a specific implementation of the Omnichain concept, especially in terms of using zero-knowledge proof technology to enhance privacy and security. Way Networks technical
architecture enables it to achieve seamless interoperability between chains while maintaining decentralization and efficiency.
In summary, Omnichain provides an overall framework for cross-chain interoperability, while Way Network provides stronger privacy protection and security for this framework through zero-knowledge
proof technology.
IX. Conclusion
This paper presents a comprehensive literature review of zero-knowledge proof (ZKP) technology and its recent developments and applications in the blockchain space. We systematically review ZKPs in
the blockchain context, survey the state-of-the-art zero-knowledge proof schemes applicable to blockchain and verifiable computation, and explore their applications in anonymous and confidential
transactions as well as privacy-focused smart contracts. The paper enumerates the pros and cons of these academic peer-reviewed schemes and methods, provides references for practical evaluation and
comparison of these schemes, and highlights the skills and knowledge that developers need to possess when choosing a suitable scheme for a specific use case.
In addition, this paper also looks forward to the future development direction of zero-knowledge proof in hardware acceleration, blockchain scalability, interoperability and privacy protection.
Through a detailed analysis of these latest technologies and development trends, this paper provides a comprehensive perspective for understanding and applying zero-knowledge proof technology,
demonstrating its great potential in improving the efficiency and security of blockchain systems. At the same time, this research lays a solid foundation for subsequent research on ZK project
This article is sourced from the internet: ArkStream Capital: A milestone in zero-knowledge proof technology development over the past 40 years
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Factors to Calculate Right Position Size in Forex - FreeForexCoach.com
These are factors that you must Consider to calculate right position size in forex for your trades.
In fact, proper position sizing keeps you from risking too much or to little on a trade and blowing up your account
Apart from knowing your entry, stop loss and exit levels, position sizing is the very most important thing every trader must know.
Your position size determines the amount of units you use to buy or sell a currency pair.
We shall further discuss the different factors to consider to calculate the right position size in forex for your trades.
1. your trading capital / Account size.
Knowing your account balance helps you to calculate the correct amount you want to risk per trade in currency.
Assuming you have opened an account with $1000. Your account balance should reflect $1000.
Your account capital can be in any currency denomination you would prefer to hold as a person. It can GBP, NZD, CHF, CYN, JPY, EUR, RAND or USD any of your choice.
Later we shall see how to calculate right position size in forex for different currency denomination.
2. Percentage risk per trade
The percentage risk per trade is the amount of money you are willing to risk from your account for each trade you take.
This is the most important step for determining Forex position size.
Set a percentage amount of your account you’re willing to risk on each trade.
Most professional traders choose to risk 1 to 2% or less of their total capital account on each trade.
Similarly, risking as less as 1% or below keeps your losses small even if you incur several consecutive losses.
At the same time it protects your account from being exposed to too much risk in case of big volatile movements in the market.
If you have a $1000 trading account. And you decide to risk $10 per trade, that is 1% of your account.
If your risk less than 1% let’s say 0.5% , $5. Or may be 2%
In case you risk 2% from your account for every trade you take, then it would cost you ($1000×2%) =$20 per trade.
What matters here is to risk a percentage you are comfortable with
Choose how much you’re willing to risk on every trade, and keep it consistent. If you choose 1% as your account risk per trade, then you should risk only 1% on every trade.
3. Stop loss in pips
The stop loss in pips is the distance between your entry level point and the stop loss level point.
The stop loss closes out the trade automatically in case you were wrong about the direction of the market.
This helps you to limit big losses from your account.
Stop loss levels may vary with different traders and on different trades due to different trading strategies and market volatility.
Before entering any trade, consider both your entry point and your stop loss location.
Depending on your strategy, decide where to place your stop for your trade.
Measure the distance in pips between your stop loss and your entry price. This will be the number of pips you have at risk for that trade.
If your entry point for a buy on the EUR/USD pair is at 1.22938 and you place your stop loss below entry at 1.22550.
Then your stop loss in pip should be the difference between entry level and stop loss level.
That is; (1.22938 – 1.22550) = 38.8 pips.
This means your trade has to move 38.8 pips against you to be considered a failed trade.
Your stop loss should not be too close to your entry other wise you will be stopped out on a short notice.
Also, you should not put it too wide to avoid big losses in case you are wrong about the market.
Once you know how far away your entry point is from your stop loss, in pips, you can calculate your position size for that trade.
4. pip value per pip
Here you need to first identify the currency in which your account is in.
The currency pair you are trading and the number of units traded/lot size.
We calculate the pip value basing on the quote currency in the currency pair.
How to calculate position size.
Position size = (Account size ×% risk per trade)/ (stop loss in pips × pip value)
What you should know is that position size varies with different lot sizes or account type.
The size on a standard account cannot be the same as that of a mini account nor that of a micro account.
Different examples on Calculate Right Position Size in Forex.
What you should note first is that for all pairs where the USD is a quote/counter currency in the pair, e.g EUR/USD, pip value is the same as indicated;
1000 lot (micro) is worth $0.1 per pip movement. A 10,000 lot (mini) is worth $1. Whereas 100,000 lot (standard) is worth $10 per pip movement.
If the USD is not the quote currency then these pip values will vary slightly. Learn more.
Let’s now look at some examples on how to calculate right position size in forex;
1. When your account capital is not in USD; in EUR
Let’s say your account balance is €1000.
If you are trading a GBP/USD going long, risking 2% per trade and your stop loss is 150 pips
First, you need to convert the amount risked per trade from Euros to US dollar first.
Risk per trade = (1000× 0.02) = €20
Since we need to convert Euros to US dollars, we must know the exchange rate on the EUR/USD.
If the exchange rate is 1.18974.
I.e: USD 1 = EURO 1.18974
FOR €20, (20/1.18974) =$ 16.8
Therefore the risk per trade in USD = $16.8
Position size = Amount risked per trade/( stop loss in pips x pip value)
If it was a Standard account;
Remember, for a standard account (100,000) 1 pip = $10, for all pairs where the USD is a quoted
Substituting into our formula:
= $16.8/(150 x $10)
Position size = 0.01
In case of a mini, 10000 units = $1,
=$16.8/(150 x$ 1)
Position size = 0.11
And if it was a micro, 1000 units = $0.1,
=$16.8/(150 x $0.1)
Position size = 1.12
NOTE: The numerator and the denominator should be of the same currency.
2. Suppose the conversion currency was the base currency in the pair.
For instance trading USD/CHF with our €1000 account using 2% risk with 150 stop loss pips
First we must Convert the EURO to CHF first to get the value of €20 in CHF,
If the exchange rate of EUR/CHF is 1.4500.
Risk per trade = (20 x 1.4500) =CHF 29
Position size = Amount risked per trade/( stop loss in pips x pip value)
If it was a Standard account;
Remember, for a standard account (100,000) 1 pip = $10, for all pairs where the USD is a quoted
Substituting into our formula:
= 29/(150 x 10)
Position size = 0.019
If it was a mini, 10000 units = $1,
=29/(150 x 1)
Position size = 0.19
And if it was a micro, 1000 units = $0.1,
=29/(150 x 0.1)
Position size = 1.93
Note that it is the counter currency you use in conversion of of your account denomination
3. When your account capital is in USD
For instance if the account balance in $10,000.
If you are risking 2% each trade using a stop loss of 200 pips on EUR/USD.
First determine the value of risk amount per trade = (2% ×10,000) =$200
Since the quote currency is the same as the account denomination, you don’t have to convert the risked amount.
So you just substitute in our formula.
Position size = Amount risked per trade/( stop loss in pips x pip value)
If it was a Standard account;
Remember, for a standard account (100,000) 1 pip = $10, for all pairs where the USD is a quoted
Substituting into our formula:
= $200/(200 x $10)
Position size = 0.1
If it was a mini, 10000 units = $1,
=$200/(200 x$ 1)
Position size = 1.0
And if it was a micro, 1000 units = $0.1,
=$200/(200 x $0.1)
Position size = 10.0
I will insist, don’t make things hard for you to trade.
All you need to know is how much you are willing to risk for each trade you take, your stop loss in pips and the average lot value per pip.
You can then easily calculate the proper position size to use on any trade!
There are several ways greed can impact your Forex trading success. Firstly, greed can make you abandon your well crafted trading strategy in favor of impulsive and speculative actions. Instead of
adhering to predetermined entry and exit points based on technical or...
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How to traverse binary tree by depth (Part I) ?How to traverse binary tree by depth (Part I) ?
How to traverse binary tree by depth (Part I) ?
In this article, I will introduce the traversal of binary tree in CSharp.
There are depth first traversal which includes:
• pre-order traversal: root node first, then left node, then right node
• in-order traversal: left node first, then root node, then right node
• post-order traversal: left node first, then right node, then root node
and breath first traversal which is level-order traversal.
In this post I will introduce the depth first traversal, and later for breath first traversal.
So, before we traverse a binary tree, we need to create it before. And before we create a tree, we need to create tree node. I assume the data hold the value, and the left node and right node are its
public class Node<T>
public Node<T> LNode { get; set; }
public Node<T> RNode { get; set; }
public T Data { get; set; }
public Node(T data)
Data = data;
Then, I need to create a binary tree.
static Node<string> BinTree()
Node<string>[] binTree = new Node<string>[11];
binTree[0] = new Node<string>("A");
binTree[1] = new Node<string>("B");
binTree[2] = new Node<string>("C");
binTree[3] = new Node<string>("D");
binTree[4] = new Node<string>("E");
binTree[5] = new Node<string>("F");
binTree[6] = new Node<string>("G");
binTree[7] = new Node<string>("H");
binTree[8] = new Node<string>("I");
binTree[9] = new Node<string>("J");
binTree[10] = new Node<string>("K");
binTree[0].LNode = binTree[1];
binTree[0].RNode = binTree[2];
binTree[1].LNode = binTree[3];
binTree[1].RNode = binTree[4];
binTree[2].LNode = binTree[5];
binTree[2].RNode = binTree[6];
binTree[3].RNode = binTree[7];
binTree[4].LNode = binTree[8];
binTree[5].LNode = binTree[9];
binTree[5].RNode = binTree[10];
return binTree[0];
Once I have the tree, I can think about creating the different binary tree traversals. Firstly, let’s see the pre-order traversal.
The principe is the we get the root value, then its left node value and finally its right node value.
Remember: Left node is always before Right node.
static void PreOrder<T>(Node<T> node)
if (node != null)
In the previous method, we’ve used recursive method to get the node’s child, and its child’s child etc.
Then, we can see the in-order traversal. It just change the order of the three nodes.
The principe is the we get the its left node value, then root value, and finally its right node value.
static void InOrder<T>(Node<T> node)
if (node != null)
Finally, the post-order traversal. The principe is the we get the its left node value, then its right node value, and finally its root value.
static void PostOrder<T>(Node<T> node)
if (node != null)
Now, we just need to call the different implementations and get the results.
public static void Main()
Node<string> tree = BinTree();
PreOrder<string>(tree); //result: A B D H E I C F J K G
InOrder(tree); //result: B D H E I A C F J K G
PostOrder<string>(tree); //result: B D H E I C F J K G A
So here, we’ve arrived at the end of this post, I hope this does help to you. Enjoy coding! | {"url":"https://www.sunjiangong.com/2013/07/10/how-to-traverse-binary-tree-by-depth-part-I.html","timestamp":"2024-11-09T04:10:30Z","content_type":"text/html","content_length":"72967","record_id":"<urn:uuid:960f4238-4bd9-439e-a1c0-4cfbe6b1e3bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00504.warc.gz"} |
1. Create the data table
From the Welcome or New Table dialog, choose to create an XY data table, choose tutorial data sets, and select the sample data "Enzyme kinetics -- Michaelis-Menten" from the enzyme kinetics section.
2. Inspect the data
The sample data will be partly covered by a floating note explaining how to fit the data (for people who are not reading this help page). You can move the floating note out of the way, or minimize
The data are in triplicate. Some values are missing, and Prism always handles these just fine.
3. View the graph
Prism automatically created a graph and gave it the same name as the data table. Click on the Michaelis-Menten graph in the graphs section.
Since this is the first time you are viewing the graph, Prism will pop up the Change Graph Type dialog. Select the third choice, to plot individual replicates rather than mean and error bars.
The graph Prism makes automatically is fairly complete. You can customize the symbols, colors, axis labels, position of legend, etc.
4. Choose nonlinear regression
Click the Analyze button and choose Nonlinear regression from the list of XY analyses.
Even faster, click the shortcut button for nonlinear regression.
5. Choose a model
On the Fit tab of the nonlinear regression dialog, open the equation folder, Enzyme Kinetics - Substrate vs. Velocity. Then choose the Michaelis-Menten equation.
Learn more about the principles of enzyme kinetics and about fitting Michaelis-Menten curves.
For this example, leave all the other settings to their default values.
Click OK to see the curves superimposed on the graph.
6. Inspect the graph
7. Inspect the results
The goal of nonlinear regression is to find the best-fit values of the parameters. These are reported at the top of the table. You can't really interpret the best-fit values without knowing how
precise they are, and this is reported both as standard errors and confidence intervals.
8. Go back and perform the replicates test
The replicates test assesses the adequacy of the fit by comparing the scatter among the triplicates with the scatter of points around the curve. It is not calculated by default, so the results do not
appear in the results of step 7.
You don't have to do the fit over again. Instead click the button in the upper left corner of the results table to return to the nonlinear regression dialog.
Go to the Diagnostics tab, and check the option to perform the replicates test. Note that you can also check an option to make your settings here become the default for future fits.
The P value is small (0.013). This means that the scatter of the data from the curve is greater than you'd expect from the variation among triplicates. This suggests that you might want to consider
fitting an alternative model, which we do in the next example. | {"url":"https://www.graphpad.com/guides/prism/latest/curve-fitting/reg_example_enzyme_kinetics.htm","timestamp":"2024-11-11T15:08:18Z","content_type":"text/html","content_length":"46464","record_id":"<urn:uuid:0c852dc3-48b7-4162-8fc0-a39b02c57789>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00752.warc.gz"} |
Three-Manifold Mutations Detected by Heegaard Floer Homology
2014 Theses Doctoral
Three-Manifold Mutations Detected by Heegaard Floer Homology
Given a self-diffeomorphism h of a closed, orientable surface S with genus greater than one and an embedding f of S into a three-manifold M, we construct a mutant manifold by cutting M along f(S) and
regluing by h. We will consider whether there exist nontrivial gluings such that for any embedding, the manifold M and its mutant have isomorphic Heegaard Floer homology.
In particular, we will demonstrate that if h is not isotopic to the identity map, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of HF-hat
of M differs from that of its mutant. We will also show that if the gluing map is isotopic to neither the identity nor the genus-two hyperelliptic involution, then there exists an embedding of S into
a three-manifold M such that the total rank of HF-hat of M differs from that of its mutant.
• Clarkson_columbia_0054D_11918.pdf application/pdf 473 KB Download File
More About This Work
Academic Units
Thesis Advisors
Lipshitz, Robert
Ph.D., Columbia University
Published Here
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PPT - Chapter 15 PowerPoint Presentation, free download - ID:421316
1. Chapter 15 Chapter 15 Maximum Likelihood Estimation, Likelihood Ratio Test, Bayes Estimation, and Decision Theory Bei Ye, Yajing Zhao, Lin Qian, Lin Sun, Ralph Hurtado, Gao Chen, Yuanchi Xue, Tim
Knapik, Yunan Min, Rui Li
2. Section 15.1 Maximum Likelihood Estimation
3. Maximum Likelihood Estimation (MLE) • Likelihood function • Calculation of MLE • Properties of MLE • Large sample inference and delta method • 2.Calculation of Maximum Likelihood Estimation •
2.Calculation of Maximum Like
4. Likelihood Function 1.1 Parameter space Θ X1 ,…, Xn : i.i.d. observations θ: an unknown parameter Θ: The set of all possible values of θ 1.2 Joint p.d.f. or p.m.f. of X1 ,…, Xn
5. Likelihood Function 1.3 Likelihood Function of θ For observed χ1,…,χn: • The joint p.d.f. or p.m.f. is a function of χ1,…,χnfor given θ . • The likelihood function is a function of θ for given
6. Example : Normal Distribution Suppose χ1,…,χn is a random sample from a normal distribution with p.d.f.: a vector parameter: ( ) • Likelihood Function:
7. Calculation of MLE 2.1 Maximum Likelihood Estimation: Need to find which maximizes the likelihood function . • Simple Example: • Two independent Bernoulli trials with success probabilityθ. • θ is
known : 1/4 or 1/3 (Θ). • The probabilities of observing χ= 0, 1, 2 successes can be calculated. Let’s look at the following table.
8. Probability of ObservingχSuccesses The MLE is chosen to maximize for observed χ. χ=0:χ=1 or 2: Calculation of MLE
9. Calculation of MLE 2.2 Log-likelihood function: Setting the derivative of L(θ) equal to zero and solving for θ. • Note: The likelihood function must be differentiable and then this method can be
10. Properties of MLE MLE: optimality properties in large samples The concept of information due to Fisher
11. Properties of MLE 3.1 Fisher Information: • Alternative expression
12. Properties of MLE • for an i.i.d. sample:
13. Properties of MLE • for k-dimensional vector parameter
14. Properties of MLE 3.2 Cramér-Rao Lower Bound • A random sample X1, X2, …, Xn from p.d.f f(x|θ). • Let be any estimator of θ with where B(θ) is the bias of If B(θ) is differentiable in θ and if
certain regularity conditions holds, then • (Cramér-Rao inequality) • The ratio of the lower bound to the variance of any estimator of θ is called the efficiency of the estimator. • An estimator
has efficiency = 1 is called the efficient estimator.
15. 4.1 Large Sample Inferences To make Large sample inference on unknown parameter θ (Single Parameter), we need to estimate : I(θ) is estimated by: This estimate does not require evaluation of the
expected value. An approximate large sample CI on θ is: Large Sample Inferences and Delta Method
16. 4. Large Sample Inferences and Delta Method 4.2 Delta Method for Approximating the Variance of an Estimator To estimate a nonlinear function h(θ). Suppose that : and is a known function of θ
Delta Method:
17. Section 15.2 Likelihood Ratio Test
18. Likelihood Ratio (LR) Test • Background of LR test • Neyman-Pearson Lemma and Test • Examples • Generalized Likelihood Ratio Test
19. Background of LR test Egon Sharpe Pearson1895-1980English mathematician • Jerzy Splawa Neyman, 1894-1981 Polish-American mathematician.
20. Neyman-Pearson lemma We want to find a rejection region R such that the error of both type I and type II error are as small as possible. Suppose have joint p.d.f Consider the ratio Then a best
critical region of is Where is a constant such that
21. What is Likelihood Ratio (LR) test • A ratio is computed between the maximum probability of a result under null and alternative hypothesis. where the numerator corresponds to the maximum
probability of an observed result under the null hypothesis; denominator under the alternative hypothesis. • Test idea: if observe x, then condition is evidence in favor of the alternative; the
opposite inequality is evidence against the alternative. • Hence, the decision of rejecting null hypothesis was made based on the value of this ratio.
22. The Test • Let be a random sample with p.d.f • Hypothesis: • Test statistic: • Reject when the likelihood ratio exceeds
23. Characteristics of LR Test • Most powerful test of significance level α; Maximize • Very useful and widely applicable, esp. in medicine to assist in interpreting diagnostic tests . • Exact
distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine. • The computations are difficult to perform by hand.
24. Example 1: Test on Normal Distribution Mean The likelihoods under and are:
25. Example 1 continued Likelihood ratio is Reject when the ratio exceeds a constant , which is chosen for a specific significance level: The test is independent of . It is the most powerful level
test for all .
26. A Numerical Example Suppose a random sample with size and Test versus with Where So we reject if .
27. Generalized Likelihood Ratio Test • Neyman-Pearson Lemma shows that the most powerful test for a Simple vs. Simple hypothesis testing problem is a Likelihood Ratio Test. • We can generalize the
likelihood ratio method for the Composite vs. Composite hypothesis testing problem.
28. Hypothesis • Suppose H0 specifies that θ is in Θ0 and H1 specifies that θ is in Θ0c. Symbolically, the hypotheses are:
30. Test Statistics • Note that λ ≤ 1. • An intuitive way to understand λ is to view: • the numerator of λ as the maximum probability of the observed sample computed over parameters in the null
hypothesis • the denominator of λ as the maximum probability of the observed sample over allpossible parameters. • λ is the ratio of these two maxima
31. Test Statistics • If H0 is true, λ should be close to 1. • If H1 is true, λ should be smaller. • A small λ means that the observed sample is much more likely for the parameter points in the
alternative hypothesis than for any parameter point in the null hypothesis.
32. Reject Region & Critical Constant • Rejects Ho if λ < k, where k is the critical constant. • k < 1. • k is chosen to make the level of the test equal to the specified α, that is, α = PΘo ( λ ≤ k
33. A Simple Example to illustrate GLR test • Ex 15.18 (GLR Test for Normal Mean: Known Variance) • For a random sample x1, x2……, xnfrom an N (μ, σ2) distribution with known σ2, derive the GLR test
for the one-sided testing problem: Ho: µ ≤ µ0 vs. H1: µ > µ0 where µ0 is specified.
35. Solutions • If , then the restricted MLE of µ under H0 is simply . • If , then the restricted MLE of µ under H0 is , because in this case, the maximum of the likelihood function under H0 is
attained at .
36. SolutionsThus, the numerator & denominator of the likelihood ratio are showing below, respectively
38. Solution • Clearly, we do not reject H0 when λ = 1, i.e., when . • Therefore, the condition λ < k is equivalent to subject to . • In other words, we reject H0 if is large, which leads to the
usual upper one sided z-test.
39. Section 15.3 Bayesian Inference
40. Bayesian Inference • Background of Bayes • Bayesian Inference defined • Bayesian Estimation • Bayesian Testing
41. Background of Thomas Bayes • Thomas Bayes • 1702 – 1761 • British mathematician and Presbyterian minister • Fellow of the Royal Society • Studied logic and theology at the University of Edinburgh
• He was barred from studying at Oxford and Cambridge because of his religion
42. Background of Bayes • Baye’s Theorem • Famous probability theorem for finding “reverse probability” • The theorem was published posthumously in a paper entitled “Essay Towards Solving a Problem
in the Doctrine of Chances”
43. Bayesian Inference • Application to Statistics – Qualitative Overview • Estimate an unknown parameter • Assumes the investigator has some prior knowledge of the unknown parameter • Assumes
the prior knowledge can be summarized in the form of a probability distribution on , called the prior distribution • Thus, is a random variable
44. Bayesia • Application to Statistics – Qualitative Overview (cont.) • The data are used to update the prior distribution and obtain the posterior distribution • Inferences on are based on the
posterior distribution
45. Bayesian Inference • Criticisms by Frequentists • Prior knowledge is not accurate enough to form a meaningful prior distribution • Perceptions of prior knowledge differ from person to person •
This may cause inferences on the same data to differ from person to person.
46. Some Key Terms in Bayesian Inference… In the classical approach the parameter, θ, is thought to be an unknown, but fixed, quantity. In the Bayesian approach, θ is considered to be a quantity
whose variation can be described by a probability distribution which is called prior distribution. • prior distribution – a subjective distribution, based on experimenter’s belief, and is
formulated before the data are seen. • posterior distribution – is computed from the prior and the likelihood function using Bayes’ theorem. • posterior mean – the mean of the posterior
distribution • posterior variance – the variance of the posterior distribution • conjugate priors - a family of prior probability distributions in which the key property is that the posterior
probability distribution also belongs to the family of the prior probability distribution
47. 1.5.3.1 Bayesian Estimation Now lets move on to how we can estimate parameters using Bayesian approach. Now let’s move on to how we can estimate parameters using Bayesian approach. (Using text
notation) Let be an unknown parameter based on a random sample, from a distribution with pdf /pmf Let be the prior distribution of Let be the posterior distribution If we apply Bayes Theorem(Eq.
15.1), our posterior distribution becomes : Note that is the marginal PDF of X1,X2,…Xn
48. Bayesian Estimation(continued) As seen in equation 15.2, the posterior distribution represents what is known about after observing the data . From earlier chapters, we know that the likelihood of
a variable is So, to get a better idea of the posterior distribution, we note that: posterior distribution likelihood prior distribution i.e. For a detailed practical example of deriving the
posterior mean and using Bayesian estimation, visit: http://www.stat.berkeley.edu/users/rice/Stat135/Bayes.pdf
49. Example 15.25 Let x be an observation from an distribution where μ is unknown and σ2 is known. Show that the normal distribution is a conjugate prior on μ. We can ignore the factor because it
will cancel from both the numerator and denominator of the expression for . Similarly, any terms not involving μ can be canceled from the numerator and denominator.
50. Example 15.25 (continue) Thus, we see that is proportional to Where and It follows that has the form of the normal distribution. Specifically, is distribution (the normalizing constant comes from
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18 Plot Activities: DIYs, Graphing Practice, Task Cards, Games, And More - Teaching Expertise
Are you tired of seeing your students’ eye glaze over when you try to explain different types of math plots? Do you want to add some fun and hands-on experiences for your students? Look no further!
We’ve got 18 hands-on activities that you can implement in the math classroom to get your students excited about their learning! Now, you can make learning about plotting more engaging than ever
1. Use Money
We know that students learn best when they can connect their learning to real-life situations. Using coins to create line plots is the perfect way to engage students and encourage them to apply their
learning to real-life problems. This line plot activity uses money earned from a lemonade sale and asks students to graph the earnings.
Learn More: Teaching with a Mountain View
2. Sticky Notes Line Plot
Have you ever thought about using sticky notes and a project to practice line plots? This activity involves just that! Project a poll on the board with a statement such as “my birthday is in”. Then,
have students place their sticky notes above their answers.
Learn More: True Life Math Teachers
3. Using Straws and Paper
Use a straw and paper balls to create a scatter plot. Students will use straws and blow air to move the paper balls across the graph. When the students are finished, they will copy the scatter plot
on a paper graph.
Learn More: The Teacher Studio
4. Scatter Plot with Oreos
Use cookies to play a “Battleship” sort of game. All you need is a grid and cookies. Ask your students to place the cookies somewhere on the grid. Taking turns, each student will guess the coordinate
until the cookie “ship” is sunk.
Learn More: Homeschooling My Kinetic Kids
5. Real Life Coordinate Graphing
Create a grid on your classroom floor and give your students a list of points to plot. They can then move objects on the grid or act as the pieces themselves.
Learn More: Apples and Bananas Education
6. Use Stickers to Create Line Plots
This fun activity involves students measuring their feet and then using stickers to graph their classmate’s foot sizes on a line plot.
Learn More: The Teaching Studio
7. Conversation Hearts Stem and Leaf Plot
Use conversation hearts to create a stem and leaf plot for any data. It could be class height, their favorite colors, or anything they’d like! Simple ideas like this are so much fun for students!
Learn More: In the 5th Grade with Teacher Julia
8. Task Cards
Task cards are a great way to engage all of your students and to get them thinking about their learning. Just be sure to have a list of the correct answers so students can self-check their work when
Learn More: Teaching with a Mountain View
9. Create a Line Plot on the Floor
Create your very own line plot on your classroom floor. Using sticky notes or manipulatives, you can create a line plot lesson plan that your students will love.
Learn More: School and the City
10. Raisin Box Line Plot
This lesson is great for elementary classrooms! All you need is a box of raisins for each student and a board/wall for the line plot. Students will count how many raisins are in their box and will
then use their box to create a line plot.
Learn More: Glitter in Third
11. Dice Roll Line plot
Dice are such an amazing resource to have for math class. Using dice, have students add the values of their answers. After finding the sum, they can graph their answers on a line plot.
Learn More: Classroom Freebies
12. Cubes Line Plot
Stacking cubes are another great tool to have in your math classroom. You can use these cubes for many things, but stacking them to create a line plot is a great way to give your students a visual
Learn More: My Teaching Pal
13. Use Poster Paper
A piece of poster paper can be a great resource to help illustrate students’ learning and understanding. You can have students graph a scatter plot, a stem and leaf plot, or even a line plot. After
students create their plots, you can hang them around the classroom for students to reference.
Learn More: Team J’s Second Grade Fun
14. Coordinate Grid
This activity involves having students plot points on a coordinate in order to create a picture. Once all of the points are graphed, students can color the picture in.
Learn More: Mrs. Thompson’s Treasures
15. Connect Fourp
Connect four is a classic game that all students love! With an accompanying coordinate grid, have your students plot the point of each chip/ball they place in the grid.
Learn More: Panicked Teacher
16. Coordinate City
Have students use grid paper to create a “blueprint” of a city. You can give the students a legend, such as how many feet each square represents. Make sure students plot the points of each building
as they create them.
Learn More: Middle School Math Man
17. Scatter Plot BINGO
Use this awesome resource to play coordinate bingo with your students. Call out each coordinate and have the learners place something on that point (it can be candy, a small toy, etc.). When someone
gets 6 in a row, they will yell BINGO!
Learn More: Flap Jack Educational Resources
18. Candy Graphing
Who doesn’t love candy? Using M&M’s, students can create a line plot based on the colors they have. Students can then plot the points using the data they gathered when creating their line plots.
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Quantitative Methods
2014/2015 KAN-CFIVO1001U Quantitative Methods
English Title
Quantitative Methods
Course information
Language English
Course ECTS 7.5 ECTS
Type Mandatory
Level Full Degree Master
Duration One Semester
Course period Autumn
Timetable Course schedule will be posted at calendar.cbs.dk
Study board Study Board for MSc in Economics and Business Administration
Course coordinator
• Peter Raahauge - Department of Finance (FI)
Main academic disciplines
• Finance
• Statistics and mathematics
Last updated on 04-07-2014
Learning objectives
When presented with a problem dealt with in the course, the student should be able to solve the problem using the appropriate software in a way similar to how the problem was solved during the
course. When presented with a problem, which is similar but different from the problems dealt with in the course, the student should be able to solve the problem, based on a theoretical understanding
of the problems dealt with in the course.
Quantitative Methods:
Exam ECTS 7,5
Examination form Written sit-in exam
Individual or group Individual
Assignment type Written assignment
Duration 4 hours
Grading scale 7-step scale
Examiner(s) One internal examiner
Exam period December/January
Limited aids, see the list below and the exam plan/guidelines for further information:
• Additional allowed aids
Aids allowed to bring • Books and compendia brought by the examinee
to the exam • Notes brought by the examinee
• Allowed calculators
• Allowed dictionaries
Same examination form as the ordinary exam
Make-up exam/re-exam If the number of registered candidates for the make-up examination/re-take examination warrants that it may most appropriately be held as an oral examination, the programme
office will inform the students that the make-up examination/re-take examination will be held as an oral examination instead.
Description of the exam procedure
Open book exam with limited Internet access.
Course content and structure
The course provides the quantitative tools necessary for following courses like Investments and Empirical Finance successfully.
The course takes a hands-on approach to the material, and a central part of course is worked examples (exercises with guiding solutions), which the students are supposed to implement using
appropriate IT-tools like Excel, VBA, and/or R. The course uses a "flipped classroom" approach, see http://en.wikipedia.org/wiki/Flip_teaching: The theoretical foundation for the worked examples is
explained using screencasts available on the Internet for viewing on demand. The students work with the worked examples. The scheduled classes are used for personalized guidance with respect to
theory and implementation of the worked examples.
Theoretical topics covered in the worked examples include:
Analysis: Functions and their properties, Differentiation and Taylor series approximations, Equation solving, Optimization, Integration
Linear Algebra: Vector and matrix algebra, Linear equation systems
Statistics: Random variables and probability distributions, Inference, Hypothesis testing, Regression models, Panel data, Monte Carlo methods
To the extent the theoretical topics are known from prerequisite courses, the topics will be elaborated (ex: two- or N-dimensional stochastic distributions), addressed in alternative ways (ex: Monte
Carlo "proof" of formulas for stochastic variables), and/or used to introduce software functionality.
IT topics covered in the worked examples, to the extend time permits, include:
Excel: Data import and data types, Formulas, references and names, Tables, Graphs, Solver add-in, Data analysis add-in
VBA: Functions, Subs, Local variables, Arrays, For-loops and If-statements, Debugging
R: RStudio IDE, Data import, Data types and selected functions, Scripts, Plots, For-loops and If-statements, Functions, Debugging, Random numbers and Monte Carlo analysis, Linear algebra,
Optimization, Integration with Excel
Teaching methods
Flipped classroom with on-line lectures, worked examples (exercises with guiding solutions), and classes with personalized guidance.
Student workload
On-line lectures 33 hours
Preparation for on-line lectures 33 hours
Working with examples incl. reading and personalized guidance 140 hours
Exam 4 hours
Expected literature
Mandatory literature
• Sydsæter and Hammond: Essential Mathematics for Economic Analysis; 3rd ed., 2008, Prentice Hall (or similar).
• Braun and Murdoch: A First Course in Statistical Programming with R; 1st ed., 2007, Cambridge University Press (or similar).
• David Skovmand: Supplementary Notes on: Linear Algebra, Probability and Statistics for Empirical Finance, 2013, downloadable.
• Robert L. McDonald: An Introduction to VBA in Excel, 2000, downloadable.
• Notes and Worked Examples.
Supplementary literature:
• Reference books on Excel and VBA. Popular bestsellers like 'Excel 2013 Bible' and 'Excel 2013 Power Programming with VBA' by John Walkenbach are each very extensive (>1000 pages) and available
on-line at CBS for free.
• Selected sections of Financial Markets and Investments, Claus Munk, 2014, downloadable, will help motivating the curriculum.
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Combining Substances in a Certain Ratio to Give a Set Density
Substances are often combined to make new substances. Metal alloys are made by mixing metals together in certain proportions to give certain properties of strength, hardness, density or some other
quality. If it is required that the density be a certain value then we can calculate the proportions in which the metals should be combined (this calculation applies to all mixing of substances, not
just metals).
Suppose that metal 1 has a density
Suppose that metal 2 has a density
The density of the mixture will be
Suppose the ratio is
so that
Suppose that the two metals have densities of
Suppose the ratio is
so that
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• Juan Bermejo-Vega, Freie Universitaet Berlin, Berlin, Germany: Architectures for quantum simulation showing quantum supremacy
Abstract: One of the main aims in the field of quantum simulation is to achieve what is called "quantum supremacy", referring to the experimental realization of a quantum device that
computationally outperforms classical computers. In this work, we show that one can devise versatile and feasible schemes of two-dimensional dynamical quantum simulators showing such a quantum
supremacy, building on intermediate problems involving IQP circuits. In each of the schemes, an initial product state is prepared, potentially involving an element of randomness as in disordered
models, followed by a short time evolution under a translationally invariant Hamiltonian with nearest-neighbor interactions and a mere sampling measurement in a fixed basis. The final state
preparation in each scheme is fully efficiently certifiable. We discuss experimental necessities and possible physical architectures, inspired by platforms of cold atoms in optical lattices and a
number of others, as well as specific assumptions that enter the complexity-theoretic arguments. This work shows that benchmark settings exhibiting a quantum advantage may require little control
in contrast to universal quantum computing.
Based on arXiv:1703.00466
• Lorenzo Catani, University College London, London, UK: Spekkens' toy model as a unifying framework for state-injection schemes with contextuality as resource
Abstract: The idea of this project is to show that Spekkens' toy model - a non-contextual phase-space inspired hidden variable model with a restriction on what an observer can know about the
reality - can be used to unify frameworks of state-injection schemes, in all dimensions, where contextuality is an injected resource to reach universal quantum computation. I will apply this
idea, formalised in the notion of "Spekkens' circuits", to the popular examples of Howard et al. regarding qudits (odd prime dimensions) and Delfosse et al. regarding rebits. At the end I will
also compare our approach to the recent one by Bermejo-Vega et al. . Our framework, in addition to this application in quantum computation, would also answer the open question about which are the
maximal sub theories of Spekkens' model that are consistent (operationally equivalent) with QM.
• Nicolas Delfosse, UC Riverside + Caltech, CA, USA: Optimal decoding of surface codes for qubit loss, and a little bit more
Abstract: Surface codes are among the best candidates to ensure the fault-tolerance of a quantum computer. In order to avoid the accumulation of errors during a computation, it is crucial to have
at our disposal a fast decoding algorithm to quickly identify and correct errors as soon as they occur. I will describe a decoding algorithm for dealing with qubit loss that is optimal both in
terms of performance and speed. Then, I will talk about realistic applications of this decoding strategy.
Based on joint work with Gilles Zemor, https://arxiv.org/abs/1703.01517 .
• Iman Marvian, MIT, USA: Symmetry-Protected Topological Entanglement
Abstract: I propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the "SPT-entanglement",
is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is
a connected region surrounded by A, B and the boundaries of the system. In the case of 1-dimensional systems I prove that in the limit where A and B are large and far from each other compared to
the correlation length, the SPT-entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the non-trivial phases.
Moreover, I show that the SPT-entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher
dimensions as well. Also, I show that there is an intriguing connection between SPT-entanglement and the Fourier transform of the string order parameters, which are the traditional tool for
detecting SPT phases. This leads to a new algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, I discuss implications of
these results in the context of measurement-based quantum computation.
• Akimasa Miyake, University of New Mexico, Albuquerque, USA: Measurement-based quantum computation and genuine 2D symmetry-protected topological orders
Abstract: After a brief introduction to symmetry-protected topological orders, I discuss novel algebraic structures of measurement-based quantum computation when the resource multipartite
entanglement is genunine (or stronger form of) 2D symmetry-protected topologically ordered states. This talk is based on recent works (arXiv:1508.02695, arXiv:1612.08135, arXiv:1703.11002) in
collaboration with Jacob Miller.
• Hendrik Poulsen Nautrup, University of Innsbruck, Austria: Fault-tolerant Interface between quantum memories and processors
Abstract: Topological error correction codes are promising candidates to protect quantum computations from the deteriorating effects of noise. While some codes provide high noise thresholds
suitable for robust quantum memories, others allow straightforward gate implementation needed for data processing. To exploit the particular advantages of different topological codes for
fault-tolerant quantum computation, it is necessary to be able to switch between them. In my talk I present a practical solution, subsystem lattice surgery, which requires only two-body nearest
neighbor interactions in a fixed layout in addition to the indispensable error correction. This method can be used for the fault-tolerant transfer of quantum information between arbitrary
topological subsystem codes in two dimensions. As an example which is of practical interest, I consider a simple interface, a quantum bus, between noise resilient surface code memories and
flexible color code processors.
• Cihan Okay, University of Western Ontario, London, Canada: Topological methods and contextuality
Abstract: I will talk about some applications of topological methods in quantum computation. More specifically the topological part will involve group cohomology and various constructions
suitable for studying contextuality in quantum mechanics. The aim of the talk will be towards finding possible applications of structural results for such constructions which arise in topology.
For a recent work along these lines see arXiv:1701.01888.
• Robert Raussendorf, University of British Columbia, Vancouver, Canada
• David Stephen, University of British Columbia, Vancouver, Canada
• Emily Tyhurst, University of British Columbia, Vancouver, Canada: Separating state-independent and state-dependent contextuality in the context of Mermin's square
Abstract: Connections between negativity in quasiprobability distributions and quantum contextuality as a resource for computation are well-established in local Hilbert space dimension greater
than two. However, for qubits the separation between state-independent and state-dependant contextuality complicates matters. In this talk I will speak about the canonical example of Mermin's
square, and a simple contextual hidden variable model that allows for a classical simulation of the system. The simulation method deliberately separates costs due to state-independent
contextuality, provided by two different hidden variable models; and the costs due to state-dependent contextuality, provided by a quasiprobability distribution over the hidden variable models.
The quasiprobability distribution is in part due to work by Howard and Campbell in arXiv:1609.07488 [quant-ph]
• Mark van Raamsdonk, University of British Columbia, Vancouver, Canada: Locally Maximally Entangled States of Multipart Quantum Systems
Abstract: For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement, i.e. the reduced density
matrix for each elementary subsystem is a multiple of the identity matrix. In this talk, we first show that information about the representations of arbitrary finite and compact groups can be
used to construct a special class of ``stabilizer'' LME states. We then review how the space of LME states up to local unitary transformations has two very natural geometrical descriptions, one
as a symplectic manifold (i.e. with the structure of a phase space in Hamiltonian classical mechanics) and one as a complex manifold. The equivalence of these descriptions shows that the space of
LME states up to local unitary transformations is actually equivalent to the space of all states with ``generic'' entanglement up to SLOCC equivalence. Using this geometrical viewpoint, we are
able to provide necessary and sufficient conditions on the subsystem dimensions (d_1, d_2, ... , d_n) for the existence of LME states and compute the dimension of the space of such states when
they exist.
• Dongsheng Wang, University of British Columbia, Vancouver, Canada: Topological qubits from valence bond solids
• Tzu-Chieh Wei, University of Stony Brook, Stony Brook, USA: Symmetry-protected topologically ordered states for universal quantum computation
Abstract: Measurement-based quantum computation (MBQC) is a model for quantum information processing utilizing local measurement on suitably entangled states for the implementation of quantum
gates. The cluster state on the 2D square lattice was first discovered to enable universal quantum computation. However, complete characterization for universal resource states is still missing.
Recent development in condensed matter physics on symmetry-protected topological (SPT) order has provided an intriguing link and new perspective for the quest of novel resource states. The 2D
AKLT states are special points in the so-called valence-bond solid phase, which is regarded as a SPT phase, but it requires translational invariance to be imposed, in addition to the internal
spin rotation symmetry. Here I will show that certain types of fixed-point wave functions in generic nontrivial 2D SPT phases (without requiring translational invariance) are indeed universal for
MBQC. (These fixed-point results can be generalized to higher dimensions.) Moreover, I will discuss whether we can extend the universality beyond the fixed points by examples. What would be a
potential breakthrough is to establish an entire SPT phase supporting universal MBQC, but this is still an open question.
• Kohei Kishida, University of Oxford, Oxford, UK
Juan Bermejo-Vega (FU Berlin, Germany), Robert Raussendorf (UBC), Tzu-Chieh Wei (Stony Brook, USA) | {"url":"https://asqc2.qi.ubc.ca/asqcProgram.html","timestamp":"2024-11-06T22:01:08Z","content_type":"text/html","content_length":"13704","record_id":"<urn:uuid:78b0f6cd-51d0-4054-9e3d-a933b6381869>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00832.warc.gz"} |
Quadratics Unit
My introduction to quadratics has evolved over the years. Recently I settled on having my students do as much graphing as possible and allowing them to make the connections they need to make in
regards to the equation of a quadratic and it's resulting graph. I have found that allowing my students to roll their sleeves up and start graphing allows them to become more comfortable with
quadratics. This pays dividends when it comes time to solve quadratic equations by factoring, completing the square or by using the quadratic formula.
The order of things goes something like this:
Intro to Quadratics
This worksheet usually comes after we have exhausted ourselves on linears and introduces the idea that there are other relationships and, no, they don't all form a line. We do this by measuring
various circular objects and calculating the area and circumference. We then graph
circumference vs. radius
area vs. radius. Graph the Magic Number
This worksheet is a little less organic, but still results in a quadratic relationship.
Stretch Factor
What happens when we graph change the "a" value? I give students a series of "Silent Board Games" (borrowed from CPM), where students are given incomplete input/output tables and are required to find
the patterns to complete them. Once they are completed, they graph the parabolas on the given coordinate plane. The idea is for them to make the connections between the input/output table, the
equation and the graph. Usually students are pretty quick at recognizing they can describe what the graph would look like just by observing the equation.
Graph Given the Vertex
In this worksheet, I give students random points on the plane and have them graph a parabola with a stretch factor of 1 or -1. By this time I expect them to have a grasp of the relationship between
the points on a parabola and it's vertex. This allows them to see that if they can graph one, they can graph them all.
Vertical Shift/Horizontal Shift
Both of these worksheets are in the same format as the stretch factor worksheet. I have students complete input output tables and graph. By this time, they are looking for relationships between the
equation and the visual interpretation of the graph.
GeoGebra Investigation
This is pretty intensive and I probably need to break it up into smaller more manageable labs. I would like to throw that out there for discussion. The main ideas that I wanted my students to explore
• parabolic symmetry
• line of symmetry is the average of the x intercepts
• x value of the vertex and the l.o.s. are the same
• relationship between a, b and the l.o.s.
• similarities and differences between quadratic functions and equations
• solve quadratic equations by graphing
• determining the number of solutions by using the discriminant
Find the vertex
This last worksheet has students take different quadratics and use the equation to find the vertex and line of symmetry. From there they will graph using only the vertex and the stretch factor. I
then allow them to use GeoGebra to check their answers.
(if you're interested)
2 comments:
keninwa said...
Could you please put a link to your old blog in your profile or somewhere in this blog? I updated Google Reader, and now I want to go back and look at an old posting and can't find it. Thanks.
David Cox said...
You got it. It's right above my mug. I'm looking to import those posts here, but we'll see. | {"url":"https://coxmath.blogspot.com/2010/01/quadratics-unit.html?showComment=1264807266578","timestamp":"2024-11-04T21:50:38Z","content_type":"text/html","content_length":"70955","record_id":"<urn:uuid:c9641e1e-40e7-40c6-84d3-5d95c3164606>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00319.warc.gz"} |
Salinas Chess Club Invents A New Game Called 'Name That Move'! - ChessKid.com
Salinas Chess Club Invents A New Game Called 'Name That Move'!
The Future Citizens Foundation has a chess club at the Taylor Farms Center for Learning that meets two afternoons a week and on Saturdays. The Center for Learning provides supervised, healthy
activities for elementary school children in the Salinas, CA area. The chess sessions have between 8 and 16 kids on any day and beginners are always welcome.
I volunteer as the chess coach and give every child that attends chess club a free ChessKid account. Students who are working on the site frequently are given a Gold membership.
Together with the ChessKid King and Queen Level students, we invented a game called "Name That Move"!
At the start of every chess club we solve “mate in 1” puzzles. The puzzles are set up before the kids come in, and everyone attempts to solve them. The beginners use the puzzles to learn what
checkmate is and the more advanced students write down the answers using chess notation. Many of the puzzles that we use are from László Polgár's book, CHESS: 5334 Problems, Combinations, and Games,
1994, Tess Press.
The mate in 1 puzzles are enlarged and laminated on different colors of paper for ease of use.
After a few months we had a collection of cards with puzzles on them. Players all draw a card with a mate in 1 puzzle and call out the name of the correct move when they think they know it.
The cards do not have numbers on the ranks or letters marked on the files, so the players have to identify the square while calling out the mate in 1 correct move. This makes it fun to practice
learning chess notation and the names of the squares.
We don’t take turns -- whoever thinks they have the answer just says, “I got it!” If they are wrong, another player can then say, “I got it!” The pace moves quickly!
If the answer is correct, the player draws another card. If it is wrong, they can try again. Whoever collects the most cards at the end wins!
It is a fun, fast game. Try to "Name That Move" in your chess club and see what your players think!
MacGregor Eddy, Volunteer Chess Coach
Taylor Farms Center for Learning Chess Club
(Editor's note: Another good way to practice learning the names of the squares is the "Vision" feature on ChessKid, where you can turn off the notation for an even larger challenge.) | {"url":"https://www.chesskid.com/learn/articles/name-that-move-a-game-for-fun-and-to-improve-vision","timestamp":"2024-11-11T01:00:07Z","content_type":"application/xhtml+xml","content_length":"13967","record_id":"<urn:uuid:1c762bbf-5c92-4cff-b3c6-baa178475cf4>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00040.warc.gz"} |
What is Math Anxiety? How to Overcome It | Conquer Your Exam
Do you laugh at jokes trashing mathematics with a passion? Do you, every year, develop a certain vendetta against your math textbook — or teacher? Are you a person who hates math? Most importantly,
how do you overcome that? Math anxiety is real and overcoming it can be hard.
First, we will discuss what math anxiety really is, and where it originates. After that, we’ll look at techniques for getting over any math anxiety you may be feeling right now. Finally, there’s a
quick self-test for you to measure your own math anxiety. Let’s get to it.
What is Math Anxiety? Is Fear of Mathematics Real?
A new term has come into use called ‘math anxiety.’ Don’t be fooled by the name, though — math anxiety is not an anxiety disorder, and can not be diagnosed by a physician. ‘Math anxiety’ simply
refers to the stress some students experience when interacting with math. This is actually great, though, because while real anxiety disorders can be difficult and complicated to deal with, ‘math
anxiety,’ if faced head-on, can be eradicated.
What exactly is ‘math anxiety’? ‘Math anxiety’ refers to students’ fear, stress toward, or perceived hatred of mathematics. It can begin early on, and is usually fostered in students as young as
elementary school by a variety of factors: public embarrassment, poor teachers, and pop culture, to name a few. We’ll get in to that quite soon. First, let’s look at some common symptoms:
What are Common Symptoms of Math Anxiety?
Common symptoms of ‘math anxiety’ are feelings of hopelessness or frustration toward math (“I will never learn!”), feelings of embarrassment or shame (“Why can I never get it right?” “I suck at math
and always will”).
These feelings, in turn, lead one to avoid studying or doing math homework (“I’m never going to need it in life anyway”). And that makes one fall behind.
Students who don’t study tend to do worse than students who do — it’s not a matter of who’s smart and who isn’t, but a matter of who puts in the work. However, students with feelings of math anxiety
who do not study will tend to do worse in math classes, which further feeds their math anxiety.
In this way, math anxiety is a self-supporting cycle: you don’t like math, or feel anxious thinking about it, so you zone out in class or avoid doing homework. Not paying attention or studying makes
you do worse on tests, which makes you feel more anxious. And so on.
Ultimately, math anxiety can lead to global avoidance of mathematics, which essentially means that students avoid taking math classes or engaging in situations which require maths — say, they would
opt for a more conceptual, rather than mathematical, physics class. This can be fine — but it can also prove a major obstacle to any STEM major, nearly all of which require math classes. Not to
mention that half the SAT and a fourth of the ACT are based on math. Furthermore, many colleges like to see students challenging themselves — not avoiding math classes at all costs.
Don’t let that scare you, though — if you have math anxiety, there are ways to fix it. You are smart and resilient. You will get into college and succeed. You are not doomed.
Now let’s look at the basis of this math anxiety you’re fighting.
How Does Math Anxiety First Appear?
Frankly, math anxiety is rarely the fault of the student — how could it be, when it first appears in kids as young as kindergarteners? There is no technical research on math anxiety (we’ll get to
that), but basically it forms from a few specific seeds:
Pop Culture
First, there is a major vendetta against math pervading all pop culture. Maybe elementary students aren’t as aware of pop culture as their far older peers, but even young students are witness to
their parents or teachers labelling math as ‘hard’ and ‘tricky.’ Who hasn’t heard a trusted friend or adviser say “I wasn’t smart enough for math”? This leads kids who have no idea whether they are
good at math or not to believe that ‘math is hard,’ which naturally makes them want to avoid it — and to see their own flaws whenever they try.
Second, a lot of teachers simply don’t like math. Even if teachers don’t tell their students explicitly, their lack of enthusiasm can subliminally influence students to dislike math, too.
Math is hard, but math is easier to measure than any other discipline, especially early on. History and elementary-level science are pretty concept-based, and reading comprehension is very important,
but difficult to measure quantitatively. As American schools are turned more and more toward testing, math takes the focal point. Some schools’ funding — and teachers’ salaries — are even dependent
on their students’ test scores.
There are a lot of good reasons for testing students’ math ability — namely, it holds teachers accountable that their students do not fall behind. If a student is falling behind, they can be given
help early on. But there are also negative effects of this testing — teachers teach to the test, rather than giving students a thorough understanding; and students can be ranked, which naturally
makes those in the lower strata feel ashamed. It sucks to be publicly humiliated; it sucks even more when you’re seven and never had a chance to excel at the subject you’re now being criticized for.
Students put in low-performance groups can develop a complex from a very young age that they are ‘bad at math,’ even if that is not the case at all.
Math as Magic
Nearly all subjects build on themselves: you learn the alphabet, then how to read simple words, and then how to analyze text and write essays. Your very basic ‘Columbus discovered America’ changes to
‘Columbus discovered people already living in America,’ and changes again at higher levels to ‘Columbus committed genocide.’ You learn the solar system in elementary school, and ten years later, you
use calculus and the laws of gravitation to calculate planets’ orbits.
None of these examples are as cut-and-dry as the steps of math: you learn how to count, and by counting on your fingers, you add. You add the same number a lot of times, and that’s multiplication.
You multiply the same number by itself many times, and that’s exponents. If you understand the steps, it’s not magic.
The problem is, math is hard, and there are test brackets to reach, so a lot of teachers begin teaching math as though it is magic. Rather than understanding long subtraction, you are taught
‘borrowing’ and good, accurate techniques — but if they do not register in your mind as logical steps, it ultimately doesn’t help. If math is magic that must be handed down by some higher power, of
course you don’t like it — you have no control or understanding of it.
Of course, anyone reading this article understands how subtraction works, even if you don’t remember the vocabulary around it (and who needs to?). But there are probably holes in your math knowledge
— maybe your trigonometry teacher never explained sines and cosines to your satisfaction, or the log function still confuses you. Maybe you skipped the theorem on how to find derivatives, and just
use the quick way, and you trust that it’s accurate, but you couldn’t re-create what Newton did. We’ve all been there.
Just remember two things:
Just because you don’t understand something does not mean you’re stupid, or lacking in any way.
Avoiding the problem won’t make it go away.
Now, one more note on math anxiety, and then we’ll go to solutions!
What Does the Research Say About Math Anxiety?
Let’s be honest: there is little to no solid, accredited research by higher institutions concerning math anxiety. That said, there are a lot of surveys and statistics on math anxiety — some of which
are quite professionally conducted and reviewed in educator circles. Teachers recognize that many of their students struggle with math, and they are investigating the reasons behind why students fear
mathematics, then seeking solutions. What have they found? Read on and see!
How Do You Alleviate Math Anxiety?
To overcome your math anxiety, you have to break the destructive cycle of fear-avoidance-fear. Most of us can’t grab our fear in a fist and crush it; that means you’d do best to tackle avoidance. It
isn’t easy, but with time and persistence, it will get easier, to the point where you don’t feel any math stress.
Here are some strategies for reducing math anxiety:
Study math.
Go over your notes, or do your homework. Make sure that you understand what you are doing; seeing yourself succeed, even if it is just on a homework assignment, will improve your self-confidence and
reduce your stress toward math.
Find a support system.
This can be a teacher, parent, or even friend. Most of us know at least one other student who is skilled at math, or a kindly teacher who can help explain. Even if you and your current math teacher
don’t click, think of previous teachers you’ve had, or even a teacher you haven’t had with a good reputation. Maybe your adviser can help you. There’s probably someone out there, and you can use them
as a resource when you don’t understand.
Organize your notes.
Disorganization creates stress. To be truthful, math is difficult to take notes for — ‘lecture’ isn’t as straightforward as it is with most humanities or science classes, and class is generally full
of practice problems. Find a way to organize your notes — via highlights, underlining, or separating your practice problems on a different page — so that you can access formulas and information
Attack your tests.
When you know there is a test coming up, create a study plan. Clearly delineate which concepts you know, and which you need to work on. Plan when you will study, either with a planner or calendar, or
something more informal (think texting yourself “math after dinner Thursday night”). Then study those concepts which need work, and seek out a friend or advisor if you need someone else to explain.
By the time the test rolls around, you should feel ready — then walking out of the test room, rather than feeling defeated, you’ll be strutting with confidence.
Overcome your own self-talk.
The biggest producer of math anxiety isn’t teachers, or tests, or other students — it’s you. That’s not an accusation. The fact is, your friends don’t actually care if you’re good at math; your
teachers aren’t judging you in particular when they teach dozens of students. It’s your instincts that fear shame or humiliation, and your subconscious urging you to avoid that shame by avoiding math
altogether. It’s not your fault, but it is something you can recognize in yourself, and take responsibility for.
Ultimately, if you succeed on a test, you should feel proud. You worked hard, learned the concepts, and owned that test. But if you are struggling, you must look at your own behavior; what can you do
to make yourself succeed?
Other Tips for Reducing Math Stress
Apart from the tips above, here are a few pointers:
Don’t let math feel like magic.
If you don’t understand something, ask questions — either in front of the class, or in private. If a certain method simply doesn’t make sense to you, ask the teacher to break it down, or go through
the proof with you. Make sure you understand why you are doing what you are doing.
This is a double-edged sword, because some textbooks suck. But at higher levels, even poorly written textbooks have lists of formulas and mathematical proofs for those formulas, which can help aid
understanding. So, read your textbook when you need a little extra help.
Prove things to yourself.
Forgot the laws of exponents? That’s fine! Use small numbers and prove how it works yourself. Like so:
a^n*m = (a^n)^m ????
2^2*3 = 2^6 = 2*2*2*2*2*2 = 64, and (2^2)^3 = (4)^3 = 4*4*4 = 64
Therefore, a^n*m = (a^n)^m !!!!
You can always prove laws of mathematics, and doing so makes you feel empowered. You own that math!
Test Yourself for Math Anxiety (Self-Test) — this section should be an adaptation of Ellen Freedman’s test
Still don’t want to do that math homework in your bag? Don’t worry, we’ve got more for you to read:
Rate these questions on a scale of how much they apply to you. Then add them up (and face your hatred of anything mathematical head on!)
I feel my mood drop when I step in to math class. 1 2 3 4 5
I dislike presenting problems on the board in front of the class. 1 2 3 4 5
I am afraid to ask questions. 1 2 3 4 5
I do not like being called on. 1 2 3 4 5
I feel like I am not prepared for what we are learning. 1 2 3 4 5
I tend to zone out in math class. 1 2 3 4 5
The thought of testing makes me feel stressed. 1 2 3 4 5
I don’t know how to study for math tests. 1 2 3 4 5
(a quick interlude, because this is very common: practice problems!! There are plenty in the back of your book, and on KhanAcademy, which is free)
I think I understand, but struggle to work the problems on my own. 1 2 3 4 5
I’m afraid of falling behind the rest of the class. 1 2 3 4 5
Okay, now add them up! (Betcha you can do it without a calculator!)
40-50 Ouch, you and math aren’t the best of friends. That’s okay! Face it and study for your next test!
30-39 Getting there. You’ll be alright.
20-29 This is pretty normal.
10-19 Why are you reading this? You’re fine!
Wrapping Things Up: Identifying and Coping with Math Anxiety
What are your takeaways?
Math anxiety in students is very normal. It is not an anxiety disorder, and it can be overcome by focusing on studying, cultivating good habits, and building self-confidence.
A big key to math anxiety is fear of failure, which is quite often unfounded — you are better at math than you think.
Math is not magic! Make sure you understand what you are doing, and why it works.
That’s it! Now stop reading and get to your homework!
Did you enjoy this post? Then you’ll love our other high school study tips. Check them out below:
> How to Get Good Grades in Math
> The 7 Most Common Careless Mistakes on the SAT Math Test (And How to Prevent Them) | {"url":"https://www.conqueryourexam.com/what-is-math-anxiety/","timestamp":"2024-11-12T23:14:29Z","content_type":"text/html","content_length":"368840","record_id":"<urn:uuid:e03b9434-606a-4e72-8842-eb73cdfa7f83>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00104.warc.gz"} |
The frequency and temperature characteristics of ceramic capacitor - Xuansn CapacitorThe frequency and temperature characteristics of ceramic capacitor
1 Frequency Characteristics of Ceramic Capacitor
🍉1.1 Q value and frequency characteristics of ceramic capacitor
The capacitance of the first type of ceramic dielectric capacitors (such as COG) is substantially invariant with frequency over the entire usable frequency range.
Q value and resonant frequency are important indicators when high-frequency/ultra-high-frequency capacitors are used in bad resonant circuits. High-frequency/ultra-high-frequency capacitors with
excellent performance have good performance in this regard, such as “Murata”‘s COG dielectric. Ultra-high frequency ceramic capacitors with a capacitance below 10pF have a Q value of more than 1000
meters below 400MHz. In fact, this Q value decreases with the increase of frequency, which can be explained by the increase of loss number with frequency. When the frequency is high to a certain
extent, the Q value will drop sharply (the Q value starts to drop sharply after about 6.8pF to above 1.5G), as shown in Figure 3.23, which is also consistent with the increase of ESR with the
increase of frequency.
Here, “ultra-high frequency” mainly means that it can work at ultra-high frequency, and of course it can also work at various frequencies below ultra-high frequency. As can be seen from the figure,
the characteristic shifts to the left as the capacitance increases. In fact, the discussion of larger capacitance will be meaningless at ultra-high frequency. For example, the 1000pF capacitive
reactance is only 0.318Ω at 1GHz, while the inductance of the 1cm lead is 10nH, and the resonance frequency with the 1000pF capacitor is about 50MHz, which is 1 at 1GHz. /20, that is to say, a
capacitor of 1000pF is used for a resonant circuit, and its resonant frequency generally does not exceed 50MHz. At a frequency of 1MHz, it can only be used for filtering or bypassing. This is the Q
value and it is meaningless.
🍇1.2 Resonant frequency, ESR and impedance frequency characteristics
Any capacitor has its own resonance frequency, that is, the frequency at which its own capacitance and parasitic inductance form series resonance. In the same package, the parasitic inductance is
basically the same. Naturally, the larger the capacitance, the lower the resonant frequency, as shown in Figure 3.24, and the smaller the package size, the higher the resonant frequency.
The ESR of the first type of ceramic dielectric capacitors increases with frequency, as shown in Figure 3.24, and as the frequency decreases, the ESR characteristics gradually flatten.
The ESR of the first type of dielectric, such as COG dielectric capacitors, decreases with the increase of capacitance, as shown in Figure 3.25. The reason is obvious. As the capacitance increases,
the area of the plate also increases. Under the same packaging conditions, only It can be obtained by increasing the number of layers of polar plates. Since the ESR of each layer of polar plates is
basically the same, the number of parallel polar plates increases, and the ESR will inevitably decrease.
The relationship between ESR, capacitance and frequency of C0G dielectric ceramic capacitors is shown in Figure 3.26. The impedance frequency characteristics are shown in Figure 3.27.
Characteristics can be divided into three parts: capacitive part, resonant part, inductive part. In the capacitive part, the capacitor exhibits capacitor characteristics, which is consistent with: Xc
=(1πƒ·C)-1, and the impedance decreases with the increase of frequency, as shown in the left half of the curve in Figure 3.27. In the resonance part, the inductive reactance of the parasitic
inductance of the capacitor increases with the frequency to a level close to the capacitive reactance. Since the inductive reactance and the capacitive reactance have opposite signs, the actual
impedance of the capacitor in this frequency band is smaller than the capacitive reactance of the capacitor. When the inductive reactance is equal to When the capacitive reactance is in the resonance
state, the capacitive reactance is canceled by the inductive reactance, leaving only the ESR, as shown in the steep drop of impedance in the characteristic curve in Figure 3.27. As the frequency
further increases, the inductive reactance begins to be larger than the capacitive reactance, and the capacitor begins to gradually behave as an inductive characteristic, as shown in the rising part
on the right side of the characteristic curve in Figure 3.27.
The impedance frequency characteristics of ceramic capacitor the second type of dielectric capacitors are shown in Figure 3.28. Similar to the first type of dielectric capacitors, the characteristics
of ceramic capacitor can also be divided into three parts: capacitive part, resonant part, and inductive part. The shape of the characteristics of ceramic capacitor curve is also basically the same.
Different from the first type of dielectric capacitors, the capacitance of the second type of dielectric capacitors is generally much larger than that of the first type of capacitors, and the
frequency band where the characteristic curve is located is lower than that of the first type of dielectric capacitors. For example, the resonance frequency of a capacitor with a capacitance of 10nF
is about 50MHz, the resonance frequency of a capacitor with a capacitance of 100nF is reduced to less than 20MHz, and the resonance frequency of a capacitor with a capacitance of 10UF is reduced to
Similar to the first class of dielectric capacitors, as the capacitance increases, the ESR also decreases. The difference is that the frequency characteristics of ESR have changed. Taking the 50V/
10UF capacitor of X5R dielectric as an example (the right side of Figure 3.28), the ESR in the low frequency band decreases with the increase of frequency, about 100kHz, in the range of 100kHz to
1MHz. The ESR of 1MHz is reduced to the lowest value, and the ESR of the frequency band above 1MHz increases with the increase of frequency.
As can be seen from the figure on the right side of Figure 3.28, the ESR of the 50V/10UF Type II ceramic capacitor is only 5mΩ, which is also an extremely low ESR value among various electrical
🍓1.3 Frequency characteristics of loss factor
The relationship between loss factor and frequency: The loss factor of the first type of ceramic dielectric capacitor increases with the increase of frequency. The relationship between the loss
factor of the first type of COG dielectric, capacitance and frequency is shown in Figure 3.29. Increasing the capacitive reactance decreases, so that the loss factor of the capacitor increases with
the increase of the capacitance at the same frequency. The dissipation factor of capacitors of the same capacitance increases with frequency. In fact, the dielectric loss of COG media does not vary
with frequency within the application frequency. The reason why the loss factor increases with frequency is that when the loss factor is measured under the condition that the terminal voltage of the
capacitor is constant, as the frequency increases, the capacitor current increases, so the loss generated in the capacitor’s ESR also increases. When the frequency is higher than a certain value, the
loss generated by the ESR becomes the main loss, and the loss factor at this time will increase linearly with the frequency.
The variation characteristics of the loss factor of the second type of ceramic dielectric capacitors with frequency are basically similar to those of the first type of dielectrics, and will not be
repeated here.
2 Temperature characteristics of ceramic capacitors
For ceramic capacitors, in addition to temperature affecting capacitance, there are insulation resistance, dissipation factor, etc.
🍒2.1 Insulation resistance of ceramic capacitor
The insulation resistance of X7R dielectric ceramic capacitor changes relatively greatly with temperature, as shown in Figure 3.30.
The insulation resistance of X7R dielectric ceramic capacitors decreases from 4000s (or Ω.F) at about +15°C to a little over 120s at +100°C; the insulation resistance of Y5V dielectric ceramic
capacitor changes with temperature than that of X7R, as shown in Figure 3.31 .
Insulation resistance drops from 2700s (or Ω.F) around +20°C to a little over 300s at +80°C. The insulation resistance of the two is basically the same, but lower than that of C0G. From the
relationship between insulation resistance and temperature, in high temperature applications, attention should be paid to whether the insulation resistance of the capacitor meets the requirements.
🍑2.2The relationship between the dissipation factor and temperature of the second type of ceramic dielectric capacitors
The dissipation factor of the second type of ceramic dielectric capacitor is significantly higher than that of the first type of dielectric, and it varies more with temperature. The dissipation
factor of X7R dielectric ceramic capacitors decreases as the temperature rises, from about 4.5% at -55°C to 1% at +125°C, and it hardly changes with temperature between 50 and 70°C. The dissipation
factor of Y5V dielectric ceramic capacitors decreases with temperature, from about 12% at -20°C to less than 1% at +85°C, of which it hardly changes with temperature between 50 and 85°C. When the
temperature is lower than normal temperature, the loss factor of X7R is obviously smaller than that of Y5V, and the loss factor of X7R is smaller than that of Y5V at normal temperature. | {"url":"https://capacitorsfilm.com/the-frequency-and-temperature-characteristics-of-ceramic-capacitors/","timestamp":"2024-11-04T01:12:02Z","content_type":"text/html","content_length":"107046","record_id":"<urn:uuid:b4270897-01c0-4e3b-85fb-3e3dc710419d>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00209.warc.gz"} |
Interactive Engineering Diagrams: Concept and UI
• How to succeed with this course?
• Interactive engineering diagrams using the user interface
• Interactive engineering diagrams
• Basic steps in a data pipeline
• Why interactive engineering diagrams?
• Engineering Diagram Parsing and its Algorithm
• The partial match parameter is a boolean, i.e., either true or false. If true, the algorithm allows partial matching of entities in the engineering diagrams. Note that the default value is always
false.By default, all tokens for entities returned as matches must be found in the diagram (a token is a substring of either consecutive letters or consecutive digits). However, if the partial
match is enabled (set to true), the algorithm also finds matches based on unique subsets of tokens.Note that the algorithm will still prefer matches where all tokens are found.
• End of course | {"url":"https://learn.cognite.com/interactive-engineering-diagrams-concept-and-ui-1","timestamp":"2024-11-01T23:37:22Z","content_type":"text/html","content_length":"199800","record_id":"<urn:uuid:89ec5116-8769-496d-8190-3efa4b2e6df4>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00712.warc.gz"} |
How Does The Calculator Work? | Unlocking the Mystery
Discover the inner workings of calculators as we demystify the question: How does the calculator work? From basic functions to complex operations, explore how a calculator operates step by step.
Understand the intricacies of how a calculator works, providing insights into its functionality, key mechanisms, and the fascinating world of mathematical computation.
Whether you’re curious about how a calculator works or seeking insights into its operations, this article unveils the answers you’ve been looking for. Explore the magic behind the question, “How does
a calculator work?”
How Does a Calculator Work?
1. Basic Operations:
At the core of a calculator’s functionality lies its ability to perform basic arithmetic operations such as addition, subtraction, multiplication, and division. Each keypress initiates a series of
electronic processes to execute these fundamental calculations.
2. Input and Display:
When you enter a number or equation into the calculator, the input is processed electronically. The calculator’s display, whether it’s an LCD screen or a more modern digital display, then showcases
the entered numbers or the calculated result.
3. Mathematical Functions:
Calculators are equipped with a variety of mathematical functions beyond basic operations. These include square roots, powers, logarithms, and more. The calculator’s internal programming allows it to
execute these functions efficiently.
4. Memory Storage:
Calculators often feature memory functions that allow users to store and recall numbers during calculations. This is particularly useful for handling multi-step calculations or for storing constants.
5. Modern Electronic Calculators:
Unlike early mechanical calculators, modern electronic calculators use microprocessors to handle calculations. This advanced technology enables faster and more complex mathematical operations.
6. Battery Power:
While some calculators rely on solar cells for power, many use batteries. These batteries provide the necessary energy to drive the internal circuits and power the display.
How a Calculator Works Step by Step:
1. User Input:
The process begins when a user inputs numbers, mathematical operations, or functions into the calculator using the designated keys.
2. Key Recognition:
The calculator’s microprocessor recognizes the key inputs and translates them into electronic signals.
3. Electronic Processing:
The electronic circuitry processes these signals, performing the corresponding mathematical operations based on the user’s input.
4. Display Output:
The calculated result or intermediate steps are then displayed on the calculator’s screen for the user to view.
5. Memory Functions:
If the user utilizes memory functions, the calculator stores and retrieves values as needed during calculations.
6. Power Source:
The calculator draws power from its energy source, whether it’s batteries or solar cells, to sustain the electronic processes.
How does the calculator work – FAQs:
How does a calculator know how to do math?
Calculators have internal microprocessors programmed with algorithms that enable them to perform mathematical operations. The user’s input initiates these algorithms, allowing the calculator to
execute the specified calculations.
How does a calculator work step by step?
The calculator processes user input through electronic circuits, performs mathematical operations based on the input, and displays the calculated result. Memory functions and power sources contribute
to the overall functionality.
How do calculators work without batteries?
Some calculators, especially those with solar cells, can operate without batteries by harnessing energy from ambient light. However, many calculators use batteries as a reliable power source.
Do calculators use logic gates?
Yes, calculators utilize logic gates within their electronic circuitry. Logic gates process binary information, allowing the calculator to perform the necessary mathematical operations through a
series of logical steps.
In conclusion, the workings of a scientific calculator are a symphony of electronic processes, logic, and efficient algorithms. From basic addition to complex mathematical functions, these devices
have evolved over time to become indispensable tools for anyone dealing with numbers. Understanding how calculators operate provides a fascinating glimpse into the intersection of technology and
mathematics, highlighting the innovation that has shaped these everyday devices.
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Towards Accurate Scene Text Detection with Bidirectional Feature Pyramid Network
Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu 610041, China
School of Computer Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China
Author to whom correspondence should be addressed.
Submission received: 24 February 2021 / Revised: 3 March 2021 / Accepted: 7 March 2021 / Published: 16 March 2021
Scene text detection, this task of detecting text from real images, is a hot research topic in the machine vision community. Most of the current research is based on an anchor box. These methods are
complex in model design and time-consuming to train. In this paper, we propose a new Fully Convolutional One-Stage Object Detection (FCOS)-based text detection method that can robustly detect
multioriented and multilingual text from natural scene images in a per pixel prediction approach. Our proposed text detector employs an anchor-free approach, unlike state-of-the-art text detectors
that do not rely on a predefined anchor box. In order to enhance the feature representation ability of FCOS for text detection tasks, we apply the Bidirectional Feature Pyramid Network (BiFPN) as the
backbone network, enhancing the model learning capacity and increasing the receptive field. We demonstrate the superior performance of our method on multioriented (ICDAR-2015, ICDAR-2017 MLT) and
horizontal (ICDAR-2013) text detection benchmark tasks. Moreover, our method has an f-measure of 88.65 and 86.32 for the benchmark datasets ICDAR 2013 and ICDAR 2015, respectively, and 80.75 for the
ICDAR-2017 MLT dataset.
1. Introduction
Scene text detection is both fundamental and challenging in the field of machine vision, and plays a critical role in subsequent text recognition tasks. Current mainstream text detection methods [
], rely on predefined anchor frames to extract high-quality word candidate regions. Despite the success of these methods, they are associated with the following limitations: (1) text detection
results are highly sensitive to the size, orientation, and the number of predefined anchor boxes. (2) Due to the variable size, shape, and orientation of text in natural scenes, it is difficult to
capture all the text instances via the predefined anchor boxes. (3) A large number of anchor boxes are required in order to improve text detection performance, resulting in complex and time-consuming
calculations. For example, in order to improve the accuracy of DeepText [
], Zhong empirically designed four scales and six aspect ratios, resulting in 24 prior bounding boxes at each sliding position. The number of anchors is 2.6 times more than those of Faster R-CNN [
Fully convolutional networks (FCNs) [
] recently have been very successful in the dense prediction task, including semantic segmentation [
], keypoint detection [
], and depth estimation [
]. Several scene text detection methods [
] treat text detection as a semantic segmentation problem and use the FCN for pixel-level text prediction. Liao et al. [
] proposed a novel binarization module called differentiable binarization (DB), which enabled the segmentation network to set the threshold of binarization adaptively, greatly improving the
performance of text detection. Such methods typically diverge from anchor boxes to anchor-free frameworks by using corner/center points. This facilitates computational efficiency and generally
improves the performance over anchor box-based text detectors. Since only coarse text blocks can be detected from the saliency map, a complex postprocessing step is required to extract the precise
bounding boxes [
]. Recently, Tian et al. [
] proposed a fully convolutional one-stage object detector (FCOS) pipeline to solve this issue and achieve state-of-the-art results in general object detection tasks.
In this paper, we design a simple yet efficient anchor-free method based on FCOS, a one-stage fully convolutional text detection framework with a weighted bidirectional pyramid feature network
(BiFPN) for the scene text detection of natural images. Our proposed method’s architecture is shown in
Figure 1
. In order to enhance the feature representation ability, we employ EfficientNet as a new backbone network. Experiments demonstrate the superior performance of our proposed approach compared to
state-of-the-art methods for benchmark (ICDAR-2013, ICDAR-2015, ICDAR-2017 MLT) text detection datasets.
2. Related Work
Anchor-based text detector.
Text detection methods based on region proposals use a general object detection framework and often employ regression text boxes to get the region text information [
]. For example, in [
], the GoogleNet [
] inception structure was employed to improve Faster R-CNN [
]. As a result, an initial region proposal network (InceptionRPN) was generated, which acquired text candidate regions, removed background regions using a text detection network, and voted on the
detected overlapping regions to determine the optimal result. Jiang et al. [
] proposed a rotational region convolutional network (R2CNN) to detect arbitrarily-oriented text in scene images. A novel connectionist text proposal network (CTPN) was proposed in [
] in order to locate text lines in scene images. In [
], a vertically regressed proposal network (VRPN) was proposed to match text regions using multiple neighboring small anchors. While in [
], Ma et al. presented the rotation region proposal network (RRPN) to detect arbitrarily oriented text. This paper aims to generate tilted proposals with angular information about the text
orientation. The angle information is then adjusted and bounding box regression is performed to make the proposals more accurately fit the orientation of the text region.
Previous research has adopted bounding boxes or quadrangles as a text description approach. For example, the approach presented in [
] was based on a single shot MultiBox detector (SSD) [
] object detection framework, which used a quadrilateral or rotated rectangle representation to replace the rectangular box. Reference [
] proposed an end-to-end two-stage scene text detection network architecture, named the quadrilateral region proposal network (QRPN), that can accurately locate scene texts with quadrilateral
boundaries. In [
], the authors proposed the rotation-sensitive regression detector (RRD) framework to perform classification and regression on different features extracted by two different designs of network
branches. Deng et al. [
] proposed a new two-stage algorithm. In the first stage, the method predicts text instance locations by detecting and linking corners instead of traditional anchor points. In the second stage, the
authors designed a pooling layer called dual-Roi pooling, which embeds data augmentation inside a regional sub-network.
Anchor-free text detector.
Anchor-free-based approaches treat text as a distinct object and leverage efficient object detection architectures (e.g., YOLOv1 [
], SSD [
], CornerNet [
], and DenseBox [
]) to detect words or text lines directly from natural images. YOLOv1 [
] does not use anchor boxes, but rather predicts bounding boxes at points close to the center of the object, resulting in a low recall. CornerNet [
] is a recently introduced single-stage anchor-free detector that detects bounding box corner pairs and groups them together to make the final detected bounding box. However, CornerNet needs a
complex postprocessing procedure to cluster corner pairs that belong to the same instance. An additional distance metric also needs to be learned when grouping. Another family of anchor-free
detectors is that based on DenseBox [
] (e.g., UnitBox [
]). These detectors are considered unsuitable for generic object detection due to difficulties in handling overlapping bounding boxes and relatively low recall values. FCOS [
] is a single-stage anchor-free detector recently proposed to obtain detection accuracy comparable to traditional anchor-based detectors. Unlike YOLOv1, FCOS utilizes all points in the ground truth
bounding box to predict the bounding box, while the detected low-quality bounding boxes are restrained by the proposed “centerness” branch. In this paper, we introduce a method based on FCOS, and
integrate the Bidirectional Feature Pyramid Network (BiFPN) into the FCOS framework. Experiments demonstrate the ability of BiFPN to enhance the model learning capacity and increase the receptive
3. Our Approach
3.1. Bidirectional Feature Pyramid Network
Mainstream text detection architectures employ pyramid feature combination steps (e.g., feature pyramid network (FPN)) to enrich features with high-level semantic information. The traditional FPN
generally enriches the feature maps from the final output of a single path architecture in a top-down manner. Despite their great success, such methods are limited by several factors: (1) the design
does not incorporate high-level context with the former level features, retaining the spatial detail and semantic information in the network path. (2) Input features vary with resolution, resulting
in inconsistent contributions to the output feature. Tan et al. [
] recently proposed a bidirectional pyramid network (BiFPN) that fused multiscale features for object detection. The framework contained two key modules: cross-scale connections and weighted feature
fusion. Unlike the one-way information flow of the traditional top-down FPN, BiFPN included a bottom-up path aggregation network and an additional edge from the original input to the output node. We
employed five levels of feature maps defined as {P3, P4, P5, P6, P7} (
Figure 2
), with each feature level P3, P4, P5, P6, and P7 having 8, 16, 32, 64, and 128 strides, respectively.
The fast feature fusion of the BiFPN, as follows:
$P 3 o u t = C o n v w 1 ⋅ P 3 i n + w 2 ⋅ R e s i z e P 4 i n w 1 + w 2 + ϵ , P 4 t d = C o n v w 1 ⋅ P 4 i n + w 2 ⋅ R e s i z e P 5 i n w 1 + w 2 + ϵ , P 4 o u t = C o n v w 1 ⋅ P 4 i n + w 2 ⋅ P
4 t d + w 3 ⋅ R e s i z e P 3 o u t w 1 + w 2 + w 3 + ϵ , P 7 o u t = C o n v w 1 ⋅ P 7 i n + w 2 ⋅ R e s i z e P 6 o u t w 1 + w 2 + ϵ ,$
$P n t d$
is the middle result of the nth layer on the top-down path;
$P n out$
is the output result of the nth layer on the bottom-up path; Resize is an upsampling or downsampling operation for resolution matching; Conv is depthwise separable convolution [
] for feature fusion, and here we use weighted normalized feature fusion [
≥ 0 is guaranteed by applying a Relu after each w
; and
= 0.0001.
3.2. FCOS for Text Detection
The majority of state-of-the-art text detectors such as Deep-text [
], TextBoxes [
], TextBoxes++ [
] and ABCNet [
] use a predefined anchor box, which requires elaborate parameter tuning and complex calculations for box IoUs during training. With no anchor box, the FCOS [
] can predict the 4D vector and class labels for each spatial location on the feature map layer directly. The 4D vector describes the relative offsets (
, and
) from the four sides (left, top, right, and bottom) of a location bounding box (
Figure 3
Anchor-Free Text Detection Head.
$F i ∈ ℝ H × W × C$
be the feature maps at layer
of a backbone CNN and s be the total stride until the layer. The ground-truth bounding boxes for an input image are defined as {Bi}, where
$B i = x 0 i , y 0 i , x 1 i y 1 i ∈ ℝ 4$
. Here
$x 0 i , y 0 i$
$x 1 i y 1 i$
denote the coordinates of the left-top and right-bottom corners of the bounding box. We can map each location p(x,y) on feature map Fi back onto the input image using
$⌊ s 2 ⌋ + x s , ⌊ s 2 ⌋ + y s$
. This is close to the center of the receptive field of location p. Anchor-based text detectors consider the location on the input image as the anchor box center and regress the target bounding box
with these anchor boxes as references. In contrast, following [
], we treated the location as a training sample instead of an anchor-box and regressed the target bounding box at the location.
In our framework, p was treated as a positive sample if it fell into any ground-truth box. In addition to the classification label, we also defined the 4D real vector
= (
) as the regression targets at the location, where
are the distances from the location to the four edges of the bounding box (as shown in
Figure 3
). We simply selected the bounding box with the minimal area as the regression target. More specifically, if location (
) was associated with bounding box Bi, the training regression targets for the location can be determined as Equation (2).
$l * = x − x 0 i , t * = y − y 0 i r * = x 1 i − x , b * = y 1 i − y$
After the execution of the feature extraction backbone, the anchor-free text detection head predicted the text location in the images of nature.
Figure 4
presents the network architecture of the text box detection network. Similar to [
], the input features of the backbone network were fed into the three convolutional layers for the final text/nontext classification and the quadrilateral bounding box regression branches. Note that
the proposed method has at least 9× fewer network output variables than popular anchor-based text detectors [
] that use preset anchor boxes. Following [
], we also employed the centerless branch to eliminate low-quality text prediction bounding boxes.
4. Experiments
4.1. Datasets
We evaluated our proposed method on several standard benchmark tasks including ICDAR 2013 (IC13) [
] and ICDAR 2015 (IC15) [
] for multioriented text detection and ICDAR 2017 MLT (MLT17) [
] for multilingual text detection. IC13 [
] inherits from ICDAR 2003 [
], with 229 and 233 natural images for training and testing, respectively. IC15 [
], the first incidental scene text dataset, was built for the Incidental Scene Text challenge in the ICDAR-2015 Robust Reading Competition and contains 1000 images for training and 500 images for
validation/testing. The 17,548 text instances (annotated by the 4 vertices of the quadrangle) are usually skewed or blurred since they are acquired without prior preference or intention. IC15
provides word-level English annotations. MLT17 [
] consists of 18,000 images containing text in 9 different languages: Arabic, Bengali, Chinese, English, French, German, Italian, Japanese, and Korean. A total of 9000 images were used for training
the model (7200 for training and 1800 for validation), and the other half for testing.
In order to compare with the state-of-the-art methods, we performed the comparison on three popular public datasets. Specifically, we used the official evaluation tools in the public datasets ICDAR
2013, ICDAR 2015, while for ICDAR 2017, we used the evaluation tools provided by the authors [
4.2. Implementation Details
We used EfficientNet-B1 [
] as the backbone networks for our proposed model, with the hyper-parameters following those of EfficientDet [
]. In particular, our network was trained using the stochastic gradient descent (SGD) across 80 K iterations with the initial learning rate of 0.01 and a minibatch of 16 images. The learning rate was
reduced by a factor of 10 at iterations 50 K and 70 K. Furthermore, the weight decay and momentum were set as 0.0005 and 0.9, respectively. We pretrained the weights on ImageNet [
] for the initialization of our backbone networks, while the newly added layers were initialized by applying random weights with a gaussian distribution of mean 0 and standard deviation of 0.01. For
the ICDAR-2017 MLT dataset, we used the training and validation data (i.e., 9000 training images), while for both ICDAR 2013 and ICDAR 2015, we employed the pretrained model from ICADAR-2017 MLT,
with the provided training images applied for finetuning. We implemented our method in Py torch and performed the training on a RTX TITAN GPU system.
Loss Function:
The loss function employed during training is defined as follows:
$L = L c l s + L box + L center ,$
where the classification loss
$L c l s$
, box regression loss
$L box$
, and centerness loss
$L center$
are equal to those in [
4.3. Results and Comparison
We compared the performance of our approach with that of the state-of-the-art methods using the ICDAR 2013 (IC13), ICDAR 2015 (IC5), and ICDAR 2017 MLT (MLT17) benchmark datasets (
Table 1
Table 2
Table 3
Comparison with Prior Works.Table 1
Table 2
Table 3
demonstrate that our approach outperformed the other methods for all three datasets with just the use of single-scale and single-model testing. For example, our approach achieved values of 93.1,
84.6, and 88.65 for the precision, recall, and F-measure, respectively, on the challenging ICDAR-2013 dataset (
Table 1
). Equivalent values for the ICDAR-2015 dataset were 87.6, 84.91, and 86.23, respectively, surpassing the other methods despite their use of extra training data (
Table 2
). The same trend was observed for the MLT17 dataset, with values of 83.41, 78.26, and 80.75, respectively (
Table 3
). Our proposed method achieved better results than the other methods on these three challenging text detection benchmarks.
Figure 5
depicts the qualitative detection results.
Our framework had the following advantages.
• Scene text detection was designed as a proposal-free and anchor-free pipeline, which did not require the manual design of an anchor box or the heuristic adjustment of an anchor box, reducing the
number of parameters and simplifying the training process.
• Compared to the anchor box-based methods, our one-stage text detector, which avoided the use of RPN networks and IOU-based proposal filtering, greatly reduced the computation.
• As a result, our text detection framework, was simpler and more efficient. Our framework could be easily extended to other vision tasks, providing a new solution for the detection task of scene
text recognition.
BiFPN is better than FPN.
We compared BiFPN with FPN on the three benchmark datasets. As shown in
Table 1
Table 2
Table 3
, the integration of BiFPN into the proposed approach improved the F-measure by 2.7, 2.2 and 2.5% compared to the proposed approach with the traditional FPN. In our experiments, we observed that
BiFPN had better feature extraction ability than FPN. The main reason for this was BiFPN’s two-way feature fusion, which could better preserve text features.
5. Conclusions
In the current paper, we proposed a new FCOS based text detection approach that includes an anchor-box and proposal-free one-stage text detector. Our method can robustly detect text from natural
scene images, which is simpler, more efficient, and more scalable than anchor-based methods. Moreover, we demonstrated the ability of the bidirectional feature pyramid network (BiFPN) as the new
backbone network of FCOS to significantly enhance the feature representation of FCOS, effectively improving text detection in natural scene pictures. Our proposed method achieved better results than
other state-of-the-art methods on these three challenging text detection benchmarks (ICDAR 2013, ICDAR 2015 and ICDAR 2017 MLT). In the future, we will continue to focus on feature fusion methods to
further improve detection capabilities. In addition, due to the simplicity of our framework, we are interested in extending our framework to scene text recognition tasks.
Author Contributions
Conceptualization, D.C., J.D. and Y.Z.; methodology, D.C., J.D. and Y.Z.; software, D.C., J.D. and Y.Z.; validation, D.C., J.D. and Y.Z.; formal analysis, D.C. and Y.Z.; resources, D.C., J.D. and
Y.Z.; data curation, D.C., J.D. and Y.Z.; writing—original draft preparation, D.C., and J.D.; writing—review and editing, D.C. and Y.Z.; All authors have read and agreed to the published version of
the manuscript.
This research was funded by Science & Technology Department of Sichuan Province, grant number 2020ZHZY0002.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
This research was supported by the Science & Technology Department of Sichuan Province (Grant No. 2020ZHZY0002). The authors sincerely thank Teddy Zhang who provided valuable comments in writing this
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Architecture of the proposed fully convolutional one-stage (FCOS)-based text detector, compromising a backbone, bidirectional feature pyramid network (BiFPN) and FCOS box head for text
Figure 3. The proposed method predicts a 4D vector (l, t, r, b), where l, t, r, b are the distances from the location to the four sides of the bounding box.
Figure 4. Architecture of text box prediction network, where H and W are the height and width of the feature maps.
Work P R F
FASText [48] 84 69 77
DeepText [1] 85 81 83
WeText [49] 82.6 93.6 87.7
TextBoxes [2] 89 83 86
R2CNN [26] 92 81 86
SegLink [50] 87.7 83 85.3
SLPR [51] 90 72 80
RGC [52] 89 77 83
Proposed + FPN 89.6 81.7 85.92
Proposed + BiFPN 93.1 84.6 88.65
Work P R F
WordSup [53] 77.03 79.33 78.16
SSTD [54] 80 73 77
EAST [55] 83.27 78.3 80.72
FOTS [56] 88.8 82 85.3
TextBoxes++ [3] 87.8 78.5 82.9
[57] 94.1 70.7 80.7
Proposed + FPN 85.6 82.43 83.99
Proposed + BiFPN 87.6 84.91 86.23
Work P R F
FOTS [56] 80.95 57.51 67.25
FOTS * [56] 81.80 62.30 70.70
SPCNET [58] 80.60 68.60 74.10
[59] 88.60 73.90 80.50
[57] 83.80 55.60 72.40
PSENet [60] 77.01 68.40 72.45
Pixel-Anchor [61] 79.54 59.54 68.10
Pixel-Anchor * [61] 83.90 65.80 73.76
[21] 80.00 69.80 74.30
Proposed + FPN 80.12 76.42 78.23
Proposed + BiFPN 83.41 78.26 80.75
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MDPI and ACS Style
Cao, D.; Dang, J.; Zhong, Y. Towards Accurate Scene Text Detection with Bidirectional Feature Pyramid Network. Symmetry 2021, 13, 486. https://doi.org/10.3390/sym13030486
AMA Style
Cao D, Dang J, Zhong Y. Towards Accurate Scene Text Detection with Bidirectional Feature Pyramid Network. Symmetry. 2021; 13(3):486. https://doi.org/10.3390/sym13030486
Chicago/Turabian Style
Cao, Dongping, Jiachen Dang, and Yong Zhong. 2021. "Towards Accurate Scene Text Detection with Bidirectional Feature Pyramid Network" Symmetry 13, no. 3: 486. https://doi.org/10.3390/sym13030486
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Year 12 Maths Extension 1 | Dymocks Tutoring
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MathSciDoc: An Archive for Mathematician
In this paper, we introduce a motivic version of To¨en’s derived Hall algebra. Then we point out that the two kinds of Hall algebras in the sense of To¨en and Kontsevich–Soibelman, respectively, are
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Multiple COUNTIFS plus OR
Hi gurus! Here is our question of the week:
I have a formula with multiple COUNTIFS, but I also need an OR...with a COUNTIF and CONTAINS. But this doesn't seem to be working in my favor! (Getting the Unparseable error.) Here's what I have so
=(COUNTIFS({Sheet1}, "2020", {Sheet1}, "3") + COUNTIFS({Sheet2}, "2020", {Sheet2}, "3") ...COUNTIFS({Sheet5}, "2020", {Sheet5}, "3")), OR(COUNTIF({Sheet1}, CONTAINS("20I", @cell))))
Any tips?
• I think your issue is the ordering of the formula. I can't quite tell what you're trying to get the OR to do here, but the OR should be at the point in a formula where you need to make a
distinction between two different events. Where you have it, at the end, there is nothing for it to resolve to.
Can you describe what you're trying to get the OR to do in this formula and then that might lead to a solution. Please use the @mention so that I will get a notification when you post your reply.
• I think you are correct, but I wasn't able to get that to work either.
I need to count the cells that fall into the large COUNTIFS category. If a cell doesn't fall into that category, I need to look in the "20I" criteria to count it and include in the overall
I hope this makes sense!
• That does make sense. I think you'll need to include the IF in each of the COUNTIF statements then. So it would read "If the counts in the large category contain a result, then use that number,
otherwise, look at a different range. Something like this should work:
=IF(COUNTIFS({Sheet1}, "2020", {Sheet1}, "3") > 0, COUNTIFS({Sheet1}, "2020", {Sheet1}, "3"), COUNTIF({Sheet1}, CONTAINS("20I", @cell)))
Then you would just repeat that with a + to add your other calculations for your other sheets.
• Wow--that is going to be a huge equation! Is there no easier/cleaner way to accomplish this?
• Not that I can think of. This is the only way I know how to check for 1 condition, and if it isn't met run a different calculation.
You could use helper columns to break up your formula into smaller chunks if the size of the formula is a concern. You'd essentially create a column for each sheet formula and then your main
column would just add all the helper columns together.
So I think it is a question of more columns or a long formula.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/83437/multiple-countifs-plus-or","timestamp":"2024-11-05T08:39:52Z","content_type":"text/html","content_length":"424358","record_id":"<urn:uuid:467fc0e5-b982-4540-980a-8be138994561>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00441.warc.gz"} |
Graphing y = y=2x^2-16x+29
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$4 x - 16 = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 4$$
The values of the extrema at the points:
(4, -3)
Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest
deviation from:
Minima of the function at points:
$$x_{1} = 4$$
The function has no maxima
Decreasing at intervals
$$\left[4, \infty\right)$$
Increasing at intervals
$$\left(-\infty, 4\right]$$ | {"url":"https://calculator-online.org/graph/e/y_equal_two_x_squared_minus_16x_pls_twenty_minus_nine","timestamp":"2024-11-05T12:24:26Z","content_type":"text/html","content_length":"54416","record_id":"<urn:uuid:814c81de-fbfa-4c34-bc1e-aae3c4073a36>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00643.warc.gz"} |
Gold Leaching of Pyrite Concentrate - 911Metallurgist
Gold Leaching of Pyrite Concentrate
The ore consisted of quartz, in which, above the 250-ft. level, the iron-minerals were largely oxidized and some free gold was visible; below that level few traces of oxidation occurred, and pyrite
constituted the principal mineralizer in the quartz, together with occasional pockets of galena and a few eccentric specks of covellite. The 20-stamp mill was equipped with plates for amalgamation,
and three Standard tables for concentration. The tails from the tables were elevated by a centrifugal pump to a launder, by which they were conveyed to a 60-ton cyanide-mill for further treatment.
The cyanide-mill was erected at so great a distance from the stamp-mill that a separate crew of workers and a separate power-plant were necessary. No arrangement had been made for the treatment of
the concentrates as produced; although later, an adobe roasting-furnace had been erected near the stamp-mill, as well as a small cyanide-plant, for the treatment of the roasted product.
The treatment of the mill-tails in the remote cyanide-plant was unprofitable. The concentrates could not be shipped for treatment, on account of the high freight-cost and high smelter-charges ; and,
as the law of Arizona prohibited the use of wood as a fuel for roasting ore, the adobe furnace was useless. As a natural conclusion, it seemed evident that, if possible, an entire change of programme
was necessary, involving two requisites : 1. A very clean concentration of the mill-tails, producing a final material containing very low values; 2. A local treatment of the raw concentrates.
The concentrating-plant was unfitted to attain the first requisite ; the Standard tables affording a very fair rough concentration, but making a very poor separation of middlings and slimes. Two Frue
vanners were installed, the inefficient middling-pumps from the tables were thrown on the scrap- heap, and the plant started operation as follows: The tails from 10 stamps passed from the plates to
one Standard table, and from this upon a second table. The tails from the other 10 stamps were similarly conducted upon a Standard table, but from this into a small spitzkasten, situated above the
distributor of Frue vanner No. 1, from which the thickened pulp passed to the vanner for concentration, while the overflow passed to the slime-sump, mentioned later. The slimes from each of the
tables were conducted by launder into a sump, from which they were elevated by a 2-in. Byron Jackson centrifugal pump into a set of 3 large spitzkasten arranged in a series, the overflow from the
first passing into the second, etc., the final overflow, consisting of fairly clean water, passing into a supply-tank for use as feed-water for the batteries. The more or less de-watered slimes from
the large spitzkasten were conducted by launders to a small spitzkasten, which discharged a product containing still less water upon Frue vanner No. 2 for concentration, the overflow from this last
spitzkasten also passing back to the slime-sump. In the case of the first 10 stamps, the middlings from the first table passed with the tails to the second table, from which the final
middling-product was caught in a separate receptacle and returned to the battery. The middlings from the second 10 stamps passed with the tails from the table to the vanner. By this arrangement, only
that water was wasted which was necessary to carry off the final tails and that which was necessary for the vanner-supply. It was found that the final tails were of very low grade; the result being
somewhat surprising, considering the smallness of the concentration-plant for an output of 70 tons per day, but being doubtless referable to the comparatively simple character of the ore.
A futile attempt had formerly been made to extract the values of the table-concentrates by cyanidation without roasting, and about 60 tons remained in one of the cyanide tanks, from which a portion
of the values had been extracted, but which yet contained about 6 oz. of silver and 2.25 oz. of gold per ton. An analysis of these concentrates suggested the following approximate constitution: SiO2,
17.64; CaO, 4.95; FeS2, 52.73; Fe2O3, 25.31; Pb, not det.; Cu, none; As, none; total, 100.63 per cent.
Continued leaching of these concentrates, for about 21 days, with solutions of various strengths, removed about 75 per cent, of the remaining values, but with a large consumption of cyanide, when the
extraction practically ceased. The material was thoroughly aerated by turning over with shovels, and one-half placed in an adjoining tank; both portions were again subjected to leaching with cyanide,
but with no compensatory extraction.
In the above treatment of the concentrates, the first solutions drawn off were of a brilliant claret color; and, as no copper had been found in the analysis, the reason for the phenomenon was
somewhat obscure. A repeated test of the concentrates again showed no copper, but an analysis of the material, precipitated from the colored solution by addition of acid, proved it to consist of
cupric ferro-cyanide. Further consideration discovered the source of the copper in the residue left by evaporation in one of the solution-tanks; this had been taken up by the new solution and
produced the color mentioned. Analysis of samples of freshly-made concentrates showed them to contain about 0.2 per cent, of copper; enough to assist nicely in the zinc-precipitation.
After various experiments upon the raw concentrates, it was found that if the material was ground to 100-mesh and agitated for 32 hr., about 85 per cent, of the gold and 70 per cent, of the silver
could be extracted, at an expense of about 6.5 lb. of cyanide per ton. This result promised a very fair return (especially in a case where no other procedure was available); and the system was put in
effect in the mill, with the happy, and somewhat unusual, result that the mill-practice has yielded very much better returns than those obtained in the laboratory experiments ; a fact which is
probably due to the rise in temperature produced during the grinding of the pulp.
The yield of concentrates from the mill was about 2 tons per day. For grinding so small a product, the purchase of a ball-or tube-mill was out of the question; so also was the use of one of the
stamp-batteries for the purpose; and about the least expensive machine which could be thought of, which would have a sufficient capacity and which would probably fulfill the other requirements, was
one of the old-style amalgamating pans, such as are used in silver-mills. A second-hand, 5-ft. pan, with wooden sides, was bought in, installed in the mill, and arranged to discharge into each of two
leaching-tanks belonging to the small cyanide-plant formerly mentioned. The pan was charged with about one ton of solution carrying 6 lb. of cyanide, 6 lb. of lime and 1.5 tons of raw concentrates;
it was set in motion at the rate of about 75 rev. per min., and continued to grind for 8.5 hr.; about 2 lb. more of cyanide being added during the day, as the strength failed. At the end of the
period, the material was found to be finely ground, and was discharged into the leaching-tank; it was also found that, at the close of the grinding, the temperature of the mass had risen about 40° F.
above the outside temperature. A sample of the ground pulp was taken, filtered, washed and assayed, with the somewhat surprising result that an extraction was shown of 90.7 per cent, of the values.
Since the initial test, the operation has been carried on continuously with similar results; the only variation occurring when the grinding was affected by the clogging of the mullers by foreign
matter. Since this occurrence, the concentrates have been passed through an 8-mesh screen, and no further difficulty has been noticed. After the grinder is discharged into the leaching-tank, and
after the solution has settled to some extent, it is customary to cover the material in the tank with dry middlings from the table, for the purpose of facilitating percolation; in this manner,
filtration goes on with sufficient rapidity. When one leaching-tank is filled, it is continuously leached with cyanide solution, while the other tank is being filled; it is then washed and discharged
through a bottom-discharge door. The capacity of each tank is about 15 tons; so that a tank has about 4 days of extra leaching after it is filled; it then has 3 days’ washing before discharge. During
a campaign of 6 months, in which time various grades of concentrates have been treated, the average extraction has been about 94 per cent, of the total values. The grinding and cyanide-plant require
little attention, except in the matter of charging the grinder and discharging it, and the whole plant is easily operated, without additional cost, by the man who has charge of the tables and
vanners. The pan consumes about 8 h.p., and, as stated, the total cyanide consumption is about 8 lb. per ton; adding the cost of 6 lb. of lime, the total cost of the treatment of the concentrates,
including the values left in the tails, does not materially exceed $5 per ton.
The concentrates, as they are taken from the tables and vanners, are given as much chance as possible to aerate and oxidize, since it is found to be the case that partly oxidized material grinds more
quickly and gives up its values with a notably less amount of cyanide. In the extraction of the final values from the old concentrates upon which tests were first made, it is found necessary to grind
them for 2.5 hr. only. Muller-shoes last about 3 months; while the dies last much longer. The zinc-precipitation is good, and the solutions have not yet become too foul for use; most of the copper is
slagged off in remelting the bullion.
In the first attempts at the use of the grinding-pan, it was the intention to make the pan work continuously, by overflowing and allowing the continual discharge of finely-ground ore-particles into
the leaching-tank. It was found, however, that the rapid motion of the mullers carried coarse as well as fine to the surface, and the scheme was abandoned. It was next attempted to allow a continuous
discharge through pipes set midway up the sides of the pan, sizing with cyanide solution by spitzlutten, and return of the coarse material to the grinder. This also was abandoned, and the
discharge-opening at the bottom of the pan was threaded and a valve put in. The charge in the pan is now ground, the valve connected with a discharge-pipe, the valve then opened and the contents of
the pan discharged.
To show the grinding-efficiency of the pan, a series of sizing- tests were made upon concentrates as they came from the tables, and upon similar material after having been ground for 8 hr., as
In crushing to the above degree of fineness, the material seemed to be fairly amenable to cyanidation, while not too fine for good filtration; so that 8-hr. grinding is continued. | {"url":"https://www.911metallurgist.com/blog/pyrite-concentrate-gold-leaching/","timestamp":"2024-11-08T04:35:51Z","content_type":"text/html","content_length":"165042","record_id":"<urn:uuid:62d4b008-d699-4310-9afe-56881d7d377a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00020.warc.gz"} |
How much it costs to run your own server 24/7?
How much money it costs to run your personal computer as a server 24/7?
Assuming a month has 30 days:
24 hours * 30 days = 720 hours
Desktop CPU TDP (Thermal design power) can be 65W, 95W or much higher; assuming total power 400W:
400W * 720h / 1000 = 288kWh
Price: depends on your location. California is expensive, 40 cents to 80 cents per kWh; other states in the US can be 10 cents to 30 cents. If 40 centts / kWh:
288 kWh * $0.5 = $144
Plug in your numbers and compare that with a cloud VM, then decide if it makes sense for you to run you own server. | {"url":"https://www.hackingnote.com/en/hardware/operating-cost/","timestamp":"2024-11-07T17:07:06Z","content_type":"text/html","content_length":"80305","record_id":"<urn:uuid:b23feb78-15a0-4c9d-99e9-335093085aef>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00201.warc.gz"} |
Quantum Hall Physics in String Theory
by Oren Bergman (hep-th/0401106, 8 pages, 2 figures)
Deformation Quantization: Observable Algebras, States and Representation Theory
by Stefan Waldmann (hep-th/0303080, 32 pages)
Threebranes in F-theory
by Alastair Paulin-Campbell (hep-th/0205060, 126 pages)
Strings, quantum gravity and non-commutative geometry on the lattice
by J. Ambjorn (hep-lat/0201012)
Aspetti non perturbativi della Teoria delle Stringhe
by G. D'Appollonio (hep-th/0111284)
Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space
by Cosmas K. Zacho (hep-th/0110114, 22 pages, 2 figures)
Heterotic, Open and Unoriented String Theories from Topological Membrane
by Pedro Castelo Ferreira (hep-th/0110067, 122 pages, 34 figures)
Aspects non perturbatifs de la theorie des supercordes
by B. Pioline (hep-th/9806123, 181 pages, 14 figures) | {"url":"https://www.stringwiki.org/w/index.php?title=Other&oldid=1697","timestamp":"2024-11-09T21:50:21Z","content_type":"text/html","content_length":"17243","record_id":"<urn:uuid:246282ff-1962-473f-aaca-c08869a11aca>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00225.warc.gz"} |
Curley Effect - iGeek
Curley Effect
From iGeek
James Michael Curley, a four-time mayor of Boston, used wasteful redistribution to his poor Irish constituents to buy votes, while he used incendiary rhetoric/policies to encourage richer citizens to
emigrate from Boston, thereby shaping the electorate in his favor. Boston stagnated, the people suffered, but the chickens kept voting for Colonel Sanders (e.g. Curley kept winning elections). The
Curley effect is inefficient redistributive policies pushed by incumbent politicians to shape the electorate through emigration of their opponents or reinforcement of class identities. In other
words, California (and the lefts) political model.
We can't communicate effectively if we don't agree on what words or terms mean.
Cultural Marxists
decided that since they uusally can't win through honesty, logic, history and facts, they could win by twisting/perverting meanings (especially in popular culture and colleges), to distort every
discussion into a debate on pedantics, or use truthspeak as a litmus test for who is properly indoctrinated/compliant. This section isn't intended as a comprehensive dictionary, but just to stop that
gaslighting, by defining what I (and often history/society means or should mean) when using a term. Not what the far left is trying to re-invent terms into.
The land of intolerance, hypocrisy and progressivism... but I repeat myself. This article lists some examples of the intolerance, incompetence, and progressivism that has come to exemplify the Golden
State. (NOTE: While the Golden State once referred to the color of the dried grass hills so common in California, it now refers to the vagrant urine covered streets of San Francisco or L.A.) | {"url":"https://www.igeek.com/Curley_Effect","timestamp":"2024-11-03T18:26:55Z","content_type":"text/html","content_length":"23159","record_id":"<urn:uuid:c40a3ad4-fdab-480e-8cdd-08475463c293>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00410.warc.gz"} |
Re: how to avoid numeric error
• To: mathgroup at smc.vnet.net
• Subject: [mg20364] Re: [mg20309] how to avoid numeric error
• From: BobHanlon at aol.com
• Date: Sun, 17 Oct 1999 02:45:39 -0400
• Sender: owner-wri-mathgroup at wolfram.com
8.551437202665365431742222523666`12.6535*^864 -
Despite its large magnitude, the magnitude of the imaginary part is small
compared to the magnitude of the real part (differ by 12 orders of magnitude).
10^864*Chop[(-5.2)^1208./10^864] == (-5.2)^1208
Extending the concept of Chop
relativeChop[x_ , delta_:10^-10] /; (Abs[x] == 0) = 0;
relativeChop[x_, delta_:10^-10] :=
Module[{mag = Abs[x]}, mag*Chop[x/mag, delta]]
relativeChop[(-5.2)^1208.] == (-5.2)^1208
Bob Hanlon
In a message dated 10/16/1999 12:38:51 AM, atulksharma at yahoo.com writes:
>I am at a loss to explain this behavior, though I perhaps
>misunderstand how Mathematica implements it's machine precision routines.
>This is a simple example of a problem that cropped up during evaluation
>constants of integration
>in a WKB approximation, where I would get quite different results depending
>on how the constant was evaluated. I have localized the discrepancy to
>term of the form shown below:
>testParameters =
> {x1 -> 5.2, x2 -> 0.3, x3 -> 0.002, x4 -> -0.00025}
>(-x1)^(-(x2 + x3)/x4) /. testParameters
>In this case, as it turns out, x1 = -5.2, which is a floating point number,
>and the exponent = 1208 (which may be integer or floating point, but is
>floating point in this case).
>I assumed that the result would be evaluated to machine precision in either
>since x1 is a float regardless. However, depending on whether the exponent
>is integer or not, I get two different results, with a large imaginary
>8.55143720266536543174145`12.6535*^864 -
> 2.48026735232231456274073`12.6535*^852*I
>I assume that this has some simple relationship to machine precision and
>round-off error, but am I wrong in assuming that x1 should determine the
>numeric precision of the entire operation?
>I am using Mathematica 3.01.1 on a PC/Win95 platform.
>I also encountered another problem, which bothers me because it's so
>insidious. In moving a notebook from one machine to another by floppy (work
>to home), a parsing error occurred buried deep inside about 30 pages of
>code. A decimal number of the form 1.52356 was parsed as 1.5235 6 with
>space inserted and interpreted as multiplication. The same error occured
>the same place on several occasions (i.e. when I start getting bizarre
>results, I know to go and correct this error).
>I know these sound minor, but they have a large effect on the solution
>could easily go undetected. Thanks in advance. | {"url":"https://forums.wolfram.com/mathgroup/archive/1999/Oct/msg00231.html","timestamp":"2024-11-08T18:07:30Z","content_type":"text/html","content_length":"32823","record_id":"<urn:uuid:7baf2832-3faf-44d8-ade2-f423179fdb30>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00689.warc.gz"} |
Darwin's God
Information Organization
recent paper
, authored by Winston Ewert, uses a dependency graph approach to model the relationships between the species. This idea is inspired by computer science which makes great use of dependency graphs.
Complicated software applications typically use a wealth of lower level software routines. These routines have been developed, tested, and stored in modules for use by higher level applications. When
this happens the application inherits the lower-level software and has a dependency on that module.
Such applications are written in human-readable languages such as Java. They then need to be translated into machine language. The compiler tool performs the translation, and the build tool assembles
the result, along with the lower level routines, into an executable program. These tools use dependency graphs to model the software, essentially building a design diagram, or blueprint which shows
the dependencies, specifying the different software modules that will be needed, and how they are connected together.
Dependency graphs also help with software design. Because they provide a blueprint of the software architecture, they are helpful in designing decoupled architectures and promoting software reuse.
Dependency graphs are also used by so-called “DevOps” teams to assist at deployment time in sequencing and installing the correct modules.
What Ewert has shown is that, as with computer applications which inherit software from a diverse range of lower-level modules, and those lower-level modules likewise feed into a diverse range of
applications, biology’s genomes likewise reveal such patterns. Genomes may inherit molecular sequence information from a wide range of genetic modules, and genetic modules may feed into a diverse
range of genomes.
Superficially, from a distance, this may appear as the traditional evolutionary tree. But that model has failed repeatedly as scientists have studied the characters of species more closely.
Dependency graphs, on the other hand, provide a far superior model of the relationships between the species, and their genetic information flow.
Ten Thousand Bits?
Did you know Mars is going backwards? For the past few weeks, and for several weeks to come, Mars is in its retrograde motion phase. If you chart its position each night against the background stars,
you will see it pause, reverse direction, pause again, and then get going again in its normal direction. And did you further know that retrograde motion helped to cause a revolution? Two millennia
ago, Aristotelian physics dictated that the Earth was at the center of the universe. Aristarchus’ heliocentric model, which put the Sun at the center, fell out of favor. But what Aristotle’s
geocentrism failed to explain was retrograde motion. If the planets are revolving about the Earth, then why do they sometimes pause, and reverse direction? That problem fell to Ptolemy, and the
lessons learned are still important today.
Ptolemy explained anomalies such as retrograde motion with additional mechanisms, such as epicycles, while maintaining the circular motion that, as everyone knew, must be the basis of all motion in
the cosmos. With less than a hundred epicycles, he was able to model, and predict accurately the motions of the cosmos. But that accuracy came at a cost—a highly complicated model.
In the Middle Ages William of Occam pointed out that scientific theories ought to strive for simplicity, or parsimony. This may have been one of the factors that drove Copernicus to resurrect
Aristarchus’ heliocentric model. Copernicus preserved the required circular motion, but by switching to a sun-centered model, he was able to reduce greatly the number of additional mechanisms, such
as epicycles.
Both Ptolemy’s and Copernicus’ models accurately forecast celestial motion. But Copernicus was more parsimonious. A better model had been found.
Kepler proposed ellipses, and showed that the heliocentric model could become even simpler. It was not well accepted though because, as everyone knew, celestial bodies travel in circles. How foolish
to think they would travel along elliptical paths. That next step toward greater parsimony would have to wait for the likes of Newton, who showed that Kepler’s ellipses were dictated by his new,
highly parsimonious, physics. Newton described a simple, universal, gravitational law. Newton’s gravitational force would produce an acceleration, which could maintain orbital motion in the cosmos.
But was there really a gravitational force? It was proportional to the mass of the object which was then cancelled out to compute the acceleration. Why not have gravity cause an acceleration
Centuries later Einstein reported on a man in Berlin who fell out of a window. The man didn’t feel anything until he hit the ground! Einstein removed the gravitational force and made the physics even
simpler yet.
The point here is that the accuracy of a scientific theory, by itself, means very little. It must be considered along with parsimony. This lesson is important today in this age of Big Data. Analysts
know that a model can always be made more accurate by adding more terms. But are those additional terms meaningful, or are they merely epicycles? It looks good to drive the modeling error down to
zero by adding terms, but when used to make future forecasts, such models perform worse.
There is a very real penalty for adding terms and violating Occam’s Razor, and today advanced algorithms are available for weighing the tradeoff between model accuracy and model parsimony.
This brings us to common descent, a popular theory for modeling relationships between the species. As we have discussed many times here, common descent fails to model the species, and a great many
additional mechanisms—biological epicycles—are required to fit the data.
And just as cosmology has seen a stream of ever improving models, the biological models can also improve. This week a very important model has been proposed in a new paper, authored by Winston Ewert,
in the Bio-Complexity journal.
Inspired by computer software, Ewert’s approach models the species as sharing modules which are related by a dependency graph. This useful model in computer science also works well in modeling the
species. To evaluate this hypothesis, Ewert uses three types of data, and evaluates how probable they are (accounting for parsimony as well as fit accuracy) using three models.
Ewert’s three types of data are: (i) Sample computer software, (ii) simulated species data generated from evolutionary / common descent computer algorithms, and (iii) actual, real species data.
Ewert’s three models are: (i) A null model in which entails no relationships between
any species, (ii) an evolutionary / common descent model, and (iii) a dependency graph model.
Ewert’s results are a Copernican Revolution moment. First, for the sample computer software data, not surprisingly the null model performed poorly. Computer software is highly organized, and there
are relationships between different computer programs, and how they draw from foundational software libraries. But comparing the common descent and dependency graph models, the latter performs far
better at modeling the software “species.” In other words, the design and development of computer software is far better described and modeled by a dependency graph than by a common descent tree.
Second, for the simulated species data generated with a common descent algorithm, it is not surprising that the common descent model was far superior to the dependency graph. That would be true by
definition, and serves to validate Ewert’s approach. Common descent is the best model for the data generated by a common descent process.
Third, for the actual, real species data, the dependency graph model is astronomically superior compared to the common descent model.
Let me repeat that in case the point did not sink in. Where it counted, common descent failed compared to the dependency graph model. The other data types served as useful checks, but for the data
that mattered—the actual, real, biological species data—the results were unambiguous.
Ewert amassed a total of nine massive genetic databases. In every single one, without exception, the dependency graph model surpassed common descent.
Darwin could never have even dreamt of a test on such a massive scale.
Darwin also could never have dreamt of the sheer magnitude of the failure of his theory. Because you see, Ewert’s results do not reveal two competitive models with one model edging out the other.
We are not talking about a few decimal points difference. For one of the data sets (HomoloGene), the dependency graph model was superior to common descent by a factor of 10,064. The comparison of the
two models yielded a preference for the dependency graph model of greater than ten thousand.
Ten thousand is a big number.
But it gets worse, much worse.
Ewert used Bayesian model selection which compares the probability of the data set given the hypothetical models. In other words, given the model (dependency graph or common descent), what is the
probability of this particular data set? Bayesian model selection compares the two models by dividing these two conditional probabilities. The so-called Bayes factor is the quotient yielded by this
The problem is that the common descent model is so incredibly inferior to the dependency graph model that the Bayes factor cannot be typed out. In other words, the probability of the data set given
the dependency graph model, is so much greater than the probability of the data set given the common descent model, that we cannot type the quotient of their division.
Instead, Ewert reports the logarithm of the number. Remember logarithms? Remember how 2 really means 100, 3 means 1,000, and so forth?
Unbelievably, the 10,064 value is the logarithm (base value of 2) of the quotient! In other words, the probability of the data on the dependency graph model is so much greater than that given the
common descent model, we need logarithms even to type it out. If you tried to type out the plain number, you would have to type a 1 followed by more than 3,000 zeros!
That’s the ratio of how probable the data are on these two models!
By using a base value of 2 in the logarithm we express the Bayes factor in bits. So the conditional probability for the dependency graph model has a 10,064 advantage of that of common descent.
10,064 bits is far, far from the range in which one might actually consider the lesser model. See, for example, the Bayes factor Wikipedia page, which explains that a Bayes factor of 3.3 bits
provides “substantial” evidence for a model, 5.0 bits provides “strong” evidence, and 6.6 bits provides “decisive” evidence.
This is ridiculous. 6.6 bits is considered to provide “decisive” evidence, and when the dependency graph model case is compared to comment descent case, we get 10,064 bits.
But it gets worse.
The problem with all of this is that the Bayes factor of 10,064 bits for the HomoloGene data set is the very best case for common descent. For the other eight data sets, the Bayes factors range from
40,967 to 515,450.
In other words, while 6.6 bits would be considered to provide “decisive” evidence for the dependency graph model, the actual, real, biological data provide Bayes factors of 10,064 on up to 515,450.
We have known for a long time that common descent has failed hard. In Ewert’s new paper, we now have detailed, quantitative results demonstrating this. And Ewert provides a new model, with a far
superior fit to the data.
Guess Who Wins?
The title of John Farrell’s article in Commonweal from earlier this year is a dead giveaway. When writing about the interaction between faith and science, as Farrell does in the piece, the title “The
Conflict Continues” is like a flashing red light that the mythological Warfare Thesis is coming at you.
Sure enough, Farrell does not disappoint. He informs his readers that the fear that science could “make God seem unnecessary” is “widespread today among religious believers,” particularly in the US
where “opposition to belief in evolution remains very high.”
Indeed, this fear has “haunted the debate over the tension between religion and science for centuries.” Farrell notes that Edward Larson and Michael Ruse point out in their new book On Faith and
Science, that the “conflict model doesn’t work so well. But that seems to be a minor speed bump for Farrell. He finds that:
The idea that the world operates according to its own laws and regularities remains controversial in the evolution debate today, as Intelligent Design proponents attack the consensus of science
on Darwinian evolution and insist that God’s direct intervention in the history of life can be scientifically demonstrated.
Farrell also writes that Isaac Newton, driven by concerns about secondary causes, “insisted God was still necessary to occasionally tweak the motions of the planets if any threatened to wander off
Farrell’s piece is riddled with myths. Secondary causes are not nearly as controversial as he would have us believe. He utterly mischaracterizes ID, and Newton said no such thing. It is true that
Newton suggested that the Creator could intervene in the cosmos (not “insisted”).
And was this the result of some radical voluntarism?
Of course not. Newton suggested God may intervene in the cosmos because the physics of the day (which by the way he invented), indicated that our solar system could occasionally have instabilities.
The fact that was running along just fine, and hadn’t yet blown up, suggested that something had intervened along the way.
Newton was arguing from science, not religion. But that doesn’t fit the Epicurean mythos that religion opposes naturalism while science confirms it. The reality is, of course, the exact opposite.
Pop Quiz: Who Said It?
There are many fundamental problems with evolutionary theory. Origin of life studies have dramatically failed. Incredibly complex biological designs, both morphological and molecular, arose abruptly
with far too little time to have evolved. The concept of punctuated equilibrium is descriptive, not explanatory. For example, the Cambrian Explosion is not explained by evolution and, in general,
evolutionary mechanisms are inadequate to explain the emergence of new traits, body plans and new physiologies. Even a single gene is beyond the reach of evolutionary mechanisms. In fact, the
complexity and sophistication of life cannot originate from non-biological matter under any scenario, over any expanse of space and time, however vast. On the other hand, the arch enemy of
evolutionary theory, Lamarckian inheritance, in its variety of forms, is well established by the science.
Another Darwin’s God post?
No, these scientific observations are laid out in a new peer-reviewed, scientific paper.
Origin of Life
Regarding origin of life studies, which try to explain how living cells could somehow have arisen in an ancient, inorganic, Earth, the paper explains that this idea should have long since been
rejected, but instead it has fueled “sophisticated conjectures with little or no evidential support.”
the dominant biological paradigm - abiogenesis in a primordial soup. The latter idea was developed at a time when the earliest living cells were considered to be exceedingly simple structures
that could subsequently evolve in a Darwinian way. These ideas should of course have been critically examined and rejected after the discovery of the exceedingly complex molecular structures
involved in proteins and in DNA. But this did not happen. Modern ideas of abiogenesis in hydrothermal vents or elsewhere on the primitive Earth have developed into sophisticated conjectures with
little or no evidential support.
In fact, abiogenesis has “no empirical support.”
independent abiogenesis on the cosmologically diminutive scale of oceans, lakes or hydrothermal vents remains a hypothesis with no empirical support
One problem, of many, is that the early Earth would not have supported such monumental evolution to occur:
The conditions that would most likely to have prevailed near the impact-riddled Earth's surface 4.1–4.23 billion years ago were too hot even for simple organic molecules to survive let alone
evolve into living complexity
In fact, the whole idea strains credibility “beyond the limit.”
The requirement now, on the basis of orthodox abiogenic thinking, is that an essentially instantaneous transformation of non-living organic matter to bacterial life occurs, an assumption we
consider strains credibility of Earth-bound abiogenesis beyond the limit.
All laboratory experiments have ended in “dismal failure.” The information hurdle is of “superastronomical proportions” and simply could not have been overcome without a miracle.
The transformation of an ensemble of appropriately chosen biological monomers (e.g. amino acids, nucleotides) into a primitive living cell capable of further evolution appears to require
overcoming an information hurdle of superastronomical proportions, an event that could not have happened within the time frame of the Earth except, we believe, as a miracle. All laboratory
experiments attempting to simulate such an event have so far led to dismal failure.
Diversity of Life
But the origin of life is just the beginning of evolution’s problems. For science now suggests evolution is incapable of creating the diversity of life and all of its designs:
Before the extensive sequencing of DNA became available it would have been reasonable to speculate that random copying errors in a gene sequence could, over time, lead to the emergence of new
traits, body plans and new physiologies that could explain the whole of evolution. However the data we have reviewed here challenge this point of view. It suggests that the Cambrian Explosion of
multicellular life that occurred 0.54 billion years ago led to a sudden emergence of essentially all the genes that subsequently came to be rearranged into an exceedingly wide range of
multi-celled life forms - Tardigrades, the Squid, Octopus, fruit flies, humans – to name but a few.
As one of the authors writes, “the complexity and sophistication of life cannot originate (from non-biological) matter under any scenario, over any expanse of space and time, however vast.” As an
example, consider the octopus.
First, the octopus is an example of novel, complex features, rapidly appearing and a vast array of genes without an apparent ancestry:
Its large brain and sophisticated nervous system, camera-like eyes, flexible bodies, instantaneous camouflage via the ability to switch colour and shape are just a few of the striking features
that appear suddenly on the evolutionary scene. The transformative genes leading from the consensus ancestral Nautilus (e.g., Nautilus pompilius) to the common Cuttlefish (Sepia officinalis) to
Squid (Loligo vulgaris) to the common Octopus (Octopus vulgaris) are not easily to be found in any pre-existing life form.
But it gets worse. As Darwin’s God has explained, The Cephalopods demonstrate a highly unique level of adenosine to inosine mRNA editing. It is yet another striking example of lineage-specific design
that utterly contradicts macroevolution:
These data demonstrate extensive evolutionary conserved adenosine to inosine (A-to-I) mRNA editing sites in almost every single protein-coding gene in the behaviorally complex coleoid Cephalopods
(Octopus in particular), but not in nautilus. This enormous qualitative difference in Cephalopod protein recoding A-to-I mRNA editing compared to nautilus and other invertebrate and vertebrate
animals is striking. Thus in transcriptome-wide screens only 1–3% of Drosophila and human protein coding mRNAs harbour an A-to-I recoding site; and there only about 25 human mRNA messages which
contain a conserved A-to-I recoding site across mammals. In Drosophila lineages there are about 65 conserved A-sites in protein coding genes and only a few identified in C. elegans which support
the hypothesis that A-to-I RNA editing recoding is mostly either neutral, detrimental, or rarely adaptive. Yet in Squid and particularly Octopus it is the norm, with almost every protein coding
gene having an evolutionary conserved A-to-I mRNA editing site isoform, resulting in a nonsynonymous amino acid change. This is a virtual qualitative jump in molecular genetic strategy in a
supposed smooth and incremental evolutionary lineage - a type of sudden “great leap forward”. Unless all the new genes expressed in the squid/octopus lineages arose from simple mutations of
existing genes in either the squid or in other organisms sharing the same habitat, there is surely no way by which this large qualitative transition in A-to-I mRNA editing can be explained by
conventional neo-Darwinian processes, even if horizontal gene transfer is allowed.
In the twentieth century Lamarckian Inheritance was an anathema for evolutionists. Careers were ruined and every evolutionist knew the inheritance of acquired characteristics sat right along the flat
earth and geocentrism in the history of ideas. The damning of Lamarck, however, was driven by dogma rather than data, and today the evidence has finally overcome evolutionary theory.
Indeed there is much contemporary discussion, observations and critical analysis consistent with this position led by Corrado Spadafora, Yongsheng Liu, Denis Noble, John Mattick and others, that
developments such as Lamarckian Inheritance processes (both direct DNA modifications and indirect, viz. epigenetic, transmissions) in evolutionary biology and adjacent fields now necessitate a
complete revision of the standard neo-Darwinian theory of evolution or “New Synthesis " that emerged from the 1930s and 1940s.
Indeed, we now know of a “plethora of adaptive Lamarckian-like inheritance mechanisms.”
There is, of course, nothing new in this paper. We have discussed these, and many, many other refutations of evolutionary theory. Yet the paper is significant because it appears in a peer-reviewed
journal. Science is, if anything, conservative. It doesn’t exactly “follow the data,” at least until it becomes OK to do so. There are careers and reputations at stake.
And of course, there is religion.
Religion drives science, and it matters.
Numerous, Successive, Slight Modifications
Proteins are a problem for theories of spontaneous origins for many reasons. They consist of dozens, or often hundreds, or even thousands of amino acids in a linear sequence, and while many different
sequences will do the job, that number is tiny compared to the total number of sequences that are possible. It is a proverbial needle-in-the-haystack problem, far beyond the reach of blind searches.
To make matters worse, many proteins are overlapping, with portions of their genes occupying the same region of DNA. The same set of mutations would have to result in not one, but two proteins,
making the search problem that much more tricky. Furthermore, many proteins perform multiple functions. Random mutations somehow would have to find those very special proteins that can perform double
duty in the cell. And finally, many proteins perform crucial roles within a complex environment. Without these proteins the cell sustains a significant fitness degradation. One protein that fits this
description is centrobin, and now a new study shows it to be even more important than previously understood.
Centrobin is a massive protein of almost a thousand amino acids. Its importance in the division of animal cells has been known for more than ten years. An important player in animal cell division is
the centrosome organelle which organizes the many microtubules—long tubes which are part of the cell’s cytoskeleton. Centrobin is one of the many proteins that helps the centrosome do its job.
Centrobin depletion causes “strong disorganization of the microtubule network,” and impaired cell division.
Now, a new study shows just how important centrobin is in the development of the sperm tail. Without centrobin, the tail, or flagellum, development is “severely compromised.” And once the sperm is
formed, centrobin is important for its structural integrity. As the paper concludes:
Our results underpin the multifunctional nature of [centrobin] that plays different roles in different cell types in Drosophila, and they identify [centrobin] as an essential component for
C-tubule assembly and flagellum development in Drosophila spermatogenesis.
Clearly centrobin is an important protein. Without it such fundamental functions as cell division and organism reproduction are severely impaired.
And yet how did centrobin evolve?
Not only is centrobin a massive protein, but there are no obvious candidate intermediate structures. It is not as though we have that “long series of gradations in complexity” that Darwin called for:
Although the belief that an organ so perfect as the eye could have been formed by natural selection, is enough to stagger any one; yet in the case of any organ, if we know of a long series of
gradations in complexity, each good for its possessor, then, under changing conditions of life, there is no logical impossibility in the acquirement of any conceivable degree of perfection
through natural selection.
Unfortunately, in the case of centrobin, we do not know of such a series. In fact, centrobin would seem to be a perfectly good example of precisely how Darwin said his theory could be falsified:
If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down. But I
can find out no such case.
Darwin could “find out no such case,” but he didn’t know about centrobin. Darwin required “a long series of gradations,” formed by “numerous, successive, slight modifications.”
With centrobin we are nowhere close to fulfilling these requirements. In other words, today’s science falsifies evolution. This, according to Darwin’s own words.
Religion drives science, and it matters.
Bacterial Resistance to Antibiotics
Rachel Gross’ recent article about evolutionist’s public outreach contains several misconceptions that are, unfortunately, all too common. Perhaps most obvious is the mythological Warfare Thesis that
Gross and her evolutionary protagonists heavily rely on. Plumbing the depths of ignorance, Gross writes:
Those who research the topic call this paradigm the “conflict mode” because it pits religion and science against each other, with little room for discussion. And researchers are starting to
realize that it does little to illuminate the science of evolution for those who need it most.
“Those who research the topic call this paradigm the ‘conflict mode’”?
This is reminiscent of Judge Jones endorsement of Inherit the Wind as a primer for understanding the origins debate, for it is beyond embarrassing. Exactly who are those “who research the topic” to
which Gross refers?
Gross is apparently blithely unaware that there are precisely zero such researchers. The “conflict mode” is a long-discarded, failed view of history promoted in Inherit the Wind, a two-dimensional,
upside-down rewrite of the 1925 Monkey Trial.
But ever since, evolutionists have latched onto the play, and the mythological history it promotes, in an unabashed display of anti-intellectualism. As Lawrence Principe has explained:
The notion that there exists, and has always existed, a “warfare” or “conflict” between science and religion is so deeply ingrained in public thinking that it usually goes unquestioned. The idea
was however largely the creation of two late nineteenth-century authors who confected it for personal and political purposes. Even though no serious historians of science acquiesce in it today,
the myth remains powerful, and endlessly repeated, in wider circles
Or as Jeffrey Russell writes:
The reason for promoting both the specific lie about the sphericity of Earth and the general lie that religion and science are in natural and eternal conflict in Western society, is to defend
Darwinism. The answer is really only slightly more complicated than that bald statement.
Rachel Gross is, unfortunately, promoting the “general lie” that historians have long since been warning of. Her article is utter nonsense. The worst of junk news.
But it gets worse.
Gross next approvingly quotes Brigham Young University associate professor Jamie Jensen whose goal is to inculcate her students with Epicureanism. “Acceptance is my goal,” says Jensen, referring to
her teaching of spontaneous origins in her Biology 101 class at the Mormon institution.
As we have explained many times, this is how evolutionists think. Explaining their anti-scientific, religious beliefs is not enough. You must believe. As Jensen explains:
By the end of Biology 101, they can answer all the questions really well, but they don’t believe a word I say. If they don’t accept it as being real, then they’re not willing to make important
decisions based on evolution — like whether or not to vaccinate their child or give them antibiotics.
Whether or not to give their child antibiotics?
As we have discussed many times before, the equating of “evolution” with bacterial resistance to antibiotics is an equivocation and bait-and-switch.
The notion that one must believe in evolution to understand bacterial resistance to antibiotics is beyond absurd.
It not only makes no sense; it masks the monumental empirical contradictions that bacterial antibiotic resistance presents to evolution. As a university life science professor, Jensen is of course
well aware of these basic facts of biology.
And she gets paid to teach people’s children?
Religion drives science, and it matters.
There You Go Again
Why are evolutionists always wrong? And why are they always so sure of themselves? With the inexorable march of science, the predictions of evolution, which evolutionists were certain of, just keep
on turning out false. This week’s failure is the much celebrated notion that the eukaryote’s power plant—the mitochondria—shares a common ancestor with the alphaproteobacteria. A long time ago, as
the story goes, that bacterial common ancestor merged with an early eukaryote cell. And these two entities, as luck would have it, just happened to need each other. Evolution had just happened to
create that early bacterium, and that early eukaryote, in such a way that they needed, and greatly benefited from, each other. And, as luck would have it again, these two entities worked together.
The bacterium would just happen to produce the chemical energy needed by the eukaryote, and the eukaryote would just happen to provide needed supplies. It paved the way for multicellular life with
all of its fantastic designs. There was only one problem: the story turned out to be false.
The story that mitochondria evolved from the alphaproteobacteria lineage has been told with great conviction. Consider the Michael Gray 2012 paper which boldly begins with the unambiguous truth claim
that “Viewed through the lens of the genome it contains, the mitochondrion is of unquestioned bacterial ancestry, originating from within the bacterial phylum α-Proteobacteria (Alphaproteobacteria).
There was no question about it. Gray was following classic evolutionary thinking: similarities mandate common origin. That is the common descent model. Evolutionists say that once one looks at
biology through the lens of common descent everything falls into place.
Except that it doesn’t.
Over and over evolutionists have to rewrite their theory. Similarities once thought to have arisen from a common ancestor turn out to contradict the common descent model. Evolutionists are left
having to say the similarities must have arisen independently.
And big differences, once thought to show up only in distant species, keep on showing up in allied species.
Biology, it turns out, is full of one-offs, special cases, and anomalies. The evolutionary tree model doesn’t work.
Now, a new paper out this week has shown that the mitochondria and alphaproteobacteria don’t line up the way originally thought. That “unquestioned bacterial ancestry” turns out to be, err, wrong.
The paper finds that mitochondria did not evolve from the currently hypothesized alphaproteobacterial ancestor, or from “any other currently recognized alphaproteobacterial lineage.”
The paper does, however, make a rather startling claim. The authors write:
our analyses indicate that mitochondria evolved from a proteobacterial lineage that branched off before the divergence of all sampled alphaproteobacteria.
Mitochondria evolved from a proteobacterial lineage, predating the alphaproteobacteria?
That is a startling claim because, well, simply put there is no evidence for it. The lack of evidence is exceeded only by the evolutionist’s confidence. Note the wording: “indicate.”
The evolutionist’s analyses indicate this new truth.
How can the evolutionists be so sure of themselves in the absence of literally any evidence?
The answer is, because they are evolutionists. They are completely certain that evolution is true. And since evolution must be true, the mitochondria had to have evolved from somewhere. And the same
is true for the alphaproteobacteria. They must have evolved from somewhere.
And in both cases, that somewhere must be the earlier proteobacterial lineage. There are no other good evolutionary candidates.
Fortunately this new claim cannot be tested (and therefore cannot be falsified), because the “proteobacterial lineage” is nothing more than an evolutionary construct. Evolutionists can search for
possible extant species for hints of a common ancestor with the mitochondria, but failure to find anything can always be ascribed to extinction of the common ancestor.
This is where evolutionary theory often ends up: failures ultimately lead to unfalsifiable truth claims. Because heaven forbid we should question the theory itself.
Religion drives science, and it matters.
Pure Junk
Evolutionists do not have a clear understanding of how photosynthesis arose, as evidenced by a new paper from Kevin Redding’s laboratory at Arizona State University which states that:
After the Type I/II split, an ancestor to photosystem I fixed its quinone sites and then heterodimerized to bind PsaC as a new subunit, as responses to rising O2 after the appearance of the
oxygen-evolving complex in an ancestor of photosystem II. These pivotal events thus gave rise to the diversity that we observe today.
That may sound like hard science to the uninitiated, but it isn’t.
The Type I/II split is a hypothetical event for which the main evidence is the belief that evolution is true. In fact, according to the science, it is astronomically unlikely that photosynthesis
evolved, period.
And so, in typical fashion, the paper presents a teleological (“and then structure X evolved to achieve Y”) narrative to cover over the absurdity:
and then heterodimerized to bind PsaC as a new subunit, as responses to rising O2 …
First, let’s reword that so it is a little clearer: The atmospheric oxygen levels rose and so therefore the reaction center of an early photosynthesis system heterodimerized in order to bind a new
protein (which helps with electron transfer).
This is a good example of the Aristotelianism that pervades evolutionary thought. This is not science, at least in the modern sense. And as usual, the infinitive form (“to bind”) provides the
telltale sign. In other words, a new structure evolved as a response to X (i.e., as a response to the rising oxygen levels) in order to achieve Y (i.e., to achieve the binding of a new protein,
But it gets worse.
Note the term: “heterodimerized.” A protein machine that consists of two identical proteins mated together is referred to as a “homodimer.” If two different proteins are mated together it is a
“heterodimer.” In some photosynthesis systems, at the core of the reaction center is a homodimer. More typically, it is a heterodimer.
The Redding paper states that the ancient photosynthesis system “heterodimerized.” In other words, it switched, or converted, the protein machine from a homodimer to a heterodimer (in order to bind
PsaC). The suffix “ize,” in this case, means to cause to be or to become. The ancient photosynthesis system caused the protein machine to become a heterodimer.
Such teleology reflects evolutionary thought and let’s be clear—this is junk science. From a scientific perspective there is nothing redeeming here. It is pure junk.
But it gets worse.
These pivotal events thus gave rise to the diversity that we observe today.
Or as the press release described it:
Their [reaction centers’] first appearance and subsequent diversification has allowed photosynthesis to power the biosphere for over 3 billion years, in the process supporting the evolution of
more complex life forms.
So evolution created photosynthesis which then, “gave rise to” the evolution of incredibly more advanced life forms. In other words, evolution climbed an astronomical entropic barrier and created
incredibly unlikely structures which were crucial for the amazing evolutionary history to follow.
The serendipity is deafening.
Religion drives science, and it matters.
As Though They Were Planted There
In the famed Cambrian Explosion most of today’s animal phyla appeared abruptly in the geological strata. How could a process driven by blind, random mutations produce such a plethora of new species?
Evolutionist Steve Jones has speculated that the Cambrian Explosion was caused by some crucial change in DNA. “Might a great burst of genetic creativity have driven a Cambrian Genesis and given birth
to the modern world?” [1] What explanations such as this do not address is the problem of how evolution overcame such astronomical entropic barriers. Rolling a dice, no matter how creatively, is not
going to design a spaceship.
The Cambrian Explosion is not the only example of the abrupt appearance of new forms in the fossil record, and the other examples are no less easy for evolution to explain. Nor has the old saw, that
it’s the fossil record’s fault, fared well. There was once a time when evolutionists could appeal to gaps in the fossil record to explain why the species appear to arise abruptly, but no more. There
has just been too much paleontology work, such as a new international study on dinosaurs published this week, confirming exactly what the strata have been showing all along: new forms really did
arise abruptly.
The new study narrows the dating of the rise of dinosaurs in the fossil record. It confirms that many dinosaur species appeared in an “explosion” or what “we term the ‘dinosaur diversification event
(DDE)’.” It was an “explosive increase in dinosaurian abundance in terrestrial ecosystems.” As the press release explains,
First there were no dinosaur tracks, and then there were many. This marks the moment of their explosion, and the rock successions in the Dolomites are well dated. Comparison with rock successions
in Argentina and Brazil, here the first extensive skeletons of dinosaurs occur, show the explosion happened at the same time there as well.
As lead author Dr Massimo Bernardi at the University of Bristol explains, “it’s amazing how clear cut the change from ‘no dinosaurs’ to ‘all dinosaurs’ was.”
There just isn’t enough time, and it is another example of a failed prediction of the theory of evolution.
1. Steve Jones, Darwin’s Ghost, p. 206, Random House, New York, 2000.
h/t: The genius.
A Hall of Mirrors
A new paper from Andreas Wagner and co-workers argues that a key and crucial driver of evolution is changes to the interaction between transcription factor proteins and the short DNA sequences to
which they bind. In other words, evolution is driven by varying the regulation of protein expression (and a particular type of regulation—the transcription factor-DNA binding) rather than varying the
structural proteins themselves. Nowhere does the paper address or even mention the scientific problems with this speculative idea. For example, if evolution primarily proceeds by random changes to
transcription factor-DNA binding, creating all manner of biological designs and species, then from where did those transcription factors and DNA sequences come? The answer—that they evolved for some
different, independent, function; itself an evolutionary impossibility—necessitates astronomical levels of serendipity. Evolution could not have had foreknowledge. It could not have known that the
emerging transcription factors and DNA sequence would, just luckily, be only a mutation away from some new function. This serendipity problem has been escalating for years as evolutionary theory has
repeatedly failed, and evolutionists have applied ever more complex hypotheses to try to explain the empirical evidence. Evolutionists have had to impute to evolution increasingly sophisticated,
complex, higher-order, mechanisms. And with each one the theory has become ever more serendipitous. So it is not too surprising that evolutionists steer clear of the serendipity problem. Instead,
they cite previous literature as a way of legitimizing evolutionary theory. Here I will show examples of how this works in the new Wagner paper.
The paper starts right off with the bold claim that “Changes in the regulation of gene expression need not be deleterious. They can also be adaptive and drive evolutionary change.” That is quite a
statement. To support it the paper cites a classic 1975 paper by Mary-Claire King and A. C. Wilson entitled “Evolution at two levels in humans and chimpanzees.” The 1975 paper admits that the popular
idea and expectation that evolution occurs by mutations in protein-coding genes had largely failed. The problem was that, at the genetic level, the two species were too similar:
The intriguing result, documented in this article, is that all the biochemical methods agree in showing that the genetic distance between humans and the chimpanzee is probably too small to
account for their substantial organismal differences.
Their solution was to resort to a monumental shift in evolutionary theory: evolution would occur via the tweaking of gene regulation.
We suggest that evolutionary changes in anatomy and way of life are more often based on changes in the mechanisms controlling the expression of genes than on sequence changes in proteins. We
therefore propose that regulatory mutations account for the major biological differences between humans and chimpanzees.
In other words, evolution would have to occur not by changing proteins, but by changing protein regulation. What was left unsaid was that highly complex, genetic regulation mechanisms would now have
to be in place, a priori, in order for evolution to proceed.
Where did those come from?
Evolution would have to create highly complex, genetic regulation mechanisms so that evolution could occur.
Not only would this ushering in of serendipity to evolutionary theory go unnoticed, it would, incredibly, be cited thereafter as a sort of evidence, in its own right, showing that evolution occurs by
changes to protein regulation.
But of course the 1975 King-Wilson paper showed no such thing. The paper presupposed the truth of evolution, and from there reasoned that evolution must have primarily occurred via changes to protein
regulation. Not because anyone could see how that could occur, but because the old thinking—changes to proteins themselves—wasn’t working.
This was not, and is not, evidence that changes in the regulation of gene expression can be “adaptive and drive evolutionary change,” as the Wagner paper claimed.
But this is how the genre works. The evolution literature makes unfounded claims that contradict the science, and justifies those claims with references to other evolution papers which do the same
thing. It is a web of deceit.
Ultimately it all traces back to the belief that evolution is true.
The Wagner paper next cites a 2007 paper that begins its very first sentence with this unfounded claim:
It has long been understood that morphological evolution occurs through alterations of embryonic development.
I didn’t know that. And again, references are provided. This time to a Stephen Jay Gould book and a textbook, neither of which demonstrate that “morphological evolution occurs through alterations of
embryonic development.”
These sorts of high claims by evolutionists are ubiquitous in the literature, but they never turn out to be true. Citations are given, and those in turn provide yet more citations. And so on, in a
seemingly infinite hall of mirrors, where monumental assertions are casually made and immediately followed by citations that simply do the same thing.
Religion drives science, and it matters.
Pre Adaptation
In contrast [to trait loss], the gain of genetically complex traits appears harder, in that it requires the deployment of multiple gene products in a coordinated spatial and temporal manner.
Obviously, this is unlikely to happen in a single step, because it requires potentially numerous changes at multiple loci.
If you guessed this was written by an Intelligent Design advocate, such as Michael Behe describing irreducibly complex structures, you were wrong. It was evolutionist Sean Carroll and co-workers in a
2007 PNAS paper.
When a design person says it, it is heresy. When an evolutionist says it, it is the stuff of good solid scientific research.
The difference is the design person assumes a realist view (the genetically complex trait evinces design) whereas the evolutionist assumes an anti-realist view (in spite of all indications, the
genetically complex trait must have arisen by blind causes).
To support their position, evolutionists often appeal to a pre adaptation argument. This argument claims that the various sub components (gene products, etc.), needed for the genetically complex
trait, were each needed for some other function. Therefore, they evolved individually and independently, only later to serendipitously fit together perfectly and, in so doing, form a new structure
with a new function that just happened to be needed. As Richard Dawkins once put it:
The bombardier beetle’s ancestors simply pressed into different service chemicals that already happened to be lying around. That’s often how evolution works.
The problem, of course, is that this is not realistic. To think that each and every one of the seemingly unending, thousands and thousands, of genetically complex traits just happened to luckily
arise from parts that just happened to be lying around, is to make one’s theory dependent on too much serendipity.
Religion drives science, and it matters.
The Politicization of Science
Twitter CEO Jack Dorsey recently tweeted that Peter Leyden’s and Ruy Teixeira’s article, “The Great Lesson of California in America’s New Civil War,” is a “Great read.” The article both urges and
forecasts a blue-state takeover of America where our current political divide gives way to a Democrat dominion. This new “Civil War” is to begin this year and, like the last one will have an economic
cause. Unfortunately, the thinking of Leyden and Teixeira is steeped in scientific ignorance which drives their thesis.
According to Leyden and Teixeira both the last, and now upcoming, Civil Wars are about fundamentally different economic systems that cannot coexist. In the mid nineteenth century it was an agrarian
economy dependent on slaves versus a capitalist manufacturing economy dependent on free labor. Today, the conflict is between (i) the red states which are dependent on carbon-based energy systems
like coal and oil, and (ii) the blue states that are shifting to clean energy and weaning themselves off of carbon. Granting this dubious thesis, why are these two economies so irreconcilable?
Because of global warming and the terrible natural disasters it brings:
In the era of climate change, with the mounting pressure of increased natural disasters, something must give.
You read that right. Leyden’s and Teixeira’s thesis is driven by anthropogenic global warming, or AGW, which they sprinkle throughout the article. Red states are bad because they deny it, blue states
are good because they face the truth and reckon with it with progressive policies. After all, it is “the scientific consensus that climate change is happening, that human activity is the main cause,
and that it is a serious threat.”
It must be nice to go through life with such certainty. Ignorance, as they say, is bliss.
We can begin with the most obvious mistake. While it certainly revs people up to hear that global warming is “a serious threat,” we have little evidence for this. Even those “consensus” scientists
agree that we are not justified in claiming the sky is falling. And, no, in spite of what you may have heard, the recent hurricanes were probably not products of global warming.
But what about that scientific consensus that Leyden and Teixeira speak of? Doesn’t that make their case?
Unfortunately, Leyden and Teixeira are the latest example of how historians have utterly failed. In spite of their best efforts, historians, and especially historians of science, have not been able
to disabuse people of the myths of science.
In science, as in politics, majorities are majorities until they aren’t. A scientific consensus can occur both for theories that end up enshrined in museums and for theories that end up dumped in the
trash bin.
Once upon a time the scientific consensus held the Earth was the center of the universe. Only later did the scientific consensus shift to the Sun as the center of the universe.
Both were wrong.
What Mr. Nelson taught you in seventh grade history class was right after all: If you don’t understand history you will repeat its mistakes. And Leyden and Teixeira are today’s poster children of
such naiveté.
A scientific consensus for a theory means just one thing: That the majority of scientists accept the theory. Nothing more, nothing less. The problem with science, as Del Ratzsch once explained, is
that it is done by people.
What we do know about AGW is that the data have been massaged, predictions have failed, publications have been manipulated, enormous pressure to conform has been applied, and ever since Lynn White’s
1966 AAAS talk the science has been thoroughly politicized.
None of this means that AGW is false, but the theories that end up in textbooks and museums don’t usually need enormous social and career pressures for sustenance.
As it stands scientists have been walking back the hype (it’s climate change, not global warming anymore), and trying to explain the lack of a hockey stick temperature rise (the ocean is temporarily
absorbing the heat); insiders are backing out (see here and here), and new papers are showing current temperatures have not been so out of the ordinary (e.g., here).
AGW is certainly an important theory to study. And perhaps it is true. But its track record of prediction is far more important than the number of people voting for it.
The idea that AGW is the driver behind a new Civil War in America to start, err, later this year is simply absurd. I’m less concerned about Leyden’s and Teixeira’s political desires as I am about the
mythologies they are built on.
Religion drives science, and it matters.
Not Even Wrong
Astrophysicist Ethan Siegel may not have been aware of the phosphorous problem when he wrote his article on fixing the Drake Equation which appeared at Forbes last week. But he certainly should have
known about origin of life problem. His failure to account for the former is a reasonable mistake, but his failure to account for the latter is not.
The Drake Equation is simply the product of a set of factors, estimating the number of active, technologically-advanced, communicating civilizations in our galaxy—the Milky Way. Siegel brings the
Drake Equation up to date with a few modifications.
He is careful to ensure that his final result is not too large and not too small. Too large an estimate would contradict the decades-long SETI (search for extraterrestrial intelligence) project
which, err, has discovered precisely zero radio signals in the cosmos that could be interpreted as resulting from an intelligent civilization. Too small an estimate would signal an end to Siegel’s
investigations of extraterrestrial intelligence.
What is needed is a Goldilocks numbers—not too large and not too small. Siegel optimistically arrives at a respectable 10,000 worlds in the Milky Way “teeming with diverse, multicellular, highly
differentiated forms of life,” but given the length of time any such civilization is likely to exist, there is only a 10% chance of such a civilization existing co-temporally with us.
Ahh, just right. Small enough to avoid contradicting SETI, but large enough to be interesting.
But Siegel’s value of 25% for the third factor, the fraction of stars with the right conditions for habitability, seems much too high give new research indicating phosphorus is hard to come by in the
The problem, it seems, is that phosphorus (the P in the ubiquitous energy-carrying ATP molecule you learned about in high school biology class) is created only in the right kind of supernovae, and
there just isn’t enough to go around. As one of the researchers explained:
The route to carrying phosphorus into new-born planets looks rather precarious. We already think that only a few phosphorus-bearing minerals that came to the Earth—probably in meteorites—were
reactive enough to get involved in making proto-biomolecules. If phosphorus is sourced from supernovae, and then travels across space in meteoritic rocks, I'm wondering if a young planet could
find itself lacking in reactive phosphorus because of where it was born? That is, it started off near the wrong kind of supernova? In that case, life might really struggle to get started out of
phosphorus-poor chemistry, on another world otherwise similar to our own.
This could be trouble for Siegel. The problem in his goal-seeked 10% result he has committed to specific values. The wiggle room is now gone, and new findings such as the phosphorus problem will only
make things worse. Siegel’s 10% result could easily drop by 10 orders of magnitude or more on the phosphorus problem alone.
That would be devastating, but it would be nothing compared to a realistic accounting for the origin of life problem. That is Siegel’s fifth factor and he grants it a value of 1-in-10,000. That is,
for worlds in habitable zones, there is a 1/10,000 probability of life arising from non-life, at some point in the planet’s history.
That is absurd. Siegel pleads ignorance, and claims 1-in-10,000 is “as good a guess as any,” but of course it isn’t.
We can begin by dispelling the silly proclamations riddling the literature, that the origin of life problem has been essentially solved. As the National Academy of Sciences once declared:
For those who are studying the origin of life, the question is no longer whether life could have originated by chemical processes involving nonbiological components. The question instead has
become which of many pathways might have been followed to produce the first cells [1]
Fortunately the National Academy of Sciences has since recanted that non-scientific claim, and admitted there is no such solution at hand. Such scientific realism can now be found elsewhere as well.
The origin of life problem has not been solved, not even close. But that doesn’t mean we are left with no idea of how hard the problem is, and that 1-in-10,000 (i.e., 10^-4) is “as good a guess as
any,” as Siegel claims. Far from it. Even the evolution of a single protein has been repeatedly shown to be far, far less likely than 10^-4.
As for something more complicated than a single protein, one study estimated the chances of a simple replicator evolving at 1 in 10^1018. It was a very simple calculation and a very rough estimate.
But at least it is a start.
One could argue that the origin of life problem is more difficult than that, or less difficult than that. But Siegel provided no rational at all. He laughably set the bounds at 1-in-ten and
one-in-a-million, and then with zero justification arbitrarily picked 1-in-10,000.
In other words, Siegel set the lower and upper limits at 10^-1 and 10^-6, when even a single protein has been estimated at about 10^-70, and a simple replicating system at 10^-1018.
Siegel’s estimate is not realistic. With zero justification or empirical basis, Siegel set the probability of the origin of life at a number that is more than 1,000 orders or magnitude less than what
has been estimated.
Siegel’s estimate was not one thousand times too optimistic, it was one thousand orders of magnitude too optimistic. It was not too optimistic by three zeros; it was too optimistic by one thousand
zeros. Siegel is not doing science. He is goal-seeking, using whatever numbers he needs to get the right answer.
Religion drives science, and it matters.
A Pattern Problem
A few years ago Paul Nelson debated Joel Velasco on the topic of design and evolution. Nelson masterfully demonstrated design in nature. For his part Velasco also provided an excellent defense of
evolution. But the Epicurean claim that the world arose via random chance is not easy to defend, and Velasco’s task would be challenging. Consider, for example, the orphans which Nelson explained are
a good example of taxonomically-restricted designs. Such designs make no sense on evolution, and though Velasco responded with many rebuttals, none were very convincing. Since that debate the orphan
problem has become worse, as highlighted by a new study of brochosomes.
The term orphan refers to a DNA open reading frame, or ORF, without any known similar sequence in other species or lineages, and hence ORFan or “orphan.” Since orphans are unique to a particular
species or lineage, they contradict common ancestry’s much celebrated nested hierarchy model.
The Nelson-Valasco Debate
Velasco addressed the orphan problem with several arguments. First, Velasco reassured the audience that there isn’t much to be concerned with here because “Every other puzzle we’ve ever encountered
in the last 150 years has made us even more certain of a fact that we already knew, that we’re all related.”
Second, Velasco argued that the whole orphan problem is contrived, as it is nothing more than a semantic misunderstanding—a confusion of terms. These are nothing more than open reading frames without
significant similarity to any known sequence.
Third, Velasco argued that many of the orphans are so categorized merely because the search for similar sequence is done only in “very distantly related” species.
Furthermore, and fourth, Velasco argued that orphans are really nothing more than a gap in our knowledge. For the more we know about a species, the more the orphan problem goes away. And which
species do we know the most about? Ourselves of course. And we have no orphans: “Well what about humans, we know a lot about humans. How many orphan genes are in humans? What do you think? Zero.”
In fact, and fifth, Velasco argued that while new orphans are discovered with each new genome that is decoded, the trend is slowing and is suggestive that in the long run relatives for these orphans
will be found: “In fact if you trend the absolute number going up, as opposed to the percentage of orphan genes in organisms, that number is going down.”
So to summarize Velasco’s position, the orphan problem will be solved so don’t worry about, but actually orphans are not a problem at all but rather a semantic misunderstanding, but on the other hand
the orphan problem is a consequence of incomplete genomic data, but actually on the other hand the problem is a consequence of insufficient knowledge about the species, and in any case even though
the number of known orphans keeps on rising, they will eventually go away because the orphans, as a percentage of the overall genomic data (which has been exploding exponentially) is going down.
This string of evolution arguments reminds us of the classic dog-owner’s defense: He’s not my dog, he didn’t bite you, and besides you hit the dog first anyway. Not surprisingly, each of Velasco’s
arguments fails, as I explained here.
In fact, there are many orphans, and while function can be difficult to identify, it has been found for many orphans. As science writer Helen Pilcher explained:
In corals, jellyfish and polyps, orphan genes guide the development of explosive stinging cells, sophisticated structures that launch toxin-filled capsules to stun prey. In the freshwater polyp
Hydra, orphans guide the development of feeding tentacles around the organism’s mouth. And the polar cod’s orphan antifreeze gene enables it to survive life in the icy Arctic.
Up to a third of genomes have been found have been found to be unique, as this review explains:
Comparative genome analyses indicate that every taxonomic group so far studied contains 10–20% of genes that lack recognizable homologs in other species. Do such ‘orphan’ or
‘taxonomically-restricted’ genes comprise spurious, non-functional ORFs, or does their presence reflect important evolutionary processes? Recent studies in basal metazoans such as Nematostella,
Acropora and Hydra have shed light on the function of these genes, and now indicate that they are involved in important species-specific adaptive processes.
And this is yet another failed prediction of evolution, as this paper explains:
The frequency of de novo creation of proteins has been debated. Early it was assumed that de novo creation should be extremely rare and that the vast majority of all protein coding genes were
created in early history of life. However, the early genomics era lead to the insight that protein coding genes do appear to be lineage-specific. Today, with thousands of completely sequenced
genomes, this impression remains.
Why then was Velasco so confident and almost nonchalant in his argumentation? Why was he so assured that, one way or another, the orphan problem was not a problem? And why did he believe there are
zero orphans in humans, and so it merely is a matter of studying biology, and the orphans will go away?
Lander Orphan Study
It could be due to a significant 2007 study from Eric Lander’s group which rejected most of the large number (several thousands) of orphans that had been tentatively identified in the human genome.
The study confidently concluded that “the vast majority” of the orphans were “spurious”:
The analysis here addresses an important challenge in genomics— determining whether an ORF truly encodes a protein. We show that the vast majority of ORFs without cross-species counterparts
[i.e., orphans] are simply random occurrences. The exceptions appear to represent a sufficiently small fraction that the best course is would be [sic] consider such ORFs as noncoding in the
absence of direct experimental evidence.
The authors went on to propose that “it is time to undertake a thorough revision of the
human gene catalogs by applying this principle to filter the entries.”
That peer-reviewed paper, in a leading journal, was well received (e.g., Larry Moran called it an “excellent study”) and it certainly appeared to be authoritative. So it is not surprising that
Velasco would be confident about orphans. For all appearances, they really were no problem for evolution.
There was just one problem. This was all wrong.
There was no scientific evidence that those human sequences, identified as orphans, were “spurious.” The methods used in the Lander study were full of evolutionary assumptions. The results entirely
hinged on evolution. Although the paper did not explicitly state this, without the assumption of evolution no such conclusions could have been made.
This is what philosophers refer to as theory-ladenness. Although the paper authoritatively concluded the vast majority of the orphans in the human genome were spurious, this was not an empirical
observation or inference, as it might seem to some readers. Their data (and proposed revisions to human gene catalogs), methods, and conclusions were all laden, at their foundation, with the theory
of evolution.
So Velasco’s argument was circular. To defend evolution he claimed there were zero orphans in the human genome, but that “fact” was a consequence of assuming evolution is true in the first place. If
the assumption of evolution is dropped, then there is no evidence for that conclusion.
Since the Nelson-Velasco debate the orphan problem has just gotten worse. Consider, for example, brochosomes which are intricate, symmetric, secretory granules forming super-oily coatings on the
integuments of leafhoppers. Brochosomes develop in glandular segments of the leafhopper’s Malpighian tubules.
The main component of brochosomes, as shown in a recent paper, is proteins. And these constituent proteins, as well as brochosome-associated proteins, are mostly encoded by orphan genes.
As the paper explains, most of these proteins “appear to be restricted to the superfamily Membracoidea, adding to the growing list of cases where taxonomically restricted genes, also called orphans,
encode important taxon-specific traits.”
And how did all these orphan genes arise so rapidly? The paper hypothesizes that “It is possible that secreta exported from the organism may evolve especially rapidly because they are not strongly
constrained by interactions with other traits.”
That evolutionists can so easily reach for just-so stories, such as this, is yet another example of how false predictions have no consequence for evolutionary theory. Ever since Darwin evolutionists
have proclaimed how important it is that the species fall into the common descent pattern. This has especially been celebrated at the molecular level.
But of course the species fall into no such pattern, and when obvious examples present themselves, such as the brochosome proteins, evolutionists do not miss a step.
There is no empirical content to this theory. Predictions hailed as great successes and confirmations of the truth of evolution suddenly mean nothing and have no consequence when the falsification
becomes unavoidable.
Religion drives science, and it matters.
h/t: El Hombre
What Your Biology Teacher Didn’t Tell You
Jerry Coyne’s website (Why Evolution Is True) has posed study questions for learning about evolution. Evolutionists have responded in the “Comment” section with answers to some of the questions (see
here, here, and here). But when I posted a few relevant thoughts, they were quickly deleted after briefly appearing. That’s unfortunate because those facts can help readers to understand evolution.
Here is what I posted:
Well the very first question is question begging:
“Why is the concept of homology crucial for even being able to talk about organic structure?”
It isn’t. We are “able to talk about organic structure” without reference to homology. In fact, if you are interested in biology, you can do more than mere talk. Believe it or not you actually
can investigate how organic structure works, without even referencing homology. The question reveals the underlying non-scientific Epicureanism at work. This is not to say homology is not an
important concept and area of study. Of course it is. But it is absurd to claim it is required even merely to talk about organic structure. Let’s try another:
“What is Darwin’s explanation for homology?”
Darwin’s explanation for homology is that it is a consequence of common descent. He repeatedly argues that homologous structures provide good examples of non-adaptive patterns as well as
disutility, thus confirming common descent by virtue of falsifying the utilitarianism-laden doctrine of creation. See for example pp. 199-200, where Darwin concludes:
“Thus, we can hardly believe that the webbed feet of the upland goose or of the frigate-bird are of special use to these birds; we cannot believe that the same bones in the arm of the monkey, in
the fore leg of the horse, in the wing of the bat, and in the flipper of the seal, are of special use to these animals. We may safely attribute these structures to inheritance.”
Pure metaphysics, and ignoring the enormous problem that non-adaptive patterns cause for evolutionary theory. Oh my. Well, let’s try another:
“How does Darwin’s account of serial homology (the resemblance of parts within an organism, for example, the forelimbs to the hindlimbs, or of a cervical vertebra to a thoracic vertebra) depend
on the repetition of parts or segmentation?”
Hilarious. It’s a wonderful example of teleology, just-so-stories, and metaphysics, so characteristic of the genre, all wrapped up in a single passage (pp. 437-8). Darwin goes into a typical rant
of how designs and patterns (serial homologies in this case) absolutely refute utilitarianism. “How inexplicable are these facts on the ordinary view of creation!,” he begins. Pure metaphysics.
He then provides a just-so story about how “we may readily believe that the unknown progenitor of the vertebrata possessed many vertebræ,” etc., and that like any good breeder, natural selection
“should have seized on a certain number of the primordially similar elements, many times repeated, and have adapted them to the most diverse purposes.”
Seized on? Wow, that natural selection sure is good—long live Aristotelianism. Gotta love this mythology.
Action Potentials
Are there long, gradual, pathways of functional intermediate structures, separated by only one or perhaps a few mutations, leading to every single species, and every single design and structure in
all of biology? As we saw last time, this has been a fundamental claim and expectation of evolutionary theory which is at odds with the science.* If one mutation is rare, a lot of mutations are
astronomically rare. For instance, if a particular mutation has a one-in-a-hundred million (one in 10^8) chance of occurring in a new individual, then a hundred such particular mutations have a one
in 10^800 chance of occurring. It’s not going to happen. Let’s have a look at an example: nerve cells and their action potential signals.
[* Note: Some evolutionists have attempted to get around this problem with the neutral theory, but that just makes matters worse].
Nerve cells have a long tail which carries an electronic impulse. The tail can be several feet long and its signal might stimulate a muscle to action, control a gland, or report a sensation to the
Like a cable containing thousands of different telephone wires, nerve cells are often bundled together to form a nerve. Early researchers considered that perhaps the electronic impulse traveled along
the nerve cell tail like electricity in a wire. But they soon realized that the signal in nerve cells is too weak to travel very far. The nerve cell would need to boost the signal along the way for
it to travel along the tail.
After years of research it was discovered that the signal is boosted by membrane proteins. First, there is a membrane protein that simultaneously pumps two potassium ions into the cell and three
sodium ions out of the cell. This sets up a chemical gradient across the membrane. There is more potassium inside the cell than outside, and there is more sodium outside than inside. Also, there are
more negatively charged ions inside the cell so there is a voltage drop (50-100 millivolt) across the membrane.
In addition to the sodium-potassium pump, there are also sodium channels and potassium channels. These membrane proteins allow sodium and potassium, respectively, to pass through the membrane. They
are normally closed, but when the decaying electronic impulse travels along the nerve cell tail, it causes the sodium channels to quickly open. Sodium ions outside the cell then come streaming into
the cell down the electro-chemical gradient. As a result, the voltage drop is reversed and the decaying electronic impulse, which caused the sodium channels to open, is boosted as it continues on its
way along the nerve cell tail.
When the voltage goes from negative to positive inside the cell, the sodium channels slowly close and the potassium channels open. Hence the sodium channels are open only momentarily, and now with
the potassium channels open, the potassium ions concentrated inside the cell come streaming out down their electro-chemical gradient. As a result the original voltage drop is reestablished.
This process repeats itself as the electronic impulse travels along the tail of the nerve cell, until the impulse finally reaches the end of the nerve cell. Although we’ve left out many details, it
should be obvious that the process depends on the intricate workings of the three membrane proteins. The sodium-potassium pump helps set up the electro-chemical gradient, the electronic impulse is
strong enough to activate the sodium channel, and then the sodium and potassium channels open and close with precise timing.
How, for example, are the channels designed to be ion-selective? Sodium is about 40% smaller than potassium so the sodium channel can exclude potassium if it is just big enough for sodium. Random
mutations must have struck on an amino acid sequence that would fold up just right to provide the right channel size.
The potassium channel, on the other hand is large enough for both potassium, and sodium, yet it is highly efficient. It somehow excludes sodium almost perfectly (the potassium to sodium ratio is
about 10000), yet allows potassium to pass through almost as if there were nothing in the way.
Nerve cells are constantly firing off in your body. They control your eyes as you read these words, and they send back the images you see on this page to your brain. They, along with chemical
signals, control a multitude of processes in our bodies, and there is no scientific reason to think they gradually evolved, one mutation at time.
Indeed, that idea contradicts everything we know from the science. And yet this is what evolutionists believe. Let me repeat that: evolutionists believe nerve cells and their action potential designs
evolved one mutation at time. Indeed, evolutionists believe this is a proven fact, beyond all reasonable doubt.
It would be difficult to imagine a more absurd claim. So let’s have a look at the details of this line of thinking. Here is a recent paper from the Royal Society, representing the state of the art in
evolutionary thinking on this topic. The paper claims to provide a detailed explanation of how early evolution produced action potential technology.
Sounds promising, but when evolutionists speak of “details,” they have something slightly different in mind. Here are several passages from the paper which reveal that not only is there a lack of
details, but that the study is thoroughly unscientific.
We propose that the next step in the evolution of eukaryote DCS [membrane depolarization (through uncontrolled calcium influx), contraction and secretion] coupling has been the recruitment of
stretch-sensitive calcium channels, which allow controlled influx of calcium upon mechanical stress before the actual damage occurs, and thus anticipate the effects of membrane rupture.
The recruitment of calcium channels? And exactly who did the recruiting? Here the authors rely on vague terminology to paper over a host of problematic details of just how random mutations somehow
performed this recruiting.
To prevent the actual rupture, the first role of mechanosensory Ca++ channels might have been to pre-activate components of the repair pathway in stretched membranes.
“To prevent”? Let’s spell out the logic a little more clearly. The authors are hypothesizing that these calcium channels evolved the ability to pre-activate the repair pathway “to prevent” actual
rupture. By spelling out the logic a bit more clearly, we can see more easily the usual teleology at work. The evolution literature is full of teleology, and for good reason. Evolutionists are unable
to formulate and express their ideas without it. The ever-present infinitive form is the tell-tale sign. Aristotelianism is dead—long live Aristotelianism.
As another anticipatory step, actomyosin might have been pre-positioned under the plasma membrane (hence the cortical actomyosin network detected in every eukaryotic cell) and might have also
evolved direct sensitivity to stretch … Once its cortical position and mechanosensitivity were acquired, the actomyosin network could automatically fulfil an additional function: cell-shape
maintenance—as any localized cell deformation would stretch the cortical actomyosin network and trigger an immediate compensatory contraction. This property would have arisen as a side-effect (a
‘spandrel’) of the presence of cortical actomyosin for membrane repair, and quickly proved advantageous.
An “anticipatory step”? “Pre-positioning”? Actomyosin “evolved” sensitivity to stretch? The position and mechanosensitivity “were acquired”? The network could “fulfil an additional function”? Sorry,
but molecular machines (such as actomyosin) don’t “evolve” anything. There is more teleology packed into these few sentences than any medieval tract. And for good measure the authors also add the
astonishing serendipity that this additional function “would have arisen as a side-effect.” That was lucky.
Once covering the cell cortex, the actomyosin network acquired the ability to deform the cell by localized contraction.
The actomyosin network “acquired the ability” to deform the cell by localized contraction? Smart move on the part of the network. But may we ask just how did that happen?
Based on the genomic study of the protist Naegleria which has a biphasic life cycle (alternating between an amoeboid and a flagellated phase), amoeboid locomotion has been proposed to be
ancestral for eukaryotes. It might have evolved in confined interstitial environments, as it is particularly instrumental for cells which need to move through small, irregularly shaped spaces by
exploratory deformation.
Amoeboid locomotion evolved “as it is particularly instrumental.” No infinitive form but this is no less teleological. Things don’t evolve because they are “instrumental.” What the authors fail to
inform their readers of is that this would require an enormous number of random mutations.
One can hypothesize that, if stretch-sensitive calcium channels and cortical actomyosin were part of the ancestral eukaryotic molecular toolkit (as comparative genomics indicates), membrane
deformation in a confined environment would probably trigger calcium influx by opening of stretch-sensitive channels, which would in turn induce broad actomyosin contraction across the deformed
part of the cell cortex, global deformation and cell movement away from the source of pressure.
The concept of a “molecular toolkit” is standard in evolutionary thought, and another example teleological thinking.
One can thus propose that a simple ancestral form of amoeboid movement evolved as a natural consequence of the scenario outlined above for the origin of cortical actomyosin and the
calcium–contraction coupling; once established, it could have been further elaborated.
Amoeboid movement evolved “as a natural consequence,” and “once established” was “further elaborated”? This is nothing more than teleological story-telling with no supporting evidence.
It is thus tempting to speculate that, once calcium signalling had gained control over primitive forms of amoeboid movement, the same signalling system started to modify ciliary beating, possibly
for ‘switching’ between locomotor states.
Calcium signaling “gained control” and then “started to modify” ciliary beating “for ‘switching’ between locomotor states”? The “for switching” is yet another infinitive form, and “gained control” is
an active move by the calcium signaling system. Pure, unadulterated, teleology.
Possibly, in ancestral eukaryotes calcium induced a relatively simple switch (such as ciliary arrest, as still seen in many animal cells and in Chlamydomonas in response to high Ca++
concentrations), which was then gradually modified into more subtle modulations of beating mode with a fast turnover of molecular actors mediated by differential addition, complementation and
“Calcium induced a relatively simple switch”? Sorry, ions don’t induce switches, simple or otherwise. And the switch “was then gradually modified into more subtle modulations”? Note how the passive
voice obviates those thorny details. The switch “was modified” conveniently omits the fact that such modification would have to occur via random mutation, one mutation at a time.
Alternatively, control of cilia by calcium could have evolved convergently—but such convergence would then have been remarkably ubiquitous, as there seems to be no eukaryotic flagellum that is
not controlled by calcium in one way or another.
“Could have evolved convergently”? And exactly how would that happen? At least the authors then admit to the absurdity of that alternative.
Unfortunately, they lack such sensibility for the remainder of the paper. As we saw above, the paper is based on a sequence of teleological thinking. It falls into the evolutionary genre where
evolution is taken, a priori, as a given. This going in assumption underwrites vast stretches of teleological thought, and cartoon-level story telling. Not only is there a lack of empirical support,
but the genre is utterly unscientific, as revealed by even a mildly critical reading.
And needless to say, the paper does absolutely nothing to alleviate the problem we began with. The many leaps of logic and reasoning in the paper reveal all manner of monumental changes evolution
requires to construct nerve cells and the action potential technology. We are not looking at a narrative of minute, gradual changes, each contributing to the overall fitness. Many, many simultaneous
mutations are going to be needed. Even a conservative minimum number of 100 simultaneous mutations leads to the untenable result of a one in 10^800 chance of occurring.
It’s not going to happen. Religion drives science, and it matters.
Mutations are rare and good ones are even more rare. One reason mutations are rare is because there are sophisticated error correction mechanisms in our cells. So according to evolution random
mutations created correction mechanisms to suppress random mutations. And that paradox is only the beginning. Because error correction mechanisms, as with pretty much everything else in biology,
require many, many mutations to be created. If one mutation is rare, a lot of mutations are astronomically rare. For instance, if a particular mutation has a one-in-a-million (one in 10^6) chance of
occurring in a new individual, then a hundred such particular mutations have a one in 10^600 chance of occurring. It’s not going to happen.
How do evolutionists reckon with this scientific problem?
First, one common answer is to dismiss the question altogether. Evolution is a fact, don’t worry about the details. Obviously this is not very compelling.
Second, another common answer is to cast the problem as a strawman argument against evolution, and appeal to gradualism. Evolutionists going back to Darwin have never described the process as “poof.”
They do not, and never have, understood the process as the simultaneous origin of tens or hundreds, or more mutations. Instead, it is a long, slow, gradual process, as Darwin explained:
If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down. But I
can find out no such case […] Although the belief that an organ so perfect as the eye could have been formed by natural selection, is enough to stagger any one; yet in the case of any organ, if
we know of a long series of gradations in complexity, each good for its possessor, then, under changing conditions of life, there is no logical impossibility in the acquirement of any conceivable
degree of perfection through natural selection
The Sage of Kent could find “no such case”? That’s strange, because they are ubiquitous. And with the inexorable march of science, it is just getting worse. Error correcting mechanisms are just one
example of many. Gradualism is not indicated.
What if computer manufacturers were required to have a useful, functional electronic device at each step in the manufacturing process? With each new wire or solder, what must emerge is a “long series
of gradations in complexity, each good for its possessor.”
That, of course, is absurd (as Darwin freely confessed). From clothing to jet aircraft, the manufacturing process is one of parts, tools, and raw materials strewn about in a useless array, until
everything comes together at the end.
The idea that every single biological structure and design can be constructed by one or two mutations at a time, not only has not been demonstrated, it has no correspondence to the real world. It is
just silly.
What evolution requires is that biology is different, but there is no reason to believe such a heroic claim. The response that multiple mutations is a “strawman” argument does not reckon with the
reality of the science.
Third, some evolutionists recognize this undeniable evidence and how impossible evolution is. Their solution is to call upon a multiverse to overcome the evidence. If an event is so unlikely it would
never occur in our universe, just create a multitude of universes. And how many universes are there? The answer is, as many as are needed. In other words, when confronted with an impossibility,
evolutionist simply contrive a mythical solution.
Forth, another common response that evolutionists make is to appeal to the fitness of the structure in question. Biological designs, after all, generally work pretty well, and therefore have high
fitness. Is this not enough to prove that it evolved? For evolutionists, if something helps, then it evolves. Presto.
To summarize, evolutionists have four different types of responses to the evidence, and none of the responses do the job.
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..:: CoDe Maverick ::..
Windows management instrumentation (WMI) can be used to access system management data across remote machines. You can use this to get status and configuration information on windows machines
listening on the network. The classes found in the System.Management namespace helps developers to write code to access these information quickly.
The following example shows how to access LogicalMemoryConfiguration data on a remote machine.
using System;
using System.Net;
using System.Management;
namespace WMIonRemoteMachine
class Program
static void Main(string[] args)
//Specify the Adminstrator's Username and Password
ConnectionOptions co = new ConnectionOptions();
co.Username = "Administrator";
co.Password = "password#xyz";
//Connect to the default namespace on Remote Machine
ManagementScope scope = new ManagementScope(@"\\[REMOTE MACHINE]\root\cimv2", co);
SelectQuery query = new SelectQuery("SELECT * FROM Win32_LogicalMemoryConfiguration");
ManagementObjectSearcher searcher = new ManagementObjectSearcher(scope, query);
foreach (ManagementObject mObj in searcher.Get())
foreach (System.Management.PropertyData property in mObj.Properties)
System.Console.WriteLine(property.Name.PadLeft(25,' ') + ": " + property.Value);
AvailableVirtualMemory: 1405028
Caption: Logical Memory Configuration
Description: Logical Memory Configuration
Name: LogicalMemoryConfiguration
SettingID: LogicalMemoryConfiguration
TotalPageFileSpace: 2523064
TotalPhysicalMemory: 1046512
TotalVirtualMemory: 3569576
The username and password supplied in the above code should belong to an account that is a member of Administrator Group on the remote machine. If the ConnectionOptions is not set then the namespace
residing on the Local System is accessed.
The default WMI namespace or schema "\root\cimv2" is queried to retrieve common system management information. WMI implements the Common Information Model (CIV) schema proposed by Distributed
Management Task Force (DMTF).
Remote connections in WMI are affected by Firewall. It blocks all data requests from remote machines. If a connection fails, an exception of type System.Runtime.InteropServices.COMException is thrown
in System.Management with an error message "The RPC server is unavailable. (Exception from HRESULT: 0x800706BA)". So make sure that the firewall settings are configured to allow these connections.
You have a Grid bound to a datasource. You do all the right stuff by putting the correct formatting expressions in place for the BoundColumn. But when the page is rendered the columns are not
formatted. I'm sure everybody would have encountered this at some point or the other while rendering data in ASP.NET 2.0 GridView.
The fix is straight, set the HtmlEncode property of the Bound Columns to "false".
<asp:BoundField DataField="Number" DataFormatString="{0:c}" HtmlEncode="false" HeaderText="Number" />
<asp:BoundField DataField="Date" DataFormatString="{0:MM/dd/yyyy}" HtmlEncode="false" HeaderText="Date" />
By default this property is set to True for security reasons. When the page is rendered, the output HTML is encoded to prevent cross site scripting (XSS) attacks. So make sure to turn off
HtmlEncoding on those columns that you want to display formatted data.
Sometimes we want to view the HTML rendered by the web server for debugging purposes. We right-click on the browser (IE), select "View Source" and a neat little notepad opens up with the HTML. This
is valid as long as the pages are rendered in a conventional way where the entire page is posted back for each subsequent requests.
In AJAX, where the pages are rendered asynchronously, "View Source" would only show you the HTML for the page that was originally rendered but does not show you any updates to that page modified
through AJAX callbacks.
Enter the following javascript in the address bar to view the outer HTML actually generated through AJAX Callbacks.
Here is a list of articles that I've published elsewhere. This post will be updated frequently...
If you're a .NET lover and moreover worked with GDI+ then you would definitely want to mess around with images. I've written a short article, Generating ASCII Art from an Image using C#, and
published it on c-sharpcorner with complete source code. This is a quick and simple Image-To-ASCII generator with zoom-in/zoom-out feature that converts JPEGs and Bitmaps of any resolution to cool
and fascinating ASCII arts. Have fun playing with the code!
Windows Management Instrumentation (WMI) is the base for management data and operations on Windows Operating system. And System.Management namespace is the WMI namespace in .NET framework. The
ManagementObjectSearcher class is one of the first level class objects contained within this namespace. This class can be used to retrieve a collection of ManagementObject based on a specified Win32
query. For example, it can be used to enumerate all Disk Drives, Disk Partitions, Network Adapters, Network Connections, Processes etc.
To enumerate all the disk drives and their associated properties we first instantiate a new ManagementObjectSearcher object which takes as input a WMI query. The Get() method on this object returns a
collection of management objects that match this query.
string mosQuery = "SELECT * FROM Win32_DiskDrive";
System.Management.ManagementObjectSearcher query = new System.Management.ManagementObjectSearcher(mosQuery);
foreach (System.Management.ManagementObject mObj in query.Get())
foreach (System.Management.PropertyData property in mObj.Properties)
System.Console.WriteLine(property.Name + "__::" + property.Value);
Sample Output (Just one object in the Collection)
Caption__::FUJITSU MHV2060BH
Description__::Disk drive
Manufacturer__::(Standard disk drives)
MediaType__::Fixed hard disk media
Model__::FUJITSU MHV2060BH
Or, you can also use this query,"SELECT * FROM Win32_Service WHERE Started = False", to enumerate a list of services which are not started.
The Win32_DiskDrive/Win32_Service is a WMI object that is being queried here. You can replace the above query with one of those that are listed below. I've compiled a list of management queries by
inspecting the objects using WMI object browser. You can play with them or implement the same in your applications as per your requirements...
//mosQuery = "SELECT * FROM Win32_Account";
//mosQuery = "SELECT * FROM Win32_BIOS";
//mosQuery = "SELECT * FROM Win32_BootConfiguration";
//mosQuery = "SELECT * FROM Win32_Bus";
//mosQuery = "SELECT * FROM Win32_CacheMemory";
//mosQuery = "SELECT * FROM Win32_CDROMDrive";
//mosQuery = "SELECT * FROM Win32_ComputerSystem";
//mosQuery = "SELECT * FROM Win32_DesktopMonitor";
//mosQuery = "SELECT * FROM Win32_DeviceMemoryAddress";
//mosQuery = "SELECT * FROM Win32_DiskDrive";
//mosQuery = "SELECT * FROM Win32_DiskPartition";
//mosQuery = "SELECT * FROM Win32_DMAChannel";
//mosQuery = "SELECT * FROM Win32_Environment";
//mosQuery = "SELECT * FROM Win32_Fan";
//mosQuery = "SELECT * FROM Win32_IDEController";
//mosQuery = "SELECT * FROM Win32_IRQResource";
//mosQuery = "SELECT * FROM Win32_Keyboard";
//mosQuery = "SELECT * FROM Win32_LoadOrderGroup";
//mosQuery = "SELECT * FROM Win32_LogicalDisk";
//mosQuery = "SELECT * FROM Win32_LogicalMemoryConfiguration";
//mosQuery = "SELECT * FROM Win32_LogicalProgramGroup";
//mosQuery = "SELECT * FROM Win32_MemoryArray";
//mosQuery = "SELECT * FROM Win32_MemoryDevice";
//mosQuery = "SELECT * FROM Win32_MotherBoardDevice";
//mosQuery = "SELECT * FROM Win32_NetworkAdapter";
//mosQuery = "SELECT * FROM Win32_NetworkConnections";
//mosQuery = "SELECT * FROM Win32_NTEventLogFile";
//mosQuery = "SELECT * FROM Win32_NTLogEvent";
//mosQuery = "SELECT * FROM Win32_OperatingSystem";
//mosQuery = "SELECT * FROM Win32_PCMCIAController";
//mosQuery = "SELECT * FROM Win32_PnPEntity";
//mosQuery = "SELECT * FROM Win32_PointingDevice";
//mosQuery = "SELECT * FROM Win32_PortableBattery";
//mosQuery = "SELECT * FROM Win32_PortResource";
//mosQuery = "SELECT * FROM Win32_POTSModem";
//mosQuery = "SELECT * FROM Win32_Printer";
//mosQuery = "SELECT * FROM Win32_Process";
//mosQuery = "SELECT * FROM Win32_Processor";
//mosQuery = "SELECT * FROM Win32_SCSIController";
//mosQuery = "SELECT * FROM Win32_SerialPort";
//mosQuery = "SELECT * FROM Win32_Service";
//mosQuery = "SELECT * FROM Win32_share";
//mosQuery = "SELECT * FROM Win32_SoundDevice";
//mosQuery = "SELECT * FROM Win32_SystemDriver";
//mosQuery = "SELECT * FROM Win32_SystemUsers";
//mosQuery = "SELECT * FROM Win32_TemperatureProbe";
//mosQuery = "SELECT * FROM Win32_TimeZone";
//mosQuery = "SELECT * FROM Win32_USBController";
//mosQuery = "SELECT * FROM Win32_USBHub";
//mosQuery = "SELECT * FROM Win32_UserAccount";
//mosQuery = "SELECT * FROM Win32_VideoController";
Happy Coding !!!
All those cool splash screens that you see during application-startup can be created using Windows Forms in .NET 2.0/1.1 easily. Read my previous post on creating visually attractive custom shaped
forms in .NET. You can enhance these forms with fading effects that could be "cool" and appealing.
The trick is with the Form's Opacity property which governs its overall transparency levels. A double value of 1.00, which is its default, sets the Form to be Opaque. Whereas, any value less than
that would make it transparent. In mathematical terms, a form is 60% opaque when its opacity value is 0.60. And it is completely invisible when the value is 0.00.
Fading Effect
The code below does the trick of fading-out. Insert this code in the InitializeComponent method or in a button-click event wherever you would want a "fading-effect" to be fired.
for (int i = 100; i >= 0; i -= 10)
this.Opacity = (double)i / 100;
As you can see, between each levels of transparency, I'm specifying a delay of 100 milliseconds. This would give the real fading effect to a form. The Application.DoEvents() method forces a repaint
without blocking.
Have you ever wanted to create an application with a GUI that looked just like one of those skins in Windows Media Player 10 ? In .NET you don't have to write a single line of code to generate one
such form. Earlier, creating a non-rectangular form was the toughest part which involved complex low-level coding by trapping various Form-handles and calling so many system API's.
I'll show you how to create a Form just like the one as shown in the figure.
Start by creating a transparent Bitmap image using MS Paint or other Image editors.
Add this image to the Form by setting its BackgroundImage property.
Set the BackgroundImageLayout to None.
Set the FormBorderStyle to None.
Set the TransparencyKey to White. This color will appear transparent on the Form.
Thats it, run your application to find a more attractive window. But hold on, its not over yet. This form has one serious limitation. You can't drag and move this form around because you've alread
set the FormBorderStyle property to None and the borders & title bar are missing. The title bar functionalities need to be explicitly added to the code to handle all the basic functions like Close,
Minimize, Maximize and Move.
Adding a Close Handler
Drag and drop a Button control from the ToolBox over the form. Resize and rename this as btnClose with its Text property 'X'. Set the control's BackColor property to the one that blends with your
Form's color. Double click on it and add the following code to the Form's Close EventHandler.
private void btnClose_Click(object sender, EventArgs e)
Adding Drag/Move functionality
When the left-button of a mouse is clicked on the form, we first capture the thickness, in pixels, of the border for the window. We then create a point that is relative to the Form's Border and the
mouse's current position. This Point becomes the new reference that is available globally.
private Point _OffsetPoint;
private void CustomForm_MouseDown(object sender, System.Windows.Forms.MouseEventArgs e)
if (e.Button == MouseButtons.Left)
int formBorderWidth = SystemInformation.FrameBorderSize.Width;
int formBorderHeight = SystemInformation.FrameBorderSize.Height;
_OffsetPoint = new Point(-(e.X + formBorderWidth), -(e.Y + formBorderHeight));
The SystemInformation Class can be used to get information about the current system environment like Windows display element sizes, OS settings and other Hardware installed on the machine. The
FrameBorderSize property gives the thickness of the border for the window.
When a window is dragged, the new position of the mouse determines the Form's new location. This is achieved by the Point Class' Offset Method. Setting the form's Location property with this new
point will translate the form to this point.
private void CustomForm_MouseMove(object sender, System.Windows.Forms.MouseEventArgs e)
if (e.Button == MouseButtons.Left)
Point mousePos = Control.MousePosition;
mousePos.Offset(_OffsetPoint.X, _OffsetPoint.Y);
Location = mousePos;
This is just slice of what you can do with Visual Studio.NET and the .NET framework. Have fun with it !!
Recently I needed to replace a string in whole bunch of HTML template files that I had been working on. I found a couple of text editors which attempted to do this only after loading the entire lot
of files in the memory. When the number of files were in hundreds they failed miserably. I also tried to load the files into a project in Visual Studio 2005 IDE and use its Find & Replace in Current
Project feature. This failed to find texts with line breaks eventually forcing me to drop this idea. After spending an hour of intense googling to find the right tool, I decided to make my own Search
& Replace application in C# and I did make it in 15mins. This application can search and replace texts in all files and subfolders filtered by their extensions.
Instead of using the string.Replace() method provided by the string object, I wrote a custom method which finds and replace strings in a loop. This gives me more flexibility to keep track of the
number of replacements as well as the files that were actually affected. The Directory.GetFiles() method returns the complete paths of all files (after applying the filter) in the specified folder
and all its subfolders.
Here is The code....
private void btnReplace_Click(object sender, EventArgs e)
//Enter some text that you want to search and replace
string find = txtFind.Text;
int replaced = 0;
//Get all the files from the root directory filtered by a filter text.
string[] fileList = Directory.GetFiles(@"C:\Documents and Settings\tganesan\Desktop\FindnReplace", "*.txt", SearchOption.AllDirectories);
//Loop through each file, call the ReplaceText() method
//and replace the file if something was replaced.
foreach (string file in fileList)
StreamReader sr = new StreamReader(file);
string content = sr.ReadToEnd();
if(ReplaceText(ref content, txtFind.Text, txtReplace.Text, ref replaced))
StreamWriter sw = new StreamWriter(file);
//TODO: Add the files to a collection that were affected.
MessageBox.Show("Total replacements = " + replaced);
/// <summary>
/// This method loops through the File content
/// and replaces the text if found.
/// </summary>
/// <param name="content"></param>
/// <param name="oldValue"></param>
/// <param name="newValue"></param>
/// <param name="replaced"></param>
/// <returns></returns>
private bool ReplaceText(ref string content,string oldValue, string newValue, ref int replaced)
Boolean isReplaced = false;
int startIndex = 0;
while (startIndex != -1)
startIndex = content.IndexOf(oldValue,startIndex);
if (startIndex != -1)
content = content.Remove(startIndex, oldValue.Length);
content = content.Insert(startIndex,newValue);
replaced += 1;
isReplaced = true;
return isReplaced;
Don't forget to take a backup of files before running this code.
This exception was thrown while filtering out rows from a DataTable with an incorrect filter expression. See the sample code shown below...
DataTable dt = GetDataTable();
DataRow[] drs = dt.Select("ID='" +param + "'");
private DataTable GetDataTable()
DataTable dt = new DataTable();
dt.Columns.Add(new DataColumn("ID", typeof(Int16)));
DataRow dr;
for (int i = 0; i <= 100; i++)
dr = dt.NewRow();
dr["ID"] = i;
return dt;
The GetDataTable() method just returns a sample DataTable with one column of Type Int16. The param in the filter expression is of string type. The expression works fine as long as the param string is
a number, like 0, 1, 2... since ID is an Int16 column. The expression can be framed either as "ID = '2'" or "ID = 2". When param is an empty string or null, the above exception is thrown as these
types of strings cannot be converted into a type that is equivalent to the comparing column (Int16).
So the next time when you use a filter expression to filter out rows from a DataTable ensure that you use the right Data Types.
In this feed I'll show you how to Split a file into user-specified chunks and eventually merge them all together. You will find this very helpful if you have very large text files, greater than a GB,
that cannot be viewed in your "lousy Notepad". These large text files could be one of the crucial log files from your enterprise applications that may accrue data, if left un-attended, over time.
The code example I have shown below is generalized to split any file irrespective of their format.
private void btnSplit_Click(object sender, EventArgs e)
string inputFile = txtInputFile.Text; // Substitute this with your Input File
FileStream fs = new FileStream(inputFile, FileMode.Open, FileAccess.Read);
int numberOfFiles = Convert.ToInt32(txtChunks.Text);
int sizeOfEachFile = (int)Math.Ceiling((double)fs.Length / numberOfFiles);
for (int i = 1; i <= numberOfFiles; i++)
string baseFileName = Path.GetFileNameWithoutExtension(inputFile);
string extension = Path.GetExtension(inputFile);
FileStream outputFile = new FileStream(Path.GetDirectoryName(inputFile) + "\\" + baseFileName + "." + i.ToString().PadLeft(5, Convert.ToChar("0")) + extension + ".tmp", FileMode.Create, FileAccess.Write);
int bytesRead = 0;
byte[] buffer = new byte[sizeOfEachFile];
if ((bytesRead = fs.Read(buffer, 0, sizeOfEachFile)) > 0)
outputFile.Write(buffer, 0, bytesRead);
private void btnMerge_Click(object sender, EventArgs e)
string outPath = txtInputFolder.Text; // Substitute this with your Input Folder
string[] tmpFiles = Directory.GetFiles(outPath, "*.tmp");
FileStream outputFile = null;
string prevFileName = "";
foreach (string tempFile in tmpFiles)
string fileName = Path.GetFileNameWithoutExtension(tempFile);
string baseFileName = fileName.Substring(0, fileName.IndexOf(Convert.ToChar(".")));
string extension = Path.GetExtension(fileName);
if (!prevFileName.Equals(baseFileName))
if (outputFile != null)
outputFile = new FileStream(outPath + baseFileName + extension, FileMode.OpenOrCreate, FileAccess.Write);
int bytesRead = 0;
byte[] buffer = new byte[1024];
FileStream inputTempFile = new FileStream(tempFile, FileMode.OpenOrCreate, FileAccess.Read);
while ((bytesRead = inputTempFile.Read(buffer, 0, 1024)) > 0)
outputFile.Write(buffer, 0, bytesRead);
prevFileName = baseFileName;
The split method is straightforward, you set the count of number of files to be split, and the size of each file is allocated equally. Each file is named after its parent, numbered and tailed with an extension of ".tmp". If you're splitting a Text file with no intention of merging them at a later time, you can replace the ".tmp" extension with ".txt".
The Merge method above, is in fact a "Merge All" method. It merges all the files with extensions ".tmp" in the specified directory and re-creates the parent file back. That's the reason why I'm
retaining the original fileName and their extensions.
The Directory.GetFiles() method returns an array of all the file paths from the directory in ascending order. If you have fileNames like Testfile1.txt, Testfile2.txt, .... testfile100.txt then their
order in the string array would be Testfile1.txt, Testfile10.txt, Testfile100.txt, Testfile2.txt, Testfile20.txt, Testfile3.txt,.... as the numbers are just strings. The Merge would fail eventually
because of this merge-order. This can be addressed if you LeftPad the number with 0's, preferably a padding of 5 characters, while splitting. The fileNames now would be something like these,
Testfile00001.txt, Testfile00002.txt...Testfile00100.txt.
All the "tmp" files are deleted from the folder after they are merged. | {"url":"https://codemaverick.blogspot.com/2007/","timestamp":"2024-11-02T12:30:58Z","content_type":"application/xhtml+xml","content_length":"91039","record_id":"<urn:uuid:50325edd-9ddb-4d34-9556-a83591633475>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00369.warc.gz"} |
Binary Search and Linked List for Table ADT - Tlue AftabBinary Search and Linked List for Table ADT
In this article, we’ll learn about binary search on the array. And also get the idea about the linked list. We’ve already discussed the effect of sorted and unsorted arrays on the operations of Table
ADT (abstract data type). Here, we’re concerned about more implementations of Table ADT i.e. through binary search and linked list. The main reason of studying these things is to gain maximum
efficiency in terms of speed and space.
Binary Search
• Binary search is like looking up a phone number or a word in the dictionary.
• Start in middle of book.
• If the name you’re looking for, comes before names on the page, search in the first half.
• Otherwise, look into the second half.
Figure-1: Example of Binary Search
Algorithm for Binary Search
if(value == middle_element)
value is found
else if(value < middle_element)
search left half of list with the same method
search right half of list with the same method
Let’s look at the examples of binary search. Consider “a” is an sorted array in ascending order. We’ll discuss three cases in these examples:
• Case 1: value == a[mid]
• Case 2: value > a[mid]
• Case 3: value < a[mid]
Examples of Binary Search
Example No. 1 (Case 1)
Case 1: val == a[mid]
val = 10
low = 0, high = 8
mid = (0 + 8) / 2 = 4
Figure-2: Example No. 1 (Case 1)
In Figure-2, we’re searching for value (val) “10”. Through median formula [(low+high)/2], we jump to the middle position of the array. Fortunately, we found, what we were looking for, there. The
program will stop here and return the value that we found at a specific position.
Example No. 2 (Case 2)
Case 2: val > a[mid]
val = 19
low = 0, high = 8
mid = (0 + 8) / 2 = 4
new low = mid + 1 = 5
Figure-3: Example No. 2 (Case 2)
If the value is greater than the value at the mid. Then, we’ll look into the upper half of the array. This makes searching very fast, as we don’t have to look whole array rather a small section of
the array. We do this by bringing “low” next to the mid. We continue doing this, until “val” is greater than the element in the middle of array.
Example No. 3 (Case 3)
Case 3: val < a[mid]
val = 7
low = 0, high = 8
mid = (0 + 8) / 2 = 4
new high = mid – 1 =5
Figure-4: Example No. 3 (Case 3)
In this, we’ve to look the lower half of the array, If val < a[mid]. We change the position of “high” and bring it to the one step back from the mid. It can be well understood through figures.
Let, we’re looking for val=7, which is less than mid(10). We’ll continue parsing the array until we don’t reach “7”.
Figure-5: Case 3
Figure-6: Case 3
Figure-7: Case 3
In Figure-7, we finally found 7 at the position 2.
Binary Search _C++ Code
Now, you can understand the binary search code easily.
int isPresent(int *arr, int val, int N){
int low = 0;
int high = N - 1;
int mid;
while ( low <= high ){
mid = ( low + high )/2;
if (arr[mid] == val)
return 1; // found!
else if (arr[mid] < val)
low = mid + 1;
high = mid - 1;
return 0; // not found
Efficiency of Binary Search
See Figure-8, the search divides a list into two small sub-lists till a sub-list is no more divisible.
Figure-8: Bisections of the array
When we divide the array of N items into two halves continuously, then:
After 1 bisection, No. of items:
After 2 bisections:
For i bisections, we are left with following no. of items:
Which is at one point of time is only one element of the array.
Computing the value of i from this, gives us:
Linked List
We study the next implementation when we find the problem in previous one. 😊
Binary search made the find operation very fast in array. But in case of insertion or removal, we need to perform shifting of elements. If it’s in middle, then all the elements next to it will be
shifted. Worse is at the beginning where shifting will be performed on all the elements of array. So, it’s not good in case of speed for insertion and removal.
The data structure which takes contiguous space in the memory, can cope the above problem. This data structure is the linked list.
Binary search only works for the array.
Nodes of the linked list are scattered over the memory. So, binary search won’t work with them.
TableNodes are stored consecutively (unsorted or sorted).
• Insert: In case of unsorted linked list, adding the node to front will take constant time. But in case of sorted, you’ve to traverse the linked list for right position of new node.
• Find: All the keys are scanned through whether they are sorted and unsorted. So, time of find is proportional to N.
• Remove: We’ve to perform find first. Later, remove and the links are readjusted accordingly.
Figure-9: Linked List
Fixed size of the array becomes a constraint and it can’t let more elements in it. While linked list has no such constraint. But the find operation in linked list becomes very slow.
We can optimize the linked list to the skip list to get rid of the problem of slow searching.
We can operate table ADT through binary search for fast find operation. And linked list to avoid the constraint of limited elements. Binary search gets the advantage of fast searching over linked
list. But fails in terms of insertion and removal. While linked list take over binary search in terms of insertion and removal. Moreover, linked list can be optimized for fast find operation through
skip list.
REFERENCE: CS301 Handouts (page 430 to 438) | {"url":"https://tlueaftab.com/binary-search-and-linked-list-for-table-adt/","timestamp":"2024-11-06T10:28:18Z","content_type":"text/html","content_length":"299676","record_id":"<urn:uuid:1faad37a-c834-473e-8a9d-87f5eeb63c13>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00871.warc.gz"} |
How do you write a rule for the nth term of the arithmetic sequence a_7=21, a_13=42? | HIX Tutor
How do you write a rule for the nth term of the arithmetic sequence #a_7=21, a_13=42#?
Answer 1
${a}_{n} = \frac{7}{2} \left(n - 1\right)$
Arithmetic sequence, #a_n = a_1 + (n-1)d#, d=common difference.
given that, #a_7 = 21# # a_1 + (7-1)d = 21# # a_1 + 6d = 21 ->#A
#a_13 = 42# #a_1 + 12d = 42->#B
B-A, #6d =21# #->d =21/6 = 7/2#
Plug in #d = 7/2# in A or B. # a_1 + 6(7/2) = 21 # # a_1 + 21 = 21 # #->a_1 = 0#
therefore, #a_n = 0 + (n-1)7/2# #a_n = 7/2(n-1)#
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Answer 2
To write a rule for the nth term of an arithmetic sequence, you need to first find the common difference ((d)) between consecutive terms. Once you have the common difference, you can use the formula
for the nth term of an arithmetic sequence:
[a_n = a_1 + (n - 1) \times d]
Given that (a_7 = 21) and (a_{13} = 42), you can find the common difference by subtracting the two terms and dividing by the difference in their indices:
[d = \frac{a_{13} - a_7}{13 - 7}]
Once you have the common difference, you can use it to find the first term ((a_1)) by substituting any known term into the formula. Then, you can write the rule for the nth term using the formula
above, where (a_n) represents the nth term, (a_1) is the first term, and (d) is the common difference.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Goldmund AG has an unlevered cost of capital is 12% and expe
Goldmund AG has an unlevered cost of capital is 12% and expects unlevered free cash flow of $7 million each year
Subject:FinancePrice: Bought3
Goldmund AG has an unlevered cost of capital is 12% and expects unlevered
free cash flow of $7 million each year. The firm also has outstanding debt of $17.5 million and expects to maintain this level of debt permanently. Goldmund's corporate tax rate is 30%.
a) What is the value of Goldmund without leverage?
b) What is the present value of Goldmund's tax shield?
c) What is the value of Goldmund with leverage?
it is ok to write down on paper and send me the picture
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Student S Paper Modified. Used Only for Educational Purposes
Student’s Paper Modified. Used only for educational purposes.
Partition Reports
Partition sections are major parts of a Classification Report. Each classification’s description is a partition of the classification.
See the following Classification Report of Public High Schools in XXXX, XXXX (in the left column of the table) and the Partition Report of each High School: PS #1, PS #3, and PS #3 (in the right
column of the table). Each partition could stand alone as a full Partition Report.
Scroll Down.
Classification of Public High Schools in XXXX, XXXX
Classify of 3 Public Schools, According to Student Teacher Ratio and Grades
Public School One (PS #1)Teacher-Student Ratio
1. Seventy-five (75%) percent of the classes has one teacher to every twenty students.
Classes have accelerated courses.
Classes employ a seminar style of teaching.
All students enrolled in classes are students, with a 4.0 GPA.
1. Twenty-percent (25%) of the classes has one teacher to every thirty students.
Classes have grade appropriate courses.
Classes employ a traditional of teaching.
Students enrolled have between a 4.0 GPA and a 2.0 GPA.
• Five (5%) of the students enrolled have a 4.0 GPA;
• Sixty percent (60%) of the students enrolled have a 3.0-4.9 GPA;
• Thirty-five (35 %) of the students enrolled have a 2.0-2.9 GPA.
1. Five percent (5%) of the classes has one teacher and one teacher’s aide to every ten students.
Classes have developmental courses.
Classes employ a one-to-one style of teaching.
Students enrolled have between a 3.0 GPA and a GPA below 2.0.
• Ten percent (10%) of the students enrolled have a 3.0 GPA;
• Twenty (20%) of the students enrolled have a 2.0-2.9 GPA,
• Ninety (90%) of the students enrolled have below a 2.0 GPA.
1. PS #1 ranked in the top 10% of the city’s academic standards for the academic year 2011-2012.
90% of the 10% showed content competence
85% of the 10% showed performance competence.
70% of the 10% showed proficiency competence.
1. PS #1 is SOL fully accredited.
Scores were as follows:
• Language Arts = 85%
• Mathematics = 80%
• Science = 70%
• History = 75%
Public School Two (PS #2)Teacher-Student Ratio
1. Twenty percent (20%) percent of the classes has one teacher to every twenty students.
Classes have accelerated courses.
Classes employ a seminar style of teaching.
All students enrolled in classes are students, with a 4.0 GPA.
2, Seventy percent (70%) of the classes has one teacher to every thirty
Classes have grade appropriate courses.
Classes employ a traditional of teaching.
Students enrolled have between a 4.0 GPA and a 2.0 GPA.
• Seven (7%) of the students enrolled have a 4.0 GPA;
• Seventy percent (70%) of the students enrolled have a 3.0-4.9 GPA;
• Twenty-three percent (23%) of the students enrolled have a 2.0-2.9 GPA.
1. Ten percent (10%) of the classes has one teacher and one teacher’s aide to every ten students.
Classes have developmental courses.
Classes employ a one-to-one style of teaching.
Students enrolled have between a 3.0 GPA and a GPA below 2.0
• Ten percent (10%) of the students enrolled have a 3.0 GPA;
• Ten (10%) of the students enrolled have a 2.0-2.9 GPA,
• Eighty (80%) of the students enrolled have below a 2.0 GPA.
1. PS #2 ranked in the top 20% of the city’s academic standards for the academic year 2011-2012.
85% of the 20% showed content competence
75% of the 20% showed performance competence.
70% of the 20% showed proficiency competence.
1. PS #2 is SOL fully accredited.
Scores were as follows:
• Language Arts = 8%
• Mathematics = 86%
• Science = 73%
• History = 78%
Public School Three (PS #3)Teacher-Student Ratio
1. Ten percent (10%) of the classes has one teacher to every twenty students.
Classes have accelerated courses.
Classes employ a seminar style of teaching.
All students enrolled in classes are students, with a 4.0 GPA.
1. Seventy-five percent (75%) of the classes has one teacher to every thirty students.
Classes have grade appropriate courses.
Classes employ a traditional of teaching.
Students enrolled have between a 4.0 GPA and a 2.0 GPA.
• Five (5%) of the students enrolled have a 4.0 GPA;
• Thirty-five (25%) of the students enrolled have a 3.0-4.9 GPA;
• Seventy percent (70%) of the students enrolled have a 2.0-2.9 GPA.
1. Fifteen percent (15%) of the classes has one teacher and one teacher’s aide to every ten students.
Classes have developmental courses.
Classes employ a one-to-one style of teaching.
Students enrolled have between a 3.0 GPA and a GPA below 2.0
• Ten percent (10%) of the students enrolled have a 3.0 GPA;
• Twenty (20%) of the students enrolled have a 2.0-2.9 GPA,
• Ninety (90%) of the students enrolled have below a 2.0 GPA.
1. PS #3 ranked in the top 15% of the city’s academic standards for the academic year 2011-2012.
90% of the 15% showed content competence
85% of the 15% showed performance competence.
70% of the 15% showed proficiency competence.
1. PS #3 is SOL fully accredited.
Scores were as follows:
• Language Arts = 74%
• Mathematics = 75%
• Science = 70%
• History = 70%
Partition of Public School One (PS #1)
Partition of Public School #1, According to Student –Teacher Ratio and Grades
Teacher-Student Ratio
Seventy-five (75%) percent of the classes has one teacher to every twenty students.
Classes have accelerated courses.
Classes employ a seminar style of teaching.
All students enrolled in classes are students, with a 4.0 GPA.
Twenty-percent (25%) of the classes has one teacher to every thirty students.
Classes have grade appropriate courses.
Classes employ a traditional of teaching.
Students enrolled have between a 4.0 GPA and a 2.0 GPA.
• Five (5%) of the students enrolled have a 4.0 GPA;
• Sixty percent (60%) of the students enrolled have a 3.0-3.9 GPA;
• Thirty-five (35 %) of the students enrolled have a 2.0-2.9 GPA.
Five percent (5%) of the classes has one teacher and one teacher’s aide to every ten students.
Classes have developmental courses.
Classes employ a one-to-one style of teaching.
Students enrolled have between a 3.0 GPA and a GPA below
• Ten percent (10%) of the students enrolled have a 3.0 GPA;
• Twenty (20%) of the students enrolled have a 2.0-2.9 GPA,
• Ninety (90%) of the students enrolled have below a 2.0 GPA.
PS #1 ranked in the top 10% of the city’s academic standards for the academic year 2011-2012.
90% of the 10% showed content competence
85% of the 10% showed performance competence.
70% of the 10% showed proficiency competence.
PS #1 is SOL fully accredited.
Scores were as follows:
• Language Arts = 85%
• Mathematics = 80%
• Science = 70%
• History = 75%
Partition of Public School Two (PS #2)Teacher-Student Ratio
Twenty percent (20%) percent of the classes has one teacher to every twenty students.
Classes have accelerated courses.
Classes employ a seminar style of teaching.
All students enrolled in classes are students, with a 4.0 GPA.
Seventy percent (70%) of the classes has one teacher to every thirty
Classes have grade appropriate courses.
Classes employ a traditional of teaching.
Students enrolled have between a 4.0 GPA and a 2.0 GPA.
• Seven (7%) of the students enrolled have a 4.0 GPA;
• Seventy percent (70%) of the students enrolled have a 3.0-4.9 GPA;
• Twenty-three percent (23%) of the students enrolled have a 2.0-2.9 GPA.
Ten percent (10%) of the classes has one teacher and one teacher’s aide to every ten students.
Classes have developmental courses.
Classes employ a one-to-one style of teaching.
Students enrolled have between a 3.0 GPA and a GPA below 2.0
• Ten percent (10%) of the students enrolled have a 3.0 GPA;
• Ten (10%) of the students enrolled have a 2.0-2.9 GPA,
• Eighty (80%) of the students enrolled have below a 2.0 GPA.
PS #2 ranked in the top 20% of the city’s academic standards for the academic year 2011-2012.
85% of the 20% showed content competence
75% of the 20% showed performance competence.
70% of the 20% showed proficiency competence.
PS #2 is SOL fully accredited.
Scores were as follows:
• Language Arts = 8%
• Mathematics = 86%
• Science = 73%
• History = 78%
Partition of Public School Two (PS #3Teacher-Student Ratio
Ten percent (10%) of the classes has one teacher to every twenty students.
Classes have accelerated courses.
Classes employ a seminar style of teaching.
All students enrolled in classes are students, with a 4.0 GPA.
Seventy-five percent (75%) of the classes has one teacher to every thirty students.
Classes have grade appropriate courses.
Classes employ a traditional of teaching.
Students enrolled have between a 4.0 GPA and a 2.0 GPA.
• Five (5%) of the students enrolled have a 4.0 GPA;
• Thirty-five (25%) of the students enrolled have a 3.0-4.9 GPA;
• Seventy percent (70%) of the students enrolled have a 2.0-2.9 GPA.
Fifteen percent (15%) of the classes has one teacher and one teacher’s aide to every ten students.
Classes have developmental courses.
Classes employ a one-to-one style of teaching.
Students enrolled have between a 3.0 GPA and a GPA below 2.0
• Ten percent (10%) of the students enrolled have a 3.0 GPA;
• Twenty (20%) of the students enrolled have a 2.0-2.9 GPA,
• Ninety (90%) of the students enrolled have below a 2.0 GPA.
PS #3 ranked in the top 15% of the city’s academic standards for the academic year 2011-2012.
90% of the 15% showed content competence
85% of the 15% showed performance competence.
70% of the 15% showed proficiency competence.
PS #3 is SOL fully accredited.
Scores were as follows:
• Language Arts = 74%
• Mathematics = 75%
• Science = 70%
• History = 70%
Dr. Elizabeth Lohman, Tidewater Community College, CC-By | {"url":"https://docsbay.net/doc/1707/student-s-paper-modified-used-only-for-educational-purposes","timestamp":"2024-11-13T05:00:55Z","content_type":"text/html","content_length":"23451","record_id":"<urn:uuid:bce67c00-4365-4ac6-be55-dca0fa41cebd>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00649.warc.gz"} |
Introduction to Weekly Positional Matchups (POSAFPA) - Dynasty Nerds
Welcome to the Positional Matchups introduction. Here, I will discuss how I derive positional matchups from fantasy points allowed and, ultimately, how to use them. In the weekly article, I will
explain which teams give up fantasy points to what positions. I will then highlight potentially exploitable matchups and trends. I’ll provide a full table for upcoming matchups every week in a nice
and easy-to-apply format. So, let’s get to it.
To understand things, it’s best to start with fantasy points allowed. The concept is easy enough. In Week 5, the Packers allowed Raiders running backs to score a total of 14.3 PPR fantasy points. On
its own, this number is rather meaningless because we don’t know how much Raiders running backs typically score. If they typically score 20 points, we would consider this a poor outing, and if they
typically score 10 points, then this could be viewed as a success. This is where the “Schedule-Adjusted” portion of the report comes into play.
Over the first four weeks, Raiders running backs have scored an average of 11.45 points per game, which means they outperformed expectations this past week. Another way to put this is that the
Packers allowed 2.85 points over average this week. Through the magic of spreadsheets, I calculate this for every team and every position each week and compile that into Schedule-Adjusted Fantasy
Points Allowed.
With the theorycraft over, we can start to use this information. Points-allowed tables are nothing new and can be found all over. Schedule-adjusted points allowed, like we are doing, are a bit harder
to calculate, so it’s not as widespread. Let’s take a look at the table for the first five weeks.
Fantasy Points Allowed
There is a pretty widespread when it comes to points allowed, and we can use this information to our benefit. It’s straightforward enough for quarterbacks: if your QB plays the Chargers, you can
expect 4.7 extra points that week. But there are still a few issues. One, this can still be influenced by the schedule.
Let’s say Team A plays opponents whose RBs average 20 points a game, but Team A allows 25 points that week. Then let’s say Team B plays opponents whose RBs average 5 points a game, but Team B allows
10 points. Both would show up as 5 points allowed over average despite Team B allowing double the average points, but Team A only allowed a quarter over average.
Another shortcoming is that outside of single-starter positions, it is difficult to figure out how these points will be dispersed across a team. A team’s RBs might score 6 points extra in a matchup,
but it’s unlikely you know your player’s usual share off the top of your head. We can make this easier to use by converting it to a percentage.
Percentage of Fantasy Points Allowed
Now, here’s something we can use. Not only will this represent the schedule in a more unbiased fashion, but it’s far easier to apply. Let’s look at an example of Dameon Pierce. Over the first five
weeks, Dameon Pierce has scored 6.7, 5.5, 14.9, 11.8, and 9.2 PPR points per game. If you have extraordinary mental math capabilities, you’d know that is a 9.6 points per game average. I do not. I
would look at those numbers and ballpark him around 10 points a game. Next week, he plays the Saints. According to the above chart, the Saints allow 23% fewer fantasy points to running backs compared
to their average. Again, in the interest of simplicity, I would round this to about 25%. If his average is 10 points and he will score 25% less, we can estimate that he’ll total about 7.5 PPR points
this upcoming week.
Congratulations, you just created your first player projection! It might be rough, and we might be a little liberal with the rounding, but we’re talking about the NFL. We’re already knee-deep into
bad statistical practices compared to the real mathematicians out there. That’s okay, though, because we’re not predicting the next big market crash or the trajectory of an asteroid. We’re just
trying to beat Matt from Engineering, who sniped Zay Flowers from you in the draft.
But we aren’t done just yet. How can we improve this even further? Simple, cross-reference it with the schedule so you don’t need to remember the 16ish matchups every week. It’s relatively simple
compared to the other formulas but makes it that much easier to set lineups. So, without further ado, I present the finished product.
The Percentage of Schedule-Adjusted Fantasy Points Allowed! I once dreamt of calling it POSAFPA (pronounced poh-saf-puh), but I recognize that we are already inundated with acronyms. That said, it’s
a good unique identifier and search term. For simplicity’s sake, though, I will refer to it as Positional Matchups from here on out.
Positional Matchups: The Final Product
It’s the same info as the previous table but easier to use (have you caught on to the theme yet). Now that I’ve given you a nice, easy-to-use tool, let me explain how you should use it.
First things first, this isn’t an end-all-be-all tool. While this takes into account positional matchups, there are other things to consider when setting lineups. My lineup decisions are typically a
combination of player volume, positional matchup, the Vegas over/under on the game, the implied game script, the weather, if applicable, and a hefty dose of information from people smarter than I.
This is, however, a significant component and most often my decision-maker when staring down two similar volume options. For the other info mentioned, I recommend looking at our other Sit-Start
articles here.
There are two other things I’d like to note and they both relate to how the NFL uses position designations. First is that players with dual or nontraditional roles are not as influenced by matchups.
Running backs used primarily as pass catchers are one example. Wide receivers who get carries and rushing QBs are also less susceptible to negative matchups.
The second is that while WR is a single position, it is, in fact, several roles. If a wide receiver primarily plays on the perimeter and faces a shutdown corner, they might be in for a rough matchup,
but the slot receiver might fare just fine. Unfortunately, we are simply limited by the information available to us. Pairing positional matchups with shadow corner reports can give you insight into
which players may be the exception that week.
I’ll give an example of what this is good for. Let’s say you’re looking to fill your flex position and deciding between DK Metcalf (WR-SEA) or Brian Robinson (RB-WAS). For DK Metcalf, find SEA in the
first column, then follow that row over to the WR column, and you’ll see his positional matchup is 10%. That’s a moderately good matchup. Next for Robinson, find WAS in the first column and follow
that row to the RB column, and you’ll see his matchup as -32%, which is a strongly negative matchup. Given both players are averaging just over 14 PPR points per game, I would be leaning DK Metcalf
So there you have it. Look out for the weekly report to include matchups to target/avoid. I will also highlight team trends for future matchups and provide a few examples of players in borderline
situations. Until then, good luck.
3 Responses
1. So, if I am understanding this correctly, the numbers say that Demmondre Lenoir DB SF would be a better start than Trent McDuffie DB KC?
2. Interesting read…what is your twitter handle so I can follow your content?
1. @moncalff
Thanks for the reminder to mention it in my articles. | {"url":"https://www.dynastynerds.com/introduction-to-weekly-positional-matchups-posafpa/","timestamp":"2024-11-05T20:12:42Z","content_type":"text/html","content_length":"223858","record_id":"<urn:uuid:44a4824f-6b85-45ae-a9bf-daba8caa4f92>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00860.warc.gz"} |
Looking for experts to compare Naive Bayes with other classifiers in R – any leads? | Pay Someone To Take My R Programming Assignment
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I’m looking for anyone interested in doing activities on R R (the R R Talk team), and getting feedback from users of R, in general. Thanks! Thanks! So, here is the blog: Inventorms Guide to R R – The
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If you’re looking for a quick measure of accuracy or user-generated ranking you can use SAS. SAS is an open-source Python package designed to help developers and implement search engines with data
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observing 1 × 50 element = 0.5, each element appearing in 50 samples at a time (just like for the other six examples above). For the performance review of the tests of the algorithms, we first split
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From this, we calculated the $C$-size of the data and a range of maximum possible numbers of element weight estimates from this data using the following formula: $$C = \max \left\{ {1 / \log v} \
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v} + 1 \right).$$ In the next step, the standard deviation of the weights computed by the split-probability model (i.e., weighted by the weights that yielded first the class × number of elements)
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conclusions, and a description of whyLooking for experts to compare Naive Bayes with other classifiers in R – any leads? It is easier than ever to make one of your own. Instead the way you choose is
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so we have 1000 models, for $y=1$ and we can do this using linear regression. For example, if $y$ is variable $x_{1}$ and $y=-7$, then the predicted outcome will be $(26(1-p))28-6 \geq -6\sqrt{36\log
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log(1-p)\right\} }{\ln\left\{ 0\right\}}$, and so we want to evaluate the expectation. We are talking about summing, as vector $\vec{y} = \vec{\alpha}_{0}+\dots +\vec{\alpha}_{k+1}+\vec{\alpha}_{k} +
\vec{\alpha}_{k+1}+\vec{\alpha}_{k+2} +\ldots +\vec{\alpha}_{k+m-1}$. So the expectation with the number of models is $\frac{x_{2}}{x_{1}}\log(1-p)+\dots+\frac{x_{k}\log(1-p)-1}{\pi(1-p)}$, so it’s
up to you to run everything. 2) Try to use a random seed method, as suggested by some R scholar, here and on Stack Overflow, to look at simulated data and calculate correctly the probabilities of the
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Pareto-Normal model of a large population. 3) Define other variables such as $y \sim \sigma$, the population of people with your model | {"url":"https://rprogrammingassignments.com/looking-for-experts-to-compare-naive-bayes-with-other-classifiers-in-r-any-leads","timestamp":"2024-11-10T18:41:21Z","content_type":"text/html","content_length":"199085","record_id":"<urn:uuid:9aec0372-fa49-4442-a924-03fff0085f05>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00878.warc.gz"} |
Math.NET Numerics
Math.NET Numerics aims to provide methods and algorithms for numerical computations in science, engineering and every day use. Covered topics include special functions, linear algebra, probability
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Math.NET Numerics is part of the Math.NET initiative and is the result of merging dnAnalytics with Math.NET Iridium, replacing both. Available for free under the MIT License. It targets Microsoft
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See NuGet & Binaries for a complete list of our NuGet packages, Zip files and the release archive.
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let's say we have a matrix \(\mathrm{A}\) and want to find an orthonormal basis of the kernel or null-space of that matrix, such that \(\mathrm{A}x = 0\) for all \(x\) in that subspace.
using MathNet.Numerics.LinearAlgebra;
using MathNet.Numerics.LinearAlgebra.Double;
Matrix<double> A = DenseMatrix.OfArray(new double[,] {
Vector<double>[] nullspace = A.Kernel();
// verify: the following should be approximately (0,0,0)
(A * (2*nullspace[0] - 3*nullspace[1]))
Even though the core of Math.NET Numerics is written in C#, it aims to support F# just as well. In order to achieve this we recommend to reference the MathNet.Numerics.FSharp package in addition to
MathNet.Numerics, which adds a few modules to make it more idiomatic and includes arbitrary precision types (BigInteger, BigRational).
open MathNet.Numerics.LinearAlgebra
let m = matrix [[ 1.0; 2.0 ]
[ 3.0; 4.0 ]]
let m' = m.Inverse()
It also works well in the interactive F# environment (REPL) which can be launched with fsharpi on all platforms (including Linux). As a start let's enter the following lines into F# interactive.
Append ;; to the end of a line to run all code up to there immediately and print the result to the output. Use the tab key for auto-completion or #help;; for help. For convenience our F# packages
include a small script that sets everything up properly:
#load "../packages/MathNet.Numerics.FSharp/MathNet.Numerics.fsx"
open MathNet.Numerics
open MathNet.Numerics.LinearAlgebra
let m : Matrix<float> = DenseMatrix.randomStandard 50 50
(m * m.Transpose()).Determinant()
Let's use Visual Basic to find the polynomial roots \(x\) such that \(2x^2 - 2x - 2 = 0\) numerically. We already know there are two roots, one between -2 and 0, the other between 0 and 2:
Imports MathNet.Numerics.RootFinding
Dim f As Func(Of Double, Double) = Function(x) 2*x^2 - 2*x - 2
Bisection.FindRoot(f, 0, 2) ' returns 1.61803398874989
Bisection.FindRoot(f, -2, 0) ' returns -0.618033988749895
' Alternative to directly compute the roots for this special case:
FindRoots.Quadratic(-2, -2, 2)
You need a recent version of Mono in order to use Math.NET Numerics on anything other than Windows. Luckily there has been great progress lately to make both Mono and F# available as proper Debian
packages. In Debian testing and Ubuntu 14.04 (trusty/universe) you can install both of them with APT:
sudo apt-get update
sudo apt-get install mono-complete
sudo apt-get install fsharp
If you don't have NuGet yet:
sudo mozroots --import --sync
curl -L https://nuget.org/nuget.exe -o nuget.exe
Then you can use NuGet to fetch the latest binaries in your working directory. The -Pre argument causes it to include pre-releases, omit it if you want stable releases only.
mono nuget.exe install MathNet.Numerics -Pre -OutputDirectory packages
# or if you intend to use F#:
mono nuget.exe install MathNet.Numerics.FSharp -Pre -OutputDirectory packages
In practice you'd probably use the Monodevelop IDE instead which can take care of fetching and updating NuGet packages and maintain assembly references. But for completeness let's use the compiler
directly this time. Let's create a C# file Start.cs:
using System;
using MathNet.Numerics;
using MathNet.Numerics.LinearAlgebra;
class Program
static void Main(string[] args)
// Evaluate a special function
// Solve a random linear equation system with 500 unknowns
var m = Matrix<double>.Build.Random(500, 500);
var v = Vector<double>.Build.Random(500);
var y = m.Solve(v);
Compile and run:
# single line:
mcs -optimize -lib:packages/MathNet.Numerics.3.0.0-alpha8/lib/net40/
-r:MathNet.Numerics.dll Start.cs -out:Start
# launch:
mono Start
Which will print something like the following to the output:
DenseVector 500-Double
-0.181414 -1.25024 -0.607136 1.12975 -3.31201 0.344146
0.934095 -2.96364 1.84499 1.20752 0.753055 1.56942
0.472414 6.10418 -0.359401 0.613927 -0.140105 2.6079
0.163564 -3.04402 -0.350791 2.37228 -1.65218 -0.84056
1.51311 -2.17326 -0.220243 -0.0368934 -0.970052 0.580543
0.755483 -1.01755 -0.904162 -1.21824 -2.24888 1.42923
-0.971345 -3.16723 -0.822723 1.85148 -1.12235 -0.547885
-2.01044 4.06481 -0.128382 0.51167 -1.70276 ...
See Intel MKL for details how to use native providers on Linux.
val m : obj
val m' : obj
Multiple items
val float : value:'T -> float (requires member op_Explicit)
<summary>Converts the argument to 64-bit float. This is a direct conversion for all primitive numeric types. For strings, the input is converted using <c>Double.Parse()</c> with InvariantCulture
settings. Otherwise the operation requires an appropriate static conversion method on the input type.</summary>
<param name="value">The input value.</param>
<returns>The converted float</returns>
[<Struct>] type float = System.Double
<summary>An abbreviation for the CLI type <see cref="T:System.Double" />.</summary>
<category>Basic Types</category>
type float<'Measure> = float
<summary>The type of double-precision floating point numbers, annotated with a unit of measure. The unit of measure is erased in compiled code and when values of this type are analyzed using
reflection. The type is representationally equivalent to <see cref="T:System.Double" />.</summary>
<category index="6">Basic Types with Units of Measure</category> | {"url":"https://numerics.mathdotnet.com/","timestamp":"2024-11-09T01:25:14Z","content_type":"text/html","content_length":"23581","record_id":"<urn:uuid:104a9549-b7e6-4d16-bd79-8a2166688ca5>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00028.warc.gz"} |
Statistics Class 10 Notes
Statistics Class 10 Notesadmin2023-01-27T16:41:28+05:30
Statistics Class 10 Notes
Statistics Class 10 Notes
CBSE Class 10 statistics notes are comprehensive notes which cover the latest syllabus of CBSE and NCERT. The statistics notes are well-prepared to help students learn the topics, concepts, and
problems covered in each chapter. Thanks to these valuable notes, students don’t have to worry about buying different books to prepare during the test period. Every important formula and concept
presented in the chapter is covered in the revision notes.
Quick revision notes are available to you if you need an overview of a chapter. These notes will undoubtedly save you time on those hectic exam days. The revision notes for class 10 statistics help
you revise the whole chapter quickly. The major benefit of the Statistics Class 10 Notes is that they go over the key ideas and illustrate diagrams with real-world situations, which motivates pupils
to do well on tests.
Below, we’ve included a brief overview of the CBSE Notes for Class 10, which concisely cover each chapter’s major topics.
Overview of CBSE Class 10 Statistics Revision Notes
Statistics is a branch of mathematics that studies extracting meaningful information from the provided data. Statistics is used to infer collected data in addition to involving the collection,
organisation, analysis, and interpretation of data. It aids in deriving significant conclusions from the data and in using those conclusions to make wise judgments. So let’s examine these class 10
statistics notes in more detail.
• Although dividing the data into grouped frequency tables is briefly described in statistics up to the 10th level, class 10 statistics notes clearly explain how to measure the central tendency.
• The value that best captures the traits of the complete dataset, taking into account each and every value in the collection of data, is known as the central tendency. The Mean, Median, and Mode
are the three central tendency measurements. These measures of central tendency are defined in the Class 10 Statistics notes.
• In addition to learning about important concepts in the chapters, students may efficiently get prepared for exams. By downloading the Statistics Notes for Class 10 PDF, students may quickly
become familiar with the chapters.
• These are the Statistics class 10 notes that a team of knowledgeable professors has created. You may quickly review the whole chapter using the revision notes. Revising notes on exam days is one
of the finest strategies advised by professors.
• It includes all the topics given in NCERT class 10 Mathematics textbook.
• With the help of these notes, students will be able to gain an understanding of the chapters, making them excellent study tools.
• Each subject has been covered in-depth by experts to guarantee that students fully comprehend the material.
• The class 10 statistics notes are accessible in PDF format, allowing students to review them whenever they want and wherever they feel most comfortable.
• Numerous details are included in the Class 10 Statistics chapters that students may only fully understand after reviewing the chapters once more. Statistics Notes include examples, tables, and
graphs for a better understanding of students. The Class 10 Statistics notes are useful in this situation.
• With the help of these notes, students will be able to gain an understanding of the chapters, making them excellent study tools.
CBSE Class 10 Chapter-wise Statistics Notes
The Mean, Median, and Mode are the three central tendency measurements. These measures of central tendency are defined as follows in the Class 10 Statistics notes.
• The mean of a data set is the product of the total number of observations divided by the total number of observations.
• The observation with the highest frequency is the mode of collection of data.
• The value that reflects the centre observation in a set of data is called the median.
Revision Notes of Statistics Class 10
• In this, we will discuss important statistical concepts, such as grouped data, ungrouped data and the measures of central tendencies like mean, median and mode, methods to find the mean, median
and mode, and the relationship between them with more examples.
• The gathering, categorisation, and representation of any type of data are topics covered in the mathematical field of statistics to facilitate analysis and comprehension.
• Bar graphs, pie charts, histograms, and frequency polygons are only a few of the several types of data representations covered in the class 10 statistics course of the CBSE curriculum.
• The calculation of central tendencies of ungrouped random data is also covered in the Class 10 Statistics Notes. However, calculating the central tendency of grouped data is thoroughly
illustrated in the Statistics Class 10 explanation.
• Class 10 statistics notes cover three measures of central tendency Mean, Median, and Mode.
• Mean: The mean of a data set is the product of the total number of observations divided by the total number of observations. It is the average of “n” numbers, which is calculated by dividing the
sum of all the numbers by n.
• Mode: The observation with the highest frequency is the mode of collection of data. The number which appears most frequently in the series then it is said to be the mode of n numbers.
• Median: The value that reflects the centre observation in a data set is called the median. If we arrange the numbers in an ascending or descending order, then the middle number in the series will
be the median. The median is the average of the two middle numbers if the series number is even. The Median is defined as the measure of central tendency, which obtains the value of the
middlemost observation in the data.
• Ungrouped data: Ungrouped data is data in its original or raw form. The observations are not classified into groups.
• Grouped data: In grouped data, observations are organised in groups. For example, a class of students got different marks in a school exam.
• Frequency: Frequency is the number of times a particular observation occurs in data. For example, if four students scored between 90 and 100, then the marks scored between 90 and 100 have a
frequency of 4.
• Class Interval Data can be grouped into class intervals so that all observations in that range belong to that class. Class width = upper class limit – lower class limit
• Class 10 Notes: Statistics Notes cover Mean, Median, Mode, and Empirical relationship between mean, median, and mode.
• The Mean of a given data set can be found by three different methods, namely: Direct Method, Assumed mean method, and Step Deviation Method.
• Empirical formula statistics class 10: It gives the relationship between the three measures of central tendency. The empirical formula is written as 3 Median = Mode 2 Mean. This formula can be
used to measure the values of other central tendencies; two measures of central tendency are known.
• The median can be used as a measure of central tendency in Class 10 Statistics explanation issues if the individual values of each observation are insignificant.
• Mean of grouped data: The first step in calculating the mean of grouped data is to identify the midpoint (also known as a class mark) of each interval or class. The frequencies of the relevant
classes must then be multiplied by these midpoints. The mean value is determined by dividing the total number of values by the sum of the products.
• Mean of Grouped Data With Class-Interval: There are three ways to determine the mean when the data is classified into class intervals. These are Direct Method, Assumed mean method, and Step
Deviation Method.
• Direct Method: In this method, a midpoint that reflects the whole class is used. The term for it is the class mark. The top limit and lower limit are averaged to produce it.
• Method of Deviation or Assumed Mean: We may use this strategy to simplify our computations if we need to compute huge numbers. In this approach, we pick one of the x’s and use it as the assumed
mean, or “a.” The deviation is then calculated as the difference between the expected mean and each x. The rest of the procedure is identical to the direct procedure.
• Step Deviation Method: To simplify our computations, we divide the values of d in this technique by a number “h”.
• Median of Grouped Data: Finding the cumulative frequency and n/2 will help us determine the median of a set of grouped data. The next step is to identify the median class, which is the category
for cumulative frequencies that are close to or larger than n/2.
• Mode of grouped data: The value of the observation with the highest frequency is known as the mode of the, which is the value among the observations that happen most frequently.
• Distribution of cumulative frequencies of less than type: The number of observations that are less than or equal to a specific observation is shown by the cumulative frequency of the less than
• Distribution of cumulative frequencies for more than type: The number of observations greater than or equal to a certain observation is shown by the cumulative frequency of the more than type.
Benefits of Revising 10th Science Notes
Understanding the subjects and getting high marks on the board examinations depend greatly on revision. It is crucial to have revision notes for each chapter while reviewing, especially in areas like
statistics, where there are several formulae and information. The following are some of the most significant advantages of the CBSE class 10 statistics notes:
• To fully understand the topics covered in the chapter and increase confidence, students must use these statistics class 10 notes while preparing for the exam.
• The revision notes are prepared based on the CBSE curriculum and as per NCERT guidelines. It covers almost all the Statistics concepts and formulas so that you will not have to refer to other
additional books.
• Students can use these notes to help with last-minute preparation. The notes are given in their most condensed and focused form.
• Because of the clear and simple language, students can quickly remember the ideas and concepts discussed in each chapter.
• These NCERT statistics notes for class 10 equip students to respond to any kind of inquiry, whether subjective or objective.
• To help students quickly learn all of the ideas and concepts, class 10 statistics notes include diagrams, flowcharts, and bullet points.
• Class 10 Statistics notes are available in a PDF format that helps students to revise the notes anytime and at any place as per their comfort.
• Class 10 Statistics notes include all facts and formulas that are important for the students to solve the question given in the chapter quickly and conveniently.
• When studying for tests and having to swiftly review the entire syllabus, statistics notes help students quickly and easily retain all the previously acquired elements from the chapter.
• Class 10 notes assist students in becoming comfortable with the subjects and recognising the chapter’s main ideas.
• The activities of referring to many sources of study information for chapter preparation during revision are reduced by class 10 statistics notes.
• One of the best strategies for swiftly reviewing subjects before an exam is to take notes. Statistics Notes helps you to revise formulas and essential concepts before an exam in a quick way so
that you do not forget them during the exam.
Our class 10 statistics notes will facilitate quick material review and help students do well on their coming examinations. | {"url":"https://deekshalearning.com/cbse/cbse-class-10-maths-notes-chapter-14-statistics/","timestamp":"2024-11-05T05:34:27Z","content_type":"text/html","content_length":"596490","record_id":"<urn:uuid:49c9782e-9dd6-433b-92e0-d9aaa7afb570>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00034.warc.gz"} |
Cartographer's Toolkit
Archive for February, 2011
Posted by Gretchen in Statistics on February 28, 2011
I’ve been reading through Daniel B. Carr and Linda Williams Pickle’s recent book on micromaps – “Visualizing Data Patterns with Micromaps.”
Even without the micromaps the charting recommendations are quite useful. In fact, there was one chart-type in particular that I wanted to copy for my recent buildout study data but I couldn’t figure
out how to copy it in Excel. So I wrote to Carr and asked him how they produced it. Basically, it looks like a scatterplot except that it has the x and y axes flipped so that the thing being measured
is labeled on the y-axis and the measurements are labeled on the x-axis. The primary thing that attracts me to this layout is the fact that you don’t have to turn your head sideways to read the
labels as you would when they are on the x-axis.
You know how it is much easier to browse the book store or library when the books are lined up so you can see them head-on rather than crooking your neck to read the horizontal spines, right? Same
thing with graphs. Movie rental stores, back in the days of the VHS tape, originally organized their tapes so that just the spines showed. This saves a lot of space. However, the stores got bigger
and they changed their strategy so that the tapes were showing face-out to the isle. This must have increased their sales because it became the accepted practice after a while.
Anyway, Carr quickly wrote back and told me that I was a fool for still using Excel and really why the heck would anyone not use R?! No, he was really much nicer than that, actually, but I do
understand his exasperation. I’ve been meaning to learn R for quite sometime. My brother tells me that I have no excuse for not learning it because it is “so easy.” So I really should. But then that
didn’t stop me from continuing to look for a work around.
And I found a pretty cool work around! In fact, it’s Carr’s own software found here on the National Cancer Institute site. I was able to download the software, upload my own data into it and have a
good-looking graph within 10 minutes. The software does support some changes in presentation like colors, labels, and the like but if you really need to customize a graph like this you’ll have to use
R. In fact, Carr has R scripts for most of the graphs in his book and has made them available here.
Posted by Gretchen in Education on February 24, 2011
I’ve just noticed a slew of new books on the market that seem like they’d be a great fit for cartographers. They are in my cart on Amazon because I haven’t quite finished the three books I’ve got on
the nightstand right now. Just by way of explanation – those tutorial books are great and I always feel like every once in a while I should skim through books like that in case there’s something new
that I haven’t been aware of.
On nightstand now, i.e., still plowing through:
Hopefully on nightstand soon, i.e., when the other three are all read or at least skimmed:
Posted by Gretchen in Statistics on February 23, 2011
The area-weighted root mean square error is very useful for determining how closely variables match one another while taking into account a normalization factor – in this case, area. We often
normalize by area in GIS and cartography in order to better compare one analysis unit with another. Think about a map of U.S. states: the states are such vastly different sizes that most variables,
such as incidence and population, are not comparable from state to state if you simply use the raw number. Instead, you must divide the variable by the size of the state to provide an adequate
comparison across states.
In yesterday’s post about using ArcMap in a creative way to make a scatterplot, you can see that area is a significant factor in my research on watersheds. For some context, these are small basins in
the Pacific Northwest that we are analyzing to determine what their current impervious surface percentage is with two different datasets. One dataset is actual imperviousness as measured by 1-meter
NAIP imagery analysis. The other is a derived dataset using landuse codes from tax assessor parcels to predict what current imperviousness is. Those predictions are actually based on that initial
dataset, the 1-meter imperviousness, where we came up with average coefficients for how much impervious, on average, is in each landuse group.
To figure out how close they come to being the same, the closer the better, I plotted the values in a scatterplot. However, it would be nice to get an actual measure, and that’s where the
area-weighted root mean square error (or RMSE) comes in.
To calculate it, I added up the total area in all the basins first. Then for each basin you determine the difference between the two variables, in other words subtract the value for one from the
other. I did all this in Excel. You square those differences in another column, then in another column multiply that answer by the area of the basin. You could definitely do all this in one column, I
just liked to see them separately.
Sum that last column, divide the sum with the total area of all the basins, then take the square root of that value. I presented the value as a percentage, so I multiplied this by 100.
When I used 5-meter impervious data I got an area-weighted RMSE of 1.97% and when I used the 1-meter impervious data I got an area-weighted RMSE of 1.05%. That’s really great because it means that
the 1-meter data gets me more precision for the model. It still doesn’t definitely tell me how close to accurate I am getting, however, so that’s the next thing for me to explore. There’s always
*I’d like to thank the preeminent William Huber for suggesting these analytical procedures a few years ago when I first started doing buildout studies. Note to other solo-consultants: hiring experts
in statistics and other fields to review and advise is a small expense to pay to ensure that your project is of top quality.
Posted by Gretchen in Creativity on February 22, 2011
This isn’t your typical mapping task, but I am currently evaluating the effectiveness of a model that uses 1-meter impervious surface data and I needed a scatterplot. Did you know that you can
actually make one in ArcGIS? It’s definitely a creative use of the software as it really isn’t a function that you just choose from a drop-down menu. What you are doing is essentially plotting your
data points in x,y space.
I had a dataset of basins (aka watersheds) that had a measured imperviousness percentage based on the 1-meter impervious data – basically just an intersect between the two datasets and a summary
procedure. Note you could easily do a raster zonal analysis instead, but my data were already in vector. I also had data on the percent impervious of the basins based on a model. In an ideal world
the two would match. In other words, y=x.
So I had two variables: the actual impervious and the modeled impervious. Add to that a third variable: size of basin, and that’s all that needs to be on the scatterplot. I figured size of basin is
important because it would make sense that larger basins would have less error than smaller basins. So I outputed those three variables into a table and then imported the table as an x,y dataset
where x was the actual and y was the modeled.
I guess I should mention that I took the square root of the percentages first, before plotting them on the graph. This preserves the relationships that I’m measuring while reining in the values, so
to speak. Then I symbolized by graduated symbols. These were hollow circles to help visualize overlap. I created an y=x line too, then fitted the graph axes where they should go.
Overall I’m quite pleased, both with the visualization and the results. The results show that the data do get pretty close to the y=x line for the most part. There’s quite a bit of scatter in both
directions with the very small basins. However, these basins are very small. I think it is safe to say the model does not predict imperviousness in very small basins well.
I [DEL:may:DEL] did write an entry tomorrow on calculating the area-weighted root mean square error. That is sure to be extra-exciting!
Posted by Gretchen in News on February 21, 2011
Firstly, I’m not so sure about that last post. The order of a choropleth legend seems trivial to me, for one. For another, as someone on twitter pointed out, we are mostly used to seeing the high
value on the bottom and as someone else on twitter mentioned, low to high is a common way to present numbers.
Second, the New York Times has an interesting article, “Freelance Scholars: A Nomadic Lot” that talks about one scholar’s decision to quit her professorship to move to Rome and build a cartographic
history of the urban development of Rome. She wrote a book on the subject titled The Waters of Rome.
Third, the GISCI has updated their homepage to include information about the poster contest. Don’t forget, submissions are allowed all during March, with the last day being March 31.
Posted by Gretchen in Best Practices on February 17, 2011
Earlier I posted about Choropleth Limitations. Choropleth maps have legends that show how the colors (or shades of gray, perhaps) match with the values they represent. There are a few different
options for the presentation of these legends. In ArcMap, the legend defaults to a vertical style with the highest value (largest number) on the bottom of the list and the lowest value is at the top
of the list like this:
I often prefer to push the colors together for a continuous color scheme like this by changing the ArcMap setting to 0 for “patches (vertically).” I also sometimes prefer to change the numbers to
Courier New or some other monospaced typeface since numbers line up nicely when they are placed vertically. Like this:
However, I’m currently reading through a book by Daniel B. Carr and Linda Williams Pickle called “Visualizing Data Patterns with Micromaps” that asserts that the high value should be on top since
that is how we are used to reading the y axis on graphs. This would change the legend to look like this:
I like to change it up occassionally and create a horizontal legend. I would think that Carr and Pickle would be okay with this since it reads from left to right the same way you would read the x
axis on a graph, from low to high: | {"url":"http://blog.gretchenpeterson.com/archives/date/2011/02","timestamp":"2024-11-12T22:55:19Z","content_type":"application/xhtml+xml","content_length":"54821","record_id":"<urn:uuid:ae91d931-991e-4341-ab95-77fc6e9f3d5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00497.warc.gz"} |
Statistical timing optimization of combinational logic circuits
High performance circuit design is becoming increasingly important in VLSI design. The most important problem faced in the design of these circuits is to meet a certain performance level. In the past
few years CAD algorithms and tools have been well developed that improve the performance of logic circuits in the sense that the worst case delay is minimized. However, manufacturers recognize that
worst case delay models are typically pessimistic and the manufactured ICs will have a range of performances reflecting the manufacturing variations. Thus, the real problem that needs to be solved in
the performance optimization of these circuits is to maximize the percentage of fabricated circuits that will achieve a certain performance level, as opposed to minimizing the worst case delay which
has been the focus thus far. In this paper we develop methods to improve the statistical timing behavior of a combinational logic circuit, given probability distributions for the gate and wire
delays. This work uses a statistical timing analysis technique developed earlier to drive timing optimization in the right direction to achieve a prescribed goal with the least area overhead.
Original language English (US)
Title of host publication Proceedings - IEEE International Conference on Computer Design
Subtitle of host publication VLSI in Computers and Processors
Editors Anon
Publisher Publ by IEEE
Pages 77-80
Number of pages 4
ISBN (Print) 0818642300
State Published - 1993
Event Proceedings of the 1993 IEEE International Conference on Computer Design: VLSI in Computers & Processors - Cambridge, MA, USA
Duration: Oct 3 1993 → Oct 6 1993
Publication series
Name Proceedings - IEEE International Conference on Computer Design: VLSI in Computers and Processors
Other Proceedings of the 1993 IEEE International Conference on Computer Design: VLSI in Computers & Processors
City Cambridge, MA, USA
Period 10/3/93 → 10/6/93
All Science Journal Classification (ASJC) codes
• Hardware and Architecture
• Electrical and Electronic Engineering
Dive into the research topics of 'Statistical timing optimization of combinational logic circuits'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/statistical-timing-optimization-of-combinational-logic-circuits","timestamp":"2024-11-08T13:11:42Z","content_type":"text/html","content_length":"49069","record_id":"<urn:uuid:54095077-f0c7-4c05-9445-74ce939fb150>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00288.warc.gz"} |
Running With Power: What It Can Tell Us About Our Human Limits
In our books The Secret Of Running (www.thesecretofrunning.com) and The Secret Of Cycling (www.thesecretofcycling.com) we have described our unified theory for the performance in running and cycling.
Our running model is based on the premise that the power produced by the “human engine” (i.e. the leg muscles and the heart-lung system) must be equal to the sum of the power required to surmount the
running resistance Pr, the air-resistance Pa and the climbing resistance Pc, as indicated in the figure below.
This means that in practice a runner is slowed down by the air-resistance and—in case of hills—the climbing resistance. We have modeled Pr, Pa and Pc, using the laws of physics and in analogy to
similar models used in cycling.
Next, we have modeled the power of the human engine P, using the laws of physiology. The result is a complete model which enables us to calculate the speed of a runner, depending on his running power
and running economy and on the environmental conditions (wind, temperature, altitude, air-pressure, hills, footing).
In this paper, we will briefly describe the model and some interesting results. Meanwhile, we have tested the model in many situations (running, cycling and both in the lab and in races) and found
the results very convincing and consistent. Finally, we have observed that the Stryd power data match our model calculations perfectly.
The physics of running
We have applied the laws of physics to running and derived the equations, presented in the box:
As an example we use a c-value (the running economy or Energy Cost of Running) of 0.98 kJ/kg/km, a body weight m of 70 kg and a speed v of 20 km/h. Pr then becomes 0.98*70*20/3.6 = 381 Watts. We can
compare this to the air-resistance with the example of an air density ρ of 1.205 kg/m3 (which is the case at sea level at a temperature of 20 °C and an air-pressure of 1,013 mbar), an air-resistance
coefficient cdA of 0.24 m2 (we have derived this number from literature), windless weather (so vw = 0) and the same speed of 20 km/h. Pa then becomes 0.5*1.205*0.24*(20/3.6)3 = 25 Watts.
This means that even at a level course (so Pc = 0), and in windless weather, some 7 percent of the power of the runner is lost due to the air-resistance. In record attempts pacers are frequently used
to shield the elite runners and reduce the air-resistance (by some 20 percent). We have used our model to calculate how big the advantage of pacers is for world record performances.
According to our calculations, Kenenisa Bekele owes some 21 seconds of his phenomenal 10,000 m world record to the reduced air-resistance from his pacers. The air-resistance is eliminated altogether
on a treadmill, so in theory Kenenisa Bekele could run even two minutes faster at the 10,000 m on a treadmill! The table below shows the present world records and the results of our calculations
without pacers (so with increased air-resistance) and on a treadmill (so without any air-resistance).
The recent sub-two-hour marathon attempt of Eliud Kipchoge confirmed the importance of reducing the air-resistance. We have calculated that the reduction in air-resistance during this attempt (by the
combination of the rotating group of pacers and the wind-breaking time screen on the car) was virtually perfect at 37.5 percent. According to our calculations, his 2:00:25 is equivalent to 2:02:18 in
a normal race (which would be still an impressive new world record).
Another example of the power of our model calculations is presented in the figure below. Here, we have calculated the impact of the air-resistance on the 100m world record of Usain Bolt. We found
that his Berlin world record of 9.58 is equivalent to 9.36 at the altitude of Mexico City (where the air-pressure is much lower). Theoretically, Usain might have run even 9.18 seconds when all
factors would have been ideal (altitude, temperature, tail wind of 2 m/s)!
The physiology of running
We have applied the laws of physiology to running and derived the table below that specifies the power limits of the four power producing processes in the human muscles. We consider these numbers as
the maximum of human power for male elite athletes (who have very little body fat, for women the numbers are some 11 percent lower due to their higher fat percentage).
They are based on the fundamental biochemistry of the conversion processes (i.e. the maximum conversion speed and the energy production per unit of time) and on a gross metabolic efficiency of 25
percent (i.e. 25 percent of the metabolic energy is transferred into mechanical work, this number is considered the maximum for elite athletes in running and cycling).
Next, we have analyzed the impact of endurance time on the “fuel mix” in the muscles and the power produced. Sprinters use mainly ATP, 400 – 800 m runners use mainly the anaerobic conversion of
glycogen, but distance runners rely on the aerobic conversion of glycogen and fatty acids.
This means that as distance/endurance time increases, less power can be produced so the speed is reduced. We have proven that this shift in the fuel mix is the cause of the well-known “Riegel’s
formula” which describes the reduction in speed with distance and endurance time. The figure below shows our results for the fuel mix at various endurance times:
Running power and FTP of the world records
We have used our model to calculate the running power P to run at the speed of the world records. The table below confirms that at increasing distance and endurance time the running power P is
The table also shows the so-called Functional Threshold Power (FTP), which is defined as the power that can be maintained during one hour. We have recalculated P to FTP using Riegel’s formula as
explained above.
The table clearly shows that most world records are equivalent to an FTP of around 6.35 Watts/kg. The FTPs at the records at 15, 25 and 30K are a little lower, which is probably due to the fact that
these distances are run much less frequent. The FTP at the record at the 1,500 m is significantly higher, which is explained by the fact that the anaerobic processes play a more significant role
The limits of human power
From the biochemical data we have derived the limits of human power at various endurance times, see the table and figure below:
The table shows that the biochemical limit of the power that can be maintained for one hour (defined as the FTP) is 6.41 Watts/kg. This is quite close to the equivalent FTP of 6.35 Watts/kg of the
world records in distance running that we noted above.
Meanwhile, we have applied our unified model to many elite performances in many sports (running, cycling, ice-speed skating) and we have consistently found an FTP of around 6.35 Watt/kg to represent
the upper limit of human performance. The only time that we got higher values were for the performances of EPO-doped cyclists.
Conclusions and outlook
We have derived a new and unified theory on running, based on the laws of physics and physiology.
Our running model can be used to calculate the speed of a runner as a function of his running power (P in Watts/kg), running economy (ECOR in kJ/kg/km) and the environmental conditions (wind,
temperature, hills, air-pressure, altitude, footing). The model can also be used to analyze the limit of human performances and we found remarkable similar values of the FTP across various sports.
Meanwhile, the first running powermeters have come on the market. We have tested the Stryd footpod, both in the lab and in the field and found that the results match our model calculations perfectly.
Obviously, these powermeters provide runners now with the means to optimize their training and racing. For the first time, runners now have—on a daily basis—the numbers to try to improve their
running power P and running economy ECOR and to use their running power optimally, so constantly throughout the race.
We realize that this will not be easy because for us (and for most people) running has been habituated over many years. We will not be able to change our running habits and running form overnight.
But with time and concrete data, we are confident we will be able to get better.
We hope that many readers will join us in this effort. Let’s share our data and conclusions on how we can improve our running! We are curious to the reactions and experiences of the readers, we
welcome you to share these at www.thesecretofrunning.com.
Thank you to the co-author Ron van Megen | {"url":"https://www.trainingpeaks.com/blog/running-with-power-what-it-can-tell-us-about-our-human-limits/","timestamp":"2024-11-06T20:54:44Z","content_type":"application/xhtml+xml","content_length":"157710","record_id":"<urn:uuid:ede395eb-03be-47d9-a653-8be5bc1cb76f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00520.warc.gz"} |
20.3: Applying Kirchhoff’s rule to model circuits
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In this section, we show how to model a circuit using Kirchhoff’s rules. In general, one can consider a circuit to be fully modeled if one can determine the current in each segment of the circuit. We
will show how one can apply the same procedure to model any circuit that contains batteries and resistors. The procedure is as follows:
1. Make a good diagram of the circuit.
2. Simplify any resistors that can easily be combined into effective resistors (in series or in parallel).
3. Make a new diagram with the effective resistors, showing battery arrows, and labeling all of the nodes so that loops can easily be described.
4. Make a guess for the directions of the current in each segment.
5. Write the junction rule equations.
6. Write the loop equations.
7. This will lead to \(N\) independent equations that one can solve for the \(N\) different currents in the circuit.
8. Once you have determined all of the currents, if some of them are negative numbers, switch the direction of those currents in the diagram (they will be negative if you guessed the direction
We will illustrate the procedure on the circuit shown in Figure \(\PageIndex{1}\), for which we would like to know the current through each resistor and each battery. The circuit contains 5 resistors
(\(R_1\)-\(R_5\)), 2 real batteries (with ideal voltages \(\Delta V_1\) and \(\Delta V_2\)), and 2 additional resistors to model the internal resistances of the real batteries (\(r_1\), \(r_2\))
Figure \(\PageIndex{1}\): A circuit that can be simplified and then solved with Kirchoff's rules.
How many different currents are in the circuit shown in Figure \(\PageIndex{1}\)?
1. 3
2. 4
3. 5
4. 6
Simplifying the resistors (step 2): In this circuit, resistors \(r_2\), \(R_1\) and \(R_2\) are in series, so that they can be combined into an effective resistor, \(R_6\):
\[\begin{aligned} R_6=r_2+R_1+R_2\end{aligned}\]
With this simplification, we obtain the circuit illustrated in Figure \(\PageIndex{2}\)
Figure \(\PageIndex{1}\) have been combined into the effective resistor, \(R_{6}\), to simplify the circuit.
Next, we note that resistors \(R_4\) and \(R_5\) are in parallel and can be easily combined into a resistor, \(R_7\):
\[\begin{aligned} R_7=\frac{R_4R_5}{R_4+R_5}\end{aligned}\]
which leads to the circuit illustrated in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{2}\) have been combined into the effective resistor, \(R_{7}\), to simplify the circuit.
Finally, we note that \(r_1\) and \(R_7\) are in series and can be combined into an effective resistor, \(R_8\):
\[\begin{aligned} R_8=r_1+R_7=r_1+\frac{R_4R_5}{R_4+R_5}\end{aligned}\]
leading to the simplified circuit illustrated in Figure \(\PageIndex{2}\), which we have labeled with nodes and battery labels.
Figure \(\PageIndex{3}\): have been combined into the effective resistor, \(R_{8}\), to simplify the circuit.
Guessing the directions of the currents (step 4): Before we can write the equations from Kirchhoff’s rules, we need to label the currents in the circuit diagram. In general, it is not always obvious
in which way the currents will go, so we make a guess that we can fix later if we guessed wrong.
In order to guess the current directions, choose one point on the circuit and move along a segment. Label the current in that segment and continue moving through the circuit, splitting up the current
when a junction is encountered. Make sure to only have one current per segment. We guess the currents as follows, referring to Figure \(\PageIndex{5}\):
• We start at point \(a\) and move upwards to point \(f\). We will call the current in that segment, \(I_1\).
• Since there is no junction, the current \(I_1\) continues through the resistor \(R_8\) to point \(e\).
• There is a junction at point \(e\), so we split the current \(I_1\) into currents \(I_2\) (towards point \(d\)), and \(I_3\) (downwards to point \(b\)).
• We follow current \(I_2\) first; \(I_2\) flows from \(e\) to \(d\), then down to \(c\), through the battery \(\Delta V_2\), and to point \(b\), where there is again junction.
• We follow current \(I_3\), which just flows down to the junction at point \(b\), where it “meets up” with current \(I_2\).
• Currents \(I_2\) and \(I_3\) both flow into the junction at point \(b\), and the current flowing out of the junction, through the battery \(\Delta V_1\), and towards point \(a\) is, again, \(I_1
\), since this current then flows up to point \(f\).
• All segments of wire have a labeled current, so we are done guessing currents.
You can proceed in an analogous way for any circuit. The final circuit, with currents labeled, is shown in Figure \(\PageIndex{5}\):
Figure \(\PageIndex{1}\).
We can now proceed with using Kirchhoff’s rules to solve for the values of the currents in the circuit. It is useful to note that there are 3 unknown currents in this circuit; we thus hope that
Kirchhoff’s rules will give us 3 independent equations.
Applying the junction rule (step 5): In the circuit from Figure \(\PageIndex{5}\), there are two junctions (at points \(b\) and \(e\)), so we will get two equations from the junction rule. To apply
the junction rule, the sum of the currents coming into the junction must be equal to the currents going out of the junction:
\[\begin{aligned} \text{incoming currents}&=\text{outgoing currents}&\\[4pt] I_2+I_3 &= I_1 \quad &\text{(junction $b$)}\\[4pt] I_1 &= I_2+I_3 \quad &\text{(junction $e$)}\\[4pt]\end{aligned}\]
Note that the two equations are not independent (in fact, they are the same). It is generally the case that if there \(N\) junctions, one will obtain less than \(N\) independent equations (usually, \
(N-1\) equations will be independent). In this case, the two junctions only gave us one equation.
Applying the loop rule (step 6): This circuit contains 3 different loops: \(abcdefa\), \(abefa\), and \(bcdeb\), which will lead to 3 equations from the loop rule. We expect that these equations will
not be independent, since this would lead to 4 equations and 3 unknowns when combined with the junction rule equation. Let us start with the loop \(abcdefa\):
• From \(a\) to \(b\), we trace through the battery in the opposite direction from the battery arrow: \(-\Delta V_1\).
• From \(b\) to \(c\), we trace through the battery in the same direction as the battery arrow: \(+\Delta V_2\).
• From \(c\) through \(d\) and through to \(e\) we go through the resistor \(R_6\) in the opposite direction from the current, \(I_2\), in that resistor: \(+I_2R_6\).
• From \(e\) to \(f\), we go through the go through the resistor \(R_8\) in the opposite direction from the current, \(I_1\), in that resistor: \(+I_1R_8\).
• And we are back at the starting point, so the sum of the above terms is equal to zero.
which gives the equation:
\[\begin{aligned} -\Delta V_1+\Delta V_2+I_2R_6+I_1R_8=0\quad\text{(loop abcdefa)}\end{aligned}\]
Similarly, for the loop \(abefa\), we obtain:
\[\begin{aligned} -\Delta V_1+I_3R_3+I_1R_8=0\quad\text{(loop abefa)}\end{aligned}\]
and for loop \(bcdeb\):
\[\begin{aligned} \Delta V_2+I_2R_6-I_3R_3=0\quad\text{(loop bcdeb)}\end{aligned}\]
Although it appears that we have obtained 3 additional equations, only two of these are independent. For example, if you sum the second and third equations (loops \(abefa\), and \(bcdeb\)), you
simply obtain the first equation (loop \(abcdefa\)). In general, if there are \(N\) different loops, one will obtain less than \(N\) independent equations (usually \(N-1\) independent equations, as
we did here).
At this point, after choosing one of the junction equations, and two of the loop equations, we have 3 independent equations that we can solve for the 3 unknown currents^1:
\[\begin{aligned} I_1 &= I_2+I_3 \quad &\text{(junction $e$)}\\[4pt] -\Delta V_1+\Delta V_2+I_2R_6+I_1R_8&=0\quad&\text{(loop abcdefa)}\\[4pt] -\Delta V_1+I_3R_3+I_1R_8&=0\quad&\text{(loop abefa)}\
It is only a matter of some simple math to solve for the 3 unknowns from these 3 equations (which we carry out in the example below).
Referring to the circuit in Figure \(\PageIndex{6}\), what is the voltage across the real terminal of the battery with ideal voltage \(\Delta V_1\) (the voltage between points \(a\) and \(b\))? What
is the current through resistor \(R_5\)?
Figure \(\PageIndex{5}\), with values filled in.
Since this circuit is the same that we just analyzed, we know that it can be simplified into the circuit shown in Figure \(\PageIndex{7}\), with resistors:
\[\begin{aligned} R_6&=r_2+R_1+R_2=(1\Omega)+(3\Omega)+(4\Omega)=8\Omega\\[4pt] R_8&=r_1+\frac{R_4R_5}{R_4+R_5}=(1\Omega)+\frac{(2\Omega)(2\Omega)}{(2\Omega)+(2\Omega)}=2\Omega\end{aligned}\]
Figure \(\PageIndex{6}\).
From above, we know that this leads to the following three equations:
\[\begin{aligned} I_1 &= I_2+I_3 \quad &\text{(junction $e$)}\\[4pt] -\Delta V_1+\Delta V_2+I_2R_6+I_1R_8&=0\quad&\text{(loop abcdefa)}\\[4pt] -\Delta V_1+I_3R_3+I_1R_8&=0\quad&\text{(loop abefa)}\
In order to solve these types of equations, it is usually convenient to place the battery voltages on the right hand side, and the resistor voltages on the left hand side. Although it is generally
bad practice to fill numbers into the equations before solving them, it is almost always a good idea when solving the \(N\) equations for the \(N\) currents. Furthermore, in order to make the
equations legible, it is also useful to not write in the units (which is very bad practice in general!). Thus, filling in the values for the resistors and the battery voltages, moving the voltages to
the right hand side, we obtain the following system of equations:
\[\begin{aligned} I_1-I_2-I_3&=0 \quad &\text{(junction $e$)}\\[4pt] 2I_1+8I_2&=8 \quad&\text{(loop abcdefa)}\\[4pt] 2I_1+4I_3&=12 \quad&\text{(loop abefa)}\end{aligned}\]
Subtracting the second equation from the third equation (to eliminate \(I_1\)):
\[\begin{aligned} 4I_3-8I_2&=4\\[4pt] \therefore I_3&=1+2I_2\end{aligned}\]
Substituting this into the junction equation:
\[\begin{aligned} I_1-I_2-I_3&=0\\[4pt] I_1-I_2-1-2I_2&=0\\[4pt] \therefore I_2=\frac{1}{3}(I_1-1)\end{aligned}\]
Finally, substituting this into the equation from loop \(abcdefa\), allows us to determine \(I_1\) and the other two currents:
\[\begin{aligned} 2I_1+8I_2&=8\\[4pt] 2I_1+8\left(\frac{1}{3}(I_1-1) \right)&=8\\[4pt] \therefore I_1&=\frac{16}{7}=2.29\text{A}\\[4pt] \therefore I_2&=\frac{1}{3}(I_1-1)=0.43\text{A}\\[4pt] \
therefore I_3&=1+2I_2=1.86\text{A}\\[4pt]\end{aligned}\]
In this case, the currents are all positive, so the diagram in Figure \(\PageIndex{7}\) is correct and we do not need to reverse the direction of any of the currents.
We can now determine the potential difference across the real terminals of the battery \(\Delta V_1\). The current through the battery is \(I_1=2.29\text{A}\), which cause a voltage drop, \(\Delta V_
{r1}\), across its internal resistance, \(r_1\) of:
\[\begin{aligned} \Delta V_{r1}=I_1r_1=(2.29\text{A})(1\Omega)=2.29\text{V}\end{aligned}\]
The voltage across the real terminals of the battery is then:
\[\begin{aligned} \Delta V_{real}=\Delta V_1-\Delta V_{r1}=(12\text{V})-(2.29\text{V})=9.7\text{V}\end{aligned}\]
The current through the resistor \(R_5\) (Figure \(\PageIndex{6}\)) requires a little more thought, since we calculated the current, \(I_1\) through the effective resistor \(R_8\), which we must now
“break apart”. Figure \(\PageIndex{8}\) shows the components of \(R_8\).
Figure \(\PageIndex{7}\). The current, \(I_{1}\), coming from the battery goes through \(r_{1}\) and then splits up.
The current \(I_1\), that goes through the \(\Delta V_1\) battery also goes through the \(r_1\) internal resistance of the battery. That current then splits up into currents, \(I_4\) and \(I_5\), to
go through the resistors \(R_4\) and \(R_5\). Although it should be obvious that half of \(I_1\) will go through each resistor (since these are equal), we can determine this from applying Kirchhoff’s
rules to the combination of resistors in Figure \(\PageIndex{8}\):
\[\begin{aligned} I_1&=I_4+I_5 \quad&\text{(junction)}\\[4pt] I_5R_5-I_4R_4&=0\quad&\text{(clockwise loop)}\end{aligned}\]
From the loop equation, we have:
\[\begin{aligned} I_5=\frac{R_4}{R_5}I_4=I_4\end{aligned}\]
since \(R_4=R_5=2\Omega\). Since \(I_4=I_5\), the junction equation gives:
\[\begin{aligned} I_5=\frac{1}{2}I_1=1.15\text{A}\end{aligned}\]
By solving for \(I_4\) and \(I_5\), we have now determined all of the currents through all of the segments of the original circuit in Figure \(\PageIndex{6}\).
In this example, we showed how one can use a simplified circuit to solve the current through the effective resistors in the simplified circuit. Once those currents are known, we showed that it is
straightforward to determine the currents through individual resistors that have been combined into effective resistors.
Solving a circuit can be daunting, especially if the diagram is drawn in an unfamiliar way. While the circuits in this chapter are designed to be as easy to read as possible, many circuits are much
more strange. For example, here is a circuit which you may come across:
Figure \(\PageIndex{9}\): A weird looking circuit.
The circuit in Figure \(\PageIndex{9}\) May look like it is a difficult circuit to solve, but the diagram can be re-drawn to reveal the simplicity of the circuit:
Figure \(\PageIndex{10}\): A much less weird looking circuit.
What used to be a strange kite shape is now just a parallel circuit, which can be further simplified by calculating the effective resistance:
\[\begin{aligned} R_{eff} &= (R_1^{-1}+R_2^{-1}+(R_3+R_4)^{-1})^{-1}\end{aligned}\]
Which gives a series circuit with only one resistor:
Figure \(\PageIndex{11}\): A simple circuit.
Circuits can be drawn in many unique or potentially confusing ways, but knowing how to read the circuit and re-draw it can help make the diagram more legible and the circuit easier to solve.
1. The 3 unknowns do not necessarily have to be the currents, and could be any combination of the currents, battery voltage and resistors. As long as there at most 3 unknown quantities, this circuit
can be solved. | {"url":"https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/20%3A_Electric_Circuits/20.03%3A_Applying_Kirchhoffs_rule_to_model_circuits","timestamp":"2024-11-12T13:16:52Z","content_type":"text/html","content_length":"155275","record_id":"<urn:uuid:a2a00526-f45a-43c4-be1f-dbd8df872876>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00801.warc.gz"} |
Implementing the Spaceship Operator for Optional
Comparison operators can get complicated. Barry Revzin explores how the new operator <=> helps.
In November 2017, the C++ Standards Committee added operator<=> , known as the spaceship operator [ P0515 ], to the working draft for what will eventually become C++20. This is an exciting new
language feature for two reasons: it allows you to write one function to do all your comparisons where you used to have to write six , and it also allows you to write zero functions – just declare
the operator as defaulted and the compiler will do all the work for you! Exciting times.
The paper itself presents many examples of how to implement the spaceship operator in various situations, but it left me with an unanswered question about a particular case – so I set out trying to
figure out. This post is about the journey of how to implement operator<=> for optional<T> . First, thanks to John Shaw for helping work through all the issues with me. And second, the resulting
solution may not be correct. After all, I don’t even have a compiler to test it on. So if you think it’s wrong, please let me know (and please post the correct answer in this self-answered
StackOverflow question [ StackOverflow ]).
First, the specs. optional<T> has three categories of comparisons, all conditionally present based on the facilities of the relevant types:
• optional<T> compares to optional<U> , where valid (6 functions).
• optional<T> compares to U , where valid (12 functions). I’m sceptical of this particular use-case, but this post is all about implementing the spec.
• optional<T> compares to nullopt_t (12 functions). This case is trivial to implement, since several of the operations are simply constants (e.g. operator>=(optional<T>, nullopt_t) is true ). But,
that’s still 12 trivial-to-implement functions.
In all cases, the semantics are that a disengaged optional is less than any value, but all disengaged values are equal. The goal is to take advantage of the new facilities that the spaceship operator
provides us and reduce the current load of 30 functions to just 3.
We’ll start with the optional on optional comparison. There are four cases to consider: both on, left on only, right on only, and both off. That leads us to our first approach (Listing 1).
template <typename T>
class optional {
// from here on out, assuming that heading
// exists ...
template <typename U>
constexpr auto operator<=>(
optional<U> const& rhs) const
-> decltype(**this <=> *rhs)
using R = decltype(**this <=> *rhs);
if (has_value() && rhs.has_value()) {
return **this <=> *rhs;
} else if (has_value()) {
return R::greater;
} else if (rhs.has_value()) {
return R::less;
} else {
return R::equal;
Listing 1
The spaceship operator returns one of five different comparison categories:
• strong_ordering
• weak_ordering
• partial_ordering
• strong_equality
• weak_equality
Each of these categories has defined named numeric values. In the paper, the categories are presented in a way that indicates the direction in which they implicitly convert in a really nice way, so
I’m just going to copy that image as Figure 1 (all credit to Herb Sutter).
Likewise, their table of values is shown in Table 1 below.
Category Numeric values Non-numeric values
-1 0 1
strong_ordering less equal greater
weak_ordering less equivalent greater
partial_ordering less equivalent greater unordered
strong_equality less equal non-equal
weak_equality equivalent non-equivalent
Table 1
Just carefully perusing this table, it’s obvious that our first implementation is totally wrong. strong_ordering has numeric values for less, equal, and greater… but the rest don’t! In fact, there is
no single name that is common to all 5. By implementing it the way we did, we’ve reduced ourselves to only supporting strong orderings.
So if we can’t actually name the numeric values, what do we do? How can we possibly do the right thing?
Here, we can take advantage of a really important aspect of the comparison categories: convertibility. Each type is convertible to all of its less strict versions, and each value is convertible to
its less strict equivalents. strong_ordering::greater can become:
• weak_ordering::greater or
• partial_ordering::greater or
• strong_equality::nonequal or
• weak_equality::nonequivalent
And the way we can take advantage of this is to realize that we don’t really have four cases, we have two: both on, and not that. Once we’re in the ‘not’ case, we don’t care about the values anymore,
we only care about the bools. And we already have a way to do a proper 3-way comparison: <=> ! (See Listing 2.)
template <typename U>
constexpr auto operator<=>(optional<U>
const& rhs) const
-> decltype(**this <=> *rhs)
if (has_value() && rhs.has_value()) {
return **this <=> *rhs;
} else {
return has_value() <=> rhs.has_value();
Listing 2
The shapeship operator for bool s gives us a strong_ordering , which is convertible to everything. So that part is guaranteed to work and do the right thing (I encourage you to work through the cases
and verify that this is indeed the case).
But this still isn’t quite right. The problem is actually <=> (thanks, Captain Obvious?). You see, while a < b is allowed to fallback to a <=> b < 0 , the reverse is not true. a <=> b is not allowed
to call anything else (besides b <=> a ). It either works, or it fails. By using the spaceship operator directly on our values, we’re actually reducing ourselves to only those modern types that
support 3-way comparison. Which, so far, is no user-defined types. Moreover, <=> doesn’t support mixed-integer comparisons, so even for those types that come with built-in spaceship support (that’s a
fantastic phrase), we would effectively disallow comparing an optional<int> to an optional<long> . So, this operator in this particular context isn’t very useful.
So what are we to do? Re-implement 3-way comparison ourselves manually? Nope, that’s what the library is for! Along with language support for the spaceship operator, C++20 will also come with several
handy library functions and the relevant one for us is std::compare_3way() . This one will do the fallback: it prefers <=> , but if not will try the normal operators and is smart enough to know
whether to return strong_ordering or strong_equality . And it’s SFINAE-friendly. Which means for our purposes, we can just drop-in replace our too-constrained version with it (see Listing 3).
template <typename U>
constexpr auto operator<=>(optional<U>
const& rhs) const
-> decltype(compare_3way(**this, *rhs))
if (has_value() && rhs.has_value()) {
return compare_3way(**this, *rhs);
} else {
return has_value() <=> rhs.has_value();
Listing 3
And I think we’re done.
Now that we’ve figured out how to do the optional-vs-optional comparison, comparing against a value is straightforward. We follow the same pattern for the value-comparison case, we just need to know
what to return in the case where the optional is disengaged. Semantically, we need to indicate that the optional is less than the value. Again, we can just take advantage that all the comparison
category conversions just Do The Right Thing and use strong_ordering::less (see Listing 4).
template <typename U>
constexpr auto operator<=>(U const& rhs) const
-> decltype(compare_3way(**this, rhs))
if (has_value()) {
return compare_3way(**this, rhs);
} else {
return strong_ordering::less;
Listing 4
We just replaced 12 functions (that, while simple, are certainly non-trivial to get right) with 10 lines of code . Mic drop.
All that’s left is the nullopt_t comparison, which is just a simple comparison (Listing 5).
constexpr strong_ordering operator<=>(nullopt_t ) const
return has_value() ? strong_ordering::greater
: strong_ordering::equal;
Listing 5
Putting it all together, and Listing 6 is what we end up with to cover all 30 std::optional<T> comparisons.
template <typename T>
class optional {
// ...
template <typename U>
constexpr auto operator<=>(optional<U>
const& rhs) const
-> decltype(compare_3way(**this, *rhs))
if (has_value() && rhs.has_value()) {
return compare_3way(**this, *rhs);
} else {
return has_value() <=> rhs.has_value();
template <typename U>
constexpr auto operator<=>(U const& rhs) const
-> decltype(compare_3way(**this, rhs))
if (has_value()) {
return compare_3way(**this, rhs);
} else {
return strong_ordering::less;
constexpr strong_ordering
operator<=>(nullopt_t ) const
return has_value() ? strong_ordering::greater
: strong_ordering::equal;
Listing 6
Not bad for 25 lines of code?
Let me just reiterate that I’m not sure if this is the right way to implement these operators. But that’s the answer [ StackOverflow ] I’m sticking with until somebody tells me I’m wrong (which, if I
am, please do! We’re all here to learn).
Needless to say, I’m very much looking forward to throwing out all my other comparison operators. Just… gotta wait a few more years.
Bonus level
Here’s what I think a comparison operator would look like for std::expected<T, E> . The semantics here are that the values and errors compare against each other, if they’re the same. If they’re
different types, the value is considered greater than the error. Although, for the purposes of this exercise, the specific semantics are less important than the fact that we get consistent semantics.
And I think the right way to implement consistent semantics is as shown in Listing 7.
template <typename T, typename E>
class expected {
// ...
template <typename T2, typename E2>
constexpr auto operator<=>(expected<T2, E2>
const& rhs) const
-> common_comparison_category_t<
if (auto cmp = has_value() <=>
rhs.has_value(); cmp != 0) {
return cmp;
if (has_value()) {
return compare_3way(value(), rhs.value());
} else {
return compare_3way(error(), rhs.error());
Listing 7
common_comparison_category is a library metafunction that gives you the lowest common denominator between multiple comparison categories (which hopefully is SFINAE-friendly, but I’m not sure). The
first if check handles the case where the value-ness differs between the two expected objects. Once we get that out of the way, we know we’re in a situation where either both are values (so, compare
the values) or both are errors (so, compare the errors). Just thinking of how much code you have to write today to accomplish the same thing makes me sweat…
[P0515] ‘Consistent comparison’ (2017) http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2017/p0515r3.pdf
[StackOverflow] ‘Implementing operator<=> for optional<T>’ https://stackoverflow.com/questions/47315539/implementing-operator-for-optionalt
I’m a C++ developer for Jump Trading, member of the C++ Standards Committee, also ‘Barry’ on StackOverflow. On the side, I’m also an avid swimmer and do data analysis for SwimSwam magazine. And take
care of my adorable Westie and life mascot, Hannah. | {"url":"https://www.accu.org/journals/overload/26/147/revzin_2563/","timestamp":"2024-11-12T07:10:29Z","content_type":"text/html","content_length":"43583","record_id":"<urn:uuid:70e82db6-20ea-4d90-9c28-fc7c4448c202>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00298.warc.gz"} |
WolfQuest: Anniversary Edition - All Den Locations
WolfQuest: Anniversary Edition – All Den Locations
Where to Find All Dens
Slough Creek Map
The minimum of how many you need to find is four, but there’s still more dens to find. I hope this helps you in finishing that quest and finding the perfect home!
Slough Creek Dens
Den One
Open, Rock
Den Two
Wooded, Burrow
Den Three
Open, Tree
Den Four
Open, Rock
Den Five
Open, Burrow
Den Six
Wooded, Burrow
Den Seven
Open, Tree
Den Eight
Open, Burrow
Den Nine
Wooded, Burrow
Den Ten
Open, Tree
Den Eleven
Open, Tree
Den Twelve
Wooded, Burrow
Den Thirteen
Open, Rock
Den Fourteen
Open, Burrow
Den Fifteen
Open, Burrow
Den Sixteen
Open, Burrow
Den Seventeen
Open, Rock
Den Eighteen
Wooded, Tree
Den Nineteen
Wooded, Burrow
Den Twenty
Wooded, Rock
Den Twenty-One
Open, Rock
Den Twenty-Two
Open, Rock
Den Twenty-Three
Wooded, Burrow
Den Twenty-Four
Open, Tree
Den Twenty-Five
Open, Burrow
Den Twenty-Six
Open, Burrow
Den Twenty-Seven
Open, Rock
Den Twenty-Eight
Open, Burrow
Den Twenty-Nine
Open, Burrow
Den Thirty
Open, Rock
Lost River Map
The minimum of how many you need to find is four, but there’s still more dens to find. I hope this helps you in finishing that quest and finding the perfect home!
Lost River Dens
Den One
Open, Rock
Den Two
Wooded, Tree
Den Three
Open, Rock
Den Four
Wooded, Burrow
Den Five
Wooded, Burrow
Den Six
Open, Burrow
Den Seven
Wooded, Burrow
Den Eight
Open, Burrow
Den Nine
Wooded, Burrow
Den Ten
Wooded, Rock
Den Eleven
Open, Burrow
Den Twelve
Open, Burrow
Den Thirteen
Wooded, Tree
Den Fourteen
Wooded, Rock
Den Fifteen
Open, Burrow
Den Sixteen
Open, Tree
Den Seventeen
Wooded, Tree
Den Eighteen
Wooded, Tree
Den Nineteen
Wooded, Tree
Den Twenty
Open, Burrow
Den Twenty-One
Wooded, Burrow
Den Twenty-Two
Open, Burrow
Den Twenty-Three
Open, Rock
Den Twenty-Four
Open, Rock
Den Twenty-Five
Open, Rock
Den Twenty-Six
Wooded, Tree
Den Twenty-Seven
Wooded, Burrow
Den Twenty-Eight
Wooded, Burrow
Den Twenty-Nine
Wooded, Tree
Den Thirty
Open, Burrow
Den Thirty-One
Open, Rock
10 Comments
1. Lol, I guess I managed to pick den 25, the furthest away, for my first time.
2. youre missing at least 1 den in the lower left area of this pic but still great ref!
3. thanks for this i been hunting for dens and i
never got pups but i might get pups now 😀
4. I do not know the exact den location but where i am it is telling me there is a den nearby, and it should be south of the river on the FAR left side of the map! Im going to try to find it now!
5. There’s actually one missing its south of 1st one next to the river bottom west side of the map. Stumbled upon it by accident there might be other dens around the den locations above.
6. the maps for spring and winter maps are at the WolfQuest Wiki/ you can search Wolfquest wiki and itll pop up, it has loads of stuff on it.
7. den 17 is a package deal it comes with 3 anger wolves!
8. Found a den! North of 25, 21 and 22
9. I’ve found another den, its straight above the second meadow. I’ll add a pic down below! you can use the pic if you want idc, and if I find more I’ll post em
□ Thanks! | {"url":"https://gameplay.tips/guides/9320-wolfquest-anniversary-edition.html","timestamp":"2024-11-15T04:05:33Z","content_type":"text/html","content_length":"124460","record_id":"<urn:uuid:f12134cc-2d61-4a44-bed9-23ec958d62f6>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00645.warc.gz"} |
Parallel communicating one-way reversible finite automata system. - Citegraph
In this paper, we discuss the computational power of parallel communicating finite automata system with 1-way reversible finite automaton as components. We show that unlike the multi-head one way
reversible finite automata model (where we are still not sure whether it accepts all the regular languages) parallel communicating one-way reversible finite automata systems can accept all the
regular languages. Moreover for every multi-head one way reversible finite automaton there exist a parallel communicating one-way reversible finite automata system which accepts the same language. We
also make an interesting observation that although the components of the system are reversible the system as a whole is not reversible. On the basis of which we conjecture that parallel communicating
one-way reversible finite automata systems may accept languages not accepted by multi-head one way reversible finite automata. | {"url":"https://www.citegraph.io/paper/5cede0edda562983788cc256","timestamp":"2024-11-03T22:03:58Z","content_type":"text/html","content_length":"30511","record_id":"<urn:uuid:33af58b0-eba9-45a5-8faa-b2a85ffe03d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00169.warc.gz"} |
Student Score Anna 98 Ben 95 Carrie 92 David 82 Eddie 75 Fiona 95 Greg 92 Holly 30 Isabella 87 Jack 84 Kayla 94 Lance 73 Madison 92 Natalie 77 Olivia...
Answered You can hire a professional tutor to get the answer.
Student Score Anna 98 Ben 95 Carrie 92 David 82 Eddie 75 Fiona 95 Greg 92 Holly 30 Isabella 87 Jack 84 Kayla 94 Lance 73 Madison 92 Natalie 77 Olivia...
One of these test scores is an outlier. Whose score is it? Show calculations to demonstrate that the score is an outlier. ________________
First, answer all of these questions for the entire data set WITH THE OUTLIER INCLUDED:
What is the mean of the data set? _________
What is the median of the data set? __________
Draw a box plot of the data set:
Now REMOVE THE OUTLIER FROM THE DATA SET and answer the questions:
What is the mean of the data set? _________
What is the median of the data set? __________
Draw a box plot of the data set:
How did removing the outlier affect the median? How did it affect the mean? Which measure of center was most affected by the outlier? Which measure of center do you feel gives a more accurate
representation of the "typical" student's performance?
Show more
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Ask a Question | {"url":"https://studydaddy.com/question/student-score-anna-98-ben-95-carrie-92-david-82-eddie-75-fiona-95-greg-92-holly","timestamp":"2024-11-03T18:55:21Z","content_type":"text/html","content_length":"26912","record_id":"<urn:uuid:c389271b-ffc9-4856-ac05-1e594ac5da3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00190.warc.gz"} |