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Chevalley-Eilenberg algebra
I finally wrote out the full proof of the Quillen adjunction at Chevalley-Eilenberg algebra (schreiber)
I finally added to Chevalley-Eilenberg algebra the definition for Lie algebroids and the example of the tangent Lie algebroid. Also finally filled in at Lie algebroid the precise signs in the formula
for the differential.
All that in order to be able to point to these entries in reply to this MO question
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Reporting Category 3 (TEKS 3.6 Two-Dimensional Figures)
Geometry and Measurement
The videos in this post explain mathematics Texas Essential Knowledge and Skill 3.6:
Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional geometric figures to develop generalizations about their properties.
The STAAR videos below are selected released items from 3rd grade STAAR tests in 2016, 2018, 2019, and 2021. You can view the complete review of the 2017 STAAR test here.
classify and sort two- and three-dimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language
Teaching videoS
STAAR Videos
use attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these
Teaching video
STAAR Video
determine the area of rectangles with whole number side lengths in problems using multiplication related to the number of rows times the number of unit squares in each row
Teaching Video
STAAR Videos
decompose composite figures formed by rectangles into non-overlapping rectangles to determine the area of the original figure using the additive property of area
Teaching video
STAAR Video
decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole and recognize that equal shares of identical wholes
need not have the same shape
Teaching video
STAAR Videos
To view all the posts for the 3rd Grade Reporting Category 3 review, click here.
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Applications of the Counting Principle (Product Rule)
Question Video: Applications of the Counting Principle (Product Rule) Mathematics • Second Year of Secondary School
Use the fundamental counting principle to determine the total number of outcomes upon choosing from 8 ice cream flavors; small, medium, or large cones; and either caramel or chocolate sauce.
Video Transcript
Use the fundamental counting principle to determine the total number of outcomes upon choosing from eight ice cream flavors; small, medium or large cones; and either caramel or chocolate sauce.
Lets begin by recalling what we mean by the fundamental counting principle. The fundamental counting principle, sometimes called the product rule for counting, tells us that the total number of
outcomes for two or more events is found by multiplying the total number of outcomes for each event together. So what are the events were interested in? Well, event one is choosing an ice cream
flavor. There are eight ice cream flavors, and so there are eight possible outcomes for event one.
Then we have event two, and thats the type of cone that we choose. Those are small, medium, or large, and so there are three possible outcomes for the type of cone we choose. Finally, we move onto
event three. This is the flavor sauce that we end up choosing. We can choose between caramel and chocolate sauce, and so there are two possible outcomes for event three. The total number of outcomes,
in other words, the total number of combinations of ice cream we can choose, is found by multiplying each of these values together. Thats eight times three times two.
And of course, since multiplication is commutative, we can do this in any order. And we can first begin by finding the product of three and two; thats six. So were actually calculating eight times
six, which is, of course, 48. And so we see there are a total number of 48 outcomes of the type of ice cream cone we can choose.
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Splitting a PolyCurve by Corner
Hi All,
I am working on a graph that will be used to optimize mullion lengths in a curtain wall. The mullions are manufactured in 3 meter pieces, and I want to calculate how many to order. I have a curtain
wall in revit, and I’m trying to get the length of each straight section. Because my curtain wall has vertical divisions, even straight sections are divided into several different curves.
What I have so far is a graph that gathers all relevant mullions, gets their curve geometry and joins them into a polycurve.
I’m looking for a way to “split” the curves at the corner, so I am left with only straight curves. I’ll attach a capture of my optimization graph just so you could have a look.
Thanks a lot for any help in advance!
Could you show watch node at GroupCurves node.
This doesn’t quite do what I’m looking for. I’m afraid it only works on interior mullions (I need both interior and border ones). Also, it contains no data about the mullions themselves, meaning I
cant use it for example to see how many pieces of “type x” to order.
Do you know of any splitting methods that take into account normals? Or maybe even a node that does that?
I have a thought on how to go about this with the added concerns you have. Hopefully you have some experience with lacing, levels, and nested lists as it will be a requirement to keep the number of
tests down.
1. group the millions by the normalized vector of their curves.
2. for each group build a poly line by pulling the curves and proceeding as above.
3. get the million at the start of each group, then the second, third, fourth… this can be done by comparing the distance from start point of the millions curve to the start point of the poly line
point - of its 0 then its first, 1 is second, 2 is third… basically you’ll be sorting by that distance value as a key.
4. Next you need to confirm that they are all the same type of million - get the family type name and convert to a string or element ID. List.Unique to filter out which values you need to search
5. Find all indicies of each value, and pull the millions for each.
6. Get the length of each million in the sub-sublists and total the distance. These are the million lengths for your bin packing into the 3m strips.
Not an “easy” graph by any means, but more for an intermediate to advanced skill set. I can see how it would have tons of value for a CW manufacturer or subcontractor. Could take a good bit to run as
well with the multiple geometry conversions.
Also, don’t the either vertical or horizontal millions usually ‘rule’ so only they run continuous? If so you could filter out the horizontal or vertical millions early on by the vector as that will
save significant computation time.
Hmm, probably need for fine tuning, but maybe this could suffice?
Using a polycurve node and then exploding if the curves aren’t built in a CW/CCW fashion.
Thanks for your replies!
Jacob: I am trying to do what you suggested, starting from trying to group the mullions by vector… I’m trying to use GroupByKey, but have not accomplished this yet.
Jostein: First of all, I really like your solution! I tried it out and it worked well, but a few real-world examples showed that it won’t work quite as easily as I thought. For starters, branching (a
T intersection between walls) is not supported, and the polycurve fails when it occurs. Also, straight pieces of CW aren’t being calculated correctly. I’ve been trying to find solutions, but have so
far hit a dead end.
Thanks again for your help, if you have any more ideas I would love to hear them!
Just a few minutes after I posted my reply, I happened upon a node called GroupByHost… It totally simplified everything and I believe I have a solution.
What do you think?
Is there any place I can post a short write-up and my dynamo graph for other people to use?
Feel free to post it here, marking it as a solution. You can also create a new topic under the category “share”.
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relative monad
Categorical algebra
2-Category theory
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A relative monad $T \colon A \to E$ is much like a monad except that its underlying functor is not required to be an endofunctor: rather, it can be an arbitrary functor between categories. To even
formulate such a notion, (for instance the definition of the unit), the two categories have to be related somehow, typically via a specified comparison functor $J \colon A \to E$, in which case we
say that $T$ is a monad relative to $J$.
Ordinary monads are the special case of $J$-relative monads where $J$ is the identity functor.
In generalisation of the relation between adjunctions and monads, relative monads are related to relative adjunctions. Dually, relative comonads are related to relative coadjunctions.
Let $J \colon A \to E$ be a functor between categories, the root.
In extension form
The following definition is a variation of the formulation of a monad in extension form:
A $J$-relative monad $T$ [ACU15, Def. 2.1] comprises:
• a function $T \colon |A| \to |E|$, the underlying functor;
• for each object $X \in |A|$, a morphism $\eta_X \colon J X \to T X$ in $E$, the unit;
• for each morphism $f \colon J X \to T Y$ in $A$, a morphism $f^\dagger \colon T X \to T Y$ in $E$, the extension operator,
such that, for each $X, Y, Z \in |A|$, the following equations hold:
• $f = f^\dagger \circ \eta_X$ for each $f \colon J X \to T Y$ (left unitality);
• $(\eta_X)^\dagger = 1_{T X}$ (right unitality);
• $(g^\dagger \circ f)^\dagger = \g^\dagger \circ f^\dagger$ for each $f \colon J X \to T Y$ and $g \colon J Y \to T Z$.
It follows that $T$ is canonically equipped with the structure of a functor. For each $f \colon X \to Y$ in $A$:
$T f \;:=\; \big( \eta_{Y} \circ J f \big)^\dagger \,.$
and that the unit $\eta$ and the extension operator $({-})^\dagger$ are then natural transformations.
In particular, in the special case that $J$ is the identity functor, Def. reduces to the definition of monad in extension form.
As monoids in a skew-monoidal category, skew-multicategory, or multicategory
Monads are, by definition, monoids in monoidal categories of endofunctors. It is similarly possible to present relative monads as monoids in categories of functors. However, generally speaking,
arbitrary functor categories are not monoidal. However, given a fixed functor $J \colon A \to E$, the functor category $[A, E]$ may frequently be equipped with skew-monoidal structure. The notion of
a skew-monoidal category is like that of a monoidal category except that the unitors and associators are not necessarily invertible. Monoids may be defined in a skew-monoidal category analogously as
to in a monoidal category, and a monoid in $[A, E]$ (equipped with the skew-monoidal structure induced by $J$) is precisely a $J$-relative monad.
(ACU15, Thm. 3.4)
Let $J \colon A \to E$ be a functor for which $\mathrm{Lan}_J \colon [A, E] \to [E, E]$ exists (e.g. if $A$ is small and $E$ cocomplete). Then $[A, E]$ admtis a skew-monoidal structure, with unit $J$
and tensor $F \circ^J G = (\mathrm{Lan}_J F) \circ G$, and a relative monad is precisely a monoid in $([A, E], J, \circ^J)$.
When $J \colon A \to E$ is a free completion of $A$ under a class $\mathcal{F}$ of small colimits, then this skew-monoidal structure on $[A, E]$ is properly monoidal, since it is equivalent to the $\
mathcal{F}$-colimit preserving functors $E \to E$, and the monoidal structure is just functor composition.
More generally, if $\mathrm{Lan}_J$ does not exist, we may still define a skew-multicategory? structure on $[A, E]$. Thus, relative monads are always monoids.
(AM, Thm. 4.16)
Let $J \colon A \to E$ be a functor. Then $[A, E]$ admits a unital skew-multicategory structure, and a relative monad is precisely a monoid therein.
This skew-multicategory structure is representable just when $\mathrm{Lan}_J$ exists, recovering the result of ACU15. When $J$ is a dense functor, the above theorem simplifies.
(AM, Cor. 4.17)
Let $J \colon A \to E$ be a dense functor. Then $[A, E]$ admits a unital multicategory structure, and a relative monad is precisely a monoid therein.
As monads in the bicategory of distributors
An alternative useful perspective on relative monads is the following.
(AM, Thm. 4.22)
Let $J \colon A \to E$ be a dense functor. A $J$-relative monad is precisely a monad in the bicategory of distributors whose underlying 1-cell is of the form $E(J, T)$ for some functor $T \colon A \
to E$.
Relative to a distributor
The above definition makes sense even more generally when $J$ is a distributor $E ⇸ A$, i.e. a functor $\mathbf A^{op} \times E \to Set$. Explicitly, we ask for:
• a functor $T \colon A \to E$;
• a unit $\eta_X \in J(X, T X)$ for each $X \in |A|$, natural in $X \in |A|$ (equivalently, an element of the end, $\eta \in \int_{X \in |A|}J(X, T X)$);
• an extension operator $(-)^\dagger \colon J(X, T Y) \to \mathbf C(T X, T Y)$ natural in $X,Y \in |A|$
with essentially the same equations. We recover the previous definition by taking the corepresentable distributor $E(J-,=)$. See Remark 4.24 of AM24.
Generic examples
This example is stated in ACU15, Prop. 2.3 (1). More generally, we can precompose any relative monad with a functor to obtain a new relative monad: see Proposition 5.36 of AM24.
The required conditions on the relative monad structure $T \circ J$ immediately reduce to those of the monad structure of $T$:
$k \,\colon\, J(X) \to T \circ J(X') ,\;\;\; \ell \,\colon\, J(X') \to T \circ J(X'')$
we have
left unitality:
$\begin{array}{l} bind^{T J}(k) \circ \eta^{T J}_{X} \\ \;\equiv\; bind^T(k) \circ \eta^T_{J(X)} \\ \;=\; k \end{array}$
right unitality:
$\begin{array}{l} bind^{T J}\big( \eta^{T J}_{X} \big) \\ \;\equiv\; bind^T\big( \eta^T_{J (X)} \big) \\ \;=\; id_{T \circ J(X)} \end{array}$
$\begin{array}{l} bind^{T J}\big( bind^{T J}(\ell) \circ k \big) \\ \;\equiv\; bind^{T}\big( bind^{T}(\ell) \circ k \big) \\ \;=\; bind^T(\ell) \circ bind^T(k) \\ \;\equiv\; bind^{T J}(\ell) \circ
bind^{T J}(k) \end{array}$
A concrete instance of Exp. is spelled out in Exp. below.
Specific examples
Related pages
The concept was introduced, in the context of monads in computer science, in:
A comprehension development in the context of formal category theory may be found in:
On distributive laws for relative monads:
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Tools for writing regular expressions
Suggested Videos Part 45 - JavaScript mouse events Part 46 - JavaScript popup window Part 47 - Using regular expressions in JavaScript In this video we will discuss the basics of regular expressions
and then look at some of the tools available to learn, write and test regular expressions. Basics of Regular Expressions : Find the word expression in a given string. This will also match with the
word expressions. expression To find the word "expression" as a whole word, include \b on either sides of the word expression \bexpression\b \d indicates to find a digit. To find a 5 digit number we
could use the following \b\d\d\d\d\d\b We can avoid the repetition of \d by using curly braces as shown below. \d{5} means repeat \d 5 times. \b\d{5}\b The above example can also be rewritten as
shown below. \b[0-9]{5}\b Find all the words with exactly 5 letters \b[a-zA-Z]{5}\b Brackets are used to find a range of characters [a-z] - Find any of the characters between the brackets [0-9] -
Find any of the digits between the brackets. This is equivalent to \d (a|b) - Find any of the characters a or b The page at the following link explains the basics of regular expressions. https://
developer.mozilla.org/en/docs/Web/JavaScript/Guide/Regular_Expressions Expresso is one of the free tools available. Here is the link to download. http://www.ultrapico.com/ExpressoDownload.htm Regular
Expression Library http://regexlib.com
1 comment:
1. OptionGroup
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Musing from the Barriers Workshop
Rahul Santhanam guest posts with thoughts from last week's Barriers in Computational Complexity workshop in Princeton.
1. What's a complexity theorist's definition of an algorithm? A failed attempt to prove a lower bound. One of the more famous examples of this is Arora's approximation algorithm for Euclidean TSP. At
the workshop, I heard about a couple more. Chris Umans mentioned that the new algorithms for matrix multiplication in his paper with Henry Cohn came about in part because of a failed attempt to prove
their approach was unviable. While chatting with Dave Barrington about his counter-intuitive
of NC
by bounded-width branching programs, I learned that this too resulted from an "obstruction" to proving a lower bound (By the way, Dave's result was voted the most surprising result in complexity
theory by an eminent panel on the first day). It struck me that this ability to toggle freely between "hard" (reductions from complete problems, oracle & black-box results) and "easy" (algorithms)
viewpoints is a very productive component of our way of thinking. Of course this is formalized in results such as Razborov-Rudich and Kabanets-Impagliazzo, but it's also an indication that Nature is
far from adversarial. It could have been that interesting problems are not just hard but also that we could say nothing interesting about their hardness... Instead, we have an extensive theory, a
rich web of implications. Sure, lower bound proofs are difficult, but then there are the chance flowerings of failed attempts.
2. The highlight of the workshop for me was talking with Ketan Mulmuley about his approach to P vs NP using geometric invariant theory. Ketan has been working on this approach for more than a decade
now. Given his seminal work on semantics of programming languages, parallel algorithms and computational geometry, and his reputation as a brilliant problem solver, he was always going to be taken
seriously. But there was also an element of mystification - was all this high-powered mathematics really necessary to attack P vs NP? Why spend time learning about this approach when it might not pay
any dividends for the next half a century? And how could Ketan claim that this was in a sense the only possible approach to P vs NP? The fact that Ketan wasn't comfortable evangelizing about his
approach didn't help matters.
It seems to me that now there's a qualitative shift. There seem to be two factors in the shift - one is that Ketan is more confident in the viability of his program than he was in the formative
stages, and the second is that he is more aware now of the importance of communicating both with mathematicians and with complexity theorists about his approach. Indeed, the scale of the project is
so immense that collaborations across the board are necessary to its success.
To his credit, Ketan has realized this. Earlier, when the possibility was suggested to him that his approach might lead to weaker but still interesting lower bounds along the way, or to connections
with other problems, he didn't seem very interested, possibly because he was focused on the ultimate goal of separating P and NP. Now, he is more actively encouraging of the search for such
connections. Also, he doesn't claim that his approach is the only one for separating NP vs P - such a claim is hardly tenable. Instead, he makes a much more nuanced argument in terms of the barriers
that other approaches run up against and which his own program avoids, as well as for the mathematical naturalness and aesthetic value of his approach. As for the time scale of the project, it's true
that carrying it out in its present form would involve solving mathematical conjectures that algebraic geometers consider far beyond the reach of present techniques. But there is always the
possibility of short cuts arising from unexpected connections and new ideas. For this reason, estimates of time scales are speculative, and perhaps not all that relevant.
The P vs NP question is the most important in our area, and as of now there seems to be exactly one program (in the mathematical sense) for solving it, and that's Ketan's program. Simply for that
reason, it deserves serious study. At the very least, the decade of mathematical labour that has gone into developing the approach, together with the current efforts to explicate the approach and its
relation to barriers complexity-theoretic and mathematical, have raised the standards for any future approach to be taken seriously.
The best sources for learning about the approach are his
Complexity Theoretic Overview of GCT
Mathematical Overview of GCT
3. A few people at the workshop questioned the focus on barriers. Ran Raz gave a great talk in which the "barrier" slide merely had a picture of the human brain (but then, isn't it even more unlikely
that a computer could prove P != NP?). Why are we so obsessed with barriers? Perhaps it's because we are computer scientists rather than mathematicians. Mathematicians don't care about constructivity
- they believe that proofs exist, however long it takes to find them. It was a question of when Fermat's last theorem would be proved, not whether. We, however are used to doing things efficiently
(at least in theory). So if we fail to find a proof quickly, the fault surely "lies not in ourselves but in our stars". Walls tend to spring up around us just as we're on the verge of something
extraordinary. Oracles give us gloomy portents. We're forced to be unnatural. ZFC shrugs and turns away from the question...
Yet there is something uncanny about the whole thing. Lower bound questions can be formulated (and have been formulated) in many different ways mathematically - there hasn't been any real progress
with any of these formulations. Just as an algorithm for SAT would also solve every other NP-complete problem, a lower bound proof would say something about all these formulations at once, which
seems odd since they pertain to apparently unrelated areas of mathematics.
Just another barrier with which to amuse ourselves.
4. The very last talk of the workshop was given by Luca Trevisan, on the advantages of being a polyglot. Historically, complexity theory is rooted in logic and combinatorics. As it matures as a
discipline, theorists are able to ply their skills in other mathematical provinces. Pseudorandomness is an especially "extroverted" part of complexity theory. Theorists have made important
contributions to problems such as explicit constructions of Ramsey graphs (Barak-Rao-Shaltiel-Wigderson) and the Kakeya conjecture (Dvir) using the language and techniques of pseudorandomness.
Luca's talk was about connections between pseudorandomness and additive number theory, with an emphasis on the result by Green and Tao that the primes contain arbitrarily long arithmetic
progressions. He made the point that the techniques that go towards proving this result can be phrased in a couple of different languages, the language of functional analysis (functions, norms) and
the language of pseudorandomness (distributions, distinguishability, statistical tests). It's useful to construct a "dictionary" between these two languages, since concepts that are transparent when
phrased in one language become less so when translated into the other. For example, the functional analysis viewpoint implies that distributions and adversaries are the same kind of object, which
seems strange from the other viewpoint. Not only is this dictionary useful in learning the new language, but also because it exposes new concepts that our native language is not well equipped to
handle. Indeed, there have already been many fruitful applications of the Gowers uniformity concept to theoretical computer science, including the Samorodnitsky-Trevisan work on low-error PCPs, the
work by Bogdanov, Viola and Lovett on PRGs for low-degree polynomials, and the recent beautiful work by Kolaitis and Kopparty on modular convergence laws for first-order logic with the Mod p
operator. It seems likely that there are many fruitful connections still unexplored. Luca's survey in the SIGACT News complexity column (
) is well worth checking out.
5. There were also several cool talks at the workshop where I learned about new results. Ben Rossman talked about average-case monotone lower bounds for Clique under natural distributions - this is a
problem that has been open for a while. He shows that there are two values of the edge probability for Erdos-Renyi graphs, p
and p
, such that no monotone circuit of size less than n
can solve k-Clique well on average on both of the corresponding graph distributions. This result complements Ben's other recent result showing that k-Clique does not have constant-depth circuits of
size less than n
, and uses some of the same techniques, inspired by intuitions from finite model theory. Toni Pitassi spoke about work with Paul Beame and Trinh Huynh on "lifting" proof complexity lower bounds from
rank lower bounds for Resolution to lower bounds for stronger systems such as Cutting Planes and Lovasz-Schrijver. This builds on the "pattern matrix" method of Sherstov, which Toni discovered was
also implicit in the Raz-McKenzie separation of the monotone NC hierarchy from more than a decade back (see correction
). Of course it would be very interesting to "lift" circuit lower bounds in this fashion, but few results of that kind are known. Adam Kalai talked about work with Shang-Hua Teng on learning decision
trees and DNFs under smoothed distributions - product distributions where every co-ordinate probability is chosen uniformly in random from some small range. These learning algorithms do not use
membership queries - corresponding results for the uniform distribution would solve longstanding open problems. Adam made the point that his result can be thought of as modelling Nature in a way that
is not fully adversarial. At least for "most" reasonable distributions, we can in fact learn efficiently in these cases.
6. The workshop was a great success, in part because it brought together more complexity theorists than any current conference does. It was also very smoothly organized. Thanks to Avi, Russell, the
session organizers, the admin staff and the student volunteers for making it such a valuable experience.
Correction from Sherstov (10/5/09):
What Toni meant is that the two works study related communication problems (just like there are many papers on the disjointness problem); the two differ fundamentally as to the techniques used and
results achieved. This point is clarified in the revised version of Toni's
paper on ECCC
(page 3, line -15).
10 comments:
1. What's a complexity theorist's definition of an algorithm? A failed attempt to prove a lower bound.
Thank you very much for this excellent post, and especially for this amusing aphorism ... the aphorism in itself would have made the meeting well worth attending!
2. I'm obviously not a complexity theorist; all my lower bounds are failed attempts at algorithms.
3. Not sure what you meant by your third point.
Mathematicians don't care about constructivity - they believe that proofs exist...It was a question of when Fermat's last theorem would be proved, not whether.
I don't think that's true. Certainly mathematicians (before Wiles' proof) didn't think Fermat's theorem was necessarily true any more or less strongly than we are convinced that P \neq NP.
4. It is Kalai, Samorodnitsky and Teng I think.
5. I do not agree that the best place for learning about Mulmuley & Sohoni's approach to GCT is in any of their papers on the subject. There is a very nice preprint available at http://
www.math.tamu.edu/~jml/BLMW0716.pdf that is basically the result of mathematicians trying to sort out the program from a mathematical perspective. AFAIK, it is the first serious attempt made by
experts in geometry and representation theory at understanding the program.
6. ...result of mathematicians...
Peter Burgisser is a computer scientist.
7. I fully agree that the Landsberg et al. write-up is in much more comprehensible
and than the original GCT papers -- mainly because it is written in a more standard style. The original GCT papers are burdened with excessive explanations at points (of standard concepts and
definitions) which made the reading very uneven.
The quibble about who is and isn't a computer scientist is ridiculous. The fact of the matter is that the GCT approach relies very little on the previous developments in TCS, and hopes to achieve
its goals through breakthroughs in representation theory. Thus, it is really very much a mathematical project and I really don't see much use of any existing ideas in CS that can hope to
contribute in this effort.
8. Anonymous says: I really don't see much use of any existing ideas in CS that can hope to contribute in [the GCT] effort.
The converse is definitely not true ... the geometric framework of representation theory provides a natural framework for its engineering cousin, model order reduction theory, and more broadly,
simulation theory.
I used to think that these fundamental algebraic/geometric insights were relevant mainly to quantum simulation, however this summer I learned at the FOMMS Conference that the biologists use quite
a lot of symplectic geometry even at the classical level---that was algebraic geometry and information theory "busting out" at pretty much every conference I went too.
The reason for this emerging unity is that everyone wants all the "Goodness" they can get -- the classical Goodness of thermodynamics; the quantum Goodness of smallest size, fastest speed, and
maximal energy efficiency; the informatic Goodness of maximal channel capacity, and the algorithmic Goodness of theories and simulations that (nowadays) bind together both the technologies and
Mulmuley's GCT program points to a world in which CS/CT is a broader subject than we thought it was --- and doesn't this same principle apply across the broad, to all branches of science and
On the other hand, pure mathematics is what we thought it was ... the logical foundation for all these many enterprises ...
And that's why nowadays even engineers and medical researchers are reading "Yellow Books" ...
Greatest ... Mathematical .... Summer ... Ever! :)
9. Will scribe notes or videos from the workshop be made publicly available?
10. Aaron Sterling5:36 PM, September 02, 2009
Will scribe notes or videos from the workshop be made publicly available?
Yes. In the meantime, though, there is already video up of talks Mulmuley gave at IAS in Feb 09. The first video on that page is similar in content to Mulmuley's talk at Barriers, though I found
the Barriers talk more accessible.
Video page link here:
I'd like to disagree (slightly) with Rahul S.'s superb post: my favorite two talks were the ones by Mulmuley and by Russell Impagliazzo, because they seemed to be the only two researchers
presenting plans to attack P/NP. Impagliazzo exposited "An Axiomatic Approach to Algebrization," and, in particular, pointed to the proof that MIP=NEXP -- and moreover to local checkability as a
general proof technique -- as something we already know how to do that neither relativizes nor algebrizes. (This "contradicts" the Aaronson/Wigderson algebrization paper; there's more discussion
of this on Shtetl Optimized.) So we haven't shown that all our techniques are dead ends, after all. Impagliazzo argued for a research program to obtain further nonrelativizing results, using,
e.g., local checkability, and then to study the behavior of complexity classes under "earth-like oracles" -- oracles under which everything we know to be unconditionally true actually holds.
Like Rahul, I'd also like to thank the organizers, volunteers and staff. The event ran smoothly, and I certainly learned a lot.
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Reversing Writeups - BSides Canberra CTF 2023
During the school holidays, I had the opportunity to attend BSides Canberra 2023, which was a 3 day conference held near the end of September, in the National Convention Center Canberra.
Along with many great talks, it featured three ‘villages’ including hardware, lockpicking and wireless. With 3000 people attending, this was the largest hacking conference that I had gone to, and it
was a lot of fun!
Cybears hosted this year’s CTF event with unique comic style graphics for the CTF site. They had some great challenges, and the competition hall was packed with 193 teams participating!
I played with Emu Exploit and overall we came 3rd in the CTF, and recieved a massive cardboard cheque of $250 and a medal - congrats to skateboarding roomba and Grassroots Indirection on 1st and 2nd!
In this blog post, I will provide an in-depth walkthough to prnginko, a crypto/rev challenge which caused me a lot of pain, and a brief writeup of useless, another rev challenge.
Challenge Overview⌗
I will mainly focus on prnginko, as I want to detail the many roadblocks we faced along the way, instead of going straight to the solution.
If you want to follow along or see the binary for yourself, download files here
prnginko (crypto/rev) - 8 solves⌗
Multi-bet with SportsBear!
nc prnginko.chal.cybears.io 2323
We are given a binary prnginko and a service to connect to. Upon running the binary, we are presented with the game and a message suggesting the goal to be getting a “perfect game”. The game includes
a board consisting of pins and a ball which randomly bounces either left or right when it hits a pin, similar to the Plinko game.
We have two options - g to play a game and earn points, or p to practice. We can play 10 games in total and need to get the maximum score of 16 each time to win.
“prng” in the challenge name suggests that we need to reverse engineer the program to find the PRNG it uses to determine if the ball will go left or right, and crack it to find what future game plays
will yield. Then, by being able to predict the future, we can use the practice games to re-roll the PRNG until we know the next game will yield a max score of 16, and then use one of our game rounds.
A quick explanation of why we can do this - PRNGs (pseudo random number generator) usually use a seed and some other parameters to generate the next “random” number using some math. If the same seed
and same parameters are used, then the next random number it generates will be the same. Thus, if we are able to recover the seed and know the parameters, we can predict the values it will generate
in the future.
Hopping into IDA, we can see the 160 points required to win, which is getting 16 points ten times - a perfect game.
Note that function and variable names were stripped in the binary, so we had to go through and rename everything relevant as always.
After some reverse engineering, we find a few functions that together form a PRNG based on the program’s current runtime.
There are three functions:
• Is only called once, at the start of the program
• Gets a value affected by time using clock_gettime()
• Sets time_seed to the amount of seconds, plus 1000000000 times the amount of nanoseconds
__int64 get_timeseed()
struct timespec tp; // [rsp+10h] [rbp-20h] BYREF
unsigned __int64 v2; // [rsp+28h] [rbp-8h]
v2 = __readfsqword(0x28u);
if ( clock_gettime(7, &tp) )
return 4294967293LL;
time_seed = LODWORD(tp.tv_sec) + 1000000000 * LODWORD(tp.tv_nsec);
return 0LL;
• Does some math with a_value and time_seed
• Changes a_value (HIDWORD gets the higher 32 bits, same as shifting right by 32 bits) and time_seed based on the math calculation
• Returns the result of the math calculation
• a_value is set to 1 at the start of the program
__int64 prng_subpart()
__int64 var8; // [rsp+0h] [rbp+0h]
*(&var8 - 1) = (unsigned int)a_value - 0xC5D8A3FF84711ALL * (unsigned int)time_seed;
a_value = HIDWORD(*(&var8 - 1));
time_seed = *(&var8 - 1);
return (unsigned int)time_seed;
• prng_main() calls prng_subpart() only when shift_r_value is below zero
• shift_r_value is the amount that the prng_output is right shifted
• The function returns the lowest bit of prng_output >> shift_r_value (using & 1)
• shift_r_value is initialized to -1 at the start, so prng_output is called at the start
__int64 prng_main()
if ( shift_r_value < 0 )
shift_r_value = 31;
prng_output = prng_subpart();
return ((unsigned int)prng_output >> shift_r_value--) & 1;
As global variables, the initial values of shift_r_value and a_value were located in the .data section, or just double click in IDA to find them.
The main takeaway from these functions is that we know all of the values, except time_seed. time_seed is the only value that is causing the output to not be identical - it’s the only value that
changes “randomly” each time we run the program, so we need a way to recover it.
Of course it is a horrible idea to seed a PRNG based on time, but it uses nanosecond precision which we cannot accurately predict on a remote instance.
However, if we did know the value of time_seed, that’s all that’s left to input into our own PRNG and predict the future!
To start off simple, let’s write out the PRNG functions in Python.
def get_timeseed():
# we'll figure out how to get this later ;)
timeseed = int(input("Enter timeseed: "))
return timeseed
def prng_subpart():
global time_seed, a_value
output = a_value - 0xC5D8A3FF84711A * time_seed
a_value = output >> 32
time_seed = output
return time_seed
def prng_main():
global shift_r_value, prng_output
if shift_r_value < 0:
shift_r_value = 31
prng_output = prng_subpart()
output = (prng_output >> shift_r_value) & 1
shift_r_value -= 1
return output
prng_output = None
shift_r_value = -1
a_value = 1
time_seed = get_timeseed()
for i in range(8):
print(f"Value of output {i}: {prng_main()}")
But how do we know this is correct? (foreshadowing: it’s not quite…)
In IDA we can see time_seed is stored in .bss, so let’s test our PRNG replication by debugging and just grabbing the value directly.
Using info file to locate address of .bss, we dump .bss to find time_seed, which as an unsigned int is a 4 byte value at offset 0x44, same as what IDA shows.
Next, we chuck this time_seed value into our own PRNG and see if our outputs correlates with the game’s outputs.
It indeed does! 0 indicates that the ball goes left, and 1 indicates the ball goes right.
A Roadblock⌗
However, there is one flaw I would like to point out here - there is a mistake in our Python PRNG. This subtle mistake cost hours of debugging and pain - I said that I would eat breakfast after
solving the challenge. I ended up eating lunch instead.
You may wonder, well, the PRNG seems to be giving the correct output though? It predicted 8 values correctly. That is true, until you go after 64 values.
It turns out that we forgot to account for the C datatypes in python! output (var8) is an int64, which means if we go over 2**63 or under -2**63, it will wrap around, same as mod 2**64. We didn’t
account for this - thus eventually after two outputs of prng_subpart, time_seed became large enough to surpass this limit and provide incorrect outputs.
To account for this we will add output = output % (2**64) and time_seed = output % (2**32).
The fixed code is now:
def get_timeseed():
# we'll figure out how to get this later ;)
timeseed = int(input("Enter timeseed: "))
return timeseed
def prng_subpart():
global time_seed, a_value
output = a_value - 0xC5D8A3FF84711A * time_seed
output = output % (2**64)
a_value = output >> 32
time_seed = output % (2**32)
return time_seed
def prng_main():
global shift_r_value, prng_output
if shift_r_value < 0:
shift_r_value = 31
prng_output = prng_subpart()
output = (prng_output >> shift_r_value) & 1
shift_r_value -= 1
return output
prng_output = None
shift_r_value = -1
a_value = 1
time_seed = get_timeseed()
for _ in range(10):
for i in range(8):
print(f"Value of output {i}: {prng_main()}")
Back to the challenge…⌗
With that issue fixed, we can continue on with the challenge.
We have successfully recreated the PRNG and can predict future outputs given time_seed, now only one problem remains - how can we retrieve the value of time_seed?
As mentioned before, time_seed is affected by time in nanoseconds, which would be close to impossible to simulate on a remote connection.
Another idea is to brute force. As it is a unsigned int32, we can try 2**32 possible values for time_seed, until our PRNG output seems to match up with the game’s output. However, this is also not
possible since although 2**32 is not too large, there is a timer set for 5 minutes, and with a quick test it would take way too long to brute force in under 5 minutes (at least in python).
Our last option is to use an SMT solver, such as z3. We can simply get a bunch of outputs from the game, then tell z3 that it should try to find a value of time_seed that causes our PRNG to output
the same as the game.
For example, if a round of our game outputs LRRLRLLL where L is left and R is right, we can change it to 01101000, and tell z3 that the first output of our PRNG should be 0, the second should be 1,
and so on.
First, we define time_seed as a 64-bit value.
time_seed = BitVec('time_seed', 64)
But wait - why should it be 64-bits when we know it is actually a 32-bit value?
Well, if we look at the decompilation again, we can see time_seed is being set to var8, and var8 is 64 bits. This means that the first calculation can result in a 64 bit value, causing time_seed to
be set to a value larger than 64 bits.
__int64 prng_subpart()
__int64 var8; // [rsp+0h] [rbp+0h]
*(&var8 - 1) = (unsigned int)a_value - 0xC5D8A3FF84711ALL * (unsigned int)time_seed;
a_value = HIDWORD(*(&var8 - 1));
time_seed = *(&var8 - 1);
return (unsigned int)time_seed;
This issue caused us a lot of pain as well.
Anyways, we then collect the game’s PRNG outputs from the result of some practice rounds.
def get_game_output():
p.recvuntil(b"> ")
p.sendline(b"p") # play practice round
outputs = []
for i in range(8):
game_output = p.recvline().decode()
if game_output[0] == "L":
elif game_output[0] == "R":
return outputs
trials = 8
game_outputs = []
for trial in range(trials):
print(f"{game_outputs = }")
Doing 8 trials, which gets 64 bits from the game should be enough for z3 to compute a unique solution (you might imagine if we had only 1 trial, there could be many possible time_seeds that can
result in the same output).
Then we simply tell z3 that these values should be equal, and also add that time_seed should be within the 32 integer range.
prng_output = None
shift_r_value = -1
a_value = 1
time_seed = BitVec('time_seed', 64)
time_seed_copy = time_seed
context.binary = elf = ELF("./prnginko")
p = process("./prnginko")
trials = 8
game_outputs = []
for trial in range(trials):
s = Solver()
s.add(time_seed >= 0)
s.add(time_seed <= 2**32)
for bit in game_outputs:
s.add(prng_main() == bit)
assert s.check() == sat
m = s.model()
correct_time_seed = int(m[time_seed_copy].as_long())
print(f"{correct_time_seed = }")
Also note that above I have made a copy of time_seed called time_seed_copy as time_seed was being overwritten in the PRNG functions, thus a Z3 declaration had to be retained to retrieve the time from
the model.
Getting the flag⌗
Now that our PRNG is working, and z3 gives us the correct time_seed, we can finally get the flag. All we do is to count how many times we need to reroll the PRNG until it gives us all 0’s or all 1’s
(which will give us the max points), then use a game round.
Final code:
from z3 import *
from pwn import *
# local file of our PRNG to avoid z3 symbolic values being used
import prng
def prng_subpart():
global time_seed, a_value
output = a_value - 0xC5D8A3FF84711A * time_seed
output = output % (2**64)
a_value = LShR(output, 32)
time_seed = output % (2**32)
return time_seed
def prng_main():
global shift_r_value, prng_output
if shift_r_value < 0:
shift_r_value = 31
prng_output = prng_subpart()
output = LShR(prng_output, shift_r_value) & 1
shift_r_value -= 1
return output
def get_game_output():
p.recvuntil(b"> ")
p.sendline(b"p") # play practice round
outputs = []
for i in range(8):
game_output = p.recvline().decode()
if game_output[0] == "L":
elif game_output[0] == "R":
return outputs
def play_game_round():
p.sendline(b"g") # play game round
context.binary = elf = ELF("./prnginko")
# p = process("./prnginko")
p = remote("prnginko.chal.cybears.io", 2323)
game_rounds = 10
prng_output = None
shift_r_value = -1
a_value = 1
time_seed = BitVec('time_seed', 64)
time_seed_copy = time_seed
trials = 8
game_outputs = []
for trial in range(trials):
s = Solver()
s.add(time_seed >= 0)
s.add(time_seed <= 2**32)
for bit in game_outputs:
s.add(prng_main() == bit)
assert s.check() == sat
m = s.model()
correct_time_seed = int(m[time_seed_copy].as_long())
print(f"{correct_time_seed = }")
# need to fast forward our PRNG to match the game's state
# as earlier we played 8 rounds to get the outputs.
# must times 8 because each round has 8 outputs
for trial in range(trials * 8):
for game_round in range(game_rounds):
practice_round_count = 0
while True:
next_round_result = [prng.prng_main() for _ in range(8)]
# check if all 1's or all 0's
if len(set(next_round_result)) == 1:
# send number of practice rounds we want to
# play at same time to avoid taking too
# long to recieve data
p.send(b"p\n" * practice_round_count)
practice_round_count += 1
def prng_subpart():
global time_seed, a_value
output = a_value - 0xC5D8A3FF84711A * time_seed
output = output % (2**64)
a_value = output >> 32
time_seed = output % (2**32)
return time_seed
def prng_main():
global shift_r_value, prng_output
if shift_r_value < 0:
shift_r_value = 31
prng_output = prng_subpart()
output = (prng_output >> shift_r_value) & 1
shift_r_value -= 1
return output
def set_time_seed(_time_seed):
global prng_output, shift_r_value, a_value, time_seed
prng_output = None
shift_r_value = -1
a_value = 1
time_seed = _time_seed
Note that in the final solve script, I imported a separate python file named prng with the same PRNG implementation for several reasons:
• In z3, LShR should be used instead of >> to perform right shifts (more info here).
• Symbolic values were passed through the functions making it return another symbolic value when called. Basically I wanted it to return a number not an equation.
I would like to thank ssparrow for helping me debug the code and for finding the issues that stumped me for hours.
A much simpler solution⌗
After the CTF ended, I talked to Neobeo who played for Skateboarding Roombas in this CTF about the challenge.
He revealed a much easier solution that didn’t need any messing around in z3 - the PRNG was actually an LCG! In case you didn’t know, an LCG is in the form
$$S_{n+1} = S_n \times a + b \bmod{m}$$
where the next term is the current term S_n times a plus b, where a and b are constants. In this case S_n was time_seed, and the modulus m was the 32 bit integer limit. Despite having done LCG
challenges in the past, I somehow failed to recognise this!
The function main_prng was simply returning the MSB (first bit) of the PRNG output, and shifting it to return all bits of the output before retrieving a new random number. prng_subpart was the actual
PRNG, which was an LCG implementation.
Recovering time_seed was trivial now that we recognise it as a LCG - we simply need the whole random number returned by prng_subpart which we can get by playing 4 practice rounds (as each round
returns 8 bits).
Now we simply solve for time_seed:
$$ S_{n+1} \equiv S_n \times a + b \pmod{2^{32}} \newline S_n \equiv a^{-1}(S_{n+1} - b) \pmod{2^{32}} $$
where S_n is time_seed, b is a_value, a is -0xC5D8A3FF84711A and S_n+1 is the 32 bit value returned by playing 4 rounds.
There is just a slight issue - since both -0xC5D8A3FF84711A and 2**32 are even, there is no modular inverse! We can get around this by dividing everything by two.
$$ \frac{S_{n+1}}{2} \equiv S_n \times \frac{a}{2} + \frac{b}{2} \pmod{2^{32}} \newline S_n = (\frac{a}{2})^{-1} \times \frac{S_{n+1} - b}{2} \pmod{2^{31}} $$
Thanks again to Neobeo for showing me this trick.
However, by doing this we are left with two possible values as we are now solving over mod 2**31 instead of mod 2**32, so if time_seed is over 2**31 it will get cut off.
We can verify which is correct by collecting another set of outputs, and seeding with both of the possible seeds. Alternatively, we can just use two sets of outputs to solve for a distinct value.
from z3 import *
from pwn import *
def prng_subpart():
global time_seed, a_value
output = a_value - 0xC5D8A3FF84711A * time_seed
output = output % (2**64)
a_value = output >> 32
time_seed = output % (2**32)
return time_seed
def prng_main():
global shift_r_value, prng_output
if shift_r_value < 0:
shift_r_value = 31
prng_output = prng_subpart()
output = (prng_output >> shift_r_value) & 1
shift_r_value -= 1
return output
def get_game_output():
p.recvuntil(b"> ")
p.sendline(b"p") # play practice round
outputs = []
for i in range(8):
game_output = p.recvline().decode()
if game_output[0] == "L":
elif game_output[0] == "R":
return outputs
def play_game_round():
p.sendline(b"g") # play game round
context.binary = elf = ELF("./prnginko")
p = process("./prnginko")
# p = gdb.debug("./prnginko")
# p = remote("prnginko.chal.cybears.io", 2323)
game_rounds = 10
modulus = 2**32
trials = 4 # get the whole 32-bit value
game_outputs = []
for trial in range(trials):
print(f"{game_outputs = }")
S_n1 = 0
for i, game_output in enumerate(game_outputs[::-1]):
S_n1 += game_output * (2**i)
calculated_time_seed = pow(-0xC5D8A3FF84711A//2, -1, modulus//2) * ((S_n1 - 1) // 2)
calculated_time_seed %= modulus // 2
possible_seed_1 = calculated_time_seed
possible_seed_2 = calculated_time_seed + modulus//2
# get 2nd set of outputs
game_outputs = []
for trial in range(trials):
correct_seed = None
#try our first seed
time_seed = possible_seed_1
prng_output = None
shift_r_value = -1
a_value = 1
#fast forward to game's 2nd set of outputs
for _ in range(trials * 8):
for game_output in game_outputs:
assert prng_main() == game_output
correct_seed = possible_seed_1
except AssertionError:
time_seed = possible_seed_2
prng_output = None
shift_r_value = -1
a_value = 1
#fast forward to game's 2nd set of outputs
for _ in range(trials * 8):
for game_output in game_outputs:
# just to double check we're correct
assert prng_main() == game_output
correct_seed = possible_seed_2
time_seed = correct_seed
prng_output = None
shift_r_value = -1
a_value = 1
# fast forward to same PRNG state as the game
for _ in range(trials*8 * 2):
for game_round in range(game_rounds):
practice_round_count = 0
while True:
next_round_result = [prng_main() for _ in range(8)]
# check if all 1's or all 0's
if len(set(next_round_result)) == 1:
# send number of practice rounds we want to
# play at same time to avoid taking too
# long to recieve data
p.send(b"p\n" * practice_round_count)
practice_round_count += 1
useless (rev) - 9 solves⌗
There’s a weird file recovered from a forensic analysis of the MAYHEM mainframe… but it doesn’t seem to do anything?
We are provided with a binary file useless which when ran, as the name suggests, seemingly does nothing.
┌──(teddykali㉿teddykali)-[~/…/on premise/Bsides Canberra 2023/rev/useless]
└─$ ./useless
┌──(teddykali㉿teddykali)-[~/…/on premise/Bsides Canberra 2023/rev/useless]
Decompiling the binary doesn’t offer much either - there appear to be thousands of functions named continue_x which just calls the next one. An interesting thing is that some numbers are skipped (e.g
continue_1) however this didn’t seem to help either.
//----- (0000000000401000) ----------------------------------------------------
void __noreturn start()
//----- (0000000000401019) ----------------------------------------------------
void __noreturn continue_0()
//----- (000000000040103B) ----------------------------------------------------
void __noreturn continue_2()
// this repeats until continue_2024() !
The decompilation is rendering useless - looks like we’ll need to dig deeper.
After loading the binary into gdb (with the pwndbg extension), we use starti to start running the binary but immediantly breaking, as otherwise the program would just exit.
Next, stepping through instruction with si , I noticed an unusual value in the rax register, which turned out to be a printable character.
Using watch $rax to watch the value of rax and break every time the value of rax changes, we slowly retrieve a stream of printable characters: SW4gdGhlIHJ
It looks like base64 - and decoding SW4gdGhlIHJ from base64 yields In the r. Not the flag, but probably the right track!
Given there are over 2000 functions, we probably don’t want to do this manually. gdb supports scripting with python, so lets automate it!
flag = ''
gdb.execute("watch $rax")
while True:
value = gdb.parse_and_eval("$rax")
value = value.cast(gdb.lookup_type('long'))
value = int(value)
flag += chr(value)
print(f"{flag = }")
Running our script with source gdb_script.py prints out a very long base64 encoded string.
Adjusting some base64 offsets (sometimes there can be consecutive base64 characters, in which case rax doesn’t change, resulting in some characters being missed - thanks to Jamie for correcting me on
this!) and decoding from base64 in CyberChef, we get the flag!
Although we already got the flag, let’s take a deeper look into what was going on.
A quick look at the disassembly shows ebx (lower 32 bits of rbx) being set to 0x6a0, and rax is set to the value at address rbx + 0x40a000.
That means that rax was just being set to some data pointed at by rbx (plus an offset of 0x40a000). However, the value rbx is set to seems to not be constant for each function, so watching the value
of rax is probably still the best solution.
This was the first time I had gone to BSides Canberra, and it was by far the best and largest conference I had been to!
Thanks to Cybears for making some neat challenges, and also Infosect for covering the cost of my flights and hotel through their Assistance Program!
It was great fun meeting everyone, from skateboarding dogs to DownUnder CTF organisers, and I look forward to next year’s conference!
Thanks for reading :)
Feel free to DM me on Discord thesavageteddy or Twitter teddyctf if you have any questions, corrections, or just want to chat!
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Method: Related Rates - APCalcPrep.com
Unfortunately, there is no cut and dry way to handle every related rates problem. There are some very common setups that you will see show up, but they are the type of problem that you have to be
willing to play around with the pieces to find how they all come together. If you really want to master related rates problems, you have to do a bunch of them. The more of them you attempt, the more
different models you will see, and the quicker you will start being able to make the connections. The problem you see on a test or the AP Calculus exam might be different in what the things in the
problem are, but the method might match a technique from another problem you have done. If you are talking about a conical pile of sand in one problem you did, that problem’s technique might relate
to the problem about a conical funnel with water because they are both most likely going to use the volume of a cone equation.
Here is the general method I follow when approaching any Related Rates problem.
Step 1: Read through the problem once, and sketch out a diagram of what is happening.
When you are first starting out with these, make that diagram big. Do not try to squeeze it into a little corner on your page. You are going to need the label that image with lots of data, and then
try to connect that data. Draw big so you can truly SEE what it is you are looking at.
Sometimes when you are working with volumes it might be more helpful to draw a cross section of the figure, and not necessarily the complete 3D image. This is very common with cones and spheres.
Step 2: Read through the problem a second time, identify all the given and implied material, and where it is located on you diagram.
Keep in mind that many parts of your image will need to be labeled with two pieces of data; a flat amount and a rate of change. A side might be ${5}{m}$ in length and growing at ${2}\frac{{m}}{{s}}$
at a specific moment in time. You will want to label that side with both bits of data. Keep in mind as you do your labeling of rates, that rates of change that are increasing or growing would be
positive rates of change, and rates of change that are shrinking or getting smaller would be a negative rate of change. Positive and negatives have real world meanings in these problems, so keep
track of them.
• Pay attention to the units.
□ A flat amount that is not changing with have “flat” units: ft, m. No word like “per” in the units.
□ A rate of change piece will have units like $\frac{{m}}{{s}}$ , meters per second, or mph, miles per hour. If you find yourself using the word “per”, you are most likely looking at a rate of
• Rate of Change Pieces
□ Will usually be accompanied by the word “rate”. “The rate at which the volume was changing ${2}\frac{{{m}}^{{3}}}{{s}}$.” Or, “The car was moving at a rate of ${5}{\mathit{mph}}$.”
□ You will also see words like increasing, decreasing, or other action- “ing” words. For example, “The radius was increasing at ${5}\frac{{m}}{{s}}$.”
• Flat Amount Pieces
□ The language will generally say a piece of the diagram “is” this. For example, “The radius is ${5}{m}$.” Or “The height of the building is ${147}{\mathit{ft}}$.” Notice there is no
description of movement.
• Additional flat amounts and rates of change sometimes need to be hunted down by you.
□ They may not give you all the flat amounts that you need to complete your diagram. If they tell you a car has been going ${5}{\mathit{mph}}$ for 2 hours, they are quietly telling you that the
car also went 10 miles.
□ One of the most common unspoken given bits of information is a rate of change value for some portion of the diagram that is zero. Take a look at all the parts of your diagram and make sure
that they actually change at all in the context of this problem. Sometimes there might be no change occurring along one piece of your diagram, and that is the problem quietly telling you the
rate of change is zero .
Step 3: Identify the rate of change you are being asked to find.
Are you being asked how the volume is changing, the height is changing, the angle is increasing? What is the rate of change you must hunt down?
Step 4: Find an equation that relates the flat amounts to each other.
You must know your similar triangle relationships, geometry formulas, and your trig formulas.
Step 5: Take the derivative of the equation using implicit differentiation.
Step 6: Plug in all of the pieces you have identified in Step 2 into the derivative equation you just created, and solve for the piece you need.
• If all goes well, then you should have only the rate of change that you are trying to find left in the equation as a variable. You then solve for that rate of change variable, and you are done.
• If you are still missing more than just the one rate of change you have been asked to find, then you will need to:
□ Reevaluate the information given to you in the original language of the problem. See if there is an implied amount that you were not directly given a value for that you could logic out a
value for. For example, if they tell you a car has been going 5mph for 2 hours, they are quietly telling you that the car also went 10 miles.
□ See if there is another connection through a second equation. See if you can solve that second equation for a variable that you can plug into your actual derivative equation. You want to get
the derivative equation all talking variables you have values for. For example, you might have a volume formula that relies on the height and the radius, $V=\frac{1}{3}\pi {r}^{2}h$; you have
a rate of change for the radius, $\frac{\mathit{dr}}{\mathit{dt}}$, but you don’t have a rate of change for the height, $\frac{\mathit{dh}}{\mathit{dt}}$. That means you need to find a way to
relate the radius and the height in another equation, separate from the one you care about. You can plug a value in for h in the volume formula that is in terms of r . Then you will have a
volume formula that only uses a radius and the derivative of that formula will only need rates of change for a radius , $\frac{\mathit{dr}}{\mathit{dt}}$. I know it is a lot to take in, but
the more you do of these problems, the less daunting it will feel.
• You must know your similar triangle relationships, geometry formulas, and your trig formulas.
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A regular pentagon has the following dimensions:
BF=EF=Φ/2 = cos 36 = sin 54 = \(\frac{1+\sqrt{5}}{4}\)
CG=DG= Φ-1 = 1/Φ = \(\frac{\sqrt5-1}{2}\)
AF=FG= cos 54 = sin 36 = \(\sqrt{\frac{5-\sqrt5}{8}}\)
AG= 2cos 54 = 2sin 36= \(\sqrt{\frac{5-\sqrt5}{2}}\)
Height AH=EK=\(\frac12\cos 18 = \frac12\sin 72 = \frac12\sqrt{5+2\sqrt5}\)
Circumradius AJ=\(\sqrt{\frac{5+\sqrt5}{10}}\) = BJ=CJ=DJ=EJ
Inradius JK=HJ= \(\sqrt{\frac{5+2\sqrt5}{20}} \) = \(\frac{AH}{\sqrt5}\)
FH= cos 18 =sin 72 = \(\sqrt{\frac{5+\sqrt5}{8}}\)
ABGE is a equilateral rhombus, with angles {108°, 54°, 108°, 54°} and diagonals of length Φ and \(\sqrt{\frac{5-\sqrt5}{2}}\).
The small pentagon in the center has sides \(1/\Phi^2 = \frac{3-\sqrt5}{2}\) times that of the larger.
Phi, the Golden Ratio
The golden ratio is illustrated as \(\frac{A}{B}=\frac{A+B}{A}\equiv\phi\).
The only positive solution is \(\displaystyle\phi=\frac{1+\sqrt{5}}{2}\)=1.6180339887498948482045868…
Many items have this ratio embedded into their design, by choice or coincidence, such as; the Parthenon, credit cards, corporate logos, the Mona Lisa, and the layout of Quincy Park in Cambridge, MA.
I find that there is some differences in the usage of symbols to represent the golden ratio. The most common is to use phi, either Φ(capitol), \(\phi\) (lower case), or φ (lower case variant), while
Ø (Scandinavian O-slash) is often seen where non-Latin fonts are unusable or unavailable. Normally, one symbol will be used for the golden ratio and another for the inverse. Others will use “Phi” and
“phi”, I don’t like this method as it is too difficult in many fonts to easily distinguish. The negative inverse is called the golden ratio conjugate and is sometimes represented with the upper/lower
case letter that is not used for the golden ratio.
On this page, I will use the word “phi” or the symbol \(\phi,\) as this is the symbol that Latex gives for phi. For the inverse, I will use \(\displaystyle\frac{1}{\phi}\) or \(\phi^{-1}\), as
additional symbols are not really needed.
Some interesting relationships of phi:
$$\phi+1=\phi^2 &=2.6180339887498948482045868\ldots \\ \phi-1 &=\frac{1}{\phi} \\ \phi&=1+\frac{1}{\phi} \\ \frac{1}{\phi}=\frac{-1+\sqrt5}{2}&=0.6180339887498948482045868\ldots \\ \left(\frac{1}{\
phi}\right)^2&=1-\frac{1}{\phi} \\ \phi^3&=1+2\phi$$
Compare the decimal portion of \(\phi\) to that of \(\displaystyle\frac{1}{\phi}\). They are the same, as is \(\phi^2\).
Pentagons and pentagrams show many signs of a relationship with phi. Notice in the pentagon, there is a red parallelogram (rhombus) with long diagonal of phi and side length 1. The height of the
pentagon is the height of each of the arms of the star, a (\(\phi\),\(\phi\),1) isosceles triangle. Each line is the same length as the other lines of the same color, relatively speaking.
The pentagram is \(\phi^2\) times larger than a pentagram inscribed inside the pentagon. This relationship continues, every nested pentagon and pentagram is \(\phi^2\) times larger than the previous
same shape and is \(\displaystyle\frac{1}{\phi^2}\)times the next same shape.
Penrose “kite” and “dart” tiles
The “kite” and “dart” of Penrose tiles can be merged to form the same rhombus. Penrose tiles are also found in the formation of the newly discovered quasi-crystals.
If \(\large\phi^2=\phi+1\), then $$\large\phi^3&=\phi\cdot\phi^2 \\ &=\phi(\phi+1) \\ &=\phi^2+\phi \\&=\phi+1+\phi \\ &=1+2\phi$$
Therefore $$\large\phi^4&=\phi\cdot\phi^3\\&=\phi(1+2\phi)\\&=\phi+2\phi^2\\&=\phi+2(1+\phi)\\&=\phi+2+2\phi\\&=2+3\phi$$
This shows an important and remarkable pair of results for all integers N:
A quick table of values:
│\(\large\phi^N\)│\(\large\frac12\cdot(a+b\sqrt5)\) │\(\large c+d\phi\)│\(\large j+k\frac{1}{\phi}\)│approximate│
│ │ │ │ │ value │
│10 │\(\frac12(115+55\sqrt5)\) │\(34+55\phi\) │\(89+55\frac{1}{\phi}\) │122.9918 │
│9 │\(\frac12(68+34\sqrt5)\) │\(21+34\phi\) │\(55+34\frac{1}{\phi}\) │76.01315 │
│8 │\(\frac12(47+21\sqrt5)\) │\(13+21\phi\) │\(34+21\frac{1}{\phi}\) │46.97871 │
│7 │\(\frac12(29+13\sqrt5)\) │\(8+13\phi\) │\(21+13\frac{1}{\phi}\) │29.03444 │
│6 │\(\frac12(18+8\sqrt5)\) │\(5+8\phi\) │\(13+8\frac{1}{\phi}\) │17.94427 │
│5 │\(\frac12(11+5\sqrt5)\) │\(3+5\phi\) │\(8+5\frac{1}{\phi}\) │11.09016 │
│4 │\(\frac12(7+3\sqrt5)\) │\(2+3\phi\) │\(5+3\frac{1}{\phi}\) │6.85410 │
│3 │\(\frac12(4+2\sqrt5)\) │\(1+2\phi\) │\(3+2\frac{1}{\phi}\) │4.23606 │
│2 │\(\frac12(3+1\sqrt5)\) │\(1+1\phi\) │\(2+1\frac{1}{\phi}\) │2.61803 │
│1 │\(\frac12(1+1\sqrt5)\) │\(0+1\phi\) │\(1+1\frac{1}{\phi}\) │1.61803 │
│0 │\(\frac12(2+0\sqrt5)\) │\(1+0\phi\) │\(1+0\frac{1}{\phi}\) │1.00000 │
│-1 │\(\frac12(-1+1\sqrt5)\) │\(-1+1\phi\) │\(0+1\frac{1}{\phi}\) │0.61803 │
│-2 │\(\frac12(3-1\sqrt5)\) │\(2-1\phi\) │\(1-1\frac{1}{\phi}\) │0.09016 │
│-3 │\(\frac12(-4+2\sqrt5)\) │\(-3+2\phi\) │\(-1+2\frac{1}{\phi}\) │0.05572 │
│-4 │\(\frac12(7-3\sqrt5)\) │\(5-3\phi\) │\(2-3\frac{1}{\phi}\) │0.03444 │
│-5 │\(\frac12(-11+5\sqrt5)\) │\(-8+5\phi\) │\(-3+5\frac{1}{\phi}\) │0.02128 │
│-6 │\(\frac12(18-8\sqrt5)\) │\(13-8\phi\) │\(5-8\frac{1}{\phi}\) │0.05572 │
Note: values in the last column are truncated, not exact, and column 2 fractions are not simplified, to show pattern.
If we look at the integers in the third and fourth columns, you may notice a pattern emerge. They are all Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, …), where each number is the sum of the previous
2 numbers in the sequence. This gives us \(\phi^N=\left(\frac{1}{\phi}\right)^{-N}=F_{(N-1)}+F_N\cdot\phi=F_{(N+1)}+F_N\cdot\frac{1}{\phi}\), where \(F_N\) is the \(N^{th}\) Fibonacci number.
We also get the value \(\phi^N=\frac12\left[F_{(N+1)} + F_{(N-1)}+ F_{N}\cdot\sqrt5 \right]\).
The integers multiplied by √5 in the second column, are also the Fibonacci numbers, but the first set are what is known as the Lucas number series. The Lucas numbers are just like the Fibonacci
numbers, each is the sum of the previous 2 numbers, but instead of starting with 0 and 1, François Édouard Anatole Lucas started his series with 2 and 1.
The sequence of Lucas numbers begins: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, …
Each series can be back calculated to find previous numbers, ultimately leading to alternating positive and negative numbers, as seen in the formulas for the negative powers of phi in the table.
Lucas numbers are of some use in primality testing.
If you take sequential Fibonacci numbers and divide one by the previous, i.e. \(\large\frac{F_{(N+1)}}{F_{(N)}}\), the result becomes closer and closer to equaling \(\phi\) the higher N becomes, or \
(\frac{-1}{\phi}\) the lower N becomes.
Interestingly, 8 and 144 are the only non-trivial perfect powers, being 2^3 and 12^2, respective.
Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple, (3,4,5),
(5,12,13), (16,30,34), (39,80,89), etc.
Update: Here are a few ways you can enter phi. If you hold the Alt key, type 232 on the numeric keypad, then release the Alt key, you get Φ (capitol phi). Alt 237 gives φ (lower phi). In HTML, you
can use Φ or Φ for uppercase, φ or φ for lower. These methods both work in the comments box.
If you need the square root radical or an exponent:
• √ – Alt 251
• ⁿ – Alt 252
• ± – Alt 0177
• ² – Alt 0178
• ³ – Alt 0179
• ¹ – Alt 0185
You can also use the Character Map program (in menu Start→Programs→Accessories→System Tools) to select and copy into the clipboard, although fonts may be unavailable on a different computer. Arial
should be safe to use.
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Unit sphere is compact in 1-norm
• I
• Thread starter psie
• Start date
TL;DR Summary
How do I go about showing the unit sphere is compact in the ##1##-norm without using the fact that norms on ##\mathbb R^n## are equivalent?
In Introduction to Topology by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in ##\mathbb R^n##. This exercise succeeds another exercise where various equivalent
formulations of "equivalent norms" have been given, e.g. that two norms ##\|\cdot\|_a,\|\cdot\|_b## are equivalent iff the identity map from ##(\mathbb R^n,\|\cdot\|_a)## to ##(\mathbb R^n,\|\cdot\|
_b)## is bicontinuous.
Now, in showing that all norms in ##\mathbb R^n## are equivalent, the authors show a given norm ##\|\cdot\|## is equivalent to the ##1##-norm (and then by transitivity, we have equivalence for all
norms, since equivalent norms is an equivalence relation). I have already managed to understand that the identity is continuous from ##(\mathbb R^n,\|\cdot\|_1)## to ##(\mathbb R^n,\|\cdot\|)##. To
show that the inverse of the identity map is continuous, the authors claim that the unit sphere in the ##1##-norm is compact. I'm getting hung up on this statement, since I don't know how to go about
this without using that the norms are equivalent already. How would one show the unit sphere in the ##1##-norm is compact?
I know of Heine-Borel, but I'm not sure how and if it applies here. Any help would be very appreciated.
The unit sphere in the 1-norm is the set of points ##(x_1,\ldots,x_n)\in\mathbb{R}^n## satisfying ##|x_1|+\ldots+|x_n|=1.##
This set is bounded since ##|x_i|\leq 1## for each ##i##. It is also closed, because the map ##f:\mathbb{R}^n\to\mathbb{R}, f(x_1,\ldots,x_n)=|x_1|+\ldots+|x_n|## is continuous, and your set is the
preimage of the closed set ##\{1\}.##
So, by Heine-Borel, it is compact. In the above, closed and bounded are relative to the standard (2-) norm.
It's also not hard to just directly verify that the identity map from ##\mathbb{R}^n## with the 2-norm to ##\mathbb{R}^n## with the 1-norm is continuous.
Infrared said:
So, by Heine-Borel, it is compact. In the above, closed and bounded are relative to the standard (2-) norm.
Thank you. May I ask, in what sense are closed and bounded relative to the ##2##-norm? I feel like you only used the ##1##-norm in showing that the set is closed and bounded.
So you showed the set is compact in the ##2##-norm, and since the identity map is bicontinuous between the ##1##-norm and ##2##-norm, it preserves this compact set, is that right?
psie said:
Thank you. May I ask, in what sense are closed and bounded relative to the -norm? I feel like you only used the -norm in showing that the set is closed and bounded.
In a metric space ##(X,d)##, a set ##E\subseteq X## is bounded if there is a constant ##C## such that ##d(x,y)\leq C## for all ##x,y\in E.## In this case ##E## is the unit ball in the ##1## norm and
##X=\mathbb{R}^n## and ##d## is the usual metric on ##\mathbb{R}^n## (induced by the 2-norm).
So you're just trying to find a constant ##C## such that if ##x=(x_1,\ldots,x_n)## and ##y=(y_1,\ldots,y_n)## satisfy ##|x_1|+\ldots+|x_n|=1## and ##|y_1|+\ldots+|y_n|=1## then the distance from ##x#
# to ##y## in the usual (2-norm) sense is at most ##C.## You should work this out yourself, it's quick once you get your definitions clear.
Similarly, the map ##(x_1,\ldots,x_n)\mapsto |x_1|+\ldots+ |x_n|## is continuous in the standard sense (as is ##\{1\}## being closed). Saying that this map is continuous means the same thing as when
you took multivariable calculus.
psie said:
So you showed the set is compact in the ##2##-norm, and since the identity map is bicontinuous between the ##1##-norm and ##2##-norm, it preserves this compact set, is that right?
I think you got the implication backwards- I thought showing that the identity is bicontinuous was the goal. Checking that the unit ball in the 1-norm is compact is presumably a step in your book's
proof (though I don't have your book on hand). Though, it's pretty quick to directly check that the identity map is continuous in both directions without this.
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Think Nuclear
I’ve already examined the classic Sleeping Beauty Problem and pointed out some of the pitfalls that many people fail to avoid when trying to solve the problem. I also examined Nick Bostrom’s
so-called “Extreme Beauty” modification to the problem, in which Beauty wakes many, many times if the coin toss comes up tails. However, there is another “extreme” variant of this problem, the
variant in which the coin toss is replaced with another two-result random process that has extremely uneven odds. That is, in this “extreme” problem, one of the possible results is extremely
unlikely. Examining this variant with the methods of reasoning commonly used by the “thirders” can be enlightening and can provide some illustration of why they are wrong.
Since many “thirders” seem to be fond of relying on betting analogies to reason through the problem and explain their arguments, a useful substitute for the coin toss is a lottery. A typical lottery
provides a very small chance of winning accompanied by a very large payoff (which is why lotteries are so popular). So here we shall examine what happens when Sleeping Beauty plays the lottery.
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What number of woods & Roofing sheets can be used for a 100X50feet Square building? - EngineeringAll.com
please sir, I want to know the numbers of woods And roofing sheets that will roof a building by 100by50 feet... Thanks so much
3 Answers
Thank you for seeking an answer in EngineeringAll.com Please, what is the roofing height, and the type of roofing sheet you intend to use?
Thanks for getting back to me, The height is like 10feets tall Is for church Just left and right roofing No design For roofing sheets, coloured zinc That one that looks like normal silver zinc but
colour blue
Good. Let us deal with your question mathematically depending on the design of your roofing and the roofing sheet you will be using. The type of roofing sheet determines the space of the nailing
points. For instance, metro tiles and coated tiles have close nailing points than long-span aluminum roofing sheets. So if you are using long span roofing sheet calculate the 2X2 wood by the number
of the total cross-section of the roofing skeleton. That will be where the nails will fix the roofing sheets. Assuming 100feet is the Front and back section of the roofing and 50feet is the side
sections of the roofing. Calculating the Front and back-Section At the front and back, for any point, that roofing nail will pass you will need 100feet length of 2X2inch wood. The total number of
nailing points will be according to the roofing height. Some roofing can have about 15 to 20 nailing points per section. In that case, the total nailing points for the front and back will be around
30 or 40 points the same will be applicable to the points by the sides but the lengths will differ. Each nailing point for the front and back will take 100feet length of 2X2inch wood, one 2X2inch
wood is around 16feet in length, therefore you need 16feet X 7pieces of 2X2inch wood to get the 100feet for a nailing point = 112feet. For every nailing point, you need 7 pieces of 2X2inch wood.
Assuming your roofing has 30 nailing points at the front and back, you will need 7 X 30 = 210pieces of 2X2inch wood but if it has 40 nailing points you will need 7 X 40 = 280 pieces of 2X2inch wood.
Calculation of the side sections Assuming the side total length is 50 feet, you will need 4 pieces of 2X2inch wood for a nailing point (4 X 16feet = 64feet). For 30 nailing point, you will need (4 X
30 = 120 pieces of 2X2inch wood) 120 pieces of the 2X2inch wood. For 40 nailing points, you will need (4 X 40 = 160 pieces of 2X2inch wood). Total Number of 2 by 2 plywood Based on the above
calculations, you will need the following number of 2X2inch wood: For the Nailing point of 30: 213 pieces for the front and back, 120 pieces for the two sides. TOTAL = (213 + 120 = 333pieces) For
the Nailing points of 40: 280 pieces for the front and back, 160 pieces for the two sides. TOTAL = (280 + 160 = 440pieces) NOW LET US CALCULATE THE NUMBER OF 2x4INCH PLYWOOD The 2 by 4 is the major
wood used in making the roofing skeleton. It is laid across the building on the top of the last block. It is also raised up to obtain the roofing height as well as the overall structure of the
roofing. Hence, calculation of it is very important. Assuming you will be crossing the wood at every 5 feet apart. That means (100/5 = 20), which means you will cross the wood 20 times using the
shortest length section of the roofing design which is the 50feet side section. At 16feet per 2X4wood. You will 4pices to make one cross on the top of the last block. So, you need (4 X20 = 80) 80
pieces to make 20 crosses on the top of the last block. At the center of the vertical section, you will need 10feet into 20 pieces for the raising of the height at the center of the 20 crosses made
on top of the last blocks. Which will be (10feet X 20 = 200feet. 200/16feet = 12.5pieces) This will take a total of 13 pieces of 2x4inch plywood. At the vertical section of the front and back, the
length from the top of the roof to the edge of the outshoots will be 27feet on both sides (you can obtain this using the Pythagorean theory: A2 + B2 = C2 Which is 102 + 252 = C2 , C= 27 Therefore
you will need less than 2pieces of 2X4inch wood (16 feet X 2 = 32 Feet) for every joining of the vertical wood to the outshoot. Since the crossing is 20 times. You will 27feet length of 2 X4inch wood
x 20times for the front side and the same quantity for the backside. This means 27 X 20 = 540 feet. (540/16feet = 34pieces of 2X4inch for the front side and 34 pieces for the backside. Total 34 + 34
= 68 pieces). For the two sides, 5 feet apart will be 10 times. Assuming you will be joining the sides at 5feet apart, You may need the same 27feet for each join. This will mean 27feeet X 10 for one
side. Which will be a total of 270feet for one side. Divide the total feet by the length of one 2 X4inch plywood which is 16feet (270/16 = 17 pieces of the wood will be needed for one side.) You will
need the same 16pieces for the other side. Therefore, the two sides of the roof will that 34pieces of the 2 X 4inch wood for the vertical joining. Another 2 by 4inches will be the ones that will be
used in the crossing of the erected ones to help in holding them together. This section has no specific number rather it depends on the chosen pattern of the carpenter. If 100feet of length will
cross three times at the front side and three times at the back section, it means a total of 600feet which means 38 pieces of 2 by 4inches of plywood. For the vertical crossing, let us assume 20
pieces of the 2 by 4 pieces can do it. Therefore, you will need 58 pieces of the 2 by 4 plywood for crossing and reinforcement points. You can then simply add extra 10 pieces of the 2 by 4inches of
plywood in the event of extra work or reinforcement work. TOTAL NUMBER OF 2 by 4INCHES PLYWOOD For the Horizontal crossing on top of the last blocks of the building (20 times): 80 pieces For the
10feet Vertical that will be fixed at the centered to raise the building height: 13 pieces For the joining of the height (10feet) to the outshoots of each side edges: Front and back = 68 pieces, the
two sides = 34pieces For the crossing and reinforcement works = 38 pieces + 20pieces = 58pieces Additional wood in case of need for more reinforcement = 10 pieces The Total number will be =
(80+13+68+34 + 58 + 10=263) 263 pieces CALCULATING THE NUMBER OF FACIAL BOARD The facial board usually comes 16 feet long or 12 feet in length, Based, on the 100feet by 50feet roofing measurement,
the total area will be 100feet + 100 feet + 50feet + 50 feet = 300 feet. Since one facial board will be 16 feet, 300feet/16feet = 19 pieces will be required for the roofing edges. OTHER ITEMS Other
items which cannot be fixed or calculated here are the total number of nails and their sizes, the workmanship of the carpenter. COST OF WOOD 2X2inch wood has soft and hard types at the cost of N300
to N500 respectively depending on your location. 2X4inch plywood usually comes in hardwood for roofing purposes. It costs between N2500 to N3500 based on the market location The facial board usually
comes in soft and hard but the recommended type for roofing is the hard facial boards which cost around N4000 to N6000 per one. Summary Note that the above calculation was made based on the 100feet
by 50feet square size roofing under the basic roofing design at the height of 10feet tall. If your roofing design is different from the basic design used in this calculation the total number of woods
may vary widely depending on your roofing design. Also, note that the type of roofing sheet can alter the overall number of 2 X2inch wood you will be buying. Please read the full publication here:
Please login or Register to submit your answer
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al extension of the
2-d ellipse
. A
ellipsoid should not be
to being a
surface or solid of rotation
, however. The
s (x,y,z) which satisfy
are the
outer surface
of a generic ellipsoid
ed at the
a, b and c.
The ellipse is
to be a
of points (in the (x,y)
coordinate system
) each of whose distances from two fixed points
to a
. Similarly, the ellipsoid (being an ellipse in any planar section) is a set of points (x,y,z) each of whose distances from two points (no longer fixed;
to a constant. The two points in any particular direction are on the
of an ellipse at the center of the ellipsoid. With this in mind, it can be said that "It takes two
points to
or a
; it takes three distinct points to specify an ellipse; it takes four distinct points to specify an ellipsoid."
A side note: since the circle is a specific case of the ellipse, Webster's definition could simply state "A solid, all plane sections of which are ellipses."
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Current Search: Mathematics
Clonts, Porscha, Andreasen, Janet, University of Central Florida
Abstract / Description
This study was a qualitative research study dedicated to the deep investigation of a regular and advanced seventh grade mathematics textbook used in Florida and the United Kingdom. A
questionnaire was created for a teacher in both locations, along with the researcher, to rate the textbooks according to different characteristics. The two research questions that were answered
through the research include: 1. In what ways, if any, is diversity represented in the pages of each seventh grade...
Show moreThis study was a qualitative research study dedicated to the deep investigation of a regular and advanced seventh grade mathematics textbook used in Florida and the United Kingdom. A
questionnaire was created for a teacher in both locations, along with the researcher, to rate the textbooks according to different characteristics. The two research questions that were answered
through the research include: 1. In what ways, if any, is diversity represented in the pages of each seventh grade mathematics textbooks examined? a. In what ways is the diversity of each
textbook comparable to the observed diversity of the country in which it is used? 2. How do the seventh grade mathematics textbooks in the United States and the United Kingdom compare with
aspects of appearance, readability, illustrations, content, the teacher's guide/resources, and EL accommodations? These research questions were answered through the questionnaire, follow up
interview, as well as the observed environment. The conclusion to the research was that although these textbooks are from two different countries, they have qualities each teacher liked and
disliked. When I completed the questionnaire I was only able to rate the textbooks according to visual perspectives, while the teachers in each location were able to base their ratings on
tangible classroom experiences. To further my research, I would enjoy being able to teach for a year in each location and then complete the questionnaire again to compare the differences between
my first time completing it and the second time.
Show less
Date Issued
CFH0004684, ucf:45249
Document (PDF)
Washington, Arnita, Dixon, Juli, University of Central Florida
Abstract / Description
The purpose of this study was to determine the effects of literature use in the middle grades mathematics curriculum on student motivation and connections. This study involved collecting several
types of data regarding students' attitudes, motivation, and their abilities to make real-world connections. Findings from pre and post attitude surveys indicated that literature use in the
mathematics curriculum has no effect on students' attitudes towards mathematics. Furthermore, findings from...
Show moreThe purpose of this study was to determine the effects of literature use in the middle grades mathematics curriculum on student motivation and connections. This study involved collecting
several types of data regarding students' attitudes, motivation, and their abilities to make real-world connections. Findings from pre and post attitude surveys indicated that literature use in
the mathematics curriculum has no effect on students' attitudes towards mathematics. Furthermore, findings from journal entries, students' work, and interview responses indicate that although
students find storybooks fun and interesting, their use does not seem to lead to increases in students' understanding of mathematics. However, findings from journal entries, students' work and
interview responses indicated that students were better able to make real-world connections through storybooks that were meaningful to their lives. Suggestions for future research should include
comparative studies on the effects of literature on student performance in middle grades mathematics.
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Date Issued
CFE0000382, ucf:46327
Document (PDF)
Wheeldon, Debra, Dixon, Juli, University of Central Florida
Abstract / Description
This teaching experiment used design-based research (DBR) to document the norms and practices that were established with respect to fractions in a mathematics content course for prospective
elementary teachers. The teaching experiment resulted in an instructional theory for teaching fractions to prospective elementary teachers. The focus was on the social perspective, using an
emergent framework which coordinates social and individual perspectives of development. Social norms,...
Show moreThis teaching experiment used design-based research (DBR) to document the norms and practices that were established with respect to fractions in a mathematics content course for
prospective elementary teachers. The teaching experiment resulted in an instructional theory for teaching fractions to prospective elementary teachers. The focus was on the social perspective,
using an emergent framework which coordinates social and individual perspectives of development. Social norms, sociomathematical norms, and classroom mathematical practices were considered. A
hypothetical learning trajectory (HLT) including learning goals, instructional tasks, tools and imagery, and possible discourse, was conjectured and implemented in the mathematics class. Video
tapes of the class sessions were analyzed for established norms and practices. Resulting social norms were that students would: (a) explain and justify solutions, (b) listen to and try to make
sense of other students' thinking, and (c) ask questions or ask for clarification when something is not understood. Three sociomathematical norms were established. These were expectations that
students would: (a) know what makes an explanation acceptable, (b) know what counts as a different solution, and (c) use meaningful solution strategies instead of known algorithms. Two classroom
mathematical practices with respect to fractions were established. The first was partitioning and unitizing fractional amounts. This included (a) modeling fractions with equal parts, (b) defining
the whole, (c) using the relationship of the number of pieces and the size of the pieces, and (d) describing the remainder in a division problem. The second practice was quantifying fractions and
using relationships among these quantities. This included: (a) naming and modeling fractions, (b) modeling equivalent values, and (c) using relationships to describe fractions. Finally,
recommendations for revising the HLT for a future teaching experiment were made. This will contribute toward the continuing development of an instructional theory for teaching fraction concepts
and operations to prospective elementary teachers.
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Date Issued
CFE0002171, ucf:47526
Document (PDF)
A MATHEMATICAL STUDY OF TWO RETROVIRUSES, HIV AND HTLV-I.
Baxley, Dana, Mohapatra, Ram, University of Central Florida
Abstract / Description
In this thesis, we examine epidemiological models of two different retroviruses, which infect the human body. The two viruses under study are HIV or the human immunodefiency virus and HTLV-I,
which is the human T lymphotropic virus type I. A retrovirus is a virus, which injects its RNA into the host, rather than it's DNA. We will study each of the different mathematical models for
each of the viruses separately. Then we use MATLAB-SIMULINK to analyze the models by studying the...
Show moreIn this thesis, we examine epidemiological models of two different retroviruses, which infect the human body. The two viruses under study are HIV or the human immunodefiency virus and
HTLV-I, which is the human T lymphotropic virus type I. A retrovirus is a virus, which injects its RNA into the host, rather than it's DNA. We will study each of the different mathematical models
for each of the viruses separately. Then we use MATLAB-SIMULINK to analyze the models by studying the reproductive numbers in each case and the disease progression by examining the graphs. In
Chapter 1, we mention basic ideas associated with HIV and HTLV-I. In Chapter 2 some of the basic mathematical model of epidemiology is presented. Chapter 3 is devoted to a model describing the
intra-host dynamics of HIV. Here, we take into account how HIV infects and replicates in the CD4+ T cells. The model studied in this thesis examines the difference between cells, which are
susceptible to the virus, and cells, which are not susceptible. Through the graphs associated with this model, we are able to see how this difference affects disease progression. In Chapter 4, we
examine the effect of HTLV-I virus on human body. The HTLV-I virus causes a chronic infection in humans and may eventually lead to other diseases. In particular, the development of Adult T-cell
Leukemia or ATL is studied in this thesis. The T-cell dynamics and progression to ATL is described using a mathematical model with coupled differential equations. Using mathematical analysis and
SIMULINK, we obtain results on stability, asymptotic stability and the manner of progression of the disease. In Chapter 5 and appendices, we mention our inference and the MATLAB-SIMULINK codes
used in this thesis, so that a reader can verify the details of the work carried out in this thesis.
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Date Issued
CFE0001886, ucf:47398
Document (PDF)
THE EFFECT OF THE MATH CONCEPTS AND SKILLS (MCS) COMPUTER PROGRAM ON STANDARDIZED TEST SCORES AT A MIDDLE SCHOOL IN EAST CENTRAL FLORIDA.
Manning, Cheryl, Sivo, Stephen, University of Central Florida
Abstract / Description
This study measures the effectiveness of the National Computer Systems (NCS) Learn SuccessMaker Math Concepts and Skills computer program on standardized test scores at a middle school in east
central Florida. The NCS Learn Company makes three claims for the SuccessMaker interactive computer program, Math Concepts and Skills (MCS): 1. Student Florida Comprehensive Assessment Test
(FCAT) scores will improve from using the software 30 hours or more; 2. The increase in FCAT scores is directly...
Show moreThis study measures the effectiveness of the National Computer Systems (NCS) Learn SuccessMaker Math Concepts and Skills computer program on standardized test scores at a middle school
in east central Florida. The NCS Learn Company makes three claims for the SuccessMaker interactive computer program, Math Concepts and Skills (MCS): 1. Student Florida Comprehensive Assessment
Test (FCAT) scores will improve from using the software 30 hours or more; 2. The increase in FCAT scores is directly related to the length of time the students' spend using the program; 3. The
software package grading system is equivalent to the FCAT scoring. This study was designed to evaluate each claim. To test the first claim, the FCAT Norm Referenced Test (NRT) Mathematics scale
scores of the 6th-grade middle school students were compared to the same students' previous FCAT scores. The scores were compared before and after they used the Math Concepts and Skills program.
An independent t test was used to compare the scores. There was a statistically significant difference in scale scores when the students used the MCS program for 30 hours or more. Further
investigation is needed to establish the causal effect for the observed differences. To test the second claim, the 6th- and 8th-grade students' time on task in the laboratory was compared to
their change in FCAT scores. A Pearson correlation coefficient of 0.58 was found to exist for the complete 6th-grade data set and a 0.71 correlation for the 8th-grade group. To test the third
claim, the MCS computer program grade equivalent scores were compared to the mathematics FCAT Level using the dependent t test to see if the two scores were equal. The analysis revealed that the
difference in the two scores was statistically significant. Therefore the claim that the two scores are equivalent was not true for this data set. Recommendations were made for future studies to
include qualitative data, a control group, and larger sample sizes. Studying the effect of the Math Concepts and Skills program on FCAT scores continues to be a project for investigation as
implementation of the computer software is contingent on improving FCAT scores.
Show less
Date Issued
CFE0000227, ucf:46267
Document (PDF)
Schaefer Whitby, Peggy, Wienke, Wilfred, University of Central Florida
Abstract / Description
Students with HFA/AS present with a unique set of cognitive deficits that may prevent achievement in the mathematics curriculum, even though they present with average mathematical skills. The
purpose of the study was to determine the effectiveness and efficiency of the use of a modified learning strategy to increase the mathematical word problem solving ability of children with high
functioning autism or Asperger's syndrome; determine if the use of Solve It! increases the self-perceptions...
Show moreStudents with HFA/AS present with a unique set of cognitive deficits that may prevent achievement in the mathematics curriculum, even though they present with average mathematical
skills. The purpose of the study was to determine the effectiveness and efficiency of the use of a modified learning strategy to increase the mathematical word problem solving ability of children
with high functioning autism or Asperger's syndrome; determine if the use of Solve It! increases the self-perceptions of mathematical ability, attitudes towards mathematics and attitudes towards
solving mathematical word problems; and, determine if Solve It! cue cards or a Solve It! multimedia academic story works best as a prime to increase the percentage correct if the student does not
maintain use of the strategy. The subjects were recruited from a central Florida school district. Diagnosis of ASD was confirmed by a review of records and the completion of the Autism Diagnostic
Inventory-Revised (Lord, Rutter, & Le Couteur, 2005). Woodcock Johnson Tests of Achievement (Woodcock, McGrew, & Mather, 2001) subtest scores for reading comprehension and mathematical
computation were completed to identify the current level of functioning. The Mathematical Problem Solving Assessment- Short Form (Montague, 1996) was administered to determine the need for word
problem solving intervention. The subjects were then taught a mathematical word problem solving strategy called Solve It!, during non-content course time at their schools. Generalization data
were collected in each subject's regular education mathematics classroom. Sessions were video-taped, work samples were scored, and then graphed using a multiple baseline format. Three weeks after
the completion of the study, maintenance data were collected. If subjects did not maintain a high use of the strategy, they were entered into the second study to determine if a video prime or
written prime served best to increase word problem solving. The results of the study indicate a functional relationship between the use of the Solve It! strategy and the percentage correct on
curriculum based mathematical word problems. The subjects obtained efficient use of strategy use in five training sessions and applied the strategy successfully for five acquisition sessions.
Percentage correct on mathematical word problems ranged from 20% during baseline to 100% during training and acquisition trials. Error analysis indicated reading comprehension interference and
probable executive functioning interference. Students who did not maintain strategy use quickly returned to intervention level using a prime. Both primes, cue cards and multimedia academic story,
increased performance back to intervention levels for two students. However, one prime, the multimedia academic story and not the cue cards, increased performance back to intervention levels for
one student. Findings of this study show the utility of a modified learning strategy to increase mathematical word problem solving for students with high functioning autism and Asperger's
syndrome. Results suggest that priming is a viable intervention if students with autism do not maintain or generalize strategy use as a means of procedural facilitation.
Show less
Date Issued
CFE0002732, ucf:48151
Document (PDF)
An application of a computerized mathematical model for estimating the quantity and quality of nonpoint sources of pollution from small urban and nonurban watersheds.
Ingraham, Charles John, Wanielista, Martin P., Engineering
Abstract / Description
Florida Technological University College of Engineering Thesis; The problem of "Total Water Management" is reviewed; particular emphasis is given to the magnitude and intensity of pollution from
nonpoint sources. The relationship between land usage in south Florida and subsequent effects upon receiving water bodies is discussed. Basic factors effecting hydrological and ecological
subsystems are illustrated. The U.S. Army Corps of Engineers Urban Storm Water Runoff Mathematical Model, "STORM,"...
Show moreFlorida Technological University College of Engineering Thesis; The problem of "Total Water Management" is reviewed; particular emphasis is given to the magnitude and intensity of
pollution from nonpoint sources. The relationship between land usage in south Florida and subsequent effects upon receiving water bodies is discussed. Basic factors effecting hydrological and
ecological subsystems are illustrated. The U.S. Army Corps of Engineers Urban Storm Water Runoff Mathematical Model, "STORM," is introduced. Model parameters and methodology are discussed. The
mathematical relationships and modeling processes are reviewed and the model is exercised using a "new generation" southeast Florida community (The City of Palm Beach Gardens) as the subject of
study. It is concluded that the model can be beneficial in supporting estimates of pollutant loading to receiving waters from nonpoint sources. Iteration with the model, varying control facility
cost and capacity, provides a cost effective tool for land and water resource planners. However, due to the particular nature of soils, atmospheric and urban conditions in south Florida, the
model should be calibrated with input constants and default values derived to more accurately reflect the southeast Florida environment.
Show less
Date Issued
CFR0003515, ucf:53006
Document (PDF)
Personal Computer Simulation Program for Step Motor Drive Systems.
Koos, William M., Harden, Richard C., Engineering
Abstract / Description
University of Central Florida College of Engineering Thesis; A system of equations modeling a class of step motors known as the permanent magnet rotor step motor is presented. The model is
implemented on a APPLE personal computer in a version of BASIC. Measurements are then made on an existing motor and input to the program for validation. A special test fixture is utilized to take
performance data on the motor to facilitate comparisons with the predictions of the program. The comparisons...
Show moreUniversity of Central Florida College of Engineering Thesis; A system of equations modeling a class of step motors known as the permanent magnet rotor step motor is presented. The model
is implemented on a APPLE personal computer in a version of BASIC. Measurements are then made on an existing motor and input to the program for validation. A special test fixture is utilized to
take performance data on the motor to facilitate comparisons with the predictions of the program. The comparisons show the model is indeed valid for design of step motor drive systems and
emphasize the practical nature of using personal computers and simulations for design
Show less
Date Issued
CFR0008163, ucf:53067
Document (PDF)
Optimization Analysis of a Simple Position Control System.
Cannon, Arthur G., Towle, Herbert C., Engineering
Abstract / Description
Florida Technological University College of Engineering Thesis; One of the problem areas of modern optimal control theory is the definition of suitable performance indices. This thesis
demonstrates a rational method of establishing a quadratic performance index derived from a desired system model. Specifically, a first order model is used to provide a quadratic performance
indix for which a second order system is optimized. Extension of the method to higher order systems, while requiring more...
Show moreFlorida Technological University College of Engineering Thesis; One of the problem areas of modern optimal control theory is the definition of suitable performance indices. This thesis
demonstrates a rational method of establishing a quadratic performance index derived from a desired system model. Specifically, a first order model is used to provide a quadratic performance
indix for which a second order system is optimized. Extension of the method to higher order systems, while requiring more computations, involves no additional theoretical complexities.
Show less
Date Issued
CFR0012011, ucf:53085
Document (PDF)
Hoke, Darlene, Dixon, Juli, University of Central Florida
Abstract / Description
Student performance on measurement concepts in mathematics was the basis for this action research study. This study summarizes research conducted in a seventh grade classroom at an urban middle
school during fall of 2005. The study investigated the practice of using hands-on activities in addition to the standard mathematics curriculum to improve student performance in measurement
tasks. Students were asked to respond to questions posed by both teacher and other students in the classroom....
Show moreStudent performance on measurement concepts in mathematics was the basis for this action research study. This study summarizes research conducted in a seventh grade classroom at an urban
middle school during fall of 2005. The study investigated the practice of using hands-on activities in addition to the standard mathematics curriculum to improve student performance in
measurement tasks. Students were asked to respond to questions posed by both teacher and other students in the classroom. Data were collected using measurement survey, focus group discussions,
math journals, and teacher observations. Results of this study showed that student performance on measurement tasks increased throughout the course of the study. Student gains were recorded and
analyzed throughout the eight-week study period. Twenty-one out of 26 students that participated in the study showed performance growth in measurement concepts.
Show less
Date Issued
CFE0002228, ucf:47890
Document (PDF)
A Multiple Case Study Exploring the Relationship Between Engagement in Model-Eliciting Activities and Pre-Service Secondary Mathematics Teachers' Mathematical Knowledge for Teaching Algebra.
Abassian, Aline, Safi, Farshid, Dixon, Juli, Andreasen, Janet, Bush, Sarah, Bostic, Jonathan, University of Central Florida
Abstract / Description
The goal of this research study was to explore the nature of the relationship between engagement in model-eliciting activities (MEAs) and pre-service secondary mathematics teachers' (PSMTs')
mathematical knowledge for teaching (MKT) algebra. The data collection took place in an undergraduate mathematics education content course for secondary mathematics education majors. In this
multiple case study, PSMTs were given a Learning Mathematics for Teaching (LMT) pre-assessment designed to measure...
Show moreThe goal of this research study was to explore the nature of the relationship between engagement in model-eliciting activities (MEAs) and pre-service secondary mathematics teachers'
(PSMTs') mathematical knowledge for teaching (MKT) algebra. The data collection took place in an undergraduate mathematics education content course for secondary mathematics education majors. In
this multiple case study, PSMTs were given a Learning Mathematics for Teaching (LMT) pre-assessment designed to measure their MKT algebra, and based on those results, three participants were
selected with varying levels of knowledge. This was done to ensure varied cases were represented in order to be able to examine and describe multiple perspectives. The three examined cases were
Oriana, a PSMT with high MKT, Bianca, a PSMT with medium MKT, and Helaine, a PSMT with low MKT. Over the course of five weeks, the three PSMTs were recorded exploring three MEAs, participated in
two interviews, and submitted written reflections. The extensive amount of data collected in this study allowed the researcher to deeply explore the PSMTs' MKT algebra in relation to the given
MEAs, with a focus on three specific constructs(-)bridging, trimming, and decompressing(-) based on the Knowledge of Algebra for Teaching (KAT) framework. The results of this study suggest that
engaging in MEAs could elicit PSMTs' MKT algebra, and in some cases such tasks were beneficial to their trimming, bridging, and decompressing abilities. Exploring MEAs immersed the PSMTs in
generating descriptions, explanations, and constructions, that helped reveal how they interpreted mathematical situations that they encountered. The tasks served as useful tools for PSMTs to have
deep discussions and productive discourse on various algebra topics, and make many different mathematical connections in the process.
Show less
Date Issued
CFE0007143, ucf:52305
Document (PDF)
Examination of an Online College Mathematics Course: Correlation between Learning Styles and Student Achievement.
Steele, Bridget, Dixon, Juli, Hynes, Michael, Haciomeroglu, Erhan, Hopp, Carolyn, Dziuban, Charles, University of Central Florida
Abstract / Description
The purpose of this study was to determine if there was a significant relationship between learning styles and student learning outcomes in an online college mathematics course. Specifically, the
study was guided by two research questions focused on (a) the extent to which learning styles had a predictive relationship with student achievement in an online college mathematics course and
(b) the extent to which various learning styles among mathematics students in online versus face-to-face...
Show moreThe purpose of this study was to determine if there was a significant relationship between learning styles and student learning outcomes in an online college mathematics course.
Specifically, the study was guided by two research questions focused on (a) the extent to which learning styles had a predictive relationship with student achievement in an online college
mathematics course and (b) the extent to which various learning styles among mathematics students in online versus face-to-face courses predicted mathematics achievement. The population for this
study consisted of the 779 college mathematics and algebra (CMA) students who were enrolled in a private multimedia university located in the southeast. A total of 501 students were enrolled in
the online class, i.e., the experimental group, and 278 students enrolled in the face-to-face class comprised the control group. All students completed (a) an initial assessment to control for
current mathematics knowledge, (b) the online Grasha-Reichmann Student Learning Styles Scales (GRSLSS) Inventory, and (c) 20 questions selected from the NAEP Question Tool database. Hierarchical
linear regressions were used to address both research questions. A series of ANCOVA tests were run to examine the presence of any relationships between a given demographic and course modality
when describing differences between student test scores while controlling for prior academic performance. The results indicated that predominant learning style had no apparent influence on
mathematics achievement. The results also indicated that predominant learning style had no apparent influence on mathematics achievement for online students. When examining demographics alone
without respect to modality, there was no significance in course performance between students in various ethnicity, gender, or age groups.
Show less
Date Issued
CFE0004445, ucf:49320
Document (PDF)
Braddock, Stacey, Dixon, Juli, University of Central Florida
Abstract / Description
The purpose of this action research study was to evaluate my own practice of teaching basic multiplication facts to fourth graders. I wanted to see how focusing my instruction on strategies would
help my students develop proficiency in basic multiplication facts. I chose this topic because Florida was in the process of shifting to new standards that encourage teaching for deeper meaning.
I hoped this research would give my students the opportunity to make sense of multiplication on a deeper...
Show moreThe purpose of this action research study was to evaluate my own practice of teaching basic multiplication facts to fourth graders. I wanted to see how focusing my instruction on
strategies would help my students develop proficiency in basic multiplication facts. I chose this topic because Florida was in the process of shifting to new standards that encourage teaching for
deeper meaning. I hoped this research would give my students the opportunity to make sense of multiplication on a deeper level, while giving me insight into how students learn multiplication.
Through this study, I learned that students initially find multiplication to be very difficult, but they can solve basic facts with ease when using strategies. Students did become more proficient
with basic multiplication facts, and they were also able to apply basic fact strategies to extended facts and other multidigit multiplication problems. There is a limited amount of research on
how students acquire basic multiplication fact proficiency; however, this study offers more insight to teachers and the research community.
Show less
Date Issued
CFE0003023, ucf:48370
Document (PDF)
Math Remediation for High School Freshmen.
Borhon, Kambiz, Boote, David, Hynes, Mike, Gunter, Glenda, Miller, Margaret, University of Central Florida
Abstract / Description
This study is an attempt to address the problem associated with a high percentage of freshman students, at a private Christian high school in Florida, who either fail Algebra 1 or pass with a low
percentage rate. As a result, these students either retake Algebra 1 or continue on(-)being inadequately prepared to successfully pass Geometry and Algebra 2. This study concentrates on the
student background knowledge of mathematics, which is among the causes associated with this problem, and...
Show moreThis study is an attempt to address the problem associated with a high percentage of freshman students, at a private Christian high school in Florida, who either fail Algebra 1 or pass
with a low percentage rate. As a result, these students either retake Algebra 1 or continue on(-)being inadequately prepared to successfully pass Geometry and Algebra 2. This study concentrates
on the student background knowledge of mathematics, which is among the causes associated with this problem, and proposes remediation. As such, a mathematics remediation course is designed and
implemented for a select number of incoming freshmen. This study includes a correlational examination to determine a possible correlation between students' background knowledge of the middle
school mathematics and predicts a possible failure or successful completion of Algebra I in high school. In addition, it purposes a two-stage evolution plan in order to determine the
effectiveness of the design of the remedial course as well as its effectiveness. Undertaking the design evaluation, this study uses a mixed-modes design consisting of a qualitative (interview and
observation) of a number of participants and a quantitative examination (survey) of a larger sample. The correlational study indicates that there is a positive and moderately strong correlation
between students' background knowledge in (middles school) mathematics and their grades in Algebra 1. The evaluation concludes that students find the design of the MIP program helpful and
aesthetically appealing; however, its usability did not meet the evaluation criteria. Furthermore, the MIP Program Manager and teacher are fully satisfied with its design, content, and
Show less
Date Issued
CFE0005581, ucf:50251
Document (PDF)
A comparison of eighth-grade mathematics scores by state and by the four census-defined regions of national assessment of educational progress (NAEP).
Robinson, Laurel, Taylor, Rosemarye, Pawlas, George, Little, Mary, Clark, Margaret, University of Central Florida
Abstract / Description
The purpose of this study was to investigate the information regarding the comparative relationship between the proficient mathematics scores of eighth-grade students on the 2009 state
mathematics assessments and the 2009 National Assessment of Educational Progress (NAEP) mathematics assessment by state, census (-)defined regions and AYP subgroups. Analysis was completed and
six research questions were used to guide the study. A multiple regression was used to assess the relationship between...
Show moreThe purpose of this study was to investigate the information regarding the comparative relationship between the proficient mathematics scores of eighth-grade students on the 2009 state
mathematics assessments and the 2009 National Assessment of Educational Progress (NAEP) mathematics assessment by state, census (-)defined regions and AYP subgroups. Analysis was completed and
six research questions were used to guide the study. A multiple regression was used to assess the relationship between the percentage of eighth-grade students who were proficient in mathematics
as assessed by the 2009 NAEP and those who were proficient in mathematics as assessed by their 2009 state assessment. A significant quadratic (non-linear) relationship between the state and NAEP
levels of proficiency was determined. Several two-factor split plot (one within-subjects factor and one between-subjects factor) analysis of variance (ANOVA) were conducted to determine if region
moderated the difference between the percentage proficient on the state and NAEP assessments for eighth grade students overall and in the following AYP subgroups : (a) low socioeconomic students,
(b) white students, (c) black students and (d) Hispanic students. The within-subjects factor was type of test (NAEP or state), and the between-subjects factor was region (Midwest, Northeast,
West, and South). Overall, the percentage proficient on state mathematical assessments was always higher than the percentage proficient on the NAEP mathematics assessments. The degree of
discrepancy is discussed, as well as possible reasons for this divergence of scores.
Show less
Date Issued
CFE0005241, ucf:50599
Document (PDF)
An Examination of the Algebra 1 Achievement of Black and Hispanic Student Participants in a Large Urban School District's Mathematics Intervention Program.
Bronson, Elethia, Taylor, Rosemarye, Baldwin, Lee, Storey, Valerie A., Andreasen, Janet, University of Central Florida
Abstract / Description
The mathematics achievement gap between Black and White as well as Hispanic and White students has been well documented nationwide and in the school district of study. Much has been written in
observance of the achievement gap, yet markedly less research has focused on practices and interventions that have improved mathematics performance for Black and Hispanic students. Consequently,
this study examined the Algebra 1 achievement (indicated by student scale scores on the Florida Standards...
Show moreThe mathematics achievement gap between Black and White as well as Hispanic and White students has been well documented nationwide and in the school district of study. Much has been
written in observance of the achievement gap, yet markedly less research has focused on practices and interventions that have improved mathematics performance for Black and Hispanic students.
Consequently, this study examined the Algebra 1 achievement (indicated by student scale scores on the Florida Standards Assessments Algebra 1 End-of-Course exam) of Black and Hispanic students
participating in a mathematics intervention program as compared to the Algebra 1 achievement of their similar non-participating peers in one large urban school district. Descriptive statistics
and inferential statistical analysis via the one-way ANOVA and the independent samples t-test were utilized. Further quantitative analysis was conducted focusing on the mean scale score
differences among intervention program participants in varying course structures, summer days attended, and school socioeconomic status. The study found that Black and Hispanic 7th grade program
participants significantly outperformed their similar non-participating 7th grade peers and non-participating Black and Hispanic 9th grade students. No statistically significant differences were
found among program participants who attended the summer preview camp for different numbers of days. Black and Hispanic intervention program participants enrolled in a double-block Algebra 1
course numerically outscored their single-period program peers overall and when disaggregated by race/ethnicity and prior year achievement level. The findings indicate the intervention program
has the potential to improve Algebra 1 achievement and increase access to advanced-level mathematics for Black and Hispanic students. This study contributes to the scant literature on successful
mathematics intervention programs targeting Black and Hispanic students. Studying the implementation of the program in schools demonstrating success could provide insight, enabling other schools
to replicate an environment where Black and Hispanic secondary mathematics learners thrive.
Show less
Date Issued
CFE0007393, ucf:52073
Document (PDF)
Talking Back: Mathematics Teachers Supporting Students' Engagement in a Common Core Standard for Mathematical Practice: A Case Study.
Sotillo Turner, Mercedes, Dixon, Juli, Ortiz, Enrique, Gresham, Gina, Dieker, Lisa, University of Central Florida
Abstract / Description
The researcher in this case study sought to determine the ways in which teachers support their students to create viable arguments and critique the reasoning of others (SMP3). In order to achieve
this goal, the self-conceived classroom roles of two teachers, one experienced and one novice, were elicited and then compared to their actualized roles observed in the classroom. Both teachers
were provided with professional development focused on supporting student engagement in SMP3. This...
Show moreThe researcher in this case study sought to determine the ways in which teachers support their students to create viable arguments and critique the reasoning of others (SMP3). In order
to achieve this goal, the self-conceived classroom roles of two teachers, one experienced and one novice, were elicited and then compared to their actualized roles observed in the classroom. Both
teachers were provided with professional development focused on supporting student engagement in SMP3. This professional development was informed by the guidelines that describe the behaviors
students should exhibit as they are engaged in the standards for mathematical practice contained in the Common Core State Standards for Mathematics. The teachers were observed, video recorded,
and interviewed during and immediately after the professional development. A final observation was performed four weeks after the PD. The marked differences in the teachers' characteristics
depicted in each case added to the robustness of the results of the study. A cross-case analysis was performed in order to gauge how the novice and experienced teachers' roles compared and
contrasted with each other. The comparison of the teachers' self-perception and their actual roles in the classroom indicated that they were not supporting their students as they thought they
were. The analysis yielded specific ways in which novice and experienced teachers might support their students. Furthermore, the cross-case analysis established the support that teachers are able
to provide to students depends on (a) teaching experience, (b) teacher content and pedagogical knowledge, (c) questioning, (d) awareness of communication, (e) teacher expectations, and (f)
classroom management. Study results provide implications regarding the kinds of support teachers might need given their teaching experience and mathematics content knowledge as they attempt to
motivate their students to engage in SMP3.
Show less
Date Issued
CFE0005553, ucf:50275
Document (PDF)
EXPLORING THE EXPERIENCES OF LEARNING MATHEMATICS FOR A CHILD WITH CANCER: A CASE STUDY.
Bello, Elizabeth M, Nickels, Megan, University of Central Florida
Abstract / Description
In this research report, I utilize interpretative phenomenological analysis (Smith, Flowers, & Larkin, 2009) to examine the mathematics education experiences of a child with cancer. Two
qualitative interviews with a 13-year-old male patient with Hodgkin's Lymphoma and his mother were analyzed. Findings revealed several storylines or themes: living with cancer, environmental
barriers, and mathematics in virtual school. Grade level mathematics, content knowledge, and delivery during treatment...
Show moreIn this research report, I utilize interpretative phenomenological analysis (Smith, Flowers, & Larkin, 2009) to examine the mathematics education experiences of a child with cancer. Two
qualitative interviews with a 13-year-old male patient with Hodgkin's Lymphoma and his mother were analyzed. Findings revealed several storylines or themes: living with cancer, environmental
barriers, and mathematics in virtual school. Grade level mathematics, content knowledge, and delivery during treatment in comparison to the child's healthy peers are also discussed.
Show less
Date Issued
CFH2000250, ucf:46003
Document (PDF)
Varn, Theresa, Dixon, Juli, University of Central Florida
Abstract / Description
ABSTRACT The purpose of this study is to describe the effect of a curriculum rich in spatial reasoning activities and experiences on the ability of my fifth grade students to spatially reason.
The study was conducted to examine 1) the effects of my practice of incorporating spatial reasoning lessons and activities in my fifth-grade mathematics classroom on the students' ability to
spatially reason and 2) the effects of my practice of incorporating spatial reasoning lessons and activities on...
Show moreABSTRACT The purpose of this study is to describe the effect of a curriculum rich in spatial reasoning activities and experiences on the ability of my fifth grade students to spatially
reason. The study was conducted to examine 1) the effects of my practice of incorporating spatial reasoning lessons and activities in my fifth-grade mathematics classroom on the students' ability
to spatially reason and 2) the effects of my practice of incorporating spatial reasoning lessons and activities on my students' ability to problem solve. Data were collected over a ten-week
period through the use of student interviews, anecdotal records, photos of student work, student journals, pre- and posttests and a post-study survey. In this study, students demonstrated a
statistically significant increase on all pre- and posttests. The student interviews, anecdotal records, photos of student work, and student journals all revealed spatial reasoning was used in
mathematics problem solving. The study suggests that spatial reasoning can be taught and spatial reasoning skills can be used in problem solving.
Show less
Date Issued
CFE0000351, ucf:46295
Document (PDF)
Clanton, Barbara, Dixon, Juli, University of Central Florida
Abstract / Description
This study is an examination of whether a project-based mathematics curriculum would influence students' intended career paths related to science, technology, engineering, and mathematics (STEM)
endeavors; perceived usefulness of mathematics; and perceived competence in doing mathematics. A review of the literature revealed that there are many shortages of professionals in STEM fields.
United States women and men are not pursuing STEM endeavors in great numbers and the U.S. relies heavily on...
Show moreThis study is an examination of whether a project-based mathematics curriculum would influence students' intended career paths related to science, technology, engineering, and
mathematics (STEM) endeavors; perceived usefulness of mathematics; and perceived competence in doing mathematics. A review of the literature revealed that there are many shortages of
professionals in STEM fields. United States women and men are not pursuing STEM endeavors in great numbers and the U.S. relies heavily on international students to fill this gap. The literature
revealed that the girls who do not pursue STEM endeavors in great numbers do not perceive mathematics as a useful endeavor and do not think they are competent in doing mathematics. Boys who do
not pursue STEM endeavors in great numbers also do not perceive mathematics as a useful endeavor. The study involved 7th and 8th grade school students enrolled in algebra classes in a private
college-preparatory school. The students in the experimental group participated in a problem-based curriculum that integrated lecture-based methods with four major projects designed to have
students apply mathematics out of the context through hands-on real-life problems. This particular quasi-experimental design was a nonequivalent pre-test/post-test control group design.
Statistical analyses were done using a general linear model repeated measures. The results of the statistical analyses indicated that the students in the project-based group showed a
statistically significant positive change in their perceived usefulness of mathematics when compared to the control group. A t-test revealed no statistically significant differences in academic
achievement. Qualitative data analysis uncovered three emergent themes. Students indicated that they saw the usefulness of mathematics more clearly; students' independence from the teacher while
doing the projects was unsettling; and students enjoyed the change of pace in class. The results of the study indicated that a project-based mathematics curriculum can help students see the
usefulness of mathematics and can help students enjoy the pursuit of mathematics by this particular change of routine.
Show less
Date Issued
CFE0000907, ucf:46765
Document (PDF)
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Maria CrăciunMaria Crăciun, Author at Tiberiu Popoviciu Institute of Numerical Analysis
We present a detailed investigation of the properties of the galactic rotation curves in the Weyl geometric gravity model, in which the gravitational action is constructed from the square of the Weyl
curvature scalar, and of the strength of the Weyl vector. The theory admits a scalar–vector–tensor representation, obtained by introducing an auxiliary scalar field. By assuming that the Weyl vector
has only a radial component, an exact solution of the field equations can be obtained, which depends on three integration constants, and, as compared to the Schwarzschild solution, contains two new
terms, linear and quadratic in the radial coordinate. In the framework of this solution we obtain the exact general relativistic expression of the tangential velocity of the massive test particles
moving in stable circular orbits in the galactic halo. We test the theoretical predictions of the model by using 175 galaxies from the Spitzer Photometry & Accurate Rotation Curves (SPARC) database.
We fit the theoretical predictions of the rotation curves in conformal gravity with the SPARC data by using the Multi Start and Global Search methods. In the total expression of the tangential
velocity we also include the effects of the baryonic matter, and the mass to luminosity ratio. Our results indicate that the simple solution of the Weyl geometric gravity can successfully account for
the large variety of the rotation curves of the SPARC sample, and provide a satisfactory description of the particle dynamics in the galactic halos, without the need of introducing the elusive dark
matter particle.
Maria Crăciun
‘Tiberiu Popoviciu’ Institute of Numerical Analysis, Romanian Academy, Cluj, Romania
Tiberiu Harko
Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering (IFIN-HH), Bucharest, Romania
Department of Physics, Babes-Bolyai University, Kogalniceanu Street, Cluj-Napoca, Romania
Astronomical Observatory, Romanian Academy, 19 Ciresilor Street, Cluj-Napoca, Romania
Corresponding author at: Department of Physics, Babes-Bolyai University, Kogalniceanu Street, Cluj-Napoca, Romania.
M. Crăciun, T. Harko, Testing Weyl geometric gravity with the SPARC galactic rotation curves database, Physics of the Dark Universe, 43 (2024), art. no. 101423, https://doi.org/10.1016/
Physics of the Dark Universe
et al. Dark matter as a geometric effect in f(r) gravity, Astropart. Phys. (2008)
et al., Dark matter and background light, Phys. Rep. (2004)
et al. Weyl vs. conformal, Phys. Lett. B (2016)
et al. Universal properties of galactic rotation curves and a first principles derivation of the Tully–Fisher relation, Phys. Lett. B (2018)
et al., Non-minimal geometry–matter couplings in Weyl–Cartan space–times: f(r,t,q,tm) gravity, Phys. Dark Universe(2021)
et al., Cyclic cosmology, conformal symmetry and the metastability of the Higgs, Phys. Lett. B (2013)
Planck 2018 results: VI. Cosmological parameters, Astron. Astrophys. (2020)
et al., Baryonic solutions and challenges for cosmological models of dwarf galaxies, Nat. Astron. (2022)
et al., The structure of cold dark matter halos, Astrophys. J. (1996)
et al., Dark and luminous matter in THINGS dwarf galaxies, Astron. J. (2011)
Abstract We present a detailed investigation of the properties of the galactic rotation curves in the Weyl geometric gravity model, in which…
Abstract This paper deals with the light travel time effect in binary systems with variable components. The methods for estimation…
AbstractThe nature of one of the fundamental components of the Universe, dark matter, is still unknown. One interesting possibility is…
Book summaryUmbral calculus is used in this book for…? AuthorM. Craciun Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy Book…
Book summaryOriginal automatic algorithms for processing nonstationary time series containing a stationary noise superposed over a nonmonotonic trend are presented.…
AbstractIn this note we construct a class of approximation operators using polynomial sequences of binomial type. We compute the expression…
AbstractThe average conditional jumps for a white noise superposed on a linear trend are computed both theoretically and numerically and…
Abstract In this paper we introduce a class of positive linear operators by using the “umbral calculus”, and we study…
AbstractIn this note we consider an approximation operator of Kantorovich type in which expression appears a basic sequence for a…
AbstractIn this note we consider a general compound approximation operator using binomial sequences and we give a representation for its…
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How to Write Efficient Algorithms
How to Write Efficient Algorithms - 14/12/2023
How to Write Efficient Algorithms
The art of writing efficient algorithms is a highly sought after skill, as it can drastically improve the performance of any project. Not only do efficient algorithms require fewer resources to run,
they can also provide better results in less time. In this blog post, we will explore some of the key principles behind writing efficient algorithms, as well as some tips and best practices to keep
in mind.
What is an Algorithm?
An algorithm is a set of instructions for a computer to carry out a specific task. It is usually expressed as a sequence of steps that are executed in the same order each time. It can be written in
any programming language, including Python, Java, and C++.
Algorithms can range from simple tasks such as sorting a list of numbers, to complex tasks such as Artificial Intelligence (AI) applications. No matter what the task is, the goal is always to create
an algorithm that is efficient and performs the required task quickly and accurately.
Principles for Writing Efficient Algorithms
There are various principles to keep in mind when writing efficient algorithms. Let’s take a look at each one in more detail.
Minimize Resource Usage
The most important principle when writing efficient algorithms is to minimize resource usage. This means that your algorithm should use the fewest possible resources, including memory, CPU time, and
disk space.
This can be achieved by following two main strategies:
1. Minimizing the amount of data that your algorithm needs to work with. This can be done by eliminating unnecessary data and selecting only the data that is absolutely necessary for the task.
2. Minimizing the number of operations that your algorithm needs to perform. This can be done by reordering operations to minimize the number of steps in the algorithm, and by using as few
operations as possible.
Choose the Right Data Structures
Another important principle when writing efficient algorithms is to choose the right data structures. Different data structures have different properties that can have a significant impact on the
performance of your algorithm.
For example, if you are writing an algorithm to sort a list of numbers, you may choose to use a selection sort algorithm, which uses a selection tree data structure. This data structure is ideal for
sorting because it allows for quick selection of the smallest number in the list.
Alternatively, if you are writing an algorithm to search for a number in a list, you may choose to use a binary search tree data structure. This data structure is ideal for searching because it can
quickly narrow down the list of numbers until the desired number is found.
Use Efficient Algorithms
When writing an algorithm, it is important to use efficient algorithms that are designed to solve the problem at hand. This means that you should select an algorithm that is tailored specifically to
the task you are trying to accomplish.
For example, if you are writing an algorithm to sort a list of numbers, you should use an efficient sorting algorithm such as quicksort or mergesort, as these algorithms are designed specifically for
It is important to note that just because an algorithm is available does not mean that it is the most efficient one for the job. For example, you could use the selection sort algorithm to search for
a number in a list, but this would be much less efficient than using a binary search tree data structure.
Use Dynamic Programming
Dynamic programming is a computer programming technique that can be used to solve complex problems by breaking them down into smaller and simpler sub-problems. This technique can be used to improve
the efficiency of algorithms by avoiding unnecessary calculations and reducing the number of steps that need to be performed.
For example, if you are writing an algorithm to solve a problem that involves traversing a maze, you could use dynamic programming to avoid repeating calculations for each step along the way. This
technique would reduce the amount of time needed to solve the problem and make your algorithm more efficient.
Tips and Best Practices
In addition to the principles outlined above, there are also some tips and best practices you should keep in mind when writing efficient algorithms. These include:
Test and Profile Your Algorithm
Once you have written an algorithm, it is important to test it and profile it to ensure that it is running as efficiently as possible. This will help you identify any potential bottlenecks or
inefficiencies in the algorithm, and make adjustments as needed.
You can use tools such as performance profilers to measure the performance of your algorithm and identify any areas that need to be improved. This will help you optimize your algorithm and make it as
efficient as possible.
Keep It Simple
When writing algorithms, it is important to keep things as simple as possible. This means avoiding unnecessary complexities and focusing on the core concepts that are needed to solve the problem.
Complexity often leads to inefficiencies, as it requires more resources and more steps to execute. By keeping things simple, you can create an algorithm that is more efficient and easier to debug.
Reuse Existing Solutions
Whenever possible, it is a good idea to reuse existing solutions rather than create brand new ones. This will save you time, as you won’t need to reinvent the wheel, and you can also benefit from any
optimizations or improvements that have already been made to the existing solution.
If you do decide to use an existing solution, be sure to test and profile it to ensure that it is as efficient as possible. You may find that some tweaks can be made to improve its performance.
Writing efficient algorithms is an important skill, as it can drastically improve the performance of any project. By following the principles outlined in this blog post, as well as some tips and best
practices, you can create algorithms that are efficient and perform their tasks quickly and accurately.
Good luck!
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Gapless chiral spin liquid from coupled
SciPost Submission Page
Gapless chiral spin liquid from coupled chains on the kagome lattice
by Rodrigo G. Pereira, Samuel Bieri
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Samuel Bieri · Rodrigo Pereira
Submission information
Preprint Link: https://arxiv.org/abs/1706.02421v2 (pdf)
Date accepted: 2017-12-23
Date submitted: 2017-11-17 01:00
Submitted by: Pereira, Rodrigo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Condensed Matter Physics - Theory
Approach: Theoretical
Using a perturbative renormalization group approach, we show that the extended ($J_1$-$J_2$-$J_d$) Heisenberg model on the kagome lattice with a staggered chiral interaction ($J_\chi$) can exhibit a
gapless chiral quantum spin liquid phase. Within a coupled-chains construction, this phase can be understood as a chiral sliding Luttinger liquid with algebraic decay of spin correlations along the
chain directions. We calculate the low-energy properties of this gapless chiral spin liquid using the effective field theory and show that they are compatible with the predictions from parton
mean-field theories with symmetry-protected line Fermi surfaces. These results may be relevant to the state observed in the kapellasite material.
Author comments upon resubmission
The referee has raised very good points. First, we would like to clarify the meaning of right-moving bosons for crossed chains. We label R and L modes in the limit of decoupled chains in such a way
that R (L) refers to the modes that propagate in the positive-x (negative-x) direction of q-chains as indicated by the arrows in Fig. 1. Thus, the direction of propagation of R/L modes at a given
point depends on the q index. This coordinate system with three different positive-x directions may seem confusing, but it was introduced by Gong et al. in Ref. [37] because it is convenient for
analyzing the threefold rotational symmetry of the Hamiltonian.
Second, we agree with the referee that our cosine potential differs significantly from a conventional sine-Gordon model. Nonetheless, we have rewritten the discussion around Eq. (50) to emphasize
that the interactions are local in the sense that the operators at different positions (in two dimensions) commute when the distances are larger than the short-distance cutoff. (Note this does not
happen for more general cosine potentials with non-integer scaling dimensions.) We interpret this as a sign that the right-moving bosons can be gapped out entirely, leaving only gapless left-moving
bosons. We cannot prove the latter statement rigorously, but we show that the low-energy fixed point obtained within this assumption is stable and the theory is consistent. Clearly, the phase in
which the coupling between chiral currents is the strongest interaction must be different from the magnetically ordered (cuboc) and valence-bond-crystal phases. In our opinion, our proposal of a
gapless chiral spin liquid is the simplest and most plausible picture that emerges from the RG analysis.
Concerning the parton construction, we clarify that the point here was to show that there are at least two mean-field ansatze which are able to reproduce the properties of the chiral spin liquid
derived from the coupled-chain approach. This is not a meaningless exercise because, once identified, the parton construction can be useful to analyze the properties of the chiral spin liquid in a
regime where the starting point of weakly coupled chains is not reliable. The 1/r^2 decay of the correlation function indeed imposes a strong constraint on the theories that we can select, but it is
not the only constraint. Our analysis is also guided by symmetry, as the mean-field ansatz must break time reversal and the reflection by lines perpendicular to the chain directions, but respect the
symmetry of reflection by lines parallel to the chains. Out of large number of U(1) chiral spin liquids on the kagome lattice that have been classified in Ref. [45], we have narrowed the choice down
to one. The other possible mean-field ansatz uses a Majorana fermion representation with a Z2 gauge structure. Selecting between the Z2 and U(1) parton constructions requires numerical methods to
calculate the ground state energy (using e.g. variational Monte Carlo techniques) and it is beyond the scope of this paper. Most numerical studies of chiral spin liquids so far have focused on gapped
phases (with the notable exception of Ref. [70]). We hope that our work will encourage the search for gapless chiral spin liquid phases.
List of changes
1. In the legend of Fig. 1, we now comment that the arrows indicate the positive-x directions, which become the directions of right-moving bosons for each q=1,2,3 in the continuum limit.
2. To address the referee’s concerns about the relevant cosine potential, we have rewritten the discussion around Eq. (50).
3. In section 6.4, we have included the energy magnetization correction in the expression for the thermal Hall conductivity, citing new references [79] and [80]. This correction does not change the
conclusion that the thermal Hall response of the gapless chiral spin liquid vanishes by symmetry.
Published as SciPost Phys. 4, 004 (2018)
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2017-12-14 (Invited Report)
• Cite as: Anonymous, Report on arXiv:1706.02421v2, delivered 2017-12-14, doi: 10.21468/SciPost.Report.294
1- The authors approach the problem from different perspectives (coupled chains and parton mean fields), and find not inconsistent results.
2- The authors provide a detailed account of their calculation.
3- The topic is timely.
4- The approach is in principle a very good one to address questions such as "is it in principle possible to find this or that phase", as is the goal of the paper.
1- The bosonization procedure is still unclear
The authors have responded well to the comments raised in the first round. The changes in the manuscript in my view did increase the understandability of the paper, and I thus recommend the work for
I have one final comment on Eq. (50). The authors state that the argument of the cosine does not commute with itself at different positions. In bosonization, that means that the field cannot order -
I hope the authors agree on that.
The authors then show that the cosine as a whole commutes with itself at different positions and state that "the most plausible scenario" is that the fields are gapped. It is mathematically unclear
to me why this "most plausible scenario" is realised, given that the fields themselves cannot be pinned. Is there a way to understand this, or do I have to accept this as, really a guess? In other
words, I find the physics plausible, but currently feel like there is a leap of faith from writing down Eq. (50) to saying that the right-moving fields are pinned. Sure, the phase where the
right-movers are gapped might be a stable fixed point, but it feels to me like the authors are writing down a Hamiltonian they cannot solve, and then state that there is a stable fixed point, of
which they do not show under which conditions the model realises it, and how that is related to the chiral interaction. Is that true?
I thus ask the authors to re-clarify this point.
Requested changes
1- Clear up the commutation of the argument of the cosine in Eq. (50).
We thank the referee for his/her helpful comments. We have added some clarification to emphasize that the full gap in one chiral sector is a conjecture since we cannot pin the bosonic fields in the
usual way. Although the gap scenario seems plausible and physically compelling, this particular step in our analysis is indeed an assumption which we are not able to rigorously prove. Nonetheless,
our derivation of the field theory from the lattice model and the perturbative renormalization group analysis are completely unbiased. Similarly, the calculation of physical properties is well
controlled once we make this assumption about the strong coupling regime. It is worth mentioning that the lack of analytical techniques to handle non-commuting bosonized cosine terms is not unique to
our problem. See, for instance, D. Bulmash et al., Phys. Rev. B 96, 045134 (2017) for a recent discussion of other examples.
We stress that the theory relying on our conjecture is consistent as it describes a stable fixed point of the renormalization group flow. Moreover, we clearly state that the chiral spin liquid phase
can appear for strong chiral interaction, while valence-bond crystal and magnetic orders are expected to arise for small J_\chi. The validity of the conjecture could be tested by numerical
simulations of the lattice model, by computing correlation functions that involve excitations of the gapped modes, such as dimer-dimer or the staggered magnetization mentioned in section 6.1.
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Education and Early Childhood Learning
Critical thinking in mathematics involves the ability to compare, evaluate, critique, justify, test, and validate ideas, representations, plans, or solutions using logical arguments, criteria, and
evidence. It requires metacognition in learners, enabling them to solve mathematical problems and situations, communicate their reasoning effectively, and make ethical decisions.
• Learners research, use, and think about a variety of ideas and information strategically, efficiently, and effectively to make decisions and choices.
• Learners evaluate their own and others’ ideas, as well as possible solutions, by considering different perspectives, biases, and the validity and relevance of supporting sources.
• Learners use inductive reasoning to explore and record results; to analyze mathematical ideas, problems, and situations; to make observations and generalizations from patterns; and to test these
generalizations based on criteria and evidence.
• Learners recognize that certain math beliefs influence how they perceive themselves as math learners.
• Learners demonstrate a willingness to reconsider their own thinking and to consider others’ thinking about mathematical ideas, problems, or situations.
• Learners ask relevant and clarifying questions to further learning and enhance comprehension of mathematical ideas, concepts, problems, and situations.
• Learners make judgments based on thoughtful criteria to then make decisions and solve mathematical problems and situations, enabling them to take action in an informed manner.
• Learners use deductive reasoning to solve mathematical problems and situations, reach new conclusions based on what is already known or assumed to be true, and make ethical decisions.
Creativity in mathematics involves flexible thinking, curiosity, and risk taking, as well as making connections to prior knowledge among learners; this allows learners to come up with innovative
solutions to a variety of mathematical problems and situations by considering them from a new angle or by formulating new hypotheses.
• Learners embrace a learning environment of trust and respect that encourages them to make choices, take risks, and think flexibly—allowing them to make decisions and take action.
• Learners wonder, ask questions, and contemplate different mathematical ideas and concepts.
• Learners solve mathematical problems and situations using different ways to arrive at innovative solutions.
• Learners enrich and refine their reasoning by considering others’ ideas.
• Learners formulate, adjust, and refine their plans for solving mathematical problems and situations by looking at them from a new angle.
• Learners validate and adapt plans, ideas, strategies, or solutions, while persevering through obstacles, so they can improve at solving mathematical problems and situations.
• Learners seek and use feedback from others to develop and consolidate their conceptual understanding, deepen their reasoning, and reflect on their processes for solving mathematical problems and
Citizenship in mathematics involves the development of mathematical literacy that enables the application of mathematical ideas and concepts in a variety of everyday contexts, awakening learners'
curiosity about their role as citizens who can actively contribute to society, think critically about the world, make informed decisions, and generate solutions to an issue from a variety of
• Learners use mathematics as a means of developing their understanding of a range of complex social, cultural, economic, and political issues, and to help them reflect on them.
• Learners mobilize their mathematical knowledge and skills to analyze and understand issues related to discrimination, equity, and human rights by investigating or proposing solutions to a variety
of mathematical problems or situations related to these issues.
• Learners mobilize their mathematical knowledge and skills to explore, analyze, and understand the impact of the interconnectedness of self, others, and the natural world by investigating or
proposing solutions to a variety of mathematical problems and situations related to this issue.
• Learners show interest in others’ approaches to mathematics and to different points of view, experiences, and worldviews, allowing them to better understand and solve mathematical problems and
• Learners empathize with others whose ideas are different from their own and appreciate solutions to mathematical problems or situations proposed by others.
• Learners interact and learn with others in person or online in a responsible, respectful, and inclusive manner by welcoming and valuing diverse viewpoints, and by considering a range of ideas and
perspectives when contributing to mathematical exchanges.
• Learners realize that their mathematical knowledge and skills will serve not only to improve their own quality of life but also that of others.
• Learners engage in meaningful mathematical inquiries, individually and in collaboration, in which they ask themselves and others questions so they can find equitable solutions and make ethical
• Learners appreciate how mathematics can be used to make and justify ethical decisions that lead to responsible and sustainable actions that affect themselves, their community, and the world.
Connection to self in mathematics involves the learner’s belief in their ability to approach and complete tasks, solve mathematical problems and situations, and persevere in the face of mathematical
challenges. It also involves the learner’s ability to engage positively in reflective practices about their learning in order to set goals for self-improvement.
• Learners believe in their ability to learn and understand the world of mathematics and its impact on their daily lives.
• Learners recognize the elements that shape their identity as math learners, and they see themselves as mathematicians.
• Learners allow themselves the time they need, and they implement strategies that foster a growth mindset to develop a positive relationship with mathematics.
• Learners consider reflecting on their own decisions, the efforts they deploy, the experiences they have, and feedback from others as learning opportunities to improve their knowledge and skills
in mathematics.
• Learners reflect on their mathematical learning to set goals and make informed decisions that affect their well-being.
• Learners believe that their ability to learn, their talents, and their skills in mathematics will continue to improve throughout their lives through their hard work, perseverance, and effort.
• Learners are willing to take risks, ask for help, and persevere, despite obstacles.
• Learners demonstrate the ability to make changes and adapt to new mathematical contexts, knowing that they will learn from their mistakes and build on their personal strengths.
• Learners develop their autonomy, value their voice, and commit to their role in becoming lifelong mathematics learners.
Collaboration in mathematics involves adhering to a culture of exchanging ideas and viewpoints in order to improve, both collectively and individually, and to learn from and with others to develop
and apply new ideas in mathematics.
• Learners collaborate with others, value diverse points of view, and consider a range of ideas and perspectives when contributing to mathematical exchanges.
• Learners participate actively and fully in learning experiences by sharing thinking and learning strategies with others to confirm or extend understandings of mathematical ideas; they
respectfully voice their opinions, ideas, and conjectures.
• Learners value the contributions of others, making room for different points of view that will foster mathematical exchanges.
• Learners practise active listening, question their own and others’ mathematical ways of thinking, and ask questions of others to deepen their understanding of mathematical concepts and ideas.
• Learners show a willingness to compromise and change their opinions when presented with convincing arguments during mathematical exchanges.
• Learners make sense of mathematical concepts and ideas by co-constructing their understanding with others.
• Learners support others and take responsibility for their roles throughout the learning process and in the execution of mathematical tasks.
Communication in mathematics involves the learners’ ability to share their mathematical ideas, reasoning, and solutions in a variety of ways, including orally, in writing, concretely, graphically,
and symbolically, and in various contexts. It enables learners to clarify and validate their ideas and reasoning, and to challenge their attitudes and beliefs about mathematics.
• Learners express their mathematical ideas and emotions about mathematics, taking into account non-verbal cues and adjusting what they say according to the context.
• Learners present their mathematical ideas visually, orally, in writing, graphically, or symbolically, taking into account the conventions related to the mode of communication used, their
audience, and the types of communication contexts, while using clear, precise mathematical language.
• Learners understand how their words and actions shape their identity as mathematical learners and shape their relationships with others.
• Learners look for oral, non-verbal, or visual cues during exchanges to improve their understanding of terminology, what others are saying, ideas presented, and various solutions to mathematical
problems and situations.
• Learners seek to understand different points of view and different solutions to a mathematical problem or situation by observing, practising active listening, and asking clarifying questions,
thereby creating a culture of mutual communication.
• Learners recognize and accept that the ways they learn and represent their understanding may be different from those of others.
• Learners make sense of mathematical ideas, problems, and situations, and deepen their understanding by making connections among their own language, mathematical terminology, and mathematical
• Learners contribute to mathematical exchanges and express their thoughts and emotions about mathematical ideas in a positive and respectful way, whether in person or online.
• Learners defend their points of view and their mathematical reasoning while accepting the points of view and reasoning of others in a constructive and responsible way; they understand how these
exchanges benefit themselves as much as they do other members of their learning community.
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Statistics Worksheet
Statistics is a tool we use to better understand data. We are literally overwhelmed with the amount of statistically information or methods for gathering it we come in contact every day. This
includes data from your favorite sports such as batting percentage, goals against average, completion rates, and we could go on for days. This data is tracked to help us make good decisions and
predict the outcome of events. Being able to identify what needs to be measured and what it means has made careers for decades. When you are trying to analyze data or looking for a great way to
display data to make it more recognizable, we look to statistics. Statistics usually looks to best understand uncertainty and variation. Both which work together, as more variation approached,
uncertainty is heighted with it. Probability is a commonly discussed topic in this field. We find probability to change as often as does the parameters or terms it is placed under. To have an
educated guess as how to determine all of these values Statisticians must know the as much about the control of the environment is contained.
We feature a wide range of relevant topics that help students learn how to better analyze data and make projections based on their analysis. You will learn the common methods such as central
tendency, but we will also show you how to structure that data to help others better understand your work. We have to remember that it is just as important to be able to interpret data as it is to be
able to communicate what you see in the data. We feature worksheets that work on simple to complex probability, central tendencies of data, understanding the nature of combinations or reading through
regressions and tree diagrams.
Get Free Worksheets In Your Inbox!
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Heating or Cooling Time
There are many occasions where it might be helpful to know how much time it will take to heat or cool your system to a certain temperature. Or, you may want to calculate how much power is required to
heat or cool a given volume of fluid in a certain amount of time.
Thankfully, there is a fairly simple equation you can use as long as you know the mass of the bath fluid, it’s specific heat capacity, the temperature differential, and either power or time.
That said, using this equation isn’t entirely reliable, as there are various factors that could throw off the calculation. In this post, we take a look at the equation for calculating heating or
cooling time and the reasons you should look for a system with slightly more power than you think you need.
Calculating Heating or Cooling Time
You can use the same basic equation when calculating heating or cooling time, although there is a little more work involved for calculating cooling time. When heating, the power applied is constant,
but when cooling, the power (or the cooling capacity) is variable depending on the temperature.
Calculating Heating Time
To find out how much time it will take to heat a bath to a certain temperature, you can use the following equation:
t = mcΔT / P
• t is heating or cooling time in seconds
• m is the mass of the fluid in kilograms
• c is the specific heat capacity of the fluid in joules per kilogram and per Kelvin
• ΔT is the temperature differential in degrees Celcius or Fahrenheit
• P is the power at which energy is supplied in watts or joules per second
Similarly, to calculate the power needed to heat or cool a bath to a certain temperature in a given time, you can use this equation:
P = mcΔT / t
While these equations are fairly straightforward to follow, there can be some confusion when it comes to which units to use. Instead, you could use an online calculator to help.
This calculator is nice and simple and lets you calculate time, power, or energy consumed, but it’s only good for calculations involving water. If you need to deduce heating time for other fluids,
this calculator is more suitable as it lets you enter the specific heat capacity of the substance you’re using. It has two options enabling you to calculate either power required or time required.
The Process Heating Services calculator.
Calculating Cooling Time
To calculate cooling time, you can use the same equation as above. The question is what value should you use for power. The cooling capacity (or cooling power) is different depending on the
temperature. Cooling capacity decreases at lower setpoint temperatures because there’s a smaller temperature differential between the chiller liquid and refrigerant. Heat transfer is reduced so
cooling capacity is lowered.
For example, here are the cooling capacity specifications for the PolyScience 45 L Refrigerated & Heated Circulating Baths.
You have a few options here depending on how to accurately you want your calculation to be:
• Use a conservative estimate by assuming the lower power up to the next listed temperature. For instance, taking the specifications above, you could assume that the cooling capacity is 250 W for
all temperatures between -20°C and 0°C and 800 W for all temperatures between 0°C and 20°C.
• Potentially underestimate but with more accuracy by taking the average power between various temperatures.
• Use a quick and dirty (and likely less accurate) method by only considering the cooling capacity at the midpoint temperature.
• Opt for an alternative quick method that uses an average of cooling capacity values at various points in the temperature range (the points would need to include the upper and lower ends of the
temperature range for this to be viable).
What if your minimum temperature is below the lowest temperature cooling capacity specification provided? This generally should not be a concern as cooling capacity values are typically provided for
a temperature at or below the minimum temperature of the unit.
If you’re trying to cool to a lower temperature, it may be too low, meaning the unit won’t be able to provide the cooling capacity you need. However, if the specs don’t provide the cooling capacity
at a temperature that is close to the minimum temperature of the unit, you can ask the manufacturer or us to provide the information you need.
Factors to Consider When Calculating Heating or Cooling Time
As mentioned, there are several reasons your calculations may not deliver a realistic result. As such, if you’re using this equation to determine heating or cooling time, you should assume that the
process will take a little longer than expected. Similarly, if you’re using the calculation to determine how much power you need to achieve a given heating or cooling time, you should assume some
additional power will be required.
Here are the factors you need to consider:
1. Ambient Heat Gain or Loss
Ambient heat loss gain or loss is inevitable, even in a closed system. A cooled system can absorb heat from the ambient air or system components, decreasing its cooling capacity. In a heating system,
you may lose heat to the ambient air or to components of the system, for example, as it runs through tubes or pipes.
Insulating your system and controlling the ambient temperature can help, but there may still be an unknown amount of heat gain or loss.
2. Loss of Fluids to Evaporation
If you’re working with an open system, you may lose some fluids to evaporation during the heating or cooling process. The amount of evaporation that occurs will depend on several factors, including:
• Which fluid you’re using: Lower boiling point fluids such as ethanol, methanol, and water can evaporate easily.
• The surface area of the bath: The larger the surface area, the higher the rate of evaporation.
• The temperature range you are using: The higher the temperature, the higher the rate of evaporation.
Heat loss occurs through evaporation, and when you’re wasting heat energy, the time taken to heat the bath will increase. In addition, as a result of fluid loss, the mass value (m) in the equation
won’t be accurate, potentially throwing off results. If you’re using a blend of two or more fluids and one component of a blend evaporates quicker than others, the ratio will be altered, leading to
inaccuracy in the specific heat capacity (c).
Evaporation is difficult to predict and account for accurately (and if you are good enough with thermodynamics to be comfortable doing this, you probably wouldn’t be reading this article). As such,
your best bets are to either estimate the evaporation rate through an empirical test and then factor that in mathematically using the heat of evaporation, or simply add a factor of safety.
3. Maintenance Issues
In heating systems, it’s common for scale to build up on the elements of a water bath due to mineral deposits. Left unchecked, this buildup can have an impact on the efficiency with which heat is
transferred from the element to the fluid. With scale buildup insulating the element, more energy is required to heat the system to the desired temperature.
When heating, this will increase the time it will take to reach the desired temperature in a system of given power. If you’re looking at power, it will increase the amount of power required to reach
the desired temperature in a certain amount of time.
For cooling systems, cooling capacity can be impacted by maintenance issues too. In water-cooled condensers, corrosion, scale buildup, or biological growth can inhibit heat transfer, lowering the
cooling capacity. In air-cooled condensers, dust and debris buildup on fan blades and fins can decrease air flow, having a similar effect of lowering the cooling capacity.
Performing regular maintenance on your unit, including cleaning the various components, flushing the fluid, and using a corrosion inhibitor can help.
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Forecast conditional variances from conditional variance models
V = forecast(Mdl,numperiods,Y0) returns a numeric array containing paths of minimum mean squared error (MMSE), consecutive forecasted conditional variances V of the fully specified, univariate
conditional variance model Mdl, over a numperiods forecast horizon. The model Mdl can be a garch, egarch, or gjr model object. The forecasts represent the continuation of the presample data in the
numeric array Y0.
Tbl2 = forecast(Mdl,numperiods,Tbl1) returns the table or timetable Tbl2 containing the paths of MMSE conditional variance variable forecasts of the model Mdl over a numperiods forecast horizon.
forecast uses the table or timetable of presample data Tbl1 to initialize the response series. (since R2023a)
To initialize the forecast, forecast selects the response variable named in Mdl.SeriesName or the sole variable in Tbl1. To select a different response variable in Tbl1 to initialize the forecasts,
use the PresampleResponseVariable name-value argument.
[___] = forecast(___,Name,Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. forecast returns the output
argument combination for the corresponding input arguments. For example, forecast(Mdl,10,Y0,V0=v0) initializes the conditional variances for the forecast using the presample data in v0.
Specify Numeric Presample Response Data to Forecast GARCH Model Conditional Variances
Forecast the conditional variance of simulated data over a 30-period horizon. Supply a vector of presample response data.
Simulate 100 observations from a GARCH(1,1) model with known parameters.
Mdl = garch(Constant=0.02,GARCH=0.8,ARCH=0.1);
rng("default") % For reproducibility
[v,y] = simulate(Mdl,100);
Forecast the conditional variances over a 30-period horizon. Specify the simulated response data. Plot the forecasts.
vF = forecast(Mdl,30,y);
hold on
title("Forecasted Conditional Variances")
legend("Simulated presample","Forecasts")
hold off
Forecasts converge asymptotically to the unconditional innovation variance.
Forecast EGARCH Model Conditional Variances
Forecast the conditional variance of simulated data over a 30-period horizon.
Simulate 100 observations from an EGARCH(1,1) model with known parameters.
Mdl = egarch(Constant=0.01,GARCH=0.6,ARCH=0.2, ...
rng("default") % For reproducibility
[v,y] = simulate(Mdl,100);
Forecast the conditional variance over a 30-period horizon. Specify the simulated data as presample responses. Plot the forecasts.
VF1 = forecast(Mdl,30,y);
hold on
title("Forecasted Conditional Variances")
legend("Simulated responses","Forecasts")
hold off
Forecast GJR Model Conditional Variances
Forecast the conditional variance of simulated data over a 30-period horizon.
Simulate 100 observations from a GJR(1,1) model with known parameters.
Mdl = gjr(Constant=0.01,GARCH=0.6,ARCH=0.2, ...
rng("default") % For reproducibility
[v,y] = simulate(Mdl,100);
Forecast the conditional variances over a 30-period horizon. Specify the simulated presample responses. Plot the forecasts.
vF = forecast(Mdl,30,y);
hold on
title("Forecasted Conditional Variances")
hold off
Compare Conditional Variance Forecasts of NYSE Returns
Since R2023a
Forecast the conditional variance of the average weekly closing NASDAQ returns from fitted GARCH(1,1), EGARCH(1,1) and GJR(1,1) models.
Load the U.S. equity indices data Data_EquityIdx.mat.
The timetable DataTimeTable contains the daily NASDAQ closing prices, among other indices.
Compute the weekly average closing prices of all timetable variables.
DTTW = convert2weekly(DataTimeTable,Aggregation="mean");
Compute the weekly percent returns and their sample mean.
DTTRet = price2ret(DTTW);
DTTRet.Interval = [];
DTTRet.NASDAQ = DTTRet.NASDAQ*100;
T = height(DTTRet)
meanRet = mean(DTTRet.NASDAQ)
hold on
title("Daily NASDAQ Returns");
ylabel("Return (%)");
The variance of the series seems to change. This change is an indication of volatility clustering. The conditional mean model offset is very close to zero.
When you plan to supply a timetable, you must ensure it has all the following characteristics:
• The selected response variable is numeric and does not contain any missing values.
• The timestamps in the Time variable are regular, and they are ascending or descending.
Remove all missing values from the timetable, relative to the NASDAQ returns series.
DTTRet = rmmissing(DTTRet,DataVariables="NASDAQ");
numobs = height(DTTRet)
Because all sample times have observed NASDAQ returns, rmmissing does not remove any observations.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTTRet,"weeks")
areTimestampsRegular = logical
areTimestampsSorted = issorted(DTTRet.Time)
areTimestampsSorted = logical
areTimestampsRegular = 1 indicates that the timestamps of DTTRet represent a regular weekly sample. areTimestampsSorted = 1 indicates that the timestamps are sorted.
Fit GARCH(1,1), EGARCH(1,1), and GJR(1,1) models to the data. By default, the software sets the conditional mean model offset to zero.
MdlGARCH = garch(1,1);
MdlEGARCH = egarch(1,1);
MdlGJR = gjr(1,1);
EstMdlGARCH = estimate(MdlGARCH,DTTRet,ResponseVariable="NASDAQ");
GARCH(1,1) Conditional Variance Model (Gaussian Distribution):
Value StandardError TStatistic PValue
_________ _____________ __________ ___________
Constant 0.0030629 0.0011827 2.5897 0.0096065
GARCH{1} 0.86501 0.02911 29.715 4.8912e-194
ARCH{1} 0.11835 0.024582 4.8144 1.4765e-06
EstMdlEGARCH = estimate(MdlEGARCH,DTTRet,ResponseVariable="NASDAQ");
EGARCH(1,1) Conditional Variance Model (Gaussian Distribution):
Value StandardError TStatistic PValue
_________ _____________ __________ __________
Constant -0.081262 0.030237 -2.6875 0.0071983
GARCH{1} 0.95557 0.01335 71.579 0
ARCH{1} 0.2768 0.052237 5.299 1.1645e-07
Leverage{1} -0.10519 0.025542 -4.1185 3.8142e-05
EstMdlGJR = estimate(MdlGJR,DTTRet,ResponseVariable="NASDAQ");
GJR(1,1) Conditional Variance Model (Gaussian Distribution):
Value StandardError TStatistic PValue
_________ _____________ __________ __________
Constant 0.0069063 0.0020036 3.447 0.0005668
GARCH{1} 0.78545 0.043862 17.907 1.0334e-71
ARCH{1} 0.090637 0.034313 2.6415 0.0082543
Leverage{1} 0.18663 0.054402 3.4305 0.0006025
Forecast the conditional variance for 20 weeks using the fitted models. Use the observed returns as presample innovations for the forecasts.
fh = 20;
DTTVFGARCH = forecast(EstMdlGARCH,fh,DTTRet, ...
DTTVFEGARCH = forecast(EstMdlEGARCH,fh,DTTRet, ...
DTTVFGJR= forecast(EstMdlGJR,fh,DTTRet, ...
The forecasted conditional variance variables are called Y_Variance in each returned timetable.
Plot the forecasts along with the conditional variances inferred from the data.
DTTVGARCH = infer(EstMdlGARCH,DTTRet,ResponseVariable="NASDAQ");
DTTVEGARCH = infer(EstMdlEGARCH,DTTRet,ResponseVariable="NASDAQ");
DTTVGJR = infer(EstMdlGJR,DTTRet,ResponseVariable="NASDAQ");
plot(DTTRet.Time(end-100:end),DTTVGARCH.Y_Variance(end-100:end), ...
title("GARCH(1,1) Conditional Variances")
plot(DTTRet.Time(end-100:end),DTTVEGARCH.Y_Variance(end-100:end),"r", ...
title("EGARCH(1,1) Conditional Variances")
plot(DTTRet.Time(end-100:end),DTTVGJR.Y_Variance(end-100:end),"r", ...
title("GJR(1,1) Conditional Variances")
Plot conditional variance forecasts for the next 500 weeks after the sample.
fh = 500;
DTTVF1000GARCH = forecast(EstMdlGARCH,fh,DTTRet, ...
DTTVF1000EGARCH = forecast(EstMdlEGARCH,fh,DTTRet, ...
DTTVF1000GJR= forecast(EstMdlGJR,fh,DTTRet, ...
legend("GARCH Forecast","EGARCH Foecast","GJR Forecast",Location="northeast")
title("Long-Run Conditional Variance Forecast")
The forecasts converge asymptotically to the unconditional variances of their respective processes.
Input Arguments
numperiods — Forecast horizon
positive integer
Forecast horizon, or the number of time points in the forecast period, specified as a positive integer.
Data Types: double
Y0 — Presample response data y[t]
numeric column vector | numeric matrix
Presample response data y[t] used to infer presample innovations ε[t], and whose conditional variance process σ[t]^2 is forecasted, specified as a numpreobs-by-1 numeric column vector or a
numpreobs-by-numpaths numeric matrix. When you supply Y0, supply all optional data as numeric arrays, and forecast returns results in numeric arrays.
numpreobs is the number of presample observations.
Y0 can represent a mean 0 presample innovations series with a variance process characterized by the conditional variance model Mdl. Y0 can also represent a presample innovations series plus an offset
(stored in Mdl.Offset). For more details, see Algorithms.
Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation. numpreobs must be at least Mdl.Q to initialize the
conditional variance model. If numpreobs > Mdl.Q, forecast uses only the latest Mdl.Q rows. For more details, see Time Base Partitions for Forecasting.
Columns of Y0 correspond to separate, independent paths.
• If Y0 is a column vector, it represents a single path of the response series. forecast applies it to each forecasted path. In this case, all forecast paths Y derive from the same initial
• If Y0 is a matrix, each column represents a presample path of the response series. numpaths is the maximum among the second dimensions of the specified presample observation matrices Y0 and V0.
Data Types: double
Tbl1 — Presample data
table | timetable
Since R2023a
Presample data containing the response variable y[t] and, optionally, the conditional variance variable σ[t]^2 used to initialize the model for the forecast, specified as a table or timetable, the
same type as Tbl1, with numprevars variables and numpreobs rows. You can select a response variable or conditional variance variable from Tbl1 by using the PresampleResponseVariable or
PresampleVarianceVariable name-value argument, respectively.
Each selected variable is single path (numpreobs-by-1 vector) or multiple paths (numpreobs-by-numpaths matrix) of presample response or conditional variance data. Each row is a presample observation,
and measurements in each row occur simultaneously. numpreobs must be one of the following values:
• Mdl.Q when Tbl1 provides only presample responses
• max([Mdl.P Mdl.Q]) when Tbl1 also provides presample conditional variances
If you supply more rows than necessary, forecast uses the latest required number of observations only.
If Tbl1 is a timetable, all the following conditions must be true:
• Tbl1 must represent a sample with a regular datetime time step (see isregular).
• The datetime vector of sample timestamps Tbl1.Time must be ascending or descending.
If Tbl1 is a table, the last row contains the latest presample observation.
Although forecast requires presample response data, forecast sets default presample conditional variance data in one of the following ways:
• If numpreobs ≥ max([Mdl.P Mdl.Q]) + Mdl.P, forecast infers presample conditional variances from the presample response data (see infer).
• Otherwise:
□ If Mdl is a GARCH(P,Q) or GJR(P,Q) model, forecast sets all required conditional variances to the unconditional variance of the conditional variance process.
□ If Mdl is a EGARCH(P,Q) model, forecast sets all required conditional variances to the exponentiated, unconditional mean of the logarithm of the EGARCH(P,Q) variance process.
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: forecast(Mdl,10,Y0,V0=[1 0.5;1 0.5]) specifies two different presample paths of conditional variances.
V0 — Presample conditional variances σ[t]^2
positive column vector | positive matrix
Presample conditional variances σ[t]^2 used to initialize the conditional variance model, specified as a numpreobs-by-1 positive column vector or numpreobs-by-numpaths positive matrix. Use V0 only
when you supply the numeric array of presample response data Y0.
Rows of V0 correspond to periods in the presample, and the last row contains the latest presample conditional variance.
• For GARCH(P,Q) and GJR(P,Q) models, numpreobs must be at least Mdl.P to initialize the variance equation.
• For EGARCH(P,Q) models, numpreobs must be at least max([Mdl.P Mdl.Q]) to initialize the variance equation.
If numpreobs exceeds the minimum number, forecast uses only the latest observations.
Columns of V0 correspond to separate, independent paths.
• If V0 is a column vector, forecast applies it to each forecasted path. In this case, the conditional variance model of all forecast paths V derives from the same initial conditional variances.
• If V0 is a matrix, it must have numpaths columns, the same number of columns as Y0.
forecast sets default presample conditional variance data in one of the following ways:
• If the number of rows of Y0 numpreobs ≥ max([Mdl.P Mdl.Q]) + Mdl.P, forecast infers V0 from Y0 (see infer).
• Otherwise:
□ If Mdl is a GARCH(P,Q) or GJR(P,Q) model, forecast sets all required conditional variances to the unconditional variance of the conditional variance process.
□ If Mdl is a EGARCH(P,Q) model, forecast sets all required conditional variances to the exponentiated, unconditional mean of the logarithm of the EGARCH(P,Q) variance process.
Data Types: double
PresampleResponseVariable — Variable of Tbl1 containing presample response paths y[t]
string scalar | character vector | integer | logical vector
Since R2023a
Variable of Tbl1 containing presample response paths y[t], specified as one of the following data types:
• String scalar or character vector containing a variable name in Tbl1.Properties.VariableNames
• Variable index (integer) to select from Tbl1.Properties.VariableNames
• A length numprevars logical vector, where PresampleResponseVariable(j) = true selects variable j from Tbl1.Properties.VariableNames, and sum(PresampleResponseVariable) is 1
The selected variable must be a numeric matrix and cannot contain missing values (NaN).
If Tbl1 has one variable, the default specifies that variable. Otherwise, the default matches the variable to name in Mdl.SeriesName.
Example: PresampleResponseVariable="StockRate"
Example: PresampleResponseVariable=[false false true false] or PresampleResponseVariable=3 selects the third table variable as the presample response variable.
Data Types: double | logical | char | cell | string
PresampleVarianceVariable — Variable of Tbl1 containing presample conditional variance paths σ[t]^2
string scalar | character vector | integer | logical vector
Since R2023a
Variable of Tbl1 containing presample conditional variance paths σ[t]^2, specified as one of the following data types:
• String scalar or character vector containing a variable name in Tbl1.Properties.VariableNames
• Variable index (integer) to select from Tbl1.Properties.VariableNames
• A length numprevars logical vector, where PresampleVarianceVariable(j) = true selects variable j from Tbl1.Properties.VariableNames, and sum(PresampleVarianceVariable) is 1
The selected variable must be a numeric vector and cannot contain missing values (NaN).
To use presample conditional variance data in Tbl1, you must specify PresampleVarianceVariable.
Example: PresampleVarianceVariable="StockRateVar"
Example: PresampleVarianceVariable=[false false true false] or PresampleVarianceVariable=3 selects the third table variable as the presample conditional variance variable.
Data Types: double | logical | char | cell | string
• NaN values in numeric presample data sets Y0 and V0 indicate missing data. forecast removes missing data from the presample data sets following this procedure:
1. forecast horizontally concatenates Y0 and V0 such that the latest observations occur simultaneously. The result can be a jagged array because the presample data sets can have a different
number of rows. In this case, forecast prepads variables with an appropriate amount of zeros to form a matrix.
2. forecast applies list-wise deletion to the combined presample matrix by removing all rows containing at least one NaN.
3. forecast extracts the processed presample data sets from the result of step 2, and removes all prepadded zeros.
List-wise deletion reduces the sample size and can create irregular time series.
• For numeric data inputs, forecast assumes that you synchronize the presample data such that the latest observations occur simultaneously.
• forecast issues an error when any table or timetable input contains missing values.
Output Arguments
V — Paths of MMSE forecasts of conditional variances σ[t]^2 of future model innovations ε[t]
numeric column vector | numeric matrix
Paths of MMSE forecasts of conditional variances σ[t]^2 of future model innovations ε[t], returned as a numperiods-by-1 numeric column vector or a numperiods-by-numpaths numeric matrix. forecast
returns V only when you supply the input Y0.
V represents a continuation of V0 (V(1,:) occurs in the next time point after V0(end,:)).
V(j,k) contains the j-period-ahead forecasted conditional variance of path k.
forecast determines numpaths from the number of columns in the presample data sets Y0 and V0. For details, see Algorithms. If each presample data set has one column, then V is a column vector.
Tbl2 — Paths of MMSE forecasts of conditional variances σ[t]^2 of future model innovations ε[t]
table | timetable
Since R2023a
Paths of MMSE forecasts of conditional variances σ[t]^2 of future model innovations ε[t], returned as a table or timetable, the same data type as Tbl1. forecast returns Tbl2 only when you supply the
input Tbl1.
Tbl2 contains a variable for all forecasted conditional variance paths, which are in a numperiods-by-numpaths numeric matrix, with rows representing periods in the forecast horizon and columns
representing independent paths, each corresponding to the input presample response and conditional variance paths in Tbl1. forecast names the forecasted conditional variance variable in Tbl2
responseName_Variance, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is StockReturns, Tbl2 contains a variable for the corresponding forecasted conditional variance paths with
the name StockReturns_Variance.
Tbl2.responseName_Variance represents a continuation of the presample conditional variance process, either supplied by Tbl1 or set by default (Tbl2.responseName_Variance(1,:) occurs in the next time
point, with respect to the periodicity Tbl1, after the last presample conditional variance).
Tbl2.responseName_Variance(j,k) contains the j-period-ahead forecasted conditional variance of path k.
If Tbl1 is a timetable, the following conditions hold:
• The row order of Tbl2, either ascending or descending, matches the row order of Tbl1.
• Tbl2.Time(1) is the next time after Tbl1.Time(end) relative the sampling frequency, and Tbl2.Time(2:numobs) are the following times relative to the sampling frequency.
More About
Time Base Partitions for Forecasting
Time base partitions for forecasting are two disjoint, contiguous intervals of the time base; each interval contains time series data for forecasting a dynamic model. The forecast period (forecast
horizon) is a numperiods length partition at the end of the time base during which forecast generates forecasts V from the dynamic model Mdl. The presample period is the entire partition occurring
before the forecast period. forecast can require observed responses (or innovations) Y0 or conditional variances V0 in the presample period to initialize the dynamic model for forecasting. The model
structure determines the types and amounts of required presample observations.
A common practice is to fit a dynamic model to a portion of the data set, then validate the predictability of the model by comparing its forecasts to observed responses. During forecasting, the
presample period contains the data to which the model is fit, and the forecast period contains the holdout sample for validation. Suppose that y[t] is an observed response series. Consider
forecasting conditional variances from a dynamic model of y[t] numperiods = K periods. Suppose that the dynamic model is fit to the data in the interval [1,T – K] (for more details, see estimate).
This figure shows the time base partitions for forecasting.
For example, to generate forecasts Y from a GARCH(0,2) model, forecast requires presample responses (innovations) Y0 = ${\left[\begin{array}{cc}{y}_{T-K-1}& {y}_{T-K}\end{array}\right]}^{\prime }$ to
initialize the model. The 1-period-ahead forecast requires both observations, whereas the 2-periods-ahead forecast requires y[T – K] and the 1-period-ahead forecast V(1). forecast generates all other
forecasts by substituting previous forecasts for lagged responses in the model.
Dynamic models containing a GARCH component can require presample conditional variances. Given enough presample responses, forecast infers the required presample conditional variances. This figure
shows the arrays of required observations for this case, with corresponding input and output arguments.
• If the conditional variance model Mdl has an offset (Mdl.Offset), forecast subtracts it from the specified presample responses to obtain presample innovations. Subsequently, forecast uses to
initialize the conditional variance model for forecasting.
• forecast sets the number of sample paths to forecast numpaths to the maximum number of columns among the specified presample response and conditional variance data sets. All presample data sets
must have either numpaths > 1 columns or one column. Otherwise, forecast issues an error. For example, if Y0 has five columns, representing five paths, then V0 can either have five columns or one
column. If V0 has one column, then forecast applies V0 to each path.
[1] Bollerslev, T. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics. Vol. 31, 1986, pp. 307–327.
[2] Bollerslev, T. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics. Vol. 69, 1987, pp. 542–547.
[3] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[4] Enders, W. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, 1995.
[5] Engle, R. F. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica. Vol. 50, 1982, pp. 987–1007.
[6] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance. Vol. 48, No. 5,
1993, pp. 1779–1801.
[7] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[8] Nelson, D. B. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica. Vol. 59, 1991, pp. 347–370.
Version History
Introduced in R2012a
R2023a: forecast accepts input data in tables and timetables, and returns results in tables and timetables
In addition to accepting input presample data in numeric arrays, forecast accepts input data in tables or regular timetables. When you supply data in a table or timetable, the following conditions
• forecast chooses the default series on which to operate, but you can use the specified optional name-value argument to select a different variable.
• forecast returns results in a table or timetable.
Name-value arguments to support tabular workflows include:
• PresampleResponseVariable specifies the variable name of the response paths in the input presample data Tbl1 to initialize the response series for the forecast.
• PresampleVarianceVariable specifies the variable name of the conditional variance paths in the input presample data Tbl1 to initialize the conditional variance series for the forecast.
R2019a: Models require specification of presample response data to forecast conditional variances
forecast now has a third input argument for you to supply presample response data.
Before R2019a, the syntaxes were:
You could optionally supply presample responses using the
name-value argument.
There are no plans to remove the previous syntaxes or the Y0 name-value argument at this time. However, you are encouraged to supply presample responses because, to forecast conditional variances
from a conditional variance model, forecast must initialize models containing lagged variables. Without specified presample responses, forecast initializes models by using reasonable default values,
but the default might not support all workflows. This table describes the default values for each conditional variance model object.
Model Object Presample Default
garch All presample responses are the unconditional standard deviation of the process.
egarch All presample responses are 0.
gjr All presample responses are the unconditional standard deviation of the process.
Update Code
Update your code by specifying presample response data in the third input argument.
If you do not supply presample responses, forecast provides default presample values that might not support all workflows.
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Studying the effect of spin removal in the Ising model - NHSJS
Studying the effect of spin removal in the Ising model
Praagya Agrawal
The Ising model is an exactly solvable statistical system at one or two dimensions that exhibits phase transitions and critical phenomena^1. This paper explores the effect of the removal of lattice
sites from the centre and the corners on the system being modelled. To simulate the Ising model, the Metropolis Algorithm is used. This paper will show that the number of sites and the location of
the sites are important parameters, since sites at some locations have a higher interaction.
The Ising model is the simplest representation that can model the essential features of real systems, such as the critical point^2.
The Ising model was proposed in the 1920s, and named after Ernst Ising. It is used to numerically solve the phase transitions that occur when changes in a certain parameter result in a large-scale
change in the entire system being simulated.
The mathematical framework of the Ising model starts by creating a lattice that contains a set of equally spaced points. These points are referred to as lattice sites. Lattice sites share bonds, and
if two sites are connected by a bond, they are called nearest neighbours. Ising models can be one-dimensional, two-dimensional, or three-dimensional.
Lattice sites are denoted by configuration of the system. Fig 1, 2, and 3 show lattices formed by their lattice sites in one-dimensional, two-dimensional, and three-dimensional Ising model
Ernst Ising’s study focused on simulating ferromagnetism, as this paper will. In this system, at 2 or 3 dimensions, each lattice site represents an atom, which has a spin that can be either “up” or
“down.” The net spin of the system determines its magnetic properties. The simulation used in this paper, however, focuses on the net spin itself and not the type of magnetic properties.
An important component of the Ising model is to find the total energy of a system, which is given by its Hamiltonian. For this, we assume that only two factors affect the total energy: (1)
interactions of nearest-neighbour lattice sites and (2) interactions between an external field and the lattice sites^4. The Hamiltonian is then given by
where ^5.
The energy of the
Where ^6)
In the model, the partition function, given by
where ^7
This probability also suggests that the probability of being in a state with a lower Hamiltonian (and thus energy) is greater than the probability of being in a state with a higher Hamiltonian, which
means that the system prefers lower energy levels.
The Ising model is capable of showing the effect of magnitude of the parameters of the system: initial energy of the lattice
This paper analyses the effect of the removal of lattice sites on the system, by observing the changes in the evolution of spin and energy, with parameters
It is difficult to manually mathematically solve the system since there are
The Monte Carlo algorithm used is the Metropolis Algorithm, which is used to simulate a series of states of the system and reach an equilibrium. It starts with an initial lattice with a given size
and configuration, along with its net spin and energy. The average spin is calculated by simply adding the spin of every individual lattice site and dividing it by the number of lattice sites. The
net energy is calculated using the Hamiltonian. Then, it picks a random site in the lattice and proposes a flip in its spin (i.e., -1 to +1 or +1 to -1), by creating a temporary lattice with the spin
reversed, and calculates the change in energy ^8. In this paper, ^9.
First, the paper will analyze the effect of removing particles from the centre of the lattice on the evolution of average spin and net energy on a system with no external magnetic field. The size
chosen is
The results shown below are averages from 10 simulations.
Figure 6| Effect of the removal of the central lattice sites on average spin.
Figure 7| Effect of the removal of the central lattice sites on net energy.
Figure 8| Effect of the removal of the non-central lattice sites on average spin.
Figure 9| Effect of the removal of the non-central lattice sites on net energy.
Figure 10| Effect of the removal of the lattice sites from the entire lattice on average spin.
Figure 11| Effect of the removal of lattice sites from the entire lattice on net energy.
The figures suggested that it might be possible to predict the average spin and net energy of the configuration
The results suggest that the net energy of
The percentage maximum variation of average spin and net energy when
From these results we can observe that the location of the removed sites does have an impact, and sites seem to have different interactions. However,
While the Ising model undoubtedly has applications in theoretical and computational physics, its uses spread beyond that. D. Stauffer showed that the Ising model can compute socioeconomic systems,
such as business confidence, racial segregation, and the replacement of the dominant language spoken in an area^10). C. Daskalakis, N. Dikkala, G. Kamath also showed that the Ising model has
applications in both synthetic and real-world social networks^11). The results of this paper can perhaps be taken forward by applying removal of sites from such systems, representing exiting or
temporary inactivity of agents in socioeconomics systems or social networks.
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What is the limit of (t^2 - 64) / (2t^2 + 17 t + 8) as t approaches -8? | HIX Tutor
What is the limit of # (t^2 - 64) / (2t^2 + 17 t + 8)# as t approaches -8?
Answer 1
Factor both the numerator and the denominator to see what you can eliminate.
#=>lim_(t -> -8) ((t + 8)(t - 8))/((2t^2 + 16t + t + 8))#
#=>lim_(t->-8) ((t + 8)(t - 8))/((2t(t + 8) + 1(t + 8))#
#=>lim_(t->-8) ((t + 8)(t - 8))/((2t + 1)(t + 8))#
#=>lim_(t->-8) (t - 8)/(2t + 1)#
#=> (-8 - 8)/(2 xx -8+ 1)#
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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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Time Complexity Analysis using Big O Notation - The Productive Nerd
Time Complexity Analysis using Big O Notation
In the intricate realm of algorithmic efficiency lies the cornerstone of Time Complexity Analysis, a fundamental concept crucial for gauging the performance of algorithms. Through the lens of Big O
notation, we delve into a realm where efficiency meets precision, unraveling the intricacies of computational efficiency and scalability.
While words like “big o notation” and “time complexity” may sound daunting, they serve as the compass guiding us through the labyrinth of algorithmic performance. Join us as we unlock the power of
Big O notation, deciphering its significance and unraveling its practical applications in evaluating algorithmic efficiency and scalability.
Understanding Time Complexity Analysis
Time complexity analysis refers to the study of algorithms based on the amount of time they take to run as a function of the input size. It provides insights into how the algorithm’s efficiency
scales with increasing input, crucial for optimizing performance and resource utilization.
By understanding time complexity analysis, developers can make informed decisions about choosing the most efficient algorithm for a given problem. It involves evaluating how the algorithm’s runtime
grows relative to the size of the input, typically expressed using Big O notation, a standardized way to describe an algorithm’s complexity.
Through this analysis, developers can compare different algorithms and identify the most efficient solution. It enables them to predict how algorithms will perform on large datasets, aiding in
selecting the optimal approach for a specific problem based on its expected scale. A clear grasp of time complexity analysis is fundamental to writing efficient and scalable code.
Introduction to Big O Notation
In the realm of algorithmic analysis, understanding the concept of Big O Notation holds paramount importance. Here’s a breakdown of what it entails:
• Big O Notation serves as a vital tool in gauging the efficiency of algorithms by providing a standardized approach to evaluate their time complexity. It aids in predicting the worst-case scenario
of an algorithm’s runtime.
• Commonly used notations within the Big O framework, such as O(1), O(log n), O(n), O(n^2), among others, offer a succinct and systematic way to express the rate of growth of an algorithm’s runtime
concerning the input size.
• The significance of Big O Notation lies in its ability to offer a simplified yet comprehensive overview of how an algorithm scales with input size. By categorizing algorithms into specific
complexity classes, it enables developers to make informed decisions on algorithm selection based on performance considerations.
Explanation and Significance
Big O notation is a vital concept in algorithmic analysis that quantifies the efficiency of algorithms in terms of their time and space complexity. By expressing the upper bound of an algorithm’s
growth rate using mathematical functions, Big O notation offers a standardized way to compare algorithmic efficiencies, aiding developers in selecting optimal solutions for various computational
problems. This notation simplifies the evaluation process by focusing on the algorithm’s most significant factors that affect its performance, such as the input size and the worst-case scenario.
Understanding the significance of Big O notation enables developers to make informed decisions when designing and optimizing algorithms. It provides a common language for discussing algorithmic
efficiencies across different domains, fostering collaboration and sharing of best practices within the programming community. Moreover, by analyzing the time complexity of algorithms through Big O
notation, programmers can anticipate performance bottlenecks, optimize critical sections of their code, and ultimately deliver more efficient and scalable solutions to end-users. As such, mastering
Big O notation is essential for any programmer striving to write efficient and maintainable code.
In conclusion, the explanation and significance of Big O notation lie at the core of algorithmic analysis and design. By providing a concise and standardized method for evaluating the efficiency of
algorithms, Big O notation empowers developers to optimize their code for better performance and scalability. Embracing this fundamental concept is key to mastering time complexity analysis and
enhancing the overall quality of software solutions in the ever-evolving landscape of technology.
Commonly Used Notations in Big O
In the realm of algorithmic analysis, Big O Notation is a fundamental tool for quantifying the efficiency of algorithms. Commonly used notations within Big O provide insights into the behavior of
algorithms in terms of time complexity. Here are some prevalent notations widely encountered in the context of time complexity analysis:
• O(1) – Represents constant time complexity where the execution time remains constant regardless of the input size.
• O(log n) – Denotes logarithmic time complexity commonly observed in algorithms like binary search.
• O(n) – Reflects linear time complexity where the execution time increases linearly with the input size.
• O(n^2) – Symbolizes quadratic time complexity typical in algorithms using nested loops for iteration.
Understanding these common notations is crucial for effectively evaluating and comparing the performance of algorithms. By recognizing these patterns, developers can make informed decisions regarding
algorithm selection based on their specific requirements and constraints.
Evaluating Time Complexity with Big O
When evaluating time complexity with Big O notation, it is essential to compare algorithms efficiently. Consider the following points:
• Analyze the growth rates of different algorithms to identify the most efficient one in terms of time complexity.
• Use Big O to categorize algorithms based on their efficiency and scalability.
• Utilize practical examples to demonstrate how Big O notation simplifies the comparison of algorithmic behaviors.
Understanding the significance of Big O notation in evaluating time complexity is crucial for efficient algorithm design and optimization. By applying Big O analysis, developers can make informed
decisions that enhance the performance and scalability of their algorithms.
Comparing Algorithms Using Big O
Comparing algorithms using Big O involves assessing their efficiency as input size grows. For example, if Algorithm A has O(n^2) complexity and Algorithm B has O(n), B is more efficient for large
datasets. Big O helps determine the algorithm that scales better in different scenarios, aiding in optimal algorithm selection.
Practical Examples in Time Complexity Analysis
When considering practical examples in time complexity analysis using Big O notation, let’s delve into a scenario where we compare the time complexities of two sorting algorithms: Bubble Sort and
Merge Sort. Bubble Sort, with a time complexity of O(n^2), proves inefficient with large datasets due to its quadratic nature. However, Merge Sort, boasting a time complexity of O(n log n), excels in
handling significant amounts of data efficiently by dividing and conquering the sorting process.
Another example lies in analyzing the time complexity of searching algorithms, such as Linear Search and Binary Search. Linear Search, with a complexity of O(n), sequentially scans through elements
until finding the target. Conversely, Binary Search, with a time complexity of O(log n), efficiently narrows down search space by recursively halving it, making it particularly efficient with sorted
Considering these practical examples in time complexity analysis offers insights into how different algorithms perform varying tasks and how their efficiencies can be quantified using Big O notation.
By understanding and evaluating these examples, one can make informed choices when designing algorithms for real-world applications, ensuring optimal performance based on the expected input sizes and
Notable Features of Big O Analysis
Big O Analysis offers a standardized approach to measure the efficiency of algorithms in terms of their worst-case scenarios. This notation simplifies complex algorithms into easily comparable forms,
aiding developers in making informed decisions based on the scalability and performance of their code. By providing a clear hierarchy of algorithmic efficiencies, Big O Notation allows for the
prioritization of optimization efforts, ensuring that resources are allocated efficiently to enhance overall system performance.
One key feature of Big O Analysis is its scalability across various algorithmic complexities and sizes of input data. It enables developers to anticipate how code performance will behave as the input
size grows, helping in designing algorithms that can handle large datasets efficiently. Additionally, Big O Notation serves as a universal language for discussing and evaluating algorithmic
efficiencies, facilitating communication and collaboration among developers, researchers, and tech professionals worldwide.
Another notable feature of Big O Analysis is its ability to abstract away implementation-specific details, focusing solely on the fundamental operations that contribute most significantly to an
algorithm’s time complexity. This abstraction allows developers to analyze and compare algorithms independently of programming languages or hardware constraints, providing a broad perspective on
algorithm efficiency that transcends specific technical environments. Overall, the clarity and standardization offered by Big O Notation make it a powerful tool for optimizing algorithm performance
and driving innovation in computational problem-solving.
Best Practices in Utilizing Big O Notation
When utilizing Big O notation in time complexity analysis, adhere to best practices to enhance algorithmic efficiency and understandability:
• Choose the simplest notation that accurately represents the upper bound of an algorithm’s complexity.
• Consider the worst-case scenario to provide a comprehensive evaluation of algorithmic performance.
• Recognize that Big O notation helps in comparing algorithms independently of hardware or implementation specifics.
• Strive for clarity in notation usage to aid in effective communication and comprehension among developers.
By following these best practices in utilizing Big O notation, developers can make informed decisions when designing algorithms, fostering efficiency and scalability in their codebase.
Big O Notation in Real-World Applications
In real-world applications, the practical significance of Big O notation lies in its ability to provide a standardized framework for assessing the efficiency of algorithms in terms of time
complexity. For instance, when developing software for large-scale systems, understanding the time complexity of algorithms becomes paramount in optimizing performance.
By utilizing Big O notation, software engineers can make informed decisions regarding algorithm selection based on their computational efficiencies. This approach enables them to choose algorithms
that are best suited for specific tasks, ultimately leading to improved overall system performance. For example, in the context of optimizing search algorithms for databases, choosing an algorithm
with a lower Big O complexity can significantly reduce search time.
In the realm of real-world applications such as data processing, machine learning, and network optimization, the use of Big O notation allows professionals to gauge the scalability and efficiency of
algorithms when dealing with large datasets or complex computations. This aids in streamlining processes and enhancing overall productivity in various industries where algorithmic efficiency is
paramount for performance.
Overall, the seamless integration of Big O notation into real-world applications empowers developers and engineers to make data-driven decisions that impact the performance and scalability of
systems. By understanding and implementing Big O analysis, organizations can boost their operational efficiency and deliver optimized solutions that align with the demands of modern computational
Advantages and Limitations of Big O Analysis
Big O Analysis provides a systematic approach to evaluating algorithmic efficiency, aiding developers in understanding how their code performs as input size grows. It offers a standardized way to
compare algorithms, enabling informed choices during algorithm selection and optimization processes. This advantage of Big O Notation allows for efficient decision-making in algorithm design.
On the flip side, Big O Analysis has limitations in that it simplifies complexities to general trends, overlooking finer details. While it offers a high-level perspective on algorithm performance, it
may not capture variations in real-world scenarios where constants or lower-order terms significantly impact runtime. Thus, relying solely on Big O may lead to overlooking practical nuances in
algorithm implementation.
Despite its limitations, embracing Big O Analysis empowers developers to make informed decisions on algorithmic choices based on scalability and efficiency. By acknowledging both the advantages and
limitations of Big O Notation, developers can strike a balance between theoretical analysis and practical considerations in optimizing algorithm performances for real-world applications.
Enhanced Techniques Beyond Big O
Beyond Big O notation, advanced techniques such as Omega and Theta provide a more nuanced analysis of algorithm efficiency. Omega denotes the best-case scenario, indicating the lower bound of the
algorithm’s running time. In contrast, Theta represents the tight bounds where the algorithm’s complexity is both the upper and lower limits, offering a more precise estimation than Big O alone.
Moreover, analyzing algorithms through Big Omega and Theta allows for a comprehensive understanding of performance across different scenarios. This approach is particularly useful when assessing
real-world applications where algorithms may exhibit varying efficiencies under different inputs.
By incorporating these enhanced techniques, developers gain deeper insights into algorithmic behavior, enabling them to make informed decisions when selecting the most suitable algorithm for a
specific task. Understanding the interplay between Big O, Omega, and Theta elevates the analysis beyond a simplistic view, providing a more holistic perspective on algorithmic efficiency and
Evolution of Time Complexity Analysis
The Evolution of Time Complexity Analysis has seen significant advancements in the field of algorithm design and analysis. Initially, time complexity assessment focused primarily on worst-case
scenarios, commonly represented by Big O Notation. However, with the evolution of computing technology and the increasing complexity of algorithms, a more nuanced approach has emerged.
Modern Time Complexity Analysis considers not only worst-case scenarios but also average-case and best-case scenarios, providing a more comprehensive view of algorithm performance. This evolution has
led to the development of more sophisticated analysis techniques beyond traditional Big O Notation, such as Omega and Theta Notations, offering a more refined understanding of algorithmic efficiency.
Furthermore, the Evolution of Time Complexity Analysis has witnessed the integration of empirical studies and probabilistic analysis into algorithm evaluation methodologies. By incorporating
real-world data and statistical techniques, researchers can provide more accurate predictions of algorithm performance in practical applications, enriching the overall understanding of time
In conclusion, the Evolution of Time Complexity Analysis highlights the continuous adaptation of analytical methods to meet the challenges posed by ever-evolving technologies and computational
demands. This evolution underscores the dynamic nature of algorithmic analysis and the necessity of embracing diverse approaches to accurately gauge the efficiency of algorithms in modern computing
Mastering Time Complexity Analysis
To master Time Complexity Analysis, it is vital to delve deeply into advanced algorithmic techniques beyond basic Big O notation. This involves exploring complexities like Omega and Theta, offering a
comprehensive understanding of algorithm efficiency. Additionally, understanding space complexity alongside time complexity is crucial for a holistic analysis. Mastery also entails applying these
concepts to real-world scenarios, honing problem-solving skills in algorithm design.
Furthermore, mastering Time Complexity Analysis involves dissecting complex algorithms to determine their efficiency accurately. Practicing the analysis of various algorithmic scenarios enhances the
ability to optimize code for better performance. By mastering techniques beyond Big O, such as amortized analysis or logarithmic complexities, one can refine their algorithmic expertise and tackle
diverse computational challenges effectively.
Moreover, staying abreast of evolving algorithmic trends and advancements is integral to mastering Time Complexity Analysis. Constant learning and adaptation to new complexities and optimization
strategies ensure proficiency in algorithmic problem-solving. Engaging with the algorithmic community, participating in coding competitions, and exploring research papers can further enhance one’s
mastery in Time Complexity Analysis. Ultimately, continuous practice, exploration, and application of advanced algorithmic concepts are key to mastering Time Complexity Analysis effectively.
Big O notation, a fundamental concept in time complexity analysis, provides a standardized way to describe the efficiency of an algorithm. By expressing the upper bound of the algorithm’s execution
time, Big O notation helps in comparing different algorithms based on their performance characteristics. This notation simplifies the intricate process of assessing the scalability and efficiency of
algorithms, making it easier to understand the behavior of an algorithm as the input size grows.
In evaluating time complexity using Big O notation, algorithms are classified into categories based on their growth rates. For instance, O(1) signifies constant time complexity, O(n) represents
linear complexity, and O(n^2) indicates quadratic complexity. By analyzing the asymptotic behavior of algorithms through Big O notation, developers can make informed decisions regarding algorithm
selection, optimization, and design improvements to enhance the overall performance of their applications.
Practical examples illustrating the application of Big O notation in time complexity analysis demonstrate how varying algorithmic approaches produce different efficiency levels. Understanding how to
interpret and apply Big O notation in real-world scenarios equips developers with the knowledge needed to optimize algorithm performance and build scalable systems. By following best practices and
leveraging the insights gained from Big O analysis, developers can enhance the efficiency and effectiveness of their algorithms in diverse computational tasks.
In conclusion, mastering time complexity analysis through Big O notation is a crucial skill for any programmer seeking optimal algorithmic efficiency. Understanding the significance of Big O and its
practical applications empowers developers to make informed algorithmic choices, leading to more efficient and scalable solutions in real-world scenarios.
It is through the evaluation of time complexity with Big O that developers can navigate the trade-offs between speed and resource consumption in algorithm design. By embracing best practices and
leveraging Big O notation effectively, programmers can unlock the potential for enhanced algorithmic performance in both theoretical analyses and practical implementations.
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30 Best 12th Grade Math Tutors - Wiingy
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12th Grade Math topics you can learn
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12th-grade math skills & concepts to know for better grades
Here are the important topics in 12th grade math:
• Limits and Continuity
• Derivatives and Differentiation
• Applications of Derivatives
• Integration and Antiderivatives
• Techniques of Integration
• Applications of Integrals
• Parametric and Polar Equations
• Sequences and Series
• Taylor and Maclaurin Series
• Differential Equations
• Multivariable Calculus
• Vector Calculus
Advanced Algebra
• Matrices
• Determinants
• Systems of Equations and Inequalities
• Vectors and Vector Spaces
• Complex Numbers and Roots of Unity
• Linear Algebra
Advanced Geometry
• Analytic Geometry
• Three-Dimensional Geometry
• Non-Euclidean Geometry
Advanced Trigonometry
• Advanced Trigonometric Identities
• Trigonometric Equations
• Inverse Trigonometric Functions
• Complex Numbers in Trigonometry
Statistics and Probability
• Probability Distributions
• Statistical Inference
• Hypothesis Testing
• Regression Analysis
• Multivariate Statistics
Discrete Mathematics
• Combinatorics
• Graph Theory
• Number Theory
• Logic and Set Theory
Advanced Topics
• Mathematical Modeling
• Numerical Analysis
• Differential Geometry
• Abstract Algebra
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How To Do A Smooth Transform Scale With Code Examples
In this article, we will look at how to get the solution for the problem, How To Do A Smooth Transform Scale With Code Examples
How does a transform scale work?
The scale() CSS function defines a transformation that resizes an element on the 2D plane. Because the amount of scaling is defined by a vector, it can resize the horizontal and vertical dimensions
at different scales. Its result is a <transform-function> data type.
transition: transform 100ms ease-in-out;
transform: scale(1.1);
How do you scale down an image in CSS?
Resize Image in CSS
• Use the max-width and max-height Properties to Resize the Image in CSS.
• Use the object-fit Property to Resize the Image in CSS.
• Use the auto Value for Width and the max-height Property to Resize the Image in CSS.
What does smooth transition mean?
The phrase smooth transition refers to any transition which passes smoothly, without incident. Here's a list of synonyms for smoothly. Contexts ▼ In a fluid motion, without bumping or jerking.
Easily, without much difficulty or problems.
How do I make a smooth animation?
The 12 Animation Tips and Tricks to Master
• Use Squash & Stretch to Avoid Stiff Movement.
• Add Anticipation to Your Movement.
• Make Sure All Movement Has Follow Through.
• Add Arcs to Create Natural Movement.
• Ease In and Out of Your Movement.
• Use Your Frames to Create Intentional Timing.
• Make Use of Secondary Action.
How do I make my website fit my screen size in HTML?
You should set body and html to position:fixed; , and then set right: , left: , top: , and bottom: to 0; . That way, even if content overflows it will not extend past the limits of the viewport.
Caveat: Using this method, if the user makes their window smaller, content will be cut off.
Are CSS animations expensive?
Continuous animations can consume a significant amount of resources, but some CSS properties are more costly to animate than others. The harder a browser must work to animate a property, the slower
the frame rate will be.
How do you transform smoothly in CSS?
Simple. Just list out all of the transform functions that you want to use within the transform property. There is a catch to using multiple functions: the order in which you assign the functions can
greatly change the end results. The browser performs the calculations for each function in order, from right to left.
How do you use transition scale in CSS?
scale. The scale value allows you to increase or decrease the size of an element. For example, the value 2 would transform the size to be 2 times its original size. The value 0.5 would transform the
size to be half its original size.
How does transform translation work?
The translate() CSS function repositions an element in the horizontal and/or vertical directions. Its result is a <transform-function> data type.
Mirror Inverse Program In Php With Code Examples
In this article, we will look at how to get the solution for the problem, Mirror Inverse Program In Php With Code Examples How do you reverse an array without using another array? Reverse an array of
characters without creating a new array using java. If you want to reverse int array, you have to change public static void reverseArray(String[] array) as public static void reverseArray(int[]
array) and String temp as int temp . Save this answer. <?php // PHP implementation of the approach //
Try Laravel Vite With Code Examples
In this article, we will look at how to get the solution for the problem, Try Laravel Vite With Code Examples What is Laravel mix? Laravel Mix, a package developed by Laracasts creator Jeffrey Way,
provides a fluent API for defining webpack build steps for your Laravel application using several common CSS and JavaScript pre-processors. In other words, Mix makes it a cinch to compile and minify
your application's CSS and JavaScript files. laravel new breeze-test --git cd breeze-test compose
Jquery Insertafter With Code Examples
In this article, we will look at how to get the solution for the problem, Jquery Insertafter With Code Examples What is jQuery container? Description. "$("#container p")" Selects all elements matched
by <p> that are descendants of an element that has an id of container. $( "<p>Test</p>" ).insertAfter( ".inner" ); Below is a list of different approaches that can be taken to solve the Jquery
Insertafter problem. <div class="container"> <h2>Greetings</h2> <div class="inner">Hello</div> <div cla
Plt Axis Label Font Size With Code Examples
In this article, we will look at how to get the solution for the problem, Plt Axis Label Font Size With Code Examples How do I change the font in Matplotlib? How to change Font in Matplotlib? # set
the font globally. plt.rcParams. update({'font.family':'sans-serif'}) # set the font name for a font family. plt.rcParams. # the current font family. print(plt.rcParams['
font.family']) # list of fonts in sans-serif. plt.rcParams['font.sans-serif'] plt.xlabel
Word Table To Json With Code Examples
In this article, we will look at how to get the solution for the problem, Word Table To Json With Code Examples How do I create a JSON file in Word? Convert DOCX to JSON Format via Java Open DOCX
file using Document class. Convert DOCX to HTML by using Save method. Load HTML document by using Workbook class. Save the document to JSON format using Save method. Stupid Answer : use
"wordhtml.com" to convert and get rid of things. Then use any html to json converter. Can you convert a Word Doc
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Central Algebras #
In this file we define the predicate Algebra.IsCentral K D where K is a commutative ring and D is a (not necessarily commutative) K-algebra.
Main definitions #
• Algebra.IsCentral K D : D is a central algebra over K iff the center of D is exactly K.
Implementation notes #
We require the K-center of D to be smaller than or equal to the smallest subalgebra so that when we prove something is central, there we don't need to prove ⊥ ≤ center K D even though this direction
is trivial.
Central Simple Algebras #
To define central simple algebras, we could do the following:
class Algebra.IsCentralSimple (K : Type u) [Field K] (D : Type v) [Ring D] [Algebra K D] where
[is_central : IsCentral K D]
[is_simple : IsSimpleRing D]
but an instance of [Algebra.IsCentralSimple K D] would not imply [IsSimpleRing D] because of synthesization orders (K cannot be inferred). Thus, to obtain a central simple K-algebra D, one should use
Algebra.IsCentral K D and IsSimpleRing D separately.
Note that the predicate Albgera.IsCentral K D and IsSimpleRing D makes sense just for K a CommRing but it doesn't give the right definition for central simple algebra; for a commutative ring base,
one should use the theory of Azumaya algebras. In fact ideals of K immediately give rise to nontrivial quotients of D so there are no central simple algebras in this case according to our definition,
if K is not a field. The theory of central simple algebras really is a theory over fields.
Thus to declare a central simple algebra, one should use the following:
variable (k D : Type*) [Field k] [Ring D] [Algebra k D]
variable [Algebra.IsCentral k D] [IsSimpleRing D]
variable [FiniteDimensional k D]
where FiniteDimensional k D is almost always assumed in most references, but some results does not need this assumption.
Tags #
central algebra, center, simple ring, central simple algebra
For a commutative ring K and a K-algebra D, we say that D is a central algebra over K if the center of D is the image of K in D.
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Irrefutable Patterns in Binding Positions¶
Since Agda 2.6.1, irrefutable patterns can be used at every binding site in a telescope to take the bound value of record type apart. The type of the second projection out of a dependent pair will
for instance naturally mention the value of the first projection. Its type can be defined directly using an irrefutable pattern as follows:
proj₂ : ((a , _) : Σ A B) → B a
And this second projection can be implemented with a lamba-abstraction using one of these irrefutable patterns taking the pair apart:
Using an as-pattern makes it possible to name the argument and to take it apart at the same time. We can for instance prove that any pair is equal to the pairing of its first and second projections,
a property commonly called eta-equality:
eta : (p@(a , b) : Σ A B) → p ≡ (a , b)
eta p = refl
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Top Three Technical Indicators For Crypto Trading - Academy
Top Three Technical Indicators For Crypto Trading - Reading time: about 6 minutes
For all the information that a crypto trading chart presents, traders often need to delve a little deeper to make sense of the price data. Hidden in the many data points on a typical chart are
patterns that can help traders to make predictions about future trends. To identify these patterns, traders use tools known as technical indicators. In this article, we’ll explain what technical
indicators are, why they are a key part of a crypto trader’s arsenal, and the most commonly used technical indicators.
＞ What are technical indicators?
In This
＞ Why are technical indicators important in crypto trading?
＞ Most commonly used technical indicators
What are technical indicators?
Technical indicators are essentially complex calculations that use existing trading data such as price and volume to chart predictions about the direction that a token might take.
Technical indicators generally fall into one of two categories: leading or lagging indicators. Leading indicators cast forward to provide predictions around price action and what might happen in the
future. Lagging indicators typically focus on historical data to confirm price movements that have happened in the past.
Why are technical indicators important in crypto trading?
Technical indicators help traders to identify price trends as well as support and resistance levels. A successful crypto trading strategy might incorporate various technical indicators to anticipate
potential opportunities, as well as optimal entry and exit points. These calculations can significantly increase a crypto trader’s chances of success in the markets.
It should be noted that using technical indicators and technical analysis is a separate discipline from fundamental analysis, which instead looks at external economic and financial influences on a
token’s price.
Most commonly used technical indicators
Here are three of the most common technical indicators and how they can be used to supercharge crypto trading.
1. Moving Averages (MA)
A moving average helps smooth out price action by calculating the average token price over a specified period. This indicator is plotted on the trading chart as a line and comes in different
variations, such as simple moving averages (SMAs), exponential moving averages (EMAs), and weighted moving averages (WMAs).
Simple moving averages are calculated by taking the sum of the closing prices of a token over a specific time period and then dividing it by that time period. For example, to calculate a 50-day SMA
would add up the closing prices of the token for the past 50 days and then divide the sum by 50. The 50-day and 200-day are the most typical time periods used for the moving average. Crypto day
traders might find the 20-day moving average more useful for analyzing price movement over the short term.
Exponential moving averages give more weight to recent prices, hence are more responsive to current price action. In other words, they are calculated using a formula that places a greater weight on
the most recent prices.
Similar to EMAs, weighted moving averages place a greater weight on the most recent prices using a weighting specified by the user unlike EMAs, that use a specific formula.
Moving averages are often used as a lagging indicator, as they provide results based on price movements that have already taken place. If a moving average is trending upwards, the token’s price
movement can be classified as bullish, and downward-trending as bearish.
2. Relative Strength Index (RSI)
The Relative Strength Index (RSI) is a technical indicator which can be used to show if a token is overbought or oversold. Devised by Welles Wilder in 1978, this indicator is calculated using the
average gains and losses of a token over a specified number of periods. It is plotted on a scale from 0 to 100.
RSI is calculated by first averaging the gain and loss over the specified number of periods. The average gain is calculated by taking the sum of the gains over the number of periods, and dividing
that sum by the number of periods. The average loss is calculated in a similar manner with the sum of the losses divided by the number of periods. The RSI is then calculated using the following
RSI = 100 - (100 / (1 + (average gain / average loss)))
The RSI is considered overbought when it is above 70 and oversold when it is below 30. These levels can be adjusted based on the particular token or market being analyzed. Most crypto traders use the
RSI over a 14-day period.
The RSI can be used to identify potential trend reversals, as well as to confirm the strength of a trend. It can also be used in combination with other technical indicators to generate buy and sell
3. Bollinger Bands
Bollinger Bands consist of a set of three lines plotted on a chart. Named after John Bollinger, these three lines comprise the simple moving average (SMA) combined with standard deviations to
indicate volatility.
The middle line is a SMA of the security’s price, while the upper and lower bands are plotted at a certain number of standard deviations above and below the SMA. The standard deviation is a measure
of volatility and the bands are used to help identify periods of high and low volatility.
Bollinger Bands are plotted on a chart using the following formula:
Upper band = SMA + (number of standard deviations x standard deviation)
Lower band = SMA - (number of standard deviations x standard deviation)
Points on the graph where the bands contract indicate low current volatility with the likelihood of high volatility in the near future, potentially signalling a breakout in price action. These
movements are known as Bollinger’s ‘squeeze’ and ‘bounce.’
Other useful buy and sell signals come in the form of prices touching the upper and lower bands. If the closing price touches the lower band, it is generally considered a buy signal, while a closing
price touching the upper band should be taken as a sell signal.
The default setting for Bollinger Bands is a 20-period SMA with the upper and lower bands plotted at two standard deviations above and below the SMA. However, these parameters can be adjusted based
on the token being analyzed.
Trading technical indicators are important because they provide traders with valuable information about the current market conditions and potential future price movements. While the world of
technical analysis might seem intimidating at first, knowing what a technical indicator is and how to use the three indicators discussed in this article will give you a leg up in your crypto trading.
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Harnessing the Power of Logarithmic Plots in MATLAB – TheLinuxCode
Harnessing the Power of Logarithmic Plots in MATLAB
As an engineer or scientist, you know that visualizing data in the right way is critical for identifying meaningful patterns and relationships hidden within massive datasets. When numbers swing
wildly across many orders of magnitude—whether they represent sensor readings, computational outputs, or any other observations—trying to plot all those values on a standard linear chart becomes
futile. Like attempting to fit a blue whale into a fish tank!
This common data visualization challenge is exactly why logarithmic plots prove so invaluable. By compressing an exponential scale down into linear spacing, logarithmic plots (“log plots” for short)
make it possible to visualize data spanning tiny fractions to astronomically huge values on a single limited graph.
MATLAB offers many powerful tools for working with log-transformed data, but the semilogy() function provides the simplest way to get started. Keep reading as I walk you through hands-on examples of
how to visualize data on a logarithmic y-axis in MATLAB, revealing insights that would be obscured on conventional linear charts!
Why Use Logarithmic Plots?
Before jumping into MATLAB syntax, you may be wondering…when and why should I logarithmically transform my data instead of taking the standard visualization approach?
Log plots shine when data ranges across multiple orders of magnitude. For example, if you needed to analyze sensor measurements that spanned from 0.00002 to 500,000—a range of 10^-5 to 10^6. Trying
to visualize that on a linear plot would squeeze all the small values into one indecipherable mush on the bottom edge.
By logarithmically spacing the y-axis, you achieve consistent separation between data points. This lets you spot patterns across the full spectrum of readings.
To quantify the popularity: among engineers and scientists, over 30% of statistical visualizations incorporate logarithmically scaled axes, whether in MATLAB, Python, R, Excel, or other tool. Log
scales help uncover signal amidst noise.
Common use cases include:
• Wide-ranging scientific calculations
• Sensor readings collecting tiny fractions & huge peaks
• Audio processing spanning decibels
• Economical data over exponential growth periods
• And anywhere else you need to spot trends across values differing by 1000x, 1 millionx, etc!
Now let’s explore how to easily generate those revelatory log plots in MATLAB…
MATLAB’s Semilogy() Function for Log Y-Axis
The most straightforward way to visualize data on a logarithmic scale in MATLAB is by using the semilogy() function. In one line of code, it transforms the y-axis to display a base-10 logarithmic
scale, while keeping the x-axis as standard linear spacing.
semilogy() accepts vector inputs similarly to MATLAB’s conventional plot function. But the visualized output stretches out exponential trends, revealing details invisible on linear charts!
Here is an overview of key semilogy() parameters:
semilogy(X, Y)
semilogy(X, Y, ‘Linespec‘)
semilogy(Y, ‘Linespec‘)
Let‘s break this down:
• X and Y are vector inputs holding the x- and y-axis data values respectively
• Log scale displays on y-axis while x-axis remains linear
• Can customize visualization through Linespec arguments
• Omitting X vectors sets implicit index coordinate scaling
You’ll see concrete examples of various invocation patterns next. But first, to compare…
If instead you wanted both x and y axes with log scaling, MATLAB provides the loglog() function. And for just logarithmic x-scaling, there is semilogx().
But when you simply need to stretch out the y-values across orders of magnitude for easier analysis, semilogy() is your tool of choice!
Now let‘s visualize some data…
Basic Y-Axis Log Scale Plot
Starting simple, here is an example workflow for generating a basic semilogy() plot in MATLAB:
X = 1:0.1:5;
Y = power(X, 3);
semilogy(X, Y)
grid on
With just 3 lines of code, we:
1. Define an X vector input for the x-axis
2. Calculate a cubic Y vector for the y-axis
3. Plot Y logarithmically by passing into semilogy()
Note how the initial Y values visually spread apart vs crunching down in the corner on a conventional linear plot. This reveals more insight into the full trend shape, especially in subsequent
examples plotting multiple data sets on shared axes.
Now let’s move on to…
Visualizing Multiple Datasets
Simply passing matrix inputs instead of vectors allows visualizing multiple data sets together on the log-scaled y-axis.
For example:
X = 1:0.1:5;
Y1 = power(X, 3);
Y2 = power(X, 5);
semilogy(X, [Y1; Y2], ‘LineWidth‘, 1.5);
legend(‘Cubic Data‘, ‘Quintic Data‘);
grid on;
Generates this multi-line plot:
With matrices, as long as one of the dimensions matches (here, the common X vector), MATLAB happily plots everything together on the semilog y-axis.
I also used some Linespec formatting for thicker plot lines, a legend, and gridlines. But it’s that simple to overlay multiple data sets!
This helps you visually compare trends. Like seeing how the quintic function values rise faster than the cubic, thanks to their equal spacing in log-scale. Understanding these patterns across a wide
domain becomes difficult on linear plots.
Now, what if your data includes…
Complex Numbers on Log Scaled Plots
The semilogy() function can even generate logarithmic plots for complex numbers! MATLAB displays the real components on the x-axis, with imaginary components stretch out on the log-scaled y-axis.
For example, let‘s plot this complex exponential data:
complexY = logspace(3 + 5i, 6 + 9i);
grid on
Here the real part scales from 3 to 6 moving left-to-right, while the imaginary component stretches from 5i to 9i bottom-to-top on the logarithmic axis.
This allows, for example, analyzing attenuation patterns of acoustic signals through Fourier Transform conversions into frequency-domain complex numbers. The log spacing reveals phase patterns.
Next let‘s explore…
Customizing Log Plot Appearance
The full syntax for semilogy() includes an optional Linespec parameter for customizing plot apperance:
semilogy(X, Y, ‘LineWidth‘, 2, ‘Marker‘, ‘^‘, ‘Color‘, ‘r‘);
As you can see, thicker red plot line, triangle markers, and a grid added with grid on
Many stylistic adjustments possible, including:
LineSpec Parameter Description Options
Color Plot line color ‘b‘, ‘r‘, [0,0.5,1] rgba vector
LineStyle Line dash style ‘-‘, ‘–‘, ‘-.‘, ‘:‘
LineWidth Thickness of plot line numeric, 1-10 (pts)
Marker Data point marker symbold ‘.‘, ‘o‘, ‘x‘, ‘^‘, ‘*‘
MarkerSize Size of markers numeric size
Note semilogx() and loglog() also utilize this parameter for flexible customization!
Now that you know how to generate and customize logarithmic plots, when might you use them versus standard linear plots?
Log Scale vs Linear: When Does Each Excel?
Consider this code that plots the same data using default linear scaling versus logarithmic semilogy():
X = 1:0.1:100;
Y = power(10, X);
% Linear y-axis
plot(X, Y);
xlabel(‘Linear Scale‘);
yticks([0, 25, 50, 75, 100]);
% Log y-axis
semilogy(X, Y);
xlabel(‘Log Scale‘);
grid on;
Comparing the visualization, linear spacing compresses the large Y values into a dense region after ~50, losing detail. The log transform plot shows much clearer separation throughout the range.
Key differences:
• Log plots excel at revealing patterns across very large & small values
• Linear plots better for understanding precise amounts and differences
• Logarithmic axis distortion can falsely imply relationships within data
Whether linear or logarithmic scaling provides better insight depends on the specific analysis needs. But it‘s always worth trying both!
In summary:
Log Plots Linear Plots
Visual comparisons across axes Precise value inspection
Separates extreme highs/low values Actual relationship mapping
Fits large exponential ranges Negative/zero support
Now we‘ve covered…
Additional Tips for Log Scale Plotting
Here some final tips, tricks, and reminders for effective data visualization using logarithmic scales:
• For log-scaled x-axis, use MATLAB‘s semilogx()
• Log-log plots come from loglog() function
• Always start exploration with linear plot for baseline
• Add gridlines on log plots to gauge spacing with grid on
• Increase numerical precision to avoid losing detail
• Try different log bases like natural log if needed
The key is leveraging logarithmic transforms purposefully where they excel—allowing impossible-otherwise numerical comparisons across many orders of magnitude in the same limited graph area.
MATLAB‘s semilogy() makes tapping into this visualization superpower dead simple. Whether you are analyzing sensor readings, financial projections, or any other data spanning a wide gamut, I hope you
find this introduction helpful for making sense of exponentially growing haystacks to find those insightful needles!
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Blog: Logs for the Monero Research Lab Meeting Held on 2019-07-15
Logs for the Monero Research Lab Meeting Held on 2019-07-15
July 15, 2019
<sarang> Let's start
<sarang> GREETINGS
<suraeNoether> howdy everyone
<suraeNoether> anyone else here? :P
<sarang> I suppose we can continue anyway
<sarang> ROUNDTABLE
<sarang> suraeNoether: care to begin?
<suraeNoether> Sure. First, dEBRUYNE: I got an aswer from Jerry Brito re your question about bitlicense
<hyc> hi
<sarang> Can you repeat the question?
<sarang> (for our logs)
<suraeNoether> yes: dEBRUYNE was wondering if I could ask jerry brito about the possibilities of how Monero can work with the NYDFS bitlicense
<suraeNoether> the example of Zcash being something that recently been listed on coinbase, etc, indicating that the NYDFS gave their blessing somehow
<sarang> This is Zcash's compliance brief: https://z.cash/wp-content/uploads/2019/04/Zcash-Regulatory-Brief.pdf
<sarang> You may find it useful
<suraeNoether> it turns out that we have it backwards: exchange businesses or money transmitting business need to get valided through the NYDFS, and the reason that zcash was listed on coinbase had
more to do with how much contact zcash has had with the coinbase team
<suraeNoether> so, rather than having coincenter talk to NYDFS, what we need to do is start having meetings with people at coinbase, or gemini, or whichever platform we are discussing
<suraeNoether> dEBRUYNE: does that make sense?
<dEBRUYNE> Yes, thanks
<dEBRUYNE> sarang: Also -> https://www.dfs.ny.gov/about/press/pr1805141.htm
<sarang> Sure, it doesn't really make sense to have a protocol validated by a regulator anyway
<suraeNoether> right
<sarang> Wait, what?
<suraeNoether> okay, moving past regulation
<sarang> That press release specifically identifies assets
<sarang> I don't really know what that means
<sarang> This is why I am neither a regulator nor a lawyer :/
<suraeNoether> well, let's move on and discuss it in a bit
<sarang> Perhaps they go to regulators with a specific version or something, I dunno
<sarang> sure
<suraeNoether> a konferenco post morto update
<sarang> s/morto/mortem ?
<suraeNoether> latin or esperanto?
<sarang> -____-
<suraeNoether> lol
<suraeNoether> so anyway, i spent the past week doing a few things wrapping up the konferenco, including organizing the budget projected vs actuals
<suraeNoether> and writing these four guide documents. THESE ARE INTENDED TO BE LIVING DOCUMENTS, UPDATED REGULARLY BY KONFERENCO ORGANIZERS.
<suraeNoether> they are not commandments in stone.
<suraeNoether> https://github.com/b-g-goodell/mrl-skunkworks/tree/master/Konferenco
<suraeNoether> they are to be debated and argued
<suraeNoether> sarang and i were debating funding structures earlier
<sarang> vigorously
<suraeNoether> KonGuide.docx is a general guide for maybe how things can go in the future
<suraeNoether> i recommend even if you disagree with my budgeting/finance recommendations (with respect to the CCS or something), move past that and read the organizational part of the document
<sarang> One note… using markdown instead of docx is much better for version history on git
<sarang> (and displays natively via github)
<suraeNoether> if someone wants to convert it, i'd love that
<suraeNoether> i've been braindumping into libreoffice
<suraeNoether> KonGuideKO.docx is designed for konferenco organizers
<suraeNoether> this includes a list of things to do to get ready for the konferenco, including checklists at the end
<suraeNoether> KonGuideSC.docx is designed for the "steering committee" which will probably have whoever is financially liable for the konferenco sitting on it. they make final budget decisions and
sign contracts.
<suraeNoether> KonGuideCC.docx is designed for the "content committee" which will be deciding on speakers and inviting them, and organizing the schedule
<sarang> What are a couple/few things (briefly) you would have done differently, in hindsight?
<suraeNoether> well, budgetarily, this was a nightmare. there were three very large sources of red on the budget sheet that should have been addressed
<sarang> If you would have been able to more regularly cash out the CCS (or done it in chunks), would that have solved the problem?
<suraeNoether> firstly, the original CCS request was designed to ask for 60,100$ but by the time I actually received it, it was worth $28,500 or so. waiting until it was done in one big chunk and
then transferring it to me introduced so much time into the equation for price that volatility ate a lot of the money.
<sarang> Not good for the organizers or donors
<sarang> (they don't know the eventual value of their donations)
<suraeNoether> one way to rectify that could be regularly withdrawing from the funding as it goes, another way would be to have funding take place in stages
<suraeNoether> secondly, our turnout was much lower than we had all hoped
<sarang> What if you raised money based on when different things needed to be purchased? Like the venue, or food, or A/V support, etc.
<sarang> Then donors have specific things they can donate do, as opposed to more vague "this month's MonKon funding stage"
<sarang> s/do/to
<suraeNoether> so what happens if you drum up money by the payment deadline for venue but not A/V? it's a tricky question.
<suraeNoether> i don't pretend to ahve all the answers
<suraeNoether> second source of funding problems: we had 58 general admission tickets, 4 student tickets, 11 platinum tickets, 27 speaker tickets, 13 sponsor tickets, and 3 media passes. our original
budget was based on 230 attendees and 20 speakers. So, our ticket sales were disappointing in that regard.
<sarang> Well presumably you would not be the one stuck with all this, and be able to focus more on research or MKon content instead
<suraeNoether> but that was exacerbated since we were paying for flights and hotels for speakers, and the increased size of the speaker list caused increased requisite costs, too
<suraeNoether> thirdly, and most fatally, i think, was the increased cost in A/V
<sarang> Having quality recordings was huge
<sarang> video views were pretty high
<suraeNoether> our original proposal was 1/3 what we ended up paying (and that doesn't count any of the time or labor or equipment donated by parasew, marcvvs, and sgp)
<suraeNoether> ^ bingo
<sarang> And it meant that anyone could watch for free
<suraeNoether> i think the A/V costs from this year is a good benchmark for future years, I don't think we got screwed on A/V, but our costs were very high in this area because of it
<sarang> A/V is expensive, hands down
<sarang> but it seems one of the best returns to the community
<suraeNoether> so, long story short: the market murdered me, the ticket sales murdered me, and A/V murdered me, but i'm still alive despite thrice being murdered
<sarang> you have 6 lives left
<suraeNoether> nah, i was murdered twice already, i'm down to 4
<sarang> Ignoring all the budgeting, I'd say it was a big success
<suraeNoether> I agree. final budget will be posted later this week once i've octuply checked everything.
<suraeNoether> nioc ^ check out our numbers from above. total attendance was like 117 before staff was included
<sarang> that's not half bad for a first run
<nioc> so my quick that totaled up 120 was not bad :)
<suraeNoether> a few brief comments for the four guides i've written: you can do what you like, if you are planning on hosting a Konferenco Wien or a Konferenco Beijing or whatever, do what you like.
But make sure all of your funding and structure details are 100% clear in your CCS. Sarang thinks some of my ideas about profit for these events are not fair to the community, so consider the whole
set of documents worth
<suraeNoether> arguing over and debating.
<nioc> NYC during blockchain week and MCC will get you 3x
<suraeNoether> nioc yeah, but in terms of *ticket sales* we had like 71 or 72 or something like that
<suraeNoether> nioc yeah but it will 4x or 5x all our expenses
<dEBRUYNE> Playing devils advocate, but the funds could've been hedged
<dEBRUYNE> There are plenty of markets that allow short selling of xmr
<sarang> There could have been more defined payouts
<sarang> suraeNoether: anything else to discuss?
<sarang> Or any questions for him?
<suraeNoether> dEBRUYNE: yeah, I received the funds on 2-5-19 and by that point the damage had been done. that would be handled by the CCS guys
<suraeNoether> sarang: defined payouts wouldn't have helped
<suraeNoether> the market crashed basically welllllll before we needed any of the money
<sarang> I see
<dEBRUYNE> At what time was the donation completed though?
<dEBRUYNE> Because at that point the price should've been hedged
<suraeNoether> dEBRUYNE: my recollection is around xmas, but i could be misrecollecting
<suraeNoether> luigi1111 may know
<ArticMine> Yes if the expenses were in USD
<suraeNoether> this question occurred to me yesterday and i forgot to write it down
<dEBRUYNE> Price moved from ~50 to ~70 (from christmas to may), so that doesn't seem right
<sarang> Shall we move on from this topic for now?
<suraeNoether> dEBRUYNE: i had only gains from the time that i was holding crypto. i just received 591 XMR worth $28,509 at the time, whereas when I posted the request it was for 591 XMR worth
$60,100. The question is the gap in time between funding-completed and the time it hit the Konferenco wallet on Feb 2
<nioc> there were donations till at least Dec 16
<suraeNoether> i'm fine with waiting for specific dates from luigi or whoever can tell us
<suraeNoether> and moving on
<suraeNoether> sarang, how about you tell us about something more research related?
<sarang> Heh ok
<sarang> I have a few things
<luigi1111> I don't have info on completed funding dates
<sarang> First, I ran a timing/space analysis for the RCT3 sublinear transaction protocol
<luigi1111> not sure if there's a way to get it. surely can manually somehow
<sarang> https://github.com/SarangNoether/skunkworks/blob/sublinear/rct3.md
<sarang> I'm working up some proof-of-concept code for its spend proving system presently (not done)
<sarang> I also worked up a proof of concept for a two-layer Lelantus prover that sacrifices size and verification time for shorter prove time
<sarang> Interesting, but probably not relevant to our use case
<luigi1111> I thought sponsors were going to cover some of the shortfall or something since we knew back then 591 wasn't enough
<sarang> https://github.com/SarangNoether/skunkworks/blob/lelantus/lelantus/layer.py
<dEBRUYNE> suraeNoether: I guess we can discuss this later. One more thing I wanted to ask though, the zcash donation was made in may
<dEBRUYNE> Was that on top of the 28.5k then? Given that you received that earlier
<luigi1111> in truth, ccs isn't particularly well suited for people or projects that are sensitive to volatility. there may be mitigations of course
<suraeNoether> nope, they donated directly to the CCS so that was included
<suraeNoether> luigi1111: the goal was to get corporate sponsorships
<suraeNoether> luigi1111: we got some
<suraeNoether> luigi1111: we did not get enough to cover the shortfall
<luigi1111> I see
<luigi1111> what is the shortfall?
<suraeNoether> sarang sorry to interrupt you: good work on lelantus. have you worked out the tradeoff between our current size vs. verf time compared to a lelantus version with a faster prover?
<suraeNoether> luigi1111: i'll be posting budget later this week
<luigi1111> ok sounds good
<sarang> The faster Lelantus prover makes sense for Zcoin, who want >O(10K) ring members
<sarang> and, to be fair, you can batch away much of the verification loss
<sarang> for O(100-100K) ring members it's likely not really a problem in practice
<sarang> but it's still damn clever
<suraeNoether> i'd be interested in seeing some hard numbers, like the value N such that for >O(N) ring members, the shorter prove time is worthwhile
<sarang> whoops, O(100-1K)
<sarang> define "worthwhile"…
<sarang> all depends on what the max prove time (and the corresponding computational complexity) is that you're willing to accept
<suraeNoether> or how much verf time/space you are willing to sacrifice
<sarang> Zcoin has non-public numbers for this (can't share yet)
<suraeNoether> k
<sarang> However, it's pretty impressive… like, on the order of 10x improvement for large rings (in proving time)
<sarang> I don't think the integration into the rest of the Lelantus prover is completed yet, FWIW
<sarang> there's some info you need to extract from the 1-of-N proof for balance purposes that I haven't worked out
<sarang> On the RCT3 side, this week I should have working code for that transaction protocol
<sarang> I'm checking a bunch of their math (might have errors, not sure yet)
<sarang> and that's about it for me
<sarang> Any particular questions?
<ArticMine> On a more mundane level of mixing I estimate we can move from 11 to 13 without touching fees
<sarang> Based on my CLSAG numbers, you mean?
<sarang> or something else?
<ArticMine> No current tech
<sarang> CLSAG will almost certainly not make it into the fall upgrade
<sarang> How so?
<suraeNoether> ArticMine: i would almost rather increase fees than ring size at this point :\
<ArticMine> There was a drop in tx size last fork
<suraeNoether> sarang i have a question: fill in the blank to complete the analogy. (lelantus protocol) : (monero protocol) :: (2-year old mid-range automobile with no damage) : ________
<sarang> While bigger rings are generally better, a marginal increase from 11 to 13 will do little to help analysis that already exists
<sarang> …
<suraeNoether> 2-year old mid-range refrigerator with no stink?
<sarang> lol
<sarang> maybe that cleaning car that the Cat in the Hat drives?
<sarang> Looks cool, pretty functional, not sure what'd happen in practice =p
<suraeNoether> how about the weird stretchy-squishy car from Willy Wonka
<sarang> lol
<suraeNoether> wait, has monero switched with lelantus in your analogy?
<gingeropolous> bumpdaringsize
<sarang> erm
<suraeNoether> bumpdaringsize.gif
<sarang> We should determine specific reasons why we would increase
<sarang> e.g. things like chain reaction or accidental dead outputs are exceedingly unlikely now
<gingeropolous> #1. 13 > 11
<sarang> things like EABCD…ZE are presumably more unaffected by such a marginal change
<suraeNoether> Does anyone have any questions for sarang about his lelantus work recently, other than stupid SAT analogies?
<ArticMine> FloodXMR is very sensitive to ring size
<sarang> (or on RCT3 for that matter)
<sarang> Yes, but we don't have correct numbers on that yet
<sarang> in terms of cost, that is
<sarang> It should be quantified before a blind increase, IMO
<suraeNoether> okay, everyone: i have to get to an appointment
<sarang> roger
<suraeNoether> remember, for the hypochondriacs: if the pain is behind and above your stomach, it could be pancreatitis and not a heart attack
<ArticMine> Ok we wait for CLSAG and then take another look at mixin
<sarang> I'm not saying we have to wait until spring
<sarang> only that I'd prefer quantified reasons for an increase to know the benefits
<sarang> Increasing from 11 to 13 won't stop a wealthy adversary from chain spamming with the current fee structure
<sarang> knowing the added protection against deanon would be useful though
<sarang> sgp_ has a useful little tool for this
<sarang> I'll grab numbers for that part at least (the flood folks are running new simulations on a private testnet)
<sarang> OK, does anyone else have work to share?
<sarang> Or other updates relevant to this channel?
<sarang> If not, we can leave the floor open to QUESTIONS while we go over ACTION ITEMS (to respect the time)
<sarang> I'll grab numbers on an 11 -> 13 ring increase, finish up RCT3 proof-of-concept stuff, and continue defcon prep
<sarang> suraeNoether can update us later when he returns
<sarang> Any last questions or comments before we adjourn?
<midipoet> i have read over konferenco talk, and have taken notes of the links…will digest. thanks suraeNoether
<sarang> OK, we are now adjourned! Thanks to everyone for joining in
<sarang> Logs will be posted to the github agenda issue shortly
Post tags : Dev Diaries, Cryptography, Monero Research Lab
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Applied Cosmography: A Pedagogical Review
by Yu. L. Bolotin, et al.
Publisher: arXiv.org 2018
Number of pages: 66
Based on the cosmological principle only, the method of describing the evolution of the Universe, called cosmography, is in fact a kinematics of cosmological expansion. The effectiveness of
cosmography lies in the fact that it allows, based on the results of observations, to perform a rigid selection of models that do not contradict the cosmological principle.
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Global interpretation of LHC indications within the Georgi-Machacek Higgs model
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Word Search Leetcode Solution - TutorialCup Backtracking
Word Search Leetcode Solution
Difficulty Level Medium
Frequently asked in Amazon Apple Bloomberg ByteDance Cisco eBay Expedia Facebook Intuit Microsoft Oracle Pinterest ServiceNow Snapchat
Views 8836
Problem Statement
Given an m x n board and a word, find if the word exists in the grid.
The word can be constructed from letters of sequentially adjacent cells, where “adjacent” cells are horizontally or vertically neighbouring. The same letter cell may not be used more than once.
board = [["A","B","C","E"],
word = "ABCCED"
board = [["A","B","C","E"],
word = "ABCB"
Approach ( Backtracking )
This is a 2D grid traversal problem, where we have to explore the grid to check if the given word can be formed using adjacent cells of the grid. But instead of performing DFS on whole grid space we
would more optimally use backtracking method. In backtracking method we will only go to that path to find the solution which matches our aim. If we know at some point that the current path will not
lead to the solution then we will backtrack and go for the next choice.
We will go around the grid by marking the current cell in our path as visited at each step. And at end of each step we will also unmark it so that we could have a clean state to try another
We would make a backtracking function which will start with a particular cell and traverse the adjacent cells of grid in DFS fashion. Because the given word can start from anywhere in the grid, we
would loop over all the cells of the grid and for each cell we will call the backtracking function starting from this current cell.
As this backtracking function is a recursive function, below are the steps to implement this recursive function:
1. In the beginning we will check if we have reached to the bottom or the base case of the recursion. If the word to be searched is empty or in other words if it’s found, we return true.
2. We check if our path is currently valid or not by checking if we have crossed the boundary of the grid or if the current cell matches the first character of the search word or not.
3. If the current step is valid then mark this cell as visited. And start exploring the four directions by calling the same backtracking function for right, down, left and up cells.
4. At end we un-visit the current cell and return the result of exploration as true or false. If any of the sub exploration results in true then we return true from here otherwise return false.
C++ Program for Word Search Leetcode Solution
#include <bits/stdc++.h>
using namespace std;
int row,col;
int dx[4] = {0, 1, 0, -1};
int dy[4] = {1, 0, -1, 0};
bool backtrack(int i,int j,vector<vector<char>>& board, string word,unsigned int ind)
if(ind>=word.size()) return true;
if(i<0 || i>=row || j<0 || j>=col || board[i][j]!=word[ind]) return false;
char t=board[i][j];
board[i][j]= '#';
for(int k=0;k<4;k++)
return true;
board[i][j] = t;
return false;
bool exist(vector<vector<char>>& board, string word)
row= board.size();
col= board[0].size();
for(int i=0;i<row;i++)
for(int j=0;j<col;j++)
return true;
return false;
int main()
vector<vector<char>> board= {
string word = "ABCCED";
cout<< "true" ;
cout<< "false" ;
return 0;
Java Program for Word Search Leetcode Solution
class Rextester{
static int row,col;
static int dx[] = {0, 1, 0, -1};
static int dy[] = {1, 0, -1, 0};
public static boolean exist(char[][] board, String word)
row= board.length;
col= board[0].length;
for(int i=0;i<row;i++)
for(int j=0;j<col;j++)
return true;
return false;
static boolean backtrack(int i,int j,char[][] board, String word,int ind)
if(ind>=word.length()) return true;
if(i<0 || i>=row || j<0 || j>=col || board[i][j]!=word.charAt(ind)) return false;
char t=board[i][j];
board[i][j]= '#';
for(int k=0;k<4;k++)
return true;
board[i][j] = t;
return false;
public static void main(String args[])
char[][] board= {
String word = "ABCCED";
Complexity Analysis for Word Search Leetcode Solution
O( N*(3^L) ) : where N is the total number of cells in the grid and L is the length of the given word to be searched. For the backtracking function initially we get 4 choices for directions but
further it reduced to 3 as we have already visited it in previous step. This depth of the this recursive function will be equal to the length of the word (L). Hence in worst case total number of
function invocation will be the number of nodes in 3-nary tree, which is about 3^L.
In worst case we call this function starting from N number of cells. Hence overall time complexity will be O( N*(3^L) ).
O(L) : where L is the length of the given word. This space is used for recursion stack.
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ECCC - Reports tagged with exact and approximate counting
Given a polynomial f(X) with rational coefficients as input
we study the problem of (a) finding the order of the Galois group of
f(X), and (b) determining the Galois group of f(X) by finding a small
generator set. Assuming the generalized Riemann hypothesis, we prove
the following complexity bounds:
1. ... more >>>
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What is positive and negative variance?
So, Correlation is the Covariance divided by the usual deviations of the 2 random variables. Of course, you could remedy for Covariance by way of the Correlation; we would simply have the Correlation
times the product of the Standard Deviations of the 2 random variables. We know, by definition, that a continuing has zero variance (again, for instance, the constant 3 is at all times 3), which
means it additionally has a regular deviation of 0 (standard deviation is the square root of variance). So, if we tried to resolve for the Correlation between a continuing and a random variable, we&#
39;d be dividing by 0 within the calculation, and we get one thing that is undefined.
This may be proved via MGFs, although we won’t discover the proof right here. Let’s begin with a qualitative framework; you can most likely already guess what Covariance ‘primarily means’. We know
that variance measures the spread of a random variable, so Covariance measures how two random random variables vary together.
What do variances indicate?
A favorable budget variance refers to positive variances or gains; an unfavorable budget variance describes negative variance, meaning losses and shortfalls. Budget variances occur because
forecasters are unable to predict the future costs and revenue with complete accuracy.
What do negative variances point out?
As exploratory data analysis, an ANOVA employs an additive knowledge decomposition, and its sums of squares indicate the variance of every element of the decomposition (or, equivalently, every set of
phrases of a linear model). The idea behind the variance-covariance is similar to the concepts behind the historical technique - besides that we use the acquainted curve as a substitute of actual
information. The advantage of the conventional curve is that we automatically know the place the worst 5% and 1% lie on the curve. They are a perform of our desired confidence and the standard
In applied statistics, there are totally different forms of variance evaluation. In venture administration, variance evaluation helps maintain control over a challenge's expenses by monitoring
planned versus actual prices. Effective variance evaluation might help an organization spot developments, issues, alternatives and threats to short-time period or long-time period success. Variance
evaluation is usually associated with explaining the difference (or variance) between precise prices and the usual costs allowed for the great output. For example, the distinction in supplies prices
could be divided into a supplies value variance and a supplies utilization variance.
These include speculation testing, the partitioning of sums of squares, experimental strategies and the additive model. The growth of least-squares methods by Laplace and Gauss circa 1800 offered an
improved technique of combining observations (over the existing practices then utilized in astronomy and geodesy). Laplace knew the way to estimate a variance from a residual (rather than a complete)
sum of squares. By 1827, Laplace was utilizing least squares strategies to address ANOVA problems regarding measurements of atmospheric tides.
• We know, by definition, that a relentless has zero variance (again, for instance, the fixed three is always 3), which means it also has a standard deviation of zero (normal deviation is the
square root of variance).
• This is definitely type of logical, as a result of it doesn’t make sense to think about a constant worth being correlated with something.
• So, Correlation is the Covariance divided by the usual deviations of the two random variables.
• Of course, you would solve for Covariance in terms of the Correlation; we might simply have the Correlation times the product of the Standard Deviations of the 2 random variables.
Types of variances
Variable overhead spending variance is the difference between actual variable overheads and standard variable overheads based on the budgeted prices. Budget variance is a periodic measure utilized by
governments, firms or people to quantify the distinction between budgeted and actual figures for a particular accounting class. A favorable price range variance refers to optimistic variances or
positive aspects; an unfavorable finances variance describes unfavorable variance, that means losses and shortfalls.
Is a negative variance always adverse?
The reason is that having less revenues than planned is not good. On the other hand, if actual expenses are less than the budgeted amount of expenses, the variance will be shown as a positive amount.
The reason is that fewer actual expenses than budgeted is favorable (or good, positive).
The complete cost, complete income, and fixed value curves can each be constructed with simple formulation. For instance, the total revenue curve is just the product of promoting value times quantity
for every output amount. The information utilized in these formulation come both from accounting information or from various estimation techniques similar to regression analysis. For example, a
business that sells tables must make annual sales of 200 tables to break-even. At present the company is promoting fewer than 200 tables and is subsequently operating at a loss.
Variance evaluation, additionally described as evaluation of variance or ANOVA, involves assessing the difference between two figures. It is a tool applied to financial and operational data that
goals to identify and determine the reason for the variance.
Unlike Variance, which is non-unfavorable, Covariance may be negative or constructive (or zero, of course). A optimistic worth of Covariance means that two random variables are likely to vary in the
identical path, a unfavorable value means that they vary in reverse directions, and a 0 signifies that they don’t differ collectively.
What are the two types of variance?
When effect of variance is concerned, there are two types of variances: When actual results are better than expected results given variance is described as favorable variance. When actual results are
worse than expected results given variance is described as adverse variance, or unfavourable variance.
We then see within the output that the first column is larger on average, which makes sense. We additionally see that the 2 columns have a tendency to maneuver together (they're each relatively
giant/small at the same time), which is smart as a result of we assigned them a optimistic Covariance of half. We can then use dmvnorm (much like dnorm) to seek out the density (consider the joint
PDF) at level ; that's, the density when the primary Normal random variable is at 1 and the second random variable is at 1. Recall that, normally, if the Covariance is zero, then random variables
aren’t essentially impartial. However, on this case, we see that a Covariance of zero does imply independence.
Material Variance
As a enterprise, they must consider rising the variety of tables they sell yearly in order to make enough money to pay fixed and variable prices. The break-even level (BEP) in economics, business—and
specifically cost accounting—is the purpose at which complete price and complete income are equal, i.e. "even". There is not any net loss or acquire, and one has "damaged even", although opportunity
prices have been paid and capital has obtained the chance-adjusted, anticipated return. In brief, all prices that have to be paid are paid, and there may be neither revenue nor loss. While the
analysis of variance reached fruition within the 20th century, antecedents prolong centuries into the past according to Stigler.
The randomization-based evaluation assumes only the homogeneity of the variances of the residuals (as a consequence of unit-remedy additivity) and uses the randomization process of the experiment.
Both these analyses require homoscedasticity, as an assumption for the conventional-mannequin evaluation and as a consequence of randomization and additivity for the randomization-based analysis. The
individual danger is easy enough (just the marginal variance of every inventory), however think extra about the interactive risks.
Analysis of variance (ANOVA) is a collection of statistical fashions and their associated estimation procedures (such as the "variation" amongst and between groups) used to investigate the
differences amongst group means in a sample. ANOVA was developed by statistician and evolutionary biologist Ronald Fisher.
img alt="what do negative variances indicate" src="https://i.
It is only possible for a agency to cross the break-even point if the dollar worth of gross sales is greater than the variable cost per unit. This implies that the promoting price of the nice should
be greater than what the company paid for the great or its components for them to cowl the initial value they paid (variable and fixed costs). Once they surpass the break-even price, the company can
begin making a profit.
Before 1800, astronomers had isolated observational errors resulting from response instances (the "private equation") and had developed strategies of reducing the errors. An eloquent non-mathematical
clarification of the additive effects model was out there in 1885.
Variance Analysis
The ANOVA relies on the law of complete variance, the place the observed variance in a selected variable is partitioned into parts attributable to totally different sources of variation. In its
simplest kind, ANOVA offers a statistical check of whether or not two or extra inhabitants means are equal, and therefore generalizes the t-test beyond two means. Notice how we defined the imply of
the first column to be 2, and the imply of the second column to be 1.
What does a positive variance indicate?
Variance measures how far a set of data is spread out. A variance of zero indicates that all of the data values are identical. A high variance indicates that the data points are very spread out from
the mean, and from one another. Variance is the average of the squared distances from each point to the mean.
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Blobs in Maya
For this post, I have been working on implementing isosurfaces as a modeling tool in Maya to create blobby things such as the one depicted in the image below.
An isosurface is a shape which is described by taking a scalar field and building a surface from all the points on the field of which value matches a given threshold level we provide. To generate a
polygonal surface we can use in computer graphics, our scalar field will be given by a 3D matrix -or grid- of numbers, which we will iterate in order to fetch the points that describe our shape.
Unlike the mathematical scalar field, which is defined in every point, our grid describes a discreet sampling of the field, for we only know the values of the field in the vertices of the grid
To build the surface from this discretized field, we generate triangles from each one of the cells conforming the grid by using the Marching Cubes algorithm. Given a grid cell, we mark each one of
the 8 vertices as above or below the threshold level we provided. From those values, the algorithm provides a set of triangles which describe the surface inside that cell (Fig 1).
Maya Implementation
This particular implementation I provide here is integrated with Maya as a Surface Plug-in. It uses Maya’s particle systems to generate the scalar field – for each particle, we define a radius that
describes the sphere of influence of that particle. The influence rapidly decays to zero as soon as we move away from the particle. Next, a grid within the particle’s bounding box is created, and
sampled at a frequency given by the Grid Res attribute in the shape node. The surface node can also be plugged the particle colors and interpolate them in the resulting triangles, producing nice
color blends. The higher the sampling frequency, the smaller the triangles and the more detailed surface we obtain. Bear in mind however that the number of cells grows proportionally to the cube of
the grid resolution, so the surface generation quickly becomes a processor-intensive task.
I started prototyping the plug-in using Maya Python, but it soon became too slow to be practical, and I had to switch to C++. While working on it, I realized that for “splashy” particle systems
such as the one depicted in the picture above, applying the triangulation to the entire grid is very wasteful, as only a small portion of the total cells (around 20% in that particular example) tend
to be touched by the particles and thus will actually produce triangles. Therefore I optimized the triangle generation by performing a first pass over the grid and tagging those cells that will
potentially be affected by the particles, skipping all the rest. On a second pass, the marching cubes is only applied to those tagged cells (colored in white in Fig 2.). Finally, Maya seems to like
copying data around when handling nodes, and the internal cell occupancy cache can generate a lot of memory fragmentation if created/deleted every frame. To avoid this, a lazy-copy wrapper around te
memory chunk is provided, so that the number of memory operations is minimized.
Finally, if you would like to try it, I have posted a version of the plug-in compiled for Maya 2011 x64 here, as well as a sample scene. For the programmers among us, the source code can be found
2 thoughts on “Blobs in Maya”
1. I just compiled your Marching cubes code for Maya 2014 win x64.
And got issue with grid res attribute, if we increase grid res to high value then output mesh gets triangle holes.
See this
And need below updates for this plugin
*Support for input curve cv, poly vertex, locator, etc.
*UV mapping (2D / 3D texture coordinates)
1. The code provided in this website is given away for free, as-is. I welcome contributions and fixes that can be shared back, but I don’t intend for these plugins to be other than companion
reference implementations for the content in the articles.
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NCERT Exemplar Problems Class 11 Mathematics Chapter 7 Permutations and Combinations
Short Answer Type Questions
Q1. Eight chairs are numbered 1 to 8. Two women and 3 men wish to occupy one chair each. First the women choose the chairs from amongst the chairs 1 to 4 and then men select from the remaining
chairs. Find the total number of possible arrangements.
Sol.First the women choose the chairs from amongst the chairs numbered 1 to 4.
Q2. If the letters of the word RACHIT are arranged in all possible ways as listed in dictionary, then what is the rank of the word RACHIT?
Sol: The alphabetical order of the letters of the word RACHIT is: A, C, H, I, R, T. Number of words beginning with A = 5!
Number of words beginning with C = 5!
Number of words beginning with H = 5!
Number of words beginning with 1 = 5!
Clearly, the first word beginning with R is RACHIT.
.•. Rank of the word RACHIT in dictionary = 4×5! + 1= 4 x120+1= 481
Q3. A candidate is required to answer 7 questions out of 12 questions, which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either
group. Find the number of different ways of doing questions.
Sol: Since the candidate cannot attempt more than 5 questions from either group, he is able to attempt minimum two questions from either group.
The possible number of questions attempted from each group will be as given in the following table:
┃Group I │5│4│3│2┃
┃Group II │2│3│4│5┃
Q4. Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the point.
Sol: There are 18 point in a plane, of which 5 points are collinear.
Q5. We wish to select 6 persons from 8 but, if the person A is chosen, then B must be chosen. In how many ways can selections be made?
Sol: Total number of persons = 8
Number of person to be selected = 6
It is given that, if A is chosen then, B must be chosen. Therefore, following cases arise:
Q6. How many committee of five persons with a chairperson can be selected form 12 persons?
Sol: Total number of persons =12
Number of persons to be selected = 5
Q7. How many automobile license plates can be made, if each plate contains two different letters followed by three different digits?
Sol: There are 26 English alphabets and 10 digits (0 to 9).
It is given that each plate contains two different letters followed by three different digits.
Q8. A bag contains 5 black and 6 red balls. Determine the number of ways in • which 2 black and 3 red balls can be selected from the lot.
Sol: The bag contains 5 black and 6 red balls.
Q9. Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
Sol: Total number of things. = n
We have to arrange r things out of n in which three particular things must occur together.
Q10. Find the number of different words that can be formed from the letters of the word TRIANGLE, so that no vowels are together.
Sol: Given word is: TRIANGLE Consonants are: T, R, N, G, L Vowels are: I, A, E
Since we have to form words in such a way that no two vowels are together, we first arrange consonants.
Five consonants can be arranged in 5! ways.
Q11. Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.
Sol: We have to form 4-digit numbers which are greater than 6000 and less than 7000.
We know that a number is divisible by 5, if at the unit place of the number there is 0 or 5.
So, unit digit can be filled in 2 ways.
The thousandth place can be filled by ‘6’ only.
The hundredth place and tenth place can be filled together in 8 x 7 = 56 ways. So, total number of ways = 56 x 2 = 112
Q12. Thereare 10persons named P[1],P[2],P[3],…,P[!0].Outof 10 persons, 5 persons are to be arranged in a line such that in each arrangement P, must occur whereas P[4] and P[5] do not occur. Find the
number of such possible arrangements.
Sol. Given that, P[1], P[2], …, P[10], are 10 persons, out of which 5 persons are to be arranged but P, must occur whereas P[4] and P[5] never occurs.
As P, is already occurring w’e have to select now 4 out of 7 persons.
.•. Number of selections = ^7C[4] = 35 Number of arrangements of 5 persons = 35 x 5! = 35 x 120 = 4200
Q13. There are 10 lamps in a hall each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
Sol: There are 10 lamps in a hall.
The hall can be illuminated if at least one lamp is switched.
.•. Total number of ways = ^10C[1]+ ^10C[2] + ^l0C[3]… + ^10C[]0
]= 2^10– 1 = 1024- 1 = 1023
Q14. A box contains two white, three black and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?
Sol: There are two white, three black and four red balls.
We have to draw 3 balls, out of these 9 balls in which at least one black ball is included.
So we have following possibilities:
┃Black balls │1│2│3┃
┃Other than black │2│1│0┃
.’. Number of selections = ^3C[1] x ^6C[2] + ^3C[2] x ^6C, + ^3C[3] x ^6C[0
]= 3×15+ 3×6+1= 45+ 18 + 1= 64
Q15. If ^nC[r-1]= 36 ^nC[r] = 84 and ^nC[r+1]= 126, then find the value of ^rC[2].
Q16. Find the number of integers greater than 7000 that can be formed with the digits 3, 5, 7, 8 and 9 where no digits are repeated.
Sol: We have to find the number of integers greater than 7000 with the digits 3,5, 7, 8 and 9.
So, with these digits, we can make maximum five-digit numbers because repetition is not allowed.
Since all the five-digit numbers are greater than 7000, we have Number of five-digit integers = 5x4x3x2x1 = 120 A four-digit integer is greater than 7000 if thousandth place has any one of 7, 8 and
Thus, thousandth place can be filled in 3 ways. The remaining three places can be filled from remaining four digits in ^4P[3] ways.
So, total number of four-digit integers = 3x ^4P[3] = 3x4x3x2 = 72 Total number of integers = 120 + 72 = 192
Q17. If 20 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, in how many points will they intersect each other?
Sol: It is given that no two lines are parallel which means that all the lines are intersecting and no three lines are concurrent.
One point of intersection is created by two straight lines.
Number of points of intersection = Number of combinations of 20 straight lines taken two at a time
Q18. In a certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64. How many telephone numbers have all six digits distinct?
Sol: If first two digit is 41, the remaining 4 digits can be arranged in ^8P[4] = 8 x 7 x 6×5 = 1680 ways.
Similarly, if first two digit is 42, 46, 62, or 64, the remaining 4 digits can be arranged in ^8P[4] ways i.e., 1680 ways.
.’. Total number of telephone numbers having all six digits distinct = 5x 1680 = 8400
Q19. In an examination, a student has to answer 4 questions out of 5 questions, questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.
Sol: It is given that 2 questions are compulsory out of 5 questions.
So, the other 2 questions can be selected from the remaining 3 questions in ^3C[2] = 3 ways.
Q20. A convex polygon has 44 diagonals. Find the number of its sides.
[Hint: Polygon of n sides has (^nC[2] – n) number of diagonals.]
Sol: Let the convex polygon has n sides.
Number of diagonals=Number of ways of selecting two vertices – Number of sides = ^nC[2] – n
It is given that polygon has 44 diagonals.
Long Answer Type Questions
Q21. 18 mice were placed in two experimental groups and one control group with all groups equally large. In how many ways can the mice be placed into three groups?
Sol: It is given that 18 mice were placed equally in two experimental groups and one control group i.e., three groups.
Each group is of 6 mice.
Q22. A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag, if (i) they can be of any colour (ii) two must be white and two
red and (iii) they must all be of the same colour.
Sol:Total number of marbles = 6 white +- 5 red = 11 marbles
(a) If they can be of any colour means we have to select 4 marbles out of 11
∴ Required number of ways = ^11C[4
](b) Two white marbles can be selected in ^6C[2
]Two red marbles can be selected in ^5C[2] ways.
∴ Total number of ways = ^6C[2] x ^5C[2] = 15 x 10 = 150
(c) If they all must be of same colour,
Four white marbles out of 6 can be selected in ^6C[4] ways.
And 4 red marbles out of 5 can be selected in ^5C[4] ways.
∴ Required number of ways = ^6C[4] + ^5C[4] = 15 + 5 = 20
Q23. In how many ways can a football team of 11 players be selected from 16 players? How many of them will
• include 2 particular players?
• exclude 2 particular players?
Sol: Total number of players = 16
We have to select a team of 11 players
So, number of ways = ^16C[11
](i) If two particular players are included then more 9 players can be selected from remaining 14 players in ^14C[9
](ii) If two particular players are excluded then all 11 players can be selected from remaining 14 players in ^14C[11]
Q24. sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII. If there are 20 students in each of these classes, in how many ways can the team
be constituted?
Sol: Total number of students in each class = 20
We have to select at least 5 students from each class.
So we can select either 5 students from class XI and 6 students from class XII or 6 students from class XI and 5 students from class XII.
∴ Total number of ways of selecting a team of 11 players = ^20C[5] x ^20C[6] + ^20C[6] x ^20C[5] = 2 x ^20C[5] x ^20C[6]
Q25. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected, if the team has
(i) no girls
(ii) at least one boy and one girl
(iii) at least three girls
Sol: Number of girls = 4;
Number of boys = 7
We have to select a team of 5 members provided that
Objective Type Questions
Q26. If ^nC[12] = ^nC[8], then n is equal to
(a) 20
(b) 12
(c) 6
(d) 30
Q27. The number of possible outcomes when a coin is tossed 6 times is
(a) 36
(b) 64
(c) 12
(d) 32
Sol: (b) Number of outcomes when a coin tossed = 2 (Head or Tail)
∴Total possible outcomes when a coin tossed 6 times = 2x2x2x2x2x 2 = 64
Q28. The number of different four-digit numbers that can be formed with the digits
2, 3, 4, 7 and using each digit only once is
(a) 120
(b) 96
(c) 24
(d) 100
Sol: (c) Given digits 2,3,4 and 7, we have to form four-digit numbers using these digits.
∴Required number of ways = ^4P[4] = 4!=4x3x2x1 = 24
Q29. The sum of the digits in unit place of all the numbers formed with the help of 3,4, 5 and 6 taken all at a time is
(a) 432
(b) 108
(c) 36
(c) 18
Sol: (b) If the unit place is ‘3’ then remaining three places can be filled in 3! ways.
Thus ‘3’ appears in unit place in 3! times.
Similarly each digit appear in unit place 3! times.
So, sum of digits in unit place = 3!(3 + 4 + 5 + 6) = 18 x 6 = 108
Q30. The total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is
(a) 60
(b) 120
(c) 7200
(d) 720
Sol: (c) Given number of vowels = 4 and number of consonants = 5 We have to form words by 2 vowels and 3 consonants.
So, lets first select 2 vowels and 3 consonants.
Number of ways of selection = ^4C[2] x ^5C[3] = 6 x 10 = 60 Now, these letters can be arranged in 5! ways.
So, total number of words = 60 x 5! = 60 x 120 = 7200
Q31. A five-digit number divisible by 3 is to be formed using the numbers 0, 1,2,4, and 5 without repetitions. The total number of ways this can be done is
(a) 216
(b) 600
(c) 240
(d) 3125
[Hint: 5 digit numbers can be formed using digits 0, 1, 2, 4, 5 or by using digits 1, 2, 3, 4, 5 since sum of digits in these cases is divisible by 3.]
Sol:(a) We know that a number is divisible by 3 if the sum of its digits is divisible by 3.
Now sum of the given six digits is 15 which is divisible by 3. So to form a number of five-digit which is divisible by 3 we can remove either ‘O’ or ‘3’. If digits 1, 2, 3,4, 5 are used then number
of required numbers = 5!
If digits 0, 1,2,4, 5 are used then first place from left can be filled in 4 ways and remaining 4 places can be filled in 4! ways. So in this case required numbers are 4 x 4! ways.
So, total number of numbers = 120 + 96 = 216
Q32. Everybody in a room shakes hands with everybody else. If the total number of hand shakes is 66, then the total number of persons in the room is
(a) 11
(b) 12
(c) 13
(d) 14
Sol: (b) Between any two person there is one hand shake.
Q33. The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is
(a) 105 (b) 15 (c) 175 (d) 185
Sol: (d) Number of ways of selecting 3 points from given 12 points = ^12C[3
]But any three points selected from given seven collinear points does not form triangle.
Number of ways of selecting three points from seven collinear points = ^7C[3]Required number of triangles = ^12C[3] – ^7C[3] = 220 -35 = 185
Q34. The number of parallelograms that can be formed form a set of four parallel lines intersecting another set of three parallel lines is
(a) 6
(b) 18
(c) 12
(d) 9
Sol: (b) To form parallelogram we required a pair of line from a set of 4 lines and another pair of line from another set of 3 lines.
Required number of parallelograms = ^4C[2] x ^3C[2] = 6×3 = 18
Q35. The number of ways in which a team of eleven players can be selected from 22 players always including 2 of them and excluding 4 of them is
(a) ^16C[11]
(b) ^16C[5]
(c) ^16C[9]
(d) ^20C[9]
Sol: (c) Total number of players = 22
We have to select a team of 11 players.
We have to exclude 4 particular of them, so only 18 players are now available. Also from these 2 particular players are always included. Therefore we have to select 9 more players from the remaining
16 players.
So, required number of ways = ^16C[9]
Q36. The number of 5-digit telephone numbers having at least one of their digits repeated is
(a) 900000
(b) 10000
(c) 30240
(d) 69760
Sol: (d) Total number of telephone numbers when there is no restriction = 10^5 Also number of telephone numbers having all digits different = ^l0P[5 ]Required number of ways = 10^5 – ^l0P[5] =
1000000 -10x9x8x7x6 = 1000000-30240 = 69760
Q37. The number of ways in which we.can choose a committee from four men and six women, so that the committee includes at least two men and exactly twice as many women as men is
(a) 94
(b) 126
(c) 128
(d) none of these
Sol: (a) Number of men = 4; Number of women = 6
It is given that committee includes at least two men and exactly twice as many women as men.
So, we can select either 2 men and 4 women or 3 men and 6 women.
∴ Required number of committee formed = ^4C[2] x ^6C[4] + ^4C[3] x ^6C[6
]= 6×15 + 4×1=94
Q38. The total number of 9-digit numbers which have all different digits is
(a) 10!
(b) 9!
(c) 9×9!
(d) 10×10!
Sol: (c) We have to form 9-digit number which has all different digit.
First digit from the left can be filled in 9 ways (excluding ‘0’).
Now nine digits are left including ‘O’.
So remaining eight places can be filled with these nine digits in ^9P[S] ways.
So, total number of numbers = 9 x ^9P[8] = 9×9!
Q39. The number of words which can be formed out of the letters of the word ARTICLE, so that vowels occupy the even place is
(a) 1440
(b) 144
(c) 7!
(d) ^4C[4] x ^3C[3
]Sol: (b) We have word ARTICLE.
Vowels are A, I, E and consonants are R, T, C, L.
Now vowels occupy three even places (2^nd, 4^th and 6^th) in 3! ways.
In remaining four places four consonants can be arranged in 4! ways.
So, total number of words = 3! x4! = 6×24= 144
Q40. Given five different green dyes, four different blue dyes and three different red dyes, the number of combinations of dyes which can be chosen taking at least one green and one blue dye is
(a) 3600
(b) 3720
(c) 3800
(d) 3600
[Hint: Possible numbers of choosing or not choosing 5 green dyes, 4 blue dyes and 3 red dyes are 2^5, 2^4 and 2^3, respectively.]
Fill in the Blanks Type Questions
True/False Type Questions
Q51. There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is ^12C[2] – ^5C[2].
Sol: False
Required number of lines = ^12C[2] – ^5C[2].+ 1
Q52. Three letters can be posted in fiv.e letter boxes in 3^5
Sol: False
Each letter can be posted in any one of the five letter boxes.
So, total number of ways of posting three letters = 5x5x5 = 125
Q53. In the permutations of n things r taken together, the number of permutations in which m particular things occur together is
Q54. In a steamer there are stalls for 12 animals and there are horses, cows and calves (not less than 12 each) ready to be shipped. They can be loaded in 3^12 ways.
In each stall any one of the three animals can be shipped.
So total number of ways of loading = 3x3x3x…xl2 times = 3^12
Q55. If some or all of n objects are taken at a time, then the number of combinations is 2^n– 1.
Sol: True
If some or all objects taken at a time, then number of combinations would be ^nC[1 ]+ ^nC[2] + ^nC[3] + … + ^nC[n] = 2^n – 1
Q56. There will be only 24 selections containing at least one red ball out of a bag containing 4 red and 5 black balls. It is being given that the balls of the same colour are identical.
Sol: False
Number of ways of selecting any number of objects from given n identical objects is 1.
Now selecting zero or more red ball from 4 identical red balls = 1 + 1 + 1 + 1 + 1=5
Selecting at least 1 black ball from 5 identical black balls =1 + T+1 + 1 + 1= 5 So, total number of ways = 5 x 5 = 25
Q58. A candidate is required to answer 7 questions, out of 12 questions which are divided into two groups, each containing 6 He is not permitted to attempt more than 5 questions from either group. He
can choose the seven questions in 650 ways.
Sol: False
A candidate can attempt questions in following maimer
Q59. To fill 12 vacancies there are 25 candidates of which 5 are from scheduled castes. If 3 of the vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of
ways in which the selection can be made is ^5C[3] x ^22C[9].
We can select 3 scheduled caste candidate out of 5 in ^5C[3] ways.
And we can select 9 other candidates out of 22 in ^22C[9]ways.
.’. Total number of selections = ^5C[3] x ^22C[9]
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Audiogon Discussion Forum
1V output for 126mV input is 18dB (not 15dB)
If your DAC produces 2V nominal, then preamp output will be 2V x 1/0.126 = 15.9V - way too much for an amp with 2V nominal input.
Perhaps your DAC has jumpers to reduce gain by 10dB (like in my Benchmark DAC), otherwise you can insert 10dB attenuator. Best place would be between amp and preamp, assuming that your preamp can
output 15.9Vrms - if not, then between DAC and preamp.
Thank you @kijanki for your helpful response. Correct, that the preamp gain is 18dB, not 15dB. I don't know which position of the attenuator will represent the unity gain on my preamp. But there
should be enough steps to attenuate the signal such that the amp input sees a reasonable voltage. But don't know if the usable range will be idea.
It sounds to me that if anything I may have to worry about too much gain rather than too little gain, right?
Almost all amps have 26 dB of gain I believe. IMHO, better to have a low gain preamp. Less noise.
@radiohead99 Voltage gain of your preamp is about 8x. In order to obtain 0dB gain (2V input, 2V output) you need to set wiper at about 50% (12 O'Clock) position (to divide by 8). Since ideal
position would be around 2 O'Clock (with extra gain for soft recordings) you are not much off. It is all assuming that you have logarithmic audio pot (most likely). Some companies use linear pot
with loading resistor (Benchmark DAC1) - a little bit different characteristic.
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Technical Overview
Technical Overview¶
Three Central Classes¶
From a bird’s eye perspective, pyMOR is a collection of generic algorithms operating on objects of the following types:
Vector arrays are ordered collections of vectors. Each vector of the array must be of the same dimension. Vectors can be copied to a new array, appended to an existing array or deleted from the
array. Basic linear algebra operations can be performed on the vectors of the array: vectors can be scaled in-place, the BLAS axpy operation is supported and inner products between vectors can be
formed. Linear combinations of vectors can be formed using the lincomb method. Moreover, various norms can be computed and selected dofs of the vectors can be extracted for empirical
interpolation. To act on subsets of vectors of an array, arrays can be indexed with an integer, a list of integers or a slice, in each case returning a new VectorArray which acts as a modifiable
view onto the respective vectors in the original array. As a convenience, many of Python’s math operators are implemented in terms of the interface methods.
Note that there is not the notion of a single vector in pyMOR. The main reason for this design choice is to take advantage of vectorized implementations like NumpyVectorArray which internally
store the vectors as two-dimensional NumPy arrays. As an example, the application of a linear matrix based operator to an array via the apply method boils down to a call to NumPy’s optimized dot
method. If there were only lists of vectors in pyMOR, the above matrix-matrix multiplication would have to be expressed by a loop of matrix-vector multiplications. However, when working with
external solvers, vector arrays will often be given as lists of individual vector objects. For this use-case we provide ListVectorArray, a VectorArray based on a Python list of vectors.
Associated to each vector array is a VectorSpace which acts as a factory for new arrays of a given type. New vector arrays can be created using the zeros and empty methods. To wrap the raw
objects of the underlying linear algebra backend into a new VectorArray, make_array is used.
The data needed to define a new VectorSpace largely depends on the implementation of the underlying backend. For NumpyVectorSpace, the only required datum is the dimension of the contained
vectors. VectorSpaces for other backends could, e.g., hold a socket for communication with a specific PDE solver instance. Additionally, each VectorSpace has a string id, defaulting to None,
which is used to signify the mathematical identity of the given space.
Two arrays in pyMOR are compatible (e.g. can be added) if they are from the same VectorSpace. If a VectorArray is contained in a given VectorSpace can be tested with the in operator.
The main property of operators in pyMOR is that they can be applied to VectorArrays resulting in a new VectorArray. For this operation to be allowed, the operator’s source VectorSpace must be
identical with the VectorSpace of the given array. The result will be a vector array from the range space. An operator can be linear or not. The apply_inverse method provides an interface for
(linear) solvers.
Operators in pyMOR are also used to represent bilinear forms via the apply2 method. A functional in pyMOR is simply an operator with NumpyVectorSpace(1) as range. Dually, a vector-like operator
is an operator with NumpyVectorSpace(1) as source. Such vector-like operators are used in pyMOR to represent Parameter-dependent vectors such as the initial data of an InstationaryModel. For
linear functionals and vector-like operators, the as_vector method can be called to obtain a vector representation of the operator as a VectorArray of length 1.
Linear combinations of operators can be formed using a LincombOperator. When such a linear combination is assembled, _assemble_lincomb is called to ensure that, for instance, linear combinations
of operators represented by a matrix lead to a new operator holding the linear combination of the matrices.
For many interface methods default implementations are provided which may be overridden with operator-specific code. Base classes for NumPy-based operators can be found in pymor.operators.numpy.
Several methods for constructing new operators from existing ones are contained in pymor.operators.constructions.
Models in pyMOR encode the mathematical structure of a given discrete problem by acting as container classes for operators. Each model object has operators, products dictionaries holding the
Operators which appear in the formulation of the discrete problem. The keys in these dictionaries describe the role of the respective operator in the discrete problem.
Apart from describing the discrete problem, models also implement algorithms for solving the given problem, returning VectorArrays from the solution_space. The solution can be cached, s.t.
subsequent solving of the problem for the same parameter values reduces to looking up the solution in pyMOR’s cache.
While special model classes may be implemented which make use of the specific types of operators they contain (e.g. using some external high-dimensional solver for the problem), it is generally
favourable to implement the solution algorithms only through the interfaces provided by the operators contained in the model, as this allows to use the same model class to solve high-dimensional
and reduced problems. This has been done for the simple stationary and instationary models found in pymor.models.basic.
Models can also implement estimate and visualize methods to estimate the discretization or model reduction error of a computed solution and create graphic representations of VectorArrays from the
Base Classes¶
While VectorArrays are mutable objects, both Operators and Models are immutable in pyMOR: the application of an Operator to the same VectorArray will always lead to the same result, solving a Model
for the same parameter will always produce the same solution array. This has two main benefits:
1. If multiple objects/algorithms hold references to the same Operator or Model, none of the objects has to worry that the referenced object changes without their knowledge.
2. The return value of a method of an immutable object only depends on its arguments, allowing reliable caching of these return values.
A class can be made immutable in pyMOR by deriving from ImmutableObject, which ensures that write access to the object’s attributes is prohibited after __init__ has been executed. However, note that
changes to private attributes (attributes whose name starts with _) are still allowed. It lies in the implementors responsibility to ensure that changes to these attributes do not affect the outcome
of calls to relevant interface methods. As an example, a call to enable_caching will set the objects private __cache_region attribute, which might affect the speed of a subsequent solve call, but not
its result.
Of course, in many situations one may wish to change properties of an immutable object, e.g. the number of timesteps for a given model. This can be easily achieved using the with_ method every
immutable object has: a call of the form o.with_(a=x, b=y) will return a copy of o in which the attribute a now has the value x and the attribute b the value y. It can be generally assumed that calls
to with_ are inexpensive. The set of allowed arguments can be found in the with_arguments attribute.
All immutable classes in pyMOR and most other classes derive from BasicObject which, through its meta class, provides several convenience features for pyMOR. Most notably, every subclass of
BasicObject obtains its own logger instance with a class specific prefix.
Creating Models¶
pyMOR ships a small (and still quite incomplete) framework for creating finite element or finite volume discretizations based on the NumPy/Scipy software stack. To end up with an appropriate Model,
one starts by instantiating an analytical problem which describes the problem we want to discretize. analytical problems contain Functions which define the analytical data functions associated with
the problem and a DomainDescription that provides a geometrical definition of the domain the problem is posed on and associates a boundary type to each part of its boundary.
To obtain a Model from an analytical problem we use a discretizer. A discretizer will first mesh the computational domain by feeding the DomainDescription into a domaindiscretizer which will return
the Grid along with a BoundaryInfo associating boundary entities with boundary types. Next, the Grid, BoundaryInfo and the various data functions of the analytical problem are used to instatiate
finite element or finite volume operators. Finally these operators are used to instatiate one of the provided Model classes.
In pyMOR, analytical problems, Functions, DomainDescriptions, BoundaryInfos and Grids are all immutable, enabling efficient disk caching for the resulting Models, persistent over various runs of the
applications written with pyMOR.
While pyMOR’s internal discretizations are useful for getting started quickly with model reduction experiments, pyMOR’s main goal is to allow the reduction of models provided by external solvers. In
order to do so, all that needs to be done is to provide VectorArrays, Operators and Models which interact appropriately with the solver. pyMOR makes no assumption on how the communication with the
solver is managed. For instance, communication could take place via a network protocol or job files. In particular it should be stressed that in general no communication of high-dimensional data
between the solver and pyMOR is necessary: VectorArrays can merely hold handles to data in the solver’s memory or some on-disk database. Where possible, we favor, however, a deep integration of the
solver with pyMOR by linking the solver code as a Python extension module. This allows Python to directly access the solver’s data structures which can be used to quickly add features to the
high-dimensional code without any recompilation. A minimal example for such an integration using pybind11 can be found in the src/pymordemos/minimal_cpp_demo directory of the pyMOR repository.
Bindings for FEnicS and NGSolve packages are available in the bindings.fenics and bindings.ngsolve modules. The pymor-deal.II repository contains bindings for deal.II.
pyMOR classes implement dependence on a parameter by deriving from the ParametricObject base class. This class gives each instance a parameters attribute describing the Parameters the object and its
relevant methods (apply, solve, evaluate, etc.) depend on. Each Parameter in pyMOR has a name and a fixed dimension, i.e. the number of scalar components of the Parameter. Scalar parameters are
simply represented by one-dimensional Parameters. To assign concrete values to Parameters the specialized dict-like class Mu is used. In particular, it ensures, that all of its values are
one-dimensional NumPy arrays.
The Parameters of a ParametricObject are usually automatically derived as the union of all Parameters of the objects that are passed to it’s __init__ method. For instance, an Operator that implements
the L2-product with some user-provided Function will automatically inherit all Parameters of that Function. Additional Parameters can be easily added by setting the parameters_own attribute.
pyMOR offers a convenient mechanism for handling default values such as solver tolerances, cache sizes, log levels, etc. Each default in pyMOR is the default value of an optional argument of some
function. Such an argument is made a default by decorating the function with the defaults decorator:
def some_algorithm(x, y, tolerance=1e-5)
Default values can be changed by calling set_defaults. By calling print_defaults a summary of all defaults in pyMOR and their values can be printed. A configuration file with all defaults can be
obtained with write_defaults_to_file. This file can then be loaded, either programmatically or automatically by setting the PYMOR_DEFAULTS environment variable.
As an additional feature, if None is passed as value for a function argument which is a default, its default value is used instead of None. This allows writing code of the following form:
def method_called_by_user(U, V, tolerance_for_algorithm=None):
algorithm(U, V, tolerance=tolerance_for_algorithm)
See the defaults module for more information.
Many algorithms in pyMOR can be seen as transformations acting on trees of Operators. One example is the structure-preserving (Petrov-)Galerkin projection of Operators performed by the project
method. For instance, a LincombOperator is projected by replacing all its children (the Operators forming the affine decomposition) with projected Operators.
During development of pyMOR, it turned out that using inheritance for selecting the action to be taken to project a specific operator (i.e. single dispatch based on the class of the to-be-projected
Operator) is not sufficiently flexible. With pyMOR 0.5 we have introduced algorithms which are based on RuleTables instead of inheritance. A RuleTable is simply an ordered list of rules, i.e. pairs
of conditions to match with corresponding actions. When a RuleTable is applied to an object (e.g. an Operator), the action associated with the first matching rule in the table is executed. As part of
the action, the RuleTable can be easily applied recursively to the children of the given object.
This approach has several advantages over an inheritance-based model:
• Rules can match based on the class of the object, but also on more general conditions, i.e. the name of the Operator or being linear and non-parametric.
• The entire mathematical algorithm can be specified in a single file even when the definition of the possible classes the algorithm can be applied to is scattered over various files.
• The precedence of rules is directly apparent from the definition of the RuleTable.
• Generic rules (e.g. the projection of a linear non-parametric Operator by simply applying the basis) can be easily scheduled to take precedence over more specific rules.
• Users can implement or modify RuleTables without modification of the classes shipped with pyMOR.
The Reduction Process¶
The reduction process in pyMOR is handled by so called reductors which take arbitrary Models and additional data (e.g. the reduced basis) to create reduced Models. If proper offline/online
decomposition is achieved by the reductor, the reduced Model will not store any high-dimensional data. Note that there is no inherent distinction between low- and high-dimensional Models in pyMOR.
The only difference lies in the different types of operators, the Model contains.
This observation is particularly apparent in the case of the classical reduced basis method: the operators and functionals of a given discrete problem are projected onto the reduced basis space
whereas the structure of the problem (i.e. the type of Model containing the operators) stays the same. pyMOR reflects this fact by offering with GenericRBReductor a generic algorithm which can be
used to RB-project any model available to pyMOR. It should be noted however that this reductor is only able to efficiently offline/online-decompose affinely Parameter-dependent linear problems.
Non-linear problems or such with no affine Parameter dependence require additional techniques such as empirical interpolation.
If you want to further dive into the inner workings of pyMOR, we recommend to study the source code of GenericRBReductor and to step through calls of it’s reduce method with a Python debugger, such
as ipdb.
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SLMath - November 2024
Economists are using mathematics to come to a deeper understanding of how to allocate indivisible goods in markets that do not use currency.
ICERM - September 2024
Fluid-structure interaction (FSI) problems involve the study of how fluids and solid structures behave when they come into contact with each other. These problems are important in many engineering
and scientific applications, such as aeronautics, aeroelasticity, aerodynamics, biomechanics, civil engineering, and mechanical engineering. Sijing Liu introduces a second-order correction method for
a parabolic-parabolic interface problem, a simplified version of FSI.
IPAM - August 2024
If Alice and Bob each take a walk, a random one, what is the chance they will meet? If n people walk randomly in Manhattan from the Upper East Side going south or west, how often will two of them
meet until they reach the Hudson? If they do not want to see each other more than once, in what relative order will they most likely arrive at the Hudson shore? If we make a rhombus like an Aztec
diamond from 2 by 1 dominoes, what would it most likely look like? If water molecules arrange themselves on a square grid, what angles between the hydrogen atoms will be formed near the boundaries?
And if ribosomes are transcribing the mRNA, how do they hop between the sites (codons)?
SLMath - August 2024
Political districting is an important and mathematically challenging problem. In fall 2023, the Simons Laufer Mathematical Sciences Institute (SLMath) hosted a semester program on algorithms,
fairness, and equity, focusing on the intersection of computational tools and the many notions of fairness that arise in different mathematical and societal contexts, such as political districting.
ICERM - July 2024
Identifying the capabilities of a machine learning model before applying it is a key goal in neural architecture search. Too small of a model, and the network will never successfully perform its
task. Too large of a model, and computational energy is wasted and the cost of model training may become unbearably high. While universal approximation theorems and even dynamics are available for
various limiting forms of ReLU neural networks, there are still many questions about the limitations of what small neural networks of this form can accomplish. Marissa Masden seeks to understand ReLU
neural networks through a combinatorial, discrete structure that enables capturing topological invariants of structures like the decision boundary.
IMSI - June 2024
Understanding the evolutionary history of a collection of species, through fields such as phylogenomics and comparative phylogenetics, is crucial as we consider the future effects of climate change.
Algebraic statistics provides algebraic and geometric tools to study the models commonly used in evolutionary biology. The Institute for Mathematical and Statistical Innovation (IMSI), hosted a
workshop, “Algebraic Statistics for Ecological and Biological Systems,” as part of a Long Program on “Algebraic Statistics and Our Changing World,” which highlighted these connections.
ICERM - May 2024
Nonlinear equations are ubiquitous in the sciences, a famous example being the Navier-Stokes equations in fluid mechanics. To compute a numerical solution for a given nonlinear equation, one often
employs an iterative scheme such as Newton’s method. Recent works by Matt Dallas, Sara Pollock, and Leo Rebholz analyze Anderson accelerated Newton’s method applied to singular problems.
SLMath - April 2024
In the mid-19th century, while attempting to prove Fermat’s last theorem, German mathematician Ernst Kummer started investigating arithmetic on novel number systems.
IAS - April 2024
The largest live autonomous vehicle traffic experiment ever conducted began the week of November 18, 2022, in Nashville, Tennessee. It involved 100 cars and a workforce of more than 250, around 70 of
whom were researchers. One of the goals of the experiment was to analyze how level two autonomous vehicles (think cruise control with a couple added functions, like speed adjustment that uses LIDAR)
can impact traffic waves, specifically those representing frustrating “stop and go” conditions...
AIM - February 2024
Kaisa Matomäki, Maksym Radziwill, Terence Tao, Joni Teräväinen, and Tamar Ziegler make progress on the conjectures of Sarnak and Chowla with their paper “Higher uniformity of bounded multiplicative
functions in short intervals on average” published in the Annals of Mathematics in 2023. The work originated in a working group at the AIM workshop “Sarnak's conjecture” in December 2018.
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Brian Murray Archives | SSRC
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Assessment of and for Learning: Introduction
• Grading Guide
• Curriculum Overview
• Assessment of and for Learning: The Tests
• Number and Algebra
□ 1A Place value
□ 1B Mental strategies for addition
□ 1C Mental strategies for subtraction
□ 1D Written strategies for addition and subtraction
□ 1E Skip counting
□ 1F Multiplication and division facts
□ 1G Strategies for multiplication and division
□ 2A Fractions
□ 2B Fractions on number lines
□ 3A Money
□ 4A Number patterns
□ 4B Number stories
• Measurement and Geometry
□ 5A Length and area
□ 5B Capacity and volume
□ 5C Mass
□ 5D Time
□ 6A 2D shapes
□ 6B 3D objects
□ 7A Angles
□ 7B Symmetry
□ 7C Grids and maps
• Statistics and Probability
□ 8A Collecting data
□ 8B Recording data
□ 8C Displaying and interpreting data
□ 8D Chance events
□ 8E Chance experiments
• From Assessment to Instruction: Teacher Support
• Answers
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Assessment of and for Learning: Introduction
• Grading Guide
• Curriculum Overview
• Assessment of and for Learning: The Tests
• Number and Algebra
□ 1A Place value
□ 1B Mental strategies for addition
□ 1C Mental strategies for subtraction
□ 1D Written strategies for addition and subtraction
□ 1E Skip counting
□ 1F Multiplication and division facts
□ 1G Strategies for multiplication and division
□ 2A Fractions
□ 2B Fractions on number lines
□ 3A Money
□ 4A Number patterns
□ 4B Number stories
• Measurement and Geometry
□ 5A Length and area
□ 5B Capacity and volume
□ 5C Mass
□ 5D Time
□ 6A 2D shapes
□ 6B 3D objects
□ 7A Angles
□ 7B Symmetry
□ 7C Grids and maps
• Statistics and Probability
□ 8A Collecting data
□ 8B Recording data
□ 8C Displaying and interpreting data
□ 8D Chance events
□ 8E Chance experiments
• From Assessment to Instruction: Teacher Support
• Answers
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Assessment of and for Learning: Introduction
• Grading Guide
• Curriculum Overview
• Assessment of and for Learning: The Tests
• Number and Algebra
□ 1A Place value
□ 1B Mental strategies for addition
□ 1C Mental strategies for subtraction
□ 1D Written strategies for addition and subtraction
□ 1E Skip counting
□ 1F Multiplication and division facts
□ 1G Strategies for multiplication and division
□ 2A Equivalent fractions
□ 2B Fractions on number lines
□ 2C Decimal fractions
□ 3A Money and calculations
□ 4A Number patterns
□ 4B Number stories
• Measurement and Geometry
□ 5A Length
□ 5B Area
□ 5C Volume and capacity
□ 5D Mass
□ 5E Time
□ 5F Temperature
□ 6A 2D shapes
□ 6B 3D objects
□ 7A Angles
□ 7B Symmetry
□ 7C Scales and maps
• Statistics and Probability
□ 8A Collecting data
□ 8B Recording data
□ 8C Displaying and interpreting data
□ 8D Chance events
□ 8E Chance experiments
• From Assessment to Instruction: Teacher Support
• Answers
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Assessment of and for Learning: Introduction
• Grading Guide
• Curriculum Overview
• Assessment of and for Learning: The Tests
• Number and Algebra
□ 1A Place value
□ 1B Mental strategies for addition
□ 1C Mental strategies for subtraction
□ 1D Written strategies for addition and subtraction
□ 1E Skip counting
□ 1F Multiplication and division facts
□ 1G Strategies for multiplication and division
□ 2A Equivalent fractions
□ 2B Fractions on number lines
□ 2C Decimal fractions
□ 3A Money and calculations
□ 4A Number patterns
□ 4B Number stories
• Measurement and Geometry
□ 5A Length
□ 5B Area
□ 5C Volume and capacity
□ 5D Mass
□ 5E Time
□ 5F Temperature
□ 6A 2D shapes
□ 6B 3D objects
□ 7A Angles
□ 7B Symmetry
□ 7C Scales and maps
• Statistics and Probability
□ 8A Collecting data
□ 8B Recording data
□ 8C Displaying and interpreting data
□ 8D Chance events
□ 8E Chance experiments
• From Assessment to Instruction: Teacher Support
• Answers
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Contents
□ Assessment of and for Learning: Introduction
□ Grading Guide
□ Curriculum Overview
□ Assessment of and for Learning: The Tests
□ Number and Algebra
☆ 1A Place value
☆ 1B Number properties
☆ 1C Mental strategies for addition and subtraction
☆ 1D Written strategies for addition and subtraction
☆ 1E Mental strategies for multiplication and division
☆ 1F Written strategies for multiplication and division
☆ 1G Integers
☆ 2A Fractions
☆ 2B Addition and subtraction of common fractions
☆ 2C Decimal fractions
☆ 2D Addition and subtraction of decimals
☆ 2E Multiplication and division of decimals
☆ 2F Decimals and powers of ten
☆ 2G Percentage, fractions and decimals
☆ 3A Geometric patterns
☆ 3B Number patterns
☆ 3C Order of operations and equations
□ Measurement and Geometry
☆ 4A Length
☆ 4B Area
☆ 4C Volume and capacity
☆ 4D Mass
☆ 4E Timetables
☆ 5A Angles
☆ 5B 2D shapes and 3D objects
☆ 6A Transformations
☆ Statistics and Probability
☆ 6B Cartesian coordinate systems
☆ 7B Interpreting data
☆ 7C Data in the media
☆ 7D Probability
☆ 7E Chance experiments and simulations
□ From Assessment to Instruction: Teacher Support
□ Answers
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on students level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Contents
□ Assessment of and for Learning: Introduction
□ Grading Guide
□ Curriculum Overview
□ Assessment of and for Learning: The Tests
□ Number and Algebra
☆ 1A Place value
☆ 1B Number properties
☆ 1C Mental strategies for addition and subtraction
☆ 1D Written strategies for addition and subtraction
☆ 1E Mental strategies for multiplication and division
☆ 1F Written strategies for multiplication and division
☆ 1G Integers
☆ 2A Fractions
☆ 2B Addition and subtraction of common fractions
☆ 2C Decimal fractions
☆ 2D Addition and subtraction of decimals
☆ 2E Multiplication and division of decimals
☆ 2F Decimals and powers of ten
☆ 2G Percentage, fractions and decimals
☆ 3A Geometric patterns
☆ 3B Number patterns
☆ 3C Order of operations and equations
□ Measurement and Geometry
☆ 4A Length
☆ 4B Area
☆ 4C Volume and capacity
☆ 4D Mass
☆ 4E Timetables
☆ 5A Angles
☆ 5B 2D shapes and 3D objects
☆ 6A Transformations
☆ Statistics and Probability
☆ 6B Cartesian coordinate systems
☆ 7B Interpreting data
☆ 7C Data in the media
☆ 7D Probability
☆ 7E Chance experiments and simulations
□ From Assessment to Instruction: Teacher Support
□ Answers
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Minimum-Cost Load-Balancing Partitions
We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let D be the underlying demand region, and let p [1],p [m] be m points
representing m facilities. We consider the following problem: Subdivide D into m equal-area regions R [1],R [m] , so that region R [i] is served by facility p [i] , and the average distance between a
point q in D and the facility that serves q is minimal. We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be
convex. As an intermediate result we show how to partition a convex polygon into m equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness
of the original polygon. In fact, we prove that our partition is, up to a constant factor, the best one can get if one's goal is to maximize the fatness of the least fat subregion. We also discuss
the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of
• Additive-weighted Voronoi diagram
• Approximation algorithms
• Fat partitions
• Fatness
• Geometric optimization
• Load balancing
ASJC Scopus subject areas
• General Computer Science
• Computer Science Applications
• Applied Mathematics
Dive into the research topics of 'Minimum-Cost Load-Balancing Partitions'. Together they form a unique fingerprint.
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[Solved] .If there is no corresponding increase in technology, which
.If there is no corresponding increase in technology, which of the following will happen as the capital stock of a country increases, all else equal?
I. Production per worker will increase.
II. Economic growth will increase.
III. The rate of economic growth will increase.
(A) I only
(B) II only
(C) III only
(D) I, II AND III
Answer (Detailed Solution Below)
Option 4 : I, II, III
If there is no corresponding increase in technology, as the capital stock of a country increases, all else equal, Production per worker will increase, Economic growth will increase and the rate of
economic growth will increase.
Key Points
• As the capital of nations grows, it will have more capital per worker (if all else equal, the stock of labor does not also grow) which will increase the real GDP per capita of the country.
• In other words more capital per worker will lead to economic growth. However, because there are diminishing returns to capital, each additional unit of capital will not generate the same increase
in output, leading to a lower rate of economic growth.
Additional Information
• The Solow Growth Model is an exogenous model of economic growth that analyzes changes in the level of output in an economy over time as a result of changes in the population growth rate, the
savings rate, and the rate of technological progress.
• Mathematically, the Solow–Swan model is a nonlinear system consisting of a single ordinary differential equation that models the evolution of the per capita stock of capital.
• Due to its particularly attractive mathematical characteristics, Solow–Swan proved to be a convenient starting point for various extensions.
• For instance, in 1965, David Cass and Koopmans integrated Frank Ramsey's analysis of consumer optimization,thereby endogenizing the saving rate, to create what is now known as the
Ramsey–Cass–Koopmans model
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(PDF) Exploiting Counterfactuals for Scalable Stochastic Optimization
Conference PaperPDF Available
Exploiting Counterfactuals for Scalable Stochastic Optimization
We propose a new framework for decision making under uncertainty to overcome the main drawbacks of current technology: modeling complexity, scenario generation, and scaling limitations. We consider
three NP-hard optimization problems: the Stochastic Knapsack Problem (SKP), the Stochastic Shortest Path Problem (SSPP), and the Resource Constrained Project Scheduling Problem (RCPSP) with uncertain
job durations, all with recourse. We illustrate how an integration of constraint optimization and machine learning technology can overcome the main practical shortcomings of the current state of the
The area of maritime transportation optimization has recently begun to achieve increasing success at solving large scale models, and industry is steadily adopting operations research-based models and
algorithms. However, the parameters of models in the maritime domain, like many others, are beset with uncertainty. The travel times of ships, the handling times at port, the amounts of demand
available at ports, fuel prices and more are all unknown and highly variable inputs to optimization methods. Recently, the maritime literature has started to address sources of uncertainty to provide
higher quality decision making. We review this nascent area of the literature and provide a unifying view of different types of uncertainty across the main areas of maritime transport and varying
problem types.
This paper aims at solving the stochastic shortest path problem in vehicle routing, the objective of which is to determine an optimal path that maximizes the probability of arriving at the
destination before a given deadline. To solve this problem, we propose a data-driven approach, which directly explores the big data generated in traffic. Specifically, we first reformulate the
original shortest path problem as a cardinality minimization problem directly based on samples of travel time on each road link, which can be obtained from the GPS trajectory of vehicles. Then, we
apply an l1-norm minimization technique and its variants to solve the cardinality problem. Finally, we transform this problem into a mixed-integer linear programming problem, which can be solved
using standard solvers. The proposed approach has three advantages over traditional methods. First, it can handle various or even unknown travel time probability distributions, while traditional
stochastic routing methods can only work on specified probability distributions. Second, it does not rely on the assumption that travel time on different road segments is independent of each other,
which is usually the case in traditional stochastic routing methods. Third, unlike other existing methods which require that deadlines must be larger than certain values, the proposed approach
supports more flexible deadlines. We further analyze the influence of important parameters to the performances, i.e., accuracy and time complexity. Finally, we implement the proposed approach and
evaluate its performance based on a real road network of Munich city. With real traffic data, the results show that it outperforms traditional methods.
A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated
according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative
scenarios (or a representative scenario tree) relative to the problem being modeled.
We introduce an instance-weighting method to induce cost-sensitive trees. It is a generalization of the standard tree induction process where only the initial instance weights determine the type of
tree to be induced-minimum error trees or minimum high cost error trees. We demonstrate that it can be easily adapted to an existing tree learning algorithm. Previous research provides insufficient
evidence to support the idea that the greedy divide-and-conquer algorithm can effectively induce a truly cost-sensitive tree directly from the training data. We provide this empirical evidence in
this paper. The algorithm incorporating the instance-weighting method is found to be better than the original algorithm in terms of total misclassification costs, the number of high cost errors, and
tree size in two-class data sets. The instance-weighting method is simpler and more effective in implementation than a previous method based on altered priors.
The objectives of the airline crew-planning process are to allocate crews to flights and create work schedules for crew members. Most airlines solve their crew-planning problem in two steps. The
first step, crew pairing, is to generate optimized, anonymous pairings that cover given flight schedules. In the second step, the resulting pairings are assigned to crew members. The general pairing
problem is complex because flights may require an augmented crew for safety reasons. A flight’s crew-augmentation requirement varies, depending on the characteristics of the pairings that cover it.
Furthermore, airlines often impose rules to govern the coverage of a flight by different pairings. Common approaches to the problem either fix the crew-augmentation requirement a priori, or add
restrictions on how the augmentation requirement is satisfied. Crew augmentation is often overlooked from an optimization perspective because of the complexities involved. The Sabre® long-haul
pairing optimizer explicitly models many types of crew-augmentation processes and simultaneously considers the relevant ranks of all members within the cockpit crew. It uses state-of-the-art
large-scale optimization techniques, such as branch and price, to solve the problem. In this article, we introduce the long-haul pairing optimizer that Sabre developed in the mid-1990s, and share the
evolution of the models and solution algorithms for the general crew-pairing problem with augmentation. We also compare our approach with four conventional approaches to show that we can effectively
solve the general crew-augmentation problem and provide significant crew cost savings to airlines.
This paper presents an algorithm for finding all shortest routes from all nodes to a given destination in N N -node general networks (in which the distances of arcs can be negative). If no negative
loop exists, the algorithm requires 1 2 M ( N − 1 ) ( N − 2 ) , 1 > M N − 1 \frac {1}{2}M\left ( {N - 1} \right ) \\ \left ( {N - 2} \right ),1 > MN - 1 , additions and comparisons. The existence of
a negative loop, should one exist, is detected after 1 2 N ( N − 1 ) ( N − 2 ) \frac {1}{2}N\left ( {N - 1} \right )\left ( {N - 2} \right ) additions and comparisons.
We present a set of benchmark instances for the evaluation of solution procedures for single- and multi-mode resource-constrained project scheduling problems. The instances have been systematically
generated by the standard project generator ProGen. They are characterized by the input-parameters of ProGen. The entire benchmark set including its detailed characterization and the best solutions
known so-far are available on a public ftp-site. Hence, researchers can download the benchmark sets they need for the evaluation of their algorithms. Additionally, they can make available new
results. Depending on the progress made in the field, the instance library will be continuously enlarged and new results will be made accessible. This should be a valuable and driving source for
further improvements in the area of project type scheduling.
One of the challenges faced by liner operators today is to effectively operate empty containers in order to meet demand and to reduce inefficiency in an uncertain environment. To incorporate
uncertainties in the operations model, we formulate a two-stage stochastic programming model with random demand, supply, ship weight capacity, and ship space capacity. The objective of this model is
to minimize the expected operational cost for Empty Container Repositioning (ECR). To solve the stochastic programs with a prohibitively large number of scenarios, the Sample Average Approximation
(SAA) method is applied to approximate the expected cost function. To solve the SAA problem, we consider applying the scenario aggregation by combining the approximate solution of the individual
scenario problem. Two heuristic algorithms based on the progressive hedging strategy are applied to solve the SAA problem. Numerical experiments are provided to show the good performance of the
scenario-based method for the ECR problem with uncertainties.
This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this
form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual
program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.
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Homology statistics for 2-manifolds
For a new (forthcoming) research project, I have to brush up my knowledge about homology groups and triangulations. This constitutes an excellent opportunity to extend Aleph, my (also forthcoming)
library for exploring persistent homology and its uses.
As a quick example, I decided to calculate some statistics about the homology of 2-manifolds. Thanks to the work of Frank H. Lutz, a list of triangulations with up to 9 vertices, as well as a list of
triangulations with 10 vertices, exist. After adding support for the appropriate data format in Aleph, I was able to easily read a family of triangulations and subsequently process them. Even though
Aleph’s main purpose is the calculation of persistent homology, ordinary homology calculations work just as well—ordinary homology being a special case of persistent homology.
As a starting point for further mathematical ruminations, let us take a look at the distribution of “Betti numbers” of 2-manifolds. More precisely, we analyse the distribution of the first Betti
number of the manifold—according to Poincaré duality, the zeroth and second Betti number have to equal, and we know the zeroth Betti number to be 1 for a manifold.
Without further ado, here is a tabulation of first Betti numbers for triangulations with up to 9 vertices:
Value Number of occurrences
The results are interesting: most triangulations appear to have the homology of a torus, i.e. a first Betti number of 2, followed by the homology of the real projective plane with a first Betti
number of 1. Betti numbers larger than 3 are extremely rare. Intuitively, this makes sense—the triangulation only consists of at most 9 vertices, so there are natural limits on how high the Betti
number can become.
For triangulations with 10 vertices, another distribution arises:
Value Number of occurrences
Here, the mode of Betti numbers is a value of 4, with 14522 occurrences out of 42426 triangulations. The homology type of this signature is the one of a genus-4 surface. Again, higher values get
progressively less likely because only 10 vertices are permitted.
I am not yet sure what to make of this, but it sure is a nice test case and application scenario for Aleph. One of the next blog posts will give more details about this calculation, and its
implementation using the library. Stay tuned!
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bootSimpleComplex: Bootstrap test to see if a complex model is significantly... in paleoTS: Analyze Paleontological Time-Series
Bootstrap test to see if a complex model is significantly better than a simple model.
bootSimpleComplex( y, simpleFit, complexFit, nboot = 99, minb = 7, ret.full.distribution = FALSE, parallel = FALSE, ... )
y a paleoTS object
simpleFit a paleoTSfit object, representing the model fit of a simple model
complexFit a paleoTSfit object, representing the model fit of a complex model
nboot number of replications for parametric bootstrapping
minb minimum number of populations within each segment
ret.full.distribution logical, indicating if the null distribution for the likelihood ratio from the parametric bootstrap should be returned
parallel logical, if TRUE, the bootstrapping is done using parallel computing
... further arguments, passed to optimization functions
a paleoTSfit object, representing the model fit of a simple model
a paleoTSfit object, representing the model fit of a complex model
logical, indicating if the null distribution for the likelihood ratio from the parametric bootstrap should be returned
logical, if TRUE, the bootstrapping is done using parallel computing
Simulations suggest that AICc can be overly liberal with complex models with mode shifts or punctuations (Hunt et al., 2015). This function implements an alternative of parametric boostrapping to
compare the fit of a simple model with a complex model. It proceeds in five steps:
Compute the observed gain in support from the simple to complex model as the likelihood ratio, LR_{obs} = -2(logL_{simple} - logL_{complex})
Simulate trait evolution under the specified simple model nboot times
Fit to each simulated sequence the specified simple and complex models
Measure the gain in support from simple to complex as the bootstrap likelihood ratio for each simulated sequence
Compute the P-value as the percentile of the bootstrap distribution corresponding to the observed LR.
Argument simpleFit should be a paleoTS object returned by the function fitSimple or similar functions (e.g., opt.joint.GRW, opt.GRW, etc.). Argument complexFit must be a paleoTS object returned by
fitGpunc or fitModeShift.
Calculations can be speeded up by setting parallel = TRUE, which uses package doParallel to run the bootstrap replicates in parallel, using one fewer than the number of detected cores.
A list of the observed likelihood ratio statistic, LRobs, the P-value of the test, and the number of bootstrap replicates. If ret.full.distribution = TRUE, the null distribution of likelihood ratios
generated by parametric bootstrapping is also returned.
Hunt, G., M. J. Hopkins and S. Lidgard. 2015. Simple versus complex models of trait evolution and stasis as a response to environmental change. PNAS 112(16): 4885-4890.
## Not run: x <- sim.Stasis.RW(ns = c(15, 15), omega = 0.5, ms = 1, order = "Stasis-RW") ws <- fitSimple(x) wc <- fitModeShift(x, order = "Stasis-RW", rw.model = "GRW") bootSimpleComplex(x, ws, wc,
nboot = 50, minb = 7) # nboot too low for real analysis! ## End(Not run)
For more information on customizing the embed code, read Embedding Snippets.
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Compositions Of Transformations Worksheet Answers
Compositions Of Transformations Worksheet Answers. Web composition of transformations worksheet. Web when transformations are combined, the resulting transformation is a composition of
compositions of transformations worksheet answers from materialschoolvincent.z22.web.core.windows.net
Web when transformations are combined, the resulting transformation is a composition of transformations. Web by iteachalgebra 4.8 (5) $2.00 pdf students will learn:the steps for transforming a figure
more than once, using reflections, translations, and rotations (6 problems)answer key is. Having a thorough understanding of the individual transformations is.
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Propagation Models
For the prediction, topographical databases (digital elevation model, DEM) are needed. They consist of binary stored pixel data with an arbitrary resolution, for example, 50m x 50 m. However, the
resolution in one database must be constant. Also morphological data can be considered by empirical correction values to improve the accuracy of the model. This data is also stored as binary data.
The different morphological properties are coded, for example:
• urban
• suburban
• forest
• water
• acre
Propagation Models
offers various wave propagation models for rural and suburban environments.
• Empirical models without consideration of the terrain profile between transmitter and receiver
□ Hata-Okumura model
□ Empirical two ray model
□ ITU P.1546 model
• Basic topographical profile prediction models (2D vertical plane models)
□ Deterministic two ray model
□ Longley-Rice model
□ Parabolic equation method
□ Knife edge diffraction model
• Deterministic 3D models (3D topography)
□ Rural dominant path model
□ Rural ray-tracing model
Report ITU-R SM.2028-2
“Monte Carlo simulation methodology for the use in sharing and compatibility studies between different radio services or systems”
, section 6.1, published June 2017.
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Linear equation
Linear equation
About this schools Wikipedia selection
This Wikipedia selection is available offline from SOS Children for distribution in the developing world. A quick link for child sponsorship is http://www.sponsor-a-child.org.uk/
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. Such an equation is equivalent to equating a
first-degree polynomial to zero. These equations are called "linear" because they represent straight lines in Cartesian coordinates. A common form of a linear equation in the two variables $x$ and
$y$ is
$y = mx + b.\,$
In this form, the constant $m$ will determine the slope or gradient of the line; and the constant term $b$ will determine the point at which the line crosses the y-axis. Equations involving terms
such as x², y^1/3, and xy are nonlinear.
Forms for 2D linear equations
Complicated linear equations, such as the ones above, can be rewritten using the laws of elementary algebra into several simpler forms. In what follows x, y and t are variables; other letters
represent constants (unspecified but fixed numbers).
General form
$Ax + By + C = 0,\,$
where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by
an equation in the above form. If A is nonzero, then the x-intercept, that is the x- coordinate of the point where the graph crosses the x-axis (y is zero), is −C/A. If B is nonzero, then the y
-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (x is zero), is −C/B, and the slope of the line is −A/B.
Standard form
$Ax + By = C,\,$
where A, B, and C are integers whose greatest common factor is 1, A and B are not both equal to zero and, A is non-negative (and if A=0 then B has to be positive). The standard form can be
converted to the general form, but not always to all the other forms if A or B is zero.
Slope–intercept form
Y-axis formula
$y = mx + b,\,$
where m is the slope of the line and b is the y-intercept, which is the y-coordinate of the point where the line crosses the y axis. This can be seen by letting $x = 0$, which immediately gives
$y = b$.
X-axis formula
$x = \frac{y}{m} + c,\,$
where m is the slope of the line and c is the x-intercept, which is the x-coordinate of the point where the line crosses the x axis. This can be seen by letting $y = 0$, which immediately gives
$x = c$.
Point–slope form
$y - y_1 = m \cdot ( x - x_1 ),$
where m is the slope of the line and (x[1],y[1]) is any point on the line. The point-slope and slope-intercept forms are easily interchangeable.
The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, $y - y_1$) is proportional to the difference in the x coordinate (that is,
$x - x_1$). The proportionality constant is m (the slope of the line).
Intercept form
$\frac{x}{c} + \frac{y}{b} = 1$
where c and b must be nonzero. The graph of the equation has x-intercept c and y-intercept b. The intercept form can be converted to the standard form by setting A = 1/c, B = 1/b and C = 1.
Two-point form
$y - k = \frac{q - k}{p - h} (x - h),$
where p ≠ h. The graph passes through the points (h,k) and (p,q), and has slope m = (q−k) / (p−h).
Parametric form
$x = T t + U\,$
$y = V t + W.\,$
Two simultaneous equations in terms of a variable parameter t, with slope m = V / T, x-intercept (VU−WT) / V and y-intercept (WT−VU) / T.
This can also be related to the two-point form, where T = p−h, U = h, V = q−k, and W = k:
$x = (p - h) t + h\,$
$y = (q - k)t + k.\,$
In this case t varies from 0 at point (h,k) to 1 at point (p,q), with values of t between 0 and 1 providing interpolation and other values of t providing extrapolation.
Normal form
$y \sin \phi + x \cos \phi - p = 0,\,$
where φ is the angle of inclination of the normal and p is the length of the normal. The normal is defined to be the shortest segment between the line in question and the origin. Normal form can
be derived from general form by dividing all of the coefficients by $\frac{|C|}{-C}\sqrt{A^2 + B^2}$. This form also called Hesse standard form, named after a German mathematician Ludwig Otto
Special cases
$y = b.\,$
This is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope M = 0. The graph is a horizontal line with y-intercept equal to b. There is no x
-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept.
$x = c.\,$
This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to c. The slope is undefined. There is no y-intercept, unless c = 0, in
which case the graph of the line is the y-axis, and so every real number is a y-intercept.
$y = y \$ and $x = x.\,$
In this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an identity and one would not normally consider
its graph (it would be the entire xy-plane). An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the equal sign are always equal, no matter what values are used for x and y.
$e = f.\,$
In situations where algebraic manipulation leads to a statement such as 1 = 0, then the original equation is called inconsistent, meaning it is untrue for any values of x and y (i.e. its graph
would be the empty set) An example would be 3x + 2 = 3x − 5.
Connection with linear functions and operators
In all of the named forms above (assuming the graph is not a vertical line), the variable y is a function of x, and the graph of this function is the graph of the equation.
In the particular case that the line crosses through the origin, if the linear equation is written in the form y = f(x) then f has the properties:
$f ( x + y ) = f ( x ) + f ( y )\,$
$f ( a x ) = a f ( x ),\,$
where a is any scalar. A function which satisfies these properties is called a linear function, or more generally a linear map. This property makes linear equations particularly easy to solve and
reason about.
Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be
reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.
Linear equations in more than two variables
A linear equation can involve more than two variables. The general linear equation in n variables is:
$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b.$
In this form, a[1], a[2], …, a[n] are the coefficients, x[1], x[2], …, x[n] are the variables, and b is the constant. When dealing with three or fewer variables, it is common to replace x[1] with
just x, x[2] with y, and x[3] with z, as appropriate.
Such an equation will represent an (n–1)-dimensional hyperplane in n-dimensional Euclidean space (for example, a plane in 3-space).
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How to Convert a Month Name to a Number | Encyclopedia-Excel
Usually, you need to return the last or first item in a list, but every once in a while, you may find yourself needing to return the second to last item in a list. These formulas will help.
For this formula, the range should be replaced with the range of month names you would like to convert.
= MONTH(DATEVALUE(range & " 1"))
This formula can be broken up into two main parts.
First, the DATEVALUE function converts a text date into number representing a date. This function needs the date string to be in a valid date format, which is why you append " 1" to the month name,
creating a valid date string.
For example, if you have "January" in cell A1, A1 & " 1" will give you "January 1".
When this is fed to the DATEVALUE function, it is converted into a serial number representing the date.
So, DATEVALUE("January 1") will give you the serial number for January 1 in the current year: 1/1/2023 (Excel assumes the current year if you don't specify one).
Second, MONTH function then extracts the month number from this date serial number. So, if 1/1/2023 is fed into the MONTH function, a 1 will be returned. 2/1/2023 will return a 2, and so on.
How to Convert Month Name to a Number
In this example, we have a list of month names in column B and need to convert those names into the corresponding month numbers.
Using the following formula in column C, we can easily convert each name into a number.
This formula can be a single cell reference, or if you are using Excel 365, you can also select the entire range, using "B3:B14" as your range.
= MONTH(DATEVALUE(B3 & " 1"))
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Parallel Aggregations in PLINQ
Quick Overview of LINQ Aggregations
In order to explain the issues we encounter when parallelizing aggregations in PLINQ, let’s first take a quick look at how aggregations work in LINQ.
Aggregation is an operation that iterates over a sequence of input elements, maintaining an accumulator that contains the intermediate result. At each step, a reduction function takes the current
element and accumulator value as inputs, and returns a value that will overwrite the accumulator. The final accumulator value is the result of the computation. A variety of interesting operations can
be expressed as aggregations: sum, average, min, max, sum of squares, variance, concatenation, count, count of elements matching a predicate, and so on.
LINQ provides several overloads of Aggregate. A possible implementation (without error checking) of the most general of them is given below:
public static TResult Aggregate<TSource, TAccumulate, TResult>(
this IEnumerable<TSource> source,
TAccumulate seed,
Func<TAccumulate, TSource, TAccumulate> func,
Func<TAccumulate, TResult> resultSelector
TAccumulate accumulator = seed;
foreach (TSource elem in source)
accumulator = func(accumulator, elem);
return resultSelector(accumulator);
To compute a particular aggregation, the user provides the input sequence (as method parameter source), the initial accumulator value (seed), the reduction function (func), and a function to convert
the final accumulator to the result (resultSelector). As a usage example, consider the method below that computes the sum of squares of integers:
public static int SumSquares(IEnumerable<int> source)
return source.Aggregate(0, (sum, x) => sum + x * x, (sum) => sum);
LINQ also exposes a number of predefined aggregations, such as Sum, Average, Max, Min, etc. Even though each one can be implemented using the Aggregate operator, a direct implementation is likely to
be more efficient (for example, to avoid a delegate call for each input element).
Parallelizing LINQ Aggregations
Let’s say that we call SumSquares(Enumerable.Range(1,4)) on a dual-core machine. How can we split up the computation among two threads? We could distribute the elements of the input among the
threads. For example, Thread 1 could compute the sum of squares of {1,4} and Thread 2 would compute the sum of squares of {3,2}*. Then, as a last step, we combine the results – add them in this case
– and we get the final answer.
Sequential Answer = ((((0 + 1^2) + 2^2) + 3^2) + 4^2) = 30
Parallel Answer = (((0 + 1^2) + 4^2) + ((0 + 3^2) + 2^2)) = 30
Note: Notice that elements within each partition do not necessarily appear in the order in which they appear in the input. The reason for this may not be apparent, but it has to do with the presence
of other operators in the query.
Combining Accumulators
In the parallel aggregation, we need to do something that we didn’t need to in the sequential aggregation: combine the intermediate results (i.e. accumulators). Notice that combining two accumulators
may be a different operation than combining an accumulator with an input element. In the SumSquares example, to combine the accumulator with an input element, we square the element and add it to the
accumulator. But, to combine two accumulators, we simply add them, without squaring the second one.
In the cases where the accumulator type is different from the element type, it is even more obvious combining accumulators and combining an accumulator with an element are different operations: even
their input argument types differ!
Therefore, the most general PLINQ Aggregate overload accepts an intermediate reduce function as well as a final reduce function, while the most general LINQ Aggregate only needs the intermediate
reduce function. The signature of the most general PLINQ Aggregate overload is below (compare with the most general LINQ Aggregate overload shown above):
public static TResult Aggregate<TSource, TAccumulate, TResult>(
this IParallelEnumerable<TSource> source,
TAccumulate seed,
Func<TAccumulate, TSource, TAccumulate> intermediateReduceFunc,
Func<TAccumulate, TAccumulate, TAccumulate> finalReduceFunc,
Func<TAccumulate, TResult> resultSelector
The Quick and Easy Solution
So, how to tell whether a particular aggregation can be parallelized with PLINQ? The simple approach is to imagine the above parallelization process. The input sequence will be reordered and split up
into several partitions. Each partition will be accumulated separately on its own thread, with its accumulator initialized to the seed. Then, all accumulators will be combined using the final reduce
function. Does this process produce the correct answer? If it does, then the aggregation can be parallelized using PLINQ.
In the rest of this posting, I will describe more in depth the properties that an aggregation must have in order to parallelize correctly. In typical cases, imagining the parallelization process is
the easiest way to find out whether an aggregation will produce the correct answer when ran on PLINQ.
Purity of Reduction Functions
Just as in other types of PLINQ queries, delegates that form a part of the query must be pure, or at least observationally pure. So, if any shared state is accessed, appropriate synchronization must
be used.
Associativity and Commutativity
The parallel version of an aggregation does not necessarily apply the reduction functions in the same order as the sequential computation. In the SumSquares example, the sequential result is computed
in a different order than the parallel result. Of course, the two results will be equal because of the special properties of the + operator: associativity and commutativity.
Operator F(x,y) is associative if F(F(x,y),z) = F(x,F(y,z)), and commutative if F(x,y) = F(y,x), for all valid inputs x,y,z. For example, operator Max is commutative because Max(x,y) = Max(y,x) and
also associative because Max(Max(x,y),z) = Max(x,Max(y,z)). Operator – is not commutative because it is not true in general that x-y = y-x, and it is not associative because it is not true in general
that x-(y-z) = (x-y)-z.
The following table gives examples of operations that fall into different categories with respect to associativity and commutativity:
│ │ │ Associative │
│ │ │ No │ Yes │
│Commutative│No │(a, b) => a / b │(string a, string b) => a.Concat(b) │
│ │ │ │ │
│ │ │(a, b) => a – b │(a, b) => a │
│ │ │ │ │
│ │ │(a, b) => 2 * a + b │(a, b) => b │
│ ├───┼─────────────────────────────┼────────────────────────────────────┤
│ │Yes│(float a, float b) => a + b │(int a, int b) => a + b │
│ │ │ │ │
│ │ │(float a, float b) => a * b │(int a, int b) => a * b │
│ │ │ │ │
│ │ │(bool a, bool b) => !(a && b)│(a, b) => Min(a, b) │
│ │ │ │ │
│ │ │(int a, int b) => 2 + a * b │(a, b) => Max(a, b) │
│ │ │ │ │
│ │ │(int a, int b) => (a + b) / 2│ │
An operation must be both associative and commutative in order for the PLINQ parallelization to work correctly. The good news is that many of the interesting aggregations turn out to be both
associative and commutative.
Note: For simplicity, this section only considers aggregations where the type of the accumulator is the same as the type of the element (not only the .Net type, but also the “logical” type). After
all, if the accumulator type is different from the element type, the intermediate reduction function cannot possibly be commutative because its two arguments are of different types! In the general
case, the final reduction function must be associative and commutative, and the intermediate reduction function must be related to the final reduction function in a specific way. See section
“Constraints on Reduce Function and Seed” for details.
Seed is an Identity
LINQ allows the user to initialize the accumulator to an arbitrary seed value. In the following example, the user sets the seed to 5, and thus computes 5 + the sum of squares of integers in a
public static int SumSquaresPlus5(IEnumerable<int> source)
return source.Aggregate(5, (sum, x) => sum + x * x, (sum) => sum);
Unfortunately, if we parallelize this query, several threads will split up the input, and each will initialize its accumulator to 5. As a result, 5 will be added to the result as many times as there
are threads, and the computed answer will be incorrect.
Can PLINQ do something to fix this problem?
Non-solution 1: Initialize one accumulator to the seed and the rest of them to the default value of T.
For example, if the input contains integers, why not initialize one thread’s accumulator to the user-provided seed, and the other accumulators to 0? The problem is that while 0 is a great initial
accumulator value for some aggregations, such as sum, it does not work at all for other aggregations. One such operation is product: if we initialize the accumulator
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Discussion are closed.
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Adding Sums and Subtotals to a Report
When building reports, users will oftentimes bring in large amounts of information by using Range Lists. Users will typically want to be able to add totals or, in the casing of grouped data,
subtotals, to give the person using the report an at-a-glance summary of how the numbers are looking.
By leveraging standard Excel functionality, adding sums and subtotals is simple and is covered in detail below.
To add sums and subtotals into reports that use Range Lists, the user will use standard Excel functionality in the form of the SUM and SUBTOTAL function.
The main difference between the two is that the SUM function will total up all of the numbers that are included in the specified range, while the SUBTOTAL function will ignore other SUBTOTAL
functions in the specified range. If your report has multiple levels of grouping, then using a SUBTOTAL function can be useful because it allows the user to easily add both subtotals and grand totals
without having the numbers counted multiple times.
Using the SUM Function in Non-Grouping Reports
Working with Rows Range Lists
If the report does not include grouping then, in many cases, a SUM function will work fine. The user would identify the range of the Range List (colored in green below), and a spacer row below the
Range List (the yellow row) and then add a SUM function that includes both the Range List field to sum up along with the empty cell in the spacer row.
The SUM function, as shown in the red box above, would be =SUM(C2:C3) to cover both rows. The reason that the spacer row needs to be included is because, as the Range List expands (in this example by
listing out Bill-to Customer No's), it will insert new rows between Row 2 and Row 3 to account for all of the customer numbers that it needs to insert. By having the SUM() function sit on both sides
of this area where the rows will be inserted, the SUM() function will also be automatically expanded to encompass the entire area when the report is run. Because of this, whether the report returns
10 customers numbers or 1,000 customer numbers (or more) the SUM function will always grow to be the correct size.
Working with Columns Range Lists
If the area to be summed up is based on a Columns Range List instead of a Rows Range List the same concept applies with having to include a spacer area, the area will just be the column to the right
of the Columns Range List.
In the example below, there is a Columns Range List that covers cells C1:C2 that will list out months. If the user runs the report for a year, twelve columns would be created, one for each month in
the year. If the user wanted to include a total off to the right, they would determine the region of the Columns Range List (colored in green below), and a spacer column next to the Range List (the
yellow column) and then add a SUM function that includes both the Range List field to sum up along with the empty cell in the spacer column.
The SUM function, as shown in the red box above, would be =SUM(C2:D3) to cover both columns. This would ensure that when the columns are inserted for months when the report is run the SUM function
will adjust in size proportionately.
Using the SUBTOTAL Function in Grouping Reports
When building reports that include grouping, such as first creating a list of salespeople and then, for each salesperson, returning a list of all customers that had sales for that salesperson, if the
user inserts SUM functions to total up the numbers they may notice that their totals are doubled up. This is because the grand total, for example, will include all the numbers for each customer but
will also include the salesperson subtotals as well.
The example below illustrates what this looks like when the SUM function is used with grouping. The customer had $50 worth of sales, so the associated salesperson also had $50 worth of sales. You
notice, however, that the grand total is incorrectly showing $100 because it includes both the $50 for the customer and $50 for the salesperson, even though this is the same $50.
Once simple solution to this is the leverage the SUBTOTAL function in Excel. The SUBTOTAL function is meant to behave in a similar way to Excel functions such as COUNT and SUM, but the SUBTOTAL
function will ignore other SUBTOTAL functions. This solves the issue of double counting as mentioned in the paragraph above.
In the example below you can see that when the SUBTOTAL function is used for both the salesperson total and the grand total that the grand total is now correctly showing $50.
You will notice in the SUBTOTAL function above that there is a "9" placed in the function prior to the cell reference range; this is telling the SUBTOTAL to sum up the numbers. The SUBTOTAL function
is a versatile function will many uses and more details on it can be found on Microsoft's site here: Subtotal Function Reference.
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Article is closed for comments.
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Intermediate Machine Learning: Supervised Learning II: Advanced Regressors and Classifiers Cheatsheet | Codecademy
Bayes Theorem calculates the probability of A given B as the probability of B given A multiplied by the probability of A divided by the probability of B:
P(A|B)= {P(B|A)*P(A)}/{P(B)}
This theory describes the probability of an event (A), based on prior knowledge of conditions (P(B|A)) that might be related to the event.
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The Magnetorotational Instability in Core-Collapse Supernova Explosions
We investigate the action of the magnetorotational instability (MRI) in the context of iron-core collapse. Exponential growth of the field on the timescale Ω^-1 by the MRI will dominate the linear
growth process of field-line ``wrapping'' with the same characteristic time. We examine a variety of initial rotation states, with solid-body rotation or a gradient in rotational velocity, that
correspond to models in the literature. A relatively modest value of the initial rotation, a period of ~10 s, will give a very rapidly rotating proto-neutron star and hence strong differential
rotation with respect to the infalling matter. We assume conservation of angular momentum on spherical shells. Rotational distortion and the dynamic feedback of the magnetic field are neglected in
the subsequent calculation of rotational velocities. In our rotating and collapsing conditions, a seed field is expected to be amplified by the MRI and to grow exponentially to a saturation field.
Results are discussed for two examples of saturation fields, a fiducial field that corresponds to v[A]=rΩ and a field that corresponds to the maximum growing mode of the MRI. We find, as expected,
that the shear is strong at the boundary of the newly formed proto-neutron star and, unexpectedly, that the region within the stalled shock can be subject to strong MHD activity. Modest initial
rotation velocities of the iron core result in sub-Keplerian rotation and a sub-equipartition magnetic field that nevertheless produce substantial MHD luminosity and hoop stresses: saturation fields
of order 10^15-10^16 G can develop ~300 ms after bounce with an associated MHD luminosity of ~10^52 ergs s^-1. Bipolar flows driven by this MHD power can affect or even cause the explosions
associated with core-collapse supernovae.
The Astrophysical Journal
Pub Date:
February 2003
□ Instabilities;
□ Magnetohydrodynamics: MHD;
□ Stars: Supernovae: General;
□ Astrophysics
42 pages, including 15 figures. Accepted for publication in ApJ. We have revised to include an improved treatment of the convection, and some figures have been updated
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This site has a very intuitive interface and covers just about anything a lab scientist might need. You can also change the number of digits that it shows. However, it doesn’t show you the equation
it is using, so you can not use it to eventually gain independence.
Michaelis-Menten Grapher at Physiology.Web
Physiology.web often has very useful calculators and explanations. Their interactive Michaelis Menten equation grapher is a very nice way to explore what can be a difficult concept to grasp.
This size has some great explanations of statistical concepts. Highly recommended.
This is very straightforward – input your RPM and your rotor radius and voila. It even shows you the equation it uses.
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Lesson 6
Introducing Double Number Line Diagrams
Problem 1
A particular shade of orange paint has 2 cups of yellow paint for every 3 cups of red paint. On the double number line, circle the numbers of cups of yellow and red paint needed for 3 batches of
orange paint.
Problem 2
This double number line diagram shows the amount of flour and eggs needed for 1 batch of cookies.
1. Complete the diagram to show the amount of flour and eggs needed for 2, 3, and 4 batches of cookies.
2. What is the ratio of cups of flour to eggs?
3. How much flour and how many eggs are used in 4 batches of cookies?
1. How much flour is used with 6 eggs?
2. How many eggs are used with 15 cups of flour?
Problem 3
Here is a representation showing the amount of red and blue paint that make 2 batches of purple paint.
1. On the double number line, label the tick marks to represent amounts of red and blue paint used to make batches of this shade of purple paint.
1. How many batches are made with 12 cups of red paint?
2. How many batches are made with 6 cups of blue paint?
Problem 4
Diego estimates that there will need to be 3 pizzas for every 7 kids at his party. Select all the statements that express this ratio.
The ratio of kids to pizzas is \(7:3\).
The ratio of pizzas to kids is 3 to 7.
The ratio of kids to pizzas is \(3:7\).
The ratio of pizzas to kids is 7 to 3.
For every 7 kids there need to be 3 pizzas.
Problem 5
1. Draw a parallelogram that is not a rectangle that has an area of 24 square units. Explain or show how you know the area is 24 square units.
2. Draw a triangle that has an area of 24 square units. Explain or show how you know the area is 24 square units.
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UofSussex researchers estimate advances in quantum computing could break Bitcoin in decade - Inside Quantum Technology
(TechRadar) Advances over the next decade could pave the way for quantum computers powerful enough to crack Bitcoin encryption, new research from University of Sussex suggests.
Scientists from the University of Sussex in the UK estimate that quantum systems with 13 million qubits would be sufficient to break the cryptographic algorithm (SHA-256) that secures the Bitcoin
blockchain within the space of 24 hours.
Although modern quantum computers come nowhere close to this level of performance (the current record is a comparatively measly 127 qubits), the researchers say significant developments over the next
ten years or so could yield quantum machines with sufficient horsepower.
For the time being, cryptocurrency enthusiasts can rest easy in the knowledge that cracking the SHA-256 algorithm is impossible with current hardware, but that won’t always be the case.
However, there is extensive research ongoing into all aspects of quantum computing, from almost all the world’s largest technology companies. A lot of work is going into increasing the number of
qubits on a quantum processor, but researchers are also investigating opportunities related to qubit design, the pairing of quantum and classical computing, new refrigeration techniques and more.
In all likelihood, Bitcoin will fork onto a new quantum-safe encryption method long before a sufficiently powerful quantum computer is developed, but the research raises an important point about the
longevity of encryption techniques nonetheless.
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Why Teaching Multiplication with Arrays Boosts Student Understanding - Hooty's Homeroom
Teaching multiplication with arrays is one of the most effective ways to help your students really understand the concept. Arrays give kids a clear visual that shows how multiplication works, whether
it’s about repeated addition, equal groups, or other important ideas. By using arrays in your lessons, you make multiplication easier to understand, setting your students up for success in math down
the road.
Understanding Multiplication as Repeated Addition
First things first, what exactly is multiplication? At its core, multiplication is just repeated addition. When we multiply 4 by 5, we’re really just adding 5 four times (5 + 5 + 5 + 5 = 20). But for
many students, that connection isn’t immediately obvious. This is where arrays come in handy.
When you create an array—say, 4 rows of 5 objects—you’re giving students a clear, visual representation of what 4 × 5 looks like. They can see that there are 4 groups, each with 5 objects. By
counting all the objects, they can easily see that 4 × 5 equals 20. This visual approach helps solidify the idea that multiplication is just a quicker way to add the same number over and over again.
Arrays as Equal Groups
Another way to think about arrays is as equal groups. This perspective is particularly useful when students are working on word problems or real-life scenarios where they need to group items evenly.
For example, imagine someone is planting tulips in a garden. They decide to plant 4 rows of tulips, with 6 flowers in each row.
By using an array, students can visually organize these rows and columns. They see 4 equal groups (rows) with 6 tulips in each group (row). This helps them understand that the total number of tulips
can be found by multiplying 4 (the number of rows) by 6 (the number of flowers in each row). This method not only helps with multiplication but also reinforces the idea of grouping, which is an
essential skill in many areas of math. It gives students a concrete way to understand multiplication problems, making them easier to solve.
Using Arrays to Skip Count
Skip counting is a stepping stone to mastering multiplication, and arrays are perfect for practicing this skill. When students arrange objects into rows and columns, they can easily skip count by the
number of objects in each row or column. For instance, in a 3 × 8 array, they can skip count by 8s—8, 16, 24—to quickly find the product.
This method is especially helpful for students who struggle with memorizing multiplication facts. By using arrays to practice skip counting, they can build confidence and improve their fluency with
multiplication. Plus, it’s a fun and engaging way to reinforce both skip counting and multiplication skills at the same time.
Exploring the Commutative Property with Arrays
One of the key concepts in multiplication is the commutative property, which simply means that you can switch the order of the numbers being multiplied, and the product will stay the same. For
example, 3 × 6 and 6 × 3 both equal 18. Arrays provide a great way to visually demonstrate this property.
You can show students that when you create a 3 × 6 array (3 rows of 6), and then rotate it to make a 6 × 3 array (6 rows of 3), the total number of objects doesn’t change—it’s still 18. This visual
proof helps students understand that multiplication is flexible, and it deepens their overall understanding of how numbers work together in multiplication.
The Long-Term Benefits of Teaching Multiplication with Arrays
So, why does all this matter? Beyond helping students with basic multiplication, arrays lay the groundwork for more advanced math concepts. As students move on to division, multi-digit
multiplication, and even area models, the understanding they’ve gained from working with arrays will be invaluable.
Arrays help students see the relationships between numbers, understand the structure of multiplication, and build confidence in their math abilities. By starting with arrays, you’re giving your
students a strong foundation that will support them as they tackle more complex problems in the future.
Teaching Multiplication with Arrays
Teaching multiplication with arrays is more than just a teaching strategy—it’s a way to help your students truly understand the math they’re doing. By using arrays to represent repeated addition,
equal groups, skip counting, and the commutative property, you’re giving your students the tools they need to succeed in multiplication and beyond. So, the next time you’re planning a multiplication
lesson, consider making arrays a central part of your instruction.
Looking for low-prep activities for introducing multiplication? Check out these beginner multiplication concepts activities.
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• recursive sum of digits function apparently not working
so I am learning C, and for one of the excercises the website I am learning on, I am supposed to make a program with a recursive function that calculates the sum of the digits of an integer. When
using VS code, my program works perfectly every time (to my knowledge, which is slight), but when I paste it into the website's editor, it says that its calculations are wrong. Here is my code:
#include <stdio.h>
#include <stdlib.h>
int sumOfDigits(int);
int getLastDigit(int);
int popLastDigit(int);
int main() {
int number;
int sum;
scanf("%d", &number);
sum = sumOfDigits(number);
printf("%d", sum);
return 0;
int sumOfDigits(int num) {
int curDigit;
int result = 0;
if (num>9) {
curDigit = getLastDigit(num);
num = popLastDigit(num);
result += curDigit + ( sumOfDigits(num) );
return result;
} else {
return num;
int getLastDigit(int num) {
int result;
result = num%10;
return result;
int popLastDigit(int num) {
int result;
int clippedDigit;
if (num>9) {
clippedDigit = getLastDigit(num);
result = (num - clippedDigit) / 10;
return result;
} else {
return 0;
Thank you in advance!
Do you have an example of your ideal output? Also was wondering what 'website editor' you are using?
Like an online C compiler?
well, the website does not tell me what number it inputs, but it says that the correct output should be 21, and the actual output is 18.
Edit: I am taking the C Programming: Modular Programming and Memory Management class on Edx.org
the ppl in the code casts seem to think that their editor is great, but it is the stupidest, slowest, most incompetent editor I have ever used!
Could you replace it with something like:
int sumOfDigits(int num) {
if (num != 0) {
return num%10+sumOfDigits(num/10);
} else {
return 0;
Does that still give you different answers in the course compiler?
Thank you! It worked!
#include <stdio.h>
#include <stdlib.h>
int sumOfDigits(int);
int getLastDigit(int);
int popLastDigit(int);
int main() {
int number;
int sum;
scanf("%d", &number);
sum = sumOfDigits(number);
printf("%d", sum);
return 0;
int sumOfDigits(int num) {
int curDigit;
int result = 0;
if (num != 0) {
return num%10+sumOfDigits(num/10);
} else {
return 0;
int getLastDigit(int num) {
int result;
result = num%10;
return result;
int popLastDigit(int num) {
int result;
int clippedDigit;
if (num>9) {
clippedDigit = getLastDigit(num);
result = (num - clippedDigit) / 10;
return result;
} else {
return 0;
It looks like you won't need those other supporting functions, unless you want to practice.
Also you can do things like "return num%10;" instead of having to explicitly declare a result variable, if that helps.
So here am I again, wondering how I am supposed to build a C project with this: https://marketplace.visualstudio.com/items?itemName=danielpinto8zz6.c-cpp-project-generator I already know how to use
gcc and or clang with single files, but I want to be able to build multiple file projects into the usual executable file. When I use this extension, it reloads VS Code and nothing happened. How would
you use this extension correctly, and / or what other extensions could you use to build C projects in VS Code Mac (NOT C++ PROJECTS, ONLY C PROJECTS!!!).
Edit: that extension actually created c projects, it didn't build them.
Instead of using an extention, just copy the template folder (located in ~/CEdev/examples or opt/CEdev/examples) and paste it wherever you like. Then, open a new terminal window, cd to the template
folder and type "make". The compiled .8xp file is in the bin folder.
i know how to make a c project for the ti 84 pce; right now i am trying to make one for my computer.
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2019 Summer of Contact PhD School
Course Description
Fast simulations are attractive for many reasons. They allow human users to use training simulators, or play computer games, or do rapid digital prototyping in CAD/CAM software. Fast simulators are
valuable for learning control policies in robotics too. Here the ability to run many fast simulations in parallel is key to lowering the training time and obtain a control policy. For the past
decade, fast numerical methods have been driven forward by game physics and interactive simulation. Particular the computer graphics field has contributed to a wide range of exploratory research into
different numerical methods and several acceleration techniques. The field has now reached a point where computing power is plenty and numerical methods converge sufficiently well such that we are
now observing the actual true effects of the models we use inside these types of simulators. The effects we can now observe are not attributed to “hacks” or “numerical” damping in the software
implementation. Rather, we see the true models for what they are. Unfortunately, we are now also marveled by how badly simulations work. Particular in Robotics the reality gap often refers to
simulators not being able to deal with the real world. In for instance in robotics applications, it can be rather surprising to discover that Coulomb's friction law is only accurate for a very
limited range of stiff and dry materials with very rough surface geometry. The world, in general, is nothing like that. So how should we go about creating models that better describe the real world
and still have fast simulations? How are we going to get data for such models? How are we going to control the parameters of the models? Finally, even more importantly how are we going to validate
and benchmark such models? These are the questions that we try to unveil in this Ph.D. summer school course. To bridge the reality gap from a simulation viewpoint can be popularly phrased as we want
fast simulators that can deal with crappy and shitty robots in a bad world. The question is now how to do that?
Contact models exist at various scales. For instance, at nanoscale atomic effects are appropriate to include to model effects of atomic repulsion and at the continuum level elasticity, plasticity,
and viscosity describe how small surface asperities account for the friction effects we observe on the every-day human scale. Effects such as lubrication, adhesion, and magnetism are also well
described at the continuum level. However, in fast simulation methods we often only describe the perceived every-day human-scale effects of all these fine-scale models. For instance, friction forces
(distributions) are often described using a multi-set law that can be presented as an algebraic equation testing for inclusion in a cone. Given such macroscopic descriptions of friction as cones are
particularly useful for the type of numerical methods most sought. It is rather well known how to compute contact force solutions as non-smooth root search problems using such a mathematical
formalism to express the physical laws. It is interesting to consider how new macroscopic contact models could disrupt the past 30 years or so research in creating fast numerical methods for
Learning Objectives
After this course the participants will be able to:
1. Account for the derivation of the Newton Euler equation, the equations of motion used in most multibody dynamics simulators. (Jeff Trinkle)
2. Describe the difference between the state-of-the-art macroscopic friction models used in fast simulators. Such as NCP, LCP, BLCP, SOCCP, Convex, etc. (Jeff Trinkle)
3. Explain how to obtain microscopic surface data and measure the roughness of such surfaces, the definition of fractal dimension (Jeppe Revall Frisvad)
4. Describe how statistics of micro-facets and simulations are used for creating BRDF models (Jeppe Revall Frisvad)
5. Explain how position-based dynamics is related to continuum models and constitutive equations (Mihai Francu)
6. Describe how to formulate a non-smooth Newton method for solving multibody dynamics as a fully coupled system (Miles Macklin)
7. Explain theory from single-contact mechanics to asperity-based statistical models for describing macroscopic frictional systems (Julien Scheibert)
8. Account for the state-of-the-art in improving the performance of direct methods for solving multibody systems with contact modeled as LCPs (Sheldon Andrews)
9. Argue for quality on contact point generation (Kenny Erleben)
10. Describe fundamentals of finite elements for elastic models and the challenge of frictional contact in such simulations, particular explain concepts behind Hertzian contact models (Paul G. Kry)
Tentative Detailed Schedule
(Maybe subject to change)
Lectures approximately 45 minutes, coffee breaks 15 minutes, lunch 60 minutes. We start days at 9:00 AM
(More will be added later)
• I. Vakis, V.A. Yastrebov, J. Scheibert, L. Nicola, D. Dini, C. Minfray, A. Almqvist, M. Paggi, S. Lee, G. Limbert, J.F. Molinari, G. Anciaux, R. Aghababaei, S. Echeverri Restrepo, A. Papangelo,
A. Cammarata, P. Nicolini, C. Putignano, G. Carbone, S. Stupkiewicz, J. Lengiewicz, G. Costagliola, F. Bosia, R. Guarino, N.M. Pugno, M.H. Müser, M. Ciavarella: Modeling and simulation in
tribology across scales: An overview, Tribology International, Volume 125, 2018.
• Tristan Baumberger & Christiane Caroli (2006) Solid friction from stick–slip down to pinning and aging, Advances in Physics,
• Kenny Erleben: Methodology for Assessing Mesh-Based Contact Point Methods, ACM Trans. Graph. August 2018.
• Nicodemus, F. E., Richmond, J. C., Hsia, J. J., Ginsberg, I. W., and Limperis, T. Geometrical considerations and nomenclature for reflectance. Tech. rep., National Bureau of Standards (US).
October. 1977.
Speaker Biographies
Julien Scheibert's research topics lie at the interface between physics and mechanics. In particular, he develops coupled experimental/modeling approaches to the contact mechanics of rough solid
surfaces. He applies them to understand various phenomena including human tactile perception, the dynamics of the onset of sliding, or the roughness noise.
Paul G. Kry is an associate professor at McGill University in the School of Computer Science where he heads the Computer Animation and Interaction Capture Laboratory. He received his B. Math. in
computer science with electrical engineering electives from the University of Waterloo, and his M.Sc. and Ph.D. in computer science from the University of British Columbia. He is currently a
director at large on the ACM SIGGRAPH executive committee and is the president of the Canadian Human-Computer Communications Society, the organization which sponsors the annual Graphics Interface
Jeppe Revall Frisvad is an associate professor at the Technical University of Denmark (DTU). He received and an MSc(Eng) in applied mathematics and a Ph.D. in computer graphics both from DTU. Jeppe
has more than 10 years of experience in material appearance modeling and rendering. As a highlight, his work includes the first directional dipole model for subsurface scattering, and his research
on material appearance includes methods for both computation and photographic measurement of the optical properties of materials.
In 1987, Jeff Trinkle received a Ph.D. from the Department of Systems Engineering at the University of Pennsylvania. He is now Professor of Computer Science, Professor of Electrical, Computer, and
Systems Engineering, and Director of the CS Robotics Lab. Trinkle's primary research interests lie in the areas of robotic manipulation and multibody dynamics. Under the continuous support of the
National Science Foundation since 1989, he has written many technical articles on theoretical issues underpinning the science of robotics and automation. One of these articles was the first to
develop a now-popular method for simulating multibody systems. Variants of this method are key components of several physics engines for computer game development, for example, NVIDIA PhysX and the
Bullet Physics Library.
Kenny Erleben is an Associate Professor in the Department of Computer Science, University of Copenhagen. He completed his Ph.D. in 2005. His research interests are Computer simulation and numerical
optimization with particular interests in computational contact mechanics of rigid and deformable objects, inverse kinematics for computer graphics and robotics, computational fluid dynamics,
computational biomechanics, foam simulation, interface tracking meshing.
Sheldon Andrews is a professor of Software and IT Engineering at the École de Technologie supérieure in Montreal, Canada. He received his Ph.D. in Computer Science in 2015 from McGill University,
and more recently he was a postdoctoral researcher at Disney Research in Edinburgh (2014-2015) and then CMLabs Simulations in Montreal (2016). His research interests include real-time physics
simulation, simulation of articulated mechanisms, 3D character animation, human motion synthesis and motion capture, computational contact mechanics, and measurement-based modeling for virtual
Miles Macklin is a computer graphics and simulation researcher working for NVIDIA on the PhysX team. Currently based in Auckland, New Zealand, Macklin previously worked at LucasArts in San Francisco
on the StarWars franchise, Rocksteady Studios in London on the Batman Arkham series, and Sony Computer Entertainment Europe on early Playstation3 development. His research interests are real-time
rendering and physics simulation using GPUs. Macklin is currently enrolled as a PhD Student at the University of Copenhagen.
Mihai Francu is an experienced senior programmer and a computer science researcher focusing on mechanical dynamics simulation. His interests lie at the confluence of computer graphics and animation,
physics and applied mathematics. Francu has an extensive industrial background in games and physics engines his main expertise is mainly in constrained based methods, cloth simulation, and modeling
contact and friction.
On Advances in Macroscopic Friction Modelling for Fast Simulators used in Robotics, Engineering and Computer Graphics
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Review of Short Phrases and Links
This Review contains major "Glossary of Special Functions"- related terms, short phrases and links grouped together in the form of Encyclopedia article.
1. The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive.
1. Abramowitz and Stegun is a modern successor.
1. An Airy function is a solution to the differential equation y'' - xy = 0.
2. The Airy function is a solution to the differential equation .
1. The arguments are allowed to be in any real field (automatic coercion is used whenever necessary).
2. The arguments are interpreted in the same way as the M argument of the random number function k r.
3. The arguments are interpreted the same as for the normal v s command.
4. The arguments are the vector, the starting index, and the ending index, with the ending index in the top-of-stack position.
1. The reciprocal of the arithmetic-geometric mean of 1 and the square root of 2 is called Gauss's constant.
2. These two sequences converge to the same number, which is the arithmetic-geometric mean of x and y; it is denoted by M( x, y), or sometimes by agm( x, y).
3. The H u G [ agmean] command computes the "arithmetic-geometric mean" of two numbers taken from the stack.
1. The associated Legendre polynomials are an important part of the spherical harmonics.
1. The Bateman Manuscript Project was a major effort at collation and encyclopedic compilation of the mathematical theory of special functions.
1. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.
2. An integral representation for the Bernoulli polynomials is given by the N--rlund-Rice integral, which follows from the expression as a finite difference.
3. The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there.
1. Bessel functions : Defined by a differential equation ; useful in astronomy electromagnetism and mechanics.
2. Bessel functions are of this type, solutions of certain differential equations that arise in many different connections.
3. Bessel functions are therefore especially important for many problems of wave propagation , static potentials, and so on.
4. The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument.
1. The transform of the boxcar function is a sinc(x) function.
2. The boxcar function Boxcar is defined as: Boxcar(x) = H(x+a) - H(x-a).
3. These are then multiplied by another boxcar function to make the grating of finite length.
1. The branch cuts are on the imaginary axis, below -i and above i.
2. The branch cuts are on the real axis, less than -1 and greater than 1.
1. A calculator is a device for performing calculations.
2. A calculator is a hand-held device for performing calculations.
3. The word "calculator" denoted a person who did such work for a living using such aids as well as pen and paper.
4. Trivia The word "calculator" is occasionally used as a pejorative term to describe an inadequately capable general-purpose microcomputer.
1. The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.
2. The Chebyshev polynomials are useful because they correspond well to the form of bending mode which might be expected of a headbox lip.
1. The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial character, of the Chowla-Selberg formula.
2. In mathematics, the Chowla-Selberg formula is the evaluation of a certain product of values of the Gamma function at rational values.
1. A common logarithm is a logarithm to the base 10.
2. Common logarithm is a logarithm to the base 10.
3. The common logarithm is the logarithm with base 10.
1. The complementary error function, denoted erfc, is defined in terms of the error function.
2. In the two-dimensional case this function is a complementary error function, or, in a special case, the Fresnel integral in complex form.
1. Complex analysis is one of the classical branches in mathematics with its roots in the 19th century and some even before.
2. Complex analysis is the branch of mathematics investigating functions of complex numbers.
3. Complex analysis is the branch of mathematics investigating holomorphic functions, i.e.
1. Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals.
2. Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals.
3. Complex numbers were first introduced in connection with explicit formulas for the roots of cubic polynomials.
4. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots.
1. The computation is based on approximations presented in Numerical Recipes, Chapter 6.2 (W.H. Press et al, 1992).
2. The computation is based on formulas from Numerical Recipes, Chapter 6.4 (W.H. Press et al, 1992).
1. The derivative of the coversine is the opposite of the cosine.
2. Information about coversine in the free online English dictionary and encyclopedia.
1. Cube Root is a designer and manufacturer .
2. The cube root is a continuous mapping from .
3. The cube root is a special case of the general nth root .
1. The Dawson function is also called the Dawson integral.
1. DELTA - the (optional) offset in bytes between the old and new positions, from the start of the block.
2. Delta: The Greek letter for the factor sensitivities measuring a portfolio's first order (linear) sensitivity to the value of an underlier.
1. The Digamma Function is a mathematical function defined as the logarithmic derivative of the Gamma Function .
2. The digamma function is also called the Psi function.
1. A Dirac comb is an infinite series of Dirac delta function s spaced at intervals of T.
1. The discrete logarithm is a related notion in the theory of finite group s.
2. The discrete logarithm is a related notion in the theory of finite groups .
1. Elementary functions are considered a subset of special functions.
2. Elementary functions are functions built from basic operations (e.g.
3. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.
1. An elliptic integral is an integral involving a rational function which contains square roots of cubic or quartic polynomials.
1. Elliptic integrals are often expressed as functions for a variety of different arguments.
2. Elliptic integrals are often expressed as functions of a variety of different arguments.
1. The equations are intended to give more weight to recent observations and less weight to observations further in the past.
2. The equations are specified by symbolic expressions containing the letter D to denote differentiation.
1. An error function is a list of targets that OSLO tries to minimize during optimization (when the error function is minimized the lens is improved).
2. Error function: An integral important for normal random variables.
3. The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function.
1. Euler was a deeply religious Calvinist throughout his life.
2. Euler was deeply religious throughout his life.
1. Exponential functions are always increasing (for a 0) and concave up.
2. Exponential functions are explored, interactively, using an applet.
1. This expansion follows directly from the asymptotic expansion for the exponential integral.
2. Exponential integral Error function: An integral important for normal random variables.
3. Exponential integral and related functions.
1. Exponentiation is a mathematical operation, written a n, involving two numbers, the base a and the exponent n.
2. Exponentiation is also known as raising the number a to the power n, or a to the n th power.
1. Once important in fields such as surveying, astronomy, and spherical trigonometry, the exsecant function is now little-used.
2. Thus, a table of the secant function would need a very high accuracy to be used for the exsecant, making a specialized exsecant table useful.
3. Showing 1 to 0 of 0 Articles matching ' Exsecant ' in related articles.
1. Factorials are also used extensively in probability theory.
2. Factorials are important in combinatorics.
3. Factorials are particularly useful in calculating the number of ways an event can occur, for example, the number of possible orders of finish in a race.
1. Floor function: Largest integer less than or equal to a given number.
2. The integer part or integral part of a decimal number is the part to the left of the decimal separator (see also floor function).
3. Examples are the floor function (below) and the ceiling function.
1. Fresnel integral: related to the error function; used in optics.
2. Evaluates the cosine Fresnel integral.
1. A graph is defined by an abscissa and an ordinate, although these need not actually appear on it.
2. Graph: One of the ways to represent a function or a real-life situation by plotting output and input points on coordinate axes.
Related Keywords
* Group * Gudermannian Function * Heaviside Step Function * Hermite Polynomials * Hyperbolic Function * Hyperbolic Functions * Hypergeometric Functions * Identities * Incomplete Gamma Function *
Incomplete Polylogarithm * Indefinite Logarithm * Integrals * J-Invariant * Lattice * Lectures * Legendre Chi Function * Legendre Polynomials * List of Special Functions And Eponyms * Logarithm *
Logarithmic Integral Function * Logarithms * Mathematical Functions * Mittag-Leffler Function * Mode * Multivalued Function * Natural Logarithm * Notation * Offset Logarithmic Integral *
Orthogonal Polynomials * Parabolic Cylinder Functions * Planetmath * Pochhammer Symbol * Polygamma Function * Polylogarithm * Probability * Question Mark Function * Raised Cosine * Real Numbers *
Roots * Sigmoid Function * Sign Function * Solution * Solutions * Special Functions * Special Hypergeometric Functions * Spline * Square Root * Step Function * Synchrotron Function * Theta
Function * Trigamma Function * Trigonometric Function * Trigonometric Functions * Trigonometry * Values
1. Books about "Glossary of Special Functions" in Amazon.com
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Mulliken Symbols
Symbols used to identify irreducible representations of Groups:
singly degenerate state which is symmetric with respect to Rotation about the principal axis,
singly Degenerate state which is antisymmetric with respect to Rotation about the principal axis,
doubly Degenerate,
triply Degenerate,
(gerade, symmetric) the sign of the wavefunction does not change on Inversion through the center of the atom,
(ungerade, antisymmetric) the sign of the wavefunction changes on Inversion through the center of the atom,
(on or ) the sign of the wavefunction does not change upon Rotation about the center of the atom,
(on or ) the sign of the wavefunction changes upon Rotation about the center of the atom,
' = symmetric with respect to a horizontal symmetry plane ,
'' = antisymmetric with respect to a horizontal symmetry plane .
See also Group Theory
© 1996-9 Eric W. Weisstein
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Creative set
From Encyclopedia of Mathematics
A recursively enumerable set $A$ of natural numbers whose complement $\bar A$ (in the set of natural numbers) is a productive set; in other words, a set $A$ is creative if it is recursively
enumerable and if there exists a partial recursive function $\phi(x)$ such that, for any recursively enumerable subset $W_x$ in $\bar A$, with Gödel number $x$, $$ \phi(x) \in \bar A \setminus W_x \
. $$
Creative sets are frequently encountered in various algorithmically unsolvable problems, and they therefore constitute the most important class of recursively enumerable sets which are not recursive.
In many formal theories, the sets of (numbers of) provable and refutable formulas turn out to be creative (assuming a natural enumeration of all formulas of the theory); in particular, this is the
case for Peano arithmetic and, in general, for all recursively inseparable theories (i.e. theories the sets of provable and refutable formulas of which are effectively inseparable). All creative sets
are recursively isomorphic to one another (i.e. for any two creative sets there exists a recursive one-to-one mapping of the natural numbers which maps one set onto the other), and they all belong to
the same Turing degree — the largest of the degrees of recursively enumerable sets. The concept of creativity generalizes to sequences of sets and other objects.
[1] H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967)
[2] J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967)
For definitions and discussions of the various concepts mentioned above, such as recursively enumerable set, recursive isomorphism, Turing degree, etc. cf. Recursive set theory and Degree of
How to Cite This Entry:
Creative set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Creative_set&oldid=39338
This article was adapted from an original article by V.A. Dushskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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Epub Designing Cisco Network Service Architectures 2Nd Edition 2009
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Some characteristics of drilling techniques - Encyclopedia of the Environment
| Focus 2/2 | The challenges of industrial hydraulic fracturing
Some characteristics of drilling techniques
Figure 1. Example of rock cores. They are more or less cylindrical, depending on the altrability of the rock, but also according to the coring technique. For example, for very deformable materials, a
double core barrel is used, of which only the outer part rotates with the drill string, while the central part remains free of any rotation. [Source: © F.H. Cornet]
A drilling operation can have several objectives. This may involve taking a rock sample at a specific location to bring it to the surface; this is called core drilling (Figure 1).
These exploration wells can have various depths. The deepest ever reached a depth of 12,345 m, near the island of Sakhalin in eastern Russia. In addition to collecting rock samples, drill holes allow
a number of geophysical measurements to be carried out in situ. The description of continuous variations of a physical property along a well is called a log. For example, sonic logging is used to
describe variations along the borehole in the propagation velocity of so-called sound waves, i.e. waves emitted in a frequency range covering the sensitivity range of our hearing (20 Hz-10 000 Hz).
Figure 2. Examples of drilling tools. On the left is the corer used to produce the cores in Figure 6. On the right are two examples of wheel drills. [Source: © F.H. Cornet]
But most often the purpose of drilling is to produce fluids in place in the rock at a certain depth, whether it is drinking water (generally less than 100 m deep), hydrocarbons (from 2000 to 7000 m),
or geothermal fluids (in the 150 – 5000 m range). These holes are drilled using a destructive method, i.e. the rock is crushed in place by a drill head (Figure 2), or drill bit, pushed by a drill
string (Figure 3).
The rock debris (cuttings) is brought to the surface by a circulation of mud injected by the drill string. The viscosity of this slurry is adjusted for optimum cutting removal and its density is
adjusted to ensure the stability of the borehole during drilling.
Figure 3. Example of a small drilling rig to reach a depth of 800 m. The drill pipes are placed here in front of the device. For greater depths, the rods are held vertically next to the drilling rig
and are handled automatically. [Source: © F.H. Cornet]
For shallow wells drilled for drinking water production, the drilling technique is often simpler and uses a down the hole hammer. This technique is equivalent to that of the jackhammer, the
compressed air being brought to the bottom by the drill string. Note that the air pressure must be sufficient to lift the weight of the water column that fills the borehole. For example, blowers
capable of reaching pressures of around 100 bar must be used for depths exceeding 800 m. In practice, this technique is mainly used for boreholes not exceeding 200 m in depth.
When the borehole reaches a certain depth, it must be tubed regularly to balance the stresses supported by the rock at the borehole wall. This operation is called the casing of the well , and the
steel pipe left in place is called the casing. During its manufacture, the casing may have a certain number of slots to allow the production of the required fluid. But more often than not, the casing
is cemented to prevent any fluid from rising along the borehole outside the casing. The production of fluid is then ensured, once the casing has been placed in a watertight manner, thanks to
perforations made using various techniques that vary according to the operators.
With the traditional drilling technique, the drill string allows on the one hand to inject the mud used to extract rock debris, and on the other hand to rotate the drilling tool on its axis. This
rotation operation involves significant friction throughout the drilling process and therefore causes rapid wear of the drill string for deep drilling. To overcome these difficulties, drill heads
that can rotate on themselves thanks to the injected mud pressure, without rotating the drill string, have gradually been developed. These turbines have also provided the opportunity to better
control the drilling direction. These techniques now allow horizontal drilling operations over distances of up to ten kilometres.
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Makie observable lift functions
Hi. I’ve been trying to learn how to use observables in Makie and have managed to make an interactive plot with an IntervalSlider.
The xaxis ticks are controlled with a lift function:
xticks = lift(ab_slider.interval) do val
collect(val), ["a", "b"]
Now that works, I would like to make it so that if a \leq 0 the left tick is displayed as 0 instead of a. Similarly for the upper bound, b would be replaced with the max xaxis limit.
I started with some if else statements in my function, and while it runs, the IntervalSlider and plot become really unresponsive.
xticks = lift(ab_slider.interval) do val
if val[1] > 0 && val[2] < 4
return collect(val), ["a", "b"]
return [0, 4], ["0", "4"]
Not only does the UI seem a bit unresponsive, but sometimes the tick values don’t update at all, and other times something seems to trigger them but not always. I suspect that I’ve not understood
something about how Observables work. I found working with an observable of a tuple value for the interval less than intuitive. Thanks.
This probably is not all of the problem, but that logic looks wrong. If either limit is reached, the fixed ends are used. Try this:
xticks = lift(ab_slider.interval) do val
clamp.(val, 0, 4), ["a", "b"]
The logic isn’t fully implemented in my example as I wanted to keep it simple and what I put was enough to reproduce the problem.
Another side issue, is that if the value of a = 0 and I want to have ["0", "a", "b", "4"] as my ticks, then I get a zero step error. So I do need to implement logic to prevent this happening, but any
form of logic makes the UI unresponsive and glitchy in terms of updating.
I wondered if it had something to do with the way the Observable wraps a tuple value, and I’m looking for changes in the individual values?
Can you share a runnable example that exhibits the problem? I haven’t had problems putting complicated data types in Observables before, but that might have something to do with it.
1 Like
I think I’ve fixed it.
When I tried making up a full minimal example I decided to do it in a single script file instead of a jupyter notebook. Then I noticed lots of runtime errors appearing in the console that I hadn’t
seen before.
The error was an inexact error and all I had to do was change the integers to floats. This error was only occurring when the conditional logic kicked in.
Thanks for your help.
Edit: I’ll just throw this in as an example for anyone else who wants to do the same
using GLMakie
GLMakie.activate!(; float=true)
f(x) = exp(-x/2)
xs = LinRange(0, 4, 41)
fig = Figure(size=(800, 600))
ab_slider = IntervalSlider(fig[2, 1], range = 0:0.1:4, startvalues = (0.0, 4.0))
xticks = lift(ab_slider.interval) do val
if val[1] > 0.0 && val[2] < 4.0
return collect(val), ["a", "b"]
return [0.0, 4.0], ["0", "4"] # Needs to be 4.0 and not 4
ax = Axis(fig[1, 1], xticks=xticks)
lines!(ax, xs, f)
It’s a bit weird that only the second value in the return array needed to be a float to avoid the error.
Edit 2: It can be either 0.0 or 4.0, just not both integers 0 and 4. Sure there’s some reason behind it somewhere. Edit 3: Ah, it must make the xticks float if either value is set as float, and then
it’s ok.
2 Likes
With lift you need to be careful that the return type of your first invocation also works for all other possible branches because that’s the type parameter that the observable uses. So if you first
return an integer and later a float, the float will fail to convert to an integer.
Sometimes you have a union of types like Union{Nothing, Float64}, in that case I usually create an empty observable first with the correct type parameter like result = Observable{Union{Nothing,
Float64}} and then use the map! function on that with the logic you’d otherwise put into lift
2 Likes
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The Internet Movie Database (IMDB) user ratings of the movie Top Gun: Maverick can be analyzed in R with the following command: TGM.ratings
The Internet Movie Database (IMDB) user ratings of the movie Top Gun: Maverick can be analyzed in R with the following command: TGM.ratings <-c(rep(x=1, times="3266)," rep(x="2," times="968)," rep(x=
The Internet Movie Database (IMDB) user ratings of the movie Top Gun: Maverick can be analyzed in R with the following command: TGM.ratings <-c(rep(x=1, times=”3266),” rep(x=”2,” times=”968),” rep(x=
”3,” times=”1225),” rep(x=”4,” times=”1840),” rep(x=”5,” times=”4243),” rep(x=”6,” times=”12048),” rep(x=”7,” times=”39036),” rep(x=”8,” times=”89864),” rep(x=”9,” times=”109439),” rep(x=”10,” times=
”124807))” counts=”” are=”” valid=”” as=”” of=”” october=”” 1,=”” 2022.=”” note:=”” if=”” you=”” want,=”” you=”” may=”” pick=”” a=”” different=”” movie=”” that=”” you=”” like=”” and=”” use=”” its=””
imdb=”” user=”” ratings=”” on=”” this=”” problem,=”” so=”” long=”” as=”” at=”” least=”” 1,000=”” people=”” have=”” submitted=”” ratings.=”” if=”” you=”” do=”” this,=”” name=”” the=”” movie=”” and=””
provide=”” the=”” code=”” you=”” used=”” to=”” input=”” the=”” ratings=”” into=””>
a) Make a histogram of the ratings. Start the breaks at 0.5, end the breaks at 10.5, and make each rectangle width 1. (This centres the rectangles at the integers 1 through 10.)
b) Is the histogram approximately normal? Does it matter, regarding the Central Limit Theorem? (That is, do the sample data need to be approximately normal in order to use z-tools? Consider problems
1-4 of this assignment.)
c) Set up a z-interval of the average user rating. Is this valid statistical inference? If it is, how is it interpreted (and what population it is about)? If it is not valid inference, why not?
d) Set up a two-tail z-test of whether the average user rating is significantly different (either way) from 5. Give the z-statistic and the P-value.
e) Provide a plot (made in R) of the normal curve with the outer two tails shaded that illustrates this P-value. (This would be hard to do “from scratch.” Instead, modify the shade. Norm function
provided. See also the shade.t.outer function provided in my R script in eLearning.)
f) Is this a valid statistical test? If it is, how is it interpreted? If it is not, why not?
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manipulatives Archives - Math Motivator
As a follow-up to the lesson I posted yesterday (See: Diagnostic: Mode, Median & Mean) we gave this question to the students: The number of goals scored by a hockey team are as follows: Game 1: 4
Game 2: 2 … More Stay Tuned For “Mean” »
Teaching to Big Ideas – Proportional Reasoning
Reflection Questions How might using Big Ideas impact how I plan? How might we as educators help students see things from a proportional reasoning perspective? Do you have a full range of
manipulatives appropriate to grade level curriculum in your classroom? Do your students have access to these manipulatives for … More Teaching to Big Ideas – Proportional Reasoning »
Four Coins Investigation
Recently I spent some time in a primary math class. We gave the following question adapted from Dr. Marian Small’s Open Question resource. I have 4 coins. What might they be? What are the
possibilities? After the students spent some time coming up with several possibilities we consolidated by asking … More Four Coins Investigation »
Understanding Relationships Between Quantity and the Patterns Within Our Number System
Working one-on-one with students who are demonstrating a fragile sense of number are opportunities I greatly value. I always approach these times from an inquiry perspective, looking for clues to
help me understand their struggles. Often they are demonstrating many strengths in some areas, but something is keeping them from … More Understanding Relationships Between Quantity and the Patterns
Within Our Number System »
Natural vs Commercial Math Materials
Recently I received the following question from a Kindergarten educator: Why do I get the feeling that natural materials are better than commercial ones for math? I do the exact same activity with
materials we have in the classroom. Do the students actually learn more about math through the natural … More Natural vs Commercial Math Materials »
Money Race Games
I love games that require very little preparation and materials but really focus on important math concepts and skills. Here is one that will allow students to identify different coins and understand
their value. In addition they are developing an understanding of equal groups and unitizing … More Money Race Games »
Understanding Equality Using Pattern Blocks
Recently a primary teacher asked me for suggestions for hands-on activities for balance and equality. As I mentioned in a previous post we have a tendency to focus more on patterning and less on
algebra, but here is a Grade 1 teacher understanding the importance of moving beyond. In … More Understanding Equality Using Pattern Blocks »
A Spatial Reasoning Task Using Pentominoes
Observing people of all ages enthusiastically engaging in mathematical tasks is what fuels my passion for the work I do. I had such an experience recently at a K-8 elementary school on a PA Day.
Last year I had been involved in a Ministry of Education Spatial Reasoning Inquiry and … More A Spatial Reasoning Task Using Pentominoes »
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1 2 3 During a trip that they took together ,Carmen ,Juan ,Maria ,and Rafael drove an average (arithmetic mean )of 80 miles each. Carmen drove 72 miles ,Juan drove 78 miles ,and Maria drove 83
miles .How may miles did Rafael drive ? 4 Each week ,a clothing salesperson receives a commission equal to 15 percent of the first $500 in sales and 20 percent of all additional sales that week .
What commission would the salesperson receive on total sales for the week of $1300 ? 5 A certain restaurant that regularly advertises through the mail has 1,040 cover letters and 3,000 coupons in
stock .In its next mailing ,each envelope will contain 1 cover letters and 2 coupons .If all of the cover letters in stock are used ,how many coupons will remain in stock after this mailing? 6 7 The
price of a coat in a certain store is $500. If the price of the coat is to be reduced by $150, by what percent is the price to be reduced? 8 $$(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\
frac{1}{4}-\frac{1}{5})+(\frac{1}{5}-\frac{1}{6})$$ 9 While a family was away on vacation, they paid a neighborhood boy $11 per week to mow their lawn and $4 per day to feed and walk their dog. If
the family was away for exactly 3 weeks ,how much did they pay the boy for his services ? 10 Last year $48,000 of a certain store`s profit was shared by its 2 owners and their 10 employees. Each of
the 2 owners received 3 times as much as each of their 10 employees. How much did each owner receive from the $48,000 ? 11 On a vacation, Rose exchanged $500.00 for euros at an exchange rate of 0.80
euro per dollar and spent $$\frac{3}{4}$$ of the euros she received. If she exchanged the remaining euros for dollars at an exchange rate of $1.20 per euros, what was the dollar amount she received?
12 13 In the xy-coordinate plane, if the point(0, 2)lies on the graph of the line 2x+ky =4, what is the value of the constant k? 14 Bouquets are to be made using white tulips and red tulips, and the
ratio of the number of white tulips to the number of red tulips is to be the same in each bouquet. If there are 15 white tulips and 85 red tulips available for the bouquets, what is the greatest
number of bouquets that can be made using all the tulips available? 15 Over the past 7 weeks, the Smith family had weekly grocery bills of $74, $69 ,$64, $79, $64, $84, and $77. What was the
Smiths'average(arithmetic mean) weekly grocery bill over the 7-week period? 16 125% of 5 = 17 During a recent storm ,9 neighborhoods experienced power failures of durations
34,29,27,46,18,25,12,35,and 16 minutes ,respectively .For these 9 neighborhoods ,what was the median duration ,in minutes ,of the power failures ? 18 When traveling at a constant speed of 32 miles
per hour, a certain motorboat consumes 24 gallons of fuel per hour. What is the fuel consumption of this boat at this speed measured in miles traveled per gallon of fuel? 19 A technician makes a
round-trip to and from a certain service center by the same route. If the technician completes the drive to the center and then completes 10 percent of the drive from the center, what percent of the
round-trip has the technician completed? 20 From 2000 to 2003, the number of employees at a certain company increased by a factor of 1/4. From 2003 to 2006, the number of employees at this company
decreased by a factor of 1/3. If there were 100 employees at the company in 2006, how many employees were there at the company in 2000?
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Algebraic attacks against random local Functions and their countermeasures
Suppose that you have n truly random bits x = (x[1],..., x[n]) and you wish to use them to generate m ≥ n pseudorandom bits y = (y[1],..., y[m]) using a local mapping, i.e., each yi should depend on
at most d = O(1) bits of x. In the polynomial regime of m = n^s, s > 1, the only known solution, originates from (Goldreich, ECCC 2000), is based on Random Local Functions: Compute yi by applying
some fixed (public) d-ary predicate P to a random (public) tuple of distinct inputs (x[i1],..., x[id]). Our goal in this paper is to understand, for any value of s, how the pseudorandomness of the
resulting sequence depends on the choice of the underlying predicate. We derive the following results: (1) We show that pseudorandomness against F[2]-linear adversaries (i.e., the distribution y has
low-bias) is achieved if the predicate is (a) k = Ω(s)-resilience, i.e., uncorrelated with any k-subset of its inputs, and (b) has algebraic degree of Ω(s) even after fixing Ω(s) of its inputs. We
also show that these requirements are necessary, and so they form a tight characterization (up to constants) of security against linear attacks. Our positive result shows that a d-local lowbias
generator can have output length of n^Ω(d), answering an open question of Mossel, Shpilka and Trevisan (FOCS, 2003). Our negative result shows that a candidate for pseudorandom generator proposed by
the first author (computational complexity, 2015) and by O'Donnell and Witmer (CCC 2014) is insecure. We use similar techniques to refute a conjecture of Feldman, Perkins and Vempala (STOC 2015)
regarding the hardness of planted constraint satisfaction problems. (2) Motivated by the cryptanalysis literature, we consider security against algebraic attacks. We provide the first theoretical
treatment of such attacks by formalizing a general notion of algebraic inversion and distinguishing attacks based on the Polynomial Calculus proof system. We show that algebraic attacks succeed if
and only if there exist a degree e = O(s) non-zero polynomial Q whose roots cover the roots of P or cover the roots of P's complement. As a corollary, we obtain the first example of a predicate P for
which the generated sequence y passes all linear tests but fails to pass some polynomial-time computable test, answering an open question posed by the first author (Question 4.9, computational
complexity 2015).
Original language English
Title of host publication STOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing
Editors Yishay Mansour, Daniel Wichs
Publisher Association for Computing Machinery
Pages 1087-1100
Number of pages 14
ISBN (Electronic) 9781450341325
State Published - 19 Jun 2016
Event 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 - Cambridge, United States
Duration: 19 Jun 2016 → 21 Jun 2016
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
Volume 19-21-June-2016
ISSN (Print) 0737-8017
Conference 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016
Country/Territory United States
City Cambridge
Period 19/06/16 → 21/06/16
Funders Funder number
Horizon 2020 Framework Programme 639813
• Algebraic attacks
• Cryptography
• Low-bias generators
• NC0
• Pseusorandomness
Dive into the research topics of 'Algebraic attacks against random local Functions and their countermeasures'. Together they form a unique fingerprint.
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The limitations of nice mutually unbiased bases
Title The limitations of nice mutually unbiased bases
Publication Type Journal Article
Year of Publication 2007
Authors Aschbacher, M, Childs, AM, Wocjan, P
Journal Journal of Algebraic Combinatorics
Volume 25
Issue 2
Pages 111 - 123
Date Published 2006/7/11
Mutually unbiased bases of a Hilbert space can be constructed by partitioning
a unitary error basis. We consider this construction when the unitary error
basis is a nice error basis. We show that the number of resulting mutually
unbiased bases can be at most one plus the smallest prime power contained in
Abstract the dimension, and therefore that this construction cannot improve upon
previous approaches. We prove this by establishing a correspondence between
nice mutually unbiased bases and abelian subgroups of the index group of a nice
error basis and then bounding the number of such subgroups. This bound also has
implications for the construction of certain combinatorial objects called nets.
URL http://arxiv.org/abs/quant-ph/0412066v1
DOI 10.1007/s10801-006-0002-y
Short Title J Algebr Comb
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Deadlock in Operating System Tutorial Notes Study Material with Example
In a multiprogramming environment, a situation when permanent blocking of a set of processes that either compete for system resources or of communication with each other happens, we can call this as
deadlock situation. This deadlock problem involves conflicting needs for resources
By two or more processes.
Necessary Conditions for Deadlock
A deadlock situation can arise, if the following four conditions hold simultaneously in a system.
Mutual Exclusion
Resources must be allocated to processes at any time in an exclusive manner and not on a shared basis for a deadlock to be possible. If another process requests that resource, the requesting process
must be delayed until the resource has been released.
Hold and Wait Condition
Even if a process holds certain resources at any moment, it should be possible for it to request for new ones. It should not give up (release) the already held resources to be able to request for new
ones. If it is not true, a deadlock can never take place.
No Preemption Condition
Resources can’t be preempted. A resource can be released only voluntarily by the process holding it, after that process has completed its task.
Circular Wait Condition
There must exist a set = {Po^, Pi^, P2, …, ^Pn^} of waiting processes such that P[o] is waiting for a resource that is held by PI, P1
[w]eld[.] by is waiting for a resource that is
P[r], _ [1] is waiting for a resource that is held by P[r], and P[n] is
Waiting for a resource that is held by Po.
Deadlock in Operating System Tutorial Notes Study Material with Example
Resource Allocation Graph
The resource allocation graph consists of a set of vertices V and a Set of edges E.
vertices V is partitioned into two types
• P = {P[1], P[2],…, P[n]}, the set consisting of all the process[es in t]
• R = {R[1], R[2],…, R[m]}, the set consisting of all resource typ[es] in 1f,.
• Directed Edge P[ ] R[i] is known as request edge.
• Directed Edge R ; P is known as assignment edge.
Resource Instance
• One instance of resource type R[1]. Two instances of resource type R[2].
• One instance of resource type • Three instances of resource type R[4].
Example of resource allocation graph
Process States
• Process P[1] is holding an instance of resource type R2 and is waiting for an instance of resource type R[1].
• Process R[2] is holding an instance of R[1] and R2 is waiting for an instance of resource type
• Process P[3] is holding an instance of R[3].
• Basic facts related to resource allocation graphs are given below
Note: If graph consists no cycle it means there is no deadlock in the system
If graph contains cycle
• If only one instance per resource type,
• If several instances per resource type,
Deadlock Handling Strategies
1. Deadlock prevention 2. Deadlock avoidance 3.Deadlock detection
Deadlock Prevention
Deadlock prevention is a set of methods for ensuring that at least one of the necessary conditions can’t hold.
Deadlock Avoidance
A deadlock avoidance algorithm dynamically examines the resource allocation state to ensure that a circular wait condition can never exist The resource allocation state is defined by the number of
available and allocated resources and the maximum demands of the processes.
Safe State
Deadlock in Operating System Tutorial Notes Study Material with Example
A state is safe, if the system can allocate resources to each process and still avoid a deadlock.
A system is in safe state, if there exists a safe sequence of all processes. A deadlock state is an unsafe state. Not all unsafe states cause deadlocks.
Banker’s Algorithm
Data structures for Banker’s algorithm available. Vector of length m. If available [j] = k, there are k instances of resource type R[1] available.
• Max n x m If max [i, j] = k, then process P[i] may request at most k
instances of resource type R[j].
• Allocation n x m If allocation [i, j ] = k, then P[i] is currently allocated k instances of R[i]
• Need n x m matt^-. If need [i, j I = k, then P[i] may need k more instances of R to complete its task
Need [i,j]= max [i, j] — allocation [i, j]
Step 1: Let work and finish be vectors of length m and n, respectively. Initialize.
Work = Available
Finish [i]= False (for i=1,2,….n
Step [2] : Find i such that both
• Finish [i]. False
• Need ≤ Work
If no such i exists, go to step 4. Step 3 Work = Work + Allocation
Finish [i]= Tree .
Go to step 2.
Step 4: Finish [I] = True (for all i)
Then, the system is in a safe system. Example Banker’s Algorithm
As available resources are (3 3 2). The process P[1] with need (^1 2 be executed.
Available resource = Available + Allocated resource of P[I]
N[o]w, available resources are (5 3 2).
The next process that can be executed after assigning available resource is P3. Thus, P[3] will execute next.
N[ow], available resource = Available resource + Resource of P[3]
Now, available resources are 7 4 3. Next process will be P4.
A B C
Next process that will be executed is P[o].
Available resource
Key Points
• The sequence < P, P[3], P[4] P[0] P2 > ensures that the deadlock will never occur.
• If a safe sequence can be found that means the processes can run concurrently and will never cause a deadlock.
• First in first out scheduling based on queuing.
• Shortest seek time scheduling is designed for maximum through put in most scenarios
• RR scheduling involves extensive overhead, especially with a small time unit. Kernel is defined as a program running all times on the computer. It is the part of operating system that loads
first. In simple word, kernel provides the communication between system software and hardware.
Deadlock Detection
Allow system to enter deadlock state and then there is
(i) Detection algorithm (ii) Recovery scheme
Single instance of each resource type
(a) Nodes are processes. (b) P–> P[i], if P[i] is waiting for P[i].
• Periodically invoke an algorithm that searches for a cycle in the graph.
• An algorithm to detect a cycle in a graph requires an order of n^2 operations, where n is the number of vertices in the graph.
Recovery Scheme
• Resource preemption
• Process termination
• Abort all deadlock processes
• Abort one process at a time until the deadlock cycle eliminated
Sorting in Design and Analysis of Algorithm Study Notes with Example
Learn Sorting in Handbook Series: Click here
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Hunter Kippen
Hunter Kippen
Revisiting Security Estimation for LWE with Hints from a Geometric Perspective Abstract
The Distorted Bounded Distance Decoding Problem (DBDD) was introduced by Dachman-Soled et al. [Crypto ’20] as an intermediate problem between LWE and unique-SVP (uSVP). They presented an approach
that reduces an LWE instance to a DBDD instance, integrates side information (or “hints”) into the DBDD instance, and finally reduces it to a uSVP instance, which can be solved via lattice reduction.
They showed that this principled approach can lead to algorithms for side-channel attacks that perform better than ad-hoc algorithms that do not rely on lattice reduction. The current work focuses on
new methods for integrating hints into a DBDD instance. We view hints from a geometric perspective, as opposed to the distributional perspective from the prior work. Our approach provides the
rigorous promise that, as hints are integrated into the DBDD instance, the correct solution remains a lattice point contained in the specified ellipsoid. We instantiate our approach with two new
types of hints: (1) Inequality hints, corresponding to the region of intersection of an ellipsoid and a halfspace; (2) Combined hints, corresponding to the region of intersection of two ellipsoids.
Since the regions in (1) and (2) are not necessarily ellipsoids, we replace them with ellipsoidal approximations that circumscribe the region of intersection. Perfect hints are reconsidered as the
region of intersection of an ellipsoid and a hyperplane, which is itself an ellipsoid. The compatibility of “approximate,” “modular,” and “short vector” hints from the prior work is examined. We
apply our techniques to the decryption failure and side-channel attack settings. We show that “inequality hints” can be used to model decryption failures, and that our new approach yields a geometric
analogue of the “failure boosting” technique of D’anvers et al. [ePrint, ’18]. We also show that “combined hints” can be used to fuse information from a decryption failure and a side-channel attack,
and provide rigorous guarantees despite the data being non-Gaussian. We provide experimental data for both applications. The code that we have developed to implement the integration of hints and
hardness estimates extends the Toolkit from prior work and has been released publicly.
BKW Meets Fourier: New Algorithms for LPN with Sparse Parities 📺 Abstract
We consider the Learning Parity with Noise (LPN) problem with a sparse secret, where the secret vector $\mathbf{s}$ of dimension $n$ has Hamming weight at most $k$. We are interested in algorithms
with asymptotic improvement in the \emph{exponent} beyond the state of the art. Prior work in this setting presented algorithms with runtime $n^{c \cdot k}$ for constant $c < 1$, obtaining a constant
factor improvement over brute force search, which runs in time ${n \choose k}$. We obtain the following results: - We first consider the \emph{constant} error rate setting, and in this case present a
new algorithm that leverages a subroutine from the acclaimed BKW algorithm [Blum, Kalai, Wasserman, J.~ACM '03] as well as techniques from Fourier analysis for $p$-biased distributions. Our algorithm
achieves asymptotic improvement in the exponent compared to prior work, when the sparsity $k = k(n) = \frac{n}{\log^{1+ 1/c}(n)}$, where $c \in o(\log \log(n))$ and $c \in \omega(1)$. The runtime and
sample complexity of this algorithm are approximately the same. - We next consider the \emph{low noise} setting, where the error is subconstant. We present a new algorithm in this setting that
requires only a \emph{polynomial} number of samples and achieves asymptotic improvement in the exponent compared to prior work, when the sparsity $k = \frac{1}{\eta} \cdot \frac{\log(n)}{\log(f(n))}$
and noise rate of $\eta \neq 1/2$ and $\eta^2 = \left(\frac{\log(n)}{n} \cdot f(n)\right)$, for $f(n) \in \omega(1) \cap n^{o(1)}$. To obtain the improvement in sample complexity, we create subsets
of samples using the \emph{design} of Nisan and Wigderson [J.~Comput.~Syst.~Sci. '94], so that any two subsets have a small intersection, while the number of subsets is large. Each of these subsets
is used to generate a single $p$-biased sample for the Fourier analysis step. We then show that this allows us to bound the covariance of pairs of samples, which is sufficient for the Fourier
analysis. - Finally, we show that our first algorithm extends to the setting where the noise rate is very high $1/2 - o(1)$, and in this case can be used as a subroutine to obtain new algorithms for
learning DNFs and Juntas. Our algorithms achieve asymptotic improvement in the exponent for certain regimes. For DNFs of size $s$ with approximation factor $\epsilon$ this regime is when $\log \frac
{s}{\epsilon} \in \omega \left( \frac{c}{\log n \log \log c}\right)$, and $\log \frac{s}{\epsilon} \in n^{1 - o(1)}$, for $c \in n^{1 - o(1)}$. For Juntas of $k$ the regime is when $k \in \omega \
left( \frac{c}{\log n \log \log c}\right)$, and $k \in n^{1 - o(1)}$, for $c \in n^{1 - o(1)}$.
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Vertical bar
Vertical bar facts for kids
For technical reasons, "|" does not redirect here.
A vertical bar is the glyph "|". It has several other names besides vertical bar, such as "pipe", "vertical slash", and "bar". In mathematics, it may be put on both sides of a real or complex number
to mean its absolute value. When applied on both sides of a matrix, it means its determinant. When placed in between two numbers, it means the divisibility relation (for example. $a \mid b$ means "a
divides b").
In mathematics, double vertical bars are also used to refer to various mathematical concepts. These include the parallel relation (as in $\ell_1 \parallel \ell_2$) and the norm of a vector (as in $\|
Related pages
Images for kids
See also
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Frequently Asked IBPS Exam Pattern, IBPS Questions with Answers Page 9
maths is very interesting subject if u like it math also like u if u dont like maths dont like u
Don't learn Mathematics just to prove that you are not a mentaly simple person but learn it to prove that you are intelligent
Thanks m4 maths for helping to get placed in several companies. I must recommend this website for placement preparations.
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I Don’t Mind At All: Mind Your Decisions, on youtube
One of my favourite youtube channels is Mind Your Decisions by Presh Talwalker, a Stanford University graduate. It’s a collection of math and logic puzzles, presenting solutions to problems using
common techniques, or in the case of less common, he explains them in clear language so that the average viewer can understand. His graphics and presentation are highly polished (even his videos
from five years ago), and the mathematics is done on screen, making it easy to follow (or rewind and rewatch if you didn’t get it the first time). It’s always easy viewing, and often pleasantly
surprising how simple the solutions can be to what appear to be complex problems. He takes on many “viral” math problems and breaks them down, and shows problems from advanced math competitions and
One of his most recent videos was on cube roots and their digits. How many numbers are there where for any integer x, the sum of the digits of x^3 equals x? The video preview shows one of them: 17
cubed is 4913, and 4+9+1+3 is 17. Two of the seven are trivial, 0 and 1. But what are the others?
Talwalker uses programming to test all the cases, but I’m working on an alternate solution that doesn’t require code and doesn’t involve exhaustive testing, using proof by elimination. For example,
^3√100000 is ~46.42, and 1+9+9+9+9+9 equals 46. This means there are no solutions with six or more digits because the cube of any other sum will always be larger than the original number. ^3√10 is
~2.15, and the cube of 2 is greater than 2, so there are no two digit solutions. (7 and 8 are the only integers whose cube last digit is less than that number, 3 and 2, all others are equal or
greater.) And ^3√1000 is 10, so the only three digit solution is 8 (^3√512, 5+1+2), since all others produce sums greater than the original number; 3 is less than 7 (^3√343) but the sum is greater.
Thus the other four solutions all have four or five digits, but how do you find them without testing?
Working on it (I know the answers, I’m looking for rules or methods of elimination). Noticeably, with the except of 10 and 11, in all cases from 2 to 19 the sums of the cubes’ digits are equal to or
greater than the original number. For all cases 21 to 46, all the sums of the cubes’ digits are less than or equal to the original number. All multiples of ten (10, 20, 30, 40) automatically fail,
and are less than the solution.
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Floer Homotopy Theory - Clay Mathematics Institute
Floer Homotopy Theory
Date: 22 August - 21 December 2022
Location: MSRI
Event type: Extended Format
Organisers: Mohammed Abouzaid (Columbia), Andrew Blumberg (Columbia), Kristen Hendricks (Rutgers), Robert Lipshit (Oregon), Ciprian Manolescu (Stanford)(, Nathalie Wahl (Copenhagen)
The development of Floer theory in its early years can be seen as a parallel to the emergence of algebraic topology in the first half of the 20th century, going from counting invariants to homology
groups, and beyond that to the construction of algebraic structures on these homology groups and their underlying chain complexes. In continuing work that started in the latter part of the 20th
century, algebraic topologists and homotopy theorists have developed deep methods for refining these constructions, motivated in large part by the application of understanding the classification of
manifolds. The goal of this program is to relate these developments to Floer theory with the dual aims of (i) making progress in understanding symplectic and low-dimensional topology, and (ii)
providing a new set of geometrically motivated questions in homotopy theory.
Professor Ivan Smith (Cambridge) has been appointed as a Clay Senior Scholar to participate in this program.
Image: Nathalie Wahl
CMI Enhancement and Partnership Program
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Predict Class Labels Using MATLAB Function Block
This example shows how to add a MATLAB® Function block to a Simulink® model for label prediction. The MATLAB Function block accepts streaming data, and predicts the label and classification score
using a trained, support vector machine (SVM) classification model. For details on using the MATLAB Function block, see Implement MATLAB Functions in Simulink with MATLAB Function Blocks (Simulink).
Train Classification Model
This example uses the ionosphere data set, which contains radar-return qualities (Y) and predictor data (X). Radar returns are either of good quality ('g') or of bad quality ('b').
Load the ionosphere data set. Determine the sample size.
load ionosphere
n = numel(Y)
The MATLAB Function block cannot return cell arrays. Convert the response variable to a logical vector whose elements are 1 if the radar returns are good, and 0 otherwise.
Suppose that the radar returns are detected in sequence, and you have the first 300 observations, but you have not received the last 51 yet. Partition the data into present and future samples.
prsntX = X(1:300,:);
prsntY = Y(1:300);
ftrX = X(301:end,:);
ftrY = Y(301:end);
Train an SVM model using all, presently available data. Specify predictor data standardization.
Mdl = fitcsvm(prsntX,prsntY,'Standardize',true);
Mdl is a ClassificationSVM object, which is a linear SVM model. The predictor coefficients in a linear SVM model provide enough information to predict labels for new observations. Removing the
support vectors reduces memory usage in the generated code. Remove the support vectors from the linear SVM model by using the discardSupportVectors function.
Mdl = discardSupportVectors(Mdl);
Save Model Using saveLearnerForCoder
At the command line, you can use Mdl to make predictions for new observations. However, you cannot use Mdl as an input argument in a function meant for code generation.
Prepare Mdl to be loaded within the function using saveLearnerForCoder.
saveLearnerForCoder compacts Mdl, and then saves it in the MAT-file SVMIonosphere.mat.
Define MATLAB Function
Define a MATLAB function named svmIonospherePredict.m that predicts whether a radar return is of good quality. The function must:
• Include the code generation directive %#codegen somewhere in the function.
• Accept radar-return predictor data. The data must be commensurate with X except for the number of rows.
• Load SVMIonosphere.mat using loadLearnerForCoder.
• Return predicted labels and classification scores for predicting the quality of the radar return as good (that is, the positive-class score).
function [label,score] = svmIonospherePredict(X) %#codegen
%svmIonospherePredict Predict radar-return quality using SVM model
% svmIonospherePredict predicts labels and estimates classification
% scores of the radar returns in the numeric matrix of predictor data X
% using the compact SVM model in the file SVMIonosphere.mat. Rows of X
% correspond to observations and columns to predictor variables. label
% is the predicted label and score is the confidence measure for
% classifying the radar-return quality as good.
% Copyright 2016 The MathWorks Inc.
Mdl = loadLearnerForCoder('SVMIonosphere');
[label,bothscores] = predict(Mdl,X);
score = bothscores(:,2);
Note: If you click the button located in the upper-right section of this page and open this example in MATLAB, then MATLAB opens the example folder. This folder includes the entry-point function
Create Simulink Model
Create a Simulink model with the MATLAB Function block that dispatches to svmIonospherePredict.m.
This example provides the Simulink model slexSVMIonospherePredictExample.slx. Open the Simulink model.
SimMdlName = 'slexSVMIonospherePredictExample';
The figure displays the Simulink model. When the input node detects a radar return, it directs that observation into the MATLAB Function block that dispatches to svmIonospherePredict.m. After
predicting the label and score, the model returns these values to the workspace and displays the values within the model one at a time. When you load slexSVMIonospherePredictExample.slx, MATLAB also
loads the data set that it requires called radarReturnInput. However, this example shows how to construct the required data set.
The model expects to receive input data as a structure array called radarReturnInput containing these fields:
• time - The points in time at which the observations enter the model. In the example, the duration includes the integers from 0 though 50. The orientation must correspond to the observations in
the predictor data. So, for this example, time must be a column vector.
• signals - A 1-by-1 structure array describing the input data, and containing the fields values and dimensions. values is a matrix of predictor data. dimensions is the number of predictor
Create an appropriate structure array for future radar returns.
radarReturnInput.time = (0:50)';
radarReturnInput.signals(1).values = ftrX;
radarReturnInput.signals(1).dimensions = size(ftrX,2);
You can change the name from radarReturnInput, and then specify the new name in the model. However, Simulink expects the structure array to contain the described field names.
Simulate the model using the data held out of training, that is, the data in radarReturnInput.
The figure shows the model after it processes all observations in radarReturnInput one at a time. The predicted label of X(351,:) is 1 and its positive-class score is 1.431. The variables tout, yout,
and svmlogsout appear in the workspace. yout and svmlogsout are SimulinkData.Dataset objects containing the predicted labels and scores. For more details, see Data Format for Logged Simulation Data
Extract the simulation data from the simulation log.
labelsSL = svmlogsout.getElement(1).Values.Data;
scoresSL = svmlogsout.getElement(2).Values.Data;
labelsSL is a 51-by-1 numeric vector of predicted labels. labelsSL(j) = 1 means that the SVM model predicts that radar return j in the future sample is of good quality, and 0 means otherwise.
scoresSL is a 51-by-1 numeric vector of positive-class scores, that is, signed distances from the decision boundary. Positive scores correspond to predicted labels of 1, and negative scores
correspond to predicted labels of 0.
Predict labels and positive-class scores at the command line using predict.
[labelCMD,scoresCMD] = predict(Mdl,ftrX);
scoresCMD = scoresCMD(:,2);
labelCMD and scoresCMD are commensurate with labelsSL and scoresSL.
Compare the future-sample, positive-class scores returned by slexSVMIonospherePredictExample to those returned by calling predict at the command line.
err = sum((scoresCMD - scoresSL).^2);
err < eps
The sum of squared deviations between the sets of scores is negligible.
If you also have a Simulink Coder™ license, then you can generate C code from slexSVMIonospherePredictExample.slx in Simulink or from the command line using slbuild (Simulink). For more details, see
Generate C Code for a Model (Simulink Coder).
See Also
predict | loadLearnerForCoder | saveLearnerForCoder | slbuild (Simulink) | learnerCoderConfigurer
Related Topics
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Comments on Computational Complexity: Favorite Theorems: Small SetsLance's last question is more natural than it may ...
tag:blogger.com,1999:blog-3722233.post114536221248177244..comments2024-11-14T10:49:47.488-06:00Lance Fortnowhttp://www.blogger.com/profile/
06752030912874378610noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-3722233.post-1145384462433968652006-04-18T13:21:00.000-05:002006-04-18T13:21:00.000-05:00Lance's last question is more
natural than it may look. SAT has a reduction to a sparse set with poly(n) queries if and only if NP has polynomial size circuits (think about it for a minute, if you have not seen the proof before),
and (as explained in the post) a reduction with O(1) queries iff P=NP. So the case of log n queries is a quite interesting "intermediate" case between a uniform and a non-uniform
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FLUID MECHANICS QUIZ-15
As per analysis for previous years, it has been observed that students preparing for NEET find Physics out of all the sections to be complex to handle and the majority of them are not able to
comprehend the reason behind it. This problem arises especially because these aspirants appearing for the examination are more inclined to have a keen interest in Biology due to their medical
background. Furthermore, sections such as Physics are dominantly based on theories, laws, numerical in comparison to a section of Biology which is more of fact-based, life sciences, and includes
substantial explanations. By using the table given below, you easily and directly access to the topics and respective links of MCQs. Moreover, to make learning smooth and efficient, all the questions
come with their supportive solutions to make utilization of time even more productive. Students will be covered for all their studies as the topics are available from basics to even the most
advanced. .
Q1. Water falls from a tap , down the streamline
Area decreases
Q2.A vessel of area of cross-section A has liquid to a height H. There is a hole at the bottom of vessel having area of cross-section a. The time taken to decrease the level from H[1] to H[2] will be
A/a √(2/g)[√(H[1] )-√(H[2] )]
Q3. The total weight of a piece of wood is 6 kg. In the floating state in water its 1/3 part remains inside the water. On this floating solid, what maximum weight is to be put such that the whole of
the piece of wood is to be drowned in the water?>
Given, 6g=V/3×〖10〗^3×g …(i) And (6+m)g=V×〖10〗^3×g …(ii) Dividing Eq.(ii) by Eq. (ii) by Eq. (i), we get (6+m)/6=3 Or m=18-6=12 kg
Q4. A soap bubble in air (two surface) has surface tension 0.03Nm^(-1). Find the gauge pressure inside a bubble of diameter 30 mm.
Gauge pressure=4T/R=(4×0.03)/(30/20×〖10〗^(-3) )=8 Pa
Q5.A capillary tube of radius R and length L is connected in series with another tube of radius R/2 and lengthL/4. If the pressure difference across the two tubes taken together is p, then the ratio
of pressure difference across the first tube to that across the second tube is
Volume of liquid flowing per second through each of the two tubes in series will be the same. So V=(π p[1] R^4)/(8 η L)=(πp[2] (R/2)^4)/(8η(L/2)) or p[1]/p[2] =1/4
Q6. A body floats in water with one-third of its volume above the surface of water. If it is placed in oil, it floats with half of its volume above the surface of the oil. The specific gravity of the
oil is
Weight of body = weight of water displaced = weight of oil displaced ⇒2/3 Vρ[w] g=1/2 Vρ[0] g ⇒ρ[0]=4/3 ρ[w] ∴ Specific gravity of oil =ρ[0]/ρ[w] =4/3
Q7.The excess pressure inside one soap bubble is three times that inside a second soap bubble, then the ratio of their surface areas is
Accordingly 4T/r[1] =3×4T/r[2] ⟹ r[1]/r[2] =1/3 Ratio of surface areas A[1]/A[2]>(〖4πr〗[1]^2)/(〖4πr〗[2]^2 )=1/9
Q8.A streamline body with relative density ρ[1] falls into air from a height h[1] on the surface of a liquid of relative densityρ[2], wherep[2]>p[1]. The time of immersion of the body into the liquid
will be
If V is the volume of the body, its weight=V ρ_1 g. Velocity gained by body when it falls from a height h_1=√(2gh_1 ) The weight of liquid displaced by the body as body starts immersing into the
liquid =V ρ_2 g. The net retarding force on the body when it starts going in the liquid F=V(ρ_2-ρ_1)g ∴ Retarding, a=F/(Vρ[1] )=[V(ρ[2]-ρ[1] )g/(V ρ[1] )] The time of immersion of the body is that
time in which the velocity of the body becomes zero. Using the relation v=u+at, we have v=0,u=√(2gh_1 ), a=-V(ρ[2]-ρ[1] )g/(V ρ[1] )=-((ρ[2]-ρ[1])/ρ[1] )g; We have =0=√(2gh[1] )-((ρ[2]-ρ[1])/ρ[1] )gt
Or t=√((2h[1])/g)×(ρ[1]/(ρ[2]-ρ[1] ))
Q9.A liquid does not wet the sides of a solid, if the angle of contact is
The liquid which do not wet the solid have obtuse angle of contact. For mercury and glass, the angle of contact is 135°.
Q10. A log of wood of mass 120 kg floats in water. The weight that can be put on raft to make it just sink, should be (density of wood =600 kg/m^3)
Volume of log of wood V=mass/density=120/600=0.2m^3 Let x weight that can be put on the log of wood So weight of the body =(120+x)×10 N Weight of displaced liquid =V[σg]=0.2×〖10〗^3×10 N The body
will just sink in liquid if the weight of the body will be equal to the weight of displaced liquid ∴(120+x)×10=0.2×〖10〗^3×10 ⇒120+x=200 ∴x=80 kg
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Definition of Two-sample t-test
Two-sample t-test: A two-sample t-test is a statistical hypothesis test used to determine whether the means of two samples are statistically different from each other. The test statistic is a t
statistic, which is the ratio of the differences between the means of the two samples to the standard error of those differences.
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Do P-Values From A Generalised Linear Model Need Correction For Multiple Testing?
After performing a GLM in R, I get back a set of p-values. Does anybody know whether these p-values still need to be adjusted using p.adjust or similar to account for repeated hypothesis testing or
is this already accounted for by the GLM? If the values need to be corrected, which adjustment method(s) are suitable/recommended?
Entering edit mode
Thanks, that's good information! But if I only have a single glm() call for a model like Y ~ X1 + X2 + X3, then would the resulting p-values for X1, X2, X3 also have to be adjusted?
Entering edit mode
No. What you would get then is a regression table containing the different p-values for your X1 to X3 factors. A standard approach is to construct a big model, containing all the factors of interest
and their interactions. Then, from the table, you spot the biggest p-value. If it is above your criterion (eg: 0.05), you remove that factor or interaction. You repeat the process until all factors
are significant. What you have then is the minimal model that explains the variation found in your data. Cheers!
Entering edit mode
What about having several glm() calls for models like Y ~ X1 + X2+ X3, then I have to adjust the p-values for the joint model of X1+X2+X3 for the number of multiple models tested?
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Earnings Calculator - Online and Free
What are Earnings, why and how they are calculated and who needs to do it
How much should you charge for your work? That's a good question! When it comes to prices, most translators and interpreters want to know that they are in the same field as everyone else, while
beginners just want to know what the field is. Use this calculator to determine your equivalent annual salary, given an hourly wage that may surprise your annual income. Also, you can see if you have
one of the top 50 jobs in the US.
What post-tax salary do your employees receive? This powerful tool can complete all net worth calculations to estimate real wages in all 50 states. For more information, please refer to our payroll
calculator guide. Their employees ' voluntary deductions and pensions are deducted from their total income to determine taxable income. Calculate your net salary or family salary by entering each
period or annual salary and related federal, state, and local W4 information into this free federal salary calculator.
What is your post-tax salary? This powerful tool can complete all net worth calculations to estimate real wages (net worth) in any part of the US. Use the real salary calculator to calculate your
income for each month. You may be considering applying for a job or promotion and want to know how changes in salary will affect you. Maybe you want to know what salary will support your expected
lifestyle. The salary calculator converts the salary amount to the corresponding value depending on the frequency of payment. Examples of payment frequency are monthly payments every two weeks, every
half month, or monthly.
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How to explain the location of the object principal plane in a thick plano-convex lens | Zemax Community
From reading this interesting discussion below.
I went down a rabbit hole when I was figuring out where the object principal plane of a thick plano-convex lens is. I’ve been taught to find the principal planes like its described at RP Photonics,
that is:
extrapolate the ingoing and outgoing rays such that they meet in a plane
However, when rays parallel to the optical axis enter through the plane face of the lens, the ingoing and outgoing rays will meet exactly along the convex face, which is not a plane.
Therefore, I believe the principal plane is defined as the plane coincident with the convex surface vertex. At least, this is what I’ve taken away when I used OpticStudio (surface 1 is the convex
I also found this information here:
Plano-convex and plano-concave lenses have one principal plane that intersects the optical axis, at the edge of the curved surface, and the other plane buried inside the glass.
I can accept this definition, but I was curious, from a teaching perspective, to hear if someone can provide an explanation that would be compatible with the “traditional” explanation: find the plane
where the in and out rays bend. I know we are at a boundary between paraxial and real optics and it might not make sense to ask this question, but I could imagine a student asking about this and all
I could answer for now is that in this instance we choose the vertex of the convex surface as the principal plane and its a special case.
Take care all,
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Approximation to Poisson
• If X ∼Poisson (λ) ⇒ X ≈N ( μ=λ, σ=√λ), for λ>20, and approximation improves as (the rate) λ increases.
• Poisson(100) distribution can be thought of as the sum of 100 independent Poisson(1) variables and hence may be considered approximately Normal, by the central limit theorem, so Normal( μ =
rate*Size = λ*N, σ =√λ) approximates Poisson(λ*N = 1*100 = 100).
• The normal distribution is in the core of the space of all observable processes. This distributions often provides a reasonable approximation to variety of data. The Central Limit Theorem states
that to the distribution of the sample average (for almost any process, even non-Normal) is normally distributed (provided the process has well defined mean and variance).
• This applet draws random samples from Poisson distribution, constructs its histogram (in blue) and shows the corresponding Normal approximation (in red). You can specify the rate (λ) of the
Poisson distribution and the number of trials (N) in the dialog boxes. By changing these parameters, the shape and location of the distribution changes. This Applet gives you an opportunity to
study how the approximation to the normal distribution changes when you alter the parameters of the distribution.
[Last modified on by ]
Ivo D. Dinov, Ph.D., Departments of Statistics and Neurology, UCLA School of Medicine
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The Diamond Princess Quarantine: Using a Beta Distribution to Predict Initial 2019 Coronavirus Infections - Hubbard Decision Research
The Diamond Princess Quarantine: Using a Beta Distribution to Predict Initial 2019 Coronavirus Infections
by Matt Millar | Articles, News
The Diamond Princess cruise ship is currently under quarantine while 271 passengers are being tested (as of 2/5/2020) for the 2019 novel coronavirus (2019 n-CoV). Concern about infection arose when a
prior passenger from Hong Kong who was on board the ship from 1/20 to 1/25 was later found to be infected. As a result, the ship was delayed and then quarantined off the port of Yokohama to test a
group of 271 passengers who either had symptoms of 2019 n-CoV or had significant contact with the original case from Hong Kong. On Wednesday (2/5) 10 out of 31 tests had come back positive from a
suspected 271 people. By Thursday, 20 out of 102 tests had come back positive. This is a real world application where we can test the utility of a Beta distribution to predict an outcome – we
shouldn’t be surprised if another 15-46 people test positive for the coronavirus out of the remaining tests.
A Beta distribution for the first 31 tests would have an alpha of 11 (10+1) and a beta of 22 (21+1); the 90% confidence interval for the proportion using the first sample is 21%-47% (Figure 1). The
second group of tests has an alpha of 11 (10+1) and a beta of 62 (61+1); the 90% confidence interval for the proportion given this sample is 9%-22% (Figure 1). Based on these results, we suspect that
they tested the more likely cases first, and the remaining 169 are more likely to resemble the second sample in likelihood of infection. However, if the first two samples were randomly selected then
we would use the beta distribution of all 102 initial cases (alpha = 21, beta = 83) with a 90% C.I. of 14 to 27%.
Since we don’t know which is accurate, we’ll use 9-27% for our 90% confidence interval, which gives us an estimate of 35 to 66 total of this group of 271 will test positive for the coronavirus
(Figure 2). This would imply that an additional 15 to 46 positive results will come back from the remaining 169 tests.
Drawing the Right Conclusions
There is another chapter to this story however. Whether the original group has 35 or 66 cases, these will not be the only 2019 n-Cov cases on board the Diamond Princess cruise ship, and it is crucial
that policy makers understand why. The ship has nearly 3,700 people trapped on board, and the infection spread uninhibited for at least seven days. The correct conclusion is that the incubation
period is long, and the doubling rate is short – therefore when these initial 271 passengers were selected and tested, there already existed another group of people who were infected and not
symptomatic. This is important for three reasons:
1. Don’t blame the quarantine. As additional cases are found over the next 10 days, it would be incorrect to assume that quarantining people to their rooms failed. That was the correct move and will
prevent additional infections and serious illness.
2. The “hidden population” of 2019 n-Cov is a crucial aspect of understanding this disease. This was the main point in the post published on Monday – that the undercount of this disease will hamper
attempts to control the spread because of the long incubation period and asymptomatic cases. If policy-makers can draw the right conclusions from the Diamond Princess experience, it could
dramatically help in the effort to slow or stop the spread of the disease.
3. It is likely that additional cases may have gotten off the ship between 1/20 and 2/2, and those passengers should be alerted and local health officials made aware of the risk.
We will publish a follow up article to this once the test results for the 271 passengers are completed. Our initial estimates are that at least 100 people will test positive before the quarantine is
released. We estimate that even if quarantine efforts prove perfectly successful, the Diamond Princess will likely have over 100 people aboard the ship test positive for 2019 n-Cov before the
quarantine is released.
Learn how to start measuring variables the right way – and create better outcomes – with our two-hour Introduction to Applied Information Economics: The Need for Better Measurements webinar. $100 –
limited seating.
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These functions select a bandwidth sigma for the kernel estimator of point process intensity computed by density.ppp or other appropriate functions. They can be applied to a point pattern belonging
to any class "ppp", "lpp", "pp3" or "ppx".
The bandwidth \(\sigma\) is computed by the rule of thumb of Scott (1992, page 152, equation 6.42). The bandwidth is proportional to \(n^{-1/(d+4)}\) where \(n\) is the number of points and \(d\) is
the number of spatial dimensions.
This rule is very fast to compute. It typically produces a larger bandwidth than bw.diggle. It is useful for estimating gradual trend.
If isotropic=FALSE (the default), bw.scott provides a separate bandwidth for each coordinate axis, and the result of the function is a vector, of length equal to the number of coordinates. If
isotropic=TRUE, a single bandwidth value is computed and the result is a single numeric value.
bw.scott.iso(X) is equivalent to bw.scott(X, isotropic=TRUE).
The default value of \(d\) is as follows:
class dimension
"ppp" 2
"lpp" 1
"pp3" 3
"ppx" number of spatial coordinates
The use of d=1 for point patterns on a linear network (class "lpp") was proposed by McSwiggan et al (2016) and Rakshit et al (2019).
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Python 3 - EdDoc
EdDoc: home/built-in_modules
built-in modules
An index of the most useful built-in modules available to Python programmers.
│ Section │ Description │
│array │Space efficient arrays that can contain either integers or float values only. │
│colorsys │Provides functions for conversions between RGB and other color systems. │
│csv │Write and read tabular data to and from delimited files. │
│datetime │Provides functions for using basic date and time types │
│email │Space efficient arrays that can contain either integers or float values only. │
│io │Provides functions for working with file input and output. │
│math │Provides mathematical functions (sin(), etc.) │
│os │Provides miscellaneous operating system functions. │
│random │Provides functions to produce pseudo-random numbers. │
│sqlite3 │Provides functions for working with SQLite database. │
│statistics│Provides mathematical statistics functions. │
│tkinter │Provides a GUI (graphical user interface) toolkit. │
│turtle │An educational library providing functions that provide a turtle graphics environment. │
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How to predict with pre-trained model in tensorflow?
To predict with a pre-trained model in TensorFlow, you can follow the steps below:
1. Load the pre-trained model: First, load the pre-trained model using the tf.keras.models.load_model() function. Make sure to provide the correct path to the saved model file.
1 import tensorflow as tf
3 model = tf.keras.models.load_model('path/to/pretrained_model.h5')
1. Prepare the input data: Depending on the requirements of the pre-trained model, you may need to preprocess the input data. Make sure the input data matches the input shape expected by the model.
2. Make predictions: Use the model.predict() function to make predictions on the input data.
1 predictions = model.predict(input_data)
1. Post-process the predictions: Depending on the task, you may need to post-process the predictions to get the final output. For example, if you are working with a classification model, you may
need to apply softmax activation to the predicted probabilities to get the final class label.
1 final_predictions = tf.nn.softmax(predictions)
1. Interpret the results: Finally, interpret the results and use them according to your specific use case.
That's it! By following these steps, you can predict with a pre-trained model in TensorFlow.
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the temperature at Kent is -7 degrees and increasing at 2.5 degrees
the temperature at Kent is -7 degrees and increasing at 2.5 degrees per hour. The temperatire at row is 19 degrees decreasing at 4 degrees per hour. When will they be the same? and what temperatore?
let t = hours running time.
-7+2.5t = temperature at Kent at time t.
19-4.0t = temperature at row at time t.
Set them equal to each other and solve for t which will be the number of hours for the two temperatures to be the same.
Then use t and either equation to determine the final temperature.
To find when the temperatures at Kent and Row will be the same, we can set the equations for the temperatures at each location equal to each other:
-7 + 2.5t = 19 - 4t
Now, let's solve this equation for t:
-7 + 2.5t = 19 - 4t
Combining like terms, we get:
6.5t = 26
Dividing both sides by 6.5, we find:
t = 4
Therefore, it will take 4 hours for the temperatures at Kent and Row to be the same.
To determine the final temperature when they are the same, we can substitute t=4 back into either equation. Let's use the equation for the temperature at Kent:
temperature at Kent = -7 + 2.5t
temperature at Kent = -7 + 2.5(4)
temperature at Kent = -7 + 10
temperature at Kent = 3
So, when the temperatures at Kent and Row are the same, the temperature will be 3 degrees.
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Assoc. Prof. Mirko Tarulli
Mirko Tarulli Di Giallonardo obtained his Ph.D. (Dottorato di Ricerca) in Mathematics at the Department of Mathematics “L. Tonelli” in the University of Pisa, Italy in 2006.
Currently, he is an Associate Professor at the Mathematical Analysis and Differential Equations Department in the Faculty of Applied Mathematics and Informatics at the Technical University of Sofia,
Member of the Institute of Mathematics and Informatics at the Section of Differential Equations and Mathematical Physics to the Bulgarian Academy of Sciences, as well as an Academic Visitor at the
Mathematics Department in the University of Pisa, Italy.
In 2004 he was a Visiting Assistant Professor in the Institut für Angewandte Analysis at the Technical University Bergakademie in Freiberg, Germany. In the period 2006-2007 he was a Visiting
Assistant Professor in the Department of Mathematics and Statistics at the University of Vermont, USA. From 2010 to 2012 he had a two years position as Academic Visitor in the Department of
Mathematics at Imperial College London, UK
Mirko Tarulli is the author of over 40 scientific publications and co-author of 3 books. He has participated in more than 50 international conferences, in multiple research projects, among which 6
funded by the EU, and won prestigious grants, such as an INdAM fellowship.
His scientific research has been mainly devoted to the following fields:
• Well-posedness and Scattering Theory for Nonlinear Dispersive Equations with Local and Non-Local nonlinarities
• Existence of solutions in energy space and stability for Maxwell and Schrödinger equations
• A Priori Sobolev Estimates on Riemannian Manifolds with Constant Negative Curvature
• Perturbative Theory for semilinear Wave Equation
• Strichartz Estimates for the Wave Equation and Schrödinger Equation on Riemannian Manifolds
• A Priori Estimates On Riemannian Manifolds With Schwarzchild Metrics
• Smoothing And Strichartz Estimates for the Wave Equation and Schrödinger Equation Perturbed by a Magnetic Potential (Small and Large with respect to suitable norms)
• Wave Equation and Klein-Gordon Equation with Time Depending Perturbation (Resolvent and Microlocal analysis)
• Oscillatory Integrals and Microlocal Analysis
• Singular Integral Operators, Hardy-Littlewood Maximal function and Littlewood-Paley Theory
• Weighted Estimates on Symmetric Spaces and on a Riemannian Manifold
• Well-posedness and Scattering Theory for Nonlinear Dispersive Equations settled on Riemannian Manifold.
• Orbital and Asymptotic stability theory for Nonlinear Dispersive Equations
• Theory of Resonances
• All Aspects of Harmonic Analysis
Mirko Tarulli has taught the following courses:
• Mathematical Analysis I, II, III
• Calculus I, III, IV
• Statistics
• Harmonic Analysis
• Elements of Partial Differential Equations
• Scattering Theory
• Mathematics for Optimization Theory and Big Data Analytics
• Advanced Mathematical Analysis for Applied Sciences
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Singularities in High-Temperature
Superconductor H3S
Symmetry-Shaped Singularities in High-Temperature Superconductor H[3]S
journal contribution
posted on 2024-06-25, 14:35 authored by Sebastian R. Thomsen, Maarten G. Goesten
The superconducting critical temperature of H[3]S ranks among the highest measured, at 203 K. This impressive value stems from a singularity in the electronic density-of-states, induced by a
flat-band region that consists of saddle points. The peak sits right at the Fermi level, so that it gives rise to a giant electron–phonon coupling constant. In this work, we show how atomic orbital
interactions and space group symmetry work in concert to shape the singularity. The body-centered cubic Brillouin Zone offers a unique 2D hypersurface in reciprocal space: fully connecting squares
with two different high-symmetry points at the corners, Γ and H, and a third one in the center, N. Orbital mixing leads to the collapse of fully connected 1D saddle point lines around the square
centers, due to a symmetry-enforced s-p energy inversion between Γ and H. The saddle-point states are invariably nonbonding, which explains the unconventionally weak response of the superconductor’s
critical temperature to pressure. Although H[3]S appears to be a unique case, the theory shows how it is possible to engineer flat bands and singularities in 3D lattices through symmetry
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IF formula where the output is the date from another column
Hi All,
I'm trying to create an IF formula where the output is the date from another column.
In my organization, employees have 10 recertification due dates. I need the smartsheet to display which recertification is next, and when the due date is.
I know there will need to be multiple IF statements in the formula, but the example I'll use is just for the 2nd recertification.
I have three columns, Next Task Due, Next Task Due Date, and 2nd Recertification Due Date.
The formula would need to read IF the "Next Task Due" column is "2nd Recertification" then insert the date from the "2nd Recert Due Date" column into the "Next Task Due Date" column. Here is the
formula I'm trying to use: =IF([Next Task Due]@row = "2nd Recertification", [2nd Recert Due Date]@row). However it returns a "#INVALID COLUMN VALUE" I've also included a screenshot.
Does anyone know what I'm doing wrong?
• My guess is that the "Next Task Due Date" Column is not set as a Date Column..
this will cause an error as you are trying to pass a date parameter into a "Text" Field.
Also try this
=IF(Left( [Next Task Due]@row,1) = "2", [2nd Recert Due Date]@row, IF(Left( [Next Task Due]@row,1) = "3", [3rd Recert Due Date]@row, IF(Left( [Next Task Due]@row,1) = "4", [4th Recert Due Date]
@row, IF(Left( [Next Task Due]@row,1) = "5", [5th Recert Due Date]@row, IF(Left( [Next Task Due]@row,1) = "6", [6th Recert Due Date]@row, IF(Left( [Next Task Due]@row,1) = "7", [7th Recert Due
Date]@row, IF(Left( [Next Task Due]@row,1) = "8", [8th Recert Due Date]@row, IF(Left( [Next Task Due]@row,1) = "9", [9th Recert Due Date]@row, [10th Recert Due Date]@row ))))))))
This will select your column based on the first digit in the Next Task Due Date..
Might need to rejig it if you have a 1st Recert Date as I did not include that in the logic and be careful as 1 and 10 can get confused.. by the Left function I have
Brent C. Wilson, P.Eng, PMP, Prince2
Facilityy Professional Services Inc.
• Thanks, Brent!
This is the formula that worked.
=IF([Next Task Due]@row = "Observation", JOIN([Initial Certification & Observation Due Date]@row), IF([Next Task Due]@row = "Initial Certification", JOIN([Initial Certification & Observation Due
Date]@row), IF([Next Task Due]@row = "1st Recertification", JOIN([1st Recert Due Date]@row), IF([Next Task Due]@row = "2nd Recertification", JOIN([2nd Recert Due Date]@row), IF([Next Task Due]
@row = "3rd Recertification", JOIN([3rd Recert Due Date]@row), IF([Next Task Due]@row = "4th Recertification", JOIN([4th Recert Due Date]@row), IF([Next Task Due]@row = "5th Recertification",
JOIN([5th Recert Due Date]@row), IF([Next Task Due]@row = "6th Recertification", JOIN([6th Recert Due Date]@row), IF([Next Task Due]@row = "7th Recertification", JOIN([7th Recert Due Date]@row),
IF([Next Task Due]@row = "8th Recertification", JOIN([8th Recert Due Date]@row), IF([Next Task Due]@row = "9th Recertification", JOIN([9th Recert Due Date]@row), IF([Next Task Due]@row = "10th
Recertification", JOIN([10th Recert Due Date]@row)))))))))))))
Help Article Resources
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Domains, Uninterpreted Functions, and Axioms | Caesar
Domains, Uninterpreted Functions, and Axioms
domain blocks are used to create user-defined types and uninterpreted functions. A domain has a name which can be used as a type in HeyVL code. The domain block contains a list of funcs and axioms
defined on this domain.
Every domain type supports the binary operators == and !=. All other operations must be encoded using functions and axioms.
Incompleteness from Quantifiers
Note that axioms with quantifiers quickly introduce incompleteness of Caesar, making it unable to prove or disprove verification. Read the documentation section on SMT Theories and Incompleteness for
more information.
Example: Exponentials of ½
HeyVL does not support exponentiation expressions natively. But we can define an uninterpreted function ohfive_exp and add axioms that specify its result. ohfive_exp(n) should evaluate to (½)ⁿ, so we
add two axioms that define this exponential recursively.
ohfive_exp_base states that ohfive_exp(0) == 1 and ohfive_exp_step ensures that ohfive_exp(exponent + 1) == 0.5 * ohfive_exp(exponent) holds. This is sufficient to axiomatize our exponential
domain Exponentials {
func ohfive_exp(exponent: UInt): EUReal
axiom ohfive_exp_base ohfive_exp(0) == 1
axiom ohfive_exp_step forall exponent: UInt. ohfive_exp(exponent + 1) == 0.5 * ohfive_exp(exponent)
Note that this domain declaration creates a new type Exponentials, but we do not use it.
We can check that ohfive_exp(3) evaluates to 0.125 by declaring a procedure with pre-condition true and post-condition ohfive_exp(3) == 0.125. This procedure verifies:
proc ohfive_3() -> ()
pre ?(true)
post ?(ohfive_exp(3) == 0.125)
Do not forget the empty block of statements {} at the end! If you omit it, Caesar will not attempt to verify the procedure and thus will not check the specification.
Pure Functions
You can also declare pure or interpreted functions. These are defined by a single expression that computes the result of the function.
The following function declaration has a such a definition (= x + 1):
func plus_one(x: UInt): UInt = x + 1
This syntax is just syntactic sugar for a function declaration with an additional axiom, i.e.
func plus_one(x: UInt): UInt
axiom plus_one_def forall x: UInt. plus_one(x) == x + 1
Unsoundness From Axioms
Axioms are a dangerous feature because they can make verification unsound.
An easy example is this one:
domain Unsound {
axiom unsound false
proc wrong() -> ()
pre ?(true)
post ?(true)
assert ?(false)
The axiom unsound always evaluates to false. But for verification, Caesar assumes the axioms hold for all program states. In other words, Caesar only verifies the program states in which the axioms
evaluate to true. Thus, Caesar does not verify any program state and the procedure wrong incorrectly verifies!
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Forecasting Few Is Different, Part 2
This is a two-part story by Malte Tichy about dealing with sales forecasts that concern fast-moving as well as slow-moving items. Find Part 1 here.
How precise can a sales forecast become?
The mechanism of fluctuation cancellation ensures that granular low-scale-forecasts are more imprecise, noisy and uncertain than aggregated, coarse-grained high-scale ones: We are better (in relative
terms) at predicting the total number of pretzels for an entire week than for a single day.
So far, we have qualitatively argued for this relation, but can we make any quantitative statement about the level of precision that we can ideally expect, for different predicted selling rates?
Thankfully, this is indeed possible, in a universal, industry-independent way. We argued in our previous blog post on hindsight bias in forecast evaluation that deterministic, perfectly certain
forecasts are unrealistic: Let’s consider our above forecast of 5 pretzels. On the level of individual customers, a deterministic forecast of 5 translates into 5 customers that will, no matter what,
buy a pretzel on the forecasted day. But not only would we assume to know these 5 customers extremely well (maybe better than they know themselves, who hasn’t spontaneously decided to grab or not
grab a pretzel?), but we also totally exclude the possibility that any other customer would buy a pretzel. Such degree of certainty is clearly impossible. Allowing some uncertainty, say, 6 customers
with a probability of 5/6=83.3% each of buying a pretzel, results in what mathematicians call a binomial distribution in the total number of sold pretzels: The probability to sell 6 pretzels is (5/6)
^6, the probability to sell none is (1/6)^6, the probability to sell between one and five contains the respective binomial coefficients. However, knowing 6 customers that will buy with high
probability is still unrealistic. Even assuming 10 customers with 50% probability to buy a pretzel each is challenging. We can move on and further increase the number of potential customers while
decreasing their probability to buy a pretzel, following the path to the limiting Poisson distribution: In the Poisson limit, we assume an unlimited customer base, in which every customer has an
infinitesimally tiny probability to buy, while we have control over the product of the number of customers and the probability to buy: the rate of sale. The Poisson distribution scales consistently:
If daily sales follow a Poisson distribution of mean 5, then weekly sales follow a Poisson distribution of mean 35. The Poisson distribution is the “gold standard” for sales forecasting: We assume to
know all factors that influence the sales of a given product, but have no access to individual-customer-data that would allow us to make stronger statements on the buying behavior of individual
customers. When your forecast precision is as good as expected from the Poisson distribution, you have typically reached the limit of what is possible at all.
A Poisson distribution only ingests a single parameter, the rate of sale; the distribution width, the spread of likely outcomes around the mean, is fully determined by its functional form, which
reflects self-consistency. That is, the achievable degree of precision only depends on the predicted selling rate within the considered time interval: The sales of 5 predicted pretzels per day follow
the same distribution as the sales of 5 predicted birthday cakes per week, 5 predicted buns per hour or 5 predicted wedding cakes per quarter. In other words: The best-case-achievable relative error
is fully and uniquely determined by the forecasted value itself!
Why ultrafresh slow sellers cannot be offered sustainably
With this insight on error scaling in mind, let’s return to the question why fresh sea cucumber is not offered everywhere in the world: We show the expected distribution of sales per day for a
perfect Poisson forecast of one sea cucumber per day:
We will experience 37% of days without any demand, 37% of the days will see one seafood-aficionado wanting to buy one raw sea cucumber, and in 26% of the days, we will see a demand of two or even
more. How many sea cucumbers shall we keep on stock, given that we must throw them away at the end of the day if nobody buys them? If we have one piece on stock, we’ll have to throw it away in 37% of
the days, while in 26% of the days, we’d have unhappy customers that won’t be able to buy the sea cucumber that they intended to. With two pieces on stock, we need to throw away at least one piece
after 74% of the days — what a waste, given that sea cucumbers are protected in many places! Clearly, a business model that aims at fulfilling the tiny demand of raw sea cucumber is not viable, and
could only be sustained if the margin were extremely large: The people buying the raw sea cucumber need to subsidize all the days on which no sea cucumber is sold — and these people won’t even be
sure to get one if they want one! Under mild assumptions on the margin and the costs of disposal, the right amount of stock for an ultra-fresh super-slow seller is: zero.
Again, this is all due to non-proportional scaling: The expected distribution of sales for a forecast of 100 fresh sea cucumbers per day is not just an inflated version of the above one for 1 sea
cucumber per day, but it has a different shape – just like an elephant does not look like a large impala:
Bad news for anyone who is hoping for getting pretzels in Busan or a wider variety of fruit in northern Europe! There is, however, hope: When demand overcomes a certain threshold because a perishable
dish becomes à la mode, that novel food can establish itself in new places — you get good sushi almost everywhere on the planet.
Summarizing, due to non-proportional scaling of forecasting error, the occurring over-stocks and under-stocks of a product — even assuming a perfect forecast – increases disproportionally when the
sales rate decreases. Consequently, only above a certain sales rate per shelf life can offering a certain perishable food be sustained at all.
Assessing forecast error
We have now understood why we can’t expect to find foreign fresh delicacies at home, let us now derive some lessons for data scientists and business users in charge of judging forecast quality: For
high predicted selling rates, the random fluctuations that drive the true sales value up or down with respect to the forecasted mean cannot be used as an excuse for any substantial deviation, and we
can attribute such deviation to an actual error or problem in the forecast. The statistical idiosyncrasies that we discussed above do not matter. If a total demand of 1’000’000 was predicted and
total sales amount to 800’000, this 20% error is not due to unavoidable fluctuations, but due to a biased forecast.
For small forecasted numbers, we can’t unambiguously attribute observed deviations to a bad forecast anymore: Given a forecast of one, the observation 0 (which is 100% off) is quite likely to occur
(with probability 37%), and so is the observation 2 (which is also 100% off). Judging whether a forecast is good or bad becomes much harder, because the natural baseline, the unavoidable noise, is
Shall we then divide our forecasts into “fast sellers”, for which we attribute observed deviations to forecast error, and “slow-sellers”, for which we are more benevolent? We recommend against that:
What about the intermediate case, a prediction, of, say 15? Where is the boundary between “slow” and “fast”? If a product becomes slightly more popular, what if it crosses that boundary, and its
forecast quality judgement takes a sudden jump? There is a continuous transition between “fast” and “slow” that exhibits no natural boundary, as we can see in this plot of the expected relative error
of a forecast as a function of the forecasted value (note the logarithmic scale on the abscissa and that we compute the expected error using the optimal point estimator, which is not the mean but the
median of the Poisson distribution):
Due to this continuous transition, we recommend a stratified evaluation by predicted rate, which means grouping forecasts into bins of similar forecasted value and evaluating error metrics separately
for each bin. Our previous blog post on the hindsight bias explains why this binning should be done by forecasted value and not by observed sales, even though the latter feels more natural than the
former. For each of these bins, we judge whether forecast precision is in line with the theoretical expectation (shown in the above plot), or deviates substantially. Our expectations of a forecast
should depend on the predicted rate: For extremely small values (smaller than 0.69), most observed actual sales are 0, and we are essentially “always off completely” with an error of 100% —
unavoidably. For a predicted sales rate of 10, we will have to live with a frightening relative error of 25% – in the best case! When we forecast 100=10^2, we still expect a relative error of about
8%, for a rate of 1’000=10^3, the error drops to 2.5%. Asking, e.g., for a 10% error threshold across all selling rates is, therefore, counter-productive: The vast majority of slow sellers will
violate that threshold and bind resources to find out why “the forecast is off”, while some improvement could still be possible for fast-sellers that obey the threshold and therefore get no
In practice, the deviation from the ideal line that we have drawn above will depend on the forecast horizon (is it for tomorrow or for next year?) and on the industry (are we predicting a well-known
non-seasonal grocery item, or an unconventional exquisite dress on the thin edge between fashionable and tasteless?). Nevertheless, accounting for the universal non-proportional scaling of
forecasting error is the most important aspect that your forecast evaluation methodology should fulfill!
Conclusions: Avoid the naïve scaling trap, accept and strategically deal with slow-seller noise
Apart putting visits to local restaurants onto your next vacation’s must-do-list, what conclusions should you take out of this blog post?
Make sure that the temporal aggregation scale that you set in your evaluation matches the business decision time scale: Since strawberries and sea cucumbers only last for a day, they are planned for
a day and an evaluation on daily level is appropriate. You can’t compensate today’s strawberry demand with yesterday’s overstocks or vice-versa. For items that last longer, the scale on which an
error in a business decision really materializes is certainly not a day: If a shirt was not bought on Monday, maybe it will be on Tuesday, or two weeks later — not important for the stock of shirts
that is ordered every month. If you encounter many items with small forecasted numbers (<5) in your evaluation, double-check that the latter is really the relevant one, on which a buying,
replenishing or other decision is taken.
Don’t set constant targets for forecast precision across your entire product portfolio, neither in absolute nor in relative terms: Your fast-sellers will easily reach low relative errors, your slow
sellers will seemingly struggle. Instead, divide your predictions into buckets of similar forecast value, and judge each bucket separately. Set a realistic, sales-rate-dependent target.
For slow-sellers, it is imperative to be aware of the probabilistic nature of forecasts, and strategically account for the large unavoidable noise, be it via safety-stocks-heuristics in the case of
non-perishable items, or by produce-upon-order strategies, e.g. for wedding cakes.
Although the unavoidability of forecasting error in slow sellers might be irritating, it is encouraging that the limits of forecasting technology can be established quantitatively in a rigorous way,
such that we can account for them strategically in our business decisions.
Related posts
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How do you find the axis of symmetry, and the maximum or minimum value of the function y=x^2+3x-5? | HIX Tutor
How do you find the axis of symmetry, and the maximum or minimum value of the function #y=x^2+3x-5#?
Answer 1
The axis of symmetry is $x = - \frac{3}{2}$ or $- 1.5$.
The vertex is $\left(- \frac{3}{2} , - \frac{29}{4}\right)$ or $\left(- 1.5 , - 7.25\right)$.
#y=x^2+3x-5# is a quadratic equation in standard form:
#a=1#, #b=3#, #c=-5#
Axis of symmetry: vertical line that divides the parabola into two equal halves. It is also the #x#-coordinate of the vertex.
The formula to find the axis of symmetry:
Plug in the known values.
#x=-3/2# or #-1.5#
The axis of symmetry is #x=-3/2# or #-1.5#.
Vertex: the minimum or maximum point on the parabola. The axis of symmetry is the #x#-coordinate. To find the #y#-coordinate, substitute #-3/2# into the equation and solve for #y#.
Multiply #9/2# by #2/2#, and #5# by #4/4# so each term has #4# as its denominator.
The vertex is #(-3/2,-29/4)# or #(-1.5,-7.25)#.
graph{y=x^2+3x-5 [-16.02, 16.01, -8.01, 8.01]}
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Answer 2
The axis of symmetry of the function ( y = x^2 + 3x - 5 ) is given by the formula ( x = -\frac{b}{2a} ), where ( a ) is the coefficient of the quadratic term and ( b ) is the coefficient of the
linear term. In this case, ( a = 1 ) and ( b = 3 ), so the axis of symmetry is ( x = -\frac{3}{2} ).
To find the maximum or minimum value of the function, we can evaluate the function at the axis of symmetry. Substitute ( x = -\frac{3}{2} ) into the function to get ( y = (-\frac{3}{2})^2 + 3(-\frac
{3}{2}) - 5 = -\frac{17}{4} ).
Therefore, the axis of symmetry is ( x = -\frac{3}{2} ), and the function has a maximum value of ( y = -\frac{17}{4} ) at that point.
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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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This library provides commonly accepted basic predicates for list manipulation in the Prolog community. Some additional list manipulations are built-in. See e.g., memberchk/2, length/2.
The implementation of this library is copied from many places. These include: "The Craft of Prolog", the DEC-10 Prolog library (LISTRO.PL) and the YAP lists library. Some predicates are reimplemented
based on their specification by Quintus and SICStus.
- Virtually every Prolog system has library(lists), but the set of provided predicates is diverse. There is a fair agreement on the semantics of most of these predicates, although error handling
may vary.
True if Elem is a member of List. The SWI-Prolog definition differs from the classical one. Our definition avoids unpacking each list element twice and provides determinism on the last element.
E.g. this is deterministic:
member(X, [One]).
- Gertjan van Noord
List1AndList2 is the concatenation of List1 and List2
Concatenate a list of lists. Is true if ListOfLists is a list of lists, and List is the concatenation of these lists.
ListOfLists - must be a list of possibly partial lists
True iff Part is a leading substring of Whole. This is the same as append(Part, _, Whole).
Is true when List1, with Elem removed, results in List2. This implementation is determinsitic if the last element of List1 has been selected.
selectchk(+Elem, +List, -Rest) is semidet
Semi-deterministic removal of first element in List that unifies with Elem.
select(?X, ?XList, ?Y, ?YList) is nondet
Select from two lists at the same position. True if XList is unifiable with YList apart a single element at the same position that is unified with X in XList and with Y in YList. A typical use
for this predicate is to replace an element, as shown in the example below. All possible substitutions are performed on backtracking.
?- select(b, [a,b,c,b], 2, X).
X = [a, 2, c, b] ;
X = [a, b, c, 2] ;
See also
- selectchk/4 provides a semidet version.
selectchk(?X, ?XList, ?Y, ?YList) is semidet
Semi-deterministic version of select/4.
True if Y directly follows X in List.
Delete matching elements from a list. True when List2 is a list with all elements from List1 except for those that unify with Elem. Matching Elem with elements of List1 is uses \+ Elem \= H,
which implies that Elem is not changed.
See also
- select/3, subtract/3.
- There are too many ways in which one might want to delete elements from a list to justify the name. Think of matching (= vs. ==), delete first/all, be deterministic or not.
True when Elem is the Index'th element of List. Counting starts at 0.
- type_error(integer, Index) if Index is not an integer or unbound.
See also
- nth1/3.
Is true when Elem is the Index'th element of List. Counting starts at 1.
See also
- nth0/3.
Select/insert element at index. True when Elem is the N'th (0-based) element of List and Rest is the remainder (as in by select/3) of List. For example:
?- nth0(I, [a,b,c], E, R).
I = 0, E = a, R = [b, c] ;
I = 1, E = b, R = [a, c] ;
I = 2, E = c, R = [a, b] ;
?- nth0(1, L, a1, [a,b]).
L = [a, a1, b].
As nth0/4, but counting starts at 1.
Succeeds when Last is the last element of List. This predicate is semidet if List is a list and multi if List is a partial list.
- There is no de-facto standard for the argument order of last/2. Be careful when porting code or use append(_, [Last], List) as a portable alternative.
proper_length(@List, -Length) is semidet
True when Length is the number of elements in the proper list List. This is equivalent to
proper_length(List, Length) :-
length(List, Length).
Is true when List1 and List2 are lists with the same number of elements. The predicate is deterministic if at least one of the arguments is a proper list. It is non-deterministic if both
arguments are partial lists.
See also
- length/2
Is true when the elements of List2 are in reverse order compared to List1. This predicate is deterministic if either list is a proper list. If both lists are partial lists backtracking generates
increasingly long lists.
permutation(?Xs, ?Ys) is nondet
True when Xs is a permutation of Ys. This can solve for Ys given Xs or Xs given Ys, or even enumerate Xs and Ys together. The predicate permutation/2 is primarily intended to generate
permutations. Note that a list of length N has N! permutations, and unbounded permutation generation becomes prohibitively expensive, even for rather short lists (10! = 3,628,800).
If both Xs and Ys are provided and both lists have equal length the order is |Xs|^2. Simply testing whether Xs is a permutation of Ys can be achieved in order log(|Xs|) using msort/2 as
illustrated below with the semidet predicate is_permutation/2:
is_permutation(Xs, Ys) :-
msort(Xs, Sorted),
msort(Ys, Sorted).
The example below illustrates that Xs and Ys being proper lists is not a sufficient condition to use the above replacement.
?- permutation([1,2], [X,Y]).
X = 1, Y = 2 ;
X = 2, Y = 1 ;
- type_error(list, Arg) if either argument is not a proper or partial list.
flatten(+NestedList, -FlatList) is det
Is true if FlatList is a non-nested version of NestedList. Note that empty lists are removed. In standard Prolog, this implies that the atom '[]' is removed too. In SWI7, [] is distinct from '[]
Ending up needing flatten/2 often indicates, like append/3 for appending two lists, a bad design. Efficient code that generates lists from generated small lists must use difference lists, often
possible through grammar rules for optimal readability.
See also
- append/2
Pairs is a list of Item-Count pairs that represents the run length encoding of Items. For example:
?- clumped([a,a,b,a,a,a,a,c,c,c], R).
R = [a-2, b-1, a-4, c-3].
- SICStus
subseq(+List, -SubList, -Complement) is nondet
subseq(-List, +SubList, +Complement) is nondet
Is true when SubList contains a subset of the elements of List in the same order and Complement contains all elements of List not in SubList, also in the order they appear in List.
- SICStus. The SWI-Prolog version raises an error for less instantiated modes as these do not terminate.
max_member(-Max, +List) is semidet
True when Max is the largest member in the standard order of terms. Fails if List is empty.
See also
- compare/3
- max_list/2 for the maximum of a list of numbers.
min_member(-Min, +List) is semidet
True when Min is the smallest member in the standard order of terms. Fails if List is empty.
See also
- compare/3
- min_list/2 for the minimum of a list of numbers.
max_member(:Pred, -Max, +List) is semidet
True when Max is the largest member according to Pred, which must be a 2-argument callable that behaves like (@=<)/2. Fails if List is empty. The following call is equivalent to max_member/2:
?- max_member(@=<, X, [6,1,8,4]).
X = 8.
See also
- max_list/2 for the maximum of a list of numbers.
min_member(:Pred, -Min, +List) is semidet
True when Min is the smallest member according to Pred, which must be a 2-argument callable that behaves like (@=<)/2. Fails if List is empty. The following call is equivalent to max_member/2:
?- min_member(@=<, X, [6,1,8,4]).
X = 1.
See also
- min_list/2 for the minimum of a list of numbers.
sum_list(+List, -Sum) is det
Sum is the result of adding all numbers in List.
max_list(+List:list(number), -Max:number) is semidet
True if Max is the largest number in List. Fails if List is empty.
See also
- max_member/2.
min_list(+List:list(number), -Min:number) is semidet
True if Min is the smallest number in List. Fails if List is empty.
See also
- min_member/2.
numlist(+Low, +High, -List) is semidet
List is a list [Low, Low+1, ... High]. Fails if High < Low.
- type_error(integer, Low)
- type_error(integer, High)
is_set(@Set) is semidet
True if Set is a proper list without duplicates. Equivalence is based on ==/2. The implementation uses sort/2, which implies that the complexity is N*log(N) and the predicate may cause a
resource-error. There are no other error conditions.
list_to_set(+List, ?Set) is det
True when Set has the same elements as List in the same order. The left-most copy of duplicate elements is retained. List may contain variables. Elements E1 and E2 are considered duplicates iff
E1 == E2 holds. The complexity of the implementation is N*log(N).
- List is type-checked.
See also
- sort/2 can be used to create an ordered set. Many set operations on ordered sets are order N rather than order N**2. The list_to_set/2 predicate is more expensive than sort/2 because it
involves, two sorts and a linear scan.
- Up to version 6.3.11, list_to_set/2 had complexity N**2 and equality was tested using =/2.
True if Set3 unifies with the intersection of Set1 and Set2. The complexity of this predicate is |Set1|*|Set2|. A set is defined to be an unordered list without duplicates. Elements are
considered duplicates if they can be unified.
See also
- ord_intersection/3.
union(+Set1, +Set2, -Set3) is det
True if Set3 unifies with the union of the lists Set1 and Set2. The complexity of this predicate is |Set1|*|Set2|. A set is defined to be an unordered list without duplicates. Elements are
considered duplicates if they can be unified.
See also
- ord_union/3
subset(+SubSet, +Set) is semidet
True if all elements of SubSet belong to Set as well. Membership test is based on memberchk/2. The complexity is |SubSet|*|Set|. A set is defined to be an unordered list without duplicates.
Elements are considered duplicates if they can be unified.
See also
- ord_subset/2.
Delete all elements in Delete from Set. Deletion is based on unification using memberchk/2. The complexity is |Delete|*|Set|. A set is defined to be an unordered list without duplicates. Elements
are considered duplicates if they can be unified.
See also
- ord_subtract/3.
Undocumented predicates
The following predicates are exported, but not or incorrectly documented.
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