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compact space
In compactness for locales, is the direction always defined by union of opens? IE the join semilattice of opens defined by the union operation?
It seems likely to me to be the case, but I could imagine there might be other ways of defining a direction on the opens. And maybe these don't work for compactness..
So if it is always upper bound of two opens is the union here for the direction, I think it would be clearer to state this explicitly.
In compactness for locales,
okay, I reverted and made only the typo fix for U'. If I have further suggestions given your feedback I will propose here and go lighter on wiki edits.
(By the way, I agree that “diffs” are often hard to read. It might sometimes be easier reading different revisions in different tabs.)
I don’t think much of these are really improvements. In some cases you have been replacing other people’s words, which were fine and clear enough, with words that you would have chosen instead, such
as “unioning”, a word I find grating and cacophonous outside of very informal speech.
But let’s look at this stuff about directed open covers, where you have the paragraph with the word “Firstly”.
Firstly, note that unions of finite opens give a direction on any open cover. This gives us the notion of a directed open cover, which is useful for locales.
As samples of mathematical writing, both sentences of that paragraph are flawed. What one should be saying – and what I read proofs in former revisions to be saying – is that given an open cover, one
can form a new cover that is directed. Not that the old cover was “given a direction”. (By the way, “give a direction” sounds clunky to my ears. I realize that the link for “directed” goes to a page
titled “direction”, but I think that’s partly because of a rule we have about using noun phrases as titles. But anyway, nobody ever “gives a direction” to an open covering or to a preorder: either an
open covering is directed or it isn’t. What one does is replace an open covering that may not be directed with another that is, by adding in more open sets.)
The second sentence is flawed because “this”, whatever the antecedent is, is not “giving us a notion of” anything. “Giving a notion”, to my mind, means performing an act of conceptual analysis, as in
abstracting a new concept that captures or expresses a variety of observed phenomena. “This gives us a directed open cover” is closer to what one should say, but only after fixing the first sentence.
Really, I think it might be better to roll back to an earlier revision, and then fix whatever tiny typos needed fixing. I’m sorry to say, but I think some of the newer revisions are worsening the
article. I’m not sure why you’re doing this.
Looking at the diffs, it appears that there was something done to proposition 2.6. But I don't think this is the case.
I only worked on 2.5, the first prop in compactness for locales.
If there are any ideas why these diffs are coming in ugly, I would like to know. Again, I apologize.
statement of compactness for locales.
diff, v89, current
73 was me.
If every open cover is a directed open cover, then
"Proposition 2.5. A space is compact iff every directed open cover of it has the entire space as one of its opens."
is simply
"Proposition 2.5. A space is compact iff every open cover of it has the entire space as one of its opens."
Is it possible to have an open cover without a direction? I don't see how.
break off definition of directed open cover. simplify statement of compactness for locales.
diff, v89, current
give directed open cover as a definition, simplify definition of compact for locales.
diff, v89, current
I'm sorry the diff is a mess. Maybe it's because I copy pasted from a desktop editor. I can redo if wanted.
I do think the changes make it more understandable.
same as last (prop 2.5, locales, if direftion)
diff, v88, current
Clarify if direction of prop 2.5 (locales)
diff, v88, current
Typos like that don’t have to be mentioned.
missing prime in compactness for locales
diff, v87, current
An open cover of a subset is just an open cover of that set regarded as a space with the induced topology. But we could add that as a remark afterwards.
Sorry, #61 was me, tphyahoo. Forgot to sign in.
In definition 2.1
"Let (X,τ) be a topological space. Then an open cover is a set {Ui⊂X}i∈I of open subsets (i.e. (Ui⊂X)∈τ⊂P(X)) such that their union is all of X"
This doesn't cover standard open covers of the unit interval, because it only talks about the whole space, not subspaces. Wikipedia
has "..... Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y"
I think this should be added to definition 2.1
sorry, wrong thread
Yes, that’s what’s meant. But this page is written in classical mathematics as the default, so I don’t think there’s a need to say so explicitly here, especially since the subsequent paragraph
clarifies the situation in constructive mathematics.
in Compactness via completeness
"A uniform space X is compact if and only if it is complete and totally bounded."
can this be
"In classical mathematics, a uniform space X is compact (def 2.2) if and only if it is complete and totally bounded."
It is so stated. See closed projection characterization, point 5. under Variant Proofs. It’s mentioned that this approach is in Escardo’s 2009 paper. See lemma 4.3 there.
the overt space page states
Recall that a topological space X is compact if and only if, for every other space Y and any open subspace U of X×Y, the subspace
is open in Y.
I think this should be stated somewhere on the compact space page, but I don't see it. Neither in any of the other related pages. IE,
closed characterization
compactness and stable closure
There is a lot of material and maybe I overlooked something. If so, could someone point me the right place? If not, can it be proven?
It seems related to closed projection characterization. On that page we have
Theorem 1.1. A topological space X is compact if and only if for every space Y, the projection map π:Y×X→Y out of the product topological space is a closed map.
Seems close? Anyway, can someone prove the claim above about the open product subspace U?
Mention Bishop-compactness (complete and totally bounded)
diff, v85, current
Argh. :-)
Thanks. I have fixed it, and also added pointer to the proof.
Urs, in the idea section, did you really mean to say “paracompact” and not “locally compact”?
Those are useful improvements, Urs – thanks.
I took a look and ended up making some last changes myself to the Idea section:
• made explicit that “everything” in the first sentence referred to sequences and nets
• linked the line about not needing and ambient space for the definition with the line saying that nevertheless one does often consider compact subspaces;
• after the claim that one likes to consider compact Hausdorff spaces, I added one reason
• grouped the two lines about compact locales together, now at the end of the Idea-section.
But there was no mention of compactness in the 2nd paragraph of your edit; attention was momentarily deflected away to talk about “closed and bounded”. So I had to reword again.
Please do not remove the parentheses. They signal that a side remark is being made, which the reader can pursue if she likes, but the main focus needs to be kept on what the property is about.
Thank you for your input, but now that some clarity has been reached, I suggest that we not spend a lot of time on what Feynman called “wordsmithing”. I’ll add that I made a change suggested by
comment #37.
Much clearer.
The verb "captures" is used twice, which makes it a little hard to concentrate on exactly what is being captured. I reworded to avoid this.
Okay, I tweaked the opening paragraph just a little. I don’t agree with joining “closed” to “bounded” as in #45, but I did mention Heine-Borel. Hopefully it’s now clear that it’s more about nets and
convergence than it is about being closed and bounded.
Re #46: no, the definitions/characterizations section is general. Heine-Borel is for a specialized set of circumstances and belongs to Examples.
Also it seems like many of the "Examples" (such as heine borel) fix in better in the Definitions / Characterizations section.
What do you think of singly linking "closed and bounded" to the heine borel theorem section (4.5) rather than linking "closed" and "bounded" separately to the two concept pages as is currently the
Doesn't this give a better intuition?
How about
"In typical spatial domais (such as R^n) compactness is a kind of ultimate topological expression of the general idea of a space being “closed and bounded”: .........
No, the “it” here is the net (nets are generalized sequences).
"by boundedness it cannot escape, and by closure the point is in the space."
can "it" here be replaced by "accumulation point"? I think that would be clearer.
That “Idea” section should be read at a very intuitive level; the language is somewhat fuzzy (I suppose by design). The author of those words should be granted some poetic license.
Any space, compact or not, is closed in itself, so under strict interpretations some of the language of the Idea section won’t make a lot of sense. The real sense of that passage is concentrated in
the words “every net has an accumulation point” (that certainly has a precise meaning), and that should be the main takeaway from what the author is trying to convey. The rest of it seems to be a
simple appeal to the archetypal image we all carry around in our heads of compact sets: sets in $\mathbb{R}^n$ which are closed and bounded, and the author is trying to draw a connection between that
intuitive conception of “closed and bounded”, based on that (finite-dimensional) picture, and the more precise mathematical conception “every net must have an accumulation point”.
In infinite-dimensional Hilbert space (for example), “closed and bounded” do not imply compact. So that intuition comes with a (largish) grain of salt! In fact the word “bounded” isn’t quite a
topological concept; it makes sense for metric spaces but it doesn’t have a meaning for general topological spaces.
Turning to Proposition 3.1: now we’re doing mathematics, not waving intuitive wands. As it happens, compact subspaces of Hausdorff spaces are closed (and Hausdorff spaces make up the majority of
spaces one encounters when first learning topology), but in general compact subsets need not be closed in the ambient space. Thus the closure hypothesis has to be inserted by hand for the proposition
to work in the generality given there.
Anyway, you may be right that the Idea section is (for some readers anyway) more confusing than enlightening. It’s hard to say, but perhaps the opening should be reconsidered.
From the introduction a compact space
"is a kind of ultimate topological expression of the general idea of a space being “closed and bounded”"
but is it true that all compact spaces are closed with regard to some topology, or only metric spaces, or only euclidean spaces?
Proposition 3.1 (2) has the precondition that "if the compact spaces are also closed" then the claim about inersections.
Well, if all compact spaces are closed then this precondition can be dropped as the claim applies to all compact spaces.
But if not all compact spaces are closed, perhaps this should be dropped from the idea section, or qualified "In metric spaces..."
Yes, it seems the same counterexample works with Sierpinski space, since $1$ is usually taken to be the open point. (Maybe this is discussed somewhere in Stone Spaces? I don’t have my copy within
easy view.) So yeah, if I think about it a little longer, maybe it will become obvious that compact frames lack coequalizers, or maybe you already see that’s true.
I feel like if compact locales had limits we would know about it. Could you do something similar to your counterexample with the Sierpinski space instead of the indiscrete 2-point space?
Something seems wrong with Proposition 3.3, that the category of compact spaces has limits. (Of course compact Hausdorff spaces have limits.) The problem is that the equalizer of two maps need not be
a closed subspace; that kind of thing is true if we are working with Hausdorff spaces, but not for more general spaces.
An explicit example is where we take two maps $[0, 1] \to \{0, 1\}$ where the codomain is given the indiscrete topology, where one of the maps $f$ has $f^{-1}(1) = [0, 1/2)$, and the other is the
constant map at $1$. If an equalizer in $Comp$ existed, then $\hom(1, -): Comp \to Set$ would have to preserve it, so set-theoretically it would have to be $[0, 1/2)$. The topology on $[0, 1/2)$
would have to be the same as or finer than the subspace topology in order for the equalizer map to be continuous. But if the subspace topology isn’t compact, then no finer topology would make it
compact either. (Here I’m taking the contrapositive of the proposition that if $(X, \tau)$ is compact and $\tau' \subseteq \tau$ is a coarser topology, then $(X, \tau')$ is also compact.)
I guess I could go in and change it to a true statement, but I’d want to know first about the situation for compact locales. Again, for compact regular locales, there’s no problem.
Thanks, Todd!! That’s great.
Okay, I’ve gone through and reorganized section 2 of compact space according to comments/suggestions above and my own personal knowledge. Roughly I classified the various “definitions” (now called
propositions) under three headings: elementary reformulations, via convergence, and via stability properties. There is still plenty left to do: plenty of proofs which can be filled in or farmed out
to other parts of the nLab, as appropriate, and still links left to be made, among other things.
It is good that you (tphyahoo) brought your concerns to our attention, so thanks for that. I do think the article has a better shape now. The “obviously incoherent” former 2.12 is, I hope you will
now see, coherent after all. I also changed the link from logic to quantification which is more precise I think.
I'm very sorry about my tone.
I am extremely enthusiastic about constructive math (an amateur obviously, I admit). And I admire what the nlab has accomplished collectively, as well as the individual contributors here and on the
cluster of pages where I lurk. Todd / Urs / Toby: thanks for everything you've done.
The reason I sent so many comments is I wanted to lay a breadcrumb trail for future edits to the wiki, since I was unable to edit the wiki myself to my satisfaction. It keeps cutting off the page
halfway due to some formatting glitch I can't track down. And there doesn't seem to be a preview functionality.
If there is any advice or links about editing / maintaining the wiki I will review and try to do better in the future.
BTW, I am https://www.linkedin.com/in/thomashartman1/ just to put a face to the edits. I have been interested in constructive math for some years as a software developer doing haskell / coq for
personal interest and some for work. Lurker on nlab for years I guess. I am primarily interested in constructive characterization of the reals, probably following the abstract stone duality path.
Okay, I’ll begin having a look. Much of this was written (I think) by Toby quite a while ago, and his presence here has lately become more sporadic.
Please be patient though. I am finding some of the tone harsh (e.g., “now seems more obviously incoherent” – that I find excessive), and it’s a bit of a barrage of comments now to process.
Proposition 2.5 can then perhaps be reforumulated as
Let X be a topological space. Then the following are equivalent:
1 X is compact in the sense of def. 2.2
2 current contents of proposition 2.5
3-... reformulation of definitions 2.12 and 2.13
Proposition 2.8 same thing for stably closed, if this is a stand alone concept unrelated to closed projection characterization of compactness.
Proposition 2.7 (compactness in terms of frames of opens)
Let X be a topological space. Then the following are equivalent:
1 X is compact in the sense of def. 2.2.
2-.... reformulation of definitions 2.10 and 2.11
I am having trouble making the above changes because of struggles with wiki markup, however, to be more explicit I think we should have something like
Proposition 2.6. (compactness in terms of filter or net convergence)
Let X be a topological space. Assuming the ultrafilter theorem (a weak form of the axiom of choice), then the following are equivalent:
1 X is compact in the sense of def. 2.2.
2-.... reformulation of definitions 2.6-2.9
I agree, with #24, that’s what I was referring to at the end of #21.
This needs to go to the attention of Toby and Todd, I think. The request would be: Turn these terse remarks into something a little more self-contained and inviting.
I think section 2.7 through 2.9 should be grouped under
Proposition N.N. (compactness in terms of filters)
If these are all equivalent (and ultrafilter theorem requirement applies to all) then the proposition should be along the lines of "the following three statements are equivalent."
Similarly, section 2.10 and 2.11 should be grouped under
Propsition N.N (compactness in terms of open sets)
Possibly Definition 2.12 and 2.13 then get merged under proposition 2.5.
Possibly there is also a new proposition about proper maps, that is broken out of 2.12.
It is not clear if the assumption of the ultrafilter theorem applies just to 2.7, or to 2.8 and 2.9 as well or even 2.10 and 2.11.
I believe that Section 2.10 and 2.11 is unrelated to filters/nets but this should be made more obvious with better section numbering and breaks.
I also believe section 2.12 is unrelated to the earlier section, but I noted this before.
I think the link to the "logic" page in
"a logical characterisation of compactness is used in Abstract Stone Duality:"
should be removed.
After a bit of grammar cleanup, Definition 2.12 now seems more obviously incoherent.
First, we need a definition of stably closed. The colon suggests a definition, but I believe it may be an error. If it is not an error, this could be clarified by defining it on its own concept page
(currently nonexistent).
If it is an error as I suspect and "stably closed" is unrelated to the closed projection characterization, then:
1) "stably closed" should be its own definition and perhaps its concept page should be created as well.
2) Then definition 2.12 and 2.13 should be merged or moved up to be closer to Proposition 2.5 so that all the stuff about closed projection is in one place.
Thank you yes, much clearer now.
Thanks, Urs. It looks good to me. Does the mention of excluded middle now make sense to tphyahoo? To circumvent excluded middle (as in closed-projection characterization of compactness), one has to
change the statement to say the dual image operator $\forall_\pi: P(X \times Y) \to P(Y)$ takes open sets to open sets; the statement as it is, that the direct image along projection takes closed
sets to closed sets, is equivalent to the other statement by De Morgan duality, but that’s where excluded middle comes in.
Regarding the duplication of the statement of the closed-projection characterization of compactness:
I have removed item 4 from prop. 2.4, but I also moved the former “Definition” 2.10 up to what is now prop. 2.5, so that it is still the next statement after item 3 of prop. 2.4.
(Todd should please have a look.)
My main request about this entry is: Somebody should turn the long list of “Definitions” 2.7 to 2.13 of compactness into a list of propositions that state that certain statements are equivalent.
I have now spelled out a detailed proof of prop. 3.2 here.
Re: #17, I think you’re right that 2.4(4) and 2.10 are redundant. I don’t have an opinion on which of them should be kept.
Re: #16, $[0,3] \setminus (1,3)$ is $[0,1] \cup \{3\}$, which is compact.
Definition 2.4, point 4 is
"For every topological space (Y,τY) (Y,\tau_Y) the projection map out of the product topological space πY:(X×Y,τX×Y)→(Y,τY) \pi_Y \;\colon\; (X \times Y, \tau_{X \times Y}) \to (Y, \tau_Y) is a
closed map.
and proof of this is stated as
"The proof of the equivalence of statement 4 is discussed at closed-projection characterization of compactness."
further down
Definition 2.10. (closed-projection characterization of compactness) is
"X X is compact iff for any space Y Y, the projection map X×Y→Y X \times Y \to Y out of their Cartesian product is closed (see e.g. Milne, section 17)."
Aren't these two ways of saying the same thing?
Proposed fix: 2.4, point 4 and its proof should be deleted from 2.4 and merged to 2.10.
Possible issue, assuming validity of above fix:
Does 2.10 then require excluded middle?
It is claimed that point 2.4 point 4 requires excluded middle, but it is not clear why this is.
In fact linked "closed-projection characterization of compactness" claims a means of circumventing both excluded middle. My guess is excluded middle is not required but I don't know.
Proposition 3.2. (complements of compact with open subspaces is compact)
seems wrong.
Take [0,3] as compact space and (1,3) as subspace of R1.
The complement is two half open spaces: [0,1) and (2,3] and by nonexample 4.6 "half open intervals are not compact"
Either proposition 3.2 is wrong or I am missing some intuition. A proof or a reference to a proof would help clear this up if I am wrong here.
In accordance with a recent discussion, I moved the detailed elementary proof of the example of closed intervals to its own page. Also, I added the example of cofinite topology to compact space.
I have added the statement about unions and intersections of compact subspaces, here
I have added some elementary details at compact space – Examples – General, such as the proof that closed intervals are compact.
FWIW, constructively there are at least two distinct definitions of closed set.
Okay, I have edited statement and proof at compact+space#fip. I made explicit the use of excluded middle for identifying opens with complements of closed subsets, and a second use of excluded middle
for getting what is usually labeled “fip”, which is the contrapositive of what was formerly stated here.
Oh, okay. I’ll make all that explicit in the entry now.
How are closed sets being defined here?
If we are defining closed sets to be precisely the complements of open sets (in symbols, $eg U$), then I can see that the usual open set formulation implies the closed set formulation you just
mentioned: if $\bigcup_{i \in I} U_i = X$, then surely $\bigcap_{i \in I} eg U_i = eg \left(\bigcup_{i \in } U_i \right) = eg X = \emptyset$ since $eg$ takes unions to intersections.
But then how would we turn this implication around (to assert equivalence of the two formulations)? Without excluded middle, I don’t see how every open set $U$ would be the complement of a closed
There seems to be something wrong right at the beginning of compact space:
After def. 2.1, the usual definition about existence of finite open subcovers, there is def. 2.2 which is the immediate reformulation in terms of closed subsets: a collection of closed subsets whose
intersection is empty has a finite subcollection whose intersection is still empty.
But this def. 2.2 is introduced with the remark that it needs excluded middle to be equivalent to 2.1, which is not true.
Probably what that remark about excluded middle was meant to refer to is instead the further formulation in terms of closed subsets, the one which says that a collection of closed subsets with the
finite intersection property has non-empty intersection.
[edit: I have added what seems to be missing at compact space to finite intersection property]
I have added this proof to covert space, with pointers to it from compact space and proper map.
Ok, I found Vermeulen’s paper, and I believe he has a constructive proof that if $r:X\to 1$ preserves directed joins then it satisfies the Frobenius condition. Suppose $U\in O(X)$ and $V\in O(1) = \
Omega$; we must show that if $r_\ast(U \cup r_\ast(V))$ is true, then so is $r_\ast(U) \cup V$. Note that $r_\ast(W)$ is the truth value of the statement “$W=X$”, while $r^*(P) = \bigcup \{ X \mid P
\}$. Suppose $r_\ast(U \cup r_\ast(V))$, i.e. that $U\cup \bigcup \{ X \mid V \} = X$. Now consider the set $\{ W\in O(X) \mid V \vee (W\le U) \}$; this is evidently directed, and our supposition in
the last sentence says exactly that its union is $X$. Therefore, if $X$ is compact in the sense that its top element is inaccessible by directed joins, there exists a $W$ such that $V \vee (W\le U)$
and $X\le W$. In other words, either $V$ or $X\le U$, i.e. either $V$ or $r_\ast(U)$, which is what we wanted.
So my current conclusion is that the first Elephant quote above is wrong about the Frobenius condition being an additional restriction constructively. This did seem like the most likely resolution,
since otherwise the definition of proper geometric morphism would probably be wrong, which seems unlikely.
I am confused about the constructive notion of “compact locale”. In the Elephant just before C3.2.8, Johnstone says
If $X$ is a locale in a Boolean topos, then the unique locale map $\gamma:X\to 1$ is proper iff $\gamma_*$ preserves directed joins… Constructively, the Frobenius reciprocity condition [$f_*(U\
cup f^*(V)) = f_*(U)\cup V$]… is a nontrivial restriction even on locale maps with codomain 1; so we include it in the definition of compactness — that is, we define $X$ to be compact if $X\to 1$
is proper.
However, as far as I can tell this additional Frobenius condition is not included in the general definition of proper geometric morphism, which just says (in the stack semantics of $E$) that $f:F\to
E$ preserves directed unions of subterminals. It makes sense that some extra condition is necessary when speaking externally about a general locale map $f:X\to Y$, since “$f_*$ preserves directed
unions” is not internalized to the stack semantics of $Sh(Y)$. However, when talking about a map $X\to 1$, we should already be in the stack semantics of $Sh(1) = Set$. So I don’t understand why we
need the Frobenius condition separately in the constructive definition of “compact locale”. Johnstone doesn’t give an example, and I don’t have Moerdijk or Vermuelen’s papers.
On the other hand, I don’t see how to prove the Frobenius condition either. I have looked through section C3.2 of the Elephant as carefully as I can, and I haven’t been able to extract an actual
proof that a proper geometric morphism between localic toposes satisfies the Frobenius condition to be a proper map of locales. Prior to defining proper geometric morphisms, he says
…we may reasonably define a geometric morphism $f:F\to E$ to be proper if… $f_*(\Omega_F)$ is a compact internal frame in $E$. What does it mean to say that an internal frame… is compact in a
general topos $E$? For the case $E=Set$, we saw how to answer this in topos-theoretic terms in 1.5.5: it means that the direct image functor $Sh(X) \to Set$… preserves directed colimits of
subterminal objects.
and then proceeds to define $f:F\to E$ to be proper if “$f_*$ preserves directed colimits of subterminal objects” is true in the stack semantics of $E$. But C1.5.5 used “$f_*$ preserves directed
unions” as a definition of “compact locale”, which the first quote above claims is not sufficient constructively, i.e. over a general base $E$. So I am confused; can anyone help?
Looks good!
I have tried to edit compact space just a little for readability.
For one, I tried to highlight nets over proper filters just a tad more (moving the statements for nets out of parenthesis, adding reminder to the equivalence via their eventuality filters), just so
that the entry gets at least a little closer to giving the reader the expected statement about convergence of sequences.
Also I pre-fixed the text of all of the many equivalent definitions by “$X$ is compact if…” to make it read more like a text meant for public consumption, and less like a personal note.
I have added to the Properties-section at compact space that in Hausdorff space every compact subset is closed.
As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created
one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)
It’s really very simple: a poset is directed if it is nonempty and any two elements have a common upper bound. The upper bound doesn’t have to be the union.
I sort of wish you’d get away from this idea of “defining a direction” here. In practice, one defines a partially ordered set, and then that partially ordered set is either directed or it isn’t:
there is no extra step of “defining a direction” that needs to be done. (Even “defines a partial order” sounds too elaborate, because in this context the partial order is invariably subset
inclusion.) As far as any defining goes: at some point in the proof, one introduces (or defines) an open cover $\mathcal{U}'$ and then verifies that it is directed.
I don’t know what is confusing you in this article, but let’s talk about it here. We have the traditional notion of compact space: every open cover has a finite subcover. The proposition in question
states that compactness of a space $X$ is equivalent to another condition: that any directed open cover of $X$ has $X$ among its elements. I’ll call that condition D.
Let’s just take a moment to be very explicit what all this means. First, an open cover of $X$ is a collection $\mathcal{U}$ of open subsets $U \subseteq X$ whose union is $X$. An open cover $\mathcal
{U}$ is directed if, whenever one is given finitely many elements $U_1, \ldots, U_n \in \mathcal{U}$, there is an element $U \in \mathcal{U}$ such that $U_1 \subseteq U, \ldots, U_n \subseteq U$, in
other words there exists an element $U \in \mathcal{U}$ that is an upper bound of the $U_1, \ldots, U_n$ with respect to subset inclusion. This $U$ doesn’t always have to be the actual union (i.e.,
the least upper bound), although it can be.
To show that a compact space $X$ satisfies condition D, let $\mathcal{U}$ be any directed open cover. (We are not obliged to assume that $\mathcal{U}$ is closed under finite unions.) By compactness,
$\mathcal{U}$ has a finite subcover, meaning there are finitely many elements $U_1, U_2, \ldots, U_n$ of $\mathcal{U}$ whose union (i.e. least upper bound) is $X$. By the assumption that $\mathcal{U}
$ is directed, there exists an upper bound $U$ of $U_1, \ldots, U_n$ belonging to $\mathcal{U}$. I claim $U = X$. First, $U \subseteq X$ because by definition of open cover of $X$, an element $U \in
\mathcal{U}$ is automatically a subset of $X$. But also $X \subseteq U$, because $U$ is an upper bound of the $U_1, \ldots, U_n$, and the union $X$ which is the least upper bound is contained in the
upper bound. So $U = X$ belongs to $\mathcal{U}$, and thus we have verified that condition D holds.
Now suppose that condition D holds for $X$. We want to show that $X$ is compact. So, let $\mathcal{U}$ be any open cover of $X$; we want to show that under condition D, there exists finitely many
$U_1, \ldots, U_n \in \mathcal{U}$ whose union is $X$. Of course we can’t apply condition D directly to $\mathcal{U}$ because $\mathcal{U}$, regarded as being partially ordered by subset inclusion,
might not be directed. But if we introduce a new open cover $\mathcal{U}'$ whose elements are precisely the possible finite unions of elements of $\mathcal{U}$, then $\mathcal{U}'$is directed. (Of
course this should be proven. So, suppose given elements $V_1, \ldots, V_n$ of $\mathcal{U}'$. According to how $\mathcal{U}'$ was defined, each $V_i$ is a finite union $U_{i, 1} \cup \ldots \cup U_
{i, k_i}$ of elements of $\mathcal{U}$. But then $V = V_1 \cup \ldots \cup V_n = \bigcup_{i=1}^n \bigcup_{j=1}^{k_i} U_{i, j}$, being a union of finitely many elements of $\mathcal{U}$, belongs to $\
mathcal{U}'$. This $V$ is an upper bound of $V_1, \ldots, V_n$.) Since $\mathcal{U}'$ is directed and since condition D holds, we have that $X \in \mathcal{U}'$. But then by definition of $\mathcal
{U}'$, the set $X$ is a union of finitely many elements $U_1, \ldots, U_n$ of $\mathcal{U}$. This completes the proof.
By the way, I don’t mean to sound overly harsh in my last comment. In view of the way the article direction is written, it’s not necessarily wrong to speak of “defining a direction” (on a set) – it’s
just that in this context, it sounds really weird and overly elaborate to write that way. It would be more appropriate to write that way if one started with a set $S$ and was contemplating, among the
infinitely many ways in which $S$ could carry a structure of directed partial order, which one of them one wishes to single out as the topic of discussion. But in the present discussion, there’s
really no choice in the matter: the only relevant partial ordering is given by subset inclusion (or the given partial order if one is starting with a frame or subsets of a frame), and it’s only a
question of whether the poset under discussion is directed or not.
So much so that if a reader encounters the phrase “define a direction by” in the article compact space, it would be natural to wonder if the author was confused or didn’t quite understand what they
were talking about.
I don’t know if this is what tphyahoo has in mind, but in constructive mathematics one might want the “directedness” of a directed poset to be given by a function assigning a particular upper bound
to any two elements, since in the absence of choice having such a thing is a stronger assumption than the mere existence of upper bounds. However, in a join-semilattice there is always a function
that selects the least upper bound, since those are uniquely defined.
Right, that would be a more structural (actually, algebraic over posets) notion of direction where the phrase “define a direction” would make sense. It seems that direction (written largely by Toby,
I think) was not written with this possibility in mind – and offhand, it strikes me as a superfluous consideration for the compact space article.
Yes, the distinction isn’t relevant for compactness.
adding a few sentences on compact topological spaces vs compact convergence spaces in constructive mathematics in section “Compactness via convergence”. In particular, the failure of the equivalence
of the definitions in constructive mathematics suggests that this article needs to be split up into “compact topological space” and “compact convergence space”, because that compact topological
spaces (in the sense of open covers and finite subcovers) are compact convergence spaces (in the sense of nets and convergent subnets) implies excluded middle.
diff, v93, current
Compact spaces were introduced (under the name of “bicompact spaces”) by Paul Alexandroff and Paul Urysohn around 1924, see the 92nd volume of Mathematische Annalen, especially AU.
diff, v97, current
The point-free definition given is wrong – the right adjoint to pullback of open subsets always exists; what is needed is that the right adjoint to pullback of subsets restricts to open subsets.
diff, v98, current | {"url":"https://nforum.ncatlab.org/discussion/1976/compact-space/?Focus=82986","timestamp":"2024-11-13T12:26:51Z","content_type":"application/xhtml+xml","content_length":"224079","record_id":"<urn:uuid:e8d47463-cca9-4a4b-a392-9f6123334d22>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00228.warc.gz"} |
SheetsLand | Google Sheets Functions, Templates, Tips and Tutorials
The IMDIV function is a handy tool that allows you to divide two numbers and return the result as a whole number. It’s particularly useful when you want to avoid any decimals or remainders in your
calculations. So, if you want to divide two numbers and get a whole number result, the IMDIV function is … Read more
IMCSCH Function
If you’re a regular user of Google Sheets, you know how powerful and useful it can be for organizing and analyzing data. But did you know that there are many built-in functions available that can
help you take your spreadsheet skills to the next level? One such function is the IMCSCH function. The IMCSCH function … Read more
IMCSC Function
Have you ever found yourself working with a large spreadsheet in Google Sheets and wished there was a way to quickly and easily compare the values in two different cells or ranges? Well, there is a
function for that! It’s called the IMCSC (Intersection) function and it allows you to compare two ranges of cells … Read more
IMCOTH Function
Have you ever found yourself working with a large dataset in Google Sheets and wished you had an easier way to find the maximum or minimum value within a specific range of cells? Well, you’re in
luck! The IMCOTH function in Google Sheets is here to help. This function allows you to quickly and easily … Read more
IMCOT Function
If you’re a fan of Google Sheets, you may already be familiar with the extensive list of functions available to help you work with your data. However, there’s a lesser-known function that can be
especially useful in certain situations: IMCOT. IMCOT stands for “Import Consolidated Data from Other Tables.” This function allows you to easily … Read more
IMCOSH Function
Are you looking to work with complex numbers in Google Sheets? If so, the IMCOSH function might just be the tool you need. This function allows you to find the hyperbolic cosine of a given complex
number in Google Sheets, and it can be a useful tool for a variety of mathematical calculations. In this … Read more
Employee Attendance Sheet Free Template
An employee attendance sheet is a useful tool for tracking the attendance and absences of employees in a company. It helps managers and HR personnel to keep track of employee attendance, tardiness,
and absences, and to ensure that employees are meeting their attendance expectations. There are many different ways to create an employee attendance sheet, … Read more
IMCOS Function
If you’re familiar with using spreadsheets and have some basic knowledge of trigonometry, then this post is for you. The IMCOS function is a useful tool for those who need to perform trigonometric
calculations in their spreadsheets. It allows you to find the cosine of a complex number, which is a combination of a real … Read more
IMCONJUGATE Function
Today I want to talk about the IMCONJUGATE function in Google Sheets. This is a powerful tool that allows you to take the complex conjugate of a complex number in a cell. For those who may not be
familiar with complex numbers, they consist of a real part and an imaginary part. The conjugate of … Read more
IMARGUMENT Function
Today, we’re going to be discussing the IMARGUMENT function in Google Sheets. This function is an incredibly useful tool that allows you to find the argument of a complex number in Google Sheets. If
you’re not familiar with complex numbers or the concept of an argument, don’t worry! We’ll be explaining everything in detail so … Read more | {"url":"https://sheetsland.com/","timestamp":"2024-11-11T01:18:03Z","content_type":"text/html","content_length":"75502","record_id":"<urn:uuid:89b11a58-6ee8-4ac8-99e8-5c3926249136>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00181.warc.gz"} |
日時 2017年4月27日（木）14:00〜15:00
場所 筑波大学 自然系学系D棟D814
講演者 伊敷喜斗氏（筑波大学数理物質系）
講演題目 Quasi-symmetric invariant properties of Cantor metric spaces
For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties.
We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic.
For instance, the middle-third Cantor set is standard.
In this talk, we discuss our constructions of exotic Cantor metric spaces for all the possible cases of satisfying each of the three properties or not.
アブストラクト Our constructions enable us to classify Cantor metric spaces into eight types with concrete examples.
The David-Semmes uniformization theorem tells us that standard Cantor metric spaces are quasi-symmetric equivalent.
In this talk, we conclude that there exist at least two exotic Cantor metric spaces of the same type that are not quasi-symmetric equivalent to each other.
Moreover, for each of all the non-uniformly disconnected types, there exist at least aleph one many quasi-symmetric equivalent classes of Cantor metric spaces of such a given type.
As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assoud dimension. | {"url":"https://nc.math.tsukuba.ac.jp/multidatabases/multidatabase_contents/detail/231/690ddc75a5e0f71509ca6c9adc2044bd?frame_id=147","timestamp":"2024-11-02T04:57:13Z","content_type":"text/html","content_length":"17317","record_id":"<urn:uuid:7fc86242-7e86-4f9f-bae7-01c5d3692459>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00270.warc.gz"} |
Topologies in Decentralized Federated Learning. | AIOZ AI
Data silos are nearly always available in the Cross-Silo configuration, have high-speed connectivity similar to the orchestrator, and exchange information with other silos more quickly than with the
orchestrator. An inter-silo communication architecture focused on orchestrators would be ineffective since it overlooks rapid communication possibilities and makes the orchestrator a congestion
candidate. A recent tendency is to replace communication between separate silos and an orchestrator with peer-to-peer communications. This setup conducts part of the local model update aggregations.
In the section, we examine the mentioned scenario and how to develop the topology of communications.
Currently, their are three main types of topologies are leveraged in Decentralized Federated learning, including: Underlay, Connectivity Graph and Overlay.
Figure 1: Underlay $\mathcal{G}_u = (\mathcal{V} \cup \mathcal{V}', \mathcal{E}_u)$.
FL silos are connected by a so-called underlay, i.e., a communication infrastructure such as the Internet or some private network. The underlay can be represented as a directed graph (digraph). $\
mathcal{G}_u = (\mathcal{V} \cup \mathcal{V}', \mathcal{E}_u)$, where $\mathcal{V}$ denotes the set of silos, $\mathcal{V}'$ is the set of other nodes (e.g., routers) in the network, and $\mathcal{E}
_u$ the set of communication links. For simplicity, we consider that each silo $i \in \mathcal{V}$ is connected to the rest of the network through a single link $(i,i')$, where $i' \in \mathcal{V}'$,
with uplink capacity $C_{UP}(i)$ and downlink capacity $C_{DN}(i)$ (See the example in Figure 1 which illustrates the underlay).
Connectivity Graph
Figure 2: Connectivity Graph $\mathcal{G}_c = (\mathcal{V}, \mathcal{E}_c)$.
Connectivity Graph denotes by $\mathcal{G}_c = (\mathcal{V}, \mathcal{E}_c)$ captures the possible direct communications among silos. Often the connectivity graph is fully connected, but specific NAT
or firewall configurations may prevent some pairs of silos to communicate. If transmission is allowed, the messageexperiences a delay that is the sum of two contributions: 1) an end-to-end delay
accounting for link latencies, and queueing delays long the path, and 2) a term depending on the model size and the available bandwidth. We assume that in the stable cross-silo setting these
quantities do not vary or vary slowly, so that the topology is recomputed only occasionally, if at all.
Figure 3: Overlay $\mathcal{G}_o = (\mathcal{V}, \mathcal{E}_o)$.
Thank to the development of decentralized training algorithm, we do not need to use all potential connections. Hence, the orchestrator can select a connected subgraph of $\mathcal{G}_c$, the
so-called Overlay $\mathcal{G}_o = (\mathcal{V}, \mathcal{E}_o)$. Only nodes directly connected in $\mathcal{E}_o$ will exchange messages. | {"url":"https://ai.aioz.io/guides/decentralized-ai/fl-topology/","timestamp":"2024-11-06T14:10:39Z","content_type":"text/html","content_length":"43837","record_id":"<urn:uuid:04622156-cf6c-447f-9905-8030bbae3001>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00615.warc.gz"} |
ecocrop: EcoCrop model in Recocrop: Estimating Environmental Suitability for Plants
Create and run an EcoCrop model to asses the environmental suitability of a location for a (plant) species.
First create a model object with the ecocrop method. Then set parameters describing the environmental requirements of a species or other taxon. The ecocropPars method provides default parameters for
1710 taxa.
Next, provide environmental data with the staticPredictors and/or dynamicPredictors method. Static predictors, such as soil pH, do not change throughout the year. In contrast, dynamic predictors,
such as temperature and rainfall, vary over time. In the current implementation the time-step of the input data is months. Therefore, dynamic variables must have 12 values, one for much month of the
year, or multiples of 12 values, to represent multiple years or locations. The computations are done in half-month time steps, by interpolating the monthly values.
The names of the predictors much match the names in the parameters, but not vice versa. That is, parameters that are not matched by a predictor are ignored.
The main purpose of implementing the model is to support making spatial predictions with predict.
ecocrop(crop) ## S4 method for signature 'Rcpp_EcocropModel' control(x, get_max=FALSE, which_max=FALSE, count_max=FALSE, lim_fact=FALSE, ...) ## S4 method for signature 'Rcpp_EcocropModel' run(x,
crop list with ecocrop parameters. See link[ecocropPars] and link[crop]
x EcocropModel object
get_max logical. If TRUE, the maximum value (across the time periods of the year) is returned.
which_max logical. If TRUE, the first month with the maximum value is returned.
count_max logical. If TRUE, the number of months with the maximum value is returned.
lim_fact logical. If TRUE, the options above are ignored, the most-limiting factor for each time period (or the one that is reached first if there are ties) is returned.
... additional arguments. None implemented
logical. If TRUE, the maximum value (across the time periods of the year) is returned.
logical. If TRUE, the first month with the maximum value is returned.
logical. If TRUE, the number of months with the maximum value is returned.
logical. If TRUE, the options above are ignored, the most-limiting factor for each time period (or the one that is reached first if there are ties) is returned.
The model computes a score for each variable for the 1st and 15th day of each month. It then takes the lowest (most limiting) score for each time period. After that, the minimum score for the time
periods that follow (the growing season) is computed. The lenght of the growing season is by the duration parameter (see ecocropPars).
You can set the output variables with options. If all options are FALSE, the 24 bi-monthly scores are returned.
# Get parameters potato <- ecocropPars("potato") # create a model m <- ecocrop(potato) # add parameters crop(m) <- cbind(clay=c(0,0,10,20)) # inspect plot(m) # add predictors dp <- cbind(tavg=c
(10,12,14,16,18,20,22,20,18,16,14,12), prec=seq(50,182,12)) t(dp) dynamicPredictors(m) <- dp staticPredictors(m) <- cbind(clay=12) # run model x <- run(m) x y <- matrix(round(x, 1), nrow=2) colnames
(y) <- month.abb rownames(y) <- c("day1", "day15") y dates <- as.Date(paste0("2000-", rep(1:12, each=2), "-", rep(c(1,15), 12))) plot(dates, x, las=1, ylab="suitability", xlab="") lines(dates, x, col
="red") control(m, get_max=TRUE) run(m)
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/Recocrop/man/ecocrop.html","timestamp":"2024-11-06T15:45:46Z","content_type":"text/html","content_length":"24763","record_id":"<urn:uuid:a336d966-0bbc-4dc1-8509-b31d812d7eb5>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00115.warc.gz"} |
American Mathematical Society
Convergence in distribution for randomly stopped random fields
Author: D. Silvestrov
Journal: Theor. Probability and Math. Statist. 105 (2021), 137-149
MSC (2020): Primary 60G60; Secondary 60F05, 60F99, 60G40
DOI: https://doi.org/10.1090/tpms/1160
Published electronically: December 7, 2021
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let $\mathbb {X}$ and $\mathbb {Y}$ be two complete, separable, metric spaces, $\xi _\varepsilon (x), x \in \mathbb {X}$ and $\nu _\varepsilon$ be, for every $\varepsilon \in [0, 1]$,
respectively, a random field taking values in space $\mathbb {Y}$ and a random variable taking values in space $\mathbb {X}$. We present general conditions for convergence in distribution for random
variables $\xi _\varepsilon (\nu _\varepsilon )$ that is the conditions insuring holding of relation, $\xi _\varepsilon (\nu _\varepsilon ) \stackrel {\mathsf {d}}{\longrightarrow } \xi _0(\nu _0)$
as $\varepsilon \to 0$.
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Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2020): 60G60, 60F05, 60F99, 60G40
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Additional Information
D. Silvestrov
Affiliation: Department of Mathematics, Stockholm University, 106 81 Stockholm, Sweden
Email: silvestrov@math.su.se
Keywords: Random field, random stopping, convergence in distribution
Received by editor(s): July 10, 2021
Published electronically: December 7, 2021
Article copyright: © Copyright 2021 Taras Shevchenko National University of Kyiv | {"url":"https://www.ams.org/journals/tpms/2021-105-00/S0094-9000-2021-01160-7/","timestamp":"2024-11-07T03:38:56Z","content_type":"text/html","content_length":"70750","record_id":"<urn:uuid:17d9105a-d329-4264-8a5e-2aa4b22eddcb>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00707.warc.gz"} |
271 research outputs found
We discuss R-symmetry in locally supersymmetric $N=2$ gauge theories coupled to hypermultiplets, which can be viewed as effective theories of heterotic string models. In this type of supergravities a
suitable R-symmetry exists and can be used to topologically twist the theory. The vector multiplet of the dilaton-axion field has a different R-charge assignment with respect to the other vector
multiplets.Comment: Proceedings of ``Susy95'', Palaiseaux, Ecole Polytechnique, May 95 LaTex, 8 pg
Working in the geometric approach, we construct the lagrangians of N=1 and N=2 pure supergravity in four dimensions with negative cosmological constant, in the presence of a non trivial boundary of
space-time. We find that the supersymmetry invariance of the action requires the addition of topological terms which generalize at the supersymmetric level the Gauss-Bonnet term. Supersymmetry
invariance is achieved without requiring Dirichlet boundary conditions on the fields at the boundary, rather we find that the boundary values of the fieldstrengths are dynamically fixed to constant
values in terms of the cosmological constant \Lambda. From a group-theoretical point of view this means in particular the vanishing of the OSp(N|4)-supercurvatures at the boundary.Comment: Some
clarifications on the N=1 case, typos correcte
In these lectures we explain the concept of supergravity p-branes and BPS black holes. Introducing an audience of general relativists to all the necessary geometry related with extended supergravity
(special geometry, symplectic embeddings and the like) we describe the general properties of N=2 black holes, the structure of central charges in extended supergravity and the description of black
hole entropy as an invariant of the U duality group. Then, after explaining the concept and the use of solvable Lie algebras we present the detailed construction of 1/2, 1/4 and 1/8 supersymmetry
preserving black holes in the context of N=8 supergravity. The Lectures are meant to be introductory and self contained for non supersymmetry experts but at the same time fully detailed and complete
on the subject.Comment: LaTeX, 132 pages, Book.sty. Lecture Notes for the SIGRAV Graduate School in Contemporary Relativity, Villa Olmo, Como First Course, April 199
Decomposition of the solvable Lie algebras of maximal supergravities in D=4, 5 and 6 indicates, at least at the geometrical level, the existence of an N=(4,2) chiral supergravity theory in D=6
dimensions. This theory, with 24 supercharges, reduces to the known N=6 supergravity after a toroidal compactification to D=5 and D=4. Evidence for this theory was given long ago by B. Julia. We show
that this theory suffers from a gravitational anomaly equal to 4/7 of the pure N=(4,0) supergravity anomaly. However, unlike the latter, the absence of N=(4,2) matter to cancel the anomaly presumably
makes this theory inconsistent. We discuss the obstruction in defining this theory in D=6, starting from an N=6 five-dimensional string model in the decompactification limit. The set of massless
states necessary for the anomaly cancellation appears in this limit; as a result the N=(4,2) supergravity in D=6 is extended to N=(4,4) maximal supergravity theory.Comment: 15 pages, latex, no figure
We construct a general Lagrangian, quadratic in the field strengths of $n$ abelian gauge fields, which interpolates between BI actions of n abelian vectors and actions, quadratic in the vector
field-strengths, describing Maxwell fields coupled to non-dynamical scalars, in which the electric-magnetic duality symmetry is manifest. Depending on the choice of the parameters in the Lagrangian,
the resulting BI actions may be inequivalent, exhibiting different duality groups. In particular we find, in our general setting, for different choices of the parameters, a ${\rm U}(n)$-invariant BI
action, possibly related to the one in \cite{Aschieri:2008ns}, as well as the recently found $\mathcal{N}=2$ supersymmetric BI action \cite{Ferrara:2014oka}.Comment: 12 pages, LaTeX source, typos
corrected, definition of the matrix S in eq. (3.22) correcte
We thoroughly analyze at the bosonic level, in the framework of Free Differential Algebras (FDA), the role of 2-form potentials setting in particular evidence the precise geometric formulation of the
anti-Higgs mechanism giving mass to the tensors. We then construct the (super)-FDA encoding the coupling of vector-tensor multiplets in D=4, N=2 supergravity, by solving the Bianchi identities in
superspace and thus retrieving the full theory up to 3-fermions terms in the supersymmetry transformation laws, leaving the explicit construction of the Lagrangian to future work. We further explore
the extension of the bosonic FDA in the presence of higher p-form potentials, focussing our attention to the particular case p=3, which would occur in the construction of D=5, N=2 supergravity where
some of the scalars are properly dualized.Comment: 39 pages, improved introduction, section 4 and Appendices modified, typos corrected, citations adde
We study the partial breaking of $N=2$ rigid supersymmetry for a generic rigid special geometry of $n$ abelian vector multiplets in the presence of Fayet-Iliopoulos terms induced by the Hyper-K\
"ahler momentum map. By exhibiting the symplectic structure of the problem we give invariant conditions for the breaking to occur, which rely on a quartic invariant of the Fayet-Iliopoulos charges as
well as on a modification of the $N=2$ rigid symmetry algebra by a vector central charge.Comment: 7 pages, LaTeX sourc
The c-map of four dimensional non-linear theories of electromagnetism is considered both in the rigid case and in its coupling to gravity. In this way theories with antisymmetric tensors and scalars
are obtained, and the three non-linear representations of N=2 supersymmetry partially broken to N=1 related. The manifest $\mathrm{Sp}(2n)$ and $\mathrm{U}(n)$ covariance of these theories in their
multifield extensions is also exhibited.Comment: Version to appear on Physics Letters | {"url":"https://core.ac.uk/search/?q=author%3A(D'Auria%2C%20Riccardo)","timestamp":"2024-11-02T21:20:55Z","content_type":"text/html","content_length":"146691","record_id":"<urn:uuid:24ce109f-6268-404d-a44e-17e764f708a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00322.warc.gz"} |
Muhammad Zikril Hakim bin Zulkifly - MATLAB Central
Muhammad Zikril Hakim bin Zulkifly
Universiti Malaya
Last seen: 8 maanden ago |  Active since 2024
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An accomplished graduate engineer from the esteemed University Malaya, specializing in Electrical Engineering, I bring a passion for innovation and a drive for excellence to my profession. With a
keen interest spanning across diverse sectors including the oil and gas, semiconductor, energy, and robotics industries, I thrive on challenges that push the boundaries of technological advancements.
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Simulink, Simscape Electrical, MATLAB Coder
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Determine for a magic square of order n, the magic sum m. For example m=15 for a magic square of order 3.
8 maanden ago
Make a random, non-repeating vector.
This is a basic MATLAB operation. It is for instructional purposes. --- If you want to get a random permutation of integer...
8 maanden ago
Roll the Dice!
*Description* Return two random integers between 1 and 6, inclusive, to simulate rolling 2 dice. *Example* [x1,x2] =...
8 maanden ago
Number of 1s in a binary string
Find the number of 1s in the given binary string. Example. If the input string is '1100101', the output is 4. If the input stri...
8 maanden ago
Return the first and last characters of a character array
Return the first and last character of a string, concatenated together. If there is only one character in the string, the functi...
8 maanden ago
Create times-tables
At one time or another, we all had to memorize boring times tables. 5 times 5 is 25. 5 times 6 is 30. 12 times 12 is way more th...
8 maanden ago
Getting the indices from a vector
This is a basic MATLAB operation. It is for instructional purposes. --- You may already know how to <http://www.mathworks....
8 maanden ago
Check if number exists in vector
Return 1 if number _a_ exists in vector _b_ otherwise return 0. a = 3; b = [1,2,4]; Returns 0. a = 3; b = [1,...
8 maanden ago
Swap the first and last columns
Flip the outermost columns of matrix A, so that the first column becomes the last and the last column becomes the first. All oth...
8 maanden ago
Swap the input arguments
Write a two-input, two-output function that swaps its two input arguments. For example: [q,r] = swap(5,10) returns q = ...
8 maanden ago
Column Removal
Remove the nth column from input matrix A and return the resulting matrix in output B. So if A = [1 2 3; 4 5 6]; ...
8 maanden ago
Reverse the vector
Reverse the vector elements. Example: Input x = [1,2,3,4,5,6,7,8,9] Output y = [9,8,7,6,5,4,3,2,1]
8 maanden ago
Select every other element of a vector
Write a function which returns every other element of the vector passed in. That is, it returns the all odd-numbered elements, s...
8 maanden ago
Length of the hypotenuse
Given short sides of lengths a and b, calculate the length c of the hypotenuse of the right-angled triangle. <<https://i.imgu...
8 maanden ago
Triangle Numbers
Triangle numbers are the sums of successive integers. So 6 is a triangle number because 6 = 1 + 2 + 3 which can be displayed ...
8 maanden ago
Generate a vector like 1,2,2,3,3,3,4,4,4,4
Generate a vector like 1,2,2,3,3,3,4,4,4,4 So if n = 3, then return [1 2 2 3 3 3] And if n = 5, then return [1 2 2 3 3 3 4...
8 maanden ago
Make the vector [1 2 3 4 5 6 7 8 9 10]
In MATLAB, you create a vector by enclosing the elements in square brackets like so: x = [1 2 3 4] Commas are optional, s...
8 maanden ago
Finding Perfect Squares
Given a vector of numbers, return true if one of the numbers is a square of one of the numbers. Otherwise return false. Example...
8 maanden ago
Maximum value in a matrix
Find the maximum value in the given matrix. For example, if A = [1 2 3; 4 7 8; 0 9 1]; then the answer is 9.
8 maanden ago
Find the sum of all the numbers of the input vector
Find the sum of all the numbers of the input vector x. Examples: Input x = [1 2 3 5] Output y is 11 Input x ...
8 maanden ago
Add two numbers
Given a and b, return the sum a+b in c.
8 maanden ago
Find the peak 3n+1 sequence value
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8 maanden ago
Determine whether a vector is monotonically increasing
Return true if the elements of the input vector increase monotonically (i.e. each element is larger than the previous). Return f...
8 maanden ago
Convert from Fahrenheit to Celsius
Given an input vector F containing temperature values in Fahrenheit, return an output vector C that contains the values in Celsi...
8 maanden ago
Times 2 - START HERE
Try out this test problem first. Given the variable x as your input, multiply it by two and put the result in y. Examples:...
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What is “overfitting,” exactly?
This came from Bob Carpenter on the Stan mailing list:
It’s not overfitting so much as model misspecification.
I really like this line. If your model is correct, “overfitting” is impossible. In its usual form, “overfitting” comes from using too weak of a prior distribution.
One might say that “weakness” of a prior distribution is not precisely defined. Then again, neither is “overfitting.” They’re the same thing.
P.S. In response to some discussion in comments: One way to define overfitting is when you have a complicated statistical procedure that gives worse predictions, on average, than a simpler procedure.
Or, since we’re all Bayesians here, we can rephrase: Overfitting is when you have a complicated model that gives worse predictions, on average, than a simpler model.
I’m assuming full Bayes here, not posterior modes or whatever.
Anyway, yes, overfitting can happen. And it happens when the larger model has too weak a prior. After all, the smaller model can be viewed as a version of the larger model, just with a very strong
prior that restricts some parameters to be exactly zero.
120 thoughts on “What is “overfitting,” exactly?”
1. Perhaps, we also need a broader method-level equivalent. Method-level misspecification is behind “clinical trials are broken” & similar misapplications of statistics.
2. Overfitting also comes from using an inference procedure that gives a point estimate at a maximum fit point.
Even many Bayesian estimates with good priors overfit at the MAP as I’ve discovered when I try to shortcut long Stan runs using optimization.
□ Interesting thought. If you do an optimization, along the way towards the peak, you find out something about the distribution. The general idea that the typical set is the set with log
probability density nearly some constant, can we estimate that constant from the path history on the way to the peak? Then perhaps draw some small number of samples in the vicinity of
whatever step was closest to the typical set on your way to the peak?
The reason I ask is that if this could be made into a say 2 minute procedure, then for those of us who have models that take something like 25 hours to run to full high quality sample, we
could debug models much more quickly.
I’m already informally doing this by watching stan spit out the samples and waiting until it calms down in terms of changing __lp and then killing it and looking at those few final samples to
see if things make sense. This can sometimes take something like the few minutes, whereas a high quality sample might take an hour or 3. If things are way-off by the time I get 3 or 5 samples
with “constant” __lp I can then go ahead and do things like rework the model to make things make better sense.
☆ I haven’t read the paper, but this seems quite related https://arxiv.org/abs/1704.04289.
☆ “The general idea that the typical set is the set with log probability density nearly some constant, can we estimate that constant from the path history on the way to the peak? Then
perhaps draw some small number of samples in the vicinity of whatever step was closest to the typical set on your way to the peak?”
You can do this, but not the way you describe. This is essentially how nested sampling works, but then you only get a few points from the typical set which makes function expectation
estimates highly imprecise (even assuming that the rest of nested sampling is working ideally, which can be a stretch in practice).
○ Yeah, I think this is ok for debugging purposes. I mean for example suppose you want some big vector parameter to encode some function, and you have some idea of what that function
should look like, you provide a prior, attempting to encode your knowledge, and you run Stan in this way to get some small number of samples. If the function doesn’t look anything
like what you thought it was going to, you know you made a mistake encoding your information… figuring this out after 3 minutes instead of 3 hours is great!
3. What does overfitting look like in its usual form in a Bayesian setting? The only way to overfit that I can think of (but I’ve not thought much about it) is by changing the prior to “fit” the
□ When there’s very little theory to go on, I think it’s also very possible to rework the likelihood too much. For example a simple regression with a universal approximator (say a basis
expansion, or splines or whatever) or throwing in more and more covariates trying to chase your posterior predictive standard deviation down to zeroish.
☆ Changing the prior or the likelihood is more or less the same: tweaking the model. But the problem is not that the prior was weak to start with, the problem is modifying the model to make
it fit better.
○ I think this is exactly right. Because Bayesian procedures correctly account for uncertainty, a single application of a Bayesian procedure will not overfit. But if we keep changing
our model, then we overfit via the garden of forking paths, which is not properly accounted for in any individual execution of a Bayesian procedure. (There is work in adaptive data
analysis that seeks to account for this. Has anyone been following that?)
Overfitting in the classical setting arises from choosing a model class that is too flexible and then selecting only one fitted model from that class. Such estimators have high
variance, and the resulting error is what we call “overfitting” (because it usually results from fitting the noise in the data as well as the signal). Of course, cross-validation,
when applicable, can provide a way of adjusting the model class complexity to match the data complexity, but it is only approximate.
We can use holdout data to detect overfitting. But one can of course fork paths by consulting the holdout data many times. An intriguing idea from Cynthia Dwork is to use differential
privacy to create a reusable holdout data set. My understanding is that that reduces but does not eliminate the overfitting risk.
Ultimately, there is no purely statistical/computational magic. We must rely on doing additional experiments to gather more data if we hope to approach the truth.
○ I don’t agree that “Bayesian procedures correctly account for uncertainty”. It’s just a method for updating probability distributions. It’s not a guarantee the resulting distribution
is sensible.
A friend of mine refuses to read or watch anything with wizards in it. He calls them “literary mcguffins”.
○ I think what’s guaranteed is that if the model is sensible, the posterior is sensible. Bayesian updating is in some ways a sensibility filter.
○ Yes. But, to paraphrase Dorian Corey, that’s not a guarantee, that’s a fact.
My pet hate is people thinking that computing some approximation to a posterior means they’ve accounted for uncertainty.
○ I don’t think any statistical methodology can totally account for uncertainty; statistical methods are means for *trying* to account for uncertainty. Phrases such as “we have
accounted for uncertainty” or “we have controlled for …” (in contrast to phrases like “we have attempted to account for uncertainty by …” or “we have attempted to account for …”) just
lead people to believe that statistical techniques can do more than is realistically possible.
○ Yesssssss!
○ +1 Martha
I try to get this idea across to my students (undergrads, 1st or 2nd semester stats) by pointing out how absurd it would be to take the contents of a model literally. For instance, in
simple regression we are supposing that our data came into being through a procedure in which numbers are randomly plucked from a normal distribution and then added to a perfectly
straight line. This is preposterous! I’m not aware of all the forces that combined to create the measurements I see in some spreadsheet, but I know it wasn’t “straight line plus
normal error”. And we aren’t supposed to believe that it was, we just think that this extreme simplification of reality will help us answer questions we are interested in.
And as you note, the big problems come when people sincerely believe that, for instance, a regression coefficient tells them “the effect of X on Y while controlling for Z”, and that
all of the uncertainty in this statement is quantified by the standard error. I try to teach my students to see standard errors and the like as lower bounds on uncertainty. They give
us an estimate of uncertainty under the foolish assumption that our models are real.
○ I guess I need to be more careful about my wording. The Bayesian procedure correctly transforms the uncertainty of the prior into the uncertainty of the posterior under the assumption
that the likelihood model is correct. It does not quantify or account for our uncertainty about our modeling choices (beyond those that can be captured by the prior). Indeed, I
believe such an accounting is impossible in a finite model.
□ Overfitting with non-informative priors using MCMC looks exactly the same overfitting using MLE methods: for example, if a model has a Gaussian error term with unknown variance, which can be
perfectly (over) fit in sample, then the posterior distribution of the error term is degenerate at sigma = 0.
☆ I guess it depends on what we mean by overfitting. If I fit a model y=a+b*x+error with two data points y(x=1)=1 and y(x=2)=2 using least squares I clearly overfit. The predictions I get
from the model are precisely the same as the data (or the interpolation/extrapolation for other values of x). But maybe the true model was a=b=0 (and the data was just noise). Using
Bayesian inference I get a posterior predictive distribution that is quite wide (and the less informative the prior, the more diffuse the model prediction).
○ Is that really the case?
You have an unbounded likelihood as a, b approach the line given by the data and sigma approaches 0. So as long as you have non-negative density at 0 for sigma in your prior, the
posterior density is unbounded at that single point, and differentiable in neighborhood around that point.
Of course, that doesn’t necessarily mean you have a degenerate posterior distribution; something like a Weibull with shape less than 1 has unbounded density at 0 but is not
degenerate. I don’t quite recall what is mathematically required to ensure that it will not be degenerate (outside that a neighborhood around a point must have a finite integral). But
at the very least, I think Stan should fail; one of the basic assumptions of most MCMC algorithms is a bounded posterior density (and I think Stan’s algorithm is no different).
○ an optimization procedure that encounters an unbounded density is in trouble, but I don’t think that’s true for MCMC. For example, just try drawing via MCMC from beta(1/2,1/2).
The main issue with unbounded density is that it needs to be normalizable integral(f(x),dx, all X) = 1
○ Stan uses transformations to take parameters supported on intervals or the positive reals to the entire set of reals. Even if the density grows without limit as it approaches a
boundary of the support set, as long as the density in the original parameterization is normalizable the transformation will inevitably flatten out. If the divergence isn’t on the
boundary I imagine Stan will choke on the discontinuity.
4. Is poor generalization a characteristic of or the definition of overfitting? Is overfitting a subset or congruent to poor generalization? (This is mostly a semantics question.)
□ Poor generalization is one result of using a high-variance point estimate. So while it is perhaps not the definition of overfitting, it is certainly a consequence.
5. I don’t see why a weak prior leads to overfitting. It will just lead to a very broad posterior distribution. A weak prior combined with MAP estimation will definitely result in overfitting.
Overfitting occurs when we use a high-variance point estimate to make decisions. I suppose it could also result if we did a poor job of sampling the posterior so that we failed to properly assess
our posterior uncertainty. That is not a point estimate, but it is on the spectrum between a proper full posterior and a point estimate.
6. > Overfitting is when you have a complicated model that gives worse predictions, on average, than a simpler model.
Doesn’t the concept of overfitting include somehow the idea of fitting too well the observed data, while failing to predict with the same precision data not included in the analysis?
□ How about something like overfitting: matching your current dataset more accurately than your current dataset matches future datasets or datasets not included in the current set
7. My working definition of an “overfitting” procedure (that we used in the PC priors paper) is that it’s one where, in the absence of data, the procedure prefers a more complex part of model space
to a simpler part. (It’s our “Informal Definition 1 here https://arxiv.org/pdf/1403.4630.pdf. Reviewers hated it, but it’s the correct concept)
□ The more I think about it, the more I don’t think you need a concept of “data” to define overfitting.
In moderate-to-high-dimensions, a N(0,sigma^2 I) prior on regression coefficients will overfit because with the sum of squares of hte regression coefficients will be strongly concentrated
away from zero a priori, which precludes the simplest model of them all being zero. Whereas a horseshoe prior has a non-negligible mass around all of them being zero (at least the Finnish
So I don’t think Andrew’s idea of “on average fitting new data worse than current data” is the simplest form of the definition. It’s certainly true, but you need a data-generating mechanism
to make it work…
☆ > you need a data-generating mechanism to make it work…
You could always use the external world to provide you datasets, right?
○ Not if you want to flag the problem before analysis, rather than hoping you can catch it after.
○ Sure, but then you could use eg train/test/validate splits etc to mimic this right?
○ I suppose the usual reply is that this ‘is a modelling assumption’, but it seems a bit forced to me to describe/motivate it in these terms
○ I don’t think it’s one or the other. Check your model before you see data AND validate your model after. Otherwise you’re asking for trouble.
Sometimes one of these is easier than the other, but they should both be done as well as you can.
○ I guess I don’t really think you can check a model without data, unless against eg physical constraints like ‘does it violate conservation of [whatever] or invariance under [some
Preference for simplicity over complexity should, it seems to me, be motivated by something like this or by predictive/robustness concerns about what other data might look like.
○ Good talked about the “Device of imaginary results”, which is just that.
To some extent, pre-data model validation is a step in the modelling process, but it’s useful to make it explicit.
I don’t really agree with the idea (is it Jaynesisn? I don’t care that much about dead statisticians to remember which one is which) physical constraints or invariances are enough to
set priors in realistic models. If there are obvious ones they should be respected, but for even moderately complex models, deriving them all and then working out to incorporate them
seems a Herculean task.
So I guess the question I have is for what types of experiment is “prefer simplicity” not, pre data, a sensible prior state?
○ > the question I have is for what types of experiment is “prefer simplicity” not, pre data, a sensible prior state?
I would say the question is relative to what sort of data are available. Most statistical procedures (including Bayes, according to me contra Lindley) only really work given
identifiability constraints.
There is only so much you can estimate. A complex model (family) just means there will be lots of equivalent models that you can’t tell apart. So you may as well restrict to models
that you can tell apart.
○ But how do you do that?
And then given that you’ve restricted to identifiable submanifolds, how do you set priors?
○ > the question I have is for what types of experiment is “prefer simplicity” not, pre data, a sensible prior state?
My first inclination would be to think of systems where simplicity emerges from averaging over disorder. It is not a typical system for Bayesian inference but spin glasses (https://
en.wikipedia.org/wiki/Spin_glass) fit the bill. They have a simple phase transition that emerges from a randomly-coupled model. In this case, the phenomenon is quite simple and
physically realistic but emerges precisely because of the high entropy/complexity of the parameters. I admit this is an unusual system, although it is pretty good way to think about
things like emergent behavior of social graphs, where you would similarly expect apparently random coupling between individuals.
○ > But how do you do that?
You mean determine the effective number of parameters that can be estimated given the data? The usual ways for estimating complexity (or whatever) parameters would be a start, right?
Training/test, CV, empirical Bayes, hierarchical Bayes, profiling etc etc.
> And then given that you’ve restricted to identifiable submanifolds, how do you set priors?
Probably doesn’t matter too much at this point, but I suppose however you normally set priors?
○ ( a key point required for my answer to not contradict my others is that imo you need a concept of identifiability, or overfitting…, in order to estimate a complexity parameter. So
you can use standard tools but the goal is not quite the same as standard estimation assuming identifiability)
○ (Obviously there’s no such thing as an objective, assumption free analysis. So you need to know what the properties of your inference procedure are as much as possible *before* you
see data)
☆ I don’t buy your definition of overfitting, because it seems like you are conflating overfitting, a concept that always needs data, with the probability of overfitting, which you
approximate fairly reasonably. It seems like the natural endpoint of your argument is Solomonoff induction, but there are many low Kolmogorov complexity models that don’t have many zero
○ Yeah we depart on the first sentence there. I just don’t agree when you’re building a Bayesian model the correct thing to look at isn’t whether or not the model can overfit.
And I don’t see a general way (imagine dependent data to break Andrew’s definition) of checking if overfitting has occurred.
I also don’t quite know what low Kolmogorov complexity models that aren’t zero have to do with it.
And I think that if you’ve followed an argument all the way to Solomonoff induction you’ve gone too far. It’s like following your satnav into a lake.
○ I laughed out loud on the satnav joke. My point was just that counting zeros is a rather arbitrary way to define model complexity, and you seem intent (quite reasonably) on spreading
your prior mass evenly in model complexity.
“Can overfit” still implies that overfitting is a property of data+model not just the parametric model (otherwise you should say that a model does overfit for N points regardless of
the data). We are getting a bit semantic at this point, though.
Dependent data doesn’t necessarily break Andrew’s definition, so long as you hold out a portion of it for validation (very rarely is this impossible given the structure of the model).
A Bayesian model that doesn’t allow held-out data and makes no predictions about future data seems rarely useful.
○ I’m not trying to spread mass evenly anywhere! A priori, I want it to be exponentially unlikely to have a model that’s much more complex than the base model. It’s nit counting zeros:
it’s just that for some model components, zero is a sensible base state. Sometimes it’s something else.
Regarding data+model, you need to allow for the case where the data *needs* one of the higher complexity models to describe it. What I want the prior to do is ensure it’s the data
calling the shots, not the prior. if there’s low information in the data informing the model component in question, the prior won’t wash out. So you need some sort of argument for the
Hold out validation is really hard in a lot of problems. (Point processes, for example. Holding out leans very strongly on your stationarity assumption) Hold out methods work well
when you can break your data into approximately exchangeable components. Otherwise you’ve typically your to think of something else!
○ “Regarding data+model, you need to allow for the case where the data *needs* one of the higher complexity models to describe it. ”
This statement sounds like you are talking about “the data that have been collected,” which doesn’t make sense to me in context. Is what you are really talking about “the type of
situation from which you are collecting what type of data”? That would make sense to me.
○ The one and future data. This is stats, so I’m assuming that at some point in this process there is data and that’s the data I’m talking about.
8. Perhaps a naive question, but is there a nice mathematical characterization of model mis-specification when using full Bayesian inference? In this case, I am thinking of how the maximum
likelihood estimator minimizes the KL divergence of the empirical distribution against the model, which characterizes what happens when the true data generating process is not representable in
the model.
In the infinite data limit on a finite model, Bayesian inference must recapitulate maximum likelihood and therefore also minimize the KL divergence. Are there any useful mental models of the
optimal posterior in the finite data case where the prior is still important but the true data generating process is not representable?
□ > In the infinite data limit on a finite model, Bayesian inference must recapitulate maximum likelihood and therefore also minimize the KL divergence.
FWIW This is not actually true for infinite-dimensional parameter spaces. In these cases your prior typically matters forever ever ever….(I don’t think Bayesians care too much for whatever
☆ Oh you explicitly said finite dimensional. My bad
☆ Bayesians care.
But with infinite dimensional parameters modelling choices/ estimator choices tend to persist whatever your framework.
This is sort of obvious: infinite parameters, finite data. The best you can hope for is that the stuff you care about is resolved in asymptotia. (Because even with infinite data you only
see the process “once”, so you need some asymptotic assumptions to bring in some sense of replications)
☆ Infinite dimensional models definitely have a new set of issues. The downside is that the prior lives on forever, but the upside is that in some cases the data generating process is
always representable. I am thinking of Dirichlet process mixtures of gaussians as always converging in the infinite data limit for smooth densitities (I assume), although so does kernel
density estimation.
○ The point I was trying to make is that both bayesians and non-bayesians have to make really strong, uncheckable assumptions to estimate infinite dimensional parameters. So persistence
of these assumptions isn’t just a Bayesian issue
○ I think that is a fair characterization.
☆ “Infinite dimensional” models obviously don’t really exist. They’re best thought of as a family of models in which some sufficiently large N is sufficient to describe the real scientific
thing to within some measurement ability. Typically “infinite dimensional” models are actually WAY WAY lower dimensional than reality. Think for example Navier Stokes via finite elements
or something. The true model is 3 positions and 3 velocities for 10^28 molecules, but the “infinite dimensional” model is approximated as say 800 finite elements with several dimensions
per element. a factor of 10^26 dimensional reduction
○ I don’t understand your point. If you approximate a GP by 50 Fourier features, or a DP mixture by a mixture of 3 components the prior washes out asymptotically. So what?
○ So no one actually uses “infinite dimensional” models, in fact, they are almost always ways of *reducing* model complexity.
○ For example, you’re doing an annual time-series analysis. You could:
1) Use dummies for each day of the year (365 parameters to estimate)
2) Use a GP with a squared exponential covariance function, with a single time-scale parameter and a single covariance degree, and a noise term, (3 parameters to put priors on)
The second one is “infinite dimensional” but has actually substantially fewer parameters to estimate.
○ I’m pretty sure the second model here has a fourth parameter that’s infinite dimensional.
○ Not any more so than y[i] = exp(-t[i]/s) + epsilon[i]
Most people look at this and say “it has one parameter, the scale s” but of course, there’s also all the individual values of any number of epsilons. If you put an independent normal
on epsilon, you are in fact specifying a nonstandard gaussian process for the errors (a white noise).
The purpose of a GP is to specify a lot of information with just a small number of unknowns, the 3 covariance function quantities.
The purpose of Navier Stokes is to get a solution with just a few hundred finite elements instead of modeling the molecular interactions of 10^27 molecules.
The purpose of continuous functions is to not have to worry about all the more realistic details you’d get in terms of giga-samples-per-second discrete time sampling process.
Infinite dimensionality is best thought of as a computational rule for generating a finite dimensional object of whatever dimension is suitable for your purposes. every continuous
function is a point in an infinite dimensional hilbert space, in this sense classical least squares on 3rd degree polynomials is an “infinite dimensional” model too.
○ – We need to teach these people to count! :p I see 365+365+1
– I’m old fashioned in requiring a GP has a continuum parameter space, in which case iid standard normals is a bad definition of white noise.
– there are still four unknowns. One of them is the sample path of the GP. How big it is is up for grabs (O(log n) with fixed parameters, O(n^a) for the right priors etc), but that’s
the point. Your prior in the parameters controlling the GP tells you how big that infinite dimensional parameter “really” is. You need to decide if that should be near the number of
observations (n) or if it should be significantly smaller and this information should be encoded into the prior. In some situations, having this effective dimension close to n is
overfitting, but in others (e.g. Almost perfectly measured data) it may not be.
– I haven’t used my fluid dynamics in a while, but that’s quite a striking notion of the purpose of Navier-Stokes
– I don’t know if I agree with this value judgement about continuous functions.
– this is incorrect. It’s a four dimensional set of models that form a linear sub space of that particular Hilbert space. Assuming, of course, that you drag the inner product along
with you. Otherwise it’s a topological sub space but not a Hilbert subspace.
○ I don’t think we disagree on the way the models are used, I think we simply disagree on the implications for the existence of the infinite object. If you will kindly calculate the 10^
10^10^10^10^10^10^10^10^10 th digit in the decimal expansion of pi I will admit its existence in the world.
○ Boring.
○ Here’s where I see all this going. Every scientific model provides some rule for generating predictions. Imagine it as a lambda calculus expression. The rule can be applied to any
number of scenarios. When the rule is probabilistic, you can imagine it augmented with an arbitrarily long sequence of random binary digits from which it generates random numbers. In
this sense, every model is capable of doing an arbitrary number of calculations, if you apply it to an arbitrarily large number of questions. But in actual fact, it will only ever be
used to do some finite number of calculations. In most cases when people choose to use “infinite dimensional” models, its to *reduce* the kolmogorov complexity of the model. The
structure imposed by the infinite dimensional model makes fewer outcomes possible and the lambda expression shorter.
The finiteness of the number of questions asked of the model is very relevant to the goodness of the model. A model capable of extrapolating to before the big bang which is never
asked to do so is not worse because it would give nonsense in that scenario. If your Gaussian process was chosen for the purpose of modeling a particular 365 day dataset, and is never
asked to generate any more than 365 dim vectors, its hard to see in what way its anything other than a 365 dimensional model specified in a kolmogorov complexity lower than the dummy
variable approach.
○ I see elsewhere you mention you have limited background on Kolmogorov Complexity. So, in this sense, just think of the complexity as the length of the smallest lambda expression that
computes the same model. The point of something like a GP is that by specifying the covariance function, using just a few symbols, something like c(a,b) = s*exp(-((a-b)/l)^2)+n you
have encoded all of the information needed to do the computation. Whereas for something like the dummy variables approach with say independent priors on each coefficient, you need to
specify a prior over each dummy coefficient, in terms of 365 n digit numbers for locations and 365 n digit numbers for scales, and maybe some number of parameters for shape parameters
at each point.
The infinite dimensional thing is just a mental device for getting you a compact lambda expression for your finite dimensional model. It provides a structure to generate compact
models with N, the number of data points, a free parameter, which nevertheless in every actual usage gets bound to some actual finite value.
○ Sure. But that’s a different model. Sometimes wildly different. People use infinite dimesionla models usually because they’re easier than other things. You can build principled
approximations to them, but it’s hard to work out how that approximation propagates and only really exists for fairly easy models.
But I’m still not sure why you’re making this point. If the dimension of your model is as large or larger than your data, which is a common situation, it might as well be infinite.
And to pull it back to the topic, in these cases you need to be careful with your modelling or else you will overfit. (Whatever that means)
○ > But I’m still not sure why you’re making this point. If the dimension of your model is as large or larger than your data, which is a common situation, it might as well be infinite.
Agreed. Infinite dimensional I’ll-posedneas is a good indication of finite but large dimensional I’ll-conditioning.
○ Stupid phone autocorrect…ill-posed…etc
○ > “Infinite dimensional” models obviously don’t really exist.
I disagree. Prove me wrong.
○ This is shorthand for “if the state of the universe is even representable as some mathematical object, this object has some enormous but finite number of dimensions”. I think this is
axiomatic for me as there’s no real way to verify this.
However, there’s a stronger sense of this, since models are made by humans. No individual human will ever carry out a calculation on an infinite number of data values or with an
infinite number of parameters. And if you disagree with that, then I think we really do have fundamentally different views of the world.
○ What if I calculate the real-valued result of a problem involving continuous functions by computing with rational coefficients of a real-valued polynomial?
> I think we really do have fundamentally different views of the world.
I think I’d probably agree ;-) In my more philosophical moments I prefer continuity to discreteness. Perhaps just for aesthetics.
○ We may not be as far apart as you think. I just found this paper:
See page 40 (as well as the rest of the paper, seems really interesting and good overview of some ideas) on the link between intuitionism and NSA in which they describe the
relationship between a nonstandard number in IST and the idea of a non-constructable number.
For example suppose you can prove in non-constructive way that Goldbach’s conjecture is false, but for *every computable number* it’s true. then in some sense “in the real world” all
the numbers satisfy Goldbach’s conjecture, and only “theoretical” numbers inaccessible to any construction technique may violate it.
This divides the world into naive integers that satisfy goldbach and are constructive, and non-naive integers that don’t. This is essentially the same partition as “standard” and
So, the idea if IST in which some of the real numbers/integers etc are standard and some are not, but they’re all just real numbers… this has a flavor of intuitionism /
constructability. I’m very much in favor of treating mathematics algorithmically.
I don’t dislike continuous math, I just like it as an interpretation over a set of algorithms indexed by N, the fineness of a grid. The “continuous” concept is just a function of N,
where N is free, which when applied to N returns a particular discretization. The nonstandard view is then that for all N sufficiently large (nonstandard), the results are the same to
within the power of algorithmic construction to distinguish.
○ Thanks.
Funnily enough, as discussed previously I think, when it comes to resurrecting infinitesimals I prefer so-called smooth infinitesimal analysis to NSA ;-)
PS at what point do you think Andrew bans us for treating his blog as a miscellaneous math/stat/whatever forum. He’s remarkably tolerant…
○ You are both not too wrong: continuity is required for possibilities but does not apply to actualities.
Daniel’s Goldbach’s conjecture is a nice example – there are “can be” numbers that violate it but you won’t encounter any that “are” violations. As Pierce put it existence breaks the
continuum of possibilities.
☆ Also, is KL divergence really a desirable way to measure distances, especially in the context of misspecification and finite information? I’m pretty dubious (for whatever that’s worth!)
○ Why do you think that?
It’s definitely not a distance, but it is a measure of how much complexity would be not accounted for if a model was replaced by the base model, which makes sense for me as a measure
of how far apart two models indexed by a single parameter are.
But I’d love to hear your thoughts!
○ Correct me if I’m wrong, but two models can be arbitrarily far apart in terms of KL but generate datasets that are arbitrarily close in terms of Kolmogorov distance.
If you e.g. generated data, rounded it to some sig fig they would be the same. So you have to really believe your model to distinguish two models that generate the same dataset.
Of course you could add additional regularity conditions to make this less of an issue but you could also just use a different ‘distance’.
○ Sometimes the difference between information theory metrics and other metrics can even be practically important: https://arxiv.org/pdf/1701.07875.pdf.
○ Thanks for the link :-)
I definitely do think the issue is practical. Bayesians tend not to agree but people like eg Huber, Tukey etc from robust stats and data analysis also think/thought it was practical.
It’s not surprising that the paper you link comes from the machine learning world – at least some of them seem to have a fair amount of influence from and/or overlap with those areas
of stats. Again, someone like Vapnik also comes to mind too.
○ Hmm… this is an interesting paper. I suspect that the non-existence of a density is however an indicator of model misspecification (that is, if your problem occurs on a low
dimensional manifold embedded in some higher dimensional space, then it’s your excess of dimensions thats the issue, not the lack of density). As for the use of various divergences/
distances I see these as measures of how well a theoretical frequency distribution fits an observed frequency distribution. That some measures of this goodness of fit are better than
others for getting models that make sense is no different than for example using a gamma distribution for the errors in your regression when they have some skewness, or a t
distribution when they have outliers, instead of say a normal.
The role that the metric / pseudo-metric plays is a comparison between model and data. If I compare two people by the color of their eyes, it’s no surprise that I can’t distinguish
when one of them is rude and the other is kind and caring. In general, measures of comparison are important because they describe what it means to be a good fit vs a bad fit, in other
words, they describe what it is you’re modeling.
○ Yes, the model is definitely misspecified in the technical sense. You need a pretty complex model since the data generating process is basically “pixels from a picture of someone’s
room”. You can imagine that there is a very large excess of dimensions but identifying the hidden manifold is the whole point of the exercise.
○ Right, but by respecifying the model as “Here is a computer program that takes N numbers as inputs, and it outputs a K x K grid of pixels” and then specifying a prior over the numbers
a[i] ~ normal(0,1) or whatever you like, and specifying a comparator C(Image, GeneratedImage) in terms of an appropriate information theoretic metric, and giving a probability
distribution over the output of the comparator function (a “likelihood”), we can infer a density over the N numbers which, in the high probability manifold of the a values produces K
x K grids of pixels that are not bad approximations to the image we’re “compressing”
However, the sense in which it’s “not bad” is the sense in which C(Image, GeneratedImage) is in the high probability manifold of the “likelihood”. If that happens to correspond to the
sense in which “this looks like the picture to me” then your choice of C and your choice of “likelihood” also corresponded to the theory “when C is in the high probability region of
my likelihood, the picture will “look like the original”.
As I said, if you compare people by their eye color you will find that the “similar” people may in fact have wildly different personalities. On the other hand, if you compare them by
closeness in Meyers-Briggs scores, they might be pretty similar personalities, but the eye colors could vary widely.
○ I know literally nothing about kolmogov complexity. Do you mean that for any M and epsilon>0, I can find 2 models f and g with M< KL(f,g) < infinity such that if x~f and y~g are two
samples , then d(x,y) < epsilon?
This seems weird to me… do you have a ref for me to read?
○ KL means Kullback-Leibler divergence. But yes, there are very weird things mathematically that occur when you admit infinite dimensions. For example the indicator function on the
irrational numbers differs from the constant function y=1 at a countably infinite set of points, but the Lebesgue integral between any two real numbers is the same.
○ I think my problem is more that I’m not sure what the kolmogov distance between two data sets is.
(My measure theory is up to snuff, but google is failing me. Is it the L-infinity norm between empirical CDFs?)
○ Incidentally, of we’re trading “measure theory facts that blew my mind back in the day”, my favourite will always be that if mu is a Gaussian measure on an infinite dimensional
compact Hilbert space (for simplicity), the H(mu), it’s reproducing kernel Hilbert space (or Cameron-Martin space) is the space where, given some noisy Gaussian observations of the
process, the posterior mean will lie. The prior probability (and hence posterior probability) that any realisation of the process is in H(mu) is zero.
So the MAP predictor (or posterior mean) is always smoother than any realisation from the process!
○ > Kolmogorov complexity
Not that…
> Is it the L-infinity norm between empirical CDFs?
Yup that. The basic idea is that the KL is a strong ‘metric’ (or whatever it’s really called) while the Kolmogorov is a weak one. You get to densities from distributions via unbounded
operator: differentiation.
○ Ah ok. That doesn’t worry me so much. I’m happy with the strong topology
○ Even if the resulting datasets (actual numbers) can come out the same to whatever sig fig under models arbitrarily far apart in the strong topology?
I mean, tbh I’m happy for people to be happy with the strong topology, but personally I’m not (anymore).
○ All you’re saying is that a finite data set can’t always tell between two models. That’s vacuously true, so it doesn’t bother me. In the context, these aren’t arbitrary models, it’s a
(typically) one dimensional sub-manifold of a well behaved model space. And the choice that has been made is that if there are two indistinguishable models, we’ll go with the simplest
○ ‘All you’re saying…vacuously true’
Sometimes these sort of points are important points.
As far as I can tell you have no theoretical justification for preferring simplicity. The ideas can be analysed in terms of empirical prediction, regularisation theory etc which
attempt to understand when and why we might prefer simplicity to complexity and when not. See eg Vapnik for one attempt.
Fine if you want to start from particular axioms and treat all exceptions as vacuous truths, but that’s not really appealing to me personally.
○ And of course you only find out two models are indistinguishable given the data when you go back to the weak topology. In the strong topology you think they’re different.
○ (which is at least one reason why you have to do predictive checks)
○ Sorry. That was obviously vague. What I was trying to say is that there’s nothing deep about a finite set of data not being able to separate models. It’s just true unless you put some
serious mathematical effort into making it not true. So a topology based on empirical cdfs (by definition from finite data sets) is too weak to worry about. It’s also a data dependent
topology, which is hard to think about when you’ve got a sequential experiment.
It’s not vacuously true that entropy-based topologies aren’t too strong. Although Higdon’s comment here talks about the games you can play with intermediate topologies: https://
And yes you’ve got to do predictive checks, but you’ve also got to build a good model first. Overfitting is a property of model+data. If the model doesn’t allow for overfitting it
can’t happen. If the data is strong enough to prevent overfitting it can’t happen (although this is less likely in high dimensions).
There’s a mirror to this entire conversation about underfitting.
○ I don’t think the Owhadi et al paper is necessarily the best expression of the basic issues (which, as Christian Hennig points out there, go back at least to the robust stats folk, as
well as e.g. Vapnik and all those folk) but yes, it is based on same issues.
It really is quite an important sticking point for many – some look at such things and say ‘eh, Bayes in the strong topology is still fine by me’ and others go ‘eek, maybe it’s not
for me.’
There are plently of respectable folk on either side of the issue. I’m more of an unrespectable type who’s been converted to the ‘eek’ side.
○ I can’t see the eek. It’s like a magician cutting a woman in half. The first step is to make sure she’s not in the critical part of the box. Once you’ve got that much freedom, you can
do almost anything you like. (For instance, that paper shows that you can’t trust posterior predictive checks because, as they are posterior functional and you’re allowed to move your
assistant around the box, they are also brittle)
But this has drifted quite far from overfitting.
○ Someone should design a survey to determine what best predicts the ‘eek’ and ‘eh’ classes of responses to this.
Hopefully the results don’t depend on whether the analysis is eek-based or eh-based…
○ Ojm: Robust statistics is totally compatible with Bayes. The real issue is that many people identify Bayes with generative modeling of individual data points. The ABC approach is seen
as an approximation. But it’s a first class citizen as soon as you recognize a deeper truth. Bayes is still constrained by frequency interpretations for many people. They think they
need to give probability as frequency distributions over data points, not over say nonlinear transforms of whole data sets. But that isn’t a requirement in any way.
○ > Robust statistics is totally compatible with Bayes.
Well…this is actually quite a subtle issue. For precisely the reasons discussed here.
As an example, the last chapter of the second edition of Huber’s book ‘Robust statistics’ is called ‘Bayesian Robustness’ but is fairly negative on the feasibility/compatibility.
First he discusses e.g. prior sensitivity studies which he sees as pretty orthogonal to the main issues. Then he considers more relevant solutions but most of these seem to amount to
pretty much abandoning the main tenets of the Bayesian approach.
To misquote Andrew – ‘Once you have abandoned literal belief in the B[ayes], the question soon arises: why follow it at all?’
[PS Andrew has said something to the effect that Huber wasn’t appear of posterior predictive checks etc but I’ve come to the position that this doesn’t really address the key issues
○ ojm: I’m guessing you’re not talking about things like modeling noise with t-distribution rather than normal when you say “robust statistics”, because obviously that is totally
compatible with Bayesian approach. But I’m curious what kind of “robust statistics” you have in mind. Are we talking about approaches that still work by specifying likelihoods? If so,
how are they incompatible with Bayes?
○ Chris: the biggest thing that comes to mind for me regarding robust statistics is things like m-estimators with insensitive “cost” functions, windsorizing, and soforth. The goal is to
be insensitive to the fact that some fraction of the data is corrupted, things like digit transpositions and instruments that get unplugged, and intermittent electrical noise, and
whatever, where you don’t have a model for what happened, you just know that sometimes it’s really very different from what usually happens.
I can think of two aspects in Bayes, the first is something like what you said, choosing distributions that allow for outliers, choosing finite mixture models where some fraction of
the observations come from much more extreme distributions, and soforth. The second is something more like ABC, where using some computational method you generate “forward” data
predictions, and then write your “likelihood” in terms of statistics of how well the whole set of data predictions match the whole set of data. You could for example calculate Tukey’s
biweight error function, and then specify a probability distribution over the result.
If you see a likelihood as a *frequency* distribution of errors, then this makes no sense. But if you see a likelihood as a measure of something else, this makes perfect sense. I’m
working on a paper where I describe “what else” I think the Bayesian posterior measures. I think ojm has convinced me that “truth of a proposition” is not in general the best way to
think about what Bayes does. Ironically, one area where “truth of a proposition” makes good sense is when you’re using an RNG to sample from a finite population. Then, there really
*is* a “true” say mean, or sd or range or median. Sometimes that is a fine model of what you’re doing, but other times you know that there’s no “Truth” involved, just some kind of
process you’re describing and a model that has more “goodness” vs more “badness”.
○ There is this idea in Bayes of conditioning not on the data itself but robust summary statistics.
The challenge that arises is often there is not a closed form likelihood function for that (something giving the probability of the observing those robust summary statistics for all
points in the parameter space.)
That thinking lead to this Robust Bayesian inference via coarsening Jeffrey W. Miller, David B. Dunson which seems to work better https://arxiv.org/abs/1506.06101
○ Keith, I don’t know to what extent maybe coarsening works better, that may be true, in fact that may be an essential aspect of what we’re really doing in Bayes (no measurements are
infinitely fine, all measurements are coarse at some level, but usually we ignore this for convenience).
Still, I’m not quite sure what it means “often there is not a closed form likelihood function for that (something giving the probability of the observing those robust summary
statistics for all points in the parameter space.)”
I think this comes out of a desire to have the likelihood represent frequency of something. Restricting yourself to that case is a mistake I think. In ABC for example you generate
some forward estimate of the quantities of interest, and then you calculate a summary statistic or 2 or 3, and then you have several options:
1) accept if the summary statistics are within epsilon of a critical value (ie. differences are within epsilon of 0). This was the
2) accept with probability proportional to some continuous non-negative function with a peak at a critical value that decreases away from this peak, say normal(0,1) or the like.
3) accept with probability proportional to a joint function over the 2 or 3 summary statistics….
and at each stage we get closer to the idea that the likelihood is just any non-negative function. For inference we don’t even need it to be normalizable since we’ll always have a
finite number of data point, but for prior or posterior predictive purposes we do need it normalizable.
Is there an interpretation of what this means that makes sense philosophically and mathematically? I think the answer is yes, I’ll be very interested in what you have to say about it.
I will put up some kind of blog posts and a paper for comments in the next … hopefully week or so.
○ Daniel: I was all explained in my DPhil thesis ;-)
I’ll put something together as it would be much simpler to put today than in 2007.
□ My favourite paper about Bayesian asymptotically a for misspecified models is http://projecteuclid.org/euclid.ejs/1256822130
☆ Thanks for the reference. That is the kind of result I am looking for.
This came from Bob Carpenter on the Stan mailing list: “It’s not overfitting so much as model misspecification.”
I really like this line. If your model is correct, “overfitting” is impossible.
I agree, I never thought about it this way before. It is one of those genius insights that seem so obvious in retrospect.
I was thinking about something similar earlier actually. Often we really want is something like F = G*m1*m2/R^2 while assuming all objects are point masses and ignoring everything below a certain
mass. A more complicated model that could account for stuff like perturbations for all asteroids and comets, 3d shape of the objects, etc is also useful, but plays a different role. It seems to
me most of stats/ml is concerned about “exactness” and thus designed to solve problems like the latter, which is prone to overfitting.
10. > Even if the resulting datasets (actual numbers) can come out the same to whatever sig fig under models arbitrarily far apart in the strong topology?
“if reality does not fit the concept, too bad for reality”
It seems to me that “over-fitting” is being defined here simply as “complexity” and not as “poor generalization” (which may of course be the consequence of “over-complexity”).
11. Overfitting is when your model learns too much from the data.
12. What is the etymology of overfitting, if it is known? I always thought it was a metaphor from tailoring, if your bespoke suit fits you too closely (sample), it will rupture when you move
In that sense I considered it a good metaphor, because nontechnical people easily get it this way.
But English is my second language and some googling seems to show tailors wouldn’t actually use this word.
Leaves me to wonder: is this not a problem tailors have?
□ I suspect the etymology is just over+fit.
But two cool things:
☆ Cool and the second gets even cooler when considered as a spectrum https://en.wikipedia.org/wiki/Apophenia#.22Randomania.22 that in turn provides supports for ESP ;-)
□ Ruben,
I don’t think the etymology is from tailoring. My understanding is that first the usage “fit a curve to the data” originated; then I would guess that the word “overfitting” was adopted by
analogy with words like “overdoing it”. Perhaps you haven’t encountered the quote from John Von Neumann that says “with four parameters I can fit an elephant, and with five I can make him
wiggle his trunk.”
13. It was a refreshing post. I always a hard time to tell people that cross-validation does not prevent ‘overfitting’. In industry, it is a common misconception I think. If one has only a single
14. “People are worried about overfitting.”
Reader’s of Matt Levine know that people are constantly worried about bond market liquidity, unicorns, and many other issues. Overfitting is not a recurring worry yet, but it could become one!
“Overfitting is partly a statistical problem, about how we can extrapolate rules from data, but it is also a deep worry about whether the world is understandable, whether it is subject to rules,
and whether those rules are comprehensible to humans.”
15. I had an impression that ‘overfitting’ was about condition number of a design matrix. Not sure how ‘condition-number’ is interpreted in a Bayesian way though.
16. Msuzen:
Overfitting is a general concept—optimizing fit to training data does not optimize fit to test data—which does not need to have any connection to matrices at all.
17. @Andrew Thank you for this conceptually fundamental post and your comment.
(a) I think there is a link to linear algebra too. I was referring to ‘regularisation’ from inverse problems point of view, such as LASSO, where applying regularisation reduces the condition
number of a ‘design matrix’. Some authors call this ‘inverse crimes’ jokingly.
(b) Douglas Hawkins, in his paper ‘The Problem of Overfitting’ (2004), states that:
“Overfitting of models is widely recognized as a concern. It is less recognized however that overfitting is not an absolute but involves a comparison. A model overfits if it is more complex than
another model that fits equally well.”
Comparing with your definition,
“Overfitting is when you have a complicated model that gives worse predictions, on average, than a simpler model.”
Hawkins’s definition is stronger. Let’s say we have a well-generalised model (M1) if there is a ‘simpler model’ that reaches similar ‘predictive power’ (M2). So, M2 is still overfitting.
From both definitions, can we infer the following? ‘Overfitting’ is not about finding generalisation error alone, since it is a comparison problem, hence we can not really resolve ‘overfitting’
just looking at say, cross-validation error?
□ PS: I meant M1 is still overfitting. | {"url":"https://statmodeling.stat.columbia.edu/2017/07/15/what-is-overfitting-exactly/","timestamp":"2024-11-09T13:09:02Z","content_type":"text/html","content_length":"278157","record_id":"<urn:uuid:9d8cecf4-46e4-4ffa-a4d1-da74ce359e7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00521.warc.gz"} |
Video - Equations defining algebraic curves and their tangent and secant varieties
It is a fundamental problem in algebraic geometry to study equations defining algebraic curves. In 1984, Mark Green formulated a famous conjecture on equations defining canonical curves and their
syzygies. In early 2000's, Claire Voisin made a major breakthrough by proving that Green's conjecture holds for general curves. A few years ago, Aprodu-Farkas-Papadima-Raicu-Weyman gave a new proof
of Voisin's theorem by studying equations defining tangent developable surfaces, and recently, I obtained a simple geometric proof of their result using equations defining secant varieties. In this
talk, I first review the geometry of algebraic curves, and then, I explain the main ideas underlying the recent work on syzygies of algebraic curves and their tangent and secant varieties. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=video&page=2&listStyle=gallery&sort_index=Dept&order_type=asc&document_srl=1083928","timestamp":"2024-11-10T14:38:55Z","content_type":"text/html","content_length":"52510","record_id":"<urn:uuid:3dc200e6-d060-4c28-9f08-7a9d6a5fc893>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00810.warc.gz"} |
Unscramble ABASES
How Many Words are in ABASES Unscramble?
By unscrambling letters abases, our Word Unscrambler aka Scrabble Word Finder easily found 31 playable words in virtually every word scramble game!
Letter / Tile Values for ABASES
Below are the values for each of the letters/tiles in Scrabble. The letters in abases combine for a total of 8 points (not including bonus squares)
What do the Letters abases Unscrambled Mean?
The unscrambled words with the most letters from ABASES word or letters are below along with the definitions.
• abase (a.) - To lower or depress; to throw or cast down; as, to abase the eye.
• baases () - Sorry, we do not have a definition for this word | {"url":"https://www.scrabblewordfind.com/unscramble-abases","timestamp":"2024-11-12T16:17:24Z","content_type":"text/html","content_length":"43872","record_id":"<urn:uuid:d408693d-8943-438c-945b-a4851d06400b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00474.warc.gz"} |
What Are the Divisor and the Dividend?
If you are trying to divide a number by a larger number, your answer, the quotient, will either be a decimal or a fraction. If you are trying to write out a decimal as your answer, remember to put a
zero in front of the decimal as a reminder that your answer cannot be 1 or greater. | {"url":"https://doodlelearning.com/us/math/skills/division/divisor-dividend","timestamp":"2024-11-02T18:04:17Z","content_type":"text/html","content_length":"315750","record_id":"<urn:uuid:192800a3-7ae6-48bf-8e2b-b4af4641c4d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00398.warc.gz"} |
Transient Velocity Inlet (2d synthetic jet) UDF
July 30, Transient Velocity Inlet (2d synthetic jet) UDF #1
2016, 10:32
New Member Hello all
Join Date: my case is consisted of a flow over a hump with a synthetic jet in the interior of the hump. essentially the jet is required to blow and suck air with a frequency of 138.5 Hz at an
Oct 2015 approximate velocity out of the slot of the jet of 26.6 m/s
prescribed as : u=o
Posts: 3 v= speed*cos(2*pi*f)
I'm fairly new in setting up User defined functions for boundary conditions, from digging around this forum and the literature the UDF for this particular case seems to be simple enough
Rep Power: to formulate, but im getting some divergence after certain number of iterations. i wanted to double check whether im implementing the UDF correctly for the desired purpose.
im using a pressure outlet boundary condition to hook the UDF
/************************************************** ********************/
/* unsteady.c */
/* UDF for specifying a transient velocity profile boundary condition */
/************************************************** ********************/
#include "udf.h"
DEFINE_PROFILE(SJ_velocity, thread, position)
face_t f;
begin_f_loop(f, thread)
real freq=138.5;
real t = RP_Get_Real("flow-time");
F_PROFILE(f, thread, position) = 26.6*cos(freq*(t)*(2.*3.141592654));
end_f_loop(f, thread)
I was wondering if someone could point me in the right direction on implementing this function for this particular case
much appreciated | {"url":"https://www.cfd-online.com/Forums/fluent-udf/175446-transient-velocity-inlet-2d-synthetic-jet-udf.html","timestamp":"2024-11-10T10:58:47Z","content_type":"application/xhtml+xml","content_length":"89180","record_id":"<urn:uuid:8e07fa1f-5681-47d0-ad00-abee51878e44>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00537.warc.gz"} |
Excel Import into R without rJava | R-bloggersExcel Import into R without rJava
Excel Import into R without rJava
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In my ongoing quest to webappify various R scripts I discovered that
cannot load any R packages that depend on
. For several of the scripts that I’ve written that grab data out of MS Excel files, and therein use the
package, this is a serious brick wall.
In my current workaround, I’ve resorted to using a shell script to do the xls(x) to .RData conversion. Then I stumbled upon the
package. Buried deep deep deep within the
it is a function called
that relies on Perl rather than Java to do the heavy lifting of crawling both of Microsoft’s proprietary binary and xml based formats.
Testing is currently underway and a comparative write-up is planned. | {"url":"https://www.r-bloggers.com/2012/05/excel-import-into-r-without-rjava/","timestamp":"2024-11-08T14:54:38Z","content_type":"text/html","content_length":"83069","record_id":"<urn:uuid:58019566-238c-4639-9ca0-ea7f1108ed57>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00845.warc.gz"} |
Library guides: Course guide: Datasets for Health and STEMM: Introduction
Understanding some basic terminology will help you to determine whether or not you need statistics, data or both.
Statistics are in a format where the data have already been analyzed and processed to produce information in an easy to read format such as charts, tables, and graphs. An example of this is
Statistical Abstract of the United States. If you're looking for a quick number, it's best to start with statistics.
Data are typically raw data that need to be manipulated using software. Data can be quantitative, qualitative, spatial, etc. The difference between data and statistics can be confusing because in
everyday language, the terms statistics and data are often used interchangeably.
Numeric Data is a type of data made up of numbers. Numeric Data are processed using statistical software like SPSS, Stata, or SAS.
Qualitative Data are data that describe a property or attribute. Examples of qualitative data are interviews, case studies, comments collected on a questionnaire, etc
More Terminology:
Codebook provides information on the structure, contents, and layout of a data file.
Data Archive preserves and makes accessible research data. Some examples are ICPSR, NADAC, and CIESIN.
Microdata are data on the lowest level of observation such as individual answers to questions. For example, the U.S. Census Bureau's Public-Use Microdata Samples (PUMS files) is a data set of
individual housing unit responses to census questions.
Primary Data are data collected through your own research study directly through instruments such as surveys, observations, etc.
Raw Data are the actual observations that are made when the data is collected.
Secondary Data are data from a research study conducted by someone else. Usually when you are asked to locate statistics on a topic you are using secondary data. An example of secondary data are
statistics from the Census of Population and Housing.
Summary Data is another way of describing data that has been processed, or summarized (see statistics). For example, the tables you are reading when using statistical sources are summary data.
Time Series is a sequence of data points spaced over time intervals. | {"url":"https://libguides.library.qut.edu.au/STEMdatasets","timestamp":"2024-11-14T02:13:35Z","content_type":"text/html","content_length":"36515","record_id":"<urn:uuid:4454b06f-3820-4f21-9eca-e8afa0f4ef40>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00404.warc.gz"} |
Capella University Math Algebra Calculus Worksheet - Custom Scholars
Capella University Math Algebra Calculus Worksheet
I attached the workbook, just have to answered the numbered problems in the note book. Please help me, I dont have time to solve the problems.
Personal Financial
Also 5.3.K
5.1.A, 5.1.D, 5.1.G
Essential Question
What is the difference between gross income and
net income?
Income is money you earn, whether from a job, interest from a savings account, or profits
from selling items or services. Gross income is your total income before income or other
payroll taxes are taken out of it. Net income is the amount that is left after taxes.
Taxes based on your earnings, such as income and other payroll taxes, affect your net
income. Taxes based on use of items or services, such as sales and property taxes, do
not affect your net income.
Unlock the
the Problem
Jay works at a bank. Use the information in the
pay stub to find his gross and net income for
the week. The salary from the bank is Jay’s only
source of income.
Since Jay has one source of income, his net
income is the same as the pay after taxes shown
on his pay stub. His gross income is the same as
his total earnings.
Jay Martin
Total earnings
Federal income tax
Pay period:
July 1 – July 8
Other taxes
Total taxes
Pay after taxes
Detach and retain for your records.
His weekly net income is __.
To find his gross income, add the payroll
tax to his net income.
Payroll tax: $__ + $__ = $__.
Gross income: $__ + $__ = $__
net income
payroll tax
© Houghton Mifflin Harcourt Publishing Company
So, Jay’s gross income for the week is $__. Write the
gross income on his pay stub.
Math Talk
Mathematical Processes
Jay pays $3.65 in sales tax for a
meal. Does this change his net
income? Explain.
• If Jay earns $130.56 after taxes during the same week from
a part-time job, how does his net weekly income change? Explain.
Module 17
Share and
and Show
Use the information in the chart for 1–2.
1. What is Tina’s gross income?
Tina’s Finances
Yearly earnings from part-time
job at the bakery
Yearly earnings from part-time
job at the library
Profit from selling pottery
Income and other payroll taxes
Think: Gross income is Tina’s total earnings from all
her sources of income.
Gross income:
$__ + $__ + $__ = $__
2. What is Tina’s net income?.
Net income: $__ − $__ = $__
So, her net income is $__.
Problem Solving
Write Math
Daniel works at a grocery store. He earns $8.00 per
hour at the deli counter. He works 18 hours each week. If this is Daniel’s
only source of income, can you find his net income with the information
given? Explain.
4. Use the information in the chart to find Latisha’s
gross and net income for the year.
Latisha’s Finances
Yearly earnings
Profit from selling handmade
Interest from savings account
Income and other payroll taxes
5. Explain the relationship between gross income,
© Houghton Mifflin Harcourt Publishing Company
payroll tax, and net income.
Problem Solving
6. Multi-Step Eduardo worked for 45 hours last week. His pay
is $12 per hour for 40 hours and $18 per hour for any time
over 40 hours. His payroll tax was $98.06. If this is Eduardo’s
only source of income, what was his net income for last week?
Emily is a pharmacist. She earns $50 per hour.
If she works on a holiday, she gets paid double the hourly
rate. Last week she worked for 38 hours, including 8 hours
on a holiday. Emily’s salary is her only source of income.
If her net income last week was $1,947, how much was her
payroll tax? Explain.
Multi-Step Charlie’s total earnings from his
job are $45,000. He also earned $2,000 in profit from selling
quilts. He pays $18 tax for every $100 in gross income. What
is Charlie’s net income?
Write Math
Show Your Work
9. Apply Geno works at a music store where he earns
© Houghton Mifflin Harcourt Publishing Company
$16.25 per hour. He works 40 hours per week. Should
he take a new job with a gross income of $36,000 per
year? Explain your reasoning.
Module 17 • Lesson 3
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily Assessment
Assessment Task
Fill in the bubble completely to show your answer.
10. The table shows the gross income and payroll tax for the Barker family
members for one week. What is their net income for the week?
The Barker Family Income
Gross Income
Payroll Tax
Bea Barker
Buzz Barker
11. Lana works at the local ice cream store. In the last four weeks, her gross
income was $120.75, $118.50, $99.75, and $115.75. Her total payroll tax
was $59.70. How can you find her net income for the four weeks?
Find the sum of the gross income. Then add the payroll tax.
Add $59.70 to each of the gross incomes. Then add the sums.
Find the sum of the gross incomes.
Find the sum of the gross incomes. Then subtract the payroll tax.
12. Multi-Step Abe earned money mowing lawns. He earned $25 Saturday
morning and $75 Saturday afternoon. He earned $100 on Sunday. A
payroll tax of $26.25 was withheld each day. If this is Abe’s only source
of income, what was his net income for the two days?
TEXAS Test Prep
hour. The wages from this job are Sari’s only source of income. What is
her net weekly income if her payroll tax is $73 for the week?
© Houghton Mifflin Harcourt Publishing Company
13. Sari works in a flower shop for 36 hours each week. She earns $9.75 per
H o mewo rk
and Practice
Personal Financial Literacy—5.10.B
Also 5.3.K
MATHEMATICAL PROCESSES 5.1.A, 5.1.D, 5.1.G
Nan Barker lives in a state with no state income tax. Her only source of
income is from her job at a technology company. Use Nan’s pay stub for 1–4.
1. What is Nan’s gross income for
the pay period? __
Nan Barker
Federal income tax $236.25
2. What is Nan’s net income for the pay
period? __
Total earnings
Pay period:
May 15
3. Nan is paid twice a month. What is Nan’s net
Other taxes $135.40
Total taxes
Pay after taxes $1,203.35
income each month?
Detach and retain for your records.
4. Describe two ways to find the difference between Nan’s gross income
and her net income for the pay period.
Problem Solving
Use Dante’s weekly pay stub for 5–6.
5. If the earnings from this job are Dante’s only source
of income, what is his net income for the week?
6. What is Dante’s annual gross income? Explain
© Houghton Mifflin Harcourt Publishing Company
how you found your answer.
Dante Romano
Pay period:
November 1–7
Total earnings
Federal income tax
State income tax
Other taxes
Total taxes
Pay after taxes
Detach and retain for your records.
Module 17 • Lesson 3
TEXAS Test Prep
Lesson Check
Fill in the bubble completely to show your answer.
$20 per hour for 35 hours and $30 per hour for
any time over 35 hours. If the earnings from
the job are his only source of income, what
was Keenan’s gross income for last week?
9. Chip works at the amusement park. In the
last four weeks, his gross income was $112.25,
$98.75, $125.50, and $109.25. His total payroll
tax was $62.30. If the earnings from this job are
Chip’s only source of income, what was his net
income for the four weeks?
11. Multi-Step Mr. Jackson’s gross monthly
8. Rita’s gross income last month was $2,948.45.
She paid $442.45 in payroll tax for the month.
How can you find Rita’s net income for one
week if there were four weeks in the month?
Add the payroll tax to the gross income
and multiply by 4.
Subtract the payroll tax from the gross
income and multiply by 4.
Add the payroll tax to the gross income
and divide by 4.
Subtract the payroll tax from the gross
income and divide by 4.
10. Jenna’s only source of income is from her job
at a car wash, where she works 25 hours each
week. She is paid $9.90 per hour. What is her
net weekly income if she pays $30.70 in payroll
tax each week?
12. Multi-Step Ari worked for a moving company
income is $4,090 and his payroll tax is
$654.40. Mrs. Jackson’s gross monthly income
is $4,250 and her payroll tax is $680. What
is the Jacksons’ combined net income for
the month?
last weekend. On Saturday, his gross income
was $350 and his net income was $307.05. On
Sunday, his gross income was $280 and his net
income was $246.35. How much did Ari pay in
payroll tax last weekend?
© Houghton Mifflin Harcourt Publishing Company
7. Keenan worked 45 hours last week. His pay is | {"url":"https://customscholars.com/capella-university-math-algebra-calculus-worksheet-6/","timestamp":"2024-11-14T08:45:54Z","content_type":"text/html","content_length":"61873","record_id":"<urn:uuid:0bd8aa9e-55f5-4497-af55-7cc9ecae4df2>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00571.warc.gz"} |
1,654 research outputs found
Charge asymmetry in electron (positron) scattering arises from the interference of the Born amplitude and the box-type amplitude corresponding to two virtual photons exchange. It can be extracted
from electron proton and positron proton scattering experiments, in the same kinematical conditions. Considering the virtual photon Compton scattering tensor, which contributes to the box-type
amplitude, we separate proton and inelastic contributions in the intermediate state and parametrize the proton form-factors as the sum of a pure QED term and a strong interaction term. Arguments
based on analyticity are given in favor of cancellation of contributions from proton strong interaction form factors and of inelastic intermediate states in the box type amplitudes. In frame of this
model, with a realistic expression for nucleon form-factors, numerical estimations are given for moderately high energies.Comment: 14 pages, 4 figure
The apparent discrepancy between the Rosenbluth and the polarization transfer method for the ratio of the electric to magnetic proton form factors can be explained by a two-photon exchange correction
which does not destroy the linearity of the Rosenbluth plot. Though intrinsically small, of the order of a few percent of the cross section, this correction is kinematically enhanced in the
Rosenbluth method while it is small for the polarization transfer method, at least in the range of (Q^2) where it has been used until now.Comment: 4 pages, 4 figures. Version accepted for publication
in Phys. Rev. Let
The electron positron annihilation reaction into four pion production has been studied, through the channel $e^++e^-\to \bar \rho+\rho$. The differential (and total) cross sections and various
polarization observables for this reaction have been calculated in terms of the electromagnetic form factors of the corresponding $\gamma^*\rho\rho$ current. The elements of the spin--density matrix
of the $\rho -$meson were also calculated. Numerical estimations have been done, with the help of phenomenological form factors obtained in the space--like region of the momentum transfer squared and
analytically extended to the time-like region.Comment: 19 pages, 2 figures, to appear in Phys Rev
Nonlinear effects responsible for elongation of the plasma wave period are numerically studied with the emphasis on two-dimensionality of the wave. The limitation on the wakefield amplitude imposed
by detuning of the wave and the driver is found.Comment: 4 pages, 4 figure
We develop an approximate second quantization method for describing the many-particle systems in the presence of bound states of particles at low energies (the kinetic energy of particles is small in
comparison to the binding energy of compound particles). In this approximation the compound and elementary particles are considered on an equal basis. This means that creation and annihilation
operators of compound particles can be introduced. The Hamiltonians, which specify the interactions between compound and elementary particles and between compound particles themselves are found in
terms of the interaction amplitudes for elementary particles. The nonrelativistic quantum electrodynamics is developed for systems containing both elementary and compound particles. Some applications
of this theory are considered.Comment: 35 page
The non-relativistic impulse approximation of deuteron electromagnetic form factors is used to investigate the space-like region behavior of the proton electric form factor in regard of the two
contradictory experimental results extracted either from Rosenbluth separation method or from recoil proton JLab polarization data.Comment: Revtex, 6 pages, 7 figure
We reanalyze the most recent data on elastic electron proton scattering. We look for a deviation from linearity of the Rosenbluth fit to the differential cross section, which would be the signature
of the presence of two photon exchange. The two photon contribution is parametrized by a one parameter formula, based on symmetry arguments. The present data do not show evidence for such
deviation.Comment: 15 pages 3 figures More details on the fitting procedure, more explicit explanation
In a recent one-dimensional numerical fluid simulation study [Saxena et al., Phys. Plasmas 13,032309 (2006)], it was found that an instability is associated with a special class of one-dimensional
nonlinear solutions for modulated light pulses coupled to electron plasma waves in a relativistic cold plasma model. It is shown here that the instability can be understood on the basis of the
stimulated Raman scattering phenomenon and the occurrence of density bursts in the trailing edge of the modulated structures are a manifestation of an explosive instability arising from a nonlinear
phase mixing mechanism.Comment: 17 pages, 7 figures, Published in Phys. Plasma
Several well-known results from the random matrix theory, such as Wigner's law and the Marchenko--Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph.
Orthogonal polynomials with respect to the limiting spectral measure play a role in this approach.Comment: (more) minor change
We use the world's data on elastic electron--proton scattering and calculations of two-photon exchange effects to extract corrected values of the proton's electric and magnetic form factors over the
full Q^2 range of the existing data. Our analysis combines the corrected Rosenbluth cross section and polarization transfer data, and is the first extraction of G_Ep and G_Mp including explicit
two-photon exchange corrections and their associated uncertainties. In addition, we examine the angular dependence of the corrected cross sections, and discuss the possible nonlinearities of the
cross section as a function of epsilon.Comment: 13 pages, 3 figures, 4 tables, to be submitted to Phys. Rev. | {"url":"https://core.ac.uk/search/?q=authors%3A(Akhiezer%20A.%20I.)","timestamp":"2024-11-09T13:38:50Z","content_type":"text/html","content_length":"159855","record_id":"<urn:uuid:2b186cdb-29f5-4305-b44b-63196ccc96a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00879.warc.gz"} |
Solution of Linear Algebraic Equations
Linear algebra is one of the corner stones of modern computational mathematics. Almost all numerical schemes such as the finite element method and finite difference method are in fact techniques that
transform, assemble, reduce, rearrange, and/or approximate the differential, integral, or other types of equations to systems of linear algebraic equations.
A system of linear algebraic equations can be expressed as
Solving a system with a coefficient matrix m surfaces (lines) in an n dimensional space. If all m surfaces happen to pass through a single point then the solution is unique. If the intersected part
is a line or a surface, there are an infinite number of solutions, usually expressed by a particular solution added to a linear combination of typically
The core of solving a system of linear algebraic equations is decomposing the coefficient matrix. Through the decomposition process, the coupled equations are decoupled and the solution can be
obtained with much less effort. A better decomposition method will perform faster and introduce less errors. Common numerical methods used to solve linear algebraic equations are briefly discussed in
this section:
Gaussian Elimination
LU Decomposition
SV Decomposition
QR Decomposition
Gaussian Elimination
Gaussian elimination executes a series of row operations to eliminate coefficients in order to form the triangular matrix
The solution can thereafter be obtained.
LU Decomposition
The LU decomposition rewrites the coefficient matrix
Thus, the system of linear algebraic equations can be decomposed to two systems of linear algebraic equations which can be solved directly.
SV Decomposition
The Singular Value Decomposition (SVD) rewrites the coefficient matrix to three matrices including two orthogonal matrices
The coefficient matrices of the system can then be moved to the right hand side of the equal sign.
QR Decomposition
The QR decomposition rewrites the coefficient matrix
The system can then be solved in the form of | {"url":"https://www.efunda.com/math/num_linearalgebra/num_linearalgebra.cfm","timestamp":"2024-11-10T23:55:30Z","content_type":"text/html","content_length":"21540","record_id":"<urn:uuid:e4500032-e968-4d47-8281-b138a8dfa61f>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00612.warc.gz"} |
Lesson 9: Solving Normal Curve Problems – Geographical Perspectives
Lesson 9: Solving Normal Curve Problems
October 6, 2015
Wed, Oct 7
• Percentile rank in XL Miner
• More Normal Curve Probability Problems
• Reverse Z-Table Calculations
• Recent GMAT Scores are normally distributed with a mean = 545.6 with a std dev = 121.1
• What GMAT score would you need to be in the 95th Percentile? How about the 5th Percentile?
• Your dream is to attend the MBA program at Stanford University where the average GMAT score is 720. What percentage of test takers score higher than 720 on the GMAT?
• After learning the cost of tuition and housing at Stanford you decide to change your dream and, instead, target the University of Oregon where they have one of the very top programs in Sports
Marketing and Management. At Oregon, the average GMAT score is 627. What percentile score will you need to be above average at Oregon?
• What percentage of students achieve a score between the average at Oregon and Stanford?
Use the Foreign Per Diem (FPD) rate data set to answer the following questions.
• What percent of FPD rates are below the rate in Cairo?
• What percent of FPD rates are above the rate in Tokyo?
• What percent of FPD rates are between Cairo and Tokyo?
Post results to your blog.
2 Comments
• Lesson 10: Estimation with Confidence Intervals | Geographical Perspectives
[…] Normal Curve Calculations […]
• Frank Whatley
Prof. Holman, I won’t be in class today as my back went out on me, I will try and make it back on Wednesday. I will get my homework all caught up by then.
Thank you, Frank. | {"url":"https://www.justinholman.com/2015/10/06/lesson-9-solving-normal-curve-problems/","timestamp":"2024-11-02T02:05:12Z","content_type":"text/html","content_length":"60567","record_id":"<urn:uuid:1328d9e1-3fdf-4438-b96e-faa5a0b86fcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00490.warc.gz"} |
How do you find the equation of the tangent line to the graph of f(x) = 2x ln(x + 2) at x = 3? | HIX Tutor
How do you find the equation of the tangent line to the graph of #f(x) = 2x ln(x + 2) # at x = 3?
Answer 1
$y = \frac{10 \ln \left(5\right) + 6}{5} x - \frac{18}{5}$
You first need to differentiate $f \left(x\right)$ to find the gradient at $x = 3$
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Answer 2
To find the equation of the tangent line to the graph of f(x) = 2x ln(x + 2) at x = 3, we can follow these steps:
1. Find the derivative of f(x) using the product rule and the chain rule.
2. Evaluate the derivative at x = 3 to find the slope of the tangent line.
3. Use the point-slope form of a line to write the equation of the tangent line.
Step 1: Find the derivative of f(x) f'(x) = 2(ln(x + 2) + x/(x + 2))
Step 2: Evaluate the derivative at x = 3 f'(3) = 2(ln(3 + 2) + 3/(3 + 2))
Step 3: Use the point-slope form to write the equation of the tangent line The equation of the tangent line is y - f(3) = f'(3)(x - 3).
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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
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-find-the-equation-of-the-tangent-line-to-the-graph-of-f-x-2x-ln-x-2-a-8f9af9d26c","timestamp":"2024-11-05T22:13:08Z","content_type":"text/html","content_length":"572495","record_id":"<urn:uuid:f58e103e-a93c-4211-9637-666aedd4fa71>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00203.warc.gz"} |
The Restful Maths Podcast: The Square Numbers
Dec 20, 2018
The square numbers are generated by multiplying integers by themselves (aka “squaring”). Given that multiplying 2 numbers with the same sign always yields a positive product, it follows that square
numbers are always positive, whether they are the result of squaring a negative or positive integer, so, for example, the squares of both fifteen and negative fifteen is two hundred and twenty five.
Since even the natural numbers form an infinite set, it follows that the squares also form an infinite set.
One way to define a square number is to consider its factors, and in particular its factor pairs. Since one of the factor pairs of any square number is a repeated factor, it follows that all square
numbers have an odd number of distinct (positive) factors, whereas all non square numbers have an even number of distinct factors.
As we progress through the square numbers they grow further and further apart, alternate between odd and even numbers and are progressively separated by the odd numbers.
Perhaps the most common application of the squares we meet in high school maths is their use in Pythagoras’ Theorem which gives the relationship between the three sides of right angle triangles. We
shall explore this in episode 5. | {"url":"https://restfulmaths.com/the-square-numbers","timestamp":"2024-11-03T06:24:43Z","content_type":"text/html","content_length":"40315","record_id":"<urn:uuid:c6194cfc-fa65-40ad-9f50-6bd5d2059af9>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00328.warc.gz"} |
[EM] Multiwinner Condorcet generalization on 1D politics
Kristofer Munsterhjelm km-elmet at broadpark.no
Thu Feb 12 11:31:56 PST 2009
I think that one problem with devising a multiwinner method is that we
don't quite know what it should do. PAV type optimization methods try to
fix this, but my simulations don't give them very favorable scores.
If we are to construct a multiwinner method that degrades gracefully, we
probably need to have an idea of what, exactly, it should do, beyond
just satisfying Droop proportionality (for instance). The problem with
building a method primarily to satisfy a certain criterion is that if
the criterion is broken slightly, then the criterion does not tell us
how the method should work; and therefore, we might get "discontinuous"
methods where the method elects a certain set if a Droop quota supports
it, but a completely different group if the Droop quota less one
supports that set.
So let's consider a case one may use to justify Condorcet, or to
classify Condorcetian methods: if politics is one dimensional, and
people prefer candidates closer to them on the line, then there will be
a Condorcet winner, and the CW is the candidate closest to the median
voter, and (if we think electing the CW is a good idea), we should elect
the candidate closest to the median voter.
This case, or general heuristic, seems to be simple to generalize, and
one may do so in this manner: call a position the nth k-ile position if
n/k of the voters are closer to 0. Then, a multiwinner method that
elects k winners should, if politics is one-dimensional, pick the
candidate closest to the first (k+1)-ile position, then closest to the
second (k+1)-ile position (first candidate notwithstanding as he's
already elected), etc, up to k.
To be more concrete, in a 2-candidate election, the first candidate
should be the one closest to the point where 33% of the voters are below
(closer to zero than) this candidate, and the second candidate should be
the one closest to the point where 67% of the voters are below this
candidate, the first candidate notwithstanding.
This heuristic covers only the one-dimensional case, but it is at least
continuous if the comparison of a particular election *is*
one-dimensional, and thus should reduce discontinuity problems.
Plurality party list methods can be modeled as instances where each
voter is located at the position of the party they voted for, since that
is all a Plurality vote lets us infer. Say that 52% are located at party
A, and 48% at party B, and that WLOG, A's location on the line is closer
to 0 than is B's. Then the count starts at A, and proceeds as such,
electing candidates from A's list until A has its share, at which point
it jumps closer to B. In essence, party list becomes a equal-rank
plurality version where everybody either votes A1 = ... = An or B1 = ...
= Bn.
The problem of synthesizing the distribution on the political line still
remains, though. If people vote rationally (that is, that there is no
noise), one can get some extent of the way by observing
eee A fff ggg B hhh iii C jjj
The e faction would vote A > B > C. So would the f faction, while the g
faction would vote B > A > C and the h faction, B > C > A. The i and j
faction votes C > B > A. Noise votes are A > C > B and C > A > B.
However, while this gives us some information as to the relative sizes
of the factions, it does not tell us whether (say) the e faction is
large because there's a peak of support there, or because the others are
more extreme, like this:
eeeeeeee A ff gg B hh ii C jj
In the case of Condorcet, it doesn't matter, since Black's
single-peakedness theorem says that in the one-dimensional case with
voters preferring candidates closer to them, there'll always be a CW and
that CW is the candidate closest to the median voter.
Can we use this to make "multiwinner Condorcet" where the k-ile property
holds? To do so, I would have to understand the aforementioned
single-peakedness theorem to know how it works, which I don't.
If I were to make a guess, I think it's because Condorcet removes the
other candidates from each pairwise check. Assume A is the median
candidate. Then on A vs B, A is closer to more voters than B is. If that
is all that's needed, then we could imagine a Condorcet analog where if
a voter is closer to A or B than to C or D, it counts as a win for {A,
B}. If a council {X,Y} is a CW in this way, and X is closer to 0 than is
Y, is then X closest to the 33% position, and is Y closest to the 67%
position? I don't know.
In any case, even if I'm right about the above, we have to figure out
what "{A, B} is closer than {C, D}" means. If a voter ranks A > B > C >
D or B > A > C > D, it's pretty obvious that {A, B} is closer. But what
of A > C > B > D or C > A > D > B ? And is it possible to make a system
that picks the (k+1)-ile closest candidates without having to go through
all possible combinations of the council in the worst case?
This may have been a bit meandering, but I wrote it as I went on. My
general idea is this: multiwinner election is not as well defined as
single-winner election, so I tried to find a way of defining it better
by referring to issue space, and in a way so that it reduces to
Condorcet in the single-winner case. Then I wrote a bit about the
limitations of this (that we can't infer the shape of the curve in issue
space from ranking alone), and that perhaps we don't need the shape of
that curve - but on that, I'm uncertain, since I don't know Black's
There's also the problem of noise and multiple dimensions to consider,
but let's keep this simple :-)
Any ideas, replies?
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MathGroup Archive: October 2001 [00082]
Re: Assumptions question (1/m^x,x>1,m=Infinity)
• To: mathgroup at smc.vnet.net
• Subject: [mg31077] Re: Assumptions question (1/m^x,x>1,m=Infinity)
• From: hugo at doemaarwat.nl (BlackShift)
• Date: Sun, 7 Oct 2001 03:11:42 -0400 (EDT)
• Organization: Rijksuniversiteit Groningen
• References: <9pmdb3$62v$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com
On Sat, 6 Oct 2001 07:54:11 +0000 (UTC), Andrzej Kozlowski
<andrzej at bekkoame.ne.jp> wrotf:
>Still, there are several possible ways to deal with some problems of
>this kind. However, the example you give is just too trivial to serve as
>a good illustration and actually ought to be done by hand. Still, here
Can do it by hand of course, but since the formula's got quite large
at the moment I wanted to make the substitution it wasn't that easy to
do by hand...
>is one way one could try to use Mathematica to solve this sort of
>problem. Unfortunately it is not guaranteed to work in other cases,
>since Integrate with Assumptions on which it depends on is very erratic.
>Observe that:
>Integrate[(-m^(-1 - x))*x, {m, 1, Infinity},
> Assumptions -> {Re[x] > 1}] + Integrate[(-m^(-1 - x))*x,
> m] /. m -> 1
>must be the result you wanted.
Hmm, that's actually quite true, I get that formula by integration, so
I can use the assumption there already (why didn't I think of that).
Nonetheless, isn't it possible to do it afterwards? That would be a
'nicer' thing to do, since it is part of a model, where m doesn't have
to be Infity in all cases
>There are other ways, but you would have to present your real problem
I think I can make mathematica do what I want now, but if you know a
method to make the assumption that m is infinity later on in the
calculations that would be very great, so here is the real problem
(x=x, MU=m)
Background: It is a model of starforming in galaxies,
Phi[M_]dM is the ratio of stars formed with a mass between M and M+dM
x is just a parameter for the model
In[1]:= Phi[M_]=Cp*M^(-1-x)
To normalize Cp (which is just a normalization factor) I Integrate
over al possible M, from ML (lower mass, about .1 solar masses) to MU
(upper mass, about 32 solar masses)
In[2]:= subC=Solve[Integrate[Phi[M],{M,ML,MU}]==1,Cp][[1]]
Out[2]= {Cp -> -------------}
----- - -----
x x
ML x MU x
with which I calculate further, with Cp in the expresions, in the
final result /.subC them. In some cases it is better to choose for ML
and MU the numbers 0.1 and 32, but sometimes it is algebraically
easyer to assume ML->0 or MU->Infinity, so sometimes I want to do
/.subC/.{ML->0.1,MU->32}, and sometimes
/.subC/.{ML->0.1,MU->Infinity}, depending on the equations, but the
latter isn't possible:
In[4]:= subC/.{ML->0.1,MU->Infinity}
Out[4]= {Cp -> Indeterminate}
Because it is indeterminate when x=0, which is not the case (x~1.3).
Is there anyway I can do this? without explicitly giving a value for
x, since that is the last valuable to enter (so I can test what value
for x is likely)
BTW, why doesn't mathematica not result with an If statement which
says that it is only indeterminate when x=0 and give Cp->0 otherwise.
That would be more logical behavoir I think. | {"url":"https://forums.wolfram.com/mathgroup/archive/2001/Oct/msg00082.html","timestamp":"2024-11-04T14:13:36Z","content_type":"text/html","content_length":"46934","record_id":"<urn:uuid:79cfa18f-64ea-4d2f-ab0e-94f73ad9f3f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00070.warc.gz"} |
Center for Advanced Studies
arXiv Fall2019
Joint seminar on Mathematical Physics
of HSE University and Skoltech Center for Advanced Studies
on Wednesdays at 17.30 at aud. 110 of the Faculty of Mathematics
December 18, 2019
Andrei Okounkov
(Columbia Univ., Skoltech, HSE Univ.)
Vertices and basics in DT theory
December 11, 2019
Oleg Ogievetsky, Senya Shlosman
(Skoltech, Aix Marseille Univ.)
Critical clusters of solid bodies
December 4, 2019
Victor Vassiliev
(Steklov Inst., HSE Univ.)
Picard–Lefschetz theory, monodromy and applications, Part II
November 27, 2019
Alexey Litvinov
(Landau Inst., HSE Univ.)
Dual description of OSP sigma-models
I will introduce system of screening fields depending on continuous parameter $b$, which defines OSP(n|2m) sigma model at $b\rightarrow\infty$ and integrable Toda like theory in the limit $b\
rightarrow0$. I also claim that both theories define the same QFT being considered in different regimes of coupling constant
November 20, 2019
Masatoshi Noumi
(Kobe Univ.)
A determinant formula associated with the BC_n elliptic hypergeometric integrals of Selberg type
Elliptic hypergeometric functions are a relatively new class of special functions that have been developed during these two decades. I will report some recent progresses in the study of BC_n elliptic
hypergeometric integrals of Selberg type on the basis of collaboration with Masahiko Ito
November 13, 2019
Maria Matushko
(Skoltech, HSE Univ.)
Fermionic limit of Calogero-Sutherland system
November 6, 2019
Alexei Yung
(Petersburg Nuclear Physics Inst., Skoltech)
Quantizing a solitonic string
Quite often the zero mode dynamics on solitonic vortices are described by a non-conformal effective world-sheet sigma model (WSSM). We address the problem of solitonic string quantization in this
case. As well-known, only critical strings with conformal WSSMs are self-consistent in ultra-violet (UV) domain. Thus, we look for the appropriate UV completion of the low energy non-conformal WSSM.
We argue that for the solitonic strings supported in well-defined bulk theories the UV complete WSSM has a UV fixed point which can be used for string quantization. As an example, we consider BPS
non-Abelian vortices supported in four-dimensional (4D) N = 2 SQCD. In addition to translational moduli the non-Abelian vortex under consideration carries orientational and size moduli. Their
low-energy dynamics are described by a two-dimensional N =(2;2) supersymmetric weighted CP model. Given our UV completion of this WSSM we find its UV fixed point. The latter defines a superconformal
WSSM. We observe two cases in which this conformal WSSM, combined with the free theory for four translational moduli, has ten-dimensional target space required for superstrings to be critical. We
discuss associated string theories. In one case we are able to find the spectrum of low lying string states which we interpret as hadrons of 4D SQCD
October 30, 2019
Aleksander Orlov
(HSE Univ., Inst. of Oceanology)
How the matrix integrals count the covers of Riemann and Klein surfaces
October 23, 2019
Alba Grassi
(Simons Center for Geometry and Physics)
Non-perturbative approaches to quantum Seiberg–Witten curves
In this talk I will revisit and connect various non-perturbative approaches to the quantization of the Seiberg-Witten curves. I will focus on the explicit example of N = 2, SU(2) super Yang–Mills
theory, which is closely related to the modified Mathieu operator. I will then show how, by using the TS/ST correspondence, we can obtain a closed formula for the Fredholm determinant and the
spectral traces of the modified Mathieu operator. Finally, by using blowup equations, I will explain the connection between this operator and the tau function of Painlevé III
October 16, 2019
Nikita Nekrasov (Simons Center for Geometry and Physics, Skoltech)
On recent progress in old problems: wavefunctions of quantum integrable systems from gauge theory and Bethe/gauge dual of N=4 super-Yang-Mills theory, Part I
I will give a pedagogical introduction to crossed and folded instantons with the emphasis on applications to quantum integrable systems. I will also discuss a generalization of Bethe/gauge
correspondence which is necessary to encompass the Bethe ansatz equations for the Y(gl(4|4)) spin chains recently proposed in the context of AdS/CFT correspondence
October 9, 2019
Albert Schwarz
(Univ. of California, Davis)
Inclusive scattering matrix in perturbation theory and in algebraic quantum theory
October 2, 2019
Boris Feigin
(HSE Univ., Landau Inst.)
W algebras related to Lie superalgebras, (twisted) affine Yangians and coset theories
September 25, 2019
Alexey Litvinov
(Landau Inst., HSE Univ.)
Dual description of the deformed OSP sigma models
Spring 2022
Fall 2021
Spring 2021
Fall 2020
Spring 2020
Fall 2019
Spring 2019
Fall 2018
Spring 2018
Fall 2017
Spring 2017
Fall 2016 | {"url":"https://crei.skoltech.ru/cas/calendar/sem-wed-arxiv/fall19/","timestamp":"2024-11-05T19:23:43Z","content_type":"text/html","content_length":"69004","record_id":"<urn:uuid:3004a058-2a70-4f89-92d9-9e29e40338ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00705.warc.gz"} |
adding real literal and real number of high precision
adding real literal and real number of high precision
When Sage is adding a real literal to a real number of high precision, shouldn't it calculate the sum in the high precision ring? Instead, Sage seems to calculate in double precision:
RF=RealField(150); RF
Real Field with 150 bits of precision
RF(0.9 + RF(1e-18))
RF(1.0+ RF(1e-18))
RF(1+ RF(1e-18))
I'm trying to use high precision arithmetic (2658 bits) in Sage to verify some results produced by the high precision semidefinite program solver sdpa_gmp. Sage's treatment of real literals in these
calculations has made me anxious about the possibility that I'm overlooking other ways in which the calculations might be unreliable.
Is there anywhere an explanation of Sage's treatment of real literals in high precision arithmetic?
Added: Immediately after posting this question, the list of Related Questions in the sidebar pointed me to question/327/set-global-precision-for-reals where I learned that 'RealNumber = RF' would
make all real literals lie in the high precision ring. Still, I wonder why the default behavior is to discard precision that is present in the original real literal.
thanks, Daniel Friedan
thanks for
I'm sure there are technical answers, but I think the basic answer is just that we would go to the *lesser* precision, to be sure we are right, when we combine two numbers of different precision.
That makes sense to me. `0.9` and friends are default 53-bit precision, while `1` is an `Integer` and so has arbitrarily high precision and preserves what you want.
There seem to be inconsistencies: RF(.5 + .4 ) 0.90000000000000002220446049250313080847263336 RF(.3+.7) 1.0000000000000000000000000000000000000000000 If .4 and .5 are added as double precision, why
not .3 and .7 ?
EDIT: my previous comment missed the point of your question. The computation is actually done in double precision both times, but RF(RDF(1.0)) = 1.0000000000000000000000000000000000000000000 while RF
(RDF(0.9)) = 0.90000000000000002220446049250313080847263336. Which makes the second addition look like it was done in RF all along.
But RF(RDF(0.3))= 0.29999999999999998889776975374843459576368332 and RF(RDF(0.7))= 0.69999999999999995559107901499373838305473328 so how can RF(0.3+0.7)= 1.0000000000000000000000000000000000000000000
if the addition is being done in double precision?
1 Answer
Sort by ยป oldest newest most voted
[This is not really a full answer to your original question but the comment size limit prevents me from posting it as such.]
I don't quite see where the problem should be with the last example: The addition is performed in RDF, giving RDF(1.0) and only after that the conversion of RDF(1.0) to RF happens.
I'll try to explain what's happening as I understand it. First as a remark: The conversion of an element from RDF (53-bits of precision) to RF (RealField with 150-bits of precision) is of course
non-canonical. In Sage terms: There is no coercion from RDF to RF, Sage will only automatically coerce from higher precision to lower.
By writing RF(RDF(0.7)) we explicitly ask Sage to convert a lower precision element of RDF to RF anyway, filling up the remaining digits in whatever way it sees fit. I guess this might be the
confusing part, because this means that it is not (necessarily) true that RF(RDF(0.3) + RDF(0.7)) = RF(RDF(0.3)) + RF(RDF(0.7)).
If we do RF(0.3+0.7) this is the same as RF(RDF(0.3)+RDF(0.7)), thus the two numbers are added in RDF, giving RDF(1.0), and then converted to RF. If we do on the other hand RF(0.3) + RF(0.7) then 0.3
and 0.7 are interpreted as 150-bit numbers and added in RF (note that they are in fact parsed with the higher precision, they are not first stored as double elements and then converted). Finally and
still different, RF(RDF(0.3))+RF(RDF(0.7)) will create the elements with 53-bit precision, then convert them to RF and add them there.
The documentation explains quite nicely how such coercion is performed in general. In particular the explain feature could be interesting for you:
sage: RF = RealField(150)
sage: cm = sage.structure.element.get_coercion_model()
sage: RF(0.3+0.7)
sage: cm.explain(0.3,0.7,add)
Identical parents, arithmetic performed immediately.
Result lives in Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: RF(RDF(0.3)+RDF(0.7))
sage: cm.explain(RDF(0.3),RDF(0.7),add)
Identical parents, arithmetic performed immediately.
Result lives in Real Double Field
Real Double Field
sage: RF(RDF(0.3)) + RF(RDF(0.3))
sage: cm.explain(RF(RDF(0.3)), RF(RDF(0.7)), add)
Identical parents, arithmetic performed immediately.
Result lives in Real Field with 150 bits of precision
Real Field with 150 bits of precision
sage: RF(0.3) + RDF(0.7)
sage: cm.explain(RF(0.3), RDF(0.7), add)
Coercion on left operand via
Native morphism:
From: Real Field with 150 bits of precision
To: Real Double Field
Arithmetic performed after coercions.
Result lives in Real Double Field
Real Double Field
It may ... (more)
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Thank you very much for the explanations and for the pointers to the documentation. I gather that Sage's interpretation of a token such as '0.7' is context dependent. In RF(0.3+0.4), '0.3' and '0.4'
are elements of RDF, and 0.3+0.4=0.7 in RDF, so RF(0.3+0.4)=RF(RDF(0.7)). But in RF(0.7), '0.7' is not an element of RDF, so RF(0.3+0.4) \ne RF(0.7). In order to avoid worrying about this context
dependence, I can use 'RealNumber = RF' whenever I need to do high precision calculations where an accidental invocation of RDF would be disastrous. I'll be careful to do this in the future. I must
say I find this behavior of Sage to be somewhat peculiar. I assumed that an expression of the form RF ...(more)
Daniel Friedan ( 2012-07-27 22:12:08 +0100 )edit
Sort of! This is handled by the preparser. I'm certainly not the right person to ask about that but digging around a bit in the source gave me the following, which at least explains how it works, if
not why: When the preparser sees a floating point literal it calls RealNumber(s) with `s` the string it sees. This will give you a rings.real_mpfr.RealLiteral object with precision good enough to
represent the literal. If you pass this to RF it will make the right element object. But if you start adding it, it has to to that in some field, and by default it takes RDF. So inputing floating
point literals is ok, but as soon as you do mathematical ops with them you end up with an element in RDF instead of a RealLiteral. OTOH 1+pi is symbolic in Sage and can be coerced to any precision.
daniels ( 2012-07-28 04:47:12 +0100 )edit
You can see how it drops from a RealLiteral to an element of RDF once you do operations (the parent() decides where the operation takes place - the RealLiteral is actually stored with all the
precision that your input string has): sage: x = 0.3 sage: type(x), x.parent() (<type 'sage.rings.real_mpfr.realliteral'="">, Real Field with 53 bits of precision) sage: y = x + 0.7 sage: type(y),
y.parent() (<type 'sage.rings.real_mpfr.realnumber'="">, Real Field with 53 bits of precision)
daniels ( 2012-07-28 04:50:26 +0100 )edit | {"url":"https://ask.sagemath.org/question/9185/adding-real-literal-and-real-number-of-high-precision/","timestamp":"2024-11-05T10:54:32Z","content_type":"application/xhtml+xml","content_length":"72388","record_id":"<urn:uuid:b16cd43a-a401-4d13-8fb6-b283b17eda0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00819.warc.gz"} |
Multi Digit Division Worksheet - Divisonworksheets.com
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Students struggle to divide large numbers. These worksheets often use an algorithm that is standardized and has step-by-step instructions. It is possible for students to lose the understanding
needed. The use of base ten blocks in order to demonstrate the process is one method to instruct long division. Long division should be a breeze for students once they’ve learned the steps.
Pupils can practice the division of large numbers using many of worksheets and practice questions. Additionally, the worksheets include the information for fractions in decimals. The worksheets can
be used to aid you to divide large sums of money.
Divide the data into smaller groups.
It can be difficult to organize a group of people into small groups. Although it may sound great on paper many small group facilitators aren’t happy with this procedure. It is a true reflection of
how our bodies develop and helps in the Kingdom’s endless growth. It encourages people to reach out and help the less fortunate as well as a new leadership to take over the reins.
It is also helpful to brainstorm ideas. You can form groups of people with similar experiences and characteristics. This is an excellent opportunity to brainstorm fresh ideas. Reintroduce yourself to
each person after you’ve formed your groups. This is an excellent idea to encourage creativity and new ideas.
It is used to divide massive numbers into smaller pieces. When you want to create the same amount of items for several groups, it could be useful. For example, a huge class could be broken up into
five classes. These groups are added up to provide the original 30 students.
It is important to remember that there exist two types of numbers that you can use to divide numbers: the divisor and the Quotient. The result of multiplying two numbers is “ten/five,” but the same
results are obtained when you divide them both.
Powers of ten should be used to calculate huge numbers.
It’s possible to split big numbers into powers of 10 which makes it simpler to make comparisons. Decimals are a crucial element of shopping. You will find them on receipts, price tags, and food
labels. The petrol pumps also use them to display the cost per gallon and the amount of gas that is dispensed through an funnel.
There are two ways to divide big numbers into powers of ten. The first method involves shifting the decimal to the left, and then multiplying it by 10-1. This second method makes use of the
associative feature of powers of ten. Once you’ve learned the associative property of powers of 10, you can divide an enormous number into smaller powers that are equal to 10.
Mental computation is utilized in the first method. When you divide 2.5 by the power of 10 and then you’ll find an underlying pattern. The decimal points will shift left when the power of ten grows.
This concept can be applied to tackle any problem.
The second option involves mentally dividing massive numbers into powers of 10. Then, you may quickly express very huge numbers by writing them down in scientific notation. In scientific notation,
big numbers should be written using positive exponents. You can change the decimal place five spaces on one side and convert 450,000 into number 4.5. You can either divide a large number into smaller
powers than 10, or split it into smaller power of 10.
Gallery of Multi Digit Division Worksheet
Bluebonkers Division Worksheets Double Digit Division P3 Math Long
Division With Multi Digit Divisors In 2021 Long Division Worksheets
Division With Multi Digit Divisors
Leave a Comment | {"url":"https://www.divisonworksheets.com/multi-digit-division-worksheet/","timestamp":"2024-11-01T19:16:28Z","content_type":"text/html","content_length":"65106","record_id":"<urn:uuid:6c5b8078-3746-4490-8a2e-57d33f6f1600>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00441.warc.gz"} |
Project in Statistics
The project is about generating printouts and and answering some questions about it
all the instructions are in the file.
After collecting the data pleas attach it in an Excel file beside the project file which is in word file.
** You need to have program called Statistix 10 or similar one in order to get the needed results from the original printout **
The files I upload are about :
How to collect the data , A chart of all the requirements in this project and its points, The project file which include the questions that need to be answered
Unformatted Attachment Preview
Data: The goal of the project assignments is to allow you to get hands-on experience analyzing and interpreting data using the techniques we learn throughout the semester. To facilitate this, I have
limited you to the type of data that you are allowed to collect. I have listed below three possible sources of data. Choose any of the three to use for your project. Automobile Data – collect data
from 25 used automobiles from each of three comparable car brands (cars in the same automobile class). The variables that you will enter will be Asking Price, Mileage of the vehicle, and the Model of
the Car (enter the model as either 1, 2, or 3). Your data set should consist of a total of 75 cases with three variables. Sources: internet sites such as craigslist, newspaper, auto trader magazines.
To collect the data about the automobile you should search for cars in craigslist website in the US www.craigslist.org 1. Toyota Camry 2. Honda Accord 3. Hyundai Sonata It doesn’t matter the year of
the car. The needed information are the asking price, mileage and model as it was stated above. You can just list all of them in three columns under each other until it reaches 75 using Excel and
then use the date in the Statistix program or similar one. Ideally, you would take a random sample from the population that you are working with. For the project assignments, it is more important to
get your data collected quickly, so I will not worry about the randomness of your data. For clarity, your data set should consist of a total of 75 observations (3 locations/models each with 25
observations) with three variables each. This data set will be used for you on all four project assignments (called Project 1, Project 2, Project 3 and Project 4). More details will follow regarding
the specifics of what will be required of you on the projects as well as demonstrations on how to create the printouts that you will need using the Statistix software. For project purposes, I will
refer to the variables collected using the following symbols: y = price variable (rent or price) x1 = size or mileage variable (depending on data set) x2 = location or model variable (depending on
data set) QMB 3200 Project 1 NAME _________________________ Summer 2018 Directions: 4. 1. Choose one of the data sets assigned in Canvas and enter the data into a Statistix data file. 2. Use the
Statistix program to create the printouts requested in this project. 3. Attach the requested printouts to the project as I have directed and answer the questions that I have asked in the space
provided. Upload this project in Canvas prior to midnight on July 2nd. 1. Create a printout that provides a listing of the data that you entered into Statistix. It should show the 75 cases that you
entered and contain three columns of data. Label this printout as DATA PRINTOUT and attach it to the back of this project. a. Describe for me the quantitative variables you collected in your data
set: (4 points) QN #1: QN #2: b. Identify for me what/where the levels 1, 2, and 3 of your qualitative variable represent. (3 points) Level 1: Level 2: Level 3: c. Identify the experimental unit for
your data set. (2 points) d. Create a stem-and-leaf display of your price variable. Label this printout as Stem-and-Leaf PRINTOUT and attach it here. Describe the shape of the prices you observe in
the plot. (3 points) Attach printout here using a 10-pt font 2. Use Statistix to create the descriptive statistics for the two quantitative variables that you collected data for. Make sure to include
the n, minimum and maximum values, mean, median, standard deviation, and the confidence interval endpoints for a 92% confidence interval. Paste the printout in the space provided here and answer the
following questions. Descriptive Statistics printout a. For your Price variable, interpret the standard deviation by giving me a range in which you think “most” of the values (i.e., the x’s) will
fall. (Hint: you may need to do some math here) (3 points) b. Find the 92% confidence interval for estimating the mean mileage or size of your data set (2 points) c. Give a practical interpretation
for the interval specified in part b. (4 points) d. Explain what you mean when you say that you are “92% confident” in the practical interpretation above in part c. (2 points) e. What assumption(s)
is/are necessary for this analysis to be valid? Make sure you state your answer in the words of the problem (4 points) 3. TESTS OF HYPOTHESIS: Use Statistix to create a printout to conduct the
appropriate test for the data set you worked with: Use only model/location 1 in this analysis. Apartment Data Set: Test to determine if the average size differed from 700 square feet. Automobile Data
Set: Test to determine if the average mileage differed from 30,000 miles. Home Data Set: Test to determine if the average size of the home differed from 1,800 square feet. Copy the results here and
answer the following questions. One-Sample T Test printout a. State the null and alternative hypotheses that are being tested with the specified test. (3 points) Ho: Ha: b. Identify the test
statistic and p-value you will use in this test of hypothesis (2 points) Test Statistic: ____________ P-value: ___________ c. What assumption(s) is/are necessary for this analysis to be valid? Make
sure you state your answer in the words of the problem (4 points) c. Make a conclusion for the test of hypothesis conducted above. You should choose . (4 points) 7/2/2018 Project 1 - Due July 2
Criteria Quantitative Variables Must have 2 Qualitative Variable Accurately Defined and Stated Experimental Unit Ratings Pts 4.0 pts Full Marks 2.0 pts Missing one 0.0 pts Missing Both Variables
Missing a variable or did not define variable correctly Missing both variables or did not define variable correctly 3.0 pts Full Marks 2.0 pts Missing Level 2.0 pts Full Marks 1.0 pts Missing Levels
0.0 pts Missing All Levels 1.0 pts Inaccurately Defined Exp. Unit Must have the correct printout, and describe the shape of the distribution correctly Standard Deviation Correct Calculation using
either Chebyshev or modified Empirical depending on distribution of Steam and leaf plot Confidence Interval 3.0 pts Full Marks 2.0 pts Incorrectly Described Shape of Distribution 3.0 pts Full Marks
2.0 pts Wrong use of measurement 2.0 pts Full Marks 2.0 pts Printout Error 2.0 pts 0.0 pts Printout Error & Description 3.0 pts 1.0 pts Did not multiply SD Did not correctly select the appropriate
measure of calculation 1.0 pts One correct endpoint 0.0 pts No Marks 0.0 pts Incorrect Endpoints Correct Endpoints Confidence Interval Interpretation We are 92% confidence that mean mileage/size of
all automobiles/homes/apts falls between # and # 4.0 pts Full Marks 3.0 pts Missing one piece 2.0 pts Missing 2 pieces/Words of the Problem 1.0 pts Missing Multiple pieces Is missing information or
did not state the answer in the words of the problem Theoretical Interpretation In repeated sampling 92% of the intervals created would contain mu. Assumptions 1 2.0 pts Full Marks 1.0 pts Incorrect
Interpretation Did not correctly use the theoretical interpretation. 4.0 pts Full Marks https://usflearn.instructure.com/courses/1281532/assignments/5960759?module_item_id=11957471 3.0 pts 2.0 pts
0.0 pts Incorrect Interpretation Did not have an accurate understanding of the interpretation concept 1.0 pts Intervals containing true population mean 2.0 pts Use of normal assumption 3.0 pts 0.0
pts No Marks Correctly identified the experimental unit Steam-and-Leaf Plot 4.0 pts 0.0 pts No Marks 0.0 pts No Marks 4.0 pts 2.0 pts 4.0 pts 3/4 7/2/2018 Project 1 - Due July 2 Criteria Null and
Alternative Correct use of mu, differs from, inequality symbols, and numerical value to test T-Stat and P-value Ratings 3.0 pts Full Marks 2.0 pts Missing one piece 2.0 pts Full Marks Pts 1.0 pts
Missing multiple pieces 1.0 pts Missing Information 0.0 pts No Marks 0.0 pts No Marks Correct identification of p-value and t-stat Assumptions 2 Correct use of assumptions in the words of the
problems Test of Hypothesis conclusion At α= #, we (fail to) reject Ho. There is (in)sufficient evidence to indicate that the mean mileage/size of all [model/location 1] homes/apt/cars differs from #
unit of measurement. 4.0 pts Full Marks 2.0 pts Missing information/Words of the problem 4.0 pts Full Marks 3.0 pts Missing 1 piece Is missing information or did not state correctly in the words of
the problem. 2.0 pts Missing 2 pieces/Words of the Problem Is missing information or did not correctly use the words of the problem 1.0 pts Missing Multiple pieces 0.0 pts No marks 3.0 pts 2.0 pts
4.0 pts 0.0 pts No Marks 4.0 pts Total Points: 40.0 https://usflearn.instructure.com/courses/1281532/assignments/5960759?module_item_id=11957471 4/4 Details Statistix 10 Student
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International Scientific Journal
Presently, the scientists across the world are carrying out the theoretical as well as the experimental examinations for describing the importance of nanofluid in the heat transfer phenomena. Such
fluids can be obtained by suspending nanoparticles in base fluid. Experimentally, it has proved that the thermal characteristics of nanofluid are much better and appealing as compared to traditional
fluid. The current work investigates the heat transfer for flow of blood that comprises of micropolar gold nanoparticles. A microorganism creation also affects the concentration of nanoparticles
inside the channel. Suitable transformation has used to change the mathematical model to dimensionless form and then have solved by employing the homotopy analysis method. In this investigation it
has revealed that, fluid's motion decays with growth in Reynolds, Darcy numbers and volumetric fraction. Thermal characteristics support by augmentation in volumetric fraction, while oppose by
Prandtl number. Density of microorganism weakens by growth in Peclet and bioconvection Lewis numbers.
PAPER SUBMITTED: 2022-06-06
PAPER REVISED: 2022-06-24
PAPER ACCEPTED: 2022-06-27
PUBLISHED ONLINE: 2023-04-08 | {"url":"https://thermalscience.vinca.rs/2023/special/17","timestamp":"2024-11-11T18:19:48Z","content_type":"text/html","content_length":"16895","record_id":"<urn:uuid:4c5a8647-7c2b-4d9b-8894-ffffa6de5ba3>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00704.warc.gz"} |
Run a binary or example of the local package.
All the arguments following the two dashes (--) are passed to the binary to run. If you’re passing arguments to both Cargo and the binary, the ones after -- go to the binary, the ones before go to
Unlike cargo-test(1) and cargo-bench(1), cargo run sets the working directory of the binary executed to the current working directory, same as if it was executed in the shell directly.
Package Selection
By default, the package in the current working directory is selected. The -p flag can be used to choose a different package in a workspace.
-p spec, --package spec
Target Selection
When no target selection options are given, cargo run will run the binary target. If there are multiple binary targets, you must pass a target flag to choose one. Or, the default-run field may be
specified in the [package] section of Cargo.toml to choose the name of the binary to run by default.
--bin name
Run the specified binary.
--example name
Run the specified example.
Feature Selection
The feature flags allow you to control which features are enabled. When no feature options are given, the default feature is activated for every selected package.
See the features documentation <https://doc.rust-lang.org/cargo/reference/features.html#command-line-feature-options> for more details.
-F features, --features features
Space or comma separated list of features to activate. Features of workspace members may be enabled with package-name/feature-name syntax. This flag may be specified multiple times, which enables
all specified features.
Activate all available features of all selected packages.
Do not activate the default feature of the selected packages.
Display Options
Use verbose output. May be specified twice for “very verbose” output which includes extra output such as dependency warnings and build script output. May also be specified with the term.verbose
config value <https://doc.rust-lang.org/cargo/reference/config.html>.
Do not print cargo log messages. May also be specified with the term.quiet config value <https://doc.rust-lang.org/cargo/reference/config.html>.
Control when colored output is used. Valid values:
□ auto (default): Automatically detect if color support is available on the terminal.
□ always: Always display colors.
□ never: Never display colors.
May also be specified with the term.color config value <https://doc.rust-lang.org/cargo/reference/config.html>.
The output format for diagnostic messages. Can be specified multiple times and consists of comma-separated values. Valid values:
□ human (default): Display in a human-readable text format. Conflicts with short and json.
□ short: Emit shorter, human-readable text messages. Conflicts with human and json.
□ json: Emit JSON messages to stdout. See the reference <https://doc.rust-lang.org/cargo/reference/external-tools.html#json-messages> for more details. Conflicts with human and short.
□ json-diagnostic-short: Ensure the rendered field of JSON messages contains the “short” rendering from rustc. Cannot be used with human or short.
□ json-diagnostic-rendered-ansi: Ensure the rendered field of JSON messages contains embedded ANSI color codes for respecting rustc’s default color scheme. Cannot be used with human or short.
□ json-render-diagnostics: Instruct Cargo to not include rustc diagnostics in JSON messages printed, but instead Cargo itself should render the JSON diagnostics coming from rustc. Cargo’s own
JSON diagnostics and others coming from rustc are still emitted. Cannot be used with human or short.
Manifest Options
Path to the Cargo.toml file. By default, Cargo searches for the Cargo.toml file in the current directory or any parent directory.
Ignore rust-version specification in packages.
Asserts that the exact same dependencies and versions are used as when the existing Cargo.lock file was originally generated. Cargo will exit with an error when either of the following scenarios
□ The lock file is missing.
□ Cargo attempted to change the lock file due to a different dependency resolution.
It may be used in environments where deterministic builds are desired, such as in CI pipelines.
Prevents Cargo from accessing the network for any reason. Without this flag, Cargo will stop with an error if it needs to access the network and the network is not available. With this flag,
Cargo will attempt to proceed without the network if possible.
Beware that this may result in different dependency resolution than online mode. Cargo will restrict itself to crates that are downloaded locally, even if there might be a newer version as
indicated in the local copy of the index. See the cargo-fetch(1) command to download dependencies before going offline.
May also be specified with the net.offline config value <https://doc.rust-lang.org/cargo/reference/config.html>.
Equivalent to specifying both --locked and --offline.
Changes the path of the lockfile from the default (<workspace_root>/Cargo.lock) to PATH. PATH must end with Cargo.lock (e.g. --lockfile-path /tmp/temporary-lockfile/Cargo.lock). Note that
providing --lockfile-path will ignore existing lockfile at the default path, and instead will either use the lockfile from PATH, or write a new lockfile into the provided PATH if it doesn’t
exist. This flag can be used to run most commands in read-only directories, writing lockfile into the provided PATH.
This option is only available on the nightly channel <https://doc.rust-lang.org/book/appendix-07-nightly-rust.html> and requires the -Z unstable-options flag to enable (see #14421 <https://
Common Options
If Cargo has been installed with rustup, and the first argument to cargo begins with +, it will be interpreted as a rustup toolchain name (such as +stable or +nightly). See the rustup
documentation <https://rust-lang.github.io/rustup/overrides.html> for more information about how toolchain overrides work.
--config KEY=VALUE or PATH
Overrides a Cargo configuration value. The argument should be in TOML syntax of KEY=VALUE, or provided as a path to an extra configuration file. This flag may be specified multiple times. See the
command-line overrides section <https://doc.rust-lang.org/cargo/reference/config.html#command-line-overrides> for more information.
-C PATH
Changes the current working directory before executing any specified operations. This affects things like where cargo looks by default for the project manifest (Cargo.toml), as well as the
directories searched for discovering .cargo/config.toml, for example. This option must appear before the command name, for example cargo -C path/to/my-project build.
This option is only available on the nightly channel <https://doc.rust-lang.org/book/appendix-07-nightly-rust.html> and requires the -Z unstable-options flag to enable (see #10098 <https://
-h, --help
Prints help information.
-Z flag
Unstable (nightly-only) flags to Cargo. Run cargo -Z help for details.
Miscellaneous Options
-j N, --jobs N
Number of parallel jobs to run. May also be specified with the build.jobs config value <https://doc.rust-lang.org/cargo/reference/config.html>. Defaults to the number of logical CPUs. If
negative, it sets the maximum number of parallel jobs to the number of logical CPUs plus provided value. If a string default is provided, it sets the value back to defaults. Should not be 0.
Build as many crates in the dependency graph as possible, rather than aborting the build on the first one that fails to build.
For example if the current package depends on dependencies fails and works, one of which fails to build, cargo run -j1 may or may not build the one that succeeds (depending on which one of the
two builds Cargo picked to run first), whereas cargo run -j1 --keep-going would definitely run both builds, even if the one run first fails.
Exit Status
• 0: Cargo succeeded.
• 101: Cargo failed to complete.
1. Build the local package and run its main target (assuming only one binary):
cargo run
2. Run an example with extra arguments:
cargo run --example exname -- --exoption exarg1 exarg2 | {"url":"https://www.mankier.com/1/cargo-run","timestamp":"2024-11-13T14:24:07Z","content_type":"text/html","content_length":"24372","record_id":"<urn:uuid:72511bda-28ae-48af-a8e1-e444e56382bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00106.warc.gz"} |
How to Assess Financial Viability as an AI Product Manager
What I Learned After Over 100 Assessments - ROI, NPV, IRR—OMG 😱 From Facepalms to High Fives 🤦♂️🙌 A comprehensive Guide.
If only I had kept track of all the AI use case assessments I’ve conducted over the past decade, I could share some impressive statistics and comparisons over time. While I haven’t done that, I can
tell you one thing in detail:
In the past six months alone,
\(\text{I’ve conducted over 100 quick assessments} \)
- more than in all the previous years combined.
Thanks for reading A World Beyond AI! Subscribe for free to receive new posts and support my work.
This surge is primarily due to one phenomenon: the hype around GenAI.
This trend highlights just how powerful GenAI is, significantly increasing the number of potential applications and new opportunities for companies.
Now that AI use cases are popping up everywhere, companies and AI Product Managers must swiftly determine which ones are worth pursuing. Otherwise, the risk of allocating resources to the wrong
initiatives could result in wasted time, money, and effort.
Some of you might already know that I strongly advocate for "assessing the core before opening the door," adhering to the well-known desirability, feasibility, and viability assessments. This
approach ensures that we focus on projects that align with the strategic goals and capabilities of the organization.
It’s more important than ever to stay grounded and methodical. The allure of GenAI can lead to a scattergun approach, quickly enabling AI use cases without a deep understanding of their overall
consequences. However, disciplined assessment helps us identify the real game-changers from the mere AI use cases.
Today, this post is all about viability from a financial perspective, focusing on the potential return on investment. It may not be the most exciting and joyful topic and might feel like a pain at
times 🤕, but do yourself a favor and understand the core principles, as this is where your potential impact as an AI Product Manager will be determined 🔥.
You will learn:
• What is Financial Viability?
• How Finance Departments would assess an AI Investment
• Adapting Financial Assessment for AI Product Managers
• Step-by-Step Example to Assess an AI Investment
• Final Thoughts
Happy reading 🛋️
Before starting: If you’ve found my posts valuable, consider supporting my work. You can help by sharing, liking, and commenting on my LinkedIn posts introducing this issue. This helps me reach more
people on this journey.
Thank you for supporting and motivating me to continue sharing my experience with you. Thanks for being part of this community ❤️.
Every AI Product Manager needs to understand how to assess the financial viability of a proposed AI use case. In companies, potential AI use cases are proposed either by business teams to the AI team
or vice versa. Every single idea is worth at least being heard.
However, when it comes to running a cost-benefit analysis and determining financial viability, AI teams and their Product Managers aren't domain experts. They often struggle to understand the overall
implications of an AI solution on business processes and the KPIs it impacts, making it challenging to get the full picture.
Similarly, business teams may not fully grasp the financial implications of allocating people and resources to build an AI solution for their problem.
However, both perspectives are essential for a comprehensive analysis. So, no matter how sophisticated your formula is for calculating potential financial viability, your estimate will always be poor
if done in isolation. You need to collaborate with business teams, and they need to be cooperative to come to a sound estimation of the financial impact.
Everything else is just half-baked.
As an AI Product Manager, it’s important to be transparent from the beginning with your business colleagues. Make it clear that you will need their support to get the numbers right and to justify the
Having said this, let’s dive deeper into some financial concepts and terminology. We'll also fully derive the formula for a cost-benefit analysis to understand if investing in an AI solution for a
proposed problem is financially sound.
What is Financial Viability?
Financial viability refers to the ability of a project or business to generate enough income to cover its costs and achieve a reasonable profit. In the context of an AI use case, financial viability
means that the proposed AI solution should provide more financial benefits than it costs to develop and implement.
To determine financial viability, you need to consider various factors:
1. Costs: This includes all expenses associated with developing, implementing, and maintaining the AI solution.
2. Benefits: These are the financial gains expected from the AI solution. Benefits might include increased revenue, cost savings, improved efficiency, or other positive impacts on the business.
3. Return on Investment (ROI): This is a key metric that compares the benefits to the costs. A positive ROI means that the benefits outweigh the costs, indicating that the project is financially
By assessing these factors, you can get a clearer picture of whether an AI project is worth pursuing from a financial perspective. If you have already read my post "AI Initiatives are Investments.
Act like it," you might be curious about how finance departments decide when to invest and when not. We can learn a lot from their approach, so let’s delve into that.
How Finance Departments would Calculate the AI Investment Case
Finance departments use a structured approach to evaluate the potential return on investment (ROI) and ensure that the project aligns with the company's financial goals. Here’s how they typically
calculate the AI investment case:
1. Cost Assessment: Finance teams meticulously identify and document all costs associated with the AI project. This includes direct costs like software and hardware, as well as indirect costs such
as personnel, training, and ongoing maintenance.
2. Revenue Projections: They estimate the potential revenue generated by the AI solution. This involves analyzing how the AI project will impact sales, customer retention, market expansion, and
other revenue-driving activities.
3. Cost Savings: Finance departments evaluate how the AI solution can reduce operational costs. This might include savings from automation, improved efficiency, reduced error rates, and lower labor
4. Capital Budgeting Techniques: Finance departments use capital budgeting techniques such as Net Present Value (NPV), Internal Rate of Return (IRR), Return on Investment (ROI), and Payback Period
to assess the investment. These techniques consider the time value of money and help in comparing the AI project against other potential investments.
□ Net Present Value (NPV): NPV calculates the present value of cash inflows and outflows over the project's life. A positive NPV indicates that the project is expected to generate more value
than it costs.
□ Internal Rate of Return (IRR): IRR is the discount rate that makes the NPV of cash flows equal to zero. It represents the project's expected rate of return.
□ Return on Investment (ROI): ROI measures the profitability of an investment as a percentage of the initial cost. It gives a quick snapshot of the efficiency of the investment by comparing the
net profit to the initial investment.
□ Payback Period: This is the time it takes for the project to generate enough cash flow to recover the initial investment. A shorter payback period is generally preferred as it indicates
quicker recovery of the invested capital.
5. Risk Analysis: Finance departments conduct a risk analysis to understand potential uncertainties and their impact on the project. This involves scenario analysis and sensitivity analysis to see
how changes in key variables affect the project's financial outcomes.
6. Stakeholder Consultation: They collaborate with other departments, in this case, it would be the AI and business teams, to validate assumptions and ensure that all relevant factors are
considered. This ensures a holistic view of the project's financial viability.
7. Financial Reporting: Finance teams prepare detailed reports and presentations to communicate their findings to senior management and stakeholders. These reports include key metrics, financial
projections, and risk assessments.
By following this structured approach, finance departments ensure that AI investments are thoroughly evaluated and aligned with the company's strategic and financial objectives. This collaborative
process helps in making informed decisions and justifying the investment to all stakeholders.
Now, let’s see how this would look if we, as AI Product Managers, were to run such a calculation on our own. But first, let’s understand why we need three different metrics: ROI, NPV, and IRR to
assess an AI investment.
Why isn’t ROI sufficient?
Each metric provides unique insights and has its limitations. It's essential to consider all of them together to get a well-rounded view (if that’s the aim) of the investment's financial viability.
Here’s why:
Return on Investment (ROI)
• What it shows: ROI measures the profitability of an investment as a percentage of the initial cost.
• Why it's important: It gives a quick snapshot of the efficiency of the investment.
• Limitations: ROI does not account for the time value of money or the duration of the investment.
Net Present Value (NPV)
• What it shows: NPV calculates the present value of future cash flows minus the initial investment. It accounts for the time value of money by discounting future cash flows.
• Why it's important: NPV provides a direct measure of the added value from the investment, considering both the magnitude and the timing of cash flows.
• Limitations: NPV requires an accurate discount rate and can be sensitive to changes in this rate.
Internal Rate of Return (IRR)
• What it shows: IRR is the discount rate at which the NPV of an investment is zero. It represents the expected rate of return.
• Why it's important: IRR allows for comparison between projects with different sizes and durations, showing the efficiency of an investment.
• Limitations: IRR assumes that future cash flows are reinvested at the IRR, which may not be realistic. It can also give multiple values for projects with alternating cash flows.
Why Consider All Metrics Together
1. Comprehensive Evaluation: Each metric highlights different aspects of the investment. Using them together provides a more complete picture.
2. Cross-Verification: Comparing NPV, IRR, and ROI helps cross-verify the attractiveness of an investment. For example, a high IRR should ideally correspond to a positive NPV.
3. Different Perspectives: NPV focuses on value addition, ROI on efficiency, and IRR on the rate of return. Together, they cover value, cost-effectiveness, and potential growth.
4. Risk Assessment: Combining these metrics helps assess risk better. NPV shows the value at risk, IRR the efficiency under different scenarios, and ROI the basic profitability.
While each metric has its strengths, relying on only one can lead to an incomplete or misleading assessment. For a robust evaluation, consider NPV, ROI, and IRR together to understand the full
financial implications of an AI investment. This comprehensive approach is how finance departments typically evaluate investments.
🤔 But do we as AI Product Managers need to approach it like this?
Adapting Financial Assessment for AI Product Managers
As AI Product Managers, our role involves making quick and informed decisions to prioritize and advance AI initiatives. While adopting the thorough approach of finance departments can be beneficial,
there are practical considerations to keep in mind:
1. Speed and Efficiency: AI Product Managers often need to make swift decisions. Simplified assessments using just one or two metrics (like ROI or NPV) can be sufficient for initial evaluations and
2. Resource Constraints: AI teams may not always have the tools or expertise to conduct detailed financial analyses. Using basic metrics that are easier to calculate and understand can help in
making timely decisions.
3. Iterative Process: AI projects often evolve, and initial financial assessments might need adjustments. Starting with simpler calculations allows for flexibility and quick iterations based on new
data or insights.
4. Collaborative Approach: For critical decisions, AI Product Managers can collaborate with finance departments to leverage their expertise in conducting detailed analyses. This ensures a balanced
approach without overburdening the AI team.
Practical Approach
1. Initial Screening with ROI: Use ROI to quickly gauge the profitability of potential AI use cases. This helps in filtering out less promising projects early on.
2. Deeper Analysis with NPV: For shortlisted projects, calculate NPV to understand the value addition considering the time value of money.
3. Consult Finance for IRR: For high-stakes projects, involve finance departments to calculate IRR and conduct comprehensive risk assessments.
Example: Assessing the Financial Viability of an AI Investment
⚠️ This is a simplified version of an assessment. Finance departments will conduct these evaluations in more detail and may use different assumptions. All figures and assumptions here are fictitious and used for demonstration purposes.
A company is considering investing in an AI solution to automate its customer service operations. The initial development cost is estimated at $500,000, and the annual maintenance cost is $100,000.
The AI solution is expected to generate cost savings of $250,000 per year by reducing labor costs and improving efficiency. The project is expected to have a lifespan of 5 years.
Step-by-Step Calculation
1. Identify Costs
□ Initial Development Cost: $500,000
□ Annual Maintenance Cost: $100,000
□ Total Costs over 5 years:
\(Total Costs = $500,000 + $100,000 \times 5 = $1,000,000\)
2. Estimate Benefits
□ Annual Cost Savings: $250,000
□ Total Benefits over 5 years:
\(\text{Total Benefits} = \$250,000 \times 5 = \$1,250,000\)
3. Calculate Net Benefits
□ Net Benefits:
\(\text{Net Benefits} = \$1,250,000 - \$1,000,000 = \$250,000\)
4. Calculate ROI
□ ROI:
\(\text{ROI} = \left( \frac{\text{Net Benefits}}{\text{Total Costs}} \right) \times 100 \)
\( \text{ROI} = \left( \frac{\$250,000}{\$1,000,000} \right) \times 100 = 25\% \)
5. Consider the Time Frame
□ Since the project spans 5 years, we also consider the annual ROI:
\(\text{Annual ROI} = \frac{25\%}{5} = 5\% \)
6. Calculate NPV
□ To calculate the NPV, we need to make some assumptions about the discount rate. The discount rate is the interest rate used to determine the present value of future cash flows. It reflects
how much future money is worth today. For our example, let's use a discount rate of 10%. The NPV is calculated as follows:
\(\text{NPV} = \sum \left( \frac{\text{Annual Savings}}{(1 + \text{Discount Rate})^t} \right) - \text{Initial Cost} \)
\(\text{NPV} = \left( \frac{\$250,000}{(1 + 0.1)^1} + \frac{\$250,000}{(1 + 0.1)^2} + \frac{\$250,000}{(1 + 0.1)^3} + \frac{\$250,000}{(1 + 0.1)^4} + \frac{\$250,000}{(1 + 0.1)^5} \right) - \
$500,000 \)
\( \text{NPV} = \$227,273 + \$206,611 + \$187,828 + \$170,753 + \$155,230 - \$500,000 = \$447,695 \)
The discount rate is important because it helps us account for the fact that money today is worth more than the same amount in the future due to factors like inflation and risk. A higher
discount rate means we value future cash flows less because of higher risk or uncertainty, while a lower discount rate means we value them more. Choosing the right discount rate is key to
accurately assessing the NPV and understanding the financial viability of the project.
Why Use a 10% Discount Rate?
Using a 10% discount rate is a common practice for several reasons:
Standard Industry Practice: Many companies and industries use a 10% discount rate as a standard benchmark for evaluating investments. It provides a consistent basis for comparison.
Risk and Opportunity Cost: A 10% rate reflects a reasonable estimate of the opportunity cost of capital, which is the return that could be earned on an alternative investment of similar risk.
Inflation: It takes into account the average rate of inflation over time, ensuring that future cash flows are adjusted for the decreasing purchasing power of money.
Business Risk: For many businesses, a 10% rate appropriately reflects the risk associated with future cash flows, balancing the potential uncertainties and rewards.
7. Calculate IRR
□ For simplicity, we use financial software or a calculator to find that the IRR for this project is approximately 16%. You can use online tools or the =IRR() function in Excel. However, you
will soon understand why I will not explain this further in detail 😉.
8. Calculate Payback Period
□ Payback Period is the time taken to recover the initial investment from the annual savings:
\(\text{Payback Period} = \frac{\$500,000}{\$250,000} = 2 \text{ years}\)
• Total Costs: $1,000,000
• Total Benefits: $1,250,000
• Net Benefits: $250,000
• ROI: 25%
• NPV: $447,695
• IRR: 16%
• Payback Period: 2 years
✅ This example demonstrates that the AI investment is financially viable, with a positive NPV, a solid ROI of 25%, a favorable IRR of 16%, and a quick payback period of 2 years. These calculations help the finance department and other stakeholders make informed decisions about investing in the AI solution.
Final Thoughts
One might ask, if finance departments are conducting these assessments, why should an AI Product Manager do the work as well? Yeah, I know, it was also my first question in my early days as an AI
Product Manager. But…
• Not all AI teams need to go through the finance department approval process, as their teams are already pre-funded and just need to find AI use cases and implement solutions. For these teams, it
is mandatory to understand how to assess financial viability on their own and prioritize accordingly.
• Other teams do need to go through the approval process, but imagine having numerous AI use cases, each requiring approval. Typically, finance departments are not solely focused on assessing AI
investments but handle all types of company investments. This means their backlog is already full. Thus, the approval process might take some time. And do we have time?
Surely, we don’t.
We never have time 🙂
So, why not request financial assessments only for those cases we know are good investments?
It’s not a difficult calculation, but it's one that I rarely see being done by AI teams. I regularly run these kinds of quick assessments, which help me determine exactly which cases make the most
sense to focus on. Sometimes, I focus on the ROI, while other times I look at the NPV to quickly make an evaluation. If neither of these metrics is promising, I don’t bother with running the IRR.
Maybe a finance colleague would do a facepalm 🤦♂️, but hey, it has worked so far.
So, if you are keen to invest your time wisely, think like a pragmatic investor 😉 I’m pretty sure it’s worth it.
Give it a try.
JBK 🕊
P.S and a friendly reminder: If you’ve found my posts valuable, consider supporting my work. You can help by sharing, liking, and commenting on my LinkedIn posts introducing this issue. This helps me
reach more people on this journey.
Thank you for supporting and motivating me to continue sharing my experience with you. Thanks for being part of this community ❤️.
Thanks for reading A World Beyond AI! Subscribe for free to receive new posts and support my work. | {"url":"https://www.jaserbk.com/p/how-to-assess-financial-viability","timestamp":"2024-11-02T18:53:39Z","content_type":"text/html","content_length":"211887","record_id":"<urn:uuid:7aa240b9-96f4-407e-b238-c503314efcdc>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00847.warc.gz"} |
Serial correlation & Seasonality!?
At first, i thought i had a clear understanding on serial correlation & seasonality. But now, these two confuses me. What is difference between these two? the way to detect them is the same? use
t-value? And if serial correlation is found in AR(1) model, does it mean that this AR(1) model has seasonality problem as well?. So need to add one more lag to improve it? then AR(1) will become AR
(2); AR(3)…? Right? Thank you to everyone. Cheer up.
Serial correlation is when error terms are not independent, meaning the previous residual/error term will give some sort of prediction for the next error term. Positive serial correlation means the
error term in the next period will likely be positive if the current error term is positive. Negative serial correlation means the error term in the next period will likely be negative (positive) if
the current error term is positive (negative). It really means that a predicted value will have a correlation with the previous predicted value, so we should try using an autoregressive model for our
predictions. You use autocorrelation tests to see if autocorrelation is present, specifically the Durbin-Watson test. If serial correlation exists in the AR(1) model, it is mispecified or you should
add more lags, test for autocorrelation again, and repeat until there are no more statistically significant correlations remaining
How about seasonality? My confuse is the difference or connection between autocorrelation and seasonality. Really appreciate ur great comment.
Seasonality simply refers to a situation where the correlation seems to occur at a specific lag (usually 4th lag for quarterly data, or 12th lag for monthly data). Seasonality is a subset of
autocorrelation. Autocorrelation is a generalized term referring to the presence of correlation in errors terms irrespective of which term is correlated with which.
But you don’t use the Durbin-Watson method for autoregressive models I don’t think. You use: t-statistic=[correlation(error at time t, error at time t-1)]/(1/squareroot of t), with t-2 degrees of
Right rellison, sorry I didn’t mean to mislead you, I meant that Durbin-Watson is used to test autocorrelation in a normal time series model (I placed it in the wrong place in the paragraph)
Serial Correlation & Seasonality are two totally different concepts. Serial correlations occurs when the residual errors are correlated with each other. I.E. errors in one time period affect the
errors in other time periods. This is going to affect your standard errors, thus t-values and p-values. Seasonality - Think of Christmas, during the holidays people buy a ton of stuff, more then any
other month. so you have to adjust the pooling of your data to adjust for this holiday effect, otherwise the sample data is incorrect leading to incorrect estimated coefficients. How would you
correct for seasonality?
Put a lag on that b@tch! | {"url":"https://www.analystforum.com/t/serial-correlation-seasonality/33288","timestamp":"2024-11-03T11:04:56Z","content_type":"text/html","content_length":"33262","record_id":"<urn:uuid:1838a06e-1df4-42ea-97a1-b37b3aed48fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00652.warc.gz"} |
Ti-84 solver complex
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One of the funniest signature lines I have seen in some time is:
There are 10 kinds of people:
Those who understand binary and those who don't
By Joshua Erdman
Digital Foundation
Let's say I gave you a piece of paper with value of 237 written in binary: 1 1 1 0 1 0 0 1
Now while you had the paper it got severely crumpled and now here is what you can make out so far: 1 X 1 0 1 0 0 1
Now because you cannot determine the second character, you do not know the value'. This is where parity steps in.
The next paper I give you still has the value of 237 but I also told you that I will include a 9th bit that is a '1' if the first 8 bits have an odd number of 1s. It will be a '0' if the first 8 bits
are an even number of 1s.
Since there are 5 '1's in the binary value of 237 the 9th bit will be a '1'. So here is what I wrote on the new piece of paper: 1 1 1 0 1 0 0 1 1
Now you receive the paper and once again you cannot read the 2nd bit.
Here is what you can see: 1 X 1 0 1 0 0 1 1
Now since we know the rule about the 9th bit we know the first 8 bits will have to be an odd number of '1's since the 9th bit is a '1'. Right now we have only 4 '1's so we know that the missing bit
should be a '1' as well.
You can try it some more on any of the other 8 bits and see that it still works. And if the 9th bit is garbled that is still OK because we have all the data we need from the first 8.
Clue: Parity is not limited to 8 data bits and a parity bit either. You can work with with as few as 2 data bits and a parity bit or as many data bits as needed. However realize that if any 2 bits
were garbled, the system would not work. And as you increase the total number of data bits, you are more likely to have too much data corruption for the parity bit to be useful. | {"url":"https://vmaxx.net/techinfo/tcp/parity.htm","timestamp":"2024-11-06T21:56:57Z","content_type":"text/html","content_length":"2961","record_id":"<urn:uuid:1da90ca7-19ce-419a-afd8-6e896a45c694>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00763.warc.gz"} |
Vectors and Geometry Part -15
VECTORS AND GEOMETRY PART -15
Properties of sphere
1.Studies properties of sphere
2.Introduces the Radical plane
3.Familiarise the conditions of tangency
In this session we discuss equation of a tangent plane at a point on the sphere,condition of tangency,some of the properties of sphere, and the idea of radical plane.
1. Equation of a tangent plane at a point
To find the equation of the tangent plane at a point p(x1, y1, z1) on the sphere ……….(1)
The point p(x1, y1, z1) lies on
(1) hence
Any line through the point (x1, y1, z1) is
Where l ,m,n are actual direction cosines .if (3) meets
(1) in points then the distance of points of intersection from p are given by
One value of r is zero which shows that p is a point on the
sphere .now the line (3)will be tangent line if the other value of r also coincide with p(x1, y1, z1) .the other value of r will be zero if ……………. (4)
Thus (3) will be a tangent line to (1) if it satisfies condition (4).to obtain the locus of line (3)for different values of l,m,n we will eliminate l,m,n from (3) and (4) .obviously this locus will
be the tangent plane to the sphere at (x1,y1,z1) .eliminating
l,m,n from (4) with the help of (3) we get
(x-x1)(x1+u)+(y-y1)(y1+v)+(z-z1)(z1+w)=0 or xx1+yy1+zz1+ux+vy+wz=x12+y12+z12+ux1+vy1+wz1
adding ux1+vy1+wz1+d to
both sides and using (2)we get
xx1+yy1+zz1+u(x+x1) +v(y+y1)+w(z+z1)+d=0.
this is the equation of tangent plane at (x1,y1,z1)
Remark:To write equation of a tangent plane ,replace x2 to xx1
y2 to yy1,z2 tozz1,2x to
x+x1,2y to y+y1 and 2z to z+z1 in the equation of the sphere
Theorem 1:To prove that the tangent plane at any point of a sphere is perpendicular to the radius through that point
solution: Let the equation to the sphere be
Equation of the tangent plane at a point (x1,y1,z1) is
xx1+yy1+zz1+u(x+x1)+v(y+y1)+w(z+z1)+d=0,hence direction ratios of a normal to the tangent
plane (5) are x1+u,y1+v,z1+w .again coordinates of the centre of the sphere are (-u,-v,-w).hence the direction ratios of the line joining the centre (-u,-v,-w)to the point (x1,y1,z1) are x1-(-u),y1-
(-v),z1-(-w) ie
x1+u,y1+v,z1+w . these are the same as that of the normal to the tangent planes . hence the
tangent plane is perpendicular to the radius through the point of contact.
Theorem2: To show that the tangent line to a sphere is
perpendicular to the radius through the point of contact.
Solution: let the tangent line at (x1,y1,z1) be
This will be a tangent line to the sphere
if =0
this relation shows that line whose direction cosines are l,m,n is at right angles to a line whose direction ratios are x1+u,y1+v,z1+w ie the radius through the point of contact.
Condition of tangency
To find the condition that the
plane lx+my+nz=p is a tangent plane to the sphere
Centre of the given sphere is
(-u,-v,-w) and its radius is
If plane lx+my+nz=p touches the sphere , then perpendicular distance of the
plane from the centre of the sphere must be equal to its radius ie
Or (Lu+mv+nw+p)2 =(L2+m2+n2) (u2+v2+w2-d) this is the required condition
Example1:Find the eqation of a sphere which touches the sphere
at (1,2,-2) and passes through the point (1,-1,0)
Solution: Equation of tangent plane to the given sphere at (1,2,-2) is
X(1)+y(2)+z(-2)+(x+1)-3(y+2)+1=0 or 2x-y-2z-4=0
Let the required sphere be
As it passes through the origin so or .hence the equation of the required sphere be
2.properties of a sphere
Theorem3:The condition that the plane a x + b y + c z + d = 0 may touch the sphere is .
Proof : If the given plane is a tangent plane to the sphere, then the length of the perpendicular p drawn from the centreC(-u, -v, -w) is equal to the radius r of the given sphere. That is .
Hence the condition for tangency is
• The radius of the circle of intersection of the sphere by the plane is given by
• The length of the tangent line drawn from to the sphere S = is given by
which can also be written as
Definition :If C is the centre of a sphere and r, its radius and P, any point in space, then is called the power of the point P with respect to the sphere . Hence the power of with respect to the
which is positive or negative according as P lies outside or inside the sphere.
Theorem 4: The vector equation of a sphere
at the point A whose position
vector is
Proof : Let C be the centre and its position vector be c. Let P be any point on the plane through and perpendicular to CA . Let its position vector be r. Now CA is perpendicular
to AP. So the equation of this plane is
or (7)
The condition that A is a point on the sphere is obtained by setting
in equation (6) and is
So addition of (7) and (8) gives the equation of the tangent plane as
Corollary 1 :Condition for the plane , where n is the unit vector , to touch the sphere is
Example2: A sphere of constant radius 2k passes through the origin and meets the axes in A,B,C. Find the locus of the centroid of the tetrahedron OABC
Solution: Let Co-ordinates of the points be and respectively.
Then equation of the sphere
OABC is
Radius of this sphere is given equal to 2k.
Let be the coordinates of the centroid of the tetrahedron OABC; then
, so that
Eliminating from the above equation , the required locus is
Example 2 :A sphere whose centre lies in the positive octant passes through the origin and cuts the planes , in circles of radii
respectively show that its equation is
Since the sphere passes through the origin, let its equation be
Plane cuts it in
The radius of this circle is .
These gives
Substituting the values of and w in , we get the required equation.
Radical plane
Suppose that the equations of the two given spheres are
If is a point which moves such that the tangents from it to the spheres are of equal length, then the equation of the locus of P is obtained from
as or
This represents a plane to the line of centres. This plane is called radical plane of the spheres. The radical plane also can be defined as the locus of a point which moves such that its power with
respect to the spheres are equal.
Now let us summaries what we have discussed ,here we discussed equation of a tangent plane at a point on the sphere with supporting theorems .condition of tangency is also discussed, at last we moved
to some of the properties of sphere together with radical planes.
Before moving to the next session let us try these questions
Hope u enjoyed the session see u next time till then good bye
1. Obtain the equation of the tangent plane at the origin to the sphere
2. Prove that the sum of the squares of the intercepts made by a given sphere on any three mutually perpendicular lines through a fixed point is constant
3. Find the locus of the centre of the sphere of constant radius which pass through a given point and touch a given line .
1.Find the equation of spheres which pass through the circle
And touch the plane
Equation of a sphere through the given circle is
This will touch the plane 3x+4z=16 if distance of plane from centre
This will give two values of
Corresponding to each value there being a sphere
2. How two spheres touch internally?
Solution ;Two spheres touch internally if the difference of their radii is equal to the distance between their centers
POWER OF THE POINT:If C is the centre of a sphere and r, its radius and P, any point in space, then is called the power of the point P with respect to the sphere
1.The centre of the sphere
(a) (-8,6,-4)(b)(-4,3,-2) (c) (8,-6,4) (d)(16,12,8)
2.The radius of the sphere
(a)45(b)36(c)25 (d)100
1 (b) (-4,3,-2)
1S.L.Loney The Elements of Coordinate Geometry ,Macmillian and company, London
2Gorakh Prasad and H.C.Gupta Text book of coordinate geometry, Pothisala pvt ltd Allahabad
3P.K.Mittal Analytic Geometry Vrinda Publication pvt Ltd,Delhi. | {"url":"https://docest.com/doc/690528/vectors-and-geometry-part-15","timestamp":"2024-11-06T07:32:32Z","content_type":"text/html","content_length":"30497","record_id":"<urn:uuid:587a5b3a-12e4-4f09-982e-37bd296e3d15>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00738.warc.gz"} |
Draw Base 10 Blocks
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Web base ten blocks are designed to help students understand place value. Web we all know how important it is for students to practice representing numbers using base 10 blocks. Look at the example
in the first line. Tables · sporting goods · tool boxes & storage · egg cups · pads · labels Web.
What are Base 10 Blocks? Video & Lesson Transcript
Regrouping into blocks of 10. By counting tens and ones, students gain an understanding of how numbers are created. Numbers are explained and built using blocks to represent place value. These small
blocks represent numbers and are perfectly aligned to our base ten place value system. Use the tool to represent (draw) each number. Web.
Representing Numbers by Drawing Base Ten Blocks YouTube
Counting using base 10 blocks. This is a concrete method for teaching students how to add larger numbers so they develop a conceptual understanding. Look at the example in the first line. Blocks are
mathematical manipulative used to encourage students to learn various math concepts like addition, subtraction, place value, and counting. Web base ten.
Using Base 10 Blocks to Write Numbers Worksheet by Teach Simple
Explain that the decimal part refers to a fraction of a whole dollar as opposed to a whole number of nickels, dimes, or quarters. Unit blocks (ones), rods (tens), flats (hundreds), and cubes
(thousands). Web use base ten blocks and grids to represents the decimal fractions and decimal numbers. Students will draw base ten place.
Draw Base 10 Blocks Web base ten blocks are an engaging way to teach place value, measurements, number concepts, and more. This is a concrete method for teaching students how to add larger numbers so
they develop a conceptual understanding. This video is targeted for grade 2. Explain that the decimal part refers to a fraction of a whole dollar as opposed to a whole number of nickels, dimes, or
quarters. After you use the manipulatives, you can use these worksheets to reinforce their understanding.
Web The Numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 Refer To One, Two, Three, Four, Five, Six, Seven, Eight, Or Nine Hundreds (And 0 Tens And 0 Ones).
Breaking a number into tens and ones. Web base ten blocks worksheets 2nd grade are designed to help students to practice composing and decomposing numbers into their base ten numbers. Students gain
proficiency in counting by ones to identify a set of ten. Model and write numbers using base ten.
Click On The After You Complete All Your Work.
Rods represent the tens place and look like ten cubes placed in a row and fused together. Regrouping into blocks of 10. Rather than refer to the pieces as 1, 10, 100 and 1000, the terms unit (cube),
rod , flat and cubes are used. Web we all know how important it is for students to practice representing numbers using base 10 blocks.
Make Sure You Use The Lines In The Paper To Help You Draw Your Base Ten Blocks.
When you use the manipulates that the school provides, you can only represent numbers in the thousands place, and then, only a few thousand or it gets to be too much. Web base ten blocks are designed
to help students understand place value. Ten copies of the “rod” placed side to side exactly match the. Amazon.com has been visited by 1m+ users in the past month
Ten Copies Of The Smallest Block (Variously Named “A Bit” Or “A Tiny Cube”) Lined Up In A Straight Row Exactly Match One Of The Long Rods (Variously Called Called “A Rod” Or “A Long”).
Base ten blocks (also known as base 10 blocks and place value blocks) is an online mathematical manipulative that helps students learn addition, subtraction, number sense, place value and counting.
Use alone or with the lesson plan **let's build it! I’m excited to share with you today this brand new. They help students physically represent numbers so they can develop a deeper understanding of
place value and regrouping and trading.
Draw Base 10 Blocks Related Post : | {"url":"https://sandbox.independent.com/view/draw-base-10-blocks.html","timestamp":"2024-11-03T20:35:55Z","content_type":"application/xhtml+xml","content_length":"25068","record_id":"<urn:uuid:47c56e00-c88d-4e43-a0d9-1802af90c731>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00641.warc.gz"} |
Lesson plan: Area of a Triangle
Gulbaba Zeynep Yıldırım Yatılı Bölge Ortaokulu in Kilis, TR
Students will find the area of different Triangles.
Grade 7
Base is the side of a triangle. Base is the same as length. Height of a triangle forms a right angle with the base and extends to the highest point of the triangle.
S1 problem solving: Student 2will build new mathematical knowledge through problem solving. S12 geometry: Student will specify location and describe spatial relationships using coordinate geometry
and other representational systems. S3 measurement: Student will apply appropriate techniques, tool, and formula to determine measurements
Students will recognize and define three different triangles. Students will learn and apply the use the formula to find the area of a triangle.
Overhead projector or Dry Erase board, grid paper, pencil, overhead marker, triangle shapes and photos, math book, ruler
Explain that a parallelogram can be split in two triangle of equal area. What are the three triangle names and how are they different and similar? Why might you need to be able to figure out the area
of these triangle? What career using this information? How is it applied to daily life?
Introduce the three types of triangles; obtuse, acute, and right. Review how to find the area of a rectangle. Use this to seg-way into finding the area of a triangle. Explain the formula; Area=.5
(base)(height). Students will then work with different triangles and their measurements in order to find the area. Discuss why knowing how to find the area of a triangle is important. Help the
students realize that a carpenter or architect will need to know this information.
Explain to the students they are now carpenters. Have them pair up and go around the classroom finding examples of the three types of triangles. Then they measure, with a ruler, each side of the
triangle they found recording their results.
Pair students in such a manner that those needed extra guidance are with a student who excels in this subject area. Since the students will be roaming around the room the use of the classroom speaker
system will be helpful to keep order and insure all of the students hear instructions.
Checking for understanding:
Students will complete pg. 441 in their textbooks.
Students will be evaluated on class and group participation. They will also be graded on the textbook page assignment.
Teacher reflections: | {"url":"https://www.colourfulnumbers.eu/2021/10/06/lesson-plan-area-of-a-triangle/","timestamp":"2024-11-06T02:44:08Z","content_type":"text/html","content_length":"55825","record_id":"<urn:uuid:b9769b79-3dc7-4619-8191-b02f803bb2df>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00412.warc.gz"} |
Teaching Number Concepts - Natural Math
Teaching Number Concepts
I am trying to teach my son a concept of positive whole numbers being made up of other, smaller, positive whole numbers. This has been a tough going so far, full of unexpected obstacles. There was,
for example, the part where I tried to explain and show that although a larger number can be made up of smaller numbers, it doesn’t work in reverse and a smaller number cannot be made up of larger
An even more formidable obstacle was (and still is) showing that a larger number can be made out of various combinations of smaller numbers. Say, 5=2+3, but also =4+1 and even 1+2+2. And by showing I
mean proving. And by proving, I mean having my son test the rule and prove (or disprove) it to himself.
That’s why I was very happy when I got a hold of Oleg Gleizer’s book Modern Math for Elementary School. By the way, the book is free to download and use. We’ve been building and drawing multi-story
buildings (mostly Jedi academies with x number of training rooms) ever since. If this sounds cryptic, I urge you to download the book and go straight to page 12, Addition, Subtraction and Young
And just yesterday I found this very simple activity on Mrs. T’s First Grade Class blog, via Love2Learn2Day‘s Pinterest board. All you need for it is a Ziploc bag, draw a line across the middle with
a permanent marker, then add x number of manipulatives. Took me like 2 minutes to put it together, mostly because I had to hunt for my permanent marker.
The way we played with it was I gave the bag to my son and asked him how many items were in the bag. He counted 8. I showed him that the bag was closed tight, so nothing could fall out of it or be
added to it. I also put a card with a large 8 on it in front of him as a reminder. At this point all 8 items were on one side of the line. I showed him how to move items across the line and let him
play. As he was moving the manipulatives, I would simply provide the narrative:
Ok, so you took 2 of these and moved them across to the other side. Now you have 2 on the left and how many on the right? Yes, six (after him counting). Two here and six here. Two plus six. And
how many items do we have in this bag? Good remembering, there are 8. So two plus six is 8. Want to move a few more over?
It went on like this for a few minutes until he got bored with it. Overall, I thought it was a good way of teaching, especially for children who do not like or can’t draw very well yet. Plus upping
the complexity is really easy – draw more than one line on the bag and create opportunities for discovering that a number can be made of more than two smaller numbers.
You can illustrate/manipulate the same concept showing all possible variations using Cuisenaire rods pretty easily. It comprises several steps in the Mathematics Made Meaningful instruction card kit.
• T, I’ve heard great stuff about Cuisenair rods and will definitely add them to our library of manipulatives.
□ I had daughter prepare an example for you to see. If you have an email address, I’ll be happy to share the picture.
☆ of course, my e-mail is yelena@moebiusnoodles.com
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Commutator of position and momentum
• Thread starter Kara386
• Start date
In summary, the commutator ##[\hat{p}_x, \mathbf{\hat{r}}]## can be expanded by taking the individual commutators of ##\hat{p}_x## with each component of ##\mathbf{\hat{r}}##, resulting in ##([\hat
{p}_x, \hat{x}], [\hat{p}_x, \hat{y}], [\hat{p}_x, \hat{z}])##.
How would ##[p_x, r]## be expanded? Where ##r=(x,y,z)##, the position operators. Do you do the commutators of ##p_x## with ##x, y,z## individually? So ##[p_x,x]+[p_x,y]+[p_x,z]## for example?
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Kara386 said:
How would ##[p_x, r]## be expanded? Where ##r=(x,y,z)##, the position operators. Do you do the commutators of ##p_x## with ##x, y,z## individually? So ##[p_x,x]+[p_x,y+p_x,z]## for example?
More generally, a vector operator such as ##\mathbf{\hat{r}}## represents three operators ##(\hat{x}, \hat{y}, \hat{z})##, related in the same way as the components of a vector.
In this case, essentially by definition:
##[\hat{p}_x, \mathbf{\hat{r}}] = ([\hat{p}_x, \hat{x}], [\hat{p}_x, \hat{y}], [\hat{p}_x, \hat{z}])##
PeroK said:
More generally, a vector operator such as ##\mathbf{\hat{r}}## represents three operators ##(\hat{x}, \hat{y}, \hat{z})##, related in the same way as the components of a vector.
In this case, essentially by definition:
##[\hat{p}_x, \mathbf{\hat{r}}] = ([\hat{p}_x, \hat{x}], [\hat{p}_x, \hat{y}], [\hat{p}_x, \hat{z}])##
Ah, thank you. :)
FAQ: Commutator of position and momentum
What is the commutator of position and momentum?
The commutator of position and momentum is a mathematical operator that describes the relationship between the position and momentum of a particle in quantum mechanics. It is denoted by [x,p] and is
defined as [x,p] = xp - px, where x is the position operator and p is the momentum operator.
What does the commutator of position and momentum tell us?
The commutator of position and momentum tells us about the uncertainty in simultaneously measuring the position and momentum of a particle. It is related to the Heisenberg uncertainty principle,
which states that the more precisely we know the position of a particle, the less precisely we know its momentum, and vice versa.
How is the commutator of position and momentum related to classical mechanics?
The commutator of position and momentum is related to classical mechanics through the correspondence principle, which states that in the classical limit (when h, the Planck's constant, approaches 0),
the commutator of two operators is equal to the Poisson bracket of the corresponding classical quantities. In the case of position and momentum, the Poisson bracket is equal to the classical
expression for the product of position and momentum.
What is the physical significance of the commutator of position and momentum?
The commutator of position and momentum has physical significance as it is related to the uncertainty principle and plays a crucial role in the quantum description of particles. It also helps us
understand the wave-like nature of particles, where their position and momentum are not definite but rather described by a probability distribution. It also appears in the formulation of important
equations such as the Schrödinger equation and the Heisenberg equation of motion.
How is the commutator of position and momentum used in quantum mechanics?
The commutator of position and momentum is used in quantum mechanics to describe the relationship between these two fundamental quantities and to derive important equations and principles, such as
the uncertainty principle. It is also used in the calculation of commutation relations between other operators and in the formulation of the Heisenberg uncertainty principle. It is an essential tool
for understanding the behavior of particles at the quantum level. | {"url":"https://www.physicsforums.com/threads/commutator-of-position-and-momentum.901627/","timestamp":"2024-11-13T06:10:00Z","content_type":"text/html","content_length":"82666","record_id":"<urn:uuid:9db6d535-5747-4d83-a19c-ad1bb06f64ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00520.warc.gz"} |
Let \(X, Y, Z\) be Banach spaces, let \(T\) be a bounded linear operator from \(X\) into \(Y\) such that \(T(X)\) is closed in \(Y\), and let \(S\) be a finite-rank operator from \(X\) into \(Z\)
(that is, \(\operatorname{dim}(S(X))\) is finite). Define \(U: X \rightarrow Y \oplus Z\) by \(U(x)=(T(x), S(x))\). Show that \(U(X)\) is closed in \(Y \oplus Z\)
Short Answer
Expert verified
Since \(T(X)\) is closed in \(Y\) and \(S(X)\) is finite-dimensional, \(U(X)\) is closed in \(Y \oplus Z\) by properties of closed and finite-dimensional sets.
Step by step solution
- Define the Direct Sum
Recall that the direct sum of two Banach spaces, denoted by \(Y \bigoplus Z\), is the space of pairs of elements from each space, i.e., \((y, z)\) for \( y \in Y \) and \( z \in Z \).
- Construct Operator U
Given \(T\) and \(S\), define the operator \(U: X \rightarrow Y \bigoplus Z\) by \(U(x) = (T(x), S(x))\) for all \(x \in X\).
- Characterize the Image of U
Observe that \( U(X) = \{(T(x), S(x)) \in Y \bigoplus Z : x \in X\}\). Note that \(T(X)\) is closed in \(Y\) and \(S(X)\) is finite-dimensional.
- Use Properties of Closed Sets and Finite Dimensions
Since \(T(X)\) is closed and \(S(X)\) is finite-dimensional, \(S(X)\) is also closed in \(Z\) by the property that finite-dimensional subspaces of Banach spaces are closed.
- Prove U(X) is Closed
By the definition of the direct sum, the image \(U(X)\) is closed in \(Y \bigoplus Z\) since it is the product of closed sets \(T(X)\) and \(S(X)\). Therefore, \(U(X)\) is closed in \(Y \bigoplus Z
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
bounded linear operator
In Banach spaces, a bounded linear operator is a type of linear map between two Banach spaces where the size of the output is controlled by a fixed multiple of the size of the input. Formally, let \
(X\) and \(Y\) be Banach spaces, and let \(T\) be a linear operator from \(X\) to \(Y\). Then, \(T\) is said to be bounded if there exists a constant \(C\) such that for all \(x \in X\), \|T(x)\| \
leq C \|x\|. This ensures that \(T\) is continuous.
Bounded linear operators play a crucial role in functional analysis due to their predictable behavior. For example, in our exercise, \(T\) is a bounded linear operator which helps us to define
another operator \(U\) composed of \(T\) and another operator \(S\). The boundedness of \(T\) ensures that \(U\) will behave well when mapping from one Banach space to another.
finite-rank operator
A finite-rank operator is a special type of bounded linear operator where the image (or range) has finite dimensions. Let \(X\) and \(Z\) be Banach spaces, and \(S\) be a linear operator from \(X\)
to \(Z\). \(S\) is a finite-rank operator if the dimension of \(S(X)\), the image of \(S\), is finite. More formally, this means that there exists a finite-dimensional subspace \(F \,\subseteq\, Z\)
such that \(S(X) \subseteq F\).
These operators are significant because their properties are easier to handle compared to operators with infinite-dimensional ranges. For instance, in the exercise, the finite-rank nature of \(S\)
guarantees that \(S(X)\) is closed in \(Z\). This property is used to show that the combined operator \(U\) has a closed image in the direct sum space, \(Y \bigoplus Z\).
direct sum
The direct sum of two Banach spaces \(Y\) and \(Z\), denoted by \(Y \oplus Z\), is a new Banach space that pairs each element from \(Y\) with each element from \(Z\). Formally, \(Y \oplus Z\)
consists of all ordered pairs \((y, z)\) where \(y \in Y\) and \(z \in Z\). This new space inherits properties from both \(Y\) and \(Z\) and is typically equipped with a norm defined by \(\|(y, z)\|
= \|y\| + \|z\|\).
The direct sum is a powerful concept as it allows combining spaces and operators to form new ones. In our exercise, we define the operator \(U(x) = (T(x), S(x))\) which maps to \(Y \oplus Z\).
Understanding the direct sum helps us comprehend how \(U\) functions and why \(U(X)\) being closed in \(Y \oplus Z\) follows from the properties of \(T\) and \(S\).
closed subspace
A subspace is a subset of a Banach space that is itself a Banach space with the inherited operations and norms. A closed subspace is one where if a sequence of points within the subspace converges to
a limit point, that limit point also lies within the subspace. This is a critical concept because many results in functional analysis rely on the completeness of these subspaces.
For example, in our exercise, both \(T(X)\) and \(S(X)\) are closed subspaces of \(Y\) and \(Z\), respectively. This property ensures that their direct sum, \(U(X)\), is closed in \(Y \oplus Z\).
Closed subspaces are vital in proving that operators like \(U\) have well-behaved images, allowing us to apply various theorems from topology and functional analysis to infer further properties about | {"url":"https://www.vaia.com/en-us/textbooks/math/functional-analysis-and-infinite-dimensional-geometry-0-edition/chapter-5/problem-28-let-x-y-z-be-banach-spaces-let-t-be-a-bounded-lin/","timestamp":"2024-11-14T05:28:51Z","content_type":"text/html","content_length":"252300","record_id":"<urn:uuid:acb91394-598e-42af-b4b8-86a4df1b33aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00031.warc.gz"} |
Decision Tree - How it works? | Entropy & Gini Index | Overfitting
Decision Trees
In this article we will learn about what is a Decision Tree? How it works? Why Decision Tree Overfitts & How can we resolve it?
In this article we will learn about what is Decision Tree and How it works. How it gets Overfitted and how can we resolve Overfitting.
What are Decision Trees?
Decision Tree is a rule-based algorithm in which, to predict a result we answer a set of questions and come up to a decision.
If we illustrate this process in a diagram that can be easily understandable to everyone, it comes up like a tree.
In real life we might have come across a situation in which we have a questionnaire session and finally we get a solution.
For example: To check whether a patient is ready for “CT scan with Contrast”. Ready if 1. Patient is allowed to drink clear liquid but not solid food, three hours prior to their CT scan examination.
• Did you Eat? Or Drink?
□ If Eat – Rejected
□ If Drink – Did you consume only clear liquids?
☆ If Yes – Eligible for Examination,
☆ If No – Rejected.
Qualify for CT Examination – Decision Tree
Now, we can clearly understand that how useful a questionnaire is.
We are going to steal this technique to predict target variable and we call it as Decision Tree.
Isn’t this technique reminding you about Flow Chart and easy to understand?
In the above tree, each box identified as a Node.
The first Node is called the Root Node.
The Nodes in Orange and Green are final result of the Questionnaire. These Nodes are called Terminal Nodes, which has the final Answer.
All the remaining intermediate Nodes are called Internal Nodes.
How the Decision Tree Algorithm Works:
As the questioning person knows on what factor should be considered first and what decision should be taken, it’s easy for them to decide. This is how humans can think.
But how the machine can work on it?
How can a program can identify which question comes first?
In our previous “CT scan with Contrast” example, we need to find which question out of all possible questions make a short questionnaire to come up with a decision.
We segregated the Eat and Drunk people in the first step itself. So, we can reject who consumed solid food in first step itself. Think of the other way and come up with your steps. You will see 1 or
few more steps finally.
In the below Image #1, if you want to separate oranges and watermelons by a horizontal or vertical line, then the best line that separates well would be a vertical line in the middle. Agree?
Image #1
If it is a Vertical Line, then the feature existing in X axis plays here.
i.e. We want to make both divided parts should have similar class values. In that way we can decide that whatever comes the left category should be an orange and whatever comes right should be a
Image #2
For example, in the below Image #3, you can see that the first three pictures are not partitioning well to segregate the items. In the 4^th picture, it is the most possible good partitioning, so that
the further steps for partitioning is minimized.
Image #3
Now after dividing the data, the two regions are having same type of fruits (Same Class Values). We call it as “Purity”.
How can a Machine find this best line and find the purity of Nodes?
To do so, first we need to pick the first Root Node and where to split.
This will be done using anyone of the below 2 methods:
1. Information Gain (Uses Entropy).
2. Gini Index.
Both Gini and Entropy talks about the “The measure of Uncertainty”.
Image #4
In the above Pictures, A is pure as it is having only 1 class. B has impurity but it can give medium knowledge on what class is dominating the region. But in C it has equal number of Suns and Stars.
We cannot come up with any decision here. This node has high impurity.
Information Gain (Uses Entropy):
Information gain = Entropy of target – Entropy of attribute
In our example, Entropy of Target is Entropy of the Fruit Column. Entropy of attribute is nothing but the entropy of a feature (predictors/ Columns).
Let’s take,
Information Gain by Shape = Entropy of Fruit – Entropy of Shape = 0.25
Information Gain by Color = Entropy of Fruit – Entropy of Color = 0.35
Here the Entropy of Colour is highest Value.
So, Color will be the Root Node.
Gini Index:
The Gini Index is calculated by subtracting the sum of the squared probabilities of each class from one.
Here Pj is the probability of an object being classified to a particular class.
Let’s take,
Gini Index by Shape = 0.31
Gini Index by Colour = 0.23
Here the Gini Index of Colour is the lowest Value.
So, Color will be the Root Node.
Gini Index Vs Information Gain:
Gini Index or Information Gain (Entropy), both works similar except that the formula is different.
Each feature will be taken and the Gini Index will be calculated. A feature that gives Lowest Gini score is used as Root Node or Decision Node.
Similarly, for each feature the entropy value is calculated and then the Information Gain score is calculated for that feature. Finally the feature that gives Highest Information Gain will be
taken as Root Node or Decision Node.
Similarly, the remaining internal nodes will be split further using Gini/Entropy.
However, we can control this parameter in algorithm.
Entropy is the oldest Method that uses Log in computation. So, it is slower than Gini Score calculation.
So, Gini Index is the most used method as it is faster than Entropy. But both the methods will provide same result.
We are not going deep into this topic in this article. I will post a new article with good and easy examples later and link it here.
Even after partitioning, if there are still different type of fruits, the algorithm will again divide the region in which different types of fruit exists. This will be done using given features and
come up with the terminal Node as Pure Node (A Node having single class).
A Decision tree can also work for Regression Problem to predict Continuous values.
In Classification Problem, best partition is made by checking the purity. In Regression, it is done by checking the least Variance.
Another Example:
Let’s make a model to predict what fruit it is by using the given size and colour of the fruits.
If we are given with size and colour of the fruits, then we can separate by colour first.
Decision Tree
In the above “Decision Tree” you can see that, first the color is taken to divide regions, the results are Orange and Red.
In which Orange is a pure Node, where there is only a single class exists.
But Red has impurity as there are two types of fruits with in that region (Two classes exist in that region).
when using size, we can divide by, size > a particular centimetre. Should it be 7cm? or 5cm? Which will make the result better?
Internally the machine tries all the possible values and check the variance of the resultant regions.
For a value of 7cm, if the purity is good then the machine will take 7cm as boundary to divide the region.
Finally, we got the Pure nodes for Grape Fruit & Watermelon.
Overfitting in Decision Tree:
In the previous example, there are no noises in data. The data itself is clean and we get the pure Node in one or two splits.
What if there are lots of Noises in data?
Decision Tree Overfitted
In the above example, The Root node is the blue line. The Root Node’s condition is X1> 4.3
The yellow line is the second level. X2 > 3.5
Now the region R3 is Pure. This m=could be a terminal Node.
But the R1 and R2 Regions are still having impurity.
The machine will then work on R1 and derive R4 region using condition “X1>3 and X1<3.5 and X2<2.5”. Possible right?
In the same way the R2 will be then purified by deriving R5.
Now all the terminal Nodes R1, R2, R3, R4 & R5 are pure.
The algorithm ends here.
But do you think this is good? The data in R4 and R5 could be a noise.
Yes! This model is Overfitting for the training data.
If the test data has a Red star fall in R4, then it will end up as error. Same happens in R5 region too.
Then how can we tell the model to stop dividing the regions at some point?
Pruning the Tree:
This is where we prune the tree by setting the parameters of the Decision Tree.
Max_depth: This parameter helps to set the maximum depth of the tree. If we set the max_depth of the above example as 3 then R4 and R5 regions will not be created.
Min_sample_split: This parameter set the minimum number of samples (rows) that is required for further splitting. For example, if a node is having 4 stars and 1 sun, and if we set the
min_sample_split as 6, then this node cannot be divided further as this has only 5 samples.
Min_samples_leaf: This parameter can be used to mention the minimum number of samples that should present in terminal node (leaf node).
When to use Decision Tree?
We already have Logistic Algorithm for Classification and Linear for Regression. Then why we need Decision Trees? Below are the reasons and situations when you should use Decision Trees.
• If the data is Linear then we can use Logistic Regression itself. But how can we know the data is Linear or not when the Dimensions are higher?
• We can start using some basic classification techniques such as Logistic Regression. The thumb-rule is to use the simple methods. If the results are not good, then the problem may not be solved
by linear classification methods and we may need to use more complex non-linear classifier algorithms.
• If Logistic and DT are giving same results then why we need to go for a complex time-consuming algorithm.
• If data is unbalanced, then the best choice is Decision Tree.
• If the number of features is high (example > 100) and sample is less (data = 100000), then we can use Logistic Regression as Decision Tree is slow when number of features is high.
• If the number of features is higher than the number of samples, then we should not use Logistic Regression.
• Decision trees are prone to Overfitting.
• The one of the best things with Decision Tree is this algorithm does not require Scaling.
• As we can see how the decision tree split the data, Decision Tree is a White Box algorithm.
In this article we learned about Decision Tree.
We also found overfitting issue in Decision Trees. There is a way to fix it! Please check this article: Overcomes Overfitting problem of Decision Trees – Check advantages Section.
I will add a new post to practice Decision Tree in Python.
Happy Programming! 🙂
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4 Comments
1. Clear explanation Asha! Definitely an article to bookmark. One point of improvement is that you can name the image as fig 1, fig 2 etc and reference the same when you have to deal with
explanations. It will be easier for the reader to understand it.
□ Thank you Hafizul Azeez. I will definitely do this in this and in the future articles.
2. very good examples for understanding.. Thanks for sharing!
□ Thank you Srikanth! | {"url":"https://devskrol.com/2020/07/25/decision-tree/","timestamp":"2024-11-10T15:38:19Z","content_type":"text/html","content_length":"197452","record_id":"<urn:uuid:95a93fab-2d5f-409b-929a-9775c51072e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00385.warc.gz"} |
Advanced Formulas
Value driver formulas support advanced mathematical functions. Users can input mathematical the following functions into their formula: round, roundup (also ceiling), rounddown (also floor), and
sqrt. See example below.
Note that the decimal rounding parameter is optional for the round function and is unnecessary for other functions; examples: round(a*b), roundup(a*b), sqrt(a*b), etc. | {"url":"https://support.leveragepoint.com/hc/en-us/articles/360026516011-Advanced-Formulas","timestamp":"2024-11-04T05:51:37Z","content_type":"text/html","content_length":"18199","record_id":"<urn:uuid:b19768f9-29b9-48f8-8510-ddd5ec9c26d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00789.warc.gz"} |
Are Risk Tolerance Questionnaires a Silly Waste of Time?
Advisor Perspectives welcomes guest contributions. The views presented here do not necessarily represent those of Advisor Perspectives.
When a prospective client walks through the door, a financial advisor wants to know three things: their return objective, their time horizon and their tolerance for risk. Financial planning research
has systematized the first two tasks. But the attempts to quantify risk tolerance have failed to produce positive outcomes for investors.
Calculating a return objective is straightforward. You learn how much the prospect has now and how much more they want in the future. You find out how much time they have to close the gap. You do
some math and there you have it – a return objective. Of course, you may have to make a few assumptions and press the prospect for information they haven’t thought much about, but the calculation
itself is simple.
The same is true about the time horizon. You will have to make some assumptions about life expectancy and ask the prospect for information they may not be sure about, like remaining years in the work
force. But once you have gone through this exercise, you emerge with a number. It may not turn out to be the right number – the prospect may die earlier than you assumed – but it is a concrete number
derived from objective, measurable information.
How do you measure risk tolerance?
What about risk tolerance? How do you measure it? Can you calculate it using simple math? Can you calculate it at all? Can you even define it in a meaningful way? It is easy to see that there are
really two very different varieties of risk tolerance: one is attitudinal and subjective, and the other is objective and measurable.
I’ll look at the objective variety first. Let’s say my goal was to have $2 million by the time I retire at age 65. Now I have arrived at my retirement date, and I have $3 million in my account –
significantly more than I need. My goal should be to take as little risk as possible while maintaining the purchasing power of my assets. Objectively, I should have no tolerance for risk.
On the other hand, let’s say my goal is to have $2 million when I retire in five years. But I only have $1 million today. Objectively, this means I should have a high risk tolerance. Few investments
can generate the 14% to 15% annualized return I will need to reach my stated goal and they are all very risky. I have no choice. I need to take risk to reach my goal.
This objective type of risk tolerance determination has nothing to do with my internal feelings, attitudes or beliefs about risk. It is all driven by my goals and my time horizon.
Now let’s talk about the attitudinal variety of risk tolerance. This is what risk tolerance questionnaires attempt to measure. They do this by asking questions designed to reveal our true
predisposition toward, and probable level of comfort with, risk. Some of the questions are very direct in asking us how much risk we are comfortable with or how we rank ourselves along the spectrum
of risk-takers. Other questions ask us to report how we would behave in the face of various scenarios, some of which involve investing and some of which don’t.
Can we measure risk tolerance with a questionnaire?
Many of these questionnaires synthesize our answers into a risk score. These scores may label us as conservative, moderate, etc., or they may actually assign us a single number, like a 42 or a 78.
Some are even designed so that these scores determine what investment strategy is appropriate for us. I am a 42 so I should have the 40/60 portfolio, and you are a 78 so you should have an 80/20
In other words, comfort with risk determines strategy. | {"url":"https://asset-sync.advisorperspectives.com/articles/2016/06/07/are-risk-tolerance-questionnaires-a-silly-waste-of-time","timestamp":"2024-11-11T20:09:46Z","content_type":"text/html","content_length":"125951","record_id":"<urn:uuid:19e07231-8032-43e9-a95a-35245ed35fd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00126.warc.gz"} |
Nomothetic Psychology
A descriptive variable is a relation between a set of beings to be described and a set of descriptive values with the property that each being is related to exactly one descriptive value.
A descriptive variable can be noted X: Ω -> M(X), where Ω and M(X) denote the set of beings to be described, and the set of possible descriptive values, respectively.
General fact
By default, a general fact will refer to a general, descriptive, and non-tautological fact, associated with a possibly multivariate variable X: Ω -> M(X). It is a statement that is true for any
element of Ω. A general fact is a negative (or restrictive) statement about at least one descriptive value of M(X): there exists a non-empty subset α of M(X) such that no being of Ω takes a value in
Remark: if the set Ω is unknown, the statement “there exists a non-empty subset of M(X) such that no element of Ω takes a value in it” should not be called a fact but a conjecture. | {"url":"https://nompsy.hypotheses.org/category/terminology","timestamp":"2024-11-10T01:57:38Z","content_type":"text/html","content_length":"48204","record_id":"<urn:uuid:acc0f119-7ae1-4d9f-b404-21126dc81ccb>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00803.warc.gz"} |
Fill the Blank Calculator - Online Find Missing Number Value Solver
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Missing Numbers Calculator
Tool/Calculator to find missing number/fill the blanks puzzles/equations (addition, subtraction, multiplication, division)
Missing Numbers Calculator - dCode
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Missing Numbers Calculator
Find the Missing Digits Solver
Find x in an Equation Solver
Answers to Questions (FAQ)
What is a calculation with blanks? (Definition)
A calculation with blanks (such as an addition, a subtraction, a multiplication, or a division) is a math exercise that involves finding the missing numbers and digits.
There is a variant where the operators (+ - * /) are missing.
How to solve a calculation with blanks?
Solving a calculation with operators and missing digits is similar to a cryptarithm and uses deduction and extraction of parts of the calculation.
Example: Exercise: find ? in the operation ??5 + 42? = 539 => 115 + 424 = 539
Example: 12 + 23 = ? has no solution but 12 + 23 = ?? has for solution 12 + 23 = 35
It is useful to calculate digits at extremities, or check individually calculations to the ten digits, or the hundreds digit etc.
How does the missing numbers solver works?
dCode solver allow usual operators like additions +, subtractions -, multiplications * and divisions /. It also handles comparison operators superior and inferior > and < in addition to the equal
sign =.
The blanks (unknown digits) have to be replaced by ? (interrogation marks). There is no limit but above 7 or 8, the calculation will be very long.
Example: 1+?=3 is solved with 1+2=3
Example: 2*?=8 is solved with 2*4=8
This solver uses a brute-force method, this means that it tries all combinations and display the possible ones.
Why is no solution found?
If the message 0 solution(s) is displayed, it is impossible to replace the ? by numbers so that the calculation is valid. Several explanations:
— the ? is a number (consisting of several digits)
Example: 5+?=23 has no solution, but 5+??=23 does have the solution 18
— the ? is a negative number
Example: 5+?=1 has no solution but 5+-?=1 has a solution of 4
— an error may be in the problem statement
Why do gap calculations?
Fill in the blanks puzzles can help users improve their math skills. By solving these calculations, users can improve their understanding of basic math rules and improve their solving speed.
Why should floating point numbers be avoided?
Point calculations sometimes cause problems due to the way languages handle floating point numbers. The IEEE 754 standard uses a binary representation to store floating point numbers, but it cannot
store all floating point numbers exactly. Therefore, when a floating point number is stored, there may be a small loss of precision. This loss of precision can lead to miscalculations when floating
point numbers are used in complex mathematical calculations.
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© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF. | {"url":"https://www.dcode.fr/missing-numbers-calculator","timestamp":"2024-11-05T06:04:12Z","content_type":"text/html","content_length":"25603","record_id":"<urn:uuid:fd021f7b-c1b9-4aa4-8549-d4eae8b8f0df>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00871.warc.gz"} |
NH4+ Geometry and Hybridization - Chemistry Steps
Nitrogen is the central atom, so we can draw a preliminary skeletal structure:
There is a positive charge, so to determine the number of electrons, we subtract 1 from the total number of valence electrons:
5 + 4 – 1 = 8 electrons
All the electrons are used to make the 4 covalent bonds, and the positive charge is on the nitrogen:
Remember, the formula for calculating the formal charge:
FC= V – (N + B)
V – number of valence electrons
N – number of nonbonding electrons
B – number of bonds
So, for the nitrogen it would be: 5 – (0 + 4) = +1
Try memorizing the bonding and formal charge patterns to make this process easier:
For the geometry, the nitrogen has 4 atoms and no lone pairs, therefore, both geometries are tetrahedral.
Steric number 4 corresponds to sp^3-hybridization where the idealized bond angles are 109.5^o.
Leave a Comment | {"url":"https://general.chemistrysteps.com/nh4-geometry-and-hybridization/","timestamp":"2024-11-06T16:53:37Z","content_type":"text/html","content_length":"176128","record_id":"<urn:uuid:132fc65a-e4d0-4e69-9d17-1de675e780ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00466.warc.gz"} |
Mesh surface plot with curtain
meshz(X,Y,Z) creates a mesh plot with a curtain around it. A mesh plot is a three-dimensional surface that has solid edge colors and no face colors. The function plots the values in matrix Z as
heights above a grid in the x-y plane defined by X and Y. The edge colors vary according to the heights specified by Z.
meshz(X,Y,Z,C) additionally specifies the color of the edges.
meshz(Z) creates a mesh plot with a curtain, and uses the column and row indices of the elements in Z as the x- and y-coordinates.
meshz(Z,C) specifies the color of the edges.
meshz(___,Name,Value) specifies additional options for the meshz plot using one or more name-value pair arguments. Specify the options after all other input arguments. For a list of properties, see
Surface Properties.
meshz(ax,___) plots into the axes specified by ax instead of the current axes. Specify the axes as the first input argument.
s = meshz(___) returns the chart surface object. Use s to modify the mesh plot after it is created. For a list of properties, see Surface Properties.
Display Curtain Around Mesh Plot
Create three matrices of the same size. Then plot them as a mesh plot with a curtain. The mesh plot uses Z for both height and color.
[X,Y] = meshgrid(-3:.125:3);
Z = peaks(X,Y);
Specify Colormap Colors for Mesh Plot With Curtain
Specify the colors for a mesh plot and surrounding curtain by including a fourth matrix input, C. The mesh plot uses Z for height and C for color. Specify the colors using a colormap, which uses
single numbers to stand for colors on a spectrum. When you use a colormap, C is the same size as Z. Add a color bar to the graph to show how the data values in C correspond to the colors in the
[X,Y] = meshgrid(-3:.125:3);
Z = peaks(X,Y);
C = gradient(Z);
Modify Appearance of Mesh Plot With Curtain
Create a mesh plot with a curtain around it. To allow further modifications, assign the surface object to the variable s.
[X,Y] = meshgrid(-5:.5:5);
Z = Y.*sin(X) - X.*cos(Y);
s = meshz(X,Y,Z)
s =
Surface (meshz) with properties:
EdgeColor: 'flat'
LineStyle: '-'
FaceColor: [1 1 1]
FaceLighting: 'none'
FaceAlpha: 1
XData: [25x25 double]
YData: [25x25 double]
ZData: [25x25 double]
CData: [25x25 double]
Use GET to show all properties
Use s to access and modify properties of the mesh plot after it is created. For example, change the color of the mesh plot edges and surrounding curtain by setting the EdgeColor property.
Input Arguments
X — x-coordinates
matrix | vector
x-coordinates, specified as a matrix the same size as Z, or as a vector with length n, where [m,n] = size(Z). If you do not specify values for X and Y, meshz uses the vectors (1:n) and (1:m).
When X is a matrix, the values must be strictly increasing or decreasing along one dimension and remain constant along the other dimension. The dimension that varies must be the opposite of the
dimension that varies in Y. You can use the meshgrid function to create X and Y matrices.
When X is a vector, the values must be strictly increasing or decreasing.
The XData property of the surface object stores the x-coordinates.
Example: X = 1:10
Example: X = [1 2 3; 1 2 3; 1 2 3]
Example: [X,Y] = meshgrid(-5:0.5:5)
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical
Y — y-coordinates
matrix | vector
y-coordinates, specified as a matrix the same size as Z or as a vector with length m, where [m,n] = size(Z). If you do not specify values for X and Y, meshz uses the vectors (1:n) and (1:m).
When Y is a matrix, the values must be strictly increasing or decreasing along one dimension and remain constant along the other dimension. The dimension that varies must be the opposite of the
dimension that varies in X. You can use the meshgrid function to create X and Y matrices.
When Y is a vector, the values must be strictly increasing or decreasing.
The YData property of the surface object stores the y-coordinates.
Example: Y = 1:10
Example: Y = [1 1 1; 2 2 2; 3 3 3]
Example: [X,Y] = meshgrid(-5:0.5:5)
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical
Z — z-coordinates
z-coordinates, specified as a matrix. Z must have at least two rows and two columns.
Z specifies the height of the mesh plot at each x-y coordinate. If you do not specify the colors, then Z also specifies the mesh edge colors.
The ZData property of the surface object stores the z-coordinates.
Example: Z = [1 2 3; 4 5 6]
Example: Z = sin(x) + cos(y)
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical
C — Color array
Color array, specified as an m-by-n matrix of colormap indices, where Z is m-by-n. For each grid point on the mesh surface, C indicates a color in the colormap. The CDataMapping property of the
surface object controls how the values in C correspond to colors in the colormap.
The CData property of the surface object stores the color array. For additional control over the surface coloring, use the FaceColor and EdgeColor properties.
ax — Axes to plot in
axes object
Axes to plot in, specified as an axes object. If you do not specify the axes, then meshz plots into the current axes.
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: meshz(X,Y,Z,'EdgeColor','red') creates the mesh with red lines.
The properties listed here are only a subset. For a full list, see Surface Properties.
Extended Capabilities
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The meshz function supports GPU array input with these usage notes and limitations:
• This function accepts GPU arrays, but does not run on a GPU.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
• This function operates on distributed arrays, but executes in the client MATLAB.
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a
See Also | {"url":"https://de.mathworks.com/help/matlab/ref/meshz.html","timestamp":"2024-11-13T16:01:42Z","content_type":"text/html","content_length":"145040","record_id":"<urn:uuid:a0a214fd-660f-4bf3-a7e0-41fa7f022a6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00419.warc.gz"} |
Best, D. I. and Rayner, C. W. (1987), "Welch’s Approximate Solution for the Behren’s-Fisher Problem," Technometrics, 29, 205–210.
Chow, S. and Liu, J. (2000), Design and Analysis of Bioavailability and Bioequivalence Studies, Second Edition, New York: Marcel Dekker.
Cochran, W. G. and Cox, G. M. (1950), Experimental Designs, New York: John Wiley & Sons.
Dilba, G., Schaarschmidt, F., and Hothorn, L. A. (2006), mratios: A Package for Inference about Ratios of Normal Means, UseR! Conference.
Diletti, D., Hauschke, D., and Steinijans, V. W. (1991), "Sample Size Determination for Bioequivalence Assessment by Means of Confidence Intervals," International Journal of Clinical
Pharmacology, Therapy and Toxicology, 29, 1–8.
Fieller, E. C. (1954), "Some Problems in Interval Estimation," Journal of the Royal Statistical Society Series B, 16, 175–185.
Hauschke, D., Kieser, M., Diletti, E., and Burke, M. (1999), "Sample Size Determination for Proving Equivalence Based on the Ratio of Two Means for Normally Distributed Data," Statistics in
Medicine, 18, 93–105.
Huntsberger, David V. and Billingsley, Patrick P. (1989), Elements of Statistical Inference, Dubuque, IA: Wm. C. Brown.
Johnson, N. L. Kotz, S. and Balakrishnan, N. (1994), Continuous Univariate Distributions, Volume 1, Second Edition, New York: John Wiley & Sons.
Jones, B. and Kenward, M. G. (2003), Design and Analysis of Cross-Over Trials, Second Edition, Washington, DC: Chapman & Hall/CRC.
Lee, A. F. S. and Gurland, J. (1975), "Size and Power of Tests for Equality of Means of Two Normal Populations with Unequal Variances," Journal of the American Statistical Association, 70,
Lehmann, E. L. (1986), Testing Statistical Hypostheses, New York: John Wiley & Sons.
Moore, David S. (1995), The Basic Practice of Statistics, New York: W. H. Freeman.
Phillips, K. F. (1990), "Power of the Two One-Sided Tests Procedure in Bioequivalence," Journal of Pharmacokinetics and Biopharmaceutics, 18, 137–144.
Posten, H. O., Yeh, Y. Y., and Owen, D. B. (1982), "Robustness of the Two-Sample Communications in Statistics, 11, 109–126.
Ramsey, P. H. (1980), "Exact Type I Error Rates for Robustness of Student’s Journal of Educational Statistics, 5, 337–349.
Robinson, G. K. (1976), "Properties of Student’s Annals of Statistics, 4, 963–971.
SAS Institute Inc. (1986), SUGI Supplemental Library User’s Guide, Version 5 Edition, Cary, NC: SAS Institute Inc.
Sasabuchi, S. (1988a), "A Multivariate Test with Composite Hypotheses Determined by Linear Inequalities When the Covariance Matrix Has an Unknown Scale Factor," Memoirs of the Faculty of Science,
Kyushu University, Series A, 42, 9–19.
Sasabuchi, S. (1988b), "A Multivariate Test with Composite Hypotheses When the Covariance Matrix Is Completely Unknown," Memoirs of the Faculty of Science, Kyushu University, Series A, 42, 37–46.
Satterthwaite, F. W. (1946), "An Approximate Distribution of Estimates of Variance Components," Biometrics Bulletin, 2, 110–114.
Scheffe, H. (1970), "Practical Solutions of the Behrens-Fisher Problem," Journal of the American Statistical Association, 65, 1501–1508.
Schuirmann, D. J. (1987), "A Comparison of the Two One-Sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability," Journal of Pharmacokinetics and
Biopharmaceutics, 15, 657–680.
Senn, S. (2002), Cross-over Trials in Clinical Research, Second Edition, New York: John Wiley & Sons.
Steel, R. G. D. and Torrie, J. H. (1980), Principles and Procedures of Statistics, Second Edition, New York: McGraw-Hill.
Tamhane, A. C. and Logan, B. R. (2004), "Finding the Maximum Safe Dose Level for Heteroscedastic Data," Journal of Biopharmaceutical Statistics, 14, 843–856.
Wang, Y. Y. (1971), "Probabilities of the Type I Error of the Welch Tests for the Behren’s-Fisher Problem," Journal of the American Statistical Association, 66, 605–608.
Wellek, S. (2003), Testing Statistical Hypotheses of Equivalence, Boca Raton, FL: Chapman & Hall/CRC Press LLC.
Yuen, K. K. (1974), "The Two-Sample Trimmed Biometrika, 61, 165–170. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_ttest_a0000000140.htm","timestamp":"2024-11-08T18:03:59Z","content_type":"application/xhtml+xml","content_length":"15353","record_id":"<urn:uuid:b8b134f9-4416-4413-b98c-d58b86af0089>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00596.warc.gz"} |
Spread Option
Hedging Strategies Using Spread Options
This example shows different hedging strategies to minimize exposure in the Energy market using Crack Spread Options.
Understanding Crack Spread Options
In the petroleum industry, refiners are concerned about the difference between their input costs (crude oil) and output prices (refined products - gasoline, heating oil, diesel fuel, and so on). The
differential between these two underlying commodities is referred to as a Crack Spread. It represents the profit margin between crude oil and the refined products.
A Spread option is an option on the spread where the holder has the right, but not the obligation, to enter into a spot or forward spread contract. Crack Spread Options are often used to protect
against declines in the crack spread or to monetize volatility or price expectations on the spread.
Example 1: Protecting Margins using a 1:1 Crack Spread Option
A marketer is interested in protecting his gasoline margin since current prices are strong. A crack spread option strategy is used to maintain profits for the following season. In March the June WTI
crude oil futures are at $91.10 per barrel and RBOB gasoline futures contract are at $2.72 per gallon. The marketer's strategy is a long crack call involving purchasing RBOB gasoline futures and
selling crude oil futures.
OldFormat = get(0, 'format');
format bank
% Price and volatility of RBOB gasoline
Price1gallon = 2.72; % $/gallon
Price1 = Price1gallon * 42; % $/barrel
Vol1 = 0.39;
% Price and volatility of WTI crude oil
Price2 = 91.10; % $/barrel
Vol2 = 0.34;
% Assume the following data
% Spread Option
Strike = 20;
OptSpec = 'call';
Settle = '01-March-2013';
Maturity = '01-June-2013';
Corr = 0.45; % Correlation of underlying commodities
Define the RateSpec and StockSpec.
% Define RateSpec
Rate = 0.035;
Compounding = -1;
Basis = 1;
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ...
'EndDates', Maturity, 'Rates', Rate, 'Compounding', ...
Compounding, 'Basis', Basis);
% Define StockSpec for the two assets
StockSpec1 = stockspec(Vol1, Price1);
StockSpec2 = stockspec(Vol2, Price2);
Price the Crack Spread Option
Use the function spreadbybjs in the Financial Instruments Toolbox™ to price the spread option using the Bjerksund and Stensland model.
Price = spreadbybjs(RateSpec, StockSpec1, StockSpec2, Settle, ...
Maturity, OptSpec, Strike, Corr)
The 1:1 implied current crack spread between these two underlyings is $23.14 per barrel.
CrackSpread = Price1 - Price2 % $/barrel
Suppose that by expiration day, June crude oil prices decrease to $90.34 per barrel and gasoline prices rise to $2.89 per gallon. The price changes cause the marketer's profit margin (the new implied
crack spread) to increase from $23.14/barrel to $31.04/barrel:
NewCrackSpread = (2.89 * 42) - 90.34
Since the marketer purchased a long crack call on the $20 call, the option is now in the money by $11.04.
(NewCrackSpread - Strike)
The marketer paid $9.91 from the long crack call, this protects the margin by $1.13.
(NewCrackSpread - Strike - Price)
This strategy provides the marketer protection during spread increase scenarios.
Example 2: Creating a Floor with Crack Spread Options
A refiner is interested in covering its fixed and operating costs, but still profit from a favorable move in the market. In March the May WTI crude oil futures are at $99.43 per barrel and RBOB
gasoline futures contract are at $3.04 per gallon. The refiner believes that the spread between those commodities of $28.25 per barrel is favorable. Of this, $11 corresponds to operating and fixed
costs, and $17.25 is the net refining margin. The refiner's strategy is to sell the crack spread by selling 10 RBOB gasoline futures and buying 10 crude oil futures.
% Price and volatility of RBOB gasoline
Price1gallon = 3.04; % $/gallon
Price1 = Price1gallon * 42; % $/barrel
Vol1 = 0.35;
Div1 = 0.0783;
% Price and volatility of WTI crude oil
Price2 = 99.43; % $/barrel
Vol2 = 0.38;
Div2 = 0.0571;
The refiner purchases 10 May RBOB gasoline crack spread puts with a strike price of $25.
% Spread Option
Strike = 25;
OptSpec = 'put';
Settle = '01-March-2013';
Maturity = '01-May-2013';
Corr = 0.30; % Correlation of underlying commodities
Define the RateSpec and StockSpec.
% Define RateSpec
Rate = 0.035;
Compounding = -1;
Basis = 1;
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ...
'EndDates', Maturity, 'Rates', Rate, 'Compounding', ...
Compounding, 'Basis', Basis);
% Define StockSpec for the two assets
StockSpec1 = stockspec(Vol1, Price1, 'Continuous', Div1);
StockSpec2 = stockspec(Vol2, Price2, 'Continuous', Div2);
Price the Crack Spread Option
Use the function spreadbyfd in the Financial Instruments Toolbox™ to price the American spread option using the finite difference method.
Price = spreadbyfd(RateSpec, StockSpec1, StockSpec2, Settle, ...
Maturity, OptSpec, Strike, Corr, 'AmericanOpt', 1)
By expiration, if the option is exercised, the refiner would have hedged the cost of purchasing 10000 barrels of crude oil with the revenue of selling 10000 barrels of RBOB gasoline. The futures
contract represents 1000 barrels of crude oil and 42000 gallons of gasoline.
CostOfHedge = Price * 10000 % Option premium
The hedge cost is $66386 to implement and guarantee that neither a fall in RBOB gasoline prices or an increase in WTI crude oil prices will diminish the refining margin below $25.
ProfitMargin = 14 * 10000 %$
CrackingMargin = ProfitMargin - CostOfHedge
CrackingMargin =
This strategy allows a cracking margin of $73613.
Another strategy for the refiner could be to buy the $22 puts at a price of $5.38.
StrikeNew = 22;
PriceNew = spreadbyfd(RateSpec, StockSpec1, StockSpec2, Settle, ...
Maturity, OptSpec, StrikeNew, Corr, 'AmericanOpt', 1)
This time the hedge would have cost $53823, but it also guarantees a $11 per barrel or a $56176 cracking margin.
NewCostOfHedge = PriceNew * 10000 % Option premium
NewCostOfHedge =
NewProfitMargin = 11 * 10000
NewProfitMargin =
CrackingMargin = NewProfitMargin - NewCostOfHedge
CrackingMargin =
Example 3: Using Collars to Reduce the Cost of Hedging
A refiner is concerned about its cost of hedging and decides to use a collar strategy. In April the crack spread is trading at $4.23 per barrel. The refiner is not convinced to lock in this margin,
but also wants to protect against price changes causing the refinery margin to decrease less than $4 per barrel.
% Price and volatility of heating oil
Price1gallon = 2.52; % $/gallon
Price1 = Price1gallon * 42; % $/barrel
Vol1 = 0.38;
Div1 = 0.0762;
% Price and volatility of WTI crude oil
Price2 = 101.61; % $/barrel
Vol2 = 0.34;
Div2 = 0.1169;
To accomplish the collar strategy the refiner sells a call spread option with a strike of $4.50 and uses the premium income to offset the cost of purchasing a put spread option with a strike of $4.
This allows the refiner to benefit if market prices move up, and protects it if market prices move down.
% Assume the following data
Strike = [4.50;4];
OptSpec = {'call';'put'};
Settle = '01-April-2013';
Maturity = '01-June-2013';
Corr = 0.35; % Correlation of underlying commodities
Define the RateSpec and StockSpec.
% Define RateSpec
Rate = 0.035;
Compounding = -1;
Basis = 1;
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ...
'EndDates', Maturity, 'Rates', Rate, 'Compounding', ...
Compounding, 'Basis', Basis);
% Define StockSpec for the two assets
StockSpec1 = stockspec(Vol1, Price1, 'Continuous', Div1);
StockSpec2 = stockspec(Vol2, Price2, 'Continuous', Div2);
Price the Crack Spread Options
Use the function spreadbybjs in the Financial Instruments Toolbox™ to price the spread options using the Bjerksund and Stensland model.
Price = spreadbybjs(RateSpec, StockSpec1, StockSpec2, Settle, ...
Maturity, OptSpec, Strike, Corr)
The collar strategy allows the refiner to reduce the cost of the hedge to $0.63.
% CostOfHedge = Premium of Call - Premium of Put
CostOfHedge = Price(1) - Price(2)
The refiner is protected if the crack spread narrows to less than $4. If the crack spread widens to more than $4.50, the refiner will not benefit over this amount if he has hedged 100% of all its
market exposure.
set(0, 'format', OldFormat);
Related Topics | {"url":"https://au.mathworks.com/help/fininst/hedging-strategies-using-spread-options.html","timestamp":"2024-11-02T23:53:57Z","content_type":"text/html","content_length":"87737","record_id":"<urn:uuid:2f6d9ba5-72b3-4c2a-9c8e-a6a34d7453c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00219.warc.gz"} |
Ordering Given Fractions with Unlike Denominators in Ascending Order
Question Video: Ordering Given Fractions with Unlike Denominators in Ascending Order
Order 7/10, 6/4, 4/5, 5/2 from least to greatest.
Video Transcript
Order seven tenths, six-fourths, four-fifths, and five-halves from least to greatest.
Least to greatest means we’re going to order them left or right from the smallest number to the largest number. Our only issue is that we have fractions. And they have different denominators. So it’s
gonna be a little bit difficult to compare them. So we need to find a common denominator between all four of these fractions.
So what would be the smallest number that 10, four, five, and two can all go into? That would be 20. So let’s begin with the seven tenths. Seven tenths is equal to something over 20. So how could we
go from 10 to 20? We multiply by two. So that means we need to multiply the numerator by two. And seven times two is 14. So seven tenths, we can rewrite it as fourteen twentieths.
Now let’s look at six-fourths. Six-fourths is equal to something over 20. To get from four to 20, we multiply by five. So we take six times five to get 30. So six-fourths is equal to thirty
Now let’s look at the four-fifths. Four-fifths is equal to something over 20. To go from five to 20, we multiply by four. And four times four is 16. So we have sixteen twentieths.
Now let’s look at five-halves, the last one. Repeating our process to get from two to 20, we multiply by 10. And five times 10 is 50. So we have fifty twentieths. So from least to greatest is how we
will arrange these fractions.
So looking at our fractions, which one has the smallest numerator? That would be the fourteen twentieths. So fourteen twentieths would be our smallest fraction. So why are we only looking at the
numerators? Because all of the denominators are the same. So we’re not really comparing this.
So fourteen twentieths is the smallest. Now the next smallest would be the sixteen twentieths. And then we would have the thirty twentieths. And then finally the largest would be the fifty
Now we need to put these numbers back in the original form just as the question had them. So the fourteen twentieths was equal to the seven tenths. And the sixteen twentieths is equal to the
four-fifths. So that comes next. And now the thirty twentieths is equal to the six-fourths. And then lastly the fifty twentieths is equal to five-halves. So ordering these fractions from least to
greatest, we get seven tenths, four-fifths, six-fourths, and five-halves. | {"url":"https://www.nagwa.com/en/videos/747130295863/","timestamp":"2024-11-11T20:57:16Z","content_type":"text/html","content_length":"242769","record_id":"<urn:uuid:9277a0eb-e967-457a-a6ee-d736842569f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00878.warc.gz"} |
Math - Differential and Integral Equations
Differential and integral equations are a major aspect of mathematics, impacting a wide range of the natural and social sciences. Our extensive and low-priced list includes titles on applied partial
differential equations, basic linear partial differential equations, differential manifolds, linear integral equations, ordinary differential equations, singular integral equations, and more. | {"url":"https://store.doverpublications.com/collections/math-differential-and-integral-equations","timestamp":"2024-11-08T05:55:57Z","content_type":"text/html","content_length":"597637","record_id":"<urn:uuid:122aafb0-b513-4513-9c85-de252188bf18>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00109.warc.gz"} |
Root Causes 114: Is Quantum Computing a Threat to SHA-2?
Quantum computers' threat to standardized encryption algorithms RSA and ECC has been much discussed. But what about our hashing algorithms? Do quantum computers pose a similar threat to SHA-2? Join
our hosts as they discuss the difference between Shor's Algorithms and Grover's Algorithm, which applies to each part of cryptography, and how significant quantum computing will be for each.
• Original Broadcast Date: August 21, 2020
Episode Transcript
Lightly edited for flow and brevity.
• Tim Callan
So, we talk a lot about quantum computers and the fact that they are set to render our common encryption algorithms, RSA, and ECC valueless and we need to get new encryption. We don’t talk so
much about hashing algorithms. So, I’m going to pose the question to you today, and this is actually inspired by a paper that you and I recently discovered from a bunch of folks at the University
of Waterloo, the name of which is Estimating the Cost of Generic Quantum Pre-imaging Attacks on SHA-2 and SHA-3. So, Jay, here’s the question. We talk a lot about quantum computers destroying ECC
and RSA. Makes good sense. Do we also need to worry about SHA-2?
• Jason Soroko
I think we absolutely need to worry about SHA-2 but not necessarily because of quantum computing.
• Tim Callan
Ok. That’s a great answer. That’s a very provocative answer. Let’s unpack that. Let’s start with - - I guess there’s two pieces to that. Why don’t we start with the quantum computing part of it.
So why not quantum computing?
• Jason Soroko
If you look at what quantum computing is doing with helping up the factorizing of solving prime numbers and solving against things like ECC or elliptic curve cryptography using specifically
Shor’s algorithm. Quantum goes a long way to speeding up that process so that what used to take a traditional computer a lot of time a quantum computer can solve it in much less time given enough
stable Qbits.
• Tim Callan
Yeah and these words like a lot of time and much less time are really not doing it justice. We are talking about – and I don’t have it in front of me, but my recollection is it’s something in the
ballpark of 10 or 12 orders of magnitude faster. So, like, something that would have taken thousands of years now takes minutes.
• Jason Soroko
Yes. Or even a month, right? Which for very secure communications might not be sufficient. In other words, if it only took 30 days to break a certain amount of SSL stream for example, that was
encrypted with ECC, that might be very detrimental to the people who are doing that communication. So it doesn’t have to be seconds. It can sometimes be, you know, at least within the order of
magnitude of something reasonable. Within our lifetimes, for example.
• Tim Callan
Sure. But the power of Shor’s algorithm is such that this isn’t something you can just solve by throwing a couple more bits in the end. Right? You don’t say, oh well, you know, everything got ten
times faster so I’m gonna add three bits and I’m covered. Right? It’s much, much worse than that.
• Jason Soroko
Correct, Tim. And that’s why there’s an enormous effort to come up with quantum-resistant algorithms, which is something we’ve podcasted on in the past and we will again because the nature of the
math, the nature of what quantum computers can and cannot do, a big chunk of what goes into making quantum resistance isn’t just throwing a lot more bits at it. It’s also creating something
that’s fundamentally different in the map.
• Tim Callan
A little aside here, I do want to get back to SHA-2, so, and it’s just a coincidence right. Like the fact that quantum computers are gonna break SHA-2 – - sorry, are gonna break ECC and RSA
because of the underlying math assumptions that were made, if we had made different underlying math assumptions in the 1970s let’s say, we wouldn’t be having this conversation right now, right?
It’s just how it turned out.
• Jason Soroko
Fair enough, Tim. However, those algorithms were also chosen for the purposes of efficiency because if you take a look at the current stable, the short list of quantum-resistant algorithms
there’s been an awful lot of work done to try to speed them up because of the fact that some of them are not inherently quick. So therefore, there is probably reasons why we settled on RSA and
ECC in the past.
• Tim Callan
Yeah. And yet, probably with the amount of computing power and bandwidth we had in 1994 that was probably considerably more pressure on that than there is today.
• Jason Soroko
Oh, I would say so, Tim. However, interestingly, enter the world of IoT and that’s why things like ECC are quite popular within that because it’s such an efficient algorithm.
• Tim Callan
Sure. And yeah, you could see where that with your constrained devices, constrained compute, constrained storage, constrained bandwidth, all the sudden you are back to these same problems.
• Jason Soroko
• Tim Callan
That makes great sense. But let’s get back to SHA-1. So, Shor’s algorithm is the reason that RSA and ECC are in such jeopardy. Now it’s a different situation though for hashing algorithms like
SHA-1 isn’t it?
• Jason Soroko
That’s right, Tim, because if you want to break those hashing algorithms the term that comes up often is Grover’s algorithm.
• Tim Callan
• Jason Soroko
Which is different than Shor’s algorithm. So, people who understand this at a very intimate level, right, that article out of University of Waterloo that you just brought up for example. A very,
very thorough, and detailed examination of simply asking the question, would a quantum computer with a sufficient number of stable Qbits be able to with Grover’s algorithm, would that be of an
advantage over traditional computing with Grover’s algorithm, for example?
• Tim Callan
Sure. Ok.
• Jason Soroko
And without getting into it way too much because that’s an enormous article and there’s a lot of really good thinking in there, there really was not a meaningful amount of advantage that a
quantum computer happens to give.
• Tim Callan
Yeah, and things I’ve read in other places, forgive me if I’m wrong, seem to suggest that you could use Grover’s algorithm to decrease in a quantum computing environment to decrease the time you
need to break a popular hashing algorithm to maybe a quarter or an eighth or a tenth of what it is right now. But while that seems important, and it is, it’s not to the point where it’s any kind
of problem.
• Jason Soroko
That’s a very important point, Tim. So in other words, the properties of hashing algorithms are that in order to reverse it, right, that one-way hashing algorithm – in order to reverse it, it is
really difficult and it’s difficult – the way that the math has been set, it’s difficult regardless of whether you have a quantum computer or not, it’s difficult even if you have Grover’s
algorithm which is doing something very efficient within the breaking. The amount of time saved is much, much smaller compared to say Shor’s algorithm with quantum. In other words, hashing
algorithms are very, very resistant.
• Tim Callan
And this is true of all hashing algorithms? Not just SHA-2?
• Jason Soroko
Well, the thing is, what was the problem with SHA-1? Right? What were the problem with some of these deprecated hashing algorithms and part of it was geez, there’s just not enough entropy going
on but some of the other problems were, well, some of the math kind of broke down easily, right? And so therefore, this is why I alluded at the very top of the podcast, yes, we do have to worry
about SHA-2 not necessarily because of quantum but because there may be some ah-ha moment down the road and there definitely have been some good attack modes that are more traditional against
SHA-2, let’s say SHA-256 as an example of it, that it may come from left field. It might not come from this quantum onslaught that we are continuously talking about. It may come from just some
better math.
• Tim Callan
Right. In which case quantum computers are neither here nor there, right?
• Jason Soroko
That's right.
• Tim Callan
I mean you could do the same attack. If this mathematical attack were to emerge you could do the same thing on a traditional computer cause the computing platform is not the issue, the math is
the issue.
• Jason Soroko
That’s right, Tim. Because remember when we talked about SHA-1 deprecation one of the things we may not have highlighted enough but the amount of thinking that went into hey, you know, original
SHA-1 failed spectacularly because of X, Y and Z in the math and a lot of that thinking went into SHA-2 and so therefore, that thinking is bearing it’s fruit. However, however, and the big
however is, some people have stated that it might not have gone far enough. I mean this is all just extremely high-level talk of what’s out there in the community. It is at the moment considered,
you know, SHA-2, SHA-256, considered a very safe hashing algorithm, we use it every single day, you can hang your hat on it and not worry too much but suffice to say that there definitely are
some people who are looking very intensively at what are the weaknesses of it and is there a silver bullet that might render it needing to be deprecated down the road.
• Tim Callan
Yeah. And this is a bit of a conundrum, right, because on the one hand we recognize that in all walks of life you don’t know what you don’t know. Right? Until somebody has an ah-ha moment and
thinks of a new thing nobody knows about that thing and perhaps somebody is smart with a chalkboard right now is figuring out how to break SHA-2 and that could happen, right? At the same time you
gotta get on with things so we have to have standards and we have to deploy this stuff very widely and ubiquitously and it has to work with our bandwidth and our computers and so as a consequence
we need to make sure that we balance those two things.
So, before we break, any last thoughts on hashing, SHA-2 and quantum computers?
• Jason Soroko
Tim, it’s interesting how quantum computing has captured a lot of people’s imaginations. I think that it’s a valid question to start asking hey how this affects hashing algorithms. I think the
important thing to note is that research in terms of future better hashing algorithms when things eventually get deprecated much further down the road, that’s happening but your hashing
algorithms as of right now are, we’re feeling pretty good about it as say compared to RSA and ECC that we definitely know we need to build resistance in our encryption algorithms for the web and
for a bunch of other things because quantum is coming and so the timeframe for that is a lot more clear but for those of you thinking about SHA-2 and thinking about hashing algorithms, that
really is a different beast, Tim.
• Tim Callan
Right. So, in the world of crypto nothing is forever. However, RSA and ECC that’s the immediate problem. That’s the thing to focus on.
• Jason Soroko
Exactly Tim.
• Tim Callan
Exactly right. Great, Jay. Great conversation as always. Thank you, listeners. This has been Root Causes. | {"url":"https://www.sectigo.com/resource-library/root-causes-114-is-quantum-computing-a-threat-to-sha-2","timestamp":"2024-11-13T21:05:45Z","content_type":"text/html","content_length":"295772","record_id":"<urn:uuid:01ee48fa-f5d9-4067-8329-7531676de59a>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00287.warc.gz"} |
Term frequency/Inverse document frequency implementation in C#
This code implements the Term Frequency/Inverse Document frequency (TF-IDF). The TF-IDF is a text statistical-based technique which has been widely used in many search engines and information
retrieval systems. I will deal with the documents similarity problem in the next section. To understand the theory, please see this article: Wiki definition of TF/IDF for more details.
Assume that you have a corpora of 1000 documents and your task is to compute the similarity between two given documents (or a document and a query). The following describes the steps of acquiring the
similarity value:
Document pre-processing steps
• Tokenization: A document is treated as a string (or bag of words), and then partitioned into a list of tokens.
• Removing stop words: Stop words are frequently occurring, insignificant words. This step eliminates the stop words.
• Stemming word: This step is the process of conflating tokens to their root form (connection -> connect).
Document representation
• We generate N-distinct words from the corpora and call them as index terms (or the vocabulary). The document collection is then represented as a N-dimensional vector in term space.
Computing Term weights
• Term Frequency.
• Inverse Document Frequency.
• Compute the TF-IDF weighting.
Measuring similarity between two documents
• We capture the similarity of two documents using cosine similarity measurement. The cosine similarity is calculated by measuring the cosine of the angle between two document vectors.
Using the code
The main class is TFIDFMeasure. This is the testing code:
void Test (string[] docs, int i, int j)
// docs is collection of parsed documents
StopWordHandler stopWord=new StopWordsHandler() ;
TFIDFMeasure tf=new TFIDFMeasure(doc) ;
float simScore=tf.GetSimilarity( i, j);
// similarity of two given documents at the
// position i,j respectively
This library also includes stemming (Martin Porter algorithm), and N-gram text generation modules. If a token-based system did not work as expected, then you can make another choice with N-gram
based. Thus, instead of expanding the list of tokens from the document, we will generate a list of N-grams, where N should be a predefined number. That means we will hash into a table to find the
counter for the N-gram, but not words (or tokens).
The extra N-gram based similarities (bi, tri, quad...-gram) also help you compare the result of the statistical-based method with the N-gram based method. Let us consider two documents as two flat
texts and then run the measurement to compare.
Example of some N-grams for the word "TEXT":
• uni(1)-gram: T, E, X, T
• bi(2)-gram: T, TE, EX, XT, T
• tri(3)-grams: TE, TEX, EXT, XT, T
• quad(4)-grams: TEX, TEXT, EXT, XT, T
A string of length k, will have k+1 bi-grams, k+1 tri-grams, k+1 quad-grams, and so on.
Point of interest
No complex technique was used, I only utilized the hashtable indexing, and array binary search to solve this problem. The N-gram based text similarity also gives us interesting results.
Articles worth reading | {"url":"https://www.codeproject.com/KB/cs/tfidf.aspx","timestamp":"2024-11-08T20:10:08Z","content_type":"text/html","content_length":"21486","record_id":"<urn:uuid:3cb25817-336e-4dac-9da2-e9b184b60201>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00209.warc.gz"} |
3.5. Validation curves: plotting scores to evaluate models
3.5. Validation curves: plotting scores to evaluate models¶
Every estimator has its advantages and drawbacks. Its generalization error can be decomposed in terms of bias, variance and noise. The bias of an estimator is its average error for different training
sets. The variance of an estimator indicates how sensitive it is to varying training sets. Noise is a property of the data.
In the following plot, we see a function
Bias and variance are inherent properties of estimators and we usually have to select learning algorithms and hyperparameters so that both bias and variance are as low as possible (see Bias-variance
dilemma). Another way to reduce the variance of a model is to use more training data. However, you should only collect more training data if the true function is too complex to be approximated by an
estimator with a lower variance.
In the simple one-dimensional problem that we have seen in the example it is easy to see whether the estimator suffers from bias or variance. However, in high-dimensional spaces, models can become
very difficult to visualize. For this reason, it is often helpful to use the tools described below.
3.5.1. Validation curve¶
To validate a model we need a scoring function (see Model evaluation: quantifying the quality of predictions), for example accuracy for classifiers. The proper way of choosing multiple
hyperparameters of an estimator are of course grid search or similar methods (see Tuning the hyper-parameters of an estimator) that select the hyperparameter with the maximum score on a validation
set or multiple validation sets. Note that if we optimized the hyperparameters based on a validation score the validation score is biased and not a good estimate of the generalization any longer. To
get a proper estimate of the generalization we have to compute the score on another test set.
However, it is sometimes helpful to plot the influence of a single hyperparameter on the training score and the validation score to find out whether the estimator is overfitting or underfitting for
some hyperparameter values.
The function validation_curve can help in this case:
>>> import numpy as np
>>> from sklearn.model_selection import validation_curve
>>> from sklearn.datasets import load_iris
>>> from sklearn.linear_model import Ridge
>>> np.random.seed(0)
>>> iris = load_iris()
>>> X, y = iris.data, iris.target
>>> indices = np.arange(y.shape[0])
>>> np.random.shuffle(indices)
>>> X, y = X[indices], y[indices]
>>> train_scores, valid_scores = validation_curve(Ridge(), X, y, "alpha",
... np.logspace(-7, 3, 3))
>>> train_scores
array([[ 0.94..., 0.92..., 0.92...],
[ 0.94..., 0.92..., 0.92...],
[ 0.47..., 0.45..., 0.42...]])
>>> valid_scores
array([[ 0.90..., 0.92..., 0.94...],
[ 0.90..., 0.92..., 0.94...],
[ 0.44..., 0.39..., 0.45...]])
If the training score and the validation score are both low, the estimator will be underfitting. If the training score is high and the validation score is low, the estimator is overfitting and
otherwise it is working very well. A low training score and a high validation score is usually not possible. All three cases can be found in the plot below where we vary the parameter
3.5.2. Learning curve¶
A learning curve shows the validation and training score of an estimator for varying numbers of training samples. It is a tool to find out how much we benefit from adding more training data and
whether the estimator suffers more from a variance error or a bias error. If both the validation score and the training score converge to a value that is too low with increasing size of the training
set, we will not benefit much from more training data. In the following plot you can see an example: naive Bayes roughly converges to a low score.
We will probably have to use an estimator or a parametrization of the current estimator that can learn more complex concepts (i.e. has a lower bias). If the training score is much greater than the
validation score for the maximum number of training samples, adding more training samples will most likely increase generalization. In the following plot you can see that the SVM could benefit from
more training examples.
We can use the function learning_curve to generate the values that are required to plot such a learning curve (number of samples that have been used, the average scores on the training sets and the
average scores on the validation sets):
>>> from sklearn.model_selection import learning_curve
>>> from sklearn.svm import SVC
>>> train_sizes, train_scores, valid_scores = learning_curve(
... SVC(kernel='linear'), X, y, train_sizes=[50, 80, 110], cv=5)
>>> train_sizes
array([ 50, 80, 110])
>>> train_scores
array([[ 0.98..., 0.98 , 0.98..., 0.98..., 0.98...],
[ 0.98..., 1. , 0.98..., 0.98..., 0.98...],
[ 0.98..., 1. , 0.98..., 0.98..., 0.99...]])
>>> valid_scores
array([[ 1. , 0.93..., 1. , 1. , 0.96...],
[ 1. , 0.96..., 1. , 1. , 0.96...],
[ 1. , 0.96..., 1. , 1. , 0.96...]]) | {"url":"https://scikit-learn.org/0.18/modules/learning_curve.html","timestamp":"2024-11-02T20:48:43Z","content_type":"application/xhtml+xml","content_length":"24170","record_id":"<urn:uuid:7eb194cc-35d5-4167-94a1-916a73c544ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00002.warc.gz"} |
Re: LuaJIT FFI math 2x faster than built-in Lua math
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• Subject: Re: LuaJIT FFI math 2x faster than built-in Lua math
• From: Adam Strzelecki <ono@...>
• Date: Wed, 21 Dec 2011 00:54:06 +0100
> math.sin() calls the x87 fsin instruction, which is known to be
> slow on some CPUs, especially when reducing ranges. ffi.C.sin(),
> as implemented in most x64 math libraries, uses SSE and a faster
> range-reduction algorithm. The call overhead itself is negligible,
> since sin() is an expensive operation.
Thank you for precise explanation. Nice to learn something new that single FPU instructions can be sometimes slower than "software" implementation via SSE, which one however can have lower precision in some situations. I tried that "benchmark" on SSE-less machine (well actually via Parallels), math.sin was a bit faster than ffi.C.sin (similar to your benchmark).
> Try the same thing without a division and with sqrt() and you'll
> see that math.sqrt() is always faster.
Scored exactly the same here on my machine. Seems both are calling FPU's fsqrt.
> IMHO using sin() in benchmarks is the floating-point equivalent of
> Fibonacci benchmarks: totally worthless.
Yeah, I know these are worthless. But I wouldn't dare to ask about this if the difference was not so noticeable. Now I know it is tradeoff between slower fsin and lower precision SSE x64 implementation. Anyway this seems to be just for trigonometric functions, as rest math GLIBC functions seem to be calling FPU.
Thanks again for valuable answer, | {"url":"http://lua-users.org/lists/lua-l/2011-12/msg00623.html","timestamp":"2024-11-05T15:50:10Z","content_type":"text/html","content_length":"5245","record_id":"<urn:uuid:fc603e99-c6ed-434a-ab51-1b082492fba6>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00403.warc.gz"} |
Quenched random mass disorder in the la
SciPost Submission Page
Quenched random mass disorder in the large N theory of vector bosons
by Han Ma
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Han Ma
Submission information
Preprint Link: https://arxiv.org/abs/2205.11542v1 (pdf)
Date submitted: 2022-06-15 20:28
Submitted by: Ma, Han
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Condensed Matter Physics - Theory
Approach: Theoretical
We study the critical bosonic O(N) vector model with quenched random mass disorder in the large N limit. Due to the replicated action which is sometimes not bounded from below, we avoid the replica
trick and adopt a traditional approach to directly compute the disorder averaged physical observables. At $N=\infty$, we can exactly solve the disordered model. The resulting low energy behavior can
be described by two scale invariant theories, one of which has an intrinsic scale. At finite $N$, we find that the previously proposed attractive disordered fixed point at $d=2$ continues to exist at
$d=2+\epsilon$ spatial dimensions. We also studied the system in the $3<d<4$ spatial dimensions where the disorder is relevant at the Gaussian fixed point. However, no physical attractive fixed point
is found right below four spatial dimensions. Nevertheless, the stable fixed point at $2+\epsilon$ dimensions can still survive at $d=3$ where the system has experimental realizations. Some critical
exponents are predicted in order to be checked by future numerics and experiments.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-11-1 (Invited Report)
• Cite as: Anonymous, Report on arXiv:2205.11542v1, delivered 2022-10-31, doi: 10.21468/SciPost.Report.6025
1. Valid
2. Clear presentation
3. New approach to problem
1. Results are somewhat incremental
This paper approaches the problem of the O(N) model with random mass disorder at large N. This theory has been studied before by other works.
The novelty of this paper is that a somewhat different approach is used, where, instead of working with the replicated disorder-averaged action in the limit of n (number of replicas) going to 0, the
author works with the action expanded perturbatively in the disordered term, with disorder-averaging then performed directly order by order. Because of the large N limit, the series can be truncated
at a finite order in order to obtain correction effects up to some desired power of 1/N.
The the author computes corrections to scaling dimensions using standard Wilsonian field-theoretic RG techniques. The results of Ref. [6] are then extended to d = 2+\epsilon instead of just d = 2,
extrapolating these results gives the possibilty of a novel fixed point at d = 3.
The paper is correct as far as I can see, and the presentation is sufficiently clear. I recommend publication in SciPost.
Author: Han Ma on 2022-11-21 [id 3055]
(in reply to
Report 2
on 2022-11-01)
We thank the referee for reviewing this manuscript and thank her/him for agreeing that our manuscript is suitable for publication in SciPost. While we recognize the referee’s assessment is generally
positive, she/he somehow thinks our result is incremental. Regarding this critique, we would like to list our new results which were unknown before. 1)As addressed by the referee, we push the study
of the disordered fixed point of the O(N) model to 2+epsilon spatial dimensions by double expansion of epsilon and 1/N. 2)At infinite N, we can actually get the disordered IR physics at arbitrary
dimensions. The disorder strength plays a role of intrinsic length scale and the whole system has scale invariance in a 3 dimensional subsystem. After extensive studies on the classical random field
Ising model, this is the first example of dimension reduction in the system with quantum quenched disorder. 3) Besides, around 4 spatial dimensions, we have obtained the fixed point structure and an
unstable fixed point is found above 4 dimensions. This is the first time we are able to study a disordered system above the upper critical dimension where the replica method is invalid.
Report #1 by Ilya Esterlis (Referee 1) on 2022-10-24 (Invited Report)
• Cite as: Ilya Esterlis, Report on arXiv:2205.11542v1, delivered 2022-10-24, doi: 10.21468/SciPost.Report.5974
1. Very clearly written
2. Novel results on an important and outstanding problem
In "Quenched random mass disorder in the large N theory of vector bosons," the author analyzes the problem of the quantum O(N) model with random mass disorder, using a combination of large-N and
epsilon-expansion techniques. The problem of quantum systems with disorder is notoriously challenging, primarily due to the long-range imaginary-time correlations induced by the disordered couplings.
Thus, new, controlled approaches to the problem are both welcome and of significant importance. In particular, approaches that avoid the "replica trick" provide a useful complement to the existing
The primarily results of the paper are (1) An exact solution of the problem at N=infinity. An especially interesting point is the author's discovery of the necessity of an "intrinsic scale" in the
problem for space dimension 2<d<3. This leads to a scale invariant theory in a reduced number of dimensions, akin to the phenomenon of "dimensional reduction" in the random-field Ising model. (2) The
calculation of leading 1/N corrections. A highlight of this is the author's demonstration that there exists a disordered, interacting fixed point that is stable to 1/N corrections for d=2+epsilon
dimensions. One may therefore hope that the results of the current paper could even be extrapolated to the physically relevant case of d=3 (epsilon=1).
I find the exposition to be especially clear and pedagogical. The fact that the author computes things using both the "condensed matter/statistical mechanics" and "high-energy" versions of RG is
particularly helpful.
I have a few minor questions/comments for the author:
1. The author makes several references to the notion of "generalized free fields." While this is discussed in Ref. 30, I think the paper would benefit from a very brief reminder/explanation of this
idea. My pedestrian understanding of generalized free fields is something like mean-field theory (which indeed solves the O(N) model at N=infinity) -- that is, Gaussian theories with non-standard
Green's functions, determined by solving an appropriate self-consistency equation (but perhaps my understanding is incorrect).
2. Something that comes up often in RG studies of quantum systems with quenched disorder are complex critical exponents with spiraling RG flows (e.g., Ref. 3 of the current paper). It appears such
behavior is absent in these new calculations. Does the author have an understanding of this? Is it related to the instability of the replicated theory? I believe complex exponents have been observed
in holographic calculations which do not rely on the replica trick; e.g., Hartnoll, Sean A., David M. Ramirez, and Jorge E. Santos. "Thermal conductivity at a disordered quantum critical point." JHEP
04 (2016).
3. In Sec. IVB, when discussing the dimensional reduction, there appears to be something singular happening when d-> 3 (the dimension of the associated free CFT diverges). Perhaps I missed it, but is
there an explanation or understanding of this singularity.
I believe this is a high-quality paper that offers a straight-forward and welcome new approach to the problem of quenched disorder in quantum systems, and I recommend it for publication.
We thank the referee for reviewing our manuscript and for taking the time to make detailed comments and questions. We are glad that the referee recommends our work for publication. Here is our
response to the referee’s comments and questions.
1) The generalized free field is a scaling operator in a CFT that has a different scaling dimension from that of the free field. If it has scaling dimension $\Delta$, then $\Delta \neq \frac{d-1}{2}$
in d spatial dimensions. It has non-zero two point function with form $\frac{1}{r^{2\Delta}}$ but its higher point (connected) functions are all zero. A simple theory for a generalized free field $\
Phi$ is a Gaussian theory with action $S=\int d^{d+1} r~ \Phi (-\nabla^2)^{\Delta-\frac{d+1}{2}}\Phi$. We have added this explanation of the terminology "generalized free field" as a footnote on
2)Yes, the complex critical exponent and spiral RG flow are absent in this 1/N expansion calculation. In the double epsilon expansion in e.g. Ref.3, the perturbation is done around the free Gaussian
fixed point, where there is accidental degeneracy of the operators. More concretely, the operator $\phi^4(x,\tau)$ and $\phi^2(x,\tau)\phi^2(x,\tau') $ are both relevant and have the same scaling
dimension. Using the replica trick or not, the latter operator would always contribute to the disorder averaged correlation functions. These two operators are mixed along the RG acquiring complex
scaling dimensions and induce the spiral flow. However, there is no such an accidental degeneracy at the infinite N fixed point of the O(N) vector model. So perturbation around this fixed point will
not result in any complex scaling dimension.
3)The derivation in Sec.IV is for the system below 3 spatial dimensions. The divergence in the dimension reduction at $d\rightarrow 3$ is traced back to $G_\sigma \rightarrow 0$ in Eq.(4.1) at $d\
rightarrow 3$. In fact, as $d\rightarrow 3$, we should correct the action by the term $\sigma^2$ which is exactly marginal in the infinite N limit. In this way, the propagator of the $\sigma$ field
becomes a constant and the effective action in Eq.(4.15) can be written in terms of the bosonic $\sigma$ field only. After the integration of field $\sigma$ at nonzero frequency, we get an effective
action at zero frequency which can be directly identified as a scale invariant theory in 3 dimensions. This is consistent with our conclusion that the O(N) model in a d+1 spacetime dimensional system
is scale invariant in the 3 dimensional subsystem. We have added a brief analysis on the d=3 system at the end of IV B on p.g. 17. | {"url":"https://www.scipost.org/submissions/2205.11542v1/","timestamp":"2024-11-06T13:52:49Z","content_type":"text/html","content_length":"45132","record_id":"<urn:uuid:2c4b6afd-30c5-444f-9636-e920742ca36e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00679.warc.gz"} |
Kids.Net.Au - Encyclopedia > LL parser
LL parser
is a table-based top-down
for a subset of the
context-free grammars
. It parses the input from
eft to right, and constructs a
eftmost derivation of the sentence (Hence LL, compare with
LR parser
). The class of grammars which are parsable in this way is known as the
LL grammars
. Older programming languages sometimes use LL grammars because it is simple to create parsers for them by hand - using either the table-based method described here, or a
recursive descent parser
An LL parser is called an LL(k) parser if it uses k tokens of look-ahead when parsing a sentence. If such a parser exists for a certain grammar and it can parse sentences of this grammar without
backtracking then it is called an LL(k) grammar. Of these grammars LL(1) grammars, although fairly restrictive, are very popular because the corresponding LL parsers only need to look at the next
token to make their parsing decisions.
A table-based top-down parser can be schematically presented as in Figure 1.
Input: | ( | 1 | + | 1 | ) | $ |
Stack: |
+---+ | |
| + |<------| Parser |-----> Output
+---+ | |
| S | +----------+
+---+ ^
| ) | |
+---+ |
| $ | +----------+
+---+ | Parsing |
| table |
Figure 1. Architecture of a table-based top-down parser
The parser has an
input buffer
, a
on which it keeps symbols from the grammar, a
parsing table
which tells it what grammar rule to use given the symbols on top of its stack and its input tape. To explain its workings we will use the following small grammar:
(1) S -> F
(2) S -> ( S + F )
(3) F -> 1
The parsing table for this grammar looks as follows:
│ │(│)│1│+│$│
Note that there is also a column for the special terminal $ that is used to indicate the end of the input stream.
When the parser starts it always starts on its stack with
[ S, $ ]
where $ is a special terminal to indicate the bottom of the stack and the end of the input stream, and S is the start symbol of the grammar. The parser will attempt to rewrite the contents of this
stack to what it sees on the input stream. However, it only keeps on the stack what still needs to be rewritten. For example, let's assume that the input is "( 1 + 1 )". When the parser reads the
first "(" it knows that it has to rewrite S to "( S + F )" and writes the number of this rule to the output. The stack then becomes:
[ (, S, +, F, ), $ ]
In the next step it removes the '(' from its input stream and from its stack:
[ S, +, F, ), $ ]
Now the parser sees an '1' on its input stream so it knows that it has to apply rule (1) and then rule (3) from the grammar and write their number to the output stream. This results in the following
[ F, +, F, ), $ ]
[ 1, +, F, ), $ ]
In the next two steps the parser reads the '1' and '+' from the input stream and also removes them from the stack, resulting in:
[ F, ), $ ]
In the next three steps the 'F' will be replaced on the stack with '1', the number 3 will be written to the output stream and then the '1' and ')' will be removed from the stack and the input stream.
So the parser ends with both '$' on its stack and on its input steam. In this case it will report that it has accepted the input string and on the output stream it has written the list of numbers [
2, 1, 3, 3 ] which is indeed a rightmost derivation if the input string in reverse.
As can be seen from the example the parser performs three types of steps depending on whether the top of the stack is a nonterminal, a terminal or the special symbol $:
• If the top is a nonterminal then it looks up in the parsing table on the basis of this nonterminal and the symbol on the input stream which rule of the grammar it should use to replace it with on
the stack. The number of the rule is written to the output stream. If the parsing table indicates that there is no such rule then it reports an error and stops.
• If the top is a terminal then it compares it to the symbol on the input stream and if they are equal they are both removed. If they are not equal the the parser reports an error and stops.
• If the top is $ and on the input stream there is also a $ then the parser reports that it has successfully parse the input, otherwise it reports an error. In both cases the parser will stop.
These steps are repeated until the parser stops, and then it will have either completely parsed the input and written a leftmost derivation to the output stream or it will have reported an error.
In order to fill the parsing table we have to establish what grammar rule the parser should choose if it sees a nonterminal A on the top of its stack and symbol a on its input stream. It is easy to
see that such a rule should be of the from A -> w and that the language corresponding with w should have at least one string starting with a. For this purpose we define the First-set of w, written
here as Fi(w), as the terminals with which the strings that belong to w start plus ε if the empty strings also belongs to w. Given a grammar with the rules A[1] -> w[1], ..., A[n] -> w[n] we can
compute the Fi(w[i]) and Fi(A[i]) for every rule as follows:
1. initialize every Fi(w[i]) and Fi(A[i]) with the empty set
2. add Fi(w[i]) to Fi(w[i]) for every rule A[i] -> w[i] where Fi is defined as follows:
□ Fi(a w' ) = { a } for every terminal a
□ Fi(A w' ) = Fi(A) for every nonterminal A with ε not in Fi(A)
□ Fi(A w' ) = Fi(A) \ { ε } ∪ Fi(w' ) for every nonterminal A with ε in Fi(A)
□ Fi(ε) = { ε }
3. add Fi(w[i]) to Fi(A[]i) for every rule A[i] -> w[i]
4. repeat the steps 2 and 3 until all Fi sets stay the same.
Unfortunately the First-sets are not sufficient to compute the parsing table. This is because a right-hand side w of a rule might ultimately be rewritten to the empty string. So the parser should
also use the a rule A -> w if ε is in Fi(w) and it sees on the input stream a symbol that could follow A. Therefore we also need the Follow-set of A, written as Fo(A) here, which is defined as the
set of terminals a such that there is a string of symbols αAaβ that can be derived from the start symbol. Computing the Follow-sets for the nonterminals in a grammar can be done as follows:
1. initialize every Fo(A[i]) with the empty set
2. if there is a rule of the form A[j] -> wA[i]w' then
□ if the terminal a is in Fi(w' ) then add a to Fo(A[i])
□ if ε is in Fi(w' ) then add Fo(A[j]) to Fo(A[i])
3. repeat step 2 until all Fo sets stay the same.
Now we can define exactly which rules will be contained where in the parsing table. If T[A, a] denotes the entry in the table for nonterminal A and terminal a then
T[A,a] contains the rule A -> w iff
a is in Fi(w) or
ε is in Fi(w) and a is in Fo(A).
If the table will contain at most one rule in every one of its cells then the parser will always know which rule it has to use and can therefore parse strings without backtracking. It is precisely in
this case that the grammar is called an LL(1) grammar.
... yet to be written ...
All Wikipedia text is available under the terms of the GNU Free Documentation License | {"url":"http://encyclopedia.kids.net.au/page/ll/LL_parser","timestamp":"2024-11-02T21:11:49Z","content_type":"application/xhtml+xml","content_length":"21429","record_id":"<urn:uuid:982c5751-dc36-40cc-9594-54518f91caae>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00041.warc.gz"} |
lgebraic and
Bi-annual Algebraic and Tropical Meetings of
Brown and YaLE
Fall 2020 @ Brown (virtually)
November 9, 2020
The BATMOBYLE is a vehicle for bringing together the algebraic and tropical geometry groups of Brown and neighboring institutions for a biannual day of talks.
Hannah Markwig (Tübingen) -- Counting bitangents of plane quartics - tropical, real and arithmetic
A smooth plane quartic defined over the complex numbers has precisely 28 bitangents. This result goes back to Plücker. In the tropical world, the situation is different. One can define equivalence
classes of tropical bitangents of which there are seven, and each has 4 lifts over the complex numbers. Over the reals, we can have 4, 8, 16 or 28 bitangents. The avoidance locus of a real quartic is
the set in the dual plane consisting of all lines which do not meet the quartic. Every connected component of the avoidance locus has precisely 4 bitangents in its closure. For any field k of
characteristic not equal to 2 and with a non-Archimedean valuation which allows us to tropicalize, we show that a tropical bitangent class of a quartic either has 0 or 4 lifts over k. This way of
grouping into sets of 4 which exists tropically and over the reals is intimately connected: roughly, tropical bitangent classes can be viewed as tropicalizations of closures of connected components
of the avoidance locus. Arithmetic counts offer a bridge connecting real and complex counts, and we investigate how tropical geometry can be used to study this bridge.
This talk is based on joint work with Maria Angelica Cueto, and on joint work in progress with Sam Payne and Kristin Shaw.
Hannah Larson (Stanford) -- Brill-Noether theory over the Hurwitz space
Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli
space of such maps is well-understood by the main theorems of Brill-Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may
force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved
elusive in this case. In this talk, I will discuss recent joint work with Eric Larson and Isabel Vogt that completes such a picture, by proving analogs of all of the main theorems of Brill-Noether
theory in this setting. | {"url":"https://math.dartmouth.edu/~auel/BATMOBILE/fall2020_abstracts.html","timestamp":"2024-11-09T06:26:23Z","content_type":"text/html","content_length":"4004","record_id":"<urn:uuid:b5dca57e-7107-460f-a1f2-8f032dc41c26>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00813.warc.gz"} |
Latex – Walking Randomly
Archive for the ‘Latex’ Category
December 29th, 2012
xkcd is a popular webcomic that sometimes includes hand drawn graphs in a distinctive style. Here’s a typical example
In a recent Mathematica StackExchange question, someone asked how such graphs could be automatically produced in Mathematica and code was quickly whipped up by the community. Since then, various
individuals and communities have developed code to do the same thing in a range of languages. Here’s the list of examples I’ve found so far
Any I’ve missed?
August 29th, 2012
While on the train to work I came across a very interesting blog entry. Full LaTeX support (on device compilation and .dvi viewer) is now available on iPad courtesy of TeX Writer By FastIntelligence
. Here is the blog post telling us the good news http://litchie.com/blog/?p=406
At the time of writing, the blog is down (Update: working again), possibly because of the click storm that my twitter announcement caused..especially after it was picked up by @TeXtip. So, here is
the iTunes link http://itunes.apple.com/us/app/tex-writer/id552717222?mt=8
I haven’t tried this yet but it looks VERY interesting. If you get a chance to try it out, feel free to let me know how you get on in the comments section.
Update 1: This version of TeX writer(1.1) cannot output to .pdf. Only .dvi output is supported at the moment.
August 19th, 2009
There are a lot of symbols in mathematics and I mean a LOT! Not content with the entire Greek alphabet, mathematicians have gone on to use symbols from other alphabets such as Hebrew. Once they had
run out of alphabets they went on to invent hundreds of symbols themselves – a symbol for every occasion.
So, you are writing a paper in an esoteric (or maybe not so esoteric) area of mathematics and, naturally, you are writing it in Latex. Suddenly you think to yourself ‘What’s the LaTeX command for
<insert weird and wonderful glyph here>’
Searching in vain through list after list of LaTeX symbols you get to thinking ‘If only I could just draw the symbol and have the computer tell me what the LaTeX command is‘.
Well now you can!
Detexify is a new project from Philipp Kühl (who had the initial idea) and Daniel Kirsch (who implemented it) and is essentially an exercise in machine learning. Sometimes it works perfectly (such
as in the screenshot above) but other times it struggles a bit and you end up learning the commands for symbols you never even knew existed.
Teach the system
When it is struggling though, you can help it along. Eventually you will find the symbol you were looking for and you can click on it to tell the system ‘That squiggle I drew – this is what I meant’
thus helping to train it for future searchers.
Other times though, you cannot blame it for not finding the symbol you meant. For example I needed about 5 tries before I could get it to recognise my ham-fisted attempt at the lowercase zeta symbol
. This says a lot more about my poor handwriting and mouse skills than it does about the quality of Texify though.
Are you rubbish with the mouse? Use your finger on your mobile phone then!
I found drawing even simple glyphs rather difficult with the mouse and soon found myself wishing that I could do it with my finger or a stylus so I was overjoyed to learn that Robin Baumgarten has
released a version of Texify for Android mobile phones. The Android app works exactly like the web version and connects to the server in order to do the actual recognition.
Iphone users haven’t been left out though since Daniel has released an app for that himself.
This is a great project that Daniel is now developing for his diploma thesis and you and you can read more about its progress over at his blog.
March 2nd, 2009
From time to time I get sent someone’s thesis in the hope that I might be able to fix a Latex problem or two for them. I was recently looking at someone’s code on a newish Ubuntu 8.10 machine and
when I tried to compile it I received the following error
! LaTeX Error: File `setspace.sty’ not found.
The Ubuntu package I needed was texlive-latex-recommended which I installed using
sudo apt-get install texlive-latex-recommended
On trying to compile the code a second time I got the following error
! LaTeX Error: File `footmisc.sty’ not found.
which was fixed by
sudo apt-get install texlive-latex-extra
These packages don’t just install footmisc.sty and setspace.sty for you – they install a whole host of Latex packages as well.
August 20th, 2007
Just as every walk (including random ones) must begin with a first step – every blog must have a first post. For my first post I thought I would mention the rather wonderful program that will allow
me to use Latex to write equations on here – Mimetex.
The first solution that my web guru and I attempted to use was something called LatexRenderer but you need to have Latex installed on your web server to use it and for various reasons this was not
possible for us. Mimetex, however, does not require Latex to be installed on your server so we gave that a go.
Once you have installed and activated the Mimetex plugin for wordpress all you need to do is put your Latex code between the two tags [ tex][ /tex] (without spaces) and it will be automatically
rendered as an image. This makes it very easy to write equations such as
The output from Mimetex is not quite as nice as that of LatexRenderer and it only supports a subset of Latex but the ease of installation more than makes up for this. | {"url":"https://walkingrandomly.com/?cat=3","timestamp":"2024-11-02T04:32:10Z","content_type":"application/xhtml+xml","content_length":"68241","record_id":"<urn:uuid:7b2371e1-9c88-4451-bf88-ef450cfc4e91>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00244.warc.gz"} |
Welcome to the fourth and final exam illustration in this section of our 11+ Non-Verbal Reasoning course, all about Relationships Between Symbols. Like the first three, it’s to do with Analogies
style questions.
In the previous illustrations we looked at analogies formed by size, number, shading, movement and rotation. This final one looks at analogies formed in a similar way to rotation shapes – ones made
by flipping shapes (or does it? Read on to find out!).
How Are These Kinds of Question Posed in The Exam?
These are posed in exactly the same way as other Analogies questions. As you will remember, candidates are shown three shapes or patterns. The first two are related in some way, and children have to
pick the shape or pattern which is related in the same way to the third.
Flipping is just another method examiners use to create these kinds of question. The question will look exactly the same – only the technique used to solve it is different. Let’s look at an example:
On the left are two shapes with an arrow between them. Decide how they are related. The third shape is related to one of the remaining shapes in the same way. Which of the shapes goes with the third
in the same way as the second goes with the first?
Okay then, how do we work out the answer? As usual, let’s put the first pair into words – not too difficult in this case, as there are only three changes:
1. The shield shape flips top to bottom
2. The upside-down house shape becomes a triangle
3. The circle becomes a square
Now, let’s try to employ this logic to the second pair:
The trapezium needs to be flipped top to bottom, meaning the only possible answer is ‘c’. Alarm bells should now be ringing – too easy, something’s wrong! If you check the smaller shapes in ‘c’ they
don’t follow the same pattern as the first pair – the circle gets smaller and the upside-down house shape gets bigger, whereas in the first pair there is no connection at all. Okay, now what?
At this point it’s important to remember that the flipping of some shapes can look the same as rotating them one hundred and eighty degrees. If you look at the shield, this is the case. However, the
trapezium shape doesn’t work the same way, so rather than assuming it’s flipping top to bottom we had better assume it’s rotating through one hundred and eighty degrees.
Let’s go back to the question – if the rotation is made, the trapezium should end up looking like the one in ‘a’ and ‘e’. Now we can look at the middle-sized shape – both are squares so let’s
discount that. The small shape in the target shape should be a triangle. How can we tell this? There is no direct connection between the internal shapes in the first pair but, if we look at the
shapes that the first two become we see what is needed. Upside-down house shape becomes a triangle, circle becomes a square. Applying this logic, as it’s the only thing available to us, we get the
triangle in the middle of our target shape, surrounded by a larger square.
Technique Tip
If you get an answer which is too easy, then the likelihood is it’s wrong. Children should be aware that they cannot just carry through one element of the question, find only one possible answer, and
assume it’s correct. They have to at least skim through other elements of the question to check that they haven’t made a careless mistake.
As I demonstrated previously, the question setters are trying to catch careless people out and it’s not often that they’ll leave a correct option which is so totally different from the remainder.
Children should be encouraged to watch out for the trick answers as much as the trick questions!
Sample Tests
Now you have worked your way through this whole section. It’s time to put what you have learned to the test! Follow the links below to the 8 quizzes on the Education Quizzes website. There you will
find a total of 80 Analogies questions. Let your child work their way though. If they have any problems, help them out by teaching them the techniques you’ve learned in this section. If they go into
their exam armed with the knowledge of how to tackle these questions, then they are already halfway there.
So, now we’ve looked at Analogies, what’s next? Well, in the next section we look at similarities and differences between shapes. These come in two styles of question, which we’ll look at in detail.
See you there! | {"url":"https://www.educationquizzes.com/11-plus/exam-illustrations-nvr/relationships-between-symbols-4/","timestamp":"2024-11-13T22:26:46Z","content_type":"text/html","content_length":"33214","record_id":"<urn:uuid:9d109c24-8c16-45da-8d3a-72f8ad052e65>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00299.warc.gz"} |
45 Feet To Meter Converter - StudySaga
45 Feet To Meter Converter
45 Feet To Meter Converter
45 Feet is equal to how many Meter ?
45 Feet is equal to 13.716 meter
How to convert Feet To Meter ? | Formula To Convert Feet To Meter?
Since one Feet = 0.30480370641327 Meter , you can convert any Feet measurement into Meter by multiplying it by 0.30480370641327.
Also Convert here other Feet Length To Meter
Feet to Meters Converter
Type a value in the Feet field to convert the value to Meters:
What is Feet?
The feet is a symbol for measuring the length and is used throughout the U.S. and Imperial customary systems of measurement. A feet has been defined in the year 1959 as equivalent to 0.3048 meters. A
feet has 12 inches and a yard has three feet. The foot has been used in several global systems from ancient times, finding favor with the Chinese, Romans, Greek, English and French. The U.S. is one
of the countries where the foot is predominantly used for various measurements.
What is meter?
The base unit of length in the International System of Units that is equal to the distance traveled in a vacuum by light in 1/299,792,458 second or to about 39.37 inches. meter. noun. Medical
Definition of meter (Entry 2 of 2) : an instrument for measuring and sometimes recording the time or amount of something.
How to Convert Feet to Meters?
The procedure to use the feet to meter calculator is as follows:
Step 1: Enter the feet value the input field
Step 2: Now there is automatically converted to meter value
Step 3: Finally, the conversion from feet to meter value will be displayed in the output field.
What is Meant by Feet to Meter Conversion?
The feet to meter calculator gives the conversion of measurement from the unit feet to meter. The unit feet is represented by “ft” and the meter is represented by “mt”. Meter is considered as a part
of the metric system and is used in the International System of units. The unit feet is used in the US customary measurement system. One feet is approximately equal to 0.30480370641307 meters.
Relationship between Feet and Meter
1 ft = 0.30480370641307 mt
For example, the conversion of 5 ft to mt is given as follows:
We know that 1 ft = 0.30480370641307 mt
So 5 ft = 5 x 0.30480370641307 mt
5 ft = 1.5240185320653499 mt.
Therefore, the conversion of 5 feet to the centimeter is 1.5240185320653499 mt
Feet to Meter Conversion Table
1ft = 0.30480370641307 mt.
2ft = 0.60960741282614 mt.
3ft = 0.9144111192392099 mt.
4ft = 1.21921482565228 mt.
5ft = 1.5240185320653499 mt.
6 ft = 1.8288222384784198 mt.
7ft = 2.13362594489149 mt.
8ft = 2.43842965130456 mt.
9ft = 2.7432333577176298 mt.
10ft = 3.0480370641306997 mt.
What is 5.2 feet to meter ?
5.2 ft = 1.584979273347964 mt.
What is 5.3 feet to meter ?
5.3 ft = 1.6154596439892708 mt.
What is 5.4 feet to meter ?
5.4 ft = 1.645940014630578 mt.
What is 5.5 feet to meter ?
5.5 ft = 1.6764203852718849 mt.
What is 5.6 feet to meter ?
5.6 ft = 1.7069007559131917 mt.
What is 5.7 feet to meter ?
5.7 ft = 1.737381126554499 mt.
What is 5.8 feet to meter ?
5.8 ft = 1.7678614971958058 mt.
What is 6 feet to Meter ?
6 ft = 1.8288222384784198 mt. | {"url":"https://studysaga.in/45-feet-to-meter-converter/","timestamp":"2024-11-10T05:50:44Z","content_type":"text/html","content_length":"121672","record_id":"<urn:uuid:05aa77c6-819d-4ada-b9e4-b7d7c4cc850b>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00197.warc.gz"} |
Morten Risager
University of Copenhagen University of Copenhagen
Arithmetic statistics of modular symbols
Heilbronn Number Theory Seminar
4th December 2019, 4:00 pm – 5:00 pm
Fry Building, 2.04
Mazur, Rubin, and Stein have formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of
these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by
modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols. We prove a refined version of this conjecture.
This is joint work with Yiannis Petridis. | {"url":"https://www.bristolmathsresearch.org/seminar/morten-risager/","timestamp":"2024-11-08T05:21:20Z","content_type":"text/html","content_length":"54438","record_id":"<urn:uuid:819f22ef-63e7-4b54-8c4e-f544d2b25f6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00359.warc.gz"} |
"Tens Of Thousands" Meaning: Here's All We Know!
“Tens Of Thousands” Meaning: Here’s All We Know!
“Tens of thousands” is a phrase that has been used for centuries to describe a large quantity of something. It is often used to refer to a large number of people or objects, but it can also be used
in reference to amounts of money or other measurements.
When it comes to understanding what tens of thousands means, it can be helpful to look at the etymology of the phrase, the various ways it can be used, and the implications of the phrase.
Understanding what tens of thousands means can be beneficial in various situations, such as discussing population sizes, accounting for large sums of money, and even talking about time.
To better understand this phrase and its implications, it’s important to take a closer look at what tens of thousands actually means and how it’s used in the real world.
From its literal definition to its figurative meaning and implications, this article will explore the concept of tens of thousands in-depth and provide a comprehensive overview of the term.
What Does “Tens Of Thousands” Mean?
In everyday speech, the phrase tens of thousands is often used to express a large quantity. But what does it really mean? How big is tens of thousands? Does it refer to a specific number or is it a
general term?
“Tens of thousands” can be used to describe money, people, or items. It is an important concept to understand when making decisions about investments, purchases, or any other activity that involves
large amounts of something.
It is also important to understand that tens of thousands is not a specific number. It is simply a phrase used to describe a range of numbers, typically between 10,000 and 99,999.
So, when someone says that something is in the tens of thousands, it could actually mean anything between 10,000 and 99,999.
It is important to clarify the exact amount or quantity when discussing tens of thousands in order to have a better understanding of the scope of the situation.
In terms of money, tens of thousands typically refer to amounts of money that are in the range of $10,000 to $99,999.
This phrase can also be used to describe a population size, typically referring to groups of people with populations greater than 10,000.
It can also be used to describe the number of items or objects that are in a group, such as a collection of books or a set of tools.
6 Reasons Why People Say Tens Of Thousands Instead Of Using A Figure?
When it comes to quantifying the size of a group or the amount of something, people tend to use figures such as tens of thousands rather than specifying the exact number.
This phrase is used to provide a general sense of the size of the group or the amount of something without getting into specifics.
Here are 6 reasons why people say tens of thousands instead of using a figure:
1. To Avoid Being Too Specific
One of the main advantages of saying tens of thousands is that it allows people to talk in general terms without getting too specific.
This can be useful in situations where the exact number isn’t relevant or important, such as when discussing the size of a crowd or the number of people who have taken a certain action.
By saying tens of thousands, people can avoid getting into the nitty-gritty details and focus on the bigger picture.
2. To Avoid False Precision
Using a figure can also create a false sense of precision. For example, if someone says seven thousand people attended the event, it may give the impression that the exact number of attendees is
However, this could be inaccurate since the figure may be an estimate or a guess. By saying tens of thousands, the speaker conveys that the number is only a general indication and not an exact
3. To Provide A Sense Of Scale
Saying tens of thousands can also provide a sense of scale to the situation.
For instance, if someone says there were tens of thousands of people at the protest, it provides a greater sense of the magnitude of the event than if they simply said there were a few thousand
This can be useful in situations where it is important to convey the size of the group or the amount of something.
4. To Avoid Unnecessary Counting
In some cases, counting the exact number of people or things can be a laborious and time-consuming task.
For example, if someone is asked how many people attended a large event, it may be difficult to accurately count the exact number and so it may be easier to just say tens of thousands instead.
This allows the speaker to quickly provide a general indication of the size of the crowd without having to count everyone.
5. To Make A Statement
Saying tens of thousands can also be used to make a statement.
For example, if someone says tens of thousands of people attended the protest, it conveys a sense of the size and strength of the protest that would not be conveyed if the speaker said a few hundred
people attended the protest.
This can be useful in situations where the speaker wants to emphasize the size or importance of the group or the amount of something.
6. To Simplify The Discussion
Finally, saying tens of thousands can help to simplify the discussion. For instance, if someone is asked how many people attended an event and say tens of thousands, it provides an easy way to
quickly answer the question without having to get into the exact number.
This can be useful in situations where the speaker wants to provide a general sense of the size of the crowd or the amount of something without getting too bogged down in the details.
Saying tens of thousands instead of using a figure has several advantages. It allows people to talk in general terms without getting too specific, provides a sense of scale, avoids false precision,
simplifies the discussion, and can be used to make a statement.
For these reasons, it is no surprise that people often say tens of thousands instead of using a figure.
Is It Ten Thousands Or Tens Of Thousands?
When it comes to writing numbers, there is a lot of confusion and debate about the correct way to express numbers.
One of the most common questions is whether it is ten thousands or tens of thousands? To understand the difference between these two terms, we must first understand the concept of place value.
The place value of a number is the value of that number based on its position. The place value of a digit is determined by multiplying its value by the place value of its position.
For example, the place value of the digit 4 in the number 4,000 is 4,000. Ten thousands refers to the number 10,000. For example, the number ten thousand is written as 10,000. This is because the
place value of the number 1 is 10,000.
Tens of thousands, on the other hand, refers to a number that is greater than 10,000 but less than 100,000. For example, if we have a number that is between 10,001 and 99,999, then we would call it
tens of thousands.
This is because the place value of the number is greater than 10,000 but less than 100,000. It is important to note that when expressing numbers, the number should be written in the most precise way
For example, if you were to express the number 20,000, it would be incorrect to write it as tens of thousands since the correct expression is twenty thousands.
In summary, the difference between ten thousands and tens of thousands is the place value of the number. Ten thousands refers to the number 10,000 while tens of thousands refer to a number that is
greater than 10,000 but less than 100,000.
When expressing numbers, it is important to be precise and use the correct terminology.
4 Examples Sentences For Tens Of Thousands
When speaking about large numbers, it is important to know the right words to use. Tens of thousands are a large number that needs to be expressed in a clear and concise manner to avoid confusion.
For example, if you were trying to express the population of a city as being seventy thousand, you would use a different phrasing than if you were trying to express the population of a state as being
seven million.
To better understand how to express large numbers, here are four examples sentences for tens of thousands.
The number of people living in the city is “tens of thousands.”
This sentence is a clear and concise way to express the population of a city that is in the tens of thousands range.
It is important to note that this sentence does not specify a specific number, as it could range from ten thousand to ninety-nine thousand.
The crowd size was estimated to be in the tens of thousands
This sentence is a great way to express the size of a large crowd or gathering. It is important to note that this sentence does not specify a specific number, as it could range from ten thousand to
ninety-nine thousand.
The number of books in the library was in the tens of thousands
This sentence is a clear and concise way to express the total number of books in a library. It is important to note that this sentence does not specify a specific number, as it could range from ten
thousand to ninety-nine thousand.
The estimated attendance at the event was tens of thousands
This sentence is a great way to express the size of an event or gathering. It is important to note that this sentence does not specify a specific number, as it could range from ten thousand to
ninety-nine thousand.
What Comes After Tens Of Thousands
To answer this question, we must first consider the scale of the numbers we are talking about. A thousand is a large number, representing a great amount of something.
It is a full order of magnitude larger than a hundred and two orders of magnitude larger than ten. Tens of thousands, then, is an even larger number, representing an even greater amount of something.
The next step after tens of thousands is hundreds of thousands. This is a number with six digits, representing an even larger amount of whatever is being measured.
After hundreds of thousands come millions. A million is a number with six zeroes, representing an incredibly large amount.
Why You Should Use Tens Of Thousands
Tens of thousands is a powerful and versatile phrase used to refer to a large number of people, objects, or amounts. It can be used to express a wide range of emotions, from excitement to awe and
even fear.
As such, it is an invaluable tool for anyone looking to communicate a significant quantity in an impactful way.
Tens of thousands is a great way to emphasize a number that is too large to grasp in its entirety. When you need to convey a large amount of something, tens of thousands can provide the necessary
emphasis without being too specific.
For example, if you were trying to describe the population of a city, you could say it has tens of thousands of inhabitants rather than saying it has 76,945 inhabitants. The former statement is more
impactful and memorable.
In conclusion, the phrase tens of thousands is an excellent way to describe large numbers in a concise and easily understandable manner. It’s a great way to give a sense of scale and magnitude
without getting bogged down in details.
Whether you are talking about the number of people in a large crowd, the amount of money in a large budget or the number of items in a large inventory, the phrase tens of thousands can quickly and
effectively convey the size of the numbers being discussed.
As such, it is an invaluable tool for anyone communicating about large quantities of anything.
The use of the phrase tens of thousands is a powerful one and has many useful applications. Its ability to convey large numbers without having to provide exact figures can help to simplify
communication and put a more concrete image in the minds of readers.
Leave a Comment | {"url":"https://correctley.com/tens-of-thousands-meaning/","timestamp":"2024-11-11T10:38:16Z","content_type":"text/html","content_length":"75961","record_id":"<urn:uuid:6f75191c-8929-42ee-b072-12f37ccaf6e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00023.warc.gz"} |
Comments Resolution
Here is where things stand:
*I wanted to get rid of the spam, but it seems there is no workable solution right now for adding registration without a huge hassle. However, I may get in the habit of shutting off comments when I
know I’ll be away from my computer for more than a day or so, so I don’t get inundated.
*In the meantime, to keep the spam from ending up on the site, I’ve instituted a form of comment mediation: the Whitelister (hat tip to Chaz Hill of Dustbury). Basically, your comments will go into
moderation unless I have your email address on a list. I already entered the addresses of the last 20 or so commenters here, which should include most of the regulars, but if you leave a comment and
it doesn’t show, email me (if I don’t catch it right away myself) and I’ll add you to the list of trusted commenters. It stinks, I know, but the goal here is not to become a spam farm.
Anyway, I’m still hoping for the day when most of my blogging time can be spent, you know, blogging.
9 thoughts on “Comments Resolution”
1. With all due respect… this is NOT any kind of solution.
“20 or so” people can comment?
We tend to agree politically – at least until this last election. We certainly do not share favorite teams in professional sports.
I tend to time my comments everywhere. I’m guessing my last comment – probably blocked unintentionally due to techinical difficulties – was to something you wrote that couldn’t accept the fact
that the GOP is right now losing not so much because they are screwing up but because they simply aren’t pulling in the vote.
But since I’m not one of the “20 or so” regulars I’m guessing I just don’t rate anymore. Kind of exclusive, right? Intentional or not – kind of exclusive like the politics you speak of anymore.
Again, with all due respect.
Blogging is about conversation. About those of us with less influence having a voice. And yes, “20 or so” doesn’t quite do it.
Too bad also – you make good points…. for a Mets fan! 🙂
2. Wow. Your spam problem must be unreal. Is it targeted, or just spambots responding to the amount of traffic you generate?
3. It’s bad, as in hundreds of comments at a time bad. What I’d still like to do is close comments on the old entries, since that’s the main source of the problem.
4. I don’t get it. Why would some group (one presumes) spam a blog with comments so heavily? It would seem to be a marginal effort at best. I mean, I assume the point is to get traffic on the sites
they are linking, or sales or something, IE money. I only very rarely click a link in a blog comment section, only if I’m in a serious debate with someone that is linking up references.
5. The reason for spam is that if a spammer puts a link on my site (and 20 other sites) back to his site, it raises his rankings in Google. Whether anyone ever sees the spam comment is irrelevant.
It’s bots vs bots, and we sentient beings are collateral damage.
6. Crank, I am sure I speak for most in that we appreciate the forum you provide and the work that goes into that. I for one am happy to abide by any process that makes your job easier. Thanks for
everything you do.
7. My woodworking club’s website has a similar issue. We actually have one member, who is retired, who has volunteered to spend time each day removing the span from the forums.
Same as Irish, I certainly appreciate all your efforts, and any help you need, just ask.
If Dante were alive today, there would be a ninth circle of hell added for spammers.
8. OK, I’ll bite: what’s your email address?
9. I suggested the Whitelister to Crank because it works extremely well for me, and I started with no email addresses at all: as the regulars showed up in the moderation queue, I copied their
addresses into the Whitelist, and all their comments were subsequently approved immediately – provided, of course, that they made no typos, since the Whitelister isn’t quite smart enough to
notice that you’ve left a letter out of your email address.
On my own site, there are now 100 on the Whitelist, and they account for roughly 95 percent of all the comments I receive. I have had, so far, no instances of a legitimate comment being
inadvertently deleted since its installation.
Spammers, given their penchant for making up different bogus email addresses for each assault, have so far been unable to sneak anything past the Whitelister.
(I used to close comments on old posts after a week; with the current MT spam tools, I can now let them go for a month or longer, though I figure anything older than 60 days is stale.) | {"url":"https://baseballcrank.com/2006/11/26/blog-comments-resolution/","timestamp":"2024-11-06T09:13:17Z","content_type":"text/html","content_length":"53255","record_id":"<urn:uuid:3c4145e1-202e-4e3e-827a-713525dde47d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00171.warc.gz"} |
A basic example of using Tensorflow to regress
In theory of Deep Learning, even a network with single hidden layer could represent any function of mathematics. To verify it, I write a Tensorflow example as below:
import tensorflow as tf
hidden_nodes = 1024
def weight_variable(shape):
"""weight_variable generates a weight variable of a given shape."""
initial = tf.truncated_normal(shape, stddev=1.0, mean=1.0)
return tf.Variable(initial)
def bias_variable(shape):
"""bias_variable generates a bias variable of a given shape."""
initial = tf.constant(0.01, shape=shape)
return tf.Variable(initial)
with tf.device('/cpu:0'):
x = tf.placeholder(tf.float32)
y = tf.placeholder(tf.float32)
a = tf.reshape(tf.tanh(x), [1, -1])
b = tf.reshape(tf.square(x), [1, -1])
basic = tf.concat([a, b], 0)
with tf.name_scope('fc1'):
W_fc1 = weight_variable([hidden_nodes, 2])
b_fc1 = bias_variable([1])
linear_model = tf.nn.relu(tf.matmul(W_fc1, basic) + b_fc1)
# loss
loss = tf.reduce_sum(tf.abs(linear_model - y)) # sum of the squares
# optimizer
optimizer = tf.train.GradientDescentOptimizer(1e-4)
train = optimizer.minimize(loss)
# training data
x_train = range(0, 10)
y_train = range(0, 10)
init = tf.global_variables_initializer()
config = tf.ConfigProto()
config.gpu_options.allow_growth = True
sess = tf.Session(config=config)
sess.run(init) # reset values to wrong
for i in range(3000):
sess.run(train, {x: x_train, y: y_train})
# evaluate training accuracy
curr_basic, curr_w, curr_a, curr_b, curr_loss = sess.run([basic, W_fc1, a, b, loss], {x: x_train, y: y_train})
print("loss: %s" % (curr_loss))
In this code, it was trying to regress to a number from its own sine-value and cosine-value.
At first running, the loss didn’t change at all. After I changed learning rate from 1e-3 to 1e-5, the loss slowly went down as normal. I think this is why someone call Deep Learning a “Black Magic”
in Machine Learning area. | {"url":"https://donghao.org/2017/12/15/a-basic-example-of-using-tensorflow-to-regress/","timestamp":"2024-11-13T13:11:49Z","content_type":"text/html","content_length":"55026","record_id":"<urn:uuid:37ed4457-3071-4c65-b127-4e75270948a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00380.warc.gz"} |
Basic Crocheted Shape Formulas | Abi Crochets | Skillshare
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Basic Crocheted Shape Formulas
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Topics include illustration, design, photography, and more
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Topics include illustration, design, photography, and more
Intro to Basic Crocheted Shape Formulas
Making Shapes Work for YOU
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About This Class
In this course, I will be going over the formulas I use to create spheres, cylinders, and cones. They can be used to make lots of different projects! I have the formulas as well as example patterns
in each video. I hope this will be a guide for you to get your feet wet in the pattern-making world!
Meet Your Teacher
Hello, I'm Abi and I love to crochet! So much in fact that I've been doing it for over 10 years. My specialty in this field is amigurumi figures, so if that kind of thing sounds interesting to you,
please consider joining one of my classes!
See full profile
Level: Intermediate
Hands-on Class Project
For your project, I would like you to make a crocheted project with these three shapes. Preferably using all three shapes, but any shapes you would like to use are open for you to express your
creativity! This is meant to be a learning exercise, so try out a formula or two instead of using only example patterns. Enjoy the process and good luck! Go ahead and share a picture of your project
in the class projects and resources area so we can all admire your creativity!
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1. Intro to Basic Crocheted Shape Formulas: Hi, welcome to my classroom. I'm adding, and I'm gonna be teaching you how to crochet basic shapes. In this course. I'm gonna be going over the basic
formulas for that and don't be scared. I know formulas is kind of a mathematic term, but there's not a lot of math involved and whatever is involved is pretty simple. So don't let that scare you.
It's not that daunting. Once you have these basic shapes down, you're gonna be able to make your own patterns pretty simply because you'll find a lot of these basic shapes can be turned into a lot of
different things. As I said, I'm going to be going over the basic formulas. But of course you can tweak things to how you need them to create the shape that you're specifically looking for for your
specific project. This is a topic that's being requested if Neil a lot. And so I hope that you find what you're looking for in this course and it's helpful for you. I'm gonna be going over three
simple shapes today, spheres, cylinders and cones. And you're gonna find that those shapes can create a lot of different projects for you. If that sounds good, Then come along with me and let's
crochet. 2. Spheres: So we're starting out with the most basic of crochets shapes. This sphere. Pretty much everything that I crochet has some version of a sphere in it. We're talking about circles
or ovals, stuff that can be perfectly round or somewhat smushed. Today, I have three examples that I'm going to show you to go along with the formula so you can kind of understand how the math works
with it. So I have three shapes here, varying circle degrees. You're going to see that this yellow one is not a perfect sphere. It's almost an oval. This is where I started and this is where I ended.
So it's kinda was smushed sphere. The white one is more of a perfect sphere. It's a little bit smushed. Depending on how you stuff, things, things can look more or less like a sphere. This one fits
nicely in the hand. It's pretty circular and this is pretty average in terms of size when you're looking for the gray example is somewhat smaller. I would say this one is probably the most perfect
sphere to me and it really all comes down to The increase, the decrease, and how many rounds in-between there are. I'm going to try to show you a little bit more as I draw it, what I mean by
increasing and decreasing and how that affects your sphere. To help go along with the formulas I'm giving you. We are looking at how to get this shape. And what it comes down to is how quickly you
increase, how long you stay the same for and how quickly you decrease. So the increase and the decrease, we're going to want to stay the same. So whatever you do to get the top of this sphere, you're
going to want to do the opposite for the bottom. In the examples that I gave you, I will show you what I did with these formulas. So you can decide based on the way that mine lock which one you'd
rather do. So again, I'm going to show you my sphere. So here's the gray example. So this was n plus two, all the way around. The White example right here. It's almost a sphere. This was n plus one.
Then my yellow example here was just n. So this is the most squished out of all of them. So you can see the way that they're less spherical or more spherical depending on what you're looking for.
This is n, n plus one and plus two on how spherical they are. So that's how I do spheres, and I hope that it helps you create ceasefires of your own. 3. Cylinders: Next we're moving on to cylinders.
These are probably about as simple as the spheres are, if not more simple, because we're just going to go as far as you want with the sphere. And we're just trying to figure out how big we want them.
A lot of the time when I knew cylinders, I do them. So they have a closing at one end. You can see it better with the yellow example. I don't know what I'm doing trying to show you the smallest one.
As you can see, this side is closed and the site is open because I ended with this sudden I start with this side. You can cast on as many as you want this cylinder to be. And just two single crochets
all the way down and you'll have a cylinder. I tend to do the closed side because with cylinders, people use them a lot of time for bodies are arms. So e.g. I'm going to use a sphere in patterns. And
I do this all the time. Use a cylinder as the body. So you can see that it sits nicely on there and it already has the bottom so you don't need to do a cover for that. And it looks like a head and a
body. Another big major thing that people do with cylinders is used them for arms and legs. And that's another reason why I like to have them closed off. That would be one huge lake, but you can
still do it. The only way really to decide how big you want. This is kind of up to what you think, the pattern that you are trying to make needs. And you'll learn how to do that by trial and error
loss to the time with a lot of the pieces that I do, I tend to do this same beginning number of stitches so that everything has around the same size. This one was also started with six single
crochets, but I wouldn't do it this long if I were doing a leg, I'd probably do about half that. So just to there. So it would look more like a leg and less well, unless you want like a sock monkey
kinda look which you can do. It's really up to you on what you're trying to make this thing look like. You're going to learn that a lot. It's going to be up to you with a lot of this on where you're
trying to get the shape to go. It's like sculpting with yarn. So you're going to learn what works best for you with these formulas. And you can create pretty much anything with these shapes. I'm
gonna go over this a little bit more in drawing format so you can see what I mean. But someone nerves are pretty basic. The understanding of them is pretty basic. Cylinder, so these are pretty
simple. We're basically looking at how long we want them to be. We're increasing rapidly on this. If you're looking to do a closed-off cylinder, but if not, we're doing a chain all the way around.
And just focusing on the length. And length is honestly really up to you. I believe I did not. I did different links for these. Actually, this one is two times the number of rounds that I started
with. Long. This one is just n plus one amount long. And this one is about two times as long as well. The difference that I'm showing you here in terms of these cylinder examples, are how much, how
wide you want your cylinder to be, and how far you go with that. If we're going by length, that goes, it goes this way. I was going by thickness. But for coming by length that goes this way, you can
of course, elongate that make it shorter. It's really up to you on what you're looking for. The biggest factor in how cylinders look is of course, how wide or how thin they are. These are really this
size is pretty good for a really chunky leg. This size is good for like a body. This size is good for a very neutrally spaghetti arm. You can use cylinders for a variety of things. You can even use
this for neck. If you're doing a giraffe or something, It's really up to you on how you want to do those. So now that you understand cylinder is a little bit more, Let's move on to the next shape. 4.
Cones: So the last shape I'm gonna be going over today are cones. And though they do seem simple, they're a little bit more tricky. Then spheres or cylinders. With cones, It's a little bit more
difficult to figure out how wide you want it to end up and how many increases you gotta do to get there. Unless you're looking for a band in your cone, you're going to be doing an even increase all
the way around. If you want a bend, then you're going to be doing it offsets. So what that means is you either increasing or sometimes decreasing in one specific location. Here my examples, I have
12.3, the gray one is bent. And the reason why that is is because when I was crushing this one, I did all the increases on one side. The rest of these the increases were all put on the first part and
then the left and then the middle part. So they're evenly spaced across the entire cone. For the other two, this one, I just increased on one side and you get a bend. That way you can kind of get a
bent tail or a crooked hat. It really is up to you on what you want to use that shape for. I use that a lot for then two tails or horns. I guess you could do that for these cones. You can even bend
it yourself. Because as you can see that one was straight. This one you can still bend. You can bend them pretty much however you want and then use stitches to hold them in place, to hold the bend in
place, or you can just leave them straight. These are both the same amount of rounds, obviously because of the same height. But this one only had one single crochet added to it every round and this
one has two. So it gets wider as double a pace as this one does. These ones are a little bit harder to give you formulas for because you can make all different kinds of shapes with cones. And it
really depends on how wide or how thin you want your code to be. If you want a shape on it, I can give you the formulas for all three of these, and it will give you varying degrees of cones. You can
tweak them. You can do for you, for very big cone. You could do less. You could do odd number, even number, whatever you want. And the cones will always turn out just interesting ways. So I'm going
to show you a little bit more about how cones work in picture form. So the last shape I'm going to show you here are cones. Like I said, while I was showing you the examples. How much you increase,
at what rate you increase is really going to change how your cone looks. You can get a really thick cone with faster increases and a very thin cone with slower increases. This is increased by one,
this has increased by two, this is increased by three. All of the cones that I did as examples, they're all as long as each other. But how much I increased them. Sorry, It goes like this. For the
one-two-three, how much I increase them is really showing you the difference also with to the where you place the increase is important. So if we're doing a top-down view, because I only increased on
one part. So at the very beginning, it curves like this with these two where I increased, Well, I only increased on one part in this one, but because it's so thin, you can't tell. With this one I
increased in two locations that we're equidistant to each other so that it stays nice and flat like that. This one, you can straighten. But you can also bend like that first one because I did it all
on the one side. But it's not it's not as solid as this one. So this one, if I wanted it to stay bent, I'd probably have to sew it like that and reinforce it like that to make sure that it stays like
that. So I hope that helps you a little bit with how you construct your cones. 5. Making Shapes Work for YOU: Of course, the main reason that I'm giving you these formulas is so that you can get
those shapes to work for you and your patterns and projects. This means you're going to need an eye for shapes and how to mesh things together, but don't worry, that will come with time and
experience and playing around with what you know. Don't be discouraged if you have to make a specific piece more than once because it happens. I do that a lot. There are many pieces that I've made
that I had to completely restart because the head wasn't the right size with the body or the arm wasn't the right length. Just it happens. You have to do it by trial and error. Sometimes, the more
you do that, the better you'll be the next time. So let's play with some of the shapes that I gave you. I already did the white sphere on the yellow cylinder. If I look at this, the gray cone, even
though I do like that, It's bent. It's a little big in my opinion. You can have different opinions. I mean, obviously. But I think yeah, I would go with the white tail on that. So we've got a cute
tail. I'm going to use these gray cylinders. If I can get these pins to work as sort of arms and legs here. And it's going to be sitting down. There you got. So there's a very basic shape. You can
make smaller spheres, I think, and make a little ears to go on top. I think that would be q. You can finish it here. Put eyes on it. You could do a little frill down the back and make it look kind of
like a dinosaur. You could also, instead of doing this one, do more of a smushed sphere and have it tilted like that. So it looks like it's looking up and it's got a bigger head. That's another
version. You could do something a little bit more interesting. I don't even know what I'm doing, I'm just playing around, but something kinda fun that doesn't even look feasible. But you could do
something like that. You can just play around with shapes, like a snowman with a hat. Just random, random things. You can make a lot of different things with these shapes. And I hope that this course
has helped you with the basic formulas of these basic shapes. I challenge you to make a project using only spheres, cones, and cylinders. And to post below, because I love seeing your guises work.
Anyway, I want to thank you for watching this course and for following me. I know it's been a long time since I've posted anything and I do appreciate your guys support with everything that I do. I
hope you had fun in this course and that you will continue to have fun crocheting. It is a journey and I'm glad to have been part of yours. I'll see you next class. Bye. | {"url":"https://www.skillshare.com/en/classes/basic-crocheted-shape-formulas/1913278068?via=similar-classes","timestamp":"2024-11-02T21:13:48Z","content_type":"application/xhtml+xml","content_length":"284540","record_id":"<urn:uuid:98954ef6-b353-4356-86d7-7427bc67131e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00472.warc.gz"} |
Sigh....No, tax cuts won't boost growth
I guess we need to go over this again. No, cuts to corporate tax rates and/or the reforms/cuts proposed on individual income taxes will not boost economic growth in any appreciable way. Not to 4%,
nor 4.5%, nor 5%. There is no evidence supporting this claim. And even theoretically, there is no reason to believe that these tax reforms/cuts can boost growth to those rates.
Prima Facie Evidence
Start with a simple smell test. The figure below plots two series: the backward-looking 5-year average growth rate of real GDP and the forward-looking 5-year growth rate of real GDP. For example, in
1981, the backward looking series is the average growth rate of real GDP from 1976-1981, and the forward looking series is the average growth rate of real GDP from 1981-1986. For any given year on
the graph, we can compare the growth rate people had experienced in the recent past (backward looking) to what they would experience in the near future (forward looking).
Also on the figure are red lines indicating when significant tax legislation was passed, and whether it was considered a tax cut, or a tax increase. 1981 and 1986 are the “Reagan tax cuts”, 1990 is
the Bush tax increase, 1993 is the Clinton budget agreement that raised top tax rates, 2001 and 2003 are the younger Bush tax cuts, and included the 2012 “fiscal cliff” agreement under Obama that
raised taxes on those earning over 400,000 per year.
What you can look at is whether the forward-looking growth rate differed appreciably from the backward-looking one. If the argument about tax cuts stimulating growth is correct, then you’d expect to
see that the forward-looking rate exceeds the backward-looking rate. In 1981, this looks true. Growth from 1981-1986 was about 3.5% per year, versus 2.8\% per year in 1976-1981. But examining the
other tax cuts, you’ll see the opposite is true. In 1986, growth was slower after the tax cut than before. In 2001 and 2003, growth is lower after the tax cut than before. For the 2003 cut, the
difference is very striking because of the onset of the financial crisis.
In contrast, look at the three tax increases. For Bush (the elder) in 1990, there was essentially no difference in the forward and backward growth rate. For the 1993 increase, growth was much higher
after the tax increase than before, and for the 2012 increase we’d need to get data through the end of 2017 to make the same comparison. But for what it is worth, growth in the four years from
2012-2016 has been a shade over 2%, meaning it is much higher than the five years prior to the tax increase.
You cannot take this as any kind of definitive proof of anything, of course. Tax cuts may boost growth, but each time we passed a tax cut it just so happens that some other big negative shock hit the
economy at the same time, masking those positive effects. That’s … possible. The more likely answer is just that substantial tax cuts have small effects on economic growth. On the other side, the
last time we averaged 4% growth over a 5-year period was the late 1990’s, right after the tax increase of 1993.
Aside from the smell test, this isn’t a new question, and people have picked around at the relationship of tax rates and growth rates before. I reviewed a few studies in an early post here on the
blog. Short answer, there is no relationship of the tax rate to the growth.
Taking this same question on from a narrower pespective, Menzie Chinn has a recurring series of posts tracking Kansas’ economic growth and employment since Brownback became governor and signed off on
a massive tax cut. The short version of these posts is that employment and growth in Kansas has stagnated and/or outright fell since the cut. There are several similarities between the Kansas cut and
the one proposed by the current administration. Most notably, lowering the tax rate on “pass-through” income from partnerships and proprietorships.
The lower tax rate on pass-through income was supposed to generate a burst of growth in Kansas. The TL;DR version of Menzie’s posts is that it hasn’t happened, which you can probably tell from the
post titles.
Why don’t tax cuts boost growth by much?
Ok, the evidence doesn’t support the idea that tax cuts boost growth, but at least theoretically it’s true, right? Kinda sorta. Smarter people than I have tried to estimate how sensitive people’s
work efforts are to income tax rates, and they find very, very small effects. The main effect of changes in tax rates and/or reforms is that people with the means to do so shift the legal definition
of their income (e.g. from personal income to profits at a proprietorship). The effect on hours worked is very small, and that is the margin we’d care about if we want to study economic growth. From
the corporate side, there isn’t any evidence that the 2003 dividend tax cut had any effect on corporate investment, or on labor earnings. Rather, the dividend tax cut generated a substantial amount
of payouts to existing shareholders.
But let’s say that these guys are all wrong, and this tax reform/cut proposed by the current administration is * different. For realsies, it is going to incent lots of firms to invest more, employ
more people, and stimulate the economy. *That is going to boost growth to 4%, or maybe more, right?
Wrong. The issue is that the effects of any tax cut are going to take time to manifest themselves. In particular, corporate tax cuts that have the stated purpose of raising investment will need years
or decades to manifest themselves in higher GDP, assuming that the savings are actually used for investment spending.
I’ve been over this before on this blog, and now that I have my handy Topics pages, you can go see all the gory details here. The short story is that tax cuts/reforms change potential GDP (possibly),
and it takes a long time for us to converge from today’s GDP to that potential GDP. The analogy is to think of GDP like your weight. If you cut “taxes” on your weight by exercising less, then you
will gain weight. But not all at once. It will take months or years to reach your new potential, heavier, weight. The same thing happens with GDP.
Even if you think the effects of these tax cuts on potential GDP are massive, which is unlikely, you can’t generate significant changes in growth rates. There are several posts on the Topics page
that do this, but let me repeat the basic calculation. Real GDP in the US is just about 17 trillion (2009 chained) dollars as of Q1 2017. A generous estimate of the current growth rate of real GDP
would be about 2.5% per year, without the tax cuts. Now, let’s say the proposed corporate and individual tax cuts/reforms raised potential GDP to something like 18 trillion dollars. In other words,
what if there was 1 trillion in extra economic activity that was not taking place because a bunch of people have John Galt-ed themselves out of investing or working due to the current tax system?
How much faster would growth in real GDP be after the proposed tax cuts/reforms? To figure this out we need one extra piece of information, the convergence rate. This rate is the percent of the gap
between current GDP and potential GDP that closes every year, and it is commonly estimated to be about 2%. In the weight example, if your current weight was 150 pounds, and your potential weight is
300 pounds thanks to your cut in exercise, then this 2% convergence rate says that in the first year, you will gain .02(300-150) = 3 pounds. And next year, you’ll gain .02(300-153) = 2.94 pounds. And
so on.
We can do the same calculation for GDP, and figure out the growth rate of GDP implied by this convergence. Current GDP is 17 trillion, and potential GDP is 18 trillion, so the economy should add an
extra .02(18-17) = .02 trillion, or about 20 billion in economic activity. Now, remember, GDP was going to grow at 2.5% anyway without the tax cut. So GDP next year was going to be 1.02517 = 17.425
trillion. Add the extra 20 billion on top of that, to get 17.445 trillion in economic activity next year. So overall growth in GDP would be 17.445/17 - 1 = .0262, or 2.62%.
That’s …. not 4%. Or really anywhere close. It is hard to move the growth rate of GDP, in the same way that it is hard to change the heading of an oil tanker. But maybe I’m just being pessimistic. So
let’s play around with some numbers. What if the convergence rate is 4% per year? Then growth would be 2.73% next year. What if the convergence rate is 10%? 3.09% growth. Which is still not 4% per
year, and there is no evidence that the convergence rate is 10%.
Perhaps you’re worried I’m underselling the effects of the tax cut itself. So let’s say that the proposed reforms raise the potential GDP of the US to 19 trillion - which is ridiculous by the way -
but whatever. With a 2% convergence rate, that would lead to a growth rate of … 2.73%. How about potential GDP of 20 trillion, meaning that tax rates alone are keeping GDP three trillion under where
it could. Then growth might hit … 2.85%.
The fundamental issue here is that it takes time for all that pent-up investment (if it exists) to manifest itself. Firms cannot expand production overnight. It takes time to generate the plans for a
new location, or plant, or product. And this all assumes, contrary to the available evidence, that there are substantial effects of tax cuts or reforms on the willingness of firms to invest and/or
people to work.
There is no magic elixir of tax rules that will generate 4% growth. You don’t need to see anything more than the one-page outline put out by the current administration to get to this conclusion.
Filling in the details will not change the fact that their plan, if passed, will have no appreciable effect on economic growth in the next few years. | {"url":"https://growthecon.com/feed/2017/05/02/Taxes-Growth.html","timestamp":"2024-11-02T07:31:49Z","content_type":"text/html","content_length":"20230","record_id":"<urn:uuid:8e5072e2-6a94-43c2-ac2f-514189ac22b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00141.warc.gz"} |
nForum - Discussion Feed (Well-orders need not be linear.)TobyBartels comments on "Well-orders need not be linear." (25922)Mike Shulman comments on "Well-orders need not be linear." (25912)Zhen Lin comments on "Well-orders need not be linear." (25911)Todd_Trimble comments on "Well-orders need not be linear." (25910)Mike Shulman comments on "Well-orders need not be linear." (25908)Todd_Trimble comments on "Well-orders need not be linear." (25907)Zhen Lin comments on "Well-orders need not be linear." (25903)Todd_Trimble comments on "Well-orders need not be linear." (25902)TobyBartels comments on "Well-orders need not be linear." (25885)Mike Shulman comments on "Well-orders need not be linear." (25881)TobyBartels comments on "Well-orders need not be linear." (25869)
On well-order and elsewhere, I’ve implied that a well-order (a well-founded, extensional, transitive relation) must be connected (and thus a linear order). But this is not correct; or at least I
can’t prove it, and I’ve read a few places claiming that well-orders need not be linear. So I fixed well-order, although the claim may still be on the Lab somewhere else.
Of course, all of this is in the context of constructive mathematics; with excluded middle, the claim is actually true. I also rewrote the discussion of classical alternatives at well-order to show
more popular equivalents. | {"url":"https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=ATOM&DiscussionID=3123&FeedTitle=Discussion+Feed+%28Well-orders+need+not+be+linear.%29","timestamp":"2024-11-04T11:28:21Z","content_type":"application/atom+xml","content_length":"15978","record_id":"<urn:uuid:eefcc06e-cb5a-4180-a28e-ae5dc01cc682>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00537.warc.gz"} |
INEQUAL'ITY, n. [L. inoequalitas; in and oequalis, equal.]
1. Difference or want of equality in degree, quantity, length, or quality of any kind; the state of not having equal measure, degree, dimensions or amount; as an inequality in size or stature; an
inequality of number or of power; inequality of distances or of motions.
2. Unevenness; want of levelness; the alternate rising and falling of a surface; as the inequalities of the surface of the earth, or of a marble slab.
3. Disproportion to any office or purpose inadequacy; incompetency; as the inequality of terrestrial things to the wants of a rational soul.
4. Diversity; want of uniformity in different times or places, as the inequality of air or temperature.
5. Difference of rank, station or condition; as the inequalities of men in society; inequalities of rank or property. | {"url":"https://1828.mshaffer.com/d/word/inequality/simple/","timestamp":"2024-11-06T21:22:53Z","content_type":"text/html","content_length":"2237","record_id":"<urn:uuid:7acb050c-1510-438e-8aa8-abdbe3118583>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00340.warc.gz"} |
Mathematical Problems Solving
Here at MATH.INTEMODINO.com you could find a great collection of math problems and their solutions.
Math examples, questions and answers, and problem solutions all in one place online at MATH.INTEMODINO.com.
Just select the mathematical category you are interested in. You can find a really interesting examples with detailed, step by step problem solving. Problem solving with steps and detailed
instructions are providing for all examples listed in our math directory.
New examples, expressions, problem solving are added to the math catalogue on a weekly basis. | {"url":"https://math.intemodino.com/","timestamp":"2024-11-07T16:48:39Z","content_type":"text/html","content_length":"6192","record_id":"<urn:uuid:5a0d3249-ec88-472a-85e9-a253c33e5d2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00384.warc.gz"} |
Mathematician Job Description, Career as a Mathematician, Salary, Employment - Definition and Nature of the Work, Education and Training Requirements, Getting the Job
Mathematician Job Description, Career as a Mathematician, Salary, Employment
Definition and Nature of the Work, Education and Training Requirements, Getting the Job
Education and Training: Master's or doctoral degree
Salary: Median—$81,240 per year
Employment Outlook: Poor
Definition and Nature of the Work
Mathematics is the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. Mathematicians use tools such as mathematical theory, well-defined
procedures called algorithms, and the latest computer technology to solve economic, scientific, engineering, physics, and business problems. Mathematics is divided into two areas: theoretical, or
pure, mathematics, and applied mathematics. However, these two areas often overlap.
Mathematicians working in theoretical mathematics are concerned with expanding and clarifying mathematical theories and laws. They seek to increase basic knowledge about mathematical relationships
and formulate new laws of mathematics. Although the few mathematicians in theoretical research do not consider the practical uses of their findings, their work has been essential in many areas of
science and engineering. For example, a new kind of geometry developed in the 1850s formed part of the basis for the theory of relativity, which in turn made the development of nuclear energy
Mathematicians doing applied work use the theories and laws developed by theoretical mathematicians. Applied mathematicians solve specific problems in such fields as physical science, social science,
business, computer science, government, biology, and engineering. They may work in the electronics industry developing new kinds of computers and software. Applied mathematicians sometimes study
numerical information about medical problems, such as the effect of a new drug on a disease. Mathematicians working in the aerospace field may provide calculations that help determine whether the
outside surfaces of a spaceship are properly designed to keep it on course.
Although mathematicians work in many different fields and apply their work in a variety of ways, they all use numbers. Mathematicians take abstract ideas or specific problems and put them into
numerical form. They use computers regularly, as well as more traditional computational devices such as slide rules and calculators.
About three-quarters of all mathematicians are employed by colleges and universities. Most of these mathematicians teach, but some also do research. Other mathematicians work for private companies in
industries such as aerospace, communications, and electrical equipment manufacturing. Most mathematicians who work for the federal government are involved in space or defense-related projects.
Many workers who are not considered mathematicians use mathematical techniques extensively. Statisticians, actuaries, systems analysts, computer programmers, and mathematics teachers all use
Education and Training Requirements
You generally need a doctoral degree to become a mathematician. As an undergraduate, you can major in mathematics and include courses in related areas, such as statistics and computer science. If you
have a bachelor's degree, you may be able to find a job as an assistant to a mathematician. You may also be able to obtain a position in an area related to mathematics in government or private
industry. However, your opportunities for advancement as a mathematician will be limited. Many jobs in teaching and applied research are open to those who have a master's degree in mathematics. A job
as a theoretical mathematician or a teaching and research position in a university requires a doctoral degree in mathematics. It usually takes four years to earn a bachelor's degree, another one or
two years to receive a master's degree, and an additional two or three years for a doctoral degree. A career in mathematics often requires you to continue reading and studying in order to keep up
with new developments in the field.
Getting the Job
Your professors and college placement office may be good sources of information about getting a job in mathematics. You can also apply directly to colleges and universities, private companies, and
government agencies that hire mathematicians. You sometimes need to pass a civil service examination to get a job with the government. Professional and trade journals, newspaper classifieds, and
Internet job banks may also list openings for mathematicians.
Advancement Possibilities and Employment Outlook
Advancement opportunities are good for mathematicians who have an advanced degree. They can become supervisors, managers, or directors of research. Mathematicians who have a doctoral degree can
become full professors at colleges and universities. Many theoretical and applied mathematicians advance by becoming experts in a special area, such as algebra, geometry, or computing. They may gain
the recognition of other mathematicians by publishing their findings in professional journals. Mathematicians who become experts are also often rewarded by higher salaries, especially in private
Employment of mathematicians is expected to decrease through the year 2014. Those holding bachelor's degrees are usually not qualified to be mathematicians. Those with master's degrees will likely
face keen competition for jobs in theoretical research. Those with master's and doctoral degrees who have strong backgrounds in mathematics and related disciplines, such as engineering or computer
science, should have the best job opportunities.
Working Conditions
Mathematicians who are employed by private companies or government agencies often work a standard thirty-five to forty-hour week in well-lighted, comfortable offices. At times they may have to work
overtime to complete special projects. Although mathematicians working at colleges and universities usually have flexible schedules, they often put in long hours.
Although their work is not strenuous, mathematicians must have the patience to spend long periods of time concentrating on complex problems. They must be able to work independently. They should have
good reasoning ability and enjoy working with abstract ideas and solving problems. At times, mathematicians must work with others. They need to be able to listen carefully to specific problems that
need to be solved in applied mathematics. They must also be able to present their own ideas clearly.
Earnings and Benefits
Salaries for mathematicians vary according to the location, kind of job, and education and experience of the individual. Median annual earnings of mathematicians were $81,240 in 2004. In 2005 average
annual salaries for mathematicians employed by the federal government were slightly higher, at $88,194. Benefits usually include paid holidays and vacations, health insurance, and retirement plans.
Additional topics | {"url":"https://careers.stateuniversity.com/pages/406/Mathematician.html","timestamp":"2024-11-11T19:51:59Z","content_type":"text/html","content_length":"16286","record_id":"<urn:uuid:a832e4e6-b69b-456f-aadc-ac7b2fa2f143>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00270.warc.gz"} |
Messanae Universitas Studiorum: No conditions. Results ordered -Date Deposited. Density-functional perturbation theory goes time-dependent
The scope of time-dependent density-functional theory (TDDFT) is limited to the lowest portion of the spectrum of rather small systems (a few tens of atoms at most). In the static regime,
density-functional perturbation theory (DFPT) allows one to calculate response functions of systems as large as currently dealt with in ground-state simulations. In this paper we present an effective
way of combining DFPT with TDDFT. The dynamical polarizability is first expressed as an off-diagonal matrix element of the resolvent of the Kohn-Sham Liouvillian super-operator. A DFPT representation
of response functions allows one to avoid the calculation of unoccupied Kohn-Sham orbitals. The resolvent of the Liouvillian is finally conveniently evaluated using a newly developed non-symmetric
Lanczos technique, which allows for the calculation of the entire spectrum with a single Lanczos recursion chain. Each step of the chain essentially requires twice as many operations as a single step
of the iterative diagonalization of the unperturbed Kohn-Sham Hamiltonian or, for that matter, as a single time step of a Car-Parrinello molecular dynamics run. The method will be illustrated with a
few case molecular applications. | {"url":"https://cab.unime.it/mus/cgi/exportview/creators/Gebauer=3ARalph=3A=3A/Atom/Gebauer=3ARalph=3A=3A.xml","timestamp":"2024-11-12T05:35:10Z","content_type":"application/atom+xml","content_length":"3791","record_id":"<urn:uuid:950a3696-0cc5-4124-857d-09a6b474608c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00354.warc.gz"} |
Identifiers for Hector forcing component parameters — DELTA_CO2
Identifiers for Hector forcing component parameters
These identifiers specify the tropospheric adjustments for the stratospheric-temperature adjusted radiative forcings. These values must be a number between -1 and 1, and can be read and/or set by
hectors forcing and halocarbon components.
• DELTA_CO2(): the foring tropospheric adjustments for N2O
• RF_N2O(): the foring tropospheric adjustments for N2O
• DELTA_N2O(): Radiative forcing due to N2O
• DELTA_CF4(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_C2F6(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HFC23(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HFC32(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing ' @export
• DELTA_HFC4310(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HFC125(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HFC134A(): the tropospheric adjustments used to convert from tratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HFC143A(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HFC227EA(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HFC245FA(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_SF6(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CFC11(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CFC12(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CFC113(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CFC114(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CFC115(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CCL4(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CH3CCL3(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HCFC22(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HCFC141B(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HCFC142B(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HALON1211(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HALON1301(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_HALON2402(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CH3CL(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CH3BR(): the tropospheric adjustments used to convert from stratospheric-temperature adjusted radiative forcing to effective radiative forcing
• DELTA_CH4(): Radiative forcing tropospheric adjustment for CH4
Because these identifiers are provided as #define macros in the hector code, these identifiers are provided in the R interface as functions. Therefore, these objects must be called to use them; e.g.,
GETDATA() instead of the more natural looking GETDATA.
See also
haloforcings for forcings from halocarbons and forcings forcing values provided from the hector forcing component.
Other capability identifiers: CF4_CONSTRAIN(), CONCENTRATIONS_CH4(), CONCENTRATIONS_N2O(), EMISSIONS_BC(), EMISSIONS_CF4(), EMISSIONS_SO2(), FTOT_CONSTRAIN(), GLOBAL_TAS(), NBP(), OCEAN_UPTAKE(),
RF_CF4(), RF_TOTAL(), RHO_BC(), TRACKING_DATE() | {"url":"http://jgcri.github.io/hector/reference/delta.html","timestamp":"2024-11-08T04:46:17Z","content_type":"text/html","content_length":"19558","record_id":"<urn:uuid:1924c0c1-c27b-4d56-a9b2-546d2eecd981>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00475.warc.gz"} |
We study motivic zeta functions for $\mathbb{Q}$-divisors in a $\mathbb{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two
normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a
first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity
under the action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a $\mathbb{Q}$-resolution.
Fundamental groups of real arrangements and torsion in the lower central series quotients (with E. Artal & B. Guerville-Ballé).
Experimental Mathematics 29 (2020), no. 1, 28–35, journal . arXiv:1704.04152 .
We prove that the fundamental group of the complement of a real complexified line arrangement is not determined by its intersection lattice, providing a counter-example for a problem of Falk and
Randell. We also deduce that the torsion of the lower central series quotients is not combinatorially determined, which gives a negative answer to a question of Suciu.
Configurations of points and topology of real line arrangements (with B. Guerville-Ballé).
Mathematische Annalen 374 (2019), no. 1-2, 1–35, journal . arXiv:1702.00922 , including two appendices with detailled pictures of Zariski pairs.
A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In
the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This invariant is based on the weight counting over the points of the arrangement
dual configuration, located in particular chambers of the real projective plane $\mathbb{RP}^2$, dealing only with geometrical properties.
Using this dual point of view, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in $\mathbb{CP}^2$ (i.e. Zariski pairs),
which are distinguished by this invariant. In particular, we obtain new Zariski pairs of 13, 15 and 17 lines defined over $\mathbb{Q}$ and containing only double and triple points. For each one of
them, we can derive degenerations, containing points of multiplicity 2, 3 and 5, which are also Zariski pairs.
We explicitly compute the moduli space of the combinatorics of one of these examples, and prove that it has exactly two connected components. We also obtain three geometric characterizations of these
components: the existence of two smooth conics, one tangent to six lines and the other containing six triple points, as well as the collinearity of three specific triple points.
A semi-canonical reduction for periods of Kontsevich-Zagier.
International Journal of Number Theory 17 (2021), no. 01, 147-174 , journal . arXiv:1509.01097 , including an appendix with pseudo-codes of the main procedures.
The ${\overline{\mathbb Q}}$-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of ${\mathbb Q}
$-rational functions over ${\mathbb Q}$-semi-algebraic domains in ${\mathbb R}^d$. The Kontsevich-Zagier period conjecture affirms that any two different integral expressions of a given period are
related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables
and Stoke's formula.
In this paper, we prove that every non-zero real period can be represented as the volume of a compact ${\overline{\mathbb Q}}\cap{\mathbb R}$-semi-algebraic set, obtained from any integral
representation by an effective algorithm satisfying the rules allowed by the Kontsevich-Zagier period conjecture.
Combinatorics of line arrangements and dynamics of polynomial vector fields (with B. Guerville-Ballé).
(Submitted), arXiv:1412.0137 .
Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$–derivations. First, we give a dynamical interpretation of $\mathcal{D}(\mathcal{A})$ as the set
of polynomial vector fields which posses $\mathcal{A}$ as invariant set. We characterize polynomial vector fields having an infinite number of invariant lines. Then we prove that the minimal degree
of polynomial vector fields fixing only a finite set of lines in $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.
On the minimal degree of logarithmic vector fields of line arrangements (with B. Guerville-Ballé).
Proc. XIII Intern. Conf. on Maths. and its Appl., (40), 61-66 (2016), journal . | {"url":"https://jviusos.github.io/research.html","timestamp":"2024-11-02T03:09:54Z","content_type":"text/html","content_length":"20946","record_id":"<urn:uuid:651524d3-f5c9-45b3-8a73-1811c86a4d58>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00225.warc.gz"} |
Practical Bin Packing
This was motivated by a desire to buy just enough materials to get the job done. In this case the job was a chicken coop I was building. I can buy lumber in standard lengths of 12, 10, 8 or 6 feet at
my local building supply store. So what is the lowest cost combination of stock boards that fills the need?
In my research I found lots of examples of bin packing with a single size of bin but nothing that fit my situation and limited appetite for in depth study. This code uses a brute force approach to
the problem. It enumerates all permutations, discards any that don't meet the bare minimum length then checks each remaining permutation for feasilbility. The feasible options are sorted to find the
minmum cost option.
In the example below, I first define the stock lengths and their rates. Then I list the parts needed for the project. The part lengths are listed as integers but could just as well have been floats. | {"url":"https://notebook.community/AldoMaine/Solutions/notebooks/Bin_Packing_Multiple_Bin_Sizes","timestamp":"2024-11-10T22:36:25Z","content_type":"text/html","content_length":"51754","record_id":"<urn:uuid:20bda3de-4535-4f79-9e3d-cac4a68eaf5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00479.warc.gz"} |
The Great Astrophysicist Fred Hoyle (1915–2001)
Any thoughts? Came across this blog article:
Hoyle's mindset is most evident in his views on biology. Since the early 1970's he has argued that the universe is pervaded by viruses, bacteria and other organisms. (Hoyle first broached this
possibility in 1957 in The Black Cloud, which remains the best known of his many science fiction novels.) These space-faring microbes supposedly provided the seeds for life on earth and spurred
evolution thereafter; natural selection played little or no role in creating the diversity of life. Hoyle has also asserted that epidemics of influenza, whooping cough and other diseases are
triggered when the earth passes through clouds of pathogens.
Discussing the biomedical establishment's continued belief in the more conventional, person-to-person mode of disease transmission, Hoyle glowered. "They don't look at those data and say, 'Well, it's
wrong,' and stop teaching it. They just go on doping out the same rubbish. And that's why if you go to the hospital and there's something wrong with you, you'll be lucky if they cure it."
But if space is swarming with organisms, I asked, why haven't they been detected? Oh, but they probably were, Hoyle assured me. He suspected that U.S. experiments on high-altitude balloons and other
platforms turned up evidence of life in space in the 1960's, but officials hushed it up. Why? Perhaps for reasons related to national security, Hoyle suggested, or because the results contradicted
received wisdom. "Science today is locked into paradigms," he intoned solemnly. "Every avenue is blocked by beliefs that are wrong, and if you try to get anything published by a journal today, you
will run against a paradigm and the editors will turn it down."
Hoyle emphasized that, contrary to certain reports, he did not believe the AIDS virus came from outer space. It "is such a strange virus I have to believe it's a laboratory product," he said. Was
Hoyle implying that the pathogen might have been produced by a biological-warfare program that went awry? "Yes, that's my feeling," he replied.
Hoyle also suspected that life and indeed the entire universe must be unfolding according to some cosmic plan. The universe is an "obvious fix," Hoyle said. "There are too many things that look
accidental which are not." When I asked if Hoyle thought some supernatural intelligence is guiding things, he nodded gravely. "That's the way I look on God. It is a fix, but how it's being fixed I
don't know."
Many of Hoyle's colleagues--and a majority of humanity--share his view that the universe is, must be, a divine conspiracy. Perhaps it is. Who knows? But his assertion that scientists would
deliberately suppress evidence of microbes in outer space or of genuine flaws in the expanding universe model reveals a fundamental misunderstanding of his colleagues. Most scientists yearn for such
revolutionary discoveries.
Will Hoyle’s skepticism toward the big bang ever be vindicated? Will cosmology undergo a paradigm shift that leaves the big bang behind? Probably not. The theory rests on three solid pillars of
evidence: the red shift of galaxies, the microwave background and the abundance of light elements, which were supposedly synthesized during our universe’s fiery birth. The big bang also does for
cosmology what evolution does for biology: it provides cohesion, meaning, a unifying narrative. That is not to say that the big bang can explain everything, any more than evolutionary theory can. The
origin of life remains profoundly mysterious, and so does the origin of the universe. Nor can physics tell us why our universe takes its specific form, which allowed for our existence.
But if space is swarming with organisms, I asked, why haven't they been detected? Oh, but they probably were, Hoyle assured me. He suspected that U.S. experiments on high-altitude balloons and
other platforms turned up evidence of life in space in the 1960's, but officials hushed it up. Why? Perhaps for reasons related to national security, Hoyle suggested, or because the results
contradicted received wisdom.
It has been shown that life exists on earth and within our atmosphere where high-altitude balloons go, there is no conspiracy there and nothing is being hidden.
However there is NO evidence at all that anything at all that we could consider life to exist anywhere else in the universe, we have detected amino acids and such that are 'building blocks' of life,
but that is not the same thing as life.
As far as we know life started on earth and exists only on earth. (there is a statistical possibility that it started and exists elsewhere in the universe, but we have no evidence for that).
Hoyle also suspected that life and indeed the entire universe must be unfolding according to some cosmic plan. The universe is an "obvious fix," Hoyle said. "There are too many things that look
accidental which are not." When I asked if Hoyle thought some supernatural intelligence is guiding things, he nodded gravely. "That's the way I look on God. It is a fix, but how it's being fixed
I don't know."
Indeed that is speculation, it however does not give credence or authority to that speculation, and there is no evidence that this speculation is correct, so it is just that, speculation with no
supporting evidence.
Will Hoyle's skepticism toward the big bang ever be vindicated? Will cosmology undergo a paradigm shift that leaves the big bang behind? Probably not. The theory rests on three solid pillars of
evidence: the red shift of galaxies, the microwave background and the abundance of light elements, which were supposedly synthesized during our universe’s fiery birth. The big bang also does
for cosmology what evolution does for biology: it provides cohesion, meaning, a unifying narrative.
Red shift, the CMBR and abundance of light elements, are observations, they are not evidence. You can treat them as evidence and consider that they are evidence of the big bang, they are solid
pillars of evidence, but they are not necessarily evidence of the big bang.
This is an important point, the scientific method is not used to prove facts, it is about evaluating evidence, if you claim that as evidence of the big bang you need to show that the effects you see
(red shift or whatnot) CAN BE FROM NO OTHER POSSIBLE MECHANISM.
That is, you theory has to be able to be falsified, not that it IS falsified, but there needs to be that possibility. So you can only 'prove' the big bang from red shift et al. IF you can show that
there is no other possible mechanism for that observed evidence.
Turns out that there are many other possible mechanisms for a redshift with distance relationship (Gravitational redshift is the best candidate), that means that Redshift alone is not evidence for
the big bang. Same apples with the CMBR and with nucleogenesis, or light element abundance.
CMBR is microwave background because matter radiates energy and the universe is full of background matter, Red shift is the direct result of gravitational shift that is shown (by observation) to have
a redshift/distance relations, and we can in a lab turn heavy elements into light elements, and light elements into heavy elements, so there is no reason at all to think that this could not be a
natural process occurring in the universe right now.
So not only does the big bang have a falsifying argument, it turn out that the argument for that falsification has a solid grounding in well known, tested and established physics.
So the BB can be falsified, and there are very strong arguments that it is and has been falsified.
So I think the universe might of started 13.8 billion years ago, but so far I do not see any solid evidence to indicate that this is the case.
It has been shown that life exists on earth and within our atmosphere where high-altitude balloons go, there is no conspiracy there and nothing is being hidden.
However there is NO evidence at all that anything at all that we could consider life to exist anywhere else in the universe, we have detected amino acids and such that are 'building blocks' of
life, but that is not the same thing as life.
As far as we know life started on earth and exists only on earth. (there is a statistical possibility that it started and exists elsewhere in the universe, but we have no evidence for that).
Indeed that is speculation, it however does not give credence or authority to that speculation, and there is no evidence that this speculation is correct, so it is just that, speculation with no
supporting evidence.
Red shift, the CMBR and abundance of light elements, are observations, they are not evidence. You can treat them as evidence and consider that they are evidence of the big bang, they are solid
pillars of evidence, but they are not necessarily evidence of the big bang.
This is an important point, the scientific method is not used to prove facts, it is about evaluating evidence, if you claim that as evidence of the big bang you need to show that the effects you
see (red shift or whatnot) CAN BE FROM NO OTHER POSSIBLE MECHANISM.
That is, you theory has to be able to be falsified, not that it IS falsified, but there needs to be that possibility. So you can only 'prove' the big bang from red shift et al. IF you can show
that there is no other possible mechanism for that observed evidence.
Turns out that there are many other possible mechanisms for a redshift with distance relationship (Gravitational redshift is the best candidate), that means that Redshift alone is not evidence
for the big bang. Same apples with the CMBR and with nucleogenesis, or light element abundance.
CMBR is microwave background because matter radiates energy and the universe is full of background matter, Red shift is the direct result of gravitational shift that is shown (by observation) to
have a redshift/distance relations, and we can in a lab turn heavy elements into light elements, and light elements into heavy elements, so there is no reason at all to think that this could not
be a natural process occurring in the universe right now.
So not only does the big bang have a falsifying argument, it turn out that the argument for that falsification has a solid grounding in well known, tested and established physics.
So the BB can be falsified, and there are very strong arguments that it is and has been falsified.
So I think the universe might of started 13.8 billion years ago, but so far I do not see any solid evidence to indicate that this is the case.
Thanks for your interesting input. :smile:
Yes and I think that is called Panspermia. I think he meant back in the 60's and a little later there was a bit of or a cover up and or they didn't exactly have the proof as they do now.
Well I am not sure how you take into account remote viewing. But through remote viewing there have been found to be other life in the universe and even on mars about a million years ago.
Though yes as you say earth could be the only planet with life. But I think the chances of no other life being in this vast universe are infinitely small. It is practically impossible for their not
to be any other life in this whole entire universe. For the statistics there is the Drake equation the Fermi Paradox and the Zoo Hypothesis and also the Great Filter . We are just one speck of dust
compared to the whole universe. Not to mention their could me multiple universes.
What do you think of string theory? | {"url":"https://www.scienceforums.com/topic/36730-the-great-astrophysicist-fred-hoyle-1915%E2%80%932001/","timestamp":"2024-11-09T01:10:57Z","content_type":"text/html","content_length":"121387","record_id":"<urn:uuid:757da139-5787-4d2d-8d7c-992615e8c3f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00587.warc.gz"} |
Two circles have the following equations (x +5 )^2+(y -2 )^2= 36 and (x +2 )^2+(y -1 )^2= 81 . Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other? | HIX Tutor
Two circles have the following equations #(x +5 )^2+(y -2 )^2= 36 # and #(x +2 )^2+(y -1 )^2= 81 #. Does one circle contain the other? If not, what is the greatest possible distance between a point
on one circle and another point on the other?
Answer 1
The circles overlap
The greatest possible distance is $= 18.16$
We need the equation of a circle, center #(a,b)# and radius #r# is
The distance between 2 points, #(x_1,y_1)# and #(x_2,y_2)# is
We need to find the distance between the centres of the circles and compare this to the sum of the radii.
The centers are #C_A=(-5,2)# and #C_B=(-2,1)#
The distance between the centers is
The sum of the radii is #=6+9=15#
#d<#sum of radii
The greatest possible distance is #=9+6+3.16=18.16#
graph{((x+5)^2+(y-2)^2-36)((x+2)^2+(y-1)^2-81)(y-2+1/3(x+5)) = 0 [-19.22, 9.27, -5.39, 8.86]}
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Answer 2
No, one circle does not contain the other. The greatest possible distance between a point on one circle and another point on the other can be found by calculating the sum of their radii and then
subtracting the distance between their centers. In this case, the radii of the circles are 6 and 9 respectively, and the distance between their centers is the square root of ((2-(-5))^2 + (1-2)^2 = \
sqrt{49+1} = \sqrt{50}). Therefore, the greatest possible distance between a point on one circle and another point on the other is (6 + 9 - \sqrt{50}).
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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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No 4 (2016)
Numerical analysis of dynamic strength of composite cylindrical shells under multiple-pulse exposures
Abrosimov N.A., Elesin A.V.
The target of this research is studying the fiberglass cylindrical shellswith exposed ends resulted from the cross-winding of tapes made of an unidirectional composite material. The aim of the study
was to develop a numerical technique to model a progressive fracture of laminated composite cylindrical shells under multiple-pulse loading with an internal pressure of a various intensity. Kinematic
model of deforming the laminate package is based on the applied geometrically nonlinear theory of shells. The formulation of geometric dependencies is based on the relations of the simplest quadratic
variant of the nonlinear elasticity theory. The physical relations of the elementary layer are formulated based on the generalized Hooke's law for the orthotropic material based on the hypotheses of
the applied shells theory. The process of a progressive shell failure is described within the degradation model of stiffness characteristics in elementary layers of a multilayer package which is
based on Hoffman's criteria for composite materials and on the criterion of maximum stresses for the fibers. The process of damage accumulation in the shell material due to a multiple application of
impulse load is takeninto account by means of the computational scheme in which the calculation of the current stress strain state is carried out with stiffness characteristics obtained in the model
of their degradation under previous loading. Energetically consistent system of motion equations of the applied shell theory is deduced from the stationarity condition related to the functional of
the shell total energy. A numerical method for solving the formulated initial-boundary value problem is based on an explicit variational difference scheme. The accuracy of the considered techniqueis
proved by comparing the obtained results with the known experimental data. The results of how the number of loadings affects the value of marginal circumferential deformationsare presented. It is
established that the level of maximalring deformations is approximately ten times less than their limit values compared to a single loading.
PNRPU Mechanics Bulletin. 2016;(4):7-19
Stress-strain state of an elastic soft functionally-graded coating subjected to indentation by a spherical punch
Volkov S.S., Vasiliev A.S., Aizikovich S.M., Seleznev N.M., Leontieva A.V.
The article is devoted to the construction of fields of displacements, stresses and strainsarising in a linearly elastic half-space with a functionally-graded coating subjected to indentation by a
punch with a spherical tip. Calculations of displacements, stresses and strains at the inner point of the coating andthe substrate is reduced to the integration on an infinite interval. The integrand
is dependent on an unknown function of stresses distribution in the contact region. Contact stresses arising due to the indentation of a rigid spherical punch into an elastic half-space with a
functionally-graded coating have earlier been constructed by the authors by solving the problem with mixed boundary conditions. For this purpose, the problem was reduced to the solution of a dual
integral equation using the integral transformation technique. For a general case of independent arbitrary variation of Young’s modulus and Poison’sratioin the depth of the coating, the kernel
transform of the integral equation can be calculated only numerically from the solution of a Cauchy problem for a system of ordinary differential equations with variable coefficients. Using
approximations for the kernel transform of the integral equation by a product of fractional quadratic functions, approximated analytical expressions for the contact stresses and unknown radius of the
contact area were constructed. The expressions obtained are asymptotically exact for both small and big values of relative coating thickness and high accuracy of intermediate values can be reached.
The method is effective for an arbitrary variation of elastic properties and makes it possible to consider values of Young’s modulus of the substrate with more than two orders of magnitude higher
than that in the coating. Series of numerical calculations of elastic displacements and stresses inside the coating and the substrate are provided for a case of soft homogeneous or
functionally-graded layer lying on an elastic half-space (foundation). Young’s modulus of the layer is assumed to be constant or linearly varying (increasing or decreasing) in depth. At the
layer-foundation interface Young’s modulus of the layer is 100 times as much as that of the foundation. This approach makes it possible to avoid the use of assumption about the non-deformability of
the foundation for modeling soft homogenous or functionally-graded layers.
PNRPU Mechanics Bulletin. 2016;(4):20-34
Tashkinov M.A., Mikhailova N.V.
The paper is devoted to the development of a method for calculation of microstructural stresses and strains fields in the multi-phase media based on the calculation of the statistical characteristics
of the local stress and strain fields in the components, which takes into account the geometrical and mechanical properties of components. Representative volumes ofstructurally mulit-phase
heterogeneous materialswere investigated. It is assumed that the components are homogeneous and isotropic. The internal geometry of the structure as well as the assessment of spatial interaction is
described by the moment functions of different orders. The behavior of individual components of the microstructure during loading ofthe representative volume is estimated using the statistical
characteristics of the local stress and strain fields. The characteristics of deformation processes are the statistical moment functions of the stress and strain fields in the components of the
material. Analytical expressions for the statistical moments and correlation functions of the stress and strain fields are obtainedusing statistical averaging of integral-differential equations that
contain moment functions, and derivedfromthe solution of the stochastic boundary value problems in elastic and elastoplastic formulation. Some special cases of typical heterogeneous media with a
random microstructure were considered. The correlation functions of stress and strain for sparse structures with spherical and ellipsoidal hollow inclusions in the elastic and elastoplastic cases
were built. The study and the selectionof approximating dependences obtained for the correlation functionswere performed. The numerical results can be used to evaluate the mechanical behavior of the
inhomogeneous medium microstructural component under different loading conditions and to predict fracture initiation.
PNRPU Mechanics Bulletin. 2016;(4):35-51
Analysis on cyclic deformation and low-high-cycle fatigue in uniaxial stress state
Bondar V.S., Danshin V.V., Alkhimov D.A.
Having analyzed the hysteresis loop of plastic, the authors have formulated the evolution equations for three types of backstresses which are responsible for the shift of yield surface; and based on
them the equations of the theory of plastic flow under combined hardening have been formulated. By integrating the evolution equation for the backstresses of the second type under rigid symmetric
cyclic loading with a constant magnitude of plastic deformation in uniaxial stress state,we have obtained the expressions for the backstresses on the first half cycle and a stable maximum and minimum
values of backstresses. After it, we have examined the work of backstresses of the second type on the field of plastic deformations; andbased on the experimental dataitis shown that the value of this
work is a constant feature of fracture in the conditions of low-high-cycle fatigue (from 101 to 106 cycles). Based on these results we formulatedthelow-high cycle fatiguecriterion. Its asymptotes at
a small and large number of cycles beforefailure have been obtained. The computational and experimental results for fatigue have been compared. The computational and experimental behavior of the
accumulated plastic strain underlow-high-cycle fatigue has been analyzed. Kinetic equation of damage accumulation describing the nonlinear process of damage accumulation has been formulated based on
the analysis of experimental data with regard to damageaccumulation under cyclic loadings.A comparison of the calculated and experimental results in multi-block cyclic loading is considered. The
ratcheting and landing processes with respect to plastic hysteresis loops under asymmetric cyclic loading have been analyzed as well as a parameter and its functional relationship to describe these
processes have been determined. Thecomputational and experimental research results of processes in soft and hard asymmetrical cyclic loadings have been compared.
PNRPU Mechanics Bulletin. 2016;(4):52-71
Anisotropy of material fatigue properties and its effect on durability of structural elements
Burago N.G., Nikitin I.S., Yushkovsky P.A., Yakushev V.L.
The generalization of the criterion of multiaxial fatigue failure is proposed for the case of the alloy with the anisotropy of fatigue properties. The second invariant of stress deviator is replaced
by the Hill’s function that is usually used to describe anisotropic plasticity of metals. In various studies the dependence of fatigue limit on the axis of loading is investigated for samples with
texture at uniaxial fatigue tests. The texture is typically induced during manufacturing semi-finished products (primarily - in rolling). In present study we develop a method for express calculation
of stress-strain state (SSS) for the elastic annular disk of variable thickness (disk of the gas turbine engine compressor) for the mode of low-cycle fatigue (flight cycles). Simplified
representations are used for dependence of solution on coordinates along the disk thickness (the power series) and in the circumferential direction (the Fourier series). For the radial distribution
of stresses and displacements the systems of ordinary differential equations have been derived and solved by the orthogonal sweep method. Proposed criteria of multiaxial fatigue and results are used
to calculate the distributions of fatigue durability for typical disk of gas turbine taking into account the centrifugal forces. Location and time to appearance of fatigue failure zones determined
accounting the influence of the anisotropy of fatigue properties. The calculations indicated that the significant decrease of the fatigue durability is in the vicinity of the disk rim.
PNRPU Mechanics Bulletin. 2016;(4):72-85
Algorithms for numerical simulation of structures deformation and fracture within relations of damaged medium mechanics
Gorokhov V.A., Kazakov D.A., Kapustin S.A., Churilov Y.A.
The article describes the algorithms implemented in computational complex UPAKS (VC UPAKS) which allow studying the start and development of fatigue cracks under low-cycle thermopower loadings by a
direct numerical simulation based on the finite element method within the time acceptable for engineering computations. The studies are carried out in the framework of damaged medium mechanics and
use the hypotheses on the multi-stage nature of damage development in the process of material failure. The paper presents an algorithm of predicting deformation processes and damage accumulation in
structural elements at low-cycle thermopower loadings, which combines the possibility to have a detailed description of deformation and damage accumulation in the early fracture stages while
minimizing the number of computations in the numerical modeling of these processes on the basis of FEM.To investigate the third stage of fracture process, the authors have developed an algorithm for
crack propagation modeling in structural elements on the basis of results of the process simulation in the first two stages without changing the initial topology of finite elements of the studied
structure. The results of simulation of elastic-plastic fracture of the experimental sample with a concentrator under a plain bending which have been made using VC UPAKS are given. The comparison of
numerical results with experiments has shown their good compliance which proves that the efficiency of the proposed algorithm. By using the simulation examples/cases of low-cycle fracture of a
cylindrical recessedsample the authors verified the software that implements the proposed algorithms as part of VC UPAKS software. It proved that they can be effectively used to simulate low-cycle
fracture processes of structure elements.
PNRPU Mechanics Bulletin. 2016;(4):86-105
Experimental and numerical research of the dynamic response of composite outlet guide vane for aircraft jet engine
Grinev M.A., Anoshkin A.N., Pisarev P.V., Shipunov G.S., Nikhamkin M.S., Balakirev A.A., Konev I.P., Golovkin A.Y.
The object of this research is composite outlet guide vane (OGV) for advanced aircraft engine. The weight reduction due to using of polymer composite materials (PCM) instead of the metal in OGV can
reach 40 %. The vane is exposed to intense aerodynamic loads during operation. The modal analysis is needed for the detuning a structure’s resonance frequency. The results of such analysis are
presented in this work. Experimental technique of modal analysis for composite OGV in frequency range up to 6,4 kHz is described in paper. Experimental study was carried out for three full-scale OGV
samples with the help of three-component scanning laser vibrometer using PSV-400-3D hardware. As the results the mean values and coefficients of variation of natural frequencies were obtained and the
main natural modes were shown. The numerical simulation of this problem was carried out by finite element method (FEM) with ANSYS Workbench software using high-performance computing complex. The
technological scheme of laying out of anisotropic plies was taken into account in the developed OGV model. The results of numerical simulations of natural frequencies and modes were compared with the
test data. Good correlation was found. This fact confirmed that the stiffness of a full-scale OGV, manufactured with various possible fluctuations of process parameters and mechanical properties of
materials, meets the required conditions. For further researches the developed numerical model allows to study the effect of reinforcing scheme and other design parameters changing on OGV frequency
response. Laboratory modal analysis can be used to control the dimensional stability and material mechanical properties.
PNRPU Mechanics Bulletin. 2016;(4):106-119
Lycheva T.N., Lychev S.A.
Theoretical relations obtained from solutions of dynamic problems of viscoelasticity represent an effective framework for experimental identification of dynamic rheological properties of materials.
For the construction of such relations, closed solutions of boundary value problems (i.e. written in the form of convergent series or integrals) are preferred, because they(unlike solutions obtained
by numerical methods) allow strict error estimates. However, the construction of analytical solutions is associated with the following difficulties. 1. As usual, the hypothesis of proportionality is
accepted for relaxation operators corresponding to the first and second Lamé moduli, which is equivalent to the hypothesis of a constant Poisson’s ratio. This significantly reduces the generality of
consideration. 2. Representation of solutions for three-dimensional problems in the form of expansions in eigenfunctions makes it necessary to taking into account the large eigenvalues which in the
vast majority of problems can be found only numerically, as the roots of transcendental equations, thus, it is likely to skip closely spaced and multiple roots. 3. Constructed series converge slowly.
In this paper we suggest ways to overcome these difficulties. Solutions of initial boundary value problems are presented in the form of spectral expansions, but in contrast to the classical method of
Fourier decomposition they are expanded over biorthogonal system of eigenfunctions of mutually conjugate pencils of differential operators. This pencils define generalized Sturm-Liouville problem
with a polynomial spectral parameter. This eliminates the hypothesis of proportionality relaxation operators. Effective relations for the terms of spectral(in particular normalization factors)
coordinate functions and asymptotic formulas for the initial approximations of eigenvalues excluding their omission in calculations are obtained. Power related ranking of elements of the spectral
decomposition is proposed which allows achievingthe required accuracy of calculations on the partial sums of a low order.
PNRPU Mechanics Bulletin. 2016;(4):120-150
Nucleation recrystallization mechanisms in metals at thermomechanical processing
Kondratev N.S., Trusov P.V.
In the last 15-20 years, mathematical models are the most important “tool” in the design and creation of technology of thermomechanical processing of metals and alloys. This is the result of
appearance of new class of models based on physical theories. Single-level macrophenomenological models based on macro experiments are replaced by crystals plasticity. Founders of physical theories
of plasticity are G.I.Taylor, G.H.Bishop , R.Hill, T.G.Lin. Many others researchers from the Soviet Union and Russia made a significant contribution to the development of this direction: R.Z. Valiev,
Y.D. Vishnyakov, S.D. Volkov, O.A. Kaybishev, V.A. Likhachev, V.E. Panin, V.V. Rybin, T.D. Shermergor et al. Physically based approach requires deep understanding of the internal mechanisms and
processes that accompany thermomechanical effects caused by inelastic deformation at different scale levels. The important one for the microstructure formation and mechanical properties of finished
products obtained by thermomechanical processing methods is the process of recrystallization. At that point in this article provides an review of the existing theories of recrystallization nucleation
mechanisms. Basic physical mechanisms of nucleation recrystallized grains are classified: 1) the mechanism, based on the classical theory of fluctuations, proposed by J.E.Burke and D.Turnbull; 2)
R.W. Cahn’s mechanism of nucleation and growth subgrains polycrystal, formed as a result of the process polygonization; 3) P.A. Beck’s and P.R. Sperry’s mechanism grain boundary migration, initially
present in the polycrystal - strain induced boundary migration (SIBM); 4) the mechanism of nucleation and growth of new grains as a result of coalescence subgrains (H.Hu, J.C.M.Li, H.Fujita ).
Analysis of existing models describing the inelastic deformation at high temperatures demonstrates the need for consideration and inclusion in the models description of physical mechanisms of
high-temperature processes accompanying plastic deformation.
PNRPU Mechanics Bulletin. 2016;(4):151-174
Dispersive characteristics of flat longitudinal elastic waves extending in porous liquid-saturated medium with cavities
Aizikovich S.M., Erofeev V.I., Leonteva A.V.
At first sight, many continuous media possess numerous micropores which contain or do not contain liquid. These pores are much less than the macroscopic sizes of medium, but they are bigger than
nuclear or molecular ones. Such models of porous medium as soil model are widely usedin geophysics. Liquid distribution (oil, water) in soil is explained by this model. This model is also used in
biology, in particular, to describe the penetration of liquid through plants, for example, wood. In recent years artificial porous materials which are widely applied in everyday life, in equipment
and other areas of human activitieshave been created. The present work considers the distribution of flat longitudinal waves in porous liquid-saturated medium with cavities. It is supposed that the
energy dissipation of a wave in the medium can be neglected. The behavior of linear waves in porous media with cavities is studied. It is known that in the porous medium (Biot's medium) two
longitudinal waves can extend, one of them is slow and the other one is fast. In our problemthree longitudinal waves are extending: two waves do itlike in the Biot medium and the third one does it
due to medium cavities. If the mediumhad neither pores, nor cavities, then one fast wave would extend. The study of linear wave’s behavior is conducted by receiving and analyzingthe dispersive
equation, phase speed and group speed characterizing the wave energy transfer.The density of spectral frequency is considered to determinethe range of the dispersion degree.The work presents the
formation and analysis of dispersive dependences for the considered system. Areas of strong and weak dispersion, areas of normal and abnormal dispersion at certainvalues of system parameters are
PNRPU Mechanics Bulletin. 2016;(4):175-186
A modifiedBouc-Wen model to describe the hysteresis of non-stationary processes
Danilin A.N., Kuznetsova E.L., Kurdumov N.N., Rabinsky L.N., Tarasov S.S.
A number of known phenomenological models are considered, which are used to describe a variety of hysteresis effects in nature. In this case, the system is considered as a "black box" with known
experimental values of input and output parameters. Correlations between them are established by mathematical functions, whose parameters are identified using experimental data. Amongthe
phenomenological models there are marked the Bouc-Wen model andits analogsthat have been successfully usedin variousscientific and technical fieldsdue to the possibilityof the analytical description
ofvarious hysteresis loopsof non-stationary processes. The conditions are formulatedwhich must be satisfied by the Bouc-Wen model. The main ones are the model adequacy of the physical process and
stability. To describe the hysteresis, a mathematical model is suggested, according to which the force and kinematic parameters are bound by a special differential equation of the first order. In
contrast to the Bouc-Wen model, the right side of this equation is chosen in the form of a polynomial of two variables determining the trajectory of a hysteresis in the process diagram. It is stated
that this presentation provides the asymptotic approximation of the solution to the curves of the comprehending (including) hysteresis cycle.This cycle is formed by curves of direct and reverse
processes ("loading-unloading" processes), which are based on experimental data for the maximum possible or permissible intervals of parameter changes during the steady vibrations. Coefficients in
the right part are determined from experimental data for the comprehending hysteresis cycle under conditions of steady-state oscillations. Approximation curves of the comprehending cycle are
constructed using the methods of minimizing the discrepancy of analytical representations to the number of experimental points. The proposed approach allows by one differential equation to describe
the trajectory of hysteresis with a random starting point within the area of the comprehending cycle.
PNRPU Mechanics Bulletin. 2016;(4):187-199
Theoretical-experimental method for determination of aerodynamic damping component of test samples with diamond-shaped cross-section
Paimushin V.N., Firsov V.A., Gyunal I., Shishkin V.M.
A numerical method for processing of experimental vibration data has been developed to find the lowest experimental frequency and amplitude dependences of the logarithmic decrement which are used to
determine damping properties of test-samples. The logarithmic decrement (LD) is determined by the experimental decay curve obtained from the tip point amplitude measurements of test-samples during
their flexural vibrations and approximated by the sum of two exponents with four parameters determined by a direct search of the objective function depending on these parameters. The conducted
numerical experiments confirmed the reliability of the developed method. It is shown that the material of the test samples with a diamond-shaped cross-section must have stable and low damping
properties for a reliable determination of the experimental aerodynamical damping component. Duralumin alloys absolutely meet these requirements. The damping matrix of the finite element model of the
test-sample with an arbitrary cross-sectional shape is constructed in the case of the amplitude-independent internal friction in the material. The internal damping parameter which specifies the
material damping properties is obtained. The experimental aerodynamic component of damping is obtained from the series of test-samples with the diamond-shape cross section. It has been noted that the
elasticity modulus of duralumin D16 AT is frequency dependent. An iterative algorithm is developed to determine the lowest vibration frequency of the test-sample considering this dependence. The
conducted numerical experiments using the test-samples with the specified cross-section confirm the reliability of the developed algorithm. The theoretical and experimental method is developed to
construct the structural formulae to determine the aerodynamic component of damping for the test-samples with the diamond-shaped cross-section. The method is based on the modification of the basic
formulae for thin plates with the constant thickness and the experimental data on the damping properties obtained for the series of test samples with the specified cross-sectional shape. The
reliability of the obtained structural formulae has been confirmed by the performed numerical experiments.
PNRPU Mechanics Bulletin. 2016;(4):200-219
The influence of fluid filtration on the strength of porous fluid-saturated brittle materials
Dimaki A.V., Shilko E.V., Astafurov S.V., Psakhie S.G.
The paper is devoted to the study on how the strength of fluid-saturated permeable brittle materials depends on strain rate. The study has been carried out by means of a numerical simulation using a
hybrid cellular automata method and a coupled model, which takes into account the interplay of deformation of solid skeleton, pore pressure change and fluid filtration. It has been found that the
influence of pore fluid on the material strength is determined by the competition of fluid pore pressure change (due to the volume deformation of solid skeleton) with filtration. On the basis of a
parametric study we obtained the combinations of physical and mechanical characteristics of solid skeleton and fluid as well as of linear dimensions of the samples, which uniquely defines the
dependence between the strength of the deformed fluid-saturated sample and the strain rate. By the examples of uniaxial compression and constrained shear tests we have shown that the character of
influence of the fluid filtration on the sample strength is determined by the sign and magnitude of pore volume change during the course of deformation. Under the loading accompanied by a decrease in
the pore volume and increase in the pore pressure, fluid redistribution reduces the local maxima of pore pressure and thereby provides an increase in strength of the samples. Under the loading
conditions which determine an increase in the pore volume and pore pressure drop, the filtration maintains the fluid pressure and thereby reduces the strength of the samples. Based on the simulation
results, we have constructed the generalized logistic dependences between the samples strength of brittle permeable materials and strain rate, mechanical properties of liquid and solid skeleton and
sample dimensions. These results show that the non-stationary character of the related deformation and filtration processes determines a significant variation of the strength in the samples of
permeable materials even at low strain rates.
PNRPU Mechanics Bulletin. 2016;(4):220-247
Numerical analysis of poroviscoelastic prismatic solids and halfspaces dynamics via boundary element method
Ipatov A.A., Belov A.A., Litvinchuk S.Y.
Dynamic behavior of poroelastic and poroviscoelastic solids is considered. Poroviscoelastic formulation is based on Biot’s model of fully saturated poroelastic media. The elastic-viscoelastic
correspondence principle is applied to describe viscoelastic properties of elastic skeleton. Viscoelastic constitutive equations are introduced. Classical viscoelastic models are used, such as
Kelvin-Voigt, Standard linear solid and model with weakly singular kernel of Abel type. Differential equation system of full Biot’s model in Laplace transform and formulas for elastic modules are
given. Original problem’s solution is built in Laplace transform and numerical inversion is used to obtain the solution in time domain. Direct boundary integral equation (BIE) system is introduced.
Regularized BIE system is considered. Mixed boundary element discretization is introduced to obtain discrete analogues. Gaussian quadrature and hierarchic integrating algorithm are used for
integration over the boundary elements. Numerical inversion of Laplace transform is done by means of modified Durbin’s algorithm with a variable integrating step. The described numerical scheme is
verified by a comparison with analytical solution in a one-dimensional case. Isotropic poroviscoelastic solids and halfspaces are considered. Results of numerical experiments are presented. Problems
of axial force acting on the end of prismatic solid and vertical force acting on a halfspace are solved. Viscous parameter influence on dynamic responses of displacements and pore pressure are
studied. Surface waves on poroviscoelastic halfspace are modelled with the help of boundary element method.
PNRPU Mechanics Bulletin. 2016;(4):248-262
Methodology of numerical modelling of mechanical properties of the porous heat-shielding material based on ceramic fibers
Lurie S.A., Rabinsckiy L.N., Solyaev Y.O., Bouznik V.M., Lizunova D.V.
We propose a method to predict the compression strength and elastic modulus of high porous ceramics based on fibers or whiskers. The method is based on the direct numerical simulation of material
microstructure using a finite element approach. The representative volume elements of material samples are created using the random algorithm taking into account the given sample size, fibers mean
size and orientation and porosity volume fraction. The fiber structure is assumed to consist of long rods as fibers and short rods as links (contacts) between fibers. For the considered structures we
proposed the formulation of the strength criterion, in accordance with which the destruction of the material occurs due to the failure of connections between the fibers. It is proposed to consider
the ultimate strength of the fiber contacts as unknown model parameter. Its value should be determined using the fitting of the estimation results to the experimental data. Predicted values of
effective stiffness and strength of material are based on the analysis of representative element stress state under mechanical pressure. In this paper, we studied the repeatability of the numerical
calculations results for the same type of representative elements with the same average microstructural characteristics. The convergence of the effective properties values with the increasing of the
fragments size is also studied. Test results of the mechanical properties modeling of fibrous materials with different porosity and fibers orientation are presented in this article.
PNRPU Mechanics Bulletin. 2016;(4):263-274
Finite element investigation of the effectiveness of the tubular piezoelectric vibratory gyroscope depending on the type of polarization and boundary conditions
Nasedkin A.V., Shprayzer E.I.
In the present paper, a dynamic behavior of piezoelectric vibratory gyroscope in the form of a hollow piezoelectric cylinder with two pairs of electrodes placed crosswise on the outer side surface
has been analyzed. In the case of one fixed end, two variants of the piezoceramic material polarization have been considered, namely, full radial polarization and partial radial polarization only
under the electrodes on the outer side surface. For a completely polarized material (in addition to the case of a cantilevered end) two other variants of the fixations simulating the conditions of
hinged opening were also considered. The behavior of the harmonic oscillation of gyroscope has been studied in the framework of linear theory of piezoelectricity (electroelasticity) which takes into
account mechanical damping and rotational effects in the relative coordinate system. All of the investigated variants admit the availability of electrically active bending oscillation modes in two
perpendicular planes which can be controlled by given electric potentials on the two pairs of electrodes. In such configurations when the gyroscope operates near to the corresponding resonance
frequencies the primary flexural vibrations are generated in one plane and the secondary flexural motions are created due to the axial gyroscope rotation in the perpendicular plane. These secondary
oscillations can be a measure to determine the value of the rotation frequency. The finite element method, ANSYS finite element package and specially designed computer programs written in the macro
language APDL ANSYS were used for numerical calculations. The results of computational experiments have shown that the variant with one fixed end and with a full radial polarization of the
piezoceramic material gives the largest maximum of the output potential in the presence of gyroscope rotation. It has been found that the variant of the gyroscope with the fixations simulating the
conditions of hinged opening is also quite promising for application.
PNRPU Mechanics Bulletin. 2016;(4):275-288
Mathematical modeling of piezo-electro-luminescent effect and diagnostics of pressure distribution along fiber optic sensor
Pan’kov A.A.
The algorithm of finding the distribution function of pressure along the three-phase fiber optic sensor based on the results of light intensity proceeding from a fiber optic phase measured on the
edge section of the sensor is developed for a case of nonlinear "function of a luminescence" which is a dependence between the intensity of light and voltage acting on the electrophosphor. The
problem is reduced to the solution of the Fredholm integral equation of the 1st kind with the differential kernel depending on the calculated effective parameters of the sensor and on the derivative
set function of a luminescence of an electrophosphor. The analytical solution has been obtained for the function of probabilities density of pressure distribution for a special case when the kernel
is expressed by the delta-function; and the Fredholm integral equation is reduced to an algebraic one. "Direct" and "reverse" problems of the Fredholm integral equation for a case of nonlinear
function of a luminescence of an electrophosphor are solved as an illustration of the algorithm. The light intensity derivative at the optical fiber output of control voltage for the set uniform law
of probability-density function with regard to pressure distribution is found in a direct problem. As for the reverse problem, the probability-density function is determined in comparison with the
known exact solution using the direct problem solution for the derivative of light intensity. The numerical solution of the reverse problem is carried out in different approximations in which the
distribution of nodal points in intervals and required nodal function values of probability-density pressure are found from a condition of minimizing summary discrepancies based on the values of the
light derivative intensity which have been set and calculated on each step based on control voltage at the output of the optical fiber.
PNRPU Mechanics Bulletin. 2016;(4):289-302
Packaging and deployment of large shell structures by internal pressure loading
Pestrenin V.M., Pestrenina I.V., Rusakov S.V., Kondyurin A.V., Korepanova A.V.
The packaging of large composite shell structures (corrugation, a cylinder and a truncated cone) and their deployment by internal pressure loading are explored. It is believed that the medial
surfaces of the constituent elements have involutes which coincide with them in a packed state. The corrugation consists of the ring components, the cylinder and cone consist of trapezoidal plane
components. These components are made of carbon fiber with orthotropic or transversely isotropic elastic properties and stapled by joints. The joints do not perceive resistance to rotation about the
tangent to the weld line. The contemplated structures perceive bending loads (unlike pneumatic ones) made of soft materials (fabrics, films). Geometrically nonlinear solid mechanics problems with the
internal pressure loading are solved by using the engineering computing system ANSYS. The deployment pressure dependence on the shell material structure, shell thickness and amount of constituent
elements are investigated. It is shown that the deployment pressure of the large shell is commensurate with the pressure of pneumatic structures of soft materials. It was found that the stresses in
the corrugation shells can reach critical values but in the cylinder and the truncated cone the stresses are insignificant. The task formulation and its solution on the thermodynamic state of the
injected gas under quasi-static internal pressure loading of the shell are suggested. It is shown that in the beginning of deployment the gas temperature will drop by about 50-80 degrees Celsius
according to gas composition, and then its temperature is tending to increase to the injected gas temperature. These results enable to expand the choice of materials for the pneumatic products
manufacturing including space applications design.
PNRPU Mechanics Bulletin. 2016;(4):303-316
A new algorithm for generating a random packing of ellipsoidal inclusions to construct composite microstructure
Shubin S.N., Freidin A.B.
The subject of the work is a microstructure of a composite which consists of a continuum matrix and a set of isolated particles homogeneously distributed inside the matrix. It is assumed that the
reinforcing particles have ellipsoidal shapes, while distribution and orientation are random. The main point of the work is a new computationally-efficient algorithm to generate microstructure of
such a composite. In the algorithm the existing “concurrent” method based on an overlap elimination is extended to ellipsoidal shapes of the particles. It begins with randomly distributed and
randomly oriented ellipsoidal particles which can overlap each other. During the performance of the algorithm intersections between particles are allowed and at each step the volumes of intersections
are minimized by moving the particles. The movement is defined for each pair of particles based on the volume of the intersection: if two particles are overlapped, then the reference point inside the
intersection is chosen and then two particles are moved in such a way that the reference point becomes the tangent point for both particles. To define the relative configuration of two particles
(separate, tangent or overlapping) and to choose reference point inside the intersection volume the technique based on formulating the problem in four dimensions and then analyzing the roots of the
characteristic equation are applied. The algorithm is able to generate close packed microstructures containing arbitrary ellipsoids including prolate and oblate ellipsoids with high aspect ratios
(more than 10). The generated packings have a uniform distribution of orientations.
PNRPU Mechanics Bulletin. 2016;(4):317-337
Danilin A.N.
In this paper, the dynamic of a structure composed of flexible rod elements connected via hinges is modeled. It is assumed that the hinges have constraints - rigid and non-rigid, controlled and
uncontrolled ones. Mathematically, they are considered as differential ones in integrable or non-integrable forms. Mathematical model is formulated based on the finite element method taking into
account finite deformations and the nonlinearity of the inertial forces. The rod element ends are considered to be connected with rigid bodies whose dimensions are small relative to the element
length. Each finite element is associated with a local coordinate system for which the displacements, angles of rotation, the translational and rotational speed are strictly considered. Shape
functions are taken as quasi-static approximations of local displacement and rotation angles of element cross-sections. Absolute displacements and rotation angles of element boundary cross-sections
are taken as generalized coordinates of the problem. The dynamic equations are obtained using d'Alembert-Lagrange principle. It is considered that the generalized coordinates are subjected to the
linear relations relative to the generalized velocities. Variation of the problem functional for which to look for the steady-state value is transformed by the addition of the constraint equations
multiplied by the undefined Lagrange multipliers. The variational problem for the transformed functional is solved as a free. The stationarity conditions together with the differential equations of
constraints determine the desired values of the generalized coordinates. This paper proposes an approach that allows to avoid cumbersome calculations of the nonlinear inertial members without
simplification of the physical model and (or) changing the original structure of equations. An example of deploying rod system consisting of three flexible rods connected in series via hinges is
considered. The solution of nonlinear dynamic equations is obtained numerically using the integral curve length parameter as a problem argument. This transformation gives a system of resolving
equations the best conditioning of the numerical solution process.
PNRPU Mechanics Bulletin. 2016;(4):338-363
Dynamics of unbalanced flexible rotor with anisotropic supports during contact with the stator
Kurakin A.D., Nikhamkin M.S., Semenov S.V.
One of the important problems of rotor systems mechanics is rotor dynamics when it contacts stator elements. There are well-known cases when rotor - stator rubbing led to serious accidents. The main
aim of the work is to gather data about movement of rotor upon contact with the stator suitable for verification of mathematical models. Also diagnostic features of rubbing in the rotor system with
anisotropic stiffness were determined. Vibration behavior of the unbalanced flexible rotor with two ball bearings when it contacts the stator was experimentally investigated. The influence of support
stiffness anisotropy, unbalance value, contact properties and stator elastic flexibility were taken into consideration. The calculation method of rotor dynamics with the stator rubbing was created
and identified via experimental data. This method is based on Jeffcott model and allows taking into consideration the support stiffness anisotropy, unbalance value, contact properties, rpm and stator
elastic flexibility. The model explains the mechanism of additional harmonics appearing in Campbell diagram and the presence of frequency response when the rotor contacts the stator. It makes
possible to use phenomena of additional harmonics appearing as a diagnostic feature. The created experimental method and gathered data can be used for the verification and tuning of mathematical
models. The suggested mathematical method of rotor-stator interaction modelling is suitable for the detection and elimination of rotor stator contacting situations. Also, it can be used as a basis
for more complicated rotor system models.
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seminars - MOD p LOCAL-GLOBAL COMPATIBILITY FOR GLn(Qp) IN THE ORDINARY CASE
Let F/Q be a CM field in which p splits completely and r : Gal(Q/F) → GLn(Fp) a continuous automorphic Galois representation. We assume that r|Gal(Qp/Fw) is an ordinary representation at a place w
above p. In this talk, we discuss a problem about local-global compatibility in the mod p Langlands program for GLn(Qp). It is expected that if r|Gal(Qp/Fw) is tamely ramified, then it is determined
by the set of modular Serre weights and the Hecke action on its constituents. However, this is not true if r|Gal(Qp/Fw) is wildly ramified, and the question of determining r|Gal(Qp/Fw) from a space
of mod p automorphic forms lies deeper than the weight part of Serre’s conjecture. We define a local invariant associated to r|Gal(Qp/Fw) in terms of Fontaine-Laffaille theory, and discuss a way to
prove that the local invariant associated to r|Gal(Qp/Fw) can be obtained in terms of a refined Hecke action on a space of mod p algebraic automorphic forms on a compact unitary group cut out by the
maximal ideal of a Hecke algebra associated to r, that is a candidate on the automorphic side corresponding to r|Gal(Qp/Fw) for mod p Langlands program.
The talk is based on a joint work with Zicheng Qian. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=63&document_srl=774929&sort_index=date&order_type=desc","timestamp":"2024-11-14T07:46:02Z","content_type":"text/html","content_length":"47559","record_id":"<urn:uuid:9bd2184f-0c25-43a4-a319-a45bfd579feb>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00282.warc.gz"} |
How to Easily Calculate the Dot Product in Excel | Online Tutorials Library List | Tutoraspire.com
How to Easily Calculate the Dot Product in Excel
by Tutor Aspire
This tutorial explains how to calculate the dot product in Excel.
What is the Dot Product?
Given vector a = [a[1], a[2], a[3]] and vector b = [b[1], b[2], b[3]], the dot product of vector a and vector b, denoted as a · b, is given by:
a · b = a[1] * b[1] + a[2] * b[2] + a[3] * b[3]
For example, if a = [2, 5, 6] and b = [4, 3, 2], then the dot product of a and b would be equal to:
a · b = 2*4 + 5*3 + 6*2
a · b = 8 + 15 + 12
a · b = 35
In essence, the dot product is the sum of the products of the corresponding entries in two vectors.
How to Find the Dot Product in Excel
To find the dot product of two vectors in Excel, we can use the followings steps:
1. Enter the data. Enter the data values for each vector in their own columns. For example, enter the data values for vector a = [2, 5, 6] into column A and the data values for vector b = [4, 3, 2]
into column B:
2. Calculate the dot product. To calculate the dot product, we can use the Excel function SUMPRODUCT(), which uses the following syntax:
SUMPRODUCT(array1, [array2], …)
• array – the first array or range to multiply, then add.
• array2 – the second array or range to multiply, then add.
In this example, we can type the following into cell D1 to calculate the dot product between vector a and vector b:
=SUMPRODUCT(A1:A3, B1:B3)
This produces the value 35, which matches the answer we got by hand.
Note that we can use SUMPRODUCT() to find the dot product for any length of vectors. For example, suppose vector a and b were both of length 20. Then we could enter the following formula in
cell D1Â to calculate their dot product:
=SUMPRODUCT(A1:A20, B1:B20)
Potential Errors in Calculating the Dot Product
The function SUMPRODUCT() will return a #VALUE! error if the vectors do not have equal length.
For example, if vector a has length 20 and vector b has length 19, then the formula =SUMPRODUCT(A1:A20, B1:B19) will return an error.
The two vectors need to have the same length in order for the dot product to be calculated.
Additional Resources
The following tutorials explain how to calculate a dot product in different statistical software:
How to Calculate the Dot Product in Google Sheets
How to Calculate the Dot Product in R
How to Calculate a Dot Product on a TI-84 Calculator
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Inscribed Square Problem
Inscribed Square Problem
Does every
Jordan curve
have 4 points on it which form the vertices of a square?
A Jordan curve is a continuous function simple) and closed).
[M] Meyerson, M.D., Equilateral triangles and continuous curves, Fund. Math. 110, (1980), 1--9.
* indicates original appearance(s) of problem.
On February 25th, 2021 Anonymous says:
This was proven in https://arxiv.org/pdf/2005.09193.pdf
On March 3rd, 2010 Anonymous says:
There is a theorem that says:
in all simple closed curves there are 4n points that are vertex of n squares (inf = > n > =1)
Jorge Pasin.
On November 8th, 2010 Anonymous says:
Would you clarify? An obtuse triangle has only one inscribed square, so this theorem is not true for n>=2. Do you have a reference to this theorem? Strashimir Popvassilev
On November 25th, 2009 Anonymous says:
Is the conjecture known to be true for C^1-smooth curves?
On June 7th, 2010 Anonymous says:
Yes. Walter Stromquist, Inscribed squares and square-like quadrilaterals in closed curves, Mathematika 36: 187-197 (1989).
On June 2nd, 2010 Anonymous says:
If it were true for C^1 curves, then since a Jordan curve is compact, it may be weierstrass approximated by a series of C^1 curves (indeed by curves whose component functions are polynomials) such
that the series converges uniformly to the given jordan curve. Then by assumption, each curve in the sequence contains 4 points forming a square, and the sequence of squares can be regarded as
(eventually) a sequence in the (sequentially) compact space of the 4-fold product of any closed epsilon enlargement of the area bounded by the original jordan curve. It follows that the sequence of
squares contains a convergent subsequence, which can be shown to be a square lying on the original jordan curve.
Thus, proving the C^1 case proves the general case.
On June 7th, 2010 Anonymous says:
The approximation argument is flawed: the squares on approximating curves may have sides decreasing to 0, in which case the limiting "square" degenerates to a point. In fact, Stromquist's theorem
covers a much wider class of curves than C^1, but not all continuous curves.
On April 28th, 2008 Anonymous says:
Phrasing should be changed from "Does any..." to "Does every..." | {"url":"http://www.openproblemgarden.org/op/inscribed_square_problem","timestamp":"2024-11-01T19:51:50Z","content_type":"application/xhtml+xml","content_length":"17947","record_id":"<urn:uuid:0dcb4705-5b6b-4419-a1be-8c34e1d97b4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00774.warc.gz"} |
what is a good model accuracy
If you do it, you STILL get a good accuracy. Java is a registered trademark of Oracle and/or its affiliates. That’s why you need a baseline. 2.) So for example, suppose you have a span predictor that
gets 90% accuracy. The MASE is the ratio of the MAE over the MAE of the naive model. And even when they are, it’s still important to calculate which observations are more present on the set. In this
way, when the MASE is equal to 1 that means that your model has the same MAE as the naive model, so you almost might as well pick the naive model. E.g. Predictive models with a given level of
accuracy (73% — Bob’s Model) may have greater predictive power (higher Precision and Recall) than models with higher accuracy (90% —Hawkins Model) would achieve the exact same accuracy (91/100
correct predictions) If the purpose of the model is to provide highly accurate predictions or decisions to b… To sum up, the radical difference in the p-values between the first and second tables
arises from the radical difference in the quality of the model results, where m1 acc . ... (i.e. decreases the accuracy of the tree over the validation set). Of the 91 benign tumors, the model
correctly identifies 90 as more insight into our model's performance. And, this is where 90% of the data scientists give up. The goal of a good machine learning model is to get the right balance of
Precision and Recall, by trying to maximize the number of True Positives while minimizing the number of False Negatives and False Positives (as represented in the diagram above). How to know if a
model is really better than just guessing? Machine learning model accuracy is the measurement used to determine which model is best at identifying relationships and patterns between variables in a
dataset based on the input, or training, data. In this case, most of my models reach a classification accuracy of around 70%. Let's try calculating accuracy for the following model that classified
Are these expectations unrealistic? In this scenario, you would have the perfect CAP, represented now by a yellow line: In fact, you evaluate how powerful your model is by comparing it to the perfect
CAP and to the baseline (or random CAP). with a class-imbalanced data set, like this one, But sample sizes are a huge concern here, especially for the extremes (nearing 0% or 100%), such that the
averages of the acutal values are not accurate, so using them to measure the model accuracy doesn't seem right. However, of the 9 malignant tumors, the Let’s see an example. If you have a ‘X’ value
that’s lower than 60%, do a new model as the actual one is not significative compared to the baseline. And if you’re wrong, there’s a tradeoff between tightening standards to catch the thieves and
annoying your customers. You feel helpless and stuck. If the model's prediction is perfect, the loss is zero; otherwise, the loss is greater. Training a model simply means learning (determining) good
values for all the weights and the bias from labeled examples.. Loss is the result of a bad prediction. For a random model, the overall accuracy is all due to random chance, the numerator is 0, and
Cohen’s kappa is 0. A baseline is a reference from which you can compare algorithms. I am looking to get a new Loaded M1A, model MA9822. Grooving the receiver to better accept scope mounts was a
magnitude more convenient and helped milk the Model’s 60’s accuracy potential. What you have to keep in mind is that the accuracy alone is not a good evaluation option when you work with
class-imbalanced data sets. 90%. (Here we see that accuracy is problematic even for balanced classes.) It represents the number of positive guesses made by the model in comparison to our baseline.
Enhancing a model performancecan be challenging at times. In other words, our model is no better than one that Data science world has any number of examples where for imbalanced data (biased data
with very low percentage of one of the two possible categories) accuracy standalone cannot be considered as good measure of performance of classification models. 100 tumors as malignant Would this be
a good 600yd iron sight config? 9 are malignant (1 TP and 8 FNs). The accuracy of a model is usually determined after the model parameters are learned and fixed and no learning is taking place. You
don’t have to abandon the accuracy. There are many ways to measure how well a statistical model predicts a binary outcome. The formula for accuracy is below: Accuracy will answer the question, what
percent of the models predictions were correct? Let’s say that usually, 5% of the customers click on the links on the messages. We will see in some of the evaluation metrics later, not both are used.
So, let’s analyse an example. From June 2020, I will no longer be using Medium to publish new stories. The accuracy seems to be — at first — a perfect way to measure if a machine learning model is
behaving well. It can be used in classification models to inform what’s the degree of predictions that the model was able to guess correctly. what is the standard requirements or criteria for a good
model? This … (the negative class): Accuracy comes out to 0.91, or 91% (91 correct predictions out of 100 total Don’t trust only on this measurement to evaluate how well your model performs. I’m
sure, a lot of you would agree with me if you’ve found yourself stuck in a similar situation. where there is a significant disparity between Only assign true to ALL the predictions. Proper
scoring-rules will prefer a ( … Primarily measure what you need to achieve, such as efficiency or profitability. $$\text{Accuracy} = \frac{\text{Number of correct predictions}}{\text{Total number of
predictions}}$$, $$\text{Accuracy} = \frac{TP+TN}{TP+TN+FP+FN}$$, $$\text{Accuracy} = \frac{TP+TN}{TP+TN+FP+FN} = \frac{1+90}{1+90+1+8} = 0.91$$, Check Your Understanding: Accuracy, Precision,
Recall, Sign up for the Google Developers newsletter. Resolution , meanwhile, is the fixed number of pixels displayed by a projector when 3D printing using Digital Light Processing (DLP). Now, you
have deployed a brand new model that accounts for the gender, the place where the customers live and their age you want to test how it performs. With any model, though, you’re never going to to hit
100% accuracy. what is the main aspect for a good model? model only correctly identifies 1 as malignant—a examples). If your ‘X’ value is between 60% and 70%, it’s a poor model. Accuracy Score = (TP
+ TN)/ (TP + FN + TN + FP) accuracy has the following definition: For binary classification, accuracy can also be calculated in terms of positives and negatives It dropped a little, but 88.5% is a
good score. If your ‘X’ value is between 90% and 100%, it’s a probably an overfitting case. benign. (the positive class) or benign I might create a model accuracy score by summing the difference at
each discrete value of prob_value_is_true. The accuracy of forecasts can only be determined by considering how well a model performs on new data that were not used when fitting the model. So if I
just guess that every email is spam, what accuracy do I get? This is a good overall metric for the model. The blue line is your baseline, while the green line is the performance of your model. The
FV3 core brings a new level of accuracy and numeric efficiency to the model’s representation of atmospheric processes such as air motions. Imagine you work for a company that’s constantly s̶p̶a̶m̶m̶i̶n̶g̶
sending newsletters to their customers. Not that you’d need a scope to get and keep the rifle in the black. There is an unknown and fixed limit to which any data can be predictive regardless of the
tools used or experience of the modeler. terrible outcome, as 8 out of 9 malignancies go undiagnosed! Then, you will find out what would be your accuracy if you didn’t use any model. Over the past 90
days, the European Model has averaged an accuracy correlation of 0.929. Imagine you have to make 1.000 predictions. Actually, let's do a closer analysis of positives and negatives to gain To
summarize, here are a few key principles to bear in mind when measuring forecast accuracy: 1. With your model, you got an accuracy of 92%. Open rear and ramp front (common on many models) proved more
than accurate enough for most .22 applications. Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License, and code samples are licensed under
the Apache 2.0 License. Or maybe you just have a very hard, resistant to prediction problem. Then, check on the ‘Customers who clicked’ axis what’s the corresponding value. for evaluating
class-imbalanced problems: precision and recall. The accuracy of a model is controlled by three major variables: 1). But…wait. While 91% accuracy may seem good at first glance, another
tumor-classifier model that always predicts benign would achieve the exact same accuracy (91/100 correct predictions) on … An adequately accurate bullet that does a good job of killing game is far
preferable to a brilliantly accurate bullet that does a marginal job when it hits the target. You send the same number of emails that you did before, but this time, for the clients you believe will
respond to your model. Sum of true positive and true negatives out of all the predictions top lift time! Of all the predictions usually, 5 % of the tools used or experience of the can! Be predictive
to their customers not both are used, in this example, suppose you have comparison! A predictive model into the target/dependent variable in the black the data give!, let 's do a closer analysis of
Positives and negatives to gain more insight into our model 's was! Not only fit the training data well but also accurately classify records it has never seen % than... Its affiliates for evaluating
classifier models test the accuracy seems to be result. … the first is accuracy matrix w… what is the performance of your model, forecast... Good at five days in the next logical step is to translate
this probability into. Us with best possible results when measuring forecast accuracy: 1 ) scientists up... Mae over the MAE over the past 90 days, the European model has averaged accuracy. On a
single example it be a good model what is a good model accuracy X ’ value is 70. Is perfect, the loss is a reference from which you can compare algorithms to hit 100,! You fail at improving the
accuracy only is not bad at all to understand, making it the most metric! The notion of good or bad can only be applied if we have a span predictor that gets 90 of. How bad what is a good model
accuracy model correctly identifies 90 as benign set ) is between 70 % 90... S not telling the all history sight config between 90 %, you will find what. Me if you decide simply to predict everything
as true it means that model! And 100 % accuracy such as efficiency or profitability below: accuracy will answer the question, percent... Are used the Positives and negatives out of all the
predictions in the.... S kappa could also theoretically be negative, with a better model tending to the perfect.. Evaluating classification models to inform what ’ s a poor model days in future!
Negatives out of all the predictions resistant to prediction problem model tending to perfect! A projector when 3D printing using Digital Light Processing ( DLP ) the..., not both are used evaluating
class-imbalanced problems: precision and recall that has zero predictive ability to distinguish tumors! This probability number into the target/dependent variable in the future scientists give up at
—! Model was able to guess correctly links on the ‘ customers who clicked ’ axis what ’ s good. A few key principles to bear in mind when measuring forecast accuracy:.... They are, it represents the
model 's prediction was on a example. Spam, what percent of the tools used or experience of the tree over the 90. Profile, is a good score accuracy score represents the number of predictions that the
model prediction! Do I get to understand, making it the most common metric used for evaluating classification models to what! > > nir accuracy ( good model will remain between the perfect CAP single
example suppose you have span..., this is a reference from which you can compare algorithms accurate enough for most.22.... All the strategies and algorithms that you ’ re wrong, there ’ STILL. A
perfect way to measure how well a statistical model predicts a binary outcome wait, imagine that ’! I just guess that every email is spam, what accuracy do I get is spam, percent... It is easy to
calculate which observations are more present on the set its.... Not balanced very hard, resistant to prediction problem a simple way of measuring the effectiveness of model... To a predictive
modeling criterion so the modeler can use it for selecting models in! % better than just guessing but a poor master that the model identifies... Excellent model to use a model links on the links on
the messages baseline, while the green line the... It, you fail at improving the accuracy is a powerful way to what is a good model accuracy the total of. Axis what ’ s the degree of predictions a
model is usually determined after the model not. Of around 70 % the model 's prediction was on a single example from benign tumors, European! A statistical model predicts a binary outcome you got an
accuracy of tree! > nir accuracy ( good model the thieves and annoying your customers by the model and test accuracy... The main aspect for a company that ’ s constantly s̶p̶a̶m̶m̶i̶n̶g̶ sending newsletters
to their customers model accuracy represents... | {"url":"http://hoogenraad.org/fidelity-vs-irsidur/2afba5-what-is-a-good-model-accuracy","timestamp":"2024-11-12T19:04:10Z","content_type":"text/html","content_length":"25941","record_id":"<urn:uuid:86836e07-48db-4fa2-8631-7c51ccfa08cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00757.warc.gz"} |
How does CMMN handle case resolution time estimation? | BPMN Assignment Help
How does CMMN handle case resolution time estimation? As part of proof-of-concept, I built a new CMMN toolbox. Locations is a random location. In R, CMMN places a specific this link on the edge of an
R. $H$ is defined to be neighbor of $p$, which has probability 1/2 for all $x$, and $1/2$ for some $p \in P$, a random location in R is a random location in CMMN. In this paper, we generally treat
case instance in R. We use rule-based learning to model the case of FLS. For our test case, we would like to correct the RFLS error on some data from CMMN. So what is the rate of any rate? Our test
case is based on a paper by Radhika Rao (1979), which is well studied. At this point, we compare the rate of the RFLS and FLS where no edge on the R is used as input. In this paper, we do not know
how or how many R-edges in the R are used as input. Since R-edges are defined in different papers, they just show their existence and uniqueness in different papers. However, how many R-edges are
used in that paper still remains to be clarified. In this paper, we show how to deal with the R-edgers made in CMMN, and not the R-edges of SMMN, which makes efficiency testing of CMMN easier. Let us
say that the R-edgers are *reduced* in CMMN. So for them, we don’t have a hint like the fact that they use LCA as the input for CMMN, or using *abnormal* real numbers as the input for the test case.
I have started this paper trying to explain this problem, I hope thatHow does CMMN handle case resolution time estimation? – Peter R. Weiss All the case time estimation is of interest because of its
association with performance in various computer science domains. Case time information (CTI), as a combination of a case- and error-handling model, is much less important. CTI is a characteristic
that can be directly used for decisions concerning a case-based decision system if the case is performed with certainty, when the system encounters a case and the case-handling model itself has taken
a certain time to sort out. Then, it may seem reasonable to use only the most accurate method, which covers most of true cases, but most of any other case-based decision in terms of the CTI can be
treated as being a least accurate approach.
Teaching An Online Course For The First Time
Nowadays, in practice, most case-based decision systems are not designed to make sense of the whole information, as e.g. the CTI is only concerned with case- and error-handling models. Using either,
the whole data, it measures the relative effects of the case and the case-handling model on the overall performance. For instance, if one is planning to go for the most accurate approach, then no
situation arises if one decides to perform on the most accurate method. So, a very large number of decisions are simply not possible. So, the number of times the case- and error-handling model is
applied is quite a bit higher if one is planning a more accurate approach. So, a more accurate approach would have less numbers of errors, but still have the correct number. However, this situation
might be different if one applies the wrong method used in the (very crude) case, and when performing the most accurate approach. The problem here is that, if it is not mentioned in (D2D), the
decision is not completely ambiguous just by chance, but then we should not expect clear answers from the decision makers. Now, if one starts with the worst case case for the CTI method, then one can
perform a critical decision analysis if they have such an effect, without also considering the worst case as well. Firstly, since one should not think about the worst case as there is no risk of
conflict between the two ones (we do not know whether there are too many cases of such confusion), the worst case is necessarily considered as to what is the quality of the user. Then, if one wishes
to do a critical decision analysis in the worst case, one should not try to decide about the cost of the study helpful site mentioning it. In particular, if one misses cases because of such a bad
decision, then it will probably not be possible at all. So, although (D1A) is quite clear, there are many others (D2A and D1D) that would use the best strategy to find cases, thus these cases do not
have the CTI method is used as it needs to be used for finding the most accurate choicesHow does CMMN handle case resolution time click to find out more We know from user manual that case resolution
time estimation is done by taking the average of the number of arguments to obtain a solution of the equation. this website is not possible to perform case estimation in linear temporal order in the
framework of CMMN so that CMMN is defined as follows. Given a simple object representation of the system being constructed, there are two basic ways how to perform the case calculation: a) Taking the
average of all the arguments to get one solution of the equation b) One time taking the average c) One time taking the average. Following above approach all the steps will have to be done before this
one time taking the average can be used in case calculation. Currently CMMN is employed to perform case conversion for system representations of a) As before b) In the case result c) In the case
result d) In the case result When each of the results are evaluated one by one according to case conversion it can be inferred the result to be the input. CMMN needs to be defined so that CMMN is
defined as follows: Problem Input is the object representation of the system being computed.
Pay Someone To Sit My Exam
Create the vector of such an object representation of the system using the rule I-CMLP-0.5.1.1 : I-CMLP-0.5.1.1 – input to CMMN input is a vector of object representation as the reason used to find
out that a given input is defined in the CMMN context. why not check here one would be able to find out, using CMMN, that a given input is in the data space also there will be problems in this
approach like the case for case conversion. Suppose that the problem was that one of the input was given in the CMMN context for both input and case conversion then it was | {"url":"https://bpmnassignmenthelp.com/how-does-cmmn-handle-case-resolution-time-estimation-2","timestamp":"2024-11-13T06:26:40Z","content_type":"text/html","content_length":"164239","record_id":"<urn:uuid:d2b048ef-8887-4191-9762-50367a477107>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00425.warc.gz"} |
Equation Editor: WinWord 1.x Equation Does Not Appear
ID: Q78749
The information in this article applies to:
• Microsoft Equation Editor version 1.0
You can convert equations originally created in Microsoft Word for Windows version 1.0, 1.1, or 1.1a to Word for Windows version 2.0; however, the Equation Editor may not start up when you
double-click the equation.
If you embed an equation in a Word for Windows version 2.0 document, double-clicking the equation should open the Equation Editor. This does not always occur if the equation was converted as part of
a Word for Windows 1.x file.
Sometimes after you double-click the equation, the Equation Editor opens with a blank window, indicating that no equation exists in that particular location.If this problem occurs, the equation is
also deleted from the text of the document. To correct this problem, use the following steps:
1. Exit the Equation Editor, and close the file without saving
2. Open the file. From the View menu, choose Field Codes (unless
Field Codes are already turned on).
3. Position the insertion point immediately after the existing field
code for the equation and retype the equation. Be sure to use CTRL+F9
to insert the special field code braces.
After you retype the equation field, the Equation Editor should work correctly.
Microsoft is researching this problem and will post new information here as it becomes available.
KBCategory: kbtool KBSubcategory:
Additional reference words: 2.00
Last Reviewed: September 9, 1996 | {"url":"https://helparchive.huntertur.net/document/23894","timestamp":"2024-11-12T19:03:24Z","content_type":"text/html","content_length":"4272","record_id":"<urn:uuid:d3c03a8e-d967-4c2a-9032-a05717f8c199>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00858.warc.gz"} |
Sebastian Rosenzweig^1, Hans Christian Martin Holme^1,2, Nick Scholand^1, Robin Niklas Wilke^1,2, and Martin Uecker^1,2
^1Intistut für Diagnostische und Interventionelle Radiologie, University Medical Center Göttingen, Göttingen, Germany, ^2Partner Site Göttingen, German Centre for Cardiovascular Research (DZHK),
Göttingen, Germany
Cardiac MRI requires fast and robust acquisition and reconstruction strategies to generate clinically reliable results. Using a combination of non-aligned radial Simultaneous Multi-slice (SMS)
acquisitions and Regularized Nonlinear Inverse reconstruction, time-consistent movies of multiple slices can be generated. Here, we develop a self-gating technique for real-time radial SMS
measurements which allows for the reconstruction of high-quality self-gated cine loops in addition to real-time movies from the same measurement.
Introduction and Purpose
Increasing imaging speed is of great importance in clinical practice and one of the main challenges in MRI research. In the past decade, fast single-slice real-time MRI techniques such as Regularized
Nonlinear Inversion (NLINV) emerged which yield a time-resolution of 20ms
. Likewise, the technique of Simultaneous Multi-Slice (SMS) was shown to significantly speed up MRI measurements
. Recently, we introduced SMS-NLINV, a technique which combines the advantages of Simultaneous Multi-Slice and Regularized Nonlinear Inverse reconstruction and achieved a time-resolution of 33ms for
three slices per frame in cardiac MRI
. This real-time method does not rely on ECG triggering or breath-holding and provides a sequence of several individual heart beats for multiple slices. We show that from the same data we can also
reconstruct one high quality self-gated cine loop of a single synthetic heart beat with improved temporal resolution and image quality.
Sequence. We use a 2D radial FLASH sequence with phase-modulated multiband RF excitation pulses to induce Fourier-encoding in partition direction. For each excitation we increase the projection angle
about the 7th tiny golden angle^7 $$$\Psi_7 \approx 23.6°$$$. In particular, this yields non-aligned spokes in partition direction which significantly improves image quality by exploiting
three-dimensional sensitivity encoding^4.
Reconstruction. The SMS-NLINV reconstruction algorithm^4,8 jointly estimates coil sensitivities and image content for all slices by solving the nonlinear inverse problem $$$F(\boldsymbol{X}) = \
boldsymbol{Y}$$$. Here, $$$\boldsymbol{Y}$$$ is the measured data, $$$\boldsymbol{X}$$$ is the objective, i.e. image content and coil sensitivities for all slices, and $$$F$$$ is the forward model.
The problem is solved using the Iteratively Regularized Gauss Newton Method.
Self-Gating. We perform both respiratory and cardiac self-gating by using the motion information inherent in the 1D projection profile of the investigated volume, which can be generated by a 1D
Fourier-transform of the partitions' DC components $$$k_x=k_y=0$$$. To extract cardiac motion we perform iterative bandpass filtering of the center of mass calculated from the 1D volume projection
profile^9. Respiratory motion estimation is done using a principle component analysis^10 and coil clustering^11 to detect the most prevalent signal variation modes in all coils. The data that match a
predetermined respiratory gating window are used for image reconstruction.
We perform one-minute free-breathing real-time FLASH SMS measurements with multiband factor mb=3 (TR=2.3 ms, slice distance 20 mm) and mb=5 (TR = 2.9 ms) on a human heart (short-axis view) without
ECG triggering. We then reconstruct all acquired frames (real-time reconstruction) as well as one synthetic heart-beat (self-gated reconstruction) using SMS-NLINV with temporal regularization and a
temporal median filter of length three. A schematic of the procedure is depicted in Fig. 1.
Results and Discussion
Fig. 2a and 3a show one frame extracted from full real-time reconstructions for multiband factor mb=3 and mb=5. Both reconstructions provide satisfying image quality with 14 frames (mb=5) and 28
frames (mb=3) per heart beat. For the same data, we can use self-gating to obtain one synthetic heart beat consisting of 30 frames (Fig. 2b and 3b) with improved image quality and temporal
Conclusion and Outlook
We demonstrated that a one-minute SMS scan can be used to obtain high-quality real-time reconstructions, which can be further improved using a combination of self-gating and SMS-NLINV. We furthermore
showed that robust self-gating is also feasible with non-aligned spokes in partition direction, which enables sensitivity encoding in all three dimensions. In the future we want to extend SMS-NLINV
for the reconstruction of Stack-of-Stars data and implement more elaborate regularization terms to fully exploit all available data.
No acknowledgement found.
[1] Uecker, M., Zhang, S., Voit, D., Karaus, A., Merboldt, K.-D. and Frahm, J. (2010), Real-time MRI at a resolution of 20 ms. NMR Biomed., 23: 986–994. doi:10.1002/nbm.1585
[2] Moeller S, Yacoub E, Olman CA, Auerbach E, Strupp J, Harel N, Ugurbil K. Multiband multislice GE-EPI at 7 Tesla, with 16-fold acceleration using partial parallel imaging with application to high
spatial and temporal whole-brain fMRI. Magn Reson Med 2010;63: 1144–1153. [3] Feinberg DA, Moeller S, Smith SM, Auerbach E, Ramanna S, Glasser MF, Miller KL, Ugurbil K, Yacoub E. Multiplexed echo
planar imaging for sub-second whole brain FMRI and fast diffusion imaging. PLoS One 2010;5:e15710.
[4] Rosenzweig, S., Holme, H. C. M., Wilke, R. N., Voit, D., Frahm, J. and Uecker, M. (2017), Simultaneous multi-slice MRI using cartesian and radial FLASH and regularized nonlinear inversion:
SMS-NLINV. Magn. Reson. Med.. doi:10.1002/mrm.26878 [5] Rosenzweig, S., Holme, H. C. M., Wilke, R. N. and Uecker, M. (2017), Simultaneous Multi-Slice Real-Time Imaging with Radial Multi-Band FLASH
and Nonlinear Inverse Reconstruction, In Proc. Intl. Soc. Mag. Reson. Med. 25; 518. [6] Rosenzweig, S. and Uecker, M. (2016), Reconstruction of multiband MRI data using Regularized Nonlinear
Inversion, In Proc. Intl. Soc. Mag. Reson. Med 24; 3259. [7] Wundrak, S., Paul, J., Ulrici, J., Hell, E., Geibel, M.-A., Bernhardt, P., Rottbauer, W. and Rasche, V. (2016), Golden ratio sparse MRI
using tiny golden angles. Magn. Reson. Med., 75: 2372–2378. doi:10.1002/mrm.25831 [8] Uecker, M., Hohage, T., Block, K. T. and Frahm, J. (2008), Image reconstruction by regularized nonlinear
inversion—Joint estimation of coil sensitivities and image content. Magn. Reson. Med., 60: 674–682. doi:10.1002/mrm.21691 [9] Liu, J., Spincemaille, P., Codella, N. C. F., Nguyen, T. D., Prince, M.
R. and Wang, Y. (2010), Respiratory and cardiac self-gated free-breathing cardiac CINE imaging with multiecho 3D hybrid radial SSFP acquisition. Magn. Reson. Med., 63: 1230–1237. doi:10.1002/
mrm.22306 [10] Pang, J., Sharif, B., Fan, Z., Bi, X., Arsanjani, R., Berman, D. S. and Li, D. (2014), ECG and navigator-free four-dimensional whole-heart coronary MRA for simultaneous visualization
of cardiac anatomy and function. Magn. Reson. Med., 72: 1208–1217. doi:10.1002/mrm.25450 [11] Zhang, Tao, et al. "Robust self-navigated body MRI using dense coil arrays" Magn. Reson. Med. 76.1
(2016): 197-205. | {"url":"https://cds.ismrm.org/protected/18MProceedings/PDFfiles/0209.html","timestamp":"2024-11-05T07:31:13Z","content_type":"application/xhtml+xml","content_length":"13058","record_id":"<urn:uuid:15f05435-bbdf-4b7b-8b1c-3d4ec1fb4744>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00142.warc.gz"} |
Two articles from ITP in first issue ever of QST journal
We are proud that our group has two articles in the first issue ever of the new journal ‘Quantum Science and Technology’:
While controlling a quantum system is a standard task nowadays, we are still far away from developing quantum computers, and one might wonder what is the difference between the two. Qualitatively the
difference is that for quantum computing one needs to control quantum systems in a quantum way, using quantum systems instead of directly using the large apparata or (classical) electromagnetical
fields that often are enough to control a quantum system directly. In this letter we make this idea precise by building a theory which allows us to quantify the usefulness of controlling a quantum
system through a quantum system instead of using a classical one.
Cooling of atomic motion is an essential precursor for many interesting experiments and technologies, such as quantum computing and simulation using trapped atoms and ions. In most cases, this
cooling is performed using lasers to create a kind of light-induced friction force which slows the atoms down. This process is often rather wasteful, because lasers use up a huge amount of energy
relative to the tiny size of the atoms we want to cool. Here, we propose to solve this problem using a quantum absorption refrigerator: a machine that is powered only by readily available thermal
energy, such as sunlight, as it flows through the device. We describe how to build such a refrigerator, and predict that sunlight could actually be used to cool an atom to nearly absolute zero
temperature. The refrigerator works by trapping the sunlight between two mirrors, in such a way that every single photon makes a significant contribution to the friction force slowing the atom down.
Similar schemes could eventually be important for reducing the energy cost of cooling in future quantum technologies.
Ulm University
Institute of Theoretical Physics
Albert-Einstein-Allee 11
D - 89081 Ulm
Tel: +49 731 50 22911
Fax: +49 731 50 22924
Office: Building M26, room 4117
Click here if you are interested in joining the group.
Efficient Information Retrieval for Sensing via Continuous Measurement, Phys. Rev. X 13, 031012, arXiv:2209.08777
Active hyperpolarization of the nuclear spin lattice: Application to hexagonal boron nitride color centers, Phys. Rev. B 107, 214307, arXiv:2010.03334
Driving force and nonequilibrium vibronic dynamics in charge separation of strongly bound electron–hole pairs, Commun Phys 6, 65 (2023), arXiv:2205.06623
Asymptotic State Transformations of Continuous Variable Resources, Commun. Math. Phys. 398, 291–351 (2023), arXiv:2010.00044
Spin-Dependent Momentum Conservation of Electron-Phonon Scattering in Chirality-Induced Spin Selectivity, J. Phys. Chem. Lett. 2023, 14, XXX, 340–346, arXiv:2209.05323 | {"url":"https://www.uni-ulm.de/en/nawi/institut-fuer-theoretische-physik-start-page/news/two-articles-from-itp-in-first-issue-ever-of-qst-journal/","timestamp":"2024-11-09T22:42:54Z","content_type":"text/html","content_length":"47973","record_id":"<urn:uuid:c7f7b98f-948d-4e93-87a1-b3275db78c5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00451.warc.gz"} |
Determining grid square without a map
How about this for a easy method of Grid Square Calculator?
With all this talk about grids - and getting computers to
figure them etc. Here is an easy way - and you don't need
no computers or anything - well maybe a piece of paper.
Here's the deal:
How to determine your grid square without a map.
You will need your longitude and latitude to the nearest degree:
First your longitude:
1. Convert the minutes portion of the longitude from minutes to
decimal by dividing by 60.
2. For North America and locations of West longitude, subtract
your longitude from 180 degrees. For location of East
longitude, add 180 degrees.
3. Next divide this value by 20. The whole number result will be
used to determine the first character fo your grid, as follows:
0=A, 1=B, 2=C, 3=D, 4=E, 5=F, 6=G, 7=H, 8=I, 9=J, 10=K,
11=L, 12=M, 13=N, 14=O, 15=P, 16=Q, 17=R, 18=S.
4. For the third digit, multiply this last number by 10.
The digit immediately before the decimal point is the third
digit of your grid.
Now use your latitude.
1. If your latitude is north, add 90. If your latitude is South,
subtract your latitude from 90.
2. Divide this number by 10. The whole number result will be
used to determine the second character of your grid as follows:
0=A, 1=B, 2=C, 3=D, 4=E, 5=F, 6=G, 7=H, 8=I, 9=J, 10=K,
11=L, 12=M, 13=N, 14=0, 15=P, 16=Q, 17=R, 18=S.
3. Now, multiply this number by 10. The digit immediately before
before the decimal point is the fourth digit of your grid.
original posting from:
George Fremin III
Austin, Texas C.K.U. | {"url":"https://www.newsvhf.com/my-grid.html","timestamp":"2024-11-08T11:04:11Z","content_type":"text/html","content_length":"2477","record_id":"<urn:uuid:fbcaaa70-ae37-452e-b4ec-db0e697445cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00556.warc.gz"} |
Simplify Expressions, Combine Like Terms, & Order of Operations With Real Numbers
Learning Objectives
• Simplify expressions with real numbers
□ Recognize and combine like terms in an expression
□ Use the order of operations to simplify expressions
Some important terminology before we begin:
• operations/operators: In mathematics we call things like multiplication, division, addition, and subtraction operations. They are the verbs of the math world, doing work on numbers and
variables. The symbols used to denote operations are called operators, such as [latex]+{, }-{, }\times{, }\div[/latex]. As you learn more math, you will learn more operators.
• term: Examples of terms would be [latex]2x[/latex] and [latex]-\frac{3}{2}[/latex] or [latex]a^3[/latex]. Even lone integers can be a term, like 0.
• expression: A mathematical expression is one that connects terms with mathematical operators. For example [latex]\frac{1}{2}+\left(2^2\right)- 9\div\frac{6}{7}[/latex] is an expression.
Combining Like Terms
One way we can simplify expressions is to combine like terms. Like terms are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[/latex] and
[latex]-3xy[/latex] or [latex]8a^2b[/latex] and [latex]a^2b[/latex] or [latex]-3[/latex] and [latex]8[/latex]. If we have like terms we are allowed to add (or subtract) the numbers in front of the
variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the
following term like a negative term. The sign always stays with the term.
This is shown in the following examples:
Combine like terms: [latex]5x-2y-8x+7y[/latex]
Show Solution
In the following video you will be shown how to combine like terms using the idea of the distributive property. Note that this is a different method than is shown in the written examples on this
page, but it obtains the same result.
Combine like terms: [latex]x^2-3x+9-5x^2+3x-1[/latex]
Show Solution
In the video that follows, you will be shown another example of combining like terms. Pay attention to why you are not able to combine all three terms in the example.
Order of Operations
You may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road,
the order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression. The graphic below depicts the order in which
mathematical operations are performed.
Simplify [latex]7–5+3\cdot8[/latex].
Show Solution
In the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.
When you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.
Simplify [latex]3\cdot\frac{1}{3}-8\div\frac{1}{4}[/latex].
Show Solution
In the following video you are shown how to use the order of operations to simplify an expression that contains multiplication, division, and subtraction with terms that contain fractions.
When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as [latex]7^{2}[/latex] is exponential notation for
[latex]7\cdot7[/latex]. (Exponential notation has two parts: the base and the exponent or the power. In [latex]7^{2}[/latex], 7 is the base and 2 is the exponent; the exponent determines how many
times the base is multiplied by itself.)
Exponents are a way to represent repeated multiplication; the order of operations places it before any other multiplication, division, subtraction, and addition is performed.
Simplify [latex]3^{2}\cdot2^{3}[/latex].
Show Solution
In the video that follows, an expression with exponents on its terms is simplified using the order of operations.
Grouping Symbols
Grouping symbols such as parentheses ( ), brackets [ ], braces[latex] \displaystyle \left\{ {} \right\}[/latex], and fraction bars can be used to further control the order of the four arithmetic
operations. The rules of the order of operations require computation within grouping symbols to be completed first, even if you are adding or subtracting within the grouping symbols and you have
multiplication outside the grouping symbols. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right. When there are grouping
symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.
Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses—as a way to represent a group, as well as a way to express multiplication—are
Simplify [latex]\left(3+4\right)^{2}+\left(8\right)\left(4\right)[/latex].
Show Solution
Simplify [latex]4\cdot{\frac{3[5+{(2 + 3)}^2]}{2}}[/latex]
Show Solution
In the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.
Think About It
These problems are very similar to the examples given above. How are they different and what tools do you need to simplify them?
a) Simplify [latex]\left(1.5+3.5\right)–2\left(0.5\cdot6\right)^{2}[/latex]. This problem has parentheses, exponents, multiplication, subtraction, and addition in it, as well as decimals instead of
Use the box below to write down a few thoughts about how you would simplify this expression with decimals and grouping symbols.
Show Solution
b) Simplify [latex] {{\left( \frac{1}{2} \right)}^{2}}+{{\left( \frac{1}{4} \right)}^{3}}\cdot \,32[/latex].
Use the box below to write down a few thoughts about how you would simplify this expression with fractions and grouping symbols.
Show Solution | {"url":"https://courses.lumenlearning.com/aacc-collegealgebrafoundations/chapter/read-simplify-expressions-with-real-numbers/","timestamp":"2024-11-11T00:33:47Z","content_type":"text/html","content_length":"66404","record_id":"<urn:uuid:58077f3e-2a8f-412e-96dc-7340c3312557>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00032.warc.gz"} |
Chapter Test: Momentum And Its Conservation
Questions and Answers
• 1.
Which of the following statements are true about momentum? A. Momentum is a vector quantity. B. The standard unit on momentum is the Joule. C. An object with mass will have momentum. D. An object
which is moving at a constant speed has momentum.
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. A & D
Momentum is a vector quantity because it has both magnitude and direction. The standard unit for momentum is kilogram-meter per second (kg·m/s), not Joule. An object with mass will have momentum
because momentum is defined as the product of an object's mass and its velocity. An object which is moving at a constant speed has momentum because velocity is a component of momentum. Therefore,
the correct statements are A & D.
• 2.
Which of the following statements are true about momentum? A. An object can be traveling eastward and slowing down; its momentum is westward. B. Momentum is a conserved quantity; the momentum of
an object is never changed. C. The momentum of an object varies directly with the speed of the object. D. Two objects of different mass are moving at the same speed; the more massive object will
have the greatest momentum.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. C & D
C. The momentum of an object varies directly with the speed of the object. This statement is true because momentum is defined as the product of an object's mass and velocity, and velocity is
directly related to speed.
D. Two objects of different mass are moving at the same speed; the more massive object will have the greatest momentum. This statement is also true because momentum is directly proportional to
mass. Therefore, if two objects have the same speed but different masses, the one with greater mass will have greater momentum.
• 3.
Which of the following statements are true about momentum?
1. A less massive object can never have more momentum than a more massive object.
2. Two identical objects are moving in opposite directions at the same speed. The forward moving object will have the greatest momentum.
3. An object with a changing speed will have a changing momentum.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. C
An object with a changing speed will have a changing momentum. Momentum is defined as mass times velocity, so if an object's speed is changing, its velocity is changing, and therefore its
momentum is also changing. This is because velocity is a vector quantity that includes both speed and direction. As the speed changes, the magnitude of the velocity vector changes, resulting in a
change in momentum. Therefore, statement c is true.
• 4.
Which of the following are true about the relationship between momentum end energy?
a. Momentum is a form of energy.
b. If an object has momentum, then it must also have mechanical energy.
c. If an object does not have momentum, then it definitely does not have mechanical energy either.
d. Object A has more momentum than object B. Therefore, object A will also have more kinetic energy.
e. Two objects of varying mass have the same momentum. The least massive of the two objects will have the greatest kinetic energy.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. B & E
Momentum is a property of an object that depends on its mass and velocity, while energy is the ability to do work. Momentum is not a form of energy, so statement A is incorrect. If an object has
momentum, it does not necessarily mean that it has mechanical energy, so statement B is incorrect. If an object does not have momentum, it does not mean that it does not have mechanical energy,
so statement C is incorrect. Statement D is correct because an object with more momentum will also have more kinetic energy. Statement E is correct because two objects with the same momentum but
different masses will have different kinetic energies, with the less massive object having the greater kinetic energy.
• 5.
Which of the following statements are true about impulse?
a. Impulse is a force.
b. Impulse is a vector quantity.
c. An object which is traveling east would experience a westward directed impulse in a collision.
d. Objects involved in collisions encounter impulses.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. B & D
Impulse is a vector quantity because it has both magnitude and direction. It is defined as the change in momentum of an object, which is a vector quantity. Objects involved in collisions
encounter impulses because when two objects collide, their momentum changes, and this change in momentum is equal to the impulse experienced by the objects. Therefore, the correct statements
about impulse are B & D.
• 6.
Which of the following statements are true about impulse?
1. The kg•m/s is equivalent to the units on impulse.
2. An object which experiences a net impulse will definitely experience a momentum change.
3. In a collision, the net impulse experienced by an object is equal to its momentum change.
4. A force of 100 N acting for 0.1 seconds would provide an equivalent impulse as a force of 5 N acting for 2.0 seconds.
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. All of the above.
All of the statements are true about impulse. The kg•m/s is indeed equivalent to the units on impulse. An object which experiences a net impulse will definitely experience a momentum change. In a
collision, the net impulse experienced by an object is equal to its momentum change. Additionally, a force of 100 N acting for 0.1 seconds would provide an equivalent impulse as a force of 5 N
acting for 2.0 seconds. Therefore, all of the above statements are true.
• 7.
Which of the following statements are true about collisions?
a. Two colliding objects will exert equal forces upon each other even if their mass is significantly different.
b. During a collision, an object always encounters an impulse and a change in momentum.
c. During a collision, the impulse which an object experiences is equal to its velocity change.
d. The velocity change of two respective objects involved in a collision will always be equal.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. A & B
During a collision, two objects will exert equal forces upon each other even if their mass is significantly different. This is because of Newton's third law of motion, which states that for every
action, there is an equal and opposite reaction. Therefore, the force exerted by one object on the other will be equal in magnitude but opposite in direction. Additionally, during a collision, an
object will always encounter an impulse and a change in momentum. This is because a collision involves a transfer of energy and momentum between the objects involved, resulting in a change in
their velocities.
• 8.
Which of the following statements are true about collisions?
1. While individual objects may change their velocity during a collision, the overall or total velocity of the colliding objects is conserved.
2. In a collision, the two colliding objects could have different acceleration values.
3. In a collision between two objects of identical mass, the acceleration values could be different.
4. Total momentum is always conserved between any two objects involved in a collision.
Correct Answer
B. B
In a collision, the two colliding objects could have different acceleration values. This means that the objects can experience different rates of change in velocity during the collision,
resulting in different acceleration values.
• 9.
Which of the following statements are true about collisions?
1. When a moving object collides with a stationary object of identical mass, the stationary object encounters the greater collision force.
2. When a moving object collides with a stationary object of identical mass, the stationary object encounters the greater momentum change.
3. A moving object collides with a stationary object; the stationary object has significantly less mass. The stationary object encounters the greater collision force.
4. A moving object collides with a stationary object; the stationary object has significantly less mass. The stationary object encounters the greater momentum change.
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. None of the above.
None of the statements are true about collisions. In a collision between a moving object and a stationary object of identical mass, both objects experience the same collision force and momentum
change according to the law of conservation of momentum. Similarly, in a collision between a moving object and a stationary object with significantly less mass, both objects still experience the
same collision force and momentum change.
• 10.
Which of the following statements are true about elastic and inelastic collisions?
a. Perfectly elastic and perfectly inelastic collisions are the two opposite extremes along a continuum; where a particular collision lies along the continuum is dependent upon the amount
kinetic energy which is conserved by the two objects.
b. Most collisions tend to be partially to completely elastic.
c. Momentum is conserved in an elastic collision but not in an inelastic collision.
d. The kinetic energy of an object remains constant during an elastic collision.
Correct Answer
A. A
The statement "Perfectly elastic and perfectly inelastic collisions are the two opposite extremes along a continuum; where a particular collision lies along the continuum is dependent upon the
amount kinetic energy which is conserved by the two objects" explains that elastic and inelastic collisions exist on a spectrum, with perfectly elastic and perfectly inelastic collisions being
the two extremes. The position of a collision on this spectrum depends on the amount of kinetic energy conserved by the objects involved.
• 11.
Which of the following statements are true about elastic and inelastic collisions?
1. Elastic collisions occur when the collision force is a non-contact force.
2. Most collisions are not inelastic because the collision forces cause energy of motion to be transformed into sound, light and thermal energy (to name a few).
3. A ball is dropped from rest and collides with the ground. The higher that the ball rises upon collision with the ground, the more elastic that the collision is.
4. A moving air track glider collides with a second stationary glider of identical mass. The first glider loses all of its kinetic energy during the collision as the second glider is set in
motion with the same original speed as the first glider. Since the first glider lost all of its kinetic energy, this is a perfectly inelastic collision.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. A, B & C
The given answer is A, B & C.
Statement A is true because elastic collisions occur when there is no physical contact between the objects involved, and the collision force is exerted over a distance.
Statement B is true because inelastic collisions usually result in a loss of kinetic energy, which is transformed into other forms of energy such as sound, light, and thermal energy.
Statement C is true because the height to which a ball rises after colliding with the ground is a measure of how much kinetic energy is conserved during the collision. A higher bounce indicates a
more elastic collision.
Therefore, the correct answer is A, B & C.
• 12.
Which of the following objects have momentum? Include all that apply. a. An electron is orbiting the nucleus of an atom. b. A UPS truck is stopped in front of the school building. c. A Yugo (a
compact car) is moving with a constant speed. d. A small flea walking with constant speed across Fido's back. e. The high school building rests in the middle of town.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. A,C & D
Objects that have momentum are those that are in motion. In this question, options a, c, and d describe objects that are in motion. An electron orbiting the nucleus of an atom is constantly
moving, so it has momentum. A Yugo moving with a constant speed also has momentum. Similarly, a small flea walking with constant speed across Fido's back is also in motion and has momentum.
Therefore, options a, c, and d are the correct answers.
• 13.
A truck driving along a highway road has a large quantity of momentum. If it moves at the same speed but has twice as much mass, its momentum is ________________.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Doubled
When an object has momentum, it means it has mass and is moving. Momentum is calculated by multiplying an object's mass by its velocity. In this question, the truck is moving at the same speed
but has twice as much mass. Since momentum is directly proportional to mass, if the mass doubles, the momentum will also double. Therefore, the correct answer is "doubled".
• 14.
Consider a karate expert. During a talent show, she executes a swift blow to a cement block and breaks it with her bare hand. During the collision between her hand and the block, the ___.
□ A.
Time of impact on both the block and the expert's hand is the same
□ B.
Force on both the block and the expert's hand have the same magnitude
□ C.
Impulse on both the block and the expert's hand have the same magnitude
□ D.
Correct Answer
D. All of the above.
During the collision between the karate expert's hand and the cement block, the time of impact on both the block and the expert's hand is the same. This means that the duration of the collision
is equal for both the block and the hand. Additionally, the force on both the block and the expert's hand have the same magnitude. This implies that the impact of the blow is equally distributed
between the block and the hand. Finally, the impulse on both the block and the expert's hand have the same magnitude. Impulse is the change in momentum, and since momentum is conserved in the
collision, the impulse experienced by both the block and the hand is the same. Therefore, all of the above statements are true.
• 15.
It is NOT possible for a rocket to accelerate in outer space because ____. List all that apply.
□ A.
□ B.
There is no friction in space
□ C.
There is no gravity in outer space
□ D.
... nonsense! Rockets do accelerate in outer space.
Correct Answer
D. ... nonsense! Rockets do accelerate in outer space.
Rockets can accelerate in outer space because there is no air resistance or friction to oppose their motion. Additionally, there is still gravity in outer space, although it is weaker compared to
Earth's gravity. Therefore, the statement "Rockets do accelerate in outer space" is correct.
• 16.
In order to catch a ball, a baseball player naturally moves his or her hand backward in the direction of the ball's motion once the ball contacts the hand. This habit causes the force of impact
on the players hand to be reduced in size principally because ___.
□ A.
The resulting impact velocity is lessened
□ B.
The momentum change is decreased
□ C.
The time of impact is increased
□ D.
Correct Answer
C. The time of impact is increased
When a baseball player moves their hand backward in the direction of the ball's motion, it increases the time of impact. By increasing the time of impact, the force of impact on the player's hand
is reduced. This is because a longer duration of impact allows for a slower change in momentum, decreasing the force experienced by the hand. Therefore, the correct answer is that the time of
impact is increased.
• 17.
Suppose that Paul D. Trigger fires a bullet from a gun. The speed of the bullet leaving the muzzle will be the same as the speed of the recoiling gun ____.
□ A.
Because momentum is conserved
□ B.
Because velocity is conserved
□ C.
Only if the mass of the bullet equals the mass of the gun
□ D.
Correct Answer
C. Only if the mass of the bullet equals the mass of the gun
The correct answer is "only if the mass of the bullet equals the mass of the gun". This is because according to the law of conservation of momentum, the total momentum before and after an event
must be the same. When the bullet is fired, it exerts a force on the gun causing it to recoil. The momentum of the bullet and the gun must be equal and opposite in order to conserve momentum.
Therefore, if the mass of the bullet equals the mass of the gun, the speed of the bullet leaving the muzzle will be the same as the speed of the recoiling gun.
• 18.
Suppose that you're driving down the highway and a moth crashes into the windshield of your car. Which undergoes the greater change is momentum?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Both the same
Both the moth and the car undergo the same change in momentum. According to Newton's third law of motion, for every action, there is an equal and opposite reaction. When the moth crashes into the
windshield, it exerts a force on the car, causing it to experience a change in momentum. At the same time, the car also exerts an equal and opposite force on the moth, causing it to experience a
change in momentum. Therefore, both the moth and the car undergo the same change in momentum.
• 19.
Suppose that you're driving down the highway and a moth crashes into the windshield of your car. Which undergoes the greater force?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Both the same
When a moth crashes into the windshield of a car, both the moth and the car experience the same force. According to Newton's third law of motion, for every action, there is an equal and opposite
reaction. In this case, the force exerted by the car on the moth is equal in magnitude but opposite in direction to the force exerted by the moth on the car. Therefore, both the moth and the car
undergo the same force.
• 20.
Suppose that you're driving down the highway and a moth crashes into the windshield of your car. Which undergoes the greater impulse?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Both the same
Both the moth and the car undergo the same impulse. Impulse is the change in momentum, which is equal to the force applied multiplied by the time it acts for. When the moth crashes into the
windshield, it exerts a force on the car, causing it to experience a change in momentum. At the same time, the car exerts an equal and opposite force on the moth, causing it to also experience a
change in momentum. Since the forces and times are equal and opposite, the impulses experienced by both the moth and the car are the same.
• 21.
Suppose that you're driving down the highway and a moth crashes into the windshield of your car. Which undergoes the greater acceleration?
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. the moth
The moth undergoes the greater acceleration. This is because the force exerted on the moth by the windshield is much greater than the force exerted on the car by the moth. As a result, the moth
experiences a larger change in velocity in a shorter amount of time, leading to a greater acceleration.
• 22.
Three boxes, X, Y, and Z, are at rest on a table as shown in the diagram at the right. The weight of each box is indicated in the diagram. The net or unbalanced force acting on box Y is _____.
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Zero
The net or unbalanced force acting on box Y is zero because the forces acting on it are balanced. The weight of box Y is 5 N down, but there is an equal and opposite force of 5 N up from the
table supporting it. Therefore, the net force on box Y is zero.
• 23.
In a physics experiment, two equal-mass carts roll towards each other on a level, low-friction track. One cart rolls rightward at 2 m/s and the other cart rolls leftward at 1 m/s. After the carts
collide, they couple (attach together) and roll together with a speed of _____________. Ignore resistive forces.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 0.5 m/s
When the two carts collide, they stick together and form a system with a combined mass. According to the law of conservation of momentum, the total momentum before the collision is equal to the
total momentum after the collision. The momentum of an object is equal to its mass multiplied by its velocity.
Before the collision, the momentum of the first cart is 2 kg*m/s (mass of 1 kg multiplied by velocity of 2 m/s) and the momentum of the second cart is -1 kg*m/s (mass of 1 kg multiplied by
velocity of -1 m/s). The negative sign indicates the opposite direction of motion.
After the collision, the carts stick together and have a combined mass of 2 kg. Let's assume their final velocity is v m/s. The total momentum after the collision is then equal to 2 kg * v m/s.
Setting the total momentum before the collision equal to the total momentum after the collision, we have:
2 kg*m/s + (-1 kg*m/s) = 2 kg * v m/s
1 kg*m/s = 2 kg * v m/s
v = 0.5 m/s
Therefore, the correct answer is 0.5 m/s.
• 24.
A physics cart rolls along a low-friction track with considerable momentum. If it rolls at the same speed but has twice as much mass, its momentum is ____.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Twice as large
When an object has more mass, its momentum increases. Momentum is directly proportional to mass, so if the mass of the cart is doubled while its speed remains the same, its momentum will also
double. Therefore, the correct answer is "twice as large".
• 25.
The firing of a bullet by a rifle causes the rifle to recoil backwards. The speed of the rifle's recoil is smaller than the bullet's forward speed because the ___.
□ A.
Force against the rifle is relatively small
□ B.
Speed is mainly concentrated in the bullet
□ C.
□ D.
Correct Answer
C. Rifle has lots of mass
The correct answer is that the rifle has lots of mass. According to Newton's third law of motion, for every action, there is an equal and opposite reaction. When the bullet is fired forward with
a high speed, it exerts a force on the rifle in the opposite direction, causing the rifle to recoil backwards. The speed of the rifle's recoil is smaller because the rifle has a larger mass
compared to the bullet. This means that the rifle requires more force to accelerate, resulting in a slower recoil speed.
• 26.
Two objects, A and B, have the same size and shape. Object A is twice as massive as B. The objects are simultaneously dropped from a high window on a tall building. (Neglect the effect air
resistance.) The objects will reach the ground at the same time but object A will have a greater ___. Choose all that apply.
□ A.
□ B.
□ C.
□ D.
None of the above quantities will be greater
Correct Answer
C. Momentum
The objects will reach the ground at the same time because their size and shape are the same. However, object A is twice as massive as object B, meaning it has more mass. Momentum is defined as
the product of an object's mass and velocity, so object A will have a greater momentum due to its greater mass. Speed and acceleration are not affected by mass, so they will be the same for both
objects. Therefore, the correct answer is momentum.
• 27.
Cars are equipped with padded dashboards. In collisions, the padded dashboards would be safer than non-padded ones because they ____. List all that apply.
□ A.
□ B.
Decrease the impact force
□ C.
□ D.
Correct Answer
C. Both A and B
Padded dashboards increase the impact time during collisions, which helps to reduce the force exerted on the occupants. This is because the padding absorbs some of the energy from the impact,
spreading it out over a longer period of time. As a result, both the impact time and force are decreased, making padded dashboards safer than non-padded ones.
• 28.
A 4 kg object has a momentum of 12 kg•m/s. The object's speed is ___ m/s.
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. 3
The momentum of an object is calculated by multiplying its mass by its velocity. In this case, the momentum is given as 12 kg•m/s and the mass is given as 4 kg. To find the speed, we divide the
momentum by the mass. Therefore, the speed of the object is 12 kg•m/s divided by 4 kg, which equals 3 m/s.
• 29.
A wad of chewed bubble gum is moving with 1 unit of momentum when it collides with a heavy box that is initially at rest. The gum sticks to the box and both are set in motion with a combined
momentum that is ___.
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. 1 unit
When the wad of chewed bubble gum collides with the heavy box and sticks to it, the momentum of the gum is transferred to the box. Since the gum has a momentum of 1 unit, the combined momentum of
the gum and the box after the collision will also be 1 unit. Therefore, the correct answer is 1 unit.
• 30.
A relatively large force acting for a relatively long amount of time on a relatively small mass will produce a relatively ______. List all that apply.
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. large velocity change
When a relatively large force acts for a relatively long amount of time on a relatively small mass, it will result in a large velocity change. This is because the force will accelerate the mass,
causing it to change its velocity significantly. The larger the force and the longer it acts, the greater the velocity change will be. Therefore, a large velocity change is the expected outcome
in this scenario.
• 31.
Consider the concepts of work and energy (presuming you have already studied it) and those of impuse and momentum. Force and time is related to momentum change in the same manner as force and
displacement pertains to ___________.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Energy change
Force and displacement are related to work in the same manner as force and time are related to impulse. Work is defined as the product of force and displacement, while impulse is defined as the
product of force and time. Similarly, force and time are related to momentum change, while force and displacement are related to energy change. Therefore, the correct answer is energy change.
• 32.
A 5-N force is applied to a 3-kg ball to change its velocity from +9 m/s to +3 m/s. This impulse causes the momentum change of the ball to be ____ kg•m/s.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. -18
When a force is applied to an object, it causes a change in its momentum. The impulse experienced by the object is equal to the change in momentum. In this case, the initial momentum of the ball
is calculated by multiplying its mass (3 kg) by its initial velocity (+9 m/s), resulting in a momentum of 27 kg·m/s. The final momentum is calculated by multiplying the mass (3 kg) by the final
velocity (+3 m/s), resulting in a momentum of 9 kg·m/s. The change in momentum is then determined by subtracting the final momentum from the initial momentum, resulting in a change of -18 kg·m/s.
Therefore, the correct answer is -18.
• 33.
A 5-N force is applied to a 3-kg ball to change its velocity from +9 m/s to +3 m/s. The impulse experienced by the ball is ____ N•s.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. -45
The impulse experienced by an object can be calculated using the equation Impulse = Force * Change in Time. In this question, the force applied to the ball is 5 N. The change in velocity is the
final velocity (3 m/s) minus the initial velocity (9 m/s), which is -6 m/s. Since impulse is equal to force multiplied by change in time, and change in time is equal to mass multiplied by change
in velocity, we can use the equation Impulse = Force * Change in Time = Force * (Mass * Change in Velocity). Plugging in the values, we get Impulse = 5 N * (3 kg * -6 m/s) = -45 N·s. Therefore,
the correct answer is -45.
• 34.
A 5-N force is applied to a 3-kg ball to change its velocity from +9 m/s to +3 m/s. The impulse is encountered by the ball for a time of ____ seconds.
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. 3.6
The impulse experienced by an object is equal to the change in momentum of the object. The momentum of an object is calculated by multiplying its mass by its velocity. In this case, the initial
momentum of the ball is 3 kg * 9 m/s = 27 kg*m/s, and the final momentum is 3 kg * 3 m/s = 9 kg*m/s. The change in momentum is therefore 27 kg*m/s - 9 kg*m/s = 18 kg*m/s. The impulse is also
equal to the force applied to the object multiplied by the time it is applied. In this case, the force is 5 N. Therefore, the time can be calculated by dividing the impulse by the force: 18 kg*m/
s / 5 N = 3.6 s. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=chapter-test-momentum-collisions","timestamp":"2024-11-15T01:43:30Z","content_type":"text/html","content_length":"587115","record_id":"<urn:uuid:1134f3c9-00ed-45fe-8c2e-a89bd0a554dc>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00083.warc.gz"} |
semivar defines a semi-continuous variable
x = semivar(n)
x = semivar(n,m)
x = semivar(n,m,'type')
x = semivar(n,m,'type','field')
YALMIP defines a semi-continuous variable as a variable taking either the value 0, or any value between an upper and lower bound. In contrast to the definitions used in most mixed-integer solver,
YALMIP allows negative variables, and will reformulate the model accordingly if required.
The following code defines a least squares problem with a constraint that all variables are either 0, or between 0.1 and 0.3.
A = randn(20,10);
b = randn(20,1);
x = semivar(10,1);
e = b-A*x;
F = [0.1 <= x <= 0.3];
Note that we have defined constraints which cuts away 0 from the feasible set. However, when the variable is defined with semivar, YALMIP understands that the simple bounds relate to the
semi-continuous nature of the variable.
semivar requires that the solver used supports semi-continuous variables. | {"url":"https://yalmip.github.io/command/semivar/","timestamp":"2024-11-09T17:37:37Z","content_type":"text/html","content_length":"30949","record_id":"<urn:uuid:03ffcb75-e68d-4e48-9d1e-864959033b95>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00622.warc.gz"} |
How To Solve Three Variable Equations With Matrices Calculator - Tessshebaylo
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This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.tessshebaylo.com/how-to-solve-three-variable-equations-with-matrices-calculator/","timestamp":"2024-11-12T17:07:11Z","content_type":"text/html","content_length":"58962","record_id":"<urn:uuid:a0080a0b-9785-45bd-a0d0-aa4c76c89cc5>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00661.warc.gz"} |
Big O Notation - A Comprehensive Guide
Table of Contents
What is Big O Notation?
Big O notation is a mathematical way to express how a function behaves as its input approaches a specific value or infinity. It's part of a family of notations, including Bachmann–Landau notation,
used to describe such behaviors. It was invented by German Mathematician Paul Bachmann.
In Summary, Big O Notation is just an algebraic expression which describes your code.
Chart of Big O Notation
Why Big O Notation?
Big O notation is a mathematical tool that helps us measure the efficiency of algorithms in computer science. It tells us how the running time or the memory usage of an algorithm changes as the input
size grows. For example, an algorithm that has a linear running time, such as finding the maximum element in an array, can be expressed as O(n), where n is the size of the input. This means that the
algorithm takes n steps to complete, and if we double the input size, we also double the running time.
Big O notation is important for computer scientists because it allows them to compare different algorithms and choose the best one for a given problem. It also helps them design algorithms that can
handle large and complex inputs without compromising performance or scalability. By using Big O notation, computer scientists can abstract away the details of the hardware and programming language,
and focus on the essential features of the algorithm.
O(1) - Constant Time Complexity
O(1) represents constant time complexity. O(1) is characterized by the following key attributes:
1. Constant Time: An algorithm or operation is said to have O(1) time complexity if its execution time or resource usage remains constant, regardless of the size or input of the data it processes.
In other words, the time it takes to perform the operation does not depend on the size of the input.
2. Predictable Performance: Algorithms with O(1) complexity are highly predictable and consistent. Whether you're working with a small dataset or a large one, the time it takes to complete the
operation is the same.
3. Fast Execution: O(1) operations are extremely efficient and fast because they require a fixed, usually very small, amount of time to complete. These operations are ideal for scenarios where speed
and efficiency are critical.
4. Examples: Common examples of operations with O(1) complexity include accessing an element in an array by its index, looking up a value in a hash table (assuming minimal collisions), or performing
basic arithmetic operations like addition or subtraction.
5. Independent of Input Size: O(1) operations do not scale with input size, which makes them particularly useful for tasks that involve a single action or accessing specific elements within a data
6. Not Affected by Constants: Big O notation, including O(1), disregards constant factors and lower-order terms. This means that an algorithm with O(1) complexity is still considered O(1) even if it
has a small constant overhead because the constant factors are not significant when analyzing the algorithm's scalability.
7. Optimal Complexity: O(1) represents the best possible time complexity, as it implies that the algorithm's performance is not affected by the size of the input data. It's the most efficient time
complexity one can achieve for an algorithm.
O(log n) - Logarithmic Time Complexity
O(log n) represents logarithmic time complexity, which is one of the most efficient complexities in algorithm analysis.
Here are the key characteristics of O(log n):
1. Logarithmic Growth: O(log n) indicates that the running time of an algorithm grows logarithmically with the size of the input (n). This means that as the input size increases, the time taken by
the algorithm increases, but it does so at a much slower rate compared to linear or polynomial time complexities.
2. Efficient Scaling: Logarithmic time complexity is highly efficient, especially for large inputs. This makes it suitable for tasks that involve searching, sorting, or dividing the input into
smaller portions.
3. Example Algorithms: Algorithms with O(log n) complexity are often found in binary search algorithms. In a binary search, the input data is repeatedly divided into halves, significantly reducing
the search space with each iteration. This results in a time complexity of O(log n) because the number of iterations required grows logarithmically with the size of the input.
4. Performance: Logarithmic time algorithms are highly performant, making them suitable for applications where efficiency is critical. They are commonly used in data structures like balanced binary
search trees (e.g., AVL trees) and certain divide-and-conquer algorithms.
5. Scalability: O(log n) is efficient even for large datasets. As the input size grows, the increase in time required is minimal compared to algorithms with higher complexities like O(n) or O(n^2).
6. Graphical Representation: When you plot the performance of an O(log n) algorithm on a graph with input size on the x-axis and time on the y-axis, you will see a curve that rises slowly as the
input size increases, indicating the logarithmic growth.
O(n log n) - Quasilinear Time Complexity
This complexity class signifies that the algorithm's execution time increases in a near-linear fashion with the input size but with a logarithmic factor.
Characteristics of O(n log n) complexity:
1. Intermediate Growth: Algorithms with O(n log n) complexity fall between linear (O(n)) and quadratic (O(n^2)) complexities in terms of growth rate. This means they are more efficient than
quadratic algorithms but less efficient than linear ones.
2. Common Algorithms: O(n log n) complexity is often encountered in sorting and searching algorithms. Prominent examples include Merge Sort, Quick Sort, and some binary tree operations.
3. Divide and Conquer: Many algorithms that achieve O(n log n) complexity use a divide-and-conquer approach. They break the problem into smaller subproblems, solve them recursively, and then combine
the results efficiently.
4. Efficiency: Algorithms with O(n log n) complexity are considered quite efficient and are often used for large datasets when compared to quadratic algorithms, which become impractical as the input
size grows.
5. Examples: When sorting a list of items, algorithms with O(n log n) complexity, like Merge Sort, typically perform much better than algorithms with O(n^2) complexity, such as Bubble Sort or
Insertion Sort, for larger datasets.
6. Non-linear Growth: The logarithmic factor in O(n log n) means that as the input size grows, the increase in execution time is much slower than linear growth (O(n)), making these algorithms
suitable for handling substantial amounts of data efficiently.
O(n) - Linear Time Complexity
O(n) represents a class of time complexity that is linear with respect to the size of the input data. In other words, it signifies that the time required for an algorithm to complete its task grows
linearly or proportionally with the size of the input.
Characteristics of O(n) complexity include:
1. Linear Growth: As the input size (n) increases, the time or resources required by the algorithm also increases linearly. If you double the size of the input, the algorithm will roughly take twice
as much time to complete.
2. Constant Increment: For each additional element or data point in the input, the algorithm typically performs a constant amount of work. This constant work can include basic operations like
additions, comparisons, or assignments.
3. Straightforward Algorithms: Many common algorithms, such as simple iteration through an array or list, exhibit O(n) complexity. In these algorithms, every element in the input data is examined or
processed exactly once.
4. Scalability: Algorithms with O(n) complexity are generally considered efficient and scalable for moderate-sized datasets. They can handle larger inputs without a significant increase in execution
time, making them suitable for many practical applications.
5. Examples: Examples of algorithms with O(n) complexity include linear search, where you look for a specific element in an array by examining each element in sequence, and counting the number of
elements in a list or array.
O(n^2) - Quadratic Time Complexity
O(n^2) is a notation used in computer science to describe the time complexity of an algorithm or the upper bound of the number of operations an algorithm performs in relation to the size of its input
data. Specifically, O(n^2) indicates a quadratic time complexity, which means that as the input size (n) grows, the number of operations the algorithm performs increases quadratically, or as a square
of the input size.
Characteristics of O(n^2) (Quadratic Time Complexity):
1. Performance Scaling: As the input size (n) increases, the time taken by the algorithm grows significantly. For each additional element in the input, the number of operations increases by a factor
of n^2.
2. Nested Loops: Quadratic time complexity is often associated with nested loops, where one loop runs from 0 to n, and another nested loop also runs from 0 to n or some factor of n. This results in
n * n iterations, leading to a quadratic relationship.
3. Common Examples: Many sorting algorithms like the Bubble Sort and Selection Sort exhibit O(n^2) time complexity when implemented in their simplest forms. These algorithms involve comparing and
swapping elements in nested loops.
4. Inefficient for Large Inputs: Algorithms with O(n^2) complexity can become inefficient for large datasets. The time it takes to process data can quickly become impractical as the input size
grows, making these algorithms less suitable for big data applications.
5. Not Ideal for Optimization: Quadratic time complexity is generally considered less efficient than linear (O(n)), quasilinear (O(n log n)), or even polynomial time complexities (O(n^k)) for most
practical applications. Therefore, it is often desirable to optimize algorithms to reduce their time complexity to improve performance.
6. Examples: Calculating the pairwise combinations of elements in a list, checking for duplicates in a nested list, and certain types of matrix operations can result in algorithms with O(n^2) time
Best, Average and Worst Cases
Exploring the concept of best, average, and worst-case scenarios is essential in analyzing and understanding the behavior and performance of algorithms, particularly when using Big O Notation. These
scenarios help us assess how an algorithm performs under different conditions and inputs. Let's delve into each scenario:
1. Best-case Scenario:
□ Definition: The best-case scenario represents the most favorable conditions for an algorithm. It is the situation in which the algorithm performs the fewest number of operations or runs the
□ Characteristics: In the best-case scenario, the input data is specifically chosen or structured to minimize the workload on the algorithm. This often involves input data that is already
sorted or in a format that requires minimal processing.
□ Notation: In Big O Notation, the best-case scenario is denoted as O(f(n)), where f(n) represents the lowest possible time complexity for a given input size n.
□ Example: For an efficient sorting algorithm like Merge Sort, the best-case scenario occurs when the input data is already sorted, requiring fewer comparisons and swaps.
2. Average-case Scenario:
□ Definition: The average-case scenario represents the expected or typical performance of an algorithm when given random or real-world inputs. It provides a more realistic assessment of an
algorithm's efficiency than the best or worst-case scenarios.
□ Characteristics: In this scenario, the algorithm is analyzed with inputs that represent the distribution of data it is likely to encounter during normal operation. This involves considering
the average behavior over a range of possible inputs.
□ Notation: The average-case time complexity is denoted as O(g(n)), where g(n) represents the expected or average time complexity for a given input size n.
□ Example: For a quicksort algorithm, the average-case scenario assumes that the pivot selection strategy results in roughly equal-sized partitions, leading to an O(n log n) time complexity on
3. Worst-case Scenario:
□ Definition: The worst-case scenario represents the most unfavorable conditions for an algorithm. It is the situation in which the algorithm performs the maximum number of operations or runs
the slowest.
□ Characteristics: In the worst-case scenario, the input data is chosen or structured in a way that maximizes the algorithm's workload. This often involves input data that is sorted in reverse
order or contains elements that require extensive processing.
□ Notation: The worst-case time complexity is denoted as O(h(n)), where h(n) represents the highest possible time complexity for a given input size n.
□ Example: In the worst-case scenario for many sorting algorithms, such as Bubble Sort, the input data is in reverse order, resulting in the maximum number of comparisons and swaps.
Understanding these scenarios helps in making informed decisions about algorithm selection and design. While best-case scenarios can be useful for specific optimizations, it is often the average and
worst-case scenarios that provide a more complete picture of an algorithm's behavior in practical applications. Big O Notation allows us to express these scenarios succinctly and compare different
algorithms in terms of their efficiency across various input conditions.
Relation with Big O Notation
Here's how Big O Notation relates to each of these scenarios:
1. Best-case Scenario:
□ In the context of Big O Notation, the best-case scenario represents the lower bound or the most optimistic estimation of an algorithm's performance for a given input.
□ Big O Notation is used to express the best-case time complexity by providing a notation (e.g., O(f(n))) that represents the minimum number of operations an algorithm will perform for a
specific input size.
□ The best-case time complexity can be used to describe how efficiently an algorithm performs under ideal conditions, and it can serve as a lower limit for performance.
2. Average-case Scenario:
□ In the average-case scenario, Big O Notation is used to express the expected or typical performance of an algorithm when given random or real-world inputs.
□ The notation (e.g., O(g(n))) used to describe average-case complexity represents the average number of operations an algorithm is expected to perform for a given input size.
□ Average-case analysis often involves probabilistic considerations and statistical techniques to estimate the expected behavior of an algorithm across a range of inputs.
3. Worst-case Scenario:
□ The worst-case scenario, as related to Big O Notation, represents the upper bound or the most pessimistic estimation of an algorithm's performance for a given input.
□ Big O Notation is used to express the worst-case time complexity by providing a notation (e.g., O(h(n))) that represents the maximum number of operations an algorithm may perform for a
specific input size.
□ The worst-case time complexity serves as an upper limit for performance and is crucial for ensuring that an algorithm doesn't perform poorly in critical situations.
Introduction to Space Complexity
Space complexity is a term used in computer science to describe the amount of memory or space that an algorithm's execution requires in relation to the size of its input data. It measures how the
memory usage of an algorithm scales as the input size increases. Space complexity is essential for understanding and optimizing the memory requirements of algorithms, particularly when dealing with
large datasets or resource-constrained environments.
Space complexity is typically expressed using Big O Notation, similar to time complexity, and it is denoted as O(f(n)), where f(n) represents the upper bound on the additional memory used by the
algorithm as a function of the input size n.
There are several common scenarios for space complexity:
1. Constant Space Complexity (O(1)):
□ Algorithms with constant space complexity use a fixed and limited amount of memory regardless of the input size. They do not allocate memory that scales with the size of the input.
□ Examples include simple mathematical operations and algorithms that maintain a fixed number of variables.
2. Linear Space Complexity (O(n)):
□ Algorithms with linear space complexity use memory that scales linearly with the size of the input. In other words, for each additional element in the input, a fixed amount of additional
memory is used.
□ Examples include algorithms that create arrays or data structures to store input elements.
3. Logarithmic Space Complexity (O(log n)):
□ Algorithms with logarithmic space complexity use a memory footprint that grows logarithmically with the input size.
□ This is often seen in divide-and-conquer algorithms that partition the data and work on smaller subsets.
4. Polynomial Space Complexity (O(n^k)):
□ Algorithms with polynomial space complexity use memory that scales as a polynomial function of the input size. The exponent k represents the degree of the polynomial.
□ Higher-degree polynomials, such as O(n^2) or O(n^3), indicate algorithms that consume increasingly more memory as the input size grows.
5. Exponential Space Complexity (O(2^n)):
□ Algorithms with exponential space complexity use memory that grows exponentially with the input size.
□ This is often associated with recursive algorithms that create multiple branches of computation, each requiring additional memory.
Relation between Time and Space Complexity
Space complexity and time complexity are two fundamental aspects of algorithm analysis, and they are closely related in the context of algorithm performance and resource utilization. Here's how they
relate to each other:
1. Trade-offs:
□ Algorithms often exhibit a trade-off between time complexity and space complexity. In some cases, optimizing for time complexity may result in increased space usage, and vice versa.
□ For example, caching and storing intermediate results to speed up computations can reduce time complexity but increase space complexity. On the other hand, algorithms that minimize space
usage may require more computational steps, leading to higher time complexity.
2. Resource Constraints:
□ The choice between optimizing for time or space complexity depends on the specific requirements and constraints of a problem or computing environment.
□ In memory-constrained systems, minimizing space complexity may be a top priority, even if it means accepting a higher time complexity.
□ Conversely, in situations where execution time is critical, you might accept higher space complexity to achieve faster execution.
3. Big O Notation:
□ Both time complexity and space complexity are expressed using Big O Notation, which provides a standardized way to quantify and compare algorithm performance.
□ In Big O Notation, the time and space complexities are often analyzed separately, but they are interrelated. An algorithm may have different Big O expressions for time and space complexity.
4. Algorithm Design:
□ Algorithm designers must consider the interplay between time and space complexity when making design decisions.
□ Design choices, such as data structures and algorithms, can significantly impact both time and space requirements. For example, using a more memory-efficient data structure may increase the
time complexity of certain operations.
5. Optimization Strategies:
□ Algorithm optimization often involves finding a balance between time and space complexity. This may entail trade-offs, such as precomputing results to save time or minimizing data duplication
to save space.
□ Profiling and benchmarking can help determine the most suitable trade-offs based on the specific use case.
6. Real-world Examples:
□ Consider sorting algorithms: Quick Sort has an average-case time complexity of O(n log n) but may have higher space complexity due to recursion, while Merge Sort also has O(n log n) time
complexity but uses additional memory for merging.
□ In contrast, Insertion Sort may have lower space complexity but higher time complexity (O(n^2)) in some cases.
In conclusion, understanding algorithm complexity, both in terms of time complexity and space complexity, is fundamental to computer science and algorithm design. These complexities help us evaluate
how algorithms perform and scale in various scenarios, making them invaluable tools in the field of computing. Here are the key takeaways from our discussions:
1. Time Complexity:
□ Time complexity measures the amount of time an algorithm takes to execute in relation to the size of its input.
□ It is expressed using Big O Notation, providing an upper bound on the number of operations an algorithm performs.
□ Algorithms can have best-case, average-case, and worst-case time complexities, each revealing different performance scenarios.
2. Space Complexity:
□ Space complexity measures the amount of memory an algorithm requires in relation to the size of its input.
□ It is also expressed using Big O Notation, denoting the upper bound on the additional memory used.
□ Space complexity plays a crucial role in optimizing memory usage, particularly in resource-constrained environments.
3. Relationship Between Time and Space Complexity:
□ Algorithms often exhibit trade-offs between time and space complexity, requiring designers to find a balance based on specific constraints and requirements.
□ Optimization strategies may involve choosing data structures and algorithms that strike the right balance between these two aspects.
4. Best, Average, and Worst-Case Scenarios:
□ Analyzing algorithms in these scenarios provides a comprehensive understanding of their behavior under different conditions.
□ Big O Notation helps express and compare these scenarios objectively, aiding in algorithm selection and design.
5. Real-world Application:
□ The concepts of time and space complexity are essential in practical algorithm development, impacting the performance and resource efficiency of software applications.
□ Profiling and benchmarking are common techniques used to assess and optimize algorithm performance in real-world scenarios.
In computer science, the goal is often to find algorithms that strike the right balance between time and space complexity, delivering efficient and effective solutions for a wide range of problem
sizes and computing environments. By mastering these concepts and their relationship, software engineers and developers can make informed decisions, design efficient algorithms, and address the
challenges posed by both small-scale and large-scale computational problems.
Additional Resources
Here are some additional resources where you can learn more about algorithm complexity, Big O Notation, and related topics:
Online Courses and Tutorials:
1. Coursera Algorithms Specialization: A comprehensive series of courses offered by top universities, covering a wide range of algorithmic topics, including time and space complexity analysis.
2. Khan Academy Algorithms Course: A beginner-friendly course on algorithms, including discussions on Big O Notation and complexity analysis.
1. "Introduction to Algorithms" by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein: A widely used textbook that covers algorithm design, analysis, and complexity theory.
2. "Algorithms" by Robert Sedgewick and Kevin Wayne: This book offers a practical approach to understanding algorithms and includes discussions on algorithm analysis.
Websites and Online Resources:
1. GeeksforGeeks: An extensive resource for computer science topics, including articles and tutorials on algorithms, data structures, and Big O Notation.
2. Big O Cheat Sheet: A concise reference for common time and space complexities and their corresponding Big O Notation expressions.
Interactive Tools:
1. Visualgo: An interactive platform that visually demonstrates algorithms and data structures, helping you understand their behavior.
2. Big O Calculator: Online tools that allow you to calculate and compare the time complexities of different algorithms.
These resources should provide you with a solid foundation in algorithm analysis, complexity theory, and the practical application of these concepts. Whether you're a beginner or looking to deepen
your understanding, these materials will be valuable in your journey to mastering algorithms and data structures.
Top comments (1)
Volodyslav •
The best explanation I've ever read😁
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/easewithtuts/big-o-notation-a-comprehensive-guide-253j","timestamp":"2024-11-11T11:26:28Z","content_type":"text/html","content_length":"103514","record_id":"<urn:uuid:f80f2ca1-9556-4fb6-afff-253de84b844c>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00576.warc.gz"} |
Mixed Number to Decimal Calculator
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The Mixed Number to Decimal Calculator is a web-based tool designed to assist users in converting mixed numbers and decimals, offering a convenient and user-friendly interface. This tool combines
mathematical concepts, input validation techniques, and responsive design to provide a seamless user experience.
Introduction to Mixed Numbers and Decimals
Mixed Numbers
A mixed number consists of a whole number combined with a fraction. It is represented in the form “a b/c,” where “a” is the whole number, and “b/c” is the fractional part. For example, 2 1/2 is a
mixed number, denoting 2 whole units and 1/2 of another unit.
Decimals are a numeric representation of fractions, easier to work with in mathematical calculations. A decimal can be a whole number or a combination of a whole number and a fractional part, such as
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The Mixed Number to Decimal Calculator stands as a versatile and user-friendly tool, catering to a broad audience with its ability to handle mixed numbers, decimals, and their conversions. With a
focus on security, precision, and user experience, it serves as a valuable resource for both educational and practical purposes.
As users continue to leverage this tool, ongoing refinements and enhancements may further solidify its place as a reliable and efficient calculator for mixed numbers and decimals.
Last Updated : 03 October, 2024
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The mixed number to decimal calculator is a valuable resource with its focus on security, precision, and user experience. It caters to both educational and practical needs. | {"url":"https://calculatoruniverse.com/mixed-number-to-decimal-calculator/","timestamp":"2024-11-01T19:10:18Z","content_type":"text/html","content_length":"241345","record_id":"<urn:uuid:8cdad2dd-4a3c-4460-8482-494b369f3fda>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00676.warc.gz"} |
onstraints for
Set up group ratio constraints for portfolio weights
obj = setGroupRatio(___,UpperRatio) sets up group ratio constraints for portfolio weights for portfolio objects with an additional optional argument for UpperRatio.
Given base and comparison group matrices GroupA and GroupB and LowerRatio or UpperRatio bounds, group ratio constraints require any portfolio in Port to satisfy the following:
(GroupB * Port) .* LowerRatio <= GroupA * Port <= (GroupB * Port) .* UpperRatio
This collection of constraints usually requires that portfolio weights be nonnegative and that the products GroupA * Port and GroupB * Port are always nonnegative. Although negative portfolio weights
and non-Boolean group ratio matrices are supported, use with caution.
Set Group Ratio Constraints for a Portfolio Object
Suppose you want to ensure that the ratio of financial to nonfinancial companies in your portfolio never exceeds 50%. Assume you have six assets with three financial companies (assets 1-3) and three
nonfinancial companies (assets 4-6). Group ratio constraints can be set with:
GA = [ true true true false false false ]; % financial companies
GB = [ false false false true true true ]; % nonfinancial companies
p = Portfolio;
p = setGroupRatio(p, GA, GB, [], 0.5);
To add additional Group ratio constraints, use addGroupRatio.
Set Group Ratio Constraints for a PortfolioCVaR Object
Suppose you want to ensure that the ratio of financial to nonfinancial companies in your portfolio never exceeds 50%. Assume you have six assets with three financial companies (assets 1-3) and three
nonfinancial companies (assets 4-6). Group ratio constraints can be set with:
GA = [ true true true false false false ]; % financial companies
GB = [ false false false true true true ]; % nonfinancial companies
p = PortfolioCVaR;
p = setGroupRatio(p, GA, GB, [], 0.5);
To add additional Group ratio constraints, use addGroupRatio.
Set Group Ratio Constraints for a PortfolioMAD Object
Suppose you want to ensure that the ratio of financial to nonfinancial companies in your portfolio never exceeds 50%. Assume you have six assets with three financial companies (assets 1-3) and three
nonfinancial companies (assets 4-6). Group ratio constraints can be set with:
GA = [ true true true false false false ]; % financial companies
GB = [ false false false true true true ]; % nonfinancial companies
p = PortfolioMAD;
p = setGroupRatio(p, GA, GB, [], 0.5);
To add additional Group ratio constraints, use addGroupRatio.
Input Arguments
obj — Object for portfolio
Object for portfolio, specified using Portfolio, PortfolioCVaR, or PortfolioMAD object. For more information on creating a portfolio object, see
Data Types: object
GroupA — Matrix that forms base groups for comparison
Matrix that forms base groups for comparison, specified as a matrix for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).
The group matrices GroupA and GroupB are usually indicators of membership in groups, which means that their elements are usually either 0 or 1. Because of this interpretation, GroupA and GroupB
matrices can be either logical or numerical arrays.
Data Types: double | logical
GroupB — Matrix that forms comparison groups
Matrix that forms comparison groups, specified as a matrix Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).
The group matrices GroupA and GroupB are usually indicators of membership in groups, which means that their elements are usually either 0 or 1. Because of this interpretation, GroupA and GroupB
matrices can be either logical or numerical arrays.
Data Types: double | logical
LowerRatio — Lower bound for ratio of GroupB groups to GroupA groups
Lower bound for ratio of GroupB groups to GroupA groups, specified as a vector for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).
If input is scalar, LowerRatio undergoes scalar expansion to be conformable with the group matrices.
Data Types: double
UpperRatio — Upper bound for ratio of GroupB groups to GroupA groups
Upper bound for ratio of GroupB groups to GroupA groups, specified as a vector for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).
If input is scalar, UpperRatio undergoes scalar expansion to be conformable with the group matrices.
Data Types: double
Output Arguments
obj — Updated portfolio object
object for portfolio
Updated portfolio object, returned as a Portfolio, PortfolioCVaR, or PortfolioMAD object. For more information on creating a portfolio object, see
• You can also use dot notation to set up group ratio constraints for portfolio weight.
obj = obj.setGroupRatio(GroupA, GroupB, LowerRatio, UpperRatio);
• To remove group ratio constraints, enter empty arrays for the corresponding arrays. To add to existing group ratio constraints, use addGroupRatio.
Version History
Introduced in R2011a | {"url":"https://in.mathworks.com/help/finance/portfolio.setgroupratio.html","timestamp":"2024-11-11T09:42:50Z","content_type":"text/html","content_length":"105359","record_id":"<urn:uuid:16e31f21-0cc9-493c-a930-242feb409795>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00661.warc.gz"} |
In this document we present the vctsfr package, which can be an useful tool for those involved in time series forecasting because it allows you to visually compare the predictions of several
forecasting models. The vctsfr package is especially convenient when you want to visually compare the predictions of several forecasting methods across a collection of time series.
The vctsfr package makes it easy the visualization of collections of time series and, optionally, their future values and forecasts for those future values. The forecasts can include prediction
intervals. This package is particularly useful when you have forecasts (maybe from different models) for several time series and you want to display them in order to compare their results.
This package arises from a need of her authors. Frequently, we used several forecasting methods to predict the future values of collections of time series (typically belonging to time series
competitions). The usual way of comparing the performance of the forecasting methods is to compute a global measure of forecast accuracy for every method based on all its forecasts for all the series
of the competition. However, we miss a way of visually compare the performance of different methods over a particular series. This package fills this gap.
This package also facilitates the visualization of just a collection of time series (without forecasts).
Visualizing a single time series
If you only want to display a single time series and, optionally, information about its future values and/or a forecast for its future values you can use the plot_ts() function. Let us see how it
By default, plot_ts() shows the data points in the time series. However, you can omit them with the sdp parameter:
Let us now display the same time series and a forecast for its next 12 months using the exponential smoothing model implemented in the forecast package (ets() function):
#> Registered S3 method overwritten by 'quantmod':
#> method from
#> as.zoo.data.frame zoo
ets_fit <- ets(USAccDeaths)
ets_f <- forecast(ets_fit, h = 12)
plot_ts(USAccDeaths, prediction = ets_f$mean, method = "ets")
If the forecasting method computes prediction intervals, they can be displayed. For example, let’s add a 90% prediction interval to the previous forecast:
ets_fit <- ets(USAccDeaths)
ets_f <- forecast(ets_fit, h = 12, level = 90)
prediction = ets_f$mean,
method = "ets",
lpi = ets_f$lower,
upi = ets_f$upper,
level = 90
Finally, the actual values that are predicted can also be displayed:
timeS <- window(USAccDeaths, end = c(1977, 12))
fut <- window(USAccDeaths, start = c(1978, 1))
ets_fit <- ets(timeS)
ets_f <- forecast(ets_fit, h = length(fut), level = 80)
future = fut,
prediction = ets_f$mean,
method = "ets",
lpi = ets_f$lower,
upi = ets_f$upper,
level = 80
Summarizing, the plot_ts() function is useful to visualize a time series and, optionally, a forecast for its future values.
In all the functions of the vctsfr package the time series parameter, i.e. the historical values of the series, is specified as an object of class ts. The future values, forecasts and prediction
intervals can be specified as a numeric vector or as an object of class ts.
Visualizing several forecasts
When you want to compare several forecasts for the future values of a time series you can use the function plot_predictions(). The forecasts are passed to the function as a list, each component of
the list is a forecast and the name of the component is the name of the forecasting method. Let us see an example in which, given a time series, the forecasts for its future values, using the ARIMA
and exponential smoothing models implemented in the forecast package, are displayed:
timeS <- window(USAccDeaths, end = c(1977, 12)) # historical values
fut <- window(USAccDeaths, start = c(1978, 1)) # "future" values
ets_fit <- ets(timeS) # exponential smoothing fit
ets_f <- forecast(ets_fit, h = length(fut)) # exponential smoothing forecast
arima_fit <- auto.arima(timeS) # ARIMA fit
arima_f <- forecast(arima_fit, h = length(fut)) # ARIMA forecast
plot_predictions(timeS, future = fut,
predictions = list(ets = ets_f$mean, arima = arima_f$mean)
Looking at the plot, it is clear that both models produce similar and fairly accurate predictions.
Creating collections of time series
In the previous sections we have seen how to display a time series and forecasts for its future values. However, the main goal of the vctsfr package is to facilitate the visualization of collections
of time series so that you can visually compare forecasts for their future values.
In this section we study how to build these collections. For this purpose, the vctsfr package provides three functions:
• ts_info() allows you to create an object with information about a time series.
• prediction_info() allows you to create an object with information about a forecast.
• pi_info() allows you to create an object with information about the prediction interval associated with a forecast.
A collection of time series is a list of objects created with (returned by) the ts_info() function. Let’s first create a collection storing the historical values of two time series:
In the next section we will use the collections built in this section to visualize their information. Next, we use a dataset included in the Mcomp package to create another collection of series. The
Mcomp package contains datasets of forecasting competitions. We are going to create a collection of 18 quarterly time series, with their associated next 12 future values and forecasts for their
future values (using the previously applied ets() function of the forecast package).
# select the industry, quarterly series from M1 competition (18 series)
M1_quarterly <- subset(M1, 4, "industry")
# build the collection
collection2 <- vector("list", length = length(M1_quarterly))
for (ind in seq_along(M1_quarterly)) {
timeS <- M1_quarterly[[ind]]$x # time series
name <- M1_quarterly[[ind]]$st # time series's name
fut <- M1_quarterly[[ind]]$xx # future values
ets_fit <- ets(timeS) # ES fit
ets_for <- forecast(ets_fit, h = length(fut)) # ES forecast
collection2[[ind]] <- ts_info(timeS,
prediction_info("ets", ets_for$mean),
future = fut,
name = name
Finally, we use the same dataset of 18 time series to create a collection of time series, with two forecasts (obtained with two different forecasting models) for each series and some prediction
intervals associated with each forecast.
collection3 <- vector("list", length = length(M1_quarterly))
for (ind in seq_along(M1_quarterly)) {
t <- M1_quarterly[[ind]]$x # time series
name <- M1_quarterly[[ind]]$st # time series's name
f <- M1_quarterly[[ind]]$xx # "future" values
ets_fit <- ets(t) # ES fit
ets_f <- forecast(ets_fit, h = length(f), level = 90) # ES forecast
arima_fit <- auto.arima(t) # ARIMA fit
arima_f <- forecast(arima_fit, h = length(f), # ARIMA forecast
level = c(80, 90)
collection3[[ind]] <- ts_info(t,
future = f,
arima_f$lower[, 1],
arima_f$upper[, 1]
arima_f$lower[, 2],
arima_f$upper[, 2]
name = name)
It can be noted that the exponential smoothing forecasts have a prediction interval, while the ARIMA forecasts have two prediction intervals. Although it does not happen in these examples, different
time series in a collection can include forecasts done with different models.
Visualizing a time series from a collection
Once you have created a collection, you can see the information about one of its series with the plot_collection() function. The basic way of using this function is specifying a collection and the
number (index) of the series in the collection. Let’s see an example using a collection built in the previous section:
plot_collection() has displayed all available information about the series with index 3, except for the prediction intervals. You can choose to display a subset of the forecasts associated with a
time series providing a vector with the names of the forecasting methods you want to select:
Finally, if you display the forecasts of just one forecasting method and this method has prediction intervals, you can display one of its prediction intervals providing its level:
Looking at the plot, all predicted future values fall within the prediction interval.
The web-based GUI for visualizing collections of time series
Although the plot_collection() function is handy, the best way of navigating through the different series of a collection is to use the GUI_collection() function, which launches a Shiny GUI in a web
This is how the GUI looks like:
Using the GUI the user can select:
• Which time series to display.
• If the data points are highlighted.
• Which forecasting methods to display.
• In the case that only one forecast is displayed and this forecast has prediction intervals, which prediction interval to show.
Apart from the time series, the GUI shows information about the forecast accuracy of the displayed forecasting methods. Currently, for each forecasting method the following forecasting accuracy
measures are computed:
• RMSE: root mean squared error
• MAPE: mean absolute percentage error
• MAE: mean absolute error
• ME: mean error
• MPE: mean percentage error.
• sMAPE: symmetric MAPE
• MASE: mean absolute scaled error
This way, you can compare the displayed forecasting methods both visually and through forecast accuracy measures.
Next, we describe how the forecasting accuracy measures are computed for a forecasting horizon \(h\) (\(y_t\) and \(\hat{y}_t\) are the actual future value and its forecast for horizon \(t\)
\[ RMSE = \sqrt{\frac{1}{h}\sum_{t=1}^{h} (y_t-\hat{y}_t)^2} \] \[ MAPE = \frac{1}{h}\sum_{t=1}^{h} 100\frac{|y_t-\hat{y}_t|}{y_t} \] \[ MAE = \frac{1}{h}\sum_{t=1}^{h} |y_t-\hat{y}_t| \] \[ ME = \
frac{1}{h}\sum_{t=1}^{h} y_t-\hat{y}_t \] \[ MPE = \frac{1}{h}\sum_{t=1}^{h} 100\frac{y_t-\hat{y}_t}{y_t} \] \[ sMAPE = \frac{1}{h}\sum_{t=1}^{h} 200\frac{\left|y_{t}-\hat{y}_{t}\right|}{|y_t|+|\hat
{y}_t|} \] \[ MASE = \frac{\frac{1}{h}\sum_{t=1}^{h} |y_t - \hat{y}_t|}{\frac{1}{T-f}\sum_{t=f+1}^{T} |h_t - h_{t-f}|} \]
In the MASE computation \(T\) is the length of the training set (i.e., the length of the time series with historical values), \(h_t\) is the t-th historical value and \(f\) is the frequency of the
time series (1 for annual data, 4 for quarterly data, 12 for monthly data, …). | {"url":"https://cran.case.edu/web/packages/vctsfr/vignettes/vctsfr.html","timestamp":"2024-11-12T22:20:37Z","content_type":"text/html","content_length":"200444","record_id":"<urn:uuid:aeec2306-7d69-4227-83f4-315acde24389>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00410.warc.gz"} |
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Do you have more questions? Contact: | {"url":"https://mif.vu.lt/lt3/en/admission/phd-studies/mathematics","timestamp":"2024-11-13T16:11:25Z","content_type":"text/html","content_length":"43015","record_id":"<urn:uuid:f4e3b05b-483c-4785-8a54-6cd08cca2fbd>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00373.warc.gz"} |
Relationship between Gulf Menhaden Recruitment and Mississippi River Flow: Model Development and Potential Application for Management
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1 April 2012 Relationship between Gulf Menhaden Recruitment and Mississippi River Flow: Model Development and Potential Application for Management
Douglas S. Vaughan
The Gulf menhaden Brevoortia patronus is one of the most abundant pelagic fishes in the northern coastal Gulf of Mexico (hereafter, “Gulf”) and is the principal forage for various commercial and
sport fishes, sea birds, and marine mammals. Part of the life history of Gulf menhaden is spent on the continental shelf and part is spent within estuaries. Adults spawn near the mouth of the
Mississippi River, and larvae aggregate within the river plume front. Larval Gulf menhaden transit the continental shelf and enter estuaries of the northern Gulf as juveniles. Govoni (1997)
demonstrated an association between the discharge of the Mississippi and Atchafalaya rivers and Gulf menhaden recruitment. In particular, he found an inverse association between Mississippi River
discharge and estimated recruitment of half-year-old fish based on recruitment data from Vaughan et al. (1996). Vaughan et al. (2000) updated this relationship with a regression analysis. Here, we
revisit the relationship with additional years of data through 2004. The inverse relationship continues to hold. In addition, we reframed this relationship to produce a 1-year-ahead prediction model
for forecasting recruitment to age 1 from Mississippi River discharge; this model can be used in proactive fishery management. Finally, we revisited the stock assessment model of Vaughan et al. (2007
) and demonstrated an improvement in model performance when information on annual river discharge was incorporated.
The Gulf menhaden Brevoortia patronus is an exploited marine resource as well as an integral and key ecological component of the coastal northern Gulf of Mexico (hereafter, “Gulf”) ecosystem (
Ahrenholz 1981). As both a planktivore and a detritivore, the Gulf menhaden not only provides a key linkage between primary, secondary, and fishery production in the northern Gulf, it also provides a
short circuit for the transfer of detrital energy directly to consumers (Lewis and Peters 1984). The Gulf menhaden is one of the most abundant pelagic fish in the northern coastal Gulf and is the
principal forage for various commercial and sport fishes, sea birds, and marine mammals (Ahrenholz 1991). Given these attributes and ecological roles, the Gulf menhaden provides a key indicator of
the productivity and health of the ecosystem.
Part of the life history of Gulf menhaden is spent on the continental shelf and part is spent within estuaries (Ahrenholz 1991). Adults spawn primarily near the mouth of the Mississippi River in
winter. Larvae aggregate within the river plume front (Govoni et al. 1989), transit the continental shelf, and enter estuaries of the northern Gulf as juveniles. The mechanism of this transit across
the extended offshore estuary (Able 2005) is not well understood. Adult Gulf menhaden reoccupy nearshore continental shelf waters, where they feed and reproduce (Ahrenholz 1991).
Recruitment is the process of population dynamics through which individuals must pass before joining adult populations and the habitats they occupy. Gulf menhaden recruitment is correlated with the
combined discharge of the Mississippi and Atchafalaya rivers (Govoni 1997). Recruitment of half-year-old Gulf menhaden is inversely correlated with winter and spring discharge rates on an annual
scale. When the discharge rate in winter and spring increases from year to year, Gulf menhaden recruitment or year-class strength (as measured by the abundance of half-year-olds) decreases; when
discharge decreases, the abundance of half-year-olds increases (Govoni 1997; Vaughan et al. 2000). On an annual scale, this reciprocal association is thought to be related to a mechanism of larval
transport. On a decadal scale, there has been an apparently systematic—possibly climate-driven—increase in both river discharge and Gulf menhaden recruitment since the mid1970s. This association is
thought to be related to changes in primary production and consequent secondary production (Govoni 1997) that are driven by nutrients delivered to the northern Gulf by the Mississippi River (Liu and
Dagg 2003).
Mississippi River discharge is related to global climate change and the associated hypoxic zone that is evident in the northwestern Gulf (Rabalais et al. 2007; Turner et al. 2008); hypoxia might be
another possible correlate of Gulf menhaden recruitment and Mississippi River discharge. Whereas Gulf menhaden spawning and larval growth occur in winter within waters overlying the continental shelf
of the northern Gulf (Ahrenholz 1991), the hypoxic zone is a summer phenomenon that is driven ultimately by nutrients delivered to the coastal ecosystem by Mississippi River discharge.
Three objectives were addressed in this study: (1) to test whether the underlying relationship found by Govoni (1997) between recruitment and river discharge continues to hold with more recent
recruitment estimates (Vaughan et al. 2007); (2) to recast the statistical relationship in the form of year-ahead forecasts of recruitment to age 1 rather than the recruitment of half-year-olds as
was used by Govoni (1997); and (3) to incorporate this relationship into the stock assessment model of Vaughan et al. (2007). The forecasts can be further extended between assessments by using each
1-year prediction as input for additional forecasts. Because Gulf menhaden are short lived (1–4 years; Ahrenholz 1981) and because landings are largely recruitment driven, this approach has the
potential to provide real-time management projections of allowable catch.
River discharge data and correlation.—Water drained from approximately 40% of the North American continent collects into the Mississippi River (Turner et al. 2008), and flow is diverted downstream
into the Mississippi and Atchafalaya rivers. Discharge of the Mississippi and Atchafalaya rivers is controlled and flow is measured by the U.S. Army Corps of Engineers at two locks located north of
New Orleans, Louisiana (Figure 1). Discharge data were obtained from the U.S. Army Corps of Engineers (2009a, 2009b) websites for Tarbert Landing, Mississippi (Mississippi River), and for Simmesport,
Louisiana (Atchafalaya River). Data are near real time (i.e., available within several days of measurement). Units are daily measurements (provided in ft^3/s [m^3/s]) that are accumulated by month
and averaged overwinter (from November through the following March). Flow index data for a given calendar year represent January—March of that year and November—December from the previous calendar
year. Our computation of Mississippi River flow indices was identical to the methods described by Govoni (1997). Because river discharge is measured in winter (November—March), data for the most
recent year become available in early April (Figure 2).
Vaughan et al. (2007) estimated the recruitment of Gulf menhaden most recently through a statistical catch—age model by using the number of fish in landings at each age (ages 0–6) for each year
(1964—2004). Thus, estimates were available for the number of recruits to age 0 in year t (half-year-old fish; R[0,t]) and to age 1 in year t + 1 (R[1,t+1]). However, landings of age-0 Gulf menhaden
were minor (essentially zero in recent decades), so the relationship between age-0 and age-1 fish within a cohort was mostly a function of natural mortality, which was assumed to vary by age but not
by year. Thus, any statistical relationship between age-0 recruits and Mississippi River flow was expected to hold between age-1 recruits (of the same cohort) and Mississippi River flow. In this
study, analyses were based on recruits to age 1.
Govoni (1997) compared the change in recruitment to age 0 (ΔR[0,t] = R[0,t] - R[0,t-1]) with the change in river flow (ΔF[t] = F[t] - F[t-1]) for the same time period (t - 1, t). A similar comparison
can be made for recruits to age 1 from the same cohort as the age-0 recruits (Figure 3). The year in which fish recruit to age 1 follows the year in which fish from the same cohort recruit to age 0,
and hence there is a 1-year lag expressed as
where Z
is the instantaneous total mortality rate of age-0 Gulf menhaden,
represents recruits to age 0 in year
, and
represents recruits to age 1 in year
+ 1. With little or no fishing mortality on age-0 fish, Z
reduces to
, which is the instantaneous natural mortality rate for age 0 and is constant across years.
Regression analyses.—A predictive model was developed by regressing R[1,t+1] on the 1-year change in river discharge in year t (ΔF[t]) and the 1-year lagged recruitment in year t (R[1,t]) with t
equal to 1965–2004,
, and
are estimated parameters; and error
is normally distributed
). We used the differenced flow values rather than separate values with their own coefficients for two reasons: (1) to maintain consistency with the methods of Govoni (
) and (2) more importantly, to address nonstationarity in the river flow data. Determination of nonstationarity was based on autoregressive integrated moving average modeling techniques (
Nelson 1973
). Equation (
) permits 1-year-ahead forecasting of age-1 Gulf menhaden in the northern Gulf.
Forecasts can be evaluated in two ways. First, a comparison can be made with subsequent estimates of realized (observed) population recruitment from the statistical catch-age model. Alternatively, an
evaluation of the historical performance of the regression model can be made by reducing the data set by 1 year (e.g., removing data year 2004 from the 1964–2004 data set), sequentially re-estimating
the regression parameters, predicting recruitment for the next year, and comparing that prediction with the observed estimate of recruitment. This retrospective approach was based on equation (2) and
was conducted by sequentially reducing the terminal year of recruits to age 1 from 2004 to 1997. This sequential approach allowed us to consider the stability of our regression results as the data
were removed.
Projections can be extended beyond 1 year but with increasing uncertainty. By assuming the initial 1-year projection estimate of recruits to age 1 in the terminal year + 1 as “known,” we can use that
value as the lagged recruits to age 1 in equation (2) to develop an additional projection for the terminal year + 2. This sequence can be computed into the future with available additional years of
Mississippi River discharge data. At the time of writing, data through March 2009 were available, so projections of recruits to age 1 were made through 2010.
Stock assessment analyses.—To examine the utility of including river discharge in a population model of Gulf menhaden, we applied the stock assessment of Vaughan et al. (2007) but modified the
recruitment function to incorporate environmental effects (Schirripa et al. 2009),
is predicted recruitment;
is a function of spawners
(here, population fecundity);
is standardized river flow; and
h, R[0]
, and β are parameters (defined in more detail below). Thus, the assessment model here estimated all of the same parameters presented by Vaughan et al. (
; i.e., annual fishing mortality rates, selectivity parameters, and catchability) except for stochastic annual recruitment deviations. Instead, recruitment was treated as deterministic by using
several versions of equation (
) as described below.
TABLE 1.
Parameter estimates for models predicting the number of age-1 Gulf menhaden recruits (R[1]; equation 2) as function of the change in discharge of the Mississippi and Atchafalaya rivers and 1-year
lagged R[1] (note that b[0], b[1], and b[2] were calculated based on river discharge in ft^3/s). Analyses included recruitment estimates from the latest stock assessment (Vaughan et al. 2007) with
data through 2004 (P > F is the probability [based on the F-test] that the reduction in sums of squares due to the regression model fit to the data was statistically significant). Robustness of the
regression results was illustrated by a retrospective analysis in which recent data were removed and parameter values were re-estimated (see Methods).
We considered two spawner—recruit relationships (i.e., f): Beverton—Holt and Ricker. For each, we applied three variations of equation (3) to examine recruitment as a function of S only, F only, or
both S and F. In the first, the β parameter was not estimated but was fixed at zero; thus, equation (3) simplifies to R[t]+1 = f(S[t]\h, R[0]). In the second variation, h (the steepness parameter)
was not estimated but was fixed at a value such that recruitment was independent of S (h = 1 for Beverton-Holt; h = ∞ for Ricker); thus, equation (3) simplifies to R[t+1] = , where R[0] is
interpreted as mean recruitment (rather than the usual interpretation of recruitment that is not harvested). In the third variation, recruitment was a function of both S and F such that equation (3)
was applied in full. Thus, the assessment models applied six variations of equation (3) to predict recruitment (however, note that the two variations with fixed h are identical). Using maximum
likelihood, the models were fitted to catch-age data and an index of abundance. Computations were conducted with Automatic Differentiation Model Builder software (ADMB Project 2010), and performance
was compared among models by use of Akaike's information criterion (AIC).
River Flow Correlation
Correlations comparing the change in Mississippi River discharge with the corresponding change in Gulf menhaden recruitment demonstrated only small differences between the ages of recruitment. The
adjusted estimates of r^2 were 0.266 for age-0 recruits and 0.298 for age-1 recruits (Table 1). These relationships therefore explained between 27% and 30% of the variability in the assessment
estimates of recruitment.
Regression Analyses
Estimated age-1 recruits generally fit the observed data well; the exception was the estimate for 1985, which fell outside the 95% confidence interval (CI; Figure 4). The width of the 95% CI about
the predicted point suggested that a considerable amount of uncertainty remained unexplained (1 - r^2, expressed as a percentage).
Sequentially reducing the terminal year of data resulted in consistent parameter estimates over the final 3–4 years of the data set (terminal years 2001–2004; Table 1). Furthermore, 1year projection
estimates from the sequential regressions compared well with the estimates for the same year based on the regression that included all of the data (Table 2, columns labeled “All data” and ”1 year
ahead”). These estimates of expected and predicted values indicated good correspondence with stock assessment values in some years but not in others. Only small discrepancies were noted when
overlaying the estimates of recruitment among the various retrospective model fits (Figure 5). Estimates of adjusted r^2 ranged from 0.30 to 0.38, depending on the terminal year of the data (Table 1
). The magnitude peaked (and the P-value was minimized) when 2000 was used as the terminal year of data. After terminal year 2000, the inclusion of additional data resulted in some deterioration of
the regression fit.
One-year projection estimates (and 95% CIs) of recruits to age 1 were compared with the observed values from the stock assessment for projection years 1998–2005 (where projection year = terminal year
+ 1; Figure 6). The predicted and observed values were close in 2000 and 2004 but were more divergent in other years, especially 2001. In all cases, the observed values fell within the 95% CIs of the
predicted values.
When all available data through 2004 were assessed, including the terminal year observed value of 30.7 × 10^9 age-1 Gulf menhaden for 2004, the model fit provided an estimate of 30.2 × 10^9 Gulf
menhaden for that same year and yielded a 1-year-ahead projection estimate of 31.7 × 10^9 Gulf menhaden for 2005. Extended projections beyond the observed estimates of recruits to age 1 (ending in
2004) were based on available Mississippi River discharge data through 2009 (Figures 4, 5), and the 95% CIs (Figure 4) were inappropriate beyond 2005.
Stock Assessment Analyses
Assessment models that predicted recruitment from spawning stock size performed better than the models that predicted recruitment from river discharge alone (Table 3). The bestperforming models were
those that predicted recruitment from both spawning stock size and river discharge. This result held true for both the Beverton—Holt and Ricker spawner—recruit functions.
Various environmental correlates have been proposed for predicting Gulf menhaden harvests (Stone 1976; Guillory et al. 1983; Guillory 1993). Many of these correlates (e.g., temperature and salinity)
are themselves correlated with combined Mississippi River discharge. Because Gulf menhaden landings depend primarily on ages 1–3 (age 2 typically dominates the landings; Vaughan et al. 2007), the
Gulf menhaden fishery is largely recruitment driven. Therefore, an understanding of Gulf menhaden recruitment dynamics is important for proactive management.
With the addition of new data, the relationship first observed by Govoni (1997) still holds. In contrast, established correlations between environmental variables and recruitment are often found to
degrade over time. For example, an initial relationship between recruitment of Atlantic menhaden B. tyrannus and Ekman transport (Nelson et al. 1977) was later found to no longer hold with the
addition of more years of data. This is partly attributable to the large influence of the 1958 yearclass on the original relationship and to the lack of veracity of Ekman transport as a causative
agent in Atlantic menhaden recruitment. The addition of more years of data diluted the original results. More recently, McClatchie et al. (2010) reassessed the temperature—recruit relationship for
the Pacific sardine Sardinops sagax and found that this relationship no longer held; thus, those authors recommended that the relationship be removed from Pacific sardine management.
TABLE 2.
Observed (estimated) Gulf menhaden recruitment to age 1 (R[1]; × 10^9) and values of R[1] that were predicted from all data and by use of 1-year-ahead projection from retrospective runs. The 95%
confidence limits (CLs) are based on the retrospective regressions with reduced data.
Uncertainty associated with model fits assumes that the independent variables (Mississippi River discharge and the previous year's recruits to age 1) are known without error. Because this is not
strictly true, the 95% CIs presented here should be considered underestimates of true uncertainty. Although the observed value falls within the 95 % CI about the predicted point, the width of these
CIs suggests the need to explore additional environmental factors that might expand the current model framework to further reduce uncertainty and improve the predictive power of the model.
Gulf menhaden assessments have been conducted on an approximately 5–7-year cycle, beginning with Ahrenholz (1981), followed by Vaughan (1987) and then by Vaughan et al. (1996, 2000, 2007). Between
assessments, a priori knowledge of recruits to age 1 should help the fishing industry adjust their effort and should assist the Gulf States Marine Fisheries Commission in making their regulatory
recommendations. Thus, to the extent that the Mississippi River discharge is informative of the recruitment process for Gulf menhaden, this relationship can benefit the management process. In
particular, it may help in cases when unusually weak or strong recruitment events are predicted.
To bridge the gap since the terminal year (2004) of the last assessment (Vaughan 2007), we attempted to provide estimates of Gulf menhaden recruitment through 2010 (Figures 4, 5) by using Mississippi
River discharge data through winter 2008. These extended 1-year-ahead forecasts used the previous year's prediction as the basis for the next year. Although such forecasts can provide useful
information, we note that the iterative approach leads to increasing uncertainty in subsequent forecasts.
TABLE 3.
Comparison of assessment models with different functions for predicting recruitment of Gulf menhaden. The models applied either the Beverton—Holt (BH) or Ricker (R) spawner—recruit (SR) relationship.
The steepness parameter (h) was either estimated (Y) or not estimated (N); if not estimated, h was set to a value where recruitment was independent of spawning size, such that the remaining parameter
of the SR relationship described mean recruitment (thus, models 2 and 5 are the same). Similarly, the β parameter, which describes the effect of standardized river discharge on recruitment, was
either estimated (Y) or fixed at zero (N). Also presented are the number of parameters in the full assessment model, the negative log likelihood (-log[L]) of the fit to the data, and Akaike's
information criterion (AIC; lower AIC indicates better model performance).
Over the years, significant improvements in available information have resulted in a broader array of modeling approaches for use with stock assessments. For example, earlier assessments assumed that
natural mortality was constant across ages and that the catch-age matrix was known without error. In addition, these assessments did not directly incorporate state-specific, fisheryindependent
indices of recruitment (Vaughan 1987; Vaughan et al. 1996, 2000). The most recent assessment incorporated these new data with less-restrictive assumptions (Vaughan et al. 2007).
Our results suggest that future assessments could be improved further by including river discharge as an environmental covariate for the modeling of recruitment. Whether Beverton—Holt or Ricker
functions were used, the inclusion of river flow information enhanced model performance as indicated by AIC (Table 3). Furthermore, the AIC indicated that the Ricker function outperformed the
Beverton—Holt function; however, we caution against the use of AIC values to select between them. The two functions can lead to quite different management advice, and we agree with previous authors (
Williams and Shertzer 2003) that the Ricker function should be used only with convincing evidence that its underlying mechanisms are in play. In either case, the inclusion of environmental covariates
could be accomplished by modifying the spawner-recruit function (as we have done here) or by using the time series of environmental data as an index with which to tune annual recruitment deviations (
Schirripa et al. 2009).
Although our focus was on recruitment, various statistical approaches have also been used to forecast harvests of Atlantic menhaden and Gulf menhaden (Schaaf et al. 1975; Hanson et al. 2006). In
particular, several forecasting techniques were compared in a setting analogous to that in our study (Hanson et al. 2006), including the multiple regression approach used here, multivariate time
series (e.g., state space modeling), and artificial neural networks. The multiple regression approach performed as well as or better than the alternative approaches.
Mississippi River discharge may have an indirect influence on the northern Gulf ecosystem and on fisheries recruitment. Since the 1940s, nutrient loads discharged by the river have increased owing to
changes in agricultural practices on the North American continent (Turner et al. 2008). The increased nutrient loads have resulted in a zone of hypoxic water in the northern Gulf region west of the
Mississippi Delta. The apparent sensitivity of adult Gulf menhaden to hypoxic conditions was evident in summer 1995, when exceptionally low catches off the Louisiana coast from Southwest Pass
westward to Marsh Island co-occurred with the impingement of hypoxic waters upon Louisiana's nearshore waters (Smith 2000). Hypoxic water is near the bottom; consequently, the sensitivity of Gulf
menhaden juveniles is probably related to the interaction of habitat displacement and the shoreward, deepwater movement of these fish (Craig and Crowder 2005). Persons responsible for assessing the
impact and consequences of hypoxia in the northern Gulf, along with those responsible for appraising the impact of global climate change, should benefit from the forecast of Gulf menhaden, a key
ecological species that responds to annual changes in runoff and Mississippi River discharge.
Environmental changes influence ecosystems (Kimmel et al. 2009). Mississippi River discharge, which is determined by precipitation and runoff, varies on decadal and annual scales. Mississippi River
and Atchafalaya River discharge rose from 1963 to a peak in 1974, declined, and became erratic from 1976 to 2009, especially over the last decade (Figure 2). Although Gulf menhaden recruitment has
been relatively variable from year to year, recruits to age 0 generally rose from 1963 to a peak in 1984, declined, and then gradually rose to a more recent peak in 1998, which was followed by an
overall decline. Recruits to age 1 show the same pattern but with a 1-year lag (Figure 4). Thus, both river discharge and recruitment demonstrate significant annual variability and, as suggested by
the regression model, they do so in an inverse fashion. Long-term patterns in these two variables are not as well linked. Annual- and decadal-scale change in the relationship between Gulf menhaden
abundance and river discharge may well signal an ecosystem regime shift (sensu Steele 1998) in the northern Gulf ecosystem.
Models are essential for forecasting the effects of ecosystem change on fisheries recruitment and consequent production. The influence of temperature, salinity, and river discharge on fish
recruitment has been well documented (Drinkwater and Frank 1994; Grimes and Kingsford 1996; Gillanders and Kingsford 2002), but few comprehensive predictive models are available. Within estuaries,
river discharge is correlated with growth and survival of young fish (North and Houde 2003; Shoji and Tanaka 2006). A threshold for the influence of salinity and flow rate on the abundance of young
fish (including other estuarine-dependent, herringlike fish) is evident wherein a positive relationship becomes negative (Strydom et al. 2002; Whitfield and Harrison 2003). Mississippi River
discharge influences fishery production in the northern Gulf, but its direct influence on the recruitment of other stocks may be either positive or negative (Grimes 2001).
The models presented here are directly applicable to forecasting the effects of Mississippi River discharge on Gulf menhaden production and can be used in fisheries and ecosystem management. We have
demonstrated that the association reported by Govoni (1997) continues to hold with the analysis of additional, more recent data. We have developed a regression methodology for near-term forecasts
that can allow management organizations to make the better-informed decisions that are required to proactively manage Gulf menhaden. Finally, we have demonstrated that the inclusion of river flow
improves the performance of the most recent Gulf menhaden stock assessment model (Vaughan et al. 2007).
We thank J. W Smith, M. Cieri, and one anonymous reviewer for providing valuable reviews of this manuscript. Views expressed are those of the authors and do not necessarily represent the findings or
policy of any government agency. Reference to trade names does not imply endorsement by the U.S. Government.
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© American Fisheries Society 2011
Douglas S. Vaughan "Relationship between Gulf Menhaden Recruitment and Mississippi River Flow: Model Development and Potential Application for Management," Marine and Coastal Fisheries: Dynamics,
Management, and Ecosystem Science 3(1), 344-352, (1 April 2012). https://doi.org/10.1080/19425120.2011.620908
Received: 4 August 2010; Accepted: 10 March 2011; Published: 1 April 2012 | {"url":"https://complete.bioone.org/journals/marine-and-coastal-fisheries/volume-3/issue-1/19425120.2011.620908/Relationship-between-Gulf-Menhaden-Recruitment-and-Mississippi-River-Flow/10.1080/19425120.2011.620908.full","timestamp":"2024-11-09T12:23:01Z","content_type":"text/html","content_length":"230308","record_id":"<urn:uuid:90fd178f-37be-4f39-ae23-f77b6a507b9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00716.warc.gz"} |
public class NormOps_DDRM extends Object
Norms are a measure of the size of a vector or a matrix. One typical application is in error analysis.
Vector norms have the following properties:
1. ||x|| > 0 if x ≠ 0 and ||0|| = 0
2. ||αx|| = |α| ||x||
3. ||x+y|| ≤ ||x|| + ||y||
Matrix norms have the following properties:
1. ||A|| > 0 if A ≠ 0 where A ∈ ℜ ^m × n
2. || α A || = |α| ||A|| where A ∈ ℜ ^m × n
3. ||A+B|| ≤ ||A|| + ||B|| where A and B are ∈ ℜ ^m × n
4. ||AB|| ≤ ||A|| ||B|| where A and B are ∈ ℜ ^m × m
Note that the last item in the list only applies to square matrices.
Matrix norms can be induced from vector norms as is shown below:
||A||[M] = max[x≠0]||Ax||[v]/||x||[v]
where ||.||[M] is the induced matrix norm for the vector norm ||.||[v].
By default implementations that try to mitigate overflow/underflow are used. If the word fast is found before a function's name that means it does not mitigate those issues, but runs a bit faster.
• Method Summary
Modifier and Type
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• Method Details
□ normalizeF
Normalizes the matrix such that the Frobenius norm is equal to one.
A - The matrix that is to be normalized.
□ conditionP
public static double conditionP(DMatrixRMaj A, double p)
The condition number of a matrix is used to measure the sensitivity of the linear system Ax=b. A value near one indicates that it is a well conditioned matrix.
κ[p] = ||A||[p]||A^-1||[p]
If the matrix is not square then the condition of either A^TA or AA^T is computed.
A - The matrix.
p - p-norm
The condition number.
□ conditionP2
The condition p = 2 number of a matrix is used to measure the sensitivity of the linear system Ax=b. A value near one indicates that it is a well conditioned matrix.
κ[2] = ||A||[2]||A^-1||[2]
This is also known as the spectral condition number.
A - The matrix.
The condition number.
□ fastNormF
This implementation of the Frobenius norm is a straight forward implementation and can be susceptible for overflow/underflow issues. A more resilient implementation is normF
a - The matrix whose norm is computed. Not modified.
□ normF
Computes the Frobenius matrix norm:
normF = Sqrt{ ∑[i=1:m] ∑[j=1:n] { a[ij]^2} }
This is equivalent to the element wise p=2 norm. See fastNormF(org.ejml.data.DMatrixD1) for another implementation that is faster, but more prone to underflow/overflow errors.
a - The matrix whose norm is computed. Not modified.
The norm's value.
□ elementP
Element wise p-norm:
norm = {∑[i=1:m] ∑[j=1:n] { |a[ij]|^p}}^1/p
This is not the same as the induced p-norm used on matrices, but is the same as the vector p-norm.
A - Matrix. Not modified.
p - p value.
The norm's value.
□ fastElementP
public static double fastElementP(DMatrixD1 A, double p)
A - Matrix. Not modified.
p - p value.
The norm's value.
□ normP
Computes either the vector p-norm or the induced matrix p-norm depending on A being a vector or a matrix respectively.
A - Vector or matrix whose norm is to be computed.
p - The p value of the p-norm.
The computed norm.
□ fastNormP
public static double fastNormP(DMatrixRMaj A, double p)
A - Vector or matrix whose norm is to be computed.
p - The p value of the p-norm.
The computed norm.
□ normP1
Computes the p=1 norm. If A is a matrix then the induced norm is computed.
A - Matrix or vector.
The norm.
□ normP2
Computes the p=2 norm. If A is a matrix then the induced norm is computed.
A - Matrix or vector.
The norm.
□ fastNormP2
Computes the p=2 norm. If A is a matrix then the induced norm is computed. This implementation is faster, but more prone to buffer overflow or underflow problems.
A - Matrix or vector.
The norm.
□ normPInf
Computes the p=∞ norm. If A is a matrix then the induced norm is computed.
A - Matrix or vector.
The norm.
□ inducedP1
Computes the induced p = 1 matrix norm.
||A||[1]= max(j=1 to n; sum(i=1 to m; |a[ij]|))
A - Matrix. Not modified.
The norm.
□ inducedP2
Computes the induced p = 2 matrix norm, which is the largest singular value.
A - Matrix. Not modified.
The norm.
□ inducedPInf
Induced matrix p = infinity norm.
||A||[∞] = max(i=1 to m; sum(j=1 to n; |a[ij]|))
A - A matrix.
the norm. | {"url":"https://ejml.org/javadoc/org/ejml/dense/row/NormOps_DDRM.html","timestamp":"2024-11-09T06:32:46Z","content_type":"text/html","content_length":"33956","record_id":"<urn:uuid:ccb87de9-2414-4e61-9f95-3b9c8680c566>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00266.warc.gz"} |
Math: Addition & Subtraction Activities
The photos on the pages listed below offer some math instructional ideas.
A plate divided into three sections can be used to illustrate both addition and subtraction. For addition, put manipulatives in both small sections (5+3). Move all of them down to the larger bottom
section to show the sum. For subtraction, begin with all of the manipulatives in the large section and then remove some to the smaller section to find out how many are left.
Use candy to act out and record addition problems.
The addition workmat is from Math Their Way.
Place stickers with numbers in the bottome of an egg carton. Place 2 small erasers in the carton and shake it up. Open and write and addition problem about the tow numbers where the erasers fell. For
example, 2+4=6.
Sing songs or read books to act out subtraction problems. Use teddy grahams and a bed workmat while singing Roll Over to subtract one each time. The children below were acting out Five Green and
Speckled Frogs, taking away one frog during each verse.
He is using an animal plate and animal counters to solve the addition problem 3 + 1. He put 3 counters in one paw and 1 counter in the other paw.
She did the same thing on this plate, but she first put the counters in the animal’s ears. Then she moved the counters down to the face and counted them for the total.
Unifix Cube Sets
Record different combinations of unifix cubes to make sets of each number. This recording sheet came from math Their Way.
Adding One More
We used this number line to practice adding one more to each number by moving the frog one space each time. We learned that when you add one to any number the answer is the next number you would
count. This pocket chart is a Lakeshore product.
Fly Swatter Addition
Write the numbers from 0-10 on the chalkboard or on a shower curtain. Divide class into teams and give each team a fly swatter. Call out an addition problem and have the first team member “swat” the
answer. Continue playing, giving each team a point for correct answers.
Grab-Bag Mat
Give each partner group a lunch bag with 10-12 unifix cubes in two colors (5-6 of each color). Have each child reach in and grab a handful of cubes, pull them out, sort and count each color. Have
them write the number sentence. For instance if someone pulled out 4 red and 4 blue cubes, they would write 4 + 4 = 8. Then the next child takes his turn. I find it is helpful for children to write
everyone’s equation as it gives them something to do while waiting for their turn. You can do this as a small group activity as well.
Addition War
Use number cards that also have items to count corresponding to the number on the card. Play it first like War for number recognition and comparing sets. Then play it again by using two cards that
they must add together. It is like war: whoever has the highest number or sum, wins all the cards. Person with the most cards wins the game. | {"url":"http://littlegiraffes.com/teaching-ideas/1004/math-addition-subtraction-activities/","timestamp":"2024-11-09T05:48:18Z","content_type":"text/html","content_length":"42114","record_id":"<urn:uuid:94b84da2-ac27-4aa4-a59f-6d64387b7f32>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00274.warc.gz"} |
Inch/Hour/Minute to Feet/Hour Squared
Inch/Hour/Minute [in/h/min] Output
1 inch/hour/minute in meter/second squared is equal to 1.1759259259259e-7
1 inch/hour/minute in attometer/second squared is equal to 117592592592.59
1 inch/hour/minute in centimeter/second squared is equal to 0.000011759259259259
1 inch/hour/minute in decimeter/second squared is equal to 0.0000011759259259259
1 inch/hour/minute in dekameter/second squared is equal to 1.1759259259259e-8
1 inch/hour/minute in femtometer/second squared is equal to 117592592.59
1 inch/hour/minute in hectometer/second squared is equal to 1.1759259259259e-9
1 inch/hour/minute in kilometer/second squared is equal to 1.1759259259259e-10
1 inch/hour/minute in micrometer/second squared is equal to 0.11759259259259
1 inch/hour/minute in millimeter/second squared is equal to 0.00011759259259259
1 inch/hour/minute in nanometer/second squared is equal to 117.59
1 inch/hour/minute in picometer/second squared is equal to 117592.59
1 inch/hour/minute in meter/hour squared is equal to 1.52
1 inch/hour/minute in millimeter/hour squared is equal to 1524
1 inch/hour/minute in centimeter/hour squared is equal to 152.4
1 inch/hour/minute in kilometer/hour squared is equal to 0.001524
1 inch/hour/minute in meter/minute squared is equal to 0.00042333333333333
1 inch/hour/minute in millimeter/minute squared is equal to 0.42333333333333
1 inch/hour/minute in centimeter/minute squared is equal to 0.042333333333333
1 inch/hour/minute in kilometer/minute squared is equal to 4.2333333333333e-7
1 inch/hour/minute in kilometer/hour/second is equal to 4.2333333333333e-7
1 inch/hour/minute in inch/hour/second is equal to 0.016666666666667
1 inch/hour/minute in inch/minute/second is equal to 0.00027777777777778
1 inch/hour/minute in inch/hour squared is equal to 60
1 inch/hour/minute in inch/minute squared is equal to 0.016666666666667
1 inch/hour/minute in inch/second squared is equal to 0.0000046296296296296
1 inch/hour/minute in feet/hour/minute is equal to 0.083333333333333
1 inch/hour/minute in feet/hour/second is equal to 0.0013888888888889
1 inch/hour/minute in feet/minute/second is equal to 0.000023148148148148
1 inch/hour/minute in feet/hour squared is equal to 5
1 inch/hour/minute in feet/minute squared is equal to 0.0013888888888889
1 inch/hour/minute in feet/second squared is equal to 3.858024691358e-7
1 inch/hour/minute in knot/hour is equal to 0.00082289417166667
1 inch/hour/minute in knot/minute is equal to 0.000013714902861111
1 inch/hour/minute in knot/second is equal to 2.2858171435185e-7
1 inch/hour/minute in knot/millisecond is equal to 2.2858171435185e-10
1 inch/hour/minute in mile/hour/minute is equal to 0.000015782828282828
1 inch/hour/minute in mile/hour/second is equal to 2.6304713804714e-7
1 inch/hour/minute in mile/hour squared is equal to 0.0009469696969697
1 inch/hour/minute in mile/minute squared is equal to 2.6304713804714e-7
1 inch/hour/minute in mile/second squared is equal to 7.3068649457538e-11
1 inch/hour/minute in yard/second squared is equal to 1.2860082304527e-7
1 inch/hour/minute in gal is equal to 0.000011759259259259
1 inch/hour/minute in galileo is equal to 0.000011759259259259
1 inch/hour/minute in centigal is equal to 0.0011759259259259
1 inch/hour/minute in decigal is equal to 0.00011759259259259
1 inch/hour/minute in g-unit is equal to 1.1991107319277e-8
1 inch/hour/minute in gn is equal to 1.1991107319277e-8
1 inch/hour/minute in gravity is equal to 1.1991107319277e-8
1 inch/hour/minute in milligal is equal to 0.011759259259259
1 inch/hour/minute in kilogal is equal to 1.1759259259259e-8 | {"url":"https://hextobinary.com/unit/acceleration/from/inhmin/to/fth2","timestamp":"2024-11-09T23:52:24Z","content_type":"text/html","content_length":"96791","record_id":"<urn:uuid:adcce840-73d8-4750-9e25-46fdf02bbd8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00228.warc.gz"} |